Pricing Mortality Securities with Correlated Mortality Indexes

This article proposes a stochastic model, which captures mortality correlations across countries and common mortality shocks, for analyzing catastrophe mortality contingent claims. To estimate our model, we apply particle filtering, a general technique that has wide applications in non-Gaussian and multivariate jump-diffusion models and models with nonanalytic observation equations. In addition, we illustrate how to price mortality securities with normalized multivariate exponential titling based on the estimated mortality correlations and jump parameters. Our results show the significance of modeling mortality correlations and transient jumps in mortality security pricing. Introduction Over the last century, populations of different countries have been increasingly linked by flows of information, goods, transportation, and communication, and as a consequence the world has become more closely connected and interdependent. While the trend of globalization has substantially driven market growth and international trade, it has also helped to spread some of the deadliest infectious diseases across borders (Daulaire, 1999). Thus, it seems improper to forecast mortality for an individual national population in isolation from others. Indeed, in practice, intercountry mortality correlation has long been a serious concern for insurers that underwrite life insurance business. Mortality forecast that takes into account a country’s linkage to others is important in the sense that not only does it facilitate better understanding of mortality risk, but it also has enormous implications for pricing mortality securities. Recent financial innova1 digitalcommons.unl.edu 2 Lin, Liu, & Yu in The Journal of risk and insurance 2012 tion makes mortality securitization a viable option for insurers or reinsurers to transfer catastrophe mortality risk arising from the possible occurrence of pandemics or largescale terrorist attacks. By segregating its cash flows linked to extreme mortality risk, an insurance firm is able to repackage them into securities that are traded in capital markets (Blake and Burrows, 2001; Lin and Cox, 2005; Cox and Lin, 2007). Since the first publicly traded mortality security issued by Swiss Re in 2003, almost all mortality transactions determine the coupons and principals based on three or more population mortality indexes, with the only exception—the Tartan mortality bond sold in 2006. This indicates that insurers or reinsurers are keenly interested in transferring potential country-correlated mortality risk embedded in their business. For instance, the mortality risk of the 2003 Swiss Re mortality bond was defined in terms of an index based on the weighted average annual population death rates in the United States, the United Kingdom, France, Italy, and Switzerland (Lin and Cox, 2008). As another example, the mortality bond issued by the Nathan Ltd. in 2008 depended on the annual population death rates of four countries, namely, the United States, the United Kingdom, Canada, and Germany. Given that the existing (and possible future) mortality securities bundle multination mortality risks, mortality correlation among countries merits serious consideration in mortality securitization pricing. In the recent literature, a number of stochastic mortality models have been proposed. Despite the importance of mortality correlation, surprisingly very few papers treat correlation as an indispensable element. For example, in order to account for catastrophic mortality death shocks, Chen and Cox (2009) incorporate a jump-diffusion process into the original Lee–Carter model to forecast mortality rates and price the 2003 Swiss Re mortality bond. Yet, while the Swiss Re bond payments depended on fivecountry weighted mortality index, the authors price this bond only based on the U.S. mortality rates. Hence, it is not clear how to extend their method to multipopulation correlated mortality scenarios. Beelders and Colarossi (2004) and Chen and Cummins (2010), on the other hand, use the extreme value theory to measure mortality risk of the 2003 Swiss Re bond. However, they simply model the combined index without considering the mortality correlations among different countries. So the question of how to model multipopulation mortality correlation remains open. Bauer and Kramer (2009) borrow a credit risk modeling approach introduced by Lando (1998) to describe stochastic force of mortality. To incorporate dramatic mortality changes that are crucial in measuring mortality risk (Lin and Cox, 2008), Bauer and Kramer (2009) propose a mortality model with an affine jump-diffusion process. Then they use their mortality model to price the Tartan transaction, which is the only publicly traded mortality bond to date solely based on one-country (i.e., United States) mortality experience. However, the authors acknowledge that their model misses correlations and diversification effects across genders and populations. Given the prevalence of multicountry combined mortality indexes in the mortality security markets, it is questionable whether their model is adequate for other mortality securities. Indeed, from a practical point of view, understanding the combined index as a function of various population death rates is at least as important, if not more so, than understanding it as a function of different age classes of a single country for mortality securitization. Moreover, the existing mortality literature has demonstrated the significance of catastrophic death events in the pricing and tranche structure for a mortality risk bond (Cox, Pricing MortaL itY Securit ieS with correLated MortaL itY indexeS 3 Lin, and Wang, 2006; Bauer and Kramer, 2009; Chen and Cox, 2009). While longevity risk modeling usually simplifies the analysis by ignoring dramatic mortality changes (Cairns, Blake, and Dowd, 2006; Schrager, 2006; Kogure and Kurachi, 2010; Wills and Sherris, 2010; Yang, Yue, and Huang, 2010; Cox et al., 2012), mortality jumps must be considered in order to successfully structure and price mortality-linked securities. Thus, in this article, as the first objective, we develop a tractable mortality model, which captures the mortality correlations among countries and incorporates mortality jumps. Specifically, we extend Cox, Lin, and Wang’s (2006) model to a more general setting and disentangle transient jumps from persistent volatilities. As a departure from Cox, Lin, and Wang who model unanticipated mortality jumps as permanent shocks, we model them as transient jumps with a double-jump process. Since in most cases severe short-term events such as epidemics underlie mortality risk (Cox, Lin, and Petersen, 2010), our model provides a better fit for historical data. A prominent calibration challenge for a model accounting for correlations is that the number of pairs grows quadratically with the number of countries of interests. To address this issue, as the second objective of this article, we employ a particle filtering approach for learning about unobservable mortality shocks and