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(rP) = rf +
βP1[E(r1)
– rf] +
βP2[E(r2) –
rf]
We need to find the risk premium for these two factors:
γ2γ2 = [E(r1) –
rf] and
γ2γ2 = [E(r2) –
rf]
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(rP) = 6% + 4.22βP1 + 10.90βP2
The the expected return–beta relationship in this economy is given by the following equation:
E(rP) = rf + βP1[E(r1) – rf] + βP2[E(r2) – rf]
Let's assume
y1 = [E(r1) –
rf] and
y2 = [E(r2) –
rf]
Hence, equation changes to:
E(rP) = rf + βP1 x y1 + βP2 x y2
The risk-free rate is 6%. This mean, rf = 6%
Hence, equation becomes:
E(rP) = 6% + βP1 x y1 + βP2 x y2
Consider Portfolio A: βP1 = 1.6 and βP2 = 2.5; E(rP) = 40%
We get: 40% = 6% + 1.6y1 + 2.5y2 Or, 34% = 1.6y1 + 2.5y2 --------------------Equation (1)
Consider Portfolio B: βP1 = 2.4 and βP2 = -0.7; E(rP) = 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.
Please show the steps how the risk premiums are calculated for y1=4.22% and y2=10.9% Suppose there are two independent e...
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