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The stochastic nature of future mortality arises from both period (time-related) and cohort (year-of-birth-related) effects. Existing index-based longevity hedging strategies mitigate the risk associated with period effects, but often overlook cohort effects. The negligence of cohort effects may lead to sub-optimal hedge effectiveness, if the liability being hedged is a deferred pension or annuity which involves cohorts that are not covered by the data sample. In this paper, we propose a new hedging strategy that incorporates both period and cohort effects. The resulting longevity hedge is a value hedge, reducing the uncertainty surrounding the τ-year ahead value of the liability being hedged. The proposed method is illustrated with data from the male population of England and Wales. It is found that the benefit of incorporating cohort effects into a longevity hedging strategy depends heavily on the persistence of cohort effects and the choice of q-forwards.

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This Special Issue of Insurance: Mathematics and Economics contains 16 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity 15: The Fifteenth International Longevity Risk and Capital Markets Solutions Conference that was held in Washington DC on 12-13 September 2019. It was hosted by the Pensions Institute at City, University of London.

This Special Issue of Insurance: Mathematics and Economics contains 16 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity 15: The Fifteenth International Longevity Risk and Capital Markets Solutions Conference that was held in Washington DC on 12-13 September 2019. It was hosted by the Pensions Institute at City, University of London.

This Special Issue of Insurance: Mathematics and Economics contains 16 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity 15: The Fifteenth International Longevity Risk and Capital Markets Solutions Conference that was held in Washington DC on 12-13 September 2019. It was hosted by the Pensions Institute at City, University of London.

This Special Issue of the Insurance: Mathematics and Economics contains 16 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity 15: The Fifteenth International Longevity Risk and Capital Markets Solutions Conference that was held in Washington DC on 12-13 September 2019. It was hosted by the Pensions Institute at City, University of London.
Longevity risk and related capital market solutions have grown increasingly important in recent years, both in academic research and in the markets we refer to as the Life Market, i.e., the capital market that trades longevity-linked assets and liabilities. Mortality improvements around the world are putting more and more pressure on governments, pension funds, life insurance companies, as well as individuals, to deal with the longevity risk they face. At the same time, capital markets can, in principle, provide vehicles to hedge longevity risk effectively and transfer the risk from those unwilling or unable to manage it to those willing to invest in this risk in exchange for appropriate risk-adjusted returns or to those who have a counterpoising risk that longevity risk can hedge, e.g., life offices and reinsurers with mortality risk on their books. Many new investment products have been created both by the insurance/reinsurance industry and by the capital markets. Mortality catastrophe bonds are an early example of a successful insurance-linked security. Some new innovative capital market solutions for transferring longevity risk include longevity (or survivor) bonds, longevity (or survivor) swaps, mortality (or q-) forward contracts and reinsurance sidecars. The aim of the International Longevity Risk and Capital Markets Solutions Conferences is to bring together academics and practitioners from all over the world to discuss and analyze these exciting new developments.

This Special Issue of the Insurance: Mathematics and Economics contains 16 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity 15: The Fifteenth International Longevity Risk and Capital Markets Solutions Conference that was held in Washington DC on 12-13 September 2019. It was hosted by the Pensions Institute at City, University of London.
Longevity risk and related capital market solutions have grown increasingly important in recent years, both in academic research and in the markets we refer to as the Life Market, i.e., the capital market that trades longevity-linked assets and liabilities. Mortality improvements around the world are putting more and more pressure on governments, pension funds, life insurance companies, as well as individuals, to deal with the longevity risk they face. At the same time, capital markets can, in principle, provide vehicles to hedge longevity risk effectively and transfer the risk from those unwilling or unable to manage it to those willing to invest in this risk in exchange for appropriate risk-adjusted returns or to those who have a counterpoising risk that longevity risk can hedge, e.g., life offices and reinsurers with mortality risk on their books. Many new investment products have been created both by the insurance/reinsurance industry and by the capital markets. Mortality catastrophe bonds are an early example of a successful insurance-linked security. Some new innovative capital market solutions for transferring longevity risk include longevity (or survivor) bonds, longevity (or survivor) swaps, mortality (or q-) forward contracts and reinsurance sidecars. The aim of the International Longevity Risk and Capital Markets Solutions Conferences is to bring together academics and practitioners from all over the world to discuss and analyze these exciting new developments.

