Comparison and bounds for functionals of future lifetimes consistent with life tables

We derive a new crossing criterion of hazard rates to identify a stochastic order relation between two random variables. We apply this crossing criterion in the context of life tables to derive stochastic ordering results among three families of fractional age assumptions: the family of linear force of mortality functions, the family of quadratic survival functions and the power family. Further, this criterion is used to derive tight bounds for functionals of future lifetimes that exhibit an increasing force of mortality with given one-year survival probabilities. Numerical examples illustrate our findings.     

Modelling and forecasting mortality in Spain

Experience shows that static life tables overestimate death probabilities. As a consequence of this overestimation the premiums for annuities, pensions and life insurance are not what they actually should be, with negative effects for insurance companies or policy-holders. The reason for this overestimation is that static life tables, through being computed for a specific period of time, cannot take into account the decreasing mortality trend over time. Dynamic life tables overcome this problem by incorporating the influence of the calendar when graduating mortality. Recent papers on the topic look for the development of new methods to deal with this dynamism.     

From regulatory life tables to stochastic mortality projections: The exponential decline model

Often in actuarial practice, mortality projections are obtained by letting age-specific death rates decline exponentially at their own rate. Many life tables used for annuity pricing are built in this way. The present paper adopts this point of view and proposes a simple and powerful mortality projection model in line with this elementary approach, based on the recently studied mortality improvement rates. Two main applications are considered. First, as most reference life tables produced by regulators are deterministic by nature, they can be made stochastic by superposing random departures from the assumed age-specific trend, with a volatility calibrated on market or portfolio data. This allows the actuary to account for the systematic longevity risk in solvency calculations. Second, the model can be fitted to historical data and used to produce longevity forecasts. A number of conservative and tractable approximations are derived to provide the actuary with reasonably accurate approximations for various relevant quantities, available at limited computational cost. Besides applications to stochastic mortality projection models, we also derive useful properties involving supermodular, directionally convex and stop-loss orders.     

Mortality modelling with Lévy processes

This paper addresses the modelling of human mortality by the aid of doubly stochastic processes with an intensity driven by a positive Lévy process. We focus on intensities having a mean reverting stochastic component. Furthermore, driving Lévy processes are pure jump processes belonging to the class of α-stable subordinators. In this setting, expressions of survival probabilities are inferred, the pricing is discussed and numerical applications to actuarial valuations are proposed.     

Bayesian graduation of mortality rates: An application to reserve evaluation

This paper presents Bayesian graduation models of mortality rates, using Markov chain Monte Carlo (MCMC) techniques. Graduated annual death probabilities are estimated through the predictive distribution of the number of deaths, which is assumed to follow a Poisson process, considering that all individuals in the same age class die independently and with the same probability. The resulting mortality tables are formulated through dynamic Bayesian models. Calculation of adequate reserve levels is exemplified, via MCMC, making use of the value at risk concept, demonstrating the importance of using “true” observed mortality figures for the population exposed to risk in determining the survival coverage rate.     

Calibrating affine stochastic mortality models using term assurance premiums

In this paper, we focus on the calibration of affine stochastic mortality models using term assurance premiums. We view term assurance contracts as a “swap” in which policyholders exchange cash flows (premiums vs. benefits) with an insurer analogous to a generic interest rate swap or credit default swap. Using a simple bootstrapping procedure, we derive the term structure of mortality rates from a stream of contract quotes with different maturities. This term structure is used to calibrate the parameters of affine stochastic mortality models where the survival probability is expressed in closed form. The Vasicek, Cox-Ingersoll-Ross, and jump-extended Vasicek models are considered for fitting the survival probabilities term structure. An evaluation of the performance of these models is provided with respect to premiums of three Italian insurance companies.     

