Updating a fair value with a fresh print

Question

At the close yesterday your best estimate of an asset's fair value was $100$ , with a standard deviation of $2$ . This morning a burst of trades prints information suggesting $103$ , which you judge to have standard deviation $3$ as a signal.

What is your updated fair value, and how confident are you now?

Accepted Answer

This is the same precision-weighting rule, read as a Bayesian update: your prior belief is one signal, the new observation is a second, independent one. Treat "prior" and "observation" exactly as two estimates and blend by precision: \hat\mu = \frac{X{	ext{prior}}/\sigma{	ext{prior}}^2 + X{	ext{obs}}/\sigma{	ext{obs}}^2}{1/\sigma{	ext{prior}}^2 + 1/\sigma{	ext{obs}}^2}. Precisions: prior 1/4 = 0.25, observation 1/9 \approx 0.1111. \hat\mu = \frac{100/4 + 103/9}{1/4 + 1/9} = \frac{25 + 11.444}{0.25 + 0.1111} = \frac{36.444}{0.3611} \approx 100.92. Your updated fair value is about 100.92.…

Updating a fair value with a fresh print

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