Academic journal article Journal of Risk and Insurance

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

Academic journal article Journal of Risk and Insurance

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

Article excerpt

ABSTRACT

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 American 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. …

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