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.:
The mean and the variance of the portfolio loss distribution
can be found as follows. We note that:
The mean of the loss distribution is given by:
Likewise:
The variance of the loss distribution is:
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.