The CBD Mortality Indexes: Modeling and Applications

Abstract

Most extrapolative stochastic mortality models are constructed in a similar manner. Specifically, when they are fitted to historical data, one or more series of time-varying parameters are identified. By extrapolating these parameters to the future, we can obtain a forecast of death probabilities and consequently cash flows arising from life contingent liabilities. In this article, we first argue that, among various time-varying model parameters, those encompassed in the Cairns-Blake-Dowd (CBD) model (also known as Model M5) are most suitably used as indexes to indicate levels of longevity risk at different time points. We then investigate how these indexes can be jointly modeled with a more general class of multivariate time-series models, instead of a simple random walk that takes no account of cross-correlations. Finally, we study the joint prediction region for the mortality indexes. Such a region, as we demonstrate, can serve as a graphical longevity risk metric, allowing practitioners to compare the longevity risk exposures of different portfolios readily.

Notes

Goldman Sachs ceased to run its QxX Index in December 2009.

Section 2.2 explains why the original CBD model (but not the other five models) satisfies the new-data-invariant property.

According to the Human Mortality Database, raw population counts by single year of age for English and Welsh males are available up to age 89 only. The population counts beyond age 89 are only estimates that are derived using the extinct cohort method.

The logit transformation of a real number w is given by ln(w/(1−w)).

For instance, 98% of the customers of Lifenet Insurance (a Japanese insurance company that focuses on selling term-life insurances) in FY3/11 were below age 60. See: http://www.lifenet-seimei.co.jp.

We use χ² _v,1−α to denote the 100(1−α)th percentile of a χ² distribution with v degrees of freedom.

See: http://life.mappinglondon.co.uk/.

The data are available at http://www.llma.org. According to the LLMA's Longevity Index Technical Document, the historical death probabilities provided by the LLMA have been graduated by cubic splines. This pre-smoothed data set covers the period of 1960–2011, which is eight years shorter than that on which our baseline results are based. The difference in the sample periods may explain why the LLMA data set results in optimal AR order of four instead of five.

We refer readers to Coughlan (2009) and LLMA's website (http://www.llma.org) for details regarding q-forwards and S-forwards.

For simplicity, we ignore the potential lag in the availability of the index data.