1.3 Cornish-Fisher Approximations (original) (raw)

1.3.1 Derivation

The Cornish-Fisher expansion can be derived in two steps. Let $ \Phi$ denote some base distribution and $ \phi$ its density function. The generalized Cornish-Fisher expansion (Hill and Davis; 1968) aims to approximate an $ \alpha$-quantile of $ F$ in terms of the $ \alpha$-quantile of $ \Phi$, i.e., the concatenated function $ F^{-1}\circ \Phi$. The key to a series expansion of $ F^{-1}\circ \Phi$ in terms of derivatives of $ F$ and $ \Phi$ is Lagrange's inversion theorem. It states that if a function $ s \mapsto t$ is implicitly defined by

$\displaystyle t = c + s \cdot h(t)$ (1.6)

and $ h$ is analytic in $ c$, then an analytic function $ f(t)$ can be developed into a power series in a neighborhood of $ s=0$ ($ t=c$):

$\displaystyle f(t) = f(c) + \sum_{r=1}^{\infty} \frac{s^{r}}{r!} D^{r-1}[f' \cdot h^{r}](c),$ (1.7)

where $ D$ denotes the differentation operator. For a given probability $ c = \alpha$, $ f = \Phi^{-1}$, and $ h = (\Phi-F)\circ\Phi^{-1}$ this yields

$\displaystyle \Phi^{-1}(t) = \Phi^{-1}(\alpha) + \sum_{r=1}^{\infty} (-1)^{r} \frac{s^{r}}{r!} D^{r-1}[((F-\Phi)^{r}/\phi)\circ\Phi^{-1}](\alpha).$ (1.8)

Setting $ s=1$ in (1.6) implies $ \Phi^{-1}(t)=F^{-1}(\alpha)$and with the notations $ x = F^{-1}(\alpha)$, $ z = \Phi^{-1}(\alpha)$(1.8) becomes the formal expansion

$\displaystyle x = z + \sum_{r=1}^{\infty} (-1)^{r} \frac{1}{r!} D^{r-1}[((F-\Phi)^{r}/\phi)\circ\Phi^{-1}](\Phi(z)). $

With $ a = (F-\Phi)/\phi$ this can be written as

$\displaystyle x = z + \sum_{r=1}^{\infty} (-1)^{r} \frac{1}{r!} D_{(r-1)}[a^{r}](z)$ (1.9)

with $ D_{(r)}=(D+\frac{\phi'}{\phi})(D+2\frac{\phi'}{\phi})\ldots(D+r\frac{\phi'}{\phi})$and $ D_{(0)}$ being the identity operator.

(1.9) is the generalized Cornish-Fisher expansion. The second step is to choose a specific base distribution $ \Phi$ and a series expansion for$ a$. The classical Cornish-Fisher expansion is recovered if $ \Phi$ is the standard normal distribution, $ a$ 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 ( $ M(t)=\ensuremath{\mathrm{E}}e^{t\Delta V}$) and the moment generating function of the standard normal distribution ( $ e^{t^{2}/2}$) into a power series at 0:

$\displaystyle M(t) e^{-t^{2}/2} = \sum_{k=0}^{\infty} c_{k} t^{k}.$ (1.10)

($ c_{k}$ 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

$\displaystyle f(x) = \sum_{k=0}^{\infty} c_{k} (-1)^{k} \phi^{(k)}(x)$ (1.11)

and for the cumulative distribution function (cdf)

$\displaystyle F(x) = \Phi(x) - \sum_{k=1}^{\infty} c_{k} (-1)^{k-1} \phi^{(k-1)}(x).$ (1.12)

($ \phi$ und $ \Phi$ are now the standard normal density and cdf. The derivatives of the standard normal density are $ (-1)^{k}\phi^{(k)}(x)=\phi(x)H_{k}(x)$, where the Hermite polynomials $ H_{k}$form an orthogonal basis in the Hilbert space $ L^{2}(\mathbb{R},\phi)$ of the square integrable functions on $ \mathbb{R}$ w.r.t. the weight function $ \phi$. The Gram-Charlier coefficients can thus be interpreted as the Fourier coefficients of the function $ f(x)/\phi(x)$ in the Hilbert space $ L^{2}(\mathbb{R},\phi)$ with the basis $ \{H_{k}\}$ $ f(x)/\phi(x) = \sum_{k=0}^{\infty} c_{k} H_{k}(x).$) 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 $ \Delta V$ is already normalized ( $ \kappa_{1}=0$, $ \kappa_{2}=1$) and consider the normalized sum of independent random variables $ \Delta V_{i}$with the distribution $ F$, $ S_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n} \Delta V_{i}$. The moment generating function of the random variable $ S_{n}$ is

