Week9 Homework Assignment Solution
BKM Ch14
2.
The effective annual yield on the semiannual coupon bonds is 8.16%. If the
annual coupon bonds are to sell at par they must offer the same yield, which
requires an annual coupon rate of 8.16%.
3.
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.
4.
The bond price will be lower. As time passes, the bond price, which is now above
par value, will approach par.
5.
Yield to maturity: Using a financial calculator, enter the following:
n = 3; PV = −953.10; FV = 1000; PMT = 80; COMP i
This results in: YTM = 9.88%
Realized compound yield: First, find the future value (FV) of reinvested coupons
and principal:
FV = ($80 × 1.10 × 1.12) + ($80 × 1.12) + $1,080 = $1,268.16
Then find the rate (yrealized ) that makes the FV of the purchase price equal to
$1,268.16.
$953.10 × (1 + yrealized )3 = $1,268.16 ⇒ yrealized = 9.99% or approximately
21.
10%
The stated yield to maturity, based on promised payments, equals 16.075%.
[n = 10; PV = –900; FV = 1000; PMT = 140]
27.
Based on expected coupon payments of $70 annually, the expected yield to
maturity is 8.526%.
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.
If the yield declines by 1% to 5% (2.5% semiannual yield), the Sentinal
bond will increase in value to $107.79 [n=20; i = 2.5%; FV = 100; PMT = 3].
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 call, its potential capital gains are greater.
If rates are expected to rise, Colina is a relatively better investment. Its
higher coupon (which presumably is compensation to investors for the call
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feature of the bond) will provide a higher rate of return than the Sentinal
bond.
28.
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 a cap on possible capital gains. So, greater volatility
makes the option to call back the bond more valuable to the issuer. This
makes the bond less attractive to the investor.
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 contrast, the 8¾% coupon bond can increase in value to at most
$1,050, offering a maximum possible gain of only 0.5%. The disadvantage of
the 8¾% coupon bond, in terms of vulnerability to being called, shows up in
its higher promised yield to maturity.
b.
If an investor expects yields 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
would not be nearly enough to make the firm consider calling the bond. In
this sense, the call feature is almost irrelevant.
31 b. (iii) The yield 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 reinvest will be low. The low interest rate that makes it profitable for
the issuer to call the bond makes it a bad deal for the bond’s holder.
Choice (ii) is wrong because a bond is more apt to be called when interest
rates are low. Only if rates are low will there be an interest saving for the
issuer.
c.
(ii) is the only correct choice.
(i) is wrong because the YTM is greater than the coupon rate when a bond
sells at a discount and is less than the coupon rate when the bond sells at a
premium.
(iii) is wrong because adding the average annual capital gain rate to the
current yield does not give the yield to maturity. For example, assume a 10year bond with a 6% coupon rate paying interest annually and a YTM of 8%
per year. Its price is $865.80. The average annual capital gain is equal to
($1000 – 865.80)/10 years = $13.42 per year. Using this number results in an
average capital gains rate per year of $13.42/$865.80 = 1.55%. The current
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coupon yield is $60/$865.80 = .0693 per year or 6.93%. Therefore, the “total
yield” is: 1.55% + 6.93% = 8.48%
This is greater than the YTM.
(iv) is wrong because YTM is based on the assumption that any payments
received are reinvested at the YTM and not at the coupon rate.
d.
(iii)
e.
(ii)
f.
(iii)
BKM Ch16
1.
The percentage change in the bond’s price is:
−
2.
Duration
7.194
× ∆y = −
× 0.005 = −0.0327 = −3.27% or a 3.27% decline.
