Advances in Risk Management

YVES CRAMA, GEORGES HÜBNER AND JEAN-PHILIPPE PETERS 9

For smaller banks, however, fixing a threshold at 10,000 EUR might
drastically reduce the amount of data available for computing the capital
requirements. Athreshold of 1,000 EUR or 5,000 EUR can be more adequate.
Whatever the final choice, statistical methods used to calculate the regula-
tory capital charge for operational risk should be adapted to account for this
threshold. This issue is discussed in the following section, while an analysis
of the impact of the collection threshold on the value of the capital charge is
provided in section 1.4.

1.3.2 Impact of the collection threshold on the estimated
parameters

As noted by Frachot, Moudoulaud and Roncalli (2003):

the data collection threshold affects severity estimation in the sense that the

sample severity distribution (for example, the severity distribution of reported

losses) is different from the “true” one (for example, the severity distribution

one would obtain if all losses were reported). Unfortunately, the true distribu-

tion is the most relevant for calculating capital charge and also for being able

to pool different sources of data in a proper way. As a consequence, linking the

sample distribution to the true one is a necessary task.

Mathematically, this is a well-known phenomenon referred to as “trunca-
tion”. More precisely, the density functionf∗(x;θ) of the losses in [L;∞) can
be expressed as:

f∗(x;θ)=

f(x;θ)

1 −F(L;θ)

wheref(x;θ) is the complete (non truncated) distribution on [0;∞). The
corresponding log-likelihood function is:

l(x;θ)=

∑N

i= 1

ln

(

fi(xi;θ)

1 −F(L;θ)

)

(1.8)

where (x 1 ,...,xN) is the sample of observed losses andLis the collection
threshold. It must be maximized in order to estimateθ.
Usually, the quality of distribution fitting is assessed through goodness-
of-fit tests. All these tests are based on a comparison between the observed
cumulative distribution function and the hypothetical one. Consequently,
they should be adjusted to account for the collection threshold as well. For
instance, the Kolmogorov–Smirnov statistics becomes:

DKS= max

i=1,...,n

[∣

∣

∣

∣Fn(xi)−

F(xi;θ)

F(U;θ)

∣

∣

∣

∣

]

(1.9)