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Pricing standardized mortality securitizations: a two-population model with transitory jump effects.

ABSTRACT

Mortality dynamics are subject to jumps that are due to events such as wars and pandemics. Such jumps can have a significant impact on prices of securities that are designed for hedging catastrophic mortality risk, and therefore should be taken into account in modeling. Although several single-population mortality models with jump effects have been developed, they are not adequate for modeling trades in which the hedger's population is different from the population associated with the security being traded. In this article, we first develop a two-population mortality model with transitory jump effects, and then we use the proposed model and an economic-pricing framework to examine how mortality jumps may affect the supply and demand of mortality-linked securities.

INTRODUCTION

The trading of mortality risk often involves two populations: one that is associated with the hedger's portfolio and another that is associated with the hedging instrument. Taking the mortality bond issued by Swiss Re in December 2003 as an example, the bond is linked to a broad population mortality index; however, the exposure of the hedger (Swiss Re) is associated with some insured lives. To adequately model trades involving more than one population, a multipopulation mortality model is necessary.

Multipopulation mortality models consider the potential correlations across different populations; more importantly, they are structured in such a way that the resulting forecasts are biologically reasonable. Specifically, the models ensure that the forecasted life expectancies of two related populations do not diverge over the long run. Beyond pricing, multipopulation mortality models enable the evaluation of population basis risk, which arises from the difference in mortality experience between the hedger's population and the population associated with the hedging instrument.

In recent years, a few two-population stochastic mortality models have been proposed. Carter and Lee (1992) propose the joint-k model, which assumes that mortality dynamics for the two populations being modeled are driven by the same time-varying factor. Li and Lee (2005) and Li and Hardy (2011) propose the augmented common factor model and the cointegrated Lee-Carter model, respectively. These two models generalize the joint-k model to permit short-term deviations from the common time-varying factor. Dowd et al. (2011) develop a gravity mortality model for two-related but different sized populations. A similar model is also proposed by Jarner and Kryger (2011). Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011) introduce a general framework for two-population mortality modeling. The framework specifies the conditions under which a two-population mortality model will not result in diverging long-term forecasts.

To our knowledge, none of the existing two-population mortality models incorporates jumps that are due to interruptive events such as the Spanish flu epidemic in 1918. It is important not to ignore mortality jumps in modeling, because otherwise we may inaccurately estimate the uncertainty surrounding a central mortality projection. The incorporation of jumps is particularly important when pricing securities for hedging extreme mortality risk, because this allows us to better estimate the probability of catastrophic mortality deterioration. In this article, we fill this gap in the literature by developing a two-population mortality model with transitory jump effects. The proposed model is primarily based on the two-population Lee-Carter model, which is discussed in Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011) (1) and subsequently used by Zhou, Li, and Tan (2011) for pricing longevity securities.

Although a two-population version has yet to be developed, various single-population mortality models with jump effects have been proposed and widely applied. In a continuous-time setting, such models are developed by Biffis (2005), Cox, Lin, and Wang (2006), and Deng, Brockett, and MacMinn (2012), and in a discrete-time setting, such models are proposed by Chen and Cox (2009) and Cox, Lin, and Pedersen (2010). These models differ in the types of mortality jumps they model. For instance, the model in Cox, Lin, and Wang is used for modeling permanent mortality jumps, whereas the model in Chen and Cox (2009) is used for modeling transitory mortality jumps. They also differ in the way in which severities of jumps are modeled. For example, Deng, Brockett, and MacMinn consider double-exponential jumps, while Chen and Cox consider Gaussian jumps.

The model we propose can be regarded as a two-population generalization of the model in Chen and Cox (2009). In particular, we assume that in each year, the mortality of a population is either jump-free or subject to one transitory mortality jump, and that the severity of a mortality jump is normally distributed. This generalization is not straightforward, partly because the correlations between jump times and jump severities of the two populations in question have to be carefully modeled, and partly because when constructing the likelihood function upon which parameter estimation is based, a careful conditioning on the jump counts is required. Relative to a single-population version, our proposed model is more suitable for modeling trades involving more than one population. Moreover, by allowing the populations being modeled to be subject to different (but correlated) mortality jumps, our model may more accurately estimate the population basis risk involved in index-based mortality hedges.

Another objective of this article is to examine the impact of mortality jumps on the trading of mortality risk. To achieve this goal, we consider the economic-pricing framework proposed by Zhou, Li, and Tan (2011, Forthcoming). This pricing framework models the trade of a mortality-linked security between two counterparties, whose portfolios can be related to different populations. Besides the estimated price, this pricing framework provides a pair of demand and supply curves, which explain the effect of introducing mortality jumps on the behaviors of the counterparties. We use the proposed model and the economic pricing framework to value a mortality bond that is similar to the one issued by Swiss Re in 2003. The bond's principal is eroded when future mortality exceeds a prespecified attachment point. Given the payoff structure, it is natural to assume that the incorporation of mortality jumps will increase the chance of principal erosion and consequently imply a lower bond price (or equivalently a higher risk premium). However, we find that the inclusion of mortality jumps may affect the estimated price of a mortality securitization in different directions, depending on how the security is structured.

The remainder of this article is organized as follows. In the second section, we describe the data used in illustrations. In the third section, we present a two-population Lee-Carter model on which our proposed extension is based. In the fourth section, we present a basic bivariate stochastic process for modeling the time-varying factors in the two-population Lee-Carter model. In the fifth section, we incorporate transitory jump effects into the basic bivariate stochastic process. In the sixth section, we examine how the inclusion of mortality jumps may affect the trading of mortality-linked securities. In the seventh section, we discuss how the choice of parameter constraints, the specification of the correlation structure, and the uncertainty surrounding the model parameter estimates may affect pricing and forecasting results. Finally, we conclude the article in the eighth section.

MORTALITY DATA

All illustrations in this article are based on historical mortality data from Swedish and Finnish male populations. For both populations, we consider a sample period of 1900-2006 and a sample age range of 25-84. The required data (death and exposure counts) are obtained from the Human Mortality Database (2011).

A TWO-POPULATION MODEL

We first construct a two-population mortality model by using two classical Lee-Carter models (Lee and Carter, 1992), one for each population. Mathematically, the two-population model can be expressed as:

In (m.sup.(i).sub.x,t]] = [[alpha].sup.(i).sub.x] + [[beta].sup.(i).sub.x] [[kappa].sup.(i).sub.t], i = 1, 2, (1)

where [m.sup.(i).sub.x,t] denotes the central death rate for population i at age x in year t, [[alpha].sup.(i).sub.x] and [[beta].sup.(i).sub.x] are age-specific parameters, and [[kappa].sup.(i).sub.t] is the period effect index for population i in year t. The use of a Lee-Carter structure for two-population mortality modeling is considered in Carter and Lee (1992), Li and Hardy (2011), Li and Lee (2005), and the original version of Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011).

It is reasonable to expect the death rates in two related populations not to diverge over the long run (e.g., Wilson, 2001; White, 2002). According to Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011), the necessary conditions for nondivergence are:

1. [[beta].sup.(1).sub.x] = [[beta].sup.(2).sub.x] for all x;

2. [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] do not diverge over the long run.

Therefore, we can set [[beta].sup.(1).sub.x] = [[beta].sup.(2).sub.x] = [[beta].sub.x], and model the difference between [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] with a stationary autoregressive process. This specific structure is used in the original version of Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011) for illustrative purposes.

We can estimate the parameters in Equation (1) by the method of maximum likelihood. To derive the log-likelihood function, we assume that:

[D.sup.(i).sub.x,t] ~ Poisson ([m.sup.(i).sub.x,t][E.sup.(i).sub.x,t]), i = 1, 2,

where [D.sup.(i).sub.x,t] is the observed number of deaths for population i at age x in year t, and [E.sup.(i).sub.x,t] is the corresponding number of persons at risk. Given this assumption, the log-likelihood function for Equation (1) can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where c is a constant, [[x.sub.0], [x.sub.1]] is the sample age range, and [[t.sub.0], [t.sub.1]] is the sample period.

As with many other stochastic mortality models, the model we consider has an identifiability problem. To stipulate parameter uniqueness, we use the following constraints: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The choice of parameter constraints is further discussed in the "Choice of Parameter Constraints" section.

The maximization of the log-likelihood can be accomplished by an iterative Newton's method, in which parameters are updated one at a time. The parameter constraints are applied at the end of each iteration. We refer readers to Li, Hardy, and Tan (2009) for further details regarding maximum likelihood estimation of the Lee-Carter model.

We fit Equation (1) to historical data from Swedish and Finnish males. The estimates of [[kappa].sup.(1).sub.t] (for Swedish males) and [[kappa].sup.(2).sub.t] (for Finnish males) are presented graphically in Figure 1. In the next two sections, we describe how the joint dynamics of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] can be modeled.

A NO-JUMP PROCESS FOR [[kappa].sup.(1).sub.t] AND [[kappa].sup.(2).sub.t]

If jumps are not considered, then we can model the dynamics of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] with the approach of Cairns, Blake, Dowd, Coughian, and Khalaf-Allah (2011). In this approach, [[kappa].sup.(1).sub.t] is modeled by a random walk with drift, while the difference between [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] is modeled by a stationary first-order autoregressive process. Mathematically, the assumed dynamics of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] can be specified by the following set of equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[mu].sub.k] is a constant, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a constant whose absolute value is strictly less than 1, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a sequence of independent and identically distributed (i.i.d.) bivariate normal random vectors with mean zero and variance-covariance matrix [V.sub.Z]. (2) This specification ensures a nondivergence between [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t], as [[DELTA].sub.[kappa](t)] will revert to a constant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] over the long run. This basic process does not take mortality jumps into account. For simplicity, we call this the no-jump process.

