This paper presents the AffineMortality R package which performs parameter estimation, goodness-of-fit analysis, simulation, and projection of future mortality rates for a set of affine mortality models for use in pricing and reserving. The computational routines build on the univariate Kalman Filtering approach of Koopman and Durbin ((2000). Journal of Time Series Analysis, 21(3), 281–296.) along other numerical methods to enhance the robustness of the results. This paper provides a discussion of how the package works in order to effectively estimate and project survival curves, and describes the available functions. Illustration of the package for mortality analysis of the US male data set is provided.

Longevity risk is the risk that people on average will live longer than expected. That potential increase in life expectancy exposes corporations and pension funds to the risk of having insufficient funds to pay a more extended stream of annuity benefits. Buy-ins, buy-outs, and longevity bonds provide pension funds with insurance and financial market instruments to hedge their longevity risk. The most straightforward instruments and the most robust markets are currently for buy-ins and buy-outs. The model developed here shows that these instruments transfer value to pension holders and, other things being equal, would not be used by firms since shareholder value is reduced. The analysis, however, also shows that these instruments can be used to solve the under-investment problem created by underfunded pension plans and so increase not only the pension fund value but also the corporate stock value.

Cancer is among the leading causes of death in the world, with about 10 million deaths, one in every six deaths, related to cancer in 2020. In Taiwan, cancer insurance is the most popular commercial health product. However, the loss ratio of cancer products increases with policy year and exceeds 100% in many insurance companies. In addition, almost all cancer benefits are significantly limited in order to avoid financial insolvency. In this study, we evaluate the risk of cancer insurance from the perspective of morbidity compression. We use the data from Taiwan's National Health Insurance Research Database. Also, we apply the standardized mortality ratio and the Lee-Carter model to estimate the trend of cancer-related values. We find that cancer incidence rates gradually increase with time, which indicates that the assumption of morbidity compression is violated. On the other hand, the mortality rates of cancer patients decrease significantly annually. Thus, length of life with cancer increases, and so does the cancer insurance premium. We suggest that cancer insurance covers only the first five years of medical expenditure after the insured is diagnosed with cancer.

Pension risk transfer and longevity risk transfer are now growing secular trends. From North America to Europe, companies are de-risking pension plans in near-record volumes and have continued to boldly do so throughout the pandemic—at or near the most favorable pricing experienced in years. The arrival of funded reinsurance on both sides of the Atlantic is bringing reinsurer capital and private assets to support the steady growth in the pension risk transfer market. Additionally, we have observed that the enduring low-rate environment and quest for uncorrelated risk has the world's largest investors directing billions into life reinsurance sidecars. How have these markets thrived during the worst global outbreak in a century? Key research on the pandemic's impact on pensioner life expectancy allowed prices to be set and transactions to proceed through a time of significant uncertainty.

We study a stochastic model for a target benefit pension plan suffering from rising longevity and falling fertility. Policies for postponing retirement are carried out to hedge the payment difficulties caused by the aging population. The plan members’ contributions are set in advance while the pension payments reflect intergenerational equity by a target payment level and intergenerational risk sharing by an adjustment. The pension fund is invested in both a risk-free asset and a risky asset. Applying the stochastic optimal control methods, we derive analytic solutions for optimal investment and benefit payment strategies which minimize the benefit risk. Besides, an optimal delayed retirement age which can hedge against the aging phenomenon under certain parameters is given. Therefore, it can provide a basis for quantifying the delay of retirement time.

One complication in mortality modelling is capturing the impact of risk factors that contribute to mortality differentials between different populations. Evidence has suggested that mortality differentials tend to diminish over age. Classical methods such as the Gompertz law attempt to capture mortality patterns over age using intercept and slope parameters, possibly causing an unjustified mortality crossover at advanced ages when applied independently to different populations. In recent research, Richards (Scandinavian Actuarial Journal 2020(2), 110–127) proposed a Hermite spline (HS) model that describes the age pattern of mortality differentials using one parameter and circumvents an unreasonable crossover by default. The original HS model was applied to pension data at individual level in the age dimension only. This paper extends the method to model population mortality in both age and period dimensions. Our results indicate that in addition to possessing desirable fitting properties, the HS approach can produce accurate mortality forecasts, compared with the Gompertz and P-splines models.

