CHAPTER 6: PROBLEM SOLVING
Problem 1 (15 points) Suppose you have invested in three stocks: A, B and C. You expect that returns on the stocks depend on the following two states of the economy, with the probabilities to happen given below.
State of Economy Boom Bust
Probability of State Occurrence 0.70 0.30
Return on A stock A 7% 3%
Return on B stock B 15% 3%
Return on stock C 33% -6%
a. (5 points) What is the expected return of an equally weighted portfolio of these three stocks? b. (5 points) What is the expected return of a portfolio invested 20 percent each in A and B, and 60 percent in C ? c. (5 points) What is the standard deviation of a portfolio invested 20 percent each in A and B, and 60 percent in C ? Solution: a) To find the the expected expected return return of the portfoli portfolio, o, we need to find find the return return of the the portfolio portfolio in each state of the economy. eco nomy. This portfolio is a special case since all three assets have the same weight. Boom: E(Rp) = (0.07 +0.15 +0 .33)/3 = 0.1833 or 18.33% Bust: E(Rp) = (0.03 + 0.03 -0.06)/3 = 0.000 or 0.00%
And the expected return of the portfolio is: E(Rp) = 0.70(0.1833) + 0.30(0.00) = 0.1283 or 12.83% b)This portfolio does not have equal weight in each asset. We still need to find the return of the portfolio in each state of the economy. Boom: E(Rp) = 0.20(.07) + 0.20(0.15) + 0.60(0.33) = 0.2420 or 24.20% Bust: E(Rp) = 0.20(.03) + 0.20(0.03) + 0.60(-0.06) = -0.024 or – 2.4%
And the expected return of the portfolio is: (Rp) = 0.70(0.2420) + 0.30(-0.024) =0.1622 or 16.22% c)To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected (mean) return. σp2 = 0.70(0.2420 - 0.1622)2 + 0.30(-0.0240 - 0.1622)2 = 0.01486 σp = (0.01486)1/2 = 0.12189 or 12.189%
Problem 2 (15 points) Based on the following information, calculate the expected return and standard deviation of each of the following stock. Assume each state of the economy is equally likely to happen. What are the covariance and correlation between the returns of the two stocks?
State of Economy Bear Normal Bull
Rate of Return on stock A Rate of Return on stock B 6.3% -3.7% 10.5% 6.4% 15.6% 25.3%
a. (5 points) What is the expected return on stock A and stock B? b. (5 points) What is the variance and standard deviation for stock A and stock B? c. (5 points) What are the covariance and correlation between the returns of the two stocks? Solution: a) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is: E(R A) = .3333(.063) + .3333(.105) + .3333(.156) = .1080 or 10.80% E(R B) = .3333(–.037) + .3333(.064) + .3333(.253) = .0933 or 9.33%
b) To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, standard deviation of stock A is: 2 A
=.3333(.063 – .1080)^2 + .3333(.105 – .1080)^2 + .3333(.156 – .1080)^2 = . 001446 A = (.001446)^(1/2) = .0380 or 3.80%
And the standard deviation of stock B is: 2 B =.3333(–.037 – .0933)^2 + .3333(.064 – .0933)^2 + .3333(.253 – .0933)^2 = . 014446 B = (.01445)^(1/2) = .1202 or 12.02% c) To find the covariance, we multiply each possible state times the product of each assets’ deviation from the mean in that state. The sum of these products is the covariance. So, the covariance is: Cov(A,B) = .3333(.063 – .1080)(–.037 – .0933) + .3333(.105 – .1080)(.064 – .0933) + .3333(.156 – .1080)(.253 – .0933) Covariance = .004539 And the correlation coefficient is: A,B = Cov(A,B)/( A B) = .004539/(.0380)(.1202) A,B = .9937
Problem 3 (15 points) Based on the following information calculate the expected return and the standard deviation for the two stocks.
