Journal of Operational Risk

Modeling very large losses

Henryk Gzyl

  • Very large losses correspond to rare events and are natural outliers in loss data collections.
  • Very large losses may not be present in a data set used to determine an aggregate loss distribution.
  • A very large loss has magnitude is much larger than the VaR of the loss distribution at a high confidence level.
  • Potential very large losses can be added to a loss distribution by means of a simple procedure and the resulting total loss distribution can be used to compute all quantities of interest.

In this paper, we present a simple probabilistic model for aggregating very large losses into a loss collection.  This supposes that “standard”  losses come in various possible sizes – small, moderate and large – which, fortunately, seem to occur with decreasing frequency. Standard modeling allows us to infer a probability distribution describing their occurrence. From the historical record, we know that very large losses do occur, albeit very rarely, yet they are not usually included in the available data sets. Such losses should be made part of the distribution  for computation purposes. For example, to a bank they may helpful in the computation of economic or regulatory capital, while to an insurance company they may be useful in the computation of premiums of losses due to catastrophic events. We develop a simple modeling procedure that allows us to include  very large losses in a loss distribution  obtained from moderately sized loss data. We say that a loss is large when it is larger than the value-at-risk (VaR) at a high confidence level. The original and extended distributions  will have the same VaR but quite different values of tail VaR (TVaR).

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