Moments of a binomial loss distribution
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Suppose a portfolio has
equally-sized
exposures. Each one is independent and has a probability
of
creating a unit loss (and a probability
of
creating a zero loss), with
the same
for each exposure, meaning that the portfolio loss,
,
is distributed according to a binomial distribution,
i.e.:
![](I/ERMMTBinomialLossDistributionMoments_files/image005.png)
The mean and the variance of the portfolio loss distribution
can be found as follows. We note that:
![](I/ERMMTBinomialLossDistributionMoments_files/image006.png)
The mean of the loss distribution is given by:
![](I/ERMMTBinomialLossDistributionMoments_files/image007.png)
![](I/ERMMTBinomialLossDistributionMoments_files/image008.png)
Likewise:
![](I/ERMMTBinomialLossDistributionMoments_files/image009.png)
![](I/ERMMTBinomialLossDistributionMoments_files/image010.png)
The variance of the loss distribution is:
![](I/ERMMTBinomialLossDistributionMoments_files/image011.png)
![](I/ERMMTBinomialLossDistributionMoments_files/image012.png)
![](I/ERMMTBinomialLossDistributionMoments_files/image013.png)
Thus binomial distribution has mean
and
variance
.
As
, the
Central Limit Theorem CLT implies that the binomial distribution tends to a normal distribution
with the same mean and variance, i.e. to
where
is the
cumulative normal distribution.