Application of probable maximum loss estimates to terrorism exposures.

If an insurer were to choose an animal symbol to concentrate the minds of its risk managers on dangerous accumulations of exposure, it wouldn't be a stockbroker's aggressive bull or bear, but a scorpion or other creature with a dangerous sting in its tail. The World Trade Center disaster is the most tragic and costly realisation of tail phenomena that the insurance industry has yet experienced. As with animal stings, awareness of the danger in the tail is the best remedy.

A key concept in insurance risk management is the loss distribution; the relative likelihood of different sizes of loss. Small or moderate losses are more common than large losses, which are yet more common than huge losses. The likelihood of occurrence of a loss therefore tails off as its size becomes ever larger. Appropriately, the low frequency and high consequence end of the loss distribution is referred to as its tail. This nether region of the loss distribution is unfamiliar territory even for experienced underwriters: extreme losses are rather rare. However, for reinsurers, this is the loss region on which their business depends; losses with a return period well exceeding a human life span. But just how large can losses get? To answer this question, the concept of a probable maximum loss (PML) is useful.

Various methods exist for defining and estimating PML. Where probabilistic loss calculations are feasible, it is traditional to define PML as the loss corresponding to some long return period, or, equivalently, some low annual probability of exceedance. For example, if a return period of 200 years was chosen as a reference for PML estimation, this would correspond to an annual probability of exceedance of 0.5%. This would seem to be a perfectly sound and precise definition. However, there are some unsuspected problems which defy common sense and professional intuition. Specifically, there is the anomaly that the PML of a joint portfolio may be greater than the sum of the PMLs of its constituent parts.

Elementary illustration
Perhaps as disturbing as the presence of this anomaly is the relative ease with which examples can be found. An elementary illustration is seismic risk in two widely distant cities, each exposed to independent rare damaging earthquakes, but little shaking of note otherwise. If the return period for such damaging events is high for both cities, e.g. about 500 years, the 250-year PML for each city would be low, because the more common seismic events would be of little consequence. However the 250-year PML for the joint portfolio would be substantial, because this would correspond to either of the two cities suffering damage from one of the rare major local events. On average, one of the 500-year return period events would occur every 250 years.

If reinsurance purchase were based on PML, it might seem better to buy reinsurance separately for each of the two cities rather than jointly. This is, of course, illusory. The root cause of the anomaly is the insensitivity of this PML definition to the shape of the tail of the loss distribution. If the tail is long and thin, then very high losses in excess of the PML might be plausible. By contrast, if the tail is comparatively short, the chance of very high losses would be almost negligible.

The underlying worry with tying PML to a specific loss return period is that it can lead to artificially low assignments of PML when the loss distribution has an extended tail. This can present an accumulation problem if a sizeable exposure is assigned an artificially low PML. Of course, diversification is an effective means of resolving this problem and providing protection against large single event losses. Ironically, a return period definition of PML may deter diversification because of the summation anomaly. Concentration of exposure in one location where the loss distribution has a long tail may seem preferable, from this particular PML perspective, to splitting it into distant independent locations.

To circumvent the problems of PML tied to a specific return period, one can instead use a weighted average of tail losses. This is most simply explained in simulation terms. Suppose that a thousand years of loss experience are simulated. Ranking the years according to severity of the worst event loss, the largest loss in the fourth worst year would be the conventional 250-year return period PML. But the average of the largest losses in the four worst years would provide an estimate of an improved (so-called coherent) definition of PML. To illustrate this, suppose the largest losses in the four worst years are $10m, $15m, $25m and $30m. Then the coherent PML would be one-quarter of $80m, which is $20m. This is twice the conventional 250-year return period PML, which is $10m. Using a weighted average of tail losses as a (so-called coherent) measure of PML, the anomaly is avoided: the sum of the PMLs for two separated portfolios is never less than the PML of the joint portfolio.

How does all this relate to the World Trade Center disaster? Comprehensive risk assessments, including terrorism, have, for some while, been conducted for specific important buildings in Manhattan and London. Both in London and New York, risk-conscious corporations have, over a number of years, carried out comprehensive internal assessments of risk to their businesses. These risk assessments have been motivated by concerns for maintaining personnel security and shareholder value, rather than the cost of insurance premiums. Accordingly, extreme man-made perils such as terrorism have been considered as well as natural hazards. For prestige target buildings in central London, such as Canary Wharf, terrorism has, in recent years, ranked as one of the highest risks.

It might be thought that terrorism was a peril beyond the scope of quantitative risk assessment. This is not so. Ironically, during the Cold War, ever before any application to natural perils, scenario-based risk analysis was developed by the RAND Corporation to counter a perceived terrorist threat from the Soviet Union. To invoke the probabilistic risk methodology for the analysis of human malicious action in the US is to close a circle of practical application. By eliciting, from terrorist experts and insiders, ensembles of realistic scenarios and ranking them according to likelihood, it is possible to generate probabilistic loss curves for terrorist action. As with natural catastrophe modeling, risk estimation is achievable even if short-term forecasting is not. The purpose is not to attempt to foresee the precise mechanics of the next terrorist act, but to encompass its generic loss consequences within those of a diverse class of plausible alternative scenarios.

Determined pursuit
Charting the loss curve from terrorist action may be a task shrouded in uncertainty, but some gross features can nevertheless be resolved. In particular, what is amply clear from the publicised goals of terrorist groups is their steadfast determination to try, try and try again. Such persistence greatly extends the tail of the terrorism loss curve. Terrorist attacks mostly fail absolutely; sometimes they partially meet their objectives; and very occasionally, as with the WTC disaster, fortune sides with the terrorists. To steer a commercial jet accurately around a bend to hit a building requires such fortune. The extreme length of the tail mirrors the extreme single-mindedness of the perpetrators. As security agencies are well aware, it only takes one security lapse for a tail scenario to become an actual statistic. Given its target history, the WTC disaster was far from being a one-in-a-million event. The demise of the WTC brings back memories of the Challenger space shuttle disaster of 1986. Pre-disaster belief by NASA management in minimal mission risk was belied by the actual risk which was computed later to be greater than 1/100.

Among the distant casualties of the WTC disaster were underwriters who had overestimated the return period of such a catastrophic loss, and thereby grossly underestimated the PML. But how can terrorism PML be estimated better in the future? Given some feeling for the broad spectrum of possible terrorism scenarios, it is hard to see how a meaningful PML from terrorist action can be estimated if the PML does not allow implicitly for events far out in the tail. Any PML procedure which truncates the tail of the loss distribution is ignoring a host of hypothetical scenarios, which, far from being merely fanciful, may already exist as blueprints in terrorist hideouts. Embedded within the tail are bizarre scenarios which would lead to catastrophic levels of claims linked across many insurance sectors through the force of loss correlation. Casualty liability, in particular, is an especially tail-sensitive risk, highly correlated with property loss.

In the context of understanding the laws of Nature, Einstein said, ‘Nature may be subtle, but not malicious'. Extreme earthquakes and windstorms do occur from time to time, and they expose weaknesses in architecture and building construction, but thankfully there is no inherent law of Nature which maximises the destructive power of these events. By contrast, in as much as terrorists seek to maximize the destructive impact of their actions and their notoriety, they deliberately aim to strike deep within the tail of the loss curve. Even more than for natural perils, those who estimate a PML for terrorism must therefore be prepared to grapple with the dangerous tail of the loss distribution. Ignorance of the danger in the tail would leave risk managers blind to a major source of risk. Steps have now been made to explore the potential for terrorist losses in the US, and further investigations will be needed to scope the bounds of future liability.