From Numerical Probabilities to Linguistic Probabilities: A Theoretical Justification of Empirical Granules Used in Risk Management

In many risk management situations, instead of the exact probability values, specialists use a granule to which this probability belongs. Specifically, they use five granules, corresponding to thresholds 10%, 40%, 60%, and 90%. In this paper, we provide an explanation for such non-uniform granulation. 1 Formulation of the Problem In risk management, it is often important to group probabilities into granules. One of the main objectives of risk management is to minimize the expected loss. By definition, the expected loss is equal to n ∑ i=1 pi · xi, where xi is the loss caused by the i-th event and pi is the probability of the i-th event. In many practical situations, we have only crude estimates of potential losses xi caused by different possible events. In this case, we can only get crude approximations for the expected loss. Thus, it does not make sense to estimate the probabilities pi with high accuracy. For example, if we know that the loss is between 10 and 100 million dollars, it does not make sense to distinguish between the probabilities of 10% or 11%, it is sufficient to estimate probabilities with a similar low accuracy. In other words, instead of considering exact probabilities, it makes sense to group possible probability values into granules, so that instead of the actual values of the probability, we use, in our estimates, only the granule containing this value.