Artif Life Robotics (1999) 3:15 18 �9 ISAROB 1999

John L. Casti

Risk, simulation, and the insurance industry

Received: January 19, 1998 / Accepted: February 19, 1998

Statistics, laws, and nature

In late August 1992, Hurr icane Andrew roared through the east coast of the Uni ted States, causing nearly twenty billion dollars worth of damage. For insurance companies faced with this gargantuan bill, A n d r e w was but the tip of the iceberg in what seems to be an ever-increasing exposure to greater risks from all types of cataclysms - floods, volcanic eruptions, ear thquakes, t idal waves, hurricanes, and just about any other major force of nature. So it is natural to wonder whether the newly emerging science of complexi ty can be of any help in enabl ing us to unders tand more clearly the l ikel ihood of these kinds of disasters.

It is certainly no secret to insurance underwriters that the distil led essence of risk assessment and management lies in identifying the relat ive frequencies of different levels of disaster. Basically, the guiding principle of risk manage- ment is if you can plan for it, you can adjust the rates for your policies to accommodate it. So the hear t of the risk question comes down to the p rob lem of prediction.

The approaches scientists use for predict ing the forces of nature are of two quite different kinds. Let me term them the phenomenological and the foundational. Phenomeno- logical methods are based upon identifying statistical regu- larities in natural processes. Fo r example, when you took at the climatic pat terns depicted in the "Drought clock" shown in Fig. 1, it is hard to believe that we do not live in interest- ing times. The hot/cold and wet/dry periods clearly seem to repeat themselves in a very cyclical manner - at least when your unit of t ime is centuries ra ther than weeks or months.

J.L. Casti ( ~ ) Institute for Econometrics, OR & System Theory, Technical University of Vienna, A-1040 Vienna, Austria

Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501. USA Tel. +1-505-984-8800; Fax +81-505-982-0565

This work was presented, in part, at the Third International Sympo- sium on Artificial Life and Robotics, Oita, Japan, January 19-21, 1998

This il lustrates the statistical way of forecasting climatic change: gather lots of historical data and look for regulari- ties in the occurrence of dramat ic events.

Founda t iona l methods, on the other hand, focus upon the basic physical processes giving rise to things like earth- quakes, floods and volcanic eruptions. Using the laws of physics, chemistry, and geology, we try to build mathemat i - cal and computer models of, say, the a tmosphere that we can use to predict what nature will be doing next. The daily weather forecasts we see on television and in the newspa- pers are genera ted in exactly this manner .

Both the phenomenological and the foundat ional ap- proaches to predict ion have been dramatical ly influenced over the past few years by developments in what has come to be te rmed "the science of complexity." Put compactly, complexity theorists have discovered that everyday com- mon sense just does not always work when it comes to predict ing and/or explaining the behavior of many types of systems, and just about every natural process, ranging from the processes of weather format ion to the human hear tbeat , fall into this category, as do a lot of social and behaviora l systems like speculative markets and the outbreak of war- fare. So here I will give a brief glimpse of what these devel- opments have to say about issues affecting risk assessment.

Randomness and predictability

Suppose we are given a sequence of observat ions x0, &, x2 . . . . . Xr of some system taken at t imes t - 0, 1, 2 . . . . . T. We might think of these numbers as measurements of, say, the water levels in a reservoir or the daily closing prices of Exxon stock. If we are in the predict ion business, certainly one of the first things we would like to know is whether or not these observat ions are complete ly ran- dom. If so, then there is no pat tern to be found and no way to predict the next observat ions from the past; if not, then we at least have a fighting chance of ferret ing out a rule that will help us to do be t te r than just guess the next observation, x > ~.

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One of the principal lessons taught by the modern theory of chaotic processes is that a sequence of numbers can appear totally random, yet may in fact be the result of following a fixed, comple te ly determinist ic rule. For ex- ample, if the known numbers happened to have been gener- a ted by the simple "logistic rule," the next number equals four t imes the preceding number minus four t imes the square of the preceding number , or for the mathemat ical ly inclined, x,+l = 4x,(l - x,), t = 0, 1 . . . . . T - 1. In this case, none of the classical statistical methods would be able to discern the fact that the set of numbers arose from a com- pletely determinist ic rule; they would all render the verdict that the sequence is pure ly random. However , methods of modern nonl inear dynamics say differently.

First, we plot the number x, i against x,. If the sequence is truly random, then x, is complete ly independent of any of its predecessors. Consequently, we would expect to obtain a scatter diagram looking something like the one in Fig. 2a.

