Expected Value of Process Variance

Study guide: Exam MAS-II

The expected value of the process variance (EPV), written $v$ , is the average within-risk variance in a Bühlmann Credibility model — how noisy a single risk's own experience is, averaged over the population of risks.

$v = \mathrm{EPV} = E_\Theta\!\left[\mathrm{Var}(X \mid \Theta)\right]$

$\Theta$ is the unobserved risk parameter; conditioning on it removes the between-risk differences and leaves only process noise
Together with the Variance of Hypothetical Means $a$ it fixes the Bühlmann constant $k = v/a$ and hence $Z = n/(n+k)$
Large $v$ means one risk's own data is unreliable, so $Z$ falls and the estimate leans on the manual rate
The total variance splits into the two pieces: $\mathrm{Var}(X) = v + a$ — the conditional-variance decomposition
Estimated from data by empirical Bayes as the pooled within-risk sample variance

EPV from a Two-Type Portfolio Half the policyholders are good drivers with Poisson claim frequency $\lambda = 0.10$ ; half are bad with $\lambda = 0.40$ . Find the EPV.

Answer For a Poisson risk, $\mathrm{Var}(X \mid \Theta) = \lambda$ , so $v = E[\lambda] = 0.5(0.10) + 0.5(0.40) = 0.25$ Note $a = \mathrm{Var}(\lambda) = 0.5(0.10 - 0.25)^2 + 0.5(0.40 - 0.25)^2 = 0.0225$ , so $k = v/a = 0.25/0.0225 = 11.1$ — it takes about 11 years of experience for a driver to earn 50% credibility.