Corrected CornishFisher Expansion: Improving the Accuracy of Modified ValueatRisk
Modified ValueatRisk (mVaR) is a parametric approach to computing ValueatRisk introduced by Zangari^{1} that adjusts Gaussian ValueatRisk for asymmetry and fat tails present in financial asset returns^{2} through a mathematical technique called Cornish–Fisher expansion.
Since its publication, mVaR has been widely adopted by academic researchers, financial regulators^{3} and practitioners, who typically highlight its straightforward numerical implementation and its ease of interpretation thanks to its explicit form^{4}.
Nevertheless, it has been observed in practice that mVaR only works well for nonnormal distributions that are close to the Gaussian distribution and for tail probabilities which are not too small^{5}.
In this post, I will explain why in the light of the results of Maillard^{6} and Lamb et al.^{7}, who show that mVaR accuracy is related to the mathematics of the CornishFisher expansion.
I will also empirically demonstrate, using Bitcoin and the SPY ETF, that the method proposed by Maillard^{6} to improve mVaR accuracy makes it usable for moderately to highly nonnormal distributions as well as for small tail probabilities^{8}.
Mathematical preliminaries
ValueatRisk
The (percentage) ValueatRisk (VaR) of a portfolio of financial assets corresponds to the percentage of portfolio wealth that can be lost over a certain time horizon and with a certain probability^{9}.
More formally, the ValueatRisk $VaR_{\alpha}$ of a portfolio over a time horizon $T$ (1 day, 10 days…) and at a confidence level $\alpha$% $\in ]0,1[$ (95%, 97.5%, 99%…) can be defined^{5} as the opposite of the lower $1  \alpha$ quantile of the portfolio return^{10} distribution over the time horizon $T$
\[\text{VaR}_{\alpha} (X) =  \inf_{x} \left\{x \in \mathbb{R}, P(X \leq x) \geq 1  \alpha \right\}\], where $X$ is a random variable representing the portfolio return over the time horizon $T$.
This formula is also equivalent^{11} to
\[\text{VaR}_{\alpha} (X) =  F_X^{1}(1  \alpha)\], where $F_X^{1}$ is the inverse cumulative distribution function, also called the quantile function, of the random variable $X$.
Gaussian ValueatRisk
The previous definition of VaR is not directly usable, because it requires to specify the portfolio return distribution.
One possible approach is to approximate the portfolio return distribution by its empirical distribution, in which case the associated VaR is called historical ValueatRisk (HVaR).
Another possible approach is to approximate the portfolio return distribution by a given probability distribution, in which case the associated VaR is called parametric ValueatRisk.
When this distribution is chosen to be the Gaussian distribution $\mathcal{N}_{\mu, \sigma^2}$, that is, when $X \sim \mathcal{N} \left( \mu, \sigma^2 \right)$ with $\mu$ the location parameter and $\sigma$ the scale parameter, the associated VaR is called Gaussian ValueatRisk (GVaR) and is computed through the formula^{12}
\[\text{GVaR}_{\alpha} (X) =  \mu  \sigma z_{1  \alpha}\], where:
 The location parameter $\mu$ and the scale parameter $\sigma$ are usually^{2} estimated by their sample counterparts computed from past portfolio returns
 $z_{1  \alpha}$ is the $1  \alpha$ quantile of the standard normal distribution
Modified ValueatRisk
Approximating a portfolio return distribution by a Gaussian distribution might be appropriate in some cases, depending on the assets present in the portfolio and on the time horizon^{13}, but generally speaking, financial assets exhibit skewed and fattailed return distributions^{2}, so that it makes more sense to also consider higher moments than just the first two.
For this reason, Zangari^{1} proposed to approximate the $1  \alpha$ quantile of the portfolio return distribution by a fourth order Cornish–Fisher expansion of the $1  \alpha$ quantile of the standard normal distribution, which allows to take into account skewness and kurtosis present in the portfolio return distribution.
