Question

Problem 7-24 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 8.2% per year, with a SD of 23.2%. The hedge fund nisk premium is estimated at 13.2% with a SD of 38.2%. 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. What should be Greta's capital allocation? (Do not round your intermediate calculations. Round your answers to 2 decimal places. S&P Hedge Risk-free asset

Answer 1

Given,

S&P 500 with

risk premium = 8.2%

So, E(rE) = 8.2 + rf

SD = 23.2%

Hedge fund with

Risk premium = 13.2%

So, E(rD) = 13.2 + rf

SD = 38.2%

Let us first select the optimum portfolio from the risky assets with the investor risk aversion A = 3,

Weight of hedge fund in such portfolio(D is for hedge fund and E is for S&P 500) =

WD = (E(rD)*SD(rE)^2)/((E(rD)*SD(rE)^2)+(E(rE)*SD(rD)^2))

So, Weight of hedge fund = (0.132*0.232^2)/((0.082*0.382^2)+(0.132*0.232^2))) = 0.37

So, Weight of S&P 500 = 1-WD = 1-0.37 = 0.63

Using this, return of the portfolio = WD*E(rD) + WE*E(rE) = 0.37*(13.2+rf) + 0.63*(8.2+rf) = 10.06 + rf

and standard deviation of portfolio = WD*SD(D) + WE*SD(E) = 0.37*38.2 + 0.63*23.2 = 28.79%

Now to make an optimal complete portfolio including risk free asset with return = rf

Weight of risky portfolio in such complete portfolio is

y = E(rp-rf)/A*SD(p)^2

So, y = (.1006 + rf - rf)/(3*0.2879^2) = 0.4047

So weight of risk free asset = 1-0.4047 = 0.5953

So now weight of individual assets in risky portfolio can be calculated using previous weight of 0.4047

So weight of S&P 500 = 0.63*0.4047 = 0.2539 or 25.39%

And weight of Hedge fund = 0.4047*0.37 = 0.1508 or 15.08%

And weight of risk free asset = 0.5953 or 59.53%