The impact of parameter uncertainty in insurance pricing and reserve with the temperature‐related mortality model

• Published date: 01 July 2019
• Author: Colin O'Hare,Athanasios A. Pantelous,Malgorzata Seklecka
• DOI: http://doi.org/10.1002/for.2558
• Date: 01 July 2019

Received: 29 June 2018 Revised: 10 September 2018 Accepted: 11 October 2018

DOI: 10.1002/for.2558

SPECIAL ISSUE ARTICLE

The impact of parameter uncertainty in insurance pricing

and reserve with the temperature-related mortality model

Malgorzata Seklecka1,3 Athanasios A. Pantelous2Colin O'Hare2

1Group Accumulation Management,

Zurich Insurance Group Ltd, Fareham,UK

2Department of Econometrics and

Business Statistics, Monash Business

School, Monash University, Wellington

Road, Victoria 3800, Australia

3Department of Mathematical Sciences,

University of Liverpool, Liverpool,United

Kingdom

Correspondence

Athanasios A. Pantelous, Department of

Econometrics and Business Statistics,

Monash Business School, Monash

University, 20 Chancellors walk,

Wellington Road, Clayton,Victoria 3800,

Australia.

Email:

Athanasios.Pantelous@monash.edu

Abstract

Changes in mortality rates have an impact on the life insurance industry, the

financial sector (as a significant proportion of the financial markets is driven by

pension funds), governmental agencies, and decision makers and policymakers.

Thus the pricing of financial, pension and insurance products that are con-

tingent upon survival or death and which is related to the accuracy of central

mortality rates is of key importance. Recently, a temperature-related mortal-

ity (TRM) model was proposed by Seklecka et al. (Journal of Forecasting, 2017,

36(7), 824–841), and it has shown evidence of outperformance compared with

the Lee and Carter (Journal of the American Statistical Association, 1992, 87,

659–671) model and several others of its extensions, when mortality-experience

data from the UK are used. There is a need for awareness, when fitting the

TRM model, of model risk when assessing longevity-related liabilities, especially

when pricing long-term annuities and pensions. In this paper, the impact of

uncertainty on the various parameters involved in the model is examined. We

demonstrate a number of ways to quantify model risk in the estimation of the

temperature-related parameters, the choice of the forecasting methodology, the

structures of actuarial products chosen (e.g., annuity,endowment and life insur-

ance), and the actuarial reserve. Finally, several tables and figuresillustrate the

main findings of this paper.

KEYWORDS

actuarial pricing, forecasting methodologies, model risk, reserve, temperature-related mortality

model, uncertainty

1INTRODUCTION

Remarkably, due to recent advances in science and tech-

nology,humansareliving,onaverage,longerthanever

before. Comparing life expectancy in the middle of the

18th century with that at the beginning of the 21st century,

This paper is dedicated to the memory of our wonderful friend, colleague

and collaborator, Prof. Colin O'Hare, who passed away suddenly as the

result of an accident on 1 August, 2018.

it can be seen that life expectancy has increased by over 30

years in a period of less than 200 years. This is an impres-

sive feat, especially considering that lifespan increased by

only 25 years over the previous 10,000 years (Niu & Melen-

berg, 2014; Pitacco, Denuit, Haberman, & Olivieri, 2009).

Obviously, as a direct consequence, longevity risk is and

will continue to be a key risk and a source of uncertainty

for individuals, governments, and financial institutions

from all over the globe. Thus the appropriate choice of

Journal of Forecasting. 2019;38:327–345. wileyonlinelibrary.com/journal/for © 2018 John Wiley & Sons, Ltd. 327

328 SEKLECKA ETA L.

mortality models and their relative forecasting ability is

becoming increasingly important both within academia

and in industry (French & O'Hare, 2013, 2014; O'Hare &

Li, 2017a, 2017b).

