The simplest factor structures for a risk
model
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The simplest sort of factor-based risk model is one that has
an underlying multivariate normal distribution. Its structure then derives from
the covariance matrix underlying the multivariate normal distribution.
The simplest factor structure for such a risk model
involving different factors
has a covariance matrix which has the
form , where is
the identity matrix.
The next simplest factor structure is arguably one where the
covariance matrix is a diagonal matrix (i.e. the elements of ,
i.e. are non-zero only
when . Its correlation
matrix is therefore still the identity matrix.
The next simplest factor structure is arguably one where the
the correlation matrix, , has elements where if and
if where
is the same for
all such values of and .