Question

Greta has risk aversion of A = 4 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 4-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 10% per year, with a standard deviation of 22%. The hedge fund risk premium is estimated at 14% with a standard deviation of 37%. 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 return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim.

	S&P Portfolio Hedge Fund Portfolio Risk premiums Standard deviations Sharpe ratios

Answer 1

risk aversion of A = 4

S&P 500 risk premium is estimated at 10%  with a standard deviation of 22%.

ER(S) - R(f) = 10% = 0.10

SD(S) = 22% = 0.22

The hedge fund risk premium is estimated at 14% with a standard deviation of 37%

ER(H) - R(f) =  14%  = 0.14

SD(H) = 37% = 0.37

Now Wegiht of S&P Portfolio W(S)

Wegiht of Hedge Fund  W(H) , W(H) = 1 -  W(S)

Since Cov (S,H) = 0

W(S) = 0.2016

W(H) = 1 -  W(S) = 1 -  0.2016 = 0.7983

SD(P) = Standard Deviation of the Portfolio

We will not add covariance term in formula as covariance is zero

SD(P) = 0.2987

Now Risk Premium of the Portfolio

ER(P) = W(S) * [ER(S) - R(f)] +  W(P) * [ER(P) - R(f)] = 0.2016 * 0.10 +  0.7983 *   0.14 = 13.194%

Sharpe ratios =  Risk Premium of the Portfolio /  Standard Deviation of the Portfolio

= 13.194% / 0.2987 = 0.4416

Ans :

Weight of S&P Portfolio = 0.2016
Weight of Hedge Fund Portfolio =  0.7983
Risk premiums of Total Portfolio =   13.194%
Standard deviations Total Portfolio = 0.2987 = 29.87%
Sharpe ratios Total Portfolio = 0.4416