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)^r ]. Expressions for the first five cumulants are

[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.

Exhibit 3.27: Central moments of 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.