Under this specification, [[kappa].sup.(1).sub.t] follows a random walk with drift. The expected trajectory of [[kappa].sup.(1).sub.t] over time is therefore a (downward) sloping straight line, with slope equal to the drift term [[mu].sub.k]. The dynamics of [[kappa].sup.(2).sub.t] over time are more complicated. The complete specification of the process for [[kappa].sup.(2).sub.t] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

However, the dynamics of [[kappa].sup.(2).sub.t] can be more easily understood from the following expression:

[[kappa].sup.(2).sub.t+1] - [[kappa].sup.(2).sub.t] = [[kappa].sup.(1).sub.t+1] - [[kappa].sup.(1).sub.t] - ([[DELTA].sub.[kappa]](t+1) - [[DELTA].sub.[kappa]](t)). (3)

From this expression, we can tell that [[kappa].sup.(2).sub.t] follows the same random walk as [[kappa].sup.(1).sub.t], but is perturbed by deviations that are induced by the autoregressive process for [[DELTA].sub.[kappa]](t). Per unit time, the change in [[kappa].sup.(2).sub.t] can be decomposed into two random components, namely, [[kappa].sup.(1).sub.t+1] - [[kappa].sup.(1).sub.t], which is determined by the random walk with drift for population 1, and [[DELTA].sub.K](t + 1) - [[DELTA].sub.k](t), which is determined by the autoregressive process for [[DELTA].sub.k](t). Initially, [[kappa].sup.(2).sub.t] changes at a different speed than [[kappa].sup.(1).sub.t]. In particular, during the first time step beyond the forecast origin, the expected change in [[kappa].sup.(2).sub.t] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3) The speed of change is adjusted gradually by the autoregressive process for [[DELTA].sub.k](t). Over the long run, [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] are expected to change at the same speed [[mu].sub.[kappa]], because the stationarity in [[DELTA].sub.[kappa]](t) implies that the expectation of [[DELTA].sub.k](t + 1) - [[DELTA].sub.[kappa]](t) is zero as t tends to infinity.

We let:

[S.sub.t] = ([[kappa].sup.(1).sub.t], [[DELTA].sub.[kappa]](t))'.

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a sequence of i.i.d, bivariate normal random vectors, {[S.sub.t+1] | [S.sub.t]} is a sequence of bivariate normal random vectors. The mean and variance-covariance matrix of [S.sub.t+1]|[S.sub.t] are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [V.sub.Z], respectively. It follows that

the conditional log-likelihood function for the processes can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where bvnpdf(s, [mu], v) is the probability density function for a bivariate normal random vector with mean [mu] and variance-covariance v, and f(u|v) is a generic conditional density function of u given v. The estimates of [[mu].sub.[kappa]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [V.sub.z] can be obtained by maximizing the conditional log-likelihood function.

Alternatively, we can fit the entire model--Equation (1) and the processes for [[kappa].sup.(1).sub.t] [[kappa].sup.(2).sub.t]--by using the one-stage Bayesian approach considered by Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011). Relative to the two-stage approach, the one-stage Bayesian approach has several advantages. For example, it can handle missing data and readily produce measures of parameter risk. In this article, we use the two-stage approach for two reasons. First, the approach reduces computational effort because it avoids the need to reestimate the parameters In Equation (1) when mortality jumps are introduced to the time-series processes for [[kappa].sup.(1).sub.t] and [[DELTA].sub.K](t). Second, explanations of the technical issues (e.g., prior/posterior distributions) Involved in the single-stage Bayesian approach may distract readers from the main focus of this article.

A TRANSITORY JUMP PROCESS FOR [[kappa].sup.(1).sub.t] AND [[kappa].sup.(2).sub.t]

Specification of the Jump Process

From Figure i we observe several spikes in the period effect indexes for both populations, most notably in year 1918. Before developing a jump process for [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t], we conducted a statistical outlier analysis to identify spikes in the population trend that would likely be captured by the model. The analysis is documented in Appendix A.

The results of the outlier analysis indicate that the two populations have very different timing and frequency of jumps. Specifically, it was found that some mortality jumps affect both populations, but some only affect one of the two populations. These properties are taken into account in the model-building work, which we now present.

Let [[kappa].sup.(1).sub.t], i = 1, 2, be the observed period effect index for population i at time t. The M(i) y(i) observed index [[kappa].sup.(i).sub.t] is decomposed into the sum of two components, [[??].sup.(i).sub.t] + [N.sup.(i).sub.t][Y.sup.(i).sub.t]. The first component, [[??].sup.(1).sub.t], is the time t value of an unobserved period effect Index that is free of jumps, while the second term, [N.sup.(i).sub.t][Y.sub.(i).sub.t], represents the jump effect at time t. Both the jump count [N.sup.(i).sub.t] and the jump severity [Y.sup.(i).sub.t] depend on the population, thereby allowing the two populations to have different jump times, jump frequencies, and jump severities.

As in the no-jump process, we assume that [[??].sup.(1).sub.t] follows a random walk with drift and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] follows a stationary first-order autoregressive process. Summing up, the period effect indexes in this generalization are modeled by the following set of equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are constants, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a constant whose absolute value is strictly less than 1. The error terms [Z.sub.[kappa]](t) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] jointly follow a zero-mean bivariate normal distribution with a variance-covariance matrix [V.sub.Z]. They have no serial dependence and are independent of the jump counts and severities.

Since we work on annual mortality data, we permit a maximum of one jump in a given year for each population. It follows that the joint probability mass function for [N.sub.(1).sub.t] and [N.sup.(2).sub.t] can be specified as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [p.sub.1], [p.sub.2], and [p.sub.3] are nonnegative constants such that [p.sub.1] + [p.sub.2] + [p.sub.3] [less than or equal to] 1. According to this specification, the marginal probability of jump for population 1 is 1 - [p.sub.1] - [p.sub.3], while that for population 2 is 1 - [p.sub.1] - [p.sub.2]. We assume that the jump counts in different years are independent of one another. (4)

We allow both positive and negative jumps. The jump severities [Y.sup.(1).sub.t] and [Y.sup.(2).sub.t] can be different from each other, but they are correlated. Specifically, we assume that the jump severity vector ([Y.sup.(1).sub.t], [Y.sup.(1).sub.t]))' follows a bivariate normal distribution with mean [[mu].sub.Y] and variance-covariance matrix [V.sub.Y]. We further assume that the jump severities are not serially correlated and that they are independent of the jump counts.

Because the introduction of transitory jumps does not affect Equation (1), there is no need to reestimate the parameters in Equation (1). We do, however, need to estimate the parameters in the processes for the period effect indexes. This can be accomplished by the method of maximum likelihood. The construction of the log-likelihood function requires a careful conditioning on the jump counts [N.sup.(1).sub.t] and [N.sup.(2).sub.t]. A complete derivation of the log-likelihood function is presented in Appendix B.

The Estimated Process

We fit the transitory jump process to the period effect indexes for Swedish males and Finnish males. Recall that in the outlier analysis, we could not identify any outlier affecting Swedish males that did not affect Finnish males. To make the jump process more parsimonious, we set [p.sub.2]--the probability that Swedish males are subject to a mortality jump but Finnish males are not--to zero in the estimation process. The parameter estimates are shown in Table 1. (5)

In estimating the model parameters, we employed a two-stage approach. In the first stage, we estimate the Lee-Carter parameters by maximizing the log-likelihood specified by Equation (2). Given the first-stage estimates of [[kappa].sup.(i).sub.t], i = 1, 2, t = [t.sub.0], ..., [t.sub.1], we then estimate the jump process parameters by maximizing the conditional log-likelihood that is documented in Appendix B. It follows that when we calculate standard errors of the jump process parameters, some correction is necessary to account for the uncertainty surrounding the first-stage estimates, on which the estimation of the jump process parameters is based. We make the necessary correction on the basis of the theoretical results in Murphy and Topel (2002). Details regarding the correction methodology are provided in Appendix C.

The standard errors for the parameter estimates are shown in Table 1. Admittedly, the standard errors for the jump parameters are large. This is because over the sample period there were only a small number of extreme mortality deteriorations on which the estimation of these parameters could be based.

Evaluating the Jump Process

We also fit the no-jump process to the period effect indexes. Because the jump process and the no-jump process are nested, (6) we can use the likelihood ratio test to evaluate the benefit from introducing jumps. The maximized log-likelihood for the null model (the no-jump process) is -634.1980, while that for the alternative model (the jump process) is -503.6123. The value of the likelihood ratio test statistic is 2 x (-503.6123-(-634.1980)) = 261.1714. Under the null hypothesis that the alternative model does not provide a significantly better fit, the test statistic follows a chi-square distribution with 7 degrees of freedom. (7) The resulting p-value is 0, indicating that the jump process gives a significantly better fit than the no-jump process.

We further compare the jump process with the no-jump process by performing an out-of-sample forecast. Because the jump process is most likely used for pricing catastrophic mortality securities, which usually have short maturities, we consider the forecast in year 2009, which is 3 years from the forecast origin. The two processes yield similar central estimates of [[kappa].sup.(1).sub.2009] and [[kappa].sup.(2).sub.2009] (see Table 2). However, the density functions simulated from the jump process are significantly less dispersed but more heavy-tailed (see Figure 2).

To examine robustness relative to changes in the sample period, we reestimate both models to data over three different sample periods, namely, 1900-1986,1900-1996, and 1900-2006. The results are shown in Table 3. First, let us focus on the components that are common to both the no-jump process and the full transitory jump process. From Table 3, we observe that the majority of these components, including the autoregressive parameter [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and all elements in [V.sub.Z], are reasonably stable. However, the drift terms, [[mu].sub.[kappa]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], vary significantly as we change the length of the sample period. For instance, the estimate of [[mu].sub.[kappa]] in the jump process increases by an amount of 0.1238, or equivalently 46 percent of its standard error, if the endpoint of the sample period is changed from 2006 to 1986. The lack of robustness in the drift terms is a well-known problem (e.g., Li and Chan, 2005; Zhou and Li, 2013). Cairns (Forthcoming) coin this problem as recalibration risk and investigate its significance in the assessment of a longevity-hedging strategy.

Second, we focus on the components that are encompassed in the jump process only. From Table 3, we observe that these components are reasonably stable. The only exception is [p.sub.3], which increases by an amount of 0.0144, or equivalently 51 percent of its standard error, if we change the sample period from 1900-2006 to 1900-1986. The lack of robustness in [p.sub.3] may be explained by the results of the outlier analysis, which we document in Appendix A. In particular, out of the full sample period of 1900-2006, there are only four detected outliers that correspond to the event represented by parameter [p.sub.3].

The variations in the estimates of [p.sub.1] (in the jump process) and [[mu].sub.[kappa]] (in both processes) merit some explanation. The estimate of Pl, the probability that there is no jump in both populations in a given year, increases slightly as more recent mortality data are used. The increase in [p.sub.1] could be attributed to the fact that mortality trends in recent years are rather smooth and spike free. By contrast, the estimate of [[mu].sub.[kappa]] decreases (becomes more negative) as more recent mortality data are used. This decrease is potentially due to the recent acceleration of mortality improvement (e.g., Kannisto et al., 1994; Vaupel, 1997).

