Question

Suppose that there are two independent economic factors, F₁ and F₂. The risk-free rate is 10%, and all stocks have independent firm-specific components with a standard deviation of 40%. Portfolios A and B are both well-diversified with the following properties:

Portfolio			Beta on F1			Beta on F2			Expected Return
	A			1.6			2.0			30	%
	B			2.5			–0.20			25	%

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

Re = Rf + [β1*RP1] + [β2*RP2]

We have to find the two risk premiums.

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

30% = 10% + [1.6*RP1] + [2*RP2]

RP2 = [20% - 1.6*RP1]/2

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

25% = 10% + [2.5*RP1] - [0.2*RP2]

25% = 10% + [2.5*RP1] - [0.2*{(20% - 1.6*RP1)/2}]

15% = 2.5*RP1 - 2% + 0.16*RP1

17% = 2.66*RP1

RP1 = 17% / 2.66 = 6.39%

Therefore,

RP2 = (20% - 1.6*RP1)/2

= [20% - 1.6*6.39%]/2 = 9.77%/2 = 4.89%

The expected return-beta relationship in this economy is:

Re = Rf + [β1*RP1] + [β2*RP2]

Re = 10% + [β1*6.39%] + [β2*4.89%]