Required Returns

Bond values and required

Market Value of Bond, B0 ($)

1,300

returns (Mills Company™s

10% coupon interest rate,

1,200

10-year maturity, $1,000 par,

1,134

January 1, 2004, issue paying 1,100

Premium

annual interest)

Par 1,000

Discount

900

887

800

700

0 2 4 6 8 10 12 14 16

Required Return, kd (%)

250 PART 2 Important Financial Concepts

Time to Maturity and Bond Values

Whenever the required return is different from the coupon interest rate, the

amount of time to maturity affects bond value. An additional factor is whether

required returns are constant or changing over the life of the bond.

Constant Required Returns When the required return is different from the

coupon interest rate and is assumed to be constant until maturity, the value of the

bond will approach its par value as the passage of time moves the bond™s value

closer to maturity. (Of course, when the required return equals the coupon inter-

est rate, the bond™s value will remain at par until it matures.)

Figure 6.6 depicts the behavior of the bond values calculated earlier and pre-

EXAMPLE

sented in Table 6.6 for Mills Company™s 10% coupon interest rate bond paying

annual interest and having 10 years to maturity. Each of the three required

returns”12%, 10%, and 8%”is assumed to remain constant over the 10 years

to the bond™s maturity. The bond™s value at both 12% and 8% approaches and

ultimately equals the bond™s $1,000 par value at its maturity, as the discount (at

12%) or premium (at 8%) declines with the passage of time.

Changing Required Returns The chance that interest rates will change and

thereby change the required return and bond value is called interest rate risk.

interest rate risk

(This was described as a shareholder-specific risk in Chapter 5, Table 5.1.) Bond-

The chance that interest rates

will change and thereby change holders are typically more concerned with rising interest rates because a rise in

the required return and bond

interest rates, and therefore in the required return, causes a decrease in bond

value. Rising rates, which result

value. The shorter the amount of time until a bond™s maturity, the less responsive

in decreasing bond values, are of

is its market value to a given change in the required return. In other words, short

greatest concern.

maturities have less interest rate risk than long maturities when all other features

(coupon interest rate, par value, and interest payment frequency) are the same.

FIGURE 6.6

Time to Maturity

Market Value of Bond, B0 ($)

and Bond Values Premium Bond, Required Return, kd = 8%

1,200

Relationship among time to 1,134

1,115

maturity, required returns,

1,100

and bond values (Mills

1,052

Company™s 10% coupon inter-

Par-Value Bond, Required Return, kd = 10%

1,000 M

est rate, 10-year maturity,

$1,000 par, January 1, 2004, 952

issue paying annual interest) 901

887

Discount Bond, Required Return, kd = 12%

800

10 9 8 7 6 5 4 3 2 1 0

Time to Maturity (years)

251

CHAPTER 6 Interest Rates and Bond Valuation

This is because of the mathematics of time value; the present values of short-term

cash flows change far less than the present values of longer-term cash flows in

response to a given change in the discount rate (required return).

The effect of changing required returns on bonds of differing maturity can be

EXAMPLE

illustrated by using Mills Company™s bond and Figure 6.6. If the required return

rises from 10% to 12% (see the dashed line at 8 years), the bond™s value

decreases from $1,000 to $901”a 9.9% decrease. If the same change in required

return had occurred with only 3 years to maturity (see the dashed line at 3 years),

the bond™s value would have dropped to just $952”only a 4.8% decrease. Simi-

lar types of responses can be seen for the change in bond value associated with

decreases in required returns. The shorter the time to maturity, the less the impact

on bond value caused by a given change in the required return.

Yield to Maturity (YTM)

When investors evaluate bonds, they commonly consider yield to maturity

yield to maturity (YTM)

The rate of return that investors (YTM). This is the rate of return that investors earn if they buy the bond at a spe-

earn if they buy a bond at a cific price and hold it until maturity. (The measure assumes, of course, that the

specific price and hold it until

issuer makes all scheduled interest and principal payments as promised.) The

maturity. (Assumes that the

yield to maturity on a bond with a current price equal to its par value (that is,

issuer makes all scheduled

B0 M) will always equal the coupon interest rate. When the bond value differs

interest and principal payments

as promised.) from par, the yield to maturity will differ from the coupon interest rate.

