Question

We know the following expected returns for stock A and the market portfolio, given different states of the economy:

State (s)	Probability	E(rA,s)	E(rM,s)
Recession	0.3	-0.04	0.04
Normal	0.5	0.11	0.07
Expansion	0.2	0.19	0.11

The risk-free rate is 0.02.

Part 1

Assuming the CAPM holds, what is the beta for stock A?

Answer 1

State (s)	Probability	E(rA,s)	E(rM,s)
Recession	0.3	-0.04	0.04
Normal	0.5	0.11	0.07
Expansion	0.2	0.19	0.11

Expected Return (when probabilities of different states of economies are given) is calculated using the formula:

E[R] = p₁*R₁ + p₂*R₂ + p₃*R₃

Expected return of stock A = E[R_A] = 0.3*(-0.04) + 0.5*0.11 + 0.2*0.19 = 8.1%

Expected return on Market = E[R_M] = 0.3*0.04 + 0.5*0.07 + 0.2*0.11 = 6.9%

Risk-free rate = 0.02 = 2%

Beta of stock A = β_A

According to CAPM, expected return on stock A is give by:

E[R_A] = R_F + β_A*(E[R_M] - R_F)

8.1% = 2% + β_A*(6.9% - 2%)

8.1% - 2% = β_A*4.9%

β_A = 6.1%/4.9% = 1.2448980

Beta of stock A = 1.2448980