Financial Bounds for Insurance Claims

• DOI: http://doi.org/10.1111/j.1539-6975.2012.01495.x
• Author: Carole Bernard,Steven Vanduffel
• Published date: 01 March 2014
• Date: 01 March 2014

© The Journal of Risk and Insurance, 2014, Vol. 81, No. 1, 27–56

DOI: 10.1111/j.1539-6975.2012.01495.x

27

FINANCIAL BOUNDS FOR INSURANCE CLAIMS

Carole Bernard

Steven Vanduffel

ABSTRACT

In this article, insurance claims are priced using an indifference pricing prin-

ciple. 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 finan-

cial 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no longer valid as a classical lower bound for insurance prices in general: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 be-

tween 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 ac-

tuarial 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 B ¨

uhlmann, 1980). We

refer to Young (2004) for a review of these three approaches.

Carole Bernard is in the Department of Statistics and Actuarial Science at the University of

Waterloo. Steven Vanduffel is in the Department of Economics and Political Sciences at Vrije

Universiteit Brussel (VUB). The authors can be contacted via e-mail: c3bernar@uwaterloo.ca

and steven.vanduffel@vub.ac.be. This article received the 2011 SCOR-EGRIE YoungEconomist

Best Paper Award at the EGRIE 2011 meeting in Vienna. Both authors gratefully acknowledge

the program “Brains Back to Brussels” that funded an extended research visit of C. Bernard

at VUB in Brussels during which this article was completed. S. Vanduffel acknowledges the

financial support of the BNP Paribas Fortis Chair in Banking. C. Bernard also acknowledges

support from the Natural Sciences and Engineering Research Council of Canada. The authors

would like to also thank two anonymous referees, and seminar participants in Ulm, Rennes,

Louvain-la-Neuve, Maastricht, Brussels (ULB), Rio, and Waterloo for interestingsuggestions.

28 THE JOURNAL OF RISK AND INSURANCE

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 in-

surance 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 in-

surance 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 refer-

ence 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 CT, it is possible to construct a financial

payoff that generates the same distribution as CTat 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 princi-

ple ignoring the financial market). In other words, premium principles have a “non-

negative 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.

Specifically, we assume throughout the article that there is a financial market that

can be used to perfectly hedge financial risk but that the insurance contract also

depends on additional sources of uncertainty that could not be hedged using financial

instruments. There is alreadyan important literature related to the pricing of insurance

contracts that have hedgeable parts and nonhedgeable parts in the presence of a

financial market. The idea that the premium can be invested in the financial market

first appeared in Kahane and Nye (1975). Brockett et al. (2009) use indifferencepricing

in the presence of a financial market for weather derivatives and expected-utility

maximizers. They show how the hedging part is important and how indifference

prices could significantly differ to actuarial prices obtained using the discounted

expectation. Pelsser (2010) and W ¨

uthrich et al. (2010) extensively discuss market

consistency,which is an important issue given the current regulation in the insurance

industry about mark-to-market valuation. Recent articles on participating policies,

equity-indexed annuities, and variable annuities propose to combine the financial and

actuarial approach by assessing the risk under the real-world measure and pricing

in the risk-neutral world. See, for example, chapters 12 and 14 of Dickson, Hardy,

and Waters(2009) for pricing equity-linked insurance with deterministic or stochastic

cash flow analysis and by using specific risk measures. Recently, Graf, Kling, and

Russ (2011) select a risk minimizing asset allocation (under the real-world measure)