Rare events: anticipating the unlikely
Nuclear accident, stock market crash … the law of large numbers and the Gaussian distribution, foundations of the statistics of average magnitudes, fail to account for such rare or extreme events. Adapted tools exist… but they are not always used!
On September 15, 2008, the American bank Lehman Brothers declared bankruptcy. This event, following the subprime crisis the previous year, shook the global financial system and panicked the stock markets around the world. Governments and central banks chain together stimulus plans, but nothing helps, the world plunges into the most serious financial crisis since that of 1929. To what extent could we have predicted this cataclysm?
Such rare and catastrophic events (to stock market crashes, let’s add earthquakes, floods, nuclear accidents, etc.) fascinate by their unpredictability and the importance of the social and scientific issues they represent. They are also challenges to scientific modeling. Exceptional almost by definition, or sometimes set aside as “statistical aberrations”, rare events are nonetheless accessible to statistical theory… as long as it requires tools better suited than those of everyday life.
In many applications, statistics are reduced to the calculation of means and standard deviations, and the distribution of observations is distributed wisely according to the famous “bell curve” or Gaussian distribution.
Stats for Dummies
The statistician often represents a sample of observations (temperatures, precipitations, stock market indices, etc.) as a series of independent draws of a “random variable” noted X and representing the quantity in question. We are then interested in the mean, or mathematical expectation, of this random variable, its standard deviation (the dispersion of values around the mean) and, more generally, its probability distribution.
The law of large numbers, first established in its simplest version in the 18th century by Jacques Bernoulli, ensures that, when the sample size is large enough, the empirical mean of the sample tends towards the theoretical mean of the random variable it represents. Likewise, the empirical standard deviation converges to the standard deviation of the random variable. By ensuring that a sample provides a good estimate of a real random variable, the law of large numbers emerges as one of the pillars of statistics.
The law of large numbers is an asymptotic law: the mean of a sample only converges to the mathematical expectation if the size N of the sample is large. The difference between the two values - the statistical error — is governed by the “central limit theorem”. This theorem, the second pillar of statistics, has been known in various forms since the eighteenth century, but its final form, due to the French mathematician Paul Lévy, dates from the beginning of the twentieth century. He asserts that, if the random variable X has a finite standard deviation, then for a sample of N independent observations of X, the difference between the sample mean and the mean of X is well represented by a Gaussian distribution whose l The standard deviation is proportional to 1 / √N.
The Gaussian distribution therefore appears as a statistical description of any quantity that can be represented as the sum of a large number of small independent contributions.
These two principles are far reaching. The law of large numbers is the foundation of surveys, because it shows that it is enough to survey a sample large enough to approximate the statistical characteristics of the real population. The central limit theorem makes it possible, for its part, to control the error thus made: the difference between the empirical mean and the theoretical mean follows a Gaussian law.
For example, if an election poll covers 10,000 people and 46% of them say they are in favor of candidate A, the central limit theorem indicates that there is a 95% chance that during the vote, he will collect between 45 and 47% of the votes. Concretely, it is not necessary to survey a very large sample of the population to have a reliable estimate.
The central limit theorem is invoked to justify representation by a Gaussian distribution in fields ranging from social sciences to physics, including economics, biology or finance. The bell curve is used in all sauces. It is even used to “normalize” the distribution of marks in competitions …
However, the omnipresence of the Gaussian distribution in statistics (hence its qualifier of “normal distribution”) is perhaps due to the too great confidence of statisticians rather than to reality: not all phenomena can be reduced to a sum of small independent contributions. In fact, in economic and social sciences, the normal law seems to be the exception rather than the rule.
From the beginning of the twentieth century, economist Vilfredo Pareto, for example, was interested in Italian national wealth, and found that the richest 20% of the population held 80% of the country’s total wealth. This situation of inequality is clearly not well represented by a Gaussian distribution of wealth, where individual wealth is concentrated around average wealth and where very rich or very poor individuals are very rare.
Pareto proposed a more realistic distribution which today bears his name: the share of individuals with a wealth greater than a level u is proportional to 1 / uα. The smaller the exponent α of this power law, the greater this proportion and the more unequal the distribution of wealth. Faced with such distributions, the notions of mean and standard deviation prove to be inadequate and offer a misleading description.
The math of inequalities
Indeed, the average wealth and the standard deviation can for example remain identical while the distribution of wealth changes (which is impossible in the context of a Gaussian distribution).
