Chapter 10 – Bond Prices and Yields
CHAPTER 10
BOND PRICES AND YIELDS
1. a. Catastrophe bond: Typically issued by an insurance company. They are
similar to an insurance policy in that the investor receives coupons and par
value, but takes a loss in part or all of the principal if a major insurance
claim is filed against the issuer. This is provided in exchange for higher
than normal coupons.
f. Convertible bond: Convertible bonds may be exchanged, at the
bondholder’s discretion, for a specified number of shares of stock.
Convertible bondholders “pay” for this option by accepting a lower
coupon rate on the security.
j. Indexed bond: Indexed bonds make payments that are tied to a general
price index or the price of a particular commodity.
Chapter 10 – Bond Prices and Yields
2. Callable bonds give the issuer the option to extend or retire the bond at the call
date, while the extendable or puttable bond gives this option to the bondholder.
3. a. YTM will drop since the company has more money to pay the interest on
its bonds.
4. Semi-annual coupon = $1,000 6% 0.5 = $30.
5. Using a financial calculator, PV = –746.22, FV = 1,000, n = 5, PMT = 0.
The YTM is 6.0295%.
6. 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
8. The bond price will be lower. As time passes, the bond price, which is now above
par value, will approach par.
Chapter 10 – Bond Prices and Yields
10. a. The purchase of a credit default swap. The investor believes the bond may
11. c. When credit risk increases, the swap premium increases because of higher
12. The current yield and the annual coupon rate of 6% imply that the bond price was
at par a year ago.
13. Zero coupon bonds provide no coupons to be reinvested. Therefore, the final value of
14.
a. Effective annual rate on a three-month T-bill:
15. The effective annual yield on the semiannual coupon bonds is (1.04)2 = 8.16%. If
16.
a. The bond pays $50 every six months.
Current price:
[$50 Annuity factor(4%, 6)] + [$1000 PV factor(4%, 6)] = $1,052.42
Chapter 10 – Bond Prices and Yields
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior
written consent of McGraw-Hill Education.
b. Rate of Return = $50 + ($1,044.52－ $1,052.42)
$1,052.42 = $50 － $7.90
$1,052.42
= 0.0400 = 4.00% per six months.
17. a. Use the following inputs: n = 40, FV = 1,000, PV = –950, PMT = 40. We
will find that the yield to maturity on a semi-annual basis is 4.26%. This
implies a bond equivalent yield to maturity of: 4.26% 2 = 8.52%
18. 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. The inputs are: n = 20, FV = 1000, PV = –price, PMT = 80. The
resulting yields for the three bonds are:
Bond equivalent yield
Effective annual yield
$950 8.53%
$1,000 8.00%
$1,050 7.51%
Bond Price
The yields computed in this case are lower than the yields calculated with semi-
annual coupon payments. All else equal, bonds with annual payments are less
attractive to investors because more time elapses before payments are received. If
the bond price is the same with annual payments, then the bond’s yield to maturity is
lower.
19.
Chapter 10 – Bond Prices and Yields
The second year
20. Remember that the convention is to use semi-annual periods:
Price of a Zero-Coupon Bond = Face Value
Price
Maturity
(years)
Maturity
(half-years)
Semi-annual
YTM
Bond equivalent
YTM
$400.00 20 40 2.32% 4.63%
$500.00 20 40 1.75% 3.50%
$500.00 10 20 3.53% 7.05%
$376.89 10 20 5.00% 10.00%
$456.39 10 20 4.00% 8.00%
$400.00 11.68 23.36 4.00% 8.00%
21. Using a financial calculator, input PV = –800, FV = 1,000, n = 10, PMT = 80.
The YTM is 11.46%.
22. The reported bond price is $1,001.25
15 days have passed since the last semiannual coupon was paid, so there is an
accrued interest, which can be calculated as:
Chapter 10 – Bond Prices and Yields
The invoice price is the reported price plus accrued interest:
23. If the yield to maturity is greater than current yield, then the bond offers the
24. The coupon rate is below 9%. If coupon divided by price equals 9% and price is
less than par, then coupon divided by par is less than 9%.
