Advances in Risk Management

12 DETERMINATION OF THE CAPITAL CHARGE FOR OPERATIONAL RISK

1.4.2 Calibration of AMA

First, we consider both business lines with a collection threshold of 0.25.
Preliminary analysis indicates that frequencies of both samples are well
described by a Poisson process. As this distribution is characterized by a
single parameter which is the average of the observed frequency, we use a
Poisson (1666) and a Poisson (7841) to model frequency for Business Line 1
(hereafter BL1) and Business Line 2 (hereafter BL2), respectively.
To model severity, we start by applying a single PDF for the whole distri-
bution. BasedontheCramer–vonMisestest, themostadequatedistributions
to model BL1 and BL2 among those presented in Table 1.1 are a Weibull (4.9,
0.09) and a Weibull (8.5, 0.07), respectively. But as often encountered with
operational risk losses, even these distributions are unable to satisfactorily
capture the whole distributional form, especially at the tail level. Figure 1.2
shows the QQ-plot for both cases. Points in the tail clearly depart from the
straight line that would indicate a good fit. To circumvent this problem, we
adopt the approach described in the modeling extreme losses section by
using EVT to model the tail of the severity distributions.
To estimate the cut-off point from which observations are used to estimate
the parameters of the GPD distribution, we first take a look at the Mean
Excess Plots (see Figure 1.3). Visual inspection indicates potential “breaks”
around 400 for BL1 and 500 for BL2.
To validate this first impression, we apply the algorithm described in the
Appendix. For both business lines, we consider all the observations above
100 to be potential threshold candidates. This means thatm=59 for BL1 and
m=227 for BL2. The threshold values for which MSE is minimized are 375
and 450, for BL1 and BL2, respectively.^8 MSE associated with each tested
threshold are plotted in Figure 1.4.
Table 1.4 reports the estimation of the three parameters for the GPD.
Location parameter is estimated through the algorithm described in the
Appendix, while scale and shape parameters are estimated with (con-
strained) MLE.^9
The next step is to model the “body” of the distribution, for example,
the losses that are below the estimated extreme threshold. For BL 1, this
means all the losses between the collection threshold (0.25 in this case) and

375. For BL 2, this covers all the losses between 0.25 and 450. To do so, we
     consider the distributions presented in Table 1.1 and we use the Cramer–
     von Mises statistic as a discriminant factor to compare goodness of the fit.
     Once the severity is fully characterized, 10,000 Monte Carlo simulations
     are performed to derive the aggregate loss distribution for each business
     line. Results are summarized in Table 1.5. The regulatory capital charge
     amounts to 1.3 and 0.8 million for BL1 and BL2, respectively. Additionally,
     it is interesting to note that these values represent 8.1 and 6.1 times the