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Bond Valuation

Global Financial Management

Campbell R. HarveyFuqua School of Business

Duke Universitycharvey@mail.duke.edu

http://www.duke.edu/~charvey

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Definition of a Bond

A bond is a security that obligates the issuer to make specified interest and principal payments to the holder on specified dates. Coupon rate Face value (or par) Maturity (or term)

Bonds are sometimes called fixed income securities.

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Types of Bonds

Pure Discount or Zero-Coupon Bonds Pay no coupons prior to maturity. Pay the bond’s face value at maturity.

Coupon Bonds Pay a stated coupon at periodic intervals prior to maturity. Pay the bond’s face value at maturity.

Perpetual Bonds (Consols) No maturity date. Pay a stated coupon at periodic intervals.

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Types of Bonds

Self-Amortizing Bonds Pay a regular fixed amount each payment period over the

life of the bond. Principal repaid over time rather than at maturity.

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Bond Issuers

Federal Government and its Agencies Local Municipalities Corporations

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U.S. Government Bonds

Treasury Bills No coupons (zero coupon security) Face value paid at maturity Maturities up to one year

Treasury Notes Coupons paid semiannually Face value paid at maturity Maturities from 2-10 years

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U.S. Government Bonds

Treasury Bonds Coupons paid semiannually Face value paid at maturity Maturities over 10 years The 30-year bond is called the long bond.

Treasury Strips Zero-coupon bond Created by “stripping” the coupons and principal from

Treasury bonds and notes.

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Agencies Bonds

Mortgage-Backed Bonds Bonds issued by U.S. Government agencies that are backed

by a pool of home mortgages. Self-amortizing bonds. Maturities up to 20 years.

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U.S. Government Bonds

No default risk. Considered to be riskfree. Exempt from state and local taxes. Sold regularly through a network of primary dealers. Traded regularly in the over-the-counter market.

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Municipal Bonds

Maturities from one month to 40 years. Exempt from federal, state, and local taxes. Generally two types:

Revenue bonds General Obligation bonds

Riskier than U.S. Government bonds.

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Corporate Bonds

Secured Bonds (Asset-Backed) Secured by real property Ownership of the property reverts to the bondholders upon

default. Debentures

General creditors Have priority over stockholders, but are subordinate to

secured debt.

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Common Features of Corporate Bonds

Senior versus subordinated bonds Convertible bonds Callable bonds Putable bonds Sinking funds

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Bond Ratings

Moody’s S&P Quality of Issue

Aaa AAA Highest quality. Very small risk of default.

Aa AA High quality. Small risk of default.

A A High-Medium quality. Strong attributes, but potentiallyvulnerable.

Baa BBB Medium quality. Currently adequate, but potentiallyunreliable.

Ba BB Some speculative element. Long-run prospectsquestionable.

B B Able to pay currently, but at risk of default in thefuture.

Caa CCC Poor quality. Clear danger of default .

Ca CC High specullative quality. May be in default.

C C Lowest rated. Poor prospects of repayment.

D - In default.

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Valuing Zero Coupon Bonds

What is the current market price of a U.S. Treasury strip that matures in exactly 5 years and has a face value of $1,000. The yield to maturity is rd=7.5%.

What is the yield to maturity on a U.S. Treasury strip that pays $1,000 in exactly 7 years and is currently selling for $591.11?

1000

1 075565

.$696.=

591 111000

17. =

+ rd

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Bond Yields and PricesThe case of zero coupon bonds

Consider three zero-coupon bonds, all with » face value of F=100» yield to maturity of r=10%, compounded annually.

We obtain the following table:

Bond 1 Bond 2 Bond 3Time / Bond value 10% $90.91 $75.13 $62.09

1 100 0 02 0 03 100 04 05 100

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Suppose the yield would drop suddenly to 9%, or increase to 10%. How would prices respond?

Bond prices move up if the yield drops, decrease if yield rises Prices respond more strongly for higher maturities

The Impact of Price Responses

Yield Bond 1 Bond 2 Bond 31 Year 3 Year 5 Year

10% $90.91 $75.13 $62.099% $91.74 $77.22 $64.99

% change 0.91% 2.70% 4.46%11% $90.09 $73.12 $59.35

% change -0.91% -2.75% -4.63%

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What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the required yield to maturity is 10% compounded semiannually?

0 6 12 18 24 ... 120 Months

45 45 45 45 1045

Bond Valuation:An Example

B

45

0 051

1

1 05

1000

1 056920 20. . .

$937.

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What is the market price of a U.S. Treasury bond that has a coupon rate of 9%, a face value of $1,000 and matures exactly 10 years from today if the required yield to maturity is 10% compounded semiannually?