Parametric mortality models permit detailed analysis of risk factors for actuarial work. However, finite data volumes lead to uncertainty over parameter estimates, which in turn gives rise to mis-estimation risk of financial liabilities. Mis-estimation risk can be assessed on a run-off basis by valuing the liabilities with alternative parameter vectors consistent with the covariance matrix. This run-off approach is especially suitable for tasks like pricing portfolio transactions, such as bulk annuities, longevity swaps or reinsurance treaties. However, a run-off approach does not fully meet the requirements of regulatory regimes that view capital requirements through the prism of a finite horizon, such as Solvency II’s one-year approach. This paper presents a methodology for viewing mis-estimation risk over a fixed time frame, and results are given for a specimen portfolio. As expected, we find that time-limited mis-estimation capital requirements increase as the horizon is lengthened or the discount rate is reduced. However, we find that much of the so-called mis-estimation risk in a one-year value-at-risk assessment can actually be driven by idiosyncratic variation, rather than parameter uncertainty. This counter-intuitive result stems from trying to view a long-term risk through a short-term window. As a result, value-at-risk mis-estimation reserves are strongly correlated with idiosyncratic risk. We also find that parsimonious models tend to produce lower mis-estimation risk than less-parsimonious ones.

We propose a new neighbouring prediction model for mortality forecasting. For each mortality rate at age x in year t, mx,t, we construct an image of neighbourhood mortality data around mx,t, that is, Ꜫmx,t (x1, x2, s), which includes mortality information for ages in [x-x1, x+x2], lagging k years (1 ≤ k ≤ s). Combined with the deep learning model – convolutional neural network, this framework is able to capture the intricate nonlinear structure in the mortality data: the neighbourhood effect, which can go beyond the directions of period, age, and cohort as in classic mortality models. By performing an extensive empirical analysis on all the 41 countries and regions in the Human Mortality Database, we find that the proposed models achieve superior forecasting performance. This framework can be further enhanced to capture the patterns and interactions between multiple populations.

This paper examines the impact of uncertainties in the future trends of mortality on annuity values in Singapore's compulsory purchase market. We document persistent population mortality improvement trends over the past few decades, which underscores the importance of longevity risk in this market. Using the money's worth framework, we find that the life annuities delivered expected payouts valued at 1.019–1.185 (0.973–1.170) per dollar of annuity premium for males (females). Even in a low mortality improvement scenario, the annuities provide an expected value exceeding 0.950. This suggests that participants in the national annuity pool have access to attractively priced annuities, regardless of sex, product, and premium invested.

Even though the trend in mortality improvements has experienced several permanent changes in the past, the uncertainty regarding future mortality trends is often left unmodeled when pricing longevity-linked securities. In this paper, we present a stochastic modeling framework for the valuation of longevity-linked securities which explicitly considers the risk of random future changes in the long-term mortality trend. We construct a set of meaningful probability distortions which imply equivalent risk-adjusted pricing measures under which the basic model structure is preserved. Inspired by risk-based capital requirements for (re)insurers, we also establish a cost-of-capital pricing approach which then serves as the appropriate reference framework for finding a reasonable range for the market price of longevity risk. In a numerical application, we demonstrate that our model produces plausible risk loadings and show that a greater proportion of the risk loading is allocated to longer maturities when the risk of random future mortality trend changes is adequately modeled.

