Neil A. Doherty

Optimal Insurance in Incomplete Markets

This paper examines the theory of optimal insurance purchasing in the presence of uninsurable background risk. Existing theorems concerning the optimal level of insurance and the optimal form of an insurance contract are shown to hold only under restricted market and risk assumptions. In particular, conditions sufficient for the optimality of full coverage or sufficient for the optimality of deductible policies depend on the correlation between insurable and uninsurable risks. These results may provide a partial explanation why existing theory is often contradicted by observable behavior.

Rational Insurance Purchasing: Consideration of Contract Nonperformance

Much of the analysis of rational insurance purchasing and hedging strategies has been undertaken on the basis of the expected utility hypothesis. This decision framework has been used to reformulate the Bernoulli principle that a risk averter rationally would fully insure at a fair premium. Other applications include the analysis of insurance at unfair premiums (Mossin [1968 ] and Smith [1968]), the effects of state-dependent utility on the rational insurance purchase (Cook and Graham [1977]), the effects of risky background wealth (Doherty and Schlesinger [1983] and Mayers and Smith [1983]), and the design of the optimal insurance policy (Raviv [1979] ). Such analyses have been conducted on the assumption that insurance policies are free of insolvency (default) risk. This assumption stands in sharp contrast to the actuarial literature in which the insolvency risk, or probability of ruin, is a central focus of attention.

Insurance with Undiversifiable Risk: Contract Structure and Organizational Form of Insurance Firms

Previous explanations of the contract choice and organizational form of insurance firms do not explain, by themselves, the recent proliferation of mutuals and new contract designs. We first present risk-bearing arguments to address these phenomena. We present two forms of insurance. The first is a conventional transfer of risk whereas the second decomposes risk between idiosyncratic and nonidiosyncratic. We show that the latter form leads to more active trade in insurance markets with correlated exposures. Moreover, the decomposed form dominates the simple transfer. These results qualify and extend the work of Borch (1962) and Marshall (1974). Market responses to the recent “liability insurance crisis” are compatible with these predictions.

Adverse Selection in Insurance Markets

In this survey we present some of the more significant results in the literature on adverse selection in insurance markets. Sections 1 and 2 introduce the subject and section 3 discusses the monopoly model developed by Stiglitz (1977) for the case of single-period contracts and extended by many authors to the multi-period case.

Adverse Selection, Private Information, and Lowballing in Insurance Markets

Recent contributions to the adverse selection literature have focused on a multiperiod competitive environment. In this setting a clean distinction arises between the predictions of competing models. Cooper and Hayes extend the model of Rothschild and Stiglitz to multiple periods. The result is a self-selecting equilibrium characterized by price highballing, or systematically overcharging new business. Kunreuther and Pauly suggest that information asymmetries between competing insurance firms render such price-quantity policies infeasible. Their model results in a pooling equilibrium that is characterized by price lowballing, or systematically undercharging new business. Our evidence is consistent with the Kunreuther and Pauly model and reveals the persistence of adverse selection in the automobile insurance market. Copyright 1990 by the University of Chicago.

Adverse Selection, Commitment, and Renegotiation: Extension to and Evidence from Insurance Markets

With asymmetric information, full commitment to long-term contracts may permit markets to approach first-best allocations. However, commitment can be undermined by opportunistic behavior, notably renegotiation. We reexamine commitment in insurance markets. We present an alternative model (which extends Laffont and Tiroles procurement model to address uncertainty and competition), which involves semipooling in the first period followed by separation. This and competing models (e.g., single-period models and no-commitment models) have different predictions concerning temporal patterns of insurer profitability. A test using California data suggests that some automobile insurers use commitment to attract selective portfolios comprising disproportionate numbers of low risks.

The Measurement of Output and Economies of Scale in Property-Liability Insurance

The conflicting results from previous studies of economies of scale might arise partly from the choice of output measure. This paper discusses conceptual and econometric problems arising from the use of premium income as a proxy for output. This output measure is not independent of the firms pricing policy and its use implies potentially serious problems of simultaneous equation bias and errors in variables. It is argued that a delivery-based output measure is theoretically superior and will encounter less severe econometric problems. Cost functions estimated with the delivery-based measure do reveal significant scale economies in the Canadian property-liability insurance industry.

Moral Hazard, Basis Risk, and Gap Insurance

This article addresses the trade-off between moral hazard and basis risk. A decision maker, e.g., a primary insurer, is considered who can purchase an index hedge and a (re)insurance contract that covers the gap between actual losses and the index-linked payout, or part of this gap. The results suggest that combining insurance with an index hedge may extend the possibility set and by that means lead to efficiency gains. Naturally, the results depend heavily on the transaction costs associated with both instruments. In particular, the authors show that if the index product is without transaction costs, at least some index-linked coverage is always purchased, so long as there is positive correlation between the index and the actual losses. So under these circumstances, there is in any case a benefit from the availability of index products. Furthermore, it is shown that the index hedge would always be supplemented by a positive amount of gap insurance.

Optimal insurance without expected utility: The dual theory and the linearity of insurance contracts

Models of the insurance markets and institutions are routinely based on expected utility. Since EU is being challenged by an increasing number of decision models, we examine whether EU-based models are robust in their predictions. To do so, we rework some basic models of optimal insurance contracts and equilibrium using the “dual” theory to EU of Yaari. When there is a single, insurable source of risk, dual theory permits only corner solutions if the contract itself is linear. This contrasts sharply with EU. Nonlinearity, and thereby the possibility of interior solutions, is introduced in two ways. First, the contract itself is nonlinear, i.e., a deductible insurance policy. Or second, the decision maker is subject to some background risk such as uninsurable risky assets or default of the insurer. When decision problems are subject to nonlinearity, the predictions on optimal insurance are more similar to, though not identical with, those generated with EU.

Interest rates and insurance price cycles

Abstract Property liability insurance prices and profits appear to move in a six year cycle. The common explanation for the cycle amongst many industry analysts is that the insurance market is inherently unstable and that prices fail to converge on clearing levels. Our explanation is different. We identify spot equilibrium prices using the Capital Asset Pricing Model. But informational, regulatory and contractual lags preclude instantaneous adjustment. We therefore model the temporal movement of prices using a partial adjustment model in which actors form rational expectations. The actual movement of insurance prices does seem to track closely those estimated by the partial adjustment model. The cycle may be better viewed as a series of converging responses to changing spot prices.