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April 6, 2022

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Similar to Problems 4-7, 4-8

Math CoachUsing a Financial Calculator

Financial, or business, calculators are programmed to perform the time value of money equations we develop in this chapter and thenext. The two most common types of inexpensive financial calculators that can perform such functions are the Hewlett-Packard 10B IIBusiness Calculator and the Texas Instruments BA II (Plus or Professional). Among many useful financial shortcuts these calculatorshave five specific financial buttons. The relevant financial buttons for time value of money (TVM) calculations are listed below. The HP10B II calculator buttons look like this:

1. N (for the number of periods),

2. I/YR (for the interest rate),3. PV (for present value),

4. PMT (for a constant payment every period), and5. FV (for future value).

Notice that the TI BA II Plus financial calculator buttons appear to be very similar:

To get to the TVM menu, select APPLICATIONS and then choose FINANCE, and finally 1 TVM SOLVER on the previous screens.A common, more sophisticated and expensive calculator is the TI-83. This calculator has a menu system that includes the financialfunctions as shown:

Setting up Your CalculatorThese calculators come from the factory with specific settings. You will find it useful to change two of them. The first is to set thenumber of digits shown after the decimal point on the calculator display. The factory setting is for two digits. However, consider aproblem in which we use a 5.6 percent interest rate. The decimal version of this percentage is 0.056, which a two-digit display wouldround to 0.06. It’s less worrisome to set the calculator to display the number of digits necessary to show the right number; this is calleda floating point display. To set a floating point display for the HP calculator, press the color button, then the DISP button, and finally thedecimal (.) button. To set the display for a floating point decimal on the TI calculator, push the 2ND button, followed by the FORMATbutton, followed by the 9 button, and finally the ENTER button.

The second change you’ll want to make is to set the number of times the calculator compounds each period. The settings may bepreset to 12 times per period. Reset this to one time per period. To change the HP calculator to compound once per period, push the 1button, then the color button, and finally the P/YR button. On the TI calculator, simply push the 2ND button, the P/Y button, the numberone, and the ENTER button. These new settings will remain in the calculator until you either change them or remove the calculator’sbatteries.

Using Your Calculator

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The calculators compute time-value problems in similar ways. Enter the cash flows into the time-value buttons (PV, PMT, and FV)consistent with the way they are shown in a time line. In other words, cash inflows should be positive and cash outflows negative. Thus,PV and FV cash flows are nearly always opposite in sign. Enter interest rates (I) in the percentage form, not the decimal form. Alsoenter the number of periods in the problem (N).

Consider our earlier example of the $100 deposit for two years earning a 5 percent interest rate.

1. To set the number of years, press 2 and then the N button.2. To set the interest rate, press 5 and then the I button. (Note that interest rates are in percentage format for using a financial

calculator and in decimal format for using the equations.)

3. To enter the current cash flow: press 100, then make it negative by pressing the +/− button, then press the PVbutton.

4. We won’t use the PMT button, so enter 0 and then the PMT button.

5. To solve for future value, press the compute button (CPT) [for the TI] and then the FV button [press the FV button only for theHP].

6. Solution: the display should show FV = 110.25

Note that the answer is positive, consistent with an inflow and the time line diagram. These values remain in the TVM registers evenafter the calculator is turned off. So when you start a new problem, you should clear out old values first. For the HP calculator, clear theregisters by pressing the shift/orange key before pressing C. You can clear the BAII Plus calculator using 2ND and CLR TVM.

You’ll notice that throughout this book we use the equations in the main text to solve time value of money problems. We provide thecalculator solutions in the margins.

4.3 • PRESENT VALUE LG4-4We asked earlier what happens when you deposit $100 cash in the bank to earn 5 percent interest for one year—thebank pays you a $105 future value. However, we could have asked the question in reverse. That is, if the bank willpay $105 in one year and interest rates are 5 percent, how much would you be willing to deposit now, to receive thatpayment in a year? Here, we start with a future value and must find the present value—a different kind ofcalculation called discounting.

How much would you be willing to deposit now to receive a certain payment in a year?©Ryan McVay/Getty Images

DiscountingWhile the process of a present value growing over time into the future is called compounding, the process offiguring out how much an amount that you expect to receive in the future is worth today is discounting. Just ascompounding significantly increases the present value into the future, discounting significantly decreases the valueof a future amount to the present. Since discounting is the reverse of compounding, we can rearrange equation 4-1 tosolve for the present value of a cash flow received one year in the future.

(4-4)

discounting The process of finding present value by reducing future values using the discount, or interest, rate.

Suppose the bank is going to pay $105 in one year and interest rates are 5 percent. Then the present value of thepayment is $105/1.05 = $100. Present values are always smaller than future values (as long as interest rates aregreater than zero!), and the difference between what an investment is worth today and what it’s worth when you’resupposed to redeem it gets larger as the interest rate increases. Likewise, if the amount of time until the expectedpayment date increases, the difference will also increase in value.

Discounting Over Multiple Periods Discounting over multiple periods is the reverse process of compoundingover multiple periods. Knowing this, we can find the general equation for present value by rearranging the terms inequation 4-2 to form:

(4-5)

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The interest rate, i, which we use to calculate present value, is often referred to as the discount rate. How much is a$100 payment to be received in the future worth today? Of course, it depends on how far into the future you expectto receive the payment and the discount rate used. If you receive a $100 cash flow in five years, then its presentvalue is $78.35, discounted at 5 percent:

discount rate The interest rate used to discount future cash flow(s) to the present.

The time line looks like this:

LG4-1

If the discount rate rises to 10 percent, the present value of our $100 to be paid to us in five years is only $62.09today. At a 15 percent interest rate, the present value declines to less than half the future cash flow: $49.72. Higherinterest rates discount future cash flows more quickly and dramatically. You can see this principle illustrated inFigure 4.4.

Moving right from point A in Figure 4.4, notice that if interest rates are 0 percent, the present value will equal thefuture value. Also note from the curved lines that when the discount rate is greater than zero, the discounting topresent value is not linear through time. The higher the discount rate, the more quickly the cash flow value falls. Ifthe discount rate is 10 percent, a $100 cash flow that you would receive in 25 years is worth less than $10 today, asshown at point B in the figure. With a 15 percent discount rate, the $100 payment to be received in 33 years, at pointC, is worth less than $1 today.

the Math Coach on…

▼

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“When using a financial calculator, be sure to either clearthe time value of money buttons or enter a zero for thefactors that you won’t use to solve the problem.„

FIGURE 4-4 Present Value of a $100 Cash Flow Made in the Future

The higher the discount rate, the more quickly the cash flow value falls.

EXAMPLE 4-3Buy Now and Don’t Pay for TwoYears LG4-4

For interactive versionsof this example, log in

to Connect or go to

Suppose that a marketing manager for a retail furniture companyproposes a sale. Customers can buy now but don’t have to pay for theirfurniture purchases for two years. From a time value of moneyperspective, selling furniture at full price with payment in two years isequivalent to selling furniture at a sale, or discounted, price withimmediate payment. If interest rates are 7.5 percent per year, what isthe equivalent sale price of a $1,000 sleeper-sofa when the customertakes the full two years to pay for it?

SOLUTION:

mhhe.com/CornettM4e. The time line for this problem is:

Using equation 4-5, the present value computation is

In this case, the marketing proposal for delaying payment for two yearsis equivalent to selling the $1,000 sleeper-sofa for a sale price of$865.33, or a 13.5 percent discount. When stores promote such sales,they often believe that customers will not be able to pay the full amountat the end of the two years and then must pay high interest ratecharges and late fees. Customers who do pay on time are getting agood deal.

Similar to Problems 4-9, 4-10, 4-11, 4-12, Self-Test Problem 2

Discounting with Multiple Rates LG4-4 We can also discount a future cash flow at different interest ratesper period. We find the general form of the equation for present value with multiple discount rates by rearrangingequation 4-3:

Present value with different discount rates = Future cash flow ÷ Each period’s discounting

(4-6)

LG4-1

Suppose that we expect interest rates to increase over the next few years, from 7 percent this year, to 8 percent nextyear, to 8.5 percent in the third year. In this environment, how would we work out the present value of a future$2,500 cash flow in year 3? The time line for this problem is

time out!4-6 How are interest rates in the economy related to the way people value future cash payments?4-7 Explain how discounting is the reverse of compounding.

Using equation 4-6 shows that the present value is $1,993.90:

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4.4 • USING PRESENT VALUE AND FUTURE VALUE LG4-5Moving Cash FlowsAs managers analyze investment projects, debt management, and cash flows, they frequently find it useful to movecash flows to different points in time. While you may be planning to keep money deposited in the bank for threeyears when you will buy a car, life often has a way of altering plans. What type of car might you purchase if themoney earns interest for only two years, or for four years? How is a corporate financial forecast affected if the firmneeds to remodel a factory two years sooner than planned? Moving cash flows around in time is important tobusinesses and individuals alike for sound financial planning and decision making.

LG4-1

Moving cash flows from one point in time to another requires us to use both present value and future valueequations. Specifically, we use the present value equation for moving cash flows to an earlier point in time, and thefuture value cash flows for moving cash flows to a later point in time. For example, what’s the value in year 2 of a$200 cash flow to be received in three years, when interest rates are 6 percent? This problem requires moving the$200 payment in the third year to a value in the second year, as shown in the time line:

Since the cash flow is to be moved one year earlier in time, we use the present value equation:

When interest rates are 6 percent, a $188.68 payment in year 2 equates to a $200 payment in year 3.

What about moving the $200 cash flow to year 5? Since this requires moving the cash flow later in time by twoyears, we use the future value equation. In this case, the equivalent of $200 in the third year is a fifth-year paymentof:

Table 4.3 illustrates how we might move several cash flows. At an 8 percent interest rate, a $1,000 cash flow due in

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year 5 compounded to year 10 equals $1,469.33. We could also discount that same $1,000 cash flow to a value of$793.83 in year 2. At an 8 percent interest rate, the three cash flows ($793.83 in year 2, $1,000 in year 5, and$1,469.33 in year 10) become equivalent. Table 4.3 illustrates the movement of other cash flows given differentinterest rates and time periods.Moving cash flows from one year to another creates an easy way to compare or combine two cash flows. Would yourather receive $150 in year 2 or $160 in year 2? Since both cash flows occur in the same year, the comparison isstraightforward. But we can’t directly add or compare cash flows in different years until we consider their timevalue. We can compare cash flows in different years by moving one cash flow to the same time as the other usingthe present value or future value equations. Once you have the value of each cash flow in the same year, you candirectly compare or combine them.

Rule of 72 LG4-6 Albert Einstein is also credited with popularizing compound interest by introducing asimple mathematical approximation for the number of years required to double an investment. It’s called the Rule of72.

Rule of 72 An approximation for the number of years needed for an investment to double in value.

▼ TABLE 4.3 Equivalent Cash Flows in Time

When InterestRates Are

A Cash Flowof In Year

Can beMoved to

Year With EquationEquivalent Cash

Flow

Moving Later versus Moving Earlier

8% $1,000 5 10FV10 = PV5 × (1 + i )

5 =

$1,000 × (1.08)5 =$1,469.33

8 1,000 5 2PV2 = FV5 / (1 + i)

3 =

$1,000 / (1.08)3 =793.83

Moving Earlier

10 500 9 8PV8 = FV9 / (1 + i )

1 =

$500 / (1.10)1 =454.55

10 500 9 0PV0 = FV9 / (1 + i)

9 =

$500 / (1.10)9 =212.05

Moving Later

12 100 4 20FV20 = PV4 × (1 + i )

16 =

$100 × (1.12)16 =613.04

12 100 4 30

FV30 = PV4 × (1 + i

)26 = $100 ×(1.12)26 =

1,904.01

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(4-7)

The Rule of 72 illustrates the power of compound interest. How many years will it take to double money depositedat 6 percent per year? Using the Rule of 72, we find the answer is 12 years (= 72/6). A higher interest rate causesfaster increases in future value. A 9 percent interest rate allows money to double in just eight years (= 72/9).Remember that this rule provides only a mathematical approximation. It’s more accurate with lower interest rates.After all, with a 72 percent interest rate, the rule predicts that it will take one year to double the money (= 72/72).However, we know that it actually takes a 100 percent rate to double money in one year.

We can also use the Rule of 72 to approximate the interest rate needed to double an investment in a specific amountof time. What rate do we need to double an investment in five years? Rearranging equation 4-7 shows that the rateneeded is 14.4 percent (= 72/5) per year.

time out!4-8 In Example 4-4, could Timber, Inc., have performed its analysis by moving the $175,000 to year 2 and comparing? Would

the firm then have made the same decision?

4-9 At what interest rate (and number of years) does the Rule of 72 become too inaccurate to use?

EXAMPLE 4-4 Pay Damages or Appeal? LG4-5

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Timber, Inc., lost a lawsuit in a business dispute. The judge ordered thecompany to pay the plaintiff $175,000 in one year. Timber’s attorneyadvises Timber to appeal the ruling. If so, Timber will likely lose againand will still have to pay the $175,000. But by appealing, Timber movesthe $175,000 payment to year 2, along with the attorney’s fee of$20,000 for the extra work. The interest rate is 7 percent. Whatdecision should Timber make?

SOLUTION:

Timber executives must decide whether to pay $175,000 in one year or$195,000 in two years. To compare the two choices more directly,move the payment in year 2 to year 1 and then compare it to $175,000.Timber should choose to make the smaller payment. The computationis

The value in year 1 of a year 2 payment of $195,000 is $182,242.99,which is clearly more than the $175,000 year 1 payment. So Timbershould not appeal and should pay the plaintiff $175,000 in one year(and may want to look for another attorney).

Similar to Problems 4-23, 4-24, 4-25, 4-26, 4-27, 4-28, 4-41

finance at work //: investmentsTVM Caveat Emptor

©Trevor Lush/Blend Images LLC

Not making your payments on time can get very expensive. Reconsider the furniture selling experience in Example 4-3. The store hasgiven customers the opportunity to buy the sleeper-sofa today and not pay the $1,000 price for two years. But what happens if you forgetto pay on time? Indeed, many people do forget. Others simply haven’t saved $1,000 and can’t make the payment. The fine print in thesedeals provides the penalties for late payment. For example, the late clause might require a 10 percent annually compounded interestrate to apply to any late payment—retroactive to the sale date. Thus, being one day late with the payment will automatically incur aninterest charge for the entire two years of $210 (= $1,000 × 1.12 − $1,000), which is in addition to the original $1,000 still owed, ofcourse.

The impact of making late payments can also show up later when you apply for credit cards, auto loans, and other credit. Creditrating agencies gather information on us from companies, banks, and landlords to grade our payment history. If you consistently makelate payments on your apartment, credit card, or electric bill, these agencies will give you a poor grade. The higher your grade, called acredit score, the more likely you are to be able to get a loan and pay a lower interest rate on that loan. People with lower credit scoresmay not be able to borrow money and when they can, they will pay higher interest rates on their credit cards and auto loans. Paying ahigher interest rate can really cost you a lot of money. Remember the future value differences between interest rates illustrated in Table4.2 and Figure 4.3. Those people who have really bad credit scores can’t get loans from banks and merchants and have to rely onpayday lending places that charge enormously high rates, as highlighted in this chapter’s Personal Application Viewpoint.

Making a late payment might not seem like a big deal at the time, but it can really cost you!

Want to know more?Key Words to Search for Updates: housing bubble, subprime lending, mortgage-backed securities, AIG, Countrywide Financial

4.5 • COMPUTING INTEREST RATES LG4-7Time value of money calculations come in handy when we know two cash flows and need to find the interest rate.The investment industry often uses this analysis. Solving for the interest rate, or rate of return,2 can answer questionslike, “If you bought a gold coin for $350 three years ago and sell it now for $475, what rate of return have youearned?” The time line for this problem looks like this:

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LG4-1

In general, computing interest rates is easiest with a financial calculator. To compute the answer using the time-value equations, consider how the cash flows fit into the future value equation 4-2:

the Math Coach on…

“When using a financial calculator to compute an interest rate between two cashflows, you must enter one cash flow as a negative number. This is because youmust inform the calculator which payments are cash inflows and which are cashoutflows. If you input all the cash flows as the same sign, the calculator will showan error when asked to compute the interest rate.„

Rearranging gives

To solve for the interest rate, i, take the third root of both sides of the equation. To do this, take 1.357 to the 1/3power using the yx button on your calculator.3 Doing this leads to

If you buy a gold coin for $350 and sell it three years later for $475, you earn a 10.7 percent return per year.

LG4-6

Time is an important factor in computing the return that you’re earning per year. Turning a $100 investment into

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$200 is a 100 percent return. If your investments earn this much in two years, then they earned a 41.42 percent rateof return per year [$100 × (1.4142)2 = $200]. Table 4.4 shows the annual interest rate earned for doubling aninvestment over various time periods. Notice how compounding complicates the solution: It’s not as simple as justdividing by the number of years. Getting a 100 percent return in two years means earning 41.42 percent per year, not50 percent per year. Table 4.4 also shows the Rule of 72 interest rate estimate.

▼ TABLE 4.4 Interest Rate per Year to Double an Investment

Return AsymmetriesSuppose you bought a gold coin for $700 last year and now the market will pay you only $350. Clearly, theinvestment earned a negative rate of return. Use a financial calculator or a time-value equation to verify that this is areturn of −50 percent. You lost half your money! So, in order to break even and get back to $700, you need to earn apositive 50 percent, right? Wrong. Note that to get from $350 to $700, your money needs to double! You need a 100percent return to make up for a 50 percent decline. Similarly, you need a gain of 33.33 percent to make up for a 25percent decline. If your investment declines by 10 percent, you’ll need an 11.11 percent gain to offset the loss. Ingeneral, only a higher positive return can offset any given negative return.

time out!4-10 Say you double your money in three years. Explain why the rate of return is NOT 33.3 percent per year.4-11 Show that you must earn a 25 percent return to offset a 20 percent loss.

4.6 • SOLVING FOR TIME LG4-8Sometimes you may need to determine the time period needed to accumulate a specific amount of money. If youknow the starting cash flow, the interest rate, and the future cash flow (the amount you will need), you can solve thetime value equations for the number of years that you will need to accumulate that money. Just as with solving fordifferent interest rates, solving for the number of periods is complicated and requires using natural logarithms.4Many people prefer to use a financial calculator to solve for the number of periods.

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When interest rates are 9 percent, how long will it take for a $5,000 investment to double? Finding the solution witha financial calculator entails entering

I = 9

PV = −5,000PMT = 0

FV = 10,000

The answer is 8.04 years, or eight years and two weeks. The Rule of 72 closely approximates the answer, whichpredicts eight years (= 72/9).

time out!4-12 In Example 4-5, how long will it take your company to double its sales?4-13 In what other areas of business can these time-value concepts be used?

EXAMPLE 4-5 Growth in Staffing Needs LG4-8

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Say that you are the sales manager of a company that producessoftware for human resource departments. You are planning yourstaffing needs, which depend on the volume of sales over time. Yourcompany currently sells $350 million of merchandise per year and hasgrown 7 percent per year in the past. If this growth rate continues, howlong will it be before the firm reaches $500 million in sales? How longbefore it reaches $600 million?

SOLUTION:

You could set up the following time line to illustrate the problem:

As shown in the margin, $350 million of sales growing at 7 percent peryear will reach $500 million in five years and three months. To reach$600 million will take just two weeks short of eight years.

Similar to Problems 4-31, 4-32, Self-Test Problem 4

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the Math Coach on…Simple TVM Spreadsheet Functions

Common spreadsheet programs include time value of money functions. The functions are

Compute a future value = FV(rate,nper,pmt,pv,type)

Compute a present value = PV(rate,nper,pmt,fv,type)

Compute the number of periods = NPER(rate,pmt,pv,fv,type)

Compute an interest rate = RATE(nper,pmt,pv,fv,type)

Compute a repeating payment = PMT(rate,nper,pv,fv,type)

The five input/outputs (FV, PV, NPER, RATE, PMT) work similarly to the five TVM buttons on a business calculator. The type inputdefaults to 0 for normal situations, but can be set to 1 for computing annuity due problems (see Chapter 5). Different spreadsheetprograms might have slightly different notations for these five functions.

Consider the future value problem of Example 4-1. The spreadsheet solution is the same as the TVM calculator solution. Note thatsince the PV is listed as a positive number, the FV output is a negative number.

The inputs to the function can be directed to other cells, like the rate, nper, and pv in the this illustration. Or the input can be the actualnumber, like pmt and type.This spreadsheet solves for the interest rate in the preceding example in the text. Just like using the TVM calculator, the PV and FVmust be of opposite signs to avoid getting an error message.

See this textbook’s online student center to watch instructional videos on using spreadsheets. Also note that the solution for all theexamples in the book are illustrated using spreadsheets in videos that are available in Connect.

Get Online

©JGI/Jamie Grill/Blend Images LLC.

Log in to your Connect course for study materials including self-test problems with solutions, answers tothe Time Out quizzes, guided example videos, and more.

Your Turn…Questions

1. List and describe the purpose of each part of a time line with an initial cash inflow and a future cash outflow.Which cash flows should be negative and which positive? Why? (LG4-1)

2. How are the present value and future value related? (LG4-2)

3. Would you prefer to have an investment earning 5 percent for 40 years or an investment earning 10 percent for20 years? Explain. (LG4-3)

4. How are present values affected by changes in interest rates? (LG4-4)

5. What do you think about the following statement? “I am going to receive $100 two years from now and $200three years from now, so I am getting a $300 future value.” How could the two cash flows be compared orcombined? (LG4-5)

6. Show how the Rule of 72 can be used to approximate the number of years to quadruple an investment. (LG4-6)

7. Without making any computations, indicate which of each pair has a higher interest rate: (LG4-7)

a. $100 doubles to $200 in five years or seven years.b. $500 increases in four years to $750 or to $800.c. $300 increases to $450 in two years or increases to $500 in three years.

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8. A $1,000 investment has doubled to $2,000 in eight years because of a 9 percent rate of return. How muchlonger will it take for the investment to reach $4,000 if it continues to earn a 9 percent rate? (LG4-8)

ProblemsBASIC PROBLEMS

4-1 Time Line Show the time line for a $500 cash inflow today, a $605 cash outflow in year 2, and a 10percent interest rate. (LG4-1)

4-2 Time Line Show the time line for a $400 cash outflow today, a $518 cash inflow in year 3, and a 9percent interest rate. (LG4-1)

4-3 One Year Future Value What is the future value of $500 deposited for one year earning an 8 percentinterest rate annually? (LG4-2)

4-4 One Year Future Value What is the future value of $400 deposited for one year earning an interest rateof 9 percent per year? (LG4-2)

4-5 Multiyear Future Value How much would be in your savings account in 11 years after depositing $150today if the bank pays 8 percent per year? (LG4-3)

4-6 Multiyear Future Value Compute the value in 25 years of a $1,000 deposit earning 10 percent per year.(LG4-3)

4-7 Compounding with Different Interest Rates A deposit of $350 earns the following interest rates:a. 8 percent in the first year.b. 6 percent in the second year.c. 5.5 percent in the third year.

What would be the third year future value? (LG4-3)4-8 Compounding with Different Interest Rates A deposit of $750 earns interest rates of 9 percent in the

first year and 12 percent in the second year. What would be the second year future value? (LG4-3)4-9 Discounting One Year What is the present value of a $350 payment in one year when the discount rate is

10 percent? (LG4-4)4-10 Discounting One Year What is the present value of a $200 payment in one year when the discount

rate is 7 percent? (LG4-4)4-11 Present Value What is the present value of a $1,500 payment made in nine years when the discount

rate is 8 percent? (LG4-4)4-12 Present Value Compute the present value of an $850 payment made in 10 years when the discount

rate is 12 percent. (LG4-4)4-13 Present Value with Different Discount Rates Compute the present value of $1,000 paid in three

years using the following discount rates: 6 percent in the first year, 7 percent in the second year, and 8percent in the third year. (LG4-4)

4-14 Present Value with Different Discount Rates Compute the present value of $5,000 paid in twoyears using the following discount rates: 8 percent in the first year and 7 percent in the second year.(LG4-4)

4-15 Rule of 72 Approximately how many years are needed to double a $100 investment wheninterest rates are 7 percent per year? (LG4-6)

4-16 Rule of 72 Approximately how many years are needed to double a $500 investment when interestrates are 10 percent per year? (LG4-6)

4-17 Rule of 72 Approximately what interest rate is needed to double an investment over five years? (LG4-6)

4-18 Rule of 72 Approximately what interest rate is earned when an investment doubles over 12 years?(LG4-6)

4-19 Rates over One Year Determine the interest rate earned on a $1,400 deposit when $1,800 is paidback in one year. (LG4-7)

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4-20 Rates over One Year Determine the interest rate earned on a $2,300 deposit when $2,900 is paidback in one year. (LG4-7)

INTERMEDIATE PROBLEMS

4-21 Interest-on-Interest Consider a $2,000 deposit earning 8 percent interest per year for five years.What is the future value, and how much total interest is earned on the original deposit versus howmuch is interest earned on interest? (LG4-3)

4-22 Interest-on-Interest Consider a $5,000 deposit earning 10 percent interest per year for 10 years.What is the future value, how much total interest is earned on the original deposit, and how much isinterest earned on interest? (LG4-3)

4-23 Comparing Cash Flows What would be more valuable, receiving $500 today or receiving $625 inthree years if interest rates are 7 percent? Why? (LG4-5)

4-24 Comparing Cash Flows Which cash flow would you rather pay, $425 today or $500 in two years ifinterest rates are 10 percent? Why? (LG4-5)

4-25 Moving Cash Flows What is the value in year 3 of a $700 cash flow made in year 6 if interest ratesare 10 percent? (LG4-5)

4-26 Moving Cash Flows What is the value in year 4 of a $1,000 cash flow made in year 6 if interest ratesare 8 percent? (LG4-5)

4-27 Moving Cash Flows What is the value in year 10 of a $1,000 cash flow made in year 3 if interestrates are 9 percent? (LG4-5)

4-28 Moving Cash Flows What is the value in year 15 of a $250 cash flow made in year 3 if interest ratesare 11 percent? (LG4-5)

4-29 Solving for Rates What annual rate of return is earned on a $1,000 investment when it grows to$1,800 in six years? (LG4-7)

4-30 Solving for Rates What annual rate of return is earned on a $5,000 investment when it grows to$9,500 in five years? (LG4-7)

4-31 Solving for Time How many years (and months) will it take $2 million to grow to $5 million with anannual interest rate of 7 percent? (LG4-8)

4-32 Solving for Time How long will it take $2,000 to reach $5,000 when it grows at 10 percent per year?(LG4-8)

ADVANCED PROBLEMS

4-33 Future Value At age 30 you invest $1,000 that earns 8 percent each year. At age 40 you invest$1,000 that earns 12 percent per year. In which case would you have more money at age 60?(LG4-2)

4-34 Future Value At age 25 you invest $1,500 that earns 8 percent each year. At age 40 you invest$1,500 that earns 11 percent per year. In which case would you have more money at age 65? (LG4-2)

4-35 Solving for Rates You invested $2,000 in the stock market one year ago. Today, the investment isvalued at $1,500. What return did you earn? What return would you need to get next year to breakeven overall? (LG4-7)

4-36 Solving for Rates You invested $3,000 in the stock market one year ago. Today, the investment isvalued at $3,750. What return did you earn? What return would you suffer next year for yourinvestment to be valued at the original $3,000? (LG4-7)

4-37 Solving for Rates What annual rate of return is earned on a $4,000 investment made in year 2 whenit grows to $6,500 by the end of year 7? (LG4-7)

4-38 Solving for Rates What annual rate of return is implied on a $2,500 loan taken next year when$3,500 must be repaid in year 4? (LG4-7)

4-39 General TVM Ten years ago, Hailey invested $2,000 and locked in a 9 percent annual interest ratefor 30 years (ending 20 years from now). Aidan can make a 20-year investment today and lock in a 10percent interest rate. How much money should he invest now in order to have the same amount ofmoney in 20 years as Hailey? (LG4-2, LG4-4)

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4-40 General TVM Ten years ago, Hailey invested $3,000 and locked in an 8 percent annual interest ratefor 30 years (ending 20 years from now). Aidan can make a 20-year investment today and lock in a 10percent interest rate. How much money should he invest now in order to have the same amount ofmoney in 20 years as Hailey? (LG4-2, LG4-4)

4-41 Moving Cash Flows You are scheduled to receive a $500 cash flow in one year, a $1,000 cash flowin two years, and pay an $800 payment in three years. If interest rates are 10 percent per year, what isthe combined present value of these cash flows? (LG4-5)

4-42 Spreadsheet Problem Oil prices have increased a great deal in the last decade. The following tableshows the average oil price for each year since 1949. Many companies use oil products as a resourcein their own business operations (like airline firms and manufacturers of plastic products). Managersof these firms will keep a close watch on how rising oil prices will impact their costs. The interest ratein the PV/FV equations can also be interpreted as a growth rate in sales, costs, profits, and so on (seeExample 4-5).

a. Using the 1949 oil price and the 1969 oil price, compute the annual growth rate in oil prices duringthose 20 years.

b. Compute the annual growth rate between 1969 and 1989 and between 1989 and 2015.c. Given the price of oil in 2015 and your computed growth rate between 1989 and 2015, compute the

future price of oil in 2018 and 2025.

Average Oil Prices

Year Per Barrel Year Per Barrel Year Per Barrel

1949 $2.54 1971 $ 3.39 1993 $14.25

1950 $2.51 1972 $ 3.39 1994 $13.19

1951 $2.53 1973 $ 3.89 1995 $14.62

1952 $2.53 1974 $ 6.87 1996 $18.46

1953 $2.68 1975 $ 7.67 1997 $17.23

1954 $2.78 1976 $ 8.19 1998 $10.87

1955 $2.77 1977 $ 8.57 1999 $15.56

1956 $2.79 1978 $ 9.00 2000 $26.72

1957 $3.09 1979 $12.64 2001 $21.84

1958 $3.01 1980 $21.59 2002 $22.51

1959 $2.90 1981 $31.77 2003 $27.54

1960 $2.88 1982 $28.52 2004 $38.93

1961 $2.89 1983 $26.19 2005 $46.47

1962 $2.90 1984 $25.88 2006 $58.30

1963 $2.89 1985 $24.09 2007 $64.67

1964 $2.88 1986 $12.51 2008 $91.48

1965 $2.86 1987 $15.40 2009 $53.48

1966 $2.88 1988 $12.58 2010 $71.21

1967 $2.92 1989 $15.86 2011 $87.04

1968 $2.94 1990 $20.03 2012 $93.02

1969 $3.09 1991 $16.54 2013 $97.91

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1970 $ 3.18 1992 $15.99 2014 $93.26

2015 $48.79

4-43 Spreadsheet Problem Consider that you are the marketing manager of a firm. You need to haveapproximately one additional salesperson for every $10 million in sales. You currently have $50million in sales and have five employees handling the sales accounts. In order to plan ahead, you wantto get an idea of when you may need to hire more salespeople. Build a table that shows the sales foreach of the next 10 years for sales growth of 5%, 10%, 15%, and 20% (see Example 4-5).

Comment on when new sales staff should be hired for each growth rate.

A B C D E F G H I J K L

1 Growth Rate Today Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 Year 8 Year 9 Year 10

2 5% $50 $52.50

3 10% $50 $55.00

4 15% $50 $57.50

5 20% $50 $60.00

NotesCHAPTER 41 No one seems to know exactly what he said, when he said it, or to whom. Similar statements commonly attributed to Einstein are: (1)

compound interest is the greatest wonder of the universe, (2) compound interest is the ninth wonder of the world, and (3) it is thegreatest mathematical discovery of all time. If he did not say any of these things, he (or someone else) should have!

2 The terms interest rate and rate of return are referring to the same thing. However, it is a common convention to refer to interest ratewhen you are the one paying the cash flows and refer to rate of return when you are the one receiving the cash flows.

3 The general equation for computing the interest rate is

4 The equation for solving for the number of periods is

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Chapter fivetime value of money 2:analyzing annuity cash flows

©Stuart Monk/Shutterstock

e explained basic time value computations in the previous chapter. Those TVM equations coveredmoving a single cash flow from one point in time to another. While this circumstance doesdescribe some problems that businesses and individuals face, most debt and investment

applications of time value of money feature multiple cash flows. In fact, most situations require manyequal payments over time. Since these situations require a bit more complicated analysis, this chaptercontinues the TVM topic for applications that require many equal payments over time. For example, carloans and home mortgage loans require the borrower to make the same monthly payment for manymonths or years. People save for the future through monthly contributions to their pension portfolios.People in retirement must convert their savings into monthly income. Companies also make regularpayments. Johnson & Johnson (ticker: JNJ) will pay level semiannual interest payments through 2033 onmoney it borrowed. The Boeing Company (ticker: BA) paid a $0.42 per share quarterly dividend tostockholders for three straight years until 2012, when it switched to a $0.44 dividend. These examples

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require payments (and compounding) over different time intervals (monthly for car loans andsemiannually for company debt). How are we to value these payments into common or comparableterms? In this chapter, we illustrate how to value multiple cash flows over time, including equal anduneven payments, and how to incorporate different compounding frequencies.

LEARNING GOALS

LG5-1 Compound multiple cash flows to the future.LG5-2 Compute the future value of frequent, level cash flows.LG5-3 Discount multiple cash flows to the present.LG5-4 Compute the present value of an annuity.LG5-5 Figure cash flows and present value of a perpetuity.LG5-6 Adjust values for beginning-of-period annuity payments.LG5-7 Explain the impact of compound frequency and the difference between the annual percentage rate and the effective annual rate.LG5-8 Compute the interest rate of annuity payments.LG5-9 Compute payments and amortization schedules for car and mortgage loans.LG5-10 Calculate the number of payments on a loan.

viewpointsbusiness APPLICATIONWalkabout Music, Inc., issued $20 million in debt 10 years ago to finance its factory construction. The debt allows Walkabout to makeinterest-only payments at a 7 percent coupon rate, paid semiannually for 30 years. Debt issued today would carry only 6 percentinterest. The company’s CFO is considering whether or not to issue new debt (for 20 years) to pay off the old debt. To pay off the olddebt early, Walkabout would have to pay a special “call premium” totaling $1.4 million to its debt holders. To issue new debt, the firmwould have to pay investment bankers a fee of $1.2 million. Should the CFO replace the old debt with new debt? (See the solution atthe end of the book.)

5.1 • FUTURE VALUE OF MULTIPLE CASH FLOWS LG5-1Chapter 4 illustrated how to take single payments and compound them into the future. To save enough money for adown payment on a house or for retirement, people typically make many contributions over time to their savingsaccounts. We can add the future value of each contribution together to see what the total will be worth at somefuture point in time—such as age 65 for retirement or in two years for a down payment on a house.

Finding the Future Value of Several Cash FlowsConsider the following contributions to a savings account over time. You make a $100 deposit today, followed by a$125 deposit next year, and a $150 deposit at the end of the second year. If interest rates are 7 percent, what’s thefuture value of your deposits at the end of the third year? The time line for this problem is illustrated as

Note that the first deposit will compound for three years. That is, the future value in year 3 of a cash flow in year 0will compound 3 v(= 3 − 0) times. The deposit at the end of the first year will compound twice (= 3 − 1). In general,a deposit in year m will compound N−m times for a future value in year N. We can find the total amount at the endof three years by computing the future value of each deposit and then adding them together. Using the future valueequation from Chapter 4, the future value of today’s deposit is $100 ×  (1 + 0.07) 3 = $122.50. Similarly, the future

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value of the next two deposits are $125 ×  (1 + 0.07) 2 = $143.11 and $150 ×  (1 + 0.07) 1 = $160.50, respectively.

Putting these three individual future value equations together would yield

personal APPLICATIONSay that you obtained a mortgage for $150,000 three years ago when you purchased your home. You’ve been paying monthlypayments on the 30-year mortgage with a fixed 8 percent interest rate and have $145,920.10 of principal left to pay. Recently, yourmortgage broker called to mention that interest rates on new mortgages have declined to 7 percent. He suggested that you could savemoney every month if you refinanced your mortgage. You could find a 27-year mortgage at the new interest rate for a $1,000 fee.Should you refinance your mortgage? (See the solution at the end of the book.)

But what if you want to move in the next few years? Is it still a good idea?

The general equation for computing the future value of multiple and varying cash flows (or payments) is

(5-1)

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In this equation, the letters m, n, and p denote when the cash flows occur in time, so N − m etc. represent the lengthof time that the respective cash flow will get to earn compound interest. Each deposit can be different from theothers.

EXAMPLE 5-1 Saving for a Car LG5-1

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Say that, as a freshman in college, you will be working as a housepainter for each of the next three summers. You intend to set asidesome money from each summer’s paycheck to buy a car for yoursenior year. If you can deposit $2,000 at the end of the first summer,$2,500 at the end of the second summer, and $3,000 at the end of thelast summer, how much money will you have to buy a car at the end ofthe last summer if interest rates are 5 percent?

SOLUTION:

The time line for the forecast is

The first cash flow, which occurs at the end of the first year, willcompound for two years. The second cash flow will be invested for onlyone year. The last contribution will not have any time to grow before thepurchase of the car. Using equation 5-1, the solution is

You will have $7,830 in cash to purchase a car for your senior year.

Similar to Problems 5-1, 5-2, 5-17, 5-18, 5-43, 5-44

Future Value of Level Cash Flows LG5-2Now suppose that each cash flow is the same and occurs every year. Level sets of frequent cash flows are commonin finance—we call them annuities. The first cash flow of an annuity occurs at the end of the first year (or other timeperiod) and continues every year to the last year. We derive the equation for the future value of an annuity from thegeneral equation for future value of multiple cash flows, equation 5-1. Since each cash flow is the same, and thecash flows are every period, the equation appears as

annuity A stream of level and frequent cash flows paid at the end of each time period—often referred to as an ordinary annuity.

▼

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The term FVA is used to denote that this is the future value of an annuity. Factoring out the common level cash flow,PMT, we can summarize and reduce the equation as

(5-2)

Suppose that $100 deposits are made at the end of each year for five years. If interest rates are 8 percent per year, thefuture value of this annuity stream is computed using equation 5-2 as

We can show these deposits and future value on a time line as

Five deposits of $100 each were made. So, the $586.66 future value represents $86.66 of interest paid. As withalmost any TVM problem, the length of time of the annuity and the interest rate for compounding are very importantfactors in accumulating wealth within the annuity. Consider the examples in Table 5.1. A $50 deposit made everyyear for 20 years will grow to $1,839.28 with a 6 percent interest rate. Doubling the annual deposits to $100 alsodoubles the future value to $3,678.56. However, making $100 deposits for twice the amount of time, 40 years, morethan quadruples the future value to $15,476.20! Longer time periods lead to more total compounding and muchmore wealth. Interest rates also have this effect. Doubling the interest rate from 6 to 12 percent on the 40-yearannuity results in nearly a five-fold increase in the future value to $76,709.14. Think about it: Depositing only $100per year (about 25 lattes per year) can generate some serious money over time. See Figure 5.1. How much would$2,000 annual deposits generate?

▼ TABLE 5.1 Magnitude of Periodic Payments, Number of Years Invested, and Interest Rate on FV of Annuity

FIGURE 5-1 Future Value of a $100 Annuity at 6 Percent

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Longer time periods lead to more total compounding and much more wealth.

the Math Coach on…Annuities and the Financial Calculator

“In the previous chapter, the level payment button (PMT) in the financial calculator was always set to zerobecause no constant payments were made every period. We use the PMT button to input the annuity amount.For calculators, the present value is of the opposite sign (positive versus negative) from the future value. Thisis also the case with annuities. The level cash flow will be of the opposite sign as the future value, as the timeline on page 118 shows.

You would use the financial calculator to solve the problem of depositing $100 for five years via the followinginputs: N = 5, I = 8, PV = 0, PMT = –100. In this case, the input for present value is zero because no deposit ismade today. The result of computing the future value is 586.66.„

Future Value of Multiple AnnuitiesAt times, multiple annuities can occur in both business and personal life. For example, you may find that you canincrease the amount of money you save each year because of a promotion or a new and better job. As an illustration,reconsider the annual $100 deposits made for five years at 8 percent per year. This time, the deposit can be increasedto $150 for the fourth and fifth years. How can we use the annuity equation to compute the future value when wehave two levels of cash flows? In this case, the cash flow can be categorized as two annuities. The first annuity is a$100 cash flow for five years. The second annuity is a $50 cash flow for two years. We demonstrate this as

the Math Coach on…Solving Multiple Annuities

“The trick to solving multiple annuity problems is to disentangle cash flows into groups of level paymentsending in the future value year that we’ve designated.„

To determine the future value of these two annuities, compute the future value of each one separately, and thensimply add them together. The future value of the $100 annuity is the same as computed before, $586.66. The futurevalue of the $50 annuity, using the TVM equation for the future value of a cash stream, is

EXAMPLE 5-2Saving in the Company PensionPlan LG5-2

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

You started your first job after graduating from college. Your companyoffers a retirement plan for which the company contributes 50 percentof what you contribute each year. So, if you contribute $3,000 per yearfrom your salary, the company adds another $1,500. You get to decidehow to invest the total annual contribution from several portfoliochoices that the plan administrator provides. Suppose that you pick amixture of stocks and bonds that is expected to earn 7 percent peryear. If you plan to retire in 40 years, how big will you expect thatretirement account to be? If you could earn 8 percent per year, howmuch money would be available?

SOLUTION:

Every year, you and your employer will set aside a total of $4,500 foryour retirement. Using equation 5-2 shows that the future value of thisannuity is

Note that you can build a substantial amount of wealth ($898,358)

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through your pension plan at work. If you can earn just 1 percent moreeach year, 8 percent total, you could be a millionaire!

Similar to Problems 5-3, 5-4, Self-Test Problem 1

finance at work //: personalWho Will Save for Their Future?

©Chris Ryan/Age Fotostock

Though it seems way too early for you to think about planning for your “golden years,” financially wise people realize that it’s never tooearly to start. Unfortunately, most people save little for their retirement years. Twenty-nine percent of households age 55 and older haveno pension or retirement savings. About 23 percent have no savings, but do have a pension. Of the 48 percent that have retirementsavings, the median amount is $109,000. How far does that get you? Using a 6 percent investment return, $109,000 can generate amonthly income of only $653.51 for 30 years, at which time it is used up. That is less than $8,000 per year! The average Social Securitymonthly benefit is just over $1,318 per month, or about $15,816 per year. Few people will have the lifestyle they wanted in theirretirement years. However, that doesn’t have to be true for you if you start saving early!

This chapter illustrates that much higher amounts of wealth can be accumulated if you start early. One easy way to do this is througha retirement plan at work. Most company and government employers offer their employees defined contribution plans. (The corporateversion is called a 401(k) plan; a nonbusiness plan is usually referred to as a 403(b) plan—both named after the legislation that createdthe plans.) These plans place all of the responsibility on employees to provide for their retirement. Employees contribute from their ownpaychecks and decide how to invest. Employees’ decisions about how much to contribute and how early to start contributing have adramatic impact on retirement wealth.

Consider employees who earn $50,000 annually for 40 years and then retire. Note that if the employees contribute for 40 years, theymust start by age 25 or so—starting early is vitally important! Contributing 5 percent of their salaries ($2,500) to the 401(k) plan everyyear and having it earn a 4 percent return will generate $237,564 for retirement. A 10 percent contribution ($5,000) would create$475,128 for retirement. Finally, investment decisions that yield an 8 percent return would yield $1.3 million with a 10 percentcontribution. This is quite a range of retirement wealth generated from just three important decisions each employee must make—howmuch to contribute, how to invest the funds, and when to start! Unfortunately, too many people make poor decisions. Their first mistakeis to start making 401(k) contributions too late to allow the funds to generate significant compounding.

Saving and investing money through a defined contribution plan is a good way to build wealth for retirement. But you must followthese rules: Start early, save much, and don’t touch!

Want to know more?Key Words to Search for Updates: Employee Benefit Research Institute (go to www.ebri.org), retirement income

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Source: “Retirement Security: Most Households Approaching Retirement Have Low Savings,” Government Accountability Office, GAO-15-419, May 12, 2015. http://www.gao.gov/products/GAO-15-419.

So, the future value of both of the annuities is $690.66 (= $586.66 + $104). In the same way, we could easilycompute the future value if the last two cash flows are $50 lower ($50 each), instead of $50 higher ($150 each). Tosolve this alternative version, we would simply subtract the $104 future value instead of adding it.

EXAMPLE 5-3Growing RetirementContributions LG5-2

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

In the previous example, you are investing a total of $4,500 per year for40 years in your employer’s retirement program. You believe that withraises and promotions, you will eventually be able to contribute moremoney each year. Consider that halfway through your career, you areable to increase your investment in the retirement program to $6,000per year (your contribution plus the company match). What would bethe future value of your retirement wealth from this program ifinvestments are compounded at 7 percent?

SOLUTION:

You can compute the future value using two annuities. The first annuityis one with payments of $4,500 that lasts 40 years. The second is a$1,500 (= $6,000 – $4,500) annuity that lasts only 20 years. Wealready computed the future value of the first annuity in the previousexample: $898,358. The future value of the second annuity is

So, your retirement wealth from this program would be $959,851 (=$898,358 + $61,493).

Similar to Problems 5-19, 5-20, Self-Test Problem 1

time out!5-1 Describe how compounding affects the future value computation of an annuity.5-2 Reconsider your original retirement plan example to invest $4,500 per year for 40 years. Now consider the result if you don’t

contribute anything for four years (years 19 to 22) while your child goes to college. How many annuity equations will youneed to find the future value of your 401(k) in this situation?

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5.2 • PRESENT VALUE OF MULTIPLE CASH FLOWS LG5-3The future value concept is very useful to understand how to build wealth for the future. The present value conceptwill help you most particularly for personal applications such as evaluating loans (like car and mortgage loans) andbusiness applications (like determining the value of business opportunities).

Finding the Present Value of Several Cash FlowsConsider the cash flows that we showed at the very beginning of the chapter: You deposit $100 today, followed by a$125 deposit next year, and a $150 deposit at the end of the second year. In the previous situation, we sought thefuture value when interest rates are 7 percent. Instead of future value, we compute the present value of thesethreecash flows. The time line for this problem appears as

The first cash flow is already in year zero, so its value will not change. We will discount the second cash flow oneyear and the third cash flow two years. Using the present value equation from the previous chapter, the present valueof today’s payment is simply $100 ÷ (1 + 0.07)0 = $100. Similarly, the present value of the next two cash flows are$125 ÷ (1 + 0.07)1 = $116.82 and $150 ÷ (1 + 0.07)2 = $131.02, respectively. Therefore, the present value of thesecash flows is $347.84 (= $100 + $116.82 + $131.02).

Putting these three individual present value equations together would yield

The general equation for discounting multiple and varying cash flows is

(5-3)

In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can differ from theothers in terms of size and timing.

Present Value of Level Cash Flows LG5-4You will find that this present value of an annuity concept will have many business and personal applicationsthroughout your life. Most loans are set up so that the amount borrowed (the present value) is repaid through levelpayments made every period (the annuity). Lenders will examine borrowers’ budgets and determine how much eachborrower can afford as a payment. The maximum loan offered will be the present value of that annuity payment. Theequation for the present value of an annuity can be derived from the general equation for the present value ofmultiple cash flows, equation 5-3. Since each cash flow is the same, and the borrower pays the cash flows everyperiod, the present value of an annuity, PVA, can be written as

(5-4)

Suppose that someone makes $100 payments at the end of each year for five years. If interest rates are 8 percent peryear, the present value of this annuity stream is computed using equation 5-4 as

The time line for these payments and present value appears as

Notice that although five payments of $100 each were made, $500 total, the present value is only $399.27. As we’venoted previously, the span of time over which the borrower pays the annuity and the interest rate for discountingstrongly affect present value computations. When you borrow money from the bank, the bank views the amount itlends as the present value of the annuity it receives over time from the borrower. Consider the examples in Table5.2.

A $50 deposit made every year for 20 years is discounted to $573.50 with a 6 percent discount rate. Doubling theannual cash flow to $100 also doubles the present value to $1,146.99. But extending the time period does not impactthe present value as much as you might expect. Making $100 payments for twice the amount of time—40 years—does not double the present value. As you can see in Table 5.2, the present value increases less than 50 percent toonly $1,504.63! If the discount rate increases from 6 percent to 12 percent on the 40-year annuity, the present valuewill shrink to $824.38.

The present value of a cash flow made far into the future is not very valuable today, as Figure 5.2 illustrates. Thatswhy doubling the number of years in the table from 20 to 40 only increased the present value by approximately 30percent. Notice how the present value of $100 annuity payments declines for the cash flows made later in time,especially at higher discount rates. The $100 cash flow in year 20 is worth less than $15 today if we use a 10 percentdiscount rate; they’re worth more than double, at nearly $38 today, if we use a discount rate of 5 percent. The figurealso shows how quickly present value declines with a higher discount rate relative to a lower rate. As we showedabove, the present values of the annuities in the figure are the sums of the present values shown. Since the presentvalues for the 10 percent discount rate are smaller, the present value of an annuity is smaller as interest rates rise.

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▼

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▼ TABLE 5.2 Magnitude of the Annuity, Number of Years Invested, and Interest Rate on PV

Present Value of Multiple AnnuitiesJust as we can combine annuities to solve various future value problems, we can also combine annuities to solvesome present value problems with changing cash flows. Consider David Price’s Major League Baseball contractsigned in 2015 with the Boston Red Sox. This was the largest contract for a pitcher ever. It was reported as having a$217 million value. The contract was structured so that the Red Sox paid Price $30 million per year in 2016 through2018, $31 million in 2019, and $32 million per year in 2020 through 2022.1 However, we know that future cashflows have lower present values. So, using a 5 percent discount rate, what is the present value of Prices contract?

FIGURE 5-2 Present Value of Each Annuity Cash Flow

the Math Coach on…Using a Financial Calculator—Part 2

The five TVM buttons/functions in financial calculators have been fine, so far, for the types of TVM problems we’ve been solving.Sometimes we had to use them two or three times for a single problem, but that was usually because we needed an intermediatecalculation to input into another TVM equation.

Luckily, most financial calculators also have built-in worksheets specifically designed for computing TVM in problems with multiplenonconstant cash flows.

To make calculator worksheets as flexible as possible, they are usually divided into two parts: one for input, which we’ll refer to as theCF (cash flow) worksheet, and one or more for showing the calculator solutions. We’ll go over the conventions concerning the CFworksheet here, and we’ll discuss the output solutions in Chapter 13.

The CF worksheet is usually designed to handle inputting sets of multiple cash flows as quickly as possible. As a result, it normallyconsists of two sets of variables or cells—one for the cash flows and one to hold a set of frequency counts for the cash flows, so thatwe can tell it we have seven $1,500 cash flows in a row instead of having to enter $1,500 seven times.

Using the frequency counts to reduce the number of inputs is handy, but you must take care. Frequency counts are only good forembedded annuities of identical cash flows. You have to ensure that you don’t mistake another kind of cash flow for an annuity.

Also, using frequency counts will usually affect the way that the calculator counts time periods. As an example, let’s talk about how wewould put the set of cash flows shown here into a CF worksheet:

To designate which particular value we’ll place into each particular cash flow cell in this worksheet, we’ll note the value and the cellidentifier, such as CF0, CF1, and so forth. We’ll do the same for the frequency cells, using F1, F2, etc., to identify which CF cell thefrequency cell goes with. (Note that, in most calculators, CF0 is treated as a unique value with an unalterable frequency of 1; we’regoing to make the same assumption here so you’ll never see a listing for F0.) For this sample time line, our inputs would be

−$800 [CF0]

$150 [CF1] 1 [F1]

$200 [CF2] 1 [F2]

$ 0 [CF3] 1 [F3]

$150 [CF4] 3 [F4]

$ 75 [CF5] 2 [F5]

To compute the present value of these cash flows, use the NPV calculator function. The NPV function computes the present value of allthe future cash flows and then adds the year 0 cash flow. Then, on the NPV worksheet, you would simply need to enter the interest rateand solve for the NPV:

10% [I]

[CPT] [NPV] = −$144.61

Note a few important things about this example:

1. We had to manually enter a value of $0 for CF3. If we hadn’t, the calculator wouldn’t have known about it and would haveimplicitly assumed that CF4 came one period after CF2.

2. Once we use a frequency cell for one cash flow, all numbering on any subsequent cash flows that we enter into the calculator isgoing to be messed up, at least from our point of view. For instance, the first $75 isn’t what we would call “CF5,” is it? We’d call it“CF7” because it comes at time period 7; but calculators usually treat CF5 as “the fifth set of cash flows,” so we’ll just have to tryto do the same to be consistent.

3. If we really don’t need to use frequency cells, we will usually just leave them out of the guidance instructions in this chapter tosave space.

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The reported value for many sports contracts may be misleading in present value terms.©Akihiro Sugimoto/age fotostock

We begin by showing the salary cash flows with the time line

First create a $32 million, seven-year annuity. Here are the associated cash flows:

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Now create a –$1 million, four-year annuity:

Notice that creating the –$1 million annuity also resulted in the third annuity of –$1 million for three years. Thistime line shows three annuities. If you add the cash flows in any year, the sum is Price’s salary for that year. Nowwe can find the present value of each annuity using equation 5-4 three times.

Adding the value of the three annuities reveals that the present value of Price’s salary was $178.89 million(= $185.16m – $3.55m – $2.72m). So, the present value of Price’s contract turns out to be quiteconsiderable, but it is not the $217 million contract value advertised!

EXAMPLE 5-4 Value of Payments LG5-4

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to Connect or go tomhhe.com/CornettM4e.

Your firm needs to buy additional physical therapy equipment that costs$20,000. The equipment manufacturer will give you the equipment nowif you will pay $6,000 per year for the next four years. If your firm canborrow money at a 9 percent interest rate, should you pay themanufacturer the $20,000 now or accept the four-year annuity offer of$6,000?

SOLUTION:

We can find the cost of the four-year, $6,000 annuity in present valueterms using equation 5-4:

The cost of paying for the equipment over time is $19,438.32. This isless, in present value terms, than paying $20,000 cash. The firm shouldtake the annuity payment plan.

Similar to Problems 5-7, 5-8, Self-Test Problem 2

Perpetuity—A Special Annuity LG5-5A perpetuity is a special type of annuity with a stream of level cash flows that are paid forever. These arrangementsare called perpetuities because payments are perpetual. Assets that offer investors perpetual payments are preferredstocks and British 2½% Consolidated Stock, a debt referred to as consols.

perpetuity An annuity with cash flows that continue forever.

consols Investment assets structured as perpetuities.

The value of an investment like this is the present value of all future annuity payments. As the cash flow continuesindefinitely, we can’t use equation 5-4. Luckily, mathematicians have figured out that when the number of periods,N, in equation 5-4 goes to infinity, the equation reduces to a very simple one:

(5-5)

For example, the present value of an annual $100 perpetuity discounted at 10 percent is $1,000 (= $100 ÷ 0.10).Compare this to the present value of a $100 annuity of 40 years as shown in Table 5.2. The 40-year annuity’s valueis $977.91. You’ll see that extending the payments from 40 years to an infinite number of years adds only $22.09 (=$1,000 − $977.91) of value. This demonstrates once again how little value today is placed on cash flows paid manyyears into the future.

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5.3 • ORDINARY ANNUITIES VERSUS ANNUITIES DUE LG5-6So far, we’ve assumed that every cash flow comes in at the end of every period. But in many instances, cash flowscome in at the beginning of each period. An annuity in which the cash flows occur at the beginning of each period iscalled an annuity due.

annuity due An annuity in which cash flows are paid at the beginning of each time period.

Consider the five-year $100 annuity due. The cash flow in the beginning of year 1 looks like it’s actually a cash flowtoday. Annuity due moved the cash flow from the end of the year to the beginning, which looks like the end of theprevious year.

Note that these five annuity-due cash flows are essentially the same as a payment today and a four-year ordinaryannuity.

time out!5-3 How important is the magnitude of the discount rate in present value computations? Do significantly higher interest rates

lead to significantly higher present values?

5-4 Reconsider the physical therapy equipment example. If interest rates are only 7 percent, should you pay the up-front fee orthe annuity?

Future Value of an Annuity Due So, how do we calculate the future value of the five-year annuity due shown inthe time line? The first cash flow of an ordinary five-year annuity can compound for four years. The last cash flowdoes not compound at all. From the time line, you can see that the first cash flow of the annuity due essentiallyoccurs in year zero, or today. So the first cash flow compounds for five years. The last cash flow of an annuity duecompounds one year. The main difference between an annuity due and an ordinary annuity is that all the cash flowsof the annuity due compound one more year than the ordinary annuity. The future value of the annuity due willsimply be the future value of the ordinary annuity multiplied by (1 + i):

(5-6)

Earlier in the chapter, the future value of this ordinary annuity was shown to be $586.66. Therefore, the future valueof the annuity due is $633.59 (= $586.66 × 1.08).

finance at work //: behavioralTake Your Lottery Winnings Now or Later?

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©McGraw-Hill Education

On January 13, 2016, three lottery tickets co-won a record Powerball jackpot of $1.5 billion. So each ticket won $500 million. Thewinners had two choices for payment: They could take a much-discounted lump sum cash payment immediately or take 30 annuitypayments (one immediately and then one every year for 29 years, which is a 30-year annuity due). The annuity payment would be$16.67 million (= $500 million ÷ 30) for 30 years. The alternative immediate lump sum was $309.97 million before taxes. One way todecide between the two alternatives would be to use the time value of money concepts. The winners might have computed the presentvalue of the annuity and compared it to the lump sum cash payment.

At the time, long-term interest rates were 3.9 percent. The present value of the annuity offered was $303.11 million per winning ticket.(Compute this yourself.) Notice that winning $500 million does not deliver $500 million of value! If the decision was made from thisperspective, the group should choose to take the lump sum because it has more value than the annuity alternative, and that is what thethree winners did. In fact, most winners do. Financial advisors tend to recommend that lottery winners take the lump sum because theybelieve that the money can earn a higher return than the 3.9 percent interest rate. Of course, you pay income taxes too. The 39.6%federal tax rate brings the lump sum payment down to $187.2 million.

Good reasons arise for taking the annuity, however. To earn the higher return on the lump sum, the advisor (and the group ofowners) would have to take risks. In addition, most of the lump sum would have to be invested. But most people who choose the lumpsum end up spending much of it in the first couple of years. Stories abound about lottery winners who declare bankruptcy a few yearsafter receiving their money. Choosing the annuity helps instill financial discipline, since the winners can’t waste money today that theywon’t receive for years.

Want to know more?Key Words to Search for Updates: Powerball winners (go to www.powerball.com)

Source: Riley, Charles, Sara Sidner, and Tina Burnside, “We Have Powerball Winners!,” CNN Money, January 13, 2016,http://money.cnn.com/2016/01/13/news/powerball-winner-lottery.

Present Value of an Annuity Due What is a five-year annuity due, shown previously, worth today? Rememberthat we discount the first cash flow of an ordinary five-year annuity one year. We discount the last cash flow for thefull five years. But since the first cash flow of the annuity due is already paid today, we don’t discount it at all. Wediscount the last cash flow of an annuity due only four years. Indeed, we discount all the cash flows of the annuitydue one year less than we would discount the ordinary annuity. Therefore, the present value of the annuity due issimply the present value of the ordinary annuity multiplied by (1 + i):

(5-7)

the Math Coach on…Setting Financial Calculators for Annuity Due

“Financial calculators can be set for beginning-of-period payments. Once set, you compute future andpresent values of annuities due just as you would the ordinary annuity. To set the HP calculator, press thecolor button followed by the BEG/END button. To set the TI calculator for an annuity due, push the 2NDbutton, followed by the BGN button, followed by the 2ND button again, followed by the SET button, andfollowed by the 2ND button a third time, finally the QUIT button. To set the HP and TI calculators back to end-of-period cash flows, repeat these procedures.„

Earlier in the chapter, we discovered that the present value of this ordinary annuity was $399.27. So the presentvalue of the annuity due is $431.21 (= $399.27 × 1.08).

Interestingly, we make the same adjustment, (1 + i), to both the ordinary annuity present value and future value tocompute the annuity due value.

5.4 • COMPOUNDING FREQUENCY LG5-7So far, all of our examples and illustrations have used annual payments and annual compounding or discountingperiods. But many situations that use cash flow time-value-of-money analysis require more frequent or less frequenttime periods than simple yearly entries. Bonds make semiannual interest payments; stocks pay quarterly dividends.Most consumer loans require monthly payments. Monthly payments require monthly compounding. In this section,we’ll discuss the implications of compounding more than once a year.

time out!5-5 In what situations might you need to use annuity due analysis instead of an ordinary annuity analysis?5-6 Reconsider your retirement plan earlier in this chapter. What would your retirement wealth grow to be if you started

contributing today?

Effect of Compounding FrequencyConsider a $100 deposit made today with a 12 percent annual interest rate. What’s the future value of this deposit inone year? Equation 4-2 from the previous chapter shows that the answer is $112. What would happen if the bankcompounded the interest every six months instead of at the end of the year? Halfway through the year, the bankwould compute that the deposit has grown 6 percent (half the annual 12 percent rate) to $106. At the end of the year,the bank would compute another 6 percent interest payment. However, this 6 percent is earned on $106, not theoriginal $100 deposit. The end-of-year value is therefore $112.36 (= $106 × 1.06). By compounding twice per yearinstead of just once, the future value is $0.36 higher. Though this amount may seem negligible, you might besurprised to see how quickly the difference becomes significant.

