Pricing Mortality Securities With Correlated Mortality Indexes

• Author: Yijia Lin,Sheen Liu,Jifeng Yu
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01481.x
• Date: 01 December 2013
• Published date: 01 December 2013

© The Journal of Risk and Insurance, 2013, Vol. 80, No. 4, 921–948

DOI: 10.1111/j.1539-6975.2012.01481.x

921

PRICING MORTALITY SECURITIES WITH CORRELATED

MORTALITY INDEXES

Yiji a Lin

Sheen Liu

Jifeng Yu

ABSTRACT

This article proposes a stochastic model, which captures mortality correla-

tions across countries and common mortality shocks, for analyzing catastro-

phe mortality contingent claims. To estimate our model, we apply particle

filtering, a general technique that has wide applications in non-Gaussian and

multivariate jump-diffusion models and models with nonanalytic observa-

tion equations. In addition, we illustrate how to price mortality securities

with normalized multivariate exponential titling based on the estimated

mortality correlations and jump parameters. Our results show the signifi-

cance of modeling mortality correlations and transient jumps in mortality

security pricing.

INTRODUCTION

Over the last century,populations of different countries have been increasingly linked

by flows of information, goods, transportation, and communication, and as a conse-

quence the world has become more closely connected and interdependent. While

the trend of globalization has substantially driven market growth and international

trade, it has also helped to spread some of the deadliest infectious diseases across

borders (Daulaire, 1999). Thus, it seems improper to forecast mortality for an indi-

vidual national population in isolation from others. Indeed, in practice, intercountry

mortality correlation has long been a serious concern for insurers that underwrite life

insurance business.

Mortality forecast that takes into account a country’s linkage to others is important

in the sense that not only does it facilitate better understanding of mortality risk, but

it also has enormous implications for pricing mortality securities. Recent financial

Yijia Lin is in the Department of Finance, College of Business Administration, Univer-

sity of Nebraska-Lincoln. Sheen Liu is in the Department of Finance, College of Business,

WashingtonState University, Tri-Cities.Jifeng Yu is in the Department of Management, College

of Business Administration, University of Nebraska-Lincoln. The first author can be contacted

via e-mail: yijialin@unl.edu. The authors thank two anonymous referees for their very help-

ful suggestions and comments during the revision process. Yijia Lin gratefully acknowledges

financial support from the Research Council Grand-In-Aid Award—Maude Hammond Fling

Faculty Research Fellowship of the University of Nebraska-Lincoln.

922 THE JOURNAL OF RISK AND INSURANCE

innovation makes mortality securitization a viable option for insurers or reinsurers to

transfer catastrophe mortality risk arising from the possible occurrence of pandemics

or large-scale terrorist attacks. By segregating its cash flows linked to extreme mor-

tality risk, an insurance firm is able to repackage them into securities that are traded

in capital markets (Blake and Burrows, 2001; Lin and Cox, 2005; Cox and Lin, 2007).

Since the first publicly traded mortality security issued by Swiss Re in 2003, almost

all mortality transactions determine the coupons and principals based on three or

more population mortality indexes, with the only exception—the Tartan mortality

bond sold in 2006. This indicates that insurers or reinsurers are keenly interested in

transferring potential country-correlated mortality risk embedded in their business.

For instance, the mortality risk of the 2003 Swiss Re mortality bond was defined in

terms of an index based on the weighted average annual population death rates in the

United States, the United Kingdom, France, Italy,and Switzerland (Lin and Cox, 2008).

As another example, the mortality bond issued by the Nathan Ltd. in 2008 depended

on the annual population death rates of four countries, namely, the United States,

the United Kingdom, Canada, and Germany. Given that the existing (and possible

future) mortality securities bundle multination mortality risks, mortality correlation

among countries merits serious consideration in mortality securitization pricing.

In the recent literature, a number of stochastic mortality models have been proposed.

Despite the importance of mortality correlation, surprisingly very few papers treat

correlation as an indispensable element. For example, in order to account for catas-

trophic mortality death shocks, Chen and Cox (2009) incorporate a jump-diffusion

process into the original Lee–Carter model to forecast mortality rates and price the

2003 Swiss Re mortality bond. Yet, while the Swiss Re bond payments depended

on five-country weighted mortality index, the authors price this bond only based

on the U.S. mortality rates. Hence, it is not clear how to extend their method to

multipopulation correlated mortality scenarios.

Beelders and Colarossi (2004) and Chen and Cummins (2010), on the other hand,

use the extreme value theory to measure mortality risk of the 2003 Swiss Re bond.

However, they simply model the combined index without considering the mortality

correlations among different countries. So the question of how to model multipopu-

lation mortality correlation remains open. Bauer and Kramer (2009) borrow a credit

risk modeling approach introduced by Lando (1998) to describe stochastic force of

mortality. To incorporate dramatic mortality changes that are crucial in measuring

mortality risk (Lin and Cox, 2008), Bauer and Kramer (2009) propose a mortality

model with an affine jump-diffusion process. Then they use their mortality model to

price the Tartan transaction, which is the only publicly traded mortality bond to date

solely based on one-country (i.e., United States) mortality experience. However, the

authors acknowledge that their model misses correlations and diversification effects

across genders and populations. Given the prevalence of multicountry combined

mortality indexes in the mortality security markets, it is questionable whether their

model is adequate for other mortality securities. Indeed, from a practical point of

view, understanding the combined index as a function of various population death

rates is at least as important, if not more so, than understanding it as a function of

different age classes of a single country for mortality securitization.

Moreover,the existing mortality literature has demonstrated the significance of catas-

trophic death events in the pricing and tranche structure for a mortality risk bond