Mortality Portfolio Risk Management

• DOI: http://doi.org/10.1111/j.1539-6975.2012.01469.x
• Author: Yijia Lin,Luis F. Zuluaga,Samuel H. Cox,Ruilin Tian
• Published date: 01 December 2013
• Date: 01 December 2013

© The Journal of Risk and Insurance, 2013, Vol. 80, No. 4, 853–890

DOI: 10.1111/j.1539-6975.2012.01469.x

853

MORTALITY PORTFOLIO RISK MANAGEMENT

Samuel H. Cox

Yiji a Lin

Ruilin Tian

Luis F. Zuluaga

ABSTRACT

We provide a new method, the “MV+CVaR approach,” for managing un-

expected mortality changes underlying annuities and life insurance. The

MV+CVaR approach optimizes the mean–variance trade-off of an insurer’s

mortality portfolio, subject to constraints on downside risk. We apply the

method of moments and the maximum entropy method to analyze the ef-

ficiency of MV+CVaR mortality portfolios relative to traditional Markowitz

mean–variance portfolios. Our numerical examples illustrate the superiority

of the MV+CVaRapproach in mortality risk management and shed new light

on natural hedging effects of annuities and life insurance.

.

INTRODUCTION

Life insurance companies sell a wide variety of life insurance and annuity products.

The insurers’ liabilities for these products depend on future mortality rates. During

recent years, economic and demographic changes have made mortality projection

and risk management more important than ever. On the one hand, life expectancy for

ages 60 and older in the past two decades has improved at a much higher rate than

what pension plans and annuity providers expected. Cowling and Dales (2008) find

that companies in the United Kingdom FTSE100 index underestimated their aggregate

pension liabilities by more than £40 billion. If the firms do not take measures to control

Samuel H. Cox is the L. A. H. WarrenChair Professor at the Asper School of Business, Univer-

sity of Manitoba. Yijia Lin is in the Department of Finance, College of Business Administration,

University of Nebraska–Lincoln. Ruilin Tian is in the Department of Accounting, Finance, and

Information System, College of Business, North Dakota State University. Luis F. Zuluaga is

at the Faculty of Business Administration, University of New Brunswick. The authors can

be contacted via e-mail: scox@cc.umanitoba.ca, yijialin@unl.edu, ruilin.tian@ndsu.edu, and

lzuluaga@unb.ca, respectively. This article was presented at the 2009 American Risk and In-

surance Association annual meeting and the 2010 Financial Management Association annual

meeting. We appreciate helpful comments fromAlexander Kling and other participants at the

meetings. The authors also thank two anonymous referees for their very helpful suggestions

and comments during the revision process.

854 THE JOURNAL OF RISK AND INSURANCE

mortality downside risk, such longevity shocks are likely to cause serious financial

consequences. For example, unanticipated mortality improvement was an important

factor accounting for the failure of Equitable Life, once a highly regarded U.K. life

insurer (Ombudsman, 2008). On the other hand, population growth, urbanization,

and increased global mobility may lead to a more rapid and widespread disease.

Genetic analysts recently confirmed that today’s “bird flu” is similar to the 1918

“Spanish flu” that killed more than 40 million people. This finding spurs fears of a

worldwide epidemic (Juckett, 2006). According to Toole(2007), losses due to a severe

pandemic could amount to 25 percent of the U.S. life insurance industry’s statutory

capital. While the great majority of U.S. life insurance companies would weather such

a pandemic, it is clear that these companies should be interested in mitigating the

risk.

We proposea method that life insurance companies can use to alleviate extreme mor-

tality outcomes while maintaining a relatively efficient mean–variance relationship

for their mortality portfolios of life insurance and/or annuities. This method, the

