Gamma-Poisson Credibility for a Small Group: Prior, Predictive and What 'Best Estimate' Means
A mutual insurer writes 40 group income protection policies covering 120 policy-years in total, over which the group generated 18 claims. Industry data for the same occupational class suggest a claim frequency of 0.50 claims per policy-year with a coefficient of variation of 0.50. The pricing actuary adopts a Gamma prior for the group's unknown claim frequency with mean 0.50 and variance 0.0625, and combines it with the group's own experience. The group will carry 150 policy-years of exposure next year. The underwriter needs a best estimate of next year's claim count and the probability of zero claims, because a no-claims discount applies. An actuarial student on the team argues that, with only 18 claims, the group's experience carries almost no information and the industry rate of 0.50 should simply be used. The actuary is also asked whether a single best estimate is an adequate answer to the underwriter.
1. Derive the prior parameters, the posterior distribution and the posterior mean, and show that the posterior mean is the credibility-weighted average of the group's own rate and the prior mean.
Read the worked answer
For $\lambda\sim\text{Gamma}(\alpha,\beta)$ (rate parameterisation), mean $=\alpha/\beta$ and variance $=\alpha/\beta^2$. Given mean 0.50 and variance 0.0625: $\beta=0.50/0.0625=8$ and $\alpha=0.50\times8=4$. The stated coefficient of variation checks out: $0.25/0.50=0.5$.
Likelihood: 18 claims over 120 policy-years, so proportional to $\lambda^{18}e^{-120\lambda}$. Multiplying, the posterior is $\text{Gamma}(4+18,,8+120)=\text{Gamma}(22,128)$.
Posterior mean $=22/128=0.1719$; posterior sd $=\sqrt{22}/128=0.0366$.
Credibility form: with $E=120$, $Z=E/(E+\beta)=120/128=0.9375$, so
$$Z\bar\lambda+(1-Z)m=0.9375\times0.15+0.0625\times0.50=0.1406+0.0313=0.1719,$$
identical to the posterior mean. The observed rate is $18/120=0.15$ and the prior is pulling it up, not down, because the prior mean is more than three times the experience.
2. Compute the predictive mean and the probability of zero claims next year, and compare the predictive spread with that of a Poisson model at the posterior mean.
Read the worked answer
Predictive mean $=150\times0.1719=25.8$ claims.
The predictive density is negative binomial. With $N\mid\lambda\sim\text{Poisson}(150\lambda)$ and $\lambda\sim\text{Gamma}(22,128)$:
$$P(N=0)=E\left[e^{-150\lambda}\right]=\left(\frac{128}{128+150}\right)^{22}=(0.46043)^{22}=e^{-17.06}=3.9\times10^{-8}.$$
The plug-in value at the posterior mean is $e^{-25.8}=6.3\times10^{-12}$ - the full predictive probability is thousands of times larger because it integrates over uncertainty in $\lambda$. Both are negligible in practice, so the no-claims discount is essentially unattainable at this portfolio size whatever the model says.
Predictive variance $=\mu+e^2\operatorname{Var}(\lambda)=25.8+150^2(22/128^2)=25.8+30.2=56.0$, sd 7.5, against a Poisson sd of 5.1 at the posterior mean. Parameter uncertainty nearly doubles the standard deviation of the count. An approximate 95% predictive range is about 11 to 41 claims, which is the number the underwriter actually needs to hear.
3. Respond to the student's argument that the group's experience should be ignored, and advise on whether a single best estimate suffices.
Read the worked answer
The student is right that 18 claims are not fully credible, but wrong to conclude that they are uninformative. Under the limited-fluctuation standard, full credibility at 90% confidence and ±10% requires an expected claim count of $(1.645/0.10)^2\approx271$; 18 claims fall far short, so a pure experience rate would be far too volatile. But credibility weight is not the same as full credibility: the Bayesian weight of 0.9375 is high because the prior variance (0.0625) is about fifty times the sampling variance of the observed rate, $\hat\lambda/E\approx0.15/120=0.00125$. A diffuse prior leaves the data to dominate. The result is therefore sensitive to the prior specification, despite the impressive-looking $Z$.
Specific concerns:
- The prior mean of 0.50 is 3.3 times the observed rate; if the 'industry class' is broad and includes much riskier groups, exchangeability is doubtful, and the posterior is pulled up 15% above the raw rate. I would test sensitivity across plausible priors.
- The 40 policies are assumed homogeneous; any within-group risk mix should be handled with a random-effects or GLM structure rather than a single $\lambda$.
- The Poisson likelihood should be checked for overdispersion; if the group's claims cluster, the posterior is overprecise.
A single number is not an adequate answer. The underwriter should receive the predictive mean, the predictive distribution (sd about 7.5, 95% range roughly 11 to 41), and a statement of how the answer moves with the prior.