A quantitative analyst is working on developing a stock selection model specifically tailored for intraday trading. To evaluate the model's effectiveness in identifying correlations in stock returns, the analyst uses the annual returns of two utility stocks, referred to as stock A and stock B. The following table presents the annual returns for each of these stocks over a 5-year period:

Year Return of stock A (Ra) Return of stock B (RB)
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 has also computed the sample means of the returns for stock A (μa) and stock B (μB), which are 0.146 and 0.138, respectively. Based on this information, what is the unbiased estimate for the sample covariance between the returns of stocks A and B?

Year	Return of stock A (Ra)	Return of stock B (RB)
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

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Explanation:

D is correct. Using the formula for the sample covariance estimator but dividing by n-1 for an unbiased estimate, we get $\sigma_{AB} = \frac{1}{n-1} \sum_{i=1}^{n} (R_{A,i} - \mu_A) * (R_{B,i} - \mu_B)$ . And so we get $\sigma_{AB} = \frac{1}{4} [(0.18 - 0.146)(0.32 - 0.138) + (0.13 - 0.146)(0.22 - 0.138) + (0.04 - 0.146)(0.00 - 0.138) + (0.30 - 0.146)(0.10 - 0.138) + (0.08 - 0.146)(0.05 - 0.138)]$ . Which is expanded as $\sigma_{AB} = 0.004865$ . A is incorrect. This is the result when the two means are switched in the summation formula and the multiplier used is 1/5. B is incorrect. This is the result when the multiplier used is 1/5 instead of 1/(5-1). C is incorrect. This is the result when the two means are switched in the summation formula.

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