Cornish–Fisher expansion: Definition & Examples

The Cornish–Fisher expansion is an asymptotic approximation to a distribution's quantiles using its cumulants. Its familiar normal-based form adjusts a normal quantile for skewness and excess kurtosis. For a risk analyst, it offers a quick estimate of how a loss threshold might change when a normal model misses asymmetry or tail shape. It does not establish that the corrected threshold has its stated coverage.

An Edgeworth expansion approximates a distribution function, answering “what probability lies below this value?” Cornish–Fisher formally inverts that expansion, answering “what value corresponds to this probability?” The two have different inputs and outputs, even though their corrections use related moments. Chernozhukov, Fernández-Val and Galichon, §§1 and 3.

The correction, with conventions made explicit

Let $L$ be a loss over a fixed horizon: positive values mean losses and negative values mean gains. Write

\mu=\mathbb{E}[L],\qquad \sigma^2=\text{Var}(L)>0,\qquad X=\frac{L-\mu}{\sigma}.

Define skewness $s=\mathbb{E}X^3]$ and [excess kurtosis $k=\mathbb{E}[X^4]-3$ . The normal distribution has $s=k=0$ . If a data source reports raw kurtosis $\beta_2=\mathbb{E}[X^4]$ , use $k=\beta_2-3$ here.

For cumulative probability

p

, let

z_p=\Phi^{-1}(p)

be the standard normal quantile. Retaining the linear-skewness, excess-kurtosis and skewness-squared terms gives

\begin{aligned} z_p^{\mathrm{CF}}={}&z_p +\frac{s}{6}(z_p^2-1) +\frac{k}{24}(z_p^3-3z_p)\\ &-\frac{s^2}{36}(2z_p^3-5z_p),\\ q_p(L)\approx{}&\mu+\sigma z_p^{\mathrm{CF}}. \end{aligned}

These are the first three polynomial orders in the normal-based expansion. For a standardised mean of independent, identically distributed observations with suitable regularity, skewness scales as

n^{-1/2}

and excess kurtosis as

n^{-1}

. The

s^2

term therefore belongs at the same asymptotic order as the

k

term. Dropping it while keeping kurtosis changes the approximation. Chernozhukov, Fernández-Val and Galichon, §3.1, equations (5)–(6).

The direction of the first correction follows from inversion. At first order in $s$ , the standardised Edgeworth CDF is

F_X(x)\approx\Phi(x)-\frac{s}{6}(x^2-1)\phi(x).

Substitute $x=z_p+\delta$ and retain first-order terms. Since $\Phi(z_p)=p$ , solving $F_X(z_p+\delta)\approx p$ gives $\delta=s(z_p^2-1)/6$ . Higher-order inversion also generates the skewness-squared contribution; it is not an additional fourth-moment estimate.

A one-day loss estimate

Consider a fictional fixed-exposure portfolio, with one-day loss measured as a fraction of its starting exposure. Suppose the analyst uses

\mu=0,\qquad\sigma=0.02,\qquad s=0.5,\qquad k=1.

The fourth standardised moment is therefore $4$ , not $1$ . At $p=95\%$ , $z_p=1.644854$ :

Contribution to the standardised quantile	Value
Normal baseline	$1.644854$
Linear skewness	$+0.142129$
Excess kurtosis	$-0.020181$
Skewness squared	$-0.004696$
Cornish–Fisher total	$1.762106$

Thus

q_{0.95}^{\mathrm{CF}}(L)=0.02(1.762106)\approx0.035242=3.5242\%.

The normal baseline is

3.2897\%

. The correction increases the estimated threshold by 23.45 basis points of starting exposure. Used as a value-at-risk estimate, this is an approximate one-day 95% loss quantile, not a demonstrated 5% exceedance rate.

Fictional one-day loss quantile curves from 80% to 99% compare the normal baseline with Cornish–Fisher for mean zero, standard deviation 2%, skewness 0.5 and excess kurtosis 1. At 95%, normal loss is 3.2897% and the corrected estimate is 3.5242%; the three standardised corrections are plus 0.142129, minus 0.020181 and minus 0.004696.

