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.

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Explanation:

Detailed Explanation

When the population mean is unknown, we use sample covariance which divides by (n-1) instead of n.

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): $\bar{X}_B = \frac{0.45 + 0.20 - 0.02 + 0.02 - 0.19 + 0.02 - 0.13}{7} = 0.0871$

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

Sum = 0.0384 + 0.0164 + 0.0155 + 0.0050 + 0.0067 - 0.0084 - 0.0058 = 0.0688

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

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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