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 125% of its wealth in the S&P, financed by borrowing 25% of its wealth at the risk-free rate?

b) 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?

c) Assume investors' preferences are characterized by the utility function  . 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 expeted return and standard deviation of this optimal portfolio?

Explanation / Answer

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%


a) Expected return

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%

b) 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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