# Rare event risks

A rare event risk can be defined as an event that has a very low probability of occurring during the lifetime of a project or investment or a specified period. Examples of such rare events could be: the rupture of a pipe in a nuclear reactor, a flood caused by a tsunami, a financial market crash in a country or a part of the world, or an epidemic of an exotic disease. It is frequently reasonably easy to estimate the approximate impact of a rare event by considering possible impact scenarios, but estimating the probability of that rare event is usually extremely difficult because there are no data available.

The probability of the rare event occurring is often estimated by considering the small probability that a random variable Y exceeds some large threshold. This random variable may be a function of several other random variables {X}, and possible values for Y are generated by first generating values for each {X}. Importance sampling technique can considerably ease the processing burden, without compromising the model accuracy.

The following example illustrates a simple method of modeling a rare event in estimating the consequences of a flood caused by high-wave storms:

Problem: A vast area of land in country A is below the average sea level. Dikes are built along the sea coast in order to protect the country from floods. Extreme wave-height and still water level are two very important factors for causing flood along the sea coast. The scientists concerned with the safety of the area found out that a catastrophe can occur if the wave-height and still water level (both measured in meter) satisfy the following relation:

Catastrophe factor = 0.4 * wave-height [m.] + sea level [m.] > 6.2.

Taking into account the country's geography, the area of flooded land can be calculated using the following formula:

LogNormal(1.38 * Catastrophe factor ^2, 3.52 * Catastrophe factor ^(1/4))  [100 sq. km.].

Wave-heights and still water levels during high tide have been reliably measured without any interruption at several stations along the sea coast. The analysis of the data gave estimates for the parameters of the input distributions:

Sea level during a storm: Normal(0.2, 0.3) [m.]

Distribution fitted to wave-height during a storm: LogNormal(0.5, 0.6), Shift(+0.5) [m.]

Giving that storms occur at a Poisson expected rate of 25 per year, we need to calculate the probability that within 5 years there will be a flood that can occupy more that 7,000 sq. km.

Solution: This problem could be solved by Monte Carlo simulation, which might however take a very great number of iterations to stabilize because of the low probability of the event being modeled. A flood that occupies more than 7,000 sq. km. can occur more as a result of an extreme wave height than as a result of an extreme sea level. Since these two input variables are uncorrelated, it is easiest to calculate the probability that the wave-height exceeds some threshold (say 4 m.), truncate the variable to values beyond the threshold, simulate this rare event and then adjust the outcome with the calculated probability.

The model Floods gives an example.

The links to the Floods software specific models are provided here: Floods Floods

The above technique becomes far more complicated if not impossible with a combination of two or more input variables that can affect the outcome rare event. In this case we need to simulate the variables' entire ranges as shown in the following example:

Problem: Imagine the construction of an offshore oil platform, during which we need to evaluate the yearly probability that the oil platform will fail. The platform has 6 beams with individual but identical strength distributions and maximum yearly stresses on the platform (e.g. wind). The platform becomes unstable when two or more of these beams fail. If the stresses on each beam were independent, one could simply model one beam to determine the probability of failure P, and then the probability of two or more failures would be 1- (1-P)6 – 6P(1-P)5 (i.e. 1-[probability no beams fail] – [probability 1 beam fails]) from binomial probability theory.

However, this won't work if these maximum stresses on the individual beams are highly correlated (as they will be because the beams are subject to the same environment). The high degree of correlation between the stresses means that the probabilities of failure of individual beams are not independent (as binomial theory demands) and we therefore have to simulate. Being a (hopefully!) very low probability, the simulation would require a great number of iterations. The model Offshore Oil Platform gives an example.

 Note with the Offshore Oil Platform:The model Offshore Oil Platform shows the use of the MaxExtreme distribution. The use of this distribution is based on the following scenario: storm stresses occur randomly at an expected rate of 10 per year (which we can model with a Poisson(lambda = 10)) and the variation of intensity of individual stresses is 2.9 + Exponential (1/5) distributed, then the maximum stress in a year is MaxExtreme (7.5; 5). The MaxExtreme relationship to this Exponential distribution is explained here.

The links to the Offshore Oil Platform software specific models are provided here Offshore_Oil_Platform Offshore_Oil_Platform

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