4. A 20-year maturity 9% coupon bond paying coupons semiannually is callable in

Concept

five years at a call price of $1,050. The bond currently sells at a yield to maturity

CHECK of 8% (bond equivalent yield). What is the yield to call?

Realized Compound Yield versus Yield to Maturity

We have noted that yield to maturity will equal the rate of return realized over the life of the

bond if all coupons are reinvested at an interest rate equal to the bond™s yield to maturity. Con-

sider for example, a two-year bond selling at par value paying a 10% coupon once a year. The

yield to maturity is 10%. If the $100 coupon payment is reinvested at an interest rate of 10%,

the $1,000 investment in the bond will grow after two years to $1,210, as illustrated in Figure

9.5, Panel A. The coupon paid in the first year is reinvested and grows with interest to a sec-

ond-year value of $110, which, together with the second coupon payment and payment of par

value in the second year, results in a total value of $1,210. The compound growth rate of in-

vested funds, therefore, is calculated from

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Essentials of Investments, Companies, 2003

Fifth Edition

311

9 Bond Prices and Yields

F I G U R E 9.5

A. Reinvestment rate = 10%

$1,100 Growth of

invested funds

Cash flow: $100

1 2

Time: 0

$1,100 = $1,100

Future

100 x 1.10 = $ 110

value:

$1,210

B. Reinvestment rate = 8%

$1,100

Cash flow: $100

1 2

Time: 0

$1,100 = $1,100

Future

100 x 1.08 = $ 108

value:

$1,208

yrealized)2

$1,000 (1 $1,210

yrealized 0.10 10%

With a reinvestment rate equal to the 10% yield to maturity, the realized compound yield

equals yield to maturity.

But what if the reinvestment rate is not 10%? If the coupon can be invested at more than

10%, funds will grow to more than $1,210, and the realized compound return will exceed

10%. If the reinvestment rate is less than 10%, so will be the realized compound return. Con-

sider the following example.

If the interest rate earned on the first coupon is less than 10%, the final value of the invest-

ment will be less than $1,210, and the realized compound yield will be less than 10%. Sup-

EXAMPLE 9.5

pose the interest rate at which the coupon can be invested equals 8%. The following

calculations are illustrated in Panel B of Figure 9.5. Realized

Compound Yield

Future value of first coupon payment with interest earnings $100 1.08 $ 108

Cash payment in second year (final coupon plus par value) 1,100

Total value of investment with reinvested coupons $1,208

The realized compound yield is computed by calculating the compound rate of growth of

invested funds, assuming that all coupon payments are reinvested. The investor purchased

the bond for par at $1,000, and this investment grew to $1,208.

yrealized)2

$1,000(1 $1,208

yrealized 0.0991 9.91%

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

312 Part THREE Debt Securities

Example 9.5 highlights the problem with conventional yield to maturity when reinvestment

rates can change over time. However, in an economy with future interest rate uncertainty, the

rates at which interim coupons will be reinvested are not yet known. Therefore, while realized

compound yield can be computed after the investment period ends, it cannot be computed in

advance without a forecast of future reinvestment rates. This reduces much of the attraction of

horizon analysis the realized yield measure.

We also can calculate realized compound yield over holding periods greater than one

Analysis of bond

returns over multiyear period. This is called horizon analysis and is similar to the procedure in Example 9.5. The

horizon, based on forecast of total return will depend on your forecasts of both the yield to maturity of the bond

forecasts of bond™s

when you sell it and the rate at which you are able to reinvest coupon income. With a longer

yield to maturity

investment horizon, however, reinvested coupons will be a larger component of your final

and reinvestment

proceeds.

rate of coupons.

Suppose you buy a 30-year, 7.5% (annual payment) coupon bond for $980 (when its yield to

maturity is 7.67%) and plan to hold it for 20 years. Your forecast is that the bond™s yield to

9.6 EXAMPLE maturity will be 8% when it is sold and that the reinvestment rate on the coupons will be 6%.

At the end of your investment horizon, the bond will have 10 years remaining until expiration,

Horizon Analysis

so the forecast sales price (using a yield to maturity of 8%) will be $966.45. The 20 coupon

payments will grow with compound interest to $2,758.92. (This is the future value of a

20-year $75 annuity with an interest rate of 6%.

Based on these forecasts, your $980 investment will grow in 20 years to $966.45

$2,758.92 $3,725.37. This corresponds to an annualized compound return of 6.90%, cal-

culated by solving for r in the equation $980 (1 r)20 $3,725.37.

9.4 BOND PRICES OVER TIME

As we noted earlier, a bond will sell at par value when its coupon rate equals the market in-

terest rate. In these circumstances, the investor receives fair compensation for the time value

of money in the form of the recurring interest payments. No further capital gain is necessary

to provide fair compensation.