0 1 2 3 4 ... n

C C C C C+F

Valuing Coupon BondsThe General Formula

B

C

r r

F

rCA F r

d dn

dn n d

n

11

1 11

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Bond Yields and PricesThe case of coupon bonds

Suppose you purchase the U.S. Treasury bond described earlier and immediately thereafter interest rates fall so that the new yield to maturity on the bond is 8% compounded semiannually. What is the bond’s new market price?

Suppose the interest rises, so that the new yield is 12% compounded semiannually. What is the market price now?

Suppose the interest equals the coupon rate of 9%. What do you observe?

Note:» Coupon bonds can be regarded as portfolios of zero-coupon

bonds (how?)» What implication does this have for price responses?

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New Semiannual yield = 8%/2 = 4%

What is the price of the bond if the yield to maturity is 8% compounded semiannually?

Similarly:

If r=12%: B=$ 827.95

If r= 9%: B=$1,000.00

Valuing Coupon Bonds (cont.)

B

C

r r

F

r

n

n

1

1

1 1

B

1

0 041

1

1 0445

1 000

1 049520 20. .

*,

.$1067.

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Relationship Between Bond Prices and Yields

Bond prices are inversely related to interest rates (or yields). A bond sells at par only if its coupon rate equals the coupon

rate A bond sells at a premium if its coupon is above the coupon

rate. A bond sells a a discount if its coupon is below the coupon

rate.

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Volatility of Coupon Bonds

Consider two bonds with 10% annual coupons with maturities of 5 years and 10 years.

The yield is 8% What are the responses to a 1% price change?

The sensitivity of a coupon bond increases with the maturity?

Yield 5-year bond 10-year bond8% $1,079.85 $1,134.209% $1,038.90 $1,064.18

% Change -3.79% -6.17%7% $1,123.01 $1,210.71

% Change 4.00% 6.75%Average 3.89% 6.46%

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Bond Prices and Yields

Bond Price

F

c Yield

Longer term bonds are moresensitive to changes in interestrates than shorter term bonds.

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Consider the following two bonds:» Both have a maturity of 5 years» Both have yield of 8%» First has 6% coupon, other has 10% coupon, compounded

annually. Then, what are the price sensitivities of these bonds to a 1%

increase (decrease) in bond yields?

Why do we get different answers?

Bond Yields and PricesThe problem

Yield 6%-Bond 10%-Bond8% $920.15 $1,079.859% $883.31 $1,038.90

% Change -4.00% -3.79%7% $959.00 $1,123.01

% Change 4.22% 4.00%Average 4.11% 3.89%

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Calculate the average maturity of a bond:» Coupon bond is like portfolio of zero coupon bonds» Compute average maturity of this portfolio» Give each zero coupon bond a weight equal to the

proportion in the total value of the portfolio Write value of the bond as:

The factor:

is the proportion of the t-th coupon payment in the total value of the bond.

DurationApproximating the maturity of a bond

BC

r

C

r

C

r

C F

rtt

nn

1 221 1 1 1( ) ( )

...( )

...( )

C

B rt

t( )1

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Duration is defined as a weighted average of the maturities of the individual payments:

» This definition of duration is sometimes also referred to as Macaulay Duration.

The duration of a zero coupon bond is equal to its maturity.

Duration: A Definition

DC

B r

C

B rt

C

B rnC F

B rt

tn

n

1 221

21 1 1( ) ( )

...( )

...( )

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Calculate the duration of the 6% 5-year bond:

Calculate the duration of the 10% 5-year bond:

The duration of the bond with the lower coupon is higher» Why?

Calculating Duration

Time Payment PV(Payment)% of PV Time*%PV1 60 55.56 6.04% 0.062 60 51.44 5.59% 0.113 60 47.63 5.18% 0.164 60 44.10 4.79% 0.195 1060 721.42 78.40% 3.92

920.15 100.00% 4.44

Time Payment PV(Payment)% of PV Time*%PV1 100 92.59 8.57% 0.092 100 85.73 7.94% 0.163 100 79.38 7.35% 0.224 100 73.50 6.81% 0.275 1100 748.64 69.33% 3.47

1079.85 100.00% 4.20

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Duration: An Exercise

What is the interest rate sensitivity of the

following two bonds. Assume coupons are

paid annually.

Bond A Bond B

Coupon rate 10% 0%

Face value $1,000 $1,000

Maturity 5 years 10 years

YTM 10% 10%

Price $1,000 $385.54

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Duration Exercise (cont.)

