1.3 Cornish-Fisher Approximations
1.3.1 Derivation
The Cornish-Fisher expansion can be derived in two steps. Let
denote
some base distribution and
its density function. The generalized
Cornish-Fisher expansion (Hill and Davis; 1968) aims to approximate an
-quantile
of
in terms of the
-quantile of
, i.e., the concatenated
function
. The key to a series expansion of
in terms of derivatives of
and
is Lagrange's inversion
theorem. It states that if a function
is implicitly defined by
 |
(1.6) |
and
is analytic in
, then an analytic function
can be developed
into a power series in a neighborhood of
(
):
,$](xfghtmlimg196.gif) |
(1.7) |
where
denotes the differentation operator. For a given probability
,
, and
this yields
.$](xfghtmlimg200.gif) |
(1.8) |
Setting
in (1.6) implies
and with the notations
,
(1.8) becomes the formal expansion
With
this can be written as
$](xfghtmlimg207.gif) |
(1.9) |
with
and
being the identity operator.
(1.9) is the generalized Cornish-Fisher expansion. The second step
is to choose a specific base distribution
and a series expansion for
. The classical Cornish-Fisher expansion is recovered if
is the
standard normal distribution,
is (formally) expanded into the
Gram-Charlier series, and the terms are re-ordered as described below.
The idea of the Gram-Charlier series is to develop the ratio of the moment
generating function of the considered random variable (
) and the moment generating function of the standard normal distribution
(
) into a power series at 0:
 |
(1.10) |
(
are the Gram-Charlier coefficients. They can be derived from the
moments by multiplying the power series for the two terms on the left hand
side.) Componentwise Fourier inversion yields the corresponding series for
the probability density
 |
(1.11) |
and for the cumulative distribution function (cdf)
 |
(1.12) |
(
und
are now the standard normal density and cdf. The
derivatives of the standard normal density are
, where the Hermite polynomials
form an orthogonal basis in the Hilbert space
of the square
integrable functions on
w.r.t. the weight function
. The
Gram-Charlier coefficients can thus be interpreted as the Fourier coefficients
of the function
in the Hilbert space
with the
basis
)
Plugging (1.12) into (1.9) gives the formal Cornish-Fisher
expansion, which is re-grouped as motivated by the central limit theorem.
Assume that
is already normalized (
,
)
and consider the normalized sum of independent random variables
with the distribution
,
. The moment generating function of the random variable
is
Multiplying out the last term shows that the
-th Gram-Charlier coefficient
of
is a polynomial expression in
, involving the
coefficients
up to
. If the terms in the formal Cornish-Fisher
expansion
$](xfghtmlimg233.gif) |
(1.13) |
are sorted and grouped with respect to powers of
, the classical
Cornish-Fisher series
 |
(1.14) |
results. (The Cornish-Fisher approximation for
results from setting
in the re-grouped series (1.14).)
It is a relatively tedious process to express the adjustment terms
correponding to a certain power
in the Cornish-Fisher expansion
(1.14) directly in terms of the cumulants
, see
(Hill and Davis; 1968). Lee developed a recurrence formula for the
-th adjustment term
in the Cornish-Fisher expansion, which is implemented in the
algorithm AS269 (Lee and Lin; 1992,1993). (We write the recurrence formula here,
because it is incorrect in (Lee and Lin; 1992).)
 |
(1.15) |
with
.
is a formal polynomial
expression in
with the usual algebraic relations between the summation
``+'' and the ``multiplication'' ``
''. Once
is multiplied out
in
-powers of
, each
is to be interpreted as the Hermite
polynomial
and then the whole term becomes a polynomial in
with
the ``normal'' multiplication ``
''.
denotes the scalar that
results when the ``normal'' polynomial
is evaluated at the fixed
quantile
, while
denotes the expression in the
-algebra.
This formula is implemented by the quantlet
- q
- =
CornishFisher
(z, n, cum)
Cornish-Fisher expansion for arbitrary orders for the standard normal
quantile z, order of approximation n, and the vector of
cumulants cum.
|
The following example prints the Cornish-Fisher approximation for
increasing orders for z=2.3 and cum=1:N:
Contents of r
[1,] 2 4.2527
[2,] 3 5.3252
[3,] 4 5.0684
[4,] 5 5.2169
[5,] 6 5.1299
[6,] 7 5.1415
[7,] 8 5.255
1.3.2 Properties
The qualitative properties of the Cornish-Fisher expansion are:

- If
is a sequence of distributions converging to the standard
normal distribution
, the Edgeworth- and Cornish-Fisher approximations
present better approximations (asymptotically for
) than the
normal approximation itself.

- The approximated functions
and
are not necessarily monotone.

has the ``wrong tail behavior'', i.e., the
Cornish-Fisher approximation for
-quantiles becomes less and less
reliable for
(or
).

- The Edgeworth- and Cornish-Fisher approximations do not necessarily
improve (converge) for a fixed
and increasing order of approximation,
.
For more on the qualitative properties of the Cornish-Fisher approximation see
(Jaschke; 2001). It contains also an empirical analysis of the error of the
Cornish-Fisher approximation to the 99%-VaR in real-world examples as well as
its worst-case error on a certain class of one- and two-dimensional
delta-gamma-normal models:

- The error for the 99%-VaR on the real-world examples - which
turned out to be remarkably close to normal - was about
,
which is more than sufficient. (The error was normalized with respect to the
portfolio's standard deviation,
.)

- The (lower bound on the) worst-case error for the one- and
two-dimensional problems was about
, which corresponds to a
relative error of up to 100%.
In summary, the Cornish-Fisher expansion can be a quick approximation with
sufficient accuracy in many practical situations, but it should not be used
unchecked because of its bad worst-case behavior.