Single‐ and Cross‐Generation Natural Hedging of Longevity and Financial Risk

• Date: 01 September 2017
• Author: Luca Regis,Elisa Luciano,Elena Vigna
• Published date: 01 September 2017
• DOI: http://doi.org/10.1111/jori.12104

©2015 The Journal of Risk and Insurance. Vol.84, No. 3, 961–986 (2017).

DOI: 10.1111/jori.12104

Single- and Cross-Generation Natural Hedging of

Longevity and Financial Risk

Elisa Luciano

Luca Regis

Elena Vigna

Abstract

This article provides natural hedging strategies for life insurance and annu-

ity businesses written on a single generation or on different generations in

the presence of both longevity and interest-rate risks. Weobtain closed-form

solutions for delta and gamma hedges against cohort-based longevity risk.

We exploit the correlationbetween the mortality intensities of different gen-

erations and hedge the longevity risk of one cohort with products on other

cohorts. An application with UK data on survivorship and bond dynamics

shows that hedging is effective, even when rebalancing is infrequent.

Introduction

Longevity risk, that is, the risk of unexpected changes in survivorship, is now per-

ceived as an important threat to the safety of insurance companies and pension funds.

Most actors in the financial market are long longevity risk. This has stimulated the

transformation of contracts subject to longevity risk into an asset class, as originally

suggested by Blake and Burrows (2001). The creation of q-forwards, s-forwards,

longevity bonds, and swaps represents a step in this direction, but this asset class is

still in its infancy. In the meanwhile, insurance companies can benefit from natural

hedging, that is, from natural offsetting between the longevity risk exposures of death

benefits and life contracts, such as annuities. The importance of exploiting this natural

Elisa Luciano is at the University of Torino, Collegio Carlo Alberto and NETSPAR. Luca Regis

is at the IMT Institute for Advanced Studies Lucca and Collegio Carlo Alberto. Elena Vignais at

the University of Torino,Collegio Carlo Alberto and CeRP. Luciano can be contacted via e-mail:

elisa.luciano@unito.it. Regis can be contacted via e-mail: luca.regis@imtlucca.it. Vigna can be

contacted via e-mail: elena.vigna@unito.it . Wethank Jeff Mulholland, An Chen, Nadine Gatzert,

and participants at the VII Longevity Conference, MAF 2012 Conference, 1st EAJ Conference,

Longevity VIII Conference, and XXXVI AMASES Conference and to seminar at IMT Lucca

for helpful suggestions. We also thank three anonymous referees for valuable comments, in

particular for suggesting the “Practical Issues” and the “Hedge Effectiveness” sections. Luca

Regis gratefully acknowledges financial support from the Crisis Lab project funded by the

Italian Ministry of Education. An incomplete version of the article has been circulated under

the title “Natural Delta Gamma Hedging of Longevity and Interest Rate Risk” as ICER wp

21/2011.

961

962 The Journal of Risk and Insurance

offsetting extends beyond theory. Cox and Lin (2007) find empirical evidence that

insurers whose liability portfolios benefit from natural hedging have a competitive

advantage and charge lower premiums. Despite being safe, sound, and comparatively

cheap, natural hedging is not trivial in the presence of longevity risk because the

latter is difficult to capture per se in a parsimonious and manageable way and even

more difficult to couple with a satisfactory model of financial risk, such as interest

rate risk. However, the interactions between longevity and financial risk cannot be

avoided from the perspective of immunization in the form of liability management,

as the value of the reserves is subject to interest-rate risk, and a fortiori,fromthe

perspective of asset and liability management (ALM).

Natural hedging of longevity risk without financial risk is recently addressed by Cox

and Lin (2007), Wang et al. (2010), and Gatzert and Wesker (2012, 2013). Cox and Lin

(2007), motivated by the empirical evidence mentioned above, propose the use of

mortality swaps between annuity providers and life insurance writers. Wang et al.

(2010) propose an immunization strategy that matches the duration and convexity of

life insurance and annuity benefits. They demonstrate that this strategy is effective

in reducing longevity risk by calibrating it to U.S. mortality data. However, they

consider only liabilities, while we consider both assets and liabilities as well as

financial risk. Gatzert and Wesker (2012) use simulations to select portfolios of

policies that immunize the insurer’s solvency against changes in mortality. Gatzert

and Wesker (2013) consider the interactions among systematic, unsystematic, basis

risk, and adverse selection in determining the effectiveness of natural hedging.

Natural hedging with financial risk is studied by Stevens et al. (2011). They show

that financial risk has a clear impact on the overall initial riskiness of the annuity–life

insurance mix. The effect of natural hedging may be overestimated when financial

risk is ignored, affecting hedging possibilities . In their case, financial risk occurs only

from potential losses from assets, while in our case, it affects both the assets and the

fair value of liabilities.

We extend the existing literature in four directions. First, we model longevity and

financial risk at the same time, and we assess their impact on the fair value of the

insurer’s net liabilities or reserves. We aim at hedging changes in reserves, at the first-

and second-order approximations (delta and gamma hedging, respectively). The risk

factors to hedge against are the differences between the mortality and interest-rate in-

tensities forecasted today and their actual realizations in the future. Second, to hedge

liabilities, we let the insurer use new sales of insurance contracts, reinsurance, and

bonds, so we extend previous research by using both assets and liabilities for immu-

nization. Third, we exploit hedging within a single generation and across generations

(or across genders) to capture the fact that some productsmay be not marketed. For in-

stance, death benefits for older generations may not be marketed. Thus, we develop a

cohort-based mortality model, and we split the longevity risk factor of each generation

into common and idiosyncratic parts. Fourth, we provide all delta–gamma hedges in

closed form. This enhances the computation and comprehension of the hedge drivers.

Moreover, optimal hedges solve linear systems. Consequently, assessing whether re-

serves can be perfectly hedged (up to any chosen level of accuracy) and whether the

mix of assets and/or liabilities that achieves the hedge is unique is a trivial matter.The

framework can accommodate some important practical aspects, such as self-financing

constraints, sales constraints, and limited availability or absent products.