The resulting VaR, called modified ValueatRisk or sometimes CornishFisher ValueatRisk (CFVaR), is computed through the formula^{12}
\[\text{mVaR}_{\alpha} (X) =  \mu  \sigma \left[ z_{1\alpha} + (z_{1\alpha}^2  1) \frac{\kappa}{6} + (z_{1\alpha}^33z_{1\alpha}) \frac{\gamma}{24} (2z_{1\alpha}^35z_{1\alpha})\frac{\kappa^2 }{36} \right]\], where the location parameter $\mu$, the scale parameter $\sigma$, the skewness parameter $\kappa$ and the excess kurtosis parameter $\gamma$ are usually^{2} estimated by their sample counterparts computed from past portfolio returns
To be noted that using this formula to compute VaR is equivalent to making the assumption that the portfolio return distribution follows what could be called a CornishFisher distribution^{7} $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$, whose inverse cumulative distribution function is given by
\[F_X^{1}(u) = \mu + \sigma \left[ z_u + (z_u^2  1) \frac{\kappa}{6} + (z_u^33z_u) \frac{\gamma}{24} (2z_u^35z_u)\frac{\kappa^2}{36} \right]\], where:
 $\mu$ is a location parameter
 $\sigma$ is a scale parameter
 $\kappa$ is a skewness parameter
 $\gamma$ is an excess kurtosis parameter
 $u \in ]0,1[$
 $z_u = \Phi^{1}(u)$, with $\Phi$ the standard normal distribution function
, which is also equivalent^{7} to making the assumption that
\[X \sim \mu + \sigma \left[ Z + (Z^2  1) \frac{\kappa}{6} + (Z^33Z) \frac{\gamma}{24} (2Z^35Z)\frac{\kappa^2}{36} \right]\], where:
 $\mu$ is a location parameter
 $\sigma$ is a scale parameter
 $\kappa$ is a skewness parameter
 $\gamma$ is an excess kurtosis parameter
 $Z$ is a standard normal random variable, i.e. $Z \sim \mathcal{N} \left( 0, 1 \right)$
The lack of accuracy of modified ValueatRisk
Illustration
Figure 1 compares, over the period 01 February 1993  04 April 2023, the empirical distribution of the SPY ETF daily returns^{14} to the CornishFisher distribution $\mathcal{CF}_{\mu_s, \sigma_s, \kappa_s, \gamma_s}$ with parameters:
 $\mu_s \approx 0.000367$, the sample mean of the SPY ETF returns over the considered period
 $\sigma_s \approx 0.011921$, the sample standard deviation of the SPY ETF returns over the considered period
 $\kappa_s \approx 0.287409$, the sample skewness of the SPY ETF returns over the considered period
 $\gamma_s \approx 10.898897$, the sample excess kurtosis of the SPY ETF returns over the considered period
On this figure, it is visible that the CornishFisher distribution does not accurately approximate the empirical distribution of the SPY ETF returns.
The same also applies to the left tail of the empirical distribution of the SPY ETF returns, as can be seen in Figure 2.
On top of this poor approximation accuracy, and maybe even worse, taking a closer look at Figure 1 also reveals that the CornishFisher distribution does not seem to be monotonous. For example, quantiles between 20% and 40% are positive while quantiles between 60% and 80% are negative! This means that the CornishFisher distribution is not a proper probability distribution^{15}.
What could explain these observations, while the CornishFisher expansion is supposed, by construction, to be able to approximate the quantiles of any distribution?
Let’s dig in Maillard^{6}!
The domain of validity of the CornishFisher expansion
Maillard^{6} notes that in order for the CornishFisher expansion to result in a welldefined quantile function, the skewness parameter $\kappa$ and the excess kurtosis parameter $\gamma$ must satisfy the constraints
\[ \kappa  \leq 6 \left( \sqrt{2}  1 \right)\] \[27 \gamma^2  (216 + 66 \kappa^2) \gamma + 40 \kappa^4 + 336 \kappa^2 \leq 0\]These two constraints define the domain of validity of the CornishFisher expansion, represented in Figure 3.
When used outside of its domain of validity, the CornishFisher expansion is known to have several issues impacting its accuracy^{16}, among which nonmonotonous quantiles.