Adding Jump Effects to Other Two-Population Models

The jump process we proposed can also be imposed on some other two-population models. In what follows, we consider two other two-population mortality models, namely, the two-population age-period-cohort (APC) model in Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011) and the gravity model proposed by Dowd et al. (2011).

The two-population APC model in Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011) is based on the following structure:

ln ([m.sup.(i).sub.x,t]) = [[alpha].sup.(i).sub.x] + 1/[n.sub.a][[kappa].sup.(i).sub.t] + 1/[n.sub.a][[gamma].sup.(i).sub.t-x], i = 1,2, (4)

where [n.sub.a] = [x.sub.1] - [x.sub.0] + 1 is the total number of ages in the sample age range, [[alpha].sup.(i).sub.x] is an age-specific parameter, [[kappa].sup.(i).sub.t] is the period effect index for population i in year t, and [[gamma].sup.(i).sub.t-x] is the cohort effect index for individuals who are in population i and were born in year t - x. The assumed dynamics for [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] are exactly the same as the no-jump process we described in the fourth section. That is, the index KI1) follows a random walk with drift, and the difference [[kappa].sup.(1).sub.t] - [[kappa].sup.(2).sub.t] follows a stationary first-order autoregressive process.

The gravity model proposed by Dowd et al. (2011) is also based on the structure specified by Equation (4). It follows that if a two-stage estimation method is used, then the first-stage estimates of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] for both models will be identical. Dowd et al. use a random walk with drift to model [[kappa].sup.(1).sub.t]:

[[kappa].sup.(1).sub.t+1] = [[kappa].sup.(1).sub.t] + [[mu].sup.(1)] + [C.sup.(11)][Z.sup.(1)](t + 1), (5)

where [[mu].sup.(1)] and [C.sup.(11)] are constants, and {[Z.sup.(1)](t)} is a sequence of i.i.d, standard normal random variables. They specify the following process for [[kappa].sup.(2).sub.t]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where [[mu].sup.(2)], [C.sup.(21)], and [C.sup.(22)] are constants, [[phi].sup.[kappa]] is a constant whose absolute value is strictly less than 1, and {[Z.sup.(2)](t)} is a sequence of i.i.d, standard normal random variables. There are two rationales for using Equation (6) to model [[kappa].sup.(2).sub.t]. First, the equation introduces a dependency between [[kappa].sup.(2).sub.t] and the spread [[kappa].sup.(1).sub.t] - [[kappa].sup.(2).sub.t]. Second, it introduces a stochastic component that is common to both [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t].

It is interesting to see that the assumed dynamics of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t]. in the gravity model are also exactly the same as the no-jump process we described in the fourth section. To prove this, we subtract Equation (6) from Equation (5). This gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which implies that as in our no-jump process, the spread [[kappa].sup.(1).sub.t] - [[kappa].sup.(2).sub.t] follows a first-order autoregressive process. Furthermore, as in our no-jump process, the error term [C.sup.(11)] [Z.sup.(1)] (t + 1) in the random walk for [[kappa].sup.(1).sub.t] and the error term ([C.sup.(11)] - [C.sup.(21)])[Z.sup.(1)] (t + 1) - [C.sup.(22)] [Z.sup.(2)](t + 1) in the autoregressive process for [[kappa].sup.(1).sub.t] - [[kappa].sup.(2).sub.t] follow a bivariate normal distribution with a variance--covariance matrix that has nonzero off-diagonal elements.

Given that the assumed dynamics of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] in both models are identical to our no-jump process in the fourth section, we can readily introduce jump effects to these two models using the transitory jump process in the "Specification of the Jump Process" section and then evaluate the benefit of introducing jump effects to these two models. As in fitting our own model in the "The Estimated Process" section, we impose the constraint [p.sup.2] = 0 in the estimation process. The maximized log-likelihood for the no-jump process is -590.0970, while that for the transitory jump process is -485.6836. The value of the likelihood ratio test statistic is 2 x (-485.6836 - (-590.0970)) = 208.8268. This results in a p-value of 0, indicating that for these two alternative two-population models, the jump process still gives a significantly better fit to [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] than the no-jump process.

THE IMPACT ON MORTALITY RISK SECURITIZATION

An Illustrative Trade

Transitory mortality jumps can result in significant losses to life insurers. At the same time, they affect payouts from securities that are designed for hedging mortality risk. In this section, we use a hypothetical trade to illustrate how the incorporation of transitory jumps may affect the estimated price of a mortality-linked security.

We consider a trade between two economic agents, Agents A and B. Agent A is a life insurer that holds a portfolio of 10,000 life insurance policies. These policies are issued to Finnish males and are uniformly distributed over the age range of 25-44. For each policy, the death benefit is $0.01 payable at the end of the year of the policyholder's death. For simplicity, we assume that this portfolio is stationary, by which we mean the age composition does not change over time.

It is obvious that Agent A's financial obligation is linked to the mortality of Finnish males aged 25-44. Assuming no small sample risk, the insurance liability due at time t is [L.sub.t] = 5 [[summation].sup.44.sub.x=25][q.sup.(2).sub.x,t], where [q.sup.(2).sub.x,t] is the death probability of Finnish males at age x and t. We calculate [q.sup.(2).sub.x,t] by assuming a constant force of mortality between integral ages, which means [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To mitigate its exposure to catastrophic mortality risk, Agent A sells a 3-year mortality bond to Agent B, an investor who is interested in earning a risk premium. For liquidity considerations, the bond is linked to a mortality index that is based on the Swedish male population, which is larger than the population associated with the insurance liability. The index is the simple arithmetic average of the central death rates for Swedish males aged 25-44. Mathematically, the time t value of the index is given by:

[I.sup.t] = 1/20 [44.summation over(x=25)] [m.sup.(1).sub.x,t].

We assume that the mortality bond is traded at the end of year 2006, the last year in the sample period on which the estimation of model parameters is based. When the trade takes place, the value of [I.sub.2006] is known, but the values of [I.sub.t] for t > 2006 are not known and thus have to be forecasted.

The bond has a face value of $1. It makes a coupon payment at the end of each year at a rate of r+1.5 percent, where r is the risk-free interest rate, which is assumed to be 3 percent in our calculations. The principal repayment at maturity depends on the values of [I.sup.t] over the term of the security. Specifically, the principal repayment is specified as follows:

Principal Repayment = max (1 - [2009.summation over (t=2007)] [loss.sub.t],0),

where [loss.sub.t] is defined by:

[loss.sub.t] = max([I.sub.t] - a[I.sub.2006],0) - max ([I.sub.t] - b[I.sub.2006],0),/(a - b)[I.sub.2006]

and a and b are the attachment and exhaustion points, respectively. If the mortality index ever exceeds a[I.sub.2006], then the principal repayment will be reduced, and if the mortality index ever exceeds b[I.sub.2006], the principal repayment will be exhausted.

The choice of the attachment and exhaustion points depends on the objective of the hedger (Agent A). If the hedger intends to hedge the risk associated with the most extreme mortality deteriorations only, then it should choose larger values of a and b. By contrast, if it also wants to hedge the risk associated with less extreme events, then lower values of a and b may be chosen. For the mortality bond (Vita I) issued by Swiss Re in 2003, the values of a and b were set to 1.3 and 1.5, respectively (see Blake, Calms, and Dowd, 2006).

To suit the preferences of different investors, a mortality bond is sometimes structured in multiple tranches, each of which has different attachment and exhaustion points. For instance, the mortality bond (Tartan) issued by Scottish Re in 2006 consists of two tranches. The less risky tranche has an attachment point of a = 1.15 and an exhaustion point of b = 1.2, while the riskier tranche has an attachment point of a = 1.1 and an exhaustion point of b = 1.15. Further information about the setup of this bond can be found in Bauer and Kramer (2008).

In our illustration, we consider two sets of attachment and exhaustion points: (1) a = 1.3, b = 1.4; (2) a = 1.2, b = 1.3.

Pricing the Mortality Bond

We use the economic-pricing framework proposed by Zhou, Li, and Tan (2011, Forthcoming) to price the mortality bond. The framework models the trade between Agents A and B. An appealing feature of the economic-pricing method is that, as opposed to no-arbitrage approaches, it does not require the market prices of other mortality-linked securities as input. This can spare us from the problems associated with the lack of market price data.

The first step in this pricing framework is to derive the wealth processes for the two agents. In the derivation, it is assumed that the wealth of each agent can only be invested in either the mortality bond or a bank account, which yields a continuously compounded risk-free interest rate of r per annum. Other than the bank account, the mortality bond, and the liability, there are no sources of income or payout. It is also assumed that the agents do not trade the mortality bond with each other after inception. Throughout the derivation, we use P to denote the price of the mortality bond, [theta] to denote the quantity of the mortality bond traded, and [g.sub.t]([I.sub.t]), t = 2007, 2008, 2009, to denote the payout from each unit of the bond at time t.

Let [W.sup.A.sub.t] be the wealth of Agent A at the end of year t. The initial wealth, [W.sup.A.sub.2006] is a constant that is free of P and [theta]. One year after inception, Agent A's wealth becomes

[W.sup.A.sub.2007](P,[theta]) = ([W.sup.A.sub.2006] + [theta]P)[e.sup.r] - [theta][g.sub.2007] ([I.sub.2007]) - [L.sub.2007],

and for t = 2008, 2009, Agent A's wealth can be expressed as

[W.sup.A.sub.t](P, [theta]) = [W.sup.A.sub.t-1] (P, [theta]) [e.sup.r] - [theta][g.sub.t]([I.sub.t]) - [L.sub.t].

Note that for t > 2006, [W.sup.A.sub.t] is a function of P and [theta]. Using the above, we can easily show that Agent A's terminal wealth--the wealth when the mortality bond matures--can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [W.sup.B.sub.t] be the wealth of Agent B at the end of year t. The initial wealth, [W.sup.B.sub.2006], is a constant that is free of P and [theta]. One year after inception, Agent B's wealth becomes [W.sup.B.sub.2007](P,[theta]) = ([W.sup.B.sub.2006] - [theta]P) [e.sup.r] + [theta][g.sub.2007]([I.sub.2007]),

and for t = 2008, 2009, Agent B's wealth can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using the above, we can easily show that Agent B's terminal wealth is given by

[W.sup.B.sub.2009] (P, [theta]) = [W.sup.B.sub.2006][e.sup.3r] + [theta] ([2009.summation over (t=2007)][g.sub.t]([I.sub.t])[e.sup.r](2009)-t)] - [Pe.sup.3r]).