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Instead of compounding annually or semiannually, what might happen if compounding were quarterly? Since eachyear contains four quarters, the interest rate per quarter would be 3 percent (= 12 percent ÷ 4 quarters). The futurevalue in one year, compounded quarterly, is $112.55 (= $100 × 1.034). Again, the compounding frequency increasedand so did the future value. Table 5.3 shows the effect of various compounding frequencies. We’d like to draw yourattention to two important points in the table. First, the higher the compound frequency, the higher the future valuewill be. Second, the relative increase in value from increasing compounding frequency seems to diminish withincreasing frequencies. For example, increasing frequency from annual to semiannual increased the future value by36 cents. However, increasing frequency from daily to hourly compounding increases the future value by only 0.1cent.2

▼ TABLE 5.3 Future Value in One Year and Compounding Frequency of $100 at 12 percent

A B C D

1 Frequency Period Interest Rate Future Value Equation Future Value

2 Annual 12% $100 × 1.121$ 112.00

3 Semiannual 6 $100 × 1.062112.36

4 Quarterly 3 $100 × 1.034112.55

5 Monthly 1 $100 × 1.0112112.68

6 Daily 0.032877 $100 × 1.00032877365112.748

7 Hourly 0.00136986 $100 × 1.00001369868760112.749

The higher the compound frequency, the higher the future value will be.

When we work with annuity cash flows, the compound frequency used is the same as the timing of the cash flows.When annuity cash flows are paid monthly, then interest is also compounded monthly, as seen in Examples 5-5 and5-6.

EARs and APRs If you borrowed $100 at a 12 percent interest rate, you would expect to pay $112 in one year. Ifthe loan compounded monthly, then you would owe $112.68 at the end of the year, as Table 5.3 shows. So a 12percent loan compounded monthly means that you really pay more than 12 percent. In fact, you would pay 12.68percent. In this example, the 12 percent rate is called the annual percentage rate (APR). The 12.68 percent is called theeffective annual rate (EAR)—a more accurate measurement of what you will actually pay.

annual percentage rate (APR) The interest rate per period times the number of periods in a year.

effective annual rate (EAR) The interest rate per period times the number of periods in a year.

Lenders are legally required to show potential borrowers the APR on any loan offered. While the difference in APRand EAR is not that large in this example, it’s interesting that the law requires only the less accurate (and lower) oneto be shown. Since the EAR is a more accurate measure of what you will pay, it’s useful to know how to convert astated APR to an EAR. Equation 5-8 shows this conversion with a compounding frequency of m times per year:

(5-8)

Table 5.4 shows various EAR conversions. If compounding occurs annually, you will see that the EAR and the APRwill be the same. If compounding happens more than once a year, then the EAR will be higher than the APR. Thetable also demonstrates that the compound frequency effect grows substantially for higher interest rates or longerterm loans. Compounded quarterly, the EAR is hardly different at all from a 5 percent APR: 5.09 percent versus 5

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percent. The difference is larger when the APR is 12 percent. Compounded quarterly, the EAR is higher at 12.55percent.

the Math Coach on…Annuity Computations in Spreadsheets

“The TVM functions in a spreadsheet handle annuity payments similar to financial calculators. Thespreadsheet Math Coach in Chapter 4 shows the functions. The functions have an annuity input.

The example illustrated earlier in this chapter asks for the FV of annual $100 deposits earning 8 percent. Thespreadsheet solution is the same as the equation and calculator solutions. If you want the FV of an annuitydue, just change the type from 0 to 1.

See this textbook’s online student center to watch instructional videos on using spreadsheets.„

EXAMPLE 5-5 Car Loan Debt LG5-4

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to Connect or go tomhhe.com/CornettM4e.

Now you would like to buy a car. You have reviewed your budget anddetermined that you can afford to pay $500 per month as a carpayment. How much can you borrow if interest rates are 9 percent andyou pay the loan over four years? How much could you borrow if youagree to pay for six years instead?

SOLUTION:

The loan amount is the present value of the 48-month, $500 annuity.Note that the loan term will be 48 (= 4 × 12) months and the interestrate is 0.75 (= 9 ÷ 12) percent. Using equation 5-4, you discover thatyou can borrow up to $20,092 to buy a car:

If you are willing to borrow money for six years instead of four, thesmall change to the equation results in your ability to borrow $27,738.Although this would allow you to buy a more expensive car, it would

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also require two more years of $500 payments (an additional $12,000of payments!).

Similar to Problems 5-25, 5-26, Self-Test Problem 4

EXAMPLE 5-6Making Monthly PensionContributions LG5-7

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Reexamine your original plan to contribute to your company retirementplan. Instead of a total contribution of $4,500 per year for 40 years, youare able to contribute monthly. Given your expected 7 percent per yearinvestment return, how much money can you expect in your retirementaccount?

SOLUTION:

Now your total monthly contribution will be $375 (= $4,500 ÷ 12), whichwill continue for 480 months and earn a 0.58333 (= 7 ÷ 12) percentmonthly return. The results of equation 5-2 show that the future valueof this annuity is

When you made contributions annually, the future value was $898,358(Example 5-2). By changing to monthly contributions, your retirementnest egg increased by nearly $86,000 to $984,305!

Similar to Problems 5-51, 5-52, Self-Test Problem 3

the Math Coach on…Common Mistakes

“As you figure present and future values of annuity cash flows, check that all terms are consistent: thenumber of payments, interest rate, and payment size all need to use common terms. If your payments aremonthly, then the number of payments must reflect the number of months; the interest rate must be stated asa per-month rate, and the payment register must reflect that monthly payment.„

time out!5-7 Why is EAR a more accurate measure of the rate actually paid than APR?

5-8 What would have a smaller present value: a future sum discounted annually or one discounted monthly?

▼ TABLE 5.4 The EAR Is Higher than the APR

A B C D

1 APR Compounding Periods Equation = EAR

2 Varying the Compounding Periods

3 5% 1 (1 + 0.05/1)1 − 1 5.00%

4 5 4 (1 + 0.05/4)4 − 1 5.09

5 5 12 (1 + 0.05/12)12 − 1 5.12

6 Varying APR and Compounding Periods

7 9 4 (1 + 0.09/4)4 − 1 9.31

8 9 12 (1 + 0.09/12)12 − 1 9.38

9 12 4 (1 + 0.12/4)4 − 1 12.55

10 12 12 (1 + 0.12/12)12 − 1 12.68

Note: This compound frequency effect grows substantially for higher interest rates or longer term loans.

EXAMPLE 5-7Evaluating Credit CardOffers LG5-7

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As a college student, you probably receive many credit card offers inthe mail. Consider these two offers. The first card charges a 16 percentAPR. An examination of the footnotes reveals that this card compoundsmonthly. The second credit card charges 15.5 percent APR andcompounds weekly. Which card has a lower effective annual rate?

SOLUTION:

Compute the EAR of each card to compare them in common (andrealistic) terms. The first card has an EAR of

The EAR of the second card is

You should pick the second credit card because it has a lower effectiveannual rate. But note also that you will always be better off if you pay

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your credit card balance whenever the bill comes due.

Similar to Problems 5-15, 5-16, Self-Test Problem 3

5.5 • ANNUITY LOANS LG5-8In this chapter, we’ve focused on computing the future and present value of annuities. But in many situations, thesevalues are already known and what we really need to compare are the payments or implied interest rate—usually,the highest interest rate offered.

the Math Coach on…Common Mistakes

“As we noted in Chapter 4, when computing the interest rate, make sure that the present value and theannuity payments are of different signs (positive versus negative). Otherwise, the calculator will show an error.

„

What Is the Interest Rate?Many business and personal applications already state the cost of an investment, as well as the annuity cash flowsand time period. We need, then, to solve for the implied interest rate of this investment. Unfortunately, we have nogeneral, easy equation to solve for the interest rate. Even financial calculators use an iterating process, which causesthem to “think” a little longer before displaying the estimated interest rate result.

Consider the plight of a manager of a small doctor’s office who has the opportunity to buy a piece of imagingequipment for $100,000. The equipment will allow the office to generate $25,000 in profits for six years, at whichtime the equipment will be worn out and without value in the United States. What rate of return does this purchaseoffer the doctor’s office? The time line for this problem appears as

For the financial calculator solution, input N = 6, PV = –100000, PMT = 25000, and FV = 0. The interest rate resultis then 12.98 percent. So, if this is a high enough return relative to other uses of the $100,000, the doctor’s officeshould seriously consider purchasing the imaging machine.

EXAMPLE 5-8Computing Interest RateNeeded LG5-8

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After saving diligently throughout your entire career, you and yourspouse are finally ready to retire with a nest egg of $800,000. You needto invest this money in a mix of stocks and bonds that will allow you towithdraw $6,000 per month for 30 years. What interest rate do you

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need to earn?

SOLUTION:

Use a financial calculator and input N = 360, PV = –800000, PMT =6000, and FV = 0. The interest rate result is 0.6860 percent. Butremember, since the periods and payments are in months, the interestrate is too. It is customary to report this as an APR: 8.23 percent (=0.6860 percent × 12). However, the EAR more accurately reflects thetrue interest rate, 8.55 percent (= 1.0068612 – 1). In order for yourmoney to last for 30 years while funding a $6,000 per month income,you must earn at least an 8.23 APR per year return.

If you have uneven cash flows, use the calculator CF worksheet andthen solve with the IRR function.

Similar to Problems 5-33, 5-34, 5-35, 5-36

Finding Payments on an Amortized Loan LG5-9

©Phillip Spears/Getty Images

Many consumers and small business owners already know how much money they want to borrow and the level ofcurrent interest rates. Usually, they need to translate this information into the actual payments to determine if theycan really afford the purchase. A loan structured for annuity payments that completely pay off the debt is called anamortized loan. To compute the annuity cash flow of an amortized loan, rearrange the present value of an annuityformula, equation 5-4, to solve for the payment:

amortized loan A loan in which the borrower pays interest and principal over time.

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(5-9)

Most car loans require monthly payments for three to five years. Assume that you need a $10,000 loan to buy a car.The loan is for four years and interest rates are 9 percent per year APR. To implement equation 5-9, use an interestrate of 0.75 percent (= 9 percent/12) and 48 periods (= 4 × 12) as

So, when interest rates are 9 percent, it takes monthly payments of $248.85 to pay off a $10,000 loan in four years.

Interest rate levels and loan length strongly affect how large your payments will be. Table 5.5 shows the monthlypayments needed to pay off a mortgage debt at various interest rates and lengths of time. (Try computing thepayments yourself!) Note that as the interest rate declines, the monthly payment also declines. This is why peoplerush to refinance their mortgages after interest rates fall. A decline of 1 or 2 percent can save a homeowner hundredsof dollars every month. You will also see from the table that paying off a mortgage in only 15 years requires largerpayments, but generally saves thousands in interest.

Amortized Loan Schedules When you pay a car loan or home mortgage, you will often find it useful to knowhow much of the debt, or loan principal, you still owe. For example, consider a case wherein you bought a car twoyears ago using a four-year loan. In order to sell the car now, the loan balance will have to be paid off.Being able to compute this principal balance may influence your chances of selling the car.

loan principal The balance yet to be paid on a loan.

▼ TABLE 5.5 Monthly Payments on a $225,000 Loan

An interest-only loan allows the borrower to make payments that consist totally of interest payments, so none of thedebt is reduced. A $10,000 interest-only loan with a 9 percent APR paid monthly will cost $75 per month (=$10,000 × 0.09 ÷ 12). Amortizing this loan over four years requires monthly payments of $248.85 (see earlier carloan problem). The difference in the first month’s payment on the two loans is $173.85 (= $248.85 – $75) andrepresents the amount of the regular amortized loan’s payment that goes to reducing the principal balance. So afterthe first month’s payment, the amortized loan’s balance has fallen to $9,826.15, while the interest-only loan still hasa balance of $10,000.

In the second month, the interest incurred on the regular amortized loan is $73.70 (= $9,826.15 × 0.09 ÷ 12), so the$248.85 second-month payment represents principal payment of $175.15. These numbers are shown in theamortization schedule of Table 5.6. The table will show you that the early payments on a car loan go mostly to payingthe interest rather than reducing the principal. That interest component declines over time, and then the principalbalance declines.

amortization schedule A table detailing the periodic loan payment, interest payment, and debt balance over the life of the loan.

The amortization schedule shows that if you wish to sell the car after two years, you will have to pay the loancompany a car loan (principal) debt of $5,447.13. Of course, if you had an interest-only loan, you would still owethe full principal of $10,000 after two years. Amortization schedules are also useful for determining other things,like the total amount of interest that you will pay over the life of the loan. In this case, if you take a regular loan inwhich you pay both principal and interest, you pay $10,000 in principal and nearly $1,945 in interest during the fouryears of the loan. The interest component is an even larger component of longer-term loans, like 30-year mortgages.Depending on the interest rate charged, the first payment in a mortgage consists of 75 percent to 95 percent interest.The home mortgage principal balance falls very slowly in the first years of the loan.

▼ TABLE 5.6 Amortization Schedule over Four Years (9 percent APR)

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EXAMPLE 5-9Monthly MortgagePayments LG5-9

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Say that you have your heart set on purchasing a beautiful, old Tudor-style house for $250,000. A mortgage broker says that you can qualifyfor a mortgage for 80 percent (or $200,000) of the price. If you get a15-year mortgage, the interest rate will be 6.1 percent APR. A 30-yearmortgage costs 6.4 percent. One of the factors that will help you decidewhich mortgage to take is the magnitude of the monthly payments.What will they be?3

SOLUTION:

To pay off the mortgage in only 15 years, the payments would have tobe larger than for the 30-year mortgage. The higher payment will beeased somewhat because the interest rate is lower on the 15-yearmortgage. The interest rate would be i = 0.061 ÷ 12 = 0.0050833, or0.50833% per month. The payment for the 15-year mortgage is

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The payment for the 30-year mortgage would be

So, the payments on the 15-year mortgage are nearly $450 more eachmonth than the 30-year mortgage payments. You must decide whetherthe cost of paying the extra $450 each month is worth it to own thehouse with no debt 15 years sooner. The decision would depend onyour financial budget and the strength of your desire to be debt free.

Similar to Problems 5-39, 5-40, Self-Test Problems 1 and 4

We construct amortization schedules by showing the loan’s principal balance at the beginning of the month. This isthe same as the balance at the end of the previous month (except for the very first payment). Then we compute theinterest owed on that balance for the month. After paying that interest, what’s left of the monthly payment reducesthe loan balance for the next month. Because of these repetitive computations, spreadsheets make amortizationschedules easy to construct.

time out!5-9 How might credit card companies keep their cardholders in debt for a long time? What payment do the credit card

companies expect your friend to make so that he never pays down the debt?5-10 Can you find the interest rate if you know the annuity payments and a future value? Under what circumstances might you

want to solve this kind of problem? Which equation would you use?

Compute the Time Period You might also find it useful to know how long it will take to pay off a loan withspecific annuity payments. To find the number of periods, you can solve equation 5-9 for N—the number ofpayments—but the equation becomes quite complicated.4 Many people just use a financial calculator or spreadsheet.We can check to see if the $248.85 monthly payment would indeed pay off the $10,000, 9 percent car loan in fouryears. Finding the solution with a financial calculator entails entering I = 0.75, PV = 10000, PMT = –248.85, and FV= 0. The answer is 48 months.

Add-On Interest One method of calculating payments of a loan that is popular in payday lending is called add-oninterest. This method computes the amount of the interest payable at the beginning of the loan, which is then addedto the principal of the loan. This total is then divided into the number of payments to be made. Consider a loan of$1,000 to be paid with 9 percent add-on interest and repaid in six monthly payments.

add-on interest A calculation of the amount of interest determined at the beginning of the loan and then added to the principal.

the Math Coach on…

Using the AMORT Function in TVM Calculators

“TVM calculators have preprogrammed functions to compute the amount of principal paid part way through amortgage. For example, if you took out a 30-year, $200,000 mortgage at a 6 percent APR, how much principalhave you paid after five years? How much do you still owe? How much interest have you paid?

To answer these questions, first enter the mortgage information to compute the monthly payments. Then usethe AMORT function. This example uses the Texas Instruments BA II + as an example. The Hewlett-Packardand other TVM calculators have similar functions. The AMORT function allows you to compute the loanbalance at any time during the mortgage period. It also computes the amount of principal and interest that hasbeen paid during any time period. To answer the questions above

1. Press 2nd AMORT and P1 = 1 appears. (This refers to the first payment of the mortgage.)2. Press the down arrow ↓, P2 = appears. (This refers to the last payment made.)

3. The question refers to 5 years of payments, which is 60 months. Enter 60 and press ENTER.The calculator has now computed the loan balance after the 60th payment and the amount of principal andinterest that have been paid between the 1st and the 60th payments.

4. Press the down arrow ↓, displayed is BAL = 186,108.71, which is the loan balance.5. Press the down arrow ↓, displayed is PRN = 13,891.29, which is the principal paid in the first five years.

6. Press the down arrow ↓, displayed is INT = –58,054.78, which is the interest paid in the first five years.

Note that in the beginning of a mortgage, far more interest is paid than principal.„The total interest for this loan is computed as 9 percent of $1,000 for six months, or $45 (= 0.09 × $1,000 × ½). Thisis added to the principal for a total of $1,045. Each of the six monthly payments is then $1,045 ÷ 6 = $174.17. Bealert that the add-on interest method seriously understates the real interest rate that is being paid! If you borrow$1,000 and repay a $174.17 monthly annuity for six months, the monthly interest rate is 1.27 percent. This is a 15.27percent APR (= 1.27% × 12) and a 16.39 percent EAR (= 1.012712 − 1)—both much higher than the advertised 9percent interest rate of this loan.

EXAMPLE 5-10

Time to Pay Off a Credit CardBalance LG5-10

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Through poor financial management, your friend has racked up $5,000in debt on his credit card. The card charges a 19 percent APR andcompounds monthly. His latest bill shows that he must pay a minimumof $150 this month. At this rate, how long will it take your friend to payoff his credit card debt?

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SOLUTION:

Using the financial calculator, input I = 1.58333 (= 19/12), PV = 5000,PMT = –150, FV = 0. The answer is 48 months, or 4 years. If the friendpays the minimum payment, then it will be a long time before he will beout of debt. The credit card company is very content to continue to earnthe high return for many years—essentially, the interest on the loanand a very small portion of the principal. Your friend should pay morethan the minimum charge to reduce his debt quicker.

Similar to Problems 5-41, 5-42, Self-Test Problem 4

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Your Turn…Questions

1. How can you add a cash flow in year 2 and a cash flow in year 4? In year 7? (LG5-1)

2. People can become millionaires in their retirement years quite easily if they start saving early in employer

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401(k) or 403(b) programs (or even if their employers don’t offer such programs). Demonstrate the growth of a$250 monthly contribution for 40 years earning 9 percent APR. (LG5-2)

3. When you discount multiple cash flows, how does the future period that a cash flow is paid affect its presentvalue and its contribution to the value of all the cash flows? (LG5-3)

4. How can you use the present value of an annuity concept to determine the price of a house you can afford?(LG5-4)

5. Since perpetuity payments continue forever, how can a present value be computed? Why isn’t the present valueinfinite? (LG5-5)

6. Explain why you use the same adjustment factor, (1 + i), when you adjust annuity due payments for both futurevalue and present value. (LG5-6)

7. Use the idea of compound interest to explain why EAR is larger than APR. (LG5-7)

8. Would you rather pay $10,000 for a five-year, $2,500 annuity or a 10-year, $1,250 annuity? Why? (LG5-8)

9. The interest on your home mortgage is tax deductible. Why are the early years of the mortgage more helpful inreducing taxes than in the later years? (LG5-9)

10. How can you use the concepts illustrated in computing the number of payments in an annuity to figure how topay off a credit card balance? How does the magnitude of the payment impact the number of months? (LG5-10)

ProblemsBASIC PROBLEMS

5-1 Future Value Compute the future value in year 9 of a $2,000 deposit in year 1 and another $1,500 depositat the end of year 3 using a 10 percent interest rate. (LG5-1)

5-2 Future Value Compute the future value in year 7 of a $2,000 deposit in year 1 and another $2,500 depositat the end of year 4 using an 8 percent interest rate. (LG5-1)

5-3 Future Value of an Annuity What is the future value of a $900 annuity payment over five years if interestrates are 8 percent? (LG5-2)

5-4 Future Value of an Annuity What is the future value of a $700 annuity payment over six years if interestrates are 10 percent? (LG5-2)

5-5 Present Value Compute the present value of a $2,000 deposit in year 1 and another $1,500 deposit at theend of year 3 if interest rates are 10 percent. (LG5-3)

5-6 Present Value Compute the present value of a $2,000 deposit in year 1 and another $2,500 deposit at theend of year 4 using an 8 percent interest rate. (LG5-3)

5-7 Present Value of an Annuity What’s the present value of a $900 annuity payment over five years ifinterest rates are 8 percent? (LG5-4)

5-8 Present Value of an Annuity What’s the present value of a $700 annuity payment over six years ifinterest rates are 10 percent? (LG5-4)

5-9 Present Value of a Perpetuity What’s the present value, when interest rates are 7.5 percent, of a $50payment made every year forever? (LG5-5)

5-10 Present Value of a Perpetuity What’s the present value, when interest rates are 8.5 percent, of a $75payment made every year forever? (LG5-5)

5-11 Present Value of an Annuity Due If the present value of an ordinary, seven-year annuity is $6,500and interest rates are 7.5 percent, what’s the present value of the same annuity due? (LG5-6)

5-12 Present Value of an Annuity Due If the present value of an ordinary, six-year annuity is $8,500 andinterest rates are 9.5 percent, what’s the present value of the same annuity due? (LG5-6)

5-13 Future Value of an Annuity Due If the future value of an ordinary, seven-year annuity is $6,500 andinterest rates are 7.5 percent, what is the future value of the same annuity due? (LG5-6)

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5-14 Future Value of an Annuity Due If the future value of an ordinary, six-year annuity is $8,500 andinterest rates are 9.5 percent, what’s the future value of the same annuity due? (LG5-6)

5-15 Effective Annual Rate A loan is offered with monthly payments and a 10 percent APR. What’s theloan’s effective annual rate (EAR)? (LG5-7)

5-16 Effective Annual Rate A loan is offered with monthly payments and a 13 percent APR. What’s theloan’s effective annual rate (EAR)? (LG5-7)

INTERMEDIATE PROBLEMS

5-17 Future Value Given a 4 percent interest rate, compute the year 6 future value of deposits made inyears 1, 2, 3, and 4 of $1,100, $1,200, $1,200, and $1,500. (LG5-1)

5-18 Future Value Given a 5 percent interest rate, compute the year 6 future value of deposits made inyears 1, 2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400. (LG5-1)

5-19 Future Value of Multiple Annuities Assume that you contribute $200 per month to a retirement planfor 20 years. Then you are able to increase the contribution to $300 per month for another 30 years.Given a 7 percent interest rate, what is the value of your retirement plan after the 50 years? (LG5-2)

5-20 Future Value of Multiple Annuities Assume that you contribute $150 per month to aretirement plan for 15 years. Then you are able to increase the contribution to $350 per monthfor the next 25 years. Given an 8 percent interest rate, what is the value of your retirement plan afterthe 40 years? (LG5-2)

5-21 Present Value Given a 6 percent interest rate, compute the present value of payments made in years 1,2, 3, and 4 of $1,000, $1,200, $1,200, and $1,500. (LG5-3)

5-22 Present Value Given a 7 percent interest rate, compute the present value of payments made in years 1,2, 3, and 4 of $1,000, $1,300, $1,300, and $1,400. (LG5-3)

5-23 Present Value of Multiple Annuities A small business owner visits her bank to ask for a loan. Theowner states that she can repay a loan at $1,000 per month for the next three years and then $2,000 permonth for two years after that. If the bank is charging customers 7.5 percent APR, how much would itbe willing to lend the business owner? (LG5-4)

5-24 Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. Theowner states that he can repay a loan at $1,500 per month for the next three years and then $500 permonth for two years after that. If the bank is charging customers 8.5 percent APR, how much would itbe willing to lend the business owner? (LG5-4)

5-25 Present Value You are looking to buy a car. You can afford $450 in monthly payments for four years.In addition to the loan, you can make a $1,000 down payment. If interest rates are 5 percent APR,what price of car can you afford? (LG5-4)

5-26 Present Value You are looking to buy a car. You can afford $650 in monthly payments for five years.In addition to the loan, you can make a $750 down payment. If interest rates are 8 percent APR, whatprice of car can you afford? (LG5-4)

5-27 Present Value of a Perpetuity A perpetuity pays $100 per year and interest rates are 7.5 percent.How much would its value change if interest rates increased to 9 percent? Did the value increase ordecrease? (LG5-5)

5-28 Present Value of a Perpetuity A perpetuity pays $50 per year and interest rates are 9 percent. Howmuch would its value change if interest rates decreased to 7.5 percent? Did the value increase ordecrease? (LG5-5)

5-29 Future and Present Value of an Annuity Due If you start making $50 monthly contributions todayand continue them for five years, what’s their future value if the compounding rate is 10 percent APR?What is the present value of this annuity? (LG5-6)

5-30 Future and Present Value of an Annuity Due If you start making $75 monthly contributions todayand continue them for four years, what is their future value if the compounding rate is 12 percentAPR? What is the present value of this annuity? (LG5-6)

5-31 Compound Frequency Payday loans are very short-term loans that charge very high interest rates.You can borrow $225 today and repay $300 in two weeks. What is the compounded annual rateimplied by this 33.33 percent rate charged for only two weeks? (LG5-7)

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5-32 Compound Frequency Payday loans are very short-term loans that charge very high interest rates.You can borrow $500 today and repay $590 in two weeks. What is the compounded annual rateimplied by this 18 percent rate charged for only two weeks? (LG5-7)

5-33 Annuity Interest Rate What’s the interest rate of a six-year, annual $5,000 annuity with present valueof $20,000? (LG5-8)

5-34 Annuity Interest Rate What’s the interest rate of a seven-year, annual $4,000 annuity with presentvalue of $20,000? (LG5-8)

5-35 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a$1,000 per month contribution to grow to $75,000 in six years? (LG5-8)

5-36 Annuity Interest Rate What annual interest rate would you need to earn if you wanted a $600 permonth contribution to grow to $45,000 in six years? (LG5-8)

5-37 Add-On Interest Payments To borrow $500, you are offered an add-on interest loan at 8 percent.Two loan payments are to be made, one at six months and the other at the end of the year. Computethe two equal payments. (LG5-8)

5-38 Add-On Interest Payments To borrow $800, you are offered an add-on interest loan at 7 percent.Three loan payments are to be made, one at four months, another at eight months, and the last one atthe end of the year. Compute the three equal payments. (LG5-8)

5-39 Loan Payments You wish to buy a $25,000 car. The dealer offers you a four-year loan with a 9percent APR. What are the monthly payments? How would the payment differ if you paid interestonly? What would the consequences of such a decision be? (LG5-9)

5-40 Loan Payments You wish to buy a $10,000 dining room set. The furniture store offers you a three-year loan with an 11 percent APR. What are the monthly payments? How would the payment differ ifyou paid interest only? What would the consequences of such a decision be? (LG5-9)

5-41 Number of Annuity Payments Joey realizes that he has charged too much on his credit card and hasracked up $5,000 in debt. If he can pay $150 each month and the card charges 17 percent APR(compounded monthly), how long will it take him to pay off the debt? (LG5-10)

5-42 Number of Annuity Payments Phoebe realizes that she has charged too much on her credit card andhas racked up $6,000 in debt. If she can pay $200 each month and the card charges 18 percent APR(compounded monthly), how long will it take her to pay off the debt? (LG5-10)

ADVANCED PROBLEMS

5-43 Future Value Given an 8 percent interest rate, compute the year 7 future value if deposits of $1,000and $2,000 are made in years 1 and 3, respectively, and a withdrawal of $700 is made in year 4. (LG5-10)

5-44 Future Value Given a 9 percent interest rate, compute the year 6 future value if deposits of $1,500and $2,500 are made in years 2 and 3, respectively, and a withdrawal of $600 is made in year 5. (LG5-1)

5-45 EAR of Add-On Interest Loan To borrow $2,000, you are offered an add-on interest loan at 10percent with 12 monthly payments. First compute the 12 equal payments and then compute the EARof the loan. (LG5-7, LG5-8)

5-46 EAR of Add-On Interest Loan To borrow $700, you are offered an add-on interest loan at 9 percentwith 12 monthly payments. First compute the 12 equal payments and then compute the EAR of theloan. (LG5-7, LG5-8)

5-47 Low Financing or Cash Back? A car company is offering a choice of deals. You can receive $500cash back on the purchase or a 3 percent APR, four-year loan. The price of the car is $15,000 and youcould obtain a four-year loan from your credit union, at 6 percent APR. Which deal is cheaper? (LG5-4, LG5-9)

5-48 Low Financing or Cash Back? A car company is offering a choice of deals. You can receive $1,000cash back on the purchase, or a 2 percent APR, five-year loan. The price of the car is $20,000 and youcould obtain a five-year loan from your credit union, at 7 percent APR. Which deal is cheaper? (LG5-4, LG5-9)

5-49 Amortization Schedule Create the amortization schedule for a loan of $15,000, paid monthly

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over three years using a 9 percent APR. (LG5-9)5-50 Amortization Schedule Create the amortization schedule for a loan of $5,000, paid monthly over two

years using an 8 percent APR. (LG5-9)5-51 Investing for Retirement Monica has decided that she wants to build enough retirement wealth that,

if invested at 8 percent per year, will provide her with $3,500 of monthly income for 20 years. To date,she has saved nothing, but she still has 30 years until she retires. How much money does she need tocontribute per month to reach her goal? (LG5-4, LG5-9)

5-52 Investing for Retirement Ross has decided that he wants to build enough retirement wealth that, ifinvested at 7 percent per year, will provide him with $3,000 of monthly income for 30 years. To date,he has saved nothing, but he still has 20 years until he retires. How much money does he need tocontribute per month to reach his goal? (LG5-4, LG5-9)

5-53 Loan Balance Rachel purchased a $15,000 car three years ago using an 8 percent, four-year loan. Shehas decided that she would sell the car now, if she could get a price that would pay off the balance ofher loan. What is the minimum price Rachel would need to receive for her car? (LG5-9)

5-54 Loan Balance Hank purchased a $20,000 car two years ago using a 9 percent, five-year loan. He hasdecided that he would sell the car now, if he could get a price that would pay off the balance of hisloan. What’s the minimum price Hank would need to receive for his car? (LG5-9)

5-55 Teaser Rate Mortgage A mortgage broker is offering a $183,900 30-year mortgage with a teaser rate.In the first two years of the mortgage, the borrower makes monthly payments on only a 4 percent APRinterest rate. After the second year, the mortgage interest rate charged increases to 7 percent APR.What are the monthly payments in the first two years? What are the monthly payments after thesecond year? (LG5-9)

5-56 Teaser Rate Mortgage A mortgage broker is offering a $279,000 30-year mortgage with a teaser rate.In the first two years of the mortgage, the borrower makes monthly payments on only a 4.5 percentAPR interest rate. After the second year, the mortgage interest rate charged increases to 7.5 percentAPR. What are the monthly payments in the first two years? What are the monthly payments after thesecond year? (LG5-9)

5-57 Spreadsheet Problem Consider a person who begins contributing to a retirement plan at age 25 andcontributes for 40 years until retirement at age 65. For the first 10 years, she contributes $3,000 peryear. She increases the contribution rate to $5,000 per year in years 11 through 20. This is followed byincreases to $10,000 per year in years 21 through 30 and to $15,000 per year for the last 10 years. Thismoney earns a 9 percent return. First compute the value of the retirement plan when she turns age 65.Then compute the annual payment she would receive over the next 40 years if the wealth wasconverted to an annuity payment at 8 percent. (LG5-2, LG5-9)

5-58 Spreadsheet Problem When paying off a home mortgage, extra principle payments can have adramatic impact on the time needed to pay off the mortgage. (LG5-9)a. Create an amortization schedule for a $200,000, three-year mortgage, with a 6% APR.b. After the fifth year, add an extra $100 to each monthly payment. When is the loan paid off?

Combined Chapter 4 and Chapter 5 Problems4&5-1 Future Value Consider that you are 35 years old and have just changed to a new job. You have

$80,000 in the retirement plan from your former employer. You can roll that money into theretirement plan of the new employer. You will also contribute $3,600 each year into your newemployer’s plan. If the rolled-over money and the new contributions both earn a 7 percent return,how much should you expect to have when you retire in 30 years?

4&5-2 Future Value Consider that you are 45 years old and have just changed to a new job. You have$150,000 in the retirement plan from your former employer. You can roll that money into theretirement plan of the new employer. You will also contribute $7,200 each year into your new

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employer’s plan. If the rolled-over money and the new contributions both earn an 8 percentreturn, how much should you expect to have when you retire in 20 years?

4&5-3 Future Value and Number of Annuity Payments Your client has been given a trust fundvalued at $1 million. He cannot access the money until he turns 65 years old, which is in 25years. At that time, he can withdraw $25,000 per month. If the trust fund is invested at a 5.5percent rate, how many months will it last your client once he starts to withdraw the money?

4&5-4 Future Value and Number of Annuity Payments Your client has been given a trust fundvalued at $1.5 million. She cannot access the money until she turns 65 years old, which is in 15years. At that time, she can withdraw $20,000 per month. If the trust fund is invested at a 5percent rate, how many months will it last your client once she starts to withdraw the money?

4&5-5 Present Value and Annuity Payments A local furniture store is advertising a deal in which youbuy a $3,000 dining room set and do not need to pay for two years (no interest cost is incurred).How much money would you have to deposit now in a savings account earning 5 percent APR,compounded monthly, to pay the $3,000 bill in two years? Alternatively, how much would youhave to deposit in the savings account each month to be able to pay the bill?

4&5-6 Present Value and Annuity Payments A local furniture store is advertising a deal in which youbuy a $5,000 living room set with three years before you need to make any payments (no interestcost is incurred). How much money would you have to deposit now in a savings account earning4 percent APR, compounded monthly, to pay the $5,000 bill in three years? Alternatively, howmuch would you have to deposit in the savings account each month to be able to pay the bill?

4&5-7 House Appreciation and Mortgage Payments Say that you purchase a house for $200,000 bygetting a mortgage for $180,000 and paying a $20,000 down payment. If you get a 30-yearmortgage with a 7 percent interest rate, what are the monthly payments? What would the loanbalance be in 10 years? If the house appreciates at 3 percent per year, what will be the value ofthe house in 10 years? How much of this value is your equity?

4&5-8 House Appreciation and Mortgage Payments Say that you purchase a house for $150,000 bygetting a mortgage for $135,000 and paying a $15,000 down payment. If you get a 15-yearmortgage with a 7 percent interest rate, what are the monthly payments? What would the loanbalance be in five years? If the house appreciates at 4 percent per year, what will be the value ofthe house in five years? How much of this value is your equity?

4&5-9 Construction Loan You have secured a loan from your bank for two years to build your home.The terms of the loan are that you will borrow $200,000 now and an additional $100,000 inone year. Interest of 10 percent APR will be charged on the balance monthly. Since nopayments will be made during the two-year loan, the balance will grow at the 10 percentcompounded rate. At the end of the two years, the balance will be converted to a traditional 30-year mortgage at a 6 percent interest rate. What will you be paying as monthly mortgagepayments (principal and interest only)?

4&5-10 Construction Loan You have secured a loan from your bank for two years to build your home.The terms of the loan are that you will borrow $100,000 now and an additional $50,000 in oneyear. Interest of 9 percent APR will be charged on the balance monthly. Since no payments willbe made during the two-year loan, the balance will grow. At the end of the two years, the balancewill be converted to a traditional 15-year mortgage at a 7 percent interest rate. What will you payas monthly mortgage payments (principal and interest only)?

NotesCHAPTER 51. The contract actually contains some complications like incentives to play well and salary deferral. We ignore those complicating factors

here.

2. It is also possible to continuously compound. The future value of a continuously compounded deposit is , where ehas a value of 2.7183.

3. Most homeowners are actually most interested in their total payment, which will include hazard insurance for the home and propertytaxes. Such payments are referred to as PITI—principal, interest, taxes, and insurance. For simplicity, we use only PI payments here—principal and interest.

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4. The equation for solving for the number of periods in an annuity is:

Part Four

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chapter sixunderstanding financialmarkets and institutions

©Steve Allen/Stockbyte/Getty Images

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ow do funds flow throughout the economy? How do financial markets operate and relate to oneanother? As an individual investor or a financial manager you need to know. Your future decision-making skills depend on it. Investors’ funds flow through financial markets such as the New York

Stock Exchange and mortgage markets. Financial institutions—commercial banks (e.g., Bank ofAmerica), investment banks (e.g., Morgan Stanley), and mutual funds (e.g., Fidelity)—act asintermediaries to channel funds from individual savers or investors through financial markets. This chapterlooks at the nature and operations of financial markets and discusses the financial institutions (FIs) thatparticipate in those markets. Bonds, stocks, and other securities that trade in the markets are covered inChapters 7 and 8.

In this chapter we also examine how significant changes in the way financial institutions deliverservices played a major role in forming the severe financial crisis that began in late 2008. We examinesome of the crisis’s underlying causes, review some of the major events that occurred during that time,and discuss some resulting regulatory and industry changes that are in effect today in Appendix 6A,available in Connect or at mhhe.com/Cornett4e.

LEARNING GOALS

LG6-1 Differentiate between primary and secondary markets and between money and capital markets.LG6-2 List the types of securities traded in money and capital markets.LG6-3 Identify different types of financial institutions and the services that each provides.LG6-4 Know the main suppliers and demanders of loanable funds.LG6-5 Understand how equilibrium interest rates are determined.LG6-6 Analyze specific factors that influence interest rates.LG6-7 Offer different theories that explain the shape of the term structure of interest rates.LG6-8 Demonstrate how forward interest rates derive from the term structure of interest rates.

viewpointsbusiness APPLICATIONDPH Corporation needs to issue new bonds either this year or in two years. DPH Corp. is a profitable firm, but if the U.S. economywere to experience a downturn, the company would see a big drop in sales over the next two years as its products are very sensitiveto changes in the overall economy. DPH Corp. currently has $10 million in public debt outstanding, but its bonds are not activelytraded. What questions must DPH Corp. consider as its managers decide whether to issue bonds today or in two years? How canDPH Corp. get these bonds to potential buyers and thus raise the needed capital? (See the solution at the end of the book.)

6.1 • FINANCIAL MARKETS LG6-1Financial markets exist to manage the flow of funds from investors to borrowers as well as from one investor toanother. We generally differentiate financial markets by their primary financial instruments’ characteristics (such asbond maturities) or the market’s location. Specifically, we can distinguish markets along two major dimensions:

financial markets The arenas through which funds flow.

1. Primary versus secondary markets.2. Money versus capital markets.

Primary Markets versus Secondary Markets

Primary Markets Primary markets provide a forum in which demanders of funds (e.g., corporations such as IBM or

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government entities such as the U.S. Treasury) raise funds by issuing new financial instruments, such as stocks andbonds. Corporations or government entities continually have new projects or expanded production needs, but do nothave sufficient internally generated funds (such as retained earnings) to support their capital needs. Thus,corporations and governments issue securities in external primary markets to raise additional funds. These entitiessell the new financial instrument issues to initial fund suppliers (e.g., households) in exchange for the funds (money)that the issuer requires.

primary markets Markets in which corporations raise funds through new issues of securities.

In the United States, financial institutions called investment banks arrange most primary market transactions forbusinesses. Some of the best-known examples of U.S. investment banks include Morgan Stanley, Goldman Sachs,or Merrill Lynch (owned by Bank of America, a commercial bank). These firms intermediate between issuing parties(fund demanders) and investors (fund suppliers). Investment banks provide fund demanders with a number ofservices, including advising the company or government agency about the securities issue (such as an appropriateoffer price and number of securities to issue) and attracting initial public purchasers of the customer’s securitiesofferings. Firms that need funds are seldom expert at raising capital themselves, so they avert risk and lower theircosts by turning to experts at investment banks to issue their primary market securities.

investment banks Banks that help companies and governments raise capital.

commercial bank Depository institutions whose major assets are loans and whose major liabilities are deposits.

The initial (or primary market) sale of securities occurs either through a public offering or as a private placement toa small group of investors. An investment bank serves as a security underwriter in a public offering. In a privateplacement, the security issuer engages the group of buyers (usually fewer than 10) to purchase the whole issue.Buyers are typically financial institutions. To protect smaller individual investors against a lack of disclosure,publicly traded securities must be registered with the Securities and Exchange Commission (SEC). Privateplacements, on the other hand, can be unregistered and resold to large, financially sophisticated investors only.Large investors supposedly possess the resources and expertise to analyze a security’s risk. Privately placed bondsand stocks traditionally have been among the most illiquid securities in the securities markets; only the very largestfinancial institutions or institutional investors are able or willing to buy and hold them in the absence of an activesecondary market. Issuers of privately placed securities tend to be less well known (e.g., medium-sizedmunicipalities and corporations). Because of this lack of information and its associated higher risk, returns paid toholders of privately placed securities tend to be higher than those on publicly placed securities issues.

personal APPLICATIONJohn Adams wants to invest in one of two corporate bonds issued by separate firms. One bond yields 8.00 percent with a 10-yearmaturity; the other offers a 10.00 percent yield and a 9-year maturity. The second bond seems to be the better deal if one only looksat the interest rate. Is it necessarily the bond in which John should invest? Once he decides which bond represents the betterinvestment, how can John go about buying the bond? (See the solution at the end of the book.)

Should John consider bonds from other countries?

Figure 6.1 illustrates a time line for the primary market exchange of funds for a new issue of corporate bonds orequity. We will further discuss how companies, the U.S. Treasury, and government agencies that market primarygovernment securities, such as Ginnie Mae and Freddie Mac, go about selling primary market securities in Chapter8. Throughout this text, we focus on government securities from the buyer’s, rather than the seller’s, point of view.You can find in-depth discussions of government securities from the seller’s point of view in a public finance text.

Primary market financial instruments include stock issues from firms initially going public (e.g., allowing theirequity shares to be publicly traded on stock markets for the first time). We usually refer to these first-time issues as

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initial public offerings (IPOs). For example, on June 16, 2015, Fitbit announced a $732 million IPO of its commonstock. Fitbit used several investment banks, including Morgan Stanley, Deutsche Bank, and Bank ofAmerica Merrill Lynch, to underwrite the company’s stock. Publicly traded firms may issue additionalbonds or stocks as primary market securities. For example, on February 25, 2015, American Tower Groupannounced that it would sell an additional 23.5 million shares of common stock (at $97.00 per share) underwrittenby investment banks such as Goldman Sachs, Bank of America Merrill Lynch, Barclays, Citigroup and J.P. Morgan.The funds were used to finance the acquisition of Verizon. If the acquisition of Verizon was not completed,American Tower expected to use the net proceeds from the offering for general corporate purposes.

initial public offerings (IPOs) A first-time issue of stock by a private firm going public (e.g., allowing its equity, some of which was heldprivately by managers and venture capital investors, to be publicly traded in stock markets for the first time).

FIGURE 6.1 Primary Market Transfer of Funds

Secondary Markets Once firms issue financial instruments in primary markets, these same stocks and bonds arethen traded—that is, bought and resold—in secondary markets. The New York Stock Exchange (NYSE) and theNASDAQ are two well-known examples of secondary markets for trading stocks (see Chapters 7 and 8). In additionto stocks and bonds, secondary markets also exist for financial instruments backed by mortgages and other assets,foreign exchange, and futures and options (i.e., derivative securities, discussed later in the chapter).

secondary markets Markets that trade financial instruments once they are issued.

Buyers find sellers of secondary market securities in economic agents that need funds (fund demanders). Secondarymarkets provide a centralized marketplace where economic agents know that they can buy or sell most securitiesquickly and efficiently. Secondary markets, therefore, save economic agents the search costs of finding buyers orsellers on their own. Figure 6.2 illustrates a secondary market transfer of funds. Secondary market buyers often usesecurities brokers such as Charles Schwab or other brokerage firms to act as intermediaries as they exchange fundsfor securities (see Chapter 8). An important note: The firm that originally issued the stock or bond is not involved insecondary market transactions in any way—no money accrues to the company itself when its stock trades in asecondary market.

Secondary markets offer benefits to both investors (fund suppliers) and issuers (fund demanders). Investors gainliquidity and diversification benefits (see Chapter 10). Although corporate security issuers are not directly involvedin secondary market transactions, issuers do gain information about their securities’ current market value. Publiclytraded firms can thus observe how investors perceive their corporate value and their corporate decisions by trackingtheir firms’ securities’ secondary market prices. Such price information allows issuers to evaluate how well they areusing internal funds as well as the funds generated from previously issued stocks and bonds and provides indicationsabout how well any subsequent bond or stock offerings might be received—and at what price.

FIGURE 6.2 Secondary Market Transfer of Funds

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Secondary market trading volume can be quite large. Trading volume is defined as the number of shares of a securitythat are simultaneously bought and sold during a given period. Each seller and each buyer actually contract with theexchange’s clearinghouse, which then matches sell and buy orders for each transaction. The clearinghouse is acompany whose stock trades on the exchange, and the clearinghouse runs on a for-profit basis.

trading volume The number of shares of a security that are simultaneously bought and sold during a period.

FIGURE 6.3 Money versus Capital Market Maturities

The exchange and the clearinghouse can process many transactions in a single day. For example, on October 28,1997, NYSE trading volume exceeded 1 billion shares for the first time ever. On October 10, 2008 (at theheight of the financial crisis), NYSE trading volume topped 7.3 billion shares, the highest level to date. Incontrast, during the mid-1980s, a NYSE trading day during which 250 million shares traded was considered a high-volume day.

Money Markets versus Capital MarketsWe noted that financial markets are differentiated in part by the maturity dates of the instruments traded. Thisdistinction becomes important when we differentiate money markets from capital markets. Both of these marketsdeal in debt securities (capital markets also deal in equity securities); the question becomes one of when thesecurities come due.

Money Markets Money markets feature debt securities or instruments with maturities of one year or less (see Figure6.3). In money markets, agents with excess short-term funds can lend (or supply) to economic agents who need (ordemand) short-term funds. The suppliers of funds buy money market instruments and the demanders of funds sellmoney market instruments. Because money market instruments trade for only short periods of time, fluctuations insecondary-market prices are usually quite small. With less volatility, money market securities are thus less riskythan longer-term instruments. In the United States, many money market securities do not trade in a specific location;rather, transactions occur via telephones, wire transfers, and computer trading. Thus, most U.S. money markets aresaid to be over-the-counter (OTC) markets.

money markets Markets that trade debt securities or instruments with maturities of less than one year.

over-the-counter market Markets that do not operate in a specific fixed location—rather, transactions occur via telephones, wiretransfers, and computer trading.

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Money Market Instruments LG6-2 Corporations and government entities issue a variety of money marketsecurities to obtain short-term funds. These securities include

Treasury bills.

Federal funds and repurchase agreements.Commercial paper.

Negotiable certificates of deposit.Banker’s acceptances.

Table 6.1 lists and defines each money market security. Figure 6.4 graphically depicts the proportion of U.S. moneymarket instruments outstanding across three decades. Notice that, in 2016, federal funds and repurchase agreementscommanded the highest dollar value of all money market instruments, followed by negotiable CDs, Treasury bills,and commercial paper.

Capital Markets Capital markets are markets in which parties trade equity (stocks) and debt (bonds) instrumentsthat mature in more than one year (see Figure 6.3). Given their longer maturities, capital market instruments aresubject to wider price fluctuations than are money market instruments (see the term structure discussion below andin Chapter 7).

capital markets Markets that trade debt (bonds) and equity (stock) instruments with maturities of more than one year.

▼ TABLE 6.1 Money Market Instruments

Treasury bills: Short-term U.S. government obligations.

Federal funds: Short-term funds transferred between financial institutions, usually for no more than one day.

Repurchase agreements (repos): Agreements involving security sales by one party to another, with the promise to reverse thetransaction at a specified date and price, usually at a discounted price.

Commercial paper: Short-term unsecured promissory notes that companies issue to raise short-term cash (sometimes calledPaper).

Negotiable certificates of deposit: Bank-issued time deposits that specify an interest rate and maturity date and are negotiable—that is, traded on an exchange. Their face value is usually at least $100,000.

Banker acceptances (BAs): Bank-guaranteed time drafts payable to a vendor of goods.

FIGURE 6.4 Money Market Instruments Outstanding

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Here we see how the percentage of each money market instrument traded changes across three decades.Source: Federal Reserve Board, “Financial Accounts of the United States,” Statistical Releases, Washington, DC, various issues.www.federalreserve.gov

Capital Market Instruments Capital market securities includeU.S. Treasury notes and bonds.

U.S. government agency bonds.State and local government bonds.

Mortgages and mortgage-backed securities.Corporate bonds.

Corporate stocks.

Table 6.2 lists and defines each capital market security. Figure 6.5 graphically depicts U.S. capital marketinstruments outstanding over several decades. Note that corporate stocks (equities) represent the largest capitalmarket instrument, followed by mortgages and mortgage-backed securities and then corporate bonds. The relativesize of capital markets depends on two factors: the number of securities issued and their market prices. The 1990ssaw consistently rising bull markets; hence the sharp increase in equities’ dollar value outstanding. Stock values fellin the early 2000s as the U.S. economy experienced a downturn—partly because of 9/11 and partly because interestrates began to rise—and stock prices fell. Stock prices in most sectors subsequently recovered and, by 2007, evensurpassed their 1999 levels. Stock prices fell precipitously during the financial crisis of 2008 and 2009. As of mid-March 2009, the Dow Jones Industrial Average (DJIA) had fallen in value 53.8 percent in less than 1½ year’s time.This was greater than the decline during the market crash of 1937 and 1938, when it fell 49 percent. However, stockprices recovered along with the economy in the last half of 2009 and first half of 2010, rising 71.1 percent betweenMarch 2009 and April 2010. However, it took until March 5, 2013, for the DJIA to surpass its pre-crisis high of14,164.53, closing at 14,253.77 for the day.

▼ TABLE 6.2 Capital Market Instruments

Treasury notes and bonds: U.S. Treasury long-term obligations issued to finance the national debt and pay for other federalgovernment expenditures.

U.S. government agency bonds: Long-term debt securities collateralized by a pool of assets and insured by agencies of theU.S. government.

State and local government bonds: Debt securities issued by state and local (e.g., county, city, school) governments, usuallyto cover capital (long-term) improvements.

Mortgages: Long-term loans issued to individuals or businesses to purchase homes, pieces of land, or other real property.

Mortgage-backed securities: Long-term debt securities that offer expected principal and interest payments as collateral. Thesesecurities, made up of many mortgages, are gathered into a pool and are thus “backed” by promised principal and interest cashflows.

Corporate bonds: Long-term debt securities issued by corporations.

Corporate stocks: Long-term equity securities issued by public corporations; stock shares represent fundamental corporateownership claims.

FIGURE 6.5 Capital Market Instruments Outstanding

Source: Federal Reserve Board, “Financial Accounts of the United States,” Statistical Releases, Washington, DC, various issues.www.federalreserve.gov

Other Markets

Foreign Exchange Markets Today, most U.S.-based companies operate globally. Competent financialmanagers understand how events and movements in financial markets in other countries can potentially affect theirown companies’ profitability and performance. For example, in 2015, IBM experienced a drop in revenue of 9percent due to foreign exchange trends. Coca-Cola, which gets the majority of its sales from outside the UnitedStates, also saw revenues decrease by approximately 6 percent as the U.S. dollar strengthened relative to foreigncurrencies.

©Jack Star/PhotoLink/Getty Images

Foreign exchange markets trade currencies for immediate (also called “spot”) or some future stated delivery. When aU.S. corporation sells securities or goods overseas, the resulting cash flows denominated in a foreign currency

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expose the firm to foreign exchange risk. This risk arises from the unknown value at which foreign currency cashflows can be converted into U.S. dollars. Foreign currency exchange rates vary day to day with worldwide demandand supply of foreign currency and U.S. dollars. Investors who deal in foreign-denominated securities face the samerisk.

foreign exchange risk Risk arising from the unknown value at which foreign currency cash flows can be converted into U.S. dollars.

foreign exchange markets Markets in which foreign currency is traded for immediate or future delivery.

The actual number of U.S. dollars that a firm receives on a foreign investment depends on the exchange ratebetween the U.S. dollar and the foreign currency just as much as it does on the investment’s performance. Firms willhave to convert the foreign currency into U.S. dollars at the prevailing exchange rate. If the foreign currencydepreciates (falls in value) relative to the U.S. dollar (say from 0.1679 dollar per unit of foreign currency to0.1550 dollar per unit of foreign currency) over the investment period (i.e., the period between when aforeign investment is made and the time it comes to fruition), the dollar value of cash flows received will fall. If theforeign currency appreciates, or rises in value, relative to the U.S. dollar, the dollar value of cash flows receivedfrom the foreign investment will increase.

Foreign currency exchange rates are variable. They vary day to day with demand for and supply of foreign currencyand with demand for and supply of dollars worldwide. Central governments sometimes intervene in foreignexchange markets directly—such as China’s valuing of the yuan at artificially high rates relative to the dollar.Governments also affect foreign exchange rates indirectly by altering prevailing interest rates within their owncountries. You will learn more about foreign exchange markets in Chapter 19.

Derivative Securities Markets A derivative security is a financial security (such as a futures contract, optioncontract, or mortgage-backed security) with a value that is linked to another, underlying security, such as a stocktraded in capital markets or British pounds traded in foreign exchange (forex) markets. Derivative securitiesgenerally involve an agreement between two parties to exchange a standard quantity of an asset or cash flow at apredetermined price and at a specified date in the future. As the value of the underlying security changes, the valueof the derivative security changes.

derivative security A security formalizing an agreement between two parties to exchange a standard quantity of an asset at apredetermined price on a specified date in the future.

While derivative security contracts, especially for physical commodities like corn or gold, have existed for centuries,derivative securities markets grew increasingly popular in the 1970s, 1980s, and 1990s as traders, firms, andacademics figured out how to spread risk for more and more underlying commodities and securities by usingderivative contracts. Derivative contracts generally feature a high degree of leverage; that is, the investor only has toput up a very small portion of the underlying commodity or security’s value to affect or control the underlyingcommodity or security.

Derivative securities traders can be either users of derivative contracts (for hedging and other purposes) or dealers(such as banks) that act as counterparties in customer trades for fees. An example of hedging involves commoditiessuch as corn, wheat, or soybeans. For example, suppose you run a flour mill and will need to buy either soft wheat(Chicago) or hard red winter wheat (Kansas City) in the future. If you are concerned that the price of wheat will rise,you might lock in a price today to meet your needs six months from now by buying wheat futures on a commoditiesexchange. If you are correct and wheat prices rise over the six months, you may purchase the wheat by closing outyour futures positions, buying the wheat at the futures price rather than the higher market price. Likewise, if youknow that you will be delivering a large shipment to, say, Europe, in three months, you might take an offsettingposition in euro futures contracts to lock in the exchange rate between the dollar and the euro as it stands today—and (you hope) eliminate foreign exchange risk from the transaction.

Derivative securities markets are the newest—and potentially the riskiest—of the financial security markets. Lossesassociated with off-balance-sheet mortgage-backed securities created and held by FIs were at the very heart of thefinancial crisis. Signs of significant problems in the U.S. economy first appeared in late 2006 and early 2007 whenhome prices plummeted and defaults began to affect the mortgage lending industry as a whole, as well as other partsof the economy noticeably. Mortgage delinquencies, particularly on subprime mortgages, surged in the last quarterof 2006 through 2008 as homeowners who had stretched themselves to buy or refinance a home in the early 2000sfell behind on their loan payments. As mortgage borrowers defaulted, the financial institutions that held their

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mortgages and credit derivative securities (in the form of mortgage-backed securities) started announcing hugelosses on them. These losses reached $700 billion worldwide by early 2009. The situation resulted in the failure,acquisition, or bailout of some of the largest FIs and a near meltdown of the world’s financial and economicsystems. More recently, as the nearby Finance at Work box highlights, JPMorgan Chase experienced huge lossesfrom positions in the derivative securities markets.

time out!6-1 How do primary and secondary markets differ?6-2 What are foreign exchange markets?6-3 What are derivativeas securities?

6.2 • FINANCIAL INSTITUTIONS LG6-3Financial institutions (e.g., banks, thrifts, insurance companies, mutual funds) perform vital functions to securitiesmarkets of all sorts. They channel funds from those with surplus funds (suppliers of funds) to those with shortagesof funds (demanders of funds). In other words, FIs operate financial markets. FIs allow financial markets to functionby providing the least costly and most efficient way to channel funds to and from these markets. FIs play a secondcrucial role by spreading risk among market participants. This risk-spreading function is vital to entrepreneurialefforts, for few firms or individuals could afford the risk of launching an expensive new product or process bythemselves. Individual investors take on pieces of the risk by buying shares in risky enterprises. Investors thenmitigate their own risks by diversifying their holdings into appropriate portfolios, which we cover in Chapters 9 and10. Table 6.3 lists and summarizes the various types of FIs.

financial institutions Institutions that perform the essential function of channeling funds from those with surplus funds to those withshortages of funds.

To understand just how important FIs are to the efficient operation of financial markets, imagine a simple world inwhich FIs did not exist. In such a world, suppliers of funds (e.g., households), generating excess savings byconsuming less than they earn, would have a basic choice. They could either hold cash as an asset or invest that cashin the securities issued by users of funds (e.g., corporations, governments, or retail borrowers). In general,demanders (users) of funds issue financial claims (e.g., equity and debt securities) to finance the gap between theirinvestment expenditures and their internally generated savings, such as retained earnings. As shown in Figure 6.6, ina world without financial institutions, we would have direct transfers of funds from fund suppliers to fund users. Inreturn, financial claims would flow directly from fund users to fund suppliers.

direct transfer The process used when a corporation sells its stock or debt directly to investors without going through a financialinstitution.

FIGURE 6.6 Flow of Funds in a World without FIs

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▼ TABLE 6.3 Types of Financial Institutions

Commercial banks: Depository institutions whose major assets are loans and whose major liabilities are deposits. Commercialbank loans cover a broader range, including consumer, commercial, and real estate loans, than do loans from other depositoryinstitutions. Because they are larger and more likely to have access to public securities markets, commercial bank liabilitiesgenerally include more nondeposit sources of funds than do those of other depository institutions.

Thrifts: Depository institutions including savings associations, savings banks, and credit unions. Thrifts generally performservices similar to commercial banks, but they tend to concentrate their loans in one segment, such as real estate loans orconsumer loans. Credit unions operate on a not-for-profit basis for particular groups of individuals, such as a labor union or aparticular company’s employees.

Insurance companies: Protect individuals and corporations (policyholders) from financially adverse events. Life insurancecompanies provide protection in the event of untimely death or illness, and help in planning retirement. Property casualtyinsurance protects against personal injury and liability due to accidents, theft, fire, and so on.

Securities firms and investment banks: Underwrite securities and engage in related activities such as securities brokerage,securities trading, and making markets in which securities trade.

Finance companies: Make loans to both individuals and businesses. Unlike depository institutions, finance companies do notaccept deposits, but instead rely on short- and long-term debt for funding, and many of their loans are collateralized with somekind of durable good, such as washer/dryers, furniture, carpets, and the like.

Mutual funds: Pool many individuals’ and companies’ financial resources and invest those resources in diversified assetportfolios.

Pension funds: Offer savings plans through which fund participants accumulate savings during their working years. Participantsthen withdraw their pension resources (which have presumably earned additional returns in the interim) during their retirementyears. Funds originally invested in and accumulated in a pension fund are exempt from current taxation. Participants pay taxeson distributions taken after age 55, when their tax brackets are (presumably) lower.

In this economy without FIs, the amount of funds flowing between fund suppliers and fund users through financialmarkets would likely be quite low for several reasons:

Once they have lent money in exchange for financial claims, fund suppliers would need to continually monitor the use of their funds. Fundsuppliers must ensure that fund users neither steal the funds outright nor waste the funds on projects that have low or negative returns,since either theft or waste would lower fund suppliers’ chances of being repaid and/or earning a positive return on theirinvestments (such as through the receipt of dividends or interest). Monitoring against theft, misuse, or underuse of their fundswould cost any given fund supplier a lot of time and effort, and of course each fund supplier, regardless of the dollar value of theinvestment, would have to carry out the same costly and time-consuming process. Further, many investors do not have the financialtraining to understand the necessary business information to assess whether a securities issuer is making the best use of their funds. Infact, so many investment opportunities are available to fund suppliers, that even those trained in financial analysis rarely have the time tomonitor how their funds are used in all of their investments. The resulting lack of monitoring increases the risk of directly investing infinancial claims. Given these challenges, fund suppliers would likely prefer to delegate the task of monitoring fund borrowers to ensuregood performance to others.

Many financial claims feature a long-term commitment (e.g., mortgages, corporate stock, and bonds) for fund suppliers, but suppliers maynot wish to hold these instruments directly. Specifically, given the choice between holding cash or long-term securities, fund suppliers maychoose to hold cash for its liquidity. This is especially true if the suppliers plan to use their savings to finance consumption expendituresbefore their creditors expect to repay them. Fund suppliers may also fear that they will not find anyone to purchase their financial claim andfree up their funds. When financial markets are not very developed, or deep, in terms of the number of active buyers and sellers in themarket, such liquidity concerns arise.

liquidity The ease with which an asset can be converted into cash.

