Andrew J. G. Cairns

A quantitative comparison of stochastic mortality models using data from England and wales and the United States

Abstract We compare quantitatively eight stochastic models explaining improvements in mortality rates in England and Wales and in the United States. On the basis of the Bayes Information Criterion (BIC), we find that, for higher ages, an extension of the Cairns-Blake-Dowd (CBD) model that incorporates a cohort effect fits the England and Wales males data best, while for U.S. males data, the Renshaw and Haberman (RH) extension to the Lee and Carter model that also allows for a cohort effect provides the best fit. However, we identify problems with the robustness of parameter estimates under the RH model, calling into question its suitability for forecasting. A different extension to the CBD model that allows not only for a cohort effect, but also for a quadratic age effect, while ranking below the other models in terms of the BIC, exhibits parameter stability across different time periods for both datasets. This model also shows, for both datasets, that there have been approximately linear improvements over time in mortality rates at all ages, but that the improvements have been greater at lower ages than at higher ages, and that there are significant cohort effects.

Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk

It is now an accepted fact that stochastic mortality { the risk that actual future trends in mortality might difier from those anticipated { is an important risk factor in both life insurance and pensions. As such it afiects how fair values, premium rates, and risk reserves are calculated. This paper makes use of the similarities between the force of mortality and interest rates to show how we can model mortality risks and price mortality-related instruments using adaptations of the arbitrage-free pricing frameworks that have been developed for interest-rate derivatives. In so doing, it develops a range of arbitragefree (or risk-neutral) frameworks for pricing and hedging mortality risk that allow for both interest and mortality factors to be stochastic. The difierent frameworks that we describe { short-rate models, forward-mortality models, positive-mortality models and mortality market models { are all based on positive-interest-rate modelling frameworks since the force of mortality can be treated in a similar way to the short-term risk-free rate of interest.

Living with mortality: Longevity bonds and other mortality-linked securities

This paper addresses the problem of longevity risk — the risk of uncertain aggregate mortality — and discusses the ways in which life assurers, annuity providers and pension plans can manage their exposure to this risk. In particular, it focuses on how they can use mortality-linked securities and over-the-counter contracts — some existing and others still hypothetical — to manage their longevity risk exposures. It provides a detailed analysis of two such securities — the Swiss Re mortality bond issued in December 2003 and the EIB/BNP longevity bond announced in November 2004. It then looks at the universe of hypothetical mortality-linked securities — other forms of longevity bonds, swaps, futures and options — and investigates their potential uses. It also addresses implementation issues, and draws lessons from the experiences of other derivative contracts. Particular attention is paid to the issues involved with the construction and use of mortality indices, the management of the associated credit risks, and possible barriers to the development of markets for these securities. It suggests that these implementation difficulties are essentially teething problems that will be resolved over time, and so leave the way open to the development of flourishing markets in a brand new class of securities.

Pensionmetrics: stochastic pension plan design and value-at-risk during the accumulation phase

We estimate values-at-risk (VaR) in the accumulation phase of defined-contribution pension plans. We examine a range of asset-return models (including stationary moments, regime-switching and fundamentals models) and a range of asset-allocation strategies (both static and with simple dynamic forms, such as lifestyle, threshold and constant proportion portfolio insurance). We draw four conclusions from our investigations. First, we find that defined-contribution (DC) plans can be extremely risky relative to a defined-benefit (DB) benchmark (far more so than most pension plan professionals would be likely to admit). Second, we find that the VaR estimates are very sensitive to the choice of asset-allocation strategy. The VaR estimates are also sensitive, but to a lesser extent, to both the asset-returns model used and its parameterisation. The choice of asset-returns model is found to be the least significant of the three. Third, a static asset-allocation strategy with a high equity weighting delivers substantially better results than any of the dynamic strategies investigated over the long term (40 years) of the sample policy. This is important given that lifestyle strategies are the cornerstone of many DC plans. Fourth, conservative bond-based asset-allocation strategies require substantially higher contribution rates than more risky equity-based strategies if the same retirement pension is to be achieved.