25. The solution is obtained using Excel:
A B D E
1
2
3
4Settlement date
5Maturity date
6Annual coupon rate
7Yield to maturity
8Redemption value (% of face value)
9Coupon payments per year
10
11
12 Flat price (% of par)
13 Days since last coupon
14 Days in coupon period
15 Accrued interest
16 Invoice price B12+B15
2/22/2018
DATE(2018,2,22)
3/15/2026
DATE(2026,3,15)
C
5.50% coupon bond,
maturing March 15, 2026
Formula in Column B
101.0333
PRICE(B4,B5,B6,B7,B8,B9)
0.055
0.0534
100
2
2.430939227
(B13/B14)*B6*100/2
103.4642
160
COUPDAYBS(B4,B5,2,1)
181
COUPDAYS(B4,B5,2,1)
26. The solution is obtained using Excel:
A B C D E F G
1Annual
2coupons coupons
3
42/22/2018 2/22/2018
53/15/2026 3/15/2026
60.055 0.055
7102 102
8100 100
92 1
10
11 0.0519268 0.0518889
12
13
14
Semiannual
Settlement date
Maturity date
Annual coupon rate
Formula in cell E11:
YIELD(E4,E5,E6,E7,E8,E9)
Bond price
Redemption value (% of face value)
Coupon payments per year
Yield to maturity (decimal)
Chapter 10 – Bond Prices and Yields
27. Using financial calculator, n = 10; PV = –900; FV = 1,000; PMT = 140
28. 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) + 1100]. Therefore,
realized compound yield to maturity will be a function of r as given in the
following table:
r
Total proceeds
Realized YTM =√Proceeds/1,000 = 1
8%
$1,208
√1,208/1,000 – 1 = 0.0991 = 9.91%
10%
$1,210
√1,210/1,000 – 1 = 0.1000 = 10.00%
12%
$1,212
√1,210/1,000 – 1 = 0.1009 = 10.09%
29. April 15 is midway through the semi-annual coupon period. Therefore, the invoice
30. Factors that might make the ABC debt more attractive to investors, therefore
justifying a lower coupon rate and yield to maturity, are:
• The ABC debt is a larger issue and therefore may sell with greater
liquidity.
• The XYZ bond has a sinking fund requiring XYZ to retire part of the issue
each year. Since most sinking funds give the firm the option to retire this
amount at the lower of par or market value, the sinking fund can work to the
detriment of bondholders.
31. 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 therefore will have a greater price range.
b. Floating rate notes may not sell at par for any of these reasons:
The yield spread between one-year Treasury bills and other money market
c. The risk of call is low. Because the bond will almost surely not sell for
much above par value (given its adjustable coupon rate), it is unlikely that
the bond will ever be called.
d. The fixed-rate note currently sells at only 93% of the call price, so that
a. The bond sells for $1,124.7237 based on the 3.5% yield to maturity:
Chapter 10 – Bond Prices and Yields
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior
written consent of McGraw-Hill Education.
[n = 10; PV = –1,124.72; FV = 1,100; PMT = 40]
b. If the call price were $1,050, we would set FV = 1,050 and redo part (a) to
find that yield to call is 2.9763% semi-annually, 5.9525% annually. With a
lower call price, the yield to call is lower.
c. Yield to call is 3.0312% semiannually, 6.0625% annually:
[n = 4; PV = –1,124.7237; FV = 1,100; PMT = 40]
33. The price schedule is as follows:
Imputed interest
(Increase in constant yield value)
0 (now) 20 years $214.55
1 19 231.71 231.71 – 214.55 = 17.16
2 18 250.25 250.25 – 231.71 = 18.54
19 1 925.93
20 0 1,000 1,000 – 925.93 = 74.07
Year
Remaining
Maturity (T)
Constant Yield Value
1,000/(1.08)T
34. The bond is issued at a price of $800. Therefore, its yield to maturity is 6.8245%.
35. a. The yield to maturity of 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
Chapter 10 – Bond Prices and Yields
36. True. Under the expectations hypothesis, there are no risk premia built into
37. If the yield curve is upward sloping, we cannot conclude that investors expect short-
38.
Zero 8% Coupon 10% Coupon Formula
a. Current Prices $463.19 $1,000 $1,134.20 –PV(0.08,10,PMT,1000)
Price one year from now $500.25 $1,000 $1,124.94 –PV(0.08,9,PMT,1000)
Price Increase $37.06 $0.00 ($9.26)
Coupon Income $0.00 $80.00 $100.00
Income $37.06 $80.00 $90.74
b. Holding Period Return 8.00% 8.00% 8.00%
39. Uncertain. Lower inflation usually leads to lower nominal interest rates.
40.
a. We summarize the forward rates and current prices in the following table:
Maturity
(years)
YTM
Forward rate
Price (for part c)
1
10.0%
$909.09
2
11.0%
12.01%
$811.62
3
12.0%
14.03%
$711.78
Year 1
Chapter 10 – Bond Prices and Yields
b. We obtain next year’s prices and yields by discounting each zero’s face
value at the forward rates derived in part (a):
Maturity
(years)
Price
YTM
1
$892.78
[ = 1,000/1.1201]
12.01%
2
$782.93
[ = 1,000/(1.1201 1.1403)]
13.02%
Note that this year’s upward sloping yield curve implies, according to the
expectations hypothesis, a shift upward in next year’s curve.
