Advances in Risk Management

4 DETERMINATION OF THE CAPITAL CHARGE FOR OPERATIONAL RISK

distribution function (or CDF) associated tof(x;θ). Then, the corresponding
log-likelihood function is

l(x;θ)=

∑N

i= 1

ln(fi(xi;θ)) (1.1)

where (x 1 ,...,xN) is the sample of observed ordinary losses. The maximum
likelihood estimates of the parametersθjare obtained by solving the system
of equations

δl

δθj

= 0

Frequency distribution

The frequency distribution models the occurrence of operational loss events
recorded by the bank. This distribution is by definition discrete. It is
frequently modeled either as a homogeneous Poisson or as a (negative)
binomial distribution. The choice between these distributions may appear
important as the intensity parameter is deterministic in the first case and
stochastic in the second (see Embrechtset al., 2003).
The mass function of the Poisson distribution is

Pr(N=x)=

e−λλx

x!

(1.2)

whereλis a positive integer. It can easily be estimated asλis equal to
both the mean and the variance of the Poisson distribution. Note also
the following nice property of the Poisson distribution: if X 1 , X 2 ,...,
Xm are m independent random variables and Xi∼Poisson(λi), then
X 1 +X 2 +···+Xm∼Poisson(λ 1 +λ 2 +···+λm).
The binomial distribution is given by

Pr(N=x)=

(

m

x

)

px(1−q)m−x (1.3)

where

(

m

x

)

is the binomial coefficient defined asm(m−1)...x(!m−x+1),p∈(0, 1)

andmisapositiveinteger. ContrarytothePoissoncase, themeanisnotequal
to the variance for this distribution, as mean=mpand variance=mp(1−p).
It follows that the mean is larger than the variance for the binomial
distribution.
Finally, the negative binomial distribution has the following mass
function

Pr(N=x)=

(

x+r− 1

x

)

pr(1−p)x (1.4)