Modeling Multicountry Longevity Risk With Mortality Dependence: A Lévy Subordinated Hierarchical Archimedean Copulas Approach

• Author: Ken Seng Tan,Chou‐Wen Wang,Wenjun Zhu
• DOI: http://doi.org/10.1111/jori.12198
• Published date: 01 April 2017
• Date: 01 April 2017

©2017 The Journal of Risk and Insurance. Vol.84, No. S1, 477–493 (2017).

DOI: 10.1111/jori.12198

Modeling Multicountry Longevity Risk With

Mortality Dependence: A L´evy Subordinated

Hierarchical Archimedean Copulas Approach

Wenjun Zhu

Ken Seng Tan

Chou-Wen Wang

Abstract

This article proposes a new copula model known as the L´

evy subordinated

hierarchical Archimedean copulas (LSHAC) for multicountry mortality

dependence modeling. To the best of our knowledge, this is the first article

to apply the LSHAC model to mortality studies. Through an extensive

empirical analysis on modeling mortality experiences of 13 countries, we

demonstrate that the LSHAC model, which has the advantage of capturing

the geographical structure of mortality data, yields better fit, compared to

the elliptical copulas. In addition, the proposed LSHAC model generates

out-of-sample forecasts with smaller standard deviations, when compared

to other benchmark copula models. The LSHAC model also confirms that

there is an association between geographical locations and dependence of

the overall mortality improvement. These results yield new insights into

future longevity risk management. Finally, the model is used to price a

hypothetical survival index swap written on a weighted mortality index.

The results highlight the importance of dependence modeling in managing

longevity risk and reducing population basis risk.

Wenjun Zhu is an Assistant Professor at the School of Finance, Nankai University, Tianjin,

China. Zhu can be contacted via e-mail: zhuwenjun@nankai.edu.cn. Ken Seng Tan is the Uni-

versity Research Chair Professor at the Department of Statistics and Actuarial Science, Uni-

versity of Waterloo 200 University Avenue West, Waterloo, N2L 3G1, ON, Canada. Tancan be

contacted via e-mail: kstan@uwaterloo.ca. Chou-Wen Wang is at the Department of Finance

National Sun Yat-sen University, Kaohsiung, Taiwan Fellow of Risk and Insurance Research

Center, College of Commerce National Chengchi University, Taiwan. Wang can be contacted

via e-mail: chouwenwang@gmail.com. Zhu acknowledges funding support from the Society

of Actuaries (SoA) James C. Hickman Scholar Program and the China Scholarship Council

(CSC). Tan acknowledges the research funding from the Natural Sciences and Engineering

Research Council of Canada and the Society of Actuaries CAE Research Grant. Wang was

also supported in part by the MOST 101-2410-H-327-029- from the Ministry of Science and

Technology, R.O.C.

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478 The Journal of Risk and Insurance

Introduction

It is estimated that the human life expectancy in developed countries has been in-

creasing almost linearly over the past 150 years (Blake et al., 2013). The unanticipated

increases in life expectancy create significant financial burden to both public and pri-

vate pension plan sponsors. As a result, longevity risk, as attributed to the increase

in life expectancy, has been recognized as one of the major risks faced by insurers,

reinsurers, governments, and individuals in recent years. Hence, a traded market in

longevity-linked securities and derivatives has emerged to facilitate the development

of annuity markets and protect the long-term viability of global retirementincome pro-

vision. For example, as the first longevity-linked derivative transaction, a q-forward

contract between JPMorgan and the U.K. company Lucida was traded in January 2008.

In addition, Swiss Re launched the Kortis longevity bond to transfer USD 50 million

of longevity risk to the capital markets.

A number of two-population mortality models have been proposed (see, e.g., Li and

Lee, 2005; Cairns et al., 2011; Dowd et al., 2011;Jarner and Kryger, 2011; Li and Hardy,

2011; Zhou, Li, and Tan, 2013; Zhou et al., 2014). Many innovative contracts have

payoffs that are linked to broad-based population mortality indices; hence, a more

sophisticated stochastic mortality model for multipopulation is critical for pricing

these longevity-linked securities as well as for reducing population basis risk (Blake

et al., 2013). Chen, MacMinn, and Sun (2015) introduce factor copula into multipop-

ulation mortality modeling. They employ a two-stage procedure based on the time

series analysis and a factor copula approach. Wang, Yang, and Huang (2015) model

multicountry mortality using a dynamic copula framework.

We propose to use a new copula family called the L´

evy subordinated hierarchical

Archimedean copula (LSHAC) model for multipopulation mortality modeling. The

LSHAC model has some appealing advantages. First, it overcomes some serious draw-

backs of the classical Archimedean copulas (AC) and hierarchical Archimedean cop-

ulas (HAC). The AC, although it has the advantage of simplicity, suffers from a fully

exchangeable structure. As the HAC model has been proposed to partially overcome

the exchangeability by “nesting” two or more ACs with appropriate grouping, its

generators must fulfill the compatible conditions to ensure that the resulting HAC has

a valid multivariate distribution (Joe, 1997; Savu and Trede, 2010; McNeil, 2008). The

compatible conditions, however, can be difficult (and empirically almost impossible)

to verify, and hence restrict the practical application of the HACs (Savu and Trede,

2010). This difficulty is resolved by Hering et al. (2010) and Mai and Scherer (2012).

With a two-layer illustration, they show that as long as the HACs are constructed

from L´

evy subordinators, the compatible conditions are automatically satisfied. Zhu,

Wang, and Tan (2016) provide an estimation methodology for the LSHAC model in a

general setting. Motivated by these findings, the goal of this article is to employ the

multi-layer LSHAC to model the multipopulation mortality dependence. To the best

of our knowledge, this is the first article to model the multipopulation longevity risk

with the LSHAC model.

Another important advantage of the LSHAC is its ability to capture the relationship

between the geographical locations and mortality dependence using its hierarchical