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FE 445 - Problem Set #7
MULTIPLE CHOICE QUESTIONS:
1. Which of the following debt instruments is likely to be the most junior in a
company’s capital structure?
A. A conventional loan not secured by any collateral
B. A convertible bond
C. A securitization of the company’s receivables
D. A senior bond secured by a second lien on the company’s headquarters
Answer: B
2. You have purchased $8,000,000 (par amount) of a floating rate bond that pays
interest at a fixed spread of .70% above 6-month LIBOR. If 6-month LIBOR at the
last coupon reset for this bond was 2.05%, and the applicable period is 183 days, the
amount of interest you will receive on the next coupon payment date is closest to:
A. $111,800
B. $110,300
C. $83,350
D. $82,250
Answer: A
Solution: The full interest rate for the period is 2.05% + .70% = 2.75% p.a. applied on an
actual /360 day basis. $8,000,000 x .0275 x 183/360 = $111,833.33.
3. You observe the 12-month and 18-month zero coupon rates for U.S. Treasury
securities are 1.95% and 2.25%, respectively. Assuming arbitrage free markets and
no friction costs, the implied 6-month rate in 12 months’ time should be closest to:
A. 1.42%
B. 2.10%
C. 2.85%
D. 3.16%
Answer: C
Solution: [(1 + .0225)^(3/2) / (1 + .0195)]^2 -1 = .0285 (= 2.68%)
4. Consider a $1,000 corporate bond with a 5.5% coupon that pays interest quarterly
and matures in five years. If its yield to maturity is 3%, its price would be closest to:
A. $891.36
B. $1,030.56
C. $1,114.49
D. $1,115.68
Answer: D
Solution: FV = $1,000, PMT = $1,000 x (.055 / 4) = 13.75, n = 20, i = 3.00% / 4 = 0.75%
=> PV = $1,115.68
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5. The modified duration of a 6-year bond with a 3.5% annual coupon and a yield to
maturity of 4.80% is closest to:
A. 4.80 years
B. 4.90 years
C. 5.00 years
D. 5.50 years
Answer: C
Solution: The current price of this bond can be calculated by FV = $100, PMT = $3.5, n =
6, i = 4.80% => PV = $93.36. If we adjust the yield (“i”) up and down by .005% p.a., we
get prices of $93.335 and $98.384, respectively. The modified duration is ($93.384 $93.335) / $93.36 = .0524 per basis point => 5.24% per 1% move in yields (= 5.24 years)
6. You own a bond portfolio comprised of 25% in U.S. Treasury zero coupon bonds
(avg. duration of 7.5 years), 30% in corporate bonds (avg. duration of 5 years), 20%
in mortgage-backed securities (avg. duration of 3.5 years), and 25% in floating rate
bonds, all of which are about to reset their interest rates at market levels. The
effective duration of your portfolio is closest to:
A. 3.50 years
B. 3.80 years
C. 4.00 years
D. 4.10 years
Answer: D
Solution: The duration of the portfolio is simply the weighted average duration of its
holdings. The floating rate bonds are all about to reset and, as such, have essentially zero
duration. Thus, (.25 x 7.5) + (.3 x 5) + (.2 x 3.5) + (.25 x 0) = 4.075 years.
BRIEF ESSAYS:
1.
Which security has a higher effective annual interest rate?
a)
A 3-month T-Bill selling at $97,645 with a par value of $100,000
Answer: Effective annual rate for 3-month T-bill:
4
' 100,000 $
%
" − 1 = 1.02412 4 − 1 = 0.100 = 10.0%
! & 97,645 #
b) A coupon bond selling at par and paying a 10% coupon semi-annually
Answer: Effective annual interest rate for coupon bond paying 5%
semiannually:
(1.05)2 - 1 = 0.1025 or 10.25%
Therefore the coupon bond has the higher effective annual interest rate.
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2.
Consider a bond with a 10% coupon and a yield to maturity of 8%. If the bond’s
yield to maturity remains constant, then in one year’s time, will the bond’s price be
higher, lower, or unchanged? Explain your answer.
Answer: The bond price will be lower. As time passes, the bond price, which is
now above par value, will approach par.
3.
Fill in the Table below for the following zero-coupon bonds, all of which have par
values of $1,000.
Price
Maturity (years)
Yield to Maturity
———————————————————————$450
20
4.073%
$550
$550
$508.35
$613.91
$450
20
10
10
10
16.36
3.034%
6.161%
7%
5%
5%
4. Consider the following $1,000 par value zero-coupon bonds:
Bond
Maturity (years)
Yield to Maturity
——————————————————————
A
1
5%
B
2
6%
C
3
6.5%
D
4
7%
According to the expectations hypothesis, what is the expected 1-year interest rate
3 years from now?
Answer: If expectations theory holds, then the forward rate equals the short
rate, and the one-year interest rate three years from now would be
(1.07) 4
− 1 = .0851 = 8.51%
(1.065)3
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5.
Find the Macaulay duration of a 6% coupon bond making annual coupon payments
if it has 3 years until maturity and has a yield to maturity of 6%. What is the
Macaulay duration of this bond if its yield to maturity is 10%?
Answer:
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6.
Rank the modified durations of the following pair of bonds:
a)
Bond A is a 6% coupon bond, with a 20-year time to maturity selling at par.
Bond B is a 6% coupon bond , with a 20-year maturity selling below par.
Answer: Bond B has a higher yield to maturity than bond A since its
coupon payments and maturity are equal to those of A, while its price is
lower. (Perhaps the yield is higher because of differences in credit risk.)
Therefore, the duration of Bond B must be shorter.
b)
Bond A is a 20-year non-callable bond with a coupon rate of 6%, selling at par.
Bond B is a 20-year callable bond with a coupon rate of 7%, also selling at par.
Answer: Bond A has a lower yield and a lower coupon, both of which
cause Bond A to have a longer duration than Bond B. Moreover, A cannot
be called, so that its maturity is at least as long as that of B, which
generally increases duration.
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