Exercises on portfolio theory

1. For diversification to reduce risk,a. the returns on the individual securities should be highly correlatedb. the prices of the stocks should be stablec. the returns on the individual securities should be negatively correlatedd. one firm should offer dividends and the other should offer capital gains

ANS: C

2. An investor invests 30 percent of his wealth in a risky asset with an expected rate of return of 0.15 and a variance of 0.04 and 70 percent in a T-bill that pays 6 percent. His portfolio's expected return and standard deviation are __________ and __________, respectively. A) 0.114; 0.12 B) 0.087;0.06 C) 0.295; 0.12 D) 0.087; 0.12 E) none of the above Answer: B Difficulty: Moderate Rationale: E(rP) = 0.3(15%) + 0.7(6%) = 8.7%; sP = 0.3(0.04)1/2 = 6%.

3. An efficient portfolio1. maximizes risk for a given return2. minimizes risk for a given return3. maximizes return for a given level of risk4. minimizes return for a given level of risk

a. 1 and 3b. 1 and 4c. 2 and 3d. 2 and 4

ANS: C

Use the following to answer questions 4-5:You invest $100 in a risky asset with an expected rate of return of 0.12 and a standard deviation of 0.15 and a risk free asset with a rate of return of 0.05.

4. What percentages of your money must be invested in the risky asset and the risk-free asset, respectively, to form a portfolio with an expected return of 0.09? A) 85% and 15% B) 75% and 25% C) 67% and 33% D) 57% and 43% E) cannot be determined

Answer: D Difficulty: Moderate Rationale: 9% = w1(12%) + (1 - w1)(5%); 9% = 12%w1 + 5% - 5%w1; 4% = 7%w1; w1 = 0.57; 1 - w1 = 0.43; 0.57(12%) + 0.43(5%) = 8.99%.

5. What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard deviation of 0.06? A) 30% and 70% B) 50% and 50% C) 60% and 40% D) 40% and 60% E) cannot be determined

Answer: C Difficulty: Moderate Rationale: 0.06 = x(0.15); x = 40% in risky asset.

6. What is the expected return on a portfolio consisting of an equal amount invested in each stock?Stock Expected Return

A 15%

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B 10 C 22 D 14

7. What is the expected return on the portfolio if 50 percent of the funds are invested in stock C, 30 percent in stock A, and 20 percent in Stock D?

ANS:

6. The expected return is the sum of the individual asset's expected return weighted by its proportion in the portfolio. If equal amounts are invested in each asset, the expected return on the portfolio isA: .15 .25 = .0375B: .10 .25 = .0250C: .22 .25 = .0550D: .14 .25 = .0350

.1525 = 15.25%

7. If different amounts are invested in the various assets, the portfolio's expected return becomesA: .15 .30 = .0450C: .22 .50 = .1100D: .14 .20 = .0280

.1830 = 18.3%

By investing more in the stock that offers the highest expected return (and presumably the most risk), the investor increases the expected return of the portfolio.

Use the following to answer questions 8-10:You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio, P, constructed with 2 risky securities, X and Y. The weights of X and Y in P are 0.60 and 0.40, respectively. X has an expected rate of return of 0.14 and variance of 0.01, and Y has an expected rate of return of 0.10 and a variance of 0.0081.

8. If you want to form a portfolio with an expected rate of return of 0.11, what percentages of your money must you invest in the T-bill and P, respectively? A) 0.25; 0.75 B) 0.19; 0.81 C) 0.65; 0.35 D) 0.50; 0.50 E) cannot be determined

Answer: B Difficulty: Moderate Rationale: E(rp) = 0.6(14%) + 0.4(10%) = 12.4%; 11% = 5x + 12.4(1 - x); x = 0.189 (T-bills) (1-x) =0.811 (risky asset).

9. If you want to form a portfolio with an expected rate of return of 0.10, what percentages of your money must you invest in the T-bill, X, and Y, respectively if you keep X and Y in the same proportions to each other as in portfolio P? A) 0.25; 0.45; 0.30 B) 0.19; 0.49; 0.32 C) 0.32; 0.41; 0.27 D) 0.50; 0.30; 0.20 E) cannot be determined

Answer: C Difficulty: Difficult Rationale: E(rp) = .100.10 = 5w + 12.4(1 - w); x = 0.32 (weight of T-bills); As composition of X and Y are .6 and .4 of P, respectively, then for 0.68 weight in P, the respective weights must be 0.41 and 0.27; .6(.68) = 41%; .4(.68) = 27%

10. What would be the dollar values of your positions in X and Y, respectively, if you decide to hold 40% percent of your money in the risky portfolio and 60% in T-bills? A) $240; $360 B) $360; $240 C) $100; $240 D) $240; $160 E) Cannot be determined

Answer: D Difficulty: Moderate Rationale: $400(0.6) = $240 in X; $400(0.4) = $160 in Y.

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