Financial Risk Manager Part 1
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Tina Fer, a portfolio manager, created the following portfolio:
| Security | Security Weight (%) | Expected Standard deviation(%) |
|---|---|---|
| A | 10 | 6 |
| B | 90 | 15 |
If the standard deviation of the portfolio is 14.1%, then what is the covariance between the two securities?
Explanation:
Explanation
To find the covariance between securities A and B, we use the portfolio variance formula:
Portfolio Variance Formula: [ \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2w_A w_B \text{Cov}(A,B) ]
Given:
- Weight of A (w_A) = 10% = 0.10
- Weight of B (w_B) = 90% = 0.90
- Standard deviation of A (σ_A) = 6% = 0.06
- Standard deviation of B (σ_B) = 15% = 0.15
- Portfolio standard deviation (σ_p) = 14.1% = 0.141
Step 1: Calculate portfolio variance [ \sigma_p^2 = (0.141)^2 = 0.019881 ]
Step 2: Calculate individual variance components [ w_A^2 \sigma_A^2 = (0.10)^2 \times (0.06)^2 = 0.01 \times 0.0036 = 0.000036 ] [ w_B^2 \sigma_B^2 = (0.90)^2 \times (0.15)^2 = 0.81 \times 0.0225 = 0.018225 ]
Step 3: Set up the equation [ 0.019881 = 0.000036 + 0.018225 + 2 \times 0.10 \times 0.90 \times \text{Cov}(A,B) ] [ 0.019881 = 0.018261 + 0.18 \times \text{Cov}(A,B) ]
Step 4: Solve for covariance [ 0.019881 - 0.018261 = 0.18 \times \text{Cov}(A,B) ] [ 0.00162 = 0.18 \times \text{Cov}(A,B) ] [ \text{Cov}(A,B) = \frac{0.00162}{0.18} = 0.009 ]
Therefore, the covariance between the two securities is 0.009._