Suppose you are a hedge fund. You are trying to form a portfolio out of a number

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Question:

Suppose you are a hedge fund. You are trying to form a portfolio out of a number of risky assets and a risk-free asset. Throughout the question, assume you can take arbitrarily large short or leveraged positions on any (risky or risk-free) assets. For parts a)-c) of the question, assume that you are trying to form a portfolio out of two assets A, B and a risk-free asset. The expected returns and standard deviations of assets A and B are:

Asset: Expected Return: Standard Deviation:

A. 8% 20%

B 16% 15%

a) Assume that the correlation between asset A and B is 0.8. Write down the variance-covariance matrix for assets A and B.

b) Assume that the risk-free rate is 2%. Find the minimum variance portfolio weights on A and B, and the tangency portfolio weights on A and B.

c) Now suppose the risk-free rate is 20%. Expected returns, standard deviations, and correlations are still as described in the table above. As a money manager, the maximum standard deviation your clients allow on your portfolio return is 10%. So you want to maximize your portfolio expected return, conditional on portfolio SD being at most 10%. What is the optimal allocation of assets between the risk-free asset, and assets A and B (i.e. what are the optimal portfolio weights)? What is the expected return on your optimal portfolio?

d) Now, suppose you are a different hedge fund. You cannot access A and B; you can access two assets X and Y, with expected returns and standard deviations as follows:

Asset: Expected Return: Standard Deviation:

X 4% 10%

Y 10% 10%

Assume that the correlation between X and Y is 0. The risk-free rate is 7%. As in part c) of the question, you want to maximize your portfolio expected return, conditional on portfolio SD being at most 10%. What is the optimal allocation of assets between the risk-free asset, and assets X and Y (i.e. what are the optimal portfolio weights)? What is the expected return on your optimal portfolio?