The next step is to derive the demand and supply curves. We let [U.sup.A] and [U.sup.B] be the utility functions for Agents A and B, respectively. It is assumed that, given a price P, each agent would maximize its expected terminal utility by altering its demand or supply of the mortality bond; hence, given P, Agent A would supply

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

units of the mortality bond. The supply curve can be derived by calculating the value of [[theta].sup.A] at different price levels. Similarly, given P, Agent B would demand

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

units of the mortality bond. The demand curve can be derived by calculating the value of [[theta].sup.B] at different price levels.

Finally, the price of the mortality bond is estimated by searching for the value of P that equates [[theta].sup.A] and [[theta].sup.B]. The estimated rice is the rice at which the demand and supply are equal, or when the market clears. An iterative procedure for solving [[theta].sup.A](P) = [[theta].sup.B](P) is provided in Zhou, Li, and Tan (2011, Forthcoming).

In our calculations, we assume an exponential utility function, U(x) = 1 - [e.sup.-kx], for each agent. Parameter k in the utility function is the absolute risk aversion for all wealth levels. A larger k means that the agent is more conservative and risk averse. It is reasonable to assume that Agent A's absolute risk aversion is larger than Agent B's, because Agent A wants to hedge its mortality risk exposure while Agent B is willing to take the risk in return of a risk premium. As in Zhou, Li, and Tan (2011, Forthcoming), we assume that the absolute risk aversion for Agent A is [k.sup.A] = 1.0 and that for Agent B is [k.sup.B] = 0.5.

To estimate the price of the mortality bond, we first simulate 50,000 mortality paths from a two-population mortality model. Given the sample paths, we can readily calculate the expected terminal utilities E [[U.sup.A] ([W.sup.A.sub.2009](P, [theta]))] and E [[U.sup.B] ([W.sup.B.sub.2009](P, [theta]))]. We then estimate the demand and supply curves using Equations (7) and (8), and compute the market clearing price by solving [[theta].sup.A](P) = [[theta].sup.B](P).

On the basis of the no-jump process and the transitory jump process we fitted in the "Specification of the Jump Process" section, we obtain two sets of bond prices. The estimated prices are displayed in Table 4. As expected, the price of the mortality bond is lower if the attachment and exhaustion points are smaller. The reason in this case is because it is more likely that the index [I.sub.t] will exceed the attachment and exhaustion points, thus reducing the expected principal repayment and consequently the price of the bond.

A more interesting observation is that the effect of introducing transitory jumps depends on the attachment and exhaustion points. The no-jump model produces the highest price when a = 1.3 and b = 1.4, but the opposite is true when a = 1.2 and b = 1.3. We will explain the reasons underlying this observation later in this section when we analyze the demand and supply of the mortality bond.

Determinants of Supply and Demand

The economic-pricing method we consider is more transparent relative to standard no-arbitrage approaches, in which the price of a security is estimated by extrapolating prices of other similar securities available in the market. On top of the estimated price, the economic-pricing method provides us with a pair of demand and supply curves. By examining the response of the demand and supply curves to changes in different factors, we can form a better understanding about how and why the price of a mortality-linked security will change under different circumstances.

Let [V.sub.L] and [V.sub.H] be the accumulated values (when the bond matures) of the insurance liabilities and the payouts from one unit of the bond, respectively. These two quantities can be expressed mathematically as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

At inception (in 2006), these two quantities are random variables.

We can then rewrite the supply curve, Equation (7), as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the above, the second step follows from the assumption of exponential utility. Similarly, we can rewrite the demand curve, Equation (8), as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We observe that given the risk aversion parameters [k.sup.A] and [k.sup.B], the demand and supply curves depend only on the random variables [v.sub.H] and [v.sub.L]; hence, for fixed values of [k.sup.A] and [k.sup.B], the joint probability distribution of [v.sub.H] and [v.sub.L] exclusively determines the demand and supply curves.

We let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

respectively, be the mean vector and variance--covariance matrix of the joint distribution of ([v.sub.H], [v.sub.L])', measured in 2006. If ([v.sub.H], [v.sub.L])' follows a distribution (e.g., bivariate normal) that can be completely specified by its first two moments, then parameters [[mu].sub.H], [[mu].sub.L], [[sigma].sub.H] [[sigma].sub.L] and [rho] exclusively determine the demand and supply curves. Even if not, these five parameters should still have a very strong influence on the demand and supply curves. We can therefore better understand how the introduction of mortality jumps may affect the demand and supply curves by examining its effect on each of these five parameters.

The effects of [[mu].sub.H], [[mu].sub.L], [[sigma].sub.H], [[sigma].sub.L] and [rho] on the supply and demand of the mortality bond are described below.

* [[mu].sub.H], the expected value of [v.sub.H].

At a given price, when [[mu].sub.H] increases, Agent B (the buyer) is expected to receive more payouts from the mortality bond, while Agent A (the seller) is expected to make more payouts; hence, other things equal, an increase in [[mu].sub.H] would lead to an increase in the demand from Agent B and a reduction in the supply from Agent A.

* [[mu].sub.L], the expected value of [v.sub.L].

A change in [[mu].sub.L] would not affect the demand from Agent B, because Agent B's behavior depends only on the mortality bond. On the other hand, with other aspects being equal, an increase in [[mu].sub.L] means that Agent A is subjected to additional liabilities that are not life contingent and thus require no hedging. As a result, an increase in [[mu].sub.L] has no effect on the supply of the mortality bond from Agent A.

* [[sigma].sub.H], the standard deviation of [v.sub.H].

When [[sigma].sub.H] increases, the security becomes less attractive to Agent B, as it needs to take more risk for the same expected payoff from the bond. On the other hand, when [[sigma].sub.H] increases, Agent A can achieve the same amount of risk reduction by selling fewer units of the mortality bond. As a result, an increase in [[sigma].sub.H] would lead to a reduction in both supply and demand.

* [[sigma].sub.L], the standard deviation of [V.sub.L]. A higher [[sigma].sub.L] means that Agent A is subject to more mortality risk. Therefore, it has a stronger need to sell the mortality bond, which would lead to a greater supply. However, because Agent B's behavior depends only on the mortality bond, a change in [[sigma].sub.L] would not affect the demand from Agent B.

* [rho], the correlation between [V.sub.L] and [V.sub.L]. When [absolute value of [rho]] is higher, the mortality bond becomes a more effective hedging instrument. Therefore, at a given price, Agent A is willing to supply more. Same as [[sigma].sub.L], [rho] does not affect Agent B's behavior; hence, a change in [rho] would not affect the demand from Agent B.

The effects of these five factors on the supply and demand of the mortality bond are summarized in Table 5. In the table, "[up arrow]" means an increase, "[down arrow]" means a decrease, and "-" means that there is no change.

These five factors can be estimated from mortality paths simulated from a two population mortality model. For each simulated mortality path, we calculate the value [v.sub.H] and [v.sub.L]. This gives a joint empirical distribution of [v.sub.H] and [v.sub.L], from which we can readily obtain estimates of [[mu].sub.H], [[mu].sub.L], [[sigma].sub.H], [[sigma].sub.L], and [rho].

In the rest of this section, we study how the introduction of transitory jumps may affect these five factors and consequently the equilibrium price of the mortality bond.

The Supply and Demand Curves When a = 1.3 and b = 1.4

The demand and supply curves when a = 1.3 and b = 1.4 are shown in Figure 3. When transitory mortality jumps are introduced, the supply curve shifts upward, while the demand curve shifts downward. This leads to a reduction in the equilibrium price of the mortality bond.

To understand the changes in the demand and supply curves, we examine these described factors. These factors are estimated by using the no-jump process and the full transitory jump process. The estimates of [[mu].sub.H], [[sigma].sub.H], [[sigma].sub.L], and p are presented in Table 6. The estimates of [[mu].sub.L] are not shown because, as previously mentioned, a change in [[mu].sub.L] has no effect on the demand and supply curves.

The quantity [[mu].sub.H] represents the accumulated value of the expected payouts from the mortality bond, which is subject to a principal reduction when the mortality index exceeds the attachment point. The value of [[mu].sub.H] is therefore heavily dependent on the right tail of the distribution of the mortality index. In Figure 4, we plot the kernel smoothed density functions for [I.sub.2007]/[I.sub.2006] under different assumed models. The upper panel of Figure 4 depicts the entire density functions, while the lower panel gives a closer shot of the right tails.

Recall that the principal repayment of the mortality bond is reduced if the mortality index is higher than 1.3[I.sub.2006], and exhausted if it is higher than 1.4[I.sub.2006]. We observe from Figure 4 that beyond the attachment point a = 1.3, the value of the density function for [I.sub.2007]/[I.sub.2006] under the no-jump process is always lower than that under the hall transitory jump process. This means that the no-jump process implies a smaller probability (and magnitude) of principal reduction and hence a higher value of [[mu].sub.H].

The quantity [[sigma].sub.L] depends heavily on the volatility of [L.sub.t], the insurance liability due in year t, for t = 2007, 2008, 2009. In Figure 5, we show the kernel smoothed density functions for [L.sub.2007], based on the no-jump process and the full transitory jump process. The no-jump process produces a significantly more dispersed (but less heavy-tailed) distribution, because it incorporates the variations caused by jumps into its volatility term. As a result, the no-jump process gives a higher value [[sigma].sub.L].

The quantity [[sigma].sub.H] depends on the volatility of the principal repayment. Because the principal repayment is always 100 percent unless the index exceeds the attachment point, the value of [[sigma].sub.H] depends more heavily on the index's variability in the (right) tail than the overall volatility of the index. The dependence of [[sigma].sub.H] on the tail volatility increases with the value of the attachment point. In this example, [[sigma].sub.H] is higher when jumps are considered.

