Appendix 1: Credibility theory

Daniel Rodríguez

This appendix introduces the different models of credibility theory: classical model, Buhlmann, Buhlmann–Straub, and Buhlmann–Straub for small and large losses.

CLASSICAL CREDIBILITY MODELS

This section presents the classical credibility model. Classical credibility has the disadvantage of requiring the underlying data to be normally distributed, which is uncommon for operational risk losses. Nevertheless, it is presented for its simplicity to facilitate the understanding of all other credibility models.

Classical probability theory (also known as limited-fluctuation credibility) starts with two estimations obtained from different sources, denoted as estimates x and y. Each of these estimates has a specific variance, denoted as σx2 for x and σy2 for y. Estimate x, which generally refers to the institution´s internal loss data (ILD), is associated with a weight-named credibility factor and denoted by z, which will be between zero and one. The mean value of both estimates can be obtained by the expression:

  xy¯=z×x+(1z)×y (A1.1)

This equation shows that the value of xy¯ is between x and y values. The credibility factor, z, indicates the relative importance of the

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