Year (t) PV(A) PV(A) x t PV(B) PV(B)xt1 $90.91 $90.91 0 02 $82.64 $165.89 0 03 $75.13 $225.39 0 04 $68.30 $273.21 0 05 $683.01 $3,415.07 0 06 0 0 0 07 0 0 0 08 0 0 0 09 0 0 0 0

10 0 0 $385.54 $3,855.43Totals $1000.00 $4,170.47 $385.54 $3,855.43

Duration 4.17 10.00

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Duration Exercise (cont.)

Percentage change in bond price for a small increase in the interest rate:

Pct. Change = - [1/(1.10)][4.17] = - 3.79%

Bond A

Pct. Change = - [1/(1.10)][10.00] = - 9.09%

Bond B

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For a zero-coupon bond with maturity n we have derived:

For a coupon-bond with maturity n we can show:

» The right hand side is sometimes also called modified duration.

Hence, in order to analyze bond volatility, duration, and not maturity is the appropriate measure.» Duration and maturity are the same only for zero-coupon

bonds!

Duration and Volatility

B

r B

n

r

1

1

B

r B

D

r

1

1

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Duration and VolatilityThe example reconsidered

Compute the right hand side for the two 5-year bonds in the previous example:» 6%-coupon bond:

D/(1+r) = 4.44/1.08=4.11» 10%-coupon bond:

D/(1+r) = 4.20/1.08=3.89 But these are exactly the average price responses we found

before!» Hence, differences in duration explain variation of price

responses across bonds with the same maturity.

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Is Duration always Exact? Consider the two 5-year bonds (6% and 10%) from the example

before, but interest rates can change by moving 3% up or down:

This is different from the duration calculation which gives:» 6% coupon bond: 3*4.11%=12.33%<12.39%» 10% coupon bond: 3*3.89%=11.67%<11.73%

Result is imprecise for larger interest rate movements» Relationship between bond price and yield is convex, but» Duration is a linear approximation

Yield 6%-Bond 5-year bond8% $920.15 $1,079.85

11% $815.21 $963.04% Change -11.40% -10.82%

5% $1,043.29 $1,216.47% Change 13.38% 12.65%Average 12.39% 11.73%

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The Term Structure of Interest Rates

The term structure of interest rates is the relationship between time to maturity and yield to maturity:

Yield

Maturity1 2 3

5.00

5.75

6.00

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Spot and Forward Rates

A spot rate is a rate agreed upon today, for a loan that is to be made today. (e.g. r1=5% indicates that the current rate for a one-year loan is 5%).

A forward rate is a rate agreed upon today, for a loan that is to be made in the future. (e.g. 2f1=7% indicates that we could contract today to borrow money at7% for one year, starting two years from today).» r1=5.00%, r2=5.75%, r3=6.00%

» We can either:– Invest $100 for three years , or:– Invest $100 for two years, and contract (today) at the

one year rate, two years forward

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Forward RatesA first look at arbitrage

Which investment strategy is optimal:» Invest $100 for three years:

$100*(1.06)3=» Invest $100 for two years, and invest the proceeds at the

two-year forward rate:

$100*(1.0575)2(1+2f1)=

» Hence the first strategy is optimal if 2f1<6.50%, the second if

2f1>6.50%.

Hence 2f1=6.50% (Why?)

» More generally: (1+rn+t)n+t=(1+rn)n(1+nft)

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When should you borrow?

Suppose you wish to borrow $20,000 in two years in order to borrow a car, and you know you can repay the loan in three years? You have two options:

I. 1. Borrow $17,884 now at 6%, repay $20,000*(1.06)3

=$21,300.35 in three years .

2. Invest the proceeds from the loan for two years at 5.75% to have $17,884*(1.0575)2=$20,000 in two years.

II. Wait for two years, borrow at the prevailing one year loan rate in 1 year?

<forget about the cut the bank gets> When would you follow strategy I (lock in the current rate) rather

than wait (strategy II)?

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When to borrow (cont.)

If you lock in the current rate, then you secure a borrowing rate of:

$ 21,300.35/$20,000=1.065, i. e. 6.5%» This is exactly the forward rate we calculated above

– Why? Hence, you would borrow and lock in rates now, if you expect

that the one-year interest rate is going to be higher than 6.5% i 1999.» When would you set the cut-off rate for waiting higher?

(lower?) If everybody invests this way, then the forward rate equals the

expected future spot rate.» Why?

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Summary

Bonds can be valued by discounting future cash flows at the yield to maturity

Bond prices changes inverse with yield Price response of bond to interest rates depends on term to

maturity.» Works well for zero-coupon bond

Coupon bonds are like portfolios of zero-coupon bonds» Need duration as “average maturity” for coupon bonds» Only an approximation

The term structure implies terms for future borrowing:» Forward rates» Compare with expected future spot rates