A Bayesian Joint Model for Population and Portfolio-Specific Mortality

Abstract

Insurance companies and pension funds must value liabilities using mortality rates that are appropriate for their portfolio. These can only be estimated in a reliable way from a sufficiently large historical dataset for such portfolios, which is often not available. We overcome this problem by introducing a model to estimate portfolio-specific mortality simultaneously with population mortality. By using a Bayesian framework, we automatically generate the appropriate weighting for the limited statistical information in a given portfolio and the more extensive information that is available for the whole population. This allows us to separate parameter uncertainty from uncertainty due to the randomness in individual deaths for a given realization of mortality rates. When we apply our method to a dataset of assured lives in England and Wales, we find that different prior specifications for the portfolio-specific factors lead to significantly different posterior distributions for hazard rates. However, in short-term predictive distributions for future numbers of deaths, individual mortality risk turns out to be more important than parameter uncertainty in the portfolio-specific factors, both for large and for small portfolios.

Introduction

“…The difficulty in using safety margins and other scenarios in the pricing of annuities and other mortality related insurance products has proven to be difficult, to the point that liquid markets for such products are few. Notable examples of stochastic modelling are the formulations by Hahn andChristiansen (2016) andvan Berkum et al (2017), who both develop full probability models for age-specific mortality of one or more populations, and compute the posterior distributions of the relevant risk measures using Markov Chain Monte Carlo techniques.…”

A Bayesian non-parametric model for small population mortality

“…This assumption is also used in the subsequent literature (see e.g. Olivieri and Pitacco 2012, Salhi et al 2015, van Berkum et al 2017. This assumption is motivated by the following arguments:…”

“…As an alternative to the common θ assumption, Olivieri andPitacco (2012) andvan Berkum et al (2017) assume that θ is a vector of i.i.d. random variables, each with the prior distribution γ (ν, ν).…”

“…The second class of models are Bayesian (see e.g. Dowd et al 2011, Cairns et al 2011b, Antonio et al 2015, van Berkum et al 2017. While they are flexible, and account for parameter uncertainty in a natural way, these models are currently rather computationally intensive.…”

A Bayesian Joint Model for Population and Portfolio-Specific Mortality

“…Therefore, an insurance company or pension fund cannot use mortality projections for the whole population without making any adjustments. The difference between mortality in a population and a portfolio is often called basis risk, see, for example, Barrieu et al (2012).…”