Question

Please show the steps how the risk premiums are calculated for y1=4.22% and y2=10.9%

Suppose there are two independent economic factors, M₁ and M₂. The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 50%. Portfolios A and B are both well diversified.

Portfolio	Beta on M1	Beta on M2	Expected Return (%)
A	1.6	2.5	40
B	2.4	-0.7	10

What is the expected return–beta relationship in this economy? (Do not round intermediate calculations. Round your answers to 2 decimal places.)

rev: 04_04_2019_QC_CS-164824

Explanation

E(r_P) = r_f + β_P1[E(r₁) – r_f] + βP₂[E(r₂) – r_f]

We need to find the risk premium for these two factors:
γ2γ2 = [E(r₁) – r_f] and
γ2γ2 = [E(r₂) – r_f]

To find these values, we solve the following two equations with two unknowns:

40% = 6% + 1.6 γ1γ1 + 2.5 γ2γ2

10% = 6% + 2.4 γ1γ1 + (–0.7) γ2γ2

The solutions are: γ2γ2 = 4.22% and γ2γ2 = 10.90%

Thus, the expected return-beta relationship is:

E(r_P) = 6% + 4.22β_P1 + 10.90β_P2

Answer 1

The the expected return–beta relationship in this economy is given by the following equation:

E(r_P) = r_f + β_P1[E(r₁) – r_f] + βP₂[E(r₂) – r_f]

Let's assume

y₁ = [E(r₁) – r_f] and
y₂ = [E(r₂) – r_f]

Hence, equation changes to:

E(r_P) = r_f + β_P1 x y₁ + β_P2 x y₂

The risk-free rate is 6%. This mean, r_f = 6%

Hence, equation becomes:

E(r_P) = 6% + β_P1 x y₁ + β_P2 x y₂

Consider Portfolio A: β_P1 = 1.6 and β_P2 = 2.5; E(r_P) = 40%

We get: 40% = 6% + 1.6y₁ + 2.5y₂ Or, 34% = 1.6y₁ + 2.5y₂ --------------------Equation (1)

Consider Portfolio B: β_P1 = 2.4 and β_P2 = -0.7; E(r_P) = 10%

We get: 10% = 6% + 2.4y₁ - 0.7y₂ Or, 4% = 2.4y₁ - 0.7y₂ --------------------Equation (2)

We have to solve for two variables y₁ and y₂ based on these two equations.

0.7 x Equation (1) + 2.5 x Equation (2) gives us:

0.7 x 34% + 2.5 x 6% = (0.7 x 1.6 + 2.5 x 2.4)y₁ + (0.7 x 2.5 - 2.5 x 0.7)y₂

Or, 33.80% = 7.12y₁

Hence, y₁ = 33.80% / 7.12 = 4.75%

Hence, y₂ based on equation (1) = (34% - 1.6y₁) / 2.5 = (34% - 1.6 x 4.75%) / 2.5 = 10.56%

This is the method to solve it. The answers given in the question are slightly different, may be because of some rounding off somewhere. But the method is as illustrated above.

Hope this helps you understand the mathematics.