Consider the following information about two stocks where the probability of an

Question

Consider the following information about two stocks where the probability of an economic
boom is 35%.
Economic State Return A (RA) Return B (RB)
Boom 40% 8%
Recession –7% 15%
a. Calculate the expected return for stock A and stock B (not the portfolio).
b. Calculate the standard deviation of stock A and stock B (not the portfolio).
c. Calculate the correlation between stock A and stock B.

8. Consider the following information about two stocks where the probability of an economic boom is 35%. Economic State Boom Return B (Rs) Return A (RA) a. Calculate the expected return for stock A and stock B [not the portfoliol b. Calculate the standard deviation of stock A and stock B (not the portfolio). c. Calculate the correlation between stock A and stock B. d. Calculate the total risk (standard deviation) of a portfolio, where 14 of your money is invested in stock A and 3.4 of your money is invested in stock B. (Hint: use both the method with the formula for the risk of a portfolio i.e., using the covariance) and the method of calculating the variance (and standard deviation) fram the portfolio retuns.

Explanation / Answer

a) Expected return of Stock A = 40*0.35+ -(7*0.65) = 9.45% Expected return of Stock B = 8*0.35+15*0.65 = 12.55% b) Standard deviation of Stock A: Economic state Probability [p] Return [r] E[r]= p*r r-E[r]=d d^2 p*d^2 Boom 0.35 40 14.00 30.55 933.3025 326.655875 Recession 0.65 -7 -4.55 -16.45 270.6025 175.891625 9.45 502.547500 Standard deviation = [p*d^2]^0.5 = 502.5475^0.5 = 22.42% Standard deviation of Stock B: Economic state Probability [p] Return [r] E[r]= p*r r-E[r]=d d^2 p*d^2 Boom 0.35 8 2.80 -4.55 20.7025 7.245875 Recession 0.65 15 9.75 2.45 6.0025 3.901625 12.55 11.147500 Standard deviation = [p*d^2]^0.5 = 11.1475^0.5 = 3.34% c) Economic state Probability [p] Deviations from mean for Stock A Deviations from mean for Stock B Product of deviations p*product of deviations Boom 0.35 30.55 -4.55 -139.0025 -48.650875 Recession 0.65 -16.45 2.45 -40.3025 -26.196625 Co-variance = -74.8475 -74.8475 Correlation = Covariance/(SD of A*SD of B) = -74.8475/(22.42*3.34)= -1.00 Answer-Perfect negative correlation d) SD of portfolio using equation = (0.25^2*22.42^2+0.75^2*3.34^2+2*0.25*0.75*22.42*3.34*- 1)^0.5 = 3.10% Answer Formula for Portfolio SD of two assets = [Wa^2*Sda^2+Wb^2*SDb^2+2*Waq*Wb*Sda*SDb*Cor(a,b)]^0.5 Where Wa and Wb are the weights of the two assets. SDa,SDb their standard deviations and Cor(a,b), their correlation. SD of portfolio using portfolio returns: Economic state Probability [p] Return [r] E[r]= p*r r-E[r]=d d^2 p*d^2 Boom 0.35 16 5.60 4.22 17.8084 6.232940 Recession 0.65 9.5 6.18 -2.28 5.1984 3.378960 11.78 9.611900 Standard deviation = [p*d^2]^0.5 = 9.6119^0.5 = 3.10%

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