Academic journal article Journal of Risk and Insurance

Risk Transformations, Deductibles, and Policy Limits

Academic journal article Journal of Risk and Insurance

Risk Transformations, Deductibles, and Policy Limits

Article excerpt

ABSTRACT

Rothschild and Stiglitz (1970, 1971) pioneered the study of how an increase in risk affects the demand for a risky asset. Gollier (1995) first identified a necessary and sufficient condition for unambiguous comparative statics for demand under transformations of the asset's probability distribution function. In this article, the authors examine the necessary and sufficient conditions for unambiguous comparative statics for insurance demand for coverage deductibles and policy limits, when the loss variable undergoes certain classes of transformations. The authors show that, if a transformation is both mean-preserving above (below) the optimal deductible (policy limit), and probability-preserving above (below) the optimal deductible (policy limit), then the necessary and sufficient condition is given by first-order stochastic dominance below (above) the optimal point.

INTRODUCTION

Rothschild and Stiglitz (1970, 1971) pioneered the study of how an increase in risk affects a risk-averse decision maker's demand for a risky asset. Since then, several researchers (Dreze and Modigliani, 1972; Diamond and Stiglitz, 1974; Dionne and Eeckhoudt, 1987; and Briys, Dionne, and Eeckhoudt, 1989) have found conditions on the individual's utility function that can generate unambiguous comparative statics with a mean-preserving transformation (MPT) of the asset's probability distribution function. Others (Eeckhoudt and Hansen, 1980, 1983; Meyer and Ormiston, 1983, 1985; Black and Bulkley, 1989; and Dionne and Gollier, 1992) have found constraints on the increase in risk that can provide clear prediction.

Gollier (1995) first identified a necessary and sufficient condition for unambiguous comparative statics for demand under transformations of an asset's distribution. This least-constraining condition is called "greater central riskiness" in the case of a linear payoff function.

The context of the present work is most closely related to that of Eeckhoudt, Gollier, and Schlesinger (1991), who studied increases in risk for deductible insurance and found separate sufficient conditions for both increases and decreases in insurance demand. These authors followed upon earlier work by Schlesinger (1981), who provided an analysis of the demand for deductible insurance. More recently, Meyer and Ormiston (1999) have provided further analysis of deductible insurance, with particular attention to second-order condition issues.

In the present research, models of both insurance deductibles and policy limits are studied. Like Gollier (1995), the authors develop necessary and sufficient conditions for unambiguous comparative statics in both cases.

For insurance deductibles, it is found that if the transformation of the loss distribution preserves the mean, as well as the total probability, above the optimal deductible, then the necessary and sufficient condition is given by first-order stochastic dominance below this point. Alternatively, for policy limits, the authors find that if the transformation of the loss distribution preserves the mean, as well as the total probability, below the optimal policy limit, then the necessary and sufficient condition is first-order stochastic dominance above this point.

THE CASE OF DEDUCTIBLES

Insurance deductibles are used frequently in primary property, liability, and health insurance, as well as excess-of-loss and stop-loss reinsurance. The purpose of the deductible mechanism is generally to reduce problems with moral hazard by requiring that the insured take some responsibility for loss payments. Deductibles are also used to reduce the insurer's expense loading by eliminating the claim settlement expenses associated with smaller claims, for which settlement expenses are disproportionately large.

Assume that a risk-averse insured with initial wealth W and (twice-differentiable, increasing, and concave downward) utility function u(*) faces a random loss X [member of] [0, L] , where X ~ F(*). …

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