Folding in a third signal

Question

You have three reads on an asset's fair value. Your private model says $102$ with standard deviation $4$ . The market trades around $100$ with an effective standard deviation of $2$ . A closely related instrument implies $101$ with standard deviation $3$ .

What is your best combined estimate of fair value?

Accepted Answer

Precision weighting extends to any number of independent signals: each contributes its estimate weighted by its precision 1/\sigma^2, and you normalize by the total precision: \hat\mu = \frac{\sumi Xi/\sigmai^2}{\sumi 1/\sigmai^2}. Compute the three precisions: private 1/16 = 0.0625, market 1/4 = 0.25, related 1/9 \approx 0.1111. Numerator: \frac{102}{16} + \frac{100}{4} + \frac{101}{9} = 6.375 + 25 + 11.222 = 42.597. Denominator (total precision): 0.0625 + 0.25 + 0.1111 = 0.4236. So \hat\mu = \frac{42.597}{0.4236} \approx 100.56. Your fair value is about 100.56. The market, being the most pr…

Folding in a third signal

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