Insurance demand in the presence of loss‐dependent background risk

• Published date: 01 December 2023
• Author: Sebastian Hinck,Petra Steinorth
• Date: 01 December 2023
• DOI: http://doi.org/10.1111/jori.12426

Received: 22 February 2022

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Revised: 3 April 2023

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Accepted: 5 April 2023

DOI: 10.1111/jori.12426

ORIGINAL ARTICLE

Insurance demand in the presence

of loss‐dependent background risk

Sebastian Hinck|Petra Steinorth

Institute for Risk Management and

Insurance, Universität Hamburg,

Hamburg, Germany

Correspondence

Petra Steinorth, Institute for Risk

Management and Insurance, Universität

Hamburg, Hamburg, Germany.

Email: petra.steinorth@uni-hamburg.de

Abstract

We analyze insurance demand when insurable losses

come with an uninsurable zero‐mean background risk

that increases in the loss size. If the individual is risk

vulnerable, loss‐dependent background risk triggers a

precautionary insurance motive and increases optimal

insurance demand. Prudence alone is sufficient for

insurance demand to increase in two cases: the case of

fair insurance and the case where the smallest possible

loss exceeds a certain threshold value (referred to as

the large loss case). We derive conditions under which

insurance demand increases or decreases in initial

wealth. In the large loss case, prudence determines

whether changes in the background risk lead to more

insurance demand. We generalize this resultto arbitrary

loss distributions and find conditions based on decreas-

ing third‐degree Ross risk aversion, Arrow–Pratt risk

aversion, and Arrow–Pratt temperance.

KEYWORDS

background risk, insurance demand, loss‐dependent background

risk, prudence, risk vulnerability

JEL CLASSIFICATION

D81, G52

Journal of Risk and Insurance. 2023;90:991–1026. wileyonlinelibrary.com/journal/JORI

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991

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

distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

© 2023 The Authors. Journal of Risk and Insurance published by Wiley Periodicals LLC on behalf of American Risk and Insurance

Association.

1|MOTIVATION

Insurance is a powerful tool to transfer undesirable risks to third parties. In reality, insurance

contracts are often incomplete in the sense that they do not perfectly indemnify each possible loss

in any state of the world. This incompleteness has been long recognized in the insurance economics

literature since Doherty and Schlesinger (1983b).

In this study, we investigate insurance demand in the presence of an uninsurable loss‐

dependent background risk. We assume that an individual's loss consists of two components:

the insurable component denoted by

X

and the uninsurable component denoted by

ηX

. The

total loss faced by the individual is given by Xη+

X

, but the indemnity is solely based on

X

as

ηX

cannot be treated by insurance. We assume that

ηX

has a zero mean. This rules out any

wealth effects due to the uninsurable component.

ηX

can be interpreted as an observation error

between the actual loss size and the insured loss as discussed under the term approximate

insurance by Gollier (1996). Intuitively, we consider the situation in which only estimated

losses are insurable, but not actual losses.

Applications in which such an uninsurable component arises include the following:

•The insurer is not able to perfectly observe the loss. This can be the case when the loss

evolves over long periods of time and the indemnity is paid instantly, exposing the decision‐

maker to price risk.

•The insurer uses a simplified indemnification process, basing the indemnity on estimations

rather than actual losses, and insurance policies which by design do not condition on the

actual loss. Fixed indemnity insurance like hospital cash benefits serves as an example here.

•The insured loss is subject to an uninsurable exchange rate risk.

In all these examples, the indemnity is solely based on estimated losses

X

rather than actual

losses Xη+

X

. In the exchange rate example,

ηX

describes the random exchange rate

component, where

η

Xη=

X, and

η

is the random exchange rate factor. At the same time, all of

these examples have in common that the error

ηX

typically increases in loss size. Therefore, in

this study, we assume that the distribution of

ηX

undergoes (second‐order) increases in risk in

the sense of Rothschild and Stiglitz (1970) as the loss

X

increases. Consequently, the larger the

insurable loss

X

is in absolute terms, the riskier is the uninsurable loss component. For this

reason, we coin

ηX

loss‐dependent background risk. Due to loss‐dependent background risk, the

decision‐maker is exposed to some risk even when she takes out maximum coverage.

