Time-Simultaneous Fan Charts: Applications to stochastic life table forecasting

When it comes to product pricing and reserving, actuaries often need life tables that include a forecast of future longevity improvement. However, the production of such tables is not straightforward, because the demographic future of any human population is a result of complex and only partially understood mechanisms, and is highly uncertain. In recent years, actuaries have been understandably concerned about error in the mortality assumptions they make. Part of their response is a new wave of work that is focused on the forecasting of uncertainty in longevity improvement, rather than producing a single mortality projection that will almost surely be wrong. This goal is accomplished by using stochastic mortality models, which have uncertainty embedded within them, as reflected in historical changes. Given a fitted stochastic mortality model, we can express the uncertainty associated with future death rates in terms of confidence or prediction intervals. Recently, a group of researchers has proposed using fan charts to display prediction intervals for future mortality rates. These charts are highly parallel to the wellknown inflation fan charts, which have been produced periodically by the Bank of England since 1996. A fan chart depicts prediction intervals at different levels of confidence simultaneously. In particular, it shows the central 10% prediction interval with the heaviest shading, surrounded by the 20%, 30%, ..., 90% prediction intervals with progressively lighter shading. We can therefore interpret the degree of shading as the likelihood of the outcome – the darker the shading, the more likely the outcome. Mortality fan charts are highly useful to actuaries, because they provide guidance on how to determine appropriate margins for adverse deviations. Existing mortality fan charts are based on isolated pointwise prediction intervals. By pointwise we mean that the interval reflects uncertainty in a quantity at a single point of time, but it does not account for any dynamic property of the time-series. However, in actuarial practice, rather than a single death rate at a particular time point, what practitioners need is the entire trajectory of mortality rates for the birth cohort in question. Specifically, of their interest would be questions like “Within what bounds would the trajectory of cohort mortality rates likely to remain with a certain degree of confidence?” From a statistical viewpoint, a band of pointwise intervals might lead to invalid inference concerning the time trajectory. In particular, unless all trajectories develop very orderly, a band of pointwise confidence intervals would understate the actual uncertainty associated with a random mortality trajectory. In this paper, we overcome this limitation by introducing the concept of time-simultaneous fan charts. In more detail, instead of pointwise intervals, a time-simultaneous fan chart is derived from a prediction band with a prescribed probability of covering the whole time trajectory. We present two numerical methods for producing time-simultaneous fan charts. These methods can be applied to common stochastic mortality models, including the generalized Cairns-Blake-Dowd model. We illustrate the method with mortality data from the populations of Australia and New Zealand.