Modeling multi-country mortality dependence and its application in pricing survivor index swaps—A dynamic copula approach

This paper introduces mortality dependence in multi-country mortality modeling using a dynamic copula approach. Specifically, we use time-varying copula models to capture the mortality dependence structure across countries, examining both symmetric and asymmetric dependence structures. In addition, to capture the phenomenon of a heavy tail for the multi-country mortality index, we consider not only the setting of Gaussian innovations but also non-Gaussian innovations under the Lee–Carter framework model. As tests of the goodness of fit of different dynamic copula models, the pattern of mortality dependence, and the distribution of the innovations, we used empirical mortality data from Finland, France, the Netherlands, and Sweden. To understand the effect of mortality dependence on longevity derivatives, we also built a valuation framework for pricing a survivor index swap, then investigated the fair swap rates of a survivor swap numerically. We demonstrate that failing to consider the dynamic copula mortality model and non-Gaussian innovations would lead to serious underestimations of the swap rates and loss reserves.     

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.     

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.     

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.     

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.     

基于DEJD模型的人口寿命预测及SM债券定价

人口老龄化背景下的长寿风险,将会给国家养老保障体系带来极大的经济负担.如何度量和管理长寿风险,己成为近年来世界各国关注和研究的焦点.本文基于我国人口死亡率数据,在Lee-Carter模型的基础上,引入DEJD模型刻画时间序列因子的跳跃不对称性,并证实了 DEJD模型比Lee-Carter模型在拟合时间序列因子时更为有效...     

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.     

Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

Copula-AR(n)-SVL Mortality Models and Measurement of Longevity Risk

A reasonable mortality model is the key to accurately measuring longevity risks. This paper considers the dependence of mortality among different age groups and the autocorrelation and heteroscedastic structure of mortality in each age group. The multivariate Copula and AR(n)-LSV models are used to construct the mortality model. VaR, TVaR, GlueVaR are used to measure longevity risk. The results show that Copula-AR($n$)-LSV characterizes mortality trends and fluctuations better than Lee-Cater model; When mortality in China gradually decline, insurance companies will face increasing longevity risk in the future.     

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.     

A copula entropy approach to correlation measurement at the country level

The entropy optimization approach has widely been applied in finance for a long time, notably in the areas of market simulation, risk measurement, and financial asset pricing. In this paper, we propose copula entropy models with two and three variables to measure dependence in stock markets, which extend the copula theory and are based on Jaynes’s information criterion. Both of them are usually applied under the non-Gaussian distribution assumption. Comparing with the linear correlation coefficient and the mutual information, the strengths and advantages of the copula entropy approach are revealed and confirmed. We also propose an algorithm for the copula entropy approach to obtain the numerical results. With the experimental data analysis at the country level and the economic circle theory in international economy, the validity of the proposed approach is approved; evidently, it captures the non-linear correlation, multi-dimensional correlation, and correlation comparisons without common variables. We would like to make it clear that correlation illustrates dependence, but dependence is not synonymous with correlation. Copulas can capture some special types of dependence, such as tail dependence and asymmetric dependence, which other conventional probability distributions, such as the normal p.d.f. and the Student’s t p.d.f., cannot.     

Semi-parametric extensions of the Cairns–Blake–Dowd model: A one-dimensional kernel smoothing approach

Over the last few decades, there has been an enormous growth in mortality modeling as the field of mortality risk and longevity risk has attracted great attention from academic, government and private sectors. In this paper, we propose a time-varying coefficient (TVC) mortality model aiming to combine the good characteristics of existing models with efficient model calibration methods. Nonparametric kernel smoothing techniques have been applied in the literature of mortality modeling and based on the findings from Li et al.’s (2015) study, such techniques can significantly improve the forecasting performance of mortality models. In this study we take the same path and adopt a kernel smoothing approach along the time dimension. Since we follow the model structure of the Cairns–Blake–Dowd (CBD) model, the TVC model we propose can be seen as a semi-parametric extension of the CBD model and it gives specific model design according to different countries’ mortality experience. Our empirical study presented here includes Great Britain, the United States, and Australia amongst other developed countries. Fitting and forecasting results from the empirical study have shown superior performances of the model over a selection of well-known mortality models in the current literature.     

A hierarchical copula-based world-wide valuation of sovereign risk

We propose a new model for the aggregation of risks that is very flexible and useful in high dimensional problems. We propose a copula-based model that is both hierarchical and hybrid (HYC for short), because: (i) the dependence structure is modeled as a hierarchical copula, (ii) it unifies the idea of the clusterized homogeneous copula-based approach (CHC for short) and its limiting version (LHC for short) proposed in Bernardi and Romagnoli (2012, 2013). Based on this, we compute the loss function of a world-wide sovereign debt portfolio which accounts for a systemic dependence of all countries, in line with a global valuation of financial risks. Our approach enables us to take into account the non-exchangeable behavior of a sovereign debts’ portfolio clustered into several classes with homogeneous risk and to recover a possible risks’ hierarchy. A comparison between the HYC loss surface and those computed through a pure limiting approach, which is commonly used in high dimensional problems, is presented and the impact of the concentration and the granularity errors is appreciated. Finally the impact of an enlargement of the dependence structure is discussed, in the contest of a geographical area sub-portfolios analysis now relevant to determine the risk contributions of subgroups under the presence of a wider dependence structure. This argument is presented in relation to the evaluation of the insurance premium and the collateral related to the designed project of an euro-insurance-bond.     