$\displaystyle M_{n}(t) = M(t/\sqrt{n})^{n} = e^{t^{2}/2} (\sum_{k=0}^{\infty} c_{k} t^{k} n^{-k/2})^{n}. $

Multiplying out the last term shows that the $ k$-th Gram-Charlier coefficient$ c_{k}(n)$ of $ S_{n}$ is a polynomial expression in ![$ n^{-1/2} ](https://web.archive.org/web/20040920011235im/http://www.quantlet.com/mdstat/scripts/xfg/html/xfghtmlimg55.gif),involvingthecoefficients!, involving the coefficients ![](https://web.archive.org/web/20040920011235im/​http://www.quantlet.com/mdstat/scripts/xfg/html/xfghtmlimg55.gif),involvingthecoefficients! c_{i}$ up to $ i=k$. If the terms in the formal Cornish-Fisher expansion

$\displaystyle x = z + \sum_{r=1}^{\infty} (-1)^{r} \frac{1}{r!} D_{(r-1)}\left[\left(-\sum_{k=1}^{\infty} c_{k}(n) H_{k-1}\right)^{r}\right](z)$ (1.13)

are sorted and grouped with respect to powers of $ n^{-1/2} $, the classical Cornish-Fisher series

$\displaystyle x = z + \sum_{k=1}^{\infty} n^{-k/2} \xi_{k}(z)$ (1.14)

results. (The Cornish-Fisher approximation for $ \Delta V$ results from setting$ n=1$ in the re-grouped series (1.14).)

It is a relatively tedious process to express the adjustment terms $ \xi_{k}$correponding to a certain power $ n^{-k/2}$ in the Cornish-Fisher expansion (1.14) directly in terms of the cumulants $ \kappa_{r}$, see (Hill and Davis; 1968). Lee developed a recurrence formula for the $ k$-th adjustment term$ \xi_{k}$ 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).)

$\displaystyle \xi_{k}(H) = a_{k}H^{*(k+1)} - \sum_{j=1}^{k-1} \frac{j}{k} (\xi_{k-j}(H)-\xi_{k-j})*(\xi_{j}-a_{j}H^{*(j+1)})*H,$ (1.15)

with $ a_{k}=\frac{\kappa_{k+2}}{(k+2)!}$. $ \xi_{k}(H)$ is a formal polynomial expression in $ H$ with the usual algebraic relations between the summation ``+'' and the ``multiplication'' ``$ *$''. Once $ \xi_{k}(H)$ is multiplied out in $ *$-powers of $ H$, each $ H^{*k}$ is to be interpreted as the Hermite polynomial $ H_{k}$ and then the whole term becomes a polynomial in $ z$ with the ``normal'' multiplication ``$ \cdot$''. $ \xi_{k}$ denotes the scalar that results when the ``normal'' polynomial $ \xi_{k}(H)$ is evaluated at the fixed quantile $ z$, while $ \xi_{k}(H)$ denotes the expression in the$ (+,*)$-algebra.

This formula is implemented by the quantlet

q

= 2818 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 $ F_{m}$ is a sequence of distributions converging to the standard normal distribution $ \Phi$, the Edgeworth- and Cornish-Fisher approximations present better approximations (asymptotically for $ m\to\infty$) than the normal approximation itself.

$ -$

The approximated functions $ \tilde{F}$ and $ \tilde{F}^{-1}\circ\Phi$ are not necessarily monotone.

$ -$

$ \tilde{F}$ has the ``wrong tail behavior'', i.e., the Cornish-Fisher approximation for $ \alpha$-quantiles becomes less and less reliable for $ \alpha\to 0$ (or $ \alpha\to 1$).

$ -$

The Edgeworth- and Cornish-Fisher approximations do not necessarily improve (converge) for a fixed $ F$ and increasing order of approximation,$ k$.

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 $ 10^{-6}\sigma$, which is more than sufficient. (The error was normalized with respect to the portfolio's standard deviation, $ \sigma$.)

$ -$

The (lower bound on the) worst-case error for the one- and two-dimensional problems was about $ 1.0\sigma$, 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.