Even though real-world financial markets provide some liquidity services by allowing fund suppliers to trade financial securities amongthemselves, fund suppliers face price risk when they buy securities—fund suppliers may not get their principal back, let alone any returnon their investment. The price at which investors can sell a security on secondary markets such as the New York Stock Exchange (NYSE)or NASDAQ may well differ from the price they initially paid for the security. The investment community as a whole may change thesecurity’s valuation between the time the fund supplier bought it and the time the fund supplier sold it. Also, dealers, acting asintermediaries between buyers and sellers, charge transaction costs for completing a trade. So even if an investor bought a security andthen sold it the next day, the investor would likely lose money from transaction and other costs.

price risk The risk that an asset’s sale price will be lower than its purchase price.

Unique Economic Functions Performed by Financial InstitutionsBecause of (1) monitoring costs, (2) liquidity costs, and (3) price risk, most average investors may well view directinvestment in financial claims and markets as an unattractive proposition and, as fund suppliers, they will likelyprefer to hold cash. As a result, financial market activity (and therefore savings and investment) would likely remainquite low. However, the financial system has developed an alternative, indirect way for investors (or fund suppliers)to channel funds to users of funds: Financial intermediaries indirectly transfer funds to ultimate fund users. Because of

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monitoring, liquidity risk, and price risk costs, fund suppliers often prefer to hold financial intermediaries’ financialclaims rather than those directly issued by the ultimate fund users. Consider Figure 6.7, which more closelyrepresents the way that funds flow in the U.S. financial system than does Figure 6.6. Notice how financialinstitutions stand—or intermediate—between fund suppliers and fund users. That is, FIs channel funds fromultimate suppliers to ultimate fund users. Fund suppliers and users use these FIs to channel funds because offinancial intermediaries’ unique ability to measure and manage risk, and thus reduce monitoring costs, liquiditycosts, and price risk.

indirect transfer A transfer of funds between suppliers and users of funds through a financial institution.

FIGURE 6.7 Flow of Funds in a World with FIs

Financial institutions stand between fund suppliers and users.

finance at work //: marketsJP Morgan’s $2 Billion Blunder

©Monica and Michael Sweet/Getty Images

JP Morgan Chase & Co. is reeling after a huge trading bet backfired and left the bank with at least $2 billion in losses from the bad trade.This may be the end of chief executive James Dimon’s run as the so-called “King of Wall Street.” The bank’s Chief Investment Office(CIO), responsible for managing the New York company’s risk, placed a series of risky bets and trades. In an article published lastmonth, The Wall Street Journal reported that “large positions taken in that office by a trader nicknamed ‘the London whale’ had roiled asector of the debt markets. The bank, betting on a continued economic recovery with a complex web of trades tied to the values ofcorporate bonds, was hit hard when prices moved against it starting last month, causing losses in many of its derivatives positions. Thelosses occurred while J.P. Morgan tried to scale back that trade.”

In April of 2012, The Wall Street Journal reported that investors and hedge funds were trying to take advantage of trades made byChase’s London whale, Bruno Iksil, who worked out of the CIO, by making bets in the market on credit default swaps (CDSs). The CIOgroup previously had stopgaps in place to protect and prevent the company from significant losses during periods of downturn in theeconomy. However, the Journal reports that earlier in 2012, “it began reducing that position, [taking] a bullish stance on the financialhealth of certain companies and selling protection that would compensate buyers if those companies defaulted on debts. Mr. Iksil was aheavy seller of CDS contracts tied to a basket, or index, of companies.” In April of 2012, these protection costs began to go up, which

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further contributed to the bank’s losses.According to JP Morgan Chase company filings, Mr. Iksil’s group had approximately $350 billion in investment securities, about 15%

of the bank’s total assets, on December 31, 2011. Mr. Dimon said the bank has an extensive review under way of what went wrong.“These were grievous mistakes, they were self-inflicted, we were accountable and we happened to violate our own standards andprinciples by how we want to operate the company. This is not how we want to run a business.”

Mr. Dimon held a conference call with investors and analysts on May 10, stating, “In hindsight, the . . . strategy was flawed, complex,poorly reviewed, poorly executed, and poorly monitored. The portfolio has proven to be riskier, more volatile and less effective . . . thanwe thought.” Dimon resolves, “We will learn from it, we will fix it, we will move on, hopefully in the end, it will make us a better company.”Though JP Morgan Chase came through the financial crisis better off than many other financial institutions, this trading loss certainlytarnishes their reputation. Mr. Dimon reports that the loss is “slightly more than $2 billion” in the second quarter of this year.

Want to know more?Key Words to Search for Updates: JPMorgan, London whale, derivative trading losses

Sources: Fitzpatrick, Dan, Gregory Zuckerman, and Liz Rappaport, “J.P. Morgan’s $2 Billion Blunder,” The Wall Street Journal Online,May 11, 2012. JP Morgan Chase & Co. Business Update Call, May 10, 2012.

Monitoring Costs As we noted above, a fund supplier who directly invests in a fund user’s financial claims facesa high cost of comprehensively monitoring the fund user’s actions in a timely way. One solution to this problem isthat a large number of small investors can group their funds together by holding claims issued by an FI. In turn, theFI will invest in direct financial claims that fund users issue. Financial institutions’ aggregation of funds from fundsuppliers resolves a number of problems:

First, large FIs now have much greater incentive to collect information and monitor the ultimate fund user’s actions, because the FI has farmore at stake than any small individual fund supplier would have.Second, the FI performs the necessary monitoring function via its own internal experts. In an economic sense, fund suppliers appoint theFI as a delegated monitor to act on their behalf. For example, full-service securities firms such as Bank of America Merrill Lynch carry outinvestment research on new issues and make investment recommendations for their retail clients (investors), whilecommercial banks collect deposits from fund suppliers and lend these funds to ultimate users, such as corporations. Animportant part of these FIs’ functions is their ability and incentive to monitor ultimate fund users.

delegated monitor An economic agent appointed to act on behalf of smaller investors in collecting information and/or investing funds ontheir behalf.

Liquidity and Price Risk In addition to providing more and better information about fund users’ activities,financial intermediaries provide additional liquidity to fund suppliers, acting as asset transformers as follows: FIspurchase the financial claims that fund users issue—primary securities such as mortgages, bonds, and stocks—andfinance these purchases by selling financial claims to household investors and other fund suppliers as deposits,insurance policies, or other secondary securities. The secondary securities—packages or pools of primary claims—that FIs collect and then issue are often more liquid than are the primary securities themselves. For example, banksand thrift institutions (e.g., savings associations) offer draft deposit accounts with fixed principal values and (often)guaranteed interest rates. Fund suppliers can generally access the funds in those accounts on demand. Money marketmutual funds issue shares to household savers that allow the savers to maintain almost fixed principal amounts whileearning somewhat higher interest rates than on bank deposits. Further, savers can also withdraw these funds ondemand whenever the saver writes a check on the account. Even life insurance companies allow policyholders toborrow against their company-held policy balances with very short notice.

asset transformer Service provided by financial institutions in which financial claims issued by an FI are more attractive to investorsthan are the claims directly issued by corporations.

secondary securities Packages or pools of primary claims.

The Shift Away from Risk Measurement and Management and the Financial Crisis Certainly, a majorevent that changed and reshaped the financial services industry was the financial crisis of the late 2000s. As FIsadjusted to regulatory changes brought about in the 1980s and 1990s, one result was a dramatic increase in systemicrisk of the financial system, caused in large part by a shift in the banking model from that of “originate and hold” to“originate to distribute.” In the traditional model, banks take short-term deposits and other sources of funds and usethem to fund longer term loans to businesses and consumers. Banks typically hold these loans to maturity, and thushave an incentive to screen and monitor borrower activities even after a loan is made. However, the traditionalbanking model exposes the institution to potential liquidity, interest rate, and credit risk. In attempts to avoid these

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risk exposures and generate improved return-risk trade-offs, banks have shifted to an underwriting model in whichthey originate or warehouse loans, and then quickly sell them. Figure 6.8 shows the growth in bank loan secondarymarket trading from 1991 through 2015. Note the huge growth in bank loan trading even during the financial crisisof 2008 and 2009. When loans trade, the secondary market produces information that can substitute for theinformation and monitoring of banks. Further, banks may have lower incentives to collect information and monitorborrowers if they sell loans rather than keep them as part of the bank’s portfolio of assets. Indeed, most large banksare organized as financial service holding companies to facilitate these new activities.

More recently activities of shadow banks, nonfinancial service firms that perform banking services, have facilitatedthe change from the “originate and hold” model of commercial banking to the “originate and distribute” bankingmodel. Participants in the shadow banking system include structured investment vehicles (SIVs), special purposevehicles (SPVs), asset-backed commercial paper (ABCP) conduits, limited-purpose finance companies, moneymarket mutual funds (MMMFs), and credit hedge funds. In the shadow banking system, savers place their fundswith money market mutual and similar funds, which invest these funds in the liabilities of other shadow banks.Borrowers get loans and leases from shadow banks such as finance companies rather than from banks. Like thetraditional banking system, the shadow banking system intermediates the flow of funds between net savers and netborrowers. However, instead of the bank serving as the middleman, it is the nonbank financial service firm, orshadow bank, that intermediates. Further, unlike the traditional banking system, where the complete creditintermediation is performed by a single bank, in the shadow banking system it is performed through a series of stepsinvolving many nonbank, unregulated financial service firms.

FIGURE 6.8 Bank Loan Secondary Market Trading

Bank loan sales have increased dramatically over the last 20 years.

These innovations remove risk from the balance sheet of financial institutions and shift risk off the balance sheet andto other parts of the financial system. Since the FIs, acting as underwriters, are not exposed to the credit, liquidity,and interest rate risks of traditional banking, they have little incentive to screen and monitor activities of borrowersto whom they originate loans. Thus, FIs’ role as specialists in risk measurement and management is reduced.

The economy relies on financial institutions to act as specialists in risk measurement and management. The

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importance of this was demonstrated during the global financial crisis. When FIs failed to perform their critical riskmeasurement and management functions, a crisis of confidence that disrupted financial markets ensued. The resultwas a worldwide breakdown in credit markets, as well as an enhanced level of equity market volatility.

time out!6-4 List the major types of financial institutions.6-5 What three main issues would deter fund suppliers from directly purchasing securities?6-6 What events resulted in banks’ shift from the traditional banking model of “originate and hold” to a model of “originate and

distribute”?

6.3 • INTEREST RATES AND THE LOANABLE FUNDSTHEORY LG6-3We often speak of “the interest rate” as if only one rate applies to all financial situations or transactions. In fact, wecan list tens or hundreds of interest rates that are appropriate in various conditions or situations within the U.S.economy on any particular day. Let’s explore a bit how the financial sector sets these rates and how the rates relateto one another. We actually observe nominal interest rates in financial markets—these are the rates most often quotedby financial news services. As we will see in Chapters 7 and 8, nominal interest rates (or, simply, interest rates)directly affect most tradable securities’ value or price. Since any change in nominal interest rates has such profoundeffects on security prices, financial managers and individual investors spend a lot of time and effort trying to identifyfactors that may influence future interest rate levels.

nominal interest rates The interest rates actually observed in financial markets.

Of course, interest rate changes influence investment performance and trigger buy or sell decisions for individualinvestors, businesses, and governmental units alike. For example, in 2008 and 2009, the Federal Reserve,in an effort to address the severe financial crisis, unexpectedly announced that it would drop its target fedfunds rate to a range between 0 and 0.25 percent and lowered its discount window rate to 0.5 percent, the lowestlevel since the 1940s. These rates remained at historically low levels until December 2015 when the Federal Reserveraised the fed funds rate to a range of 0.25 to 0.5 percent and the discount window rate to 1 percent. This was thefirst interest rate increase since 2006.

Figure 6.9 illustrates the movement of the following key U.S. interest rates over the past 44 years:The prime commercial loan rate.The three-month T-bill rate.

The home mortgage rate.The high-grade corporate bond rate.

Figure 6.9 shows how interest rates vary over time. For example, the prime rate hit highs of over 20 percent in theearly 1980s, yet fell as low as 4.75 percent in the early 1970s. The prime rate stayed below 10 percent throughoutmuch of the 1990s, fell back further to 4.00 percent in the early 2000s, then rose to as high as 8.25 percent in themid-2000s. During the financial crisis of 2008 and 2009, the Fed took aggressive actions to stimulate theeconomy, including dropping interest rates to historic lows. As a result, the prime rate fell to 3.25 percentand stayed there until December 2015.

FIGURE 6.9 Key U.S. Interest Rates, 1972–2016

Loan rates tend to move together over time.

Source: Federal Reserve Board, website, various dates. www.federalreserve.gov

Interest rates play a major part in the determination of the value of financial instruments. For example, in September2015 the Federal Reserve unexpectedly announced that it would not raise interest rates. The financial marketsreacted significantly: The Dow Jones Industrial Average declined almost 300 points, 1.75 percent in value; theinterest rate on Treasury securities decreased (i.e., the yield on 10-year T-notes decreased 0.123 percent); gold pricesincreased 1.6 percent; and the U.S. dollar strengthened against foreign currencies. Given the impact a change ininterest rates has on security values, financial institution and other firm managers spend much time and effort tryingto identify factors that determine the level of interest rates at any moment in time, as well as what causes interest ratemovements over time.

One model that is commonly used to explain interest rates and interest rate movements is the loanable funds theory.The loanable funds theory views the level of interest rates as resulting from factors that affect the supply of anddemand for loanable funds. It categorizes financial market participants—consumers, businesses, governments, andforeign participants—as net suppliers or demanders of funds.

loanable funds theory A theory of interest rate determination that views equilibrium interest rates in financial markets as a result of thesupply of and demand for loanable funds.

Supply of Loanable Funds LG6-4The supply of loanable funds is a term commonly used to describe funds provided to the financial markets by netsuppliers of funds. In general, the quantity of loanable funds supplied increases as interest rates rise. Figure 6.10illustrates the supply curve for loanable funds. Other factors held constant, more funds are supplied as interest ratesincrease (the reward for supplying funds is higher). Table 6.4 presents data on the supply of loanable funds from thevarious groups of market participants from U.S. flow of funds data as of 2015.

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The household sector (consumer sector) is the largest supplier of loanable funds in the United States—$70.27 trillionin 2015. Households supply funds when they have excess income or want to reallocate their asset portfolio holdings.For example, during times of high economic growth, households may replace part of their cash holdings withearning assets (i.e., by supplying loanable funds in exchange for holding securities). As the total health of aconsumer increases, the total supply of loanable funds from that consumer will also generally increase.Households determine their supply of loanable funds not only on the basis of the general level of interestrates and their total wealth, but also on the risk of securities investments. The greater the perceived risk of securitiesinvestments, the less households are willing to invest at each interest rate. Further, the supply of loanable funds fromhouseholds also depends on their immediate spending needs. For example, near-term educational or medicalexpenditures will reduce the supply of funds from a given household.

FIGURE 6.10 Supply of and Demand for Loanable Funds

The demand and supply of loanable funds varies with interest rates.

▼ TABLE 6.4 Funds Supplied and Demanded by Various Groups (in trillions of dollars)

Funds SuppliedFunds

DemandedNet Funds Supplied (Funds

Supplied − Funds Demanded)

Households $70.27 $23.95 $46.32

Business—nonfinancial 22.30 58.15 −35.85

Business—financial 53.69 81.68 −27.99

Government units 26.61 18.43 8.18

Foreign participants 23.26 13.92 9.34

Source: Federal Reserve Board, “Flow of Fund Accounts,” December 2015, www.federalreserve.gov

Higher interest rates will also result in higher supplies of funds from the U.S. business sector ($22.3 trillion fromnonfinancial business and $53.69 trillion from financial business in 2015), which often has excess cash, or workingcapital, that it can invest for short periods of time in financial assets. In addition to the interest rates on theseinvestments, the expected risk on financial securities and their businesses’ future investment needs will affect theiroverall supply of funds.

Loanable funds are also supplied by some governments ($26.61 trillion in 2015). For example, some governments(e.g., municipalities) temporarily generate more cash inflows (e.g., through local taxes) than they have budgeted tospend. These funds can be loaned out to financial market fund users until needed. During the recent financial crisis,the federal government significantly increased the funds it supplied to businesses and consumers as it attempted torescue the U.S. economy from a deep economic recession (see Appendix 6A online in Connect or atmhhe.com/Cornett4e).

Finally, foreign investors increasingly view U.S. financial markets as alternatives to their domestic financial markets($23.26 trillion of funds were supplied to the U.S. financial markets in 2015). When interest rates are higher on U.S.financial securities than they are on comparable securities in their home countries, foreign investors increase their

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supply of funds to U.S. markets. Indeed the high savings rates of foreign households (such as Japanese households)has resulted in foreign market participants being major suppliers of funds to U.S. financial markets in recent years.Similar to domestic suppliers of loanable funds, foreigners assess not only the interest rate offered on financialsecurities, but also their total wealth, the risk on the security, and their future expenditure needs. Additionally,foreign investors alter their investment decisions as financial conditions in their home countries change relative tothe U.S. economy and the exchange rate of their country’s currency changes vis-à-vis the U.S. dollar (see Chapter19). For example, during the recent financial crisis, investors worldwide, searching for a safe haven for their funds,invested huge amounts of funds in U.S. Treasury securities. The amount of money invested in Treasury bills was solarge that the yield on the three-month Treasury bill went below zero for the first time ever; investors wereessentially paying the U.S. government to borrow money.

Demand for Loanable FundsThe demand for loanable funds is a term used to describe the total net demand for funds by fund users. In general,the quantity of loanable funds demanded is higher as interest rates fall. Figure 6.10 also illustrates the demand curvefor loanable funds. Other factors held constant, more funds are demanded as interest rates decrease (the cost ofborrowing funds is lower).

Households (although they are net suppliers of funds) also borrow funds in financial markets ($23.95 trillion in2015). The demand for loanable funds by households reflects the demand for financing purchases of homes (withmortgage loans), durable goods (e.g., car loans, appliance loans), and nondurable goods (e.g., education loans,medical loans). Additional nonprice conditions and requirements (discussed below) also affect a household’sdemand for loanable funds at every level of interest rates.

Businesses demand funds to finance investments in long-term (fixed) assets (e.g., plant and equipment) and forshort-term working capital needs (e.g., inventory and accounts receivable) usually by issuing debt and otherfinancial instruments ($58.15 trillion for nonfinancial businesses and $81.68 trillion for financial businesses in2015). When interest rates are high (i.e., the cost of loanable funds is high), businesses prefer to financeinvestments with internally generated funds (e.g., retained earnings) rather than through borrowed funds.Further, the greater the number of profitable projects available to businesses, or the better the overall economicconditions, the greater the demand for loanable funds.

Governments also borrow heavily in the markets for loanable funds ($18.43 trillion in 2015). For example, state andlocal governments often issue debt instruments to finance temporary imbalances between operating revenues (e.g.,taxes) and budgeted expenditures (e.g., road improvements, school construction). Higher interest rates can causestate and local governments to postpone borrowings and thus capital expenditures. Similar to households andbusinesses, governments’ demand for funds varies with general economic conditions. The federal government isalso a large borrower partly to finance current budget deficits (expenditures greater than taxes) and partly to financepast deficits. The cumulative sum of past deficits is called the national debt, which in the United States in 2015stood at a record $18.94 trillion. Thus, the national debt and especially the interest payments on the national debthave to be financed in large part by additional government borrowing.

Finally, foreign participants (households, businesses, and governments) also borrow in U.S. financial markets($13.92 trillion in 2015). Foreign borrowers look for the cheapest source of dollar funds globally. Most foreignborrowing in U.S. financial markets comes from the business sector. In addition to interest costs, foreign borrowersconsider nonprice terms on loanable funds as well as economic conditions in their home country and the generalattractiveness of the U.S. dollar relative to their domestic currency (e.g., the euro or the yen).

Equilibrium Interest Rate LG6-5The aggregate supply of loanable funds is the sum of the quantity supplied by the separate fund-supplying sectors(e.g., households, businesses, governments, foreign agents) discussed above. Similarly, the aggregate demand forloanable funds is the sum of the quantity demanded by the separate fund-demanding sectors. As illustrated in Figure6.11, the aggregate quantity of funds supplied is positively related to interest rates, while the aggregate quantity offunds demanded is inversely related to interest rates. As long as competitive forces are allowed to operate freely in afinancial system, the interest rate that equates the aggregate quantity of loanable funds supplied with the aggregatequantity of loanable funds demanded for a financial security, Q*, is the equilibrium interest rate for that security, i*,point E in Figure 6.11. For example, whenever the rate of interest is set higher than the equilibrium rate, such as iH,the financial system has a surplus of loanable funds. As a result, some suppliers of funds will lower the interest rate

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at which they are willing to lend and the demanders of funds will absorb the loanable funds surplus. In contrast,when the rate of interest is lower than the equilibrium interest rate, such as iL, there is a shortage of loanable funds inthe financial system. Some borrowers will be unable to obtain the funds they need at current rates. As a result,interest rates will increase, causing more suppliers of loanable funds to enter the market and some demanders offunds to leave the market. These competitive forces will cause the quantity of funds supplied to increase and thequantity of funds demanded to decrease until a shortage of funds no longer exists.

Factors That Cause the Supply and Demand Curves for Loanable Funds to ShiftWhile we have alluded to the fundamental factors that cause the supply and demand curves for loanable funds toshift, in this section we formally summarize these factors. We then examine how shifts in the supply and demandcurves for loanable funds determine the equilibrium interest rate on a specific financial instrument. A shift in thesupply or demand curve occurs when the quantity of a financial security supplied or demanded changes at everygiven interest rate in response to a change in another factor besides the interest rate. In either case, achange in the supply or demand curve for loanable funds causes interest rates to move. Table 6.5 recapsthe factors that affect the supply and demand for loanable funds discussed in this section, their impact on the supplyand demand for loanable funds for a specific security, and the impact on the market clearing (or equilibrium) interestrates holding all other factors constant.

FIGURE 6.11 Determination of Equilibrium Interest Rates

Interest rates always move toward the equilibrium.

Supply of Funds We have already described the positive relation between interest rates and the supply ofloanable funds along the loanable funds supply curve. Factors that cause the supply curve of loanable funds to shift,at any given interest rate, include the wealth of fund suppliers, the risk of the financial security, future spendingneeds, monetary policy objectives, and economic conditions.

Wealth As the total wealth of financial market participants (households, businesses, etc.) increases, the absolutedollar value available for investment purposes increases. Accordingly, at every interest rate, the supply of loanablefunds increases, or the supply curve shifts down and to the right. For example, as the U.S. economy grew in theearly 2010s, total wealth of U.S. investors increased as well. Consequently, the supply of funds available forinvesting (e.g., in stock and bond markets) increased at every available interest rate. We show this shift (increase) inthe supply curve in Figure 6.12(a) as a move from SS to SS". The shift in the supply curve creates a disequilibriumbetween demand and supply. To eliminate the imbalance or disequilibrium in this financial market, the equilibriuminterest rate falls, from i* to i*", which is associated with an increase in the quantity of funds loaned between fundsuppliers and fund demanders, from Q* to Q*". Conversely, as the total wealth of financial marketparticipants decreases, the absolute dollar value available for investment purposes decreases. Accordingly,

▼

at every interest rate, the supply of loanable funds decreases, or the supply curve shifts up and to the left. Thedecrease in the supply of funds due to a decrease in the total wealth of market participants results in an increase inthe equilibrium interest rate and a decrease in the equilibrium quantity of funds loaned (traded).

▼ TABLE 6.5 Factors That Affect the Supply of and Demand for Loanable Funds for a Financial Security

Panel A: The Supply of Funds

Factor Impact on Supply of FundsImpact on Equilibrium InterestRate*

Interest rate Movement along supplycurve

Direct

Total wealth Shift supply curve InverseRisk of financial security Shift supply curve DirectNear-term spending needs Shift supply curve DirectMonetary expansion Shift supply curve InverseEconomic conditions Shift supply curve Inverse

Panel B: The Demand for Funds

Factor Impact on Supply of FundsImpact on Equilibrium InterestRate

Interest rate Movement along demandcurve

Direct

Utility derived from asset purchased withborrowed funds

Shift demand curve Direct

Restrictiveness of nonprice conditions Shift demand curve InverseEconomic conditions Shift demand curve Direct

A “direct” impact on equilibrium interest rates means that as the “factor” increases (decreases), the equilibrium interest rate increases(decreases). An “inverse” impact means that as the factor increases (decreases), the equilibrium interest rate decreases (increases).

Risk As the risk of a financial security decreases (e.g., the probability that the issuer of the security will default onpromised repayments of the funds borrowed), it becomes more attractive to suppliers of funds. At every interest rate,the supply of loanable funds increases, or the supply curve shifts down and to the right, from SS to SS" in Figure6.12(a). Holding all other factors constant, the increase in the supply of funds, due to a decrease in the risk of thefinancial security, results in a decrease in the equilibrium interest rate, from i* to i*", and an increase in theequilibrium quantity of funds traded, from Q* to Q*".

Conversely, as the risk of a financial security increases, it becomes less attractive to suppliers of funds. Accordingly,at every interest rate, the supply of loanable funds decreases, or the supply curve shifts up and to the left. Holding allother factors constant, the decrease in the supply of funds due to an increase in the financial security’s risk results inan increase in the equilibrium interest rate and a decrease in the equilibrium quantity of funds loaned (or traded).

FIGURE 6.12 The Effect on Interest Rates from a Shift in the Supply Curve of or a Demand Curve for Loanable Funds

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Changes in the supply of and demand for loanable funds have varying effects.

Near-Term Spending Needs When financial market participants have few near-term spending needs, the absolutedollar value of funds available to invest increases. For example, when a family’s son or daughter moves out of thefamily home to live on his or her own, current spending needs of the family decrease and the supply of availablefunds (for investing) increases. At every interest rate, the supply of loanable funds increases, or the supply curveshifts down and to the right. The financial market, holding all other factors constant, reacts to this increased supplyof funds by decreasing the equilibrium interest rate and increasing the equilibrium quantity of funds traded.

Conversely, when financial market participants have increased near-term spending needs, the absolutedollar value of funds available to invest decreases. At every interest rate, the supply of loanable fundsdecreases, or the supply curve shifts up and to the left. The shift in the supply curve creates a disequilibrium in thefinancial market that results in an increase in the equilibrium interest rate and a decrease in the equilibrium quantityof funds loaned (or traded).

Monetary Expansion One method used by the Federal Reserve to implement monetary policy is to alter theavailability of funds, the growth in the money supply, and thus the rate of economic expansion of the economy.

When monetary policy objectives are to allow the economy to expand (as was the case in the late 2000s, during thefinancial crisis, and in the early 2010s), the Federal Reserve increases the supply of funds available in the financialmarkets. At every interest rate, the supply of loanable funds increases, the supply curve shifts down and to the right,and the equilibrium interest rate falls, while the equilibrium quantity of funds traded increases.

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Conversely, when monetary policy objectives are to restrict the rate of economic expansion (and thus inflation), theFederal Reserve decreases the supply of funds available in the financial markets. At every interest rate, the supply ofloanable funds decreases, the supply curve shifts up and to the left, and the equilibrium interest rate rises, while theequilibrium quantity of funds loaned or traded decreases.

Economic Conditions Finally, as the underlying economic conditions themselves (e.g., the inflation rate,unemployment rate, economic growth) improve in a country relative to other countries, the flow of funds to thatcountry increases. This reflects the lower risk (country or sovereign risk) that the country, in the guise of itsgovernment, will default on its obligation to repay funds borrowed. For example, the severe economic crisis inGreece in the early 2010s resulted in a decrease in the supply of funds to that country. An increased inflow offoreign funds to U.S. financial markets increases the supply of loanable funds at every interest rate, and the supplycurve shifts down and to the right. Accordingly, the equilibrium interest rate falls, and the equilibrium quantity offunds loaned or traded increases.

Conversely, when economic conditions in foreign countries improve, domestic and foreign investors take their fundsout of domestic financial markets (e.g., the United States) and invest abroad. Thus, the supply of funds available inthe financial markets decreases and the equilibrium interest rate rises, while the equilibrium quantity of funds tradeddecreases.

Demand for Funds We explained above that the quantity of loanable funds demanded is negatively related tointerest rates. Factors that cause the demand curve for loanable funds to shift include the utility derived from assetspurchased with borrowed funds, the restrictiveness of nonprice conditions on borrowing, and economic conditions.

Utility Derived from Assets Purchased with Borrowed Funds As the utility (i.e., satisfaction or pleasure) derivedfrom an asset purchased with borrowed funds increases, the willingness of market participants (households,businesses, etc.) to borrow increases and the absolute dollar value borrowed increases. Accordingly, at every interestrate, the demand for loanable funds increases, or the demand curve shifts up and to the right. For example, suppose achange in jobs takes an individual from Arizona to Minnesota. The individual currently has a convertibleautomobile. Given the move to Minnesota, the individual’s utility from the convertible decreases, while it wouldincrease for a car with heated seats. Thus, with a potential increased utility from the purchase of a new car, theindividual’s demand for funds in the form of an auto loan increases. We show this shift (increase) in the demandcurve in Figure 6.12(b) as a move from DD to DD". The shift in the demand curve creates a disequilibrium in thisfinancial market. Holding all other factors constant, the increase in the demand for funds due to an increase in theutility from the purchased asset results in an increase in the equilibrium interest rate, from i* to i*", and an increasein the equilibrium quantity of funds traded, from Q* to Q*".

Conversely, as the utility derived from an asset purchased with borrowed funds decreases, the willingness of marketparticipants (households, businesses, etc.) to borrow decreases and the absolute dollar amount borrowed decreases.Accordingly, at every interest rate, the demand for loanable funds decreases, or the demand curve shifts down and tothe left. The shift in the demand curve again creates a disequilibrium in this financial market. As competitive forcesadjust, and holding all other factors constant, the decrease in the demand for funds due to a decrease in the utilityfrom the purchased asset results in a decrease in the equilibrium interest rate and a decrease in the equilibriumquantity of funds loaned or traded.

Restrictiveness of Nonprice Conditions on Borrowed Funds As the nonprice restrictions put on borrowers as acondition of borrowing decrease, the willingness of market participants to borrow increases and the absolute dollarvalue borrowed increases. Such nonprice conditions may include fees or collateral. The lack of such restrictionsmakes the loan more desirable to the user of funds. Accordingly, at every interest rate, the demand forloanable funds increases, or the demand curve shifts up and to the right, from DD to DD". As competitiveforces adjust, and holding all other factors constant, the increase in the demand for funds due to a decrease in therestrictive conditions on the borrowed funds results in an increase in the equilibrium interest rate, from i* to i*", andan increase in the equilibrium quantity of funds traded, from Q* to Q*". Conversely, as the nonprice restrictions puton borrowers as a condition of borrowing increase, market participants’ willingness to borrow decreases, and theabsolute dollar value borrowed decreases. Accordingly, the demand curve shifts down and to the left. The shift inthe demand curve results in a decrease in the equilibrium interest rate and a decrease in the equilibrium quantity offunds traded.

Economic Conditions When the domestic economy experiences a period of growth, such as that in the UnitedStates in the mid-2000s and early 2010s, market participants are willing to borrow more heavily. For example, state

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and local governments are more likely to repair and improve decaying infrastructure when the local economy isstrong. Accordingly, the demand curve for funds shifts up and to the right. Holding all other factors constant, theincrease in the demand for funds due to economic growth results in an increase in the equilibrium interest rate andan increase in the equilibrium quantity of funds traded. Conversely, when domestic economic growth is stagnant,market participants reduce their demand for funds. Accordingly, the demand curve shifts down and to the left,resulting in a decrease in the equilibrium interest rate and a decrease in the equilibrium quantity of funds traded.

Movement of Interest Rates over TimeAs discussed in the previous section of this chapter, the loanable funds theory of interest rates is based on the supplyof and demand for loanable funds as functions of interest rates. The equilibrium interest rate (point E in Figure 6.12)is only a temporary equilibrium. Changes in underlying factors that determine the demand and supply of loanablefunds can cause continuous shifts in the supply and/or demand curves for loanable funds. Market forces will react tothe resulting disequilibrium with factors that influence a change in the equilibrium interest rate and quantity of fundstraded in that market. Refer again to Figure 6.12(a), which shows the effects of an increase in the supply curve forloanable funds, from SS to SS" (and the resulting decrease in the equilibrium interest rate, from i* to i*"), whileFigure 6.12(b) shows the effects of an increase in the demand curve for loanable funds, from DD to DD" (and theresulting increase in the equilibrium interest rate, from i* to i*").

time out!6-7 Who are the main suppliers and demanders of loanable funds?6-8 What happens to the equilibrium interest rate when the demand for (supply of) loanable funds increases?6-9 How do supply and demand together determine interest rates?

6.4 • FACTORS THAT INFLUENCE INTEREST RATES FORINDIVIDUAL SECURITIES LG6-6So far we have looked at the general determination of equilibrium (nominal) interest rates for financial securities inthe context of the loanable demand and supply theory of the flow of funds. In this section, we examine the specificfactors that affect differences in interest rates across the range of real-world financial markets (i.e., differencesamong interest rates on individual securities, given the underlying level of interest rates determined by the demandand supply of loanable funds). These factors include

Inflation.The real risk-free rate.

Default risk.Liquidity risk.

Special provisions regarding the use of funds raised by a particular security issuer.The security’s term to maturity.

We will discuss each of these factors after summarizing them in Table 6.6.

InflationThe first factor that influences interest rates is the economy-wide actual or expected inflation rate. Specifically, thehigher the level of actual or expected inflation, the higher will be the level of interest rates. We define inflation of thegeneral price index of goods and services (or the inflation premium, IP) as the (percentage) increase in the price of astandardized basket of goods and services over a given period of time. The U.S. Department of Commerce measuresinflation using indexes such as the consumer price index (CPI) and the producer price index (PPI). For example, theannual inflation rate using the CPI index between years t and t + 1 would be equal to

inflation The continual increase in the price level of a basket of goods and services.

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(6-1)

The positive relationship between interest rates and inflation rates is fairly intuitive: When inflation raises thegeneral price level, investors who buy financial assets must earn a higher interest rate (or inflation premium) tocompensate for continuing to hold the investment. Holding on to their investments means that they incur highercosts of forgoing consumption of real goods and services today, only to have to buy these same goods and servicesat higher prices in the future. In other words, the higher the rate of inflation, the more expensive the same basket ofgoods and services will be in the future.

Real Risk-Free RateA real risk-free rate is the rate that a risk-free security would pay if no inflation were expected over its holding period(e.g., a year). As such, it measures only society’s relative time preference for consuming today rather thantomorrow. The higher society’s preference to consume today (i.e., the higher its time value of money or rate of timepreference), the higher the real risk-free rate (RFR) will be.

real risk-free rate The interest rate that would exist on a risk-free security if no inflation were expected.

▼ TABLE 6.6 Factors Affecting Nominal Interest Rates

Inflation: A continual increase in the price level of a basket of goods and services throughout the economy as a whole.

Real risk-free rate: Risk-free rate adjusted for inflation; generally lower than nominal risk-free rates at any particular time.

Default risk: Risk that a security issuer will miss an interest or principal payment or continue to miss such payments.

Liquidity risk: Risk that a security cannot be sold at a price relatively close to its value with low transaction costs on short notice.

Special provisions: Provisions (e.g., taxability, convertibility, and callability) that impact a security holder beneficially oradversely and as such are reflected in the interest rates on securities that contain such provisions.

Time to maturity: Length of time until a security is repaid; used in debt securities as the date upon which the security holders gettheir principal back.

Fisher Effect Economists often refer to the relationship among real risk-free rates (RFR), expected inflation(expected IP), and nominal risk-free rates (i), described previously, as the Fisher effect, named for Irving Fisher,who identified these economic relationships early last century. The Fisher effect theorizes that nominal risk-freerates that we observe in financial markets (e.g., the one-year Treasury bill rate) must compensate investorsfor

Any inflation-related reduction in purchasing power lost on funds lent or principal due.

An additional premium above the expected rate of inflation for forgoing present consumption (which reflects the real risk-free rate issuediscussed previously).

(6-2)

Thus, the nominal risk-free rate will equal the real risk-free rate only when market participants expect inflation to bezero: Expected IP = 0. Similarly, the nominal risk-free rate will equal the expected inflation rate only when the realrisk-free rate is zero. We can rearrange the nominal risk-free rate equation to show what determines the real interestrate:1

(6-3)

It needs to be noted that the expected inflation rate is difficult to estimate accurately, so the real risk-free rate can bedifficult to measure accurately. Investors’ expectations are not always realized either.

The one-year T-bill rate in 2015 was 0.32 percent, while the CPI for the year was 0.50 percent, which implies a realrisk-free rate of –0.18 percent—that is, the real risk-free rate was actually negative. Thus, the real value ofinvestments actually decreased in that year.

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Figure 6.13 shows the nominal risk-free rate (one-year T-bill rate) versus the change in the CPI from 1962 through2015. Note that generally the T-bill rate is greater than the CPI, that is, the real risk-free rate earned on securities ispositive. It is during periods of economic slowdowns that the T-bill rate is less than the CPI; that is, real risk-freerates are negative.

Default or Credit RiskDefault risk is the risk that a security issuer may fail to make its promised interest and principal payments to itsbondholders (or its dividend in the case of preferred stockholders). The higher the default risk, the higher the interestrate that security buyers will demand to compensate them for this default (or credit) risk relative to default-risk-freeU.S. Treasury securities. Since the U.S. government has taxation powers and can print currency, the risk of itsdefaulting on debt payments is practically zero. But some borrowers, such as corporations or individuals, have lesspredictable cash flows (and no powers to tax anyone to raise funds immediately). So investors must charge issuersother than the U.S. government a premium for any perceived probability of default and the cost of potentiallyrecovering the amount loaned built into their regular interest rate premium. The difference between a quoted interestrate on a security (security j) and a Treasury security with similar maturity, liquidity, tax, and other features is calleda default or credit risk premium (DRPj). That is

default risk The risk that a security issuer will default on that security by being late on or missing an interest or principal payment.

(6-4)

where ijt = Interest rate on a security issued by a non−Treasury issuer (issuer j) of maturity m at time t.

iTt = Interest rate on a security issued by the U.S. Treasury of maturity m at time t.

Various rating agencies, including Moody’s and Standard & Poor’s, evaluate and categorize the potential defaultrisk on many corporate bonds, some state and municipal bonds, and some stocks. We cover these ratings in moredetail in Chapter 8. For example, in 2015, the 10-year Treasury rate was 2.14 percent. Moody’s Aaa-rated and Baa-rated corporate debt carried interest rates of 3.89 percent and 5.00 percent, respectively. Thus, the average defaultrisk premiums on the Aaa-rated and Baa-rated corporate debt were

Figure 6.14 presents these risk premiums for the stated creditworthiness categories of bonds from 1977 through2015. Notice from this figure and Figure 6.13 that default risk premiums tend to increase when the economy iscontracting and decrease when the economy is expanding. For example, from 2007 to 2008, real risk-free rates (T-bills—CPI in Figure 6.13) increased from 0.43 percent to 1.73 percent. Over the same period, default risk premiumson Aaa-rated bonds increased from 1.39 percent to 1.97 percent. Baa-rated bonds showed a default risk premiumincrease from 2.55 percent to 3.78 percent.

EXAMPLE 6-1Calculating Real Risk-FreeRates LG6-6

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

One-year Treasury bill rates in 2007 averaged 4.53 percent andinflation (measured by the consumer price index) for the year was 4.10percent. If investors had expected the same inflation rate as thatactually realized, calculate the real risk-free rate for 2007 according to

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the Fisher effect.

SOLUTION:4.53% − 4.10% = 0.43%

Similar to Problems 6-1, 6-2, Self-Test Problem 1

Liquidity RiskA highly liquid asset can be sold at a predictable price with low transaction costs. That is, the holder can convert theasset at its fair market value on short notice. The interest rate on a security reflects its relative liquidity, with highlyliquid assets carrying the lowest interest rates (all other characteristics remaining the same). Likewise, if a security isilliquid, investors add a liquidity risk premium (LRP) to the interest rate on the security. In the United States,most government securities sell in liquid markets, as do large corporations’ stocks and bonds. Securitiesissued by smaller companies trade in relatively less liquid markets.

liquidity risk The risk that a security cannot be sold at a predictable price with low transaction costs on short notice.

FIGURE 6.13 Nominal Interest Rates versus Inflation

Notice the difference between the nominal risk-free rate and the change in CPI over the last several decades.

Source: Federal Reserve Board and U.S. Department of Labor websites, various dates. www.federalreserve.gov and www.dol.gov.

A different type of liquidity risk premium may also exist if investors dislike long-term securities because their prices(present values, as discussed below and in Chapters 4 and 7) react more to interest rate changes than short-termsecurities do. In this case, a higher liquidity risk premium may be added to a security with a longer maturity becauseof its greater exposure to price risk (loss of capital value) on the longer-term security as interest rates change.

Special Provisions or CovenantsSometimes a security’s issuing party attaches special provisions or covenants to the security issued. Such provisionsaffect the interest rates on these securities relative to securities without such provisions attached to them. Some ofthese special provisions include the security’s taxability, convertibility, and callability. For example, investors payno federal taxes on interest payments received from municipal securities. So a municipal bondholder may demand a

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lower interest rate than that demanded on a comparable taxable bond—such as a Treasury bond (which is taxable atthe federal level but not at the state or local levels) or a corporate bond (the interest on which is taxable at the state,local, and federal levels).Another special covenant is convertibility: A convertible bond offers the holder the opportunity to exchange thebond for another type of the issuer’s securities—usually preferred or common stock—at a preset price (see Chapter7). This conversion option can be valuable to purchasers, so convertible security buyers require lower interest ratesthan a comparable nonconvertible security holder would require (all else equal). In general, special provisions thatbenefit security holders (e.g., tax-free status and convertibility) bring with them lower interest rates, and specialprovisions that benefit security issuers (e.g., callability, by which an issuer has the option to retire, or call, thesecurity prior to maturity at a preset price) require higher interest rates to encourage purchase.

Term to MaturityInterest rates also change—sometimes daily—because of a bond’s term to maturity. Financial professionals refer tothis daily or even hourly changeability in interest rates as the term structure of interest rates, or the yield curve. Theshape of the yield curve derives directly from time value of money principles. The term structure of interest ratescompares interest rates on debt securities based on their time to maturity, assuming that all other characteristics (i.e.,default risk, liquidity risk) are equal. Interest rates change as the maturity of a debt security changes; in general, thelonger the term to maturity, the higher the required interest rate buyers will demand. This addition to the requiredinterest rate is the maturity premium (MP). The MP, which is the difference between the required yield on long-versus short-term securities of the same characteristics except maturity, can be positive, negative, or zero.

term structure of interest rates A comparison of market yields on securities, assuming all characteristics except maturity are the same.

FIGURE 6.14 Default Risk Premiums on Corporate Bonds

Source: Federal Reserve Board website, various dates. www.federalreserve.gov

FIGURE 6.15 Common Shares for Yield Curves on Treasury Securities

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Three common yield curve shapes are (a) upward sloping, (b) downward sloping, and (c) a flat slope.Source: U.S. Treasury, Office of Debt Management, Daily Treasury Yield Curves, various dates. www.ustreas.gov

EXAMPLE 6-2 Determinants of Interest Rates forIndividual Securities LG6-6

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Morningstar Corp.’s eight-year bonds are currently yielding a return of6.85 percent. The expected inflation premium is 1.15 percent annuallyand the real risk-free rate is expected to be 2.25 percent annually overthe next eight years. The default risk premium on Morningstar’s bondsis 1.35 percent. The maturity risk premium is 0.50 percent on two-yearsecurities and increases by 0.05 percent for each additional year tomaturity. Calculate the liquidity risk premium on Morningstar’s eight-year bonds.

SOLUTION:

Similar to Problems 6-3, 6-4, Self-Test Problem 2

The financial industry most often reports and analyzes the yield curve for U.S. Treasury securities. The yield curvefor U.S. Treasury securities has taken many shapes over the years, but the three most common shapes appear inFigure 6.15. In graph (a), the yield curve on January 15, 2016, yields rise steadily with maturity when the yieldcurve slopes upward. This is the most common yield curve. On average, the MP is positive, as you might expect.Graph (b) shows an inverted, or downward-sloping, yield curve, reported on November 24, 2000, in which yieldsdecline as maturity increases. Inverted yield curves do not generally last very long. In this case, the yield curveinverted as the U.S. Treasury began retiring long-term (30-year) bonds as the country began to pay off the nationaldebt. Finally, graph (c) shows a flat yield curve, reported on June 4, 2007, when the yield to maturity is virtuallyunaffected by the term to maturity.

Putting together the factors that affect interest rates in different markets, we can use the following general equationto note the influence of the factors that functionally impact the fair interest rate—the rate necessary to compensateinvestors for all security risks—(ij

*) on an individual (jth) financial security.

(6-5)

where

IP = Inflation premium.RFR = Real risk-free rate.DRPj = Default risk premium on the jth security.LRPj = Liquidity risk premium on the jth security.SCPj = Special covenant premium on the jth security.MPj = Maturity premium on the jth security.

The first two factors, IP and RFR, are common to all financial securities, while the other factors can uniquelyinfluence the price of a single security.

time out!

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6-10 What is the difference between nominal and real risk-free rates?6-11 What does “the term structure of interest rates” mean?6-12 What shape does the term structure usually take? Why?

6.5 • THEORIES EXPLAINING THE SHAPE OF THE TERMSTRUCTURE OF INTEREST RATES2 LG6-7We just explained the necessity of a maturity premium, the relationship between a security’s interest rate and itsremaining term to maturity. We can illustrate these issues by showing that the term structure of interest rates cantake a number of different shapes. As you might expect, economists and financial theorists with various viewpointsdiffer among themselves in theorizing why the yield curve takes different shapes. Explanations for the yield curve’sshape fall predominantly into three categories:

1. The unbiased expectations theory.

2. The liquidity premium theory.3. The market segmentation theory.

Look again at Figure 6.15 (a), which presents the Treasury yield curve as of January 15, 2016. We see that the yieldcurve on this date reflected the normal upward-sloping relationship between yield and maturity. Now let’s turn toexplanations for this shape based on the three predominant theories noted above.

Unbiased Expectations TheoryAccording to the unbiased expectations theory of the term structure of interest rates, at any given point in time, theyield curve reflects the market’s current expectations of future short-term rates. As illustrated in Figure 6.16, theintuition behind the unbiased expectations theory is this: If investors have a four-year investment horizon, they couldeither buy current four-year bonds and earn the current (or spot) yield on a four-year bond (1R4, if held to maturity)each year, or they could invest in four successive one-year bonds [of which they know only the current one-yearspot rate (1R1)]. But investors also expect what the unknown future one-year rates [E(2r1), E(3r1), and E(4r1)] willbe. Note that each interest rate term has two subscripts, e.g., 1R4. The first subscript indicates the period in which thesecurity is bought, so that 1 represents the purchase of a security in period 1. The second subscript indicates thematurity on the security. Thus, 4 represents the purchase of a security with a four-year life. Similarly, E(3r1) is theexpected return on a security with a one-year life purchased in period 3.

According to the unbiased expectations theory, the return for holding a four-year bond to maturity should equal theexpected return for investing in four successive one-year bonds (as long as the market is in equilibrium). If thisequality does not hold, an arbitrage opportunity exists. That is, if investors could earn more on the one-year bondinvestments, they could short (or sell) the four-year bond, use the proceeds to buy the four successive one-yearbonds, and earn a guaranteed profit over the four-year investment horizon. So, according to the unbiasedexpectations theory, if the market expects future one-year rates to rise each successive year into the future,then the yield curve will slope upward. Specifically, the current four-year T-bond rate or return will exceed thethree-year bond rate, which will exceed the two-year bond rate, and so on. Similarly, if the market expects futureone-year rates to remain constant each successive year into the future, then the four-year bond rate will equal thethree-year bond rate. That is, the term structure of interest rates will remain constant (flat) over the relevant timeperiod. Specifically, the unbiased expectation theory states that current long-term interest rates are geometricaverages of current and expected future short-term interest rates. The mathematical equation representing thisrelationship is

(6-6)

therefore

▼

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(6-7)

where 1RN = Actual N-period rate today (i.e., the first day of year 1).

N = Term to maturity.1R1 = Actual 1-year rate today.E(ir1) = Expected 1-year rates for years 2, 3, 4, . . ., N in the future.

Notice that uppercase interest rate terms, 1Rt, are the actual current interest rates on securities purchased today with amaturity of t years. Lowercase interest rate terms, tr1, represent estimates of future one-year interest rates starting tyears into the future.

FIGURE 6.16 Unbiased Expectations Theory of the Term Structure of Interest Rates

Return from buying four 1-year maturity bonds versus buying one 4-year maturity bond.

the Math Coach on…

“When putting interest rates into the equation, enter them in decimal format, not percentage format.Correct: (1 + 0.0294)

Not correct: (1 +2.94)„

EXAMPLE 6-3 Calculating Yield Curves LG6-7

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Suppose that the current one-year rate (one-year spot rate) andexpected one-year T-bond rates over the following three years (i.e.,years 2, 3, and 4, respectively) are as follows:

1R1 = 2.94%, E(1r1) = 4%, E(3r1) = 4.74%, E(4r1) = 5.10%

Construct a yield curve using the unbiased expectations theory.

SOLUTION:

Using the unbiased expectations theory, current (or today’s) rates forone-, two-, three-, and four-year maturity Treasury securities should be

and the current yield to maturity curve will be upward sloping as shown:

This upward-sloping yield curve reflects the market’s expectation ofpersistently rising one-year (short-term) interest rates over the futurehorizon.3

Similar to Problems 6-5, 6-6, 6-7, 6-8, Self-Test Problem 3

Liquidity Premium TheoryThe second popular explanation—the liquidity premium theory of the term structure of interest rates—builds on theunbiased expectations theory. The liquidity premium idea is as follows: Investors will hold long-term maturitiesonly if these securities with longer term maturities are offered at a premium to compensate for future uncertainty inthe security’s value. Of course, uncertainty or risk increases with an asset’s maturity. This theory is thus consistentwith our discussions of market risk and liquidity risk, above. Specifically, in a world of uncertainty, short-termsecurities provide greater marketability (due to their more active secondary markets) and have less price risk thanlong-term securities do. As a result (due to smaller price fluctuations for a given change in interest rates), investorswill prefer to hold shorter-term securities because this kind of paper can be converted into cash with little marketrisk. Said another way, investors face little risk of a capital loss, that is, a fall in the price of the security below itsoriginal purchase price. So, investors must be offered a liquidity premium to buy longer-term securities that carryhigher capital loss risk. This difference in market and liquidity risk can be directly related to the fact that longer-termsecurities are more sensitive to interest rate changes in the market than are shorter-term securities—Chapter 7discusses bond interest rate sensitivity and the link to a bond’s maturity. Because longer maturities on securitiesmean greater market and liquidity risk, the liquidity premium increases as maturity increases.

The liquidity premium theory states that long-term rates are equal to geometric averages of current and expectedshort-term rates (like the unbiased expectations theory), plus liquidity risk premiums that increase with the security’smaturity (this is the extension of the liquidity premium added to the unbiased expectations theory). Figure 6.17illustrates the differences in the shape of the yield curve under the unbiased expectations theory versus the liquiditypremium theory. For example, according to the liquidity premium theory, an upward-sloping yield curve may reflectinvestors’ expectations that future short-term rates will be flat, but because liquidity premiums increase withmaturity, the yield curve will nevertheless slope upward. Indeed, an upward-sloping yield curve may reflectexpectations that future interest rates will rise, be flat, or even fall as long as the liquidity premium increases withmaturity fast enough to produce an upward-sloping yield curve. The liquidity premium theory can be

▼

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mathematically represented as

(6-8)

where Lt = Liquidity premium for a period t and L2 < L3 < LN.

FIGURE 6.17 Yield Curve Using the Unbiased Expectation Theory (UET) versus the Liquidity Premium Theory (LPT)

Notice the differences in the shape of the yield curve under the UET and the LPT.

Let’s compare the yield curves in Examples 6-3 (using the unbiased expectations theory) and 6-4. Notice that theliquidity premium in year 2 (L2 = 0.10%) produces a 0.05 (= 3.52% − 3.47%) percent premium on the yield tomaturity on a two-year T-note, the liquidity premium for year 3 (L3 = 0.20%) produces a 0.10 (= 3.99% − 3.89%)percent premium on the yield to maturity on the three-year T-note, and the liquidity premium for year 4 (L4 =0.30%) produces a 0.15 (= 4.34% − 4.19%) percent premium on the yield to maturity on the four-year T-note.

Market Segmentation TheoryThe market segmentation theory does not build on the unbiased expectations theory or the liquidity premium theory,but rather argues that individual investors and FIs have specific maturity preferences, and convincing them to holdsecurities with maturities other than their most preferred requires a higher interest rate (maturity premium). Themain thrust of the market segmentation theory is that investors do not consider securities with different maturities asperfect substitutes. Rather, individual investors and FIs have distinctly preferred investment horizons dictated by thedates when their liabilities will come due. For example, banks might prefer to hold relatively short-term U.S.Treasury bonds because their deposit liabilities also tend to be short-term—recall that bank customers can accesstheir funds on demand. Insurance companies, on the other hand, may prefer to hold long-term U.S. Treasury bondsbecause life insurance contracts usually expose insurance firms to long-term liabilities. Accordingly, distinct supplyand demand conditions within a particular maturity segment—such as the short end and long end of the bond market

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—determine interest rates under the market segmentation theory.

EXAMPLE 6-4Calculating Yield Curves Usingthe Liquidity PremiumTheory LG6-7

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Suppose that the current one-year rate (one-year spot rate) andexpected one-year T-bond rates over the following three years (i.e.,years 2, 3, and 4, respectively) are as follows:

In addition, investors charge a liquidity premium on longer-termsecurities such that

Using the liquidity premium theory, construct the yield curve.

SOLUTION:

Using the liquidity premium theory, current rates for one-, two-, three-,and four-year maturity Treasury securities should be

and the current yield to maturity curve will be upward sloping as shown:

Similar to Problems 6-9, 6-10, Self-Test Problem 3

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The market segmentation theory assumes that investors and borrowers generally do not want to shift from onematurity sector to another without adequate compensation—that is, an interest rate premium. Figure 6.18demonstrates how changes in supply for short- versus long-term bond market segments result in changing shapes ofthe yield to maturity curve. Specifically, as shown in Figure 6.18, the higher the demand for securities is, the higherthe yield on those securities.4 Further, as the supply of securities decreases in the short-term market and increases inthe long-term market, the slope of the yield curve becomes steeper. If the supply of short-term securities hadincreased while the supply of long-term securities had decreased, the yield curve would have a flatter slope andmight even have sloped downward. Indeed, the U.S. Treasury’s large-scale repurchase of long-term Treasury bonds(i.e., reductions in supply) in early 2000 has been viewed as the major cause of the inverted yield curve thatappeared in February 2000.

FIGURE 6.18 Market Segmentation and Determination of the Slope of the Yield Curve

The higher the demand for securities, the higher the yield on those securities.

time out!6-13 What three theories explain the shape of the yield curve?6-14 Explain how arbitrage plays a role in the unbiased expectations explanation of the shape of the yield curve.

6.6 • FORECASTING INTEREST RATES5 LG6-8We noted in the time value of money (TVM) chapters (Chapters 4 and 5) that as interest rates change, so do thevalues of financial securities. Accordingly, both individual investors and public corporations want to be able topredict or forecast interest rates if they wish to trade profitably. For example, if interest rates rise, the value ofinvestment portfolios of individuals and corporations will fall, resulting in a loss of wealth. So, interest rate forecastsare extremely important for the financial wealth of both public corporations and individuals.

Recall our discussion of the unbiased expectations theory in the previous section of this chapter. That theory

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indicated that the market’s expectation of future short-term interest rates determines the shape of the yield curve. Forexample, an upward-sloping yield curve implies that the market expects future short-term interest rates to rise. So,we can use the unbiased expectations theory to forecast (short-term) interest rates in the future (i.e., forward one-year interest rates). A forward rate is an expected, or implied, rate on a short-term security that will originate at somepoint in the future. Using the equations in the unbiased expectations theory, we can directly derive the market’sexpectation of forward rates from existing or actual rates on spot market securities.

forward rate An expected rate (quoted today) on a security that originates at some point in the future.

To find an implied forward rate on a one-year security to be issued one year from today, we can rewrite the unbiasedexpectations theory equation as follows:

(6-9)

EXAMPLE 6-5 Estimating Forward Rates LG6-8

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

In the mid-2010s, the existing or current (spot) one-, two-, three-, andfour-year zero coupon Treasury security rates were as follows:

1R1 = 0.70%, 1R2 = 0.87%, 1R3 = 1.04%, 1R4 = 1.34%

Using the unbiased expectations theory, calculate one-year forwardrates on zero coupon Treasury bonds for years 2, 3, and 4.

SOLUTION:

Similar to Problems 6-15, 6-16, Self-Test Problem 4

where 2 f1 = expected one-year rate for year 2, or the implied forward one-year rate for next year.

Saying that 2 f1 is the expected one-year rate for year 2 is the same as saying that, once we isolate the 2 f1 term, theequation will give us the market’s estimate of the expected one-year rate for year 2. Solving for 2 f1 we get

(6-10)

In general, we can find the forward rate for any year, N, into the future using the following generalized equationderived from the unbiased expectations theory:

(6-11)

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6-15 What is a forward rate?6-16 How can we obtain an implied forward rate from current short- and long-term interest rates?6-17 Why is it useful to calculate forward rates?

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Your Turn…Questions

1. Classify the following transactions as taking place in the primary or secondary markets (LG6-1):

a. IBM issues $200 million of new common stock.b. The New Company issues $50 million of common stock in an IPO.c. IBM sells $5 million of GM preferred stock out of its marketable securities portfolio.d. The Magellan Fund buys $100 million of previously issued IBM bonds.e. Prudential Insurance Co. sells $10 million of GM common stock.

