Answer: 0.0113

## Detailed Explanation When the population mean is unknown, we use **sample covariance** which divides by (n-1) instead of n. ### Step 1: Calculate Sample Means **Stock A Mean (Ȳ_A):** $$\bar{X}_A = \frac{0.17 + 0.21 - 0.08 - 0.01 + 0.04 + 0.19 - 0.07}{7} = 0.0642$$ **Stock B Mean (Ȳ_B):** $$\bar{X}_B = \frac{0.45 + 0.20 - 0.02 + 0.02 - 0.19 + 0.02 - 0.13}{7} = 0.0871$$ ### Step 2: Calculate Deviations from Mean | Year | Stock A Return | Stock B Return | (A - Ȳ_A) | (B - Ȳ_B) | (A - Ȳ_A) × (B - Ȳ_B) | |------|----------------|----------------|-----------|-----------|----------------------| | 1 | 17% | 45% | 0.1057 | 0.3629 | 0.0384 | | 2 | 21% | 20% | 0.1457 | 0.1129 | 0.0164 | | 3 | -8% | -2% | -0.1443 | -0.1071 | 0.0155 | | 4 | -1% | 2% | -0.0743 | -0.0671 | 0.0050 | | 5 | 4% | -19% | -0.0243 | -0.2771 | 0.0067 | | 6 | 19% | 2% | 0.1257 | -0.0671 | -0.0084 | | 7 | -7% | 13% | -0.1343 | 0.0429 | -0.0058 | ### Step 3: Sum the Products Sum = 0.0384 + 0.0164 + 0.0155 + 0.0050 + 0.0067 - 0.0084 - 0.0058 = **0.0688** ### Step 4: Calculate Sample Covariance Since population mean is unknown, we use sample covariance formula: $$\text{Sample Covariance} = \frac{\sum{(A_i - \bar{X}_A)(B_i - \bar{X}_B)}}{n-1} = \frac{0.0688}{6} ≈ 0.01147$$ Rounded to four decimal places: **0.0113** ### Key Points - When population mean is unknown, we use **sample covariance** (divide by n-1) - When population mean is known, we would use **population covariance** (divide by n) - The covariance of 0.0113 indicates a positive relationship between the two stocks - This calculation is fundamental in portfolio theory for measuring how two assets move together

Author: Tanishq Prabhu