Worse, if the exponent of the Pareto law is less than two, then the standard deviation of the distribution is infinite. If it is less than one, this is also the case for the mean. These indicators are therefore meaningless for Pareto laws.
This fact has led economists to define other indicators to better characterize the unequal distributions. The Gini coefficient, proposed by the Italian statistician Corrado Gini in 1912, is thus a number between 0 and 1 which measures the degree of inequality in the distribution of wealth. The closer it is to 1, the more unequal the distribution. In the case of a Pareto law with exponent α, the Gini coefficient is equal to 1 / (2α — 1). These indicators focus on the largest events in a sample, events of low occurrence, but of predominant importance. These “rare events” form tails of distributions.
Once the mean and the standard deviation no longer necessarily make sense for unequal distributions, what happens to the law of large numbers and the central limit theorem? In the 1920s, Lévy was one of the first to notice that the assumptions underlying them may cease to be valid in the event that rare events have a major influence on the sample mean.
Lévy extended the central limit theorem to these situations: he showed that for a sample of centered variables whose tail distribution behaves like a Pareto law with an exponent between 0 and 2, the empirical mean no longer behaves like a Gaussian variable, but according to a distribution that Lévy called stable law, whose tail behaves like a Pareto law with the same exponent. In other words, the limit law is a Lévy’s law, and not a normal law.
For these Lévy distributions, variance and standard deviation no longer have any meaning: they are infinite. Benoît Mandelbrot, who was a student of Lévy at the École Polytechnique, showed in 1963 that, far from being a mathematical curiosity, the situations described by Lévy’s laws were omnipresent in economics, finance and insurance: the yields of materials First, the wealth of individuals or the size of firms follow Pareto laws.
Mandelbrot characterized these situations, where the behavior of the sample is no longer correctly characterized by the mean and the standard deviation, as wild chance, as opposed to the wise chance of the Gaussian law.
Taming wild chance
This wild behavior is evident in stock index returns. While the amplitude of the fluctuations of a Gaussian sample remains of the order of its standard deviation, some stock market indices exhibit erratic amplitude variations reaching several tens of times the standard deviation! Many other situations conforming to Lévy’s laws were quickly highlighted in statistical physics.
The epithet “savage” suggests the uncontrollability of chance, but a good understanding of the hazard sometimes helps tame it. This is the case in some physical systems, as well as in situations, where, regardless of whether the underlying hazard is wise or wild, the mean or standard deviation of a sample is simply not. the relevant indicators, even if they can be estimated with precision.
This is the case with reliability studies, where, to determine the probability of failure of a system, we seek to estimate that of its “weakest link”. For example, to choose the size and characteristics of a dam, it is necessary to have a precise idea of the maximum pressure that the structure will be able to withstand, and not of the average pressure that it will undergo under normal conditions. For this, it is necessary to directly characterize the extreme values in the sample of observations. The notions of standard deviation, of mean and the theorem of the central limit are not then of much help …
Alongside Lévy’s work on the central limit theorem, a group of statisticians, including Sir Ronald Fisher and Leonard Tippett in Great Britain, Boris Gnedenko in the Soviet Union, Maurice Fréchet in France and Ludwig von Mises in Austria, as well as Emil Julius Gumbel, German mathematician who took refuge in France, developed a new branch of statistics whose goal is to study no longer the average values of samples, but their extreme values, that is to say maximum or minimum.
Just as the central limit theorem describes the behavior of the mean of a large sample, this extreme value theory describes the behavior of the maximum of a sample. These statisticians have shown that, depending on the nature of the random variable studied, the maximum value of the sample can only obey one of three different distributions: the Fréchet, Gumbel or Weibull distributions.
Like the normal law and Lévy’s law, which are natural candidates for modeling the sums of independent variables, the extreme value laws are preferred candidates for modeling the maximum values of a sample. Gumbel presented in 1941 an application of these concepts to the statistical analysis of river floods. This theory was to be applied in a dramatic context.
Taming scarcity … and the sea
The Netherlands has always lived under the threat of rising sea levels. The many dikes that dotted the country have long been enough to protect it. But on the night of January 31 to February 1, 1953, rising waters due to a storm overcame their resistance and submerged the southwest of the country, killing more than 1,800 people and thousands of cattle.
