Question

Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 7.2% per year, with a SD of 22.2%. The hedge fund risk premium is estimated at 12.2% with a SD of 37.2%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund 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 is not fully convinced by this claim.

Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 7.2% per year, with a SD of 22.2%. The hedge fund risk premium is estimated at 12.2% with a SD of 37.2%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund 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 is not fully convinced by this claim. a-1. Assuming the correlation between the annual returns on the two portfolios is 0.3, what would be the optimal asset allocation? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.) S&P Hedge a-2. What is the expected return on the portfolio? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.) Expected returnſ

Answer 1

The 1 year risk premium on the S&P portfolio is (1+7.2%)^1 â 1 = 25.97%

The 1 year risk premium on the hedge fund portfolio is (1+12.2%)^1-1 = 33.10%

S&P 1 year standard deviation is = 22.2*(1^1/2) = 11.1

Hedge fund 1 year standard deviation is 37.2*(1^1/2) = 18.6

With correlation = 0, the optimal asset allocation is

Ws & p =

25.97 * 62.35^2-33.10*(0*31.17*62.35)/(25.97*62.35^2+33.10*31.17^2-(25.97+33.10)*31.17*62.35*0)

Ws & p = 0.7584

W hedge = 1- 0.7584 = 0.2416

With these weights the expected return on portfolio = 0.7584*25.97 + 0.2416*33.10 = 27.69

Variance = (0.7584*0.3117)^2+(0.2416*0.6235)^2 = 0.07857

With risk aversion of A=3, the optimal capital allocation in risky portfolio is:

0.2769/(3*0.07857) = 1.1747

Resulting investment composition will be S&P = 0.7584 * 1.1747 = 0.8908

In hedge fund = 0.2418 * 0.8810 = 0.2130

The remaining 0.1190 will be invested in the risk-free asset