Modeling and Pricing Longevity Derivatives Using Stochastic Mortality Rates and the Esscher Transform

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.     

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.     

Longevity and adjustment in pension annuities,with application to Finland

The aim of this paper is twofold. First, the improvement in adult mortality in Finland is studied. Lee-Carter (LC) Poisson log-bilinear model is used for mortality forecasting. Secondly, the paper studies how the pension annuities are adjusted to unexpected mortality pattern. A formula for funded plan is proposed. Application is made with Finnish mortality rates predicted using the LC model.     

Do Interest Rates Follow Unit-Root Processes? Evidence from Cross-Maturity Treasury Bill Yields

Abstract. It is widely reported in the literature that interest rates follow integrated processes. Many empirical studies have, in fact, taken this result as a maintained hypothesis. This article demonstrates that the failure to reject the hypothesis that interest rates contain a unit root may be due to the severe power problem of standard test procedures in small samples. We analyze a panel of cross-maturity Treasury-bill yield series by employing a panel-based test. This test exploits cross-maturity variations of the data to improve estimation efficiency and is more powerful than standard tests for unit roots. The critical values of the test statistics are computed by Monte Carlo simulations tailored to our samples. It is found that the null hypothesis that each yield series contains a unit root can be decisively rejected. Our findings cast some doubt on previous studies that rely on the nonstationarity assumption of interest 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。     

Explaining Mortality Dynamics

Abstract

Using data for six OECD countries over the period 1950–2006, this paper studies the impact of macroeconomic fluctuations and cause of death trends on mortality dynamics in the Lee-Carter mortality forecasting model. The key results of this study are the following: (1) Periods can be identified in which the Lee-Carter mortality index k_t correlates significantly with macroeconomic fluctuations. (2) A few causes of death such as diseases of the circulatory system, influenza and pneumonia, and diabetes mellitus account for a large fraction of the variations in the Lee-Carter mortality index k_t . (3) Most cause-specific mortality rates show pronounced trends over the last few decades. These trends change the composition of deaths and alter how total mortality reacts to external factors such as macroeconomic fluctuations.     

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.     

Arbitrage, Risk Premium, and Cointegration Tests of the Efficiency of Futures Markets

This article provides a new perspective on the efficiency of futures markets in a cointegration framework. Under the conventional risk premium hypothesis, if futures and spot prices are non-stationary, they must be cointegrated if futures markets are efficient. Alternatively, the cost-of-carry model implies that there should be a cointegration relationship among spot prices, futures prices and interest rates assuming all the series contain a unit root. Market efficiency further implies specific parameter restrictions under these two models. Using data on the futures markets for gold, silver, palladium and platinum, this article first establishes that interest rates, spot and futures prices are unit root non-stationary. The evidence on cointegration is somewhat mixed: the gold futures market is consistent with the cost-of-carry model, and the silver futures market satisfies the risk premium hypothesis, but the evidence for the other two markets is inconclusive.     

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 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).     

Financial integration in the pacific basin region: RIP by PANIC attack?

We exploit advances in panel data econometrics to test whether real interest parity holds in the Pacific Basin region. We test for a unit root in the difference between either the US, Japanese or Euro area real interest rate and the real interest rates from a panel of eleven Pacific Basin economies. Unlike extant studies that test for RIP using panel data, we use Bai and Ng’s (2004) PANIC test which allows for a very general model of cross-section dependence, including the possibility of cross-unit cointegration. Ignoring the possibility of cross-unit cointegration can lead to severe size distortions and to an over-rejection of the null hypothesis of a unit root. We overturn earlier findings based on first-generation panel tests, and demonstrate that cross-unit cointegration leads to incorrect conclusions. We find that RIP holds in the Pacific region. Real interest rates converge to the US rate. We find no support for the hypothesis that Pacific Basin real interest rates converge to either the Japanese or Euro area rates.     

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.     

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.     

Mortality Regimes and Pricing

Abstract

Mortality dynamics are characterized by changes in mortality regimes. This paper describes a Markov regime-switching model that incorporates mortality state switches into mortality dynamics. Using the 1901-2005 U.S. population mortality data, we illustrate that regime-switching models can perform better than well-known models in the literature. Furthermore, we extend the 1992 Lee-Carter model in such a way that the time-series common risk factor to all cohorts has distinct mortality regimes with different means and volatilities. Finally, we show how to price mortality securities with this model.     

