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 Exp.Return (%)

A 1.6 2.5 31

B. 2.4. -0.7. 12

What is the expected return–beta relationship in this economy?

Expected return–beta relationship E(rP) =

5.00  % + ........ βP1 + ........βP2

*The answers are not 5.014 and 7.191

Answer 1

Re = Rf + [β1 x RP1] + [β2 x RP2]

We have to find the two risk premiums.

Substituting the known numbers for portfolio A in the above expression, we get:

31% = 5% + [1.6 x RP1] + [2.5 x RP2]

26% = [1.6 x RP1] + [2.5 x RP2]

RP2 = [26% - (1.6 x RP1)] / 2.5 ==========> Eq.(1)

Now, we can substitute this in the equation for portfolio B:

12% = 5% + [2.4 x RP1] - [0.7 x RP2]

7% = [2.4 x RP1] - [0.7 x {26% - (1.6 x RP1) / 2.5}]

7% = [2.4 x RP1] - 7.28% + [0.448 x RP1]

14.28% = [2.848 x RP1]

RP1 = 14.28%/2.848 = 5.01%

Therefore,

RP2 = [26% - (1.6 x RP1)] / 2.5

= [26% - (1.6 x 5.01%)] / 2.5 = 17.98% / 2.5 = 7.19%

The expected return-beta relationship in this economy is:

Re = 5% + [β1 x 5.01%] + [β2 x 7.19%]