8 492 1344 b 127 478 1208 c 150 419 1014 table 1

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13.8% 49.2% 134.4% B 12.7% 47.8% 120.8% C 15.0% 41.9% 101.4% Table 1: Compounded returns to funds A , B and C . i = A, B, C after t years. Then r t,A = . 95(1 + . 10 - . 005 - . 0005) t - 1 = . 95(1 . 0945) t - 1; r t,B = . 96(1 + . 10 - . 015 - . 0015) t - 1 = . 96(1 . 0835) t - 1 if t 2, . 99(1 + . 10 - . 015 - . 0015) t - 1 = . 99(1 . 0835) t - 1 if t > 2; r t,C = (1 + . 10 - . 025 - . 0025) t - 1 = (1 . 0725) t - 1 . The compounded returns are given in Table 1. (b) (2 points) Suppose the annual return on Fund A ’s assets is 10%. What has to be the annual return on Fund B ’s assets for the two funds to yield the same net return after 5 years? Answer: Let r b denote the annual return to Fund B ’s assets. Then . 95(1 . 0945) 5 - 1 = . 99(1 - . 015 + r b (1 - . 015)) 5 - 1 r b = 10 . 2% . (c) (2 points) Suppose the annual return on Fund A ’s assets is 10%. What has to be the annual return on Fund C ’s assets for the two funds to yield the same net return after 10 years? Answer: Let Let r c denote the annual return to Fund C ’s assets. Then . 95(1 . 0945) 10 - 1 = (1 - . 025 + r c (1 - . 025)) 10 - 1 r c = 11 . 7% . 3. Portfolio Selection Consider an investor with the following utility function over port- folios: U ( c ) = E [ r c ] - 1 2 2 c , where c denotes a portfolio, E [ r c ] its the expected return and σ c the standard deviation of the portfolio returns. The investor has access to a risk-free asset that returns 6%, and a portfolio of risky assets, p , with an expected return of 15% and standard deviation of 30%. 6
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(a) (1 point) What is the slope of the capital allocation line for this problem? Answer: The reward-to-variability ratio is E [ r p ] - r f σ p = 15% - 6% 30% = 30% . (b) (5 points) What fraction of the investor’s optimal portfolio is invested in p if A = 5? If A = 2? If A = 1? Find the value of A for which 10% of the optimal portfolio is invested in p . Answer: Given A , the fraction of wealth invested in p is given by y = E [ r p ] - r f 2 p . That is, y = . 15 - . 06 5 × . 09 = 20% if A = 5, . 15 - . 06 2 × . 09 = 50% if A = 2, . 15 - . 06 1 × . 09 = 100% if A = 1. The value of A for which y = 10% is . 15 - . 06 A × . 09 = 10% A = 10 . (c) (4 points) Suppose there exists another risky portfolio, denoted q , available to the investor. If portfolio q has an expected return of 10% and a standard deviation of 10%, what fraction of the optimal portfolio will be invested in p ? What fraction of the optimal portfolio will be invested in q ?
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  • Spring '12
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