Longevity risk and capital markets: the 2018–19 update - David Blake, Andrew J. G. Cairns

This Special Issue of the Annals of Actuarial Science contains 12 contributions to the academic literature all dealing with longevity risk and capital markets. Draft versions of the papers were presented at Longevity 14: The Fourteenth International Longevity Risk and Capital Markets Solutions Conference that was held in Amsterdam on 20-21 September 2018. It was hosted by the Pensions Institute at Cass Business School and the Netspar Network for Studies on Pensions, Ageing and Retirement .

Recently, the actuarial professions in various countries have adopted an innovative two-dimensional approach to projecting future mortality. In contrast to the conventional approach, the two-dimensional approach permits mortality improvement rates to vary with not only age but also time. Despite being an important breakthrough, the currently used two-dimensional mortality improvement scales are subject to several limitations, most notably a heavy reliance on subjective judgments and a lack of measures of uncertainty. In view of these limitations, in this paper we introduce a new model known as the heat wave model, in which short- and long-term mortality improvements are treated respectively as ‘heat waves’ that taper off over time and ‘background improvements’ that always exist. Using the heat wave model, one can derive two-dimensional mortality improvement scales that entail minimal subjective judgment and include measures of the uncertainty.

Many of the existing index-based longevity hedging strategies focus on the reduction in variance. However, solvency capital requirements are typically based on the τ-year-ahead Value-at-Risk, with τ = 1 under Solvency II. Optimizing a longevity hedge using variance minimization is particularly inadequate when the cost of hedging is nonzero and mortality improvements are driven by a skewed and/or heavy-tailed distribution. In this article, we contribute a method to formulate a value hedge that aims to minimize the Value-at-Risk of the hedged position over a horizon of τ years. The proposed method works with all stochastic mortality models that can be formulated in a state-space form, even when a non normal distributional assumption is made. We further develop a technique to expedite the evaluation of a value longevity hedge. By utilizing the generic assumption that the innovations in the stochastic processes for the period and cohort effects are not serially correlated, the proposed technique spares us from the need for nested simulations that are generally required when evaluating a value hedge.

Longevity Three: The Third International Longevity Risk and Capital Markets Solutions Conference was held in Taipei, Taiwan on 20-21 July 2007. It was hosted by National Chengchi University. Mortality improvements around the world are putting more pressure on governments, pension funds, life insurance companies as well as individuals to deal with the increasing longevity risk they face. Financial markets, on the other hand, can in principle provide vehicles to hedge longevity risk effectively. Many new investment products have been created both by the insurance/reinsurance industry and by the capital markets. Mortality catastrophe bonds are an example of a successful insurance-linked security. Some new innovative capital market solutions for transferring longevity risk include survivor bonds, reverse mortgages, longevity-linked swaps and forward contracts. The aim of the International Longevity Risk and Capital Markets Solutions Conferences is to bring together academics and practitioners from all over the world to discuss and analyze these exciting new developments. The first conference was held at Cass Business School in London in February 2005. This conference was prompted by the announcement of the Swiss Re mortality catastrophe bond in December 2003 and the EIB/BNP/PartnerRe longevity bond in November 2004. The second conference was held in April 2006 in Chicago and hosted by the Katie School at Illinois State University. In the intervening period, there were further issues of mortality catastrophe bonds, as well as the release of the Credit Suisse Longevity Index. Life settlement securitizations were also beginning to take place in the US. In the UK, new life companies backed by global investment banks and private equity firms were setting up for the express purpose of buying out the defined benefit pension liabilities of UK corporations. Goldman Sachs announced it was setting up such a buy-out company itself because the issue of pension liabilities was beginning to impede its mergers and acquisitions activities. So there was now clear evidence that a new global capital market in longevity risk transference was beginning to emerge. However, as with many other economic activities, not all progress follows a smooth path. The EIB/BNP/PartnerRe longevity bond did not attract sufficient investor interest and was withdrawn in late 2005. But a great deal was learned from this about the conditions and requirements needed to launch a successful capital market.

