**Dancing on a volcano – Extreme events can be quantitatively assessed and reliably judged upon**

Nuclear energy comes with residual risks that even the soothing assurances of the nuclear industry cannot completely hide. As unlikely a nuclear disaster after considering the mathematics of probability may be, voices are growing louder that refuse to rely on mathematically quantified model calculations for probabilities as the basis for the public debate on nuclear energy. In extreme risks such as nuclear power, once the incident actually occurs extremely large – possibly civilization-threatening – damage can be the consequence. And this argument has its merits. Multiplying a very small number (low risk) with a very large number (at high cost event), the mathematical assessment of the result has its limits. Nevertheless, we still depend on the probabilistic modeling of extreme events, and they do provide us with some useful results. But they should nevertheless be improved, what many leading mathematicians have been working on for years. On the other hand, these models should be used truthfully in the political and economic decision-making process.

It was absolutely foreseeable that an earthquake of magnitude 9 would trigger a destructive tsunami. Also the fact that such an event would make the cooling systems of nuclear power plants vulnerable, reveals itself not only from complex mathematical models. Once the cooling system in a water cooled reactors fails a nuclear meltdown occur with probability one, i.e. necessarily. In extreme events, it is such a concatenation of unusual risks, which can cause the combination of the individual events to lead to an almost apocalyptic catastrophe. Looking at each of these events independently and with the individual probabilities of each event being let us say 1% the simultaneous occurrence of all three events possesses a probability of 0.0001%. However, in an extreme case as for a nuclear meltdown, these events fall together, because their underlying causality is the same, resulting in a total probability of(in the range of) also 1%. For years, there have been mathematical models that explicitly describe such dependencies of events in extreme situations. With this in mind we can see that the probability of a core meltdown in a nuclear reactor after such an earthquake as the Tohoku in northern Japan was well over 50%. An earthquake in Japan measuring 9.0 itself is more powerful than expected, though not beyond measure. And globally it corresponds to about a 20-year event. The bottom-up, „Probabilistic Safety Analysis“ of the nuclear power industry had thus clearly been too optimistic, because they did not sufficiently take into account conditional correlation of default probabilities of individual components in extreme cases. Already a top-down analysis based on statistics of past damage would have resulted in higher probability of a nuclear meltdown by approximately a factor of 10.

We can clearly observe some analogies to the events in the global capital markets during the 2008 financial crisis which themselves corresponded to a „core meltdown“ of the system. Also here dependencies were at work that showed their devastating nature in an extreme event environment already in sufficiently complex mathematical models, however not in their simplified versions as used by the banks’ and supervisory authorities’ risk systems.

Relationships that behave differently in extreme than in normal environments can be detected in many fields: Tsunami in Indonesia, Katrina’a flooding of New Orleans, the environmental disaster in the Gulf of Mexico, all the way to the collapse of entire political systems. Everywhere we see how the existing institutional as well as our own cognitive expectancy framework is overwhelmed – however not the math. The statements of political and economic decision-makers are often based on downright outrageous simplifications, misleading misinterpretations or even interest-driven distortions of mathematical results. This helps to hide a sinister reality: Crises such as the ones mentioned are an integral part of our economic and social system!

With the term „black swan“ – a metaphor originally introduced by the philosophers John Stuart Mill and Karl Popper (which a last, however, goes back to the Roman satirist Juvenal, as he stated that a loyal wife is a „raraavis in terris, nigroquesimillimacygno (a rare bird in all countries, most like a black swan) –participants in financial market have for several years described possible, but very unlikely events. Economists continue to use models of risk management which are simplified to the level of neglect, instead of applying more complex and at the same tome more realistic descriptions of“ black swan“ events including their non-linear correlation structure. And the supervisory bodies continue to give credit to the paltry mathematical models of the financial industry. But events like the 2008 capital markets crisis and the nuclear disaster in Fukushima in March 2011 demonstrate: We are dancing on a volcano.Correspondingly we have to rely on clear information and take responsible actions.Contrary to political good will there is no lack of applicable mathematical models for that purpose.