Reference mortality K2004 of personal life insurance policies in Finland

This article presents the reference mortality model K2004 approved by the Actuarial Society of Finland and the technique that was implemented in developing it. Initially, I will present the historical development of individual mortality rates in Finland. Then, the requirements posed for a modern mortality modelling will be presented. Reference mortality model K2004 is based on total population mortality rates, which were adjusted to correspond with that portion of the population that has a life insurance policy. First, the model presents a margin of the observed life insurance mortality rate in the total population with a Lee-Carter method together with a forecast, where the downward trend in mortality rates is expected to continue at the rate illustrated since the 1960s. Then, the mortality rate has been adjusted into life insurance mortality per age so that it corresponds to the differences observed between total population and the portion of population that has a life insurance during 1991–2001. Finally, a cohort and gender-specific functional margin will be presented to obtained data.     

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.     

Nonlinear trends in real exchange rates: A panel unit root test approach

We analyze the possibility of nonlinear trend stationarity as the alternative to unit roots in 23 OECD real exchange rates, 1974–1998, by adding nonlinear time terms to the CIPS panel unit root test of Pesaran (2007). We follow a thorough bootstrapping approach and propose a technique to adjust statistical significance for the use of multiple tests over several time trend orders. The unit root null that all real exchange rates have unit roots is rejected at better than the 0.05 level. Bootstrapped results from a procedure of Chortareas and Kapetanios (2009) suggest that the hypothesis that all are stationary is reasonable. We argue that nonlinear trend stationarity is the most likely alternative hypothesis for at least some of the real exchange rates because: (1) the strongest CIPS rejection occurs when quadratic trends are specified; (2) nonlinear time terms are statistically significant at the 0.10 level; (3) the actual CIPS statistics are more consistent with CIPS sampling distributions from bootstrapped nonlinear trend stationary processes than from linear trend or mean stationary processes.     

Random walk and efficiency tests in the Asia-Pacific foreign exchange markets: Evidence from the post-Asian currency crisis data

This paper empirically tests the random walk and efficiency hypothesis for 12 Asia-Pacific foreign exchange markets. The hypothesis is tested using individual as well as panel unit root tests and two variance-ratio tests. The study covers the high (daily) and medium (weekly) frequency post-Asian crisis spot exchange rate data from January 1998 to July 2007. The inferential outcomes do not differ substantially between the unit root tests and the variance-ratio tests when using daily data but differ significantly when using weekly data. With the daily data, both types of unit root tests identify unit root components for all the series and two variance-ratio tests provide the evidence of martingale behavior for majority of the exchange rates tested. With the weekly data, panel unit root tests identify unit root component for the exchange rates and, the unit root tests on a single series basis identify unit root component for 10 foreign exchange markets. However, the variance-ratio tests reject the martingale null for the majority of the exchange rates when using weekly data.     

The Poisson Log-Bilinear Lee-Carter Model

Abstract

Life insurance companies deal with two fundamental types of risks when issuing annuity contracts: financial risk and demographic risk. Recent work on the latter has focused on modeling the trend in mortality as a stochastic process. A popular method for modeling death rates is the Lee-Carter model. This methodology has become widely used, and various extensions and modifications have been proposed to obtain a broader interpretation and to capture the main features of the dynamics of mortality rates. In order to improve the measurement of uncertainty in survival probability estimates, in particular for older ages, the paper proposes an extension based on simulation procedures and on the bootstrap methodology. It aims to obtain more reliable and accurate mortality projections, based on the idea of obtaining an acceptable accuracy of the estimate by means of variance reducing techniques. In this way the forecasting procedure becomes more efficient. The longevity question constitutes a critical element in the solvency appraisal of pension annuities. The demographic models used for the cash flow distributions in a portfolio impact on the mathematical reserve and surplus calculations and affect the risk management choices for a pension plan. The paper extends the investigation of the impact of survival uncertainty for life annuity portfolios and for a guaranteed annuity option in the case where interest rates are stochastic. In a framework in which insurance companies need to use internal models for risk management purposes and for determining their solvency capital requirement, the authors consider the surplus value, calculated as the ratio between the market value of the projected assets to that of the liabilities, as a meaningful measure of the company’s financial position, expressing the degree to which the liabilities are covered by the assets.     

基于我国国债回购市场的利率预期理论检验

本文在给出利率期限结构预期假说的定义及其推论的基础上，利用单位根和协整检验方法对上交所国债回购市场的利率数据进行了检验，结果表明国债回购利率序列均为一阶单整，由各个国债回购利率所构成的利率系统仅由一个共同的随机趋势驱动，因此得出利率预期假说在我国国债回购市场是有效的结论。本文利用向量误差修正模型对各个国债回购利率的估计结果进一步验证了这一点。