Best blend of two unbiased fair-value estimates

Question

Two independent desks each produce an unbiased estimate of a stock's fair value. Desk A's estimate has variance 4; Desk B's estimate has variance 12. You will combine them into $w \cdot A + (1-w)\cdot B$ , which stays unbiased for any weight $w$ .

What weight $w$ on Desk A minimizes the variance of the combined estimate?

Accepted Answer

The idea. Because both estimates are already centered on the truth, any blend of them is still centered, so bias is not in play at all. The only thing left to minimize is variance, and the trick is to lean on whichever estimate is more precise. Set up the variance. With independent estimates, the variance of the blend is 	ext{Var} = w^2 (4) + (1-w)^2 (12). Minimize. The weight that minimizes this puts each estimate's weight in proportion to the other's variance, that is, inversely proportional to its own variance: wA = \frac{	ext{Var}B}{	ext{Var}A + 	ext{Var}B} = \frac{12}{4 + 12} = \frac…

Best blend of two unbiased fair-value estimates

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