SEEM 3590 Investment Science YANG Chen, Fall 2024
Homework Set 5
(Due on November 29, 2024, 23:59)
Assignment
1. While studying the performance of a fund on the quarterly returns using the
Fama-French three-factor model, an investor found that the historical estimates
of the mean and standard deviation of the fund’s return are 2.2% and 5%,
respectively. She also estimated the mean and standard deviation of the market
portfolio’s return to be 4.5% and 10%, respectively, and the correlation between
the return rates of the fund and the market is 0.60. The risk-free return rate
is constant at 1% in the past. The portfolio loadings for SMB and HML are
estimated to be 0.35 and 0.45, respectively. The expected payoffs for the zero-
cost strategies associated with SMB and HML are estimated to be 0.75% and
0.85%, respectively.
(a) What is the beta of this fund?
(b) According to the Fama-French three-factor model, is the strategy of this
fund a good one? Why?
2. There are one risky asset and one risk-free asset in the market. Assume there
are three possible market states after 3 months, a, b, and c, with probability
pa, pb, pc, respectively, such that pa + pb + pc = 1 and pa > 0, pb > 0, pc > 0.
Over this 3-month period, the net rates of return of the risky asset in the a, b,
and c states are ra, rb, rc, respectively, such that ra < rb < rc. The risk-free
asset has a net rate of return of rf (constant) regardless of the state.
If rf ≥ rc, by using the risky asset and the risk-free asset, construct a zero-cost
strategy that corresponds to an arbitrage opportunity, and explain why this
strategy leads to arbitrage. Repeat this for the case if rf ≤ ra.
3. A financial institution entered into an interest rate swap with company X two
years ago to receive 4.0% fixed rate (per annum) and pay 6-month HIBOR on
a principal of $10 million in three years. Therefore, the remaining life of the
contract today is 1 year. The payments are made semiannually and the rates are
quoted with semiannual compounding. The following are the discount factors
two years ago when the swap was signed:
Time-to-Maturity (months) Discount factor
6 0.9750
12 0.9550
18 0.9360
24 0.9180
30 0.9010
36 0.8950
1
SEEM 3590 Investment Science YANG Chen, Fall 2024
How much did the financial institution pay to or receive from company X two
years ago to enter into the swap?
4. Suppose that spot interest rates with continuous compounding are as follows:
Maturity (months) Rate (% per annum)
3 4.8
6 5.2
9 5.6
12 6.0
Calculate the forward interest rates with continuous compounding for the sec-
ond, third, and fourth quarters.