2. Classify the following financial instruments as money market securities or capital market securities (LG6-2):

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a. Federal fundsb. Common stockc. Corporate bondsd. Mortgagese. Negotiable certificates of depositf. U.S. Treasury billsg. U.S. Treasury notesh. U.S. Treasury bondsi. State and government bonds

3. What are the different types of financial institutions? Include a description of the main servicesoffered by each. (LG6-3)

4. How would economic transactions between suppliers of funds (e.g., households) and users of funds (e.g.,corporations) occur in a world without FIs? (LG6-3)

5. Why would a world limited to the direct transfer of funds from suppliers of funds to users of funds likely resultin quite low levels of fund flows? (LG6-3)

6. How do FIs reduce monitoring costs associated with the flow of funds from fund suppliers to fund users? (LG6-3)

7. How do FIs alleviate the problem of liquidity risk faced by investors wishing to invest in securities ofcorporations? (LG6-3)

8. Who are the suppliers of loanable funds? (LG6-4)

9. Who are the demanders of loanable funds? (LG6-4)

10. What factors cause the supply of funds curve to shift? (LG6-5)

11. What factors cause the demand for funds curve to shift? (LG6-5)

12. What are six factors that determine the nominal interest rate on a security? (LG6-6)

13. What should happen to a security’s equilibrium interest rate as the security’s liquidity risk increases? (LG6-6)

14. Discuss and compare the three explanations for the shape of the yield curve. (LG6-7)

15. Are the unbiased expectations and liquidity premium theories explanations for the shape of the yield curvecompletely independent theories? Explain why or why not. (LG6-7)

16. What is a forward interest rate? (LG6-8)

17. If we observe a one-year Treasury security rate that is higher than the two-year Treasury security rate, what canwe infer about the one-year rate expected one year from now? (LG6-8)

ProblemsBASIC PROBLEMS

6-1 Determinants of Interest Rates for Individual Securities A particular security’s default risk premium is2 percent. For all securities, the inflation risk premium is 1.75 percent and the real risk-free rate is 3.50percent. The security’s liquidity risk premium is 0.25 percent and maturity risk premium is 0.85 percent.The security has no special covenants. Calculate the security’s equilibrium rate of return. (LG6-6)

6-2 Determinants of Interest Rates for Individual Securities You are considering an investment in 30-yearbonds issued by Moore Corporation. The bonds have no special covenants. The Wall Street Journal reportsthat one-year T-bills are currently earning 1.25 percent. Your broker has determined the followinginformation about economic activity and Moore Corporation bonds:

Real risk-free rate = 0.75%Default risk premium = 1.15%

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Liquidity risk premium = 0.50%Maturity risk premium = 1.75%

a. What is the inflation premium? (LG6-6)b. What is the fair interest rate on Moore Corporation 30-year bonds? (LG6-6)

6-3 Determinants of Interest Rates for Individual Securities Dakota Corporation 15-year bonds have anequilibrium rate of return of 8 percent. For all securities, the inflation risk premium is 1.75 percent and thereal risk-free rate is 3.50 percent. The security’s liquidity risk premium is 0.25 percent and maturityrisk premium is 0.85 percent. The security has no special covenants. Calculate the bond’s defaultrisk premium. (LG6-6)

6-4 Determinants of Interest Rates for Individual Securities A two-year Treasury security currently earns1.94 percent. Over the next two years, the real risk-free rate is expected to be 1.00 percent per year and theinflation premium is expected to be 0.50 percent per year. Calculate the maturity risk premium on the two-year Treasury security. (LG6-6)

6-5 Unbiased Expectations Theory Suppose that the current one-year rate (one-year spot rate) and expectedone-year T-bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows:

Using the unbiased expectations theory, calculate the current (long-term) rates for one-, two-, three-, andfour-year-maturity Treasury securities. Plot the resulting yield curve. (LG6-7)

6-6 Unbiased Expectations Theory Suppose that the current one-year rate (one-year spot rate) and expectedone-year T-bill rates over the following three years (i.e., years 2, 3, and 4, respectively) are as follows:

Using the unbiased expectations theory, calculate the current (long-term) rates for one-, two-, three-, andfour-year-maturity Treasury securities. Plot the resulting yield curve. (LG6-7)

6-7 Unbiased Expectations Theory One-year Treasury bills currently earn 1.45 percent. You expect that oneyear from now, one-year Treasury bill rates will increase to 1.65 percent. If the unbiased expectationstheory is correct, what should the current rate be on two-year Treasury securities? (LG6-7)

6-8 Unbiased Expectations Theory One-year Treasury bills currently earn 2.15 percent. You expect that oneyear from now, one-year Treasury bill rates will increase to 2.65 percent and that two years from now, one-year Treasury bill rates will increase to 3.05 percent. If the unbiased expectations theory is correct, whatshould the current rate be on three-year Treasury securities? (LG6-7)

6-9 Liquidity Premium Theory One-year Treasury bills currently earn 3.45 percent. You expect that one yearfrom now, one-year Treasury bill rates will increase to 3.65 percent. The liquidity premium on two-yearsecurities is 0.05 percent. If the liquidity premium theory is correct, what should the current rate be on two-year Treasury securities? (LG6-7)

6-10 Liquidity Premium Theory One-year Treasury bills currently earn 2.25 percent. You expect that oneyear from now, one-year Treasury bill rates will increase to 2.45 percent and that two years from now,one-year Treasury bill rates will increase to 2.95 percent. The liquidity premium on two-year securitiesis 0.05 percent and on three-year securities is 0.15 percent. If the liquidity premium theory is correct,what should the current rate be on three-year Treasury securities? (LG6-7)

6-11 Liquidity Premium Theory Based on economists’ forecasts and analysis, one-year Treasury bill ratesand liquidity premiums for the next four years are expected to be as follows:

Using the liquidity premium theory, plot the current yield curve. Make sure you label the axes on thegraph and identify the four annual rates on the curve both on the axes and on the yield curve itself. (LG6-

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7)6-12 Liquidity Premium Theory Based on economists’ forecasts and analysis, one-year Treasury

bill rates and liquidity premiums for the next four years are expected to be as follows:

INTERMEDIATE PROBLEMS

6-13 Determinants of Interest Rates for Individual Securities Tom and Sue’s Flowers, Inc.’s, 15-yearbonds are currently yielding a return of 8.25 percent. The expected inflation premium is 2.25 percentannually and the real risk-free rate is expected to be 3.50 percent annually over the next 15 years. Thedefault risk premium on Tom and Sue’s Flowers’ bonds is 0.80 percent. The maturity risk premium is0.75 percent on five-year securities and increases by 0.04 percent for each additional year to maturity.Calculate the liquidity risk premium on Tom and Sue’s Flowers, lnc.’s, 15-year bonds. (LG6-6)

6-14 Determinants of Interest Rates for Individual Securities NikkiG’s Corporation’s 10-year bonds arecurrently yielding a return of 6.05 percent. The expected inflation premium is 1.00 percent annuallyand the real risk-free rate is expected to be 2.10 percent annually over the next 10 years. The liquidityrisk premium on NikkiG’s bonds is 0.25 percent. The maturity risk premium is 0.10 percent on two-year securities and increases by 0.05 percent for each additional year to maturity. Calculate the defaultrisk premium on NikkiG’s 10-year bonds. (LG6-6)

6-15 Unbiased Expectations Theory Suppose we observe the following rates: 1R1 = 8%, 1R2 = 10%. If theunbiased expectations theory of the term structure of interest rates holds, what is the one-year interestrate expected one year from now, E(2r1)? (LG6-7)

6-16 Unbiased Expectations Theory The Wall Street Journal reports that the rate on four-year Treasurysecurities is 1.60 percent and the rate on five-year Treasury securities is 2.15 percent. According to theunbiased expectations theories, what does the market expect the one-year Treasury rate to be fouryears from today, E(5r1)? (LG6-7)

6-17 Liquidity Premium Theory The Wall Street Journal reports that the rate on three-year Treasurysecurities is 5.25 percent and the rate on four-year Treasury securities is 5.50 percent. The one-yearinterest rate expected in three years is, E(4r1), is 6.10 percent. According to the liquidity premiumhypotheses, what is the liquidity premium on the four-year Treasury security, L4? (LG6-7)

6-18 Liquidity Premium Theory Suppose we observe the following rates: 1R1 = 0.75%, 1R2 = 1.20%, andE(2r1) = 0.907%. If the liquidity premium theory of the term structure of interest rates holds, what isthe liquidity premium for year 2, L2? (LG6-7)

6-19 Forecasting Interest Rates You note the following yield curve in The Wall Street Journal. Accordingto the unbiased expectations theory, what is the one-year forward rate for the period beginning oneyear from today, 2f1? (LG6-8)

Maturity Yield

One day 2.00%

One year 5.50

Two years 6.50

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Three years 9.00

6-20 Forecasting Interest Rates On March 11, 20XX, the existing or current (spot) one-, two-,three-, and four-year zero coupon Treasury security rates were as follows:

Using the unbiased expectations theory, calculate the one-year forward rates on zero coupon Treasurybonds for years 2, 3, and 4 as of March 11, 20XX. (LG6-8)

ADVANCED PROBLEMS

6-21 Determinants of Interest Rates for Individual Securities The Wall Street Journal reports that thecurrent rate on 10-year Treasury bonds is 7.25 percent, on 20-year Treasury bonds is 7.85 percent, andon a 20-year corporate bond issued by MHM Corp. is 8.75 percent. Assume that the maturity riskpremium is zero. If the default risk premium and liquidity risk premium on a 10-year corporate bondissued by MHM Corp. are the same as those on the 20-year corporate bond, calculate the current rateon MHM Corp.’s 10-year corporate bond. (LG6-6)

6-22 Determinants of Interest Rates for Individual Securities The Wall Street Journal reports that thecurrent rate on 8-year Treasury bonds is 5.85 percent, the rate on 15-year Treasury bonds is 6.25percent, and the rate on a 15-year corporate bond issued by MHM Corp. is 7.35 percent. Assume thatthe maturity risk premium is zero. If the default risk premium and liquidity risk premium on an 8-yearcorporate bond issued by MHM Corp. are the same as those on the 15-year corporate bond, calculatethe current rate on MHM Corp.’s 8-year corporate bond. (LG6-6)

6-23 Determinants of Interest Rates for Individual Securities The Wall Street Journal reports that thecurrent rate on 5-year Treasury bonds is 1.85 percent and on 10-year Treasury bonds is 3.35 percent.Assume that the maturity risk premium is zero. Calculate the expected rate on a 5-year Treasury bondpurchased five years from today, E(5r5). (LG6-6)

6-24 Determinants of Interest Rates for Individual Securities The Wall Street Journal reports that thecurrent rate on 10-year Treasury bonds is 2.25 percent and the rate on 20-year Treasury bonds is 4.50percent. Assume that the maturity risk premium is zero. Calculate the expected rate on a 10-yearTreasury bond purchased 10 years from today, E(10r10). (LG6-6)

6-25 Unbiased Expectations Theory Suppose we observe the three-year Treasury security rate (1R3) to be8 percent, the expected one-year rate next year— E(2r1)—to be 4 percent, and the expected one-yearrate the following year—E(3r1)—to be 6 percent. If the unbiased expectations theory of the termstructure of interest rates holds, what is the one-year Treasury security rate, 1R1? (LG6-7)

6-26 Unbiased Expectations Theory The Wall Street Journal reports that the rate on three-year Treasurysecurities is 1.20 percent and the rate on five-year Treasury securities is 2.15 percent. According to theunbiased expectations theory, what does the market expect the two-year Treasury rate to be three yearsfrom today, E(3r2)? (LG6-7)

6-27 Forecasting Interest Rates Assume the current interest rate on a one-year Treasury bond (1R1) is 4.50percent, the current rate on a two-year Treasury bond (1R2) is 5.25 percent, and the current rate on athree-year Treasury bond (1R3) is 6.50 percent. If the unbiased expectations theory of the termstructure of interest rates is correct, what is the one-year forward rate expected on Treasury billsduring year 3, 3f1? (LG6-8)

6-28 Forecasting Interest Rates A recent edition of The Wall Street Journal reported interest rates of 1.25percent, 1.60 percent, 1.98 percent, and 2.25 percent for three-, four-, five-, and six-year Treasurysecurity yields, respectively. According to the unbiased expectation theory of the term structure ofinterest rates, what are the expected one-year forward rates for years 4, 5, and 6? (LG6-8)

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chapter sixappendix 6A:The Financial Crisis: The Failure of Financial Institution SpecialnessIn the late 2000s, the United States—and indeed the world—experienced the worst financial crisis since the GreatDepression of the 1930s. As of mid-March 2009, in less than a year and a half, the Dow Jones Industrial Average(DJIA) had fallen in value 53.8 percent, compared to 49 percent in the market crash of 1937 and 1938. Homeforeclosures reached record highs in late 2008 and continued to rise through 2009. One in 45 households (2.8million properties) were in default on their home mortgage in 2009. The investment banking industry saw the failure(or acquisition) of all but two of its major firms (Goldman Sachs and Morgan Stanley), and these two firmsconverted to commercial bank holding companies. AIG, one of the largest insurance companies in the U.S., survivedonly because of a federal government bailout. Commercial banking giant Citigroup required a massive governmentguarantee against losses and an injection of cash to prevent failure. The three major U.S. automakers facedbankruptcy without federal money. Even after having received government loans, Chrysler declared Chapter 11bankruptcy in May 2009, while General Motors followed suit a month later. As of October 2009, the U.S.unemployment rate was over 10 percent, the highest level since 1983.

This financial crisis had huge effects on financial institutions and the way they do business today and will dobusiness in the future. In the chapter, we explored the root causes of and changes brought about by the financialcrisis as they apply to specific areas of risk measurement and management within FIs. In this appendix, we reviewthe major events leading up to and throughout the financial crisis so we can see the full economywide impact andimplications for the future.

The Beginning of the CollapseSigns of significant problems in the U.S. economy first arose in late 2006 and the first half of 2007 when homeprices plummeted and mortgage delinquencies began to mount. Mortgage defaults by subprime borrowers surged inthe last quarter of 2006 through 2008 as homeowners, who a half decade earlier had stretched themselves financiallyto buy a home or refinance a mortgage, fell behind on their loan payments. Foreclosure filings jumped 93 percent inJuly 2007 over July 2006. Between August 2007 and October 2008, an additional 936,439 homes were lost toforeclosure. As the mortgage defaults continued, financial institutions that held these mortgages (and mortgage-backed securities) started announcing huge losses on them. These securitized loans, particularly securitizedsubprime mortgage loans, led to the huge financial losses that quite possibly were the root cause of the weakness ofthe U.S. economy during this time. Losses from the falling value of subprime mortgages and securities backed bythese mortgages reached over $400 billion worldwide through 2007.

In 2007, Citigroup, Merrill Lynch, and Morgan Stanley lost a combined $40 billion due mainly to bad mortgageloans. Bank of America took a $3 billion dollar loss for bad loans in just the fourth quarter of 2007, while Wachovialost $1.2 billion. UBS took a loss of $10 billion, Morgan Stanley lost $9.4 billion, Merrill Lynch lost $5 billion, andLehman Brothers took a loss of $52 million, all because of losses on investments in subprime mortgages or assetsbacked by subprime mortgages. Even mortgage-backed security insurers felt the losses. In February 2008,MBIA Inc.—one of the largest insurers of mortgage-backed securities credit risk—reported a $2 billionloss for the fourth quarter of 2007, due mainly to declines in values of mortgage-backed securities it insured.

Early on, some large financial institutions were unable to survive the mortgage crisis. For example, CountrywideFinancial, the country’s largest mortgage issuer, nearly failed in the summer of 2007 due to subprime mortgagedefaults. In an effort to add liquidity, Countrywide drew down its entire $11.5 billion line of credit with otherfinancial institutions. Such an enormous and sudden drawdown sent Countrywide’s shares down from $24.46 to$21.29 (and down 50 percent for the year) and the DJIA down 2.83 percent on fears of an increasing degradation ofthe mortgage markets and potential contagion to other financial markets. Only a $2 billion equity investment byBank of America in 2007 and then an acquisition offer in 2008 kept this financial institution alive. Another earlycasualty of the financial crisis was IndyMac Bank, the ninth-largest mortgage lender in the country in 2007, whichwas seized by the FDIC in July 2008. At that time, IndyMac had more than $32 billion in assets, making it one ofthe largest savings institutions in the U.S. In late 2007 and early 2008, with mounting defaults on its mortgages,

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IndyMac was desperate for more capital, but it could not find investors willing to put new funds into what appearedto be a failing institution. So in the summer of 2008, despite FDIC insurance coverage, depositors withdrew a totalof $1.3 billion, and the FDIC stepped in to rescue the institution. At a cost to the FDIC of between $8.5 billion and$9.4 billion, IndyMac represented the largest depository institution failure in more than 20 years.

The Failure of Bear StearnsIn the early 2000s, investment banks and securities firms were major purchasers of mortgages and mortgage-backedsecurities. Packaging loans as securities allowed them to increase their business. It followed that as mortgagedefaults increased in the mid-2000s, investment banks were particularly hard hit with huge losses on the mortgagesand securities backing them. A prime example of the losses incurred is that of Bear Stearns, at one time the fifthlargest investment bank in the United States. In the summer of 2007, two Bear Stearns funds suffered heavy losseson investments in the subprime mortgage market. The two funds filed for bankruptcy in the fall of 2007. BearStearns’s market value was hurt badly from these losses. The losses became so great that in March 2008 JPMorganChase and the Federal Reserve stepped in to rescue the bank before it failed or was sold piecemeal to variousfinancial institutions. JPMorgan Chase purchased Bear Stearns for $236 million, or $2 per share. Three days prior tothe purchase, Bear Stearns’s stock was selling for $30 per share, and it was selling for $170 less than a year earlier.

Along with brokering the sale of Bear Stearns to JPMorgan Chase, in the spring of 2008, the Federal Reserve Bank(the Fed) took a series of unprecedented steps. First, for the first time the Fed lent directly to Wall Street investmentbanks through the Primary Dealer Credit Facility (PDCF). In the first three days, securities firms borrowed anaverage of $31.3 billion per day from the Fed. Second, the Fed cut interest rates sharply, including one cut on aSunday night in March 2008. The widening regulatory arm of the Fed came amid criticism aimed at the SEC (U.S.Securities and Exchange Commission, traditionally the main regulator of investment banks) and its oversight ofBears Stearns before its collapse. The Fed was now acting as a lender to various financial institutions beyonddepository institutions.

The Crisis HitsSeptember 2008 marked a crucial turning point in the financial crisis. On September 8, the U.S. government seizedFannie Mae and Freddie Mac, taking direct responsibility for these two government-sponsored agencies. Theseagencies provided funding for about three-quarters of new home mortgages written in the United States and weredeeply involved in the market that securitizes subprime mortgages. The two firms recorded approximately $9 billionin losses in the last half of 2007 related to the market for subprime mortgage-backed securities. With theseizure, the two companies were put under a conservatorship. (Today they continue to operate withmanagement under the control of their previous regulator, the Federal Housing Finance Agency.)

On Monday, September 15, Lehman Brothers (the 158-year-old investment bank) filed for bankruptcy. MerrillLynch, rather than face bankruptcy, allowed a sale to Bank of America. AIG (one of the world’s largest insurancecompanies) met with federal regulators to raise desperately needed cash. Washington Mutual (the largest savingsinstitution in the U.S.) sought a buyer to save it from failing. A sense of foreboding gripped Wall Street. As newsspread that Lehman Brothers would not survive, FIs moved to disentangle trades made with Lehman. The Dow fellmore than 500 points, the largest drop in over seven years (see Figure 6A.1).

By Wednesday, September 17, tension had mounted around the world. Stock markets saw huge swings in value asinvestors tried to sort out who might survive (markets throughout Europe were forced to suspend trading as stockprices plunged). Money market mutual fund withdrawals skyrocketed: fund investors pulled out a record $144.5billion through Wednesday (redemptions during the week of September 8 totaled just $7.1 billion) as investorsworried about the safety of even these safest investments. Money market mutual funds participated heavily in the$1.7 trillion commercial paper market, which provided a bulk of the short-term funds to corporations. As investorspulled their money from these funds, the commercial paper market shrank by $52.1 billion for the week (throughWednesday). Without these funds available to meet short-term expenses, factories faced the real possibility ofshutting down and laying off employees. Likewise, without these short-term funds, banks faced the inability to fundshort-term lending units (such as credit card units).

As these events unfolded, financial markets froze, and banks stopped lending to each other at anything butexorbitantly high rates. Banks rely on each other for cash to meet their daily needs. Interest rates on interbankborrowing are generally low because of the confidence that the financial institutions will pay each other back.However, in mid-September, the overnight London Interbank Offered Rate (a benchmark rate that reflects the rate at

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which banks lend to one another) more than doubled (see Figure 6A.2). Confidence had broken down in August of2007 and had not been completely restored. Without funding, banks became reluctant to lend at all and creditmarkets froze further.

FIGURE 6A-1 The Dow Jones Industrial Average, October 2007–January 2010

FIGURE 6A-2 Overnight London Interbank Offered Rate (Libor in USD), 2001–2010

In mid-September 2008, the overnight London Interbank Offered Rate had more than doubled.

The Rescue Plan BeginsWith a financial crisis evident, on Thursday, September 18, the Federal Reserve and central banks around the world

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invested $180 billion in global financial markets in an attempt to unfreeze credit markets. U.S. Treasury SecretaryHenry Paulson met with Congressional leaders to devise a plan to get bad mortgage loans and mortgage-backedsecurities off balance sheets of financial institutions. After two weeks of debate (and one failed vote for passage), a$700 billion rescue plan was passed and signed into law by President George W. Bush on October 3, 2008. The billestablished the Troubled Asset Relief Program (or TARP), which gave the U.S. Treasury funds to buy “toxic”mortgages and other securities from financial institutions. The federal government was mandated to take an equitystake and executive compensation was limited in the companies that took part in the TARP program. The bill alsocalled for the administration to develop a plan to ease the wave of home foreclosures by modifying loans acquiredby the government and increased FDIC deposit insurance to $250,000 from $100,000.

The Crisis Spreads WorldwideAs the U.S. government debated the rescue plan, the financial crisis continued to spread worldwide. During the lastweek of September and first week of October 2008, the German government guaranteed all consumer bank depositsand arranged a bailout of Hypo Real Estate, the country’s second largest commercial property lender. The UnitedKingdom nationalized mortgage lender Bradford & Bingley (the country’s eighth largest mortgage lender) andraised deposit guarantees from $62,220 to $88,890 per account. Ireland guaranteed deposits and debt of its six majorfinancial institutions. Iceland rescued its third largest bank with an $860 million purchase of 75 percent of the bank’sstock, a few days later it seized the country’s entire banking system. Central governments of The Netherlands,Belgium, and Luxembourg together agreed to inject $16.37 billion into Fortis NY (Europe’s first ever cross-borderfinancial services company) to keep it afloat. However, five days later this deal fell apart and the bank was split up.The central bank in India stepped in to stop a run on the country’s second largest bank, ICICI Bank, by promising topump in cash. Central banks in Asia injected cash into their banking systems as banks’ reluctance to lend to eachother, and a run on Bank of East Asia Ltd. led the Hong Kong Monetary Authority to inject liquidity into its bankingsystem. South Korean authorities offered loans and debt guarantees to help small and midsize businesses with short-term funding. All of these actions were a result of the spread of the U.S. financial market crisis to world financialmarkets.

After the Rescue PlanIn the two months after the TARP rescue plan was enacted in the United States, the financial crisis deepened as theworld feared that the initial attempts to rescue the world’s financial system would not be sufficient. Worldwide,stock market values plunged. By mid-October the Dow had dropped 24.7 percent in less than a month, the ShanghaiComposite had dropped 30.4 percent, and the various markets in Europe fell between 20 and 30 percent. By mid-November, the Dow fell to a 5½-year low and the S&P 500 index erased its gains from the previous 10 years. Third-quarter GDP in the United States declined to −2.7 percent. Indeed, some measures of economic activity found theUnited States entered a recession as early as December 2007. The United Kingdom and Germany also saw growthdecline by 0.5 percent in the third quarter of 2008. Countries across the world saw companies scrambling for creditand cutting their growth plans. Additionally, consumers worldwide reduced their spending as the value of theirinvestments shrank. Even China’s booming economy slowed faster than had been predicted, from 10.1 percent in thesecond quarter of 2008 to 9.0 percent in the third quarter. This was the first time since 2002 that China’s growth wasbelow 10 percent and dimmed hopes that Chinese demand could help keep world economies going. In late October,the global crisis hit the Persian Gulf as Kuwait’s central bank intervened to rescue Gulf Bank, the first bank rescuein the oil-rich gulf. Until this time, the area had been relatively immune to the world financial crisis. However,plummeting oil prices (which had dropped over 50 percent between July and October) left the area’s economiessuddenly vulnerable.

Between January and November 2008, 22 U.S. banks failed, up from 3 in all of 2007. The FDIC reported that itadded 54 banks to its list of troubled institutions in the third quarter, a 46 percent increase over the second quarter.Additions to the list reflected the escalating problems in the banking industry. However, it should be noted that the171 banks on the FDIC’s problem list represented only about 2 percent of the nearly 8,500 FDIC-insuredinstitutions. Still, the increase from 117 troubled banks in the second quarter was the largest seen since late 1995. Inwhat came to be known as the “too big to fail” category, commercial banking giant Citigroup required a massivegovernment guarantee against losses (up to $306 billion) and a $20 billion injection of cash to prevent failure. Bythe middle of November it became apparent that the rescue plan enacted in early October actually would not be

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sufficient. Even more distressed financial and nonfinancial companies and consumers called for assistance.

Among the largest companies in need of a bailout were the Big Three automobile manufacturers (General Motors,Ford, and Chrysler). The leaders of these companies painted a grim picture of their financial position during twodays of Congressional hearings, warning that the collapse of the auto industry could lead to the loss of 3 million jobsnationwide. Both General Motors and Chrysler said they could collapse in weeks. However, automakers ran intoresistance from House lawmakers, who chastised the executives for fighting tougher fuel-efficiency standards in thepast and questioned their use of private jets while at the same time seeking government handouts. Fearing that theBig Three would take any bailout money and continue the same “stupid” decisions they had been making for25 years, the U.S. Senate canceled plans for a vote on a bill to take $25 billion in new auto industry loans out of the$700 billion TARP rescue fund. Lawmakers gave the three automakers until mid-December to come back withsubstantial business plans outlining what they would do with any federal funds that might be lent and how theywould restructure and improve the efficiency in their respective companies. After more days of testimony in mid-December, the U.S. Senate again failed to vote on a bailout for the automakers. General Motors and Chrysler statedthat they did not have sufficient funds to continue operations through the end of December. Then on December 19,2008, President Bush announced that $13.4 billion in federal loans would be made immediately available to GeneralMotors and Chrysler.

By the end of December, nearly $7 trillion of loans or commitments had been made (Table 6A.1 outlines the majorcommitments, loans, and investments made by the U.S. government through 2009). The U.S. Treasury had used thefirst $362 billion of TARP money: $250 billion of the $700 billion bailout money to inject capital intobanks ($125 billion of which went to the nine largest banks), another $40 billion to further stabilize insurerAIG, $25 billion to stabilize Citigroup, $20 billion to Bank of America, $20 billion used by the Fed to stabilize otherlending institutions, and $24.9 billion lent to the auto industry. Note that the Treasury had dropped the original plansto use TARP bailout money to buy troubled mortgage assets from financial institutions, stating that it was no longerthe most effective way to restart credit markets. Rather, plans were to take equity stakes in financial institutions.

▼ TABLE 6A.1 Federal Government Rescue Efforts through December 2009

Program Committed Invested Description

TARP $700.0billion$356.2billion

Financial rescue plan aimed atrestoring liquidity to financial markets

AIG 70.0b 69.8b

Auto industry financing 80.1b 77.6b

Capital Purchase Program 218.0b 204.7b

Public–Private Investment Program 100.0b 26.7b

Targeted investments to Citigroup andBank of America

52.5b 45.0b

Amount repaid $118.5billion

Federal Reserve Rescue Efforts $6.4trillion$1.5trillion

Financial rescue plan aimed atrestoring liquidity to financial markets

Asset-backed commercial papermoney market mutual fund liquidityfacility

Unlimited $0.0

Bear Stearns bailout 29.0b 26.3b

Commercial paper funding facility 1.8t 14.3b

Foreign exchange dollar swaps Unlimited 29.1b

GSE (Fannie Mae and Freddie Mac)debt purchases

200.0b 149.7b

GSE mortgage-backed securitiespurchases 1.2t 775.6b

Term asset-backed securities loan

facility 1.0t 43.8b

U.S government bond purchase 300.0b 295.3b

Federal Stimulus Programs $1.2trillion$577.8billion

Programs designed to save or createjobs

Economic Stimulus Act 168.0b 168.0b

Student loan guarantees 195.0b 32.6b

American Recovery and ReinvestmentAct

787.2b 358.2b

American International Group $182.0billion$127.4billion

Bailout to help AIG throughrestructuring, get rid of toxic assets

Asset purchases 52.0b 38.6b

Bridge loan 25.0b 44.0b

TARP investment 70.0b 44.8b

FDIC Bank Takeovers $45.4billion$45.4billion

Cost to FDIC to fund deposit losses onbank failures

2008 failures 17.6b 17.6b

2009 failures 27.8b 27.8b

Other Financial Initiatives $1.7trillion$366.4billion

Other programs designed to rescuethe financial sector

NCUA bailout of U.S.

Central Credit Union 57.0b 57.0b

Temporary Liquidity GuaranteeProgram

1.5t 308.4b

Other Housing Initiatives $745.0billion$130.6billion

Other programs intended to rescue thehousing market and prevent homeforeclosures

Fannie Mae and Freddie Mac bailout 400.0b 110.6b

FHA housing rescue 320.b 20.0b

Overall Total $11.0trillion$3.0trillion

Some Bright SpotsWhile the economy remained in crisis, some positive developments occurred between September and December2008. Oil, which rose to over $142 per barrel in July, had dropped to below $40 in late 2008. As a result, gas prices,which rose to over $4.00 per gallon in the summer of 2008, had fallen to a national average of $1.65 in December.Led by a federal government push, many banks moved to restructure delinquent mortgage loans rather thanforeclose. Fannie Mae and Freddie Mac suspended foreclosures on 16,000 homes over the 2008 holiday periodwhile they evaluated whether the borrowers would qualify for the new loan modification programs. Fannie andFreddie’s modification plan allowed mortgage restructuring, rather than foreclosure, for homeowners whosemortgages were held by one of the two companies, were at least three months behind on their payments, and whosemortgage payments were no more than 38 percent of the homeowner’s pretax monthly income. The FederalReserve’s attempt to stabilize the housing market resulted in a drop in long-term mortgage rates (30-year fixed-ratemortgage rates dipped to below 5.0 percent in late November). In a historic move, on December 17, 2008, the Fedunexpectedly announced that it would drop its target Fed funds rate to a range between 0 and 0.25 percent and lowerits discount window rate to 0.5 percent, the lowest level since the 1940s (see Figure 6A.3). Along with thisannouncement, the Fed announced that it would continue to use all its available tools to promote economic growthand preserve price stability. This was followed by interest rate cuts in many other countries including Japan and theUnited Kingdom, as well as Hong Kong and the European Central Bank.

The Crisis Continues in 2009Despite the many efforts of regulators to stem the tide of the growing recession, the U.S. and world economies

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deteriorated further at the start of 2009. The DJIA and the S&P 500 Index had their worst January ever, falling 8.84percent and 8.57 percent, respectively. Unemployment in January hit 7.6 percent, the highest level since September1992. Also in January, employers cut 589,000 jobs, the highest monthly job losses in over 34 years. Since December2007 (as the recession began) the U.S. economy lost 3.6 million jobs, half of which were lost in the periodNovember 2008 through January 2009. Gross domestic product growth was announced, showing a drop of 5.4percent in the fourth quarter of 2008. New vehicle sales in the United States fell 37 percent in January, the industry’sworst month since 1982 and the worst January since 1963. For the first time ever, more vehicles were sold in Chinathan in the United States. Worldwide, central governments tried to grapple with the building recession. The UnitedKingdom, Belgium, Canada, Italy, and Ireland were just a few of the countries to pass an economic stimulus planand/or bank bailout plan. The Bank of England lowered its target interest rate to a record low of 1 percent, hoping tohelp the British economy out of a recession. The Bank of Canada, Bank of Japan, and Swiss National Bank alsolowered their main interest rate to 1 percent or below.

The U.S. Stimulus Plan With the U.S. economy deteriorating at its swiftest rate in history, President Obamamade good on his pre-election promise to have an economic stimulus plan approved and enacted shortly after hiselection. The House passed its version of an $819 billion stimulus package on January 28, 2009. The Senate passedan $827 billion version of a stimulus plan on February 10, 2009. After more debate and compromise, both arms ofCongress agreed on and passed the economic stimulus plan, called the American Recovery and Reinvestment Act,on February 13, 2009. The plan devoted $308.3 billion to appropriations spending, including $120 billion oninfrastructure and science and more than $30 billion on energy-related infrastructure projects. Another $267 billionwent for direct spending, including increased unemployment benefits and food stamps. Finally, $212 billion was setaside for tax breaks for individuals and businesses (Table 6A.2 lists some of the major items in the stimulus plan).

FIGURE 6A-3 Federal Funds Rate and Discount Window Rate, January 1971–January 2010

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On December 17, 2008, the Fed dropped the target Fed funds rate to a range between 0 and 0.25 percent and the discount window rate to0.5 percent.

Stabilizing the Financial System In addition to the overall economic stimulus plan, the Obama administration,including Treasury Secretary Geithner, announced a separate plan that focused on the stabilization of the financialsystem. Early 2009 saw a plunge in the DJIA (falling to a low of 6,547.05 on March 9, 2009) and, particularly, themarket values of financial institutions. Banks, including Citigroup, Bank of America, and JPMorgan Chase, tradedat less than their book values as investors had little confidence in the value of their assets. Through February 13,2009, 13 U.S. banks had already failed in 2009, while 25 had failed in all of 2008 (the highest annual total since1993). The plan, announced on February 10, 2009, involved a number of initiatives, including

Injecting capital into banks.

Offering federal insurance to banks against losses on bad assets.Buying distressed mortgages from banks.

Helping homeowners avoid foreclosure.Giving the FDIC power to help troubled financial firms other than depository institutions.

▼ TABLE 6A.2 The $787 Billion Stimulus Program as Passed by the U.S. Congress, February 13, 2009

$116.1b Tax cuts and credits to low- and middle-income workers69.8b Middle income taxpayers to get an exemption from the alternative minimum tax87.0b Medicaid provisions27.0b Jobless benefits extension to a total of 20 weeks on top of regular unemployment compensation17.2b Increase in student aid

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40.6b Aid to states30.0b Modernization of electric grid and energy efficiency19.0b Payments to hospitals and physicians who computerize medical record systems29.0b Road and bridge infrastructure construction and modernization18.0b Grants and loans for water infrastructure, flood prevention, and environmental cleanup17.2b Increases in student aid

The plan also proposed expanding the Fed’s Term Asset-Backed Securities Loan Facility (TALF). The TALFcombined capital provided by the TARP with funding from the Federal Reserve in order to promote lending byincreasing investor demand for securitized loans. The TALF significantly expanded the availability and reduced thecost of term financing for investors in derivative securities. The goal of TALF was to stimulate demand for thesesecurities and thereby allow originators of securitized loans to lower the cost and increase the availability of credit toconsumers and businesses. The Fed’s TALF program initially provided financing for investors to purchase securitiesbacked by consumer loans. The Treasury wanted to expand this beyond consumer loans. For homeowners, thebanking plan called for the creation of national standards for loan modifications and for the use of tax dollars to givemortgage companies an incentive to modify mortgage loans. Further, along with the expanded TALF program, theTreasury, working with the Federal Reserve, FDIC, and private investors, created the Public–Private InvestmentFund (PPIF) to acquire real estate–related off-balance-sheet assets. By selling to PPIF, financial institutions couldreduce balance sheet risk, support new lending, and help improve overall market functioning. Finally, in lateFebruary 2009, the Obama administration announced that it would conduct a “stress test” of the 19 largest U.S.banks. This instrument would measure the ability of these banks to withstand a protracted economic slump, with anunemployment rate above 10 percent and home prices dropping another 25 percent. Results of the stress test showedthat 10 of the 19 banks needed to raise a total of $74.6 billion in capital. Within a month of the May 7, 2009, releaseof the results, the banks had raised $149.45 billion of capital.

The Economy Begins to RecoverThroughout the spring of 2009, the federal government continued to take actions to combat the stagnant economy.These included actions such as passage (in May 2009) of the Job Creation through Entrepreneurship Act to helpsmall businesses access capital and credit markets. The Cash for Clunkers Program (in June 2009) to stimulateautomobile sales was wildly popular.

By the summer and fall of 2009 the economy slowly began to recover. Pending home sales and residentialconstruction both posted significant increases in September. September marked the eighth consecutive monthlyincrease in pending home sales, which was the longest such streak since 1991, when this data began to be tracked.Home sales rose at an annual rate of 6.1 percent in September (21.2 percent ahead of their September 2008 level).Meanwhile, the Commerce Department reported that residential construction spending increased at a 3.9 percentannual rate in September. It was the third consecutive monthly increase in residential construction. TheNational Association of Realtors announced that the number of signed contracts increased for the ninthconsecutive month to the biggest year-over-year gain in the history of the index—31.8 percent higher thanSeptember 2008. These signs of life in the construction industry were an indication that the first-time HomebuyerTax Credit, put in place as part of the American Recovery and Reinvestment Act, was working. Indeed, inNovember 2009, President Obama signed into law an expanded Homebuyer Tax Credit that extended the tax creditof up to $8,000 for qualified first-time home buyers and $6,500 for repeat home buyers purchasing a principalresidence.

Third-quarter 2009 GDP increased by 2.2 percent and fourth-quarter GDP rose 5.7 percent. This increase was thefirst since the second quarter of 2008. The increases were the result of consumer spending, which increasedsignificantly. Spending on new cars and trucks was a big contributor (adding 1.45 percent to the third-quarterchange), reflecting the federal Cash for Clunkers program in effect in July and August. The increase in GDP in thefourth quarter primarily reflected increases in private inventory investment, exports, and personal consumptionexpenditures. Automobile output continued to do well, adding 0.61 percent to the fourth-quarter change in GDP.With the positive economic news, on October 14, 2009, the DJIA reached 10,000 for the first time in a year. Still,unemployment lagged, exceeding 10 percent in October 2009. But this rate was short-lived, as the unemploymentrate dropped below 10 percent in November 2009 and job losses began to decline sharply, with only 11,000American jobs lost in November 2009, compared to 741,000 jobs lost in December 2008.

There were 140 failures of banks in 2009, with assets totaling $170.9 billion. The five largest bank failures wereBankUnited ($12.5 billion in assets), Colonial Bank ($25 billion in assets), Guaranty Bank ($13 billion in assets),

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United Commercial Bank ($11.2 billion in assets), and AmTrust Bank ($12 billion in assets). The cost to the FDICfor resolving these failures was $27.5 billion. Compare this with statistics for 2008, when there were 25 bankfailures with assets totaling $373.6 billion.In December 2009, the Obama Administration announced that the long-term cost of the Troubled Asset ReliefProgram would be at least $200 billion less than previously projected, which would help bring down the projectedfederal budget deficit. Throughout 2010 and into 2012, the economy was still fragile and had certainly not recoveredfrom the extreme financial crisis, but it was stabilizing.

Finally, in July 2010, the U.S. Congress passed, and President Obama signed, the 2010 Wall Street Reform andConsumer Protection Act which sought to prevent a repeat of the market meltdown. Touted as the most extensiveproposal for the overhaul of financial rules since the Great Depression, this bill proposed a sweeping overhaul of thenation’s financial system and the rules that govern it. The bill called for the Federal Reserve to receive newoversight powers and to impose conditions designed to discourage any type of financial institution from getting toobig. The proposals put the Federal Reserve in charge of monitoring the country’s biggest financial firms—thoseconsidered critical to the health of the system as a whole. Those firms would also face new, stiffer requirements onhow much capital and liquidity they keep in reserve. The proposed overhaul also provided unprecedented powers tothe Fed to step into any financial institution—such as insurance giant AIG (whose main regulators include the NewYork State Department of Insurance and the Office of Thrift Supervision)—that is facing imminent collapse, inorder to force an orderly bankruptcy that would protect the wider economy.

More specifically, the bill set forth reforms to meet five key objectives:

1. Promote robust supervision and regulation of financial firms by establishing (i) a new Financial ServicesOversight Council of financial regulators (chaired by Treasury and including the heads of the principal federalfinancial regulators as members) to identify emerging systemic risks and improve interagency cooperation; (ii) anew authority for the Federal Reserve to supervise all firms that could pose a threat to financial stability,even those that do not own banks; (iii) stronger capital and other prudential standards for all financialfirms, and even higher standards for large, interconnected firms; (iv) a new National Bank Supervisor to superviseall federally chartered banks; (v) elimination of the federal thrift charter for thrifts not dedicated to mortgage lendingand other loopholes that allowed some depository institutions to avoid bank holding company regulation by theFederal Reserve; and (vi) the registration of advisers of hedge funds and other private pools of capital with the SEC.

2. Establish comprehensive supervision of financial markets by establishing (i) the regulation of securitizationmarkets, including new requirements for market transparency, stronger regulation of credit rating agencies, and arequirement that issuers and originators retain a financial interest in securitized loans; (ii) comprehensive regulationof all over-the-counter derivatives; and (iii) new authority for the Federal Reserve to oversee payment, clearing, andsettlement systems.

3. Protect consumers and investors from financial abuse by establishing (i) a new Consumer Financial ProtectionAgency to protect consumers across the financial sector from unfair, deceptive, and abusive practices; (ii) strongerregulations to improve the transparency, fairness, and appropriateness of consumer and investor products andservices; and (iii) a level playing field and higher standards for providers of consumer financial products andservices, whether or not they are part of a bank.

4. Provide the government with the tools it needs to manage financial crises by establishing (i) a new regime toresolve nonbank financial institutions whose failure could have serious systemic effects and (ii) revisions to theFederal Reserve’s emergency lending authority to improve accountability.

5. Raise international regulatory standards and improve international cooperation by establishing internationalreforms to support our efforts in the United States, including strengthening the capital framework, improvingoversight of global financial markets, coordinating supervision of internationally active firms, and enhancing crisismanagement tools.

NotesCHAPTER 61 Often the Fisher effect formula is written as (1 + i ) = (1 + IP ) × (1 + RFR ), which, when solved for i, becomes: i = Expected IP + RFR +

(Expected IP × RFR), where Expected IP × RFR is the inflation premium for the loss of purchasing power on the promised nominalinterest rate payments due to inflation. For small values of Expected IP and RFR this term is negligible. The approximation formula

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used here assumes these values are small.

2 This section, which contains more technical details, may be included or dropped from the chapter reading, depending on the rigor of thecourse.

3 That is, E (4 r 1) > E (3 r 1) > E (2 r 1) > 1 R 1 .

4 In general, the price and yield on a bond are inversely related. Thus, as the price of a bond falls (become cheaper), the demand for thebond will rise. This is the same as saying that as the yield on a bond rises, it becomes cheaper and the demand for it increases. SeeChapter 7.

5 This section, which contains more technical details, may be included in or dropped from the chapter reading depending on the rigor ofthe course.

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chapter sevenValuingBonds

©DNY59/Getty Images

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ow important are bonds and the bond market to a capitalist economy? Those unfamiliar with thefinancial markets may have the impression that the stock market dominates capital markets in theUnited States and in other countries. Stock market performance appears constantly on 24-hour TV

news channels and on the evening news. By contrast, we seldom hear any mention of the bond market.While bonds may not generate the same excitement that stocks do, they are an even more importantcapital source for companies, governments, and other organizations. The bond market is actually largerthan the stock market. At the end of 2015, the U.S. bond market represented roughly $39.6 trillion inoutstanding debt obligations. At the same time, the market value of all common stock issues was worthjust over half of the value of the bond market, at roughly $21.0 trillion.

Bonds also trade in great volume and frequency. During 2015, the total average daily trading in alltypes of U.S. bonds reached over $730 billion. Investors are often attracted to the stock market because itoffers the potential for high investor returns—but great risks come with that high potential return. Whilesome bonds offer safer and more stable returns than stocks, other bonds also offer high potential rewardsand, consequently, higher risk.

In this chapter, we will explore bond characteristics and their price dynamics. You will see that bondpricing uses many time value of money principles that we’ve used in the preceding chapters. ■

LEARNING GOALS

LG7-1 Describe bond characteristics.LG7-2 Identify various bond issuers and their motivation for issuing debt.LG7-3 Read and interpret bond quotes.LG7-4 Compute bond prices using present value concepts.LG7-5 Explain the relationship between bond prices and interest rates.LG7-6 Compute bond yields.LG7-7 Find bond ratings and assess credit risk’s effects on bond yields.LG7-8 Assess bond market performance.

viewpointsbusiness APPLICATIONYou are the chief financial officer (CFO) for Beach Sand Resorts. The firm needs $150 million of new capital to renovate a hotelproperty. As you discuss the firm’s plans with a credit rating agency, you learn that if 15-year bonds are used to raise this capital, thebonds will be rated BB and will have to offer a 7 percent return. How many bonds will you have to issue to raise the necessarycapital? What semiannual interest payments will Beach Sand have to make? (See the solution at the end of the book.)

7.1 • BOND MARKET OVERVIEW LG7-1Bond CharacteristicsBonds are debt obligation securities that corporations, the federal government or its agencies, or states and localgovernments issue to fund various projects or operations. All of these organizations periodically need to raise capitalfor various reasons, which was formally discussed in Chapter 6. Bonds are also known as fixed-income securitiesbecause bondholders (investors) know both how much they will receive in interest payments and when theirprincipal will be returned. From the bond issuer’s point of view, the bond is a loan that requires regular interestpayments and an eventual repayment of the borrowed principal. Investors—often pension funds, banks, and mutualfunds—buy bonds to earn investment returns. Most bonds follow a relatively standard structure. A legal contractcalled the indenture agreement outlines the precise terms between the issuer and the bondholders. Any bond’s main

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characteristics include:

bond Publicly traded form of debt.

fixed-income securities Any securities that make fixed payments.

principal Face amount, or par value, of debt.

indenture agreement Legal contract describing the bond characteristics and the bondholder and issuer rights.

The date the principal will be repaid (the maturity date).

maturity date The calendar date on which the bond principal comes due.

The par value, or face value, of each bond, which is the principal loan amount that the borrower must repay.

par value Amount of debt borrowed to be repaid; face value.

The coupon (interest) rate.

A description of any property to be pledged as collateral.Steps that the bondholder can take in the event that the issuer fails to pay the interest or principal.

Table 7.1 describes par value and other bond characteristics. Most bonds have a par value of $1,000. This is theamount of principal the issuer has promised to repay. Bonds have fixed lives. The bond’s life ends when the issuerrepays the par value to the buyer on the bond’s maturity date. Although a bond will mature on a specific calendardate, the bond is usually referenced by its time to maturity, that is, 2 years, 5 years, 20 years, and so on. In fact, themarket groups bonds together by their time to maturity and classifies them as short-term bonds, medium-term bonds,or long-term bonds, regardless of issuer. Long-term bonds carry 20 or 30 years to maturity. Of course, over time, the30-year bond becomes a 20-year bond, 10-year bond, and eventually matures. But other time periods to maturity doexist. For example, in 2011, the railroad company Norfolk Southern Corp. issued $400 million of bonds with 100years to maturity. The bonds have a coupon (interest) rate of 6 percent and mature in 2111.

time to maturity The length of time (in years) until the bond matures and the issuer repays the par value.

personal APPLICATIONYou would like to invest in bonds. Your broker suggests two different bonds. The first, issued by Trust Media, will mature in 2023. Itsprice is quoted at 96.21 and it pays a 5.7 percent coupon. The second bond suggested, issued by Abalon, Inc., also matures in 2023.This bond’s price is 101.94 and pays a 5.375 percent coupon. To help you decide between the bonds, you want to know how muchmoney it will cost to buy 10 bonds, what interest payments you will receive, and what return the bonds offer if purchased today. Also,you want to understand the differences between what the two bonds imply about their risk. (See the solution at the end of thebook.)

How do you even purchase a bond in the first place?

When interest rates economywide fall several percentage points (which often takes several years), homeownerseverywhere seek to refinance their home mortgages. They want to make lower interest payments (and sometimeswant to pay down their mortgage principal) every month. Corporations that have outstanding bond debt will alsowant to refinance those bonds. Sometimes the indenture contract (the legal contract between a bond issuer andbondholders) allows companies to do so; sometimes the indenture prohibits refinancing. Bonds that can berefinanced have a call feature, which means that the issuer can “call” the bonds back and repay the principal beforethe maturity date. To compensate the bondholders for getting the bond called, the issuer pays the principal and a callpremium. The most common call premium is one year’s worth of interest payments. In some indentures, the callpremium declines over time.

call An issuer redeeming the bond before the scheduled maturity date.

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call premium The amount in addition to the par value paid by the issuer when calling a bond.

The bond’s coupon rate determines the dollar amount of interest paid to bondholders. The coupon rate appears on thebond and is listed as a percentage of the par value. So a 5 percent coupon rate means that the issuer will pay 5percent of $1,000, or $50, in interest every year, usually divided into two equal semiannual payments. So a5 percent coupon bond will pay $25 every six months. Companies set the coupon rate as the prevailingmarket interest rate at the time of bond issue. The name coupon is a holdover from the past, when bonds wereactually issued with a coupon book. Every six months a bondholder would tear out a coupon and mail it to theissuer, who would then make the interest payment. These are sometimes referred to as bearer bonds (often a featureof spy or mystery movies), because whoever held the coupon book could receive the payments. Nowadays, issuersregister bond owners and automatically wire interest payments to the owner’s bank or brokerage account.Nevertheless, the term coupon persists today.

coupon rate The annual amount of interest paid expressed as a percentage of the bond’s par value.

▼ TABLE 7.1 Typical Bond Features

Characteristic Description Common Values

Par value The amount of the loan to be repaid. This is often referred to as theprincipal of the bond.

$ 1,000

Time to maturity The number of years left until the maturity date. 1 year to 30 years

Call

The opportunity for the issuer to repay the principal before the maturitydate, usually because interest rates have fallen or issuer’scircumstances have changed. When calling a bond, the issuercommonly pays the principal and one year of interest payments.

Many bonds are not callable. Forthose that are, a common feature isthat the bond can be called any timeafter 10 years of issuance.

Coupon rateThe interest rate used to compute the bond’s interest payment eachyear. Listed as a percentage of par value, the actual payments usuallyare paid twice per year.

2 to 10 percent

Bond price The bond’s market price reported as a percentage of par value. 80 to 120 percent of par value

the Math Coach on…Percent-to-Decimal Conversions

“When discussing interest rates or using them in calculator or spreadsheet time value of money functions,the value should be in percent (%) form, like 2.5%, 7%, and 11%. When using interest rates in formulas, thevalue needs to be in decimal form, like 0.025, 0.07, and 0.11.

To convert between the two forms of representing an interest rate, use

„

At original issue, bonds typically sell at par value, unless interest rates are very volatile. Bondholders recoup the parvalue on the bond’s maturity date. However, at all times in between these two dates, bonds might trade amonginvestors in the secondary bond market. The bond’s price as it trades in the secondary market will not likely be thepar value. Bonds trade for higher and lower prices than their par values. We’ll thoroughly demonstrate the reasonsfor bond pricing in a later section of this chapter. Bond prices are quoted in terms of percent of par value rather thanin dollar terms. Sources of trading information list a bond that traded at $1,150 as 115, and a bond that traded for$870 as 87.

bond price Current price that the bond sells for in the bond market.

▼

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Bond Issuers LG7-2For many years, bonds were considered stodgy, overly conservative investments. Not anymore! The fixed-incomeindustry has seen tremendous innovation in the past couple of decades. The financial industry has created and issuedmany new types of bonds and fixed-income securities, some with odd-sounding acronyms, like TIGRs, CATS,COUGRs, and PINEs, all of which are securities based on U.S. Treasuries. Even with all the innovation, thetraditional three main bond issuers remain: U.S. Treasury bonds, corporate bonds, and municipal bonds. Figure 7.1shows the amount of money that these bond issuers have raised each year.

EXAMPLE 7-1 Bond Characteristics LG7-1

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Consider a bond issued 10 years ago with an at-issue time to maturityof 30 years. The bond’s coupon rate is 8 percent and it currently tradesin the bond market for 109. Assuming a par value of $1,000, what is thebond’s current time to maturity, semiannual interest payment, and bondprice in dollars?

SOLUTION:

Time to maturity = 30 years − 10 years = 20 years

Annual payment = 0.08 × $1,000 = $80, so semiannual payment is $40

Bond price = 1.09 × 1,000 = $1,090

Similar to Problems 7-1, 7-2, 7-3, 7-4, Self-Test Problem 1

FIGURE 7-1 Amount of Capital Raised Yearly from Bonds Issued by Local and Federal Government and Corporations

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Local or municipal governments, the U.S. Treasury, and corporations have issued many new types of bonds and fixed-income securities overthe past two decades.Source: Securities Industry and Financial Markets Association.

Treasury Bonds Treasury bonds carry the “full-faith-and-credit” backing of the U.S. government and investorshave long considered them among the safest fixed-income investments in the world. The federal government sellsTreasury securities through public auctions to finance the federal deficit. When the deficit is large, more bonds cometo auction. In addition, the Federal Reserve System (the Fed) uses Treasury securities to implement monetary policy.Technically, Treasury securities issued with 1 to 10 years until maturity are Treasury notes. Securities issued with 10to 30 years until maturity are Treasury bonds. Figure 7.1 shows that the number of new Treasuries being offeredactually declined in the late 1990s as the federal budget deficit declined. However, this reversed in 2002 and thendramatically accelerated in 2009 after the global financial crisis and during the years of the Fed’s quantitative easingprograms.

Corporate Bonds Corporations raise capital to finance investments in inventory, plant and equipment, researchand development, and general business expansion. As managers decide how to raise capital, corporations can issuedebt, equity (stocks), or a mixture of both. The driving force behind a corporation’s financing strategy is the desireto minimize its total capital costs. Through much of the 1990s, corporations tended to issue equity (stocks) to raisecapital. Beginning in 1998 and through 2015, corporations switched to raising capital by issuing bonds to takeadvantage of low interest rates and issued $17.1 trillion in new bonds. You can see this rise in capital reflected inFigure 7.1.

Municipal Bonds State and local governments borrow money to build, repair, or improve streets, highways,hospitals, schools, sewer systems, and so on. The interest and principal on these municipal bonds are repaid in twoways. Projects that benefit the entire community, such as courthouses, schools, and municipal office buildings, aretypically funded by general obligation bonds and repaid using tax revenues. Projects that benefit only certain groupsof people, such as toll roads and airports, are typically funded by revenue bonds and repaid from user fees. Interestpayments paid to municipal bondholders are not taxed at the federal level, or by the state for which the bond isissued.

Other Bonds and Bond-Based SecuritiesTreasury Inflation-Protected Securities (TIPS) have proved one of the most successful recent innovations in the bondmarket. The U.S. Treasury began issuing this new type of Treasury bond, which is indexed to inflation, in 1997.TIPS have fixed coupon rates like traditional Treasuries. The new aspect is that the federal government adjusts thepar value of the TIPS bond for inflation. Specifically, it increases at the rate of inflation (measured by the consumerprice index, CPI). As the bond’s par value changes over time, interest payments also change. At maturity, investorsreceive an inflation-adjusted principal amount. If inflation has been high, investors will expect that the adjustedprincipal amount will be substantially higher than the original $1,000. Consider a 10-year TIPS issued on January15, 2009, that pays a 2⅛ percent coupon. The reference CPI for these bonds is 214.69971. Four years later (onJanuary 15, 2013) the reference CPI was 230.22100. So the par value of the TIPS in early 2013 was $1,072.29 (=$1,000 × 230.22100 ÷ 214.69971). Therefore, the 2⅛ percent coupon (paid semiannually) would be $11.39 =(0.02125 × $1,072.29 ÷ 2). A TIPS total return comes from both the interest payments and the inflation adjustmentto the par value.

Treasury Inflation-Protected Securities TIPS are U.S. government bonds where the par value changes with inflation.

U.S. government agency securities are debt securities issued to provide low-cost financing for desirable private-sector activities such as home ownership, education, and farming. Fannie Mae, Freddie Mac, Student LoanMarketing Association (Sallie Mae), Federal Farm Credit System, Federal Home Loan Banks, and the SmallBusiness Administration, among others, issue these agency bonds to support particular sectors of the economy.Agency securities do not carry the federal government’s full-faith-and-credit guarantee, but the government hasnever let one of its agencies fail. Because investors believe that the federal government will continue in thiswatchdog role, agency bonds are thought to be very safe and may provide a slightly higher return than Treasurysecurities do.

agency bonds Bonds issued by U.S. government agencies.

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EXAMPLE 7-2 TIPS Payments LG7-2

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

A TIPS bond was issued on July 15, 2006, that pays a 2½ percentcoupon. The reference CPI at issue was 201.95. The reference CPI forthe following interest payments were

January 2009 214.70

July 2009 213.52

January 2010 216.25

Given these numbers, what is the par value and interest payment of theTIPS on the three interest-payment dates? What is the total return fromJanuary 2009 to January 2010?

SOLUTION:

Compute the TIPS index ratio for each period as current CPI divided bythe at-issue CPI: The par value for January 2009 is $1,000 × 214.70 ÷201.95 = $1,063.13, so the interest payment is 0.025 × $1,063.13 ÷ 2 =$13.29. The answers for the next two dates are:

July 2009 Par value = $1,057.29 Interest payment = $13.22

January 2010 Par value = $1,070.81 Interest payment = $13.39

The capital gain between January 2009 and January 2010 is $1,070.81− $1,063.13 = $7.68. Adding the two interest payments together resultsin $26.61 (= $13.22 + $13.39). Thus, the total return is 3.23% = ($7.68+ $26.61 )/$ 1,063.13.

Similar to Problems 7-7, 7-8, 7-19, 7-20, 7-33, 7-34

finance at work //: personal financeBuy Treasuries Direct!

©Comstock Images/Jupiter Images

Treasury bonds are U.S. government-issued debt securities that investors trade on secondary markets. The government also issuesnonmarketable debt, called “savings bonds,” directly to investors. The common EE savings bonds, introduced in 1980, do not payregular interest payments. Instead, interest accrues and adds to the bond’s value. After a one-year holding period, they can beredeemed at many banks or credit unions. You can also purchase savings bonds and other Treasury securities (bills, notes, bonds, andTIPS) electronically through the U.S. Treasury’s website, treasurydirect.gov. You can set up an account in minutes and buy savingsbonds with cash from your bank account. You can also redeem your bonds and transfer the proceeds back to your bank account. Bondscan be purchased 24 hours a day, 7 days a week at no cost.

When bondholders redeem savings bonds, they receive the original value paid plus the accrued interest. Paper bonds sell at half ofthe face value; if investors hold them for the full 30 years, they receive the par value. Investors buy electronic bonds at face value andearn interest in addition to the par value. Unlike other bonds, savers need not report income from these interest payments to the IRSuntil they actually redeem the bonds. So savings bonds count as tax-deferred investments.

About one in six Americans owns savings bonds. Savings bonds are used for a variety of purposes, such as personal savingsinstruments or gifts from grandparents to grandchildren. After the September 11, 2001, terrorist attacks, many Americans wanted toshow support for the government. In December 2001, banks selling government EE savings bonds began printing “Patriot bond” onthem. So EE savings bonds are now often called Patriot Bonds.

Want to know more?Key Words to Search for Updates: TreasuryDirect (go to www.treasurydirect.gov)

U.S. government agencies invented one popular type of debt security: mortgage-backed securities (MBSs). Fannie Maeand Freddie Mac offer subsidies or mortgage guarantees for people who wouldn’t otherwise qualify for mortgages,especially first-time homeowners. Fannie Mae started out as a government-owned enterprise in 1938 and became apublicly held corporation in 1968. Freddie Mac was chartered as a publicly held corporation at its inception in 1970.Since 2008, both have been in government conservatorship and run by the Federal Housing Finance Agency. Toincrease the amount of money available (liquidity) for the home mortgage market, Fannie Mae and Freddie Macpurchase home mortgages from banks, credit unions, and other lenders. They combine the mortgages into diversifiedportfolios of such loans and then issue mortgage-backed securities, which represent a share in the mortgage debt, toinvestors. As homeowners pay off or refinance the underlying portfolio of mortgage loans, MBS investors receiveinterest and principal payments. After selling mortgages to Fannie Mae or Freddie Mac, mortgage lenders have“new” cash to provide more mortgage loans. This process worked well for decades until the late 2000s, whensubprime mortgages were given to people who couldn’t afford them. As you know, defaults on these loans were theunderpinnings of the financial crisis.

mortgage-backed securities Securities that represent a claim against the cash flows from a pool of mortgage loans.

We could apply the same concept to any type of loan; indeed, the financial markets have already invented manysuch pooled-debt securities. Typical examples include credit card debt, auto loans, home equity loans, andequipment leases. Like mortgage-backed securities, investors receive interest and principal from asset-backed

page 206securities as borrowers pay off their consumer loans. The asset-backed securities market is one of thefastest-growing areas in the financial services sector.

asset-backed securities Debt securities whose payments originate from other loans, such as credit card debt, auto loans, and homeequity loans.

On the bond’s maturity date, the bondholder receives the par value, which is typically $1,000. However, somecorporate bonds give the bondholder a choice between the par value and a specified number of shares of stock. Thistype of bond is referred to as a convertible bond because it can be converted to company stock. The number of sharesof stock for which the bond can be converted is specified when the bond is originally issued. Thus, the bondholderwill want to receive the shares when the stock price has risen since bond issuance, and they will want the $1,000when the stock has declined in value. Although the bondholder can convert to the stock shares anytime, investorstend to wait until the maturity date when the interest payments from the bond exceed the dividends that would bepaid from the stock shares.

convertible bond A debt security that can be converted to shares of stock or another type of security.

finance at work //: marketsMortgage-Backed Securities and Financial Crisis

©Andy Dean Photography/Alamy

In the old days, a bank with $100,000 to lend would fund a mortgage and charge a fee for originating the loan. The bank would thencollect interest on the loan over time. In the past few decades, the process changed to where that bank could sell that mortgage toinvestment banks and get the $100,000 back. The bank could then originate another mortgage and collect another fee. Bank revenuetransitioned from interest earnings to fee earnings. This worked pretty well for several decades because the bank made more profits andmore money was funneled into the community for home buyers. It is the securitization of debt that makes this possible. Financialinstitutions like Fannie Mae and investment banks bought up these mortgages, pooled them, and issued bonds against them (calledmortgage-backed securities, or MBSs) to sell to investors. In effect, buyers of the MBSs are the actual lenders of the mortgage andbanks simply earned fees for servicing the loans.

Note that this lending model gives banks and mortgage brokers the incentive to initiate as many mortgage loans as they can resell tomaximize fee income. Then in 2000 to early 2004, the Federal Reserve kept adjusting interest rates on federal funds downward and keptthem low. This both made home ownership more affordable, sparking a housing bubble, and drove investors to look for bonds that paidhigher yields. Consequently, many loans were granted to individuals with poor creditworthiness (subprime borrowers). These subprimeborrowers were charged higher interest rates. When these subprime mortgages were packed into the pool of mortgages, the MBSsoffered higher yields. Thus, there was a high demand from investors for these MBSs, which fostered more poor credit quality loanoriginations.

Then from July 2004 to July 2006, the Federal Reserve started increasing interest rates. This placed some downward pressure onhousing prices because it made homes less affordable. At the same time, most subprime mortgages originating from 2005 and 2006were written on adjustable rates, and those interest rates adjusted upward too, making the payments too high for many borrowers. Thesubprime borrowers soon began to fall behind on their monthly payments leading to foreclosures and additional downward pressure onhousing prices. The devaluation of housing prices eroded the home equity of homeowners and led to further foreclosures and furtherprice decreases. The MBSs also devalued quickly.

Who owned MBSs? It turns out that the owners of these securities were financial firms, such as investment banks, commercial

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banks, insurance companies, mutual funds, and pension funds all over the world. Their weakened financial strength led to bank failures,bailouts, and a global credit crisis.

Want to know more?Key Words to Search for Updates: subprime, MBS, financial crisis

Further Reading: John W. Schoen, “7 years on from crisis, $150 billion in bank fines and penalties,” CNBC, April 30, 2015.http://www.cnbc.com/2015/04/30/7-years-on-from-crisis-150-billion-in-bank-fines-and-penalties.html

Reading Bond Quotes LG7-3To those familiar with bond terminology, bond quotes provide all of the information needed to make informedinvestment decisions. The volume of Treasury securities traded each day is substantial. Treasury bonds and notesaverage more than a half billion dollars in trading daily. Investors exhibit much less enthusiasm for corporate ormunicipal bonds, perhaps because the markets for each particular bond or bonds with the same maturity, couponrate, and credit ratings are much thinner and, therefore, less liquid. Most bond quote tables report only a smallfraction of the outstanding bonds on any given day. Bond quotes and data can be found in The Wall Street Journaland online at places like Yahoo! Finance (finance.yahoo.com). Table 7.2 shows three bond quote examples.

