Question

Suppose there are two independent economic factors, M1 and M2. The risk-free rate is 5%, and all stocks have independent firm-specific components with a standard deviation of 52%. Portfolios A and B are both well diversified. Portfolio Beta on M1 Beta on M2 Expected Return (%) A 1.6 2.5 31 B 2.4 -0.7 12 What is the expected return–beta relationship in this economy? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer 1

E(Rp) = Rf + β(1,p)[E(R1) - Rf] + β(2,p)[E(R2) - Rf]

We need to find the risk premium Rp for each of the two factors:

Rp1 = [E(R1) - Rf]

Rp2 = [E(R2) - Rf]

For Portfolio A;

31% = 5% + 1.6(Rp1) + 2.5(Rp2)

31% - 5% = 1.6(Rp1) + 2.5(Rp2)

Rp1 = [26% - 2.5(Rp2)] / 1.6    ==========> Eq.(1)

For Portfolio B;

12% = 5% + 2.4(Rp1) - 0.7(Rp2)

Put the value of Rp1 from Eq.(1)

12% = 5% + 2.4[{26% - 2.5(Rp2)} / 1.6] - 0.7(Rp2)

12% - 5% = 39% - 3.75(Rp2) - 0.7(Rp2)

7% = 39% - 4.45(Rp2)

4.45(Rp2) = 39% - 7%

Rp2 = 32% / 4.45 = 7.19%

Put the value of Rp2 in Eq.(1);

Rp1 = [26% - (2.5 * 7.19%)] / 1.6

= [26% - 17.98%] / 1.6 = 8.02% / 1.6 = 5.01%

Thus, the expected return-beta relationship is:

E(Rp) = 5% + β(1,p)[5.01%] + β(2,p)[7.19%]