A system approach to management of catastrophic risks
Introduction
The increasing vulnerability of modern society to various “failures”, accidents, mismanagement, natural and human-made disasters, is an important characteristic of current socio-economic, technological and environmental global changes. Searching for economic efficiency without paying attention to possible risks often leads to “clustering” of individual property, production processes, installations, buildings and other values. George Dantzig has compared modern society to a busy highway [3], where a disruption in one place may cause wide spread traffic jams. Such events as Hurricane Andrew, the Kobe earthquake, the explosion of chemical tanks in Bhopal, the Chernobyl catastrophe and the ecological disaster of the Rhine after an accidental discharge of toxic chemicals at Basel caused large societal losses. Economic losses from Hurricane Andrew and the Northridge earthquake exceeded $45 billion. The Kobe earthquake (Japan) resulted in around $100 billion in property damage. Global climate and socio-economic changes may dramatically increase the severity and frequency of natural hazards in many regions. The key problem is to find ways to improve resilience and to protect society effectively against the increasing risks.
What role can the insurance industry play in encouraging prevention, preparedness and response measures, and providing financial protection against catastrophic risks without exposing itself to the danger of insolvency? Catastrophes represent new challenges for the insurance theory [1], [19]. The most significant of them is the ability to cope with dependencies among catastrophic claims. There exist also dependencies among catastrophic events, for example, weather-related natural catastrophes due to the persistence in climate [14]. We must anticipate that large and more frequent future losses would overwhelm the insurance industry as it currently exists [2], [12]. The challenge is to evaluate the role of insurance [13] coupled with other policy instruments, such as regulations, standards and new financial instruments as complements to and substitutes for reinsurance. This task requires a system approach.
From a formal point of view the control of insolvency is equivalent to the prevention of certain multidimensional jumping processes to reach critical thresholds, which is a rather general problem in risk management. To deal with dependent catastrophic losses, a geographically explicit dynamic model was developed in [5], [6], [7]. The model incorporates information on property values and their vulnerability, generators of catastrophes, risk reduction and risk spreading decisions, and stochastic optimization procedures. The aim of this paper is to discuss specific components of this model and related decision making problems.
Section 2 illustrates the importance of stochastic dynamic models and the discontinuous nature of insurance processes. 3 Catastrophe modeling, 4 Decision variables show the nonsmooth and implicit character of decision processes. Possible goals and risk functions are discussed in Section 5, which emphasizes their nonlinearity with respect to probabilities and the nested structure of the resulting stochastic optimization problems. Section 6 discusses a decision making problem involving catastrophe bonds and reinsurance contracts. Section 7 outlines the proposed adaptive Monte Carlo optimization techniques, which can also be viewed as an adaptive scenario analysis. It is pointed out that nonsmooth random goal functions may lead to inconsistencies of deterministic sample mean approximations. Section 8 illustrates numerical experiments. Concluding remarks are given in Section 9.
Section snippets
Insurability of standard risks
The concept of risk must play the same role in determining economic activities as profits and costs. This notion emphasizes the variability of outcomes, the possibility of gains and, at the same time, the chances of losses. Such “hit-or-miss” situations often lead to nonsmooth and even discontinuous decision models [9], which challenge traditional approaches. Fig. 1 shows a typical trajectory of the risk reserves of an insurance company [4,11].
Claims arrive at random time moments τ1,τ2,…, of
Catastrophe modeling
To deal with catastrophic risks from natural, technological and environmental hazards one should characterize patterns of possible disasters, their geographical location, and timing. One should also design a map of regional properties, characteristics of structures, available and implemented mitigation measures, spread of current and possible new coverage, availability of catastrophe securities, etc.
Advances in computers and mathematical modeling then make it possible to simulate a variety of
Decision variables
In the case of frequent-low consequence risks, the law of large numbers provides a simple [5] “more-risks-are-better” portfolio selection strategy: if the number of independent risks in the portfolio is larger, then the variance of aggregate claims is lower and lower premiums can be chosen. This increases the demand for insurance, the coverage of losses, and, hence, the profits of insurers.
In the case of catastrophic risks the law of large numbers does not operate and the simple
Goal and risk functions
A sequence of random catastrophes affects different locations i=1,…,N and generates dependent catastrophe losses Li(t) at different time intervals t⩾0. Without insurance and risk reduction decisions, location i faces losses Li(t). These losses are reduced or compensated after implementing the appropriate decisions. If we denote the vector of decisions by x, then Li(t) becomes a function Li(x,t) of x. As we can see from Section 4, this function may have a rather complex nonsmooth structure
A catastrophe bond versus insurance
Let us consider an important risk management situation, that illustrates the decision making problems discussed above. Catastrophe securities and bonds have been introduced to assist the insurance industry in spreading risks worldwide. The following shows that catastrophe bonds may be more attractive than similar reinsurance contracts.
Assume that a “client” (insurer, government, firm, etc.) decides to protect a “layer,” defined by decision variables (y1,y2), of possible catastrophe losses L,
Adaptive Monte Carlo optimization
The search for Pareto efficient decisions is achieved through the maximization of weighted sums of different goal functions, such as (2). A principal challenge is that F(x) in (2) is an analytically intractable function. A Monte Carlo simulation of a catastrophe ω produces only random outcomes g(x,ω) for a given set of decision variables x. Random g(x,ω) estimates Eg(x,ω). Unfortunately, f(x,g(x,ω),x,ω) cannot be used as an estimate of E(x)=Ef(x,Eg(x,ω),ω). Papers [5], [6], [7] deal with the
Numerical experiments
Fig. 4 shows a “landscape” of damaged property values (houses, lands, factories, etc.) for different locations of a region affected by an earthquake (dark part of the landscape).
In general, a catastrophic event, for instance, an earthquake, hurricane, or flood, is simulated as a random field. The distribution of losses at given locations depends on the nature of the catastrophe, the characteristics of the soil, the vulnerability of structures, etc. Trajectories of this field in a particular
Concluding remarks
Numerical experiments with different problems show the feasibility of the approaches outlined. The design of optimal risk management decisions can be based on a simulated “history” of catastrophes. Predicting catastrophes from limited historical data is often difficult or simply impossible. The optimization in the presence of uncertainty is, in a sense, a more robust task than such as the prediction: it is much easier to evaluate which one out of the two parcels is heavier than to measure their
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Visiting from EC-JRC, I-21020, Ispra, Italy.