YVES CRAMA, GEORGES HÜBNER AND JEAN-PHILIPPE PETERS 31.2 MEASURING OPERATIONAL RISK
1.2.1 Overview
Although the application of AMA is in principle open to any proprietary
model, the most popular methodology is by far the Loss Distribution
Approach (LDA), a parametric technique that consists in separately estimat-
ing a frequency distribution for the occurrence of operational losses and a
severity distribution for the economic impact of the individual losses (see for
example, Klugman, Panjer and Willmott, 1998; Frachot, Georges and Ron-
calli, 2001; or Cruz, 2002). Both distributions are then combined through an
n-convolution of the severity distribution with itself, wherenis a random
variable that follows the frequency distribution (see Frachot, Georges and
Roncalli, 2001, for details).
The output of the LDAmethodology is a full characterization of the distri-
bution of annual aggregate operational losses of the bank. This distribution
contains all relevant information for the computation of the regulatory
capital charge to cover operational risk, as this capital charge is obtained
by subtracting the expected loss from the 99.9 percent quantile of the
distribution.^2
1.2.2 Loss distribution approach
In this section, we discuss the methodological treatment of a series of internal
loss data for a single category of risk events, so as to construct a complete
distribution of operational losses.
As mentioned before, the LDA separately estimates the frequency and
severity distributions of losses. The aggregate distribution of losses is then
obtained by ann-fold convolution of the severity distribution with itself,
wherenisthe(random)numberofobservationsobtainedfromthefrequency
distribution.As an analytical solution to this problem is extremely difficult to
derive in practice, we compute this convolution by Monte Carlo simulations.
Aprecise overall characterization of both distributions is required to achieve
a satisfactory level of accuracy.
Maximum Likelihood Estimation (MLE) techniques can be used to esti-
mate the parameters of both distributions. From a statistical point of view,
theMLEapproachisconsideredtobethemostrobustandityieldsestimators
with good statistical properties (consistent, unbiased, efficient, sufficient
and unique^3 ).
More precisely, letf(x;θ) be a selected parametric density function, where
θ denotes the vector of parameters, and letF(x; θ) be the cumulative