Smoothing and forecasting mortality rates

Abstract

The prediction of future mortality rates is a problem of fundamental importance for the insurance and pensions industry. We show how the method of P-splines can be extended to the smoothing and forecasting of two-dimensional mortality tables. We use a penalized generalized linear model with Poisson errors and show how to construct regression and penalty matrices appropriate for two-dimensional modelling. An important feature of our method is that forecasting is a natural consequence of the smoothing process. We illustrate our methods with two data sets provided by the Continuous Mortality Investigation Bureau, a central body for the collection and processing of UK insurance and pensions data.

Longevity Risk

“…The upper panels contain the in-sample results for 65 year old males (left) and females (right). These graphs show that the evolution of mortality over time has some curvature, which is captured quite well and in a smooth way by the Currie et al (2004)-method (which employs so-called B-splines). Such a curvature will not be captured by the Lee and Carter (1992) model.…”

“…However, the longevity risk is quite substantial, leaving the possibility (with 95% confidence according to the model) of a wide variety of possible future mortality trends. The result of much wider prediction intervals, when changing the model from Lee and Carter (1992) to Currie et al (2004), shows the importance of taking into account model risk.…”

“…To illustrate the effect of smoothing we present in Figure 5 an application of the Currie et al (2004)-method in terms of the logarithm of the central death rate, using the same data as in case of Figs. 3-4.…”

“…Cairns et al (2006b) propose a model that builds in smoothness in mortality rates across adjacent ages in the same year. Currie et al (2004) propose a model assuming smoothness across both ages and years. Cairns et al (2007Cairns et al ( , 2008b provide an extensive comparison of these various time series based stochastic mortality models.…”

Multidimensional smoothing

“…The data set consists of the number of policy claims (deaths) and the number of years lived (exposure) for each age, 12-96 years, and each calendar year, 1951-1999; see Currie et al (2004) for more information on these data. The claims and exposures are arranged respectively in matrices Y and E, with rows indexed by age and columns by year.…”