Consider the following:
1. Strategy I: Suppose that you invest $100 in a stock. There is a 60% chance that the stock will go up in value by $10 at by the end of this year. There is a 40% chance that the stock will go down in value by $5 by the end of the year.
2. Strategy II: Suppose that you invest $10 in 10 separate stocks. Each of these stocks also has by 10% and a 40% chance of going down by 5% by the end of the year, and the performance of each of them is independent from the performance of the others.
The rate of return on each of these investment strategy can be described as
Money at end of year
Money at beginning of year − 1
Calculate the expectation and variance of rates of return for these two strategies.
Question 2 (Correlated assets 1pt)
There are two assets in the market A and B.A has an expected return of 8% and a standard deviation of 10%. B has an expected return of 10% and a variance of 0.01. The correlation between A and B is 0.25. You cannot short.
1. What is the standard deviation of B ? (0pt)
2. What is the minimum standard deviation of a portfolio consisting of A and B ? Describe that portfolio for K dollars to invest. What is the expected return of this portfolio? (0.5pt)
3. What is the minimum standard deviation for a portfolio with expected return of 10% ? (0.5pt)
Question 3 (Covariance and correlation 1.5pt)
Suppose there are three states of the world, which are all equally likely, and two different financial assets. In the first state of the world, the first asset returns 8% and the second asset returns −5%.
In the second state, the first asset returns −4% and the second asset returns 15%. In the third state, the first asset returns 12% and the second asset returns 6%.
What is the mean return on each asset? What is the variance and standard deviation of the return on each asset? Use natural units, not percentages (i.e., use 0.09 instead of 9 for 9% ).
What is the covariance of the returns of the two assets? What is the correlation between the two returns?
Consider an equal weighted portfolio of the two assets. Compute the mean return, variance and standard deviation.
Question 4 (Covariance and correlation 1.5pt)
Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 6%, and the market’s average return was 14%. Performance is measured using an index model regression on excess returns.
Stock A Stock B
Index model regression estimates 1% + 1.2 (rM − rf ) 2% + 0.8 (rM − rf )
R-square 0.576 0.436
Residual standard deviation, σ(e) 10.3% 19.1%
Standard deviation of excess returns 21.6% 24.9%
Calculate the following statistics for each stock:
1. Alpha
2. Information ratio, using market factor as the benchmark
3. Sharpe ratio
The next two questions are from Exercises for Efficiently Inefficient.
Question 5 (Hedge funds vs. mutual funds 4pt)
Consider a passive mutual fund, an active mutual fund, and a hedge fund. The mutual funds claim to deliver the following gross returns:
rpassive fund before fees
t = rstock index
t
ractive fund before fees
t = 2.20% + rstock index
t + εt
The stock index has a volatility of
√
var (rstock index
t
) = 15%. The active mutual fund has a tracking error with a mean of E (εt) = 0, a volatility of √var (εt) = 3.5%, and Cov (εt, rstock index
t
) = 0 such
that it’s beta to the stock index is 1 . The passive fund charges an annual fee of 0.10% and the active mutual fund charges a fee of 1.20%.
The hedge fund uses the same strategy as the active mutual fund to identify ”good” and ”bad” stocks, but implements the strategy as a long-short hedge fund, applying 4 times leverage. The risk-free interest rate is rf = 1% and the financing spread is zero (meaning that borrowing and lending rates are equal). Therefore, the hedge fund’s return before fees is rhedge fund before fees
t =
1% + 4 × (ractive fund before fees
t − rstock index
t
)
1. What is the hedge fund’s volatility? (0.5pt)
2. What is the hedge fund’s beta? (0.5pt)
3. What is the hedge fund’s alpha before fees (based on the mutual fund’s alpha estimate)?
(0.5pt)
4. Suppose that an investor has $40 invested in the active fund and $60 in cash (measured in thousands, say). What investments in the passive fund, the hedge fund, and cash (i.e., the riskfree asset) would yield the same market exposure, same alpha, same volatility, and same exposure to εt ? As a result, what is the fair management fee for the hedge fund in the sense that it would make the investor indifferent between the two allocations (assume that the hedge fund charges a zero performance fee)? (1.5pt)
5. If the hedge fund charges a management fee of 2%, what performance fee makes the expected fee the same as above? Ignore high water marks and ignore the fact that returns can be negative, but recall that performance fees are charged as a percentage of the (excess) return after management fees. Specifically, assume the performance fee is a fraction of the hedge fund’s outperformance above the risk-free interest rate. (1pt)
6. (Extra thinking) Comment on whether it is clear that hedge funds that charge 2-and-20 fees are ”expensive” relative to typical mutual funds. More broadly, what should determine fees for active management? (0 pt)
Question 6 (Demand pressure 1.5pt)
Suppose that a significant fraction of the population of investors needs to buy a security, say a stock ABC, for reasons unrelated to the stock’s fundamentals (its expected future earnings and dividends). For instance, suppose that an important stock market index suddenly gives a large weight to stock ABC.
1. What will happen to the stock price in a perfectly efficient market?
2. What is likely to happen to the stock price in a market with limited arbitrage?
3. In an efficiently inefficient market, where the stock price moves (as discussed in part b.), what is likely to happen to the price of another stock that is highly correlated to stock ABC (but not directly affected by the demand pressure)?
Question 7 (Bonus 1pt: Variance estimate)
Let ri for i = 1, . . . , n be independent sample of return r, a random variable with mean ̄r and variance σ2. Define the estimates
ˆ ̄r = 1
n
n∑
i=1
ri,
s2 = 1
n − 1
n∑
i=1
(ri − ˆ ̄r)2 .
Show that E [s2] = σ2, i.e. s2 is an unbiased estimate of σ2. Note that you should calculate
E
[ 1
n−1
∑n
i=1
(ri − ˆ ̄r)2]
with ˆ ̄r = 1
n
∑n
i=1 ri being random.