Assuming that interest is paid annually, the yield to maturity on a bond can

be found by solving Equation 6.3 for kd. In other words, the current value, the

annual interest, the par value, and the years to maturity are known, and the

required return must be found. The required return is the bond™s yield to matu-

rity. The YTM can be found by trial and error or by use of a financial calculator.

The calculator provides accurate YTM values with minimum effort.

The Mills Company bond, which currently sells for $1,080, has a 10% coupon

EXAMPLE

interest rate and $1,000 par value, pays interest annually, and has 10 years to

maturity. Because B0 $1,080, I $100 (0.10 $1,000), M $1,000, and

n 10 years, substituting into Equation 6.3a yields

$1,080 $100 (PVIFAk ) $1,000 (PVIFk )

d,10yrs d,10yrs

Our objective is to solve the equation for kd, the YTM.

Trial and Error Because we know that a required return, kd, of 10% (which

equals the bond™s 10% coupon interest rate) would result in a value of $1,000,

the discount rate that would result in $1,080 must be less than 10%. (Remember

that the lower the discount rate, the higher the present value, and the higher the

discount rate, the lower the present value.) Trying 9%, we get

$100 (PVIFA9%,10yrs) $1,000 (PVIF9%,10yrs)

$100 (6.418) $1,000 (0.422)

$641.80 $422.00

$1,063.80

252 PART 2 Important Financial Concepts

Because the 9% rate is not quite low enough to bring the value up to $1,080, we

next try 8% and get

$100 (PVIFA8%,10yrs) $1,000 (PVIF8%,10yrs)

$100 (6.710) $1,000 (0.463)

$671.00 $463.00

$1,134.00

Input Function Because the value at the 8% rate is higher than $1,080 and the value at the 9%

10 N

rate is lower than $1,080, the bond™s yield to maturity must be between 8% and

1080 PV

9%. Because the $1,063.80 is closer to $1,080, the YTM to the nearest whole

100 PMT

percent is 9%. (By using interpolation, we could eventually find the more precise

FV

1000 YTM value to be 8.77%.)7

CPT

Calculator Use [Note: Most calculators require either the present value (B0 in

I

this case) or the future values (I and M in this case) to be input as negative num-

Solution

bers to calculate yield to maturity. That approach is employed here.] Using the

8.766

inputs shown at the left, you should find the YTM to be 8.766%.

Semiannual Interest and Bond Values

The procedure used to value bonds paying interest semiannually is similar to that

shown in Chapter 4 for compounding interest more frequently than annually,

except that here we need to find present value instead of future value. It involves

1. Converting annual interest, I, to semiannual interest by dividing I by 2.

2. Converting the number of years to maturity, n, to the number of 6-month

periods to maturity by multiplying n by 2.

3. Converting the required stated (rather than effective) annual return for simi-

lar-risk bonds that also pay semiannual interest from an annual rate, kd, to a

semiannual rate by dividing kd by 2.

Substituting these three changes into Equation 6.3 yields

2n

I 1 1

B0 M (6.4)

t 2n

kd kd

2 i1

1 1

2 2

I

(PVIFAkd/2,2n) M (PVIFkd/2,2n) (6.4a)

2

Assuming that the Mills Company bond pays interest semiannually and that the

EXAMPLE

required stated annual return, kd, is 12% for similar-risk bonds that also pay

semiannual interest, substituting these values into Equation 6.4a yields

$100

B0 (PVIFA12%/2,2 10yrs) $1,000 (PVIF12%/2,2 10yrs)

2

WW 7. For information on how to interpolate to get a more precise answer, see the book™s home page at www.

W

aw.com/gitman

253

CHAPTER 6 Interest Rates and Bond Valuation

Table Use

B0 $50 (PVIFA6%,20periods) $1,000 (PVIF6%,20periods)

$50 (11.470) $1,000 (0.312) $885.50

Input Function

20 N

I

6

Calculator Use In using a calculator to find bond value when interest is paid

50 PMT semiannually, we must double the number of periods and divide both the

FV required stated annual return and the annual interest by 2. For the Mills Com-

1000

pany bond, we would use 20 periods (2 10 years), a required return of 6%

CPT

(12% 2), and an interest payment of $50 ($100 2). Using these inputs, you

PV

should find the bond value with semiannual interest to be $885.30, as shown at

Solution

the left. Note that this value is more precise than the value calculated using the

885.30

rounded financial-table factors.

