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The Cornish-Fisher
expansion is a formula for approximating
quantiles of a random variable
based only on its first few cumulants. In finance, it is often used in
value-at-risk (VaR) calculations.
The cumulants of a random variable X
are conceptually similar to its moments. They are defined, somewhat
abstrusely, as those values
such that the identity
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[1] |
holds for all t. Cumulants of a random variable X can—see
Stuart and Ord (1994)—be expressed in terms of its
mean
= E(X) and central moments
 .
Expressions for the first five cumulants are
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[2] |
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[3] |
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[4] |
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[5] |
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[6] |
Suppose X has mean 0 and
standard deviation 1.
Cornish and Fisher (1937) provide an expansion for
approximating the q-quantile,
 ,
of X based upon its cumulants. Using the first five cumulants, the
expansion is
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[7] |
where
is the q-quantile of a
standard normal random
variable Z. Although [7] applies only if X
has mean 0 and standard deviation 1, we can still use it to approximate
quantiles if X has some other mean
and standard deviation
 .
Simply define the normalization of X as
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[8] |
which has mean 0 and standard deviation 1. Central moments of X*
can be calculated from central moments of X with
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[9] |
Apply the Cornish-Fisher expansion to obtain the q-quantile x*
of X*. The corresponding q-quantile x of X is
then
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[10] |
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 central limit theorem
A theorem that explains why the normal distribution plays such an important role
in probability theory.
chi-squared distribution
If you square a normal random variable, the result is a
chi-squared random variable.
expected
value A parameter describing the "center of gravity" of a
distribution.
kurtosis A parameter describing the peakedness and tails of a
distribution.
normal distribution
A continuous probability distribution whose probability density
function has a "bell" shape.
quadratic VaR
measure A category of value-at-risk measures that
are
applicable to quadratic portfolios.
quantile A notion from
probability that can be used as a parameter.
skewness A parameter that
describes the lack of symmetry of a distribution.
standard deviation A
parameter describing the dispersion of a distribution. |
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Holton (2003)
describes how the Cornish-Fisher expansion is used in
value-at-risk calculations.
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