The quantity [absolute value of [rho]] is related to the correlation between the mortality rates for the two populations. In the no-jump process, the correlation between the mortality of the two populations is driven entirely by the joint distribution of the innovations, [Z.sub.k](t) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], whereas in the jump process, the correlation is driven additionally by joint distribution of the jump counts, [N.sub.t.sup.(2)] and [N.sub.t.sup.(2)], and the joint distribution of the jump severities, [N.sub.t.sup.(2)] and [N.sub.t.sup.(2)]; hence, the introduction of jumps alters the correlation structure and increases [absolute value of [rho]] in this example.

Now let us revisit the demand and supply curves. The introduction of transitory jumps increases [[sigma].sub.H] and [absolute value of [rho]] but reduces [[mu].sub.H] and [[sigma].sub.L]. According to Table 5, a lower [[mu].sub.H] and a higher [[sigma].sub.H] both exert pressure on the demand, causing the demand curve to shift downward. On the other hand, the changes in [[mu].sub.H] and [absolute value of [rho]] lead to an increase in supply, while the changes in [[sigma].sub.H] and [[sigma].sub.L] lead to a reduction. Here, the combined effect of [[mu].sub.H] and [absolute value of [rho]] is stronger, thus causing the supply curve to shift upward. Overall, the shifts in the demand and supply curves result in a smaller price and a lower quantity traded in equilibrium.

The Supply and Demand Curves When a = 1.2 and b = 1.3

Here, we repeat the analysis for the case when a = 1.2 and b = 1.3. The resulting supply and demand curves are shown in Figure 6. The estimates of the four factors are shown in Table 7. Note that there is no change to the estimates of [[sigma].sub.L] because the insurance liability is not affected by the changes in the attachment and exhaustion points.

As the attachment and exhaustion points are changed, the demand from the investor and the supply from the hedger should also change. The effects of changes in the attachment and exhaustion points are incorporated in our pricing framework through factors [[mu].sub.H], [[sigma].sub.H], and [rho]. Relative to those in the previous example, the demand curves in this example are lower. This lowering is because the changes in a and b have reduced [[mu].sub.H] and raised [[sigma].sub.H]. According to Table 5, a reduction in [[mu].sub.H] and an increase in [[sigma].sub.H] both result in a lower demand. Changes in supply due to the changes in a and b can be explained using similar arguments.

We now analyze the impact of jumps when this specific set of attachment and exhaustion points are assumed. From Figure 6 we observe that when mortality jumps are introduced, the supply curve shifts downward, while the demand curve shifts upward. This leads to an increase in the equilibrium price of the mortality bond. It is noteworthy that the changes in the demand and supply curves are exactly opposite to those in the previous example, which is based on higher attachment and exhaustion points.

We examine again the four determinants of supply and demand. As in the previous example, the jump process implies a higher value of [absolute value of [rho]] and a smaller value of [[sigma].sub.L].

However, in this example, the value [[mu].sub.H] is lower when mortality jumps are not taken into account. The reason behind this lowering can be seen from Figure 4, which shows the density functions of [I.sub.200]/[I.sub.2006] under the two mortality models. We observe from the diagram that in most of the interval between the attachment point a = 1.2 and the exhaustion point b = 1.3, the value of density function based on the no-jump process is higher. This means that the no-jump process implies a higher probability (and magnitude) of principal reduction and thus a smaller value of [[mu].sub.H].

When the attachment point is lower, the value of [[sigma].sub.H] is less dependent on the index's variability in the (right) tail but more dependent on the overall volatility of the index. The overall volatility of the index is higher when jumps are not taken into account, because, as previously mentioned, the no-jump process incorporates the variations caused by jumps into its volatility term. As a result, in this example, the value of [[sigma].sub.H] is lower when jumps are considered.

Now let us revisit the supply and demand curves in Figure 6. The introduction of transitory jumps increases [[mu].sub.H] and [absolute value of [rho]] but reduces [[sigma].sub.H] and [[sigma].sub.L]. According to Table 5, higher [[mu].sub.H] and lower [[sigma].sub.H] both have positive effects on demand, thereby causing the demand curve to shift upward. On the other hand, the changes in [[sigma].sub.H] and [absolute value of [rho]] lead to an increase in supply, while the changes in [[mu].sub.H] and [[sigma].sub.L] lead to a reduction. Here, the combined effects of [[mu].sub.H] and [[sigma].sub.L] are stronger, causing the supply curve to shift downward. Overall, the shifts in the demand and supply curves result in a higher price and a lower quantity traded in equilibrium.

Other Issues

The Choice of Parameter Constraints

In estimating the parameters in Equation (1), certain constraints must be applied to ensure that they are unique. In this article, we use the following set of constraints:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given the way we model [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t], in the long-term equilibrium, the average difference between [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] over time is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because the constant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is generally not zero, using the constraints [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 1, 2 does not mean we are assuming that the long-term equilibrium is already attained at the forecast origin [t.sub.l]. The diagram in Figure 7 illustrates the expected trajectories of [K.sup.(1).sub.t] and [K.sup.(2).sub.t] from the forecast origin [t.sub.1] to the long-term equilibrium.

There are two reasons for using this particular set of constraints. First, the constraints we use imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

thereby giving an exact fit to the mortality rates at the forecast origin [t.sub.1]. Second, as Lee and Miller (2001) and Li, Hardy, and Tan (2009) point out, the constraints we use produce significantly more accurate short-term mortality forecasts than other typical parameter constraints do. These two desirable features are highly important to our work, because our proposed model will most likely be used for pricing catastrophic mortality securities, which usually have short maturities. To explain further, let us revisit the 3-year mortality bond described in the "An Illustrative Trade" section. The bond's principal will be eroded if future mortality exceeds a prespecified attachment point. If the fit at the forecast origin and the short-term mortality forecast are biased low, then we will underestimate the probability and magnitude of principal erosion, and consequently overprice the bond.

To facilitate understanding of the impact of the choice of parameter constraints, we reestimate our proposed model with an alternative set of constraints, and recalculate the forecasting and pricing results. The alternative constraints are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[beta].sub.x] = 1 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 1.2. They are used in the original article of Lee and Carter (1992) as well as numerous other articles on stochastic mortality modeling (e.g., Cairns et al., 2009).

In Table 8, we compare the results that are based on the alternative constraints with those that are based the constraints we chose. The actual values of [m.sup.(1).sub.35] and [m.sup.(2).sub.35] at the forecast origin [t.sub.1] = 2006 are 6.3631 x [10.sup.-4] and 1.7437 x [10.sup.-3], respectively. (8) Our constraints lead to an exact fit to these values, but the estimates resulting from the alternative constraints are biased low by 39 percent and 66 percent, respectively. It also appears that the alternative constraints result in significantly biased forecasts of [m.sup.(1).sub.35] and [m.sup.(2).sub.35] in 2009, the year in which the illustrative mortality bond described in the "An Illustrative Trade" section matures. As recently mentioned, the underestimation of [m.sup.(1).sub.35] and [m.sup.(2).sub.35] over the term of the mortality bond may lead to overpricing. In this illustration, the estimated price based on the alternative constraints is approximately 4 percent higher than that based on the constraints we chose.

The exact fit at the forecast origin comes with a cost. Let [n.sub.a] = [X.sub.l]-[x.sub.0] + 1be the number of ages included the sample age range and [n.sub.y] = [t.sub.1] - [t.sub.0] + 1 be the number of years included in the sample period. When the alternative constraints are used, Equation (1) contains 3[n.sub.a] + 2[n.sub.y]-3 degrees of freedom, but when the constraints we chose are used, the degrees of freedom reduce to [n.sub.a] + 2[n.sub.y] - 3. The constraints we chose therefore make the model more stringent, which in turn results in a reduced goodness of fit. In this illustration, the maximized log-likelihood in fitting Equation (1) is -77,940 when the alternative constraints are used, but is only -115,903 when the constraints we chose are used.

Another trade-off of having an exact fit at the forecast origin is a reduced robustness, because when the constraints we chose are used, the data at the forecast origin would have a very strong influence on the resulting parameter estimates. An anomaly in the data at the forecast origin can result in an estimated model that gives a poor fit to the data in earlier years.

Parameter Uncertainty

No matter what model is used, there is always uncertainty surrounding the parameter estimates because the true value of any free parameter can never be known. In "The Estimated Process" section, we quantified parameter uncertainty by calculating the standard error of each parameter estimate using the method documented in Appendix B. In this subsection, we evaluate the impact of parameter uncertainty on forecasting and pricing.

We focus on [[mu].sub.[gamma]](1), [[mu].sub.[gamma]](2), [p.sub.1], and [p.sub.3], because these four parameters, which capture mortality jumps, are the most crucial to the pricing of short-term catastrophic mortality bonds. (9) To measure the impact of parameter uncertainty, we first take turns shocking each of these four parameters by one standard error, and then recalculating the forecasting and pricing results.

Let us first examine the forecasts of the period effect indexes in 2009, the year in which the mortality bond described in the "An Illustrative Trade" section matures.

The upper panel of Figure 8 shows the simulated density functions of [[kappa].sup.(1).sub.2009] when positive and negative shocks are applied to parameter [[mu].sub.[gamma]] (1), which measures the severity of mortality jumps in population 1. The shocks in [[mu].sub.[gamma]] (1) only have a minimal effect on the overall shape and location of the simulated density function; however, they do have an observable impact on the tails. As expected, the tails become thicker when [[mu].sub.[gamma]] (1) is raised and thinner when [[mu].sub.[gamma]] (1) is lowered. The impact of shocking [[mu].sub.[gamma]] (2) on the simulated density function of [[kappa].sup.(2).sub.2009] is similar (see the lower panel of Figure 8). Note that because [[kappa].sup.(2).sub.t] is unrelated to [[mu].sub.[gamma]] (1), shocking [[mu].sub.[gamma]] (1) has no impact on the forecast of [[kappa].sup.(2).sub.2009]. Likewise, shocking [[mu].sub.[gamma]] (2) has no impact on the forecast of [k.sup.(2).sub.2009]. (1)

We now turn to the jump frequency parameters. In our jump process, the marginal probability of jump for population 1 is 1 - [p.sub.1] - [p.sub.3], while that for population 2 is 1 - [p.sub.1] - [p.sub.2]. Because [p.sub.1] is involved in both marginal probabilities, shocking [p.sub.1] affects the forecasts for both populations. The effects of shocking [p.sub.1] on the density forecasts of [[kappa].sup.(1).sub.2009] and [[kappa].sup.(2).sub.2009] are shown in the upper and middle panels of Figure 9, respectively. Similar to shocks in the jump severity parameters, a shock in [p.sub.1] mainly affects the tails of the density functions. In particular, an increase in [p.sub.1] reduces the thickness of the tails, while a decrease in [p.sub.1] does the opposite. On the other hand, as [p.sub.3] is involved in the marginal probability for population 1 only, a shock in [p.sub.3] has no effect on the density forecast of [[kappa].sup.(2).sub.2009]. The impact of shocking P3 on the density forecast of [[kappa].sup.(1) is shown in the lower panel of Figure 9.

