Practice Question
Security A has an expected return of 12% and a standard deviation of 14%. Security B has an expected return of 10% and a standard deviation of 8%. The correlation between A and B is 1.00. If you invest 20% in A and the rest in B, what is the standard deviation of your portfolio?
Problem: Security A has an expected return of 12% and a standard deviation of 14%. Security B has an expected return of 10% and a standard deviation of 8%. The correlation between A and B is 1.00. If you invest 20% in A and the rest in B, what is the standard deviation of your portfolio?
Step 1: Define the variables
- \(E(R_A) = 12\%\)
- \(\sigma_A = 14\% = 0.14\)
- \(E(R_B) = 10\%\)
- \(\sigma_B = 8\% = 0.08\)
- \(\rho_{AB} = 1.00\)
- \(w_A = 0.20\), \(w_B = 0.80\)
Step 2: Calculate the covariance between A and B
\[ \text{Cov}(A, B) = \rho_{AB} \cdot \sigma_A \cdot \sigma_B = 1.00 \cdot 0.14 \cdot 0.08 = 0.0112 \]
Step 3: Calculate the portfolio variance
\[ \sigma_p^2 = w_A^2 \cdot \sigma_A^2 + w_B^2 \cdot \sigma_B^2 + 2 \cdot w_A \cdot w_B \cdot \text{Cov}(A, B) \] \[ \sigma_p^2 = (0.20^2 \cdot 0.14^2) + (0.80^2 \cdot 0.08^2) + (2 \cdot 0.20 \cdot 0.80 \cdot 0.0112) \] \[ = (0.04 \cdot 0.0196) + (0.64 \cdot 0.0064) + (0.32 \cdot 0.0112) \] \[ = 0.000784 + 0.004096 + 0.003584 = 0.008464 \]
Step 4: Calculate the standard deviation
\[ \sigma_p = \sqrt{0.008464} \approx 0.092 \]
Final Answer
The standard deviation of the portfolio is 9.2%.