When the coupon rate is lower than the market interest rate, the coupon payments alone

will not provide investors as high a return as they could earn elsewhere in the market. To re-

ceive a fair return on such an investment, investors also need to earn price appreciation on

their bonds. The bonds, therefore, would have to sell below par value to provide a “built-in”

capital gain on the investment.

To illustrate this point, suppose a bond was issued several years ago when the interest rate

was 7%. The bond™s annual coupon rate was thus set at 7%. (We will suppose for simplicity

that the bond pays its coupon annually.) Now, with three years left in the bond™s life, the in-

terest rate is 8% per year. The bond™s fair market price is the present value of the remaining

annual coupons plus payment of par value. That present value is

$70 Annuity factor(8%, 3) $1,000 PV factor(8%, 3) $974.23

which is less than par value.

In another year, after the next coupon is paid, the bond would sell at

$70 Annuity factor(8%, 2) $1,000 PV factor(8%, 2) $982.17

thereby yielding a capital gain over the year of $7.94. If an investor had purchased the bond at

$974.23, the total return over the year would equal the coupon payment plus capital gain, or

$70 $7.94 $77.94. This represents a rate of return of $77.94/$974.23, or 8%, exactly the

current rate of return available elsewhere in the market.

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Essentials of Investments, Companies, 2003

Fifth Edition

313

9 Bond Prices and Yields

F I G U R E 9.6

Price ($)

Premium bond Price paths of coupon

bonds in the case of

constant market

interest rates

1,000

Discount bond

Time

Maturity

0

date

<

5. What will the bond price be in yet another year, when only one year remains until Concept

maturity? What is the rate of return to an investor who purchases the bond at

CHECK

$982.17 and sells it one year hence?

When bond prices are set according to the present value formula, any discount from par

value provides an anticipated capital gain that will augment a below-market coupon rate just

sufficiently to provide a fair total rate of return. Conversely, if the coupon rate exceeds the

market interest rate, the interest income by itself is greater than that available elsewhere in the

market. Investors will bid up the price of these bonds above their par values. As the bonds ap-

proach maturity, they will fall in value because fewer of these above-market coupon payments

remain. The resulting capital losses offset the large coupon payments so that the bondholder

again receives only a fair rate of return.

Problem 9 at the end of the chapter asks you to work through the case of the high coupon

bond. Figure 9.6 traces out the price paths of high and low coupon bonds (net of accrued in-

terest) as time to maturity approaches, at least for the case in which the market interest rate is

constant. The low coupon bond enjoys capital gains, while the high coupon bond suffers cap-

ital losses.8

We use these examples to show that each bond offers investors the same total rate of return.

Although the capital gain versus income components differ, the price of each bond is set to

provide competitive rates, as we should expect in well-functioning capital markets. Security

returns all should be comparable on an after-tax risk-adjusted basis. If they are not, investors

will try to sell low-return securities, thereby driving down the prices until the total return at

the now lower price is competitive with other securities. Prices should continue to adjust un-

til all securities are fairly priced in that expected returns are appropriate (given necessary risk

and tax adjustments).

Yield to Maturity versus Holding-Period Return

We just considered an example in which the holding-period return and the yield to maturity

were equal: in our example, the bond yield started and ended the year at 8%, and the bond™s

holding-period return also equaled 8%. This turns out to be a general result. When the yield to

8

If interest rates are volatile, the price path will be “jumpy,” vibrating around the price path in Figure 9.6, and reflect-

ing capital gains or losses as interest rates fall or rise. Ultimately, however, the price must reach par value at the ma-

turity date, so on average, the price of the premium bond will fall over time while that of the discount bond will rise.

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

314 Part THREE Debt Securities

maturity is unchanged over the period, the rate of return on the bond will equal that yield. As

we noted, this should not be a surprising result: the bond must offer a rate of return competi-

tive with those available on other securities.

However, when yields fluctuate, so will a bond™s rate of return. Unanticipated changes in

market rates will result in unanticipated changes in bond returns, and after the fact, a bond™s

holding-period return can be better or worse than the yield at which it initially sells. An in-

crease in the bond™s yield to maturity acts to reduce its price, which means that the holding-

period return will be less than the initial yield. Conversely, a decline in yield to maturity

results in a holding-period return greater than the initial yield.

Consider a 30-year bond paying an annual coupon of $80 and selling at par value of $1,000.

The bond™s initial yield to maturity is 8%. If the yield remains at 8% over the year, the bond

9.7 EXAMPLE price will remain at par, so the holding-period return also will be 8%. But if the yield falls be-

low 8%, the bond price will increase. Suppose the price increases to $1,050. Then the hold-

Yield to Maturity

ing-period return is greater than 8%:

versus Holding-

$80 ($1,050 $1,000)

Period Return Holding-period return .13, or 13%

$1,000

>

6. Show that if yield to maturity increases, then holding-period return is less than ini-

Concept

tial yield. For example, suppose that by the end of the first year, the bond™s yield to

CHECK maturity is 8.5%. Find the one-year holding-period return and compare it to the

bond™s initial 8% yield to maturity.