Question

Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a 3-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of 3-year strategies. (All rates are annual, continuously compounded.) The S&P 500 risk premium is estimated at 6% per year, with a SD of 21%. The hedge fund risk premium is estimated at 9% with a SD of 38%. The return on each of these portfolios in any year is uncorrelated with its return or the return of any other portfolio in any other year. The hedge fund management claims the correlation coefficient between the annual returns on the S&P 500 and the hedge fund in the same year is zero, but Greta believes this is far from certain.

a-1.	Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? (Do not round intermediate calculations. Round your answers to 2 decimal places. Omit the "%" sign in your response.)

S&P	%
Hedge	%

a-2.	What is the expected return on the portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places. Omit the "%" sign in your response.)

Expected return

%

a-3.	What should be Greta’s capital allocation? (Do not round intermediate calculations. Round your answers to 2 decimal places. Omit the "%" sign in your response.)

S&P	%
Hedge	%
Risk-free asset	%

Answer 1

Given - The annual S&P risk premium (X) = 6% and the annual hedge fund risk premium (Y) = 9%

Also, the risk or SD of S&P (L) = 21% and the risk or SD of the hedge fund (M) = 38%

Also, the correlation between these 2 funds (r) is 0

Now, the weight in the S&P 500 (V) is given by

=(X∗M²−Y∗L∗M∗r) / (X∗M²+Y∗L²−(X+Y)∗L∗M∗r)

=(6∗38²−9∗21∗38∗0)/(6∗38²+9∗21²−(6+9)∗21∗38∗0)

= 0. 68

Therefore, weight in the Hedge Fund (W) = 1 - 0.68 = 0.32

Now, the expected risk premium on the risky portfolio (Z) consisting of the hedge fund and the S&P 500

• = V * X + W * Y
• = 0.68 * 6% + 0.32 * 9%
• = 6.96%

Also, the total risk of the risky portfolio (N)

√(V*L)2+(W*M)2+(2*V*L*W*M*r)

√(0.68*21)2+(0.32*38)2+(2*0.68*21*0.32*38*0)

= 18.75%

Now, optimal asset allocation to the risky portfolio

=Z/(A∗N2)

=6.96(3∗18.752)

=66.03%

Therefore, Greta's capital allocation is shown below:

• Risk-free asset = 100% - 66.03% = 33.97%
• S&P 500 = 66.03% * 0.68 = 44.90%
• Hedge fund = 66.03% * 0.32 = 21.12%