Question

Greta has 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 1-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 8% per year, with a standard deviation of 23%. The hedge fund risk premium is estimated at 10% with a standard deviation of 38%. 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.

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. Enter your

answers as decimals rounded to 4 places.)

a-2. What is the expected risk premium on the portfolio? (Do not round intermediate calculations. Enter your answer as decimals rounded to 4 places.)

Answer 1

a-1). Notations used: E(Rsp) = S&P500 risk premium = 8%; SDsp = S&P500 standard deviation = 23%; E(Rhf) = hedge fund risk premium = 10%; SDhf = hedge fund standard deviation = 38%; Wsp = weight of S&P500; Whf = weight of hedge fund; E(Rp) = expected risk premium of the risky portfolio; SDp = standard deviation of the risky portfolio

Since correlation between the portfolios is zero, covariance is also zero.

Wsp = E(Rsp)*SDhf^2 /[E(Rsp)*SDhf^2 + E(Rhf)*SDsp^2]

= 8%*38%^2 / [8%*38%^2 + 10%*23%^2] = 0.6859

Whf = 1 - Wsp = 1 - 0.6859 = 0.3141

E(Rp) = (Wsp*E(Rsp)) + (Whf*E(Rhf)) = (0.6859*8%) + (0.3141*10%) = 0.0863

SDp = [(Wsp*SDsp)^2 + (Whf*SDhf)^2]^0.5

= [(0.6859*23%)^2 + (0.3141*38%)^2]^0.5 = 0.1978

Sharpe ratio of the portfolio = E(Rp)/SDp = 0.0863/0.1978 = 0.4362

With a risk aversion of A = 3, amount to be invested in the risky portfolio will be

E(Rp)/(A*SDp^2) = 0.0863/(3*0.1978^2) = 0.7349

So, amount to be invested in the S&P500 portfolio = 0.7349*0.6859 = 0.5041

Amount to be invested in the hedge fund = 0.7349*0.3141 = 0.2308

Remaining will be invested in the risk-free asset which will be 1 - (0.5041+0.2308) = 0.2651

a-2). Expected risk premium on the portfolio E(Rp) = 0.7349 (as calculated above)