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Regulatory Capital and Modeling
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Frequency and severity can be defined as follows:
■
Frequency: discrete distribution, counting the number of operational risk events
per period of time, typically one year. Frequency in operational risk is most com-
monly modeled by a Poisson distribution. Poisson is the simplest frequency dis-
tribution, determined by a single parameter, y that represents both the mean and
the variance of the distribution. According to the Loss Data Collection Exercise of
the Basel Committee in the last decade (2008), nine out of ten AMA firms model
frequency with a Poisson distribution. Negative binomial distributions are usually
a better fit and are used in about 10% of the cases.
■
Severity: continuous distribution, asymmetric and heavy tailed, to account for the
dual nature of operational risk: a large number of small losses and a few very large
incidents. The most common distribution used to model severity is the lognor-
mal distribution, a log transformation of the normal distribution (Gaussian). More
heavy-tailed distributions have been increasingly used over the years, in particular
Weibull and generalized pareto distributions (GPD).
Frequency and severity distributions are then convoluted into an aggregated
loss distribution, illustrated in Figure 8.2. The most common convolution method is
Monte Carlo, where the aggregated distribution is generated by a million (or more)
random draws of severity and frequency. Other methods such as fast Fourier transform
and Panjer recursions are equation-based, requiring more coding and mathematical
ability but less computer time. For the best and most detailed review of statistical
techniques and recommended practices in operational risk modeling, please refer to
Peters et al.
3
Frequency distribution
Convolution through
Monte Carlo
si
m
ulations
Body
Tail
Op
V
aR_99.9
Severity distribution
F I G U R E 8 . 2
Frequency and severity aggregation – illustration
3
Peters, G., Shevchenko, P. and Cruz, M. (2015)
Fundamental Aspects of Operational and Insur-
ance Analytics: A Handbook for Operational Risk
, Wiley.
88
RISK ASSESSMENT
The LDA process is applied to each unit of measure (cluster of losses) that will
then be aggregated into a total distribution where the 99.9th centile will correspond to
the stand-alone capital for operational risk.
U n i t s o f M e a s u r e
The heterogeneous nature of operational risk events is a definite challenge for modeling
and reporting activities. For modeling, a solution is to group events data into clusters of
some homogeneity, called units of measure (UoMs). As an illustration, let’s consider
the following combinations of event types and business lines as possible UoMs:
■
External fraud events, split by individual business line.
■
Damage to physical assets events, grouped for all business lines.
■
Internal fraud events, per business entity (under the same senior management
and supervision).
■
Processing error events, split by individual business line.
Good UoMs constitute clusters of fairly homogenous operational risk events,
driven by the same types of factors and therefore having a similar dynamic and a
similar distribution. There is a tradeoff, though, between the homogeneity of the data
sets defined and data availability: finer segmentation leads to more homogenous data
and more granular models, but it also reduces the amount of data available to estimate
the models, leading to greater uncertainty in the results and increased complexity in
the aggregation process.
Each UoM is modeled separately and aggregated into a standalone capital for
operational risk. The choice of UoM thus has important consequences for the model
dependency structure and results. It is, unsurprisingly, the object of attention for model
validation teams and regulators alike. In principle, modelers should demonstrate that
the units of measures follow the way the business is organized and in line with the
final use of the model. This intra-risk aggregation of UoM into operational risk capital
most commonly uses copula techniques, as required by the regulator. Copulas are gen-
eralizations of correlations and can be used to model advanced dependency structures,
including tail dependence and dependence between extreme values.
P I L L A R 2 – S U P E R V I S O R Y R E V I E W P R O C E S S
S R E P , I C A A P a n d C C A R
The SREP enables the regulator to evaluate risks to which the firm is or might be
exposed and assesses the risks that the firm may pose to the financial system, should the
firm fail. The regulator also evaluates the measures taken by the firm to identify, assess
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