Chapter 14 – Bond Prices and Yields
CHAPTER 14: BOND PRICES AND YIELDS
PROBLEM SETS
1. The bond callable at 105 should sell at a lower price because the call provision is more
valuable to the firm. Therefore, its yield to maturity should be higher.
2. Zero coupon bonds provide no coupons to be reinvested. Therefore, the investor’s proceeds
3. A bond’s coupon interest payments and principal repayment are not affected by changes
in market rates. Consequently, if market rates increase, bond investors in the secondary
4. a. Effective annual rate for 3-month T-bill:
%0.10100.0102412.11
645,97
000,100 4
4
==−=−
b. Effective annual interest rate for coupon bond paying 5% semiannually:
5. The effective annual yield on the semiannual coupon bonds is 8.16%. If the annual
6. The bond price will be lower. As time passes, the bond price, which is now above par
value, will approach par.
14-2
7. Yield to maturity: Using a financial calculator, enter the following:
n = 3; PV = −953.10; FV = 1000; PMT = 80; COMP i
8. a.
Zero coupon
8% coupon
10% coupon
Current prices
$1,000.00
$1,134.20
b. Price 1 year from now
$1,000.00
$1,124.94
Price increase
$0.00
− $9.26
Coupon income
$80.00
$100.00
Pre-tax income
$80.00
$90.74
Pre-tax rate of return
8.00%
8.00%
Taxes*
$24.00
$28.15
After-tax income
$56.00
$62.59
After-tax rate of return
5.60%
5.52%
c. Price 1 year from now
$1,065.15
$1,195.46
Price increase
$65.15
$61.26
Coupon income
$80.00
$100.00
Pre-tax income
$145.15
$161.26
Pre-tax rate of return
14.52%
14.22%
Taxes**
$37.03
$42.25
After-tax income
$108.12
$119.01
After-tax rate of return
10.81%
10.49%
* In computing taxes, we assume that the 10% coupon bond was issued at par and
that the decrease in price when the bond is sold at year end is treated as a capital
loss and therefore is not treated as an offset to ordinary income.
9. a. On a financial calculator, enter the following:
You will find that the yield to maturity on a semi-annual basis is 4.26%. This
Chapter 14 – Bond Prices and Yields
14-3
b. Since the bond is selling at par, the yield to maturity on a semi-annual basis is the
c. Keeping other inputs unchanged but setting PV = –1050, we find a bond
10. Since the bond payments are now made annually instead of semi-annually, the bond
equivalent yield to maturity is the same as the effective annual yield to maturity. Using a
financial calculator, enter: n = 20; FV = 1000; PV = –price, PMT = 80.
The resulting yields for the three bonds are:
Bond Price
Bond equivalent yield =
Effective annual yield
$950
8.53%
$1,000
8.00%
$1,050
7.51%
The yields computed in this case are lower than the yields calculated with semi-annual
payments. All else equal, bonds with annual payments are less attractive to investors
11.
Price
Maturity
(years)
Bond equivalent
YTM
$400.00
20.00
4.688%
$500.00
20.00
3.526%
$500.00
10.00
7.177%
$385.54
10.00
10.000%
$463.19
10.00
8.000%
$400.00
11.91
8.000%
12. a. The bond pays $50 every 6 months. The current price is:
[$50 Annuity factor (4%, 6)] + [$1,000 PV factor (4%, 6)] = $1,052.42
Assuming the market interest rate remains 4% per half year, price six months from
b. Rate of return
42.052,1$
90.7$50$
42.052,1$
)42.052,1$52.044,1($50$−
=
−+
=
Chapter 14 – Bond Prices and Yields
14-4
13. The reported bond price is: 100 2/32 percent of par = $1,000.625
However, 15 days have passed since the last semiannual coupon was paid, so:
14. If the yield to maturity is greater than the current yield, then the bond offers the
16.
Time
Inflation in
year just
ended
Par value
Coupon
payment
Principal
repayment
0
$1,000.00
1
2%
$1,020.00
$40.80
$ 0.00
2
3%
$1,050.60
$42.02
$ 0.00
3
1%
$1,061.11
$42.44
$1,061.11
The nominal rate of return and real rate of return on the bond in each year are
computed as follows:
Nominal rate of return = interest + price appreciation
initial price
Real rate of return = 1 + nominal return
1 + inflation − 1
Second year
Third year
Nominal return
071196.0
020,1$
60.30$02.42$=
+
050400.0
60.050,1$
51.10$44.42$=
+
Real return
%0.4040.01
03.1
071196.1 ==−
%0.4040.01
01.1
050400.1 ==−
The real rate of return in each year is precisely the 4% real yield on the bond.
Chapter 14 – Bond Prices and Yields
14-5
17. The price schedule is as follows:
Year
Remaining
Maturity (T)
Constant yield value
$1,000/(1.08)T
Imputed interest
(Increase in constant
yield value)
0 (now)
20 years
$214.55
1
19
$231.71
$17.16
2
18
$250.25
$18.54
19
1
$925.93
20
0
$1,000.00
$74.07
18. The bond is issued at a price of $800. Therefore, its yield to maturity is: 6.8245%
19. a. The bond sells for $1,124.72 based on the 3.5% yield to maturity.
b. If the call price were $1,050, we would set FV = 1,050 and redo part (a) to find
c. Yield to call is 3.031% semiannually, 6.602% annually.
