Competing Risks in Critical-Illness Incidence
A life insurer is reviewing a portfolio of 600 critical-illness (CI) policies sold to policyholders aged 45–60. Over a five-year observation window the insurer recorded 72 CI diagnoses, 48 surrenders, 15 deaths before diagnosis, and 465 policies still in force at the end of the window. The actuarial student initially estimated the CI incidence curve by Kaplan–Meier, treating surrender and death as censoring. The chief actuary notes that surrender rates are strongly age-dependent and that the product allows surrender at any time with a surrender value. A reinsurer has quoted terms assuming a Cox model with a single age covariate; the fitted age coefficient is 0.041 per year with standard error 0.018, but a Schoenfeld residual plot shows a declining effect of age over duration. The insurer must decide how to present the incidence basis for pricing and whether to accept the reinsurer's proportional hazards assumption.
1. Explain whether the student’s Kaplan–Meier estimate of CI incidence is appropriate. State the estimand you would use for pricing and why.
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Kaplan–Meier treats surrender and death as ordinary right-censoring, which is valid only if the censoring mechanism is independent of the CI event time. Surrender is a policyholder choice and is likely related to health and age; if policyholders who feel unwell are more likely to surrender, or less likely, the risk set is distorted and the KM curve is biased. Death before CI is a competing event, not just censoring: a policyholder who dies can never later be diagnosed. For pricing a CI rider, two estimands are useful. The cause-specific hazard for CI, λ_CI(t), is the instantaneous CI rate among those still alive and still in force; it is appropriate for building a multi-state transition model. The cumulative incidence function, P(CI by t before surrender or death), is the probability relevant to the expected cost of the CI benefit and is obtained by Aalen–Johansen or by integrating λ_CI(s)S(s). I would fit a competing-risk model with transitions to CI, surrender and death, check the surrender assumption, and report both the cause-specific hazard and the CIF. If surrender is strongly health-related, a three-state model with surrender as a state is preferable to treating it as censoring.
2. Interpret the Cox age coefficient and the Schoenfeld residual pattern. What action would you take before accepting the reinsurer’s terms?
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The fitted coefficient means the hazard of CI is estimated to increase by a factor exp(0.041) = 1.042 per additional year of age, i.e. about 4.2% per year, with 95% CI exp(0.041 ± 1.96×0.018) = (1.006, 1.079). On its face age is significant at the 5% level, but the Schoenfeld residual pattern shows the age effect declining with duration, so the proportional hazards assumption is questionable. A declining coefficient means older age is associated with a higher early CI hazard but the effect wears off; a single HR averages over duration and can misprice both early and late durations. Before accepting the reinsurer’s terms I would (i) run a formal proportional hazards test such as Grambsch–Therneau, (ii) plot smoothed Schoenfeld residuals against time, (iii) try stratification by age band, a time-varying coefficient for age, or a parametric AFT model, and (iv) compare fitted vs observed CI incidence by duration. I would also note that with only 72 CI events the coefficient is imprecise and the reinsurer’s model may be overfitted. Terms should be accepted only if the chosen model reproduces the portfolio’s duration-specific incidence.