Statistical Inference for Lee-Carter Mortality Model and Corresponding Forecasts

Although the Lee-Carter model has become a benchmark in modeling mortality rates, forecasting mortality risk, and hedging longevity risk, some serious issues exist on its inference and interpretation in the actuarial science literature. After pointing out these pitfalls, this article proposes a modified Lee-Carter model, provides a rigorous statistical inference, and derives the asymptotic distributions of the proposed estimators and unit root test when the mortality index is nearly integrated and errors in the model satisfy some mixing conditions. After a unit root hypothesis is not rejected, future mortality forecasts can be obtained via the proposed inference. An application of the proposed unit root test to U.S. mortality rates rejects the unit root hypothesis for the female and combined mortality rates but does not reject the unit root hypothesis for the male mortality rates.     

Lee-Carter模型在中国城市人口死亡率预测中的应用与改进

由死亡率下降带来的长寿风险给社会、政治以及经济带来了新的挑战。为了更加准确地对长寿风险进行评估和管理,需要对未来死亡率趋势进行预测。本文针对我国死亡率数据样本量小以及数据存在缺失的实际情况,对Lee-Carter模型进行了改进,通过一个双随机过程对Lee-Carter模型中的时间项进行建模。在模型中考虑了样本量不足对预测结果造成的影响,使得改进后的Lee-Carter模型更加适合目前中国的人口死亡率预测。     

On the Modeling and Forecasting of Socioeconomic Mortality Differentials: An Application to Deprivation and Mortality in England

In any country, mortality rates and indices such as life expectancy usually differ across subpopulations, for example, defined by gender, geographic area, or socioeconomic variables (e.g., occupation, level of education, or income). These differentials, and in particular those related to socioeconomic circumstances, pose important challenges for the design of public policies for tackling social inequalities, as well as for the design of pension systems and the management of longevity risk in pension funds and annuity portfolios. We discuss the suitability for the modeling and forecasting of socioeconomic differences in mortality of several multiple population extensions of the Lee-Carter model, including a newly introduced relative model based on the modeling of the mortality in socioeconomic subpopulations alongside the mortality of a reference population. Using England mortality data for socioeconomic subpopulations defined using a deprivation index, we show that this new relative model exhibits the best results in terms of goodness of fit and ex post forecasting performance. We then use this model to derive projections of deprivation specific mortality rates and life expectancies at pensioner ages and analyze the impact of socioeconomic differences in mortality on the valuation of annuities.     

A Bühlmann Credibility Approach to Modeling Mortality Rates

In this article, we first propose a Bühlmann nonparametric credibility approach to forecasting mortality rates, and we then compare forecasting performances between the proposed Bühlmann approach and the Lee-Carter/Cairns-Blake-Dowd (CBD) models. Empirical results based on mortality data for both genders of Japan, the United Kingdom, and the United States with two age spans, a wide range of fitting year spans, and three forecasting periods show that the credibility approach contributes to much better forecasting performances measured by the mean absolute percentage error. Moreover, we give an informative credibility interpretation regarding the average decrements of an individual time trend for age x and a group time trend for all ages, and we discuss the effects of the slope and intercept of the linear functions for the forecasted mortality rates under the proposed Bühlmann nonparametric credibility approach and the Lee-Carter/CBD models. Finally, we also estimate the parameters of the Bühlmann credibility approach in a semiparametric framework, and we provide a stochastic version of forecasting mortality rates for the Bühlmann credibility approach.     

Measuring the Impact of Longevity Risk on Pension Systems: The Case of Italy

This article estimates the impact of longevity risk on pension systems by combining the prediction based on a Lee-Carter mortality model with the projected pension payments for different cohorts of retirees. We measure longevity risk by the difference between the upper bound of the total old-age pension expense and its mean estimate. This difference is as high as 4% of annual GDP over the period 2040–2050. The impact of longevity risk is sizeably reduced, but not fully eliminated, by the introduction of indexation of retirement age to expected life at retirement. Our evidence speaks in favor of a market for longevity risk and calls for a closer scrutiny of the potential redistributive effects of longevity risk.     

Detecting Common Longevity Trends by a Multiple Population Approach

Recently the interest in the development of country and longevity risk models has been growing. The investigation of long-run equilibrium relationships could provide valuable information about the factors driving changes in mortality, in particular across ages and across countries. In order to investigate cross-country common longevity trends, tools to quantify, compare, and model the strength of dependence become essential. On one hand, it is necessary to take into account either the dependence for adjacent age groups or the dependence structure across time in a single population setting—a sort of intradependence structure. On the other hand, the dependence across multiple populations, which we describe as interdependence, can be explored for capturing common long-run relationships between countries. The objective of our work is to produce longevity projections by taking into account the presence of various forms of cross-sectional and temporal dependencies in the error processes of multiple populations, considering mortality data from different countries. The algorithm that we propose combines model-based predictions in the Lee-Carter (LC) framework with a bootstrap procedure for dependent data, and so both the historical parametric structure and the intragroup error correlation structure are preserved. We introduce a model which applies a sieve bootstrap to the residuals of the LC model and is able to reproduce, in the sampling, the dependence structure of the data under consideration. In the current article, the algorithm that we build is applied to a pool of populations by using ideas from panel data; we refer to this new algorithm as the Multiple Lee-Carter Panel Sieve (MLCPS). We are interested in estimating the relationship between populations of similar socioeconomic conditions. The empirical results show that the MLCPS approach works well in the presence of dependence.     

