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 1-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 8% per year, with a SD of 23%. The hedge fund risk premium is estimated at 10% with a SD of 40%. 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 indeed zero, 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 risk premium on the portfolio? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.)

Expected risk premium-

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 = 40%; 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%*40%^2 / [8%*40%^2 + 10%*23%^2] = 0.7076

Whf = 1 - Wsp = 1 - 0.7076 = 0.2924

E(Rp) = (Wsp*E(Rsp)) + (Whf*E(Rhf)) = (0.7076*8%) + (0.2924*10%) = 0.0858

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

= [(0.7076*23%)^2 + (0.2924*40%)^2]^0.5 = 0.2004

Sharpe ratio of the portfolio = E(Rp)/SDp = 0.0858/0.2004 = 0.4284

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

E(Rp)/(A*SDp^2) = 0.0858/(3*0.2004^2) = 0.7124

So, amount to be invested in the S&P500 portfolio = 0.7124*0.7076 = 0.5041

Amount to be invested in the hedge fund = 0.7124*0.2924 = 0.2083

Remaining will be invested in the risk-free asset which will be 1 - (0.5041+0.2083) = 0.2876

a-2). Expected risk premium on the portfolio E(Rp) = 0.0858 (as calculated above)