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Fig. 1 The drought clock

However , when we make such a plot for the set of numbers genera ted by the logistic rule, we obtain what is called the logistic parabola shown in Fig. 2b. This is a one-dimensional curve in a two-dimensional space, suggesting that there is some very definite relat ionship linking any two adjacent numbers in the sequence; the numbers are not independent , and hence not random. This graphical p rocedure is a kind of poor man 's version of a much more sophis t icated test called the correlation dimension, which provides us with a power- ful analytical tool for distinguishing the random from what we might term the "pseudorandom."

To see how this general idea works in a specific situation, let us turn to a closely related procedure deve loped origi- nally by the hydrologist H.E. Hurst to study the movement of water reservoir levels for flood control.

N a t u r e ' s m e m o r y

Consider the water reservoir depicted in Fig. 3, and suppose the number x, above represents the inflow to the reservoir in year t. Deno te the s tandard deviat ion of the inflow over this per iod by ST, and let Rr represent the range of inflows, i.e., RT is simply the difference between the largest and smallest numbers in the sequence. Hurst discovered a r emarkab le fact about the dimensionless rat io RflST. '\

Upon calculating the R/S ratio for a very large n~mber of quite natural phenomena , ranging from flood levels \ on the Nile to t rends in global t empera ture variat ion, Hurs t found that they all seemed to obey the empirical re la t ion RflSr : T n, where H is a number now called the Hurst exponent. Fur thermore , it can be shown that when the e lements of the original da ta set are independent from one t ime per iod to the next, the value of H tends to 1/2 when the number of da ta points T is great enough. However , the surprising con- clusion of Hurs t ' s efforts is that H differs great ly from 1/2 for most natural processes. So if you believe the data, then it seems that processes as diverse as sunspot fluctuations, r iver discharges, and rainfall levels all have some kind of long- term "memory effect." More specifically, since most of these natural processes have values of H greater than 1/2, fluctuations above the mean tend to be followed by fluctuations that are also larger than average. This fact is of

Fig. 2 a Random scatter; b the Logistic parabola 1.0

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Fig. 3 A water reservoir with inflow x, and range R~ Fig. 4 A global view of the insurance industry

great interest and value in explaining why big hurricanes, big floods, and big stock market crashes tend to occur in clusters.

Current work in complex, adaptive systems has turned up many other tools that are of use in predicting dramatic events. Let me mention just two that relate more to the foundational way of prediction than to the phenomenologi- cal: the theory of catastrophes, which enables us to charac- terize mathematically shifts of stability of exactly the sort that cause earthquakes, and the theory of self-similar phe- nomena, which leads to fractal descriptions of the scale of everything from forest fires to epidemics. Both of these ideas are explored formally I and more informally 2'3 else- where. In conclusion, these examples show that complex system theory is destined to play an important role in enabling analysts to get a better handle on the problem of managing risk from natural disasters. Now let me look briefly at some of the work being done in multiagent simulation 4 to come to terms with the peculiarities of these systems.

"insurance world"

We are currently in the process of constructing a computer model of the insurance industry, using the theory of com- plex, adaptive systems to understand how the interactions among insurers, customers, and the environment lead to the emergence of prices, capacity, and structuring in the industry. This study will begin with a simplified model of the insurance industry, focusing on property catastrophe reinsurance. This model will contain all the generic ele- ments of insurance needed to later extend the approach to more specific questions and types of insurance.

The global insurance industry

As a crude first-cut, the insurance industry can be regarded as an interplay among three components: firms, which offer

insurance, clients, who buy it, and events, which determine the outcomes of the "bets" that have been placed between the insurers and their clients. The overall situation is de- picted in Fig. 4, which is a very high-level view of the insur- ance industry, but it clearly points out the fact that to understand the dynamics of the industry, it is necessary to take into account the interactions among all three compo- nents of the system. Just as with the celebrated three-body problem of celestial mechanics, the linkages among the in- teracting bodies cannot be broken without losing the es- sence of what makes the problem a "problem." So it is with the insurance industry.

By taking a lower-level, more detailed look at each of these three interacting components, we obtain different views of the industry. For example, thinking of "insurers" as being catastrophe reinsurance companies, "clients" as pri- mary insurers, and "events" as things like hurricanes, gov- ernment regulations, the capital markets, and geographic distribution of risks, we obtain a picture of the world prop- erty catastrophe reinsurance industry. However one slices these three components, what is manifestly evident is that each of the many insurance industries constitutes what in modern parlance we call a complex adaptive system.

Such systems generically take the form of a large number of interacting agents, each agent taking decisions on the basis of limited information that comes its way. Moreover, the agents have the ability to learn from their past experi- ences and thus to modify the rules that they employ as the decision process unfolds. As a consequence of these myriad interactions among the agents, the overall system displays global behavioral patterns that usually cannot be under- stood simply by knowing about the agents in isolation. Rather, these patterns "emerge" out of the interactions among the agents. Examples of such systems include price movements in speculative markets, road-traffic patterns, and the appearance and disappearance of species in an ecological ecosystem. We can comfortably add insurance industries to this list.