Next, we investigate the impact of parameter uncertainty on pricing. The impact of shocking the four parameters on the estimated trading price and quantity is summarized in Table 9. We observe that the one-standard-error shocks affect the pricing results significantly. This observation is not overly surprising, because the erosion of the bond's principal depends heavily on the tails of the density forecasts of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] over the term of the bond, and such tails, as demonstrated, are sensitive to the four parameters under consideration.

Specification of the Correlation Structure

In this subsection, we perform a further investigation of how the underlying correlation structure may affect pricing and forecasting. Specifically, we compare the results that are generated from the following five modeling approaches:

* Approach 1

Use a single-population model without jump effects for each of the two populations. There is no correlation between the random terms in the two models.

* Approach 2

Use a single-population model with transitory jump effects for each of the two populations. There is no correlation between the random terms in the two models.

* Approach 3

Use a two-population model without jump effects. The random terms [Z.sub.[kappa]](t) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the model are correlated with each other. This approach is identical to the approach described in the fourth section.

* Approach 4

Use a two-population model with transitory jump effects. A partial correlation structure is assumed in this approach. In particular, the random terms [Z.sub.[kappa]](t) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the model are correlated with each other, but there is no correlation between the jump counts [N.sup.(1).sub.t] and [N.sup.(2).sub.t] and between the jump severities [Y.sup.(1).sub.t] and [Y.sup.(2).sub.t].

* Approach 5

Use a two-population model with transitory jump effects. The model has a full correlation structure; that is, [Z.sub.[kappa]](t) is correlated with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is correlated with [N.sup.(2)).sub.t], and [Y.sup.(1).sub.t]) is correlated with [Y.sup.(2).sub.t]). This approach is identical to the approach described in the fifth section.

Here, we forecast two quantities, namely, [L.sub.2009], the 2009 value of Agent A's insurance liability, and [I.sub.2009], the 2009 value of the index to which the mortality bond described in the "An Illustrative Trade" section is linked. Note that [L.sub.2009] is related to population 2, while [I.sub.2009] is related to population 1. We are particularly interested in the projected association between these two quantities.

In Figure 10, we show, for each modeling approach, a scattered plot of 50,000 simulated pairs of [L.sub.2009] and [I.sub.2009]. We present the following comments regarding the five scattered plots.

In the scattered plot for Approach 1, the cloud of simulated values has no significant tilt, indicating that this approach does not capture any association between [L.sub.2009] and [I.sub.2009]. Moreover, the cloud shows no sign that this approach can capture extreme outcomes.

In the scattered plot for Approach 2, the cloud of simulated values contains two elongated branches that are orthogonal to each other. The existence of these two branches indicates that Approach 2 is able to capture the extreme behaviors of [L.sub.2009] and [I.sub.2009].

However, the orientation of the two branches suggests that this modeling approach fails to incorporate the potential association between the two quantities. In particular, the orientation implies that an extreme realization of [L.sub.2009] almost never comes with an extreme realization of [I.sub.2009].

In the scattered plot for Approach 3, the cloud of simulated values is somewhat tilted, thereby indicating that this modeling approach is able to incorporate some association between the two random quantities. However, the cloud does not give a clear indication that extreme behaviors of [L.sub.2009] and [I.sub.2009] are captured.

In the scattered plot for Approach 4, the cloud of simulated values has a tilted center part and two distinct branches that are orthogonal to each other. The cloud's shape suggests that Approach 4 can capture extreme outcomes and some association between the two random quantities. However, Approach 4 still fails to incorporate the possible situation of both [L.sub.2009] and [I.sub.2009] taking extreme values.

The scattered plot for Approach 5 suggests that this approach can capture what we found from historical trends and the outlier analysis (see Appendix A). As in the previous two plots, the cloud in this plot has a tilted center part, which captures the association between [L.sub.2009] and [I.sub.2009] when no extreme events occur. What makes this plot unique is the orientation of the cloud's two branches. The vertical branch captures the situation when [L.sub.2009] takes an extreme value but [I.sub.2009] does not, while the tilted branch captures the situation when both random quantities take extreme values. This plot is therefore in line with the conclusion drawn from the outlier analysis: Some mortality jumps affect one population only, but some do affect both populations.

We then study the impact on pricing by revisiting the illustrative mortality bond described in the "An Illustrative Trade" section. In Table 10, we display the trading price and quantity estimated from each of the five modeling approaches. Approaches 1 and 2 imply that the illustrative mortality bond will not be traded. This result is within our expectation, because when a correlation structure is absent, the bond's payoff and the insurer's liability will not have any association, which in turn means that there is no reason for the insurer to trade the bond. This result also warns readers that when analyzing deals involving two related populations, using independent mortality models may lead to erroneous conclusions.

Another noteworthy finding is that Approach 4, which assumes a partial correlation structure only, predicts a significantly smaller trading quantity compared to Approach 5. This finding suggests that when analyzing catastrophic mortality securities, it is important to consider the potential correlation between mortality jumps in different populations.

CONCLUDING REMARKS

Standardization is a goal of many participants in the market for mortality-related risk. Standardized mortality-linked securities are based on broad-based populations, which are generally different from populations associated with hedgers' portfolios. Therefore, when we evaluate a hedge that is constructed with a standardized instrument, a two-population mortality model is necessary.

Nevertheless, existing two-population models do not incorporate mortality jumps, which could have a significant impact on the securitization of mortality-related risk. In this article, we generalized a two-population mortality model to incorporate transitory jump effects. The proposed model permits the two populations in question to have different jump times, jump frequencies, and jump severities. This model fits significantly better to the considered data in comparison to the corresponding no-jump model.

To study the impact of introducing transitory jumps, we considered the trade of a mortality bond between a life insurer and an investor who is willing to take mortality risk for a risk premium. The principal repayment of the bond is reduced if the index to which it is linked exceeds a predetermined attachment point. When jumps are introduced, the distribution of the index becomes less dispersed but heavier tailed. Consequently, the impact on the estimated bond price depends heavily on the value of the attachment point. If the attachment point is low, then the model without jumps would imply a greater probability of principal reduction and hence a smaller bond price, but if the attachment point is high, the opposite is true.

Other than pricing mortality-linked securities, the proposed model can be used conveniently for measuring population basis risk in index-based mortality hedges. In future research, it would be interesting to apply the proposed model to various frameworks for basis risk measurement, including those proposed by Cairns et al. (Forthcoming) and Coughlan et al. (2011).

One may think that short-term catastrophic mortality could result from systematic changes in the volatility of mortality. This view is incorporated in our transitory jump process. To illustrate, we consider the volatility of mortality improvement, measured by [[kappa].sup.(i).sub.t+1] - [[kappa].sup.(i).sub.t], from time t to t + 1 in population i. If there is no jump (i.e., [N.sup.(i).sub.t+1=0]), then the volatility of [[kappa].sup.(i).sub.t+1] - [[kappa].sup.(i).sub.t] is [V.sub.Z](1, 1), but if a jump occurs (i.e., [N.sup.(i).sub.t+1]=1), then the volatility increases to [V.sub.Z](i, i) + [V.sub.Y](i, i). By following this line of thinking, one may choose to model short-term catastrophic mortality with a regime-switching model that contains two regimes, one with a higher volatility and the other with a lower volatility. Milidonis, Lin, and Cox (2011) estimated a regime-switching model to a U.S. population mortality index. Based on their estimated model, the probability that a data point is being classified in the high-volatility regime is close to one during periods when outliers are found (see Li and Chan, 2007), which supports the idea of using a regime-switching approach to model short-term catastrophic mortality. One possible avenue for future research is to extend the regime-switching mortality model in Milidonis, Lin, and Cox to a two-population version, and to compare such an extension with the transitory jump process presented in this article.

To keep the model simple, permanent jumps were not considered. Although permanent jumps are not particularly important in pricing securities for hedging short-term catastrophic shocks, they can have a significant impact on prices of securities for hedging the risk associated with long-term longevity improvements. In future research, it is warranted to develop a two-population model with permanent jump effects. Such a model would be very useful for analyzing securities such as Swiss Re's longevity trend bond, whose payoff is related to the difference in the longevity improvements experienced between two national populations. (10)

Two other simplifying assumptions were made in our analysis. First, we did not incorporate sampling risk, which results from the random variations between the lifetimes of individuals in the same portfolio. Previous studies (e.g., Li and Hardy, 2011; Cairns et al., Forthcoming) have demonstrated that the impact of sampling risk on longevity hedges for small portfolios can be significant. Second, we assumed that the portfolio of insurance contracts is stationary; that is, the size and the age distribution of the portfolio do not change over time. It would be interesting to relax these two simplifying assumptions in future extensions of our work.

Our proposed jump process can be applied to two-population models with base structures other than the Lee-Carter structure. In the "Adding Jump Effects to Other Two-Population Models" section, we applied our proposed jump process to the base structures used in Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah (2011) and Dowd et al. (2011). The results indicate that the inclusion of jumps improves the fit to [[kappa].sup.(1).sub.t]) and [[kappa].sup.(2).sub.t] even if these two alternative base structures are assumed. What we have not investigated, however, is the goodness of fit that different base structures provide. Another possible direction of future research is to investigate this issue by using methods that are similar to those used in Cairns and colleagues (Cairns et al., 2009; Cairns, Blake, Dowd, Coughlan, Epstein, et al., 2011) and Dowd et al. (2010a, 2010b).

DOI: 10.1111/j.1539-6975.2013.12015.x

APPENDIX A

In this appendix, we conduct an outlier analysis to statistically identify spikes in the population trend that would likely be captured by the model. We are particularly interested in the timings of these spikes.

We consider the following two types of outliers:

1. Additive outliers: An additive outlier (AO) affects only one single observation.

2. Temporary changes: A temporary change (TC) affects a series at a given time, with its effect decaying at an exponential rate.

