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 43%. Portfolios A and B are both well-diversified with the following properties:

Portfolio			Beta on F1			Beta on F2			Expected Return
	A			1.9			2.2			33	%
	B			2.8			–0.22			28	%

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)

Rf=?

RP1=?

RP2=?

Answer 1

E(r_p) = r_F + [β_(1,p) * rp1] + [β_(2,p) * rp2]

E(rA) = r_F + [β_(1,p) * rp1] + [β_(2,p) * rp2]

33% = 6% + [1.9 * rp1] + [2.2 * rp2]

33% - 6% = [1.9 * rp1] + [2.2 * rp2]

[1.9 * rp1] + [2.2 * rp2] = 27%

rp1 = [27% - (2.2 * rp2)] / 1.9 =======> EQ. (1)

E(rB) = r_F + [β_(1,p) * rp1] + [β_(2,p) * rp2]

28% = 6% + [2.8 * rp1] + [-0.22 * rp2]

Replace rp1 value from eq. (1);

28% - 6% = [2.8 * {27% - (2.2 * rp2)}/1.9] + [-0.22 * rp2]

22% = 39.79% - (3.24 * rp2) - (0.22 * rp2)

3.46 * rp2 = 39.79% - 22%

rp2 = 17.79% / 3.46 = 5.14%

Now, put the value of rp2 in EQ. (1)

rp1 = [27% - (2.2 * 5.14%)] / 1.9

= [27% - 11.30%] / 1.9 = 15.70% / 1.9 = 8.26%

Thus, the expected return-beta relationship is:

E(r_p) = 6% + [β_(1,p) * 8.26%] + [β_(2,p) * 5.14%]