And as can be seen in Figure 4, this is exactly what happens in the case of the SPY ETF, with the parameters $\left( \kappa, \gamma \right) \approx (0.28740, 10.898897) $ clearly outside of the domain of validity of the CornishFisher expansion.
Hopefully, there is a way to circumvent the relative narrowness of the domain of validity of the CornishFisher expansion thanks to a regularization procedure called increasing rearrangement^{17} and described in details in Chernozhukov et al.^{18}
The impact of this procedure is illustrated in Figure 5, which compares the same two distributions as in Figure 1, except that the CornishFisher distribution has been rearranged.
The rearranged CornishFisher distribution is now monotonous, as it should be, but unfortunately, it only marginally better approximates the empirical distribution of the SPY ETF returns.
So, either all hope is lost w.r.t. using mVaR with moderately nonnormal return distributions or there is another problem hidden somewhere waiting to be found…
Let’s dig a little bit further in Maillard^{6}!
CornishFisher parameters v.s. actual moments
Maillard^{6} also notes that the scale, skewness and excess kurtosis parameters $\sigma$, $\kappa$ and $\gamma$ do not match the actual standard deviation $\sigma_{CF}$, skewness $ \kappa_{CF}$ and excess kurtosis $\gamma_{CF}$ of the CornishFisher distribution $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$.
More precisely, he establishes the following relationships
\[\begin{align} \mu_{CF} &= \mu \\ \sigma_{CF} &= \sigma \sqrt{ 1 + \frac{1}{96} \gamma^2 + \frac{25}{1296} \kappa^4  \frac{1}{36} \gamma \kappa^2 } \\ \kappa_{CF} &= f_1(\kappa, \gamma) \\ \gamma_{CF} &= f_2(\kappa, \gamma) \\ \end{align}\], where:
 $\mu_{CF}$, $\sigma_{CF}$, $ \kappa_{CF}$ and $\gamma_{CF}$ are the actual mean, standard deviation, skewness and excess kurtosis of the CornishFisher distribution $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$
 $\mu$, $\sigma$, $ \kappa$ and $\gamma$ are the location, scale, skewness and excess kurtosis parameters of the CornishFisher distribution $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$
 $f_1$ and $f_2$ are non linear functions, whose explicit formulas are provided in Maillard^{6}
As a consequence, when the sample moments of a return distribution are used as plugin estimators for the CornishFisher parameters, the actual moments of the resulting CornishFisher distribution differ from these sample moments!
Do they differ enough to create a real problem, though?
Reusing the SPY ETF example:
 The sample moments of the SPY ETF empirical return distribution are:
 $\mu_s \approx 0.000367$
 $\sigma_s \approx 0.011921$
 $\kappa_s \approx 0.287409$
 $\gamma_s \approx 10.898897$
 The actual moments of the CornishFisher distribution $\mathcal{CF}_{\mu_s, \sigma_s, \kappa_s, \gamma_s}$, computed with Maillard’s relationships (1)(4), are:
 $\mu_{CF} = \mu_s \approx 0.000367$
 $\sigma_{CF} \approx 0.017732$
 $\kappa_{CF} \approx −0.639885$
 $\gamma_{CF} \approx 62.437532$
So, yes, they do differ a lot, especially the excess kurtosis!
This subtlety is the hidden problem explaining^{19} the observed lack of accuracy of modified ValueatRisk when return distributions are not close to normal^{5}. Indeed, it cannot be expected from a “wrong” CornishFisher distribution to accurately approximate anything useful.
The solution to this problem consists in inverting the relationships (1)(4) between the actual moments and the parameters of the CornishFisher distribution $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$.
In other words, we need to determine the value of the parameters $\mu$, $\sigma$, $\kappa$ and $\gamma$ of the CornishFisher distribution $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$ so that its actual moments $\mu_{CF}$, $\sigma_{CF}$, $\kappa_{CF}$ and $\gamma_{CF}$ are equal to the sample moments $\mu_{s}$, $\sigma_{s}$, $\kappa_{s}$ and $\gamma_{s}$ of the empirical return distribution, c.f. Lamb et al.^{7}.