State of Economy Recession Normal Boom
Probability of State of Economy 0.10 0.60 0.30
Rate of Return on stock A 6% 7% 11%
Rate of Return on stock B -20% 13% 33%
a. (5 points) What is the expected return on stock A and stock B? b. (5 points) What is the variance and standard deviation for stock A and stock B? c. (5 points) What is of the standard deviation of an equally weighted portfolio of these two stocks if the correlation is 0.2? Solution: a) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock asset is: E(R A) = .10(.06) + .60(.07) + .30(.11) = .0810 or 8.10% E(R B) = .10(-.2) + .60(.13) + .30(.33) = .1570 or 15.70%
b) To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation of each stock are: 2 A
= .10(.06 - .0810)^2 + .60(.07 - .0810)^2 + .30(.11 - .0810)^2 = .000369 A = (.00037)^(1/2) = .01921 or 1.921% 2 B
= .10(.-2 - .1570)^2 + .60(.13 - .1570)^2 + .30(.33 - .1570)^2 = .022161 1/2 = .14886 or 14.89% B = (.02216) c) To calculate the portfolio standard deviation, we first need to calculate the variance. To find the variance we use the formula based on the individual stocks’ variances and the covariance between the two stocks. So, the portfolio variance is: = wA2σA2 + wB2σB2 + 2wAwB σAσBρAB 2 p = (0.5^2)(0.01923^2) + (0.5^2)(0.1489^2) + 2(0.5)(0.5)(0.2)( 0.0192)( 0.1489) 2 p = 0.0059183
σ p
2
P
= (0.0059183)^(1/2) = 0.076930, or 7.6930%
Problem 4 (15 points) Consider the possible rates of return that you might obtain over the next year. You can invest in stock U or stock V .
State of Economy Recession Normal Boom
Probability of State of Economy 0.20 0.50 0.30
Rate of Return on stock U 7.0% 7.0% 7.0%
Rate of Return on stock V -5.00% 10.0% 25.0%
a. (5 points) Determine the expected return, variance, and the standard deviation for stock U and V . b. (5 points) Determine the covariance and correlation between the returns of stock U and stock V . c. (5 points) Determine the expected return and standard deviation of an equally weighted portfolio of stock U and stock V. Solution: b) The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of each stock is: E(R U) = 0.2(7.0%) + 0.5(7.0%) + 0.3(7.0) = 7.0% E(R V) = 0.2(-0.05) + 0.5(0.10) + 0.3(0.25) = 11.5%
To calculate the standard deviation, we first need to calculate the variance. Variance U2 = 0 as the expected (mean return) of stock U is 7% Standard deviation U = 0 Variance V2 = [0.2(-0.05 – 0.115) + 0.5(0.10 – 0.115) + 0.3(0.25 – 0.115)] = 0.011025 Standard deviation V = 0.1049 or 10.49%
b) Since standard deviation of stock U = 0 then the covariance and the correlation between the returns of stock U and stock V are zero. c) To calculate the portfolio standard deviation, we first need to calculate the expected return and the variance. To find the variance we use the formula based on the individual stocks’ variances and the covariance between the two stocks. Portfolio expected return E(R P) = 0.5(0.07) + 0.5(0.115) = 9.25% Portfolio variance P2 = (0.5)^2(0.0)^2 + (0.5)^2(0.105)^2 + 2(0.5)(0.5)(0)(0.105) (0.0) = 0.002756 Portfolio standard deviation P = 0.0548 or 5.48%.
Problem 5 (10 points) Security A has an expected return of 8 percent with a standard deviation of 1.5 percent. Security B has an expected return of 12 percent with a standard deviation of 2.4 percent. The two securities have a correlation coefficient of 0.20. If you invest 40 percent of your funds in Security A and 60 percent in Security B, calculate the expected return and standard deviation of the portfolio. ( Note: change the calculator to 6 decimals) a) Portfolio expected return E(R P) = wAR A + wBR B = 0.40(8%) + 0.60(12%) = 10.40% = wA2 A2 + wB2 B2 + 2wFwG A B AB = = (0.4^2)(0.015^2) + (0.6^2)(0.024^2) + 2(0.4)(0.6)(0.2)(0.015)(0.024) = 0.000278 Standard deviation P = 0.016660 = 1.67%.
b) Portfolio variance
2 p
