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 3-year strategies. (All rates are annual, continuously compounded.) The S&P 500 risk premium is estimated at 7% per year, with a SD of 20%. The hedge fund risk premium is estimated at 5% with a SD of 26%. 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. Compute the estimated 1-year risk premiums, SDs, and Sharpe ratios for the two portfolios.

Answer 1

The estimated 1-year risk premiums for the two portfolios is calculated as below:

1-Year Risk Premium (S&P 500) = (1 + S&P 500 Risk Premium Per Year)^(Years) - 1 = (1+7%)^1 - 1 = 7%

1-Year Risk Premium (Hedge Fund) = (1 + Hedge Fund Risk Premium Per Year)^(Years) - 1 = (1+5%)^1 - 1 = 5%

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The value of estimated 1-Year SDs for the two portfolios is arrived as follows:

1-Year SD (S&P 500) = (S&P 500 SD Per Year)*(Years)^(1/2) = (.20)*(1)^(1/2) = .20 or 20%

1-Year SD (Hedge Fund) = (Hedge Fund SD Per Year)*(Years)^(1/2) = (.26)*(1)^(1/2) = .26 or 26%

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The 1-Year sharpe ratios for the two portfolios are calculated as below:

1-Year Sharpe Ratio (S&P 500) = (1-Year Market Risk Premium, S&P 500)/(1-Year SD, S&P 500) = 7/20 = .3500

1-Year Sharpe Ratio (Hedge Fund) = (1-Year Market Risk Premium, Hedge Fund)/(1-Year SD, Hedge Fund) = 5/26 = .1923