A Markov Model for the Pricing of Catastrophe Insurance Futures and Spreads

This article presents a valuation theory of futures contracts and derivatives on such contracts when the underlying delivery value follows a stochastic process containing jumps of random claim sizes at random time points of catastrophe occurrence. Applications of the theory are made on insurance futures and options, new instruments for risk management launched by the Chicago Board of Trade. Several closed pricing formulas are derived based on a partial, competitive equilibrium assumption both for futures contracts and for futures derivatives, such as caps, call options, and spreads, assuming constant relative risk aversion for the representative agent. INTRODUCTION Presented here is a valuation model for futures contracts and derivatives on such contracts when the underlying delivery value is an insurance index, which follows a particular stochastic process in continuous time and with a discrete state space. The Chicago Board of Trade (CBOT) implemented a mechanism for transferring catastrophe risk from the insurance industry to private capital markets. In December 1992, CBOT launched CAT futures and CAT options creating a variety of new tools for risk management, and in September 1995, it introduced a new class of catastrophe insurance options based on new insurance indices provided by Property Claim Services, a division of American Insurance Services Group, Inc. The latter options are called PCSTM Catastrophe Insurance Options, or PCS options. PCS calculates and publishes daily the underlying index, which represents the development of specified catastrophe damages. One motivation for securitization of insurance markets may stem from the inability of traditional reinsurance markets to attract enough risk capital. Never before have natural catastrophes worldwide caused such high losses as in the 1990s. The AmeriKnut K. Aase is with the Norwegian School of Economics and Business Administration, Bergen, Norway. The author gratefully acknowledges a grant from The Norwegian School of Economics and Business Administration. Research was carried out while the author was on sabbatical leave at the Graduate School of Business, Stanford University, Stanford, Calif. 26 THE JOURNAL OF RISK AND INSURANCE can property-liability insurance industry has been hardest hit because of an accumulation of the most expensive catastrophe events in the USA. Even more crucial than the record highs may be the fact that in the long-term comparison, not only the frequency of natural catastrophes, but also the average loss per event has increased for the insurance industry. This may partly be attributed to population growth in areas such as California and the subsequent concentration of value in disaster-prone regions. The annual floods in the Central Valley of California now cause large material damage each time, whereas the floods of ten years ago, for example, were less noticed, since considerably fewer people lived in the valley then. Also, man-made constructions, in particular roads, seem to aggravate the annual problems. However, the largest U.S. catastrophe risks include earthquakes in California and hurricanes and tornadoes in Florida and the southern states.1 Based on such events, computer simulation models have been suggesting potential losses of between $50 and $100 billion, depending on the scenario. The stated capital and surplus of the entire U.S. property-liability insurance industry amounts to $200 billion, currently matching these potential losses as well as other losses, such as the consequences of the aggravated American product liability. About $20 billion of this belongs to the U.S. reinsurers. As a result, the traditional insurance system may face a gap in the area of natural catastrophe risks. In contrast, the capitalization of the international financial markets seems impressive, where wealth invested in the U.S. alone amounts to about $19 trillion, and its daily fluctuation—about 70 basis points or $133 billion on average—exceeds the maximum possible insurance loss that might arise from an earthquake catastrophe. Against this background, the search for new capacity has led to the prospect of not only trading insurance risk within the traditional insurance and reinsurance markets, but also transferring some of the risk to the more liquid financial markets. Investors in financial markets have traditionally been looking for investments uncorrelated with the market index in order to hedge and diversify, and a catastrophe index may be just such an investment. This article presents a stochastic model for PCS options as well as CAT products, which allows us to derive relative prices of catastrophe futures and derivatives in an economic model of a partial competitive equilibrium in an insurance futures market. The author does not describe the detailed institutional basics of PCS options or CAT products but only comments on certain issues that may be of importance for the development and understanding of the model.2 PCS loss indices represent the absolute USD amount of those catastrophe damages that are estimated and published by PCS for a defined region and a defined time period, whereas CAT products refer to an underlying loss ratio index. In the PCS definition, catastrophe damages are losses of more than $5 million of insured property damages that simultaneously affect a significant number of insurance companies and policyholders. The definition of CAT catastrophes depends on specific loss causes, where a minimum loss volume is not necessary. 