In the first part of the paper, we consider the wide range of extrapolative stochastic mortality models that have been proposed over the last 15 to 20 years. A number of models that we consider are framed in discrete time and place emphasis on the statistical aspects of modelling and forecasting. We discuss how these models can be evaluated, compared and contrasted. We also discuss a discrete-time market model that facilitates valuation of mortality-linked contracts with embedded options. We then review several approaches to modelling mortality in continuous time. These models tend to be simpler in nature, but make it possible to examine the potential for dynamic hedging of mortality risk. Finally, we review a range of financial instruments (traded and over-the-counter) that could be used to hedge mortality and risk. Some of these, such as mortality swaps, already exist, while others anticipate future developments in the market.

This paper introduces a new framework for modelling the joint development over time of mortality rates in a pair of related populations with the primary aim of producing consistent mortality forecasts for the two populations. The primary aim is achieved by combining a number of recent and novel developments in stochastic mortality modelling, but these, additionally, provide us with a number of side benefits and insights for stochastic mortality modelling. By way of example, we propose an Age-Period-Cohort model which incorporates a mean-reverting stochastic spread that allows for different trends in mortality improvement rates in the short-run, but parallel improvements in the long run. Second, we fit the model using a Bayesian framework that allows us to combine estimation of the unobservable state variables and the parameters of the stochastic processes driving them into a single procedure. Key benefits of this include dampening down of the impact of Poisson variation in death counts, full allowance for paramater uncertainty, and the flexibility to deal with missing data. The framework is designed for large populations coupled with a small sub-population and is applied to the England & Wales national and Continuous Mortality Investigation assured lives males populations. We compare and contrast results based on the two-population approach with single-population results.

and dynamic hedging

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By providing financial protection against the major 18th and 19th century risk of dying too soon, life assurance became the biggest financial industry…providing financial protection against the new risk of not dying soon enough may well become the next century’s major and most profitable financial industry. Peter Drucker, The Economist, 1999 1.

A framework for longevity basis risk analysis and hedge effectiveness

We consider situations where a pension plan has opted to hedge its longevity risk using an index-based longevity hedging instrument such as a q-forward or deferred longevity swap. The use of index-based hedges gives rise to basis risk, but benefits, potentially, from lower costs to the hedger and greater liquidity. We focus on quantification of optimal hedge ratios and hedge effectiveness and investigate how robust these quantities are relative to inclusion of recalibration risk, parameter uncertainty and Poisson risk. We find that strategies are robust relative to the inclusion of parameter uncertainty and Poisson risk. In contrast, single-instrument hedging strategies are found to lack robustness relative to the inclusion of recalibration risk at the future valuation date, although we also demonstrate that some hedging instruments are more robust than others. To address this problem, we develop multi-instrument hedging strategies that are robust relative to recalibration risk.

We use a case study of a pension plan wishing to hedge the longevity risk in its pension liabilities at a future date. The plan has the choice of using either a customised hedge or an index hedge, with the degree of hedge effectiveness being closely related to the correlation between the value of the hedge and the value of the pension liability. The key contribution of this paper is to show how correlation and, therefore, hedge effectiveness can be broken down into contributions from a number of distinct types of risk factor. Our decomposition of the correlation indicates that population basis risk has a significant influence on the correlation. But recalibration risk as well as the length of the recalibration window are also important, as is cohort effect uncertainty. Having accounted for recalibration risk, parameter uncertainty and Poisson risk have only a marginal impact on hedge effectiveness. Our case study shows that longevity risk can be substantially hedged using index hedges as an alternative to customised longevity hedges and that, as a consequence, index longevity hedges – in conjunction with the other components of an ALM strategy – can provide an effective and lower cost alternative to both a full buy-out of pension liabilities or even to a strategy using customised longevity hedges.