The Lee-Carter quantile mortality model

The Lee-Carter (LC) stochastic mortality model has been widely used for making future projections of mortality rates. In the framework of the LC model, the response function is non-linear in parameters. Here, we adapt this LC framework to compute conditional quantiles. The LC quantile model can be defined as quantile non-linear regression conditioned to age and the calendar year. Two strategies for estimating coefficients based on interior-point methods are described. We show that the LC quantile model provides additional information to that furnished by the traditional LC conditional mean. An application to Spanish mortality data is reported.     

Outlier analysis and mortality forecasting: The United Kingdom and Scandinavian countries

The Lee-Carter model and its variants have been extensively employed by actuaries, demographers, and many others to forecast age-specific mortality. In this study, we use mortality data from England and Wales, and four Scandinavian countries to perform time-series outlier analysis of the key component of the Lee-Carter model – the mortality index. We begin by employing a systematic outlier detection process to ascertain the timing, magnitude, and persistence of any outliers present in historical mortality trends. We then try to match the identified outliers with imperative events that could possibly justify the vacillations in human mortality levels. At the same time, we adjust the effect of the outliers for model re-estimation. A new iterative model re-estimation method is proposed to reduce the chance of erroneous model specification. The empirical results indicate that the outlier-adjusted model could achieve more efficient forecasts of variables such as death rates and life expectancies. Finally, we point out that the Lee-Carter forecasts are especially vulnerable to outliers near the forecast origin, and discuss the potential limitations of the application of the Lee-Carter model to mortality forecasting.     

“A Direct Approach to the Discounted Penalty Function”, Hansjörg Albrecher,Hans U. Gerber,and Hailiang Yang,Volume 14, No. 4, 2010

Abstract

Mortality improvements, especially of the elderly, have been a common phenomenon since the end of World War II. The longevity risk becomes a major concern in many countries because of underestimating the scale and speed of prolonged life. In this study we explore the increasing life expectancy by examining the basic properties of survival curves. Specifically, we check if there are signs of mortality compression (i.e., rectangularization of the survival curve) and evaluate what it means to designing annuity products. Based on the raw mortality rates, we propose an approach to verify if there is mortality compression. We then apply the proposed method to the mortality rates of Japan, Sweden, and the United States, using the Human Mortality Database. Unlike previous results using the graduated mortality rates, we found no obvious signs that mortality improvements are slowing down. This indicates that human longevity is likely to increase, and longevity risk should be seriously considered in pricing annuity products.     

Longevity/Mortality Risk Modeling and Securities Pricing

Securitizing longevity/mortality risk can transfer longevity/mortality risk to capital markets. Modeling and forecasting mortality rate is key to pricing mortality‐linked securities. Catastrophic mortality and longevity jumps occur in historical data and have an important impact on security pricing. This article introduces a stochastic diffusion model with a double‐exponential jump diffusion process that captures both asymmetric rate jumps up and down and also cohort effect in mortality trends. The model exhibits calibration advantages and mathematical tractability while better fitting the data. The model provides a closed‐form pricing solution for J.P. Morgan’s q‐forward contract usable as a building block for hedging.     

A multi-dimensional Bühlmann credibility approach to modeling multi-population mortality rates

In this paper, we first propose a multi-dimensional Bühlmann credibility approach to forecasting mortality rates for multiple populations, and then compare forecasting performances among the proposed approach, the CBD model, the Lee-Carter model (LC), the joint-k (JoK-LC), the co-integrated (CoI-LC), and the augmented common factor (ACF-LC) Lee-Carter models for multiple populations. Mortality data from the Human Mortality Database are fitted to the underlying mortality models for both genders of three well-developed countries (the US, the UK, and Japan) and both genders of a developed country (France) and a developing country (Poland) with an age span 25–84 and a wide range of fitting year spans. Empirical illustrations show that the proposed multi-dimensional Bühlmann credibility approach contributes to more accurate forecast results, measured by AMAPE (average of mean absolute percentage errors over all fitting year spans), than the CBD, LC, JoK-LC, CoI-LC and ACF-LC models for three forecasting year spans 2004–2013 (10-year wide), 1994–2013 (20-year wide) and 1984–2013 (30-year wide).     

Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes

In this paper we propose a general derivative pricing framework that employs decoupled time-changed (DTC) Lévy processes to model the underlying assets of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scalings; we devise the martingale structure for a DTC Lévy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying Lévy decomposition allows the introduction of asset price models consistent with the assumption of a correlated pair of continuous and jump market activity rates; we study one illustrative DTC model of this kind based on the so-called Wishart process. The theory we develop is applied to the problem of pricing not only claims that depend on the price or the volatility of an underlying asset, but also more sophisticated derivatives whose payoffs rely on the joint performance of these two financial variables, such as the target volatility option. We solve the pricing problem through a Fourier-inversion method. Numerical analyses validating our techniques are provided. In particular, we present some evidence that correlating the activity rates could be beneficial for modeling the volatility skew dynamics.     

A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration

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.     

Multivariate option pricing with time varying volatility and correlations

In this paper we consider option pricing using multivariate models for asset returns. Specifically, we demonstrate the existence of an equivalent martingale measure, we characterize the risk neutral dynamics, and we provide a feasible way for pricing options in this framework. Our application confirms the importance of allowing for dynamic correlation, and it shows that accommodating correlation risk and modeling non-Gaussian features with multivariate mixtures of normals substantially changes the estimated option prices.     

Mortality,Health, and Marriage: A Study Based on Taiwan's Population Data

Life expectancy has been increasing significantly since the start of the 20th century, and mortality improvement trends are likely to continue in the 21st century. Stochastic mortality models are used frequently to predict the expansion in life expectancy. In addition to gender, age, period, and cohort are the three main risk factors considered in constructing mortality models. Other than these factors, it is also believed that marital status is related to health and longevity, and many studies have found that married persons have a lower mortality rate than the unmarried. In this study, we have used Taiwan's marital data for the whole population (married, unmarried, divorced/widowed) to evaluate if the marital status can be a preferred criteria. Furthermore, we also want to know whether the preferred criteria will be valid in the future. We chose two popular mortality models, the Lee-Carter and age-period-cohort, to model the mortality improvements for various marital statuses. Because of a linear dependence in the parameters of the age-period-cohort model, we used a computer simulation to choose the appropriate estimation method. Based on Taiwan's marital data, we found that married persons have significantly lower mortality rates than the single, and if converting the difference into a life insurance policy, the discount amount is even larger than that for smokers/nonsmokers.     

Life Insurance Mathematics with Random Life Tables

Abstract

When the insurer sells life annuities, projected life tables incorporating a forecast of future longevity must be used for pricing and reserving. To fix the ideas, the framework of Lee and Carter is adopted in this paper. The Lee-Carter model for mortality forecasting assumes that the death rate at age x in calendar year t is of the form exp(α_x + (β_xK_t), where the time-varying parameter K_t reflects the general level of mortality and follows an ARIMA model. The future lifetimes are all influenced by the same time index K_t in this framework. Because the future path of this index is unknown and modeled as a stochastic process, the policyholders' lifetimes become dependent on each other. Consequently the risk does not disappear as the size of the portfolio increases: there always remains some systematic risk that cannot be diversified, whatever the number of policies. This paper aims to investigate some aspects of actuarial mathematics in the context of random life tables. First, the type of dependence existing between the insured life lengths is carefully examined. The way positive dependence influences the need for economic capital is assessed compared to mutual independence, as well as the effect of the timing of deaths through Bayesian credibility mechanisms. Then the distribution of the present value of payments under a closed group of life annuity policies is studied. Failing to account for the positive dependence between insured lifetimes is a dangerous strategy, even if the randomness in the future survival probabilities is incorporated in the actuarial computations. Numerical illustrations are performed on the basis of Belgian mortality statistics. The impact on the distribution of the present value of the additional variability that results from the Lee-Carter model is compared with the traditional method of mortality projection. Also, the impact of ignoring the dependence hat arises from the model is quantified.     

Option pricing and Esscher transform under regime switching

Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

Pricing temperature derivatives with a filtered historical simulation approach

In this paper, we propose pricing temperature derivatives using a filtered historical simulation (FHS) approach that amalgamates model-based treatment of volatility and empirical innovation density. The FHS approach implicitly captures the risk premium with the entire risk-neutral model (except the innovation distribution), thereby providing significantly more flexibility than existing methods that use only one designated parameter to capture the risk premium. Additionally, instead of relying on the fitted innovation distribution, the FHS approach uses empirical innovations to capture excess skewness, excess kurtosis, and other non-standard features in the temperature data, all of which are important for the correct pricing of temperature derivatives. We apply the FHS approach to pricing derivatives written on the temperature of Chicago, and demonstrate that this approach yields better in-sample and out-of-sample pricing performance than the constant market price of risk method and the consumption-based method.