FIN 532 Investment Theory
Problem Set 2
Fall 2024
1 Constructing the Minimum Variance Frontier
You are considering investing in two stocks. There are two possible states for the economy over the next year: ‘Good’ and ‘Bad’ . Each state is equally likely (that is, probability for each state is 50%). Their return in each possible state is estimated as follows:
|
State
|
Return to stock A
|
Return to stock B
|
|
Good
|
30%
|
5%
|
|
Bad
|
10%
|
10%
|
(a) What are the expected return and volatility of each stock return?
(b) What are the covariance and correlation between the two stock returns?
(c) Construct the minimum variance frontier that is possible by investing in these two stocks (assume no short selling).
(d) Suppose that a risk free rate of 5% is for borrowing or lending. Can you construct a portfolio with no risk and a return greater then the risk free rate? Explain.
2 MPT with Two Risky Assets
You have recently inherited $150,000 and have decided that you should invest the money. You have identified three funds which seem like a good fit for your investment goals: a risk free short-term bond fund (f ), a long-term bond fund (B), and a stock market index fund (S). Your research revealed the following information about the 3 funds:
|
|
Expected Return: E(r)
|
Volatility: σ(r)
|
|
Risk-Free Fund (f )
|
0.035
|
0.000
|
|
Long-Term Bond Fund (B)
|
0.060
|
0.075
|
|
Stock Market Index Fund (S)
|
0.110
|
0.18
|
|
Correlation between B and S: ρBS = 0.75
|
(a) First you consider investing 1/3 of your inheritance in each of the 3 funds. What is the expected return and volatility of this portfolio?
(b) Having taken the first two weeks of Fin-532, you know that you can construct a more efficient portfolio than simply putting an equal weight of your inheritance in each fund. You start by constructing the Mean-Variance Efficient (MVE) portfolio.
(i) What are the portfolio weights in the MVE portfolio?
(ii) What is the expected return and volatility of the MVE portfolio?
(c) After considering your investment goals and risk tolerance, you’ve decided to make sure your portfolio volatility is no greater than 12%. Given this restriction, how should you allocate the inheritance between (f , B, and S) to maximize your expected return?
3 Matrix Algebra and Portfolio Moments
The file HW2data.csv contains monthly historical returns of 5 industry portfolios from July 1926 to June 2023. (Source: Professor Kenneth French’s website.) To get you started in Matlab, first save the data file into a directory on your computer. Then create a new script file and save it in the same directory. At the begining of the script file, you can use the following code to read in the data:
T = readtable(’HW2data . csv’);
Rets = csvread(’HW2data . csv’,1,1);
Rets is a matrix with 5 columns where each column is a time-series of returns for a different industry portfolio. T is a 6 column matrix, where the first column has the month/year of the return
(a) Compute the annualized expected returns and covariance matrix. (Hint: Estimate the expected returns and covariance matrix of monthly returns first, then multiply them by 12.)
(b) Next construct the correlation matrix (recall that the correlation between asset A and B, is given by ρAB = Cov(rA , rB )/(σA σB)).
(c) Suppose the risk free rate is 1%. Compute the Sharpe ratio of each of the industry portfolios.
(d) Compute the expected return and standard deviation of an equal weighted portfolio with weight 1/5 for each industry portfolio. What is the Sharpe Ratio of this portfolio?
(e) Looking at the Sharpe Ratio of each industry portfolio as well as the correlation matrix, which of the industry portfolios looks most attractive? Which portfolio looks least attractive?
(f) Using your answer to the previous question, find a modification of the equally weighted portfolio (i.e., add some weight to one portfolio and subtract some weight from another) that delivers a higher Sharpe Ratio than both the equally weighted portfolio and any of the individual industry portfolios.