Overdispersed Claim Counts in a Motor Portfolio
A motor insurer has 2,000 policy-years of homogeneous private-car exposure. The portfolio recorded 480 claims, so the sample mean annual claim count is 0.24. The sample variance of annual counts, computed across policy-years, is 0.41. Pricing currently assumes a Poisson frequency with mean 0.24. The reinsurer has asked for evidence that the Poisson assumption is adequate before renewing a quota-share treaty. The chief actuary is reluctant to switch models because the negative binomial is harder to explain to the pricing committee, and because a small estimated dispersion parameter could create unstable tail probabilities. The data team confirms that policy-years are independent and that exposure is stable over the period. A quick check shows that the observed number of zero-claim policy-years is 1,720, compared with about 1,573 expected under Poisson. The committee wants a formal test, an alternative frequency model, and a recommendation that separates pricing from capital setting.
1. Test the Poisson assumption at the 5% level using a dispersion statistic. State the null and alternative hypotheses, the test statistic, the critical value or p-value reasoning, and the practical conclusion.
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Let X_i be annual claim count for policy-year i. Under H0: Poisson with mean mu = 0.24, so Var(X_i) = E(X_i) = 0.24. Under H1: overdispersion, Var(X_i) > 0.24.
The dispersion statistic is T = sum_i (X_i - Xbar)^2 / mu0 = (n - 1)s^2 / mu0.
Using n = 2,000, s^2 = 0.41 and mu0 = 0.24: T = 1,999 x 0.41 / 0.24 = 3,414.9.
Under H0, T is approximately chi-square with 1,999 degrees of freedom. The mean is 1,999 and the standard deviation is sqrt(2 x 1,999) = 63.2. The upper 5% critical value is about 1,999 + 1.645 x 63.2 = 2,103. The observed T = 3,415 is far above this; equivalently, z = (3,415 - 1,999)/63.2 = 22.4, so the p-value is essentially zero.
Conclusion: reject the Poisson model. The dispersion index is s^2 / Xbar = 0.41 / 0.24 = 1.71, meaning 71% more variance than Poisson. With 2,000 policy-years the test is very powerful, so the committee should focus on the effect size as well as significance: a 71% excess variance is economically material for pricing and capital.
2. Estimate the negative binomial parameters by method of moments and compute P(N = 0) under both models. Quantify the impact on a zero-claim discount for 2,000 policy-years.
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Use the negative binomial parameterization with mean mu and variance mu + mu^2 / r. Set mu = 0.24 and variance = 0.41.
0.41 = 0.24 + 0.24^2 / r 0.17 = 0.0576 / r r = 0.0576 / 0.17 = 0.3388.
Under this model, P(N = 0) = (r / (r + mu))^r. P0_NB = (0.3388 / (0.3388 + 0.24))^0.3388 = (0.3388 / 0.5788)^0.3388 = (0.5854)^0.3388 = 0.834.
Under Poisson, P0_P = exp(-0.24) = 0.7866.
The negative binomial predicts about 4.7 percentage points more zero-claim policy-years. On 2,000 policy-years that is about 95 additional zero-claim policies: 0.0474 x 2,000 = 94.8. A zero-claim discount based on Poisson would therefore under-reward the truly claim-free policies and over-reward the rest. The estimate r = 0.339 is small, so the tail is sensitive to the moment estimate; this should be checked with a GLM or Bayesian fit before being used for capital.
3. Recommend which model to use for (a) expected claim cost pricing and (b) solvency capital. Explain the main trade-off and any validation step.
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For expected claim cost pricing, the Poisson model is not adequate: it understates the variance and misprices the zero-claim discount. A negative binomial or quasi-Poisson GLM is preferable because it allows overdispersion while retaining a tractable mean structure. If the overdispersion comes from unobserved heterogeneity, a mixed model or additional risk classification may be even better than a simple negative binomial.
For solvency capital, the choice matters more because capital depends on the tail, not only the mean. The negative binomial frequency combined with a severity distribution will produce a heavier aggregate tail than Poisson. Using Poisson would understate capital. However, the estimated dispersion parameter is uncertain with only one portfolio observation, so the capital model should be validated with out-of-sample data, posterior predictive checks, or a sensitivity range for r.
The main trade-off is simplicity and explainability versus statistical adequacy. The test result is decisive enough to reject Poisson, but the committee should not treat the negative binomial as exact. A defensible approach is to use the negative binomial for pricing and a conservative range around it for capital, then revisit after another year of data.