An Option-Theoretic Model of Catastrophes Applied to Mortgage Insurance

Abstract

INTRODUCTION The ten most damaging catastrophes world-wide from 1984 to 1994 amounted to $49.5 billion in 1992 prices; the U.S. share was $30.2 billion (The Economist, 1994). Much of this loss is attributed to the destruction of physical structures. In most cases, water damage from hurricanes and floods are not covered by homeowners insurance. For earthquakes, there is typically little, if any, coverage for losses. In both cases, separate government insurance is possible, but often not required or used by the building owner. This leaves mortgage insurance companies exposed to losses resulting from a catastrophe, whenever those losses result in mortgage default. In previous work, Kau, Keenan, and Muller (1993) use an option model to address the issue of why some private mortgage insurance companies have been experiencing excessive losses. The reason shown was the failure by insurance firms to appropriately adjust mortgage insurance values with regard to the economic environment or the particular terms of the mortgage contract. Using the same option-theoretic framework, this article incorporates the risk of catastrophes and measures its impact on mortgage insurance values. In addition, this article uses recently developed techniques of options analysis to measure the probability that an insurance claim arises from a catastrophic event, as well as describing the likely extent of liability from that claim. By incorporating these same techniques, the article also addresses such other issues as the impact and value of government disaster insurance. The goal of this article, then, is to introduce catastrophic events into the pricing of mortgage insurance. These events can be purely financial, as in a sudden severe drop in building prices due to disastrous economic news for the local economy, or the catastrophic events may simply be physical destruction driven by nature. In either case, this article aims to clarify the role that catastrophe plays in affecting the liability to insurers and hence the cost of insurance. The next section presents the option model, including the theoretical changes required to achieve our goals of incorporating catastrophic events and the resulting probability of losses. Then, the effectiveness of the model is demonstrated by measuring the impact of changes in the parameters governing the chances of a catastrophe and the amount of damage that occurs. Also considered is the impact on mortgage insurance of government disaster relief designed to moderate the consequences of catastrophic damage. The model is able to ascertain the value of such implicit or actual government insurance provided against catastrophic events. Finally, a summary and conclusion emphasizes the general applicability of this new use of an option model to other types of financial and natural traumas. THE CATASTROPHIC VALUATION MODEL Measuring the impact of a catastrophic event first requires that the event be placed in an economic environment. Furthermore, because the focus here is on mortgage insurance, the nature of the mortgage contract and its influence on insurance must also be captured. Catastrophes must thus be introduced in a way consistent with the arbitrage pricing conditions that characterize the option-pricing approach to valuing mortgages and mortgage insurance. The Economic Environment Economic uncertainty is incorporated in the model by assuming that the entire term structure of interest rates is generated from the stochastic process describing r(t), the spot rate. This is to take the form (Cox, Ingersoll, and Ross, 1985), dr = [Gamma]([Theta] - r)dt + [[Sigma].sub.r] [square root of r] d[z.sub.r]. (1) Interest rates in this process revert toward a steady-state value [Theta] at rate [Gamma]. The rates are disturbed by stochastic events as dictated by the Wiener process [z.sub.r]. The parameter [[Sigma].sub.r] is the volatility of these interest rate disturbances. …