Mortality Dependence and Longevity Bond Pricing: A Dynamic Factor Copula Mortality Model With the GAS Structure

• Date: 01 April 2017
• Author: Tao Sun,Hua Chen,Richard D. MacMinn
• DOI: http://doi.org/10.1111/jori.12214
• Published date: 01 April 2017

MORTALITY DEPENDENCE AND LONGEVITY BOND

PRICING:ADYNAMIC FACTOR COPULA MORTALITY

MODEL WITH THE GAS STRUCTURE

Hua Chen

Richard D. MacMinn

Tao Sun

ABSTRACT

Modeling mortality dependence for multiple populations has significant

implications for mortality/longevity risk management. A natural way to

assess multivariate dependence is to use copula models. The application of

copula models in the multipopulation mortality analysis, however, is still in

its infancy. In this article, we present a dynamic multipopulation mortality

model based on a two-factor copula and capture the time-varying

dependence using the generalized autoregressive score (GAS) framework.

Our model is simple and flexible in terms of model specification and is

widely applicable to high dimension data. Using the Swiss Re Kortis

longevity trend bond as an example, we use our model to estimate the

probability distribution of principal reduction and some risk measures such

as probability of first loss, conditional expected loss, and expected loss. Due

to the similarity in the structure and design of CAT bonds and mortality/

longevity bonds, we borrow CAT bond pricing techniques for mortality/

longevity bond pricing. We find that our pricing model generates par

spreads that are close to the actual spreads of previously issued mortality/

longevity bonds.

INTRODUCTION

Mortality improvements around the world are putting more and more pressure on

social security systems, pension funds, life insurance companies, and individuals,

and thus call for more efficient management of longevity risk. Coping with this trend,

longevity risk-related capital market solutions have grown in recent years. Many

innovative products, such as mortality/longevity bonds, longevity swaps, buy-ins,

and buy-outs, have been adopted (Tan, Blake, and MacMinn, 2015). A better

understanding of the correlation among mortality improvements for multiple

Hua Chen is at the Temple University. Hua Chen can be contacted via e-mail: hchen@temple.edu.

Richard D. MacMinn is at the National Chengchi University and University of Texas. Richard D.

MacMinn can be contacted via e-mail: richard@macminn.org. Tao Sun is at University of

Wisconsin–La Crosse. Tao Sun can be contacted via e-mail: tsun@uwlax.edu.

© 2017 The Journal of Risk and Insurance. Vol. 84, No. S1R, 393–415 (2017).

DOI: 10.1111/jori.12214

393

populations appears to be critical for issuers, investors, insurers, pension plans, and

governments for several reasons (Cairns et al., 2011; Chen, MacMinn, and Sun, 2015).

First, almost all mortality/longevity bonds are written on a weighted index based on

the mortality experience of multiple populations. Issuers (or sponsors) need to

understand mortality correlations in order to better design the mortality/longevity

derivatives and price the premiums. Investors also need to analyze the mortality

correlations to evaluate their risk and payoff. Second, life insurers that write both life

insurance policies and annuities can naturally hedge mortality risk from insured lives

with longevity risk from annuitants. Such a natural hedge, however, is not perfect.

Therefore, insurers need to understand mortality correlations between these two

groups and manage the residual risk. Third, for mortality/longevity risk hedgers that

use capital market instruments, basis risk exists because the mortality experience of

their own population is usually different from that of the population(s) associated

with the standardized hedging instruments. A model that captures mortality

dependence among multiple populations can help them determine hedge ratios and

minimize basis risk.

1

There has been a growing lite rature on mortality mode ls for multiple populatio ns

in recent years (see, e. g., Li and Lee, 2005; Cair ns et al., 2011; Dowd et al ., 2011;

Jarner and Kryger, 2011; Li an d Hardy, 2011; Yang and Wang, 201 3; Zhou, Li, and

Tan, 2013; Zhou et al., 2014). The se models typically assume th at the forecasted

mortality experience s of two or more related popula tions are linked together an d

do not diverge over the long run. This assumption might be justified by the long-

term mortality comoveme nts. There is, however, litt le evidence to suggest mean

reversion to a constant di fference in relative mortality rates in the short run ( Hunt

and Blake, 2015). Mackenb ach et al. (2003) and Waldron (2007) also document a

divergence in mortality be tween different socioeco nomic groups in recent years .

We, therefore, develop o ur multipopulation mor tality model following a nother

stream of dependence mod eling, that is, copulas . We introduce a time-var ying

dependence structure using the generalized autor egressive score (GAS) mode l. We

use the Kortis longevity bo nd, the first longevity trend bond issued by Swiss Re in

2010, as an example to illu strate how this model can be used for mortality risk

modeling and pricing.

Copulas have been studied in both actuarial science and finance to examine

dependencies among risks (Frees and Valdez, 1998; Embrechts, Lindskog, and

McNeil, 2003). In mortality studies, copulas have been applied to model the bivariate

survival function of the lives of couples (see, e.g., Frees, Carriere, and Valdez, 1996;

Carriere, 2000; Denuit et al., 2001; Shemyakin and Youn, 2006). Surprisingly, the

application of copula models in the multipopulation mortality analysis is still in its

1

Another application of multipopulation mortality models is to forecast mortality for a small

country. Sometimes the mortality data may not be available or reliable for a small population

due to a small number of deaths, a limited number of calendar years of data, age range, or poor

data quality. Jointly estimating the small population and a larger linked population allows the

small-population mortality forecasts to be consistent with those of the larger population. Due

to the space restriction, we do not discuss this application in this article. We refer interested

readers to Dowd et al. (2011).

394 THE JOURNAL OF RISK AND INSURANCE