A typical listing for Treasury bonds appears first. Here, this Treasury bond pays bondholders a coupon of 2.750percent. On a $1,000 par value bond, this interest income would be $27.50 annually, paid as $13.750 every sixmonths per bond. The bond will mature in February of 2018—since this is fairly soon, the bond is considered ashort-term bond. Both the bid and the ask quotes for the bond appear, expressed as percentages of the bond’s parvalue of $1,000. The bid price is the price at which investors can sell the bond. A bid of 104.0156 means that aninvestor could sell for $1,040.156. Investors can buy this bond at the ask price of 104.0313, or $1,040.313. Since theprice is higher than the par value of the bond, the bond is selling at a premium to par because its coupon rate ishigher than current rates. Thus, investors call this kind of security a premium bond.

premium bond A bond selling for greater than its par value.

Notice that the ask price is higher than the bid price. The difference is known as the bid-ask spread. Investors buy atthe higher price and sell at the lower price. The bid-ask spread is thus the cost of actively trading bonds. Investorsbuy and sell with a bond dealer. Since the bond dealer takes the opposite side of the transaction, the dealer buys atthe low price and sells at the higher price. The bid-ask spread is part of the dealer’s compensation for taking on risk.An investor who bought this bond and held it to maturity would experience a $40.31 (= $1,040.31 − $1,000) capitalloss (−3.87 percent = −$40.31/$1,040.31]). The bond lost 0.0234 percent of its value during the day’s trading—achange of −$0.23 for a $1,000 par value bond. Last, the bond is offering investors who purchase it at the ask priceand hold it to maturity a 0.771 percent annual return.

Corporate bond quotes provide similar information. The table shows the quote for a Boeing bond that offersbondholders a coupon of 2.60 percent, or $13.00 semiannually (= $1,000 × 0.026 ÷ 2). The bond would beconsidered a mid-term bond (usually five years to maturity), since it matures in the year 2025. Corporate bonds arealso quoted in percentage of par value. The price quote of 98.400 indicates that the last trade occurred at a price of$984.00 per bond. Since the bond is selling for a price lower than its $1,000 par value, it’s called a discount bond. Aninvestor who bought this bond would reap a $16.00 (= $1,000 − $984.00) capital gain if the bond were held tomaturity. The Boeing bond represents an annual return of 2.79 percent for the investor who purchases the bond at$984.00.

discount bond A bond selling for lower than its par value.

▼ TABLE 7.2 Bond Quote Examples

Treasury Securities

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COUPON RATE MO/YR BID ASKED CHG ASK YLD

2.750 Feb 18 104.0156 104.0313 −0.0234 0.771

Corporate Bond

COMPANY COUPON MATURITY LAST PRICE YIELD

Boeing Co 2.60 October 2025 98.400 2.79

Municipal Bond

ISSUE COUPON MATURITY PRICE BID YLD

NYC Muni Wtr Fin Auth 4.500 06-15-37 97.570 4.66

Companies set a bond’s coupon rate when they originally issue the bond. A number of factors determine that couponrate:

The amount of uncertainty about whether the company will be able to make all the payments.

The term of the loan.The level of interest rates in the overall economy at the time.

Bonds from different companies carry different coupon rates because some, or all, of these determining factorsdiffer. Even a single company that has raised capital through bond issues many times may carry very differentcoupon rates on its various issues, because the bond issues would be offered in different years when the overalleconomic condition and interest rates differ.

Table 7.2 also shows a quote for a municipal bond issued by the New York City Municipal Water FinanceAuthority. This city government agency has raised capital by issuing municipal bonds to build reservoir facilities toprovide water to New York City. The bond pays a 4.500 percent coupon, and since it matures in 2037, it’sconsidered a long-term bond. According to Table 7.2, the bond is trading at a price just below par value—97.57percent. Most municipal bonds, unlike other bonds, feature a $5,000 face value rather than the typical par value of$1,000. So, the 97.57 percent price quote represents a dollar amount of $4,878.50 (= 0.9757 × $5,000). Thelow rate of return relative to Treasury bonds with similar maturities also has an explanation. Municipalbondholders do not have to pay federal income taxes on the interest payments that they receive from those securities.We explore this (sometimes) substantial advantage further in a later section of this chapter.

EXAMPLE7-3 Bond Quotes LG7-3

For interactiveversions of this

example, log in toConnect or go tomhhe.com/CornettM4e.

You note the following bond quotes and wish to determine each bond’s price, term, and interestpayments.

Treasury Securities

MATURITYRATE MO/YR BID ASKED CHG

9.00 Nov 20 137.5938 137.6250 −0.1563

Corporate Bond

COMPANY COUPON MATURITYLASTPRICE

LASTYIELD

Kohls Corp 7.375 Oct 15,2023

110.01 4.991

Municipal Bond

ISSUE COUPON MATURITY PRICEYLD TO

MAT

Florida StAquis &BridgeConstr

5.00 July 1,2025

106.78 4.458

SOLUTION:

The Treasury bond matures in November of 2020 and pays 9 percent interest. Investors receive cashinterest payment of $45 (= 0.09 × $1,000 ÷ 2) semiannually. Since the bond matures in less than 10years but more than 1 year, we would consider it a mid-term bond. Since no “n” appears next to thematurity date, we can also tell that the security was issued as a bond that would mature in 30 years.Investors could sell this bond for $1,375.94 (= 137.594 × $1,000) and buy it for $1,376.25 (= 1.37625× $1,000). The price fell on this particular day by $1.56 (= −0.001 5625 × $1,000). The dealer earned$0.31 (× $1,376.25 − $1,375.94) on each trade of these premium bonds.

The Kohls corporate bond pays a semiannual interest payment of $36.88 (= 0.07375 × $1,000 ÷ 2)and its price is $1,100.10 (= 1.1001 × $1,000). This premium bond’s 7.375 percent rate is likely wellabove market rates, which is why an investor would be willing to pay a premium for it.

The state of Florida issued the muni bond to fund bridge construction. With a $5,000 par value, theinterest payments are $125 (= 0.05 × $5,000 ÷ 2) every six months. The bonds are priced at$5,339.00 (= 1.0678 × $5,000).

Similar to Problems 7-9, 7-10, Self-Test Problem 1

time out!7-1 Describe the different reasons that the U.S. government, local governments, and corporations would issue bonds.7-2 What is the following bond’s price and what dollar amount will the bond pay for its semiannual interest payment?

COMPANY COUPON MATURITY PRICE YIELD

Home Depot Inc. 5.40 Mar 1, 2020 100.06 5.391

7.2 • BOND VALUATION LG7-4Present Value of Bond Cash FlowsAny bond’s value computation directly applies time value of money concepts. Bondholders know the interestpayments that they are scheduled to receive and the repayment of the par value at maturity. The current price of abond is, therefore, the present value of these future cash flows discounted at the prevailing market interest rate. Theprevailing market interest rate will depend on the bond’s term to maturity, credit quality, and tax status.

The simplest type of bond for time value of money calculations is a zero coupon bond. As you might guess from itsname, a zero coupon bond makes no interest payments. Instead, the bond pays only the par value payment at itsmaturity date. So a zero coupon bond sells at a substantial discount from its par value. For example, a bond with apar value of $1,000, maturing in 20 years, and priced to yield 6 percent, might be purchased for about $306.56. Atthe end of 20 years, the bond investor will receive $1,000. The difference between $1,000 and $306.56 (which is$693.44) represents the interest income received over the 20 years based upon the discount rate of 6 percent. Thetime line for this zero coupon bond valuation appears as

zero coupon bond A bond that does not make interest payments but generally sells at a deep discount and then pays the par value atthe maturity date.

We compute the zero’s price by finding the present value of the $1,000 cash flow received in 20 years. However, tobe consistent with regular coupon-paying bonds, zero coupon bonds are priced using semiannual compounding. Sothe formula and calculator valuation would use 40 semiannual periods at a 3 percent interest rate rather than 20periods at 6 percent. Using the present value equation of Chapter 4 results in

So the zero coupon bond’s price is indeed a steep discount to its par value. This makes sense because investorswould only buy a security that pays $1,000 in many years for a price that is much lower to make enough profit tomake up for the forgone semiannual interest payments. For comparison’s sake, instead of the 20-year zero, considera 20-year bond with a 7 percent coupon. So this 20-year maturity bond pays $35 in interest payments every sixmonths. We can think of these interest payments as an annuity stream. If the market discount rate is 6 percentannually, the time line appears as

page 210The time line shows the 40 semiannual payments (with the accompanying semiannual interest rate at 3 percent) of$35 and the par value payment at the bond’s maturity. Think through this: When bonds pay semiannualpayments, the discount rate must be a semiannual rate. Thus, the 6 percent annual rate becomes a 3 percentsemiannual rate. So we then compute the price of this bond by adding the present value of the interest paymentannuity cash flow to the present value of the future par value. A combination of the present value equations for theannuity cash flows and the value of the par redemption appear in the bond valuation equation 7-1:

(7-1)

where PMT is the interest payment, N is the number of periods until maturity, and i is the market interest rate perperiod on securities with the same bond characteristics. If this bond paid interest annually, then these variableswould take yearly period values. Since this bond pays semiannually, PMT, N, and i are all denoted in semiannualperiods. The price of this coupon bond should be

Of the $1,115.57 bond price, most of the value comes from the semiannual $35 coupon payments ($809.017) andnot the value from the future par value payment ($306.557).

Because equation 7-1 is quite complex, we usually compute bond prices using a financial calculator or computerprogram. An investor would compute the bond value using a financial calculator by entering N = 40, I = 3, PMT =35, FV = 1000, and computing the present value (PV). The calculator solution is $1,115.57.1

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EXAMPLE 7-4 Consider a 15-year bond that has a 5.5 percent coupon, paidsemiannually. If the current market interest rate is 6.5 percent, and thebond is priced at $940, should you buy this bond?

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

SOLUTION:

Compute the value of the bond using equation 7-1. Use semiannualcompounding (N = 2 × 15 = 30, i = 6.5 ÷ 2 = 3.25, and PMT = 0.055 ×$1,000 ÷ 2 = $27.50) as

So this bond’s value is $905.09, which is less than the $940 price. Thebond is overvalued in the market and you should not buy it.

Similar to Problems 7-21, 7-22, 7-23, 7-24, Self-Test Problem 1

the Math Coach on…Bond Pricing and Periods

“Since most bonds have semiannual interest payments, we must use semiannual periods to discount thecash flows. Most errors in computing a bond price occur in the adjustment for semiannual periods. The errorshappen whether you are using either the bond pricing equation or a financial calculator. To convert tosemiannual periods, be sure to adjust the three variables: number of periods, interest rate, and payments.

The number of years needs to be multiplied by 2 for the number of semiannual periods. The interest rateshould be divided by 2 for a six-month rate. Divide the annual coupon payment by 2 for the six-monthpayment. Remember to adjust all three inputs for the semiannual periods.

A coupon-paying bond’s price should hover reasonably around the par value of the bond. For a $1,000 parvalue bond, we could expect a price in the range of $700 to $1,300. If you compute a price outside this range,check to see whether you made the semiannual period adjustments correctly.„

Bond Prices and Interest Rate Risk LG7-5At the time of purchase, the bond’s interest payments and par value expected at maturity are fixed and known. Overtime, economywide interest rates change, but the bond’s coupon rate remains fixed. A rise in prevailing interest rates(also called increasing the discount rate) reduces all bonds’ values. If interest rates fall, all bonds will enjoy risingvalues. Consider that when interest rates rise, newly issued bonds offer to pay higher interest rates than the ratesoffered on existing bonds. So to sell an existing bond with its lower coupon rate, its market price must fall so thatthe buyer can expect a profit similar to that offered by newly issued bonds. Similarly, when prevailing interest ratesfall, market prices for outstanding bonds rise to bring the offered return on older bonds with higher coupon rates intoline with new issues. So market interest rates and bond prices are inversely related. That is, they move in oppositedirections.

Figure 7.2 demonstrates how the price of a 30-year Treasury bond may change over time. The 7.47 percent coupon

▼

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exactly matched prevailing interest rates when the bond was issued in 1986. Consequently, the bond sells for its parvalue of $1,000. Shortly thereafter, interest rates quickly rose to over 9 percent. As interest rates rose, bond priceshad to decline. Then in 1988, interest rates started a prolonged descent to lows not seen for decades. Note that whilea bond is issued at $1,000 and returns $1,000 at maturity, its price can vary a great deal in between. Bond investorsmust be aware that bond prices fluctuate on a day-to-day basis as interest rates fluctuate. Note that since the bondwill only pay $1,000 when it matures, the price must converge to $1,000 at the end. The determinants of marketinterest rate levels and changes are discussed in Chapter 6. Bondholders can incur large capital gains or capitallosses.The fact that, as prevailing interest rates change, the prices of existing bonds will change has a specific name in thefinancial industry—interest rate risk. Interest rate risk means that during periods when interest rates changesubstantially (and quickly), bondholders experience distinct gains and losses in their bond inventories. But interestrate risk does not affect all bonds exactly the same. Very short-term bonds experience little or no fluctuation in theirprices, and thus expose the bondholder to little interest rate risk. Long-term bondholders experience substantialinterest rate risk. Table 7.3 illustrates the impact of interest rate risk on bonds with different coupons and times tomaturity.

interest rate risk The chance of a capital loss due to interest rate fluctuations.

The first four rows show the prices and price changes for 30-year bonds with different coupon rates. Notice that thebonds with higher coupon rates also have higher prices. Bondholders as a rule find it more valuable to receive thelarge annuity payments. Also notice that a 1 percent increase in interest rates from 6 percent to 7 percent causesbond prices to fall. Bondholders with higher coupon bonds are not affected as much by interest rate increasesbecause they can take the large coupon payments and reinvest those cash flows in new bonds that offer higherreturns.

FIGURE 7-2 A Demonstration of the Price and Market Interest Rate over Time of a 30-Year Treasury Bond Issued in 1986 with a Couponof 7.47 Percent

As interest rates rise, bond prices fall. Here you can see great variance in the economy over 30 years. Long-term bondholders experiencesubstantial interest rate risk.Source: Yahoo! Finance, finance.yahoo.com.

▼ TABLE 7.3 Interest Rate Risk

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The price decline is greater for bonds with lower coupons because of reinvestment rate risk. When interest ratesincrease, bondholders’ cash flows—both periodic payments and final payoff at maturity—are discounted at a higherrate, decreasing a bond’s value. Because the cash flows from low-coupon bonds are smaller, the holder of suchbonds will have less money available from interest payments to buy the new, higher coupon bonds. Thusbondholders of lower coupon bonds have their capital tied up in assets that are not making them as much money.They face a bad dilemma: They can sell their lower coupon bonds and take a greater capital loss, using the (smaller)proceeds to buy new bonds with higher coupon rates. Or they can continue to receive the small income paymentsand hold their lower coupon bonds to maturity to avoid locking in the capital loss. Either way, they lose moneyrelative to those bondholders with higher coupon rates. You can see this illustrated in the 30-year bonds shown inTable 7.3. Reinvestment rates tend to help partially offset changing discount rates for higher coupon paying bonds.

reinvestment rate risk The chance that future interest payments will have to be reinvested at a lower interest rate.

EXAMPLE 7-5Capital Gains in the BondMarket LG7-5

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Say that you anticipate falling long-term interest rates from 6 percent to5.5 percent during the next year. If this occurs, what will be the totalreturn for a 20-year, 6.5 percent coupon bond through the interest ratedecline?

SOLUTION:

To determine the total return, compute the capital gain or loss and theinterest paid over the year. The capital gain or loss is determined fromthe change in price. The current bond price is

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The price in one year would be

So, the capital gain is $59.18 (= $1,116.97 − $1,057.79). The interestpayment during the year is $65 (= 0.065 × $1,000). If interest rates fallto 5.5 percent, then this bond should provide a total return of $124.18,which would be an 11.74 percent return (= $124.18 ÷ $1,057.79). Ofcourse, this is only an anticipated interest rate change and it may notoccur.

Similar to Problems 7-25, 7-26, 7-35, 7-36, Self-Test Problem 5

Another factor that influences the amount of reinvestment risk bondholders face is their bonds’ time to maturity. Thelast four bonds in the table all have a 5 percent coupon but have different times to maturity. Note that when interestrates increase, the bond prices of longer-term bonds decline more than shorter-term bonds. This shows that bondswith longer maturities and lower coupons have the highest interest rate risk. Short-term bonds with high couponshave the lowest interest rate risk. High interest rate risk bonds experience considerable price declines when interestrates are rising. However, these bonds also experience dramatic capital gains when interest rates are falling. While a1 percent change in market interest rates is not commonly seen on a daily or monthly basis, such a change is notunusual over the course of several months or a year.

time out!7-3 Show the time line and compute the present value for an 8.5 percent coupon bond (paid semiannually) with 12 years left

to maturity and a market interest rate of 7.5 percent.7-4 Describe the relationship between interest rate changes and bond prices.

7.3 • BOND YIELDS LG7-6Current YieldAlthough we speak about “the prevailing interest rate,” bond relationships reflect many interest rates (also calledyields). Some rates are difficult to calculate but accurately reflect the return the bond is offering. Others, like thecurrent yield, are easy to compute but only approximate the bond’s true return. A bond’s current yield is defined asthe bond’s annual coupon rate divided by the bond’s current market price. Current yield measures the rateof return a bondholder would earn annually from the coupon interest payments alone if the bond werepurchased at a stated price. Current yield does not measure the total expected return because it does not account forany capital gains or losses that will occur from purchasing the bond at a discount or premium to par.

current yield Return from interest payments; computed as the annual interest payment divided by the current bond price.

Yield to MaturityYield to maturity is a more meaningful equation for investors than the simple current yield calculation. The yield to

maturity calculation tells bond investors the total rate of return that they might expect if the bond were bought at aparticular price and held to maturity. While the yield to maturity calculation provides more information than thecurrent yield calculation, it’s also more difficult to compute, because we must compute the bond’s cash flows’internal rate of return. This calculation seeks to equate the bond’s current market price with the value of allanticipated future interest and par value payments. In other words, it is the discount rate that equates the presentvalue of all future cash flows with the current price of the bond. To calculate yield to maturity, investors must solvefor the interest rate, i, in equation 7-2, or solve for i in

yield to maturity The total return the bond offers if purchased at the current price and held to maturity.

(7-2)

Investors commonly compute the yield to maturity using financial calculators. For example, consider a 7 percentcoupon bond (paid semiannually) with eight years to maturity and a current price of $1,150. The return that the bondoffers investors, the yield to maturity, is computed as N = 16, PV = −1150, PMT = 35, and FV = 1000. Computingthe interest rate (i) gives us 2.363 percent. We must remember, however, that 2.363 percent is only the return for sixmonths because the bond pays semiannually. Yield to maturity always means an annual return. So, this bond’s yieldto maturity is 4.73 percent (2 × 2.363 percent).

Notice the link between a bond’s yield to maturity and the prevailing market interest rates used to determine abond’s price as we discussed in the previous section. We use the market interest rate to compute the bond’s value.We use the actual bond price to compute its yield to maturity. If the bond is correctly priced at its economic value,then the market interest rate will equal the yield to maturity. Thus, the relationship that we previously identifiedbetween bond prices and market interest rates applies to yields as well. This shows the inverse relationship betweenbond prices and bond yields. As a bond’s price falls, its yield to maturity increases and a rising bond priceaccompanies a falling yield. Look back at Figure 7.2 and you will see this relationship clearly.

EXAMPLE 7-6Computing Current Yield andYield to Maturity LG7-6

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

You have identified a 3.5 percent Treasury bond with four years left tomaturity and a quoted price of 96.281. Calculate the bond’s currentyield and yield to maturity.

SOLUTION:(1) First, identify that the bond’s price is $962.81 (= 96.281% × $1,000 =

0.96281 × $1,000).(2) The annual $35 in interest payments is paid in two $17.50 semiannual

payments. Therefore, the current yield of the bond is 3.64 percent (=$35 ÷ $962.81).

(3) The yield to maturity is computed using equation 7-2 and the financialcalculator as N = 8, PV = −962.81, PMT = 17.50, and FV = 1,000.Computing the interest rate (/) results in 2.263 percent and multiplyingby 2 gives the yield to maturity of 4.53 percent.

(4) Note that the current yield is less than the yield to maturity because itdoes not account for the capital gain to be earned if held to maturity.

Similar to Problems 7-13, 7-14, 7-27, 7-28, Self-Test Problem 2

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the Math Coach on…Bond Yields and Financial Calculators

“People computing a bond’s yield to maturity make three common mistakes. To avoid the first mistake,ensure that the bond price (PV) is a different sign than the interest and par value cash flows (PMT and FV).The second mistake: People forget to make the number of periods (N) and the per-period interest payment(PMT) consistent. Both should be in semiannual terms if the coupon payment is paid semiannually. Last,many people forget to multiply the resulting calculator interest rate (I) output by 2 to convert the semiannualrate back to an annual rate.„

Yield to CallThe yield to maturity computation assumes that the bondholder will hold the bond to its maturity. But remember thatsome bonds have call provisions that allow the issuers to repay the bondholder’s par value prior to its scheduledmaturity. Issuers often call bonds after large drops in market interest rates. In such cases, issuers commonly paybondholders the bond’s par value plus one year of interest payments. The reasons behind early bond redemptions areobvious. When interest rates fall, issuers can sell new bonds at lower interest rates. Companies want to refinancetheir debt—just as homeowners do—to reduce their interest payments.

Issuers gain important advantages with call provisions because they allow refinancing opportunities. Of course, thesame provisions are disadvantages for bond investors. When bonds are called, investors receive the par value andcall premium, but then investors must seek equally profitable bonds to buy with the proceeds. You will recall thatinvestors can face reinvestment risk—the available bonds aren’t as profitable because interest rates have declined.Bonds are called away at the worst time for investors. In addition, bond prices will rise as market interest rates fall,which could provide issuers opportunities to sell the bonds at a profit. But the price increases will be limited by thefact that the bond will likely be called early. As partial compensation, bond investors receive the call price, which isthe par value of the bond plus the call premium (typically one year of interest payments). The possibility that bondscan be called early dampens their upside price potential. We can even compute the price of a bond that’s likely to becalled from the equation

(7-3)

In this case, N is the number of periods until the bond can be called and i is the prevailing market rate. Theprevailing market interest rate will probably differ from the rate for a noncallable bond. The previous sectiondemonstrated via the yield curve that bonds with different maturities have different yields. A bond that matures in 20years, but is likely to be called in 5 years, will carry a yield appropriate for a five-year bond.

Now, reconsider the 20-year bond with a 7 percent coupon that we discussed previously. If the bond can be called infive years with a call price of $1,070, the appropriate discount rate happens to be 5.75 percent annually atthat time (instead of the 6 percent in the original problem). This time line would be

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The changes in this time line are only 10 semiannual payments of $35 (rather than 40 such semiannual payments), a2.875 percent semiannual discount rate, and the call price payment of $1,070. The price of this callable bond wouldbe

the Math Coach on…Spreadsheets and Bond Pricing

“Common spreadsheet programs have functions that can compute the price or yield to maturity of a bond.The functions are

Compute a bond price = PRICE(settlement,maturity,rate,yld,redemption,frequency,basis)

Compute a yield to maturity = YIELD(settlement,maturity,rate,pr,redemption,frequency,basis)

Settlement is the bond’s settlement date. This is the purchase date of the bond; typically it is today. Maturity isthe bond’s maturity date. Rate is the bond’s annual coupon rate. Pr is the bond’s price per $100 face value.Note that the par value of a bond is typically $1,000, so an adjustment is needed for this input. Redemption isthe bond’s redemption value per $100 face value. Frequency is the number of coupon payments per year. Forsemiannual, frequency = 2. Basis is the type of day count basis to use.

Consider the bond valuation problem of Example 7-4. The spreadsheet solution is the same as the TVMcalculator solution and the pricing equation.

Also consider the yield to maturity problem in Example 7-6. This spreadsheet solves for the yield to maturity.

See this textbook’s online student center to watch instructional videos on using spreadsheets. Also note thatthe solutions for all the examples in the book are illustrated using spreadsheets in videos that are alsoavailable on the textbook website.„

In this example, the callable bond would be priced at $1,106.38, which is slightly lower than an identical bond that

was not callable, priced at $1,115.57.

If a bond is likely to be called, then the yield to maturity calculation does not give investors a good estimate of theirreturn. Bondholders can use instead a yield to call calculation, which differs from the yield to maturity only in that itscalculation assumes that the investor will receive the par value and call premium at the earliest call date. Forexample, reconsider the 7 percent coupon bond (paid semiannually) with eight years to maturity, which weexamined previously. The current bond price is $1,130 (which is slightly lower than the yield to maturity bond priceof $1,150). If the bond can be called in three years at a specific call price of the par value plus one annual coupon,then what is the yield to call? The yield to call is computed as N = 6, PV = −1130, PMT = 35, and FV = 1070. Theresulting interest rate (i) is 2.26 percent. The yield to call for this bond is thus 4.52 percent (= 2 × 2.26%).

yield to call The total return that the bond offers if purchased at the current price and held until the bond is called.

Municipal Bonds and YieldMunicipal bonds (munis) seem to offer low yields to maturity compared to the return that corporate bonds andTreasury securities offer. Munis offer lower rates because the interest income they generate for investors is tax-exempt—at least at the federal level.2 Specifically, income from municipal bonds is not subject to taxation by thefederal government or the state government where the bonds are issued. As a result, municipal bond investorswillingly accept lower yields than those they can obtain from taxable bonds. Generally speaking, investors comparethe after-tax interest income earned on taxable bonds against the return earned on municipal bonds. For example,suppose an investor in the 35 percent marginal income tax bracket has $100,000 to invest in either corporate ormunicipal bonds. The $100,000 investment would earn a taxable $7,000 annually from 7 percent corporate bonds or$5,000 from tax-exempt 5 percent municipal bonds. After taxes, the corporate bond leaves the investor with $4,550[=(1 − 0.35) × $7,000]. Obviously, this is less than the tax-free income of $5,000 generated by the muni bond.

A common way to compare yields from muni bonds versus those from taxable bonds is to convert the yield tomaturity of the muni to a taxable equivalent yield, as shown in equation 7-4.

(7-4)

taxable equivalent yield Modification of a municipal bond’s yield to maturity used to compare muni bond yields to taxable bond yields.

EXAMPLE 7-7Which Bond Has a Better After-Tax Yield? LG7-6

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Imagine a time when you have a high income, placing you in the 31percent marginal tax bracket. You are interested in investing somemoney in a bond issue and have three alternatives. The first is acorporate bond with a 6.4 percent yield to maturity. The second bond isa Treasury that offers a 5.7 percent yield. The third choice is amunicipal bond priced at a yield to maturity of 4.0 percent. Which bondgives you the highest after-tax yield?

SOLUTION:

The Treasury and corporate bonds are both taxable, so we cancompare them directly with each other. The yield of 6.4 percent on thecorporate is clearly higher than the 5.7 percent yield offered by theTreasury bond. To include a comparison with the nontaxable municipalbond, compute its equivalent taxable yield as in equation 7-4:

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The municipal bond’s equivalent taxable yield of 5.80 percent is higherthan the Treasury but lower than the corporate bond.

Similar to Problems 7-15, 7-16, 7-31, 7-32, 7-37, 7-38, Self-TestProblem 3

For high-income investors (in the 35 percent marginal tax bracket) a 5 percent muni bond has an equivalent taxableyield of 7.69 percent = 0.05 ÷ (1 − 0.35)]. The 5 percent muni is more attractive for this investor than a 7 percentcorporate bond. However, for an investor with lower income (in the 28 percent marginal tax bracket) the equivalenttaxable yield is only 6.94 percent. The corporate bond provides more after-tax profit than the muni for this investor.It’s easy to see why muni bonds are popular among high-income investors (those with substantial marginal taxrates).

Summarizing YieldsIn this section, we have presented several different types of interest rates, or yields, associated with bonds. See asummary in Table 7.4. Many of these yields relate to one another. Consider the bonds and associated yields reportedin Table 7.5. Treasury bonds (1) to (3) show how coupon rates, current yield, and yield to maturity relate. When abond trades at its par value (usually $1,000), then the coupon rate, current yield, and yield to maturity are all thesame. When that bond is priced at a premium (bond 2), then both the current yield and the yield to maturity will belower than the coupon rate. They are both higher than the coupon rate when the bond trades at a discount. Noticethat yield to maturity is higher than current yield for discount bonds, and that yield to maturity is lower than currentyield for premium bonds. In other words, the current yield always lies between the coupon rate and the yield tomaturity. Both the current yield and the yield to maturity move in the opposite direction to the bond’s price.

Bonds (4) to (6) are callable corporate bonds. Recall that all the yields (current yield, yield to maturity, and yield tocall) are identical when the bond trades at par value. When interest rates fall and bond prices increase, as with bond(5), the issuing corporation has a strong incentive to call the bond after five years, as allowed in the indentureagreement. So investors should base their purchase decisions on the yield to call. When interest rates increase, bondprices decline (as bond (6) shows). In this case, investors could compute the yield to call (as shown), but theinformation isn’t useful because the company will not likely call the bond while interest rates are high.

▼ TABLE 7.4 Summary of Interest Rates

InterestRate Purpose Description

Couponrate

Computebond cashinterestpayments

The coupon rate is reported as a bond characteristic. It is reported as a percentageand is multiplied by the par value of the bond to determine the annual cash interestpayment. The coupon rate will not change through the life of the bond.

Currentyield

Quickassessmentof the interestrate a bond isoffering

Computed as the annual interest payment divided by the current price of the bond. Itmeasures the return to be expected from just the interest payments if the bond waspurchased at the current price. Since the bond price may change daily, the currentyield will change daily.

Yield tomaturity

Accuratemeasurementof the interestrate a bond isoffering

The return offered by the bond if purchased at the current price. This return includesboth the expected income and capital gain/loss if held to the maturity date. The yieldto maturity will change daily as the bond price changes.

Interest rate

Yield tocall

obtained if thebond is called

Same as the yield to maturity except that it is assumed that the bond will be called atthe earliest date it can be called.

Taxableequivalentyield

Comparisonof nontaxablebond yields totaxable bondyields

Investors must pay taxes on most types of bonds. However, municipal bonds are taxfree. To compare the muni’s nontaxable yield to maturity to that of taxable bonds,divide the yield by one minus the investor’s marginal tax rate.

Marketinterestrate

Comparisonof prices of allbonds

The interest rate determined by the bond prices of actual trades between buyers andsellers. The market interest rate will be different for bonds of different times tomaturity and different levels of risk.

Totalreturn

Determinerealizedperformanceof aninvestment

Realized return that includes both income and capital gain/loss profits.

▼ TABLE 7.5 Price, Coupon, and Yield Relationships of a 10-Year Bond

Call price = Par value + One year’s interest

The last three bonds shown in the table are municipal bonds. Recall that these bonds typically offer lower yieldsbecause the income from munis is tax exempt. It is easier to compare municipal bonds with Treasuries and corporatebonds if you compute the municipal bond’s taxable equivalent yield first. Here, we use a marginal tax rate of 35percent in the calculation. The last column of the table shows that the taxable equivalent yield of the municipalbonds is really quite competitive with corporate bond yields. Any investor with income taxed at the 35 percentmarginal tax bracket would prefer the municipal bond over the corporate bond if the muni’s taxable equivalent yieldis higher than the yield to maturity (or yield to call) of the corporate bond.

The table also shows that Treasury securities offer lower yields than corporate bonds with similar terms to maturity.The difference (or spread) between Treasury and corporate yields gives rise to a discussion of bond credit risk,which follows.

time out!7-5 Calculate the yield to maturity for a zero coupon bond with a price of $525 and 10 years left to maturity.7-6 Which is higher for a discount bond, the yield to maturity or the coupon rate? Why?

7.4 • CREDIT RISK LG7-7

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Bond RatingsWill a bond issuer make the promised interest and par value payments over the next 10, 20, or even 30 years? Creditquality risk is the chance that the bond issuer will not be able to make timely payments. To assess this risk,independent bond rating agencies, such as Moody’s and Standard & Poor’s, monitor corporate, U.S. agency, ormunicipal developments during the bond’s lifetime and report their findings as a grade or rating. The U.S.government issues the highest credit quality debt, though that consensus has recently come into doubt as the U.S.debt and budget deficit have ballooned.

credit quality risk The chance that the issuer will not make timely interest payments or even default.

bond rating A grade of credit quality as reported by credit rating agencies.

The primary “big three” bond credit rating agencies in the United States include Moody’s Investors Service,Standard & Poor’s Corporation, and Fitch IBCA Inc. Each of these credit analysis firms assigns similar ratingsbased on detailed analyses of issuers’ financial condition, general economic and credit market conditions, and theeconomic value of any underlying collateral. The Standard & Poor’s ratings are shown in Table 7.6. Theirhighest credit quality rating is AAA. Bonds rated AAA, AA, A, or BBB are considered investment gradebonds. The issuers of these securities have the highest chance of making all interest and par value paymentspromised in the indenture agreement.

investment grade High credit quality corporate bonds.

▼ TABLE 7.6 Standard & Poor’s Bond Credit Ratings

Credit RiskCreditRating Description

Investment Grade

Highestquality

AAA The obligor’s (issuer’s) capacity to meet its financial commitment on the obligation isextremely strong.

Highquality

AA The obligor’s capacity to meet its financial commitment on the obligation is very strong.

Uppermediumgrade

AThe obligor’s capacity to meet its financial commitment on the obligation is still strong,though somewhat susceptible to the adverse effects of changes in circumstances andeconomic conditions.

Mediumgrade BBB

The obligor exhibits adequate protection. However, adverse economic conditions orchanging circumstances are more likely to lead to a weakened capacity to meet its financialcommitment.

Below Investment Grade

Somewhatspeculative BB

Faces major ongoing uncertainties or exposure to adverse business, financial, or economicconditions which could lead to the obligor’s inadequate capacity to meet its financialcommitment.

Speculative B Adverse business, financial, or economic conditions will likely impair the obligor’s capacity orwillingness to meet its financial commitment.

Highlyspeculative

CCC Currently vulnerable to nonpayment, and is dependent upon favorable business, financial,and economic conditions for the obligor to meet its financial commitment.

Mostspeculative

CC Currently highly vulnerable to nonpayment.

Imminentdefault

C Bankruptcy petition has been filed or similar action taken, but payments on this obligationare being continued.

Default D Obligations are in default or the filing of a bankruptcy petition has occurred and paymentsare jeopardized.

Source: Standard & Poor’s.

The investment community considers bonds rated BB and below to be below-investment grade bonds, and someinvestors, such as pension funds or other fiduciaries, cannot purchase these securities for their portfolios. These

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bonds are considered to be speculative because they carry a significant risk that the issuer will not make current orfuture payments. Speculative bonds are sometimes called junk bonds because of this risk. In order to attract buyers,issuers sell these bonds at a considerable discount from par and a high associated yield to maturity. Agencies oftenenhance ratings from “AA” to “CCC” with the addition of a plus (+) or minus (−) sign to show relative standingwithin the major rating categories. For example, the Greek government saw its bonds upgraded from CCC+ to B−by Standard & Poor’s on January 22, 2016. The next week, Standard and Poor’s downgraded Dutch Shell Plc fromAA− to A+. These rating changes impact not only the current prices of these bonds, but also the interest rate Greeceand Royal Dutch Shell would have to pay if they issued new bonds.

junk bonds Low credit quality corporate bonds, also called speculative bonds or high-yield bonds.

Standard & Poor’s signals that it’s considering a rating change by placing an individual bond, or all of a givenissuer’s bonds, on CreditWatch (S&P). Rating agencies make their ratings information available to the publicthrough their ratings information desks. In addition to published reports, ratings are made available in many publiclibraries and over the Internet.

Credit rating agencies conduct general economic analyses of companies’ business and analyze firms’ specificfinancial situations. A single company may carry several outstanding bond issues. If these issues featurefundamental differences, then they may have different credit level risks. For example, unsecured corporate bonds, ordebentures, are backed only by the reputation and financial stability of the corporation. A senior bond has a priorityclaim over junior (more recently issued) securities in the event of default or bankruptcy. So, senior bondscarry less credit risk than junior bonds. Some bonds are secured with collateral. When you buy a car usinga loan, the car is collateral for that loan. If you don’t make the loan payments, the bank will repossess the car.Companies can also offer collateral when issuing bonds. When a firm uses collateral such as real estate or factoryequipment, the bonds are called mortgage bonds or equipment trust certificates, respectively. Bonds issued with nocollateral generally carry higher credit risk.

unsecured corporate bonds Corporate debt not secured by collateral such as land, buildings, or equipment.

debentures Unsecured bonds.

senior bonds Older bonds that carry a higher claim to the issuer’s assets.

mortgage bonds Bonds secured with real estate as collateral.

equipment trust certificates Bonds secured with factory and equipment as collateral.

Credit Risk and YieldInvestors will only purchase higher risk bonds if those securities offer higher returns. Therefore, issuers price bondswith high credit risk to offer high yields to maturity. So another common name for junk bonds is high-yield bonds.Differences in credit risk are a prime source of differences in yields between government and various corporatebonds. Figure 7.3 shows the historical average annual yields for long-term Treasury bonds and corporate bonds withcredit ratings of Aaa and Baa since 1980. Riskier low-quality bonds always offer a higher yield than the higherquality bonds. However, the yield spread between high- and low-quality bonds varies substantially over time. Theyield difference between Baa bonds and Treasuries was as high as 3.7 percent and 3.3 percent in 1982 and 2003,respectively. The spread has been as narrow as 1.3 percent and 1.4 percent in 1994 and 2006, respectively.

high-yield bonds Bonds with low credit quality that offer a high yield to maturity, also called junk bonds.

How do some corporations’ debt obligations become junk bonds? Some companies that aren’t economically soundor those that use a high degree of financial leverage issue junk bonds. In other cases, financially strong companiesissue investment grade bonds and then, over time, begin to have trouble. Eventually a company’s bonds can bedowngraded to junk status. For example, General Motors (GM) bonds were considered of the highest quality fromthe 1950s through the 1980s and much of the 1990s. On May 9, 2005, Standard & Poor’s downgraded GM bonds tojunk status. Junk bonds that were originally issued at investment grade status are called fallen angels. GM eventuallyfiled for bankruptcy protection in June 2009. By 2016, its credit rating was at the lowest end of investment grade(BBB−).

FIGURE 7-3 Yield to Maturity on Long-Term Bonds of Different Credit Risk, 1980–2015

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Looking at the historical yields for long-term Treasury and corporate bonds, notice how the yield spread between high- and low-quality bondsvaries substantially over time.

finance at work //: marketsA Greek Tragedy: Debt Crisis

©Max Shen/Getty Images

Greece joined the European monetary union in 2000, which means the euro replaced drachmas as the national currency. The euro wasa much stronger currency than the drachma because it was backed by the economic prosperity of the whole European Union. Hence,the Greek government was able to borrow from foreign investors at much lower interest rates. This contributed to the ensuing economicboom and expansion of government spending in Greece.

The Greek government has long been operating on high budget deficits and borrowing. But the problems really began in the Fall of2009 when the new elected government found that it had inherited a financial burden that was much larger than previously reported. Thebudget deficit was revised to be larger than 13 percent of the size of the economy. The revelation of these huge government deficits anddebts cast enormous doubt on the ability of the Greek government to pay its debts.

This increase in risk was reflected on December 8, 2009, by the downgrade to BBB (lowest in the Euro Zone) of the Sovereign bondof Greece by the Fitch credit rating firm. Standard & Poor’s and Moody’s both downgraded Greece Sovereigns to junk bond status inMay and June of 2010. As a consequence, Greece found that it was difficult to borrow more money and had to offer lenders yields of asmuch as 12 percent and a financial crisis developed.

In order to restore investor confidence, the Greek government has pledged to an austerity plan that cuts spending and reduces thebudget deficit. However, the success of the plans has been undermined by strong domestic opposition as illustrated by strikes and even

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riots. As a temporary solution, the European Union and the International Monetary Fund together offered a large loan package to helpout Greece. This calmed the bond market and Greek bond yields fell, but they are still high. Unfortunately, the severe economicdepression in Greece has caused further problems with its ability to pay its debt. In 2011 and 2012, Greece enacted several bondrestructurings which mandated changes in the terms. They are considered selective bond defaults. In June of 2015, Greece became thefirst developed country to miss an International Monetary Fund loan repayment.

Want to know more?Key Words to Search for Updates: Greek bonds, Greek debt crisis

time out!7-7 Explain why a change in a bond’s credit rating will cause its price to change.7-8 One company has issued two bond classes. One issue is a mortgage bond and the other is a debenture. Which issue will

have a higher bond rating and which will offer a higher yield?

Bonds that experience credit-rating downgrades must offer a higher yield. As all the future cash flows are fixed, thebond price must fall to create a higher yield to maturity. Alternatively, bonds that are upgraded experience priceincreases and yield decreases. Bond upgrades often occur during strong economic periods because corporate issuerstend to perform better financially at these times. In a weak economy, high-yield bonds lose their luster because thedefault risk rises. More credit downgrades occur during economic recessions. In general, any event that impacts thelikelihood of a firm paying back the interest payments and principle it owes will impact its credit risk. Credit risk isa determinant of the discount rate. The discount rate is also influenced by macroeconomic factors, like decisions bythe Federal Reserve Board. A change in the discount rate directly changes the value of the bond. Therefore,company performance, strength of the economy, and monetary policy all affect bond values.

7.5 • BOND MARKETS LG7-8The majority of trading volume in the bond market occurs in a decentralized, over-the-counter market. Most tradesoccur between bond dealers and large institutions (like mutual funds, pension funds, and insurance companies).Dealers bid for bonds that investors seek to sell and offer bonds from their own inventory when investorswant to buy. This is especially true for the very active Treasury securities market. However, a smallnumber of corporate bonds are listed on centralized exchanges.

FIGURE 7-4 Most Active Investment Grade Bonds, February 5, 2016

This is an example of the most actively traded investment bonds for a given day.

Source: The New York Times, http://markets.on.nytimes.com/research/markets/bonds/bonds.asp

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The NYSE operates the largest centralized U.S. bond market. The majority of bond volume at the NYSE is incorporate debt. The most actively traded investment grade bonds for the day are shown in Figure 7.4. Even the mostactive corporate bonds experience relatively low trading volume. Note that some of the bonds traded are short-term,like Anadarko Pete Corp and JPMorgan Chase & Co. Other bonds have many years to maturity, like the Anheuser-busch Inbev bond that matures in February 2046. Most of the ones shown are premium bonds (price greater than parvalue), while one is a discount bond.

Following the Bond MarketThe entire bond market encompasses a wide variety of securities with varying credit quality from different issuers.Large differences also arise among bonds in terms of their characteristics such as term to maturity and size of thecoupon. The biggest factor associated with changes in bond prices is changes in interest rates. So, one common wayto describe the direction of bond prices is simply to report the change in interest rates, since we know that interestrate changes will affect all bonds the same way. The interest rate referenced is the yield to maturity and daily yieldchange for the 10-year Treasury. Knowing how this interest rate changed today gives bond investors a good idea ofthe general price movement of all types of bonds.

Bond indexes track specific segments of the bond market. Various securities firms, such as Barclays Capital orMerrill Lynch, maintain these indexes that capture bond price and yield changes in particular segments. You canfind information about major bond indexes on the Internet and in publications like The Wall Street Journal (both inprint and online). Figure 7.5 shows indexes that track bonds by type of issuer (federal government, corporation, localgovernment, etc.) and time to maturity (short, intermediate, and long).

time out!7-9 Why can we use various interest rates to describe the performance of the entire bond market?7-10 What bond segments are measured by which bond indexes?

FIGURE 7-5 Major Bond Indexes as Reported in The Wall Street Journal and on the Internet, February 5, 2016

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Here are indexes that track bonds by type of issuer (federal government, corporation, local government, etc.) and time to maturity (short,intermediate, and long).

Source: The Wall Street Journal Online. Tracking Bond Benchmarks web page.

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Log in to your Connect course for study materials including self-test problems with solutions, answers tothe Time Out quizzes, guided example videos, and more.

Your Turn…Questions

1. What does a call provision allow issuers to do, and why would they do it? (LG7-1)

2. List the differences between the new TIPS and traditional Treasury bonds. (LG7-2)

3. Explain how mortgage-backed securities work. (LG7-2)

4. Provide the definitions of a discount bond and a premium bond. Give examples. (LG7-3)

5. Describe the differences in interest payments and bond price between a 5 percent coupon bond and a zerocoupon bond. (LG7-4)

6. All else equal, which bond’s price is more affected by a change in interest rates, a short-term bond or a longer-term bond? Why? (LG7-5)

7. All else equal, which bond’s price is more affected by a change in interest rates, a bond with a large coupon ora small coupon? Why? (LG7-5)?

8. Explain how a bond’s interest rate can change over time even if interest rates in the economy do not change.(LG7-5)

9. Compare and contrast the advantages and disadvantages of the current yield computation versus yield tomaturity calculations. (LG7-6)

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10. What is the yield to call and why is it important to a bond investor? (LG7-6)

11. What is the purpose of computing the equivalent taxable yield of a municipal bond? (LG7-6)

12. Explain why high-income and wealthy people are more likely to buy a municipal bond than a corporate bond.(LG7-6)

13. Why does a Treasury bond offer a lower yield than a corporate bond with the same time to maturity?Could a corporate bond with a different time to maturity offer a lower yield? Explain. (LG7-7)

14. Describe the difference between a bond issued as a high-yield bond and one that has become a “fallen angel.”(LG7-7)

15. What is the difference in the trading volume between Treasury bonds and corporate bonds? Give examplesand/or evidence. (LG7-8)

ProblemsBASIC PROBLEMS

7-1 Interest Payments Determine the interest payment for the following three bonds: 3½ percent couponcorporate bond (paid semiannually), 4.25 percent coupon Treasury note, and a corporate zero coupon bondmaturing in 10 years. (Assume a $1,000 par value.) (LG7-1)

7-2 Interest Payments Determine the interest payment for the following three bonds: 4½ percent couponcorporate bond (paid semiannually), 5.15 percent coupon Treasury note, and a corporate zero coupon bondmaturing in 15 years. (Assume a $1,000 par value.) (LG7-1)

7-3 Time to Maturity A bond issued by Ford on May 15, 1997, is scheduled to mature on May 15, 2097. Iftoday is November 16, 2014, what is this bond’s time to maturity? (LG7-1)

7-4 Time to Maturity A bond issued by IBM on December 1, 1996, is scheduled to mature on December 1,2096. If today is December 2, 2015, what is this bond’s time to maturity? (LG7-1)

7-5 Call Premium A 6 percent corporate coupon bond is callable in five years for a call premium of one yearof coupon payments. Assuming a par value of $1,000, what is the price paid to the bondholder if the issuercalls the bond? (LG7-1)

7-6 Call Premium A 5.5 percent corporate coupon bond is callable in 10 years for a call premium of one yearof coupon payments. Assuming a par value of $1,000, what is the price paid to the bondholder if the issuercalls the bond? (LG7-1)

7-7 TIPS Interest and Par Value A 2¾ percent TIPS has an original reference CPI of 185.4. If the currentCPI is 210.7, what is the current interest payment and par value of the TIPS? (LG7-2)

7-8 TIPS Interest and Par Value A 3⅛ percent TIPS has an original reference CPI of 180.5. If the currentCPI is 206.8, what is the current interest payment and par value of the TIPS? (LG7-2)

7-9 Bond Quotes Consider the following three bond quotes: a Treasury note quoted at 97.844, a corporatebond quoted at 103.25, and a municipal bond quoted at 101.90. If the Treasury and corporate bonds have apar value of $1,000 and the municipal bond has a par value of $5,000, what is the price of these threebonds in dollars? (LG7-3)

7-10 Bond Quotes Consider the following three bond quotes: a Treasury bond quoted at 106.438, acorporate bond quoted at 96.55, and a municipal bond quoted at 100.95. If the Treasury and corporatebonds have a par value of $1,000 and the municipal bond has a par value of $5,000, what is the priceof these three bonds in dollars? (LG7-3)

7-11 Zero Coupon Bond Price Calculate the price of a zero coupon bond that matures in 20 years if themarket interest rate is 3.8 percent. (LG7-4)

7-12 Zero Coupon Bond Price Calculate the price of a zero coupon bond that matures in 15 years if the

market interest rate is 5.75 percent. (LG7-4)7-13 Current Yield What’s the current yield of a 3.8 percent coupon corporate bond quoted at a price of

102.08? (LG7-6)

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7-14 Current Yield What’s the current yield of a 5.2 percent coupon corporate bond quoted at a price of96.78? (LG7-6)

7-15 Taxable Equivalent Yield What’s the taxable equivalent yield on a municipal bond with a yield tomaturity of 3.5 percent for an investor in the 33 percent marginal tax bracket? (LG7-6)

7-16 Taxable Equivalent Yield What’s the taxable equivalent yield on a municipal bond with a yield tomaturity of 2.9 percent for an investor in the 28 percent marginal tax bracket? (LG7-6)

7-17 Credit Risk and Yield Rank from highest credit risk to lowest risk the following bonds, with thesame time to maturity, by their yield to maturity: Treasury bond with yield of 5.55 percent, IBM bondwith yield of 7.49 percent, Trump Casino bond with yield of 8.76 percent, and Banc One bond with ayield of 5.99 percent. (LG7-7)

7-18 Credit Risk and Yield Rank the following bonds in order from lowest credit risk to highest risk allwith the same time to maturity, by their yield to maturity: Treasury bond with yield of 4.65 percent,United Airlines bond with yield of 9.07 percent, Bank of America bond with a yield of 6.25 percent,and Hewlett-Packard bond with yield of 6.78 percent. (LG7-7)

INTERMEDIATE PROBLEMS

7-19 TIPS Capital Return Consider a 3.5 percent TIPS with an issue CPI reference of 185.6. At thebeginning of this year, the CPI was 193.5 and was at 199.6 at the end of the year. What was the capitalgain of the TIPS in dollars and in percentage terms? (LG7-2)

7-20 TIPS Capital Return Consider a 2.25 percent TIPS with an issue CPI reference of 187.2. At thebeginning of this year, the CPI was 197.1 and was at 203.8 at the end of the year. What was the capitalgain of the TIPS in dollars and in percentage terms? (LG7-2)

7-21 Compute Bond Price Compute the price of a 3.8 percent coupon bond with 15 years left to maturityand a market interest rate of 6.8 percent. (Assume interest payments are semiannual.) Is this a discountor premium bond? (LG7-4)

7-22 Compute Bond Price Compute the price of a 5.6 percent coupon bond with 10 years left to maturityand a market interest rate of 7.0 percent. (Assume interest payments are semiannual.) Is this a discountor premium bond? (LG7-4)

7-23 Compute Bond Price Calculate the price of a 5.2 percent coupon bond with 18 years left to maturityand a market interest rate of 4.6 percent. (Assume interest payments are semiannual.) Is this a discountor premium bond? (LG7-4)

7-24 Compute Bond Price Calculate the price of a 5.7 percent coupon bond with 22 years left to maturityand a market interest rate of 6.5 percent. (Assume interest payments are semiannual.) Is this a discountor premium bond? (LG7-4)

7-25 Bond Prices and Interest Rate Changes A 5.75 percent coupon bond with 10 years left to maturity ispriced to offer a 6.5 percent yield to maturity. You believe that in one year, the yield to maturity willbe 5.8 percent. What is the change in price the bond will experience in dollars? (LG7-5)

7-26 Bond Prices and Interest Rate Changes A 6.5 percent coupon bond with 14 years left to maturity ispriced to offer a 7.2 percent yield to maturity. You believe that in one year, the yield to maturity willbe 6.8 percent. What is the change in price the bond will experience in dollars? (LG7-5)

7-27 Yield to Maturity A 5.65 percent coupon bond with 18 years left to maturity is offered for sale at

$1,035.25. What yield to maturity is the bond offering? (Assume interest payments are semiannual.)(LG7-6)

7-28 Yield to Maturity A 4.30 percent coupon bond with 14 years left to maturity is offered for sale at$943.22. What yield to maturity is the bond offering? (Assume interest payments are semiannual.)(LG7-6)

7-29 Yield to Call A 6.75 percent coupon bond with 26 years left to maturity can be called in 6 years. Thecall premium is one year of coupon payments. It is offered for sale at $1,135.25. What is the yield tocall of the bond? (Assume interest payments are semiannual.) (LG7-6)

7-30 Yield to Call A 5.25 percent coupon bond with 14 years left to maturity can be called in 4 years. Thecall premium is one year of coupon payments. It is offered for sale at $1,075.50. What is the yield to

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call of the bond? (Assume interest payments are semiannual.) (LG7-6)7-31 Comparing Bond Yields A client in the 39 percent marginal tax bracket is comparing a municipal

bond that offers a 4.5 percent yield to maturity and a similar-risk corporate bond that offers a 6.45percent yield. Which bond will give the client more profit after taxes? (LG7-6)

7-32 Comparing Bond Yields A client in the 28 percent marginal tax bracket is comparing a municipalbond that offers a 4.5 percent yield to maturity and a similar-risk corporate bond that offers a 6.45percent yield. Which bond will give the client more profit after taxes? (LG7-6)

ADVANCED PROBLEMS

7-33 TIPS Total Return Reconsider the 3.5 percent TIPS discussed in problem 7-19. It was issued withCPI reference of 185.6. The bond is purchased at the beginning of the year (after the interest payment),when the CPI was 193.5. For the interest payment in the middle of the year, the CPI was 195.1. Now,at the end of the year, the CPI is 199.6 and the interest payment has been made. What is the totalreturn of the TIPS in dollars and in percentage terms for the year? (LG7-2)

7-34 TIPS Total Return Reconsider the 2.25 percent TIPS discussed in problem 7-20. It was issued withCPI reference of 187.2. The bond is purchased at the beginning of the year (after the interest payment),when the CPI was 197.1. For the interest payment in the middle of the year, the CPI was 200.1. Now,at the end of the year, the CPI is 203.8 and the interest payment has been made. What is the totalreturn of the TIPS in dollars and in percentage terms for the year? (LG7-2)

7-35 Bond Prices and Interest Rate Changes A 6.25 percent coupon bond with 22 years left to maturity ispriced to offer a 5.5 percent yield to maturity. You believe that in one year, the yield to maturity willbe 6.0 percent. If this occurs, what would be the total return of the bond in dollars and percent? (LG7-5)

7-36 Bond Prices and Interest Rate Changes A 7.5 percent coupon bond with 13 years left to maturity ispriced to offer a 6.25 percent yield to maturity. You believe that in one year, the yield to maturity willbe 7.0 percent. If this occurs, what would be the total return of the bond in dollars and percentageterms? (LG7-5)

7-37 Yields of a Bond A 2.50 percent coupon municipal bond has 12 years left to maturity and has a pricequote of 98.45. The bond can be called in four years. The call premium is one year of couponpayments. Compute and discuss the bond’s current yield, yield to maturity, taxable equivalent yield(for an investor in the 35 percent marginal tax bracket), and yield to call. (Assume interest paymentsare semiannual and a par value of $5,000.) (LG7-6)

7-38 Yields of a Bond A 3.85 percent coupon municipal bond has 18 years left to maturity and has a pricequote of 103.20. The bond can be called in eight years. The call premium is one year of couponpayments. Compute and discuss the bond’s current yield, yield to maturity, taxable equivalent yield(for an investor in the 35 percent marginal tax bracket), and yield to call. (Assume interest paymentsare semiannual and a par value of $5,000.) (LG7-6)

7-39 Bond Ratings and Prices A corporate bond with a 6.5 percent coupon has 15 years left to maturity. It

has had a credit rating of BBB and a yield to maturity of 7.2 percent. The firm has recently gotten intosome trouble and the rating agency is downgrading the bonds to BB. The new appropriate discountrate will be 8.5 percent. What will be the change in the bond’s price in dollars and percentage terms?(Assume interest payments are semiannual.) (LG7-7)

7-40 Bond Ratings and Prices A corporate bond with a 6.75 percent coupon has 10 years left to maturity.It has had a credit rating of BB and a yield to maturity of 8.2 percent. The firm has recently becomemore financially stable and the rating agency is upgrading the bonds to BBB. The new appropriatediscount rate will be 7.1 percent. What will be the change in the bond’s price in dollars and percentageterms? (Assume interest payments are semiannual.) (LG7-7)

7-41 Spreadsheet Problem Say that in June of this year, a company issued bonds that are scheduled tomature three years from now in June. The coupon rate is 5.75 percent and is semiannually. The bondissue was rated AAA.a. Build a spreadsheet that shows how much money the firm pays for each interest rate payment and

when those payments will occur if the bond issue sells 50,000 bonds.

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b. If the bond issue rating would have been BBB, then the coupon rate would have been 6.30 percent.Show the interest payments with this rating. Explain why bond ratings are important to firms issuingcapital debt.

c. Consider that interest rates in the economy increased in the first half of this year. If the firm wouldhave issued the bonds in January of this year, then the coupon rate would have only been 5.40percent. How much extra money per year is the firm paying because it issued the bonds in Juneinstead of January?

7-42 Spreadsheet Problem You have a portfolio of three bonds. The long bond will mature in 19 yearsand has a 5.5% coupon rate. The midterm bond matures in 9 years and has a 6.6% coupon rate. Theshort bond matures in only 2 years and has a 4% coupon rate.a. Construct a spreadsheet that shows the value of these three bonds and the portfolio when the

discount rate is 5%. The spreadsheet can look something like this:b. Illustrate what happens when the discount rate increases by 0.5%. What do you notice about the

changes in price between the three bonds?c. Show the bond prices when the discount rate decreases by 0.5% from the discount rate in part a.

What do you notice about the price change between parts b and c?

A B C D E

1 Now Change to

2 Interest rate = 5.00% 5.50%

3 Bonds Bond Price Now Price After Change Change in $ Change in %

4 Long bond

5 Midterm bond

6 Short bond

7 Total = $0.00 $0.00

NotesCHAPTER 71. In order to focus on the valuation concepts, we present these examples with the full six months until the bond’s next interest payment.

However, bonds can be sold anytime between interest payments. When this occurs, we simply add the interest accrued since the lastpayment to the price.

2. States have differing rules about whether they tax the income from a particular municipal bond—they will generally tax income frommunis issued out of state. Further, capital gains arising from municipal bond sales may be taxed, and the income from municipal bondsmust be added to overall income when determining the Alternative Minimum Tax consesquences.

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chapter eightValuing Stocks

©Benny Marty/Shutterstock

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usinesses need capital to start up operations, expand product lines and services, and serve newmarkets. In the last chapter, we discussed debt, which is one source of financial capital upon whichbusinesses can draw. Their other source of capital is called equity, or business ownership. Public

corporations share business ownership and raise money by issuing stocks to investors. When thecompany sells this form of equity ownership to raise money, it gives up some ownership—and thus somecontrol—over the business. Investors buy stock to receive the benefits of business ownership. Mostcitizens do not have the time or expertise to operate their own businesses. Buying stock allows them toparticipate in the profits of economic activities. Access to equity capital has allowed entrepreneurs like BillGates of Microsoft and Larry Page of Google to take their companies public so that their businesses canbecome large corporations. Both the company founders and the new owners (stock investors) haveamassed much wealth over the years under this arrangement. One very important reason that investorsare willing to buy company stock as an investment is that they know that they can sell the stockduring any trading day. Investors buy and sell stocks among themselves in stock markets. Well-functioning stock markets are critical to any capitalistic economy. In this chapter, we’ll discuss stockmarket operations and stock valuation.■

LEARNING GOALS

LG8-1 Understand the rights and returns that come with common stock ownership.LG8-2 Know how stock exchanges function.LG8-3 Track the wider stock market with stock indexes and differentiate among the kinds of information each index provides.LG8-4 Know the terminology of stock trading.LG8-5 Compute stock values using dividend discount and constant-growth models.LG8-6 Calculate the stock value of a variable-growth-rate company.LG8-7 Assess relative stock values using the P/E ratio model.

viewpointsbusiness APPLICATIONAs CEO of your firm, Dawa Tech, which makes computer components, you have been able to grow its dividends by 8 percent peryear to a recent $2 per share. You expect this growth to continue. As a result, the stock price has risen to $65 and has a P/E ratio of16.25.

Tomorrow, you are scheduled to meet with some stockholders and financial analysts. To prepare for the meeting, you should knowwhat return the shareholders seem to expect and estimate where the Dawa stock price may be in three years. How will you go aboutpreparing for this meeting? (See the solution at the end of the book.)

8.1 • COMMON STOCK LG8-1Equity securities (stocks) represent ownership shares in a corporation. Common stock offers buyers the potential forcurrent income from dividends and capital appreciation from any stock price increases. Over time, some corporateprofits are reinvested in the firm, which increases the value of each shareholder’s stake in the business. At any pointin time, the market value of a firm’s common stock depends on many factors, including

common stock An ownership stake in a corporation.

The company’s profitability.Growth prospects for the future.

Current market interest rates.Conditions in the overall stock market.

Over periods of 30 to 40 years, stocks have offered investors the best opportunities to increase wealth. Since stocks

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are also susceptible to price declines and stock price fluctuations can be very volatile over short periods of time,stock investing requires a longer-term outlook.