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.

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.

This paper aims to provide accurate approximations for the quantiles of the conditional expected present value of the payments made by the annuity provider, given the future path of the Lee-Carter time index. Conditional cohort and period life expectancies are also considered. The paper also addresses some associated simulation issues, which, hitherto, have been unresolved.

The huge economic significance of longevity risk for corporations, governments, and individuals has begun to be recognized and quantified. By virtue of its size and prevalence, longevity risk is the most significant life-related risk exposure in financial terms and poses a potential threat to the whole system of retirement income provision. This article reviews the birth and development of the Life Market, the new market related to the transfer of longevity and mortality risks. We note that the emergence of a traded market in longevity-linked capital market instruments could act as a catalyst to help facilitate the development of annuity markets both in the developed and the developing world and protect the long-term viability of retirement income provision globally.

Traditionally, actuaries have modeled mortality improvement using determin- istic reduction factors, with little consideration of the associated uncertainty. As mortality improvement has become an increasingly significant source of financial risk, it has become important to measure the uncertainty in the forecasts. Proba- bilistic confidence intervals provided by the widely accepted Lee-Carter model are known to be excessively narrow, due primarily to the rigid structure of the model. In this paper, we relax the model structure by incorporating heterogeneity in each age-period cell. The proposed extension not only provides a better goodness-of-fit based on standard model selection criteria, but also ensures more conservative in- terval forecasts of central death rates and hence can better reflect the uncertainty in the forecasts. We illustrate the results using Canadian population mortality data.

In examining basis risk in index longevity hedges, it is important not to ignore the dependence between the population underlying the hedging instrument and the population being hedged. We consider four extensions to the Lee-Carter model that incorporate such dependence: Both populations are jointly driven by the same single time-varying index, the two populations are cointegrated, the populations depend on a common age factor, and there is an augmented com-mon factor model in which a population-specific time-varying index is added to the common factor model with the property that it will tend toward a certain constant level over time. Using data from the female populations of Canada and the United States, we show the augmented common factor model is preferred in terms of both goodness-of-fit and ex post forecasting per-formance. This model is then used to quantify the basis risk in a longevity hedge of 65-year old Canadian females structured using a portfolio of q-forward contracts predicated on U.S. female population mortality. The hedge effectiveness is estimated at 56% on the basis of longevity value-at-risk and 81.61% on the basis of longevity risk reduction.

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This study sets out a framework to evaluate the goodness of fit of stochastic mortality models and applies it to six different models estimated using English & Welsh male mortality data over ages 64–89 and years 1961–2007. The methodology exploits the structure of each model to obtain various residual series that are predicted to be iid standard normal under the null hypothesis of model adequacy. Goodness of fit can then be assessed using conventional tests of the predictions of iid standard normality. The models considered are: Lee and Carter’s (1992) one-factor model, a version of Renshaw and 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 the Renshaw–Haberman model, the 2006 Cairns–Blake–Dowd two-factor model and two generalized versions of the latter that allow for a cohort-effect. For the data set considered, there are some notable differences amongst the different models, but none of the models performs well in all tests and no model clearly dominates the others.

In the Lee–Carter framework, future survival probabilities are random variables with an intricate distribution function. In large homogeneous portfolios of life annuities, value-at-risk or conditional tail expectation of the total yearly payout of the company are approximately equal to the corresponding quantities involving random survival probabilities. This paper aims to derive some bounds in the increasing convex (or stop-loss) sense on these random survival probabilities. These bounds are obtained with the help of comonotonic upper and lower bounds on sums of correlated random variables.

