Answer-first summary for fast verification
Answer: 0.004865
## Explanation The unbiased estimate of sample covariance is calculated using the formula: \[\sigma_{AB} = \frac{1}{n-1} \sum_{i=1}^{n} (R_{A,i} - \mu_A)(R_{B,i} - \mu_B)\] Where: - n = 5 (number of observations) - μ_A = 0.146 (mean return of stock A) - μ_B = 0.138 (mean return of stock B) Let's calculate step by step: **Year 1:** (0.18 - 0.146)(0.32 - 0.138) = (0.034)(0.182) = 0.006188 **Year 2:** (0.13 - 0.146)(0.22 - 0.138) = (-0.016)(0.082) = -0.001312 **Year 3:** (0.04 - 0.146)(0.00 - 0.138) = (-0.106)(-0.138) = 0.014628 **Year 4:** (0.30 - 0.146)(0.10 - 0.138) = (0.154)(-0.038) = -0.005852 **Year 5:** (0.08 - 0.146)(0.05 - 0.138) = (-0.066)(-0.088) = 0.005808 **Sum of products:** 0.006188 - 0.001312 + 0.014628 - 0.005852 + 0.005808 = 0.01946 **Unbiased covariance:** σ_AB = (1/4) × 0.01946 = 0.004865 Therefore, the unbiased estimate of the sample covariance is 0.004865, which corresponds to option D.
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A quantitative analyst is constructing a stock selection algorithm that will be employed in making intraday trades and uses the annual returns of two utility stocks, stock A and stock B, to test the model's capacity to capture dependence between stock returns. The 5 years of annual returns data for each stock used in the test are shown in the following table:
| Year | Return of stock A (Rₐ) | Return of stock B (R_b) |
|---|---|---|
| 1 | 0.18 | 0.32 |
| 2 | 0.13 | 0.22 |
| 3 | 0.04 | 0.00 |
| 4 | 0.30 | 0.10 |
| 5 | 0.08 | 0.05 |
The analyst estimates that the sample means of the returns of stock A (μₐ) and stock B (μ_b) are 0.146 and 0.138, respectively. What is the unbiased estimate of the sample covariance of stocks A and B?
A
0.003828
B
0.003892
C
0.004785
D
0.004865