非整数年龄假设中的二次多项式死亡力研究

由于生命表只提供了整数年龄点上的生存函数值和死亡率值,所以在人口统计和寿险精算中计算非整数年龄处的生存函数值时,需要进行非整数年龄假设。传统的非整数年龄假设有死亡力函数和密度函数在整数年龄处有较大跳跃的缺点,而针对它们连续性的改进甚少,且没有针对死亡力函数二次多项式形式的研究。本文提出了死亡力的二次多项式形式,通过积分约束下的分段抛物插值方法给出了二次多项式死亡力形式对应的非整数年龄假设,并与前人已经提出的各种假设进行了比较,结果表明二次死亡力假设可以更精确地描述生存模型,从而使人口统计和保费、年金的计算更加精确。     

A unisex stochastic mortality model to comply with EU Gender Directive

EU Gender Directive ruled out discrimination against gender in charging premium for insurance products. This prohibition prevents the use of the standard actuarial fairness principle to price life insurance products. According to current actuarial practice, unisex premiums are calculated with a simple weighting rule of the gender-specific life tables. This procedure is likely to violate portfolio fairness principles. Up to our knowledge, in the actuarial literature there is no unisex mortality model that respects the unisex fairness principle. This paper is the first attempt to fill this gap. First, we recall the notion of unisex fairness principle and the corresponding unisex fair premium. Then, we provide a unisex stochastic mortality model for the mortality intensity that is underlying the pricing of a life portfolio of females and males belonging to the same cohort. Finally, we calibrate the unisex mortality model using the unisex fairness principle. We find that the weighting coefficient between the males’ and females’ own mortalities depends mainly on the quote of portfolio relative to each gender, on the age, and on the type of insurance products. The knowledge of a proper unisex mortality model could help life insurance companies to better understanding the nature of the risk of a mixed portfolio.     

A comparison of the Lee-Carter model and AR-ARCH model for forecasting mortality rates

With the decline in the mortality level of populations, national social security systems and insurance companies of most developed countries are reconsidering their mortality tables taking into account the longevity risk. The Lee and Carter model is the first discrete-time stochastic model to consider the increased life expectancy trends in mortality rates and is still broadly used today. In this paper, we propose an alternative to the Lee-Carter model: an AR(1)-ARCH(1) model. More specifically, we compare the performance of these two models with respect to forecasting age-specific mortality in Italy. We fit the two models, with Gaussian and t-student innovations, for the matrix of Italian death rates from 1960 to 2003. We compare the forecast ability of the two approaches in out-of-sample analysis for the period 2004-2006 and find that the AR(1)-ARCH(1) model with t-student innovations provides the best fit among the models studied in this paper.     

Valuation of contingent claims with mortality and interest rate risks

We consider the pricing of life insurance contracts under stochastic mortality and interest rates assumed not independent of each other. Employing the method of change of measure together with the Bayes’ rule for conditional expectations, solution expressions for the value of common contracts are obtained. A demonstration of how to apply our proposed stochastic modelling approach to value survival and death benefits is provided. Using the Human Mortality Database and UK interest rates, we illustrate that the dependence between interest rate and mortality dynamics has considerable impact in the value of even a simple survival benefit.     

Quadratic stochastic intensity and prospective mortality tables

We consider a quadratic stochastic intensity model with a Gaussian autoregressive factor, derive explicit formulas for predictive mortality tables and recursive updating formulas are also provided. We also explain how to use appropriately the Kalman filter to estimate the parameters of the model and to approximate the values of the underlying factor. This methodology is applied to French human mortality tables.     

Stochastic Breakeven Analysis

This paper presents a practical approach for studying the profit probability distribution defined in stochastic breakeven analysis. After presenting a procedure for calculating the mean and the second to fourth central moments of the profit distribution, the several important uses of these four moments are discussed. A major application of the four moments is in the fitting of a Pearson's curve. Using a numerical example in conjunction with a set of published tables, we then demonstrate the simplicity of our proposed approach as well as the resultant high accuracy in estimating probabilities of various profit levels.     

寿险中的破产理论及应用

本文研究了求解寿险中破产概率的简洁方法 ,得到寿险破产模型 ,设计了求解寿险中的破产概率的一种算法 ,并得到寿险破产概率的一个上界。     

Comonotonic bounds on the survival probabilities in the Lee–Carter model for mortality projection

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.     

