Practice Question
The expected return of Security A is 12% with a standard deviation of 16%. The expected return of Security B is 7% with a standard deviation of 18%. Securities A and B have a correlation of 0.7. The market return is 15% with a standard deviation of 15% and the risk-free rate is 2%. What is the Sharpe ratio of a portfolio if 45% of the portfolio is in Security A and the remainder in Security B?
To compute the Sharpe ratio of the portfolio consisting of Securities A and B, we will follow these steps:
Step 1: Calculate the Expected Return of the Portfolio
\[ E(R_p) = w_A \cdot E(R_A) + w_B \cdot E(R_B) = 0.45 \cdot 0.12 + 0.55 \cdot 0.07 = 0.0925 \text{ or } 9.25\% \]
Step 2: Calculate the Standard Deviation of the Portfolio
\[
\sigma_p = \sqrt{(0.45 \cdot 0.16)^2 + (0.55 \cdot 0.18)^2 + 2 \cdot 0.45 \cdot 0.55 \cdot 0.16 \cdot 0.18 \cdot 0.7}
\]
Calculations:
- \( (0.45 \cdot 0.16)^2 = 0.005184 \)
- \( (0.55 \cdot 0.18)^2 = 0.009801 \)
- \( 2 \cdot 0.45 \cdot 0.55 \cdot 0.16 \cdot 0.18 \cdot 0.7 = 0.017136 \)
\[
\sigma_p = \sqrt{0.005184 + 0.009801 + 0.017136} \approx \sqrt{0.032121} \approx 0.179 \text{ or } 17.9\%
\]
Step 3: Calculate the Sharpe Ratio
\[ S = \frac{E(R_p) - R_f}{\sigma_p} = \frac{0.0925 - 0.02}{0.179} \approx \frac{0.0725}{0.179} \approx 0.4045 \]
Final Answer: The Sharpe ratio of the portfolio is approximately 0.405.