Estimating operational risk capital
requirements assuming data follows a bi-exponential distribution
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Suppose a risk manager believes that an appropriate model
for a particular type of operational risk exposure involves the loss,
,
coming 50% of the time from an exponential
distribution with parameter
and
50% of the time come from an exponential distribution with parameter
.
The exponential distribution
has
a probability density function,
,
mean,
,
and variance
as
follows:
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image008.png)
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image009.png)
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image010.png)
Suppose we want method of moments (MoM) estimators for
and
and
we have loss data
for
where
the number of losses,
,
is sufficiently large to be able to ignore small sample corrections. Then the
MoM estimators can be derived as follows, where the (sample) moments used in
the estimation are
and
.
The pdfs of the individual parts,
,
and of the overall distribution,
,
are:
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image021.png)
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image022.png)
The mean of
is:
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image023.png)
where
and
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image025.png)
By integrating by parts or by noting that
for
where
and
,
we note that
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image030.png)
So method of moments estimators involve (if these
simultaneous equations have a (real) solution):
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image031.png)
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image032.png)
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image033.png)
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image034.png)
This is a quadratic equation which has the following
solutions:
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image035.png)
The two roots correspond to
and
(it
is not possible to differentiate between them given merely the data being
provided). Hence the values of
and
are:
![](I/ERMMTOperationalRiskCapitalBiExponentialMoM_files/image038.png)
In practice it is more likely that the probabilities of
drawing from the underlying exponentials are unknown. This adds an extra degree
of freedom which would introduce the need to include a further (higher) moment
into the parameter estimation process.