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.
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.
g. Serial bond: A serial bond is an issue in which the firm sells bonds with
staggered maturity dates. As bonds mature sequentially, the principal
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.
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
decreases the present value of the future cash flows.
7. 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.
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.
Using a financial calculator, FV = 1,000, n =7, PMT = 60, and i =7 gives us a
selling price of $946.11 this year.
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:
b. Effective annual interest rate on coupon bond paying 5% semiannually:
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:
Chapter 10 – Bond Prices and Yields
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:
19.
Nominal Return = Interest + Price Appreciation
Initial Price
Real Return = 1 + Nominal Return
20. Remember that the convention is to use semi-annual periods:
Price
Maturity
(years)
Maturity
(half-years)
Bond equivalent
YTM
$400.00 20 40 2.32% 4.63%
$500.00 10 20 3.53% 7.05%
$456.39 10 20 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
2/22/2015
DATE(2015,2,22)
3/15/2023
DATE(2023,3,15)
6Annual coupon rate
7Yield to maturity
9Coupon payments per year
10
11
12 Flat price (% of par)
13 Days since last coupon
14 Days in coupon period
15 Accrued interest
(B13/B14)*B6*100/2
103.4642
COUPDAYBS(B4,B5,2,1)
COUPDAYS(B4,B5,2,1)
101.0333
PRICE(B4,B5,B6,B7,B8,B9)
0.055
0.0534
C
5.50% coupon bond,
26. The solution is obtained using Excel:
A B C D E F G
1Annual
2coupons coupons
3
Settlement date
Maturity date
60.055 0.055
7102 102
8100 100
10
Redemption value (% of face value)
Coupon payments per year
11 0.0519268 0.0518889
12
13
Semiannual
Annual coupon rate
Bond price
Yield to maturity (decimal)
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%
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 call feature on the XYZ bonds makes the ABC bonds relatively more
attractive since ABC bonds cannot be called from the investor.
31. a. The floating-rate note pays a coupon that adjusts to market levels.
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
instruments of comparable maturity could be wider than it was when the
bond was issued.
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
e. The 9% coupon notes currently have a remaining maturity of fifteen years
and sell at a yield to maturity of 9.9%. This is the coupon rate that would
be needed for a newly issued fifteen-year maturity bond to sell at par.
f. Because the floating rate note pays a variable stream of interest payments
to maturity, its yield-to-maturity is not a well-defined concept. The cash
32.
a. The bond sells for $1,124.7237 based on the 3.5% yield to maturity:
Chapter 10 – Bond Prices and Yields
b. If the call price were $1,050, we would set FV = 1,050 and redo part (a) to
c. Yield to call is 3.0312% semiannually, 6.0625% annually:
33. The price schedule is as follows:
Imputed interest
(Increase in constant yield value)
0 (now) 20 years $214.55
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%.
[n = 10; PV = –800; FV = 1,000; PMT = 40] Using the constant yield method, we
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
the call price. Therefore, if yields fall, capital gains on the bond will not be
b. If an investor expects rates to fall substantially, the 4% bond offers a
greater expected return.
c. Implicit call protection is offered in the sense that any likely fall in 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-
term interest rates to rise because the rising slope could be due to either expectations
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)
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)
2
11.0%
12.01%
$811.62
3
12.0%
14.03%
$711.78
Year 1
Price: 1,000/(1 + 10%) = 909.09
Year 2
Price: 1,000/(1 + 11%)2 = 811.62
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
c. Next year, the two-year zero will be a one-year zero, and it will therefore
sell at: $1000/1.1201 = $892.78
Similarly, the current three-year zero will be a two-year zero, and it will
sell for: $782.93
Expected total rate of return:
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:
b. According to the expectations hypothesis, the forward rate equals the expected
value of the short-term interest rate next year, so the best guess would be
10.01%.
c. According to the liquidity preference hypothesis, the forward rate exceeds
42. The top row must be the spot rates. The spot rates are (geometric) averages of
the forward rates, and the top row is the average of the bottom row. For
example, the spot rate on a two-year investment (12%) is the average of the two
Chapter 10 – Bond Prices and Yields
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
4
7.0%
8.51%
$762.90
Year 1
Price: 1,000/(1 + 5%) = 952.38
Year 2
Year 3
Price: 1000/(1 + 6.50%)3 = 827.85
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%
43. 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:
P0 = $705.46
First Year
Constant yield price, P1
′ = $711.89, so imputed taxable interest over the first
year is: $711.89 – $705.46 = $6.43
Coupon received and imputed taxable interest in the year are taxed as the
ordinary income: 40% ($50 + $6.43) = $22.57
c. After-tax HPR = $50 + ($793.29 － $705.46) －$46.99
$705.46
d. Value of the bond after two years equals $798.82 [using n = 18; i = 7]
Total income from the two coupons, including reinvestment income:
e. Coupon received in first year: $50.00
Tax on coupon @ 40% – 20.00
Chapter 10 – Bond Prices and Yields
CFA 1
Answer:
a. (3) The yield on the callable bond must compensate the investor for the
risk of call.
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
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
CFA 3
Answer
Market conversion value = Value if converted into stock
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
Chapter 10 – Bond Prices and Yields
CFA 5 Answer:
a.
(2) YTM = 3.993% semiannually or 7.986% annual bond equivalent yield
(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
(2) Yield to maturity assumes that the bond is held to maturity and that all
(3) Realized compound yield (horizon yield) is affected by the forecast of