Comparing this result with the $887.00 value found earlier for annual com-

pounding (see Table 6.6), we can see that the bond™s value is lower when semian-

nual interest is paid. This will always occur when the bond sells at a discount. For

bonds selling at a premium, the opposite will occur: The value with semiannual

interest will be greater than with annual interest.

Review Questions

6“16 What basic procedure is used to value a bond that pays annual interest?

Semiannual interest?

6“17 What relationship between the required return and the coupon interest

rate will cause a bond to sell at a discount? At a premium? At its par

value?

6“18 If the required return on a bond differs from its coupon interest rate,

describe the behavior of the bond value over time as the bond moves

toward maturity.

6“19 As a risk-averse investor, would you prefer bonds with short or long peri-

ods until maturity? Why?

6“20 What is a bond™s yield to maturity (YTM)? Briefly describe both the trial-

and-error approach and the use of a financial calculator for finding YTM.

SUMMARY

FOCUS ON VALUE

Interest rates and required returns embody the real cost of money, inflationary expecta-

tions, and issuer and issue risk. They reflect the level of return required by market partici-

pants as compensation for the risk perceived in a specific security or asset investment.

Because these returns are affected by economic expectations, they vary as a function of

254 PART 2 Important Financial Concepts

time, typically rising for longer-term maturities or transactions. The yield curve reflects such

market expectations at any point in time.

The value of an asset can be found by calculating the present value of its expected cash

flows, using the required return as the discount rate. Bonds are the easiest financial assets to

value, because both the amounts and the timing of their cash flows are known with certainty.

The financial manager needs to understand how to apply valuation techniques to bonds in

order to make decisions that are consistent with the firm™s share price maximization goal.

REVIEW OF LEARNING GOALS

cial press, provide information on bonds, including

Describe interest rate fundamentals, the term

LG2

current price data and statistics on recent price be-

structure of interest rates, and risk premiums.

havior. Bond ratings by independent agencies indi-

The flow of funds between savers (suppliers) and

cate the risk of a bond issue. Various types of tradi-

investors (demanders) is regulated by the interest

tional and contemporary bonds are available.

rate or required return. In a perfect, inflation-free,

Eurobonds and foreign bonds enable established

certain world there would be one cost of money”

creditworthy companies and governments to bor-

the real rate of interest. For any class of similar-risk

row large amounts internationally.

securities, the term structure of interest rates reflects

the relationship between the interest rate, or rate of

Understand the key inputs and basic model

return, and the time to maturity. Yield curves can

LG4

used in the valuation process. Key inputs to the

be downward-sloping (inverted), upward-sloping

valuation process include cash flows (returns), tim-

(normal), or flat. Three theories”expectations the-

ing, and risk and the required return. The value of

ory, liquidity preference theory, and market seg-

any asset is equal to the present value of all future

mentation theory”are cited to explain the general

cash flows it is expected to provide over the relevant

shape of the yield curve. Risk premiums for non-

time period. The basic valuation formula for any

Treasury debt issues result from interest rate risk,

asset is summarized in Table 6.7.

liquidity risk, tax risk, default risk, maturity risk,

and contractual provision risk.

Apply the basic valuation model to bonds and

LG5

describe the impact of required return and time

Review the legal aspects of bond financing and

LG2

to maturity on bond values. The value of a bond is

bond cost. Corporate bonds are long-term debt

the present value of its interest payments plus the

instruments indicating that a corporation has bor-

present value of its par value. The basic valuation

rowed an amount that it promises to repay in the

model for a bond is summarized in Table 6.7. The

future under clearly defined terms. Most bonds are

discount rate used to determine bond value is the re-

issued with maturities of 10 to 30 years and a par

quired return, which may differ from the bond™s

value of $1,000. The bond indenture, enforced by a

coupon interest rate. A bond can sell at a discount, at

trustee, states all conditions of the bond issue. It

par, or at a premium, depending on whether the re-

contains both standard debt provisions and restric-

quired return is greater than, equal to, or less than its

tive covenants, which may include a sinking-fund

coupon interest rate. The amount of time to maturity

requirement and/or a security interest. The cost of

affects bond values. Even if the required return re-

bonds to an issuer depends on its maturity, offering

mains constant, the value of a bond will approach its

size, and issuer risk and on the basic cost of money.

par value as the bond moves closer to maturity. The

chance that interest rates will change and thereby

Discuss the general features, quotations, ratings,

LG3

change the required return and bond value is called

popular types, and international issues of corpo-

interest rate risk. The shorter the amount of time un-

rate bonds. A bond issue may include a conversion

til a bond™s maturity, the less responsive is its market

feature, a call feature, or stock purchase warrants.

value to a given change in the required return.