1 By far the most expensive insurance losses in the past 25 years were Hurricane Andrew ($16.5 billion) and the Northridge earthquake in Southern California ($12 billion); see Swiss Re, Sigma 2 (1996), Appendix 2. 2 Many institutional details and facts can be found in the bibliography at the end of this article. A MARKOV MODEL FOR THE PRICING OF CATASTROPHE INSURANCE FUTURES AND SPREADS 27 The claims in an insurance market can never really exceed the value of the properties covered by the insurance policies. Therefore, the bound for the insurance index used here is related to the total value of all the physical assets in the market. This article models the insurance index as a particular time continuous and discrete state Markov process. The latter may be considered somewhat undesirable when modeling wealth in units of a currency but may here be partly warranted by the fact that catastrophe damages are in general losses of more than a certain minimum amount.3 In this framework, catastrophic risk is priced using a representative agent competitive equilibrium with constant relative risk aversion in the market. In a previous model, see Aase [5], this was not possible because of the unboundedness of the proposed stochastic process, which was there taken to be a compound Poisson process. Only the case of a constant absolute risk aversion was then analyzed in detail. In general, a constant relative risk aversion is considered more plausible than a constant absolute risk aversion, and it is clearly desirable to have an analysis also for the former situation, which the author now attempts to provide. A continuous time, discrete Markov model might not look quite fashionable today, but in the present application it proves to be a useful framework, primarily since explicit expressions for the transition probabilities can be obtained. This article models a new, unfinished but innovative type of market structure. In the academic literature, there are some articles on catastrophe instruments by now. Cummins and Geman (1994, 1995) use arbitrage principles; Chang, Chang, and Yu (1996) use a Feynman-Kac approach to computing prices; and in the Geneva Papers, Kielholz and Durrer (1997) and Smith, Canelo, and Di Dio (1997) discuss various aspects of securitization of reinsurance. More is likely to come. This article is organized as follows. The model section presents the economic primitives and explains the catastrophe loss index. Then the pricing theory section derives a futures pricing formula and discusses risk premiums. Next, the section on the theory of catastrophe futures derivatives compares pure futures instruments to conventional ones and presents pricing formulas for futures instruments, including a futures cap, a futures call option, and a futures bull spread. The final section includes a summary and an appendix that explains how the transition probabilities are found for our stochastic model of the loss index in the model section. THE MODEL Economists have developed during the last 30 years a canonical model to deal with optimal insurance/risk-sharing and risk prevention. The author aims in this section to review the assumptions and basic results of this simple model. In a later section, the pricing principle of this theory is applied to a model of a catastrophe futures and derivatives market. This section also explains the Markov model for the loss index employed here. 3 A minimum dollar amount exists at least for PCS products. 28 THE JOURNAL OF RISK AND INSURANCE The Economic Primitives An exchange economy—an insurance economy—is considered here in a competitive equilibrium where pricing is formed via the marginal utility of a representative agent. A reinsurance syndicate might consist of I members, where each of the agents is characterized by a utility function U i and net reserves { } ( ) , 0 i t t T ≤ ≤ X , ( ) ( ) 0 i i X n = . The latter are random processes defined on the same filtered probability space ( ) , , P Ω I , satisfying the usual conditions, where { } t I is the information filtration generated by all the processes ( ) ( ) 0 , 1, 2, ..., , i s s t i I < < = X and where ,T Τ I = I is the time horizon. Each w ∈ Ω denotes a complete description of the exogenous uncertain environment from time 0 to time , T I is the sigma-field of distinguishable events at time T, and P is the common probability belief held by the agents in the economy. The functions U i are assumed to be additively separable and of the von NeumannMorgenstern form: ( ) ( ) { } ( ) ( ) 0 , , 1, 2, ..., , T i i i i t U u t dt i I Χ = Ε Χ = ∫ where E is the expectation operator and the i u functions are sufficiently smooth for a contingent capital market equilibrium of the Arrow-Debreu type to exist. An allocation ( ) (1) (2) ( ) , , ..., I Y Y Y is said to be feasible if ( ) ( ) 1 1 I I i i i t Y X X = − ≤ = ∑ ∑ . By an equilibrium we mean a collection ( ) (1) (2) ( ) ; , , ..., I Y Y Y p consisting of a price functional ( ) p i and a feasible allocation ( ) (1) (2) ( ) , , ..., I Y Y Y , such that for each ( ) , i i Y solves agent ' i s optimization problem ( ) ( ) ( ) ( ) max i i i Y L U Y ∈ subject to the budget constraint ( ) ( ) ( ) ( ) i i Y p p ≤ Χ for all 1, 2, ..., . i I = Given such an equilibrium, a utility function representing the market is a function U of the form