# Bond Valuation Global Financial Management

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Bond Valuation

Global Financial Management

Campbell R. Harvey

Fuqua School of Business

Duke University

charvey@mail.duke.edu

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

Slide2Definition 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

.

Slide3Types 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.

Slide4Types 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.

Slide5Bond Issuers

Federal Government and its Agencies

Local Municipalities

Corporations

Slide6U.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

Slide7U.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.

Slide8Agencies 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.

Slide9U.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.

Slide10Municipal 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.

Slide11Corporate 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.

Slide12Common Features of Corporate Bonds

Senior versus subordinated bonds

Convertible bonds

Callable bonds

Putable bonds

Sinking funds

Slide13Bond Ratings

Slide14Valuing 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 r

d

=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

075

56

5

.

$696

.

=

591

11

1000

1

7

.

=

+

r

d

Slide15Bond Yields and Prices

The 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:

Slide16Suppose 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

Slide17What 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

Slide18What 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 Bonds

The General Formula

Slide19Bond Yields and Prices

The 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?

Slide20New 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.)

Slide21Relationship 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.

Slide22Volatility 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?

Slide23Bond Prices and Yields

Bond Price

F

c Yield

Longer term bonds are more

sensitive to changes in interest

rates than shorter term bonds.

Slide24Consider 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 Prices

The problem

Slide25Calculate 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.

Duration

Approximating the maturity of a bond

Slide26Duration 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

Slide27Calculate 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

Slide28Duration: 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

Slide29Duration Exercise (cont.)

Slide30Duration 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

Slide31For 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

Slide32Duration and Volatility

The 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.

Slide33Is 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

Slide34The Term Structure

of Interest Rates

The

term structure of interest rates

is the relationship between time to maturity and yield to maturity:

Yield

Maturity

1

2

3

5.00

5.75

6.00

Slide35Spot and Forward Rates

A

spot rate

is a rate agreed upon today, for a loan that is to be made today. (e.g. r

1

=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.

2

f

1

=7% indicates that we could contract today to borrow money at7% for one year, starting two years from today).

r

1

=5.00%, r

2

=5.75%, r

3

=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

Slide36Forward Rates

A 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+

2

f

1

)=

Hence the first strategy is optimal if

2

f

1

<6.50%, the second if

2

f

1

>6.50%.

Hence

2

f

1

=6.50% (Why?)

More generally: (1+r

n+t

)

n+t

=(1+r

n

)

n

(1+

n

f

t

)

Slide37When 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)?

Slide38When 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?

Slide39Summary

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

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