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 one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 6.6% per year, with a SD of 21.6%. The hedge fund risk premium is estimated at 11.6% with a SD of 36.6%. 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. Calculate Greta’s capital allocation using an annual correlation of 0.3.(Do not round your intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

Risk free rate 1 year Treasury rate 2.00% (Assumed) Fund S&P 500 (S) Hedge Fund (H) Risk Premium Return 6.60% 11.60% Standard Return 21.60% 36.60% 8.60% 13.60% 10 Correlation 0.30 12 Cov (S,H) 0.02-D11*E8 E9 14 15 16 17 Calculation of weight of optimal portfolio The minimum variance portfolio of two Risky asset can be calculated as follows: 19 Wmin (S) 24 Optimal asset allocation in S&P 500, Wmin(S) Optimal asset allocation in Hedge Fund, Wmin(H) 0.83 26 0.17 27 Calculation of expected return of minimum variance portfolio Portfolio expected return can be calculated as follows: 29 Expected portfolio return, r. 1V7 32 Where w and r are the weights and return of assets A 35 36 37 38 39 Expected return of minimum variance portfolio Sum Product of portfolio weight and return 9.46% =SUMPRODUCT(D25:D26,E8:E9) Expected return of minimum variance portfolio 9.46% 41 42 43 Given the risk aversion (A) of the investor, the weight of risky assets in the portfolio is given by following equation: Where E(rp) is the expected return of the risky portfolio, rf is the risk free rate A is risk aversion of investor and ơp2 is the variance of risky portfolio 47

Formula sheet

49 Portfolio variance is given by following formula Portfolio variance *(R o (R)2(wA) (Wa)*Cov(A, B) Where wA and wB are weights of assets A and B, σΑ and Og are standard deviation of assets A and B. Given the following data: 51 53 S&P 500 (S) Hedge Fund (H Expected Return Standard Deviation Weight Covariance 8.60% 21.60% 0.83 13.60% 36.60% 0.17 57 0.0237168 59 Variance of Risky portfolio 61 4.2796 =(D57"D56)"2刊E57"E56)"2+2"D57"E57"D58 63 Hence Variance of risky portfolio is 4.27% Calculation of weight of risky asset in the portfolio Given the risk aversion (A) of the investor, the weight of risky assets in the portfolio is given by following equation: 71 72 73 74 75 Where E(rp) is the expected return of the risky portfolio, is the risk free rate, A is risk aversion of investor and ơp2 is the variance of risky portfolio. Using the following data Expected return of risky portfolio risk free rate Risk Aversion Variance of risky portfolio 9.46% 2.00% 4.27% Weight of risky assets in the portfolio can be calculated as follows: Weight of risky assets in the portfolio 79 58.24%-(D73-D74)/(D75*D76) 81 82 83 84 58.24% Hence Weight of risky assets in the portfolic Weight of risk free asset in the portfolio -1-Weight of risky asset in the portfolio 41.76% Using the weight of risky asset in the portfolio and the weight of individual fund in the risky asset, weight of fund in the portfolio can be calculated as follows: Using the following data Weight of risky assets in the portfolio Weight of S&P 500 in risky asset Weight of Hedge fund in risky asset 87 58.24% 0.83 0.17 89 91 92 93 Weight of S&P 500 in the portfolio Weight of risky asset in the portfolio weight of S&P 500 in risky asset 48.21%-D89"D90 95 Weight of Hedge fund in the portfolio Weight of risky asset in the portfolio*weight of Hedge fund in risky asset 10.03% D89"D91 100 101 102 Hence Capital allocation is as follows: Weight of risk free asset Weight if S&P 500 Weight of Hedge fund 41.76% 48.21% 10.03%