Virtually any business firm that is organized as a corporation (see Chapter 1) may choose to issue publicly tradedstock. Common stockholders vote to elect the board of directors; they also vote on various other proposals requestedby other shareholders or the management team. As owners of the firm, common stockholders are considered to beresidual claimants. This means that common stockholders have the right to claim any cash flows or value after allother claimants have received what they are owed. As a company earns cash flows, it must pay suppliers,employees, expenses, taxes, and debt interest payments. Stockholders claim the leftover (or residual) cash flow.These profits can be used to reinvest in the firm to foster growth, pay dividends to shareholders, or a combination ofthe two.

residual claimants Ownership of cash flows and value after other claimants are paid.

Stocks of growing firms are valuable. Stocks in firms that pay dividends to shareholders are also valuable. Stocksissued by firms that have greater amounts of residual cash flow are more valuable. The value is reflected in the stockprice. Therefore, stock price values arise from the company’s underlying business success. Many different investorsand analysts may estimate a stock’s fundamental value based upon some outlook or theory. But the actualstock price is determined on stock exchanges when investors seek to trade with one another. Let’s discussthis trading process and then explore how stocks are valued.

personal APPLICATIONYou are impressed with the news and entertainment firm CBC Newscorp. The per-share dividends have increased from $1.25 peryear three years ago to the recent $1.68 annual dividend. Then you discover that 15 analysts are following the firm and that theirmean growth estimate for the future is 10.1 percent. Now you want to know if the current selling price of $54 seems like a good deal ifthe appropriate required return for the stock is 13.5 percent. (See the solution at the end of the book.)

Who are these “analysts,” and where can you find their opinions?

8.2 • STOCK MARKETS LG8-2In general, people will invest significant amounts of their wealth in stocks only if they know that they can converttheir shares into cash at any time. Stock exchanges provide this liquidity, allowing buyers and sellers the means totransact stock trades with each other. This liquidity gives many people the confidence to invest in the first place andmakes stocks (as well as bonds) attractive investments relative to less-liquid assets like real estate or fine collectibles—which can be difficult to sell quickly at full value.

The most well-known stock exchange in the world is the New York Stock Exchange (NYSE). The New York StockExchange, located in New York City on the corner of Wall Street and Broad Street, is the largest U.S. stockexchange as measured by the value of companies listed and the dollar value of trading activity. The NYSE is thelargest equities marketplace in the world and is home to approximately 2,800 companies (many with multiplesecurities listed). While other exchanges may boast more companies listed, the largest companies in the world tendto list in New York. The holding company that owns the NYSE, Intercontinental Exchange, also operates Euronext,a Europe-based electronic exchange market. For decades, the American Stock Exchange, located just down thestreet, competed with the NYSE. However, in 2008, the NYSE acquired this exchange. Now, smaller companiestrade at this location, which is referred to as AMEX.

New York Stock Exchange (NYSE) Large and prestigious stock exchange with a trading floor.

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The NYSE trades millions upon millions of stock shares in a given day.©Tetra Images/Getty Images

Much of the stock buying and selling at the NYSE occurs at 17 stations, called trading posts, on the trading floor.Each post is staffed by a designated market maker (formerly known as specialists), who oversees the orderly tradingof the specific stocks assigned to that post. Brokers, located around the perimeter of the floor, act as agents for thosebuying and selling stocks. Brokers execute orders by matching buy and sell orders. Once the buy and sell ordersmatch, the transaction is completed and the trade appears on trading screens viewed by people all over the world.

trading posts Trading location on the floor of a stock exchange.

brokers Floor traders who execute orders for others and themselves.

Consider this scenario. You decide to buy shares of McDonald’s stock because of new menu items and otherinitiatives. You place a buy order for 100 shares with your broker—either with a simple phone call or through anonline brokerage service. The broker then sends the order to the NYSE electronically to the trading post assigned forMcDonald’s stock. At the trading post, the specialist makes sure the transaction is executed in a fair and orderlymanner. Your buy order competes with other orders at the point of sale for the best price and an on-floor brokerexecutes your purchase. You will receive a trade confirmation from your broker describing the trade and noting theexact amount you owe for the 100 shares of McDonald’s plus any applicable commissions. The NYSE reports thetransaction and it appears within seconds on displays across the country and around the world. Note that buy and sellorders are electronically routed from all over the world to the NYSE, which then routes trade results back. Sincemost of the trade orders are already in electronic form, why not electronically match buy and sell orders and bypassany human intervention in floor trading? Indeed, the NYSE has joined many other exchanges in becomingincreasingly electronic. Some floor market-maker firms can see a time when no human intervention will be a part offloor trading at the NYSE.

The NYSE will trade millions of McDonald’s stock shares in a given day. A stock quote for McDonald’s stock,ticker symbol MCD, is shown in Figure 8.1. On February 10, 2016, more than 5.7 million McDonald’s shares traded.The stock closed at $117.54 per share, which was $0.53 higher than the closing price of the previous day. At thisprice, McDonald’s stock is currently closer to its 52-week high of $124.83 than to its 52-week low of $87.50.

ticker symbol Unique code for a company consisting of one to five letters.

FIGURE 8-1 Read a Stock Quote, February 10, 2016, Yahoo! Finance

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If you know what to look for, reading a stock quote is not as complicated as it first may appear.Source: Yahoo! Finance.

To list its stock on the NYSE, a company must meet minimum requirements for itsTotal number of stockholders.

Level of trading volume.Corporate earnings.

Firm size.

The exchange also charges an initial list fee and an annual fee. Listing standards and fees are higher for the NYSEthan for other stock exchanges, so many firms cannot (or choose not to) list their stocks there.

Another popular stock trading system is the NASDAQ Stock Market, an electronic stock market without a physicaltrading floor. Today, NASDAQ features many of the big-name high-tech companies investors have come to know,like Apple Computer (ticker: AAPL), Intel (ticker: INTC), Microsoft (ticker: MSFT), and Qualcomm (ticker:QCOM). Many newer high-tech companies like Google (ticker: GOOG), Netflix (ticker: NFLX), and AdobeSystems Inc. (ticker: ADBE), are also listed on NASDAQ. In 2007, NASDAQ purchased OMX, which owned sevenNordic and Baltic stock exchanges, and became NASDAQ OMX. NASDAQ ranks second, behind the NYSE,among the world’s equity markets in terms of total dollar volume. NASDAQ lists approximately 3,100 domestic andforeign companies.

NASDAQ Stock Market Large electronic stock exchange.

Instead of having a trading floor, NASDAQ uses a vast electronic trading system that executes trades via computerrather than in person. Instead of one designated market maker overseeing the process for an individual stock on atrading floor, Nasdaq’s system uses multiple market makers, or dealers. Market makers use their own stock inventoryand capital to compete with other dealers to buy and sell the stocks they represent. When an investor places an orderthrough a stockbroker for a NASDAQ-listed stock, the electronic system routes the order and the investor buysshares from the dealer offering the best (lowest) price. Typical NASDAQ stocks support 10 market makers activelycompeting with one another for investor trades.

market makers Dealers and specialists who oversee an orderly trading process.

dealers NASDAQ market makers who use their own capital to trade with investors

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time out!8-1 What are three primary stock exchanges in the United States and where are they located? For which of the exchanges does

physical location matter? Why?8-2 Describe differences in trading procedures on the NYSE versus the NASDAQ. Which do you think is most fair to investors?

Why?

Table 8.1 shows trading activity on the three main stock exchanges for one day in February 2016. Note that all threeexchanges traded more than 1 billion shares during the day.

The business of providing platforms or forums for investors and speculators to trade stocks and other financial assetshas been changing rapidly. Many exchanges that previously used physical floor trading systems with specialists andopen outcry to establish stock prices are shifting to electronic systems with no trading floors. Long-standing,traditional exchanges are also merging with other domestic and international exchanges to create fewer, but larger,forums that focus not just on U.S. securities but on many more internationally focused financial assets. The widerrange of represented securities allows traders new opportunities to explore trading relationships among securitiestraded across the world. This worldwide trading will establish economically sound prices and additional financialstability around the world.

▼ TABLE 8.1 Trading on the NYSE, NASDAQ, and AMEX, February 10, 2016

NYSE AMEX NASDAQ

Advancing issues 1,624 (52%) 744 (53%) 1,269 (48%)

Declining issues 1,426 (46%) 607 (44%) 1,298 (49%)

Unchanged issues 78 (2%) 42 (3%) 94 (4%)

New highs 44 20 6

New lows 156 67 201

Total volume 4,428,773,681 1,042,526,864 2,388,659,663

Source: Yahoo! Finance.

Tracking the Stock Market LG8-3With thousands of stocks trading every minute, many stock prices rise while others fall. Table 8.1 also shows that,throughout the trading day, 1,624 stocks increased in price on the NYSE while 1,426 stocks decreased in price.While the AMEX also experienced more stocks with increases in price than declines, the NASDAQ saw moredeclines. In addition to the number of stocks advancing and declining, the table also shows the number of stocks thathit new 52-week price highs (44 listed on the NYSE) and new lows (156 on the NYSE) on that day. So, was this agood day or a bad day in the stock market?

To say anything about the general direction of the stock market, stock indexes are useful. Dozens of stock indexes aredesigned to track the overall market; many more track different market segments. The three most recognized indexesare the Dow Jones Industrial Average (DJIA), the Standard & Poor’s 500 Index (S&P 500), and the NASDAQ Composite Index.

stock index Index of market prices of a particular group of stocks. The index is used to measure those stocks’ performance.

Dow Jones Industrial Average (DJIA) A popular index of 30 large, industry-leading firms.

Standard & Poor’s 500 Index (S&P 500) A stock index of 500 large companies.

NASDAQ Composite Index A technology-firm weighted index of stocks listed on the NASDAQ Stock Exchange.

Charles H. Dow invented the first stock average in 1884. At the turn of the 20th century, railroads were the firstmajor corporations. So he began with 11 stocks, mostly railroads. Dow created a price average by simply adding up11 stock prices and dividing by the number 11. Two years later, Dow began tracking a 12-stock industrial average.This industrial average would eventually evolve into the modern DJIA, which is a price average of 30 large,industry-leading stocks that together represent roughly 30 percent of the total stock value of all U.S. equities. DJIAlevel changes describe how the largest companies that participate in the stock market performed over a given period.The DJIA was at 15,914.74, a change of −99.64 (or −0.62 percent), on the day illustrated in Table 8.1.

The Standard & Poor’s Corp. introduced its 500-stock index in 1957. Standard & Poor’s chooses companies toinclude in the S&P 500 Index to represent the 10 sectors of the economy:

1. Financial

2. Information technology3. Health care

4. Industrials5. Consumer discretionary

6. Consumer staples7. Energy

8. Telecom services9. Utilities

10. Materials

S&P uses market capitalization (a measure of company size using stock price times shares outstanding), not just stockprices, of the largest 500 U.S. firms to compute the index. These 500 firms represent roughly 80 percent of theoverall stock market capitalization (number of shares times share price). Although the DJIA is a long-time favoritewith the media and individual investors, the S&P 500 is much preferred in the investment industry because of itsbroader representation of the market as a whole. S&P 500 performance provides a standard against which most U.S.money managers and pension plan sponsors can compare their investment performance. During trading on February10, 2016, the S&P 500 lost 0.35 (-0.02 percent) to close at 1,851.86.

market capitalization The size of the firm measured as the current stock price multiplied by the number of shares outstanding.

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Investors consider the NASDAQ to be a reflection of the tech sector’s general performance.©TongRo Image Stock/Alamy

The NASDAQ Composite Index measures the market capitalization of all common stocks listed on the NASDAQstock exchange. Since the NASDAQ lists so many large, technology-oriented companies, many investors andanalysts consider this index to reflect the tech sector performance more than that of the overall stockmarket. The NASDAQ Composite gained 14.83 to 4,283.59 on February 10, 2016, a gain of 0.35 percent.It was a bit of an unusual trading day because the DJIA was down, the S&P 500 was flat, and the NASDAQ was up.

Figure 8.2 shows the levels of all these stock indexes since 1980. The DJIA (maroon line) level appears on the left-hand axis. Both the S&P 500 (green line) and the NASDAQ Composite (orange line) run from the right-hand axis.The rapid price appreciation for NASDAQ stocks during the late 1990s—the tech boom years—is unprecedented forsuch a large and widely followed market index. The NASDAQ Composite soared from 817 in March 1995 to peakon March 10, 2000, at 5,048.62, for a 518 percent total return in only five years—a 43.9 percent annual rate ofshare-price appreciation for NASDAQ stocks. The NASDAQ index performed much better than did the DJIA (19.0percent per year) or the S&P 500 (22.7 percent per year). The NASDAQ “price bubble” set the stage for one of themost dramatic stock price declines in history: The NASDAQ Composite Index plunged to 1,114.11 on October 9,2002, losing 78 percent of its value. The other index values also fell during this period, albeit not as sharply. Notethat the DJIA didn’t climb back to its 2000 high until March 2006. The S&P 500 Index finally recovered in May2007. The NASDAQ Composite did not exceed its 2000 high until April 22, 2015. The stock market has been veryvolatile during the past two decades.

time out!8-3 Discuss why the day’s market return may be different when measured by the DJIA, S&P 500 Index, and NASDAQ

Composite taken separately.8-4 Why might the “market bubble” phenomenon appear more dramatic because it occurred in the NASDAQ Composite rather

than by the DJIA or S&P 500 Index?

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8-5 If you followed the market regularly, to which index would you give the most credence? Why?

The figure also shows the stock market reaction to the financial crisis that began in 2008. There were sharp declinesin all three indexes. After closing at a new high of 14,093.08 on October 12, 2007, the DJIA then fell to close at only6,547.05 on March 9, 2009. The DJIA fully recovered in early 2013 and closed above 15,000 for the first time onMay 7, 2013, and above 18,000 on February 13, 2015. This massive stock market rise coincides with thequantitative easing programs implemented by the U.S. Federal Reserve.

FIGURE 8-2 Stock Market Index Levels since 1980

This graph comparing the DJIA, S&P 500, and NASDAQ Composite indexes gives you a picture of their magnitude, as well as their patterns.

Trading Stocks LG8-4People who wish to buy and sell stocks need to open stock brokerage accounts. Traditional, full-service stockbrokers(e.g., Morgan Stanley Smith Barney, Merrill Lynch, UBS, Edward Jones) provide clients with research and advicein addition to executing trades. Their clients pay for this research and advice: Commission fees for these servicesmay run well over $100 per trade. Discount brokerage firms (e.g., Charles Schwab, E-trade, Scottrade, TDAmeritrade) charge a much lower commission, $5 to $30 per trade, but do not provide the additional services.Investors at discount brokerages usually place trades through the brokerage’s Internet sites.

finance at work //: behavioralInvestor PsychologyTo us today, it may seem ludicrous that in the years 1634 to 1636, the people of Holland were in the midst of “Tulip Mania,” and the pricefor a single rare tulip bulb approached the equivalent of $35,000. Then the bubble burst and tulip prices quickly plunged to less than theequivalent of $1. We may call those people who invested in a $35,000 tulip bulb irrational or even “crazy.”

But this type of story seems to repeat itself throughout history. Investors paid extremely high prices for the new computer stocks inthe 1960s, the “nifty fifty” companies in the 1970s, Japanese stocks during the 1980s, and Internet stocks during the late 1990s. Themania for stocks like Iomega drove its price from an equivalent of $1 per share in January of 1995 to over $75 in just 16 months. Whenthe bubble burst, the price fell hard. Many years later, in 2008, the company was purchased by EMC Corporation for dollars per share.

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But that’s just one company. A portfolio full of Internet stocks experienced a similar price mania followed by a severe fall. Investorscreated a stock index (TheStreet.com Internet Index) designed to track Internet stocks in late 1998, but by that time, part-time investorsand veterans alike were already well into the craze. The Internet index started at 250, quickly rose to 1,270 by March of 2000, andsubsequently fell to a low of 63 in October of 2002. Of course, the tech stock bubble was followed by the real estate bubble. Inretrospect, irrational bubble-like prices are not confined to tulips and the 17th century in Holland.

Seemingly irrational behavior may not occur only during highly emotional periods of a price bubble. Recently, a growing recognitionhas arisen that “normal” investors often behave in a way that might not be described as fully rational. Investors, being human, aresubject to cognitive biases and emotions. Studies of investor behavior have discovered that investors commonly succumb topsychological biases and

Source: USDA Natural Resources Conservation Service

Trade too much.

Sell winners too soon.Refuse to realize losses.

Become overconfident—especially when trading online.Seek stocks that have already increased in price—perhaps up to their full potential price.

Consider and react to what’s happening with each stock in isolation, rather than remembering the purpose for forming an overallportfolio.

Investors who succeed in the long run are those who learn to avoid these psychological biases.

Want to know more?Key Words to Search for Updates: irrational exuberance, price bubble, mania

Buy and sell orders go through the brokerage firm to a market maker (a dealer or a specialist) at a stock exchange.The quoted bid is the highest price at which the market maker offers to pay for the stock. Investors have little choicebut to accept this selling price, because regardless of the broker used, the market maker offers the only place to sellthe stock. The quoted ask price is the lowest price at which a market maker will sell a stock—so investorsbuy at the ask price. The difference between the bid and the ask price may be only $0.01 for high-volumestocks and can be as high as $0.20 for less-often traded companies. The spread between the bid and the ask price is acost to the investor and a profit for the market maker. This profit compensates the market maker for providing amarket and liquidity for that stock.

bid The quoted price investors are likely to receive when they sell stock.

ask The quoted price investors are likely to pay when they buy stock.

Investors can place a buy or sell market order. A market order to buy stock will be filled immediately at the currentask price when routed to the stock exchange. A sell market order will be filled at the current bid price. Theadvantage of a market order is that it executes immediately at the best available price. The disadvantage of a marketorder is that the investor does not know in advance what that fill price will be. Investors can name their own pricesby using limit orders, in which investors specify the price at which they are willing to execute the buy or sell order.With a buy limit order, a trade is executed if the ask quote is at or below the price target. For a sell limit order, a

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trade is executed if the bid quote moves through the specified price. If the current quote does not meet the price citedin the limit order, the trade is not executed. The advantage of a limit order is that the investor makes the trade at thedesired price; the disadvantage is that the trade might not be executed at all.

market order A stock buy or sell order to be immediately executed at the current market price.

limit order A stock buy or sell order at a specific price. It will only be executed if the market price meets the specified price.

Consider a quote of McDonald’s stock with a bid price of $117.00 and an ask price of $117.05. An investor placinga market buy order would purchase the stock at $117.05. A market sell order would execute as the price risesthrough $117.00. Note that an investor who simultaneously bought and sold 100 shares would pay $11,705 andreceive $11,700—losing $5. An investor who places a buy limit order at $116.75 will only purchase the shares if theask price falls to $117.75 or lower. If the ask does not fall, the order will not execute. Bid and ask prices tellinvestors at what prices the stock can currently be traded in general. But being able to buy at the ask price does notguarantee that the stock should be valued at that price. We’ll discuss various ways to arrive at reasonable per-sharestock values in the next section.

time out!8-6 Explain how the difference in the bid and ask prices might be considered a hidden cost to the investor.8-7 The bid and ask prices for Amazon.com are $37.79 and $37.85. If these quotes occur when a trade order is made, at what

price would a market buy order execute? Would a limit sell order execute with a target price of $37.75?

8.3 • BASIC STOCK VALUATION LG8-5Cash FlowsIn the previous chapter, we showed how we value bonds by finding the present value of the future interest paymentsand the future par value. Stock valuation uses the same concept of finding the present value of future dividends andthe future selling price. But of course uncertainty about both price appreciation and future dividend payment streamscomplicate stock valuation. Consider the simple case of valuing a stock to be held for one year shown in the timeline.

The value of such a stock today, P0, is the present value of the dividend to be received in the first year, D1, plus thepresent value of the expected sales price in one year, P1. The interest rate used to discount the cash flows is shownas i. Using the present value equation from Chapter 4 results in

(8-1)

Whenever investors deal with future stock prices and future dividend payments, they must use expected values, notcertain ones. Companies rarely decrease their dividends; most companies’ dividends either remain constantor slowly grow. Examining a firm’s dividend history over the past few years will give clues to thatcompany’s future dividend policy. For example, The Coca-Cola Company (ticker: KO) paid a $0.155 per sharedividend for each quarter in 2006. The firm then raised the quarterly dividend to $0.17 for each quarterly dividend in2007. The company paid quarterly dividends in 2008, 2009, 2010, 2011, and 2012 of $0.19, $0.205, $0.22, $0.235,and $0.255, respectively. For 2013 through 2015, Coca-Cola raised its quarterly dividend 2.5 cents every year. So

the quarterly dividend was $0.33 in 2015, for an annual dividend of $1.32. This dividend growth seems fairly stableand predictable.

Coca-Cola increased its dividend by 3 cents per year over a five-year span.©McGraw-Hill Education/Jill Braaten, photographer

Stock prices, though, show much more volatility than dividend histories do. We face much uncertainty in trying topredict stock prices in the short term. Using a longer holding period to estimate stock value reduces some, but by nomeans all, of the uncertainty. A 2-year holding period appears like this:

The present value of the cash flows in years 1 and 2 is today’s stock value:

(8-2)

Notice that the divisor for the second term on the right-hand side of equation 8-2 is raised to the second power. Thisreflects the two years over which those cash flows must be discounted. You can do this analysis over any holdingperiod. For a holding period of n years, the value of a stock is measured by the present value of dividends over the n

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years, and the eventual sale price, Pn.

(8-3)

This formula incorporates both dividend income and capital appreciation or capital loss. It fully includes both majorcomponents of the investor’s total return from investment.

McDonald’s pays an annual dividend instead of quarterly dividends.©McGraw-Hill Education/John Flournoy, photographer

As is often the case in finance, implementing equation 8-3 presents problems for some firms in practical terms.What will the future dividends of the firm be? What will the stock price be in 3, 5, or 10 years? While it seems thatthe dividend growth of Coca-Cola will be constant, consider the actual dividends and stock price of McDonald’sCorp. since 2000 shown in Figure 8.3.

McDonald’s paid an annual dividend from 2000 to 2007. Some increases were small, like the $0.01 increase from2000 to 2001 and again to 2002. Other increases were quite large, like the $0.50 increase between 2006 and 2007.Then McDonald’s changed to the more common quarterly dividend in 2008. Since the change to quarterlypayments, McDonald’s dividend growth has been more stable and predictable through 2015. The figure also showsthat McDonald’s stock price has been very volatile. The price fell from a split adjusted $25 in 2000 to$9.50 in 2003 and then steadily climbed to $45 in 2007. The stock went sideways during the financialcrisis and then shot up to $88 in late 2011. The stock again went mostly sideways from 2012 to 2014 and then shotup again in 2015. An investor in 2000 would have had a very difficult time accurately forecasting these futuredividends and stock prices. Indeed, short-term stock price changes seem almost random. Stock valuation can reallyonly be viewed from a long-term perspective. Because predicting future dividends is uncertain at best, it’s better toproject valuation as a likely range of prices under reasonable assumptions rather than as a single price. After all, thiscomputed price is an estimate of the firm’s intrinsic value. This intrinsic value may differ from the stock pricetrading in the market. This possibility is discussed as a market efficiency topic in Chapter 10.

EXAMPLE8-1 Valuing Coca-Cola Stock LG8-5

For interactive

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▼

versions of thisexample, log in

to Connect or go tomhhe.com/CornettM4e.

In February 2016, you are valuing Coca-Cola stock to compare its value to itsmarket price. The current market price is $43.01. Given the history of Coca-Cola’s dividends, you believe that the company will pay total dividends in2016 of $1.42 (= 4 ×$0.355). Your analysis indicates that the total dividends in2017 and 2018 will be $1.54 and $1.66, respectively. In addition, you believethat the price of Coca-Cola stock at the end of 2015 will be $54.10 per share.If the appropriate discount rate is 11.0 percent, what is the value of Coca-Colastock?

SOLUTION:

To organize your data, you first create the following timeline:

Using equation 8-3, you compute the stock value as

Since your analysis shows that Coca-Cola’s stock should be valued at $43.30while it’s selling for only $43.01, the stock appears to be slightly undervalued.You believe that this might be a good time to buy some Coca-Cola stock.

Similar to Problems 8-15, 8-16, 8-27, 8-28, Self-Test Problem 1

Dividend Discount ModelsWe can extend the discounted cash flow approach in equation 8-3 for an infinite stream of dividends, n→∞, and nofinal future selling price. If stockholders receive all future cash flows as future dividends, the stock’s value to theinvestor is the present value of all these future dividends. In other words, embedded in any stock price is the value ofall future dividends. We can demonstrate this value as

(8-4)

This equation shows the general case of the dividend discount model. The dividend discount model providesa useful theoretical basis because it illustrates the importance of dividends as a fundamental stock pricedeterminant.

dividend discount model A valuation approach based on future dividend income.

FIGURE 8-3 Dividends and Stock Price of McDonald’s since 2000

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Dividends rarely decline, but stock prices often do!

But, again, finance professionals find it difficult to apply the dividend discount model because it requires that theyestimate an infinite number of future dividends. To use the model in practice, analysts make simplifyingassumptions to make the model workable. One common assumption: The firm has a constant dividend growth rate,g. If this is the case, next year’s dividend is simply this year’s dividend that grew one year at the growth rate, that is,D1 = D0 × (1 + g). In fact, we can express each dividend as a function of D0 and we can rewrite equation 8-4 as

(8-5)

So, with this version of the model, we need not forecast an infinite string of dividends; D0 and g take care of that.However, we must still compute an infinite sum of numbers. Luckily, mathematicians know equations like these;they are known as power series. This power series can be simplified to the constant-growth model, and it assumes thatthe growth rate is smaller than the discount rate (i.e., for g < i):

constant-growth model A valuation method based on constantly growing dividends.

Stock value = Next year’s dividend ÷ (Discount rate − Growth rate)

(8-6)

If g ≥ i, then the denominator would be zero or negative. Economically and mathematically, this is a nonsensicalresult. In the short run, a firm can grow very quickly. In the long run, no company can grow faster than the overalleconomic growth rate forever. You may hear the constant-growth model referred to as the Gordon growth model,after financial economist Myron J. Gordon.

EXAMPLE Constant Growth and Coca-Cola

8-2 Stock LG8-3

For interactiveversions of this

example, log in toConnect or go tomhhe.com/CornettM4e.

Assume that you are valuing Coca-Cola stock again. This time you are usingthe constant growth model, assuming a discount rate of 11.0 percent.

SOLUTION:

You have a choice of three growth rates to use. The implied projecteddividend growth rate is 7.94 percent. Past dividend growth has been 7.63percent and analysts forecast an 8.95 percent growth. Compute the stockvalue using all three growth rates.

Using a dividend growth rate of 7.94 percent and equation 8-6, the stockvalue is $47.27:

Using a dividend growth rate of 8.95 percent, the stock value is $71.22:

Using a dividend growth rate of 7.63 percent, the stock value is $42.80:

Notice how a small change in the growth rate has a large impact on the stockvalue in this model. At a current price of $43.01 per share, Coca-Cola couldbe considered undervalued or overvalued depending on the growth rate used.

Similar to Problems 8-19, 8-20, 8-29, 8-30, 8-31, 8-32, Self-Test Problem 2

Investors use several methods to estimate a firm’s growth rate for this model. They can project the dividend trendinto the future and determine the implied growth rate, compute the past growth rate, or even consider a financialanalyst’s growth rate predictions. Consider Coca-Cola’s dividend behavior. If the 2015 dividend was $1.34 and theprojected dividends will grow to $1.66 in 2018, the implied projected dividend growth rate is therefore 7.94 percent(N = 3, PV = −1.34, PMT = 0, FV = 1.66, CPT I = 7.94) annually. The growth rate in dividend changes from 2008to 2012 was 7.63 percent (N = 4, PV = −0.76, PMT = 0, FV = 1.02, CPT I = 7.63) per year. You can find analystforecasts many places online. The Yahoo! Finance web page for Coca-Cola has an Analyst Estimates link, whichshows the average analysts’ forecast for the firm’s growth in the next five years at 8.95 percent.

Preferred StockA special case of the constant-growth model occurs when the dividend does not grow but is the same every year.This zero-growth rate case describes a preferred stock. The term preferred comes from the fact that this type of stocktakes preference over common stock in bankruptcy proceedings. Preferred stockholders have a higher priority forreceiving proceeds from bankruptcy proceedings than do common stockholders. Preferred stock is largely owned byother companies, rather than by individual investors, because its dividends are mostly nontaxable income (70percent of the income is exempt from taxes) to other corporations. Preferred stockholders do not have voting rights

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like common stockholders, though, which prevents one company from controlling another through preferred stockownership.

preferred stock A hybrid security that has characteristics of both long-term debt and common stock.

An interesting characteristic of preferred stock is that it pays a constant dividend. Because the dividend does notchange, the preferred stock can be valued using the constant-growth-rate model with a zero growth rate expressed asP = D/i. What would Coca-Cola’s stock be worth if its dividend stayed at $1.34 and never grew? Using thesame 11.0 percent discount rate, the stock would be valued at $12.18 (= $1.34 ÷ 0.110). Given Coca-Cola’s current stock price of $43.01, over 71.6 percent [= ($43.01 − $12.18)/$43.01] of its stock value comes fromthe expectation that Coca-Cola’s dividend will grow. In other words, investors highly value a growing firm.

Most companies issue only common stock, but nearly 1,000 preferred stock issues still exist. Table 8.2 compares thecommon stock and preferred stock for 10 firms. Many of the preferred stocks come from the finance, energy, andreal estate sectors. Notice that the dividend yield for preferred stock is higher than for the common stock, becausepreferred stock investors should expect a return from dividend payments only. Common stockholders will alsoexpect a return from capital appreciation over time. Common stocks also trade much more frequently than preferredstocks do.

dividend yield Last four quarters of dividend income expressed as a percentage of the current stock price.

the Math Coach on…Using the Constant-Growth-Rate Model

“The distinction between the recent year’s dividends, D0, and next year’sdividends, D1, can be confusing in the constant growth-rate model. The model’sequation presents two different numerators. If you are given information aboutdividends last year or just paid, use the D0(1 + g) version of the equation. If youhave information about expected dividends or next year’s dividend, use the D1version of the equation.„

The zero-growth-rate version of the constant-growth valuation model shows that, since dividends are fixed, apreferred stock’s price changes because of changes in the discount rate, i. When interest rates throughout theeconomy change, the discount rate also changes. Preferred stock prices thus tend to act like bond prices. Wheninterest rates rise, preferred stock prices fall. When interest rates decline, preferred stock prices rise. Preferred stockis usually categorized with bonds in the fixed-income security group because it acts so much like debt securities,even though a preferred stock represents equity ownership, like common stock.

Expected ReturnStock valuation models require a discount rate, i, in order to compute the present value of the future cash flows. Thediscount rate used should reflect the investment risk level. Higher risk investments should be evaluated using higherinterest rates. For example, the previous chapter on bonds demonstrated that higher risk bonds, such as junk bonds,offer higher rates of return. Similarly, investors demand higher returns from higher risk stocks than they do fromlower risk stocks. We discuss stock risk measurement and appropriate expected returns in the next section of thisbook.

▼ TABLE 8.2 Common and Preferred Stock, February 12, 2016

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COMMON STOCK PREFERRED STOCK

Company Ticker Price

AnnualDividend(Yield%) Volume Ticker Price

AnnualDividend(Yield%) Volume

Alcoa Inc. AA $ 7.69 $0.12(1.2) 26,021,380 AAPRB $26.74 $2.69(8.1) 49,122

El Du Pont deNemours & Co.

DD $ 58.48 $1.72(2.6) 5,397,520 DDPRA $79.75 $3.50(4.5)

Ford Motor Co. F $ 11.55 $0.60(4.3) 28,080,850 FPRA $25.55 $1.88(7.3) 97,959

NationalHealthcare

NHC $ 61.29 $1.54(2.5) 23,484 NHCA $15.52 $0.80(5.1)

PG&E Corp. PCG $ 55.21 $1.82(3.4) 2,877,339 PCG.PRA $30.57 $0.75(2.6) 3,247

Public StorageInc.

PSA $233.61 $6.50(2.6) 1,040,665 PSAPRW $24.74 $1.30(5.7) 51,329

Source: New York Stock Exchange (www.nyse.com).

finance at work //: investmentsFinancial Analysts’ Predictions and Opinions

©bunhill/Getty Images

Financial analysts examine a firm’s business and financial success and assess long-term prospects and management effectiveness.They combine this microeconomic analysis with a macroeconomic view of the conditions of the economy, financial markets, and industryoutlooks. Their evaluation results in earnings predictions, stock price targets, and opinions about whether investors should buy the stock.Such recommendations can help investors decide whether to buy, hold, or sell the stock.

Analysts hired by brokerage firms and investment banks are called sell-side analysts because their firms make money by sellingstocks and bonds. These analysts publicize their predictions and opinions publicly and in company “tip sheets” that are passed along toclients. Keep in mind that sell-side analysts often have incentives to be optimistic. Pension funds and mutual funds often hire analysts togive fund managers private opinions about securities. These analysts are referred to as buy-side analysts because they are hired byinvestment firms looking for advice on what stocks to buy for their portfolios. Because this is private, little buy-side research is madepublic.

Consider the 26 sell-side analysts’ predictions reported for Coca-Cola on the Yahoo! Finance website. The average five-year shareprice growth prediction from these analysts is 8.95 percent. This is lower than the growth prediction for the industry (12.96 percent) andsector (12.79 percent). Last, analysts give opinions on whether investors should buy, sell, or hold Coca-Cola stock. Recommendationscome in five levels: Strong Buy, Buy, Hold, Underperform, and Sell. Of the 26 analysts, 4 recommend a Strong Buy, 9 recommend aBuy, and 10 recommend a Hold. Note that even though the analysts predict lower than average growth for the industry and sector, andthat the analysts provide meager price targets, only three of the analysts recommend an Underperform and none a Sell. This optimism inanalysts’ opinions is common. Knowledgeable investors know that a “hold” recommendation is as negative as most public or sell-side

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analysts get, and therefore a “hold” may actually represent a signal to sell.

Want to know more?Key Words to Search for Updates: analyst opinion, financial analyst bias

However, one method for determining what return stock investors require from a stock is to use the constant-growth-rate model. If the current stock price fairly reflects its value, then the discount rate, i, in equation 8-6 shouldbe the expected return for the stock. Solving for this expected return results in equation 8-7:

(8-7)

Note that the expected return comes again from two sources: dividend yield and expected appreciation of the stockprice, or capital gain. For example, consider that Coca-Cola’s dividend in 2016, D1, is expected to be $1.42 pershare. At a current price of $43.01, Coca-Cola offers a dividend yield of 3.30 percent (= $1.42 ÷ $43.01). Sinceanalysts believe that the firm’s stock price will grow at 8.95% percent in the future, investors expect a total return of12.25 percent (= 3.30% + 8.95%). Dividend yield can represent a substantial portion of the profits for an investor.Many people get too enamored of high growth stocks that do not pay dividends and therefore miss out on animportant source of stable returns.

growth stocks Companies expected to have above-average rates of growth in revenue, earnings, and/or dividends.

time out!8-8 Explain how valuable a firm’s (and therefore its stock’s) growth is. Demonstrate this with growth and no-growth examples.8-9 What proportion of the 12.25 percent of Coca-Cola’s expected return above comes from dividend yield?

Corporate managers conduct an important application of the expected return concept to determine the return thattheir shareholders expect of them. We will discuss this application in detail in Part Six: Capital Budgeting.

8.4 • ADDITIONAL VALUATION METHODS LG8-6Variable-Growth TechniquesSome companies grow at such a high rate that we cannot use the constant-growth-rate model to forecast their value.High growth rates might be sustainable for several years, but cannot continue forever. Consider what happens to ahigh-growth firm. Other companies will surely notice the market potential for high-growth rates and will enter thoseproduct markets to compete with the high-growth firm. The competition will soon drive down the growth rates forall companies in that product market. Companies that experience unusually high-growth tend to see that growthbecome only average in the future unless they possess some kind of entry barrier such as a patent or governmentregulation due to economies of scale.

Remember that the constant-growth-rate model does not work for companies where g > i. And of course, we do notreally expect the growth rate for these fast-growing firms to remain constant. To value these firms, we must use avariable-growth-rate technique. The variable-growth-rate method combines the present-value cash flow from equation8-3 and the constant-growth-rate model from equation 8-6.

variable growth rate A valuation technique used when a firm’s current growth rate is expected to change some time in the future.

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First, the investor chooses two different growth rates for two stages of the analysis. The first and higher growth rate,g1, is the current growth rate, which we expect to last only a few years. A few years from now, we expect the firm togrow at a slower but more sustainable rate of growth, g2. Figure 8.4 shows the cash flow time line when the firstgrowth rate applies for the first n years, followed by the second growth rate, which applies forever.

When we analyze a variable-growth-rate stock like the one in the figure, we know the recent dividend, D0, and thetwo expected growth rates. Therefore, we can calculate each of the dividends shown in general terms (i.e., D1). Forexample, the dividend in the first year (D1) is the year zero dividend that grows at g1, specifically D1 = D0 × (1 +g1). The dividend then grows at g1 again for the second year dividend, D2 = D0 × (1 + g1)

2. The dividend continuesto grow through the first stage to year n at Dn = D0 × (1 + g1)

n. Figure 8.5 shows the first-stage dividends.

At this point, the company starts to move into stage 2 at the more modest growth rate, g2, and the dividends reflectthat slower growth rate. So Dn+1 is the dividend Dn that grew at the rate g2, or Dn+1 = D0 × (1 + g1)

n × (1 + g2).Similarly, the dividend in year n + 2 is Dn+2 = D0 × (1 + g1)

n × (1 + g2)2. We can now substitute the known

dividends as presented in Figure 8.5 into Figure 8.6.

Once we have calculated all of the dividends, we can begin finding the value of the variable-growth stock byfocusing on Stage 2 of the problem. Assume that the dividends in Stage 2 are growing at a modest rate, g2, forever.As long as g2 < i, Stage 2 can use the constant-growth model, equation 8-6. Remember that the constant-growthmodel, P0 = D1/(i − g), replaces all future dividends with one value in the previous period. In previousapplications, the growth began in year 1, so the value used for all future dividends came from the year 0 dividend. Inthis case, the change in the dividend rate occurs in year n + 1, so we will use the value from year n. So, using theconstant-growth model, we can replace all the cash flows in Stage 2 with one value from year n, as

The cash flows from Figure 8.6 now appear as shown in Figure 8.7.

FIGURE 8-4 Variable Dividend Growth

Divide into two stages at the first year of the new growth rate.

FIGURE 8-5 Stage 1 Dividends

Calculate the dividends in the first stage.

▼

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FIGURE 8-6 Details of the Variable Growth Dividends

Compute the dividends in the second stage.

By replacing all the Stage 2 cash flows that continued indefinitely with one terminal price in year n, we reduce theproblem to a fixed number of cash flows. The value of this variable-growth stock is finally computed as the presentvalue of these cash flows, as solved with equation 8-3. Substituting the cash flows shown in Figure 8.7 into equation8-3 gives us the general formula for finding the value of a variable-growth stock:

General two-stage growth valuation model:

(8-8)

The practical application of the variable-growth valuation technique requires the investor to decide how long thecurrent high-growth rate will last before declining to a more stable rate.

FIGURE 8-7 New Stage 1 of the Variable Growth Dividend Technique

Replace all of the dividends to infinity with the price (terminal value) in year n. Stage 2 disappears.

The constant-growth-rate model is most useful for large, mature companies that grow in a stable manner. Thevariable-growth-rate model works well for dividend-paying companies that have an unusually fast rate of growth inthe near future but are expected to enter a more stable growth rate environment soon. But there are still many firmsthat do not fit these two descriptions of firm growth. For example, many firms pay no dividends. To value firms withno dividends, replace the dividends in the models with cash flows. When you do this, you are valuing the entirefirm, not just the stock value, because cash flows go to both stockholders and debt holders. In the case where a firmhas low or even negative cash flows, the valuation techniques in the next section can be employed.

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time out!8-10 Explain how the variable-growth-rate technique could be used for a firm whose dividend is not expected to grow for three

years and then will grow at 5 percent indefinitely.8-11 Set up and solve the McDonald’s valuation problem assuming that the first stage growth will last only two years.

EXAMPLE8-3

Variable Growth and StockValue LG8-6

For interactiveversions of this

example, log in toConnect or go tomhhe.com/CornettM4e.

The dividend has grown from $1.00 per share on November 13, 2009, to$2.80 during 2015. This represents an annual growth rate of 18.7 percent (N= 6, PV = −1.00, PMT = 0, FV = 2.80, CPT I = 18.7). You think this growthrate will continue for three years and then fall to the long-term growth rate of9.29 percent predicted by analysts. You assume a 13 percent discount rate.

SOLUTION:

Figure 8.3 shows a $2.80 (= 4 × $0.70) per share recent annual dividend.Modify equation 8-8 for a Stage 1 length of three years and then substitute i =0.13, g1 = 0.187, g2 = 0.0929, and D0 = $2.80. The valuation equation andsolution becomes

Given these parameters, the company’s stock is worth nearly $105 per share.

Similar to Problems 8-33, 8-34, Self-Test Problem 3

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The P/E Model LG8-7The valuation models that we’ve presented thus far help investors attempt to compute a stock’s fundamental valuebased upon its cash flows to the investor. Another common approach is to assess a stock’s relative value. Thisapproach compares one company’s stock valuation to other firms’ stock values to evaluate whether your targetcompany’s stock is appropriately priced. The price of a stock taken in isolation doesn’t give us a good measure ofhow expensive it is. Let’s use an analogy: At the grocery store, we are less concerned with the total price of a bag ofsugar than we are with the price per pound. Similarly, the price of the stock matters less than its price per one dollarof earnings.

relative value A stock’s priceyness measured relative to other stocks.

Consider one company that earned $5 per share in profits for the year. Its stock sells for $100. Another companyearned $2 per share and its stock price is $50 per share. At first glance, the first stock appears to be more expensivebecause its price is a high $100 compared to the lower $50 price of the second stock. However, the first companygenerated higher per-share profits than did the second company. Buying the first stock means that you purchase $5in earnings. The $100 stock price implies a cost of $20 for every $1 in earnings (= $100 ÷ $5) generated. The $50price of the second stock implies a cost of $25 for every $1 in earnings (= $50 ÷ $2). So in this regard, the secondcompany becomes more expensive. The price-earnings (P/E) ratio represents the most common valuation yardstick inthe investment industry; it allows investors to quickly compare the cost of earnings. The P/E ratio is simply thecurrent price of the stock divided by the last four quarters of earnings per share:

(8-9)

price-earnings (P/E) ratio Current stock price divided by four quarters of earnings per share.

More accurately, this figure is the trailing P/E ratio and it is often denoted as P/E0, where the 0 subscript denotes thepast (or trailing) earnings.

trailing P/E ratio The P/E ratio computed using the past four quarters of earnings per share.

Figure 8.8 shows two companies’ trailing P/E ratios: Coca-Cola and McDonald’s, as well as the Dow JonesIndustrial Average’s trailing P/E ratio over a 16-year period. The P/E ratio for the DJIA changes slowly and mostlystays in the 18 to 27 range until the recent drop to 15. Historically, the DJIA’s P/E ratio has fallen as low as singledigits and climbed to over 30. The figure also shows that the P/E ratio for McDonald’s has varied morethan has the index’s P/E ratio. The P/E ratio for Coca-Cola has experienced wild changes—as high as 63and as low as 18. Figure 8.8 shows that investors valued Coca-Cola more than McDonald’s in the late 1990s, butvalued them nearly the same by the beginning of 2008.

FIGURE 8-8 Historical P/E Ratio of the DJIA, Coca-Cola, and McDonald’s

P/E ratios of large, successful companies can vary considerably over time. Here, you see that investors valued Coca-Cola over McDonald’sfor much of the period, but there were periods when McDonald’s was valued higher.Source: Dow Jones and Company.

Variations in P/E ratios between popular companies can be quite large. For example, in February of 2016, the P/Eratio for Alphabet (the parent company of Google) was 28.7, while the ratio for Boeing Company was only 14.6.Alphabet stock is much more expensive than Boeing. But this does not mean that Boeing stock is a better deal thanAlphabet stock. Investors are willing to pay much more in relative terms for Alphabet because they expect Alphabetwill grow much faster than Boeing. Indeed, analysts predict an annual growth rate of 16.7 percent per year forAlphabet over the next five years, while they expect a growth rate of only 11.9 percent for Boeing. Remember thatExample 8-2 shows how small changes in growth can result in large stock value changes. The large difference inexpected growth between Alphabet and Boeing causes a large difference in their relative value.

We can more directly see the impact that growth can have on the P/E ratio by modifying the constant-growth model.Begin with the model, P0 = D1 ÷ (i − g). Dividing both sides by the firm’s earnings results in P0/E0 = (D1/E0) ÷ (i −g). Note that the dividend payout ratio of the firm (D/E), the discount rate (i), and the growth rate (g), taken together,determine the P/E ratio. All else held equal, larger growth rates will lead to larger P/E ratios. Also, firms that havehigher payout ratios will have higher P/E ratios. Of course, if a firm pays out a high portion of its earnings asdividends, then it may not have the cash to fund high growth. Thus, high dividend payout firms tend not to be pricedthe same as high growth firms.

The value of a stock, and thus its price, relates directly to its future success. Note that valuation models use estimatesof future dividends and growth rates. Because of this focus on the future, many people prefer to use a P/E ratio thatalso looks forward rather than trailing. A forward P/E ratio uses analyst estimates of the earnings in the next 12 monthsinstead of the past 12 months and can be denoted as P/E. The forward P/E ratio has the advantage over the trailingP/E ratio in that it incorporates investors’ expectations of the firm’s upcoming profits. A disadvantage is thatexpected earnings are harder to estimate and thus less accurate than past earnings. The media uses the trailing P/Eratio, while financial managers and investors use the forward P/E ratio more.

forward P/E ratio The P/E ratio computed using the estimated next four quarters of earnings per share.

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Investors consider Home Depot an appropriately priced stock.

Knowledgeable investors who use the P/E ratio as a relative measure of value compare it to the firm’s expectedgrowth rate. Table 8.3 shows the forward P/E ratio and analysts’ expected growth rates for the 30 Dow JonesIndustrial Average firms. Investors consider companies with high P/E ratios and high growth rates to beappropriately priced. Companies with low P/E ratios and low growth also seem to be appropriately priced. Investorsshould be concerned about firms with high P/E ratios and only single-digit growth rates. As such, Procter & Gamble,Coca-Cola, McDonald’s, and Chevron, among others, may be too expensive for their expected growth rates. Manyinvestors like to buy growth stocks. They seek companies with high growth rates. But growth stock investors arealso concerned about paying too much for a stock. While examining growth stocks, they can use the P/E ratio toassess how expensive the stock is. On the other hand, investors consider companies with low P/E ratios and highexpected growth to be undervalued, and they are often referred to as value stocks. Apple and Boeing wouldqualify as value stocks. Many investors like to buy value stocks because they feel they are getting abargain price for a stable company.

value stocks Companies considered to be temporarily undervalued.

▼ TABLE 8.3 P/E Ratios and Analyst Growth Estimates of DJIA Firms, February 12, 2016

A B C D E

1 Ticker Company Name Stock Price Forward P/E Ratio Next 5 Year’s Growth (%)

2 HD Home Depot $116.32 18.88 14.39

3 PG Procter & Gamble 80.99 20.00 6.95

4 KO Coca-Cola Company 43.11 20.83 8.95

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5 MCD McDonald’s Corporation 117.93 19.56 9.50

6 VZ Verizon Communication 50.11 12.31 4.51

7 DIS Walt Disney 91.15 14.68 11.85

8 MMM 3M Company 153.96 17.26 8.10

9 JNJ Johnson & Johnson 101.82 14.76 5.32

10 AAPL Apple, Inc. 93.99 9.39 11.93

11 UTX United Technologies 85.95 12.24 9.07

12 GE General Electric 28.26 15.96 7.97

13 WMT Walmart Stores 66.18 15.87 0.87

14 AXP American Express 52.66 9.37 8.10

15 PFE Pfizer 29.36 11.65 5.44

16 MRK Merck & Company 49.03 12.87 4.24

17 XOM Exxon Mobil 81.03 18.58 26.40

18 IBM International Business Machines 121.04 8.55 7.25

19 TRV The Travelers Company 107.49 10.79 2.94

20 DD E.I. du Pont de Nemours 58.40 16.88 9.00

21 BA Boeing Company 108.63 11.53 11.92

22 NKE Nike, Inc. 56.42 22.84 12.62

23 CSCO Cisco Systems 25.11 10.51 8.24

24 INTC Intel Corporation 28.64 10.77 10.00

25 CAT Caterpillar 63.15 17.21 0.43

26 CVX Chevron Corporation 85.43 18.18 −23.86

27 UNH UnitedHealth Group 111.82 12.74 14.03

28 MSFT Microsoft Corporation 50.50 16.45 9.51

29 GS Goldman Sachs Group 146.13 7.54 4.31

30 JPM JPMorgan Chase 57.49 8.59 7.89

31 V Visa, Inc. 70.42 21.67 16.62

Source: Yahoo! Finance Screener.

There are some cases when a P/E ratio is not useful for relative valuation. For example, sometimes, a firm will losemoney. That is, the earnings are negative. In other cases, a firm will take a large “write-off” that will temporarilysuppress earnings. In these cases, the P/E ratio would be negative or temporarily large. Therefore, other commonrelative value techniques are to utilize cash flow (CF) or book value (B) instead of earnings. The P/CF ratio is usefulwhen firms take accounting write-offs that temporarily and dramatically impact earnings. The P/B ratio is useful inall cases but is particularly useful when a firm loses money and has negative cash flows. The book value of a firm isa very stable measure of accounting value, and it is therefore useful when earnings are volatile.

EXAMPLE8-4

The P/E Ratio Model forCaterpillar LG8-7

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For interactiveversions of this

example, log in toConnect or go tomhhe.com/CornettM4e.

For interactive versions of this example, log in to Connect or go tomhhe.com/CornettM4e. Look at Table 8.3 and notice that the P/E ratio forCaterpillar seems high at 17.21 relative to the growth (0.43 percent) thatanalysts expect. Caterpillar earned $3.50 per share and paid a $3.08 dividendlast year. You decide to explore this apparent anomaly and figure out whatCaterpillar’s stock price might reach in five years.

SOLUTION:

Compute the expected future price in five years under two different scenarios.The first assumption is that Caterpillar’s P/E ratio will be the same in fiveyears as it is today. But since this P/E ratio seems a bit high, the secondscenario allows for a decline in the P/E ratio to 13. Under these two scenarios,the future price estimates are

Note that if the P/E ratio decreases from 17.21 to 13 in five years, the futureprice could be much lower than analysts expect without the change.

Similar to Problems 8-25, 8-26, 8-35, 8-36

time out!8-12 Consider two firms with the same P/E ratio. Explain how one could be described as expensive compared to the other.8-13 Compute the stock price for Goldman Sachs in five years if you expect the P/E ratio to decline to 6 and the earnings per

share is $18.58.

Estimating Future Stock PricesWe can often find it useful to estimate a stock’s future price. Consider equation 8-3’s cash flow discount valuationmodel. The model requires estimates of future dividends and a future price. How can investors estimate this futureprice? They can use the P/E ratio model for this purpose. Upon reflection, you will see that multiplying the P/E ratioby earnings results in a stock price. So, in order to estimate a future price, simply multiply the expected P/E ratio bythe expected earnings. This concept is captured in the following equation:

Future price = Future P / E ratio × Future earnings per share

(8-10)

As the formula shows, we can use assumptions about the earnings growth rate to estimate earnings in year n. Manyinvestors believe the firm’s P/E ratio in year n is best estimated using today’s P/E ratio. However, if today’s P/Eratio seems unusual compared with similar firms or even compared with a stock index, then adjustments might bewise.

Get Online

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Log in to your Connect course for study materials including self-test problems with solutions, answers tothe Time Out quizzes, guided example videos, and more.

Your Turn…Questions

1. As owners, what rights and advantages do shareholders obtain? (LG8-1)

2. Describe how being a residual claimant can be very valuable. (LG8-1)

3. Obtain a current quote of McDonald’s (MCD) from the Internet. Describe what has changed since the quote inFigure 8.1. (LG8-2)

4. Get the trading statistics for the three main U.S. stock exchanges. Compare the trading activity to that of Table8.1. (LG8-2)

5. Why might the Standard & Poor’s 500 Index be a better measure of stock market performance than the DowJones Industrial Average? Why is the DJIA more popular than the S&P 500? (LG8-3)

6. Explain how it is possible for the DJIA to increase one day while the NASDAQ Composite decreases duringthe same day. (LG8-3)

7. Which is higher, the ask quote or the bid quote? Why? (LG8-4)

8. Illustrate through examples how trading commission costs impact an investor’s return. (LG8-4)

9. Describe the difference in the timing of trade execution and the certainty of trade price between market ordersand limit orders. (LG8-4)

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10. What are the differences between common stock and preferred stock? (LG8-5)

11. How important is growth to a stock’s value? Illustrate with examples. (LG8-5)

12. Under what conditions would the constant-growth model not be appropriate? (LG8-5)

13. The expected return derived from the constant-growth-rate model relies on dividend yield and capitalgain. Where do these two parts of the return come from? (LG8-5)

14. Describe, in words, how to use the variable-growth-rate technique to value a stock. (LG8-6)

15. Can the variable-growth-rate model be used to value a firm that has a negative growth rate in Stage 1 and astable and positive growth in Stage 2? Explain. (LG8-6)

16. Explain why using the P/E relative value approach may be useful for companies that do not pay dividends.(LG8-7)

17. How is a firm’s changing P/E ratio reflected in the stock price? Give examples. (LG8-7)

18. Differentiate the characteristics of growth stocks and value stocks. (LG8-7)

19. What’s the relationship between the P/E ratio and a firm’s growth rate? (LG8-7)

20. Describe the process for using the P/E ratio to estimate a future stock price. (LG8-7)

ProblemsBASIC PROBLEMS

8-1 Stock Index Performance On March 5, 2013, the Dow Jones Industrial Average set a new high. Theindex closed at 14,253.77, which was up 125.95 that day. What was the return (in percent) of the stockmarket that day? (LG8-3)

8-2 Stock Index Performance On March 9, 2009, the Dow Jones Industrial Average reached a new low. Theindex closed at 6,547.05, which was down 79.89 that day. What was the return (in percent) of the stockmarket that day? (LG8-3)

8-3 Buying Stock with Commissions Your discount brokerage firm charges $7.95 per stock trade. Howmuch money do you need to buy 200 shares of Pfizer, Inc. (PFE), which trades at $31.40? (LG8-4)

8-4 Buying Stock with Commissions Your discount brokerage firm charges $9.50 per stock trade. Howmuch money do you need to buy 300 shares of Time Warner, Inc. (TWX), which trades at $22.62? (LG8-4)

8-5 Selling Stock with Commissions Your full-service brokerage firm charges $140 per stock trade. Howmuch money do you receive after selling 200 shares of Nokia Corporation (NOK), which trades at $20.13?(LG8-4)

8-6 Selling Stock with Commissions Your full-service brokerage firm charges $135 per stock trade. Howmuch money do you receive after selling 250 shares of International Business Machines (IBM), whichtrades at $96.17? (LG8-4)

8-7 Buying Stock with a Market Order You would like to buy shares of Sirius Satellite Radio (SIRI). Thecurrent ask and bid quotes are $3.96 and $3.93, respectively. You place a market buy order for 500 sharesthat executes at these quoted prices. How much money did it cost to buy these shares? (LG8-4)

8-8 Buying Stock with a Market Order You would like to buy shares of Coldwater Creek, Inc. (CWTR).The current ask and bid quotes are $20.70 and $20.66, respectively. You place a market buy order for 200shares that executes at these quoted prices. How much money did it cost to buy these shares? (LG8-4)

8-9 Selling Stock with a Limit Order You would like to sell 200 shares of Xenith Bankshares, Inc. (XBKS).The current ask and bid quotes are $4.66 and $4.62, respectively. You place a limit sell order at $4.65. Ifthe trade executes, how much money do you receive from the buyer? (LG8-4)

8-10 Selling Stock with a Limit Order You would like to sell 100 shares of Echo Global Logistics, Inc.

(ECHO). The current ask and bid quotes are $15.33 and $15.28, respectively. You place a limit sell orderat $15.31. If the trade executes, how much money do you receive from the buyer? (LG8-4)

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8-11 Value of a Preferred Stock A preferred stock from Duquesne Light Company (DQUPRA) pays $3.55in annual dividends. If the required return on the preferred stock is 6.7 percent, what’s the value of thestock? (LG8-5)

8-12 Value of a Preferred Stock A preferred stock from Hecla Mining Co. (HLPRB) pays $3.50 in annualdividends. If the required return on the preferred stock is 6.8 percent, what is the value of the stock? (LG8-5)

8-13 P/E Ratio and Stock Price Ultra Petroleum (UPL) has earnings per share of $1.56 and a P/E ratio of32.48. What’s the stock price? (LG8-7)

8-14 P/E Ratio and Stock Price JPMorgan Chase Co. (JPM) has earnings per share of $3.53 and a P/E ratioof 13.81. What is the price of the stock? (LG8-7)

INTERMEDIATE PROBLEMS

8-15 Value of Dividends and Future Price A firm is expected to pay a dividend of $1.35 next year and$1.50 the following year. Financial analysts believe the stock will be at their price target of $68 in twoyears. Compute the value of this stock with a required return of 10 percent. (LG8-5)

8-16 Value of Dividends and Future Price A firm is expected to pay a dividend of $2.05 next year and$2.35 the following year. Financial analysts believe the stock will be at their price target of $110 intwo years. Compute the value of this stock with a required return of 12 percent. (LG8-5)

8-17 Dividend Growth Annual dividends of ATTA Corp grew from $0.96 in 2005 to $1.76 in 2017. Whatwas the annual growth rate? (LG8-5)

8-18 Dividend Growth Annual dividends of Generic Electrical grew from $0.66 in 2012 to $1.03 in 2017.What was the annual growth rate? (LG8-5)

8-19 Value a Constant Growth Stock Financial analysts forecast Safeco Corp.’s (SAF) growth rate forthe future to be 8 percent. Safeco’s recent dividend was $0.88. What is the value of Safeco stock whenthe required return is 12 percent? (LG8-5)

8-20 Value a Constant Growth Stock Financial analysts forecast Limited Brands (LTD) growth rate forthe future to be 12.5 percent. LTD’s recent dividend was $0.60. What is the value of Limited Brandsstock when the required return is 14.5 percent? (LG8-5)

8-21 Expected Return Ecolap Inc. (ECL) recently paid a $0.46 dividend. The dividend is expected togrow at a 14.5 percent rate. At a current stock price of $44.12, what is the return shareholders areexpecting? (LG8-5)

8-22 Expected Return Paychex Inc. (PAYX) recently paid an $0.84 dividend. The dividend is expected togrow at a 15 percent rate. At a current stock price of $40.11, what is the return shareholders areexpecting? (LG8-5)

8-23 Dividend Initiation and Stock Value A firm does not pay a dividend. It is expected to pay its firstdividend of $0.20 per share in three years. This dividend will grow at 11 percent indefinitely. Using a12 percent discount rate, compute the value of this stock. (LG8-6)

8-24 Dividend Initiation and Stock Value A firm does not pay a dividend. It is expected to pay its firstdividend of $0.25 per share in two years. This dividend will grow at 10 percent indefinitely. Using an11.5 percent discount rate, compute the value of this stock. (LG8-6)

8-25 P/E Ratio Model and Future Price Kellogg Co. (K) recently earned a profit of $2.52 earnings pershare and has a P/E ratio of 13.5. The dividend has been growing at a 5 percent rate over the past fewyears. If this growth rate continues, what would be the stock price in five years if the P/E ratioremained unchanged? What would the price be if the P/E ratio declined to 12 in five years? (LG8-7)

8-26 P/E Ratio Model and Future Price New York Times Co. (NYT) recently earned a profit of $1.21

per share and has a P/E ratio of 19.59. The dividend has been growing at a 7.25 percent rate over thepast six years. If this growth rate continues, what would be the stock price in five years if the P/E ratioremained unchanged? What would the price be if the P/E ratio increased to 22 in five years? (LG8-7)

ADVANCED PROBLEMS

8-27 Value of Future Cash Flows A firm recently paid a $0.45 annual dividend. The dividend is expected

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to increase by 10 percent in each of the next four years. In the fourth year, the stock price is expectedto be $80. If the required return for this stock is 13.5 percent, what is its value? (LG8-5)

8-28 Value of Future Cash Flows A firm recently paid a $0.60 annual dividend. The dividend is expectedto increase by 12 percent in each of the next four years. In the fourth year, the stock price is expectedto be $110. If the required return for this stock is 14.5 percent, what is its value? (LG8-5)

8-29 Constant Growth Stock Valuation Waller Co. paid a $0.137 dividend per share in 2000, whichgrew to $0.55 in 2012. This growth is expected to continue. What is the value of this stock at thebeginning of 2013 when the required return is 13.7 percent? (LG8-5)

8-30 Constant Growth Stock Valuation Campbell Supper Co. paid a $0.632 dividend per share in 2013,which grew to $0.76 in 2016. This growth is expected to continue. What is the value of this stock atthe beginning of 2017 when the required return is 8.7 percent? (LG8-5)

8-31 Changes in Growth and Stock Valuation Consider a firm that had been priced using a 10 percentgrowth rate and a 12 percent required return. The firm recently paid a $1.20 dividend. The firm justannounced that because of a new joint venture, it will likely grow at a 10.5 percent rate. How muchshould the stock price change (in dollars and percentage)? (LG8-5)

8-32 Changes in Growth and Stock Valuation Consider a firm that had been priced using an 11.5percent growth rate and a 13.5 percent required return. The firm recently paid a $1.50 dividend. Thefirm has just announced that because of a new joint venture, it will likely grow at a 12 percent rate.How much should the stock price change (in dollars and percentage)? (LG8-5)

8-33 Variable Growth A fast-growing firm recently paid a dividend of $0.35 per share. The dividend isexpected to increase at a 20 percent rate for the next three years. Afterwards, a more stable 12 percentgrowth rate can be assumed. If a 13 percent discount rate is appropriate for this stock, what is itsvalue? (LG8-6)

8-34 Variable Growth A fast-growing firm recently paid a dividend of $0.40 per share. The dividend isexpected to increase at a 25 percent rate for the next four years. Afterwards, a more stable 11 percentgrowth rate can be assumed. If a 12.5 percent discount rate is appropriate for this stock, what is itsvalue? (LG8-6)

8-35 P/E Model and Cash Flow Valuation Suppose that a firm’s recent earnings per share and dividendper share are $2.50 and $1.30, respectively. Both are expected to grow at 8 percent. However, thefirm’s current P/E ratio of 22 seems high for this growth rate. The P/E ratio is expected to fall to 18within five years. Compute a value for this stock by first estimating the dividends over the next fiveyears and the stock price in five years. Then discount these cash flows using a 10 percent required rate.(LG8-5, LG8-7)

8-36 P/E Model and Cash Flow Valuation Suppose that a firm’s recent earnings per share and dividendper share are $2.75 and $1.60, respectively. Both are expected to grow at 9 percent. However, thefirm’s current P/E ratio of 23 seems high for this growth rate. The P/E ratio is expected to fall to 19within five years. Compute a value for this stock by first estimating the dividends over the next fiveyears and the stock price in five years. Then discount these cash flows using an 11 percent requiredrate. (LG8-5, LG8-7)

8-37 Spreadsheet Problem Spreadsheets are especially useful for computing stock value under

different assumptions. Consider a firm that is expected to pay the following dividends:

Year1 2 3 4 5 6

$1.20 $1.20 $1.50 $1.50 $1.75 $1.90 and grow at5% thereafter

a. Using an 11 percent discount rate, what would be the value of this stock?b. What is the value of the stock using a 10 percent discount rate? A 12 percent discount rate?c. What would the value be using a 6 percent growth rate after year 6 instead of the 5 percent rate

using each of these three discount rates?d. What do you conclude about stock valuation and its assumptions?

