4. Consider the random variables X and Y where X represents annual return on Tre

Question

4. Consider the random variables X and Y where X represents annual return on Treasury bills and Y represents annual return on stocks. Suppose (based on historical data) that the expectations, standard de- viations and correlation between these random variables are given by the numbers Ax = 5,#x = 4,Pr = 13.2,Oy = 17.6, =-.3 Suppose you decide to invest 40% of your money in treasury bills and 60% in stocks namely consider the portfolio R-AX + .6Y. What is the standard deviation of this random variable R?

Explanation / Answer

Var[aX + bY] = a2 Var[X] + b2 Var[Y] + 2abCov[X, Y]

Here X = 4

Y = 17.6

= -0.3

R = 0.4X + 0.6Y

Var( 0.4X + 0.6Y ) = 0.42 Var[X] + 0.62 Var[Y] + 2 * 0.4 * 0.6 Cov[X, Y]

Var[X] = 42 = 16

VaR(Y) = 17.62 = 309.76

Cov[X,Y] = Corr(X,Y) *x * Y = -0.3 * 4 * 17.6 = -21.12

Var( 0.4X + 0.6Y ) = 0.42 Var[X] + 0.62 Var[Y] + 2 * 0.4 * 0.6 Cov[X, Y]

Var(0.4X + 0.6Y ) = 0.16 * 16 + 0.36 * 309.76 + 0.48 * (-21.12) = 104.456

Standard deviation of R = Sqrt [ Variance] = sqrt [104.456] = 10.22

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