Estimating operational risk capital
requirements assuming data follows a bi-exponential distribution
[this page | pdf | back links]
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:
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:
The mean of is:
where 
and
By integrating by parts or by noting that 
for
where 
and
,
we note that
So method of moments estimators involve (if these
simultaneous equations have a (real) solution):
This is a quadratic equation which has the following
solutions:
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:
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.