Question

A portfolio that combines the risk-free asset and the market portfolio has an expected return of 6.7 percent and a standard deviation of 9.7 percent. The risk-free rate is 3.7 percent, and the expected return on the market portfolio is 11.7 percent. Assume the capital asset pricing model holds.

What expected rate of return would a security earn if it had a .42 correlation with the market portfolio and a standard deviation of 54.7 percent? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Expected rate of return             %

Answer 1

Risk-free asset

Return on risk-free asset = R_F = 3.7%

The standard deviation of risk-free asset = σ_F = 0 (risk-free assets have 0 standard deviation)

weight of risk-free asset in the overall portfolio = w_F

Market portfolio

Return on market portfolio = R_M = 11.7%

The standard deviation on market portfolio = σ_M

weight of market portfolio in the overall portfolio = w_M

We need to first calculate the standard deviation of the market

Overall portfolio

The overall portfolio consists of the market portfolio and risk-free asset

Return on overall portfolio = R_P = 6.7%

The standard deviation of the overall portfolio = σ_P = 9.7%

weight of risk-free asset in the overall portfolio = w_F, weight of market portfolio in the overall portfolio = w_M

Return on overall portfolio is calculated using the formula

R_P = w_F*R_F + w_M*R_M

w_F+w_M = 1

w_F = 1 - w_M

6.7% = (1-w_M)*3.7% + w_M*11.7%

6.7% = 3.7% - w_M*3.7% + w_M*11.7%

3% = w_M*8%

w_M = 3%/8% = 0.375

Variance of the portfolio is calculated using the formula:

σ_P² = w_F²*σ_F² + w_M²*σ_M² + 2*ρ*w_F*w_M*σ_F*σ_M

Since σ_F = 0

σ_P² = 0 + w_M²*σ_M² + 0 = w_M²*σ_M²

Standard deviation of the portfolio is square-root of its variance

σ_P = (w_M²*σ_M²)^1/2 = w_M*σ_M

σ_P = 9.7%, w_M = 0.375

σ_M = σ_P/w_M = 9.7%/0.375 = 25.8666666666667%

Security

correlation between the security and the market portfolio = ρ = 0.42

Standard deviation of the security = σ_S = 54.7%

We need to first calculate the beta of the security using CAPM

β_S = (ρ*σ_S)/σ_M = (0.42*54.7%)/25.8666666666667% = 0.888170103092784

Now, to calculated the expected return on the security we will use CAPM Equation:

Expected return on Security = E[R_S] = R_F+β_S*(R_M-R_F) = 3.7%+0.888170103092784*(11.7% - 3.7%) = 10.8053608247423%

Answer -> Expected rate of return = 10.81%