Mortality patterns and trajectories in closely related populations are likely to be similar in some respects, and differences are unlikely to increase in the long run. It should therefore be possible to improve the mortality forecasts for individual countries by taking into account the patterns in a larger group. Using the Human Mortality Database, we apply the Lee-Carter model to a group of populations, allowing each its own age pattern and level of mortality but imposing shared rates of, change by age. Our forecasts also allow divergent patterns to continue for a while before tapering off. We forecast greater longevity gains for the United States and lesser ones for Japan relative to separate forecasts.

The Lee-Carter mortality model provides a structure for stochastically modeling mortality rates incorporating both time (year) and age mortality dynamics. Their model is constructed by modeling the mortality rate as a function of both an age and a year effect. Recently the MBMM model (Mitchell et al. 2013) showed the Lee Carter model can be improved by fitting with the growth rates of mortality rates over time and age rather than the mortality rates themselves. The MBMM modification of the Lee-Carter model performs better than the original and many of the subsequent variants. In order to model the mortality rate under the martingale measure and to apply it for pricing the longevity derivatives, we adapt the MBMM structure and introduce a Lévy stochastic process with a normal inverse Gaussian (NIG) distribution in our model. The model has two advantages in addition to better fit: first, it can mimic the jumps in the mortality rates since the NIG distribution is fat-tailed with high kurtosis, and, second, this mortality model lends itself to pricing of longevity derivatives based on the assumed mortality model. Using the Esscher transformation we show how to find a related martingale measure, allowing martingale pricing for mortality/longevity risk-related derivatives. Finally, we apply our model to pricing a q-forward longevity derivative utilizing the structure proposed by Life and Longevity Markets Association.

When hedging longevity risk with standardized contracts, the hedger needs to calibrate the hedge carefully so that it can effectively reduce the risk. In this article, we present a calibration method that is based on matching mortality rate sensitivities. Specifically, we introduce a measure called key q-duration, which allows us to estimate the price sensitivity of a life-contingent liability to each portion of the underlying mortality curve. Given this measure, one can easily construct a longevity hedge with a small number of q-forward contracts. We further propose an extension for hedging the longevity risk associated with multiple birth cohorts, and another extension for accommodating population basis risk.

This paper looks at the development of dynamic hedging strategies for typical pen-sion plan liabilities using longevity-linked hedging instruments. Progress in this area has been hindered by the lack of closed-form formulas for the valuation of mortality-linked liabilities and assets, and the consequent requirement for simulations within simulations. We propose use of the probit function along with a Taylor expansion to approximate longevity-contingent values. This makes it possible to develop and implement computationally efficient, discrete-time Delta hedging strategies using q-forwards as hedging instruments.

A new market for so-called mortality derivatives is now appearing with survivor swaps (also called mortality swaps), longevity bonds and other specialized solutions. The development of these new financial instruments is triggered by the increased focus on the systematic mortality risk inherent in life insurance contracts, and their main focus is thus to allow the life insurance companies to hedge their systematic mortality risk. At the same time this new class of financial contracts is interesting from an investor's point of view since they increase the possibility for an investor to diversify the investment portfolio. The systematic mortality risk stems from the uncertainty related to the future development of the mortality intensities. Mathematically this uncertainty is described by modeling the underlying mortality intensities via stochastic processes. We consider two different portfolios of insured lives, where the underlying mortality intensities are correlated, and study the combined financial and mortality risk inherent in a portfolio of general life insurance contracts. In order to hedge this risk we allow for investments in survivor swaps and derive risk-minimizing strategies in markets where such contracts are available. The strategies are evaluated numerically.

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 ben-efits, 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, parame-ter 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 recali-bration risk at the future valuation date, although we also demonstrate that some hedging instruments are more robust than others. To address this problem, we de-velop multi-instrument hedging strategies that are robust relative to recalibration risk.

