Economics Chapter 5 Homework A bond is a promissory note issued by a business

Answers and Solutions: 5 – 1

Chapter 5

Bonds, Bond Valuation, and Interest Rates

ANSWERS TO END-OF-CHAPTER QUESTIONS

5-1 a. A bond is a promissory note issued by a business or a governmental unit. Treasury

bonds, sometimes referred to as government bonds, are issued by the Federal

government and are not exposed to default risk. Corporate bonds are issued by

corporations and are exposed to default risk. Different corporate bonds have different

levels of default risk, depending on the issuing company’s characteristics and on the

bondholder. Maturity dates generally range from 10 to 40 years from the time of

issue. A call provision may be written into a bond contract, giving the issuer the right

to redeem the bonds under specific conditions prior to the normal maturity date. A

bond’s coupon, or coupon payment, is the dollar amount of interest paid to each

bondholder on the interest payment dates. The coupon is so named because bonds

used to have dated coupons attached to them which investors could tear off and

redeem on the interest payment dates. The coupon interest rate is the stated rate of

interest on a bond.

c. In some cases, a bond’s coupon payment may vary over time. These bonds are called

Answers and Solutions: 5 – 2

e. Convertible bonds are securities that are convertible into shares of common stock, at a

fixed price, at the option of the bondholder. Bonds issued with warrants are similar to

convertibles. Warrants are options which permit the holder to buy stock for a stated

price, thereby providing a capital gain if the stock price rises. Income bonds pay

interest only if the interest is earned. These securities cannot bankrupt a company,

but from an investor’s standpoint they are riskier than “regular” bonds. The interest

rate of an indexed, or purchasing power, bond is based on an inflation index such as

the consumer price index (CPI), so the interest paid rises automatically when the

inflation rate rises, thus protecting the bondholders against inflation.

g. The current yield on a bond is the annual coupon payment divided by the current

market price. YTM, or yield to maturity, is the rate of interest earned on a bond if it

is held to maturity. Yield to call (YTC) is the rate of interest earned on a bond if it is

called. If current interest rates are well below an outstanding callable bond’s coupon

rate, the YTC may be a more relevant estimate of expected return than the YTM,

since the bond is likely to be called.

Answers and Solutions: 5 – 3

other debt.

i. A development bond is a tax-exempt bond sold by state and local governments whose

proceeds are made available to corporations for specific uses deemed (by Congress)

to be in the public interest. Municipalities can insure their bonds, in which an

insurance company guarantees to pay the coupon and principal payments should the

companies. Most bonds are purchased by institutional investors rather than

individuals, and many institutions are restricted to investment grade bonds, securities

with ratings of Baa/BBB or above.

j. The real risk-free rate is that interest rate which equalizes the aggregate supply of,

and demand for, riskless securities in an economy with zero inflation. The real risk–

free rate could also be called the pure rate of interest since it is the rate of interest that

would exist on very short-term, default-free U.S. Treasury securities if the expected

rate of inflation were zero. It has been estimated that this rate of interest, denoted by

k. The inflation premium is the premium added to the real risk-free rate of interest to

compensate for the expected loss of purchasing power. The inflation premium is the

average rate of inflation expected over the life of the security. Default risk is the risk

that a borrower will not pay the interest and/or principal on a loan as they become

due. Thus, a default risk premium (DRP) is added to the real risk-free rate to

compensate investors for bearing default risk. Liquidity refers to a firm’s cash and

marketable securities position, and to its ability to meet maturing obligations. A

liquid asset is any asset that can be quickly sold and converted to cash at its “fair”

value. Active markets provide liquidity. A liquidity premium is added to the real

risk-free rate of interest, in addition to other premiums, if a security is not liquid.

m. The term structure of interest rates is the relationship between yield to maturity and

term to maturity for bonds of a single risk class. The yield curve is the curve that

results when yield to maturity is plotted on the Y-axis with term to maturity on the X-

axis.

n. When the yield curve slopes upward, it is said to be “normal,” because it is like this

most of the time. Conversely, a downward–sloping yield curve is termed “abnormal”

or “inverted.”

5-3 The price of the bond will fall and its YTM will rise if interest rates rise. If the bond still

has a long term to maturity, its YTM will reflect long-term rates. Of course, the bond’s

price will be less affected by a change in interest rates if it has been outstanding a long

5-4 If interest rates decline significantly, the values of callable bonds will not rise by as much

5-5 From the corporation’s viewpoint, one important factor in establishing a sinking fund is

that its own bonds generally have a higher yield than do government bonds; hence, the

company saves more interest by retiring its own bonds than it could earn by buying

government bonds. This factor causes firms to favor the second procedure. Investors

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

5-1 With your financial calculator, enter the following:

N = 12; I/YR = YTM = 9%; PMT = 0.08  1,000 = 80; FV = 1000; PV = VB = ?

