Graduating pensioner mortality when the oldest ages are thin
A closed defined-benefit pension scheme has 8,500 pensioners aged 60 to 95. You are graduating pensioner mortality by 5-year age bands. Crude central death rates are stable up to age 79 but noisy above 85: at ages 90-94 there are 45 deaths and 180 person-years of exposure, and at 95+ there are 12 deaths and 38 person-years. Two graduations are proposed: (i) a Gompertz-Makeham GM(1,1) law with deviance 142.6 on 5 degrees of freedom, and (ii) a P-spline with 8 knots and deviance 21.4 on 2 degrees of freedom. The scheme actuary wants to use the graduated rates to value a longevity swap that runs to age 100. The trustee board asks whether the spline's much lower deviance means it is the better model, and how the graduation should be validated given the small numbers at the oldest ages.
1. Compute the crude central death rate at 90-94 and an approximate 95% confidence interval. Test the GM(1,1) fit using the deviance, stating the null hypothesis and the limitation of the test here.
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The crude rate is 45/180 = 0.25 per person-year. Treating deaths as Poisson, the standard error is sqrt(45)/180 = 0.0373, so an approximate 95% interval is 0.25 +/- 1.96(0.0373) = 0.177 to 0.323. An exact Poisson interval would be slightly asymmetric but similar.
For the deviance test, the null hypothesis is that the GM(1,1) model is correct. Under the null and standard regularity conditions, the deviance is approximately chi-square with 5 degrees of freedom. The 5% critical value is 11.07, and 142.6 is far larger, so the model is rejected. The limitation is that the chi-square approximation is unreliable when expected deaths are small, as at 95+ where there are only 12 deaths and 38 person-years. Sparse cells and overdispersion inflate the deviance. A parametric bootstrap is safer: simulate deaths from the fitted GM rates, refit, and compare the simulated deviance distribution with 142.6. If the bootstrap p-value is still tiny, the lack of fit is real, but part of it may reflect the oldest ages.
2. Explain why the P-spline's lower deviance is not sufficient evidence that it is better for valuing the longevity swap. Propose a validation and smoothing strategy.
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A lower deviance means better in-sample fit, not better prediction. With 8 knots and very small counts at older ages, the spline can chase random noise, producing implausible kinks, non-monotone hazards, or erratic extrapolation beyond 95. The GM law has a fixed functional form and therefore extrapolates to age 100 in a controlled way, even if it fits the observed age range less closely. For a longevity swap, tail behaviour and model risk matter more than a small deviance difference.
Validate by out-of-sample methods: hold out age bands or use k-fold cross-validation, and compare predicted deaths with observed deaths in the held-out bands. Use penalized likelihood with a difference penalty on adjacent spline coefficients, choosing the smoothing parameter by cross-validation or restricted maximum likelihood. Check that the graduated rates are monotone and positive, and compare with an external table. A sensible compromise is to use the spline up to age 90 and a smooth parametric tail beyond 90, with a test of consistency at the join. Report bootstrap confidence bands and show how the value of the swap changes between the GM and spline graduations.
3. Outline how you would allow for parameter uncertainty and model risk in the valuation, including a Bayesian or simulation approach.
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Use a parametric bootstrap: simulate deaths from the fitted graduation, refit both models, and repeat many times. For each replicate, project survival to age 100 and compute the swap value. This gives a distribution that reflects sampling uncertainty. Alternatively, adopt a Bayesian approach: place priors on the GM parameters, form the Poisson likelihood for deaths with exposure offsets, and sample the posterior with Metropolis-Hastings because the posterior is not conjugate. Use multiple chains, check trace plots, Rhat below 1.01 and effective sample size above 400. Then simulate survival curves from the posterior and value the swap for each draw.
For model risk, do not condition on one graduation. Present results under GM, under the penalized spline, and under a mixture with weights based on validation performance. Stress-test the tail by using a more conservative rate of improvement or a higher ultimate hazard. Show the trustee board a range and the drivers of the range, not a single number. If the swap value is highly sensitive to the 95+ extrapolation, recommend delaying the transaction or obtaining more data.