Deductible, Inflation, and Tail Risk in Aggregate Claims

Let X be current-year severity. The correct inflated payment per claim is max(0, 1.06X - 600): inflation changes the severity, while the deductible remains a fixed nominal amount. The analyst instead uses 1.06 max(0, X - 600), which is equivalent to inflating the deductible to 636. For X > 600 the correct payment exceeds the analyst’s payment by exactly 36. For 600/1.06 < X < 600, the correct payment is positive while the analyst records zero, so the error is also positive in that interval. Hence the analyst’s per-claim payment is never larger and is sometimes smaller, so expected aggregate claims are understated. Because aggregate mean is frequency times expected severity, the understatement propagates directly to the mean. The 99.5% VaR is also understated in direction, though the magnitude is not simply the same percentage: the tail is driven by large claims, where the error is a fixed 36 per claim, plus the extra claims entering above the effective deductible. The 23% normal-versus-simulation gap is a separate issue and should not be used to calibrate the deductible error.

The Pearson statistic divided by its degrees of freedom is 1,540/1,200 = 1.283. Under a correctly specified Poisson model this ratio should be near 1; the approximate standard deviation of the chi-square statistic is sqrt(2 × 1,200) = 49, so the observed excess of 340 is about 6.9 standard deviations. There is clear evidence of overdispersion, although its magnitude is moderate. Overdispersion means Poisson standard errors and predictive distributions are too narrow; the fitted means may still be reasonable if the mean structure is adequate. A normal approximation to aggregate losses is questionable because lognormal severity produces a right-skewed compound Poisson distribution, especially at 99.5%. The simulation result is more credible in direction, but 100,000 scenarios leave roughly 500 observations in the extreme tail, so Monte Carlo error at the 99.5% quantile is not negligible. A defensible approach is to correct the deductible/inflation order, fit a negative binomial or compound model that captures frequency heterogeneity and severity skew, validate convergence across seeds and larger scenario counts, compare with an analytic compound Poisson or saddlepoint approximation, and backtest the chosen quantile.

Defensible without refitting frequency: correcting the deductible and inflation order, because this is a deterministic transformation of severity; increasing the simulation size and checking seed stability; reporting Monte Carlo error around the simulated VaR; and using a skewness-aware quantile rather than a normal approximation. Also defensible is presenting the VaR alongside sensitivity to the severity assumptions, since those can be varied without changing the frequency model. Not defensible: simply scaling the normal VaR upward by 23% to match one simulation run, because that treats a model-diagnostic gap as a calibration factor without justification; changing the confidence level from 99.5% to a lower level without governance approval; or ignoring overdispersion in parameter uncertainty. If overdispersion is genuine, quasi-Poisson only changes standard errors, not fitted means, and a negative binomial or random-effects frequency model requires refitting. The actuary should separate the pure coding error (deductible order) from model inadequacy (overdispersion and normal approximation) and document both.