Our study aims to analyze the implications of loss‐dependent background risk on insurance

demand. Therefore, we derive optimal insurance demand with loss‐dependent background risk

and benchmark this coverage level to the case in which such a background risk is absent. This

approach allows us to study the comparative effect of loss‐dependent background risk on

optimal insurance demand. This comparison is meaningful in the context of salience: loss‐

dependent background risk is prone to be neglected, as the decision‐maker may not be aware of

its existence. If loss‐dependent background risk becomes salient, the decision‐maker is in a

situation where the additional risk suddenly becomes present in her perception of the decision

situation. Salience of the loss‐dependent background risk may not only change over time for

one individual, but may also differ between individuals. We address the question of whether,

and under which conditions, salience of a loss‐dependent background risk changes insurance

demand, either intrapersonal over time or interpersonal for individuals with and without

salience of the loss‐dependent background risk. Salience of this risk is also relevant from the

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HINCK and STEINORTH

perspective of the insurer in the decision problem. If the insurer neglects the fact that the

insured face the additional uninsurable risk, expectations on insurance demand are potentially

biased.

More generally, imperfections can arise for different reasons: one example of imperfect

insurance policies is the case of insurer default, in which the insurer fails to fully pay the

agreed‐upon indemnity (Doherty & Schlesinger, 1990). The insured is thus exposed to the

uninsurable risk of insurer insolvency, irrespective of the selected insurance policy. Another

example discussed in the literature is that insurance policies typically exclude losses due to

certain events like war or nuclear hazards, leaving the individual fully exposed to these kinds of

risks irrespective of the amount of insurance purchased (Doherty & Schlesinger, 1983b). These

excluded losses are examples of risks that have been coined background risk in the literature

(Gollier & Pratt, 1996). Imperfections due to loss‐dependent background risk are different from

other sources of imperfections, like, excluded losses or insurer default risk. For instance, losses

triggered by uninsurable events are typically independent of losses that can be insured, as the

events are separate. Loss‐dependent background risk, on the other hand, manifests whenever

insurable losses realize and depends on the realization of the insurable loss event. Loss‐

dependent background risk has an upside, for example, if the actual loss is smaller than the

insurable component and is therefore a special case of a basis risk. This is different to insurer

default risk, which only has a downside risk.

An important contribution of our paper is that we find that risk vulnerability determines

insurance demand even if background risk is not independent

1

: We find that a loss‐dependent

background risk leads to a higher demand for insurance if the individual is risk vulnerable.

2

Unlike in the case of an independent background risk, a loss‐dependent background risk

triggers a precautionary insurance motive for risk‐vulnerable decision‐makers that can

potentially lead to more than full coverage being optimal.

3

Our paper contributes to the literature on insurance demand in the presence of

uninsurable risk. It is most closely related to Eeckhoudt and Kimball (1992)whostudy

insurance demand with a loss‐dependent background risk that deteriorates in a third‐order

stochastic dominance sense as the insurable loss increases. They find that standardness

4

leads to more insurance demand in the presence of the loss‐dependent background risk.

The loss‐dependent background risk we consider here is a special case of the one studied in

Eeckhoudt and Kimball (1992). We show that risk vulnerability is already sufficient for

unambiguous results in our setup such that the stronger assumption of decreasing absolute

risk aversion together with decreasing absolute prudence is not needed. In the special case

of a binary loss distribution, Fei and Schlesinger (2008) show that third‐order preferences

determine how insurance demand changes if a loss‐dependent backgroundriskispresent.

We show that their result does not necessarily hold for arbitrary loss distributions, but we

also discuss assumptions on the loss distribution and the insurance policy under which

results from Fei and Schlesinger (2008) continue to hold. To further complement the work

1

In the literature on risk preferences, background risk is usually assumed to be independent of insurable risk. In the case of such an

independent background risk, risk vulnerability as introduced by Gollier and Pratt (1996) leads to a more cautious behavior. A risk‐

vulnerable decision‐maker consequently demands more insurance to better cope with an independent background risk.

2

The concept of risk vulnerability states that “adding an unfair background risk to wealth makes risk‐averse individuals behave in a

more risk‐averse way with respect to another independent risk”(Gollier & Pratt, 1996, p. 1110).

3

In the case of loss‐dependent background risk, full insurance refers to a policy that fully covers

X

, even though the realized loss may be

larger or smaller than the actual indemnity.

4

Kimball (1993) defines standardness as decreasing absolute risk aversion and decreasing absolute prudence.

HINCK and STEINORTH

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