Optimal Insurance Policy Indemnity Schedules With Policyholders’ Limited Liability and Background Risk

• DOI: http://doi.org/10.1111/jori.12247
• Date: 01 December 2019
• Author: Martin Nell,Annette Hofmann,Ole V. Häfen
• Published date: 01 December 2019

©2018 The Journal of Risk and Insurance. Vol.XX, No. XX, 1–16 (2018).

DOI: 10.1111/jori.12247

Optimal Insurance Policy Indemnity Schedules With

Policyholders’ Limited Liability and Background

Risk

Annette Hofmann

Ole V. H¨

afen

Martin Nell

Abstract

This article makes two contributions to the insurance literature by studying

optimal insurance policy indemnity schedules with policyholders’ limited

liability and background risk. First, generalizing a prominent approach by

Huberman, Mayers, and Smith (1983), it is shown that a welfaresubsidy in the

case of a ruinous loss may make the insurance premium“overly fair ” for non-

bankruptinglosses and full insurance for this event becomes optimal. Second,

introducing correlated background risk into this limited liability framework

relativizes or even turns results by Doherty and Schlesinger (1983) as to the

impact of background risk on optimal coverage into its opposite.

Introduction

Individuals often face multiple risks that can adversely affect their wealth and well-

being. Illness, unemployment, death of a loved one, motor vehicle accidents, and

natural catastrophes like floods or earthquakes are among the many risks that can

deplete an individual’s resources. The insurance literature, however, mostly studies

optimal insurance demand in the face of only one single risk that is faced by the in-

dividual (see, e.g., Arrow, 1963; Mossin, 1968; Smith, 1968). It is furthermore often

Annette Hofmann is at the School of Risk Management, Insurance, and Actuarial Science, The

Peter J. Tobin College of Business, St. John’s University, 101 Astor Place, New York,NY 10003.

Hofmann can be contacted via e-mail: hofmanna@stjohns.edu. Ole V. H¨

afen is at the Institute

for Risk and Insurance, University of Hamburg, Von-Melle-Park5, 20146 Hamburg, Germany.

H¨

afen can be contacted via e-mail: o.v.haefen@web.de.Martin Nell was at the Institute for Risk

and Insurance, University of Hamburg,Von-Melle-Park 5, 20146 Hamburg, Germany.Wegrieve

for our coauthor Martin Nell who sadly departed this world before this article’s publication.

Martin opened both of us the doors to academics. Beyond his role as challenging and inspiring

academic teacher,he quickly became a friend and mentor. His creativity and his humor will stay

in our memories. A. H. & O. V.H. The authors are indebted to the editor and two anonymous

reviewers for their very constructive comments and suggestions, which led to a much improved

version of this article. We would like to thank participants in the seminar series at St. John’s

University, in particular Steve Mildenhall, for helpful comments on an earlier version of this

article.

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assumed that the individual is acting under full liability. This may not be realistic

in the many cases where limited liability of individual policyholders describes real-

world insurance situations. Limited liability is often created by “the option of personal

bankruptcy” (Huberman, Mayers, and Smith, 1983, p. 415). Indeed, the legal frame-

work in many countries allows individuals to declare personal bankruptcy.1Several

studies (e.g., Sinn, 1982; Keeton and Kwerel, 1984) show that under limited liability,

individuals will not always demand full insurance, even if the premium equals the

expected indemnity payment (referred to as actuarially fair premium).

A widely referenced study by Huberman, Mayers, and Smith (1983, HMS henceforth)

finds that an optimal insurance policy under limited liability of the policyholder con-

tains upper limits on coverage. Given straightforward assumptions (the indemnity

schedule is nondecreasing and never overindemnifies), a risk-averse individual pur-

chases partial insurance—at least when background risk is independent of the liability

risk. This is due to the insurance premium, which incorporates the cost of an indem-

nity payment for a ruinous loss, an indemnity that is wasted from the policyholder’s

perspective due to the minimum wealth level guaranteed by the bankruptcy code.

To the best of our knowledge, no study so far examines optimal insurance policy

indemnity schedules with policyholders’ limited liability under the presence of back-

ground risk. Doherty and Schlesinger (1983), a long-standing and well-known con-

tribution to the literature on optimal indemnification, analyze optimal insurance

under the presence of (uninsurable) background risk when the policyholder acts

under full liability. They find that optimality conditions for either full insurance or

deductible policies depend on the correlation between the insurable risk and the back-

ground risk.2Unlike Doherty and Schlesinger, this article analyzes optimal insurance

1This assumption also seems appropriate for common insurance contracts involving closely

held corporations (e.g., family-owned or other enterprises) that can declare corporate

bankruptcy.Expected utility maximization is a reasonable concept for corporate decision mak-

ing given that frequently observed provisions of managerial compensation contracts tend to

be designed so that when managers increase the value of the firm, they also increase their

expected utility (see, e.g., Smith and Stulz, 1985, p. 339). Mayers and Smith (1982) give a

wide overview of possible explanations for corporate insurance demand, MacMinn (1987)

introduces bankruptcy and agency costs, and MacMinn and Han (1990) as well as Han and

MacMinn (2006) investigate the interaction between management incentives and the corpo-

rate demand for insurance. Cummins and Mahul (2004) study the case where the indemnity

schedule is subject to an upper limit, providing the insurance company with limited liability.

Froot and Stein (1998) as well as Froot (2007) show that value-maximizing financial institu-

tions may act like risk-averse decision makers whenever parts of the risk they face cannot be

hedged due to frictions in the capital market. Finally, Gollier, Koehl, and Rochet (1997) study

the risk-taking behavior of a risk-averse firm with limited liability.The authors show that the

optimal risk-taking exposure is always larger than that of a firm with unlimited liability.

2Ping and Zanjani (2015) demonstrate that a correlation between insurer background wealth

and the insurance buyer’s loss can also lead to upper limits on coverage. In their model,

the underlying reason for the upper limits stems from the insurer’s incentive to manage risk

efficiently and limit exposure to correlated losses. Our model studies the demand side by

introducing correlated background risk of the policyholder into the HMS limited liability

framework.

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