Academic journal article Journal of Risk and Insurance

Ambiguity and Insurance: Capital Requirements and Premiums

Academic journal article Journal of Risk and Insurance

Ambiguity and Insurance: Capital Requirements and Premiums

Article excerpt

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., 2010; Ranger and Niehoerster, 2012). This ambiguity is increased by 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 provide 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 ambiguity exceed prices for contracts without ambiguity and with equivalent expected losses, which is consistent with ambiguity aversion (1) and, 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 derive 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. …

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