Question

1. You are faced with the following probability distribution: Economy Recession Below Average Average Above Average Boom Probability 0.1 0.2 0.4 0.2 -30% -13% 8% 23% 45% B 15% -5% -10% 35% 18% 0.1 Assuming that you invest 40% of your money in A and 60% of your money in B, compute the followings (please show workings). a. Expected return for the portfolio b. Standard deviation for the portfolio c. Evaluate the risk reduction effect of the portfolio

Answer 1

1) - a) A(%) (P) * A B(%) (P) *B ET Economy Recession Below average Average Above average Boom Probability (P) 0.10 0.20 0.40 0.20 0.10 -2.6 4.5 Sum 6.7 Expected return A = 6.70% Expected return B = 5.3% Expected return from the portfolio = (0.4*6.7 +5.3* 0.6) 5.86%

To calculate the standard deviation of the portfolio we need to first calculate the standard deviation of individual secuirites and covariance among them (P) *A (P)* A^2 B (%) (P) *B A(%) -30 -13 Economy Recession Below average Average Above average Boom Probability (P) 0.10 0.20 0.40 0.20 0.10 -2.6 90.00 33.80 25.60 105.80 202.50 (P) *B^2 22.50 5.00 40.00 245.00 32.40 4.6 45 4.5 Sum 6.7 457.70 344.90 Standard deviation A = (457.7-6.742)^(1/2) 20.32 Standard deviation B = (344.9-5.3^2)^(1/2) 17.80 To calculate the covariance Economy Recession Below average Average Above average Boom Probability (P) 0.10 0.20 0.40 0.20 0.10 A(%) -0.3 -0.13 0.08 0.23 0.45 B(%) 0.15 -0.05 -0.1 0.35 0.18 probabilty*(Return A - 0.067)*(Return B -0.053) -0.0035599 0.0040582 -0.0007956 0.0096822 0.0048641 6.7 5.3 0.014249 Standard deviation of portfolio formula = ( weight A^2 * SD A^2 + Weight B^2 * SD B^2 +2* Weight A* Weight B * Covariance)^(1/2) (0.4^2 *0.2032^2 +0.6^2 * 0.178^2 + 2*0.6*0.4*0.014249)^(1/2) 0.1576 15.76% The portfolio overall risk has been reduced slightly although it seems there is positive correlation between the securities and the risk of the portfolio is less than the individual risk of the securities, but had the correlation been negative the portfolio risk wouldd have been reduced more.