There are other types of outliers (e.g., Tsay, 1988; Chen and Tiao, 1990). However, since the focus of this article is on short-term catastrophic mortality risk, we consider outliers that have short-term effects only.

We use the procedure proposed by Chen and Liu (1993) to identify outliers in the period effect indexes [[kappa].sup.(1).sub.t] and [[kappa].sup.(1).sub.t]. (11) In Table A1 we show the outliers identified in the period effect indexes for the two populations. (12) A negative outlier stands for an improvement in mortality, whereas a positive outlier indicates a deterioration. We may interpret the detected outliers as realized mortality jumps in the sample period.

We observe that these two populations have very different timing and frequency of jumps. For Finnish males five outliers are detected, but for Swedish males only one outlier is identified. For both populations, an outlier is found in 1918 when the Spanish flu epidemic occurred. On the other hand, there are outliers that affect one population only. For Finnish males, outliers are found in 1939, 1940, 1941, and 1944. These outliers may be attributed to Finland's participation in the Second World War, which lasted from 1939 to 1945. Similar outliers are not found in the period effect index for Swedish males, possibly because Sweden remained neutral in the war.

The result of this outlier analysis suggests that even though the trends in the two series of period effect indexes are similar in general, the jump patterns are not. It is therefore important to construct a model that is flexible enough to permit the two populations to have different timing and frequency of mortality jumps.

TABLE A1
Outliers Detected in the Period Effect Indexes, [[kappa].sup.(1).sub.t]
and [[kappa].sup.(2).sub.t], Over the Sample Period of 1900-2006

Population      Year   Magnitude   Type

Swedish males   1918    25.193      AO
                1918    49.033      AO
                1939    17.210      TC
Finnish males   1940    35.436      TC
                1941    19.620      AO
                1944    25.267      AO


APPENDIX B

Deriving the Likelihood Function for the Transitory Jump Process

We let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is easy to show that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The log-likelihood for the transitory jump model is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [f.sub.1] is the conditional density function for the random vector ([[xi].sub.t+1], [[??].sub.t+1])' given ([[xi].sub.t], [[??].sub.t])', for t [member of] {[t.sub.0], ..., [t.sub.1] - 2}, and [f.sub.2] is the density function for the random vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]'.

First, we evaluate [f.sub.1]. It is easy to see that for t [member of] {[t.sub.0], ..., [t.sub.1] - 2}, ([[xi].sub.t], [[??].sub.t], [[xi].sub.t+1], [[??].sub.t+1]])' given [N.sup.(1).sub.t], [N.sup.(2).sub.t], [N.sup.(1).sub.t+1], [N.sup.(2).sub.t+1], [N.sup.(1).sub.t+2], and [N.sup.(2).sub.t+2] follows a multivariate normal distribution. Let us denote its mean vector by [mu] and variance-covariance matrix by [SIGMA]. We partition [mu] and [SIGMA] into:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[mu].sub.i] for i = 1, 2 is a 2 x 1 vector, and [[SIGMA].sub.ij] for i = 1, 2 and j = 1, 2 is a 2 x 2 matrix.

The specifications of [mu] and [SIGMA] are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Given that ([[xi].sub.t], [[??].sub.t], [[xi].sub.t+1]], [[??].sub.t+1])' follows a multivariate normal distribution, ([[xi].sub.t+1]], [[??].sub.t+1])'|([[xi].sub.t], [[??].sub.t])' follows a a bivariate normal distribution with mean:

[M.sub.1] = [[mu].sub.2] + [[SIGMA].sub.21] [[SIGMA].sup.-1.sub.11](([[xi].sub.t], [[??].sub.t])' - [[mu].sub.1]),

and variance-covariance matrix:

[S.sub.1] = [[SIGMA].sub.22] - [[SIGMA].sub.21] [[SIGMA].sup-1.sub.11][[SIGMA].sub.12].

Let [f.sub.3] be the density function of this distribution. Using the results above, we can computer [f.sub.1] by using the following formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, we evaluate [f.sub.2.] We can express [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the random vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] follows a bivariate normal distribution with a density function [f.sub.4], a mean vector [M.sub.4] and a variance-covariance matrix [S.sub.4]. The elements in [M.sub.2] and [S.sub.2] are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can then calculate [f.sub.2] using the following formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, with the expressions for [f.sub.1] and [f.sub.2], we can calculate the log-likelihood straight-forwardly.

APPENDIX C

Standard Error Correction

We used a two-stage approach to estimate parameters in our two-population mortality model. Let [[theta].sub.1] and [[theta].sub.2] be the vectors of parameters estimated in Stages 1 and 2, respectively. We have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the first stage, we estimate parameter vector [[theta].sub.1] by maximizing the log-likelihood [l.sub.1]([[theta].sub.1]), specified by Equation (2). Let [V.sub.1] be the asymptotic variance-covariance matrix for [square root of n]([[??].sub.1] - [[theta].sub.1]) based on [l.sub.1]([[theta].sub.1]), where n is the effective sample size and the hat sign denotes an estimate. Then the standard errors for parameters in [[theta].sub.1] can be calculated from an estimate of [V.sub.1].

In the second stage, we estimate parameter vector [[theta].sub.2] by maximizing the log-likelihood [l.sub.2]([[theta].sub.1]; [[??].sub.1]), specified by in Appendix B. This log-likelihood is conditional on the estimates of parameters in [[theta].sub.1]. Let [V.sub.2] be the asymptotic variance-covariance matrix for [square root of n] ([[??].sub.2] - [[theta].sub.2]) based on [l.sub.2] ([[theta].sub.2]; [[??].sub.1]). We cannot estimate standard errors for parameters in [[theta].sub.2] directly from an estimate of [V.sub.2], because we need to also incorporate the uncertainty surrounding the first-stage estimates, on which the second-stage log-likelihood [l.sub.2] ([[theta].sub.2]; [[??].sub.1]) is based.

To estimate standard errors for parameters in [[theta].sub.2], some correction is needed. The correction we made is based on the results from Murphy and Topel (2002), which we now detail.

According to Murphy and Topel (2002), if standard regularity conditions are met, then the second-step maximum likelihood estimator of [[theta].sub.2] is consistent and asymptotically normally distributed with asymptotic variance-covariance matrix:

[V.sup.*.sub.2] = 1/n[[V.sub.2] + [V.sub.2][[CV.sub.1]C' - [RV.sub.1]C' - [CV.sub.1][R.sup.'][V.sub.2]],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Matrices [V.sub.1], [V.sub.2], C, and R can be estimated numerically. We can then estimate [V.sup.*.sub.2], from which the corrected standard error for each parameter in [[theta].sub.2] is calculated.

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(1) The two-population Lee-Carter model is discussed in an earlier version of Cairns, Blake, Dowd, Coughlan, and Khalaf-Allah, (2011). This version is available at http://www.ma. hw.ac.uk/~andrewc/papers/ajgc54.pdf

(2) We use w' to denote the transpose of a vector w, and {[w.sub.t]} to denote the sequence of [w.sub.t].

(3) In estimating the Lee-Carter parameters, we used the constraints [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 1, 2. It follows that [[DELTA].sub.[kappa]]([t.sub.1]) = 0. Putting [[DELTA].sub.[kappa]]([t.sub.1]) = 0 in the autoregressive process for [[DELTA].sub.[kappa]](t), we can easily see that the expectation of [[DELTA].sub.[kappa]]([t.sub.1] + 1) is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Finally, by letting t = [t.sub.1] and taking expectation on both sides of Equation (3), we can conclude that the expected change in [[kappa].sup.(2).sub.t] during the first time step beyond the forecast origin is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(4) In principle, jump counts could be serially correlated. For example, an outbreak of a pandemic might lead to a mortality jump earlier in one population compared to another, because it might take time for the pandemic to cover different populations. It is possible to capture the potential serial correlation of [N.sup.(1).sub.t] and [N.sup.(2).sub.t] with a discrete multiple time-series process, but this additional feature will, of course, require more parameters and thus may not be desirable given that the number of existing parameters relative to the number of data points is already large.

(5) We use w(i) to denote the ith element in a vector w, and X(i, j) to denote the (i, j)th element in a matrix X.

(6) The no-jump process is equivalent to the jump process with [p.sup.1], [p.sub.2], [p.sub.3], [[mu].sub.[gamma]] [[mu].sub.[gamma]](2), [V.sub.[gamma]] (1, 1), [V.sub.[gamma]] (1, 2), and [V.sub.[gamma]] (2, 2) equal zero.

(7) The degrees of freedom are equal to the number of effective parameters in the alternative model less the number of effective parameters in the null model (e.g., Godfrey, 1988; Klugman, Panjer, and Willmot, 2008). In this test, the alternative model (the jump process) contains 14 parameters, but the effective number of parameters is only 13, as we imposed [p.sub.2] = 0 in the estimation process. On the other hand, the null model (no-jump process) has six effective parameters. This means that there are 13 - 6 = 7 degrees of freedom. We use 107 years of data (1900-2006) and therefore we have 107 pairs of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t]. Because we conditioned on the first pair of [[kappa].sup.(1).sub.t] and [[kappa].sup.(2).sub.t] in the estimation process, the effective number of observations is 106.

(8) Age 35 is representative, because the index to which the mortality bond described in the "An Illustrative Trade" section is linked is calculated from mortality rates over the age range of 25-44, the mean of which (to the nearest integer) is 35.

(9) We do not examine [p.sub.2] here, because it was set to zero in the estimation process.

(10) In 2010, Swiss Re issued a longevity trend bond, transferring US$50 million of longevity trend risk to capital markets. The bond is based on population data and would be triggered in the event there is a large divergence in the mortality improvements experienced between male lives aged 75-85 in England and Wales and male lives aged 55-65 in the United States.

(11) The procedure can be implemented with standard statistical software for time-series analyses, such as AUTOBOX, SAS/ETS, and SCA.

(12) Li and Chan (2005) also perform an outlier analysis of the period effect indexes for Swedish and Finnish populations. Their results are not identical to ours, partly because we used a different sample period and sample age range, and partly because we considered only two types of outliers. We were able to match their results when we performed the analysis using exactly the same sample period, sample age range and outlier types as they considered.