In shock, the Dutch government soon launched a massive dam construction project. The commission responsible for reviewing safety standards demanded that the height of the dikes be fixed so that the probability of the tide exceeding them is less than 1 / 10,000. A team of scientists, armed with the recent theory of extreme values, then began to study past tides and floods in order to estimate the probability distribution of the maximum flood height. They showed that it follows a law of Gumbel, and that the required height exceeded 5 meters, a clear increase compared to the dikes of the time. On the basis of these estimates, huge dams were built; these buildings still protect the country today.
In an atmosphere of growing faith in quantitative methods, extreme value theory was subsequently applied to insurance, financial risk management and environmental risk management. Insurers use it, for example, to estimate the biggest losses they can suffer from selling insurance against natural disasters.
The disaster scenario par excellence being the explosion of a nuclear power plant, all ideas on the measurement of extreme risks sooner or later found a field of application in the field of nuclear safety. In the 1970s, probabilistic safety studies (EPS) appeared, a methodology for estimating the probability of accident scenarios in nuclear installations. The method consists of identifying the sequences of events that can lead to a catastrophic result, such as a meltdown of the reactor core; and assign probabilities to these sequences based on physical principles or acquired experience. The objective, as for the Dutch dikes, is to define the standards to be met to guarantee a probability of occurrence of the catastrophic event below a given threshold.
But in 1979, the accident at the Three Mile Island nuclear power plant in Pennsylvania, United States, was a stark reminder that in a complex system formed of multiple interrelated components like a nuclear power plant, one small error — human in it. occurrence — can cause disaster.
However, the occurrence of such errors, which are not linked to a physical factor obeying a well-identified law, is difficult, if not impossible, to quantify. Therefore, what faith should be given to the “quantitative certainties” produced by the safety studies and the resulting technical certifications?
Frank Knight, American economist and risk theorist, already distinguished in 1921 in Risk, Uncertainty and Profit the probabilizable risk, corresponding to events whose probability can be assessed with reasonable certainty, and uncertainty, which covers events whose ignores until the probability of occurrence. When we move away from physical systems to areas where human factors are more important, the share of uncertainty increases.
Incomplete observational data, overly simplifying assumptions (such as the independence of risk factors, frequently postulated in statistical models), or even omission of certain risk factors, the sources of uncertainty in risk models are numerous. They are all limitations that invite us to remain modest in our expectations of statistical models when they are not based on well-anchored physical principles.
Black swan hunting
What to do then? At most, the results of the models can be supplemented by a case-by-case analysis of extreme scenarios. This is the purpose of the stress test method, which provides a useful complement to statistical risk analyzes. The CEA and EDF have chosen this option in addition to the EPS studies to define the nuclear safety policy.
The statistics show their limits here. However, should we abandon the ambition to quantify rare events outside the physical sciences, where physical principles make it possible to calculate their probabilities?
This is what some seem to suggest, such as Nassim Taleb or Nicolas Bouleau, of the National School of Bridges and Roads, in particular in light of the apparent failure of quantitative methods of financial risk management during the financial crisis of which we have spoken. According to Nassim Taleb, it is indeed impossible to quantify the risk of rare catastrophic events of which we have never known an example in the past. These “black swans”, as he calls them, would invalidate statistical risk modeling approaches.
Avoid the crisis?
However, is the recent financial crisis a black swan like the Three Mile Island accident, where, despite precautions, unquantifiable human error thwarted the safety system? Or does it rather resemble the storm that ravaged the Netherlands in 1953, an event that was rare, but whose probability could be measured?
I favor the second option: financial institutions are far from having put into practice adequate methods to measure the risks of their portfolios. More than forty years after Mandelbrot’s work, a number of banking establishments, for ease, still use the normal law in their risk calculations: difficult then to blame the statistical models… which are not used!
Without going so far as to assert, as a journalist from the Nouvel Observateur did recently, that if the investment bank Lehman Brothers had calculated its risks with Lévy’s distributions, it would have avoided bankruptcy, it is certain that the use of more realistic models will only improve financial risk management.
But Nicolas Bouleau raises another point, of a sociological nature. The enormous social stakes of certain catastrophic events — meteorological, geological or economic — have the consequence of generating significant pressure from public decision-makers on “experts” to, if not predict these rare events, at least to quantify their risk of occurrence. The temptation is then great to prefer inappropriate statistical indicators rather than to recognize that current scientific knowledge does not provide answers to certain questions.
This is undoubtedly one aspect of the failure of financial risk management systems during the recent crisis: regardless of their accuracy, they maintained the illusion of security in the eyes of their users, who consequently declined their guard. Do not reject statistical risk modeling methods, use them with care!