Bond quotations, published regularly in the finan-

255

CHAPTER 6 Interest Rates and Bond Valuation

TABLE 6.7 Summary of Key Valuation Definitions

and Formulas for Any Asset and for Bonds

Definitions of variables

B0 bond value

CFt cash flow expected at the end of year t

I annual interest on a bond

k appropriate required return (discount rate)

kd required return on a bond

M par, or face, value of a bond

n relevant time period, or number of years to maturity

V0 value of the asset at time zero

Valuation formulas

Value of any asset:

CF1 CF2 CFn

...

V0 [Eq. 6.1]

(1 k)1 (1 k)2 (1 k)n

...

[CF1 (PVIFk,1)] [CF2 (PVIFk,2)] [CFn (PVIFk,n )] [Eq. 6.2]

Bond value:

n

1 1

B0 I M [Eq. 6.3]

kd)t kd)n

(1 (1

t1

I (PVIFAkd ,n) M (PVIFkd ,n) [Eq. 6.3a]

nually are valued by using the same procedure used

Explain yield to maturity (YTM), its calcula-

LG6

to value bonds paying annual interest, except that

tion, and the procedure used to value bonds

the interest payments are one-half of the annual in-

that pay interest semiannually. Yield to maturity

terest payments, the number of periods is twice the

(YTM) is the rate of return investors earn if they

number of years to maturity, and the required re-

buy a bond at a specific price and hold it until ma-

turn is one-half of the stated annual required return

turity. YTM can be calculated by trial and error or

on similar-risk bonds.

financial calculator. Bonds that pay interest semian-

SELF-TEST PROBLEMS (Solutions in Appendix B)

ST 6“1 Bond valuation Lahey Industries has outstanding a $1,000 par-value bond

LG5 LG6

with an 8% coupon interest rate. The bond has 12 years remaining to its matu-

rity date.

a. If interest is paid annually, find the value of the bond when the required

return is (1) 7%, (2) 8%, and (3) 10%?

b. Indicate for each case in part a whether the bond is selling at a discount, at a

premium, or at its par value.

c. Using the 10% required return, find the bond™s value when interest is paid

semiannually.

256 PART 2 Important Financial Concepts

ST 6“2 Yield to maturity Elliot Enterprises™ bonds currently sell for $1,150, have an

LG6

11% coupon interest rate and a $1,000 par value, pay interest annually, and

have 18 years to maturity.

a. Calculate the bonds™ yield to maturity (YTM).

b. Compare the YTM calculated in part a to the bonds™ coupon interest rate,

and use a comparison of the bonds™ current price and their par value to

explain this difference.

PROBLEMS

6“1 Yield curve A firm wishing to evaluate interest rate behavior has gathered yield

LG2

data on five U.S. Treasury securities, each having a different maturity and all

measured at the same point in time. The summarized data follow.

U.S. Treasury security Time to maturity Yield

A 1 year 12.6%

B 10 years 11.2

C 6 months 13.0

D 20 years 11.0

E 5 years 11.4

a. Draw the yield curve associated with these data.

b. Describe the resulting yield curve in part a, and explain the general expecta-

tions embodied in it.

6“2 Term structure of interest rates The following yield data for a number of high-

LG2

est quality corporate bonds existed at each of the three points in time noted.

Yield

Time to maturity (years) 5 years ago 2 years ago Today

1 9.1% 14.6% 9.3%

3 9.2 12.8 9.8

5 9.3 12.2 10.9

10 9.5 10.9 12.6

15 9.4 10.7 12.7

20 9.3 10.5 12.9

30 9.4 10.5 13.5

a. On the same set of axes, draw the yield curve at each of the three given times.

b. Label each curve in part a with its general shape (downward-sloping,

upward-sloping, flat).

c. Describe the general inflationary and interest rate expectation existing at

each of the three times.

6“3 Risk-free rate and risk premiums The real rate of interest is currently 3%; the

LG2

inflation expectation and risk premiums for a number of securities follow.

257

CHAPTER 6 Interest Rates and Bond Valuation

Inflation expectation

Security premium Risk premium

A 6% 3%

B 9 2

C 8 2

D 5 4

E 11 1

a. Find the risk-free rate of interest, RF , that is applicable to each security.

b. Although not noted, what factor must be the cause of the differing risk-free

rates found in part a?

c. Find the actual rate of interest for each security.

