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

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