A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities

Dynamic life tables arise as an alternative to the standard (static) life table, with the aim of incorporating the evolution of mortality over time. The parametric model introduced by Lee and Carter in 1992 for projected mortality rates in the US is one of the most outstanding and has been used a great deal since then. Different versions of the model have been developed but all of them, together with other parametric models, consider the observed mortality rates as independent observations. This is a difficult hypothesis to justify when looking at the graph of the residuals obtained with any of these methods.Methods of adjustment and prediction based on geostatistical techniques which exploit the dependence structure existing among the residuals are an alternative to classical methods. Dynamic life tables can be considered as two-way tables on a grid equally spaced in either the vertical (age) or horizontal (year) direction, and the data can be decomposed into a deterministic large-scale variation (trend) plus a stochastic small-scale variation (residuals).Our contribution consists of applying geostatistical techniques for estimating the dependence structure of the mortality data and for prediction purposes, also including the influence of the year of birth (cohort). We compare the performance of this new approach with different versions of the Lee-Carter model. Additionally, we obtain bootstrap confidence intervals for predicted q_xt resulting from applying both methodologies, and we study their influence on the predictions of e_65t and a_65t.     

Longevity risk management for life and variable annuities: The effectiveness of static hedging using longevity bonds and derivatives

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.     

PSD准则与累积展望理论不一致的讨论

Levy M和Levy H设计了一些实验,应用展望随机占优准则(PSD)给出方案的偏好序,实验结果拒绝了累积展望理论;而Wakker认为实验的数据恰恰支持了累积展望理论.试图讨论PSD准则拒绝累积展望理论的原因,PSD准则以概率的线性加权为基础,累积展望理论的重要假设是对概率进行了编译,二者存在着差异,据此,提出广义展望随机占优准则,认为在此准则下Levy M和Levy H的实验数据无法给出方案的偏好序,因此不能否认累积展望理论的正确性.     

Evaluating the performance of Gompertz,Makeham and Lee–Carter mortality models for risk management with unit-linked contracts

The paper compares the performance of three mortality models in the context of optimal pricing and hedging of unit-linked life insurance contracts. Two of the models are the classical parametric results of Gompertz and Makeham, the third is the recently developed method of Lee and Carter [Lee, R.D., Carter, L.R., 1992. Modelling and forecasting U.S. mortality. J. Amer. Statist. Assoc. 87 (14), 659–675] for fitting mortality and forecasting it as a stochastic process. First, quantile hedging techniques of Föllmer and Leukert [Föllmer, H., Leukert, P., 1999. Quantile hedging. Finance Stoch. 3, 251–273] are applied to price a unit-linked contract with payoff conditioned on the client’s survival to the contract’s maturity. Next, the paper analyzes the implications of the three mortality models on risk management possibilities for the insurance firm based on numerical illustrations with the Toronto Stock Exchange/Standard and Poor financial index and mortality data for the USA, Sweden and Japan. The strongest differences between the models are observed in Japan, where the lowest mortality for the next two decades is expected. The general mortality decline patterns, rectangularization of the survival curve and deceleration of mortality at older ages, are well pronounced in the results for all three countries.     

Survival risks,intertemporal consumption,and insurance: The case of distorted probabilities

This paper explores how to evaluate changes in survival probabilities when people do not process probabilities linearly, as is commonly assumed in the literature, but distort probabilities. We show that the valuation of risks to life depends critically on two parameters: the elasticity of the probability weighting function and the elasticity of the utility function with respect to future consumption. Using estimates from the empirical literature we derive that the bias of erroneously ignoring probability distortion in general leads to cost–benefit ratios that are too high and that generate too much priority for programs that save young lives.     

Longevity risk in portfolios of pension annuities

We analyze the importance of longevity risk for the solvency of portfolios of pension annuities. We distinguish two types of mortality risk. Micro-longevity risk quantifies the risk related to uncertainty in the time of death if survival probabilities are known with certainty, while macro-longevity risk is due to uncertain future survival probabilities. We use a generalized two-factor Lee-Carter mortality model to produce forecasts of future mortality rates, and to assess the relative importance of micro- and macro-longevity risk for funding ratio uncertainty. The results show that if financial market risk is fully hedged so that uncertainty in future lifetime is the only source of uncertainty, pension funds are exposed to a substantial amount of risk. Systematic and non-systematic deviations from expected survival imply that, depending on the size of the portfolio, buffers that reduce the probability of underfunding to 2.5% at a 5-year horizon have to be of the order of magnitude of 7% to 8% of the initial value of the liabilities.