This paper quantifies the welfare gains from Social Security when individuals face uninsurable longevity risk. While past researchers have studied this basic question, we do so from a unique perspective. In contrast to traditional macroeconomic models that abstract from specific linkages between parents and children, in our model, children are born to specific parents whose longevity is uncertain. And because parental asset holdings evolve over the life of the parent, children face uninsurable bequest income risk in addition to their own longevity risk. We find that Social Security improves ex ante expected utility by 3.4% of lifetime consumption (for the second generation). Because our baseline analysis assumes full information and optimal hedging of longevity risk, we treat these welfare gains as a conservative estimate, and we show that the gains are significantly larger when individuals fail to hedge their longevity risk.

There is considerable uncertainty regarding changes in future mortality rates. This article investigates the impact of such longevity risk on discounted government annuity benefits for retirees. It is critical to forecast more accurate future mortality rates to improve our estimation of an expected annuity payout. Thus, we utilize the Lee–Carter model, which is well-known as a parsimonious dynamic mortality model. We find strong evidence that female retirees are likely to receive more public lifetime annuity than males in the USA, which is associated with systematic mortality rate differences between genders. A cross-country comparison presents that the current public annuity system would not fully cover retiree's longevity risk. Every additional year of life expectancy leaves future retirees exposed to high risk, arising from high volatility of lifetime annuities. Also, because the growth in life expectancy is higher than the growth of expected public pension, there will be a financial risk to retirees.

In this paper, we first derive closed-form formulas for mortality-interest durations and convexities of the prices of life insurance and annuity products with respect to an instantaneously proportional change and an instantaneously parallel movement, respectively, in μ* (the force of mortality-interest), the addition of μ (the force of mortality) and δ (the force of interest). We then build several mortality-interest duration and convexity matching strategies to determine the weights of whole life insurance and deferred whole life annuity products in a portfolio and evaluate the value at risk and the hedge effectiveness of the weighted portfolio surplus at time zero. Numerical illustrations show that using the mortality-interest duration and convexity matching strategies with respect to an instantaneously proportional change in μ* can more effectively hedge the longevity risk and interest rate risk embedded in the deferred whole life annuity products than using the mortality-only duration and convexity matching strategies with respect to an instantaneously proportional shift or an instantaneously constant movement in μ only.

For life insurers in the United Kingdom (UK), the risk margin is one of the most controversial aspects of the Solvency II regime which came into force in 2016.

The risk margin is the difference between the technical provisions and the best estimate liabilities. The technical provisions are intended to be market-consistent, and so are defined as the amount required to be paid to transfer the business to another undertaking. In practice, the technical provisions cannot be directly calculated, and so the risk margin must be determined using a proxy method; the method chosen for Solvency II is known as the cost-of-capital method.

Following the implementation of Solvency II, the risk margin came under considerable criticism for being too large and too sensitive to interest rate movements. These criticisms are particularly valid for annuity business in the UK – such business is of great significance to the system for retirement provision. A further criticism is that mitigation of the impact of the risk margin has led to an increase in reinsurance of longevity risks, particularly to overseas reinsurers.

This criticism has led to political interest, and the risk margin was a major element of the Treasury Committee inquiry into EU Insurance Regulation.

The working party was set up in response to this criticism. Our brief is to consider both the overall purpose of the risk margin for life insurers and solutions to the current problems, having regard to the possibility of post-Brexit flexibility.

We have concluded that a risk margin in some form is necessary, although its size depends on the level of security desired, and so is primarily a political question.

We have reviewed possible alternatives to the current risk margin, both within the existing cost-of-capital methodology and considering a wide range of alternatives.

We believe that requirements for the risk margin will depend on future circumstances, in particular relating to Brexit, and we have identified a number of possible changes to methodology which should be considered, depending on circumstances.

We compare two contracts for managing systematic longevity risk in retirement: a collective arrangement that distributes the risk among participants, and a market-provided annuity contract. We evaluate the contracts’ appeal with respect to the retiree's welfare, and the viability of the market solution through the financial reward to the annuity provider's equityholders. We find that individuals prefer to bear the risk under a collective arrangement than to insure it with a life insurers' annuity contract subject to insolvency risk (albeit small). Under realistic capital provision hypotheses, the annuity provider is incapable of adequately compensating its equityholders for bearing systematic longevity risk.

