Managing Mortality Risk With Longevity Bonds When Mortality Rates Are Cointegrated

• Published date: 01 September 2017
• Author: Hoi Ying Wong,Mei Choi Chiu,Tat Wing Wong
• DOI: http://doi.org/10.1111/jori.12110
• Date: 01 September 2017

©2015 The Journal of Risk and Insurance. Vol.84, No. 3, 987–1023 (2017).

DOI: 10.1111/jori.12110

Managing Mortality Risk With Longevity Bonds

When Mortality Rates Are Cointegrated

Tat Wing Wong

Mei Choi Chiu

Hoi Ying Wong

Abstract

This article investigates the dynamic mean-variance hedging problem of an

insurer using longevity bonds (or longevity swaps). Insurance liabilities are

modeled using a doubly stochastic compound Poisson process in which the

mortality rate is correlated and cointegrated with the index mortality rate. We

solve this dynamic hedging problem using a theory of forward–backward

stochastic differential equations. Our theory shows that cointegration

materially affects the optimal hedging strategy beyond correlation. The coin-

tegration effectis independent of the risk preference of the insurer. Explicit so-

lutions for the optimal hedging strategy are derived for cointegrated stochas-

tic mortality models with both constant and state-dependent volatilities.

Introduction

The core business of a life insurance company (insurer) relies on the management of

mortality and longevity risks. In addition to calculating insurance premiums by means

of risk pooling, recent advances consider the transfer of insurable risks through rein-

surance and securitization. However,the latest risk transfer approach causes many is-

sues and challenges in terms of modeling and regulation (Barrieu et al., 2010). Because

financial and actuarial risks are generally regarded as independent, although there is

debate concerning their probabilistic dependency, the financial market lacked appro-

priate instruments to directly hedge against mortality-related risks until the European

Investment Bank introduced longevity bonds in November 2004 (Blake et al., 2006).

Although these bonds have not been successful thus far, due to the associated basis

risk, pricing mechanism, lack of liquidity, and lack of public understanding (Blake

et al., 2013), longevity securitization has subsequently been proposed to enrich the set

of hedging and investment securities (Dowd et al., 2006; Dawson et al., 2010; Biffis

Tat Wing Wong is at the Division of Business and Management, BNU-HKBU United Interna-

tional College. Mei Choi Chiu is at the Department of Mathematics and Information Technol-

ogy, The Hong Kong Institute of Education. Hoi Ying Wong is at the Department of Statis-

tics, The Chinese University of Hong Kong. Hoi Ying Wong can be contacted via e-mail: hy-

wong@cuhk.edu.hk. Hoi Ying Wong acknowledges the support by Research Grants Council of

Hong Kong with GRF project number 403511. The authors thank two anonymous referees for

their valuable comments and suggestions.

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988 The Journal of Risk and Insurance

and Blake, 2010). Although counterparty risk is very important, we do not address it

here (Biffis et al., forthcoming). Longevity securities and the longevity market have

raised much interesting research, representative of it are the recent developments

in the longevity risk market by Blake et al. (2015) and Biffis and Blake (2013), the

reduced-form approaches to longevity risk pricing by Deng et al. (2012) and Wang

et al. (2013), the basis risk management in longevity risk solutions by Gatzert and

Wesker (2012) and Cairns (2013), and natural hedging strategies for life insurers by

Wong et al. (forthcoming).

A longevity bond is a bond in which cash payments are proportional (or linked) to a

selected survival index, which is usually linked to a national mortality rate. Hence,

an insurer hedging with longevity bonds is exposed to the basis risk arising from the

discrepancy between the mortality rates of the underlying index and the insurer’s

portfolio. The effects of longevity basis risk on hedging effectiveness are examined

empirically by Li and Hardy (2011) and Coughlan et al. (2011). Although longevity

bond is not the most popular mortality-linked security in the market, we consider it

as the basic longevity security in our problem formulation because many longevity

securities can be replicated by longevity bonds. For instance, a longevity swap can be

viewed as a portfolio consisting a longevity bond and a risk-free bond.

A simple way to deal with the mortalities of two related populations is to assume a

fixed ratio between them. This simple practice relies on the assumption that the ratio

between the two is stable over time. Stochastic mortality models, such as the Bayesian

approach proposed by Cairns et al. (2011)and gravity model proposed by Dowd et al.

