Variance of Hypothetical Means

Study guide: Exam MAS-II

The variance of the hypothetical means (VHM), written $a$ , is the between-risk variance in a Bühlmann Credibility model — how much the risks in the portfolio genuinely differ from one another once process noise is stripped out.

$a = \mathrm{VHM} = \mathrm{Var}_\Theta\!\left(E[X \mid \Theta]\right)$

$\mu(\Theta) = E[X \mid \Theta]$ is the hypothetical mean: the long-run mean of a risk with parameter $\Theta$ . VHM is the variance of that quantity across the population
With the Expected Value of Process Variance $v$ it gives $k = v/a$ and $Z = n/(n+k)$
Large $a$ means the risks are genuinely heterogeneous, so a risk's own experience is worth a lot and $Z$ rises
$a = 0$ means every risk has the same mean — experience rating has nothing to detect, and $Z = 0$ for any $n$
The overall mean is $\mu = E[\mu(\Theta)]$ , and $\mathrm{Var}(X) = v + a$

VHM for a Gamma-Poisson Model Claim counts are Poisson with mean $\Lambda$ , where $\Lambda \sim \text{Gamma}(\alpha = 4, \theta = 0.05)$ . Find $\mu$ , $v$ , $a$ , and $k$ .

Answer $\mu(\Lambda) = \Lambda$ and $\sigma^2(\Lambda) = \Lambda$ for a Poisson. $\mu = E[\Lambda] = \alpha\theta = 4(0.05) = 0.20$ $v = E[\Lambda] = 0.20 \qquad a = \mathrm{Var}(\Lambda) = \alpha\theta^{2} = 4(0.0025) = 0.01$ $k = \frac{v}{a} = \frac{0.20}{0.01} = 20 = \frac{1}{\theta}$ For the Gamma-Poisson, $k$ is always $1/\theta$ , and $Z = n/(n + 1/\theta)$ reproduces the Bayesian posterior mean exactly.