8-38 Spreadsheet Problem Design a spreadsheet similar to the one below to compute the value of avariable growth rate firm over a five-year horizon.

a. What is the value of the stock if the current dividend is $1.30, the first stage growth is 18 percent,the second stage growth is 9 percent, and the discount rate is 11 percent?

b. What is the value of the stock if the current dividend is $1.30, the first stage growth is 2 percent, thesecond stage growth is 8 percent, and the discount rate is 9.5 percent?

c. What is the value of the stock if the current dividend is $2.50, the first stage growth is 15 percent,the second stage growth is 7 percent, and the discount rate is 10 percent?

A B C D E F

1 Inputs

2 Current dividend =

3 First-stage growth =

4 Second-stage growth =

5 Discount rate =

7 Year 1 2 3 4 5

8 Projected dividend

9 Terminal price =

10 Present value =

Part Five

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chapter ninecharacterizing risk and return

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ou can invest your money very safely by opening a savings account at a bank or by buying Treasurybills. So why would you invest your money in risky stocks and bonds if you can take advantage oflow-risk opportunities? The answer: Very low risk investments also provide a very low return.

Investors take on higher risk investments in expectation of earning higher returns. Likewise, businessesalso take on risky capital investments only if they expect to earn higher returns that at least cover theircosts, including investors’ required return. Both investor and business sentiments create a positiverelationship between risk and expected return. Of course, taking risk means that you get no guaranteethat you will recoup your investment. In the short run, higher risk investments often significantlyunderperform lower risk investments. In addition, not all forms of risk are rewarded. In this chapter, you’llsee how the risk-return relationship fundamentally affects finance theory. We focus on using historicalinformation to characterize past returns and risks. We show how you can diversify to eliminate some riskand expect the highest return possible for your desired risk level. In Chapter 10, we’ll turn to estimatingthe risks and returns you should expect in the future. ■

LEARNING GOALS

LG9-1 Compute an investment’s dollar and percentage return.LG9-2 Find information about the historical returns and volatility for the stock, bond, and cash markets.LG9-3 Measure and evaluate the total risk of an investment using several methods.LG9-4 Recognize the risk-return relationship and its implications.LG9-5 Plan investments that take advantage of diversification and its impact on total risk.LG9-6 Find efficient and optimal portfolios.LG9-7 Compute a portfolio’s return.

viewpointsbusiness APPLICATIONManagers from the production and marketing departments have proposed some risky new business projects for your firm. These newideas appear to be riskier than the firm’s current business operations.

You know that diversifying the firm’s product offerings could reduce the firm’s overall risk. However, you are concerned that takingon these new projects will make the firm’s stock too risky. How can you determine whether these project ideas would make the firm’sstock riskier or less risky? (See the solution at the end of the book.)

9.1 • HISTORICAL RETURNS LG9-1Let’s begin our discussion of risk and return by characterizing the concept of return. First, we need a method forcalculating returns. After computing a return, investors need to assess whether it was a good, average, or badinvestment return. Examining returns from the past gives us a general idea of what we might expect to see in thefuture. We should think in terms of return for the long run because a return for any one year can be quite differentfrom the average returns from the past couple of decades.

Computing ReturnsHow much have you earned on each of your investments? Two ways to determine this are to compute the actualdollar return or compute the dollar return as a percentage of the money invested.

Dollar Return The dollar return earned includes any capital gain (or loss) that occurred as well as any income thatyou received over the period. Equation 9-1 illustrates the dollar return calculation:

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(9-1)

dollar return The amount of profit or loss from an investment denoted in dollars.

For example, say you held 50 shares of Alphabet (GOOG), the parent company of Google. The stock price had amarket price of $526.40 per share at the end of 2014. Alphabet paid no dividends during 2015. At the end of 2015,Alphabet’s stock price was $758.88. For the whole of 2015, you earned a capital gain of ($758.88 − $526.40) × 50shares, or $11,624.

In Alphabet’s case, the stock price increased, so you experienced a capital gain. On the other hand, the toy and gameproducer Mattel, Inc. (MAT) started the year at $30.95 per share, paid $1.52 in dividends, and ended 2015 at $27.17.If you owned 200 shares of Mattel, you would have experienced a capital loss of −$756 (= [$27.17 − $30.95] × 200shares). This loss would have been partially offset by the $304 of dividends received. However, the total dollarreturn would still have been −$452 (= −$756 + $304). Stock prices can fluctuate substantially and cause largepositive or negative dollar returns.

personal APPLICATIONSuppose an investor owns a portfolio invested 100 percent in long-term Treasury bonds because the owner prefers low risk. Theinvestor has avoided owning stocks because of their high volatility.

The investor’s stockbroker claims that putting 10 percent of the portfolio in stocks would actually reduce total risk and increase theportfolio’s expected return. The investor knows that stocks are riskier than bonds. How can adding the risky stocks to the bondportfolio reduce the risk level? (See the solution at the end of the book.)

Is there such a thing as a high-reward, zero-risk investment?

Does your dollar return depend on whether you continue to hold the Alphabet and Mattel stock or sell it? No. Ingeneral, finance deals with market values. Alphabet stock was worth $758.88 at the end of 2015 regardless ofwhether you held the stock or sold it. If you sell it, then we refer to your gains as “realized” gains. If you continue tohold the stock, the gains are “unrealized” gains.

Percentage Return We usually find it more useful to characterize investment earnings as percentage returns sothat we can easily compare one investment’s return to other alternatives’ returns. We calculate percentage return bydividing the dollar return by the investment’s value at the beginning of the time period.

(9-2)

Because it’s standardized, we can use percentage returns for almost any type of investment. We can use beginningand ending values for stock positions, bond prices, real estate values, and so on. Investment income may be stockdividends, bond interest payments, or other receipts. The percentage return for holding the Mattel stock duringcalendar year 2015 was −7.3 percent, computed as

The return for the Alphabet position during the same period was a whopping 44.16 percent:

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Both firms belong to the S&P 500 Index, which earned 1.38 percent in 2015.

Are one-year returns typical for expectations in the long run? We look to average returns to examine performanceover time. The arithmetic average return provides an estimate for how the investment has performed over longerperiods of time. The formula for the average return is

(9-3)

where the return for each subperiod is summed up and divided by the number of subperiods. You can statethe returns in either percentage or decimal format.

percentage return The dollar return characterized as a percentage of money invested.

average returns A measure summarizing the past performance of an investment.

EXAMPLE 9-1 Computing Returns LG9-1

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

You are evaluating a stock’s short-term performance. On August 16,2010, technology firm 3PAR saw its stock price surge on news of atakeover battle between Dell and Hewlett-Packard. 3PAR stock hadclosed the previous trading day at $9.65 and was up to $18.00 by theend of the day. 3PAR had ended 2009 at $11.85 and does not pay adividend. What is the dollar return and percentage return of 300 sharesof 3PAR for the day and year to date?

SOLUTION:

For the day, realize that no income is paid. Therefore, the dollar returnis $2,505 = 300 × ($18.00 − $9.65) + 0 and the percent return is86.53% = $2,505 ÷ (300 × $9.65). The year to date (YTD) return alsodoes not include dividend income. So the dollar YTD return is $1,845 =300 × ($18.00 − $11.85). The 3PAR YTD percentage return is

Hewlett-Packard eventually won the bidding war and purchased 3PARfor $33 per share!

Similar to Problems 9-1, 9-2, 9-3, 9-4, Self-Test Problem 1

Alphabet has only been a public company for a relatively brief period, so it will not have a long history of returns.Thus, Table 9.1 shows the annual returns for Mattel and office supply store Staples, Inc., from 1991 to 2015. First,notice that over time, the returns are quite varied for both firms. The stock return for Mattel has ranged from a lowof −52.4 percent in 1999 to a high of 82.6 percent in 1991. Staples’ stock return varied between −45.7 percent(2015) to 171.6 percent (1991). Also note that the returns appear unpredictable or random. Sometimes a large

negative return is followed by another bad year, like Mattel’s returns in 2007 and 2008. Other times, a poor year isfollowed by a very good year, like 2008 and 2009 for Staples. The table also reports average annual returns forMattel and Staples of 14.8 percent and 19.4 percent, respectively. Over the years, the annual returns for these stockshave been quite different from their average returns.

▼ TABLE 9.1 Annual and Average Returns for Mattel and Staples, 1991 to 2015

A B C D E F

1 Mattel Staples Mattel Staples

2 1991 82.6% 171.6% 2004 5.4% 24.0%

3 1992 33.0% 14.9% 2005 −12.6% 9.5%

4 1993 6.9% 13.5% 2006 52.1% 9.4%

5 1994 15.5% 30.0% 2007 −10.5% −5.9%

6 1995 59.8% 46.3% 2008 −28.5% −32.5%

7 1996 10.5% 24.9% 2009 44.3% 49.5%

8 1997 46.6% 32.9% 2010 24.0% −3.3%

9 1998 −43.0% 136.4% 2011 35.6% −32.7%

10 1999 −52.4% −16.8% 2012 25.9% −4.6%

11 2000 47.4% −30.5% 2013 34.3% 0.8%

12 2001 28.8% 10.0% 2014 −32.1% 34.5%

13 2002 5.5% −5.8% 2015 −5.2% −45.7%

14 2003 −3.5% 55.0%

15 Average = 14.8% 19.4%

Note the range of returns. Few annual returns are close to the average return.

Source: Yahoo! Finance.

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Mattel’s stock returns have ranged from -52.4 percent to 82.6 percent.©McGraw-Hill Education/Mark Steinmetz, photographer

The average returns shown in this chapter are more precisely called arithmetic average returns. These averagereturns are appropriate for statistical analysis. However, they do not accurately illustrate the historical performanceof a stock or portfolio. To see this, consider the $100 stock that earned a 50 percent return one year (to $150) andthen earned a −50 percent return the next year (to $75). The arithmetic average return is therefore (50% + −50%) ÷ 2= 0%. Do you believe the average return was zero percent per year? If you started with a $100 stock and ended witha $75 stock, did you earn zero percent? No, you lost money. A measure of that performance should illustrate anegative return. The accurate measure to be used in performance analysis is called the geometric mean return,or the mean return computed by finding the equivalent return that is compounded for N periods. In thisexample, the mean return is [(1 + 0.50) × (1 + −0.50)]1/2 − 1 = −0.134, or −13.4 percent. Given the loss of $25 overtwo years, this −13.4 percent per year mean return seems more reasonable than the zero percent average return.

The general formula for the geometric mean return is

(9-4)

geometric mean return The mean return computed by finding the equivalent return that is compounded for N periods.

Performance of Asset Classes LG9-2During any given year, the stock market may perform better than the bond market, or it may perform worse. Overlonger time periods, how do stocks, bonds, or cash securities perform? Historically, stocks have performed betterthan either bonds or cash. Table 9.2 shows the average returns for these three asset classes over the period 1950 to2015, as well as over various subperiods. Over the entire period, stocks (as measured by the S&P 500 Index) earnedan average 12.6 percent return per year. This is nearly double the 6.6 percent return earned by long-term Treasurybonds. Cash securities, measured by U.S. Treasury bills, earned an average 4.4 percent return.

▼ TABLE 9.2 Annual and Average Returns for Stocks, Bonds, and T-Bills, 1950 to 2015

A B C D E

1 Stocks Long-Term Treasury Bonds T-Bills

2 1950 to 2015 Average 12.6% 6.6% 4.4%

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3 1950 to 1959 Average 20.9% 0.0% 2.0%

4 1960 to 1969 Average 8.7% 1.6% 4.0%

5 1970 to 1979 Average 7.5% 5.7% 6.3%

6 1980 to 1989 Average 18.2% 13.5% 8.9%

7 1990 to 1999 Average 19.0% 9.5% 4.9%

8 2000 to 2009 Average 0.9% 8.0% 2.7%

9 2010 Annual Return 15.1% 9.4% 0.01%

10 2011 Annual Return 2.1% 29.9% 0.02%

11 2012 Annual Return 16.0% 3.6% 0.02%

12 2013 Annual Return 32.4% −12.7% 0.07%

13 2014 Annual Return 13.7% 25.1% 0.05%

14 2015 Annual Return 1.4% −1.2% 0.21%

15 2010 to 2015 Average 13.4% 9.0% 0.06%

Returns have been very different among decades.

The table also shows each asset class’s average return for each decade since 1950. The best decade for the stockmarket was the 1950s, when stocks earned an average 20.9 percent per year. The 1990s ran a close second with a 19percent per year return. The best decade for the bond market was the 1980s, when it earned an average 13.5 percentper year return due to capital gains as interest rates fell. Stocks have outperformed bonds in every decade since 1950except the recent 2000s. Notice that the average return in the stock and bond markets has not been negative duringany decade since 1950. But average stock returns do not really paint a very accurate picture of annualreturns. Individual annual returns can vary strongly and be quite negative in any particular year. Indeed,this annual variability defines risk. The stock market return in 2008 was particularly poor because of the financialcrisis. However, not all stocks fell the same amount. Notice that Mattel and Staples declined by only 28.5 and 32.5percent while the stock market in general declined 35.5 percent. Financial company stocks fell the most during thecrisis.

time out!9-1 How important were dividend payments to the total returns that Mattel and Staples offered investors?9-2 Using the average returns shown in Table 9.2, compute how much a $10,000 investment made in each asset class at the

beginning of each decade would become at the end of each decade.

9.2 • HISTORICAL RISKS LG9-3

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When you purchase a U.S. Treasury bill, you know exactly what your dollar and percentage return are going to be.Many people find comfort in the certainty from this safe investment. On the other hand, when you purchase a stock,you do not know what your return is going to be—either in the short term or in the long run. This uncertainty isprecisely what makes stock investing risky. It’s useful to evaluate this uncertainty quantitatively so that we cancompare risk among different stocks and asset classes.

Computing VolatilityFinancial theory suggests that investors should look at an investment’s historical returns to assess how muchuncertainty to expect in the future. If you see high variability in historical returns, you should expect a high degreeof future uncertainty. Table 9.2 shows that between 2010 and 2015, the stock market experienced a range of 1.4percent return in 2015 to a 32.4 percent return in 2013. Bonds also experienced variability: −12.7 percent return in2013 to 29.9 percent return in 2011. Examining the range of historical returns provides just one way to express thereturn volatility that we can expect. In practical terms, the finance industry uses a statistical return volatility measureknown as the standard deviation of percentage returns. We calculate standard deviation as the square root of thevariance, and this figure represents the security’s or portfolio’s total risk. We’ll discuss other risk measurements inthe next chapter.

standard deviation A measure of past return volatility, or risk, of an investment.

total risk The volatility of an investment, which includes current portions of firm-specific risk and market risk.

Our process of computing standard deviation starts with the average return over the period. The average annualreturn for the stock market since 1950 is 12.6 percent. How much can the return in any given year deviate from thisaverage? We compute the actual annual deviation by subtracting the return each year from this averagereturn: Return(1950) − Average return; Return(1951) − Average return; Return(1952) − Average return, and soon. Note that many of these deviations will be negative (from a lower-than-average return that year) and others willbe positive (from a higher-than-average return). If we computed the average of these return deviations, our resultwould be zero. Large positive deviations cancel out large negative deviations and hide the variability. To really seethe size of the variations without the distractions that come with including a positive or negative sign, we squareeach deviation before adding them up. Dividing by the number of returns in the sample minus one provides thereturn variance.1 The square root of the return variance is the standard deviation:

(9-5)

Note that this equation provides an estimate of the true population standard deviation using a specific historicalsample. A large standard deviation indicates greater return volatility—or high risk. Table 9.3 shows the standarddeviations of Mattel stock returns over 25 years. The Deviation column shows the annual return minus Mattel’saverage return of 14.8 percent. The last column squares each deviation. Then we sum up these squared deviationsand divide the result by the number of observations less one (24) to compute the return variance. If wewant to use a measure that makes sense in the real world (how would you interpret a squared percentage,anyway?), we take the square root of the variance to get the standard deviation. Mattel’s standard deviation ofreturns during this sample period comes to 33.4 percent. In comparison, the standard deviation of Staples stockreturns for this same period is 48.7 percent. Since Staples’ standard deviation is higher, its stock features more totalrisk than Mattel’s stock does.

▼ TABLE 9.3 Computation of Mattel Stock Return Standard Deviation

Investors use standard deviation as a measure of risk; the higher the standard deviation, the riskier the asset.

Source: Yahoo! Finance.

EXAMPLE 9-2 Risk and Return LG9-1, LG9-3

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For interactive versions of thisexample, log in to Connect or goto mhhe.com/CornettM4e.

Find the average return and risk (as measured by standard deviation) for Mattel since 2006.Table 9.3 shows the annual returns for years 2006 to 2015.

SOLUTION:

First, compute the average annual return for the period. Using equation 9-3

Mattel has averaged a 14.0 percent return per year since 2006. To compute the risk, use thestandard deviation equation 9-5. First, find the deviations of return for each year:

Year2006 2007 2008 2009 2010 2011 2012 2013 2014

52.1%−14.0%

−10.5%−14.0%

−28.5%−14.0%

44.3%−14.0%

24.0%−14.0%

35.6%−14.0%

25.9%−14.0%

34.3%−14.0%

−32.1%−14.0%

Square those deviations:

Year 2006 2007 2008 2009 2010 2011 2012 2013

(52.1%−14.0%)2

(−10.5%−14.0%)2

(−28.5%−14.0%)2

(44.3%−14.0%)2

(24.0%−14.0%)2

(35.6%−14.0%)2

(25.9%−14.0%)2

(34.3%−14.0%)

Then add them up, divide by n − 1, and take the square root:

Mattel stock has averaged a 14.0 percent return with a standard deviation of 30.5 percent since2006.

Similar to Problems 9-15, 9-16, 9-17, 9-18, 9-33, 9-34, Self-Test Problem 2

Although analysts and investors use a stock return’s standard deviation as an important and commonmeasure of risk, it’s laborious to compute by hand. Most people use a spreadsheet or statistical software tocalculate stock return standard deviations.

Risk of Asset Classes LG9-2

We report the standard deviations of return for stocks, bonds, and T-bills in Table 9.4 for 1950 to 2015 and for eachdecade since 1950. Over the entire sample, the stock market returns’ standard deviation is 17.3 percent. As wewould expect, stock market volatility is higher than bond market volatility (11.1 percent) or for T-bills (3.0 percent).These volatility estimates are consistent with our previously stated position that the stock market carries more riskthan the bond or cash markets do. Every decade since 1950 has seen a lot of stock market volatility. The bondmarket has experienced the most volatility since the 1980s as interest rates varied dramatically.

▼ TABLE 9.4 Annual Standard Deviation of Returns for Stocks, Bonds, and T-Bills, 1950 to 2015

Stocks Long-Term Treasury Bonds T-Bills

1950 to 2015 17.3% 11.1% 3.0%1950 to 1959 19.8 4.9 0.81960 to 1969 14.4 6.2 1.31970 to 1979 19.2 6.8 1.81980 to 1989 12.7 15.1 2.61990 to 1999 14.2 12.8 1.22000 to 2009 20.4 10.3 1.92010 to 2015 11.3 16.1 0.1

Some decades experience higher risk than others in each asset class.

You will recall from Chapter 7 that since any bond’s par value and coupon rate are fixed, bond prices must fluctuateto adjust for changes in interest rates. Bond prices respond inversely to interest rate changes: As interest rates rise,bond prices fall, and if interest rates fall, bond prices rise. T-bill returns have experienced very low volatility overeach decade. Indeed, T-bills are commonly considered to be one of the only risk-free assets. Higher-risk investmentsoffer higher returns over time. But short-term fluctuations in the value of higher risk investments can be substantial.The stock market is risky—while it has offered a good annual return of 12.6 percent, that return comes withvolatility of 17.3 percent standard deviation. Many investors may intellectually understand that this high risk meansthat they may receive very poor returns in the short term. Investors really felt the full force of this risk when thestock market declined three years in a row (2000 to 2002). Some investors even decided that this was too much riskfor them and they sold out of the stock market before the 2003 rally. Other investors got out of the stock market afterit plunged to lows in March 2009. Market volatility can cause investors to make emotionally based decisions—selling at low prices.

The stock market returns’ standard deviations that appear in Table 9.4 are all considerably lower than the standarddeviations of Mattel and of Staples stocks (33.4 percent and 48.7 percent, respectively). In this case, we measure

page 270stock market return and standard deviation using the S&P 500 Index. Mattel and Staples are both included in theS&P 500 Index. Why do these two large firms have measures of total risk—standard deviations—that areat least twice as large as the standard deviations on the stock market returns? Are Mattel and Staples justtwo of the most risky firms in the Index? Actually, no. The differences in standard deviations between theseindividual companies and the entire market have much more to do with diversification. Owning 500 companies,such as all of those included in the S&P 500 Index, generates much less risk than owning just one company. Thisphenomenon appears in the standard deviation measure. We’ll discuss the effects of diversification in detail later inthis chapter.

Risk versus Return LG9-4Investors can buy very safe T-bills. Or they can take some risk to seek higher returns. How much extra return canyou expect for taking more risk? This is known as the trade-off between risk and return. The coefficient of variation(CoV) is a common relative measure of this risk-vs-reward relationship. The equation for the coefficient of variationis simply the standard deviation divided by average return. It is interpreted as the amount of risk (measured byvolatility) per unit of return:

(9-6)

As an investor, you would want to receive a very high return (the denominator in the equation) with a very low risk(the numerator). So, a smaller CoV indicates a better risk-reward relationship. Since the average return and standarddeviation for Mattel stock are 14.8 percent and 33.4 percent, its CoV is 2.26 (= 33.4 ÷ 14.8). This is better thanStaples’ CoV of 2.51 (= 48.7 ÷ 19.4). For all asset classes for the period 1950 to 2015, the stock market earned ahigher return than bonds and was also riskier. But which one had a better risk-return relationship? The CoV forcommon stock is 1.37 (= 17.3 ÷ 12.6). For Treasury bonds, the coefficient of variation is 1.68 (= 11.1 ÷ 6.6). Eventhough stocks are riskier than bonds, they involve a somewhat better risk-reward trade-off.

coefficient of variation A measure of risk to reward (standard deviation divided by average return) earned by an investment over aspecific period of time.

EXAMPLE 9-3 Risk versus Return LG9-4

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

You are interested in the risk-return relationship of stocks in eachdecade since 1950. Obtain the average returns and risks in Table 9.2and Table 9.4.

SOLUTION:

Using the coefficient of variation, the average returns, and standarddeviation of return, compute the following risk-return relationships:

Note that over short time periods, the stock risk-return relationshipvaries significantly.

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Similar to Problems 9-7, 9-8, 9-19, 9-20, 9-33, and 9-34, Self-TestProblem 3

time out!9-3 What volatility measure can we use to evaluate and compare risk among different investment alternatives?9-4 Explain why the coefficients of variation for Mattel and Staples are so much higher than the CoV for the stock market as a

whole.

9.3 • FORMING PORTFOLIOS LG9-5As we noted previously, Mattel and Staples stocks’ risk as measured by their standard deviations appear quite highcompared to the standard deviation of the S&P 500 Index. This is by no means a coincidence. Combining stocks intoportfolios can reduce many sources of stock risk. Diversification reduces risk. The S&P 500 Index, for example, tracks500 companies, which allows for a great deal of diversification.

Diversifying to Reduce RiskThink about a stock’s total risk as having two components. The first component includes risks that are both specificto that company and common to other companies in the same industry. We call this risk firm-specific risk. The stock’sother risk component is general risk that all firms—and all individuals, for that matter—face based upon economicstrength both domestically and globally. We call this type of risk market risk. These risks appear in the equation

(9-7)

Standard deviations measure total risk. Individual stocks are subject to many firm-specific risks. We can reducefirm-specific risk by combining stocks into a portfolio. Since we can reduce firm-specific risk by diversifying, thisrisk is sometimes referred to as diversifiable risk. If Staples announces lower-than-expected profits, its stock price willdecline. However, since this news is specific to Staples, the news should not affect Mattel stock’s price. On the otherhand, if the government announces a change in unemployment, both stocks’ prices will change to some degree.Macroeconomic events represent market risks because such events—unemployment claims, interest rate changes,national budget deficits or surpluses—affect all companies.

portfolio A combination of investment assets held by an investor.

diversification The process of putting money in different types of investments for the purpose of reducing the overall risk of the portfolio.

firm-specific risk The portion of total risk that is attributable to firm or industry factors. Firm-specific risk can be reduced throughdiversification.

market risk The portion of total risk that is attributable to over-all economic factors.

diversifiable risk Another term for firm-specific risk.

Suppose that you own only Mattel stock and have earned the annual returns shown in Table 9.5. Then someonesuggests that you add Staples to your Mattel stock to form a two-stock portfolio. Both Mattel and Staples stockscarry a lot of total risk. But look at the risk and return characteristics of a portfolio consisting of 50 percent Staplesstock and 50 percent Mattel stock. You start with Mattel stock, which provided an average return of 14.8 percentwith a risk of 33.4 percent. The Staples stock you are adding has more risk than Mattel. The two-stock portfolioearns an average 17.1 percent return with a standard deviation of only 32.1 percent. You added a high-risk stock to ahigh-risk stock and you ended up with a portfolio with lower risk and a higher return! This is a hallmark of mostportfolios, which pool market risk but often provide offsetting, reduced firm-specific risks overall.

▼ TABLE 9.5 Combining Stocks Can Greatly Reduce Risk

The risk-reducing power of diversification! Note that the risk of the portfolio is lower than the risk of the two stocks individually.

Source: Yahoo! Finance.

Next, add IBM stock to your Mattel and Staples stock portfolio. Figure 9.1 shows that the total risk of this three-stock portfolio declines to 25.7 percent. Note that adding Newmont Mining, Disney, and General Electric alsoreduces the total risk of the stock portfolio. As you add more stocks, the firm-specific risk portion of the totalportfolio risk declines. The total risk falls rapidly as we add the first few stocks. Diversification’s power to reduce

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firm-specific risk weakens for the later stocks added to the portfolio, because we have already eliminatedmuch of the firm-specific risk. We could continue to add stocks until the portfolio comprises all S&P 500Index firms, in which case the standard deviation of the portfolio would be 17.3 percent. At this point, virtually allof the firm-specific risk has been purged and the portfolio carries only market risk, which is sometimes callednondiversifiable risk.

nondiversifiable risk Another term for market risk.

FIGURE 9.1 Adding Stocks to a Portfolio Reduces Risk

The total portfolio risk is greatly reduced by adding the first few stocks to a portfolio.

finance at work //: personalInvestor Diversification ProblemsExperts have examined investor behavior using detailed datasets of stock brokerage accounts, employee pension plans, and the Surveyof Consumer Finances. Studies have identified many investor behaviors that are inconsistent with the principle of full diversification:

Many households own relatively few individual stocks—they held a median number of two stocks until 2001, when it rose to three. Ofcourse, many households own equity indirectly, through mutual funds or retirement accounts, and these indirect holdings tend to bemuch better diversified.

Ten to 15 percent of households with between $100,000 and $1 million in financial asset wealth own no stocks (neither directly norindirectly through funds).Investors seem to prefer securities of local firms. Many geographic regions feature companies that are heavily concentrated in fewindustries. Thus, a local preference could reduce diversification opportunities.

Many employees hold mostly their employers’ stocks (more than 50 percent of employee holdings), particularly within their 401(k)retirement savings accounts. Holding a lot of a single stock creates a “portfolio” with high total risk.

Finance professionals and the investment industry have established diversification concepts for many decades and can helpinvestors maximize their returns with appropriate risk levels. But many investors do not consult professionals; they fail to diversify andthus take unnecessary diversifiable risk.

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Ten to fifteen percent of households with between $100,00 and $1 million in financial asset wealth own no stocks.

Want to know more?Key Words to Search for Updates: diversification, pension plan choices, asset allocation

Sources: Dimmock, Stephen G., Roy Kouwenberg, Olivia S. Mitchell, and Kim Peijnenburg, “Ambiguity Aversion and HouseholdPortfolio Choice Puzzles: Empirical Evidence,” Journal of Financial Economics; in press (2016); and Hans-Martin Von Gaudecker, “HowDoes Household Portfolio Diversification Vary with Financial Literacy and Financial Advice?,” Journal of Finance 70 (2015): 489–507.

Modern Portfolio Theory LG9-6The concept that diversification reduces risk was formalized in the early 1950s by Harry Markowitz, who eventuallywon the Nobel Prize in Economics for his work. Markowitz’s modern portfolio theory shows how risk reduction occurswhen securities are combined. The theory also describes how to combine stocks to achieve the lowest total riskpossible for a given expected return. Or, said differently, it describes how to achieve the highest expected return forthe desired risk level. The combination of securities that achieves the highest expected return for a person’s desiredlevel of risk is called the investor’s optimal portfolio.

In our Mattel and Staples portfolio example, we allocated 50 percent of the portfolio to Mattel and 50 percent toStaples. Is this the best allocation for the portfolio? Consider the different allocations shown in Figure 9.2 for thetwo stocks. The graph shows the expected return (computed as average return) and risk (computed as standarddeviation) of various portfolios. It would be terrific if you could find a portfolio located in the upper left-handcorner. That is, investors would like a high expected return with low risk. One large dot shows the risk-return pointfor owning only Mattel. The other large dot shows owning only Staples. The smaller diamonds show 10/90, 25/75,40/60, 50/50, 60/40, 75/25, and 90/10 allocations of Mattel/Staples stocks.

FIGURE 9.2 Risk and Return Ramifications of Portfolio Allocations to Mattel and Staples

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Investors only value the portfolios at the top of the graph because they offer the same risk as the lower portfolios but with higher expectedreturn!

While all these portfolios are possible, not all are desirable. For example, the portfolio of 75 percent Mattel and 25percent Staples is not desirable. Other portfolios provide both higher return and lower risk. We say that one portfoliodominates the other if it has higher expected return for the same (or less) risk, or the same (or higher) expectedreturn with lower risk. The dominating portfolios appear higher and to the left in the figure. One such portfolioconsists of 25 percent Mattel stock and 75 percent Staples stock. The 50/50 portfolio (circled in the figure) is alsobetter than the 75/25 portfolio. Portfolios with the highest return possible for each risk level are called efficientportfolios. Notice that if you drew a line connecting the dots, the figure would appear like the end of a bullet. Theportfolios on the top of the bullet dominate the portfolios on the bottom; the top portfolio dots show the efficientportfolios for these two stocks.

modern portfolio theory A concept and procedure for combining securities into a portfolio to minimize risk.

optimal portfolio The best portfolio of securities for the investor’s level of risk aversion.

efficient portfolios The set of portfolios that have the maximum expected return for each level of risk.

Figure 9.3 shows efficient portfolios for combining the four stocks: Staples, Mattel, IBM, and NewmontMining. We used this portfolio to demonstrate how diversification reduces risk in Figure 9.1. Theseportfolios appear as diamonds in the figure with each diamond representing a different allocation of the four stocks.The single square represents the portfolio that consists of 25 percent in each of the four stocks. Notice that other,efficient portfolios dominate this portfolio.

FIGURE 9.3 Efficient Portfolios from Four Stocks

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The efficient portfolios dominate all of the individual stocks.

If we showed all efficient portfolios, they would appear as a line that connects the upper side of the bullet shape. Ifwe added all available securities to the graph, then all of the efficient portfolios of those securities form the efficientfrontier. Efficient frontier portfolios dominate all other possible stock portfolios. The shape of the efficient frontierimplies that diminishing returns apply to risk-taking in the investment world. To gain ever-higher expected rates ofreturn, investors must be willing to take on ever-increasing amounts of risk. The optimal portfolio for you is one onthe efficient frontier that reflects the amount of risk that you’re willing to take. Clearly, optimal portfolio choicedepends on individual risk preferences. Highly risk-averse investors will select low-risk portfolios on the efficientfrontier, while more adventuresome investors will select higher-risk portfolios. Any choice may be appropriate,given differences in individual risk preferences.

Investors can further diversify by adding foreign stocks and commodities to their portfolio. For example, a U.S.investor can lower total risk by adding stocks from emerging market countries and gold.

efficient frontier The combination of all efficient portfolios.

How Does Diversification Work? LG9-5 Will combining any two stocks greatly reduce total risk as much ascombining Staples and Mattel did? The answer is no. If two stocks are subject to exactly the same kinds of eventssuch that their returns always behave the same way over time, then we have no need to own both stocks—simplypick the one that performs better. Diversification comes when two stocks are subject to different kinds of eventssuch that their returns differ over time.

Consider the illustration in Figure 9.4. You own Stock A in Panel A of the figure. The stock features risk,as demonstrated by its price volatility over time. You would like to reduce the risk by combining yourposition in Stock A with an equal position in Stock B. In this case, the alternative stock, Stock B, moves the sameway over time as Stock A does. When Stock A goes up, so does Stock B. They also decline together. A portfolio ofboth stocks is illustrated. Notice how the portfolio has the same volatility of Stocks A and B separately. Combiningthese two stocks did not reduce volatility, or total risk.

FIGURE 9.4 Efficient Portfolios from Four Stocks

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Combining stocks that move together over time does not offer much risk reduction. Combining stocks that do not move together provides alot of risk reduction.Correlation shows that some stocks move together and some do not.

Now consider Stock C, shown in Panel B. Stock C has the same volatility as Stock A, but its price moves indifferent directions than does the price of Stock A. When Stock A’s price increases, Stock C’s price increasessometimes and decreases sometimes. As shown, a portfolio of Stock A and Stock C features much lower volatilitythan either Stock A or Stock C alone. Combining Stocks A and C reduces risk because their price movements oftencounteract one another. In short, combining stocks with similar characteristics does not provide muchdiversification and thus risk reduction. Combining stocks with many differences does providediversification and thus lowers risk.

The ways that stocks co-move over time determines how much diversification and thus risk reduction we canachieve by combining them. So what we need is some measure of co-movement to help investors form diversifiedportfolios. That measure, called correlation, is denoted by the Greek letter ρ (rho). Correlation is a statistical measurewith some very useful characteristics that makes it easy to interpret. Its value is bounded between −1 and +1. Acorrelation value of +1 means that returns from two different securities move perfectly in sync. They change lock-step up and down together. A value of −1 means that returns from two securities are perfectly inversely correlated—they move exactly opposite. A value of 0 means that the movements of the two returns over time are unrelated toone another. Investors seeking diversification look for stocks where the returns have low or negative correlationswith each other.

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correlation A measurement of the co-movement between two variables that ranges between −1 and +1.

What return correlations are common between stocks? Panel A of Table 9.6 shows the correlations between manycompanies. One high correlation shown is the 0.690 correlation between Citigroup and the Bank of New York. Thisshouldn’t be surprising, because these are two similar firms in the same industry. Combining these two stockswouldn’t reduce risk very much in a portfolio. The largest negative correlation is the correlation of −0.212 betweenNewmont Mining and the Bank of New York. These firms practice in very different industries and provide large riskreduction possibilities. Note that the correlation between Staples and Mattel is 0.198. This low correlation gives usan answer to the question of why total risk (in the form of standard deviations) fell so much when we combined thetwo stocks relative to their individual standard deviations as shown in Figure 9.2. Most of the correlations in Table9.6 are positive. Because most stocks are positively correlated, we typically add many stocks together to fullyeliminate all the firm-specific risk in the portfolio, as we showed in Figure 9.1.

▼ TABLE 9.6 Correlation between Various Stocks and Asset Classes

PANEL A: COMPANY ANNUAL RETURNS, 1991 TO 2015

Staples Mattel IBMNewmont

Mining DisneyGeneralElectric Citigroup

Staples 1Mattel 0.198 1IBM 0.314 −0.029 1NewmontMining

−0.036 −0.117 −0.087 1

Disney 0.163 0.440 0.076 −0.171 1General Electric 0.380 0.134 0.396 −0.038 0.517 1Citigroup 0.297 0.328 0.056 0.139 0.414 0.773 1Bank of NewYork

0.584 0.459 0.045 −0.212 0.606 0.671 0.690

PANEL B: ASSET CLASS ANNUAL RETURNS,1950 TO 2015

StocksLong-Term Treasury

Bonds T-Bills

Stocks 1Long-termTreasury bonds

−0.035 1

T-bills −0.063 0.137 1

LG9-2Panel B of Table 9.6 shows correlations between stocks, bonds, and T-bills. At −0.035, the correlation betweenstocks and bonds is small. The small correlation allows for the possibility of good risk reduction by adding bonds toa stock portfolio. Therefore, a well-diversified portfolio will contain both stocks and bonds.

finance at work //: marketsInternational Opportunities for DiversificationThe U.S. stock market represents nearly 47 percent of all stock value worldwide. Japanese and U.K. stock markets represent 11 percentand 8 percent of the worldwide stock market value, respectively. Many investment and diversification opportunities present themselvesinternationally! However, most people allocate very little or none of their portfolios to international securities. If worldwide opportunitiescan create greater diversification, then those who don’t invest in international stocks miss out on an important opportunity to reduce riskin their portfolios.

MSCI Barra is the leading provider of global stock market indexes. Some MSCI Barra indexes follow individual countries. In addition,MSCI Barra compiles composite indexes for groups of companies in developed markets, emerging markets, frontier markets, and bygeographic regions. Investment managers use the MSCI World Index, the MSCI EAFE (Europe, Australasia, Far East) Index, and theMSCI Emerging Markets Index as premier benchmarks to measure global stock market performance.

The following table shows the average annual returns and standard deviations for the U.S. stock market, Treasury bonds, and the

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MSCI EAFE and MSCI Emerging Markets indexes for the period 1988 to 2015. Note that both the EAFE and the Emerging Marketsindexes feature higher risk than the S&P 500 Index. The Emerging Markets return has been high, but the EAFE return has been lowcompared to the U.S. stock and bond markets.

S&P 500 Bonds EAFE Emerging Markets

Average 11.7% 9.2% 5.6% 13.3%Std. Deviation 17.6 11.9 19.4 33.7

The correlations among these markets appear as follows:

S&P 500 Bonds EAFE Emerging Markets

S&P 500 1Bonds −0.15 1EAFE 0.74 −0.41 1Emerging Markets 0.48 −0.31 0.73 1

The correlation between the S&P 500 Index and the MSCI EAFE is 0.74 and between the Emerging Markets is 0.48. These correlationsindicate that diversification might work. Even better diversification appears to be possible between the U.S. Treasury Bond Market andthe EAFE and Emerging Markets indexes—look at the negative correlations!

Source: www.msci.com.

Want to know more?Key Words to Search for Updates: international diversification, global asset allocation

Portfolio Return LG9-7 A portfolio’s return calculation is straightforward. A portfolio’s return comes directlyfrom the returns of the portfolio securities and the proportion of the portfolio invested in each security. For example,General Electric stock earned −1.9 percent. The Newmont Mining earned 17.3 percent over the same period. If youhad invested a quarter of your money in General Electric stock and three quarters of it in Newmont Miningstock, then your portfolio return would be

the Math Coach on…

“When computing portfolio returns, use the decimal format for the portfolioweights and the percentage format for the security returns. The result of equation9-8 will then be in percentage format.„

To calculate the return on a three-stock portfolio, you will need the proportion of each stock in the portfolio andeach stock’s return. We typically call these proportions weights, signified by w. So, a portfolio with n securities willhave a return of

(9-8)

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where the sum of the weights, w, must equal one.

The portfolio’s rate of return is a simple weighted sum of the returns of each stock in the portfolio. Investors chooseportfolio weights by determining how much of each stock they want in their portfolios. Ideally, investors will chooseweights for their portfolios located on the efficient frontier (shown in Figure 9.3).

EXAMPLE 9-4Computing PortfolioReturns LG9-7

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

At the beginning of the year, you owned $5,000 of Disney stock,$10,000 of Bank of New York stock, and $15,000 of IBM stock. Duringthe year, Disney, Bank of New York, and IBM returned −4.8 percent,19.4 percent, and 12.8 percent, respectively. What is your portfolio’sreturn?

SOLUTION:

First determine your portfolio weights. The three stocks make up a$30,000 portfolio. Disney makes up a 16.67 percent (= $5,000 ÷$30,000) portion of the portfolio. Bank of New York stock makes up a33.33 percent (=$10,000 ÷ $30,000) portion, and IBM a 50.0 percent(=$15,000 ÷ $30,000) portion.

The portfolio return can now be computed as

Similar to Problems 9-11, 9-12, 9-13, 9-14, 9-23, 9-24, 9-25, 9-26, 9-29, 9-30, 9-31, 9-32, Self-Test Problem 1

time out!9-5 Describe characteristics of companies that would be good to combine into a portfolio.9-6 Explain why one portfolio made up of the same companies (but not in the same proportions) as another portfolio can be

undesirable in comparison.

9-7 Combining which two companies in Table 9.6 would reduce risk the most? Combining which two would create the leastdiversification?

Get Online

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Log in to your Connect course for study materials including self-test problems with solutions, answers tothe Time Out quizzes, guided example videos, and more.

Your Turn…Questions

1. Why is the percentage return a more useful measure than the dollar return? (LG9-1)

2. Characterize the historical return, risk, and risk-return relationship of the stock, bond, and cash markets. (LG9-2)

3. How do we define risk in this chapter and how do we measure it? (LG9-3)

4. What are the two components of total risk? Which component is part of the risk-return relationship? Why?(LG9-3)

5. What’s the source of firm-specific risk? What’s the source of market risk? (LG9-3)

6. Which company is likely to have lower total risk, General Electric or Coca-Cola? Why? (LG9-3)

7. Can a company change its total risk level over time? How? (LG9-3)

8. What does the coefficient of variation measure? Why is a lower value better for the investor? (LG9-4)

9. You receive an investment newsletter advertisement in the mail. The letter claims that you should invest in astock that has doubled the return of the S&P 500 Index over the last three months. It also claims that this stockis a surefire safe bet for the future. Explain how these two claims are inconsistent with finance theory. (LG9-4)

10. What does diversification do to the risk and return characteristics of a portfolio? (LG9-5)

11. Describe the diversification potential of two assets with a −0.8 correlation. What’s the potential if thecorrelation is +0.8? (LG9-5)

12. You are a risk-averse investor with a low-risk portfolio of bonds. How is it possible that adding some

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stocks (which are riskier than bonds) to the portfolio can lower the total risk of the portfolio? (LG9-5)

13. You own only two stocks in your portfolio but want to add more. When you add a third stock, the total risk ofyour portfolio declines. When you add a tenth stock to the portfolio, the total risk declines. Adding whichstock, the third or the tenth, likely reduced the total risk more? Why? (LG9-5)

14. Many employees believe that their employer’s stock is less likely to lose half of its value than a well-diversifiedportfolio of stocks. Explain why this belief is erroneous. (LG9-5)

15. Explain what we mean when we say that one portfolio dominates another portfolio. (LG9-6)

16. Explain what the efficient frontier is and why it is important to investors. (LG9-6)

17. If an investor’s desired risk level changes over time, should the investor change the composition of his or herportfolio? How? (LG9-6)

18. Say you own 200 shares of Mattel and 100 shares of Staples. Would your portfolio return be different if youinstead owned 100 shares of Mattel and 200 shares of Staples? Why? (LG9-7)

ProblemsBASIC PROBLEMS

9-1 Investment Return FedEx Corp. stock ended the previous year at $103.39 per share. It paid a $0.35 pershare dividend last year. It ended last year at $106.69. If you owned 200 shares of FedEx, what was yourdollar return and percent return? (LG9-1)

9-2 Investment Return Sprint Nextel Corp. stock ended the previous year at $23.36 per share. It paid a $2.37per share dividend last year. It ended last year at $18.89. If you owned 500 shares of Sprint, what was yourdollar return and percent return? (LG9-1)

9-3 Investment Return A corporate bond that you own at the beginning of the year is worth $975. During theyear, it pays $35 in interest payments and ends the year valued at $965. What was your dollar return andpercent return? (LG9-1)

9-4 Investment Return A Treasury bond that you own at the beginning of the year is worth $1,055. Duringthe year, it pays $35 in interest payments and ends the year valued at $1,065. What was your dollar returnand percent return? (LG9-1)

9-5 Total Risk Rank the following three stocks by their level of total risk, highest to lowest. Rail Haul has anaverage return of 12 percent and standard deviation of 25 percent. The average return and standarddeviation of Idol Staff are 15 percent and 35 percent; and of Poker-R-Us are 9 percent and 20 percent.(LG9-3)

9-6 Total Risk Rank the following three stocks by their total risk level, highest to lowest. Night Ryder has anaverage return of 12 percent and standard deviation of 32 percent. The average return and standarddeviation of WholeMart are 11 percent and 25 percent; and of Fruit Fly are 16 percent and 40 percent.(LG9-3)

9-7 Risk versus Return Rank the following three stocks by their risk-return relationship, best to worst. RailHaul has an average return of 12 percent and standard deviation of 25 percent. The average return andstandard deviation of Idol Staff are 15 percent and 35 percent; and of Poker-R-Us are 9 percent and 20percent. (LG9-4)

9-8 Risk versus Return Rank the following three stocks by their risk-return relationship, best to worst. NightRyder has an average return of 13 percent and standard deviation of 29 percent. The average returnand standard deviation of WholeMart are 11 percent and 25 percent; and of Fruit Fly are 16 percentand 40 percent. (LG9-4)

9-9 Dominant Portfolios Determine which one of these three portfolios dominates another. Name thedominated portfolio and the portfolio that dominates it. Portfolio Blue has an expected return of 12 percentand risk of 18 percent. The expected return and risk of portfolio Yellow are 15 percent and 17 percent, andfor the Purple portfolio are 14 percent and 21 percent. (LG9-6)

9-10 Dominant Portfolios Determine which one of the three portfolios dominates another. Name thedominated portfolio and the portfolio that dominates it. Portfolio Green has an expected return of 15

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percent and risk of 21 percent. The expected return and risk of portfolio Red are 13 percent and 17percent, and for the Orange portfolio are 13 percent and 16 percent. (LG9-6)

9-11 Portfolio Weights An investor owns $6,000 of Adobe Systems stock, $5,000 of Dow Chemical, and$5,000 of Office Depot. What are the portfolio weights of each stock? (LG9-7)

9-12 Portfolio Weights An investor owns $3,000 of Adobe Systems stock, $6,000 of Dow Chemical, and$7,000 of Office Depot. What are the portfolio weights of each stock? (LG9-7)

9-13 Portfolio Return Year-to-date, Oracle had earned a –1.34 percent return. During the same timeperiod, Valero Energy earned 7.96 percent and McDonald’s earned 0.88 percent. If you have aportfolio made up of 30 percent Oracle, 25 percent Valero Energy, and 45 percent McDonald’s, whatis your portfolio return? (LG9-7)

9-14 Portfolio Return Year-to-date, Yum Brands had earned a 3.80 percent return. During the same timeperiod, Raytheon earned 4.26 percent and Coca-Cola earned −0.46 percent. If you have a portfoliomade up of 30 percent Yum Brands, 30 percent Raytheon, and 40 percent Coca-Cola, what is yourportfolio return? (LG9-7)?

INTERMEDIATE PROBLEMS

9-15 Average Return The past five monthly returns for Kohls are 4.11 percent, 3.62 percent, −1.68percent, 9.25 percent, and −2.56 percent. What is the average monthly return? (LG9-1)

9-16 Average Return The past five monthly returns for PG&E are −3.17 percent, 3.88 percent, 3.77percent, 6.47 percent, and 3.58 percent. What is the average monthly return? (LG9-1)

9-17 Standard Deviation Compute the standard deviation of Kohls’ monthly returns shown in problem 9-15. (LG9-3)

9-18 Standard Deviation Compute the standard deviation of PG&E’s monthly returns shown in problem9-16. (LG9-3)

9-19 Risk versus Return in Bonds Assess the risk-return relationship of the bond market (see Tables 9.2and 9.4) during each decade since 1950. (LG9-2, LG9-4)

9-20 Risk versus Return in T-bills Assess the risk-return relationship in T-bills (see Tables 9.2 and 9.4)during each decade since 1950. (LG9-2, LG9-4)

9-21 Diversifying Consider the characteristics of the following three stocks:The correlation between Thumb Devices and Air Comfort is −0.12. The correlation between ThumbDevices and Sport Garb is −0.21. The correlation between Air Comfort and Sport Garb is 0.77. If youcan pick only two stocks for your portfolio, which would you pick? Why? (LG9-4, LG9-5)

Expected Return Standard Deviation

Thumb Devices 13% 23%

Air Comfort 10 19

Sport Garb 10 17

9-22 Diversifying Consider the characteristics of the following three stocks:The correlation between Pic Image and Tax Help is 0.88. The correlation between Pic Imageand Warm Wear is −0.21. The correlation between Tax Help and Warm Wear is −0.19. If you can pickonly two stocks for your portfolio, which would you pick? Why? (LG9-4, LG9-5)

Expected Return Standard Deviation

Pic Image 11% 19%

Tax Help 9 19

Warm Wear 14 25

9-23 Portfolio Weights If you own 200 shares of Alaska Air at $42.88, 350 shares of Best Buy at $51.32,and 250 shares of Ford Motor at $8.51, what are the portfolio weights of each stock? (LG9-7)

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9-24 Portfolio Weights If you own 400 shares of Xerox at $17.34, 500 shares of Qwest at $8.15, and 350shares of Liz Claiborne at $44.73, what are the portfolio weights of each stock? (LG9-7)

9-25 Portfolio Return At the beginning of the month, you owned $5,500 of General Dynamics, $7,500 ofStarbucks, and $8,000 of Nike. The monthly returns for General Dynamics, Starbucks, and Nike were7.44 percent, −1.36 percent, and −0.54 percent. What is your portfolio return? (LG9-7)

9-26 Portfolio Return At the beginning of the month, you owned $6,000 of News Corp, $5,000 of FirstData, and $8,500 of Whirlpool. The monthly returns for News Corp, First Data, and Whirlpool were8.24 percent, −2.59 percent, and 10.13 percent. What’s your portfolio return? (LG9-7)

ADVANCED PROBLEMS

9-27 Asset Allocation You have a portfolio with an asset allocation of 50 percent stocks, 40 percent long-term Treasury bonds, and 10 percent T-bills. Use these weights and the returns in Table 9.2 tocompute the return of the portfolio in the year 2010 and each year since. Then compute the averageannual return and standard deviation of the portfolio and compare them with the risk and return profileof each individual asset class. (LG9-2, LG9-5)

9-28 Asset Allocation You have a portfolio with an asset allocation of 35 percent stocks, 55 percent long-term Treasury bonds, and 10 percent T-bills. Use these weights and the returns in Table 9.2 tocompute the return of the portfolio in the year 2010 and each year since. Then compute the averageannual return and standard deviation of the portfolio and compare them with the risk and return profileof each individual asset class. (LG9-2, LG9-5)

9-29 Portfolio Weights You have $15,000 to invest. You want to purchase shares of Alaska Air at $42.88,Best Buy at $51.32, and Ford Motor at $8.51. How many shares of each company should you purchaseso that your portfolio consists of 30 percent Alaska Air, 40 percent Best Buy, and 30 percent FordMotor? Report only whole stock shares. (LG9-7)

9-30 Portfolio Weights You have $20,000 to invest. You want to purchase shares of Xerox at $17.34,Qwest at $8.15, and Liz Claiborne at $44.73. How many shares of each company should you purchaseso that your portfolio consists of 25 percent Xerox, 40 percent Qwest, and 35 percent Liz Claiborne?Report only whole stock shares. (LG9-7)

9-31 Portfolio Return The following table shows your stock positions at the beginning of the year, thedividends that each stock paid during the year, and the stock prices at the end of the year. What is yourportfolio dollar return and percentage return? (LG9-7)

Company Shares Beginning of Year Price Dividend Per Share End of Year Price

US Bank 300 $43.50 $2.06 $43.43

PepsiCo 200 59.08 1.16 62.55

JDS Uniphase 500 18.88 16.66

Duke Energy 250 27.45 1.26 33.21

9-32 Portfolio Return The following table shows your stock positions at the beginning of the year, thedividends that each stock paid during the year, and the stock prices at the end of the year. What is yourportfolio dollar return and percentage return? (LG9-7)

Company Shares Beginning of Year Price Dividend Per Share End of Year Price

Johnson Controls 350 $72.91 $1.17 $85.92

Medtronic 200 57.57 0.41 53.51

Direct TV 500 24.94 24.39

Qualcomm 250 43.08 0.45 38.92

9-33 Risk, Return, and Their Relationship Consider the following annual returns of Estee Lauder and

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Lowe’s Companies:Compute each stock’s average return, standard deviation, and coefficient of variation. Which stockappears better? Why? (LG9-3, LG9-4)

Estee Lauder Lowe’s Companies

Year 1 23.4% −6.0%

Year 2 −26.0 16.1

Year 3 17.6 4.2

Year 4 49.9 48.0

Year 5 −16.8 −19.0

9-34 Risk, Return, and Their Relationship Consider the following annual returns of Molson Coors andInternational Paper:Compute each stock’s average return, standard deviation, and coefficient of variation. Which stockappears better? Why? (LG9-3, LG9-4)

Molson Coors International Paper

Year 1 16.3% 4.5%

Year 2 −9.7 −17.5

Year 3 36.5 −0.2

Year 4 −6.9 26.6

Year 5 16.2 −11.1

9-35 Spreadsheet Problem Following are the monthly returns for March 2011 to February 2016 of threeinternational stock indices: All Ordinaries of Australia, Nikkei 225 of Japan, and FTSE 100 ofEngland.

a. Compute and compare each index’s monthly average return and standard deviation.b. Compute the correlation between (1) All Ordinaries and Nikkei 225,(2) All Ordinaries and FTSE

100, and (3) Nikkei 225 and FTSE 100, and compare them.c. Form a portfolio consisting of one-third of each of the indexes and show the portfolio return each

year, and the portfolio’s return and standard deviation.

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9-36 Spreadsheet Problema. Create a spreadsheet like the one shown below. The spreadsheet should use the returns for assets A

and B to form a portfolio return using the weights for each asset shown in cells E1 and E2. Theaverage portfolio return and standard deviation should compute at the bottom of the column ofportfolio returns. When you change the weights, the portfolio returns, average, and standarddeviation should recalculate.

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b. Use the Solver function to find the weights that provide the highest return for a standard deviationof 6 percent, 7.5 percent, 9 percent, 10.5 percent, 12 percent, and 13.5 percent. Report the weightsand the return for each of these portfolio standard deviations. The Solver function is found in theData tab. (You may have to enable the function through the File tab, then Options, then Add-ins.)The solver image illustrates the maximizing of the average return for the specific constraints. Theconstraints are that the weights must be between 0 and 1, inclusive, and must sum to 1. Lastly, setthe standard deviation constraint to the desired level.

NotesCHAPTER 91. We use the denominator of N – 1 to compute a sample’s standard deviation, which is the most common for finance applications. We

would divide the standard deviation of a population simply by N.

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chapter tenestimatingrisk and return

s it possible for investors to know the exact risk they have to take? In Chapters 9 and 10, we exploremethods to find the return that individual or institutional investors require to make a particularinvestment attractive. In the previous chapter, we established a positive relationship between risk and

return using historical data. Risk and return play an undeniable role as investors seek the best return forthe least risk. But until there’s some way to forecast the future, financial managers and investors mustmake investment decisions armed only with their expectations about future risk and return. We need anexact specification that shows directly the amount of reward required for investors to take the level of riskin a given firm’s stock or portfolio of securities. In this chapter we will also see how investors get theinformation they need to make risk-reward decisions.

Investors need to know how much risk they have to take to confidently expect a 10 percent return.Managers also want to know what return shareholders require so that they can decide how to meet thoseexpectations. In Chapter 11, we’ll explore how managers conduct financial analysis to find theshareholders’ required return. If we want to specify the exact risk–return relationship, we need to developa better measure of risk for individuals and institutional investors. As we saw in Chapter 9, any firm’s totalrisk is specific to that particular firm. But the market doesn’t reward firm-specific risk because investorscan easily diversify away any single firm’s specific risks by owning other offsetting firms’ stocks to createa portfolio subject only to market or undiversifiable risk. So, we need to find just the market risk portion of

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total risk for investors. The theory to find the market risk portion of stock ownership extends modernportfolio theory. Our search to find market risk will lead us to the capital asset pricing model (CAPM),which utilizes a measure of market risk called beta. CAPM’s risk–return specification provides us apowerful tool to make better investment decisions.

LEARNING GOALS

LG10-1 Compute forward-looking expected return and risk.LG10-2 Understand risk premiums.LG10-3 Know and apply the capital asset pricing model (CAPM).LG10-4 Calculate and apply beta, a measure of market risk.LG10-5 Differentiate the levels of market efficiency and their implications.LG10-6 Calculate and explain investors’ required return and risk.LG10-7 Use the constant-growth model to compute required return.

viewpointsbusiness APPLICATIONConsider that you work in the finance department of a large corporation. Your team is analyzing several new projects the firm canpursue. To complete the analysis, the team needs to know what return stockholders require from the firm.

You are to estimate this required return. Shareholders’ expected return will depend on your company’s risk level. What informationdo you need to gather and how might you compute this return? (See the solution at the end of the book.)

Corporate finance managers and investment professionals commonly use the beta measure. But likeany theory, CAPM has its limitations. We’ll discuss the CAPM’s limitations and concerns about beta andpropose an alternate required return measure. Whether beta or any other risk–return specification isuseful relies in part on whether a stock’s price represents a fair estimate of the true company value. Stockprice validity and reliability—their general correctness— are vitally important both to investors andcorporate managers.

10.1 • EXPECTED RETURNS LG10-1In the previous chapter, we characterized risk and return in historical terms. We defined a stock’s return as the actualprofit realized while holding the stock or the average return over a longer period. We described risk simply as thestandard deviation of those returns—a term already familiar to you from your statistics classes. So, we did a goodjob describing the risk and return that the stock experienced in the past. But do those risk and return figures holdinto the future? Firms can quite possibly change their stocks’ risk level by substantially changing their business. If afirm takes on riskier new projects over time, or changes the nature of its business, the firm itself will become riskier.Similarly, firms can reduce their risk level—and hence, their stock’s riskiness—by choosing low-risk new projects.Both investors and firms find expected return, a forward-looking return calculation that includes risk measures, veryuseful to estimate future stock performance.

Expected Return and RiskWe can attribute a company’s business success over a year partly to its management talent, strategies, and otherfirm-specific activities, but overall economic conditions will also affect a firm’s level of success or failure. Considera steel manufacturer—Nucor Corp. The steel business closely follows economic trends. In a good economy, demandfor steel is strong as builders and manufacturers step up building and production. During economic

recessions, demand for steel falls off quickly. So, if we want to assess Nucor’s probabilities for success next year,we know that we must look partly at Nucor’s managerial ability and partly at the economic outlook.

personal APPLICATIONYou have just started your first job in the corporate world and need to make some retirement plan decisions. The company’s 401(k)retirement plan offers three investment choices: a stock portfolio with a beta of 1, a bond portfolio with a beta of 0.18, and a moneymarket account. For your allocation, you decide to contribute $200 per month to the stock portfolio, $100 to the bond portfolio, and$50 to the money market account.

If the expected return to the market portfolio is 11 percent, what risk level are you taking in your retirement portfolio and what returnshould you expect over the long run? (See the solution at the end of the book.)

