Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied

Question

Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a 3- 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 8% per year, with a SD of 22%. The hedge fund risk premium is estimated at 6% with a SD of 28%. 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. 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. Round your answers to 2 decimal places. Omit the "%" sign in your response.) S&P; Hedge 68.77 % 31.23 % a-2. What is the expected return on the portfolio? (Do not round intermediate calculations. Round your answer to 2 decimal places. Omit the "%" sign in your response.) Expected return a-3. What should be Greta's capital allocation? (Do not round intermediate calculations. Round your answers to 2 decimal places. Omit the%" sign in your response.) S&P; Risk-free asset

Explanation / Answer

Degree of risk aversion A= 3

Risk premium of S&P 500 = 8%

Risk premium of S&P 500 for 3 years = (1+8%)3 -1 = 25.97%

SD of S&P 500 = 22%

SD of S&P 500 for 3 years = 22% underroot (3) = 22% * 1.732 = 38.104%

Risk premium of hedge fund = 6%

Risk premium of hedge fund for 3 years = (1+6%)3 -1 = 19.1016%

SD of hedge fund = 28%

SD of hedge fund for 3 years = 28% underroot(3) = 28% * 1.732 = 48.496%

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Optimal weight in a portfolio is calculated as = Ws = [RPs - RPh + A * ( SDh2 - correlation coeff * SDh * SDs)]/ [A* (SDs2 + SDh2 - 2 * correlation coeff * SDs * SDh)]

Since correlation co-eff = 0, we have

Ws =  [RPs - RPh + A * SDh2 ]/ [A* (SDs2 + SDh2 )]

= [25.97% - 19.1016% + 3 * 48.4962]/ [3 * ( 48.4962 + 38.1042)]

= 0.774242 / 1.141133 = 0.6785 = 67.85%

Wh = 1 - Ws  = 0.3215 = 32.15%

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Expected Portfolio return = Ws * 25.97% + Wh * 19.1016% = 23.7618%

Portfolio SD = Underroot ( Ws2 * SDs2  + Wh2 * SDh2)

= 30.19%

Portion invested in risky portfolio = (Expected portfolio return - Riskfree rate)/A * SD2 of portfolio = 0.237618 - 0.07/(3 * 0.30192) = 61.3018%

Portion invested in riskfree assets = 38.6982%

Portion invested in S&P = 67.85% of 61.3018% = 41.59%

Portion invested in Hedge fund = 32.15% of 61.3018% = 19.71%

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