The climate, which plays a key role in the complex-

ity of the Earth's ecosystem, and thus changes in average

temperature, have an impact on life expectancy in vari-

ous ways. Researchers across the world have investigated

links between mortality and temperature changes by com-

paring different countries, regions or cities with diverse

climatic profiles (Gosling, Lowe, McGregor, Pelling, &

Malamud, 2009). They have mainly focused on heat and

cold waves, high and low temperatures, or additional

deaths during extreme weather conditions. Results from

these studies show that the temperature magnitude which

is attributed to increased deaths varies between popula-

tion groups and between countries (McMichael, Woodruff,

& Hales, 2006). According to Hajat, Vardoulakis, Heav-

iside, and Eggen (2014), the most vulnerable age group

is elderly people. In addition, Christidis, Donaldson, and

Stott (2010) suggest that the ability of individuals or

cohorts to adapt is a major influence on changing mor-

tality rates. At the same time, the absence of adapta-

tion can result in climate being the main contributor

to increases in heat- and cold-related mortality in com-

parison to an intermediate “comfortable” temperature

range (McMichael et al., 2006; Patz, Campbell-Lendrum,

Holloway,& Foley, 2005).

Recently, Seklecka, Pantelous, and O'Hare (2017) intro-

duced a new temperature-related mortality (TRM) model

in which an additional factor related to climate change

was included. The main structure of the TRM model is

based on the celebrated Lee and Carter (1992) model

and its numerous extensions. Furthermore, this model

includes a new parameter, which is the response to the

impact of temperature fluctuations (as a proxy of climate

change) on mortality. In that paper, to test the perfor-

mance of the TRM model using real data UK historical

mortality experience from 1974 to 2011 was considered.

The TRM model performed much better when compared

to the Lee and Carter (1992), Renshaw and Haberman

(2006), Plat (2009), and O'Hare and Li (2012) fitting for the

same dataset. Additionally, its forecasting ability was also

improved.

One of the main goals of this present paper is to exam-

ine the impact of the various parameters involved in the

TRM model on its ability to project and forecast central

mortality rates. Additionally, we check which of those

parameters are more sensitive and how their variations

may impact on insurance pricing and reserving. The sec-

ond main focus of this paper is to demonstrate the impact

of these projections on various financial calculations, and

we provide a number of ways of quantifying, both graphi-

cally and numerically, the model risk in such calculations.

Moreover, we test how the temperature change affects

insurance pricing and reserving, both of which are of

paramount importance for the sustainability of the insur-

ance industry, governmental authorities, decision mak-

ers, and policymakers (Godinez-Olivares, Boado-Penas,

& Pantelous, 2016; Pantelous & Zimbidis, 2008, 2009;

Pantelous, Zimbidis, & Kalogeropoulos, 2010; Richards

& Currie, 2009). Our results and conclusions are evalu-

ated based on historical mortality data available for the

UK from the Human Mortality Database. Data from the

Met Office were also used for the historical average tem-

peratures in the UK. Finally, forecasting ability is mea-

sured using different methodologies, such as the NAIVE,

autoregressive integrated moving average (ARIMA) pro-

cess, exponential smoothing, and cubic smoothing spline

methods.

The organization of this paper is as follows. In Section 2,

we briefly review the main characteristics and the param-

eters involved in the TRM model. Next, Section 3 focuses

on the data used from the Human Mortality Database,

and the Met Office for the UK population and presents

its preliminary analysis. In Section 4, the impact of four

new factors involved in the TRM model is discussed. The

choice of good estimates and the forecasting method-

ologies are presented in Section 5. In Section 6, insur-

ance pricing for different actuarial products is considered.

Results regarding the reserving are presented in Section

7. Finally, Section 8 concludes the whole discussion in

this paper.

2TEMPERATURE-RELATED

MORTALITY MODEL

In this section, a brief overview of the TRM model pro-

posed recently by 2017 is provided, which drives the analy-

sis of the following sections.1It should be mentioned here

that the TRM model provides a smooth expression for the

calculation of central mortality rates, mx,t,whicharearatio

of deaths, Dx,tcounts by a single age xand year t,and

exposure-to-risk, Ex,t, data in the same interval. The fitted

values of mx,t, which are calculated by the TRM model,

incorporate the trends in mortality and the trends in tem-

perature change (as a proxy of climate change). Seklecka

et al. (2017) considered monthly deaths and mid-year esti-

mated population data between 1974 and 2011 separated

1Formore details about the formulation of the TRM model, the interested

reader should consider 2017. Since the main focus of the current sequel

paper is to study the impact of uncertainty on the various parameters

involved in the TRM model, further discussion of its modeling char-

acteristics and features is, inevitably, beyond its scope and is therefore

omitted.