PORTFOLIO THEORY – EXERCISES 2
EXERCISE 1
Suppose that the conditions for the CAPM are fully respected. The market portfolio expected return is 5% and
the risk-free rate is 1%. We create a portfolio in which 80% of the wealth is placed in the security A whose beta
is 𝛽𝐴 = 0.5, 50% in the security B whose beta is 𝛽 𝐵 = 1.5, and we short the risk-free asset. What is the expected
return of the portfolio?
As the weights for the risky assets sum to 0.8 + 0.5 = 1.3, it means that the weight for the risk-free asset is
−0.3. Since the CAPM holds, the expected return of A:
𝐸[𝑅 𝐴] = 𝑅𝑓 + 𝛽𝐴 ∗ (𝐸[𝑅 𝑀] − 𝑅𝑓) = 0.01 + 0.5 ∗ (0.05 − 0.01) = 0.01 + 0.02 = 0.03
The expected return of B is:
𝐸[𝑅 𝐵] = 𝑅𝑓 + 𝛽 𝐵 ∗ (𝐸[𝑅 𝑀] − 𝑅𝑓) = 0.01 + 1.5 ∗ (0.05 − 0.01) = 0.01 + 0.06 = 0.07
Hence, the expected return of the portfolio is:
𝐸[𝑅 𝑃] = 0.8 ∗ 0.03 + 0.5 ∗ 0.07 − 0.3 ∗ 0.01 = 0.056 = 5.6%
EXERCISE 2
The market portfolio has an expected return of 5%, and the risk-free rate is 1%. Suppose that the conditions for
the CAPM are fully respected. The expected return of a portfolio in which 40% of the wealth if placed in stocks S
and 60% in an ETF that replicates the market portfolio with zero tracking error is 7%. What is the beta of the
stocks S?
First, we determine the beta of the portfolio:
𝐸[𝑅 𝑃] = 𝑅𝑓 + 𝛽 𝑃 ∗ (𝐸[𝑅 𝑀] − 𝑅𝑓)
0.07 = 0.01 + 𝛽 𝑃 ∗ (0.05 − 0.01)
0.06 = 0.04𝛽 𝑃
𝛽 𝑃 =
0.06
0.04
= 1.5
The beta of a portfolio is equal to the weighted average of the beta of its components. The beta of the ETF is
equal to 1 (because it replicates the market portfolio). Therefore, the beta of the stocks S is:
1.5 = 0.4 ∗ 𝛽𝑆 + 0.6 ∗ 1
0.9 = 0.4 ∗ 𝛽𝑆
𝛽𝑆 =
0.9
0.4
= 2.25