Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 43 / 4 / Solution

A credibility estimate combines a risk's own observed experience with a collective or prior mean, in the form $Z{\hat{\mu }}_{\mathrm{experience}}+(1-Z){\mu }_{\mathrm{collective}}$. The credibility factor $Z\in [0,1]$ is the weight given to that risk's experience. Bayesian credibility starts with a prior distribution for a latent risk parameter, updates it using observations, and estimates the conditional risk mean by its Bayesian posterior expected value under squared-error loss. In conjugate models this posterior mean can have exactly the displayed credibility form.

For the unequal policy exposures, defineConditional independence of the binomial distribution observations gives the likelihood, up to a factor not depending on $\theta$,Multiplication by the beta distribution prior therefore gives Beta-binomial conjugacy:Both posterior parameters are positive, since $0\le s\le M$ and the prior parameters are positive.

The conditional target is $\mu (\theta )=m_{k+1}\theta$. To verify the quadratic-loss rule directly, for any proposed estimate $d$ and observed data $\mathcal{D}$,The variance term does not depend on $d$, so the unique Bayes estimator under squared error loss is the posterior mean. Applying the beta mean formula givesIt is also the posterior expected future claim count, by the tower property of conditional expectation.

Let $p_0=\alpha /(\alpha +\beta )$ be the collective prior claim probability and $\hat{p}=s/M$ the observed claims per policy. ThenTherefore the exact beta-binomial credibility with unequal exposures isThe individual experience must be weighted by exposure: $s/M$ is the policy-weighted frequency, not the unweighted average of $x_i/m_i$ when the $m_i$ differ. The factor multiplies next-year expected counts after scaling both component probabilities by $m_{k+1}$.