Financial Risk Manager Part 1
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The investment committee asked the manager to present the covariance of both stocks. Using the data given in the following table, calculate the covariance if the population mean is unknown.
| Year | Stock A Return | Stock B Return |
|---|---|---|
| 1 | 17% | 45% |
| 2 | 21% | 20% |
| 3 | -8% | -2% |
| 4 | -1% | 2% |
| 5 | 4% | -19% |
| 6 | 19% | 2% |
| 7 | -7% | 13% |
Exam-Like
Community
TTanishq
A
0.0113
B
0.101
C
0.1156
D
0.00907
Explanation:
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):
Stock B Mean (Ȳ_B):
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:
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
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