Advances in Risk Management

10 DETERMINATION OF THE CAPITAL CHARGE FOR OPERATIONAL RISK

To show how spurious the estimates of the parameters or the goodness-
of-fit test can be when the collection threshold is not accounted for, consider
the generation of 10,000 random variables that follows a Weibull (0.001, 0.68)
distribution. Table 1.2 reports three cases:

InCase1, thewholeseriesisconsidered(forexample, thereisnocollection
threshold) and the parameters are estimated by the Maximum Likelihood
technique.

In Case 2, we only consider losses larger than 1,000. Parameters are also
estimated by MLE and we do not modify the likelihood function to be
optimized (for example, we ignore the collection threshold).

Table 1.2 Adjusted parameters estimation for trun-

cated distribution

Case 1 Case 2 Case 3

N 10,000 8,910 8,910

a 0.0011 0.0002 0.0012

b 0.673 0.815 0.668

KS 0.0060 0.0571 0.1110

KS* – 0.0754 0.0047

0

0.000005

0.000010

0.000015

0.000020

0.000025

100,000 200,000 300,000 400,000 500,000 600,000

Upper limit of the

wrong distribution

Threshold

Mean excess function (%) Upper limit of thetrue distribution

700,000 800,000

Wrong distribution

True distribution

Figure 1.1 Impact of the truncation on estimated distributions