Longevity bond premiums: The extreme value approach and risk cubic pricing

The purpose of this study is to analyze the securitization of longevity risk with an emphasis on longevity risk modeling and longevity bond premium pricing. Various longevity derivatives have been proposed, and the capital market has experienced one unsuccessful attempt by the European Investment Bank (EIB) in 2004. After carefully analyzing the pros and cons of previous securitizations, we present our proposed longevity bonds, whose payoffs are structured as a series of put option spreads. We utilize a random walk model with drift to fit small variations of mortality improvements and employ extreme value theory to model rare longevity events. Our method is a new approach in longevity risk securitization, which has the advantage of both capturing mortality improvements within sample and extrapolating rare, out-of- sample longevity events. We demonstrate that the risk cubic model developed for pricing catastrophe bonds can be applied to mortality and longevity bond pricing and use the model to calculate risk premiums for longevity bonds.     

Stochastic portfolio specific mortality and the quantification of mortality basis risk

In the last decade a vast literature on stochastic mortality models has been developed. However, these models are often not directly applicable to insurance portfolios because:

(a) For insurers and pension funds it is more relevant to model mortality rates measured in insured amounts instead of measured in the number of policies.

(b) Often there is not enough insurance portfolio specific mortality data available to fit such stochastic mortality models reliably.

Therefore, in this paper a stochastic model is proposed for portfolio specific mortality experience. Combining this stochastic process with a stochastic country population mortality process leads to stochastic portfolio specific mortality rates, measured in insured amounts. The proposed stochastic process is applied to two insurance portfolios, and the impact on the Value at Risk for longevity risk is quantified. Furthermore, the model can be used to quantify the basis risk that remains when hedging portfolio specific mortality risk with instruments of which the payoff depends on population mortality rates.     

Securitization, structuring and pricing of longevity risk

Pricing and risk management for longevity risk have increasingly become major challenges for life insurers and pension funds around the world. Risk transfer to financial markets, with their major capacity for efficient risk pooling, is an area of significant development for a successful longevity product market. The structuring and pricing of longevity risk using modern securitization methods, common in financial markets, have yet to be successfully implemented for longevity risk management. There are many issues that remain unresolved for ensuring the successful development of a longevity risk market. This paper considers the securitization of longevity risk focusing on the structuring and pricing of a longevity bond using techniques developed for the financial markets, particularly for mortgages and credit risk. A model based on Australian mortality data and calibrated to insurance risk linked market data is used to assess the structure and market consistent pricing of a longevity bond. Age dependence in the securitized risks is shown to be a critical factor in structuring and pricing longevity linked securitizations.     

Is the Home Equity Conversion Mortgage in the United States sustainable? Evidence from pricing mortgage insurance premiums and non-recourse provisions using the conditional Esscher transform

The purpose of this paper is to build a modeling and pricing framework to investigate the sustainability of the Home Equity Conversion Mortgage (HECM) program in the United States under realistic economic scenarios, i.e., whether the premium payments cover the fair premiums for the inherent risks in the HECM program. We note that earlier HECM models use static mortality tables, neglecting the dynamics of mortality rates and extreme mortality jumps. The earlier models also assume housing prices follow a geometric Brownian motion, which contradicts the fact that housing prices exhibit strong autocorrelation and varying volatility over time. To solve these problems, we propose a generalized Lee-Carter model with asymmetric jump effects to fit the mortality data, and model the house price index via an ARIMA-GARCH process. We then employ the conditional Esscher transform to price the non-recourse provision of reverse mortgages and compare it with the calculated mortgage insurance premiums. The HECM program turns out to be sustainable based on our model setup and parameter settings.     

Time-consistent mean–variance hedging of longevity risk: Effect of cointegration

This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.     

Swiss coherent mortality model as a basis for developing longevity de-risking solutions for Swiss pension funds: A practical approach

Pension funds in Switzerland are exposed to longevity risk possibly to a greater extent than in many other developed economies. The ground for this is a dearth of financial products to combat longevity risk, with a lack of buy-in and very limited variety of buy-out solutions available. The solutions that do exist frequently come at a very high price and many pension funds are in deficit on a buy-out basis. From our point of view creating an approach for evaluating the longevity risk faced by each pension fund and integrating it into dynamic risk budgeting strategies will help Swiss pension funds better understand the mechanism behind different longevity de-risking solutions and decide on the most suitable as well as affordable solution for them. To develop capital market solutions for longevity hedging strategies it is crucial that both hedgers (pension funds) as well as solution providers are able to quantify the longevity risk in the framework of a holistic risk management and to develop an adequate pricing approach.In this publication we present our stochastic coherent mortality model developed for Swiss pension funds based on the reference population of fifteen countries and discuss the robustness of the forecasts relative to the sample period used to fit the model, biological reasonableness of the forecasts and other modelling parameters as well as possible impact on results. The model has taken into account past single population modelling techniques and allows flexible age effect to capture the spread behaviour introduced by the target population. The augmented terms for the spread function are chosen based on their forecast accuracy and a coherent behaviour is expected in the long term. The idea behind is fairly simple and yields a design with both transparency and robustness. The model usage is not limited to Switzerland.     

