Question

New Mode Delay The returns of market portfolio and stock A under three equally likely scenarios are given in the table. Calculate the beta for stock A. Scenario Market Stock A Bust 5% 5 % Normal 15 % 15 Boom 25 % 40% 1.65 1.75 1.85 1.77 UU

Answer 1

Scenario	Probability	Market	Stock A
Bust	1/3	5%	5%
Normal	1/3	15%	15%
Boom	1/3	25%	40%

Beta of stock A is given by the formula:

Beta of stock A = βA = Cov(M, A)/σM2

First we will calculate the Variance of Market (σM2) and Covaraince between Market and stock A, Cov(M,A)

Equally likely scenarios mean that the probability of each scenario is 1/3

Probability of Bust = p1 = 1/3

Probability of Normal = p2 = 1/3

Probability of Boom = p3 = 1/3

R1,M = 5%, R2,M = 15%, R3,M = 25%

R1,A = 5%, R2,A = 15%, R3,A = 40%

Variance of Market calculation

Expected return of Market = E[RM] = p1*R1,M + p2*R2,M + p3*R3,M​​​​​​​ = (1/3)*5% + (1/3)*15% + (1/3)*25% = 15%

Expected return of Stock A = E[RM] = p1*R1,A + p2*R2,A​​​​​​​ + p3*R3,A​​​​​​​ = (1/3)*5% + (1/3)*15% + (1/3)*40% = 20%

Variance of market = σM2 = p1*(R1,M-E[R])2 + p2*(R2,M-E[R])2​​​​​​​ + p3*(R3,M-E[R])2​​​​​​​ = (1/3)*(5% - 15%)2 + (1/3)*(5% - 15%)2 + (1/3)*(5% - 15%)2 = 0.00333333333333333 + 0 + 0.00333333333333333 = 0.00666666666666667 = 1/150

Covariance Calculation

Covariance between the Market and stock A = p1*(R1,M-E[R])*(R1,A - E[RA]) + p2*(R2,M-E[R])*(R2,A - E[RA]) + p3*(R3,M-E[R])*(R3,A - E[RA]) = (1/3)*(5% - 15%))*(5%-20%) + (1/3)*(15% - 15%))*(15%-20%) + (1/3)*(25% - 15%))*(40%-20%) = 0.005 + 0 + 0.00666666666666667 = 0.0116666666666667 = 7/600

Covariance between the market and the stock A = Cov(M, A) = 7/600

Variance of Market = σM2 = 1/150

Beta of stock A is calculated using the formula:

Beta of stock A = βA = Cov(M, A)/σM2 = (7/600)/(1/150) = 1.75

Answer -> Beta of stock A = 1.75