The form of the distribution

The assumption of normality aims to simplify the measurement of risk. If security
returns are normally distributed with stable moments, then the two parameters, the
mean and the variance, are sufficient to describe it. Stability implies that the
probability of a specified portfolio loss is the same for each day.
These assumptions, however, are offset by the overwhelming evidence suggesting
the contrary which is relevant to measuring risk, namely the existence of fat tails or
leptokurtosis in the distribution that exceeds those of the normal. Leptokurtosis in
the distribution of security returns was reported as early as the 1950s. This arises
where the empirical distribution of daily changes in stock returns have more observations
around the means and in the extreme tails than that of a normal distribution.
Consequently, the non-normality in the distribution has led some studies to suggest
that attaching alternative probability distributions may be more representative of the
data and observed leptokurtosis. One such is the Paretian distribution, which has
the characteristic exponent. This is a peakness parameter that measures the tail of
the distribution. However, the problem with this distribution is that unlike normal
distribution, the variance and higher moments are not defined except as a special
case of the normal. Another distribution suggested is the Student t-distribution
which has fatter tails than that of the normal, assuming that the degrees of freedom
are less than unity. While investigations have found that t-distributions adequately
describe weekly and monthly data, as the interval length in which the security
returns are measured increases, the t-distribution tends to converge to a normal.
Other investigations have suggested that security returns follow a mixture of distributions
where the distribution is described as a combination of normal distributions
that possess different variances and possible different means.