BUS 365: Investments
Solution to Practice Problems
Bonds
1) You wish to invest $20,000 today into a portfolio of two U.S. Treasury bonds. In 5 years, you
will liquidate that portfolio and use the funds for a downpayment on a house. One of the bonds
is a 5-year note with an annual coupon rate of 4%. The other is a 10-year Treasury strip (i.e., a
zero coupon bond). Both bonds have a yield-to-maturity of 3.75%. You wish to immunize your
portfolio against interest rate risk. Also, assume that you can buy fractional numbers of bonds.
a) How should you allocate the $20,000 between the two bonds?
Answer: The duration of the 10-year bond is 10 years. The duration of the 5-year bond is
4.63 years. We want a portfolio duration of 5 years, so we need w54.63 + w1010 = 5.
Using w5+w10=1 and solving gives w5=0.931 and w10=0.069. Thus, we should invest
0.931$20,000 = $18,620 in the 5-year bond and 0.069$20,000 = $1,380 in the 10-year
bond.
b) How much money do you expect to have when you liquidate the portfolio in 5 years?
Answer: We have essentially locked in a return of 3.75% per year (the yield on the bonds),
so we expect to have $20,0001.03755 = $24,042 in 5 years.
2) A bond has 8 years to maturity, pays 6% annual coupons. The appropriate discount rate for
similar investments is currently 6%. The bond is callable in 3 years for $1010. Your expectation
is that the appropriate discount rate will be either 5% or 7% in 3 years, each with equal
probability. What is the value of the bond today?
Answer: In three years, the bond will be worth $60PVIFA5%,5+$1,000/1.055 =$1043 if
interest rates are at 5%. If interest rates are instead at 7%, the bond will be worth
$60PVIFA7%,5+$1,000/1.075 =$959 in three years. The issuer will call the bond if rates are
at 5%, and investors will receive $1010 per bond. Thus, the expected value of the bond in 3
years is ($1010+959)/2 = $984.50. The value of the bond today is then
V=$60PVIFA6%,3+$984.50/1.063 =$986.99.
3) A bond pays 6% annual coupons and matures in six years. The bond is convertible into 100
shares of stock in two years. The appropriate discount rate for the bond is 7% and you believe it
is reasonable to use this rate for all foreseeable scenarios. You believe that it is equally likely
that the stock will be worth $9 per share or $10 per share in two years. What is the value of the
bond today?
We use backward induction. In two years, the bond cash flows will be worth
$60PVIFA7%,4 + $1000/1.074 = $966.13. The holder will convert if the share price is $10
and will not convert if it is $9. The expected value of the position in two years is
0.5$966.13 + 0.5$10100 = $983.06. The value of the bond today is then
V = $60/1.07 + $60/1.072 + $983.06/1.072 = $967.13.
4) You wish to invest $100,000 today into a portfolio of two U.S. Treasury bonds. In 5 years,
you will liquidate that portfolio and use the funds for a downpayment on a house. One of the
bonds is a 4-year, 5% annual coupon note. The other is a 7-year Treasury strip (i.e., a zero
coupon bond). Both bonds have a yield-to-maturity of 6%. You wish to immunize your
portfolio against interest rate risk. (Also, assume that you can buy fractional numbers of bonds.)
a) What is the duration of each bond?
Duration of strip = 7 years (only one cash flow).
Date
1
2
3
4
TOTAL
CF
$50
$50
$50
$1050
PV(CF)
$47.17
$44.50
$41.98
$831.70
$965.35
PV(CF)T
$47.17
$89.00
$125.94
$3,326.79
$3,588.90
Duration of 4-year bond = $3,588.90/$965.35 = 3.72 years.
b) How should you allocate the $100,000 between the two bonds?
w3.72 + (1-w)7 = 5. Solving gives w = 0.602. So, we invest about $60,200 in the 4-year
bond and $39,800 in the 7-year bond.
c) How much money do you expect to have when you liquidate the portfolio in 5 years?
We are immunized, so the expected return will be about 6% and we expect to have
$100,0001.065 = $133,823 in 5 years.
5) A $1,000 face value bond pays 9% annual coupons and has 3 years to maturity. The yield-tomaturity on the bond is 8%.
a) What is the duration of the bond?
Answer:
Date
CF
PV(CF) T´PV(CF)
1
$90.00
$83.33
$83.33
2
$90.00
$77.16
$154.32
3
$1,090.00
$865.28 $2,595.83
TOTAL
$1,025.77 $2,833.49
Duration: 2.76 years
b) What is the modified duration of the bond?
Answer: Dm = D/(1+y) = 2.76/1.08 = 2.56 years
c) Suppose interest rates increase by 1.5%. What is your estimate of the value of the bond using
the duration model?
Answer: V/V = -Dmy = -2.561.5% = -3.84%
It follows that the value of the bond will be $1,025.77(1-0.0384) = $986.42
d) What is actual value of the bond after the 1.5% increase?
Answer: V = $90PVIFA9.5%,3 + $1,000/1.0953 = $987.46
e) If your answers to c) and d) are different, BRIEFLY explain why. If they are the same,
BRIEFLY explain why.
Answer: The answers differ because the concept of duration involves taking a linear
approximation to the price-yield curve, (which is convex). The duration-generated
estimate will consequently always underestimate the true value by a small amount.