Question

Consider the following information

Invest 40% of your money in Asset A

State   Probability   A          B  

Boom   .4       30%       -5%

Bust   .6       -10%       25%

What is the expected return and standard deviation for each asset?

What is the expected return and standard deviation for the portfolio?

Answer 1

Expected return of A = Prob1 * return (r1) + Prob2 * return (r2)

= 0.40 * 30% + 0.60 * -10%

= 6 %

Expected return of B = Prob1 * return (r1) + Prob2 * return (r2)

= 0.40 * -5 % + 0.60 * 25%

= 13 %

Standard devation of Asset A = [(30 - 6)² *0.4 + ( -10 -6)² * 0.6]^1/2

⁼ (230.4 + 153.6)^1/2

= 19.596

Standard devation of Asset B = [(- 5 - 13)² *0.4 + ( 25 -13)² * 0.6]^1/2

⁼ (216)^1/2

= 14.697

Expected return of portfolio = ProbA * return (rA) + ProbB * return (rB)

= 6% * 0.40 + 13% * 0.60

= 10.20%

covariance of A and B =  [(30 - 6)* (- 5 - 13) *0.4 + ( -10 -6)* ( 25 -13) *0.6]

= 288

Portfolio variance = w²_A*?²(R_A) + w²_B*?²(R_B) + 2*(w_A)*(w_B)*Cov(R_A, R_B)

= 0.40² * 384 + 0.60² * 216 + 2 * 0.4 * 0.6 * 288

= 61.44 + 77.76 + 138.24

= 277.44

Standard deviation = 16.66