Market Price of Longevity Risk for a Multi‐Cohort Mortality Model With Application to Longevity Bond Option Pricing

• Published date: 01 September 2020
• Author: Yajing Xu,Jonathan Ziveyi,Michael Sherris
• DOI: http://doi.org/10.1111/jori.12273
• Date: 01 September 2020

©2019 The Journal of Risk and Insurance. Vol.XX, No. XX, 1–25 (2019).

DOI: 10.1111/jori.12273

Market Price of Longevity Risk for a Multi-Cohort

Mortality Model With Application to Longevity

Bond Option Pricing

Yajing Xu

Michael Sherris

Jonathan Ziveyi

Abstract

We introduce a multi-cohort continuous time affine mortality model and,

along with an affine arbitrage-free term structure model, determine implied

market prices of longevity risk in the BlackRock CoRI Retirement Indexes.

These indexes provide a daily level of estimated cost of lifetime retirement

income for 20 cohorts in the United States. Individuals can invest in Black-

Rock funds that track the indexes that are quoted on the NYSE. We use our

model to derive closed-form expressions for prices of European options on

longevity zero-coupon bonds and show the impact of stochastic mortality on

long-term longevity bond option prices.

Introduction

Life insurance companies and defined benefit (DB) pension plans are exposed to

systematic longevity risk, which is the unanticipated changes in mortality rates that

are not averaged out by the law of large numbers. This risk should be reflected in

the market pricing of longevity-linked products. Changes in accounting and solvency

regulatory requirements have increased interestin the market valuation of longevity-

linked liabilities, yet there remains no generally accepted approach to incorporating

the price of systematic longevity risk.

Yajing Xu is at the SWUFE China. Xu can be contacted via email: yajing xu@swufe.edu.cn.

Michael Sherris and Jonathan Ziveyi are at the School of Risk & Actuarial Studies, Australian

Research Council Centreof Excellence in Population Ageing Research (CEPAR), UNSW Sydney.

Sherris can be contacted via email: m.sherris@unsw.edu.au. Ziveyi can be contacted via email:

j.ziveyi@unsw.edu.au.The authors would like to acknowledge the financial support of the Aus-

tralian Research Council Centre of Excellence in Population Ageing Research (CEPAR), Project

No. number CE110001029. The authors are grateful for helpful comments from participants of

the 7th China International Conference on Insurance and Risk Management (CICIRM 2016) in

Xian in July 2016, the Asia-Pacic Risk and Insurance Association (APRIA) Annual Conference

in Chengdu in August 2016, Workshop on Risk: Modelling, Optimization and Inference With

Applications in Finance, Insurance and Superannuation at UNSW Sydney in December 2017,

and the 2018 Western Risk and Insurance Association Annual Meeting.

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The development of longevity-linked securities and derivatives aims to allow the

management of this risk as well as provide market-based prices. For example, Blake

and Burrows (2001) propose the transfer of this risk to the financial market using

longevity bonds. Other proposed instruments include survivor swaps (Dowd et al.,

2006), q-forwards (Coughlan et al., 2007), and mortality-linked options (Bauer,B ¨

orger,

and Ruß, 2010). Financial markets have the capacity and experience in risk manage-

ment to take on longevity risk, which in the past has mostly been reinsured. Yet these

markets have not developed.

The pricing of longevity-linked securities depends not only on the stochastic process

for the underlying risk factors, but also the attitude of investors toward the risk of

those factors. The Life & Longevity Markets Association (LLMA) has identified the

market risk premium of longevity risk as one of the key inputs in a longevity pricing

framework.1Todetermine the market risk premium, a common practice is to use avail-

able market prices, such as life annuities, longevity-linked securities, and longevity

indices. The longevity market is however incomplete due to the lack of traded assets,

and calibration of market prices of risk remains problematic.

In an incomplete market, longevity-linked derivative pricing requires making as-

sumptions about the market risk premium of bearing longevity risk directly or im-

plicitly. Lin and Cox (2005) propose to use a Wang transform for the securitization

of longevity risk, and the market price is defined as the shift parameter in the Wang

transform to risk adjust a survival distribution based on 1996 IAM 2000 Basic Table

and annuity quotes.

A different approach is taken in Bauer and Ruß (2006) and Chigodaev, Milevsky, and

Salisbury (2016), who propose to derive parameter values for stochastic mortality

models using survival probabilities implied by annuity prices, so that the market

price of longevity risk is implicitly included in these parameter values.

Cairns, Blake, and Dowd (2006) calibrate the market price of risk to a single security,

the EIB bond, and investigate market prices of longevity risk associated with the two

factors in the discrete-time CBD model. However, the EIB bond was withdrawn for

redesign because of insufficient demand and market prices of longevity risk extracted

from this bond are questionable. Other studies, for example, Schrager (2006), consider

the use of the best-estimate survival probabilities rather than the risk-adjusted ones

implied by longevity-linked products. In other words, the market price of longevity

risk is assumed to be zero, which is a simple but unrealistic assumption.

Bisetti et al. (2017) estimate Sharpe ratios from a VAR model of U.S. stocks, bonds,

bills, and a synthetic annuity-linked security, and hence derive market prices of risk

for longevity risk using a method consistent with pricing of financial risk.

There remains no well-accepted method to calibrate and incorporate the market price

of longevity risk into mortality dynamics under a risk-neutral measure for market val-

uation that can be used in modeling and pricing of longevity-linked products. Bauer,

1LLMA (2010). Longevity Pricing Framework(www.llma.org).

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