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Problem set 3
(solutions)
Exercise 1: The universe of available securities includes two risky stock funds, A and B, and T-bills. The data for the universe are as follows: The correlation coefficient between funds A and B is - 0.2. a) Find the optimal risky portfolio, P, and its expected return and standard deviation. b) Find the slope of the CAL supported by T-bills and portfolio P. c) How much will investor with A=5 invest in funds A and B and in T-bills? Solution: a) The proportions in the optima risky portfolio are given by 𝑤! =
𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟!, 𝑟!
𝜎!^!^ 𝐸 𝑟! − 𝑟! + 𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! + 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟!, 𝑟!
𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟!, 𝑟!
𝜎!^!^ 𝐸 𝑟! − 𝑟! + 𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! + 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟!, 𝑟!
Substituting the data, the final solution is: 𝐶𝑜𝑣 𝑟!, 𝑟! = − 0. 2 × 0. 2 × 0. 6 = − 0. 024 𝑤! =
0. 6 !× 0. 1 − 0. 05 − 0. 3 − 0. 05 × − 0. 024
0. 2 !× 0. 3 − 0. 05 + 0. 6 !× 0. 1 − 0. 05 − 0. 1 − 0. 05 + 0. 3 − 0. 05 × − 0. 024
0. 36 × 0. 05 − 0. 25 × − 0. 024
0. 04 × 0. 25 + 0. 36 × 0. 05 − 0. 3 × − 0. 024
The expected return and standard deviation of this optimal risky portfolio are: A B T-bills Expected return, E(r) 0,10 0,30 0, Standard deviation, (^) σ 0,20 0,60 0,
𝐸 𝑟! = 𝑤!×𝐸 𝑟! + 𝑤!×𝐸 𝑟! = 0. 6818 × 0. 1 + 0. 3182 × 0. 3 = 0. 1636 𝑜𝑟 16 .36%
𝜎! = 𝑤!^ !𝜎!^!^ + 𝑤!^ !𝜎!^!^ + 2 𝑤!𝑤!𝜌!"𝜎!𝜎!
! ! (^) = 𝜎! = 0. 68182 × 0. 2!^ + 0. 31822 × 0. 6!^ + 2 × 0. 6818 × 0. 3182 × − 0. 024 ! ! (^) = = 0. 465 × 0. 04 + 0. 101 × 0. 36 − 0. 0104 ! ! (^) = 0. 04463 ! ! (^) = 0. 2113 𝑜𝑟 21 .13% The optimal risky portfolio, P, is not the global minimum-variance portfolio. b) The CAL of this optimal portfolio has a slope of: 𝑆! =
This CAL is the line from the risk-free rate through the optimal risky portfolio. This line represents all efficient portfolios that combine T-bills with the optimal risky portfolio. c) Given a degree of risk aversion, A, an investor will choose a proportion, y, in the optimal risky portfolio of: 𝑦∗^ =
𝐴×𝜎!^!^
5 × 0. 2113!^
Thus, the investor will invest, 50.89% of his wealth in portfolio P, and 49.11% in T-bills. Risky portfolio P, consists of 68.18% in fund A, and 31.82% in fund B. Thus, the percentage of wealth invested in fund A and fund B will be: 𝑦!! = 0. 5089 × 0. 6818 = 0. 3470 𝑦!! = 0. 5089 × 0. 3182 = 0. 1619 Exercise 2: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 9 %. The probability distribution of the risky funds is as follows: The correlation between the risky funds is 0.15. E B Expected return, E(r) 0,22 0, Standard deviation, (^) σ 0,32 0,
= 0. 099 × 0. 102 + 0. 470 × 0. 05 + 0. 00476
! ! (^) = 0. 0397 ! ! (^) = 0. 1994 𝑜𝑟 19 .94% b) c) d) 𝑤! =
𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟!, 𝑟!
𝜎!^!^ 𝐸 𝑟! − 𝑟! + 𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! + 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟!, 𝑟!
𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟! , 𝑟!
