The age pattern of transitory mortality jumps and its impact on the pricing of catastrophic mortality bonds
Introduction
The dynamics of human mortality over time are subject to short-term jumps. These jumps may be caused by influenza pandemics, most notably the Spanish flu in 1918–20 that is estimated to have infected 50% of the world’s population and led to a total mortality of 40–50 million (Crosby, 1976). More recently, the Asian flu in 1957–8 is believed to have killed approximately 1 million persons in total (Dauer and Serfling, 1961, Pyle, 1986, Potter, 2001). It is reasonable to assume that similar influenza pandemics will occur in future, because there is an unlimited reservoir of influenza subtypes. Also, for reasons such as interspecies transmission, intraspecies variation and altered virulence, the timings and severities of future pandemics (and hence mortality jumps) are unpredictable (Cox et al., 2003, Webster et al., 1997).
Mortality jumps are infrequent, but their occurrence could trigger a large number of unexpected death claims, thereby affecting the financial strength of the life insurance industry. Stracke and Heinen (2006) estimated that the worst pandemic would result in approximately € 45 billion of additional claims expenses in Germany. This amount is equivalent to five times the total annual gross profit or 100% of the policyholder bonus reserves in the German life insurance market. Toole (2007) found that in a severe pandemic scenario, additional claims expenses would consume 25% of the risk based capital (RBC) of the entire US life insurance industry. This finding means that companies with less than 100% of RBC would be at an increased risk of insolvency. In recent years, a number of reinsurers have used catastrophic mortality bonds as a risk mitigation tool. The first of such bonds was called Vita I, issued by Swiss Re in 2003 to reduce its exposure to a catastrophic mortality deterioration in five populations. With a full subscription, this bond was regarded as a huge success and led to many other catastrophic mortality bonds being issued (see Blake et al., 2006, Blake et al., 2013).
To model extreme mortality risk and value catastrophic mortality bonds, researchers have developed a number of stochastic mortality models that incorporate jump effects. These models include the contributions by Bauer and Kramer (2009), Biffis (2005), Chen (2013), Chen and Cummins (2010), Chen and Cox (2009), Chen et al., 2010, Chen et al., 2013a, Cox et al., 2006, Cox et al., 2010, Deng et al. (2012), Hainaut and Devolder (2008), Lin and Cox (2008), Lin et al. (2013) and Zhou et al. (2013a). Several features of mortality jumps have been studied in great depth. In terms of jump occurrence, Chen and Cox (2009) and Chen et al. (2010) used independent Bernoulli distributions, Cox et al. (2006) considered Poisson jump counts, whereas Lin and Cox (2008) utilized a discrete-time Markov chain. In terms of jump severity, Chen and Cox (2009) and Chen et al. (2010) made use of normal distributions, Chen and Cummins (2010) applied the extreme value theory, while Chen (2013) and Deng et al. (2012) considered double-exponential jumps. In terms of correlations across different populations, Chen et al. (2013a) used a factor-copula method, Lin et al. (2013) built a model with correlated Brownian motions, whereas Zhou et al. (2013a) considered a multinomial approach.
One feature that has not been studied extensively is the age pattern of mortality jumps, that is, how the effect of a mortality jump is distributed among different ages. Most of the existing models are either constructed for modeling aggregate mortality indexes that are based on total annual death and exposure counts, or configured in such a way that the age pattern of mortality jumps is identical to that of general mortality improvements. To discern the potential limitations of these modeling approaches, let us perform an exploratory analysis on some of the short-term mortality jumps that occurred in the US and England and Wales since 1901. We first apply the outlier detection methodology proposed by Li and Chan, 2005, Li and Chan, 2007 to find out the timings of the historical mortality outliers (jumps).1 Then for each detected mortality jump, we approximate its age pattern by computing for all age group , where is the timing of the detected jump and is the central death rate for age group at time . This quantity compares the log death rate for each age group in the year when the mortality jump occurred with the corresponding average log death rate over the six neighboring years.2 The patterns of for all detected mortality jumps are depicted in Fig. 1, from which we can conclude that the age patterns of mortality jumps are not uniform over age and exhibit certain degrees of variation. These properties cannot be reflected in models that are based on aggregate mortality indexes. Also shown in Fig. 1 are the values of (the age response parameters describing the age pattern of general mortality improvements) in the original Lee–Carter model (Lee and Carter, 1992) that is estimated to the data from each of the two populations.3 It can be seen that the patterns of and are generally different, indicating that models using the same age response parameters for mortality jumps and general mortality improvements may not be adequate.
To our knowledge, the work of Cox et al. (2010) is the only attempt so far to explicitly address the age pattern of mortality jumps, but their modeling approach is based much more heavily on expert opinions than statistical estimation. To fill this gap, in this paper we propose two variants of the Lee–Carter model with short-term jump effects. The first variant captures the age pattern of mortality jumps by a distinct collection of parameters, acknowledging the empirical fact that the age patterns of general and extreme changes in mortality rates over time are different. The second variant is a further generalization which permits the age pattern of mortality jumps to vary randomly, taking into account the correlation of jump effects among different age groups. Both model variants nest the transitory jump model developed by Chen and Cox (2009), in which mortality jumps are incorporated in the time-series process for the period effects. However, our proposed model variants demand different (and more advanced) estimation techniques, because in these model variants jump effects are involved in not only the time-series process but also other parts of the model structure.