We propose a two-regime Markov switching copula to depict the evolution of mortality dependence. One regime represents periods of high dependence and the other regime represents periods of low dependence. Each regime features a regular vine (R-vine) copula that, built on bivariate copulas, provides great flexibility for modelling complex high-dimensional dependence. Our estimated model indicates that the years of recovery from extreme mortality deterioration and the years of health care reform more likely fall into the low regime, while the years in which extreme mortality deteriorating events break out and the peaceful years without major mortality-impacting events more likely fall into the high regime. We use a case study to illustrate how the regime-switching copula can be applied to assess the effectiveness of longevity risk hedge with different beliefs about future mortality dependence evolution incorporated.

This paper updates Living with Mortality published in 2006. It describes how the longevity risk transfer market has developed over the intervening period, and, in particular, how insurance-based solutions – buy-outs, buy-ins and longevity insurance – have triumphed over capital markets solutions that were expected to dominate at the time. Some capital markets solutions – longevity-spread bonds, longevity swaps, q-forwards and tail-risk protection – have come to market, but the volume of business has been disappointingly low. The reason for this is that when market participants compare the index-based solutions of the capital markets with the customised solutions of insurance companies in terms of basis risk, credit risk, regulatory capital, collateral and liquidity, the former perform on balance less favourably despite a lower potential cost. We discuss the importance of stochastic mortality models for forecasting future longevity and examine some applications of these models, e.g. determining the longevity risk premium and estimating regulatory capital relief. The longevity risk transfer market is now beginning to recognise that there is insufficient capacity in the insurance and reinsurance industries to deal fully with demand and new solutions for attracting capital markets investors are now being examined – such as longevity-linked securities and reinsurance sidecars.

For insurance companies in Europe, the introduction of Solvency II leads to a tightening of rules for solvency capital provision. In life insurance, this especially affects retirement products that contain a significant portion of longevity risk (e.g., conventional annuities). Insurance companies might react by price increases for those products, and, at the same time, might think of alternatives that shift longevity risk (at least partially) to policyholders. In the extreme case, this leads to so-called tontine products where the insurance company’s role is merely administrative and longevity risk is shared within a pool of policyholders. From the policyholder’s viewpoint, such products are, however, not desirable as they lead to a high uncertainty of retirement income at old ages. In this article, we alternatively suggest a so-called tonuity that combines the appealing features of tontine and conventional annuity. Until some fixed age (the switching time), a tonuity’s payoff is tontine-like, afterwards the policyholder receives a secure payment of a (deferred) annuity. A tonuity is attractive for both the retiree (who benefits from a secure income at old ages) and the insurance company (whose capital requirements are reduced compared to conventional annuities). The tonuity is a possibility to offer tailor-made retirement products: using risk capital charges linked to Solvency II, we show that retirees with very low or very high risk aversion prefer a tontine or conventional annuity, respectively. Retirees with medium risk aversion, however, prefer a tonuity. In a utility-based framework, we therefore determine the optimal tonuity characterized by the critical switching time that maximizes the policyholder’s lifetime utility.

This paper investigates dynamic hedging strategies for pension and annuity liabilities that are exposed to longevity risk. In particular, we consider a hedger who wishes to minimize the variance of her hedging error using index-based longevity-linked derivatives. To cope with the fact that liquidity of longevity-linked derivatives is still limited, we consider a liquidity constrained case where the hedger can only trade longevity-linked derivatives at a frequency lower than other assets. Time-consistent, closed-form solutions of optimal hedging strategies are obtained under a forward mortality framework. In the numerical illustration, we show that lowering the trading of the longevity-linked derivatives to a 2-year frequency only leads to a slight loss of the hedging performance. Moreover, even when the longevity-linked derivatives are traded at a very low (5-year) frequency, dynamic hedging strategies still significantly outperform the static one.