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 sm=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 is totally invested in the risk-free asset?



b) 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?



c) 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?



d) 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?
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Explanation / Answer

a) The expected return of a portfolio that is totally invested in the risk free asset is caclculated as: E(R) = WA * E(RA) + Wf * E(RB) = 0 * 0.16 + 1.0 * 0.08 = 0 + 0.08 = 0.08 or 8% Therefore the expected return of a portfolio with risk free asset is 8% There is no standard deviation for the risk free asset. b) E(R) = 0.5 * 0.16 + 0.5 * 0.08 = 0.08 + 0.04 = 0.12 or 12% To calculate the standard deviation, the POrtfolio SD = WA * SDA + Wf * SDf = 0.50 * 0.10 + 0.50 * 0 = 0.05 or 5% Therefore, the portfolio SD is 5% c) Expected return for the third case: E(R) = 1.25 * 0.16 - 0.25 *0.08 = 0.2 - 0.02 = 0.18 or 18% The standard deviation is calculated as: Portfolio SD = 1.25 * 0.10 + (-0.25) * 0 = 0.125 or 12.5% Therefore the portfolio SD is 12.5% d) The standard deviation for S&P is 10% and for the entire portfolio is 20% But we know that Wf = (1-WA) Portfolio SD = WA * 0.10 + (1-WA) * 0 0.20 = WA * 0.10 WA = 0.2 or 20% Therefore weight of risk free asset = (1-0.20) = 0.80 or 80% Calculating the expected return E(R) = 0.20 * 0.16 + 0.80 * 0.08 = 0.032 + 0.064 = 0.096 or 9.6%

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