This study sets out a backtesting framework applicable to the multi-period-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: 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 of Currie (2006), which is a simplified version of Renshaw-Haberman; Cairns, Blake and Dowd’s 2006 two-factor model; and two generalised versions of the latter 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 there is little difference between the performances of five of the models. The remaining model, however, shows forecast instability. The study also finds that density forecasts that allow for uncertainty in the parameters of the mortality model are more plausible than forecasts that do not allow for such uncertainty.

This paper studies the hedging problem of life insurance policies, when the mortality and interest rates are stochastic. We focus primarily on stochastic mortality. We represent death arrival as the first jump time of a doubly stochastic process, i.e. a jump process with stochastic intensity. We propose a Delta-Gamma Hedging technique for mortality risk in this context. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its "forecast" today, the instantaneous forward intensity. We specialize the hedging technique first to the case in which survival intensities are affine, then to Ornstein-Uhlenbeck and Feller processes, providing actuarial justifications for this restriction. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk, when the forward interest rate follows a constant-parameter Hull and White process. We provide a UK calibrated example of Delta and Gamma Hedging of both mortality and interest rate 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 customized 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 customized 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 customized longevity hedges.

This paper develops a framework for developing forecasts of future mortality rates. We discuss the suitability of six stochastic mortality models for forecasting future mortality and estimating the density of mortality rates at different ages. In particular, the models are assessed individually with reference to the following qualitative criteria that focus on the plausibility of their forecasts: biological reasonableness; the plausibility of predicted levels of uncertainty in forecasts at different ages; and the robustness of the forecasts relative to the sample period used to fit the model. An important, though unsurprising, conclusion is that a good fit to historical data does not guarantee sensible forecasts. We also discuss the issue of model risk, common to many modelling situations in demography and elsewhere. We find that even for those models satisfying our qualitative criteria, there are significant differences among central forecasts of mortality rates at different ages and among the distributions surrounding those central forecasts.

We derive the optimal life-cycle portfolio choice and consumption pattern for households facing uncertain labor income, risky capital market, and mortality risk. In addition to stocks and bonds, the households have access to deferred annuities. Deferred payout life annuities are financial contracts providing life-long income to the annuitant after a specified period of time conditional on survival. We find that deferred annuities play an important role in household portfolios and generate significant welfare gains. Households with high benefits from state pensions, moderate risk aversion and moderate labor income risk purchase deferred annuities from age 40 and gradually increase their portfolio share. At retirement, deferred annuities account for 78% of total financial wealth. Households with low state pensions and high labor income risk purchase more annuities and earlier. Uncertainty with respect to future mortality rates has the same effect, i.e. household hedge against longevity risks using deferred annuities.

In life insurance, actuaries have traditionally calculated premiums and reserves using a deterministic mortality intensity, which is a function of the age of the insured only. Here, we model the mortality intensity as a stochastic process. This allows us to capture two important features of the mortality intensity: Time dependency and uncertainty of the future development. The advantage of introducing a stochastic mortality intensity is twofold. Firstly, it gives more realistic premiums and reserves, and secondly, it quantifies the risk of the insurance companies associated with the underlying mortality intensity. Having introduced a stochastic mortality intensity, we study possible ways of transferring the systematic mortality risk to other parties. One possibility is to introduce mortality-linked insurance contracts. Here the premiums and/or benefits are linked to the development of the mortality intensity, thereby transferring the systematic mortality risk to the insured. Alternatively, the insurance company can transfer some or all of the systematic mortality risk to agents in the financial market by trading derivatives depending on the mortality intensity.

