4.) Greta, an elderly investor, has a degree of risk aversion of A = 5 when appl
ID: 2383293 • Letter: 4
Question
4.) Greta, an elderly investor, has a degree of risk aversion of A = 5 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 5% per year, with a SD of 20%. The hedge fund risk premium is estimated at 12% with a SD of 40%. 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.
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.)
What is the expected return on the portfolio? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.)
What should be Greta’s capital allocation? (Do not round intermediate calculations. Enter your answers as decimals rounded to 4 places.)
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.)
Explanation / Answer
S.no.
Weight of S&P 500 in Portfolio
Weight of Hedge Fund in Portfolio
Expected Risk Premium of the portfolio (%)
Standard Deviation of the Portfolio
Utility Drawn on the basis of risk aversion
1
0
1
10
35.00
-8.375
2
0.1
0.9
9.5
31.56
-5.44375
3
0.2
0.8
9
28.28
-3
4
0.3
0.7
8.5
25.22
-1.04375
5
0.4
0.6
8
22.47
0.425
6
0.5
0.5
7.5
20.16
1.40625
7
0.6
0.4
7
18.44
1.9
8
0.7
0.3
6.5
17.50
1.90625
9
0.8
0.2
6
17.46
1.425
10
0.9
0.1
5.5
18.34
0.45625
11
1
0
5
20.00
-1
Standard Deviation of the Portfolio = (W1^2*s.d.1^2 + W2^2*s.d.2^2)^(1/2) since correlation between hedge fund and s&p 500 is zero
Utility = Expected return - .005*Degree of risk aversion*Standard deviation^2
On the basis of above table, maximum utility drawn is 1.90625 when 70% of S&P 500 and 30% of hedge fund is used in the portfolio.
on the basis of lowest risk portfolio, 80% investment should be in s&P 500 portfolio and 20% investment should be in hedge funds.
S.no.
Weight of S&P 500 in Portfolio
Weight of Hedge Fund in Portfolio
Expected Risk Premium of the portfolio (%)
Standard Deviation of the Portfolio
Utility Drawn on the basis of risk aversion
1
0
1
10
35.00
-8.375
2
0.1
0.9
9.5
31.56
-5.44375
3
0.2
0.8
9
28.28
-3
4
0.3
0.7
8.5
25.22
-1.04375
5
0.4
0.6
8
22.47
0.425
6
0.5
0.5
7.5
20.16
1.40625
7
0.6
0.4
7
18.44
1.9
8
0.7
0.3
6.5
17.50
1.90625
9
0.8
0.2
6
17.46
1.425
10
0.9
0.1
5.5
18.34
0.45625
11
1
0
5
20.00
-1
Standard Deviation of the Portfolio = (W1^2*s.d.1^2 + W2^2*s.d.2^2)^(1/2) since correlation between hedge fund and s&p 500 is zero
Utility = Expected return - .005*Degree of risk aversion*Standard deviation^2
On the basis of above table, maximum utility drawn is 1.90625 when 70% of S&P 500 and 30% of hedge fund is used in the portfolio.
on the basis of lowest risk portfolio, 80% investment should be in s&P 500 portfolio and 20% investment should be in hedge funds.
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