Original fictional calculation. The curves evaluate the displayed formula across probability levels; they are not empirical loss estimates or a fitted probability distribution.

Notice the negative kurtosis contribution at 95%. Here

z_p<\sqrt{3}

, so

z_p^3-3z_p<0

. Positive excess kurtosis does not raise every upper quantile in this truncated formula. The net change also depends on skewness and its squared term.

Sign conventions matter just as much. If $R$ is a simple return and $L=-R$ is the corresponding loss fraction, then $s_L=-s_R$ while $k_L=k_R$ . Where the continuous return CDF is strictly increasing at the relevant quantile, $q_p(L)=-q_{1-p}(R)$ . Using return skewness directly in an upper-loss-quantile calculation reverses the asymmetric correction.

A quantile curve can turn backwards

A genuine quantile function cannot decrease as

p

increases. A truncated polynomial can. This defect is documented in both the quantile-expansion literature and financial applications. Jaschke, §4.

For a concrete counterexample, take a synthetic Student-

t

variable with

4.6

degrees of freedom and rescale it to variance one. It has mean zero, skewness zero and finite excess kurtosis

10

. Inserting these actual moments gives

z_p^{\mathrm{CF}}=z_p+\frac{10}{24}(z_p^3-3z_p).

Its derivative with respect to $z_p$ is $-1/4+(5/4)z_p^2$ , which is negative near zero. In particular:

Probability	Actual standardised Student- $t$ quantile	Cornish–Fisher estimate
$40\%$	$-0.20179$	$+0.05656$
$60\%$	$+0.20179$	$-0.05656$

The approximate 40th percentile exceeds the approximate 60th percentile. These cannot be quantiles of one distribution in that order, despite the input moments coming from a valid distribution.

A unit-variance Student t distribution with 4.6 degrees of freedom has increasing actual quantiles. The Cornish–Fisher curve with skewness zero and excess kurtosis 10 decreases around the median, reporting 40th percentile plus 0.05656 and 60th percentile minus 0.05656, compared with actual values minus 0.20179 and plus 0.20179.

Original analytic counterexample, without sampled data. The actual curve is the Student- $t$ quantile multiplied by $\sqrt{2.6/4.6}$ to give standard deviation one. Shading marks the interval between the two reported probability levels.

What to check before using the number

Four finite moments are needed to form the displayed inputs; their existence alone does not establish an accurate expansion. Sample skewness and kurtosis can also be unstable, especially when a few extreme observations dominate them. The sample size used to estimate a portfolio's moments is not automatically the $n$ that makes its loss distribution a normalising sum.

Inspect the correction sizes and quantile ordering over the probability range you need. Far-tail levels make the polynomial terms large, and adding more orders need not improve the answer for a fixed distribution. Compare with a justified distributional model, numerical integration or sufficiently informative loss simulations, and assess forecast exceedances separately. Jaschke, §§3–4.

Even an increasing polynomial curve need not reproduce the moments fed into the approximation or match the underlying loss distribution. Monotonicity repairs such as increasing rearrangement address ordering; they do not by themselves validate a particular VaR forecast. Chernozhukov, Fernández-Val and Galichon, §§2–3.

For the normal baseline and standardisation used here, Normal Distribution — Mathematics Primer develops the probability calculation. A public preview is available; the full lesson is available with a free signed-in account.

Sources

Chernozhukov, V., Fernández-Val, I. and Galichon, A., Rearranging Edgeworth–Cornish–Fisher Expansions, §§1–3: inverse expansion, first correction polynomials and monotonicity repairs.
Jaschke, S. R., The Cornish-Fisher-Expansion in the Context of Delta-Gamma-Normal Approximations, version 1.41, 4 December 2001, §§3–4: construction, asymptotic ordering, tail behaviour and failure of monotonicity or convergence with increasing order.
SciPy documentation, scipy.stats.t, Methods and Notes: Student- $t$ density, quantiles and scale convention used to reproduce the analytic counterexample.

The correction, with conventions made explicit

A one-day loss estimate

A quantile curve can turn backwards

What to check before using the number

Sources

Related terms

Normal distribution

Quantile function

Value at risk