5-2 With your financial calculator, enter the following:

5-3 With your financial calculator, enter the following to find the current value of the bonds,

so you can then calculate their current yield:

N = 7; I/YR = YTM = 8; PMT = 0.09  1,000 = 90; FV = 1000; PV = VB = ?

5-4 r* = 4%; I1 = 2%; I2 = 4%; I3 = 4%; MRP = 0; rT-2 = ?; rT-3 = ?

r = r* + IP + DRP + LP + MRP.

5-5 rT-10 = 6%; rC-10 = 9%; LP = 0.5%; DRP = ?

r = r* + IP + DRP + LP + MRP.

5-6 r* = 3%; IP = 3%; rT-2 = 6.3%; MRP2 = ?

rT-2 = r* + IP + MRP = 6.3%

rT-2 = 3% + 3% + MRP = 6.3%

MRP = 0.3%.

5-7 The problem asks you to find the price of a bond, given the following facts:

Answers and Solutions: 5 – 8

5-8 With your financial calculator, enter the following to find YTM:

5-9 a.

1. 5%: Bond L: Input N = 15, I/YR = 5, PMT = 100, FV = 1000, PV = ?, PV =

$1,518.98.

Bond S: Change N = 1, PV = ? PV = $1,047.62.

b. Think about a bond that matures in one month. Its present value is influenced

primarily by the maturity value, which will be received in only one month. Even if

interest rates double, the price of the bond will still be close to $1,000. A one-year

bond’s value would fluctuate more than the one–month bond’s value because of the

Answers and Solutions: 5 – 9

5-10 a. Calculator solution:

1. Input N = 5, PV = -829, PMT = 90, FV = 1000, I/YR = ? I/YR = 13.98%.

2. Change PV = -1104, I/YR = ? I/YR = 6.50%.

5-11 N = 7; PV = -1000; PMT = 140; FV = 1090; I/YR = ? Solve for I/YR = 14.82%.

5-12 a. Using a financial calculator, input the following:

N = 20, PV = -1100, PMT = 60, FV = 1000, and solve for I/YR = 5.1849%.

Answers and Solutions: 5 – 10

5-13 The problem asks you to solve for the YTM, given the following facts:

N = 5, PMT = 80, and FV = 1000. In order to solve for I/YR we need PV.

5-14 The problem asks you to solve for the current yield, given the following facts: N = 14,

I/YR = 10.5883/2 = 5.2942, PV = −1020, and FV = 1000. In order to solve for the

current yield we need to find PMT. With a financial calculator, we find PMT = $55.00.

5-15 The bond is selling at a large premium, which means that its coupon rate is much higher

than the going rate of interest. Therefore, the bond is likely to be called—it is more likely

to be called than to remain outstanding until it matures. Thus, it will probably provide a

return equal to the YTC rather than the YTM. So, there is no point in calculating the

YTM—just calculate the YTC. Enter these values:

5-16

Price at 8%

Price at 7%

Pctge. change

10-year, 10% annual coupon

$1,134.20

$1,210.71

6.75%

10-year zero

463.19

508.35

9.75

5-year zero

680.58

712.99

4.76

30-year zero

131.37

$100 perpetuity

Answers and Solutions: 5 – 11

5-17

t

Price of Bond C

Price of Bond Z

5-18 r = r* + IP + MRP + DRP + LP.

r* = 0.02.

IP = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035.

5-19 First, note that we will use the equation rt = 3% + IPt + MRPt. We have the data needed

to find the IPs:

IP5 =

5

4% + 4% + 4% + 5% + 8%

=

5

25%

= 5%.

Answers and Solutions: 5 – 12

5-20 Basic relevant equations:

rt = r* + IPt + DRPt + MRPt + LPt.

But here IP is the only premium, so rt = r* + IPt.

We also know that It = Constant after t = 1.

We can set up this table:

5-21 a. The bonds now have an 8-year, or a 16-semiannual period, maturity, and their value

is calculated as follows:

Calculator solution: Input N = 16, I/YR = 3, PMT = 50, FV = 1000,

PV = ? PV = $1,251.22.

Answers and Solutions: 5 – 13

5-22 a. Find the YTM as follows:

N = 10, PV = -1200, PMT = 110, FV = 1000

I/YR = YTM = 8.02%.

d. Similarly from above, YTC can be found, if called in each subsequent year.

