Academic journal article Journal of Risk and Insurance

Intertemporal Insurance

Academic journal article Journal of Risk and Insurance

Intertemporal Insurance

Article excerpt

INTRODUCTION

This article develops a framework that allows insurance contracts to be priced using the techniques of intertemporal finance. From Malinvaud (1972, 1973) to Cass, Chichilnisky, and Wu (1996), a significant part of the literature on insurance has been preoccupied with explaining how insurance markets manage to function despite the individualized nature of insurance contracts: if n individuals each face the prospect of having an accident or not, the state space is of size [2.sup.n] which seems to imply an impossibly large number of contingent contracts. As Malinvaud demonstrated, this apparent paradox can be resolved through an appeal to the law of large numbers. We suggest another approach to explaining why insurance markets work: introducing the possibility that accidents do not happen all at once, as in a static model, but gradually as part of a process that unfolds over time. This article develops the discrete-time version of the theory.

In our model, insurable events are represented as a marked point process, a stochastic process in which there is never more than one accident at any given date. In this setting, the revelation over time of the true state of nature can be represented by a very simple information structure: an event tree with the same number of branches emerging from each node, one branch for each type of "accident" plus a branch corresponding to the possibility of no accident. As a consequence, market completeness in the dynamic sense of Kreps (1982) requires a number of contracts proportional to n, the number of consumers, rather than [2.sup.n].

Establishing a framework in which dynamic spanning applies to insurance markets is the heart of this article; the rest simply applies standard finance in this specialized setting. As we shall see, finance provides some fresh insights into the nature of insurance. For example, even in a world of risk-averse consumers and investors, insurance contracts can be viewed as "actuarially fair" under the right probability measure: we show that the price of an insurance contract is the discounted expected value of future payments for the insured events where expectation is taken relative to the risk-neutral probability measure introduced by Harrison and Kreps (1979) in their synthesis of financial asset pricing theory. Insurance contracts provide a particularly natural interpretation of the Harrison-Kreps result: the risk-neutral probability measure incorporates society's aversion to risk and so builds into every "actuarially fair" insurance contract a load which compensates insurers for the risk they bear.

We begin with a formal description of a discrete-time marked point process and its application to insurance markets. Using a marked point process as the basic source of uncertainty, we define an intertemporal exchange economy on the event tree generated by the accident process and characterize the competitive equilibrium involving trade in Arrow-Debreu-Radner (ADR) date-event contingent commodities on this tree. The corresponding contingent prices allow us to price insurance contracts, which are typically not standard ADR contracts, as redundant securities. With this standard competitive equilibrium in place, we then "remove" the ADR contingent commodities, leaving only the insurance contracts to deal with risk. As we then show, insurance contracts alone yield dynamically complete markets in the sense of Kreps (1982), provided there is one insurance contract for each type of accident. We then illustrate the theory with a simple example yielding a natural interpretation of the Harrison-Kreps risk-neutral measure and a dramatic demonstration how trading a few insurance contracts can dynamically complete markets even though the state space is extremely large. Although the focus is on providing a bridge to the traditional insurance literature, our ultimate intent is to use intertemporal finance to develop a theory of how to insure events such as earthquakes or hurricanes for which the traditional theory seems less relevant. …

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