20. The stated yield to maturity, based on promised payments, equals 16.075%.
21. The bond is selling at par value. Its yield to maturity equals the coupon rate, 10%. If the
first-year coupon is reinvested at an interest rate of r percent, then total proceeds at the
end of the second year will be: [$100 (1 + r)] + $1,100
Therefore, realized compound yield to maturity is a function of r, as shown in the following
table:
r
Total proceeds
Realized YTM = Proceeds/1000 – 1
8%
$1,208
1208/1000 – 1 = 0.0991 = 9.91%
10%
$1,210
1210/1000 – 1 = 0.1000 = 10.00%
12%
$1,212
1212/1000 – 1 = 0.1009 = 10.09%
Chapter 14 – Bond Prices and Yields
14-6
22. April 15 is midway through the semiannual coupon period. Therefore, the invoice price
23. Factors that might make the ABC debt more attractive to investors, therefore justifying a
lower coupon rate and yield to maturity, are:
i. The ABC debt is a larger issue and therefore may sell with greater liquidity.
ii. An option to extend the term from 10 years to 20 years is favorable if interest rates
24. a. The floating rate note pays a coupon that adjusts to market levels. Therefore, it
will not experience dramatic price changes as market yields fluctuate. The fixed
rate note will therefore have a greater price range.
b. Floating rate notes may not sell at par for any of several reasons:
Chapter 14 – Bond Prices and Yields
d. The fixed-rate note currently sells at only 88% of the call price, so that yield to
maturity is greater than the coupon rate. Call risk is currently low, since yields
25. a. The yield to maturity on the par bond equals its coupon rate, 8.75%. All else
equal, the 4% coupon bond would be more attractive because its coupon rate is far
below current market yields, and its price is far below the call price. Therefore, if
yields fall, capital gains on the bond will not be limited by the call price. In
26. a. Initial price P0 = $705.46 [n = 20; PMT = 50; FV = 1000; i = 8]
Chapter 14 – Bond Prices and Yields
14-8
Tax on explicit interest plus implicit interest in first year =
Capital gain in first year = Actual price at 7% YTM – constant yield price =
$793.29 – $711.89 = $81.40
46.705$
99.46$)46.705$29.793($50$==
−−+
d. Value of bond after two years = $798.82 [using n = 18; i = 7%]
Reinvested income from the two coupon interest payments =
e. Coupon interest received in first year: $50.00
Less: tax on coupon interest @ 40%: – 20.00
The year-1 cash flow can be invested at an after-tax rate of:
By year 2, this investment will grow to: $27.43 1.018 = $27.92
Chapter 14 – Bond Prices and Yields
14-9
CFA PROBLEMS
1. a. A sinking fund provision requires the early redemption of a bond issue. The
provision may be for a specific number of bonds or a percentage of the bond issue
b. (i) Compared to a bond without a sinking fund, the sinking fund reduces the
average life of the overall issue because some of the bonds are retired prior to the
stated maturity.
principal.
c. From the investor’s point of view, the key reason for demanding a sinking fund is
2. a. (i) Current yield = Coupon/Price = $70/$960 = 0.0729 = 7.29%
(ii) YTM = 3.993% semiannually or 7.986% annual bond equivalent yield.
On a financial calculator, enter: n = 10; PV = –960; FV = 1000; PMT = 35
Compute the interest rate.
Chapter 14 – Bond Prices and Yields
14–10
b. Shortcomings of each measure:
(i) Current yield does not account for capital gains or losses on bonds bought at
(ii) Yield to maturity assumes the bond is held until maturity and that all coupon
income can be reinvested at a rate equal to the yield to maturity.
3. a. The maturity of each bond is ten years, and we assume that coupons are paid
semiannually. Since both bonds are selling at par value, the current yield for each
bond is equal to its coupon rate.
b. If rates are expected to fall, the Sentinal bond is more attractive: since it is not
subject to call, its potential capital gains are greater.
c. An increase in the volatility of rates will increase the value of the firm’s option to
call back the Colina bond. If rates go down, the firm can call the bond, which puts
4. Market conversion value = value if converted into stock = 20.83 $28 = $583.24
Chapter 14 – Bond Prices and Yields
5. a. The call feature requires the firm to offer a higher coupon (or higher promised
yield to maturity) on the bond in order to compensate the investor for the firm’s
option to call back the bond at a specified price if interest rate falls sufficiently.
b. The call feature reduces the expected life of the bond. If interest rates fall
substantially so that the likelihood of a call increases, investors will treat the
c. The advantage of a callable bond is the higher coupon (and higher promised yield
to maturity) when the bond is issued. If the bond is never called, then an investor
earns a higher realized compound yield on a callable bond issued at par than a non-
callable bond issued at par on the same date. The disadvantage of the callable
6. a. (iii)
b. (iii) The yield to maturity on the callable bond must compensate the investor for
the risk of call.
Choice (i) is wrong because, although the owner of a callable bond receives a
premium plus the principal in the event of a call, the interest rate at which he can