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Variance of a horizon total

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A daily return is its unknown mean plus independent noise of known variance $16$ . The posterior for the mean is normal with variance $3$ . What is the posterior predictive variance of the sum of the next $4$ daily returns? Give a whole number.

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Hint

The noise is independent across days, but the error in the mean is shared by all of them.

Solution

The noise contributes $4 \times 16 = 64$ .
The shared mean is counted $4$ times in the sum, so it contributes $4^{2} \times 3 = 48$ .
The total is $64 + 48 = 112$ .

Answer: $112$

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