# 3.14 The Cornish-Fisher Expansion

The Cornish-Fisher expansion is a formula for approximating quantiles of a random variable based only on its first few cumulants. In this section, we define cumulants, specify the Cornish-Fisher expansion, and present an example.

###### 3.14.1 Cumulants

The **cumulants** of a random variable *X* are conceptually similar to its moments. They are defined, somewhat abstrusely, as those values κ* _{r}* such that the identity

[3.200]

holds for all *t*. Cumulants of a random variable *X* can—see Stuart and Ord (1994)—be expressed in terms of its mean μ and central moments μ* _{r}* =

*E*[(

*X*– μ)

*]. Expressions for the first five cumulants are*

^{r}[3.201]

[3.202]

[3.203]

[3.204]

[3.205]

###### 3.14.2 Cornish-Fisher Expansion (Five Cumulants)

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

[3.206]

where is the *q*-quantile of *Z* ~ *N*(0,1). The expansion is not like, say, the Taylor series expansion, which you can truncate at any point. You must use all terms. If you want a shorter or longer expansion, look up a version of the expansion for less or more cumulants. Using a version with more than five cumulants will not necessarily produce a better approximation.

Although [3.206] 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 μ and standard deviation σ. Simply define the **normalization** of *X* as

[3.207]

which has mean 0 and standard deviation 1. Central moments of *X** can be calculated from central moments of *X* with

[3.208]

where σ = is the standard deviation of *X*. Apply the Cornish-Fisher expansion to obtain the *q*-quantile *x** of *X**. The corresponding *q*-quantile *x* of *X* is then

[3.209]

###### 3.14.2 Example

Let’s use the Cornish-Fisher expansion to approximate the .10-quantile of the random variable *Y* defined by [3.168] in our earlier example. From the first five moments of *Y* provided in Exhibit 3.26, we calculate the central moments of *Y* and the central moments and cumulants of the normalization *Y** of *Y*. Results are indicated in Exhibit 3.27.

*Y*, central moments of

*Y**, and cumulants of

*Y** are calculated from formulas [3.11], [3.208], and [3.201] through [3.205].

The Cornish-Fisher expansion [3.206] yields the .10-quantile of *Y** as –1.123. Applying [3.209], we obtain the .10-quantile of *Y* as –5.029.

###### Exercises

Using a spreadsheet and inputs from Exhibit 3.27, reproduce the results from the example of this section.

Solution