A	B	C	D	E	F	G	H	I	J
2
3
4
5		Risk free rate = 1 year Treasury rate	0.02	(Assumed)
6
7		Fund	Risk Premium	Return	Standard Return
8		S&P 500 (S)	0.066	=D8+D5	0.216
9		Hedge Fund (H)	0.116	=D9+D5	0.366
10
11		Correlation	0.3
12
13		Cov (S,H)	=?*?S*?B
14			=D11*F8*F9	=D11*E8*E9
15
16		Calculation of weight of optimal portfolio:
17		The minimum variance portfolio of two Risky asset can be calculated as follows:
18
19
20
21
22
23
24
25		Optimal asset allocation in S&P 500, wmin(S)	=(F9^2-D14)/(F9^2+F8^2-2*D14)
26		Optimal asset allocation in Hedge Fund, wmin(H)	=1-D25
27
28		Calculation of expected return of minimum variance portfolio:
29		Portfolio expected return can be calculated as follows:
30
31
32
33
34		Where wi and ri are the weights and return of assets Ai
35
36		Expected return of minimum variance portfolio	=Sum Product of portfolio weight and return
37			=SUMPRODUCT(D25:D26,E8:E9)	=SUMPRODUCT(D25:D26,E8:E9)
38
39		Expected return of minimum variance portfolio	=D37
40
41
42		Given the risk aversion (A) of the investor, the weight of risky assets in the portfolio is given by following equation:
43
44
45
46		Where E(rp) is the expected return of the risky portfolio, rf is the risk free rate,
47		A is risk aversion of investor and ?p2 is the variance of risky portfolio.
48
49
50		Portfolio variance is given by following formula:
51		Portfolio variance = w2A*?2(RA) + w2B*?2(RB) + 2*(wA)*(wB)*Cov(A, B)
52		Where wA and wB are weights of assets A and B, ?A and ?B are standard deviation of assets A and B.
53		Given the following data:
54			S&P 500 (S)	Hedge Fund (H)
55		Expected Return	=E8	=E9
56		Standard Deviation	=F8	=F9
57		Weight	=D25	=D26
58		Covariance	=D14
59
60		Variance of Risky portfolio	=w2A*?2(RA) + w2B*?2(RB) + 2*(wA)*(wB)*Cov(A, B)
61			=(D57*D56)^2+(E57*E56)^2+2*D57*E57*D58	=(D57*D56)^2+(E57*E56)^2+2*D57*E57*D58
62
63		Hence Variance of risky portfolio is	=D61
64
65		Calculation of weight of risky asset in the portfolio:
66		Given the risk aversion (A) of the investor, the weight of risky assets in the portfolio is given by following equation:
67
68
69
70		Where E(rp) is the expected return of the risky portfolio, rf is the risk free rate,
71		A is risk aversion of investor and ?p2 is the variance of risky portfolio.
72		Using the following data:
73		Expected return of risky portfolio	=D39
74		risk free rate	=D5
75		Risk Aversion	3
76		Variance of risky portfolio	=D63
77
78		Weight of risky assets in the portfolio can be calculated as follows:
79		Weight of risky assets in the portfolio	=(E(rp)-rf)/(A*?p2)
80			=(D73-D74)/(D75*D76)	=(D73-D74)/(D75*D76)
81
82		Hence Weight of risky assets in the portfolio	=D80
83		Weight of risk free asset in the portfolio	=1-Weight of risky asset in the portfolio
84			=1-D82
85
86		Using the weight of risky asset in the portfolio and the weight of individual fund in the risky asset,
87		weight of fund in the portfolio can be calculated as follows:
88		Using the following data:
89		Weight of risky assets in the portfolio	=D82
90		Weight of S&P 500 in risky asset	=D25
91		Weight of Hedge fund in risky asset	=D26
92
93		Weight of S&P 500 in the portfolio	=Weight of risky asset in the portfolio*weight of S&P 500 in risky asset
94			=D89*D90	=D89*D90
95
96		Weight of Hedge fund in the portfolio	=Weight of risky asset in the portfolio*weight of Hedge fund in risky asset
97			=D89*D91	=D89*D91
98
99		Hence Capital allocation is as follows:
100		Weight of risk free asset	=D84
101		Weight if S&P 500	=D94
102		Weight of Hedge fund	=D97
103