If called in Year 6:

N = 6, PV = –1200, PMT = 110, FV = 1080

I/YR = YTC = 7.80%.

According to these calculations, the latest investors might expect a call of the bonds is in

Year 7. This is the last year that the expected YTC will be less than the expected YTM.

At this time, the firm still finds an advantage to calling the bonds, rather than seeing them

to maturity.

Answers and Solutions: 5 – 14

5-23 a. Real

Years to Risk-Free

Maturity Rate (r*) IP** MRP rT = r* + IP + MRP

1 2% 7.00% 0.2% 9.20%

2 2 6.00 0.4 8.40

**The computation of the inflation premium is as follows:

Expected Average

Year Inflation Expected Inflation

1 7% 7.00%

2 5 6.00

3 3 5.00

Answers and Solutions: 5 – 15

Thus, the yield curve would be as follows:

b. The interest rate on the ExxonMobil bonds has the same components as the Treasury

securities, except that the ExxonMobil bonds have default risk, so a default risk

premium must be included. Therefore,

rExxon = r* + IP + MRP + DRP.

For a strong company such as ExxonMobil, the default risk premium is virtually zero

c. LILCO bonds would have significantly more default risk than either Treasury

securities or Exxon bonds, and the risk of default would increase over time due to

Answers and Solutions: 5 – 16

SOLUTION TO SPREADSHEET PROBLEM

5-24 The detailed solution for the problem is in the file Ch05 P24 Build a Model Solution.xls

and is available on the instructor’s side of the textbook’s web site.

Mini Case: 5 – 17

MINI CASE

Sam Strother and Shawna Tibbs are vice-presidents of Mutual of Seattle Insurance

Company and co-directors of the company’s pension fund management division. A major

new client, the Northwestern Municipal Alliance, has requested that Mutual of Seattle

present an investment seminar to the mayors of the represented cities, and Strother and

Tibbs, who will make the actual presentation, have asked you to help them by answering

the following questions. Because the Boeing Company operates in one of the league’s cities,

you are to work Boeing into the presentation.

a. What are the key features of a bond?

Answer:

1. Par or face value. We generally assume a $1,000 par value, but par can be

anything, and often $5,000 or more is used. With registered bonds, which is what

Mini Case: 5 – 18

b. What are call provisions and sinking fund provisions? Do these provisions make

bonds more or less risky?

Answer: A call provision is a provision in a bond contract that gives the issuing corporation

the right to redeem the bonds under specified terms prior to the normal maturity date.

The call provision generally states that the company must pay the bondholders an

amount greater than the par value if they are called. The additional sum, which is

Mini Case: 5 – 19

c. How is the value of any asset whose value is based on expected future cash flows

determined?

Answer: 0 1 2 3 n

| | | | • • • |

CF1 CF2 CF3 CFN

PV CF1

PV CF2

Mini Case: 5 – 20

d. How is the value of a bond determined? What is the value of a 10-year, $1,000

par value bond with a 10 percent annual coupon if its required rate of return is

10 percent?

Answer: A bond has a specific cash flow pattern consisting of a stream of constant interest

payments plus the return of par at maturity. The annual coupon payment is the cash

flow: pmt = (coupon rate)  (par value) = 0.1($1,000) = $100.

For a 10-year, 10 percent annual coupon bond, the bond’s value is found as

follows:

$1,000. = $385.54 + $38.55 + . . . + $90.91 =

The bond consists of a 10-year, 10% annuity of $100 per year plus a $1,000 lump

sum payment at t = 10:

PV Annuity = $ 614.46

Mini Case: 5 – 21

e. 1. What would be the value of the bond described in part d if, just after it had been

issued, the expected inflation rate rose by 3 percentage points, causing investors

to require a 13 percent return? Would we now have a discount or a premium

bond?

Answer: With a financial calculator, just change the value of rd = I/YR from 10% to 13%, and

press the PV button to determine the value of the bond:

e. 2. What would happen to the bonds’ value if inflation fell, and rd declined to 7

percent? Would we now have a premium or a discount bond?

Answer: In the second situation, where rd falls to 7 percent, the price of the bond rises above

par. Just change rd from 13% to 7%. We see that the 10-year bond’s value rises to

$1,210.71.

Mini Case: 5 – 22

e. 3. What would happen to the value of the 10-year bond over time if the required

rate of return remained at 13 percent, or if it remained at

7 percent? (Hint: with a financial calculator, enter PMT, I/YR, FV, and N, and

then change (override) n to see what happens to the PV as the bond approaches

maturity.)