Joint survival probability via truncated invariant copula

Given an intensity-based credit risk model, this paper studies dependence structure between default intensities. To model this structure, we use a multivariate shot noise intensity process, where jumps occur simultaneously and their sizes are correlated. Through very lengthy algebra, we obtain explicitly the joint survival probability of the integrated intensities by using the truncated invariant Farlie–Gumbel–Morgenstern copula with exponential marginal distributions. We also apply our theoretical result to pricing basket default swap spreads. This result can provide a useful guide for credit risk management.     

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.     

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本文在传统Lee-Carter人口死亡率模型的框架下, 引入同出生年人群死亡率之间的相关性效应, 从而对未来死亡率的动态变化进行更加具体的刻画. 同时借鉴Lin和Cox(2005)所提出的长寿债券构造机制, 基于中国的实际人口死亡率数据, 运用多维概率扭转变换对不完全市场下长寿债券的定价结果进行比较分析.     

Modelling dependence structure with Archimedean copulas and applications to the iTraxx CDS index

In this paper we model the dependence structure between credit default swap (CDS) and jump risk using Archimedean copulas. The paper models and estimates the different relationships that can exist in different ranges of behaviour. It studies the bivariate distributions of CDS index spreads and the kurtosis of equity return distribution. To take into account nonlinear relationships and different structures of dependency, we employ three Archimedean copula functions: Gumbel, Clayton, and Frank. We adopt nonparametric estimation of copula parameters and we find an extreme co-movement of CDS and stock market conditions. In addition, tail dependence indicates the extreme co-movements and the potential for a simultaneous large loss in stock markets and a significant default risk. Ignoring the tail dependence would lead to underestimation of the default risk premium.     

应对长寿风险的分红年金:随机精算建模与应用

我国的商业养老保险作为养老金体系的重要组成部分,在实践中的发展比较缓慢,原因之一是保险公司缺乏长寿风险管理的经验。本文将探索我国商业养老保险使用分红年金管理长寿风险的可行性。研究该分红年金在给付规则和分红来源方面的特征,并基于实际数据,构建动态随机死亡率模型和随机收益率模型,采用蒙特卡洛随机模拟方法,比较分红年金和传统年金在待遇分布、资产和损失分布、破产概率等方面的特征,得出分红年金能够在精算公平原则下有效应对长寿风险,并且在待遇给付、偿付能力和盈利能力方面具有明显优势的结论。     

Forecasting mortality rate improvements with a high-dimensional VAR

Forecasting mortality rates is a problem which involves the analysis of high-dimensional time series. Most of usual mortality models propose to decompose the mortality rates into several latent factors to reduce this complexity. These approaches, in particular those using cohort factors, have a good fit, but they are less reliable for forecasting purposes. One of the major challenges is to determine the spatial–temporal dependence structure between mortality rates given a relatively moderate sample size. This paper proposes a large vector autoregressive (VAR) model fitted on the differences in the log-mortality rates, ensuring the existence of long-run relationships between mortality rate improvements. Our contribution is threefold. First, sparsity, when fitting the model, is ensured by using high-dimensional variable selection techniques without imposing arbitrary constraints on the dependence structure. The main interest is that the structure of the model is directly driven by the data, in contrast to the main factor-based mortality forecasting models. Hence, this approach is more versatile and would provide good forecasting performance for any considered population. Additionally, our estimation allows a one-step procedure, as we do not need to estimate hyper-parameters. The variance–covariance matrix of residuals is then estimated through a parametric form. Secondly, our approach can be used to detect nonintuitive age dependence in the data, beyond the cohort and the period effects which are implicitly captured by our model. Third, our approach can be extended to model the several populations in long run perspectives, without raising issue in the estimation process. Finally, in an out-of-sample forecasting study for mortality rates, we obtain rather good performances and more relevant forecasts compared to classical mortality models using the French, US and UK data. We also show that our results enlighten the so-called cohort and period effects for these populations.     

Contagion-based distortion risk measures

We propose a class of distortion measures based on contagion from an external “scenario” variable. The dependence between the scenario and the variable whose risk is measured is modeled with a copula function with horizontal concave sections. Special cases are the perfect dependence copula, which generates expected shortfall, the Marshall–Olkin family and the Placket family. As an application, we evaluate distortion measures bank liabilities with respect to a country risk scenario in the current European debt crisis.