Question

Suppose that there are two independent economic factors, F₁ and F₂. The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 36%. Portfolios A and B are both well-diversified with the following properties:

Portfolio			Beta on F1			Beta on F2			Expected Return
	A			1.2			1.6			26	%
	B			2.1			–0.16			23	%

What is the expected return-beta relationship in this economy? Calculate the risk-free rate, r_f, and the factor risk premiums, RP₁ and RP₂, to complete the equation below. (Do not round intermediate calculations. Round your answers to two decimal places.)

E(r_P) = r_f + (β_P1 × RP₁) + (β_P2 × RP₂)

Answer 1

The expected return beta relationship is computed as shown below:

The return beta relationship is computed as shown below:

Expected return on A = risk free rate + Beta on M1 x risk premium 1 + Beta on M2 x risk premium 2

Expected return on B = risk free rate + Beta on M1 x risk premium 1 + Beta on M2 x risk premium 2

26 = 6 + 1.2 x risk premium 1 + 1.6 x risk premium 2 (Equation 1)

23 = 6 + 2.1 x risk premium 1 - 0.16 x risk premium 2 (Equation 2)

Multiply 1st equation by 7 and 2^nd equation by 4. We shall get

182 = 42 + 8.4 risk premium 1 + 11.2 risk premium 2

92 = 24 + 8.4 risk premium 1 - 0.64 risk premium 2

Now we shall subtract equation 2 from equation 1 and shall get:

90 = 18 + 11.84 risk premium 2

72 / 11.84 = risk premium 2

6.081081081 or 6.08 Approximately = risk premium 2

Plugging risk premium 2 in equation 1 and solve:

26 = 6 + 1.2 risk premium 1 + 1.6 x 6.081081081

16.27027027 = 6 + 1.2 risk premium 1

10.27027027 = 1.2 risk premium 1

8.56 Approximately = risk premium 1

So, the relationship will be:

6.00% + 8.56% BP1 + 6.08% BP2.

Feel free to ask in case of any query relating to this question