Advances in Risk Management

YVES CRAMA, GEORGES HÜBNER AND JEAN-PHILIPPE PETERS 11

In Case 3, we also consider losses larger than 1,000, but we adjust the
likelihood function according to equation (1.8).

For each case, we also compute the Kolmogorov–Smirnov test. In Table
1.2, KS relates to the unmodified test (for example, not accounting for the
collection threshold), while KS* is the modified test.
The table clearly demonstrates the importance of accurately adjusting the
estimation techniques to account for the collection threshold. Without ade-
quate changes in the likelihood function and the goodness-of-fit statistics,
fallacious conclusions could be drawn as the parameters estimated in Case
2 (with KS=0.06) could be preferred to those estimated in Case 3 (with
KS=0.11). This would in turn lead to inaccurate Monte Carlo results, as
both distributions are very different. Figure 1.1 reports both distributions
and clearly shows that failing to adapt the estimation procedure to account
for truncation may have a significant impact. The estimated distribution in
Case 2 has an upper limit that is 25 percent smaller than the true distribution,
seriously impacting subsequent simulations.

1.4 EMPIRICAL ANALYSIS

1.4.1 Data

In this section, we apply the methodology outlined in the previous sections
to real operational loss data provided by a large European bank. For this
study, we focus our analysis on two complete business lines, regardless
of the loss event type.^7 For the sake of confidentiality, we call these busi-
ness lines “BL1” and “BL2”. For the same reasons, we have scaled all loss
amounts by a same constant. The summary statistics of losses are given in
Table 1.3.

Table 1.3Summary statistics for the operational loss

database

Business line 1 Business line 2

No. observations 1,666 7,841

Collection threshold 0.25 0.25

Median loss 1.35 0.94

Mean loss 118.7 20.7

Std. dev. 1,813 256

Total loss 197,707 162,034