Question

Assume that the % expected return for security A and the market M for a good, normal and bad economy (probabilities .3,.4,.3) are 20, 16, and 10 for A and 8, 4, and 12 for M. Also assume that you invest 40% in A and 60% in M. Compute the standard deviation for a portfolio of A and M.

\

1.69

3.90

3.32

3.55

Answer 1

State of economy	Probability	Return on security A	Return on market M
Good	0.3	20%	8%
Normal	0.4	16%	4%
Bad	0.3	10%	12%

Expected Return of A and M

Expected return is calculated using the formula:

Expected return = E[R] = p₁*R₁ + p₂*R₂ + p₃*R₃

Expected return on security A = E[R_A] = 0.3*20% + 0.4*16% + 0.3*10% = 15.4%

Expected return on M = E[R_M] = 0.3*8% + 0.4*4% + 0.3*12% = 7.6%

Variance is calculated using the formula:

Varaince = σ² = p₁*(R₁-E[R])² + p₂*(R₂-E[R])² + p₃*(R₃-E[R])²

Standard deviation is calculated as the square-root of variance

Standard deviation of A

Variance of A = σ_A² = 0.3*(20% - 15.4%)² + 0.4*(16% - 15.4%)² + 0.3*(10% - 15.4%)² = 0.0006348 + 0.0000144 + 0.0008748 = 0.001524

Standard deviation od A is square-root of variance of A

Standard deviation of A = σ_A = (0.001524)^1/2 = 0.0390384425918863

Standard Deviation of M

Variance of M = σ_M² = 0.3*(8% - 7.6%)² + 0.4*(4% - 7.6%)² + 0.3*(12% - 7.6%)² = 0.0000048 + 0.0005184 + 0.0005808 = 0.001104

Standard deviation of  M is square-root of variance of M

Standard deviation of M = σ_M = (0.001104)^1/2 = 0.0332264954516723

Covariance of A and M

Covariance between the return of A and M is calculated using the formula:

Cov(A,M) = p₁*(R_1,A - E[R_A])*(R_1,M - E[R_M]) + p₂*(R_2,A - E[R_A])*(R_2,M - E[R_M]) + p₃*(R_3,A - E[R_A])*(R_3,M - E[R_M]) = 0.3*(20% - 15.4%)*(8%-7.6%) + 0.4*(16% - 15.4%)*(4%-7.6%) + 0.3*(10% - 15.4%)*(12%-7.6%) = 0.0000552 + (-0.0000864) + (-0.0007128) = -0.000744

Portfolio

Weight of A in the portfolio = W_A = 40%

Weight of M in the portfolio = W_M = 60%

Standard deviation of A = σ_A = 0.0390384425918863

Standard deviation of M = σ_M = 0.0332264954516723

Cov(A,M) = -0.000744

Variance of portfolio = σ_P² = W_A^2*σ_A² + W_M²*σ_A²+ 2*Cov(A,M)*W_A*W_M = (40%)²*(0.0390384425918863)² + (60%)²*(0.0332264954516723)² + 2*(-0.000744)*40%*60% = 0.00024384 + 0.00039744 + (-0.00035712) = 0.00028416

Standard deviation of portfolio is square-root of variance

Standard deviation of portfolio =  σ_P = 0.0168570460045644 = 1.68570460045644% ~ 1.69%

Answer -> 1.69