Suppose that a fund that tracks the S&P has mean E(RM) = 16% and standard deviat

Question

Suppose that a fund that tracks the S&P has mean E(RM) = 16% and standard deviation ?M = 10%, and suppose that the T-bill rate Rf = 8%. Answer the following questions about efficient portfolios:

a) What is the expected return and standard deviation of a portfolio that has 50% of its wealth in the risk-free asset and 50% in the S&P?

b) What is the expected return and standard deviation of a portfolio that has 125% of its wealth in the S&P, financed by borrowing 25% of its wealth at the risk-free rate?

c) What are the weights for investing in the risk-free asset and the S&P that produce a standard deviation for the entire portfolio that is twice the standard deviation of the S&P? What is the expected return on that portfolio?

d) Assume investors preferences are characterized by the utility function U = E[r] 0.5A?2 . What would be the optimal allocation, i.e. the investment weights on S&P and T-bill, for an investor with a risk-aversion coefficient of A=4? What is the expected return and standard deviation of this optimal portfolio?

Explanation / Answer

We have:

Rm = 16%

SDm= 10%

Rf=8%

Part A

We have:

Wm = 0.50     Wf = 0.50

Rp = Wm x Rm + Wf x Rf

      = 0.50 x16% + 0.50 x8%

      = 12%

Part B

Wm =1.25    Wf= -0.25

Rp = Wm x Rm + Wf x Rf

      = 1.25 x16% + 0.250 x(-8%)

      = 18%

Standard deviation of portfolio = SDm x Wm

                                                = 10% x1.25

                                                = 12.50%

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