Investing mainly in my own company’s stock is safer, right? Maybe not . . .

Unfortunately, we cannot accurately predict what the economy will be like next year. Predicting economic activity islike predicting the weather—forecasts give the probability of rain or sunshine. Economists cannot say for surewhether the economy will be good or bad next year. Instead, they may forecast a 70 percent chance that theeconomy will be good and a 30 percent chance of a recession. Similarly, analysts might say that given Nucor’smanagerial talent, if the economy is good, Nucor will perform well and the stock will increase 20 percent. If theeconomy goes into a recession, then Nucor’s stock will fall 10 percent. So what return do you expect from Nucor?The return still depends on the state of the economy.

This leads us directly to a key concept: expected return. We compute expected return by multiplying each possiblereturn by the probability, p, of that return occurring. We then sum them (recall that all probabilities must add toone). Let’s place Nucor in an economy with only two states: good and recession. In this scenario, Nucor’s expectedreturn would be 11 percent [= (0.7 × 20%) + (0.3 × −10%)]. Of course, nothing is quite that simple. Economistsseldom predict simple two-state views of the economy as in the previous example. Rather, economists give muchmore detailed forecasts (such as three states: red-hot economy, average expansion, and recession). So our generalequation for a stock’s expected return with S different conditions of the economy is

(10-1)

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The result of this expected return calculation has some interesting properties. First, the expected return figureexpresses what the average return would be over time if the probabilistic states of the economy occur as predicted.For example, the 70/30 probability distribution for good/recession economic states suggests that the economy will begood in 7 of the next 10 years, earning Nucor shareholders a 20 percent return in each of those years.Shareholders would lose 10 percent in each of the three recession years. So the average return over those10 years would be 11 percent, the same as the expected return. The second interesting property: The expected returnitself will not likely occur during any one year. Remember that Nucor will earn either a return of 20 percent or −10percent. Yet its expected return is 11 percent, a value that it cannot earn because we have no economic condition forwhich the return is 11 percent. Again, this illustration seems extreme because we used only two economic states.Any real economic forecast would instead include a probability distribution of many potential economic conditions.

probability The likelihood of occurrence.

expected return The average of the possible returns weighted by the likelihood of those returns occurring.

probability distribution The set of probabilities for all possible occurrences.

We can also characterize risk via this expected return figure. The expected return procedure shows potential returnpossibilities, but we don’t know which one will actually occur, so we face uncertainty. In the last chapter, wemeasured risk using the standard deviation of returns over time. We can use the same principle to measure risk forexpected returns. What range of different expected returns will Nucor exhibit from the expected return of 11percent? In our two-state description of the economy, the deviation could be either 9 percent (= 20% − 11%) or −21percent (= −10% − 11%). We compute the standard deviation of expected returns the same way we did for historicalreturns. We square the deviations, then multiply by the probability of that deviation occurring, and then sum them allup. So Nucor’s return variance is 189.0 As a final step, we take the square root of our result toput the figure back into sensible terms. The standard deviation for Nucor is 13.75 percent (= ). The generalequation for the standard deviation of S different economic states is

(10-2)

the Math Coach on…Expected Return and StandardDeviation

“When you compute expected return and standard deviation, you’ll find it helpfulto use the decimal format for the probability of the economic state andpercentages to state the return in each state.„

Risk Premiums LG10-2Throughout the book, we have mentioned the positive relationship between expected return and risk. Consider thiskey question: You have a riskless investment available to you. The short-term government debt security, the T-bill,

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offers you a low return with no risk. Why would you invest in anything risky, when you could simply buy T-bills?The answer, of course, is that some investors want a higher return and are willing to take some risk to raise theirreturns. Investors who take on a little risk should expect a slightly higher return than the T-bill rate. People who takeon higher risk levels should expect higher returns. Indeed, it’s only logical that investors require this extra return towillingly take the added risk.The expected return of an investment is often expressed in two parts, a risk-free return and a risky contribution. Thereturn investors require for the risk level they take is called the required return:

(10-3)

required return The level of total return needed to be compensated for the risk taken. It is made up of a risk-free rate and a riskpremium.

The risk-free rate is typically considered the return on U.S. government bonds and bills and equals the real interestrate and the expected inflation premium that we discussed in Chapter 6. The risk premium is the reward investorsrequire for taking risk. How large are the rewards for taking risk? As we discussed in the previous chapter, themarket doesn’t reward all risks. The firm-specific portion of total risk for any stock can be diversified away, andsince the investor takes on such risk out of ignorance or by mistake, an efficient market will not reward anyone fortaking on this “superfluous” risk. So as we examine historical risk premiums, we do so with a diversified portfoliothat contains no firm-specific risk.

risk premium The portion of the required return that represents the reward for taking risk.

EXAMPLE 10-1 Expected Return and Risk LG10-1

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Bailey has a probability distribution for four possible states of theeconomy, as shown below. She has also calculated the return thatMotor Music stock would earn in each state. Given this information,what’s Motor Music’s expected return and risk?

Economic State Probability Return

Fast growth 0.15 25%

Slow growth 0.60 15

Recession 0.20 −5

Depression 0.05 −20

SOLUTION:

Bailey can compute the expected return using equation 10-1:

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Then Bailey can compute the expected return by computing thestandard deviation using equation 10-2:

The expected return and standard deviation are 10.75 percent and11.76 percent, respectively. We could also show these equations in atable, such as

Similar to Problems 10-1, 10-2, 10-17, 10-18, 10-23, 10-24, Self-TestProblem 1

▼ TABLE 10.1 The Realized Average Annual Risk Premium for Stocks

1950to

2015

1950to

1959

1960to

1969

1970to

1979

1980to

1989

1990to

1999

2000to

2009

2010to

2015

Riskpremium

8.2% 18.8% 4.7% 1.2% 9.3% 14.1% −1.8% 13.4%

Realized risk premiums were very different in each decade. The recent decade even had a negative risk premium!

Source: S&P 500 Index and T-bill rate data.

Table 10.1 shows the average annual return on the S&P 500 Index minus the T-bill rate for different timeperiods. The remainder after we subtract the T-bill rate is the risk premium; in this case, it’s the market riskpremium—the reward for taking general (unsystematic) stock market risk. Since 1950, the average market riskpremium has been 8.3 percent per year. Over the long run, this is the reward for taking stock market risk. The actual,realized risk premium during particular decades has varied. The average risk premium has been as high as 18.8percent for the 1950s and as low as −1.8 percent during the 2000s. The performance in the 2000s is unusual; thestock market return has been so poor that it has not beaten the risk-free rate. Investors require a risk premium fortaking on market risk. But taking that risk also means that they will periodically experience poor returns.

time out!10-1 Describe the similarities between computing average return and expected return. Also, describe the similarities between

expected return risk and historical risk.10-2 Why would people take risks by investing their hard-earned money?

10.2 • MARKET RISK LG10-3

▼

How much risk should you take to achieve the return you want over time? In the previous chapter, we demonstratedthat individual stocks and different portfolios exhibit different levels of total risk. Recall that the rewards forcarrying risk apply only to the market risk (or undiversifiable) portion of total risk. But how do investors know howmuch of the 33.4 percent standard deviation of returns for Mattel Inc. is firm-specific risk and how much of thatdeviation is market risk? The answer to this important question will determine how much of a risk premiuminvestors should require for Mattel. The attempt to specify an equation that relates a stock’s required return to anappropriate risk premium is known as asset pricing.

The Market PortfolioThe best-known asset pricing equation is the capital asset pricing model, typically referred to as CAPM. Though manytheorists formulated theories that, in the end, supported the CAPM’s effectiveness, credit for the model goes toWilliam Sharpe and John Lintner. Sharpe eventually won a Nobel Prize for his work in 1990. (Lintner died in 1983,and Nobel Prizes are not awarded posthumously.) Today, both investors and corporate finance professionals useCAPM widely. In developing the CAPM, Lintner and Sharpe sought to emphasize the individual investor’s beststrategy to maximize returns for a given amount of market risk.

CAPM starts with modern portfolio theory. Remember from the previous chapter that when you combine securitiesinto a portfolio, you can find a set of portfolios that dominate all others. The best combinations possible use all therisky securities available (but not the risk-free asset) to create efficient frontier portfolios, which would lie along acurved line in risk/return space, as shown in Figure 10.1, panel A. These portfolios represent combinations ofvarious risky securities that give the highest expected return for each potential level of risk (i.e., they lie the furthest“up and to-the-left” that we can achieve when considering all possible combinations of the available riskysecurities).

FIGURE 10.1 Maximizing Expected Return

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In MPT, investors want to be on the efficient frontier (Panel A) because it gives them the highest expected return for each level of risk.However, after adding a riskless asset (Panel B), investors can then get portfolios on the straight line (shown), which offers a higher expectedreturn for each level of risk than the efficient frontier.

The idea of a risk premium in equation 10-3 implies a risk-free investment, like T-bills. Panel B shows where therisk-free asset would appear on the capital market line (CML). The risk-free asset must lie on the y-axis preciselybecause it carries no risk. Now we draw a line from the risk-free security to a point tangent to the efficient frontier.The CML relationship appears as a line because investments show a direct risk–reward relationship. You may recallfrom your economics classes that only one tangency point will be possible between this kind of curve and a straightline. The spot where the tangency occurs is called the market portfolio, which has a special significance. The marketportfolio represents ownership in all traded assets in the economy, so this portfolio provides maximumdiversification. You can locate your optimal portfolio on this line by owning various combinations of the risk-freesecurity and the market portfolio. If most of your money is invested in the market portfolio, then you will have aportfolio on the line that lies just to the left of the market portfolio dot in the graph. If you own just a little of themarket portfolio and hold mostly risk-free securities, then your portfolio will lie on the line near the risk-free security dot. For your investments to lie on the line to the right of the market portfolio, you wouldhave to invest all your money in the market portfolio, then borrow more money at the risk-free rate and invest theseadditional funds in the market portfolio. Borrowing money to invest is known as using financial leverage. Usingfinancial leverage increases the overall risk of the portfolio, which is illustrated in this figure as a higher standarddeviation.

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market risk premium The return on the market portfolio minus the risk-free rate. Risk premiums for specific firms are based on themarket risk premium.

asset pricing The process of directly specifying the relationship between required return and risk.

capital asset pricing model (CAPM) An asset pricing theory based on a beta, a measure of market risk.

capital market line (CML) The line on a graph of return and risk (standard deviation) from the risk-free rate through the market portfolio.

market portfolio In theory, the market portfolio is the combination of securities that places the portfolio on the efficient frontier and on aline tangent from the risk-free rate. In practice, the S&P 500 Index is used to proxy for the market portfolio.

financial leverage The extent to which debt securities are used by a firm.

Notice that if you had a portfolio on the efficient frontier (labeled “old portfolio”), you could do better. Instead ofowning the old portfolio, you can put some of your money in the market portfolio and some in the risk-free securityto obtain the “new portfolio.” See how the new portfolio dominates that old one? It carries the same risk level, butoffers a higher return. In fact, notice that the line drawn between the risk-free investment and the market portfoliodominates all of the efficient frontier portfolios (except the market portfolio itself). All portfolio allocations betweenthe risk-free security and the market portfolio constitute the capital market line. All investors should want to locatetheir portfolios on the CML, rather than the efficient frontier. Portfolios on the CML offer the highest expectedreturn for any level of desired risk, which the investor controls by deciding how much of the market portfolio andhow much of the risk-free asset to hold. Risk-averse investors can put more of their money in T-bills and less intothe market portfolio. Investors willing to take on higher risk for larger returns can put more of their money in themarket portfolio.

Beta, a Measure of Market Risk LG10-4The CML demonstrates that the market portfolio is crucial. Indeed, its return less the risk-free rate represents theexpected average market risk premium. The market portfolio features no firm-specific risk; all such risk isdiversified away. So the market portfolio carries only market risk. Thus the market portfolio’s risk factor allows usto compute a measure of firm-specific risk for any individual stock or portfolio. We can now examine the questionposed at the beginning of this section: “How much of Mattel’s total risk is attributable to market risk?” The standarddeviation of returns includes all of Mattel stock’s risk—it quantifies how much the stock price rises and falls. Themarket risk portion will rise and fall along with the market portfolio. If we subtract the market risk portion from thetotal risk measure, we’re left with firm-specific risk. This part of risk rises and falls in ways unrelated to marketchanges.

Remember that portfolio theory describes a measure—correlation—that measures how two stocks move togetherthrough time. Instead of measuring how any two stocks or portfolios move together, we now want to know how astock or portfolio moves relative to market portfolio movements. This measure is known as beta ( β). Beta measuresthe comovement between a stock and the market portfolio.

beta (β) A measure of the sensitivity of a stock or portfolio to market risk.

If Mattel’s total risk level is measured by its standard deviation, σMattel, then we can find the portion of this risk thatis attributable to the market in general by multiplying Mattel’s total risk by its correlation with the marketportfolio, σMattel × ρMattel, Market. The beta computation is scaled so that the market portfolio itself has abeta of one. The scaling is done by dividing by standard deviation of the market portfolio: σMattel × ρMattel, Market ÷σMarket.

1 Stocks with betas larger than one are considered riskier than the market portfolio, while betas of less thenone indicate lower risk. Mattel has a beta of 0.65, meaning that Mattel has low sensitivity to market risk. When themarket portfolio moves, you can expect Mattel stock to move in the same direction. Technically, you should expectMattel’s realized risk premium to be 35 percent less than the realized market risk premium.

Table 10.2 shows the beta for each of the 30 companies in the Dow Jones Industrial Average. Investors considermany of these companies high risk, like Du Pont (β = 2.01) and Disney (1.56). These firms’ stocks carry highmarket risk because the demand for their products is very sensitive to the overall economy’s strength. Investorsconsider other companies safe bets with low risk, like Walmart (0.19), Verizon (0.47), and McDonald’s (0.56).Many lower-beta firms sell consumer goods that we consider the necessities of life, which we will buy whether theeconomy is in recession or expansion. The demand for these products is price inelastic and not sensitive to economicconditions. Some companies have nearly the same risk as the market portfolio, like Pfizer (1.02), Microsoft (1.02),

▼

and Intel (1.02).

▼ TABLE 10.2 Dow Jones Industrial Average Stock Betas

Company Beta Company Beta

3M Company 1.08 Intel 1.02

American Express 1.21 Johnson & Johnson 0.89

Apple 1.35 JPMorgan Chase 1.20

Boeing 1.36 McDonald’s 0.56

Caterpillar 1.09 Merck 0.75

Cisco Systems 1.18 Microsoft 1.02

Chevron 1.17 Nike 0.63

Coca-Cola 0.77 Pfizer 1.02

Disney 1.56 Procter & Gamble 0.66

Du Pont de Nemours 2.01 Travelers 1.22

Exxon Mobil 0.92 United Technologies 1.18

General Electric 1.22 UnitedHealth Group 0.62

Goldman Sachs 1.35 Verizon Communications 0.47

Home Depot 0.96 Visa 1.06

IBM 0.66 Walmart Stores 0.19

Source: Yahoo! Finance, March 2, 2016.

The Security Market Line LG10-3Beta indicates the market risk that each stock represents to investors. So the higher the beta, the higher the riskpremium investors will demand to undertake that security’s market risk. Since beta sums up precisely what investorswant to know about risk, we often replace the standard deviation risk measure shown in Figure 10.1 with beta.Figure 10.2 shows required return versus beta risk. We call the line in this figure the security market line (SML),which illustrates how required return relates to risk at any particular time, all else held equal. The SML also showsthe market portfolio’s risk premium or any stock’s risk premium.

FIGURE 10.2 The Security Market Line Uses Beta as the Risk Measure

General Electric has higher risk than the overall stock market, so it should require a higher return.

security market line Similar to the capital market line except risk is characterized by beta instead of standard deviation.

When a stock like General Electric carries a beta greater than one, then its risk premium must be larger than themarket risk premium. A stock like Johnson & Johnson carries a lower beta than does the overall market; thereforeJohnson & Johnson would offer a lower risk premium to investors.

We can use the SML to show the relationship between risk and return for any stock or portfolio. To preciselyquantify this relationship, we need the equation for the SML. The equation of any line can be defined as y = b + mx,where b is the intercept and m is the slope. In this case, the y-axis is required return and the x-axis is beta. Theintercept is Rf. You may remember that the slope is the “rise over run” between two points on the line. The risebetween the risk-free security and the market portfolio is RM − Rf and the run is 1−0. Substituting into the lineequation results in the CAPM:

(10-4)

So, we have determined a way to estimate any stock’s required return once we have determined its beta. Considerthis: We expect the market portfolio to earn 12 percent and T-bill yields are 5 percent. Then General Electric’srequired return, with a β = 1.22, is 5% + 1.22 × (12%−5%) = 13.54 percent. Table 10.3 shows the 30 Dow JonesIndustrial Average stocks’ required return, using these same market and risk-free rate assumptions. Higher-riskcompanies have higher betas, and thus require higher returns.

▼ TABLE 10.3 Required Returns for DJIA Stocks

Company Required Return Company Required Return

3M Company 12.56% Intel 12.14%

American Express 13.47 Johnson & Johnson 11.23

Apple 14.45 JPMorgan Chase 13.40

Boeing 14.52 McDonald’s 8.92

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Caterpillar 12.63 Merck 10.25

Cisco Systems 13.26 Microsoft 12.14

Chevron 13.19 Nike 9.41

Coca-Cola 10.39 Pfizer 12.14

Disney 15.92 Proctor & Gamble 9.62

Du Pont de Nemours 19.07 Travelers 13.54

Exxon Mobil 11.44 United Technologies 13.26

General Electric 13.54 UnitedHealth Group 9.34

Goldman Sachs 14.45 Verizon Communications 8.29

Home Depot 11.72 Visa 12.42

IBM 9.62 Walmart Stores 6.33

Higher beta stocks require higher expected returns.

Assumptions: market return = 12% and risk-free rate = 5%.

Source: Yahoo! Finance, March 2, 2016.

Portfolio Beta As you might expect, a stock portfolio’s beta is the weighted average of the portfoliostocks’ betas.The portfolio beta equation resembles equation 9-7, which gives the return of a portfolio:

(10-5)

portfolio beta The combination of the individual company betas in an investor’s portfolio.

EXAMPLE 10-2

CAPM and Under- or OvervaluedStock LG10-3

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Say that you are a corporate CFO. You know that the risk-free rate iscurrently 4.5 percent and you expect the market to earn 11 percent thisyear. Through your own analysis of the firm, you think it will earn a 13.5percent return this year. If the beta of the company is 1.2, should youconsider the firm undervalued or overvalued?

SOLUTION:

You can compute shareholders’ required return with CAPM as 4.5% +1.2 × (11% − 4.5%) = 12.3%. Since you think the firm will actually earnmore than this required return, the firm appears to be currentlyundervalued. That is, its price must rise more then predicted by CAPMto obtain the return you estimated in your original analysis.

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Similar to Problems 10-7, 10-8, 10-19, 10-20, Self-Test Problem 3

With this equation, you can easily determine whether adding a particular stock to the portfolio will increase ordecrease the portfolio’s total market risk. If you add a stock with a higher beta than the existing portfolio, then thenew portfolio will carry higher market risk than the old one did. Although we can find the effects on total portfoliorisk of adding particular stocks using beta, the same is not necessarily true if we use standard deviations as our riskmeasure. The new stock, however risky, might have low correlations with the other stocks in the portfolio—offsetting (negative) correlations would reduce total risk.

Finding Beta LG10-4The CAPM is an elegant explanation that relates the return you should require for taking on various levels of marketrisk. Although CAPM provides many practical applications, you need a company’s beta to use those applications.Where or how can you obtain a beta? You have two ways. First, given the returns of the company and the marketportfolio, you can compute the beta yourself. Second, you can find the beta that others have computed throughfinancial information data providers.

Many financial outlets publish company betas. Websites that provide company betas for free include MSN Moneyand Yahoo! Finance, to name just a couple. For example, in March 2016, the beta these websites listed for Disneywere: MSN Money (1.41) and Yahoo! Finance (1.56). Note that these reported betas have some differences. Toknow why differences might arise, consider how you would go about gathering information and computing betayourself.

EXAMPLE 10-3 Portfolio Beta LG10-4

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

You have a portfolio consisting of 20 percent Boeing stock (β = 1.04),40 percent Hewlett- Packard stock (β = 1.54), and 40 percentMcDonald’s stock (β = 0.34). How much market risk does the portfoliohave?

SOLUTION:

Compute a beta for the portfolio. Using equation 10-5, the portfolio betais 0.2 × 1.04 + 0.4 × 1.54 + 0.4 × 0.34 = 0.96. Note that this portfoliocarries 4 percent less market risk than the general market does.

Similar to Problems 10-11, 10-12, 10-21, 10-22, 10-27, 10-28, Self-TestProblem 2

To compute your own beta, first obtain historical returns for the company of interest and of the marketportfolio. Then run a regression of the company return as the dependent variable and the market portfolioreturn as the independent variable. The resulting market portfolio return coefficient is beta. Many importantquestions may come to mind. First, what do you use as the market portfolio? People typically use a major stockindex like the S&P 500 Index to proxy for the market portfolio. Second, what time frame should you use? You canuse daily, weekly, monthly, or even annual returns. Using monthly returns is the most common. How long a timeseries is needed? As you will recall, statistical estimates become more reliable and valid as more data are used. Butyou will have to weigh those statistical advantages against the fact that companies change their business enterprisesand thus their risk levels over time. Using data from too long ago reflects risks that may no longer apply. Generallyspeaking, using time series data of three to five years is common. Whatever decisions you make to address thesequestions, be consistent by making the same decisions for all the company betas you compute. Table 10.4 shows thespreadsheet of a stock’s beta calculation. In this case, monthly returns from five years are used for the stock returnand a market index. The SLOPE() function of the spreadsheet directly computes the regression coefficient of

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interest. The beta estimation using the spreadsheet function is 0.83.

▼ TABLE 10.4 Compute Beta Using a Spreadsheet

The spreadsheet function SLOPE() finds the statistical relationship between a stock’s return and the market return. Considering thesemonthly returns for a stock and a market index, the beta of this stock is 0.83.

finance at work //: marketsAre Stocks Really Good or Bad?One of the basic financial theories tells us how we should view the relationship between risk and expected returns. In a nutshell, risk andexpected return are positively related. A high-risk investment needs to have a high expected return, or no one would want to buy thatinvestment. With this lack of demand, the investment’s price would drop until it offers new buyers a high expected return for the future.The higher return is the reward for taking the extra risk. Similarly, low-risk investments offer low expected returns.

To quantify risk, the finance industry tends to use two measures; volatility of returns and beta. The volatility, measured by variance orstandard deviation, tells us how much a return can deviate from the average return. Beta tells us how much market risk an investmenthas. These measures are very useful in assessing the risk of an investment or portfolio and what return premium should be expected fortaking risk.

However, people do not naturally think of risk within this financial theory framework. First, investors care less about how aninvestment’s return deviates from expectations and more about how the return may be lower than expected. In other words, a higherreturn than expected is not considered risky, only a lower return, or even, gulp, a loss, is viewed as risk.

Also, real people do not think in terms of the high risk/return versus the low risk/return scale. Instead, people think in terms of betteror worse. For example, three financial economists ran an experiment in which they asked high net worth individuals for either theirexpected return predictions or their risk assessment (both on a 0 to 10 scale) of over 200 of the Fortune 500 companies. When theycompiled all the responses, they found the relationship in the figure shown below. Notice anything odd? This shows that firms with lowrisk are expected to earn a high return. People act as if expected return and risk are negatively related! This is the opposite of financialtheory. Instead of evaluating firms within the framework that high expected return goes with high risk and low return goes with low risk,people seem to think in terms of good versus bad stocks. What are the characteristics of a “good” stock? Characteristics that seem goodare high expected return and low risk. When an investor feels a stock is good, then it is attributed with high return and low risk. When aninvestor feels the stock is bad, then it is attributed with low return and high risk. Psychologists call this perception or belief an “affect.”

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Unfortunately, thinking about risk and return from the affect framework causes investors to misunderstand the underlying dynamics ofactual expected return and risk. Thus, they may make poor decisions regarding risk and return.

Want to know more?Key Words to Search for Updates: affect, measuring investment risk, risk and behavioral finance

Source: Statman, Meir, Kenneth L. Fisher, and Deniz Anginer, “Affect in a Behavioral Asset-Pricing Model,” Financial Analysts Journal,March/April 2008, Figure 3: Relationship between Expected Return Scores and Risk Scores.

Concerns about BetaConsider the estimation choices just mentioned. Say you estimate a firm’s beta using monthly data for five years andthe Dow Jones Industrial Average return as the market portfolio. Suppose that the result is a beta of 1.3. Then youtry again using weekly returns for three years and the return from the S&P 500 Index as the market portfolio’s yield,resulting in a beta of 0.9. These estimates are quite different and would create a large variation in required return ifyou plugged them into the CAPM. So, which is the more accurate estimate? Unfortunately, we may not be able todetermine which is most representative, or “true.” In general, you may estimate a little different beta using differentmarket portfolio proxies, different return intervals (like monthly returns versus annual returns), and different timeperiods. Problem 10-31 at the end of this chapter explores these differences.

In addition to these estimation problems, a company can change its risk level, and thus its beta, bychanging the way it operates within its business, by expanding into new businesses, and/or by changing itsdebt load. So even if beta is an accurate measure of what the firm’s risk level was in the past, does it apply to thefuture? Beta’s applicability will depend on the firm’s future plans.

Both financial managers and investors share these concerns about beta. In the end, beta’s usefulness depends on itsreliability. Unfortunately, beta’s empirical record is not as good as we would like. We should expect that companieswith high betas yield higher returns than companies with low betas. On average, though, this does not turn out to bethe case. A company’s beta does not appear to predict its future return very well. Since characterizing the risk–returnrelationship is so important, finance researchers have introduced other asset pricing models. One promising modeladds more risk factors to the predictive relationship other than just market risk. Firm size and book-to-market ratiohave had some success predicting returns, so new models often include factors derived from these characteristicsalong with beta as a measure of market risk.

time out!10-3 Explain why portfolios that lie on the capital market line offer better risk–return trade-offs than those that lie on the efficient

frontier.

10-4 Examine the betas in Table 10.2. Which seem about right to you and which seem to indicate too much or too little risk forthat firm?

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10.3 • CAPITAL MARKET EFFICIENCY LG10-5The risk and return relationship rests on an underlying assumption that stock prices are generally “correct”—theyare not predictably too high or too low. Imagine having a system that identified undervalued stocks with low risks(i.e., relatively high returns with a low beta). Because those stocks are undervalued, they will earn you a high return,on average, as their stock prices rise to their correct value. Note that the CAPM’s risk-return relationship would beincorrect. You would be consistently getting high returns with low risk. On the other hand, if you consistentlypicked overvalued stocks, you wouldn’t be earning enough return to compensate you for the risks you are taking.Investors move their money to the best alternatives by selling overvalued stocks and buying undervalued stocks.This causes the prices of the overvalued stocks to drop and the prices of the undervalued stocks to rise until bothstocks’ returns stand more in line with their riskiness. Thus, the risk–return relationship relies on the idea that pricesare generally accurate.

What conditions are necessary for an efficient market? Efficient, or perfectly competitive, markets featureMany buyers and sellers.

No prohibitively high barriers to entry.Free and readily available information available to all participants.

Low trading or transaction costs.

Are these conditions met for the U.S. stock market? Certainly millions of stock investors trade every day, buyingand selling securities. With discount brokers and online traders, the costs to trade are fairly minimal and present noreal barriers to enter the market. Information is increasingly accessible from many sources and trading philosophies,and commission costs and bid–ask spreads have steadily declined. With millions of the larger companies’ shares(say the S&P 500) of stock trading every day, the U.S. stock exchanges appear to meet efficiency conditions. Butother segments of the market, like those exchanges that trade in penny stocks, feature very thin trading. The prices ofthese very small companies’ stock may not be fair and these equities may be manipulated in fraudulent scams. In the1970s and 1980s, penny stock king Meyer Blinder and his firm Blinder-Robinson was known as “blind ’em and rob’em” as they practiced penny stock price manipulation to rob many small investors of their entire investments inthese small markets. These days, penny stock price manipulation is typically conducted through e-mail and Webposting scams.

efficient market A securities market in which prices fully reflect available information on each security.

penny stocks The stocks of small companies that are priced below $1 per share.

Efficient Market HypothesisOur concept of market efficiency provides a good framework for understanding how stock prices change over time.This theory is described in the efficient market hypothesis (EMH), which states that security prices fully reflect allavailable information. At any point in time, the price for any stock or bond reflects the collective wisdom of marketparticipants about the company’s future prospects. Security prices change as new information becomes available.Since we cannot predict whether new information about a company will be good news or bad news, we cannotpredict whether its stock price will go up or go down. This makes short-term stock-price movements unpredictable.But in the longer run, stock prices will adjust to their proper level as market participants gather and digest allavailable information.

The EMH brings us to the question of what type of information is embedded within current stock prices. Segmentinginformation into three categories leads to the three basic levels of market efficiency, described as:

1. Weak-form efficiency—current prices reflect all information derived from trading. This stock market information generally includescurrent and past stock prices and trading volume.

2. Semistrong-form efficiency—current prices reflect all public information. This includes all information that has already been revealedto the public, like financial statements, news, analyst opinions, and so on.

3. Strong-form efficiency—current prices reflect all information. In addition to public information, prices reflect the privately heldinformation that has not yet been released to the public, but may be known to some people, like managers, accountants, auditors,and so on.

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▼

efficient market hypothesis (EMH) A theory that describes what types of information are reflected in current stock prices.

public information The set of information that has been publicly released. Public information includes data on past stock prices andvolume, financial statements, corporate news, analyst opinions, etc.

privately held information The set of information that has not been released to the public but is known by few individuals, likelycompany insiders.

Each of the EMH’s three forms rests on different assumptions regarding the extent of information that isincorporated into stock prices at any point in time. A fourth possibility—that markets may not be efficient and pricesmay not reflect all the information known about a company—also arises.

The weak-form efficiency level involves the lowest information hurdle, stating that stock prices reflect all past priceand trading volume activity. If true, this level of efficiency would have important ramifications. A segment of theinvestment industry uses price and volume charts to make investment timing decisions. Technical analysis has alarge following and its own vocabulary of patterns and trends (resistance, support, breakout, momentum, etc.). If themarket is at least weak-form efficient, then prices already reflect this information and these activities would notresult in useful predictions about future price changes, and thus would be a waste of time. Indeed, the people whomake the most money from price charting services are the people who sell the services, not the investors who buyand use those services.

The semistrong-form efficiency level assumes that stock prices include all public information. Notice that past stockprices and volumes are publicly available information, so this level includes the weak form as a subset. Importantinvestment implications arise if markets are efficient to public information. Many investors conduct security analysisin which they obtain financial data and other public information to assess whether a company’s stock is undervaluedor overvalued. But in a semistrong-form efficient market, stock prices already reflect this information and are thus“correct.” Using only public information, you would not be able to determine whether a stock is misvalued becausethat information is already reflected in the price.

If prices reflect all public information, then those prices will react as traders hear new (or private) information.Consider a company that announces surprisingly good quarterly profits. Traders and investors will have factored theold profit expectations into the stock price. As they incorporate the new information, the stock price will quickly riseto a new and accurate price as shown in the solid black line in Figure 10.3. Note that the stock price was$35 before the announcement and $40 immediately following. If you tried to buy the stock after hearingthe news, then you would have bought at $40 and not received any benefit of the good quarterly profit news. On theother hand, if the market is not semistrong-form efficient, then the price might react quickly, but not accurately. Thedashed red line shows a reaction in a non-semistrong efficient market where the price continues to drift up well afterthe announcement. This gradual drift to the “correct” price indicates that the market initially underreacted to thenews. In this case, you could have bought the stock after the announcement and still earned a profit. The dashed blueline shows an overreaction to the firm’s better-than-expected profits announcement. If markets either consistentlyunderreact or consistently overreact to announcements that would change stock prices (earnings, stock split,dividend, etc.), then we would believe that the market is not semistrong efficient.

FIGURE 10-3 Potential Price Reaction to a Good News Announcement

Stock prices react quickly to news, but do they react accurately?

The strong-form market efficiency level presents the highest hurdle to test market reaction to information. Thestrong-form level includes information considered by the weak-form, the semistrong-form, as well as privatelyknown information. People within firms, like CEOs and CFOs, know information that has not yet been released tothe public. They may trade on this privately held, or insider, information and their trading may cause stock prices tochange as it reflects that private information. In this way, stock prices could reflect even privately knowninformation. Note that the firm managers, accountants, and auditors know several days in advance that a firm hasearned unexpectedly high quarterly profits. If the stock price already incorporated this closely held privateknowledge, then the big price reaction shown in Figure 10.3 would not occur.

So, is the stock market efficient? If it is, at what level? This has been a hotly debated topic for decades and continuesto be. It is not likely that the market is strong-form efficient. Since insider trading is punished, insider informationmust be valuable. However, much evidence suggests that the market could be weak-form or semistrong-formefficient. Of course, we also have evidence that the market is not efficient at any of the three levels. We will explorethis more in the following section.

Behavioral FinanceThe argument for the market being efficient works as follows: Many individual and professional investors constantlylook for mispriced stocks. If they find a stock that is undervalued, they will buy it and drive up its price until it’scorrectly priced. Likewise, investors would sell an overpriced stock, driving down its price until it’s correctlyvalued. With so many investors looking for market “mistakes,” it’s unlikely that any mispriced stock opportunitieswill be left in the market.

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The argument against the market being efficient is equally convincing. The market comprises many peopletransacting with one another. When someone makes trading decisions influenced by emotion or psychological bias,those decisions may not seem rational. When many people fall under such influences, their trading decisions mayactually drive stock prices away from the correct price as emotion carries the traders away from rationality. Forexample, many people believe that investors were “irrationally exuberant” about technology stocks in the late 1990s—and that their buying excitement drove prices to an artificially high level. In 2000, the excitement wore off andtech stock prices plummeted. Whenever a set of stock prices go unnaturally high and subsequently crashdown, the market experiences what we call a stock market bubble.

In the past couple of decades, finance researchers have studied behavioral finance and found that people often behavein ways that are very likely “irrational.” At times, investors appear to be too optimistic, as though they are lookingthrough rose-colored glasses. At other times they appear to be too pessimistic. Common investment decisions aren’tnecessarily optimal ones, which flies in the face of the economist’s expectation of rational economic actors. Perhaps,then, capital markets don’t represent perfectly competitive or efficient markets if buyers and sellers do not alwaysmake rational choices.

It may take many biased investors to move a stock’s price enough that it would be considered a pricing mistake.However, the important decisions in a company are typically made by just one CEO or a management team. Thus,their biases can have a direct impact on decisions involving hundreds of millions, or even billions of dollars. In otherwords, the contribution of behavioral finance to economic decision making is likely to be even more important incorporate behavior than market behavior. For example, consider the psychological concept of overconfidence. One ofthe most pervasive biases, overconfidence describes a tendency for people to overestimate the accuracy of theirknowledge and underestimate the risks of a decision. These problems can adversely affect important decisions ofinvestment (i.e., acquiring other firms) and financing (i.e., issuing new stock or bonds).

stock market bubble Investor enthusiasm causes an inflated bull market that drives prices too high, ending in a dramatic collapse inprices.

behavioral finance The study of the cognitive processes and biases associated with making financial and economic decisions.

overconfidence Overconfidence is used to describe three psychological observations. First, people are miscalibrated on understandingthe precision of their knowledge. Second, people have a tendency to underestimate risks, and third, people tend to believe that they arebetter than average at tasks they are familiar with.

time out!10-5 Can the market be semistrong-form efficient but not weak-form efficient? Explain.10-6 If the market usually overreacts to bad news announcements, what would your return be like if you bought after you heard

the bad news?

10.4 • IMPLICATIONS FOR FINANCIAL MANAGERS LG10-6Financial managers must understand the crucial relationship between risk and return for several reasons. First, whilethe relationship between risk and return is demonstrated here using the capital markets, it equally applies to manybusiness decisions. A firm’s product mix, marketing campaign combination, and research and developmentprograms all entail risk and potential rewards. Being able to understand and characterize these decisions within arisk and return framework can help managers make better decisions. In addition, managers must understand whatreturn their stockholders require at various times of firm operations. After all, a firm must receive enough revenuefrom its variously risky activities to pay its business and debt costs and reward the owners (the stockholders).Managers must thus include the return to shareholders when they analyze new business opportunities.

Firms and capital markets also interact directly. For example, a good understanding of market efficiency helpsmanagers understand how their stock prices will react to different types of decisions (like dividend changes) and

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news announcements (like unexpectedly high or low profitability). In fact, many managers own company stock andare thus compensated through programs that rely on the stock price, like restricted stock and executive stock options.Companies also periodically issue (sell) additional shares of stock to raise more capital, and these sales depend uponmarket efficiency assumptions. The firm would not want to sell additional shares if the stock price is too low (i.e.,undervalued). They would want to sell more shares at any time that they thought their shares are overvalued. Othertimes, firms repurchase (buy back) shares of stock. The firm might want to do this if its shares were undervalued,but not if its shares were overvalued. Of course, valuation is not an issue if security markets are efficient.

restricted stock A special type of stock that is not transferable from the current holder to others until specific conditions are satisfied.

executive stock options Special rights given to corporate executives to buy a specific number of shares of the company stock at afixed price during a specific period of time.

Using the Constant-Growth Model for Required Return LG10-7For decades, financial managers have used the CAPM to compute shareholders’ required return. Given recentconcerns about beta’s limitations, many see the CAPM as a less useful model for calculating appropriate returns.Some have turned instead to another model useful for computing required return—the constant-growth modeldiscussed in Chapter 8. We can arrange the terms of that model as

(10-6)

finance at work //: marketsBubble TroubleMany professionals criticize the EMH because the overall market sometimes seems too high or too low. A very dramatic example of themarket level being artificially high is the market bubble. During a market bubble, the market quickly inflates on rampant speculation andsubsequently crashes. Investors who buy near the peak of the bubble risk losing nearly all their investment.

One of the United States’ earliest stock market bubbles was the bubble and crash of 1929. Note from the figure that the DJIA startedin 1927 at around 160. By October 1929, the DJIA had reached nearly 400 and then crashed. By mid-1932, the DJIA had fallen to the40s. The sustained fall coincided with an economic depression. Panel A of the figure also shows a price bubble in gold. The price of goldwas $230 per ounce in January 1979. The late 1970s and early 1980s saw double-digit inflation throughout the economy, and manyinvestors felt that gold represented a safe and inflation-proof investment. Just one year later, the price had skyrocketed to $870. It thenfell below $300 in less than two and a half years. Could gold be in another bubble? In late 2008, gold’s price was $750; it shot to $1,500in late 2011.

See the spectacular tech bubble during the 1990s? The NASDAQ 100, which started in 1985 at 250, soared to a peak of 4,816.35 onMarch 24, 2000. It then fell to less than 1,000 in two and a half years. The rise and fall of the NASDAQ 100 seems much morepronounced than the Japanese stock bubble of the 1980s. From a January 2, 1985, start at 11,543, the Nikkei 225 soared to a closinghigh of 38,916 on December 29, 1989. The bubble then burst and the Japanese stock market plummeted. The Nikkei has yet to fullyrecover, trading today at around the 16,500 level.

EMH critics do not believe that the entire stock market, or a substantial segment of it, can be correctly valued before, during, or aftera bubble. It certainly appears to be overvalued during the time the bubble is inflating.

Want to know more?Key Words to Search for Updates: stock bubble, irrational exuberance

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Of course, this model assumes that the stock is efficiently priced. This model holds an advantage in that it usescurrent firm data (dividend, D1, and price, P0) and a simple forward estimate (growth, g) to assess what investorscurrently expect the stock to return, i. For example, Table 10.2 shows McDonald’s beta as 0.56. Using this beta,Table 10.3 shows that shareholders require only 8.92 percent return to hold McDonald’s stock, given itslow risk profile. You may find it hard to believe that McDonald’s owners (the shareholders) expect such alow return from one of the world’s most profitable firms. So perhaps this is a case in which the CAPM result isn’tvery useful. Applying the constant-growth model looks something like this: McDonald’s is expected to pay a $3.56dividend this year and the stock price currently stands at $117 per share. Financial analysts believe McDonald’s willgrow at 9.50 percent per year for the next five years. The constant growth rate model suggests that McDonald’sshareholders expect a 12.54 percent return [= ($3.56 ÷ $117) + 0.095]. So, which required return seems more likely,the 8.92 percent computed from CAPM or the 12.54 percent suggested by the constant-growth model? It is likelythat McDonald’s investors are expecting closer to the 13 percent return than the 9 percent estimate.

Financial managers need an estimate of their shareholders’ required return in order to make appropriate decisionsabout their companies’ future growth. Good financial managers will compute shareholders’ required return using asmany methods as they can to determine the most realistic value possible.

EXAMPLE 10-4 Required Return LG10-7

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

Consider that the required returns for 3M, Home Depot, and Hewlett-Packard are 12.56 percent, 12.07 percent, and 15.78 percent,respectively. These expectations may seem quite far apart, consideringthat all three firms are in the DJIA and are leaders in their marketsectors. Use the following information to compute the constant-growthmodel estimate of the required return:

ExpectedDividend

CurrentPrice

Analyst GrowthEstimate

3MCompany

$2.54 $105.70 10.4%

HomeDepot

1.56 70.10 14.5

Hewlett- 0.53 22.00 0.25

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Hewlett-Packard

0.53 22.00 0.25

SOLUTION:

You can now use equation 10-6 to find each company’s required returnas

The 3M estimates using CAPM and the constant-growth rate model aresimilar. The constant-growth rate model estimate is more than 4percent higher than the CAPM estimate for Home Depot, but far lowerfor Hewlett-Packard.

Similar to Problems 10-15, 10-16, 10-29, 10-30, Self-Test Problem 3

time out!10-7 Why is the shareholders’ required return important to corporate managers?

Get Online

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Log in to your Connect course for study materials including self-test problems with solutions, answers tothe Time Out quizzes, guided example videos, and more.

Your Turn…Questions

1. Consider an asset that provides the same return no matter what economic state occurs. What would be thestandard deviation (or risk) of this asset? Explain. (LG10-1)

2. Why is expected return considered “forward-looking”? What are the challenges for practitioners to utilizeexpected return? (LG10-1)

3. In 2000, the S&P 500 Index earned −9.1 percent while the T-bill yield was 5.9 percent. Does this mean themarket risk premium was negative? Explain. (LG10-2)

4. How might the magnitude of the market risk premium impact people’s desire to buy stocks? (LG10-2)

5. Describe how adding a risk-free security to modern portfolio theory allows investors to do better than theefficient frontier. (LG10-3)

6. Show on a graph like Figure 10.2 where a stock with a beta of 1.3 would be located on the security market line.Then show where that stock would be located if it is undervalued. (LG10-3)

7. Consider that you have three stocks in your portfolio and wish to add a fourth. You want to know if the fourthstock will make the portfolio riskier or less risky. Compare and contrast how this would be assessed usingstandard deviation versus market risk (beta) as the measure of risk. (LG10-3)

8. Describe how different allocations between the risk-free security and the market portfolio can achieveany level of market risk desired. Give examples of a portfolio from a person who is very risk averseand a portfolio for someone who is not so averse to taking risk. (LG10-3)

9. Cisco Systems has a beta of 1.25. Does this mean that you should expect Cisco to earn a return 25 percent

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higher than the S&P 500 Index return? Explain. (LG10-4)

10. Note from Table 10.2 that some technology-oriented firms (Apple) in the Dow Jones Industrial Average havehigh market risk while others (Intel and Verizon) have low market risk. How do you explain this? (LG10-4)

11. Find a beta estimate from three different sources for General Electric (GE). Compare these three values. Whymight they be different? (LG10-4)

12. If you were to compute beta yourself, what choices would you make regarding the market portfolio, the holdingperiod for the returns (daily, weekly, etc.), and the number of returns? Justify your choices. (LG10-4)

13. Explain how the concept of a positive risk–return relationship breaks down if you can systematically findstocks that are overvalued and undervalued. (LG10-5)

14. Determine what level of market efficiency each event is consistent with the following: (LG10-5)

a. Immediately after an earnings announcement the stock price jumps and then stays at the new level.b. The CEO buys 50,000 shares of his company and the stock price does not change.c. The stock price immediately jumps when a stock split is announced, but then retraces half of the gain over the

next day.d. An investor analyzes company quarterly and annual balance sheets and income statements looking for

undervalued stocks. The investor earns about the same return as the S&P 500 Index.

15. Why do most investment scams conducted over the Internet and e-mail involve penny stocks instead of S&P500 Index stocks? (LG10-5)

16. Describe a stock market bubble. Can a bubble occur in a single stock? (LG10-5)

17. If stock prices are not strong-form efficient, what might be the price reaction to a firm announcing a stockbuyback? Explain. (LG10-6)

18. Compare and contrast the assumptions that need to be made to compute a required return using CAPM and theconstant-growth model. (LG10-7)

19. How should you handle a case where required return computations from CAPM and the constant-growth modelare very different? (LG10-7)

ProblemsBASIC PROBLEMS

10-1 Expected Return Compute the expected return given these three economic states, their likelihoods, andthe potential returns: (LG10-1)

Economic State Probability Return

Fast growth 0.3 40%

Slow growth 0.4 10

Recession 0.3 −25

10-2 Expected Return Compute the expected return given these three economic states, their likelihoods,and the potential returns: (LG10-1)

Economic State Probability Return

Fast growth 0.2 35%

Slow growth 0.6 10

Recession 0.2 −30

10-3 Required Return If the risk-free rate is 3 percent and the risk premium is 5 percent, what is the required

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return? (LG10-2)10-4 Required Return If the risk-free rate is 4 percent and the risk premium is 6 percent, what is the required

return? (LG10-2)10-5 Risk Premium The average annual return on the S&P 500 Index from 1986 to 1995 was 15.8 percent. The

average annual T-bill yield during the same period was 5.6 percent. What was the market risk premiumduring these 10 years? (LG10-2)

10-6 Risk Premium The average annual return on the S&P 500 Index from 1996 to 2005 was 10.8 percent. Theaverage annual T-bill yield during the same period was 3.6 percent. What was the market risk premiumduring these 10 years? (LG10-2)

10-7 CAPM Required Return Hastings Entertainment has a beta of 0.65. If the market return is expected to be11 percent and the risk-free rate is 4 percent, what is Hastings’ required return? (LG10-3)

10-8 CAPM Required Return Nanometrics, Inc., has a beta of 3.15. If the market return is expected to be 10percent and the risk-free rate is 3.5 percent, what is Nanometrics’ required return? (LG10-3)

10-9 Company Risk Premium Netflix, Inc., has a beta of 3.61. If the market return is expected to be 13 percentand the risk-free rate is 3 percent, what is Netflix’s risk premium? (LG10-3)

10-10 Company Risk Premium Paycheck, Inc., has a beta of 0.94. If the market return is expected to be 11percent and the risk-free rate is 3 percent, what is Paycheck’s risk premium? (LG10-3)

10-11 Portfolio Beta You have a portfolio with a beta of 1.35. What will be the new portfolio beta if youkeep 85 percent of your money in the old portfolio and 5 percent in a stock with a beta of 0.78?(LG10-3)

10-12 Portfolio Beta You have a portfolio with a beta of 1.1. What will be the new portfolio beta if youkeep 85 percent of your money in the old portfolio and 15 percent in a stock with a beta of 0.5?(LG10-3)

10-13 Stock Market Bubble The NASDAQ stock market bubble peaked at 4,816 in 2000. Two and a halfyears later it had fallen to 1,000. What was the percentage decline? (LG10-5)

10-14 Stock Market Bubble The Japanese stock market bubble peaked at 38,916 in 1989. Two and a halfyears later it had fallen to 15,900. What was the percentage decline? (LG10-5)

10-15 Required Return Paccar’s current stock price is $48.20 and it is likely to pay a $0.80 dividend nextyear. Since analysts estimate Paccar will have an 8.8 percent growth rate, what is its required return?(LG10-7)

10-16 Required Return Universal Forest’s current stock price is $57.50 and it is likely to pay a $0.26dividend next year. Since analysts estimate Universal Forest will have a 9.5 percent growth rate, whatis its required return? (LG10-7)

INTERMEDIATE PROBLEMS

10-17 Expected Return Risk For the same economic state probability distribution in problem 10-1,determine the standard deviation of the expected return. (LG10-1)

Economic State Probability Return

Fast growth 0.3 40%

Slow growth 0.4 10

Recession 0.3 −25

10-18 Expected Return Risk For the same economic state probability distribution in problem 10-2,determine the standard deviation of the expected return. (LG10-1)

Economic State Probability Return

Fast growth 0.2 35%

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Slow growth 0.6 10

Recession 0.2 −30

10-19 Undervalued/Overvalued Stock A manager believes his firm will earn a 14 percent return next year.His firm has a beta of 1.5, the expected return on the market is 12 percent, and the risk-free rate is 4percent. Compute the return the firm should earn given its level of risk and determine whether themanager is saying the firm is undervalued or overvalued. (LG10-3)

10-20 Undervalued/Overvalued Stock A manager believes his firm will earn a 14 percent return next year.His firm has a beta of 1.2, the expected return on the market is 11 percent, and the risk-free rate is 5percent. Compute the return the firm should earn given its level of risk and determine whether themanager is saying the firm is undervalued or overvalued. (LG10-3)

10-21 Portfolio Beta You own $10,000 of Olympic Steel stock that has a beta of 2.2. You also own $7,000of Rent-a-Center (beta = 1.5) and $8,000 of Lincoln Educational (beta = 0.5). What is the beta of yourportfolio? (LG10-3)

10-22 Portfolio Beta You own $7,000 of Human Genome stock that has a beta of 3.5. You also own $8,000of Frozen Food Express (beta = 1.6) and $10,000 of Molecular Devices (beta = 0.4). What is the betaof your portfolio? (LG10-3)

ADVANCED PROBLEMS

10-23 Expected Return and Risk Compute the expected return and standard deviation given these foureconomic states, their likelihoods, and the potential returns: (LG10-1)

Economic State Probability Return

Fast growth 0.30 60%

Slow growth 0.50 13

Recession 0.15 −15

Depression 0.05 −45

10-24 Expected Return and Risk Compute the expected return and standard deviation given these foureconomic states, their likelihoods, and the potential returns: (LG10-1)

Economic State Probability Return

Fast growth 0.25 50%

Slow growth 0.55 11

Recession 0.15 −15

Depression 0.05 −50

10-25 Risk Premiums You own $10,000 of Denny’s Corp. stock that has a beta of 2.9. You also own$15,000 of Qwest Communications (beta = 1.5) and $5,000 of Southwest Airlines (beta = 0.7).Assume that the market return will be 11.5 percent and the risk-free rate is 4.5 percent. What is themarket risk premium? What is the risk premium of each stock? What is the risk premium of theportfolio? (LG10-3)

10-26 Risk Premiums You own $15,000 of Opsware, Inc., stock that has a beta of 3.8. You also own$10,000 of Lowe’s Companies (beta = 1.6) and $10,000 of New York Times (beta = 0.8). Assume thatthe market return will be 12 percent and the risk-free rate is 6 percent. What is the market riskpremium? What is the risk premium of each stock? What is the risk premium of the portfolio? (LG10-3)

10-27 Portfolio Beta and Required Return You hold the positions in the following table. What is the betaof your portfolio? If you expect the market to earn 12 percent and the risk-free rate is 3.5 percent, whatis the required return of the portfolio? (LG10-3)

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Price Shares Beta

Amazon.com $40.80 100 3.8

Family Dollar Stores 30.10 150 1.2

McKesson Corp. 57.40 75 0.4

Schering-Plough Corp. 23.80 200 0.5

10-28 Portfolio Beta and Required Return You hold the positions in the following table. What is the betaof your portfolio? If you expect the market to earn 12 percent and the risk-free rate is 3.5 percent, whatis the required return of the portfolio? (LG10-3)

Price Shares Beta

Advanced Micro Devices $ 14.70 300 4.2

FedEx Corp. 120.00 50 1.1

Microsoft 28.90 100 0.7

Sara Lee Corp. 17.25 150 0.5

10-29 Required Return Using the information in the table, compute the required return for each companyusing both CAPM and the constant-growth model. Compare and discuss the results. Assume that themarket portfolio will earn 12 percent and the risk-free rate is 3.5 percent. (LG10-3, LG10-7)

Price Upcoming Dividend Growth Beta

US Bancorp $36.55 $1.60 10.0% 1.8

Praxair 64.75 1.12 11.0 2.4

Eastman Kodak 24.95 1.00 4.5 0.5

10-30 Required Return Using the information in the table, compute the required return for each companyusing both CAPM and the constant-growth model. Compare and discuss the results. Assume that themarket portfolio will earn 11 percent and the risk-free rate is 4 percent. (LG10-3, LG10-7)

Price Upcoming Dividend Growth Beta

Estee Lauder $47.40 $0.60 11.7% 0.75

Kimco Realty 52.10 1.54 8.0 1.3

Nordstrom 5.25 0.50 14.6 2.2

10-31 Spreadsheet Problem As discussed in the text, beta estimates for one firm will vary depending onvarious factors such as the time over which the estimation is conducted, the market portfolio proxy,and the return intervals. You will demonstrate this variation using returns for Microsoft.

a. Using all 45 monthly returns for Microsoft and the two stock market indexes, compute Microsoft’sbeta using the S&P 500 Index as the market proxy. Then compute the beta using the NASDAQindex as the market portfolio proxy. Compare the two beta estimates.

b. Now estimate the beta using only the most recent 30 monthly returns and the S&P 500 Index.Compare the beta estimate to the estimate in part (a) when using the S&P 500 Index and all 45monthly returns.

c. Estimate Microsoft’s beta using the following quarterly returns. Compare the estimate to the onesfrom parts (a) and (b).

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10-32 Spreadsheet Problem Build a spreadsheet that automatically computes the expected market returnand risk for different assumptions about the state of the economy.

a. First, create a spreadsheet like the one shown below and compute the expected return and standarddeviation.

b. Compute the expected return and risk for the following two scenarios:

NotesCHAPTER 10

1. A mathematically equivalent equation for beta is , where cov() is the covariance between the stock andmarket portfolio returns, and var() is the variance of the market portfolio.

Part Six

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chapter elevenCalculating the cost of capital

n the previous two chapters, we discussed investors’ required return given a particular risk profile. Inthis chapter, we examine the question from the firm’s point of view: How much must the firm pay tofinance its operations and expansions using debt and equity sources? Firms use a combination of debt

and equity sources to fund their operations, projects, and any expansions they may undertake. In Chapter14, we’ll explore the factors that managers consider as they choose the optimal capital structure mix. Fornow, we’ll assume that management has chosen the optimal mix for us, and that it’s our job to implementit.

LG11-1

As we’ve seen in previous chapters, investors face different kinds of risks associated with debt,preferred stock, and equity. As a result, their required rates of return for each debt or equity source differas well. So, as the firm uses a combination of different financing sources, we must calculate the investors’average required rate of return to use as the cost of capital for evaluating decisions about investing thefirm’s capital. Since firms seldom use equal amounts of debt and equity capital sources, we will need tocalculate this as a weighted average, with weights based on the proportion of debt and equity capitalused.

As we’ll see, we can measure such a weighted-average cost of capital (WACC) in a variety of situations and

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for a number of purposes. For example, if we’re interested in determining the average rate of return thatthe firm must earn from existing operations when we don’t expect the firm’s capital structure to change,we can calculate the WACC using the firm’s current capital structure and existing component costs; however,if we’re trying to determine the average rate of return that we would need to earn from a new project inorder for it to add value to the firm, we would want to use the project’s proposed capital structure and itscomponent costs.

weighted-average cost of capital (WACC) The weighted-average after-tax cost of the capital used by a firm, with weights set equal tothe relative percentage of each type of capital used.

component costs The individual costs of each type of capital—bonds, preferred stock, and common stock.

LEARNING GOALS

LG11-1 Understand the relationship of cost of capital to the investor’s required return.LG11-2 Use the weighted-average cost of capital (WACC) formula to calculate a project’s cost of capital.LG11-3 Explain how the firm chooses among estimating costs of equity, preferred stock, and debt.LG11-4 Calculate the weights used for WACC projections.LG11-5 Identify which elements of WACC are used to calculate a project-specific WACC.LG11-6 Evaluate trade-offs between a firmwide WACC and a divisional cost of capital approach.LG11-7 Distinguish subjective and objective approaches to divisional cost of capital.LG11-8 Demonstrate how to adjust the WACC to reflect flotation costs.

viewpointsbusiness APPLICATIONMP3 Devices, Inc., is about to launch a new project to create and market a combination MP3 player-video projector. MP3 Devicescurrently uses a particular mixture of debt, common stock, and preferred shares in its capital structure, but the firm is thinking of usingthe launch of the new project as an opportunity to change that capital structure.

The new project will be funded with 40 percent debt, 10 percent preferred stock, and 50 percent common stock. MP3 Devicescurrently has 10 million shares of common stock outstanding, selling at $18.75 per share, and expects to pay an annual dividend of$1.35 one year from now, after which future dividends are expected to grow at a constant 6 percent rate. MP3’s debt consists of 20-year, 10 percent annual coupon bonds with a face value of $150 million and a market value of $165 million, and the company’s capitalmix also includes 100,000 shares of 10 percent preferred stock selling at par.

If MP3 Devices faces a marginal tax rate of 34 percent, what weighted average cost of capital should it use as it evaluates thisproject? (See the solution at the end of the book.)

One important point about the component costs to be used in the firm’s computation of the averagerequired rate of return is that dividends paid to either common or preferred stockholders are not taxdeductible, but interest paid to debt holders is. Therefore, paying a certain rate of return to stockholderscosts the firm that same rate, but paying the same interest rate to debt holders actually costs the firm lessthan the rate paid. The reason is that, since interest on debt is tax deductible, paying rate iD to debtholders is, for all intents and purposes, subsidized by the government: If the firm hadn’t paid out thatinterest rate to the debt holders, it would have had to have paid out taxes to the government on themoney it used to pay the interest. So, effectively, the firm’s effective after-tax interest cost will be equal tothe interest rate paid on debt multiplied by one minus the firm’s relevant tax rate.

For example, if a firm pays a 10 percent coupon on $1 million in debt while it is subject to a 35 percenttax rate, then each coupon payment will be equal to 0.10 × $1,000,000 = $100,000, but that $100,000 ininterest, being tax deductible, will reduce the firm’s tax bill by 0.35 × $100,000 = $35,000. So paying$100,000 in interest saves the firm $35,000 in taxes, making the effective after-tax cost of debt equal to$100,000 − $35,000 = $65,000 and the effective after-tax interest rate equal to 10% × (1 − 0.35) = 6.5%.

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■

11.1 • THE WACC FORMULA LG11-2The average cost per dollar of capital raised is called the weighted-average cost of capital (WACC). We calculateWACC using equation 11-1:

(11-1)

personal APPLICATIONMackenzie is currently finishing up her bachelor’s degree and is considering going back to grad school for a master’s degree. Shecurrently has $17,125 in student loans carrying an 8 percent interest rate from her bachelor’s degree and estimates that she will needto take out an additional $29,000 in student loans (at the same interest rate) to make it through the master’s program she’d like toattend. The IRS allows taxpayers with student loans to deduct the interest on those loans, but only up to a maximum amount of$2,500 per year. Assuming that Mackenzie will face a marginal personal tax rate of 25 percent when she graduates, what will be theaverage after-tax interest rate that she will be paying on the student loans immediately after she graduates with her master’s? (Seethe solution at the end of the book.)

How else can Mackenzie finance her graduate degree? What will happen to her after-tax interest rate?

where

Notice that we use weights based on market values rather than book values because market values reflect investors’assessment of what they would be willing to pay for the various types of securities, while book values would reflectwhat was paid for such securities at varying times in the past. Since we’re interested in coming up with the cost ofcapital for new investments in the firm or its projects, using market values here makes more sense.

Calculating the Component Cost of Equity LG11-3

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We could calculate iE using the capital asset pricing model, as discussed in Chapter 10:

(11-2)

Or, we can assume that the equity in question is a constant-growth stock such as the ones we modeled in Chapter 8.Under this assumption, we can solve the constant-growth model for iE:

(11-3)

Which way is better? Well, theoretically, both should give us the same answer, but depending on the situation, somepragmatic reasons may dictate your choice.

1. In situations where you do not have sufficient historic observations to estimate β (i.e., when the stock is fairly new), or when yoususpect that the past level of the stock’s systematic (or market) risk might not be a good indicator of the future risk, youdo not want to use the CAPM. Using calculated historic systematic risk when it is not a good estimate of βE, estimatedfuture systematic risk, will not work too well.

2. In situations where you can expect constant dividend growth, the constant-growth model is appropriate. But although you can try toadjust the model for stocks without constant dividend growth, doing so may introduce potentially sizable errors, so it is not the bestchoice for stocks that increase their dividends irregularly.

EXAMPLE 11-1 Cost of Equity LG11-2

For interactive versionsof this example, log in

to Connect or go tomhhe.com/CornettM4e.

ADK Industries’ common shares sell for $32.75 per share. ADKexpects to set its next annual dividend at $1.54 per share. If ADKexpects future dividends to grow by 6 percent per year, indefinitely, thecurrent risk-free rate is 3 percent, the expected return on the market is9 percent, and the stock has a beta of 1.3, what should be firm’s cost ofequity?

SOLUTION:

The cost of equity using the CAPM will be

The cost of equity using the constant-growth model will be

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