c. Next year, the two-year zero will be a one-year zero, and it will therefore
41.
a. The forward rate (f2) is the rate that makes the return from rolling over
one-year bonds the same as the return from investing in the two-year
maturity bond and holding to maturity:
42. The top row must be the spot rates. The spot rates are (geometric) averages of
Chapter 10 – Bond Prices and Yields
43. Forward rates and current prices are summarized in the following table:
Maturity
(years)
YTM
Forward rate
Price
1
5.0%
$952.38
2
6.0%
7.01%
$890.00
3
4
6.5%
7.0%
7.51%
8.51%
$827.85
$762.90
Year 1
Price: 1,000/(1 + 5%) = 952.38
Year 2
Price: 1,000/(1 + 6%)2 = 890.00
Forward Rate: (1 + 6%)2/(1 + 5%) – 1 = 0.0701 = 7.01%
Year 3
Price: 1000/(1 + 6.50%)3 = 827.85
Forward Rate: (1+6.50%)3/(1+6%)2 – 1 = 0.0751 = 7.51%
Year 4
Price: 1000/(1 + 7%)4 = 762.90
Forward Rate: (1+7%)4/(1+6.50%)3 – 1 = 0.0851 = 8.51%
We obtain next year’s prices and yields by discounting each zero’s face
value at the forward rates derived above:
Maturity
(years)
Price
YTM
1
$934.50
[ = 1,000/1.0701]
7.01%
2
3
$869.24
$801.04
[ = 1,000/(1.0701 1.0751)]
[ = 1,000/(1.0701 1.0751
1.0851)]
7.26%
7.68%
This year’s upward sloping yield curve implies, according to the
expectations hypothesis, a shift upward in next year’s curve.
44. a. Initial price, P0 = 705.46 [n = 20; PMT = 50; FV = 1,000; i = 8]
Chapter 10 – Bond Prices and Yields
b. Using OID tax rules, the cost basis and imputed interest under the constant
yield method are obtained by discounting bond payments at the original 8%
yield to maturity and simply reducing maturity by one year at a time:
c. After-tax HPR = $50 + ($793.29 － $705.46) －$46.99
$705.46
e. Coupon received in first year: $50.00
Tax on coupon @ 40% – 20.00
Chapter 10 – Bond Prices and Yields
You sell the bond in the second year for: P2 = $718.84, so imputed interest
over the second year = $6.95
Selling price of the bond in the second year: $798.82
CFA 1
Answer:
a. (3) The yield on the callable bond must compensate the investor for the
risk of call.
Choice (1) is wrong because, although the owner of a callable bond
CFA 2
Answer:
a. The maturity of each bond is 10 years, and we assume that coupons are paid
semiannually. Since both bonds are selling at par value, the current yield to
maturity for each bond is equal to its coupon rate.
Chapter 10 – Bond Prices and Yields
Copyright © 2019 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior
written consent of McGraw-Hill Education.
The price of the Colina bond will increase, but only to the call price of 102.
The present value of scheduled payments is greater than 102, but the call
price puts a ceiling on the actual bond price.
b. If rates are expected to fall, the Sentinal bond is more attractive: Since it is
not subject to being called, its potential capital gains are higher. If rates are
expected to rise, Colina is a better investment. Its higher coupon (which
presumably is compensation to investors for the call feature of the bond)
will provide a higher rate of return than that of the Sentinal bond.
CFA 3
Answer
Market conversion value = Value if converted into stock
= 20.83 $28 = $583.24
CFA 4
Answer:
a. The call provision 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 call price
b. The call option reduces the expected life of the bond. If interest rates fall
substantially so that the likelihood of a call increases, investors will treat
the bond as if it will “mature” and be paid off at the call date, not at the
Chapter 10 – Bond Prices and Yields
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 will earn a higher realized compound yield on a
callable bond issued at par than on a non-callable bond issued at par on the
CFA 5 Answer:
a.
(1) Current yield = Coupon/Price = $70/$960 = 0.0729 = 7.29%
(3) Realized compound yield is 4.166% (semiannually), or 8.332% annual
bond equivalent yield. To obtain this value, first calculate the future
value of reinvested coupons. There will be six payments of $35 each,
reinvested semiannually at a per period rate of 3%:
b. Shortcomings of each measure:
(1) Current yield does not account for capital gains or losses on bonds
bought at prices other than par value. It also does not account for
reinvestment income on coupon payments.