For many years, the longevity risk of individuals has been underestimated, as survival probabilities have improved across the developed world. The uncertainty and volatility of future longevity has posed significant risk issues for both individuals and product providers of annuities and pensions. This paper investigates the effectiveness of static hedging strategies for longevity risk management using longevity bonds and derivatives (q-forwards) for the retail products: life annuity, deferred life annuity, indexed life annuity, and variable annuity with guaranteed lifetime benefits. Improved market and mortality models are developed for the underlying risks in annuities. The market model is a regime-switching vector error correction model for GDP, inflation, interest rates, and share prices. The mortality model is a discrete-time logit model for mortality rates with age dependence. Models were estimated using Australian data. The basis risk between annuitant portfolios and population mortality was based on UK experience. Results show that static hedging using q-forwards or longevity bonds reduces the longevity risk substantially for life annuities, but significantly less for deferred annuities. For inflation-indexed annuities, static hedging of longevity is less effective because of the inflation risk. Variable annuities provide limited longevity protection compared to life annuities and indexed annuities, and as a result longevity risk hedging adds little value for these products.

This paper considers the problem of valuating and hedging life insurance contracts that are subject to systematic mortality risk in the sense that the mortality intensity of all policy-holders is affected by some underlying stochastic processes. In particular, this implies that the insurance risk cannot be eliminated by increasing the size of the portfolio and appealing to the law of large numbers. We propose to apply techniques from incomplete markets in order to hedge and valuate these contracts. We consider a special case of the affine mortality structures considered by Dahl [Dahl, M., 2004. Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insurance Math. Econom. 35, 113–136], where the underlying mortality process is driven by a time-inhomogeneous Cox–Ingersoll–Ross (CIR) model. Within this model, we study a general set of equivalent martingale measures, and determine market reserves by applying these measures. In addition, we derive risk-minimizing strategies and mean-variance indifference prices and hedging strategies for the life insurance liabilities considered. Numerical examples are included, and the use of the stochastic mortality model is compared with deterministic models.

Although annuities provide longevity insurance that should be attractive to households facing an uncertain lifespan, rates of voluntary annuitization remain extremely low. We evaluate the Advanced Life Deferred Annuity, an annuity purchased at retirement, providing an income commencing in advanced old age. Using numerical optimization, we show that it would provide a substantial proportion of the longevity insurance provided by an immediate annuity, at much lower cost. At plausible levels of actuarial unfairness, households should prefer it to both immediate and postponed annuitization and an optimal decumulation of unannuitized wealth. Few households would suffer significant losses were it used as a 401(k) plan default.

In this article, we consider the evolution of the post-age-60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so-called longevity risk. We introduce a two-factor stochastic model for the development of this curve through time. The first factor affects mortality-rate dynamics at all ages in the same way, whereas the second factor affects mortality-rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly.
A key component of the article is the proposal and development of a method for calculating the market risk-adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.

The deadly Spanish flu pandemic of 1918 killed more people than World War I. Given concerns over newly emerging infections, there is much interest in learning why the 1918 influenza virus was so virulent. In his Perspective, [Holmes][1] discusses two recent studies that resolve the structures of the hemagglutinin proteins of the 1918 virus and of other related strains ([ Gamblin et al .][2]; [ Stevens et al .][3]). As Holmes explains, the structures reveal that the 1918 virus hemagglutinin retains many of the characteristics of its avian ancestors that may have imbued this virus with its ferocious pathogenicity.
[1]: http://www.sciencemag.org/cgi/content/full/303/5665/1787
[2]: http://www.sciencemag.org/cgi/content/short/303/5665/1838
[3]: http://www.sciencemag.org/cgi/content/short/303/5665/1866

Late life mortality patterns are of crucial interest to actuaries assessing risk of longevity, most obviously for annuities and defined benefit pension schemes. The stability of public finances is also affected, as the governments have very substantial risk of longevity in the form of state benefits and public sector pension schemes. One important explanatory variable for late life mortality patterns is year of birth. Previous work has demonstrated various techniques for detecting such patterns, but always with long time series of mortality rates. The paper describes two alternative ways to detect such patterns, even with missing population data or the absence of a time series. The paper finds support for the idea that different birth cohorts have different rates of aging. Copyright 2008 Royal Statistical Society.