6“4 Risk premiums Eleanor Burns is attempting to find the actual rate of interest

LG2

for each of two securities”A and B”issued by different firms at the same point

in time. She has gathered the following data:

Characteristic Security A Security B

Time to maturity 3 years 15 years

Inflation expectation premium 9.0% 7.0%

Risk premium for:

Liquidity risk 1.0% 1.0%

Default risk 1.0% 2.0%

Maturity risk 0.5% 1.5%

Other risk 0.5% 1.5%

a. If the real rate of interest is currently 2%, find the risk-free rate of interest

applicable to each security.

b. Find the total risk premium attributable to each security™s issuer and issue

characteristics.

c. Calculate the actual rate of interest for each security. Compare and discuss

your findings.

6“5 Bond interest payments before and after taxes Charter Corp. has issued 2,500

LG2

debentures with a total principal value of $2,500,000. The bonds have a coupon

interest rate of 7%.

a. What dollar amount of interest per bond can an investor expect to receive

each year from Charter Corp.?

b. What is Charter™s total interest expense per year associated with this bond

issue?

c. Assuming that Charter is in a 35% corporate tax bracket, what is the com-

pany™s net after-tax interest cost associated with this bond issue?

6“6 Bond quotation Assume that the following quote for the Financial Manage-

LG3

ment Corporation™s $1,000-par-value bond was found in the Wednesday,

November 8, issue of the Wall Street Journal.

Fin Mgmt 8.75 05 8.7 558 100.25 0.63

258 PART 2 Important Financial Concepts

Given this information, answer the following questions.

a. On what day did the trading activity occur?

b. At what price did the bond close at the end of the day on November 7?

c. In what year does the bond mature?

d. How many bonds were traded on the day quoted?

e. What is the bond™s coupon interest rate?

f. What is the bond™s current yield? Explain how this value was calculated.

g. How much of a change, if any, in the bond™s closing price took place between

the day quoted and the day before? At what price did the bond close on the

day before?

6“7 Valuation fundamentals Imagine that you are trying to evaluate the economics

LG4

of purchasing an automobile. You expect the car to provide annual after-tax

cash benefits of $1,200 at the end of each year, and assume that you can sell the

car for after-tax proceeds of $5,000 at the end of the planned 5-year ownership

period. All funds for purchasing the car will be drawn from your savings, which

are currently earning 6% after taxes.

a. Identify the cash flows, their timing, and the required return applicable to

valuing the car.

b. What is the maximum price you would be willing to pay to acquire the car?

Explain.

6“8 Valuation of assets Using the information provided in the following table, find

LG4

the value of each asset.

Cash flow

Asset End of year Amount Appropriate required return

A 1 $ 5,000 18%

2 5,000

3 5,000

1 through ∞

B $ 300 15%

C 1 $ 0 16%

2 0

3 0

4 0

5 35,000

D 1 through 5 $ 1,500 12%

6 8,500

E 1 $ 2,000 14%

2 3,000

3 5,000

4 7,000

5 4,000

6 1,000

259

CHAPTER 6 Interest Rates and Bond Valuation

6“9 Asset valuation and risk Laura Drake wishes to estimate the value of an

LG4

asset expected to provide cash inflows of $3,000 per year at the end of years 1

through 4 and $15,000 at the end of year 5. Her research indicates that she

must earn 10% on low-risk assets, 15% on average-risk assets, and 22% on

high-risk assets.

a. Determine what is the most Laura should pay for the asset if it is classified as

(1) low-risk, (2) average-risk, and (3) high-risk.

b. Say Laura is unable to assess the risk of the asset and wants to be certain

she™s making a good deal. On the basis of your findings in part a, what is the

most she should pay? Why?

c. All else being the same, what effect does increasing risk have on the value of

an asset? Explain in light of your findings in part a.

6“10 Basic bond valuation Complex Systems has an outstanding issue of $1,000-

LG5

par-value bonds with a 12% coupon interest rate. The issue pays interest annu-

ally and has 16 years remaining to its maturity date.

a. If bonds of similar risk are currently earning a 10% rate of return, how much

should the Complex Systems bond sell for today?

b. Describe the two possible reasons why similar-risk bonds are currently earn-

ing a return below the coupon interest rate on the Complex Systems bond.

c. If the required return were at 12% instead of 10%, what would the current

value of Complex Systems™ bond be? Contrast this finding with your findings

in part a and discuss.