Since it has proved to be fiendishly difficult to formulate most complex, adaptive systems in conventional math- ematical terms, the method of choice for the analysis of these systems is computer simulation. We represent each

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agent as a piece of computer code, and let these chunks of code interact inside the machine to create new pieces of code by the rules of interaction. In this fashion, we create a virtual counterpart of the real-world system, its agents, and environment. This proposal suggests the creation of just such a world for the study of the global property casualty reinsurance industry.

Risk and insurance

In the "would-be world" proposed here, the agents consist of primary casualty insurers and the reinsurers. The events are then natural hazards, such as hurricanes and earth- quakes, as well as various external factors such as govern- ment regulators and the global capital markets.

The construction of Insurance World has both a purely scientific, as well as an applied, purpose. Quite indepen- dently of the specific context of the insurance industry, there are basic questions in the theory of risk and organiza- tions that Insurance World should provide a tool to illumi- nate. Some of these general scientific questions are given below.

- O p t i m a l u n c e r t a i n t y . While insurers and reinsurers talk about getting a better handle on uncertainty, so as to more accurately assess their risk and to price their product more profitably, it is self-evident that certain knowledge of natural hazards (a single-point atomic dis- tribution) would spell the end of the insurance industry. On the other hand, complete ignorance of hazards (a uniform distribution) is equally bad for the industry. This suggests that there is some optimal level of uncertainty at which the insurers - but perhaps not their clients - can operate in the most profitable and efficient fashion. What is that level? Does it vary across firms? Does it vary between reinsurers, primary insurers, and/or end consumers?

- I n d u s t r y s t ruc tu re . In terms of the standard metaphors used to characterize organizations - a machine, a brain, an organism, a culture, a political system, a psychic prison - which type(s) most accurately represents the insurance industry? And how is this picture of the organization shaped by the specific "routines" used by the decision- markers in the various components making up the organization?

- P o r t f o l i o t h e o r y . The efficient frontier concept in modern portfolio theory provides a theoretical framework for evaluating the risk-return relationship in a heteroge- neous group of assets. Traditionally, the risk is evaluated by a distribution derived from a time-series of values of the asset and the correlations among these time-series. Catastrophic losses and other types of episodic phenom- ena are difficult to place in this framework, in large part because the historical pattern of value bears little rela- tionship to the true risk on reasonable time-scales.

For some classes of assets (insurance and reinsurance in particular), the spatial and temporal correlations of the epi-

sodic risks mean that the behaviors of and interactions among asset-holders influence the overall risk. In these same types of systems, the behaviors of the asset-holders/ risk-takers also influence the expected return through both pricing of risks and common perceptions of risk based on the behavior of the rest of the market. Finally, these behaviors have a feedback on the overall flow of capital in and out of markets. All of these behavioral characteristics of the application of portfolio theory to episodic phenomena indicate the potential for dynamic processes to influence the overall structure of the market and the validity of a portfolio for this aspect of an economy. The Insurance World model will provide a framework for studying the importance of this dynamic aspect of portfolio management in the context of simulated risk- takers, asset-holders, and episodic loss events. In particular, the model will be used to shed light on the question of how an entire industry optimally balances a portfolio of risky ventures when those ventures are natural hazards like hurricanes, earthquakes, and floods, rather than man-made risks.

From the more specific viewpoint of the insurance indus- try itself, the goal of Insurance World is to understand how behaviors of both insurers and reinsurers emerge as a con- sequence of their interactions, both with each other and with the external events. The model will also provide a tool for posing more specific questions about the behavior of the property casualty insurance industry. Questions of this sort include:

- What controls the flow of capital into and out of the reinsurance firms individually, and to and from the mar- ket in general?

- W h a t is the price-setting mechanism for reinsurance contracts?

- How is the flow of capital into and out of firms with individual business strategies related to the strategies chosen by other firms?

- W h a t type of alternative mechanisms are there for providing reinsurance coverage, and how do they fare in markets composed of different players and/or conditions?

- How does the timing and magnitude of external events such as hurricanes impact on pricing and capacity?

In closing, it is worth noting that the ideas underlying Insurance World can also be used to create silicon copies of systems in many other areas of business and economics. For example, work is currently underway to create an electronic

.copy of a supermarket, its customers, and supply chain, in an effort to understand the dynamics of both customer sat- isfaction and inventory control.

References

1. Casti J (1992) Reality rules. I, II. Wiley, New York 2. Casti J (1991) Searching for certainty. Morrow, New York 3. Casti J (1994) Complexification. Harper Collins, New York 4. Casti J (1997) Would-be worlds. Wiley, New York