Multi-population mortality models: A factor copula approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short-term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA–GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.     

To borrow or insure? Long term care costs and the impact of housing

We assess the impact of housing, the availability of reverse mortgages and long-term care (LTC) insurance on a retiree’s optimal portfolio choice and consumption decisions using a multi-period life cycle model that takes into consideration individual longevity risk, health shocks and house price risk. We determine how much an individual should borrow against their home equity and how much to insure health care costs with LTC insurance. We introduce an endogenous grid method, along with a regression based approach, to improve computational efficiency and avoid the curse of dimensionality. Our results confirm that borrowing against home equity provides higher consumption in earlier years and longevity insurance. LTC insurance transfers wealth from healthy states to disabled states, but reduces early consumption because of the payment of insurance premiums. Housing is an illiquid asset that is important in meeting bequest motives, and it reduces the demand for LTC insurance for the wealthy. We show that the highest welfare benefits come from combining a reverse mortgage with LTC insurance because of strong complementary effects between them. This result highlights the benefits of innovative products that bundle these two products together.     

A common age effect model for the mortality of multiple populations

We introduce a model for the mortality rates of multiple populations. To build the proposed model we investigate to what extent a common age effect can be found among the mortality experiences of several countries and use a common principal component analysis to estimate a common age effect in an age–period model for multiple populations. The fit of the proposed model is then compared to age–period models fitted to each country individually, and to the fit of the model proposed by Li and Lee (2005).Although we do not consider stochastic mortality projections in this paper, we argue that the proposed common age effect model can be extended to a stochastic mortality model for multiple populations, which allows to generate mortality scenarios simultaneously for all considered populations. This is particularly relevant when mortality derivatives are used to hedge the longevity risk in an annuity portfolio as this often means that the underlying population for the derivatives is not the same as the population in the annuity portfolio.     

A step-by-step guide to building two-population stochastic mortality models

Two-population stochastic mortality models play a crucial role in the securitization of longevity risk. In particular, they allow us to quantify the population basis risk when longevity hedges are built from broad-based mortality indexes. In this paper, we propose and illustrate a systematic process for constructing a two-population mortality model for a pair of populations. The process encompasses four steps, namely (1) determining the conditions for biological reasonableness, (2) identifying an appropriate base model specification, (3) choosing a suitable time-series process and correlation structure for projecting period and/or cohort effects into the future, and (4) model evaluation.For each of the seven single-population models from Cairns et al. (2009), we propose two-population generalizations. We derive criteria required to avoid long-term divergence problems and the likelihood functions for estimating the models. We also explain how the parameter estimates are found, and how the models are systematically simplified to optimize the fit based on the Bayes Information Criterion. Throughout the paper, the results and methodology are illustrated using real data from two pairs of populations.     

Valuation of equity-indexed annuities with regime-switching jump diffusion risk and stochastic mortality risk

This paper extends the model and analysis of Lin,Tan and Yang(2009).We assume that the financial market follows a regime-switching jump-diffusion model and the mortality satisfies Lvy process.We price the point to point and annual reset EIAs by Esscher transform method under Merton’s assumption and obtain the closed form pricing formulas.Under two cases:with mortality risk and without mortality risk,the effects of the model parameters on the EIAs pricing are illustrated through numerical experiments.     

Modeling longevity risks using a principal component approach: A comparison with existing stochastic mortality models

This research proposes a mortality model with an age shift to project future mortality using principal component analysis (PCA). Comparisons of the proposed PCA model with the well-known models—the Lee-Carter model, the age-period-cohort model (Renshaw and Haberman, 2006), and the Cairns, Blake, and Dowd model—employ empirical studies of mortality data from six countries, two each from Asia, Europe, and North America. The mortality data come from the human mortality database and span the period 1970-2005. The proposed PCA model produces smaller prediction errors for almost all illustrated countries in its mean absolute percentage error. To demonstrate longevity risk in annuity pricing, we use the proposed PCA model to project future mortality rates and analyze the underestimated ratio of annuity price for whole life annuity and deferred whole life annuity product respectively. The effect of model risk on annuity pricing is also investigated by comparing the results from the proposed PCA model with those from the LC model. The findings can benefit actuaries in their efforts to deal with longevity risk in pricing and valuation.     

