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. (Do not round your intermediate calculations. Round "Sharpe ratios" to 4 decimal places and other answers to 2 decimal places.)

S&P Portfolio	Hedge Fund Portfolio
Risk premiums
SDs
Sharpe ratios

Answer 1

ANSWER

Step1: Computation of the weight or optimal asset allocation for the S&P 500 and a hedge fund.We have,

W_s&p =

Where,

= Standard deviation of hedge fund = 26 %

= Standard deviation of S&P 500 = 20 %

= the correlation coefficient between hedge fund & S &P = 0

Putting these value in the above equation and calculating weight of hedge fund in portfolio.We have,

W_s&p = (26)² - 26*20*0 / (26)² + (20)² - 2*26*20*0

W_s&p = 676 - 0 / ( 676 + 400 - 0)

W_s&p = 676 / 1,076 = 0.62*100 = 62 %

Weight for hedge fund is

W_hf = 1 - W_s&p

W_hf = 1 - 0.62 = 0.38*100 = 38 %

Hence,the weight in the portfolio for hedge fund and S&P are 38% and 62 % respectively.

Step2: Computation of the estimated annual risk premiums of two portfolio.We have,

Risk Aversion = 3

Risk premium for S&P 500 = 7 %

Risk premium for hedge fund = 5 %

1-year risk premium for the hedge fund = Risk Aversion x Risk premium for hedge fund

1-year risk premium for the hedge fund = 3 x 5% = 15 %

1-year risk premium for the S&P 500 = 3 x 7 % = 21 %

1-year risk premium for the portfolio of hedge fund & S &P 500 = W_hf x 1-year risk premium for the hedge fund + W_s&p x 1-year risk premium for the S&P 500

1-year risk premium for the portfolio of hedge fund & S &P 500 = 38 % x 15 % + 62 % x 21 %

1-year risk premium for the portfolio of hedge fund & S &P 500 = 5.7% + 13.02 % = 18.72 %

Hence, the estimated annual risk premium for the portfolio of hedge fund & S &P 500 is 18.72%.

Step-3: Computation of the Standard deviation of the portfolio.We have,

standard deviation = [ W_hf + W_s&p ]^1/2

Standard deviation = [ 26² x 0.38 + 20² x 0.62 ]^1/2

Standard Deviation = [ 256.88 + 248]^1/2 = 22.47 %

Hence, the Standard deviation for the portfolio of hedge fund & S &P 500 is 22.47%

Step-4: Computation of the sharpe ratio of both portfolio.We have,

Sharpe ratio = Estimated annual risk premium / Standard Deviation

For Hedge Fund:

Sharpe ratio = 15% / 26 % = 0.5769

Hence,the sharpe ratio for the hedge fund is 0.5769

For S&P 500:

Sharpe ratio = 21% / 20% = 1.05

Hence,the sharpe ratio for the S&P 500 is 1.05

For Portfolio:

Sharpe Ratio = 18.72 % / 22.47 % = 0.8331

Hence, the sharpe ratio for the portfolio of hedge fund & S &P 500 is 0.8331