Answer: Assuming that interest rates remain at the new levels (either 7% or 13%), we could

find the bond’s value as time passes, and as the maturity date approaches. If we then

plotted the data, we would find the situation shown below:

Mini Case: 5 – 23

f. 1. What is the yield to maturity on a 10-year, 9 percent annual coupon, $1,000 par

value bond that sells for $887.00? That sells for $1,134.20? What does the fact

that a bond sells at a discount or at a premium tell you about the relationship

between rd and the bond’s coupon rate?

Answer: The yield to maturity (YTM) is that discount rate which equates the present value of a

bond’s cash flows to its price. In other words, it is the promised rate of return on the

bond. (Note that the expected rate of return is less than the YTM if some probability

of default exists.) On a time line, we have the following situation when the bond sells

for $887:

Mini Case: 5 – 24

f. 2. What are the total return, the current yield, and the capital gains yield for the

discount bond? (Assume the bond is held to maturity and the company does not

default on the bond.)

Answer: The current yield is defined as follows:

.

bond theof priceCurrent

paymentinterest coupon Annual

= YieldCurrent

Mini Case: 5 – 25

g. How does the equation for valuing a bond change if semiannual payments are

made? Find the value of a 10-year, semiannual payment, 10 percent coupon

bond if nominal rd = 13%.

Answer: In reality, virtually all bonds issued in the U.S. have semiannual coupons and are

valued using the setup shown below:

1 2 N YEARS

0 1 2 3 4 2N-1 2N SA PERIODS

Mini Case: 5 – 26

You could then change rd = I/YR to see what happens to the bond’s value as r

changes, and plot the values—the graph would look like the one we developed earlier.

For example, if rd rose to 13%, we would input I/YR= 6.5 rather than 5%, and

find the 10-year bond’s value to be $834.72. If rd fell to 7%, then input I/YR = 3.5

and press PV to find the bond’s new value, $1,213.19.

We would find the values with a financial calculator, but they could also be found

with formulas. Thus:

V10-YEAR = $50 ((1- 1/(1+0.05)20)/0.065) + $1,000 (1/(1+0.05)20)

Mini Case: 5 – 27

h. Suppose a 10-year, 10 percent, semiannual coupon bond with a par value of

$1,000 is currently selling for $1,135.90, producing a nominal yield to maturity

of 8 percent. However, the bond can be called after 5 years for a price of $1,050.

h. 1. What is the bond’s nominal yield to call (YTC)?

Answer: If the bond were called, bondholders would receive $1,050 at the end of year 5. Thus,

the time line would look like this:

0 1 2 3 4 5

| | | | | |

50 50 50 50 50 50 50 50 50 50

rNOM = 2(3.765%) = 7.5301% ≈ 7.5%.

This 7.5% is the rate brokers would quote if you asked about buying the bond.

You could also calculate the EAR on the bond:

Mini Case: 5 – 28

h. 2. If you bought this bond, do you think you would be more likely to earn the YTM

or the YTC? Why?

Answer: Since the coupon rate is 10% versus YTC = rd = 7.53%, it would pay the company to

call the bond, get rid of the obligation to pay $100 per year in interest, and sell

i. Write a general expression for the yield on any debt security (rd) and define

these terms: real risk-free rate of interest (r*), inflation premium (IP), default

risk premium (DRP), liquidity premium (LP), and maturity risk premium

(MRP).

Answer: rd = r* + IP + DRP + LP + MRP.

Mini Case: 5 – 29

j. Define the nominal risk-free rate (rRF). What security can be used as an estimate

of rRF?

Answer: The real risk-free rate, r*, is the rate that would exist on default-free securities in the

absence of inflation:

k. Describe a way to estimate the inflation premium (IP) for a T-Year bond.

Answer: Treasury Inflation-Protected Securities (TIPS) are indexed to inflation. The IP for a

particular length maturity can be approximated as the difference between the yield on

Mini Case: 5 – 30

l. What is a bond spread and how is it related to the default risk premium? How

are bond ratings related to default risk? What factors affect a company’s bond

rating?

Answer: A “bond spread” is often calculated as the difference between a corporate bond’s

yield and a Treasury security’s yield of the same maturity. Therefore:

Mini Case: 5 – 31

m. What is interest rate (or price) risk? Which bond has more interest rate risk, an

annual payment 1-year bond or a 10-year bond? Why?

Answer: Interest rate risk, which is often just called price risk, is the risk that a bond will lose

value as the result of an increase in interest rates. Earlier, we developed the following

values for a 10 percent, annual coupon bond:

Mini Case: 5 – 32

n. What is reinvestment rate risk? Which has more reinvestment rate risk, a 1–

year bond or a 10-year bond?