states from discretely observed population mortality rates. The particle filter algorithm is easy to implement and fast to compute, requiring only simulation from a proposal distribution. It is able to handle a large number of correlations with relatively low computation costs. Moreover, particle filters are highly adaptable and easy to be adjusted for various applications. In particular, they are able to deal with nonlinear, non-Gaussian systems, and variables in either continuous or discrete states as well as their combinations (Heijden et al., 2004). This property of particle filtering is important for mortality modeling because multivariate jump-diffusion processes are an indispensable component of a comprehensive mortality model, which is typically not Gaussian. In addition, the particle filtering approach is a Bayesian approach through which we can deal with estimation error and prediction uncertainty unaddressed in some mortality models. As highlighted in Cairns, Blake, and Dowd (2006), a model that takes into account parameter uncertainty is able to generate more reliable forecasts. So far the particle filtering approach has many practical applications in engineering and finance. Despite its usefulness, this technique has received little attention in the mortality literature.1 In this article, we show that it is useful to estimate variables related to mortality trend, volatility, correlation, and jump, which are crucial for mortality security pricing and risk management. Aside from the question of how to appropriately model mortality stochastic process, there is an ongoing debate on how to price mortality-linked securities. If a security’s payments are contingent on the correlation of mortality risks across countries, a common feature in mortality securitization, the pricing problem will become more challenging (Chen and Cox, 2009). Mortality securities, with only a limited number issued in financial markets, are not liquidly traded. As a result, to price a mortality risk linked security, the underlying mortality risk process needs to be risk adjusted (Wills and Sherris, 2010). Dif1. To the best of our knowledge, there is only one paper by Bauer and Kramer (2009) that applies the particle filtering in the mortality context. However, Bauer and Kramer apply the particle filtering in the mortality evolution of the U.S. population without considering mortality correlation, whereas mortality correlation is one of our main focuses. 4 Lin, Liu, & Yu in The Journal of risk and insurance 2012 ferent mortality pricing methods have been proposed. Dahl (2004) applies financial risk models for mortality risk modeling and then uses market data to calibrate risk-adjusted probability measures. Lin and Cox (2005), Dowd et al. (2006), and Denuit, Devolder, and Goderniaux (2007) apply the Wang transform (Wang, 2000) to physical distributions in pricing distortion distributions. Bayraktar et al. (2009) propose to compensate a mortality risk taker according to an “instantaneous Sharpe ratio,” which is defined as the additional return in excess of the risk-free rate divided by the standard deviation of a mortality portfolio after all diversifiable risk is hedged away. In line with their Bayesian mortality models, Kogure and Kurachi (2010) present a Bayesian pricing approach, the entropy maximization principle, to risk neutralize the predictive distribution of future survival rates. However, while in these articles the transformed parameters are calibrated, they are built on a univariate setting or an independent assumption, ignoring the connections among different cohorts and populations. Thus, their pricing methods have limited implications for multivariate and correlated mortality cases. To address this pricing issue, we employ normalized multivariate exponential tilting to take into account correlations across countries for mortality securitization. Exponential tilting is an incomplete market pricing method that neutralizes statistical distributions, which is consistent with the literature on nonarbitrage pricing of contingent claims (see Buhlmann, 1980; Gerber and Shiu, 1996; Madan and Unal, 2004; Kijima, 2006; Wang, 2007; and others). It can be applied in pricing risks embedded in loan defaults, mortgage refinancing, electricity trading, weather derivatives, catastrophic insurance, and insurance-linked securities (Duffie, 1992; Karatzas and Shreve, 1992; Heston, 1993; Gerber and Shiu, 1996; Cox, Lin, and Wang, 2006; Milidonis, Lin, and Cox, 2011). Kijima (2006) and Wang (2007) extend univariate exponential tilting to multivariate cases. The need for changing multivariate probability measures arises from pricing contingent claims on multiple underlying assets or liabilities. As noted earlier, the payoffs of the existing mortality securities are contingent on several population mortality indexes. To properly compensate investors for risk arising from mortality correlations among countries, as the last objective of this article, we apply the normalized multivariate exponential tilting to price mortality securities. Specifically, we first introduce the concept of normalized exponential tilting and then formulate probability distortions for the multivariate case. For demonstration, we utilize the normalized multivariate exponential tilting to make inference about the likely market prices of risk from the pricing information contained in the existing mortality securities. To the best of our knowledge, this is the first article that not only accounts for mortality correlations but also calibrates different market prices of risk for various mortality diffusions and jumps. The article is organized as follows. In the next section, we introduce our proposed stochastic model for a set of correlated population mortality indexes with jumps. The “Estimation With Particle Filtering” section provides an overview on the particle filtering approach. The multivariate exponential tilting as an incomplete market pricing method is introduced in “Pricing Mortality Securities With Multivariate Exponential Tilting.” In “Empirical Applications,” we first estimate our proposed model by applying the particle filtering technique based on historical data. We show that mortality correlation and jump process play important roles in mortality securitization modeling. Then we price some mortality bonds issued in 2006 by applying normalized multivariate exponential tilting and compare our estimated market prices of risk to those when we do not consider morPricing MortaL itY Securit ieS with correLated MortaL itY indexeS 5 tality correlations and jumps. After a discussion of our findings, “Conclusions” concludes and provides an outlook on future research. Model Specification Our approach combines Brownian motions and compound Poisson processes. As a distinctive feature of our model, we simultaneously consider n-country population mortality death rates and their correlations. Let qt denote the observed population mortality index for country i at time t. The logarithm of qt equals yi,t = ln qt We assume that the n population mortality indexes jointly solve: