Graduation and mortality projection for a pension scheme
A pension scheme has 10,000 pensioner lives. Crude annual mortality rates have been calculated by single years of age from 60 to 91. At ages 60–64 there are 2,500 lives and 25 deaths; at ages 85–89 there are 180 lives and 18 deaths. The actuary fits two Gompertz–Makeham graduations by maximum likelihood: GM(1,1) gives a deviance of 42.1 on 30 degrees of freedom; GM(1,2) gives 28.7 on 29 degrees of freedom. The scheme's standard table has mortality rates 8% lower at ages below 70 and 5% higher at ages 80 and above than the crude rates. The trustees want a graduated table for valuing liabilities and a projection of future mortality. The actuary is also asked whether the graduated rates should be adjusted to the standard table.
1. Assess the two graduations using the deviance, degrees of freedom and the principle of parsimony. What additional tests would you perform before choosing?
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GM(1,2) has a deviance of 28.7 on 29 df, close to what would be expected if the model fitted; GM(1,1) has 42.1 on 30 df, a poorer fit. The difference is 13.4 on one extra parameter; the chi-square upper-tail probability is about 0.00025, so the extra parameter is strongly supported. That does not by itself prove the model is suitable. Check residuals by age, the sign and runs of deviations, whether the graduated rates are smooth and monotone, and whether they adhere to the standard table where required. At the oldest ages the exposure is small, with 180 lives and 18 deaths, so the chi-square approximation is questionable and a few deaths can drive the deviance. Prefer the more parsimonious GM(1,1) only if the extra parameter produces an implausible or unstable curve; otherwise GM(1,2) is defensible.
2. Discuss whether the graduated rates should be adjusted to the standard table. Include the trade-off between fidelity to the scheme's own experience and the credibility of the standard table.
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Do not mechanically apply the 8% and 5% adjustments. The scheme's own experience is the primary evidence, but at old ages it is sparse and volatile, so the standard table provides useful credibility. Use a credibility or Bayesian blending: weight own experience by an exposure-based credibility factor and the standard table by the complement, with more weight to the standard at ages 80 and above. Investigate whether the differences are real, for example from membership mix, socio-economic group or data quality, rather than random. If the standard table is used for valuation, disclose the basis and show the liability impact of using unadjusted own rates. The adjustment should be smooth across ages, not a step change at 70 and 80.
3. Outline how you would project mortality improvements and state two key judgemental choices.
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Project mortality improvements using a stochastic model such as Lee–Carter or Cairns–Blake–Dowd, fitted to national population data and, where credible, to scheme experience. Key judgemental choices include the jump-off rates, meaning the starting level of mortality, and the long-term rate of improvement; also decide whether to allow for cohort effects and how to treat parameter uncertainty. For a pension scheme, use scenario and stress testing, for example improvements 1% per annum faster than the central projection, and document the governance process. The graduated table is a current-basis table; the projection is a separate layer and is not obtained by extrapolating the Gompertz–Makeham graduation.