Practice Question
Amr has $50,000 to invest and has decided to invest $10,000 in the stock of ABC and the rest in YUL. The standard deviation of the returns of ABC is 8% while the standard deviation for YUL is 14%. He is happy to see that the correlation between the two stocks is negative and is -0.15. The standard deviation of the portfolio is closest to:
To determine the standard deviation of Amr's portfolio, we use the two-asset portfolio risk formula:
\[ \sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho_{A,B}} \]
Where:
- \( w_A = \frac{10,000}{50,000} = 0.2 \)
- \( w_B = \frac{40,000}{50,000} = 0.8 \)
- \( \sigma_A = 0.08 \)
- \( \sigma_B = 0.14 \)
- \( \rho_{A,B} = -0.15 \)
Now compute each component:
- \( w_A^2 \sigma_A^2 = (0.2)^2 \times (0.08)^2 = 0.000256 \)
- \( w_B^2 \sigma_B^2 = (0.8)^2 \times (0.14)^2 = 0.012544 \)
- \( 2 w_A w_B \sigma_A \sigma_B \rho = 2 \times 0.2 \times 0.8 \times 0.08 \times 0.14 \times (-0.15) = -0.0005376 \)
Now sum the components and take the square root:
\[ \sigma_p^2 = 0.000256 + 0.012544 - 0.0005376 = 0.0122624 \] \[ \sigma_p = \sqrt{0.0122624} \approx 0.1107 \text{ or } 11.07\% \]
Conclusion:
The standard deviation of the portfolio is approximately 11.07%. The negative correlation helped reduce overall risk.