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CHAPTER 8

Bond Valuation and Risk

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CHAPTER 8 OVERVIEW

This chapter will:

A. Explain how debt securities are priced

B. Identify the factors that affect bond prices

C. Explain how the sensitivity of bond prices to

interest rates is dependent on particular bond characteristics

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Preparing BAII Plus for use

Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’.

Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’.

Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.

Magic Tools

Time Line

PV, FV, I/Y, PMT, N

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Zero-coupon bond (ZCB) 1

Zero coupon rate, no coupon paid during bond’s life. Bond holder receives one payment at maturity, the

face value

Price of a ZCB, PZCB

NZCBZCBr

FP

1

F = face value of the bond

cost of ZCB debt capital (in decimals)

N = number of years to maturity

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Zero-coupon bond (ZCB) 2

As long as interest rates are positive, the price of a ZCB must be less than its face value.

Why? With positive interest rates, the present value of the face value (i.e., the price) has to be less than the face value.

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ZCB Problems

1) Find the price of a ZCB with 20 years to maturity, par value of $1000 and a required rate of return of 15% p.a.

N=20, I/Y=15, FV=1000, PMT=0. Price = $61.10

2) XYZ Corp.’s ZCB has a market price of $ 354. The bond has 16 years to maturity and its face value is $1000. What is the cost of debt for the ZCB (i.e., the required rate of return).

PV=-354, FV=1000, N=16, PMT=0.

Required rate of return/ Cost of debt =6.71% p.a.

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Fixed-coupon bond (FCB) 1

Firm pays a fixed amount (‘coupon’) to the investor every period until bond matures.

At maturity, firm pays face value of the bond to investor.

Face value also called par value. Unless otherwise stated, always assume face value to be $1000.

Period: can be year, half-year (6 months)

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Fixed-coupon bond (FCB) 2

FCB gives you a stream of fixed payments plus a single payment (face value) at maturity.

This cash flow stream is just an annuity plus a single cash flow at maturity.

We use the financial calculator to compute the price of the FCB.

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Fixed-coupon bond (FCB) 3

Pricing formula:

C = coupon payment paid in each periodPar = par valuer = required rate of returnn = number of period to maturity

nrparC

r

C

r

CPV

1

...11 21

Example

A bond has a par value of $1,000, pays $100 at the end of each year in coupon payment, and has thee years remaining until maturity. Assume the annualized yield required is 12%, what is the price?

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97.951

96.78272.7929.89

)12.01(

1000100

)12.01(

100

)12.01(

100321

PV

Exhibit 8.1 Valuation of a Three-Year Bond

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Find FCB price

A $1,000 par value bond has coupon rate of 5% and the coupon is paid semi-annually. The bond matures in 20 years and has a required rate of return of 10%. Compute the current price of this bond.

PMT = 25. Why?

FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV.

PV = -571.02. Thus, price = $571.02 < par value

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Useful property 1

Go back to the bond in the last problem. Suppose annual coupon rate = 10%.Verify that price = $1000 = par value

Suppose annual coupon rate = 12%Verify that price = $1,171.59 > par value.

It turns out that the following property is true.

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Useful property 2

Coupon rate < discount rate

Price < face value

selling at discount

Coupon rate = discount rate

Price = face value

selling at par

Coupon rate > discount rate

Price > face value

selling at premium

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Apply what we learnt

A 10-year annual coupon bond was issued four years ago at par. Since then the bond’s yield to maturity (YTM) has decreased from 9% to 7%. Which of the following statements is true about the current market price of the bond?

A. The bond is selling at a discountB. The bond is selling at parC. The bond is selling at a premiumD. The bond is selling at book valueE. Insufficient information

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Try one more

One year ago Pell Inc. sold 20-year, $1,000 par value, annual coupon bonds at a price of $931.54 per bond. At that time the market rate (i.e., yield to maturity) was 9 percent. Today the market rate is 9.5 percent; therefore the bonds are currently selling:

A. at a discount.

B. at a premium.

C. at par.

D. above the market price.

E. not enough information.

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Find YTM, Coupon rate

1)A $1,000 par value bond sells for $863.05. It matures in 20 years, has a 10 percent coupon rate, and pays interest semi-annually. What is the bond’s yield to maturity on a per annum basis (to 2 decimal places)?

Verify that YTM = 11.80%2) ABC Inc. just issued a twenty-year semi-annual coupon

bond at a price of $787.39. The face value of the bond is $1,000, and the market interest rate is 9%. What is the annual coupon rate (in percent, to 2 decimal places)?

Verify that annual coupon rate = 6.69%

What happens if bond pays coupon annually?

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Long FCB question

HMV Inc. needs to raise funds for an expansion project. The company can choose to issue either zero-coupon bonds or semi-annual coupon bonds. In either case the bonds would have the SAME nominal required rate of return, a 20-year maturity and a par value of $1,000. If the company issues the zero-coupon bonds, they would sell for $153.81. If it issues the semi-annual coupon bonds, they would sell for $756.32. What annual coupon rate is HMV Inc. planning to offer on the coupon bonds? State your answer in percentage terms, rounded to 2 decimal places. Verify that annual coupon rate = 7.01%

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Bond Valuation

Impact of the Timing of Payments on Bond Valuation

a. Timing affects the market price of a bond

b. Funds received sooner can be reinvested to earn additional returns

c. Most bonds have semiannual coupons

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Factors that Affect Bond Prices

Factors That Affect the Risk-Free Rate

a. Inflationary Expectations

b. Economic Growth

c. Money Supply Growth

d. Federal Government Budget Deficit

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Factors that Affect Bond Prices

Factors That Affect the Credit (Default) Risk Premiuma Changes in the Credit Risk Premium over

Time

b. Changes in Bond Ratings over Time

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Sensitivity of Bond Prices to Interest Rate Movements

Duration: measurement of the life of a bond on a present value basis. The longer the duration, the greater its sensitivity to interest rate changes.

A measure of interest rate sensitivity.

The weighted average of the times to each coupon or principal

payment made by the bond.

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Duration Formula

Time until 4th cash flow

Weight of 4th cash flow

Cash flow paid at time t

iceBondYTMCF

wwhere

wt

TwwwwwD

tt

t

T

tt

T

Pr)1(

...432

1

4321

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Examples: 3-year bond, 8%, annual coupon payments, YTM = 10%

Duration

The duration of a bond with $1,000 par value and 7% annual coupon rate, three years remaining to maturity, and a 9% yield to maturity is

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yearsDUR 8.2

)09.1(1070

)09.1(70

)09.1(70

)09.1(31070

)09.1(270

)09.1(170

321

321

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Modified duration

Macaulay's duration1 initial YTM

D* =+

YTM (in decimal)

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Using Modified duration to estimate percentage price change

Dollar change in bond price

Modified duration

Change in YTM in decimal

Initial price

yDP

P

0

initialYTMnewYTMy

Interest rate risk example

If Bond X’s duration is 8 years, its yield is 10%, what is the percentage change in price for an increase in yield of 0.2 percentage point?

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%45.10145.0002.0)1.01

8(%

P

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Interpretation

As Macaulay’s duration ↑, interest rate sensitivity ↑ , vice versa.

As modified duration ↑, interest rate sensitivity ↑ , vice versa.

Another example

Chapter 8 problem 13.

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Homework Assignment 5

Chapter 7: Questions and Applications 12,14. Chapter 8:

Questions and Applications 8,12,14.

Problems 1,2,3,5,7,11 (a b),12,17.

Now discussion.

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