Final Exam Practice Problems
FIN 532: Investments
Fall 2024
Risk Preference and Portfolio Choice
1. Given the following scenario analysis for stocks X and Y,
Bear Market Normal Market Bull Market
|
Probability
|
0.2
|
0.5
|
0.3
|
|
Stock X
|
-20%
|
18%
|
50%
|
|
Stock Y
|
-15%
|
20%
|
10%
|
(a) What are the expected rates of return for Stocks X and Y?
(b) What are the standard deviations of returns on Stocks X and Y?
(c) What is the expected return and standard deviation of a portfolio with weight 0.8 in Stock X and 0.2 in Stock Y.
2. You can invest in a risky asset with an expected rate of return of 20% per year and a standard deviation of 40% per year or a risk free asset earning 5% per year or a combination of the two. The borrowing rate is 6% per year.
(a) Draw the Capital Allocation Line. Indicate the points corresponding to (1) 50% in the risk-less asset and 50% in the risky asset; and (2) -50% in the riskless asset and 150% in the risky asset.
(b) Compute the expected rate of return and standard deviation for the two portfolios in part (a).
(c) Suppose you have a target risk level of 50% per year. How would you construct a portfolio of the risky and the riskless asset to attain this target level of risk? What is the expected rate of return of the portfolio you constructed?
3. An investor is considering 3 ETFs: a stock fund, a bond fund, and T-bill fund. The T-bill fund yields a risk-free rate of 4%. The probability distribution of the risky funds are:
|
Expected Ret
|
Std Dev
|
|
Stock 13%
Bond 8%
Correlation = 0.3
|
20%
12%
|
(a) What is the mean-variance efficient mix of stocks and bonds? What is the expected return and std of the MVE?
(b) Suppose the investor has mean variance preferences with a coefficient of risk aversion of γ = 4, what would be the optimal weights in risk-free asset and the mve?
(c) In the optimal complete portfolio, what are the optimal weights in each of these three funds?
(d) Suppose that the investors has financial wealth W = $1, 000, 000 and riskless human capital with a present value of H = $500, 000. How should the investor allocate her financial wealth to each of the three funds in order to achieve the desired weights of her total wealth you calculated in part (c)?
(e) Suppose that the investor takes your advice in part (d). Over the next year, the stock fund appreciates by 20%, the bond fund appreciates by 10% and the present value of H decreases to $400,000. Explain how the investor should rebalance her portfolio?
CAPM
1. You have the following information about hedge fund Q and the market portfolio M.
|
Stock Q
|
Market Portfolio (M)
|
|
Expected return
Standard deviation
|
?
40%
|
10%
20%
|
The correlation between fund Q and the market portfolio is ρQ,M = 0.6. The risk-free rate is 3%. For parts (a)-(c) of this question, you should assume that the CAPM holds.
(a) What is the beta of fund Q? What is the expected return of fund Q according to the CAPM?
(b) Suppose you decide to invest in a portfolio that consists of 20% in Q, 50% in the market portfolio, and 30% in the risk-free asset. What is the Sharpe ratio of this portfolio?
(c) Instead of holding the portfolio described in part (b), you decide to find a portfolio with the same expected return as the portfolio in part (b) but with the lowest possible standard deviation. What are the optimal portfolio weights in Q, M, and the risk-free asset? What is the Sharpe ratio of this portfolio?
Now suppose that the CAPM does not hold and fund Q has a positive α = 2.5%.
(d) Compute the MVE mix of Q and M.
(e) What is the α of fund Q with respect to the MVE mix you calculated in part (d).
(e) If other investors can also easily identify fund Q as having positive α, would you expect the positive α to persist? Explain why or why not?
2. Consider the following properties of the returns of stock 1, stock 2, and of the market (m): σ 1 = 0.20, σ2 = 0.30, σm = 0.15, ρ 1m = 0.4, ρ2m = 0.7 and E[rm] = 0.10. Also, suppose that the risk-free rate rf = 0.05.
(a) According to the Capital Asset Pricing Model, what should be the expected return of stock 1 and of stock 2?
(b) Suppose that the correlation between the return of stock 1 and the return of stock 2 is 0.5. What is the expected return of a portfolio that has a 40% investment in stock 1 and a 60% investment in stock 2?
(c) Assume the CAPM holds. Construct a new portfolio using the market portfolio and the risk-free asset that has the same expected return as the portfolio you considered in part b) but has the lowest standard deviation possible.