𝜎!^!^ 𝐸 𝑟! − 𝑟! + 𝜎!^!^ 𝐸 𝑟! − 𝑟! − 𝐸 𝑟! − 𝑟! + 𝐸 𝑟! − 𝑟! 𝐶𝑜𝑣 𝑟! , 𝑟!
Substituting the data, the final solution is: 𝐶𝑜𝑣 𝑟! , 𝑟! = 0. 15 × 0. 32 × 0. 23 = 0. 01104 σ E(r)^ w^ E w^ B 0,230 0,130 0,00% 100,00% 0,204 0,148 20,00% 80,00% 0,199 0,158 31,42% 68,58% 0,202 0,166 40,00% 60,00% 0,225 0,184 60,00% 40,00% 0,246 0,194 70,75% 29,25% 0,267 0,202 80,00% 20,00% 0,320 0,220 100,00% 0,00% σ E( r ) CAL(P)
P
B
E
𝑤! =
- 23 !× 0. 22 − 0. 09 − 0. 13 − 0. 09 × 0. 011
- 32 !× 0. 13 − 0. 09 + 0. 23 !× 0. 22 − 0. 09 − 0. 22 − 0. 09 + 0. 13 − 0. 09 × 0. 011 =
0. 0529 × 0. 13 − 0. 04 × 0. 011
0. 1024 × 0. 04 + 0. 0529 × 0. 13 − 0. 17 × 0. 011
The mean and standard deviation for the optimal risky portfolio are: 𝐸 𝑟! = 0. 7075 × 0. 22 + 0. 2925 × 0. 13 = 0. 1936 𝑜𝑟 19 .36% 𝜎! = 0. 70752 × 0. 32!^ + 0. 29252 × 0. 23!^ + 2 × 0. 7075 × 0. 2925 × 0. 15 × 0. 32 × 0. 23 ! ! (^) = 𝜎! = 0. 5 × 0. 102 + 0. 086 × 0. 05 + 0. 00476 ! ! (^) = 0. 06035 ! ! (^) = 0. 2457 𝑜𝑟 24 .57% e) They reward-to-variability ratio of the optimal CAL is: 𝑆! =
f) 𝐸 𝑟!" − 𝑟! 𝜎!"
×𝜎!"
0. 15 = 0. 09 + 0. 422 ×𝜎!" ⇒ 𝜎!" =
To find the proportion invested in risky portfolio is: 𝜎!" = 𝑦×𝜎! ⇒ 𝑦 =
Thus, the proportion invested in T-bills is: 1 − 𝑦 = 1 − 0. 5878 = 0. 4122 The proportion of stock fund and bond funds in the complete portfolio are: 𝑦!! = 0. 5878 × 0. 7075 = 0. 4094
𝑤!× 0. 05 − 1 − 𝑤! × 0. 1 = 0
𝑤!× 0. 05 + 𝑤!× 0. 1 = 0. 1
𝑤!× 0. 15 = 0. 1
The expected rate of return on the risk-free portfolio is: 𝐸 𝑟! = 0. 6667 × 0. 1 + 0. 3333 × 0. 15 = 0. 1167 𝑜𝑟 11. 67 Question 1: Assume that expected returns and standard deviations for all securities (including the risk-free for borrowing and lending) are known. In this case, all investors will have the same optimal risky portfolio. (True or false?) Answer: False. If the borrowing and lending rates are not identical, then depending on the tastes of the individuals (that is, the shape of indifference curves), borrowers and lenders could have different optimal risky portfolios. Question 2: The standard deviation of the portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio. (True or false?). Answer: False. Only in the special case that all assets are perfectly positively correlated, the portfolio standard deviation will reduce to the weighted average of the component-asset standard deviations. The portfolio variance will be weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weight. Question 3: Which statement about portfolio diversification is correct? a) Proper diversification can reduce or eliminate systematic risk. b) Diversification reduces the portfolio’s expected return because it reduces the portfolio’s total risk. c) As more securities are added to a portfolio, total risk would typically be expected to fall at a decreasing rate. Answer: The correct answer is (c).
