Quantifying Risk Using Loss Distributions

Abstract

Risk is unavoidable, so quantification of risk in any institution is of great importance as it allows the management of an institution to make informed decisions. Lack of risk awareness can lead to the collapse of an institution; hence, our aim in this chapter is to cover some of the ways used to quantify risk. There are several types of risks; however, in this chapter, we focus mainly on quantification of operational risk using parametric loss distributions. The main objective of this chapter is to outline how operational risk is quantified using statistical distributions. We illustrate the application of parametric loss distributions’ risk quantification using “Taxi claims data” which seems to best fit one of the loss distributions and fully illustrate how to quantify this specific data. More importantly, we also illustrate how to implement quantification of risk for two other scenarios: (i) if we assumed the underlying distribution is unknown and use the nonparametric empirical distribution approach, and (ii) when using the generalized extreme value (GEV) distribution approach. The latter two scenarios were not the main objective but were done in an effort to compare our results with some of the more commonly used techniques in real-world risk analysis scenarios.

Composite and Mixture Distributions for Heavy-Tailed Data—An Application to Insurance Claims

“…Using the 16 loss distributions outlined in Appendix A's Table A1, it is observed from the results in Table 7 that the lognormal distribution seems to be an ideal component distribution for most of the best-fitting mixture models. It seems that the conclusion by Maphalla et al [23] that the lognormal distribution is the best for taxi claims data is supported by the top mixture models with a lognormal distribution component. In an effort to conserve writing space, the corresponding parameter estimates of the top 20 models in Table 7 are provided in Table A3 in Appendix A.…”

“…The South African taxi claims data, which was kindly provided for our study by [23] (this data has been made available in the Supplementary Materials of this paper), consists of 48,043 observations and was divided by 100 for computational ease. The Danish fire loss data, however, is very popular and has a long history of applications.…”

“…This paper is motivated by the recent work by: (i) Grün and Miljkovic [8] where 256 composite models were evaluated for the Danish fire loss data (note though, the corresponding 256 mixture models have not all been considered before); and (ii) Maphalla et al [23] where the standard loss distributions with the best goodness of fit for the South African taxi claims data were found to be the lognormal and the Pareto, and the potential future research idea of modelling the South African taxi claims data using mixture models were suggested. One should note that due to the flexibility of two-component composite and mixture models, overfitting may easily occur.…”