Guaranteed renewable life insurance under demand uncertainty

• Author: Asha Sadanand,Michael Hoy,Afrasiab Mirza
• Date: 01 March 2021
• DOI: http://doi.org/10.1111/jori.12320
• Published date: 01 March 2021

J Risk Insur. 2021;88:131–159. wileyonlinelibrary.com/journal/JORI

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131

Received: 21 February 2019

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Accepted: 30 June 2020

DOI: 10.1111/jori.12320

ORIGINAL ARTICLE

Guaranteed renewable life insurance under

demand uncertainty

Michael Hoy

1

|Afrasiab Mirza

2

|Asha Sadanand

1

1

Department of Economics and Finance,

University of Guelph, Guelph, Ontario,

Canada

2

Department of Economics, University of

Birmingham, Birmingham, UK

Correspondence

Afrasiab Mirza, Department of Economics,

University of Birmingham, JG Smith

Building, Edgbaston, Birmingham, UK B15

2TT.

Email: a.mirza@bham.ac.uk

Abstract

Guaranteed renewability (GR) is a prominent feature

in many health and life insurance markets. We de-

velop a model that includes unpredictable (and un-

observable) fluctuations in demand for life insurance

as well as changes in risk type (observable) over in-

dividuals' lifetimes. The presence of demand type

heterogeneity leads to the possibility that optimal GR

contracts may have a renewal price that is either

above or below the actuarially fair price of the lowest

risk type in the population. Individuals whose type

turns out to be high risk but low demand renew more

of their GR insurance than is efficient due to the

attractive renewal price. This results in imperfect

insurance against reclassification risk. Although a

first‐best efficient contract is not possible in the pre-

sence of demand type heterogeneity, the presence of

GR contracts nonetheless improves welfare relative to

an environment with only spot markets.

KEYWORDS

demand uncertainty, guaranteed renewability, insurance,

reclassification risk

JEL CLASSIFICATION

D8; D86; G22

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This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

© 2020 The Authors. Journal of Risk and Insurance published by Wiley Periodicals LLC on behalf of American Risk and Insurance

Association

1|INTRODUCTION

Guaranteed renewability (GR), which is a prominent feature in health and life insurance markets,

provides an opportunity for individuals to insure against reclassification risk. This works as fol-

lows. Consider a set of ex ante identical individuals each of whom purchases an initial 10‐year

termlifeinsurancecontractwithaviewofpossiblypurchasingasubsequentpolicyattheendof

the term. By the end of the contract period, some insureds may have discovered that their

mortality status has changed. If this change is observable to insurers, then the price for a new

insurance contract will reflect that change in risk. Individuals recognize ex ante that their risk type

may change over time and so prefer to avoid the prospect of premium risk associated with

stochastic mortality prospects. GR contracts contain a promise to offer a subsequent insurance

policy at the expiry date of the first contract at a price that is independent of any changes in

mortality risk. The premium for the implicit insurance against reclassification risk is embedded in

the first contract (earlier period) through an extra premium assessment—a phenomenonknown as

front loading. This allows insurers to offer insurance to those individuals who turn out to be higher

risk types in the subsequent 10‐year period at a price below their actuarially fair rate. As long as

the amount of front loading is sufficient, the added profit from the first (period) contract com-

pensates for insurers' losses from the second (period) contract.

1

In our paper we focus on the implications for GR (and long‐term) insurance to ameliorate

premium risk when individuals face uncertainty over future changes in both mortality risk and

insurance needs. We consider an environment where individuals face no capital market im-

perfections (they can borrow or save at the risk‐free rate) nor other impediments such as the

existence of resettlement or viatical market opportunities that can thwart GR insurance to fully

protect against reclassification risk. We develop a two‐period model of insurance in which in-

dividuals are homogeneous in the first period and hold the same beliefs about the likelihood of

becoming a low‐or high‐demand type in the second period. An important feature of insurance

demand is how it changes over the life cycle. As noted in Hong and Rios‐Rull (2012), average

demand follows a life cycle pattern that rises from young adulthood to “around age 45 for males

and 35–40 for females”(based on 1990 data). They also show, in their figure 1(p. 3705), that there

is substantial variation in demand across individuals at all ages and especially around the peak

level of demand. This means that to have an ideal amount of coverage for premium risk in the

future, one may have to hold more insurance than is optimal early in life (i.e., for the younger part

of the life cycle where demand tends to be increasing). This turns out to be a critical factor in

determining the extent to which GR can provide insurance against reclassification risk. We allow

second‐period demand to be higher or lower than first‐period demand for either or both demand

types. Moreover, we also allow for the possibility that demand does not vanish over time. Each

individual's risk type also evolves over time in a similar manner; that is, individuals have the same

mortality risk in the first period but their mortality risk diverges in the second period. Moreover, in

period 1 individuals hold the same beliefs about the evolution of their risk type for period 2.

2

1

A similar phenomenon may be reflected in short‐term versus long‐term insurance contracts (e.g., 10 vs. 20 years) with

longer contracts providing insurance against reclassification risk through front loading to keep premiums later in the

contract sufficiently low to avoid lapsation by better risks.

2

In a similar environment but without differential demand types, Peter, Richter, and Steinorth (2015) consider the

implications of individuals learning imperfectly about their risk type over time with this information being private. Fei,

Fluet, and Schlesinger (2013) also use a model that features demand uncertainty but that does not include risk type

uncertainty nor any dynamic features of insurance demand present in our model of GR insurance.

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