6“11 Bond valuation”Annual interest Calculate the value of each of the bonds

LG5

shown in the following table, all of which pay interest annually.

Bond Par value Coupon interest rate Years to maturity Required return

A $1,000 14% 20 12%

B 1,000 8 16 8

C 100 10 8 13

D 500 16 13 18

E 1,000 12 10 10

6“12 Bond value and changing required returns Midland Utilities has outstanding a

LG5

bond issue that will mature to its $1,000 par value in 12 years. The bond has a

coupon interest rate of 11% and pays interest annually.

a. Find the value of the bond if the required return is (1) 11%, (2) 15%, and

(3) 8%.

b. Plot your findings in part a on a set of “required return (x axis)“market value

of bond (y axis)” axes.

c. Use your findings in parts a and b to discuss the relationship between the

coupon interest rate on a bond and the required return and the market value

of the bond relative to its par value.

d. What two possible reasons could cause the required return to differ from the

coupon interest rate?

260 PART 2 Important Financial Concepts

6“13 Bond value and time”Constant required returns Pecos Manufacturing has just

LG5

issued a 15-year, 12% coupon interest rate, $1,000-par bond that pays interest

annually. The required return is currently 14%, and the company is certain it

will remain at 14% until the bond matures in 15 years.

a. Assuming that the required return does remain at 14% until maturity, find

the value of the bond with (1) 15 years, (2) 12 years, (3) 9 years, (4) 6 years,

(5) 3 years, and (6) 1 year to maturity.

b. Plot your findings on a set of “time to maturity (x axis)“market value of

bond (y axis)” axes constructed similarly to Figure 6.6.

c. All else remaining the same, when the required return differs from the coupon

interest rate and is assumed to be constant to maturity, what happens to the

bond value as time moves toward maturity? Explain in light of the graph in

part b.

6“14 Bond value and time”Changing required returns Lynn Parsons is considering

LG5

investing in either of two outstanding bonds. The bonds both have $1,000 par

values and 11% coupon interest rates and pay annual interest. Bond A has

exactly 5 years to maturity, and bond B has 15 years to maturity.

a. Calculate the value of bond A if the required return is (1) 8%, (2) 11%, and

(3) 14%.

b. Calculate the value of bond B if the required return is (1) 8%, (2) 11%, and

(3) 14%.

c. From your findings in parts a and b, complete the following table, and dis-

cuss the relationship between time to maturity and changing required returns.

Required return Value of bond A Value of bond B

8% ? ?

11 ? ?

14 ? ?

d. If Lynn wanted to minimize interest rate risk, which bond should she pur-

chase? Why?

6“15 Yield to maturity The relationship between a bond™s yield to maturity and

LG6

coupon interest rate can be used to predict its pricing level. For each of the

bonds listed, state whether the price of the bond will be at a premium to par, at

par, or at a discount to par.

Bond Coupon interest rate Yield to maturity Price

A 6% 10%

B 8 8

C 9 7

D 7 9

E 12 10

261

CHAPTER 6 Interest Rates and Bond Valuation

6“16 Yield to maturity The Salem Company bond currently sells for $955, has a

LG6

12% coupon interest rate and a $1,000 par value, pays interest annually, and

has 15 years to maturity.

a. Calculate the yield to maturity (YTM) on this bond.

b. Explain the relationship that exists between the coupon interest rate

and yield to maturity and the par value and market value of a

bond.

6“17 Yield to maturity Each of the bonds shown in the following table pays interest

LG6

annually.

Bond Par value Coupon interest rate Years to maturity Current value

A $1,000 9% 8 $ 820

B 1,000 12 16 1,000

C 500 12 12 560

D 1,000 15 10 1,120

E 1,000 5 3 900

a. Calculate the yield to maturity (YTM) for each bond.

b. What relationship exists between the coupon interest rate and yield to

maturity and the par value and market value of a bond? Explain.

6“18 Bond valuation”Semiannual interest Find the value of a bond maturing in 6

LG6

years, with a $1,000 par value and a coupon interest rate of 10% (5% paid

semiannually) if the required return on similar-risk bonds is 14% annual interest

(7% paid semiannually).

