Problem Set 2
Investments, weeks 3-4
Problem 1
Suppose that the prices of zero-coupon bonds with various maturities are given in the following
tables. The face value of each bond is $1,000.
Maturity (year) Price ($)
1 925.93
2 853.39
3 782.92
4 715.00
5 650.00
a) Calculate the forward rate of interest for each year.
b) Suppose that you want to construct a 2-year maturity forward loan commencing in 3 years.
If you buy today one 3-year maturity zero-coupon bond, how many 5-year maturity zeros
would you have to sell to make your initial cash flow equal to zero?
c) What are the cash flows on this strategy in each year?
d) What is the effective 2-year interest rate on the effective 2-year-ahead forward loan?
e) Confirm that the effective 2-year interest rate equals (1 + 𝑓4 )(1 + 𝑓5 ) − 1. You therefore
can interpret the 2-year loan rate as a 2-year forward rate for the last 2 years. Show that the
effective 2-year forward rate equals
(1 + 𝑦5 )5
−1
(1 + 𝑦3 )3
Problem 2
A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual
yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6%
coupon bond making annual coupon payments also selling at a yield to maturity of 8% has
nearly identical duration – 11.79 years – but considerably higher convexity of 231.2
a) Suppose the yield to maturity on both bonds increases to 9%. What will be the actual
percentage capital loss on each bond? What percentage capital loss would be predicted by
the duration-with-convexity rule?
b) Repeat part a), but this time assume the yield to maturity decreases to 7%.
c) Compare the performance of the two bonds in the two scenarios, one involving an increase
in rates, the other a decrease. Based on the comparative investment performance, explain
the attraction of convexity.
d) In view of your answer to c), do you think it would be possible for two bonds with equal
duration but different convexity to be priced initially at the same yield to maturity if the
yields on both bonds always increased or decreased by equal amounts, as in this example?
Would anyone be willing to buy the bond with lower convexity under these circumstances?
Problem 3
A newly issued bond has a maturity of 10 years and pays a 7% coupon rate (with coupon
payments coming once annually). The bond sells at par value.
a) What are the convexity and the duration of the bond?
b) Find the actual price of the bond assuming that its yield to maturity immediately increases
from 7% to 8% (with maturity still 10 years).
c) What price would be predicted by the duration rule? What is the percentage error of that
rule?
d) What price would be predicted by the duration-with-convexity rule? What is the percentage
error of that rule?
Problem 4
Suppose that the borrowing rate that your client faces is 9%. Assume that the S&P 500 index
has an expected return of 13% and standard deviation of 25%, that Rf=5% and that your fund
has expected return of 11% and standard deviation of 15%.
a) Draw a diagram of your client’s CML, accounting for the higher borrowing rate.
Superimpose on it two sets of indifference curves, one for a client who will choose to
borrow, and one who will invest in both the index fund and a money market fund.
b) What is the range of risk aversion for which a client will neither borrow nor lend, that is,
for which y=1?
c) Solve points a) and b) for a client who uses your fund rather than an index fund.
d) What is the largest percentage fee that a client who currently is lending (𝑦 < 1) will be
willing to pay to invest in your fund? What about a client who is borrowing (𝑦 > 1)?
Problem 5
Assume that there are two risky assets listed in the market. Expected returns and risks for the
risky assets are
𝐸[𝑅] 𝜎
7.5% 22%
10% 27%
a) Determine the analytical formulation of the expected return of a generic portfolio as a
function of its standard deviation in the case in which the two risky assets are perfectly
correlated.
b) Determine the analytical formulation of the expected return of a generic portfolio as a
function of its standard deviation in the case in which the two risky assets are perfectly
negatively correlated.