Answer: Investment rate risk is defined as the risk that cash flows (interest plus principal

repayments) will have to be reinvested in the future at rates lower than today’s rate.

o. How are interest rate risk and reinvestment rate risk related to the maturity risk

premium?

Answer: Long-term bonds have high interest rate risk but low reinvestment rate risk. Short-

term bonds have low interest rate risk but high reinvestment rate risk. Nothing is

Mini Case: 5 – 33

p. What is the term structure of interest rates? What is a yield curve?

Answer: The term structure of interest rates is the relationship between interest rates, or

yields, and maturities of securities. When this relationship is graphed, the resulting

curve is called a yield curve.

Mini Case: 5 – 34

q. Briefly describe bankruptcy law. If this firm were to default on the bonds,

would the company be immediately liquidated? Would the bondholders be

assured of receiving all of their promised payments?

Answer: When a business becomes insolvent, it does not have enough cash to meet scheduled

interest and principal payments. A decision must then be made whether to dissolve

the firm through liquidation or to permit it to reorganize and thus stay alive.

If the firm is deemed to be too far gone to be saved, it will be liquidated and the

priority of claims would be as follows:

1. Secured creditors.

2. Trustee’s costs.

3. Expenses incurred after bankruptcy was filed.

4. Wages due workers, up to a limit of $2,000 per worker.

Web Solutions: 5 – 35

Web Appendix 5A

A Closer Look at Zero Coupon Bonds

Answers to Questions

5A-1 No, not all original issue discount bonds have zero coupons. Zero coupon bonds are just

one type of original issue discount bond. Any nonconvertible bond whose coupon rate is

set below the going market rate at the time of its issue will sell at a discount, and its will

be classified (for tax and other purposes) as an OID bond.

5A-3 Treasury zeros are not protected from interest rate (price) risk, because the principal is

totally susceptible to interest rate movements. You can see this by changing interest rates

and seeing what happens to the value of the zero bond. However, since Treasury zeros

generally are not callable and because there are no coupon payments to reinvest, Treasury

zeros are completely protected against reinvestment risk (the risk of having to invest cash

flows from a bond at a lower rate because of a decline in interest rates).

Web Solutions: 5 – 36

Solutions to Problems

5A-1 Year 0 1 2 3 4

Accrued value 708.43 772.19 841.69 917.44 1,000.00

Interest 63.76 69.50 75.75 82.56

Tax savings (40%) 25.50 27.80 30.30 33.02

Cash flow +708.43 +25.50 +27.80 +30.30 -966.98

Enter the following data into your calculator to determine the price of each bond:

N = 4; I/YR = 9; PMT = 0; FV = 1000; PV = ? Solve for PV = $708.43.

Web Solutions: 5 – 37

5A-2 0 1 2 3 4

Accrued value 708.43 772.19 841.69 917.44 1,000.00

Interest 63.76 69.50 75.75 82.56

Tax savings (35%) 22.32 24.33 26.51 28.90

Cash flow -708.43 -22.32 -24.33 -26.51 +971.10

Enter the following data into your calculator to determine the price of each bond:

N = 4; I/YR = 9; PMT = 0; FV = 1000; PV = ? Solve for PV = $708.43.

5A-3

Using a financial calculator, enter the following data: N = 5; I/YR = 10; PMT = 0; FV =

6000000; and then solve for PV = $3,725,527.94.

5A-4 Step 1: Find out what was paid for the bond:

PV = $1,000/(1.068)7 = $630.959.

Web Solutions: 5 – 38

5A-5 First find the yields on one-year and two-year zero coupon bonds, so you can find the

implied rate on a one-year bond, one year from now. Then use this implied rate to find its

price.

1-Year:

5A-6 0 10 50

-87.2037 1,000

 (1.05)10 = 142.0457

 1.10

156.2503

Web Solutions: 5 – 39

Web Appendix 5D

The Pure Expectations Theory and Estimation of Forward

Rates

Solutions to Problems

5D-1 rT1 = 5%; 1rT1 = 6%; rT2 = ?

5D-2 Let X equal the yield on 2-year securities 4 years from now:

(1.07)4(1 + X)2 = (1.075)6

(1.3108)(1 + X)2 = 1.5433

1 + X =

2/1

3108.1

5433.1 











X = 8.5%.

5D-3 a. (1.045)2 = (1.03)(1 + X)

1.092/1.03 = 1 + X

X = 6%.

Web Solutions: 5 – 40

5D-4 r* = 2%; MRP = 0%; r1 = 5%; r2 = 7%; X = ?

X represents the one-year rate on a bond one year from now (Year 2).