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CPS2051 Mentje G. et al.
regulator require that when dealing with operational losses, a bank should
hold capital that will protect them against a one-in-a-thousand year aggregate
loss. To determine the capital, the 99.9% Value-at-Risk (VaR) of the distribution
is calculated. One would have hoped that at least a thousand years of historical
data is available, but in reality only ten years of data may be available. For this
reason, scenario assessments by experts are used to augment the historical
data and to provide a forward looking view.
The proposed approach discussed in this paper has also been discussed
and studied elsewhere (see de Jongh et al. (2015)), specifically in the context
of operational risk and economic capital estimation.

2. Methodology
2.1 Approximating VaR
Let the random variable denotes the annual number of loss events and
assume that is distributed according to a Poisson distribution with
parameter lambda, i.e. ~() . One could use other frequency distributions
like the negative binomial distribution, but the Poisson is the most popular in
practice (see e.g. Embrechts and Hofert (2011)). Furthermore, assume that the
random variables , … , denote the loss severities of individual loss events.

1
Further, assume that these loss severities are independently and identically
distributed according to a severity distribution , i.e. , … , ~ . Then the

1
annual aggregate loss is = ∑ and the distribution of is a compound

=1
Poisson distribution that depends on and and denoted by (, ). Of
course, in practice we do not know and and have to estimate it. First we
have to decide on a model for , for example a class of distributions (, ).
and have to be estimated using statistical estimates.
The compound Poisson distribution (, ) and its VaR are difficult to
calculate analytically so that in practice, Monte Carlo (MC) simulation is often
used. This is done by generating according to the assumed frequency
distribution and then by generating , … , independent and identically

1
distributed according to the true severity distribution and calculating =
∑ . The previous process is repeated times independently to obtain

=1
, = 1,2, … , and then the 99.9% VaR is approximated by ([0.999∗]+1)

where denotes the -th order statistic and [] the largest integer contained

in . Note that three input items are required to perform it, namely the number
of repetitions and the frequency and loss severity distributions. The number
of repetitions determines the accuracy of the approximation and the larger it
is, the higher its accuracy.
In principle infinitely many repetitions are required to get the exact true
VaR. The large number of simulation repetitions involved in the MC
approaches above motivates the use of other numerical methods such as
Panjer recursion, methods based on fast Fourier transforms (see e.g. Panjer,

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