Ambiguity and Insurance: Capital Requirements and Premiums

• Author: Oliver Walker,Simon Dietz
• Date: 01 March 2019
• DOI: http://doi.org/10.1111/jori.12208
• Published date: 01 March 2019

©2017 The Journal of Risk and Insurance. Vol.86, No. 1, 213–235 (2019).

DOI: 10.1111/jori.12208

Ambiguity and Insurance: Capital Requirements

and Premiums

Simon Dietz

Oliver Walker

Abstract

Many insurance contracts are contingent on events such as hurricanes, terror-

ist attacks, or political upheavals, whose probabilities areambiguous. This ar-

ticle offers a theory to underpin the largebody of empirical evidence showing

that higher premiums are charged under ambiguity. We model a (re)insurer

that maximizes profit subject to a survival constraint that is sensitive to the

range of estimates of the probability of ruin, as well as the insurer’s attitude

toward this ambiguity.We characterize when one book of insurance is more

ambiguous than another and general circumstances in which a more am-

biguous book requires at least as large a capital holding. We subsequently

derive several explicit formulae for the price of insurance contracts under

ambiguity,each of which identifies the extra ambiguity load.

Introduction

Many (re)insurance contracts are contingent on events such as hurricanes, terrorist

attacks, or political upheavals, whose probabilities are not known with precision.

Such contracts are said to be subject to “ambiguity.” There may be several reasons

why contracts are subject to ambiguity, including a lack of historical, observational

data, and the existence of competing theories, proffered by competing experts and

formalized in competing forecasting models, of the causal processes governing events

that determine their value. For example, ambiguity is a salient feature in the insurance

of catastrophe risks such as hurricane wind damage to property in the Southeastern

United States. Here, historical data on the most intense hurricanes are limited, and

there are competing models of hurricane formation (Knutson et al., 2008; Bender et al.,

Simon Dietz is at the ESRC Centre for Climate Change Economics and Policy, Grantham Re-

search Institute on Climate Change and the Environment, and Department of Geography and

Environment, London School of Economics and Political Science. Dietz can be contacted via

e-mail: s.dietz@lse.ac.uk. Oliver Walker is at ESRC Centre for Climate Change Economics and

Policy and Grantham Research Institute on Climate Change and the Environment, London

School of Economics and Political Science and Vivid Economics Ltd., London. We are very

grateful to the editor,three anonymous referees, Pauline Barrieu, and Trevor Maynard for com-

ments on previous drafts. We would like to acknowledge the financial support of Munich Re,

the United Kingdom’s Economic and Social Research Council and the Grantham Foundation

for the Protection of the Environment. The usual disclaimer applies.

213

214 The Journal of Risk and Insurance

2010; Ranger and Niehoerster, 2012). This ambiguity is increasedby the potential role

of climate change in altering the frequency, intensity, geographical incidence, and

other features of hurricanes.

There is by now a body of evidence to show that, faced with offering a contract under

ambiguity, insurers increase their premiums, limit coverage, or are unwilling to pro-

vide insurance at all. Much of the academic evidence is survey based: actuaries and

underwriters from insurance and reinsurance companies are asked to quote prices for

hypothetical contracts in which the probabilities of loss are alternatively known or

unknown (Hogarth and Kunreuther,1989, 1992; Kunreuther, Hogarth, and Maszaros,

1993; Kunreuther et al., 1995; Cabantous, 2007; Kunreuther and Michel-Kerjan, 2009;

Cabantous et al., 2011). Their responses reveal that prices for contracts under ambi-

guity exceed prices for contracts without ambiguity and with equivalent expected

losses, which is consistent with ambiguity aversion1and, thus, in line with a much

larger body of evidence on decision making, starting with Ellsberg’s classic thought

experiments on choices over ambiguous and unambiguous lotteries (Ellsberg, 1961).

In the industry,one can find guidance that insurers should increase their “prudential

margins” (i.e., capital holdings) under ambiguity (e.g., Barlow et al., 1993) and below

we explain how this leads to higher premiums.

Yet, despite the evidence, there is seemingly little theoretical work that can explain

or formally motivate these ambiguity loadings. In this article, we seek to fill this hole

by offering a formal analysis of the connection between, on the one hand, ambiguous

information about the performance of a book of insurance and, on the other hand, the

premium charged for a new contract. We do so via the capital held against the book:

our starting point is a well-known model of the price of insurance, according to which

the objective is to maximize expected profits subject to a survival constraint (thus in

the tradition of Stone, 1973), which is imposed by managerial or regulatory fiat out

of concern for ensuring solvency or avoiding a downgrading of credit. An example

of such a constraint, imposed by regulation, is the European Union’s new Solvency

II Directive (where it is called a Solvency Capital Requirement). Our twist is that the

capital held is sensitive to the range of estimates of the probability of ruin and to the

insurer’s attitude toward ambiguity in this sense.

Based on recent contributions to the theory of decision making under ambiguity, we

characterize circumstances in which one book of insurance is “more ambiguous” than

another, and establish general conditions under which more ambiguous books entail

higher capital holdings under our capital-setting rule. We then use the rule to de-

rive pricing formulae for ambiguous contracts in a way that isolates the additional

ambiguity load, distinct from the more familiar risk load. We examine the properties

of the ambiguity load under different assumptions about the insurer’s information:

it is shown to depend on the ambiguity of the contract being priced, as well as the

insurer’s ambiguity aversion. It also depends on the relationship between the am-

biguity of the new contract and the ambiguity of the preexisting book, and under

some circumstances it can interact with the coventional risk load. We hope that these

1We give formal definitions of ambiguity, ambiguity aversion, and related concepts later.