The change in the portfolio value, , can be expressed as a sum of
independent random variables that are quadratic functions of standard normal
random variables
by means of the solution of the generalized
eigenvalue problem
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The characteristic function of a non-central
variate
(
, with standard normal
) is known analytically:
Numerical Fourier-inversion of (1.3) can be used to compute an
approximation to the cumulative distribution function (cdf) of
.
(The
-quantile is computed by root-finding in
.) The cost
of the Fourier-inversion is
, the cost of the function
evaluations is
, and the cost of the eigenvalue decomposition is
. The cost of the eigenvalue decomposition dominates the other
two terms for accuracies of one or two decimal digits and the usual number of
risk factors of more than a hundred. Instead of a full spectral decomposition,
one can also just reduce
to tridiagonal form
. (
is tridiagonal and
is orthogonal.) Then
the evaluation of the characteristic function in (1.4) involves
the solution of a linear system with the matrix
, which costs only
operations. An alternative route is to reduce
to
Hessenberg form
or do a Schur decomposition
. (
is Hessenberg and
is orthogonal. Since
has the same eigenvalues as
and they are all
real,
is actually triangular instead of quasi-triangular in the general
case, Anderson et al. (1999).
The evaluation of (1.5) becomes
, since it involves the solution of a linear system with the
matrix
or
, respectively. Reduction to tridiagonal,
Hessenberg, or Schur form is also
, so the asymptotics in the
number of risk factors
remain the same in all cases. The critical
,
above which the complete spectral decomposition
fast evaluation via
(1.3) is faster than the reduction to tridiagonal or Hessenberg
form
slower evaluation via (1.4) or (1.5) remains
to be determined empirically for given
on a specific machine.
The computation of the cumulant generating function and the characteristic function from the diagonalized form is implemented in the following quantlets:
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The advantage of the Cornish-Fisher approximation is that it is based on the cumulants, which can be computed without any matrix decomposition:
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The computation of all cumulants up to a certain order directly from
is implemented in the quantlet
VaRcumulantsDG
, while
the computation of a single cumulant from the diagonal decomposition is
provided by
VaRcumulantDG
:
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Partial Monte-Carlo (or partial Quasi-Monte-Carlo) costs
operations
per sample. (If
is sparse, it may cost even less.) The number of
samples needed is a function of the desired accuracy. It is clear from the
asymptotic costs of the three methods that partial Monte Carlo will be
preferable for sufficiently large
.
While Fourier-inversion and Partial Monte-Carlo can in principal achieve any desired accuracy, the Cornish-Fisher approximations provide only a limited accuracy, as shown in the next section.