Rui Zhou is an Assistant Professor at Warren Centre for Actuarial Studies and Research, University of Manitoba, Winnipeg, Manitoba, Canada. Johnny Siu-Hang Li holds the Fairfax Chair in Risk Management in the Department of Statistics and Actuarial Science at the University of Waterloo, Canada. Ken Seng Tan is a University Research Chair Professor in the Department of Statistics and Actuarial Science, University of Waterloo, Canada. The authors can be contacted via e-mail: rui.zhou@ad.umanitoba.ca, shli@uwaterloo.ca, and kstan@uwaterloo.ca. The authors acknowledge the financial support from the Natural Science and Engineering Research Council of Canada. The authors would also like to thank Professor Andrew Cairns and other participants at the Seventh International Longevity Risk and Capital Markets Solutions Conference for their stimulating discussions on an earlier version of this article.

TABLE 1
Estimated Parameters in the Full Transitory Jump Process for the
Period Effect Indexes

Parameter                                    Estimate (Standard Error)

[[mu].sub.[kappa]]                                 -0.7517 (0.2652)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE          -0.2365 (0.2419)
  IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE           0.9107 (0.0292)
  IN ASCII]
[V.sub.Z] (1,1)                                     4.6273 (1.0044)
[V.sub.Z] (1, 2)                                    1.2737 (0.6278)
[V.sub.Z] (2, 2)                                    3.0130 (0.7045)
[[mu].sub.y] (1)                                    4.3805 (3.9670)
[[mu].sub.y] (2)                                   17.2112 (7.2772)
[V.sub.Y] (1, 1)                                   97.9239 (60.3405)
[V.sub.Y] (1, 2)                                  152.1497 (86.9687)
[V.sub.Y] (2, 2)                                  273.5049 (144.6760)
[p.sub.1]                                           0.9033 (0.0314)
[p.sub.3]                                           0.0428 (0.0281)

Note: Parameter [p.sub.2] was set to zero in the estimation process.

TABLE 2
Central Estimates of [[kappa].sup.(1).sub.2009] and
[[kappa].sup.(2).sub.2009] From the No-Jump Process and the
Transitory Jump Process

                          [[kappa].sup.(1).sub.2009]

No-jump process                 -2.4086
Transitory jump process         -2.0109

                          [[kappa].sup.(2).sub.2009]

No-jump process                  0.1106
Transitory jump process          0.0552

TABLE 3
Estimated Parameters in the No-Jump Process and Transitory Jump
Process for the Period Effect Indexes, [[kappa].sup.(1).sub.t] and
[[kappa].sup.(2).sub.t]

                                           No-Jump Process

                                    1900-1986           1900-1996

[[mu].sub.[kappa]]               -0.6619 (0.5300)    -0.7483 (0.5040)
[MATHEMATICAL EXPRESSION NOT     -1.324 (0.8672)     -0.8680 (0.7182)
  REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT      0.8619 (0.0669)     0.8553 (0.0661)
  REPRODUCIBLE IN ASCII]
[V.sub.Z] (1, 1)                 19.62 (8.001)       19.39 (8.778)
[V.sub.Z] (1, 2)                -10.05 (5.679)      -10.86 (6.589)
[V.sub.Z] (2, 2)                 39.55 (9.622)       38.98 (10.57)
[V.sub.Y] (1, 1)
[V.sub.Y] (1, 2)
[V.sub.Y] (2, 2)
[[mu].sub.Y] (1)
[[mu].sub.Y] (2)
[p.sub.1]
[p.sub.3]

                                                     Transitory Jump
                                 No-Jump Process         Process

                                    1900-2006           1900-1986

[[mu].sub.[kappa]]               -0.8057 (0.4573)    -0.6279 (0.2983)
[MATHEMATICAL EXPRESSION NOT     -0.9801 (0.6999)    -0.5082 (0.3126)
  REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT      0.8486 (0.0675)     0.9110 (0.0316)
  REPRODUCIBLE IN ASCII]
[V.sub.Z] (1, 1)                 17.32 (7.978)        4.703 (1.204)
[V.sub.Z] (1, 2)                -11.04 (6.859)        1.657 (0.7169)
[V.sub.Z] (2, 2)                 38.18 (11.09)        3.409 (0.8606)
[V.sub.Y] (1, 1)                                    107.9 (77.63)
[V.sub.Y] (1, 2)                                    138.9 (99.09)
[V.sub.Y] (2, 2)                                    221.0 (158.7)
[[mu].sub.Y] (1)                                      5.614 (4.876)
[[mu].sub.Y] (2)                                     17.63 (8.360)
[p.sub.1]                                             0.8913 (0.0352)
[p.sub.3]                                             0.0572 (0.0387)

                                       Transitory Jump Process

                                    1900-1996           1900-2006

[[mu].sub.[kappa]]               -0.6958 (0.2877)    -0.7517 (0.2652)
[MATHEMATICAL EXPRESSION NOT     -0.2104 (0.2470)    -0.2365 (0.2419)
  REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT      0.9116 (0.0288)     0.9107 (0.0292)
  REPRODUCIBLE IN ASCII]
[V.sub.Z] (1, 1)                  4.877 (1.127)       4.627 (1.004)
[V.sub.Z] (1, 2)                  1.527 (0.7131)      1.274 (0.6278)
[V.sub.Z] (2, 2)                  3.261 (0.7587)      3.013 (0.7045)
[V.sub.Y] (1, 1)                108.7 (73.34)        97.92 (60.34)
[V.sub.Y] (1, 2)                150.8 (96.45)       152.1 (86.97)
[V.sub.Y] (2, 2)                254.0 (156.8)       273.5 (144.7)
[[mu].sub.Y] (1)                  5.200 (4.531)       4.381 (3.967)
[[mu].sub.Y] (2)                 18.12 (7.941)       17.21 (7.277)
[p.sub.1]                         0.9000 (0.0321)     0.9033 (0.0314)
[p.sub.3]                         0.0463 (0.0315)     0.0428 (0.0281)

Note: Standard errors are shown in parentheses.

TABLE 4
Prices and Quantities Traded in Equilibrium When Different Mortality
Models Are  Assumed

                          a = 1.3, b = 1.4    a = 1.2, b = 1.3

                                   Quantity            Quantity
Model                     Price     Traded    Price     Traded

No-jump process           1.0204    0.2314    0.9770    0.1802
Transitory jump process   1.0132    0.1799    0.9979    0.1542

TABLE 5
The Effects of [[mu].sub.H], [[mu].sub.L], [[sigma].sub.H],
[[sigma].sub.L], and [rho] on the Supply and Demand of the
Illustrative Mortality Bond

         [[mu].sub.H] [up arrow]   [[mu].sub.L] [up arrow]

Supply         [down arrow]                  --
Demand          [up arrow]                   --

         [[sigma].sub.H] [up arrow]   [[sigma].sub.L] [up arrow]

Supply          [down arrow]                  [up arrow]
Demand          [down arrow]                      --

         [absolute value of [rho]] [up arrow]

Supply                 [up arrow]
Demand                    --

TABLE 6
Estimates of [[mu].sub.H], [[sigma].sub.H], [[sigma].sub.L], and
[rho] When a = 1.3 and b = 1.4

Model                     [[mu].sub.H]   [[sigma].sub.H]

No-jump process              1.0223          0.1175
Transitory jump process      1.0110          0.1573

Model                     [[sigma].sub.L]    [rho]

No-jump process               0.1106        -0.3669
Transitory jump process       0.0677        -0.5832

TABLE 7
Estimates of [[mu].sub.H], [[sigma].sub.H], [[sigma].sub.L], and
[rho] When a = 1.2 and b = 1.3

Model                           [[mu].sub.H]   [[sigma].sub.H]

No-jump process                    0.9792          0.2112
Full transitory jump process       0.9960          0.1906

Model                           [[sigma].sub.L]    [rho]

No-jump process                     0.1106        -0.4980
Full transitory jump process        0.0677        -0.6034

TABLE 8
The Fitting, Forecasting and Pricing Results Based on the Constraints
We Chose and the Alternative Constraints

                                          The Constraints We Chose

Fitted value of [m.sup.(1).sub.35,2006]     6.3631 x [10.sup.-4]
Fitted value of [m.sup.(2).sub.35,2006]     1.7437 x [10.sup.-3]
Mean forecast [m.sup.(1).sub.35,2009]       6.1117 x [10.sup.-4]
Mean forecast [m.sup.(2).sub.35,2009]       1.7863 x [10.sup.-3]
Estimated price                                    0.9987

                                          The Alternative Constraints

Fitted value of [m.sup.(1).sub.35,2006]      3.9305 x [10.sup.-4]
Fitted value of [m.sup.(2).sub.35,2006]      5.8561 x [10.sup.-4]
Mean forecast [m.sup.(1).sub.35,2009]        3.6679 x [10.sup.-4]
Mean forecast [m.sup.(2).sub.35,2009]        5.9482 x [10.sup.-4]
Estimated price                                     1.0397

Note: The security being priced is the mortality bond in the "An
Illustrative Trade" section with a = 1.2 and b = 1.3.

TABLE 9
The Estimated Trading Prices and Quantities When Parameters
[[mu].sub.Y] (1), [[mu].sub.Y] (2), [p.sub.1], and [p.sub.3] Are
Shocked by One Standard Error

                                      (Price, Quantity)

Parameter Being Shocked   Increased by One SE   Decreased by One SE

[[mu].sub.Y] (1)           (0.9781, 0.1319)      (1.0155, 0.1836)
[[mu].sub.Y] (2)           (0.9978, 0.2031)      (0.9995, 0.1145)
[p.sub.1]                  (1.0229, 0.1560)      (0.9786, 0.1667)
[p.sub.3]                  (1.0209, 0.1583)      (0.9793, 0.1633)

Note: The security being priced is the mortality bond in the "An
Illustrative Trade" section with a = 1.2 and b = 1.3.

TABLE 10
The Estimated Trading Prices and Quantities When Different Modeling
Approaches Are Used

              Price               Quantity

Approach 1             No trade
Approach 2             No trade
Approach 3    0.9770               0.1802
Approach 4    0.9800               0.0026
Approach 5    0.9979               0.1542

Note: The security being priced is the mortality bond in the "An
Illustrative Trade" section with a = 1.2 and b = 1.3.
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Author:Zhou, Rui; Li, Johnny Siu-Hang; Tan, Ken Seng
Publication:Journal of Risk and Insurance
Geographic Code:1USA
Date:Sep 1, 2013
Words:16872
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