Greta is considering investing in either the S&P 500 or a hedge fund for a 3-year period. The S&P 500 has an estimated annual return of 5% with a standard deviation of 20%, while the hedge fund has an estimated annual return of 10% and standard deviation of 35%. Assuming zero correlation between the annual returns of the two investments, the optimal asset allocation that maximizes Greta's utility is 70% in the S&P 500 and 30% in the hedge fund. Based on her risk aversion of 3, the portfolio with the lowest risk is 80% in the S&P 500 and 20% in the hedge fund.

1. 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 5% per year, with a SD of 20%. The hedge fund risk premium
is estimated at 10% with a SD of 35%. 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.
Assuming the correlation between the annual returns on the two portfolios is indeed zero, what
would be the optimal asset allocation? What should be Greta's capital allocation? (Must show in
excel)
Solution
S.no. Weight of S&P 500 in Portfolio Weight of Hedge Fund in Portfolio Expected Risk
Premium of the portfolio (%) Standard Deviation of the Portfolio Utility Drawn on the basis of
risk aversion 1 0 1 10 35.00 -8.375 2 0.1 0.9 9.5 31.56 -5.44375 3 0.2 0.8 9 28.28 -3 4 0.3 0.7 8.5
25.22 -1.04375 5 0.4 0.6 8 22.47 0.425 6 0.5 0.5 7.5 20.16 1.40625 7 0.6 0.4 7 18.44 1.9 8 0.7
0.3 6.5 17.50 1.90625 9 0.8 0.2 6 17.46 1.425 10 0.9 0.1 5.5 18.34 0.45625 11 1 0 5 20.00 -1
Standard Deviation of the Portfolio = (W1^2*s.d.1^2 + W2^2*s.d.2^2)^(1/2)Â Â since
correlation between hedge fund and s&p 500 is zero Utility = Expected return - .005*Degree of
risk aversion*Standard deviation^2 On the basis of above table, maximum utility drawn is
1.90625 when 70% of S&P 500 and 30% of hedge fund is used in the portfolio. on the basis of
lowest risk portfolio, 80% investment should be in s&P 500 portfolio and 20% investment
should be in hedge funds.