Pricing Survivor Derivatives With Cohort Mortality Dependence Under the Lee–Carter Framework

• Author: Sharon S. Yang,Chou‐Wen Wang
• Published date: 01 December 2013
• Date: 01 December 2013
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01488.x

© The Journal of Risk and Insurance, 2013, Vol. 80, No. 4, 1027–1056

DOI: 10.1111/j.1539-6975.2012.01488.x

1027

PRICING SURVIVOR DERIVATIVES WITH COHORT

MORTALITY DEPENDENCE UNDER THE LEE–CARTER

FRAMEWORK

Chou-Wen Wang

Sharon S. Yang

ABSTRACT

This article introduces cohort mortality dependence in mortality modeling.

Weextend the classical Lee–Carter model to incorporate cohort mortality de-

pendence by considering mortality correlations for a cohort of people born in

the same year. The pattern of cohort mortality dependence is demonstrated

on the basis of U.S. mortality experience. We study the effect of cohort mor-

tality dependence on the pricing of survivor derivatives. For this purpose,

a survivor floor is introduced. To understand the difference between a sur-

vivor floor and other survivor securities, the valuation formulas for survivor

swaps and survivor floors are all derived in detail and the effects of co-

hort mortality dependence on pricing survivor derivatives are investigated

numerically.

INTRODUCTION

Longevity risk has become an increasingly important consideration for defined benefit

pension plans and annuity providers, because life expectancy is increasing dramat-

ically in developed countries. In 2007, exposure to improvements in life expectancy

reached $400 billion for pension fund and insurance companies in the United King-

dom and United States (see Loeys, Panigirtzoglou, and Ribeiro, 2007). Therefore,

finding a way to measure longevity risk and transferring the longevity risk away

from the pension fund or annuity provider is of great interest to plan sponsors. Rein-

surance, which represents a traditional means to transfer the longevity risk, can be

expensive and involves a potential credit risk to the counterparty. In turn, many

life insurance companies are less willing to buy reinsurance for their longevity risk.

Instead, capital market solutions such as mortality-linked securities have emerged.

Chou-WenWang is an Associate Professor in the Department of Risk Management & Insurance,

National Kaohsiung First University of Science and Technology. Sharon S. Yang is a Professor

and Chairperson in the Department of Finance, National Central University,and a Researcher in

the Risk and Insurance Research Center, College of Commerce,National Chengchi University.

The authors can be contacted via e-mail: chouwenwang@gmail.com and syang@ncu.edu.tw,

respectively. Wang was funded by NSC 98-2410-H-327-024 and Yang was funded by NSC

99-2410-H-008-019-MY3.

1028 THE JOURNAL OF RISK AND INSURANCE

Blake and Burrows (2001) were the first to advocate the use of mortality-linked se-

curities to transfer longevity risk to capital markets. They suggested that the gov-

ernments should help insurance companies hedge their mortality risks by issuing

survivor bonds whose coupon payments depend on the proportion of the population

surviving to particular ages. The longevity bond launched by the European Invest-

ment Bank (EIB) was the first securitization instrument designed to transfer longevity

risk but ultimately was not issued and remained theoretical. Furthermore, various

new securitization instruments and derivatives for longevity risk, such as survivor

swaps, survivor futures, and survivor options, have received great attention among

academics and practitioners (Blake, Cairns, and Dowd, 2006; Blake et al., 2010; Dowd

et al., 2006; Biffis and Blake, 2009). The first derivative transaction, a q-forward con-

tract, was issued in January 2008 between Lucida1and J.P. Morgan (Coughlan et al.,

2007). In addition, the first survivor swap executed in the capital markets took place

between Canada Life and a group of ILS and other investors in July 2008. In this

context, the valuation of mortality-linked securities represents an important research

topic for the development of capital market solutions for longevity risk.

The dynamics of underlying mortality indexes have important effects on valuing life

insurance or mortality-linked securities. The Lee–Carter model (Lee and Carter, 1992)

has proved an effective method for mortality forecasts, which Denuit, Devolder, and

Goderniaux (2007) use to value longevity bonds. Cairns, Blake, and Dowd (2006) also

propose a two-factor stochastic mortality model (hereafter denoted CBD model) for

higher ages and examine the pricing of longevity bonds. The Lee–Carter and CBD

models both project mortality rates based on age and period effects. Renshaw and

Haberman (2006) extend the Lee–Carter model to consider cohort effects2in mortal-

ity modeling. Cairns et al. (2009) quantitatively compare eight stochastic mortality

models and demonstrate that the CBD model (Cairns, Blake, and Dowd, 2006) that

incorporates a cohort effect fits data about English and Welsh men best, and Ren-

shaw and Haberman’s (2006) extension of the Lee–Carter model that also allows for

a cohort effect provides the best fit for data pertaining to U.S. men. Thus, the cohort

effect represents an important risk factor that governs the dynamics of mortality. The

preceding models are all discrete mortality models. In addition to these discrete mod-

els, some mortality models have been built on a continuous basis, including those

proposed by Milevsky and Promislow (2001), Dahl (2004), Biffis (2005), Dahl and

Møller (2006), and Schrager (2006). Liao, Yang, and Huang (2007) assume that mor-

tality follows a nonmean-reverting stochastic process for a single age, as proposed

by Luciano and Vigna (2005), and examine tranching in mortality-linked securities

with a product designed to transfer longevity risk. Wills and Sherris (2010) use the

continuous time dynamics of the mortality rate to price and structure a longevity

bond based on that used for a collateralized debt obligation. Thus, the existing liter-

ature focuses on longevity securitization and illustrates the structure and pricing for

longevity bonds. In this research, we attempt to price survivor derivatives for which

the claim is contingent, that is, the option-type contract. The insurer suffers exposure

to the longevity risk only when annuitants live longer than predicted by the refer-

ence mortality rates. Thus, the insurer, as the protection buyer, can hedge longevity

1A U.K.-based pension buyout insurer.

2Mortality improvements depend on the year in which the person was born.