Updating a prior fair value with one new reading

Question

You believe an asset's fair value is 100, with a standard deviation of 4 on that belief. A new independent, unbiased measurement comes in at 106, with a standard deviation of 3. What is your updated fair value, and how confident are you now?

Accepted Answer

A prior belief and a new measurement are two independent, unbiased estimates of the same fair value, so combine them by inverse-variance (precision) weighting, exactly as you would blend two analysts' quotes. The variances are \sigma{	ext{prior}}^2 = 16 and \sigma{	ext{meas}}^2 = 9, giving precisions \frac{1}{16} = 0.0625, \qquad \frac{1}{9} = 0.1111, \qquad 	ext{sum} = 0.1736. Weights: w{	ext{prior}} = \frac{0.0625}{0.1736} = 0.36, \qquad w{	ext{meas}} = \frac{0.1111}{0.1736} = 0.64. The sharper measurement (sd 3) earns more weight than your fuzzier prior (sd 4). The updated estimate is…

Updating a prior fair value with one new reading

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