Computation of the corrected Cornish–Fisher expansion using the response surface methodology: application to VaR and CVaR

Abstract

The Cornish–Fisher expansion is a simple way to determine quantiles of non-normal distributions. It is frequently used by practitioners and by academics in risk management, portfolio allocation, and asset liability management. It allows us to consider non-normality and, thus, moments higher than the second moment, using a formula in which terms in higher-order moments appear explicitly. This paper has two primary objectives. First, we resolve the classic confusion between the skewness and kurtosis coefficients of the formula and the actual skewness and kurtosis of the distribution when using the Cornish–Fisher expansion. Second, we use the response surface approach to estimate a function for these two values. This helps to overcome the difficulties associated with using the Cornish–Fisher expansion correctly to compute value at risk. In particular, it allows a direct computation of the quantiles. Our methodology has many practical applications in risk management and asset allocation.

Appendices

A Appendix: Quantile functions

The quantile function (or inverse cumulative distribution function) of the probability distribution of a random variable specifies, for a given probability, the value that the random variable will fall below, with the specified probability. In fact it is an alternative to the probability density function (pdf).

Let X be a random variable with a distribution function F, and let $\alpha \in (0,1)$. A value of x such that $F(x)=P(X \le x) = \alpha$ is called a quantile of order $\alpha$ for the distribution. Then, we can define the quantile function by:

$$\begin{aligned} q_\alpha (X)\equiv F^{-1}(\alpha )={ inf} \left\{ x \in \mathbb {R}: F(x)\ge \alpha \right\} ,\alpha \in (0,1). \end{aligned}$$

Thus, the quantile function $q_\alpha (X)$ yields the value that the random variable of the given distribution will fail to exceed, with probability $\alpha$.

B Appendix: The Cornish–Fisher procedure

The Cornish–Fisher expansion is a useful tool for quantile estimations. For any $\alpha \in (0,1)$, the upper $\alpha$th-quantile of $F_n$ is defined by $q_n(\alpha )={ inf} \left\{ x: F_n(x)\ge \alpha \right\}$, where $F_n$ denotes the cumulative distribution function of $\xi _n=(\sqrt{n}/\sigma )(\bar{X}-\mu )$, and $\bar{X}$ is the sample mean of independent and identically distributed observations $X_1,\ldots ,X_n$. If $z_\alpha$ denotes the upper $\alpha$th-quantile of N(0, 1), then the fourth-order Cornish–Fisher expansion can be expressed as follows:

$$\begin{aligned} q_n(\alpha )= z_\alpha +\frac{1}{6\sqrt{n}}(z_\alpha ^2-1)S+\frac{1}{24n}(z_\alpha ^3-3z_\alpha )(K-3)-\frac{1}{36n}(2z_\alpha ^3-5z_\alpha )S^2+o(n^{3/2}), \nonumber \\ \end{aligned}$$

(23)

where S and K are the skewness and kurtosis of the observations $X_i$, respectively.

The Cornish–Fisher expansion is useful because it allows one to obtain more accurate results compared to those acquired using the central limit theorem (CLT) approximation, which is the same as $z_\alpha$ defined in the main text. A demonstration and example of the greater accuracy provided by the Cornish–Fisher expansion compared to the CLT approximation is reported by Chernozhukov et al. (2010).

In general, relation (23) grants a non-monotonic character to $q_n(\alpha )$, which means that the true distribution’s ordering of quantiles is not preserved. Thus, the Cornish–Fisher expansion formula is valid only if the skewness and kurtosis coefficients of the distribution meet a particular constraint. This domain of validity has been studied by Maillard (2012), among others. Monotonicity requires the derivative of $z_{{ CF},\alpha }$, relative to $z_\alpha$, to be non-negative. This leads to the following constraint, which implicitly defines the domain of validity (D) of the Cornish–Fisher expansion:

$$\begin{aligned} \frac{S^2}{9}-4 \left( \frac{K-3}{8}-\frac{S^2}{6}\right) \left( 1-\frac{K-3}{8}-\frac{5S^2}{36}\right) \le 0. \end{aligned}$$

(24)

In practice, this constraint is rarely considered, because S and K are generally considered to be small in finance applications.

C Online appendix: Quality of the estimation results for case 2 to case 5

See Figs. 6, 7, 8 and 9.

Fig. 6

Errors analysis for Case 2: $0 < S\le 0.5, 5 \le K\le 40$. a$\widehat{K}_c$ on $K_c$. b$\widehat{S}_c$ on $S_c$. c Err(K). d Err(S). d Err(K) on S. e Err(S) on S. f Err(K) on K. g Err(S) on K

Fig. 7

Errors analysis for Case 3: $S\ge 0.5, K\le 5$. a$\widehat{K}_c$ on $K_c$. b$\widehat{S}_c$ on $S_c$. c Err(K). d Err(S). e Err(K) on S. f Err(S) on S. g Err(K) on K. h Err(S) on K

Fig. 8

Errors analysis for Case 4: $0.25\le S<0.5, K\le 5$. a$\widehat{K}_c$ on $K_c$. b$\widehat{S}_c$ on $S_c$. c Err(K). d Err(S). e Err(K) on S. f Err(S) on S. g Err(K) on K. h Err(S) on K

Fig. 9

Errors analysis for Case 5: $0< S<0.25, K\le 5$. a$\widehat{K}_c$ on $K_c$. b$\widehat{S}_c$ on $S_c$. c Err(K). d Err(S). e Err(K) on S. f Err(S) on S. g Err(K) on K. h Err(S) on K