We do acknowledge that parsimony is important and that the number of parameters grows as additional features are introduced. To focus on the issues we intend to investigate, we model one population only at a time and do not incorporate random changes in long-term mortality trends. Also, to preserve the tractability of the resulting log-likelihood function, we only consider Gaussian jumps, despite that other jump severity distributions such as double-exponential may produce a better fit. We overcome the challenges in statistical estimation by using the Route II estimation methodology that was recently introduced by Haberman and Renshaw (2012). This alternative estimation method is based on the first differences of the log mortality rates with respect to calendar time rather than the log mortality rates themselves. Compared to the traditional way of estimation, the advantages of the Route II methodology are twofold. First, in using the Route II approach, parameters representing the static level of mortality (i.e., the parameters) are not involved in the estimation and projection processes.4 By excluding these parameters, the model structure being estimated is more parsimonious and therefore convergence is easier to achieve. Second, in contrast to the traditional approach in which an extra step is needed to estimate the time-series process embedded in the model, the Route II methodology permits us to estimate all relevant parameters in one single estimation algorithm. So far as we aware, our paper represents the first attempt to use the Route II approach to estimate stochastic mortality models with jump effects.
It is reasonable to conjecture that the features we consider have an impact of the pricing of catastrophic mortality bonds, because two mortality jumps of the same severity but different age patterns could affect the payout from a mortality bond differently, depending on the age range with which the mortality bond is associated. To verify this conjecture, we attempt to use our proposed model variants to price a collection of hypothetical mortality bonds that are associated with different age ranges. In an incomplete market, the pricing problem is not straightforward, although it can be accomplished by insurance-based methods (Chen and Cummins, 2010, Wills and Sherris, 2010), no-arbitrage methods (Cairns et al., 2006, Chen and Cox, 2009, Chen et al., 2013a, Li, 2010, Li and Ng, 2011, Lin and Cox, 2008) or economic methods (Zhou et al., 2011, Zhou et al., 2013a, Zhou et al., 2013b, Chen et al., 2013b). The pricing method we use is the method of canonical valuation, which was first proposed by Stutzer (1996) and applied to the market of insurance-linked securities by Chen et al. (2013a), Li (2010) and Li and Ng (2011). Using the Atlas IX Capital Class B note launched in 2013 as a martingale constraint, this method identifies a risk-neutral probability measure, from which prices of the hypothetical mortality bonds can be estimated.
The rest of this paper is organized as follows. Section 2 describes the mortality data used in our illustrations. Section 3 provides the specifications of the proposed model variants. Section 4 explains how the parameters in the proposed model variants are estimated. Section 5 presents the estimation results and evaluates the goodness-of-fit. Section 6 details the pricing method we use and presents the calculated prices for a collection of hypothetical catastrophic mortality bonds. Finally, concluding remarks are provided in Section 7.
Section snippets
Mortality data
The illustrations in this paper are based on the mortality data from the unisex populations of the US and England and Wales. We obtain the required data from two official sources.
The data for the US population from 1901 to 2005 are provided by the Centers for Diseases Control and Prevention (CDC).5 These data are arranged by age groups , .
For the US population from 2006 to 2010 and English and Welsh population from 1901 to 2011, the data are obtained from the
The original Lee–Carter model
Our proposed model variants are built upon the original Lee–Carter model (Lee and Carter, 1992), which can be expressed as where denotes the central death rate for age group in year . In the model, parameter represents the static level of mortality for age group , parameter captures the variation of log mortality rates over time, parameter measures the sensitivity of to changes in , and is the error term. It is assumed that
Estimation method
As previously mentioned, the estimation method of Chen and Cox (2009) is not applicable to Models J1 and J2, in which jump effects are not simply superimposed onto the random walk with drift. To overcome the estimation challenge, we use the Route II estimation methodology documented in the paper by Haberman and Renshaw (2012), which the authors found to work well under the Lee–Carter modeling framework.
The Route II estimation method is based on the first differences of the log mortality rates
Estimation results
All three model variants are fitted to the historical mortality data from the US and English and Welsh populations. The estimated parameters and their standard errors are shown in Table 1, Table 2. The standard errors are calculated by a parametric bootstrapping procedure, which is detailed in Appendix B. Admittedly, some of the parameter estimates that are associated with jump effects have rather large standard errors. The large standard errors are not overly surprising, because over the data
Catastrophic mortality bonds
In recent years, several reinsurers have attempted to cede their exposures to extreme mortality risk by issuing catastrophic mortality bonds. These bonds have rather short times to maturity, usually three to five years. Coupon payments are typically linked to a market interest rate such as the London Interbank Offered Rate (LIBOR), while the principal repayment is not guaranteed. In particular, the principal would be reduced if the underlying index exceeds an attachment point and exhausted if
Concluding remarks
In this paper, we have investigated how the age pattern of transitory mortality jumps can be modeled explicitly. Two new variants of the Lee–Carter model have been proposed. The first variant uses a distinct collection of parameters to capture the age pattern of mortality jumps, while the second variant is a further generalization that allows the age pattern of future mortality jumps to vary randomly. When applied to historical mortality data from the US and English and Welsh populations, the
Acknowledgments
This work is supported by research grants from the Global Risk Institute, the Natural Sciences and Engineering Research Council of Canada and the Society of Actuaries Center of Actuarial Excellence Program.
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