Term assurance pricing with a short select period and sparse duration data
An insurer is pricing a 10-year renewable term assurance for 45-year-old non-smokers. A pilot portfolio contains 2,400 policies observed for four years. Deaths at durations 0-1, 1-2, 2-3 and 3-4 are 6, 9, 14 and 21. Mid-year exposed-to-risk totals, after treating 6% annual lapses as censored at mid-year, are 2,397, 2,355, 2,300 and 2,240. Crude central death rates rise steeply, and the pricing actuary must decide whether to graduate with a two-year select period plus an ultimate Gompertz law, or to fit a single Weibull distribution to all four durations and extrapolate to duration 10. The chief actuary is concerned that the duration 2-3 jump could be random noise, and that a Weibull force of mortality starts at zero, which is implausible for 45-year-olds. You are asked to assess the evidence and recommend a basis.
1. Compute the crude central death rates and approximate standard errors for the four durations. Do the data provide strong evidence of a select period lasting more than one year? Show your reasoning.
Read the worked answer
Treat deaths as Poisson with mean exposure times rate. The crude central death rates are:
- 0-1: 6/2,397 = 0.002503; SE = sqrt(6)/2,397 = 0.001022
- 1-2: 9/2,355 = 0.003822; SE = sqrt(9)/2,355 = 0.001274
- 2-3: 14/2,300 = 0.006087; SE = sqrt(14)/2,300 = 0.001627
- 3-4: 21/2,240 = 0.009375; SE = sqrt(21)/2,240 = 0.002046
The year 2-3 jump is 0.002265 with approximate SE sqrt(0.001274^2 + 0.001627^2) = 0.002066, so z = 1.10 and p is about 0.27. The year 1-2 jump is 0.001319 with SE 0.001634, z = 0.81. Neither jump is statistically strong on its own. The rates are monotone, but with only 6 to 21 deaths per duration the trend is compatible with a smooth increasing hazard. A select period is still plausible on underwriting grounds, but the data do not justify a sharp two-year select pattern with confidence. The actuary should use credibility weighting with an industry table and test the select period formally rather than relying on the apparent jump.
2. Write the log-likelihood for a Weibull model with right-censoring at mid-year for lapses, and explain how the estimated shape parameter would guide the choice against Gompertz. You do not need to fit numerically.
Read the worked answer
For a Weibull law with shape k and scale lambda, the hazard is h(t) = (k/lambda)(t/lambda)^(k-1) and the cumulative hazard is H(t) = (t/lambda)^k. For each life i, let t_i be death time if observed and censoring time if not. The log-likelihood is:
sum over deaths [ln k - ln lambda + (k-1)(ln t_i - ln lambda)] - sum over all lives (t_i/lambda)^k.
With grouped duration data, t_i is approximated by the mid-point of the interval in which death or censoring occurs. If the fitted k is around 2, the hazard is increasing but equals zero at duration 0. For a 45-year-old portfolio, zero mortality at the start of the contract is implausible. A Gompertz law h(t) = B c^t has positive hazard at t = 0 and exponential increase, while Makeham adds a constant. Compare AIC, deviance and residual patterns; if Weibull k is significantly above 1 but the fit is adequate over 0-4 years, it may still be a reasonable short-term graduation, but extrapolation to duration 10 should be checked against an external table. Do not choose Weibull merely because it has one fewer parameter.
3. Recommend a graduation basis. Include how you would test the select period and how you would allow for parameter uncertainty in the 10-year extrapolation.
Read the worked answer
Recommend a two-year select period with duration-specific multipliers for durations 0-1 and 1-2, followed by an ultimate Gompertz-Makeham law from duration 2. This respects underwriting while retaining a biologically plausible ultimate shape. Test the select period by likelihood ratio or AIC comparison between a model with separate first-two-year parameters and a constrained model in which ultimate rates apply from duration 1. Because counts are small, supplement with parametric bootstrap p-values and cross-validation by holding out a duration. Do not overfit: if the extra parameters do not improve out-of-sample prediction, use a one-year select period.
For parameter uncertainty, bootstrap policies or simulate deaths from the fitted Poisson rates, refit the graduation, and repeat. For each replicate, project survival to duration 10 and compute the present value of the term assurance. Present a distribution and prediction interval, not a single central estimate. Check sensitivity to the 6% lapse assumption and to the choice of ultimate law. In R, survreg or flexsurvreg can fit the parametric models; store the variance-covariance matrix and simulate parameters from a multivariate normal as a fast approximation, then compare with the bootstrap.