Academic journal article Journal of Risk and Insurance

Financial Bounds for Insurance Claims

Academic journal article Journal of Risk and Insurance

Financial Bounds for Insurance Claims

Article excerpt

ABSTRACT

In this article, insurance claims are priced using an indifference pricing principle. We first revisit the traditional economic framework and then extend it to incorporate a financial (sub)market as a tool to invest and to (partially) hedge. In this context, we derive lower bounds for claims' prices, and these bounds correspond to the market prices of some explicitly known financial payoffs. In particular, we show that the discounted expected value is no longer valid as a classical lower bound for insurance prices in general: it has to be corrected by a covariance term that reflects the interaction between the insurance claim and the financial market. Examples that deal with equity-linked insurance contracts illustrate the article.

INTRODUCTION

The valuation of insurance claims is at the core of actuarial science. The traditional actuarial premium principle is based on a quantity such as the expectation, the standard deviation, the variance, the quantile, or any other quantity derived from the claim distribution under the physical probability. A second approach consists of specifying a set of reasonable properties the premium principle should satisfy. Such approach is intimately connected with the axiomatic approach to risk measures (see Artzner et al., 1999). A third approach incorporates the preferences of the decision makers involved (i.e., the insurance buyer and insurance seller) in the determination of insurance prices. Such premia are then typically derived from economic indifference principles (using, for example, the expected-utility theory from von Neumann and Morgenstern, 1947; see also the zero-utility premium principle proposed by Buhlmann, 1980). We refer to Young (2004) for a review of these three approaches.

As Brockett et al. (2009) note, a "striking feature of the actuarial valuation principles is that they are formulated within a framework that generally ignores the financial market." Indeed, the different approaches proposed in the literature for pricing insurance claims usually assume that apart from the availability of a risk-free bond, there is no financial market and even if there is one it cannot be used to hedge insurance claims and to determine insurance premia. However, it is now clear that insurance claims should be priced by taking into account the financial market. First, life insurance contracts often include financial guarantees and index-linked features so that at least for these components the pricing of the contract should make reference to the financial market. Moreover, the decision makers involved in the pricing process do not only invest in risk-free bonds but use more diversified portfolios. In addition, when the insurance claim can be replicated using financial instruments, the price (premium) for it should be market consistent, effectively meaning that any good pricing rule in insurance should be such that it preserves market prices when applied to financial payoffs. Finally, Bernard, Boyle, and Vanduffel (2011) recently show that given the distribution of an insurance payoff [C.sub.T], it is possible to construct a financial payoff that generates the same distribution as [C.sub.T] at minimal (market) cost, which further suggests that there should be a link between an insurance pricing principle and pricing in financial markets.

Traditionally, the discounted expectation of the future insurance claim is a lower bound for the insurance premium (calculated through an actuarial valuation principle ignoring the financial market). In other words, premium principles have a "nonnegative loading." It is argued that a premium principle that does not satisfy this requirement can lead to the insurer's ruin (assuming the insurer faces a series of independent claims so that the law of large numbers holds). Our research shows that in presence of a financial market such no-undercut principle does not necessarily hold.

This article is related to the literature on pricing of claims in incomplete markets. …

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