Kevin Dowd

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.

After VaR: The Theory, Estimation, and Insurance Applications of Quantile-Based Risk Measures

We discuss a number of quantile-based risk measures (QBRMs) that have recently been developed in the financial risk and actuarial/insurance literatures. The measures considered include the Value-at-Risk (VaR), coherent risk measures, spectral risk measures, and distortion risk measures. We discuss and compare the properties of these different measures, and point out that the VaR is seriously flawed. We then discuss how QBRMs can be estimated, and discuss some of the many ways they might be applied to insurance risk problems. These applications are typically very complex, and this complexity means that the most appropriate estimation method will often be some form of stochastic simulation.

Currency Competition, Network Externalities and Switching Costs: Towards an Alternative View of Optimum Currency Areas

Existing models of currency competition and monetary union ignore network effects and switching costs. This paper develops a simple model that incorporates these features and shows how it can be used to shed light on observed monetary experience and current issues in international monetary relations. It explains why agents will often be reluctant to switch currencies, highlights the role of expectations, and sheds light on the nature of currency competition and the effect of legal restrictions. Copyright 1993 by Royal Economic Society.

Adjusting for risk:: An improved Sharpe ratio

Abstract This paper proposes a new rule for risk adjustment and performance evaluation. This rule is a generalization of the well-known Sharpe ratio criterion, and under normal conditions enables a manager to correctly assess alternative risky investments. The rule is superior to existing rules such as the standard Sharpe rule and the RAROC, and can make a substantial difference in estimates of required returns.

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.

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.