6“19 Bond valuation”Semiannual interest Calculate the value of each of the bonds

LG6

shown in the following table, all of which pay interest semiannually.

Coupon Years to Required stated

Bond Par value interest rate maturity annual return

A $1,000 10% 12 8%

B 1,000 12 20 12

C 500 12 5 14

D 1,000 14 10 10

E 100 6 4 14

6“20 Bond valuation”Quarterly interest Calculate the value of a $5,000-par-value

LG6

bond paying quarterly interest at an annual coupon interest rate of 10% and

having 10 years until maturity if the required return on similar-risk bonds is cur-

rently a 12% annual rate paid quarterly.

262 PART 2 Important Financial Concepts

CHAPTER 6 CASE Evaluating Annie Hegg™s Proposed Investment

in Atilier Industries Bonds

A nnie Hegg has been considering investing in the bonds of Atilier Industries.

The bonds were issued 5 years ago at their $1,000 par value and have

exactly 25 years remaining until they mature. They have an 8% coupon interest

rate, are convertible into 50 shares of common stock, and can be called any time

at $1,080. The bond is rated Aa by Moody™s. Atilier Industries, a manufacturer

of sporting goods, recently acquired a small athletic-wear company that was in

financial distress. As a result of the acquisition, Moody™s and other rating agen-

cies are considering a rating change for Atilier bonds. Recent economic data

suggest that inflation, currently at 5% annually, is likely to increase to a 6%

annual rate.

Annie remains interested in the Atilier bond but is concerned about infla-

tion, a potential rating change, and maturity risk. In order to get a feel for the

potential impact of these factors on the bond value, she decided to apply the val-

uation techniques she learned in her finance course.

Required

a. If the price of the common stock into which the bond is convertible rises to

$30 per share after 5 years and the issuer calls the bonds at $1,080, should

Annie let the bond be called away from her or should she convert it into com-

mon stock?

b. For each of the following required returns, calculate the bond™s value, assum-

ing annual interest. Indicate whether the bond will sell at a discount, at a pre-

mium, or at par value.

(1) Required return is 6%.

(2) Required return is 8%.

(3) Required return is 10%.

c. Repeat the calculations in part b, assuming that interest is paid semiannually

and that the semiannual required returns are one-half of those shown. Com-

pare and discuss differences between the bond values for each required return

calculated here and in part b under the annual versus semiannual payment

assumptions.

d. If Annie strongly believes that inflation will rise by 1% during the next 6

months, what is the most she should pay for the bond, assuming annual

interest?

e. If the Atilier bonds are downrated by Moody™s from Aa to A, and if such a

rating change will result in an increase in the required return from 8% to

8.75%, what impact will this have on the bond value, assuming annual

interest?

f. If Annie buys the bond today at its $1,000 par value and holds it for exactly

3 years, at which time the required return is 7%, how much of a gain or loss

will she experience in the value of the bond (ignoring interest already received

and assuming annual interest)?

g. Rework part f, assuming that Annie holds the bond for 10 years and sells it

when the required return is 7%. Compare your finding to that in part f, and

comment on the bond™s maturity risk.

263

CHAPTER 6 Interest Rates and Bond Valuation

h. Assume that Annie buys the bond at its current closing price of 98.38 and

holds it until maturity. What will her yield to maturity (YTM) be, assuming

annual interest?

i. After evaluating all of the issues raised above, what recommendation would

you give Annie with regard to her proposed investment in the Atilier Indus-

tries bonds?

WEB EXERCISE Go to the Web site www.smartmoney.com. Click on Economy & Bonds. Then

WW click on Bond Calculator, which is located down the page under the column

W

Bond Tools. Read the instructions on how to use the bond calculator. Using the

bond calculator:

1. Calculate the yield to maturity (YTM) for a bond whose coupon rate is

7.5% with maturity date of July 31, 2030, which you bought for 95.

2. What is the YTM of the above bond if you bought it for 105? For 100?

3. Change the yield % box to 8.5. What would be the price of this bond?

4. Change the yield % box to 9.5. What is this bond™s price?

5. Change the maturity date to 2006 and reset yield % to 6.5. What is the price

of this bond?

6. Why is the price of the bond in Question 5 higher than the price of the bond

in Question 4?

7. Explore the other bond-related resources at the site. Using Bond Market

Update, comment on current interest rate levels and the yield curve.

Remember to check the book™s Web site at

www.aw.com/gitman

for additional resources, including additional Web exercises.