“MV+CVaR approach,” combines Markowitz mean–variance (MV) portfolio theory

and conditional value at risk (CVaR) by optimizing the trade-off between mean and

variance subject to an upper bound on CVaR. Variance measures both positive and

negative deviations of portfolio values from its expected level, while CVaRfocuses on

the portfolio tail loss caused by extreme events. Although the MV+CVaR portfolios

are suboptimal relative to the Markowitz counterparts in terms of the mean–variance

efficiency, they are attractive to insurers since the MV+CVaR portfolios have lower

downside risk while achieving desirable risk–return trade-offs. In practice, life in-

surers are keenly interested in searching for an optimal risk–return relationship for

their business. At the same time they are required to meet various solvency require-

ments for possible catastrophes such as flu epidemics. Therefore, incorporating both

variance and CVaR as risk measures in business optimization such as the MV+CVaR

approach should be appealing to life insurance companies. The risk control frame-

work adopted in this article closely follows that of Rockafellar and Uryasev (2000)

and Tsai,Wang, and Tzeng (2010). In their framework, firms minimize portfolio losses

subject to CVaR constraints. In our context, we consider both variance and CVaR as

risk measures by incorporating a CVaR constraint into the classical mean–variance

setup.

Weextend Rockafellar and Uryasev (2000) and Tsai, Wang,and Tzeng (2010) in one im-

portant dimension by applying the well-developed moments method to validate the

quality of MV+CVaR portfolios. The existing literature on mortality models usually

makes distributional assumptions. Since knowledge about the underlying mortality

distributions may be limited, the assumed distributions may not represent the actual

mortality dynamics. However, the moments method is solely based on moments, not

a specific distribution. This method yields robust semiparametric upper and lower

bounds that any reasonable model with the same moments must satisfy.Once a mor-

tality portfolio has been constructed, the corresponding empirical benefit payments

can be used to estimate moments of the benefit payment ratios. In this article, we show

how such estimated moment information can be used to determine the bounds on

the underlying portfolio benefit payment ratios. In particular, the moments method

provides a mechanism to compare the downside risk of a MV+CVaR efficient mor-

tality portfolio and its MV counterpart given their moments. Our results show the

MORTALITY PORTFOLIO RISK MANAGEMENT 855

superiority of the MV+CVaR approach in mortality risk management and highlight

the natural hedging effects of life insurance and annuities.

For the numerical examples, all optimization problems are solved with MATLAB1soft-

ware. When computing the bounds of a particular portfolio, we solve the equivalent

dual problem using built-in functions from the SOS programming solver. The SOS

programming solver was developed by Prajna et al. (2004), which is a free toolbox

written in MATLAB. Our implementation is fairly general and easy to use.

The remainder of this article is organized as follows. The first section describes how to

calculate benefit payment ratios of mortality portfolios. The second section discusses

the MV+CVaRmortality optimization model. We provide two numerical examples to

illustrate the implementation of the approach. The third section describes the method

of moments. We show how to compute the semiparametric upper and lower bounds

for MV+CVaR efficient portfolios and then perform the bound analysis on those

portfolios. The fourth section demonstrates the natural hedging effect when annuities

and life insurance are considered jointly. The fifth section extends the analysis to

efficient frontiers of MV+CVaR mortality portfolios by comparing bounds of MV and

MV+CVaR optimal portfolios. In addition, we show to what extent downside risk is

reduced by changing the CVaR constraint. The sixth section is our conclusion.

MORTALITY RISK PORTFOLIOS

Focus on Mortality

Consider a life insurer underwriting new business at the current moment 0, with

the business sorted by underwriting class (x). The class symbol (x) represents the

information at time 0 that the insurer uses to determine a mortality table for lives in

the class. This information includes at least age, sex, and line of busines but could

include other information such as height, weight, blood pressure, or tobacco use

history. Actuarial textbooks usually work with a set of mortality tables that differ

only by age. The other information is suppressed, and the tables are obtained by

simply specifying the issue age x. However, in practice a company will have different

tables for each age, sex, major line of business, and so on. Thus, in our notation (x)

represents all the information the company needs to estimate a mortality table for this

class at 0. Here are our assumptions:

1. All future cash flows are discounted at risk-free interest rates. These discount

rates are known constants.

2. Life insurance policies and annuity contracts remain in force until settled at death.

There are no policy lapses.

3. Expenses are known constant multiples of policy premiums.

4. The analysis is applied only to new business without regard to the insurer’s book

of business issued before time 0.

Under these assumptions, the present value of benefits (and related expenses) paid

at times 1, 2, ...,PVB(x), and the present value of premiums (and related expenses)

1http://www.mathworks.com/