Pricing Standardized Mortality Securitizations: A Two‐Population Model With Transitory Jump Effects

• DOI: http://doi.org/10.1111/j.1539-6975.2013.12015.x
• Author: Rui Zhou,Ken Seng Tan,Johnny Siu‐Hang Li
• Date: 01 September 2013
• Published date: 01 September 2013

©The Journal of Risk and Insurance, 2013, Vol.80, No. 3, 733–774

DOI: 10.1111/j.1539-6975.2013.12015.x

Pricing Standardized Mortality Securitizations:

A Two-Population Model With Transitory Jump

Effects

Rui Zhou

Johnny Siu-Hang Li

Ken Seng Tan

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 dif-

ferent 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 de-

mand 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

Rui Zhou is an Assistant Professor at Warren Centre for Actuarial Studies and Research, Uni-

versity 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 Depart-

ment 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 Re-

search 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.

733

734 The Journal of Risk and Insurance

forecasts arebiologically reasonable. Specifically, the models ensurethat 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 pro-

posed. Carter and Lee (1992) propose the joint-kmodel, 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-kmodel 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 proposedby 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 con-

ditions under which a two-population mortality model will not result in diverging

long-term forecasts.

Toour 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)1and subse-

quently 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 ap-

plied. 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 mod-

eled. For example, Deng, Brockett, and MacMinn consider double-exponential jumps,

while Chen and Cox consider Gaussian jumps.

1The 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

Pricing Standardized Mortality Securitizations 735

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 mor-

tality 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 involv-

ing more than one population. Moreover,by allowing the populations being modeled

to be subject to different (but correlated) mortality jumps, our model may more accu-

rately 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. Toachieve this goal, we consider the economic-pricing frame-

work 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 prespecifiedattachment point. Given the payoff struc-

ture, 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 equiva-

lently 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 de-

scribe 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 secu-

rities. 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).