1+ y
1.10
a. YTM = 6%
(1)
Time until
Payment
(years)
1
2
3
(2)
Cash Flow
$60.00
$60.00
$1,060.00
Column Sums
(3)
PV of CF
(Discount
rate = 6%)
$56.60
$53.40
$890.00
$1,000.00
(4)
(5)
Weight
Column (1) ×
Column (4)
0.0566
0.0534
0.8900
1.0000
0.0566
0.1068
2.6700
2.8334
(4)
(5)
Weight
Column (1) ×
Column (4)
0.0606
0.0551
0.8844
1.0000
0.0606
0.1102
2.6532
2.8240
Duration = 2.833 years
b. YTM = 10%
(1)
Time until
Payment
(years)
1
2
3
(2)
Cash Flow
$60.00
$60.00
$1,060.00
Column Sums
(3)
PV of CF
(Discount
rate = 10%)
$54.55
$49.40
$796.39
$900.53
Duration = 2.824 years, which is less than the duration at the YTM of 6%.
3.
For a semiannual 6% coupon bond selling at par, we use the following parameters:
coupon = 3% per half-year period, y = 3%, T = 6 semiannual periods. Using Rule
8, we find:
D = (1.03/0.03) × [1 – (1/1.03)6] = 5.58 half-year periods = 2.79 years
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If the bond’s yield is 10%, use Rule 7, setting the semiannual yield to 5%, and
semiannual coupon to 3%:
D=
1.05 (1.05) + [6 × (0.03 − 0.05)]
−
= 21 − 15.4478
0.05 0.03 × [(1.05) 6 − 1] + 0.05
= 5.5522 half-year periods = 2.7761 years
5.
a.
(1)
Time until
Payment
(years)
1
5
(2)
(3)
PV of CF
Cash Flow
(Discount rate =
10%)
$10 million
$9.09 million
$4 million
$2.48 million
Column Sums $11.57 million
(4)
(5)
Weight
Column (1) ×
Column (4)
0.7857
0.2143
1.0000
0.7857
1.0715
1.8572
D = 1.8572 years = required maturity of zero coupon bond.
b.
The market value of the zero must be $11.57 million, the same as the market
value of the obligations. Therefore, the face value must be:
$11.57 million × (1.10)1.8572 = $13.81 million
8.
c.
d.
10.
a.
(i)
(i)
[9/1.10 = 8.18]
Macaulay duration
10
Modified duration =
=
= 9.26 years
1 + YTM
1.08
b.
For option-free coupon bonds, modified duration is a better measure of the
bond’s sensitivity to changes in interest rates. Maturity considers only the
final cash flow, while modified duration includes other factors, such as the
size and timing of coupon payments, and the level of interest rates (yield to
maturity). Modified duration, unlike maturity, indicates the approximate
percentage change in the bond price for a given change in yield to maturity.
c.
i. Modified duration increases as the coupon decreases.
ii. Modified duration decreases as maturity decreases.
d.
Convexity measures the curvature of the bond’s price-yield curve. Such
curvature means that the duration rule for bond price change (which is based
only on the slope of the curve at the original yield) is only an approximation.
Adding a term to account for the convexity of the bond increases the
accuracy of the approximation. That convexity adjustment is the last term in
the following equation:
∆P
1

= (− D * ×∆y) +  × Convexity × (∆y) 2 
P
2

18. (16 in the 7th ed) a.
From Rule 6, the duration of the annuity if it were to start
in 1 year would be:
4
1.10
10
−
= 4.7255 years
0.10 (1.10)10 − 1
Because the payment stream starts in five years, instead of one year, we add
four years to the duration, so the duration is 8.7255 years.
b.
The present value of the deferred annuity is:
10,000 × Annuity factor (10%,10)
= $41,968
1.10 4
Call w the weight of the portfolio invested in the 5-year zero. Then:
(w × 5) + [(1 – w) × 20] = 8.7255 ⇒ w = 0.7516
The investment in the 5-year zero is equal to:
0.7516 × $41,968 = $31,543
The investment in the 20-year zeros is equal to:
0.2484 × $41,968 = $10,425
These are the present or market values of each investment. The face
values are equal to the respective future values of the investments. The
face value of the 5-year zeros is:
$31,543 × (1.10)5 = $50,800
Therefore, between 50 and 51 zero-coupon bonds, each of par value $1,000,
would be purchased. Similarly, the face value of the 20-year zeros is:
$10,425 × (1.10)20 = $70,134
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