Advances in Risk Management

6 DETERMINATION OF THE CAPITAL CHARGE FOR OPERATIONAL RISK

Table 1.1 Severity distributions

Distribution Probability distribution function

Lognormal(μ,σ) f(x)=

1

x

√

2 π

exp

[

−(logx)^2

2

]

LogLogistic(α,β) f(x)=−

α(x/β)α−^1

β[1+(x/β)α]^2

Pareto(θ,α) f(x)=αθαx−(α+1)

Weilbull(α,β) f(x)=αβ−αxα−^1 exp

(

−

(

x

β

)α)

Kolmogorov–Smirnov test (Kolmogorov, 1933, and Smirnov, 1939) defined
by the statistics,

DKS= max

i=1,...,n

[|Fn(xi)−F(xi;θ)|] (1.5)

This test does not depend on the underlying CDF being tested. On the other
hand, it has several drawbacks: it is only available for continuous distribu-
tions, the distribution must be fully specified and it is more sensitive near
the center than at the tails, which makes it somewhat conservative. In the
operational risk framework, the first two issues are not problematic but the
last one should be a source of concern. Severity distributions are usually
heavy-tailed and a good fit at the extreme right tail of the density is crucial.
The Cramer–von Mises test (Cramer, 1928) is quite similar to the KS test,
but introduces a size-based correction. It is defined by the statistics:

CVM=n

∫∞

−∞

(Fn(x)−F(x;θ))^2 dF(x) (1.6)

which in practice can be computed as

CVM=

1

12 n

+

∑n

i= 1

(Fn(xi)−F(xi;θ))^2

whereF(xi) is the cumulative distribution function value atxi, thei-th
ordered value.
Veryoften, thefat-tailedbehaviorofoperationallossesmakestheaccurate
estimation of the (regulatory required) extreme quantiles a tricky exercise.
Consequently, heavy-tailed distributions such as the ones presented above
are sometimes unable to correctly capture the probability of occurrence of
exceptional losses (for example, the extreme right part of the severity dis-
tribution). This is especially true since loss data collection is sometimes still
in its infancy at some banks, which results in internal loss databases lacking
very large losses. Some recent studies indeed indicate that classical distribu-
tions are unable to fit the entire range of observations in a realistic manner