(2011), relax this assumption.

Drawing from the econometric literature, this article considers stochastic mortality

models with cointegration and correlation between the mortality rates of two popula-

tions. The concept of cointegration, which originates with Engle and Granger (1987),

is based on the existence of an equilibrium relation between two or more nonstation-

ary economic series for which a linear combination could either be stationary or have

a lower degree of integration than the original series. Cointegration has been found

in many asset classes, including stocks (Cerchi and Havenner, 1988), exchange rates

(Baillie and Bollerslev, 1989), international financial indices (Taylor and Tonks, 1989),

and energy products (Serletis, 1994). Its widespread usefulness resulted in a Nobel

prize in Economics for Granger in 2003. Recently,cointegration has also been incorpo-

rated into mortality modeling to forecast mortalities (Darkiewicz and Hoedemakers,

2004) and in investigations of longevity basis risk (Salhi and Loisel, 2010a, 2010b).

Njenga and Sherris (2009) offer empirical evidence of the existence of cointegrating

factors in mortality modeling and Yang and Wang (2013) show that cointegration

exists in multicountry longevity risk.

This article examines the dynamic mean-variance (MV) hedging of mortality risk us-

ing longevity bonds or,more generally, national index-based longevity securitization.

We consider MV hedging, because it has been extensively studied in the literature

and is probably the best-known hedging objective in practice. However, this article is

not a trivial extension of existing work. Rather than work with the general theory of

Managing Mortality Risk With Longevity Bonds When Mortality Rates Are Cointegrated 989

semimartingales as Jeanblanc et al. (2011) do, we concentrate on the concrete results

of hedging mortality risk under cointegration. The models considered in this article

are general enough for actuarial practice and are consistent with the doubly stochastic

mortality models in the actuarial literature (Biffis, 2005).

A theoretical approach to MV hedging requires the identification of the variance-

optimal martingale measure (VMM) for a stationary process such as that presented

in Hubalek et al. (2006). However, the cointegration concept considered in this ar-

ticle refers to a nonstationary vector process, which is error correctable.1Schweizer

(2010) notes that the VMM is generally very difficult to obtain, with the exception of

a deterministic mean-variance trade-off. Our cointegration model, however, involves

a stochastic opportunity set because the processes exhibit stochastic drift. Accord-

ingly,we employ a framework of forward–backward stochastic differential equations

(FBSDEs) to avoid difficulties with the VMM framework.

Alexander (1999) finds that taking cointegration into account has important impli-

cations for hedging. As MV hedging problems are closely related to MV portfolio

problems, recent advances in solving the portfolio problems of cointegrated risky as-

sets are incorporated into the present article. Using the theory of backward stochastic

differential equations (BSDEs), Chiu and Wong (2011a) obtain an explicit solution to

the MV portfolio selection problem by recognizing an exponential quadratic affine

form. Confined to a set of time-consistent policies, Chiu and Wong (2015) rigorously

prove that cointegration induces statistical arbitrage. Chiu and Wong (2012, 2013a,

2013b) further extend the framework to asset-liability management when risky assets

are cointegrated. Related optimization problems in mortality-related content include

those of Menoncin (2008), who solves the mortality-linked security hedging problem

by maximizing a utility function, and Biffis et al. (2010), who considers the longevity

swap in an MV hedging problem. However, no work to date has consideredhedging

longevity risk using cointegration.

We find that the existing literatureon longevity risk could be classified into two cate-

gories. The first category makes use of affine term structure mortality models to price

mortality-linked securities. The second category employs a static hedging strategy to

manage longevity risk without applying term structure models. In view of these, our

article contributes to the literature in two ways. First, we apply affine term structure

mortality models for longevity risk management and derive the dynamic hedging

strategies. Second, we enrich the affine term structure models to allow cointegration

and show the importance of cointegration technique in longevity risk management.

The aim of the optimization problem is to minimize the variance in terminal wealth

penalized by expected terminal wealth. The penalty specified by an insurer reflectsher

risk aversion and the case of zero penalty is reduced to the minimum-variance hedg-

1This is related to the concept of cointegration and error correction model. A vector of non-

stationary time series is said to be error correctable if there exists a linear combination of its

component such that it is stationary. The Granger Representation Theorem implies that such

a cointegrated time series could be written in an error-correction form, hence the name error

correctable.