Bayesian Stochastic Mortality Modelling for Two Populations

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.

Modelling and Management of Mortality Risk: A Review

In the flrst 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 flnancial 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.

A discussion of parameter and model uncertainty in insurance

Abstract In this paper, we consider the process of modelling uncertainty. In particular, we are concerned with making inferences about some quantity of interest which, at present, has been unobserved. Examples of such a quantity include the probability of ruin of a surplus process, the accumulation of an investment, the level or surplus or deficit in a pension fund and the future volume of new business in an insurance company. Uncertainty in this quantity of interest, y , arises from three sources: (1) uncertainty due to the stochastic nature of a given model; (2) uncertainty in the values of the parameters in a given model; (3) uncertainty in the model underlying what we are able to observe and determining the quantity of interest. It is common in actuarial science to find that the first source of uncertainty is the only one which receives rigorous attention. A limited amount of research in recent years has considered the effect of parameter uncertainty, while there is still considerable scope for development of methods which deal in a balanced way with model risk. Here we discuss a methodology which allows all three sources of uncertainty to be assessed in a more coherent fashion.

Backtesting Stochastic Mortality Models: An Ex-Post Evaluation of Multi-Period Ahead-Density Forecasts

Abstract This study sets out a backtesting framework applicable to the multiperiod-ahead forecasts from stochastic mortality models and uses it to evaluate the forecasting performance of six different stochastic mortality models applied to English & Welsh male mortality data. The models considered are the following: Lee-Carter’s 1992 one-factor model; a version of Renshaw-Haberman’s 2006 extension of the Lee-Carter model to allow for a cohort effect; the age-period-cohort model, which is a simplified version of Renshaw-Haberman; Cairns, Blake, and Dowd’s 2006 two-factor model; and two generalized versions of the last named with an added cohort effect. For the data set used herein, the results from applying this methodology suggest that the models perform adequately by most backtests and that prediction intervals that incorporate parameter uncertainty are wider than those that do not. We also find little difference between the performances of five of the models, but the remaining model shows considerable forecast instability.

A Gravity Model of Mortality Rates for Two Related Populations

Abstract The mortality rate dynamics between two related but different-sized populations are modeled consistently using a new stochastic mortality model that we call the gravity model. The larger population is modeled independently, and the smaller population is modeled in terms of spreads (or deviations) relative to the evolution of the former, but the spreads in the period and cohort effects between the larger and smaller populations depend on gravity or spread reversion parameters for the two effects. The larger the two gravity parameters, the more strongly the smaller population’s mortality rates move in line with those of the larger population in the long run. This is important where it is believed that the mortality rates between related populations should not diverge over time on grounds of biological reasonableness. The model is illustrated using an extension of the Age-Period-Cohort model and mortality rate data for English and Welsh males representing a large population and the Continuous Mortality Investigation assured male lives representing a smaller related population.

Longevity hedging 101: A framework for longevity basis risk analysis and hedge effectiveness

Abstract Basis risk is an important consideration when hedging longevity risk with instruments based on longevity indices, since the longevity experience of the hedged exposure may differ from that of the index. As a result, any decision to execute an index-based hedge requires a framework for (1) developing an informed understanding of the basis risk, (2) appropriately calibrating the hedging instrument, and (3) evaluating hedge effectiveness. We describe such a framework and apply it to a U.K. case study, which compares the population of assured lives from the Continuous Mortality Investigation with the England and Wales national population. The framework is founded on an analysis of historical experience data, together with an appreciation of the contextual relationship between the two related populations in social, economic, and demographic terms. Despite the different demographic profiles, the case study provides evidence of stable long-term relationships between the mortality experiences of the two populations. This suggests the important result that high levels of hedge effectiveness should be achievable with appropriately calibrated, static, index-based longevity hedges. Indeed, this is borne out in detailed calculations of hedge effectiveness for a hypothetical pension portfolio where the basis risk is based on the case study. A robustness check involving populations from the United States yields similar results.