Statistical emulators for pricing and hedging longevity risk products

We propose the use of statistical emulators for the purpose of analyzing mortality-linked contracts in stochastic mortality models. Such models typically require (nested) evaluation of expected values of nonlinear functionals of multi-dimensional stochastic processes. Except in the simplest cases, no closed-form expressions are available, necessitating numerical approximation. To complement various analytic approximations, we advocate the use of modern statistical tools from machine learning to generate a flexible, non-parametric surrogate for the true mappings. This method allows performance guarantees regarding approximation accuracy and removes the need for nested simulation. We illustrate our approach with case studies involving (i) a Lee–Carter model with mortality shocks; (ii) index-based static hedging with longevity basis risk; (iii) a Cairns–Blake–Dowd stochastic survival probability model; (iv) variable annuities under stochastic interest rate and mortality.     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

Pricing and hedging of guaranteed minimum benefits under regime-switching and stochastic mortality

This paper presents a novel framework for pricing and hedging of the Guaranteed Minimum Benefits (GMBs) embedded in variable annuity (VA) contracts whose underlying mutual fund dynamics evolve under the influence of the regime-switching model. Semi-closed form solutions for prices and Greeks (i.e. sensitivities of prices with respect to model parameters) of various GMBs under stochastic mortality are derived. Pricing and hedging is performed using an accurate, fast and efficient Fourier Space Time-stepping (FST) algorithm. The mortality component of the model is calibrated to the Australian male population. Sensitivity analysis is performed with respect to various parameters including guarantee levels, time to maturity, interest rates and volatilities. The hedge effectiveness is assessed by comparing profit-and-loss distributions for an unhedged, statically and semi-statically hedged portfolios. The results provide a comprehensive analysis on pricing and hedging the longevity risk, interest rate risk and equity risk for the GMBs embedded in VAs, and highlight the benefits to insurance providers who offer those products.     

Efficient versus inefficient hedging strategies in the presence of financial and longevity (value at) risk

This paper provides a closed-form Value-at-Risk (VaR) for the net exposure of an annuity provider, taking into account both mortality and interest-rate risk, on both assets and liabilities. It builds a classical risk-return frontier and shows that hedging strategies–such as the transfer of longevity risk–may increase the overall risk while decreasing expected returns, thus resulting in inefficient outcomes. Once calibrated to the 2010 UK longevity and bond market, the model gives conditions under which hedging policies become inefficient.     

Mortality risk modeling: Applications to insurance securitization

This paper proposes a stochastic mortality model featuring both permanent longevity jump and temporary mortality jump processes. A trend reduction component describes unexpected mortality improvement over an extended period of time. The model also captures the uneven effect of mortality events on different ages and the correlations among them. The model will be useful in analyzing future mortality dependent cash flows of life insurance portfolios, annuity portfolios, and portfolios of mortality derivatives. We show how to apply the model to analyze and price a longevity security.     

Hedging mortality/longevity risks of insurance portfolios for life insurer/annuity provider and financial intermediary

In this paper, we propose two risk hedge schemes in which a life insurer (an annuity provider) can transfer mortality (longevity) risk of a portfolio of life (annuity) exposures to a financial intermediary by paying the hedging premium of a mortality-linked security. The optimal units of the mortality-linked security which maximize hedge effectiveness for a life insurer (an annuity provider) can be derived as closed-form formulas under the risk hedge schemes. Numerical illustrations show that the risk hedge schemes can significantly hedge the downside risk of loss due to mortality (longevity) risk for the life insurer (annuity provider) under some stochastic mortality models. Besides, finding an optimal weight of a portfolio of life and annuity business, the financial intermediary can reduce the sensitivity to mortality rates but the model risk; a security loading may be imposed on the hedge premium for a higher probability of gain to compensate the financial intermediary for the inevitable model risk.     

Pricing longevity risk with the parametric bootstrap: A maximum entropy approach

In recent years, there has been significant development in the securitization of longevity risk. Various methods for pricing longevity risk have been proposed. In this paper we present an alternative pricing method, which is based on the maximization of the Shannon entropy in physics. Specifically, we propose implementing this pricing method with the parametric bootstrap (Brouhns et al., 2005), which is highly flexible and can be performed under different model assumptions. Through this pricing method we also quantify the impact of cohort effects and parameter uncertainty on prices of mortality-linked securities. Numerical illustrations based on longevity bonds with different maturities are provided.     

Parametric mortality indexes: From index construction to hedging strategies

In this paper, we investigate the construction of mortality indexes using the time-varying parameters in common stochastic mortality models. We first study how existing models can be adapted to satisfy the new-data-invariant property, a property that is required to ensure the resulting mortality indexes are tractable by market participants. Among the collection of adapted models, we find that the adapted Model M7 (the Cairns–Blake–Dowd model with cohort and quadratic age effects) is the most suitable model for constructing mortality indexes. One basis of this conclusion is that the adapted model M7 gives the best fitting and forecasting performance when applied to data over the age range of 40–90 for various populations. Another basis is that the three time-varying parameters in it are highly interpretable and rich in information content. Based on the three indexes created from this model, one can write a standardized mortality derivative called K-forward, which can be used to hedge longevity risk exposures. Another contribution of this paper is a method called key K-duration that permits one to calibrate a longevity hedge formed by K-forward contracts. Our numerical illustrations indicate that a K-forward hedge has a potential to outperform a q-forward hedge in terms of the number of hedging instruments required.