More on how to do this numerically later.
The resulting CornishFisher distribution is called the corrected CornishFisher distribution $\mathcal{cCF}_{\mu_s, \sigma_s, \kappa_s, \gamma_s}$ and the underlying CornishFisher expansion the corrected CornishFisher expansion^{4}.
Reusing one last time the SPY ETF example, we have:
 $\mu \approx 0.000367$
 $\sigma \approx 0.011217$
 $\kappa \approx 0.152059$
 $\gamma \approx 3.556476$
, and Figure 6 compares the resulting corrected CornishFisher distribution to the two distributions of Figure 5.
The approximation of the empirical return distribution by the corrected CornishFisher distribution is so accurate that these two distributions are nearly indistinguishable in this figure.
Figure 7, Figure 8 and Figure 9 compare the left tail of the three distributions from Figure 6.
A nearly perfect fit again between the empirical return distribution and the corrected CornishFisher distribution.
This example empirically demonstrates that modified ValueatRisk, when corrected using Maillard^{6} results, works well for moderately nonnormal distributions and for very small tail probabilities.
Computing the corrected CornishFisher distribution
As mentioned in the previous section, computing the corrected CornishFisher distribution requires to invert the relationships (1)(4) between the actual moments and the parameters of the CornishFisher distribution $\mathcal{CF}_{\mu, \sigma, \kappa, \gamma}$.
Because the location parameter $\mu$ is invariant by (1), and because the scale parameter $\sigma$ is easily computed thanks to (2) once the skewness parameter $\kappa$ and the excess kurtosis parameter $\gamma$ have been computed, the main mathematical challenge is to invert the system of nonlinear equations (3)(4).
The domain of validity of the corrected CornishFisher expansion
Before thinking about how to invert these equations numerically, we first need to make sure that they are invertible theoretically.
Lamb et al.^{7} prove that this is the case when the actual skewness $\kappa_{CF}$ and the actual excess kurtosis $\gamma_{CF}$ belong^{20} to what could be called the domain of validity of the corrected CornishFisher expansion^{21}, represented in Figure 10.
Lamb et al.^{7} also establish that the resulting skewness parameter $\kappa$ and excess kurtosis parameter $\gamma$ belong to the domain of validity of the CornishFisher expansion, which ensures that the resulting corrected CornishFisher distribution is a proper distribution.
To be noted that the domain of validity of the corrected CornishFisher expansion (Figure 10) is much wider than the domain of validity of the CornishFisher expansion (Figure 3).
This is extremely important in applications, because the actual skewness $\kappa_{CF}$ and the actual excess kurtosis $\gamma_{CF}$ of the corrected CornishFisher distribution typically correspond to the sample skewness $\kappa_s$ and to the sample excess kurtosis $\gamma_s$ of a given distribution^{22}, so that the corrected CornishFisher distribution is valid in practice for a much wider range of skewness and excess kurtosis than the noncorrected CornishFisher distribution.
The inversion procedure
At least two algorithms have been analyzed in the literature to compute the corrected CornishFisher parameters from the actual moments:
 A kind of interpolation algorithm, using the response surface methodology, in AmedeeManesme et al.^{4}
 A modified Newton method, in Lamb et al.^{7}
Implementation in Portfolio Optimizer
Portfolio Optimizer implements a proprietary algorithm to compute the parameters of the corrected CornishFisher distribution, whose general description is:
 Determine the actual mean $\mu_s$, standard deviation $\sigma_s$, skewness $\kappa_s$ and excess kurtosis $\gamma_s$ of the corrected CornishFisher distribution
These are either directly provided in input of the endpoint (e.g. /assets/returns/simulation/montecarlo/cornishfisher/corrected
) or
computed from an empirical distribution of returns (e.g. /portfolio/analysis/valueatrisk/cornishfisher/corrected
).
 If the skewness $\kappa_s$ and the excess kurtosis $\gamma_s$ belong to the domain of validity of the corrected CornishFisher expansion, a robust iterative numerical method is then used to compute the skewness and excess kurtosis parameters $\kappa$ and $\gamma$.
Once these parameters are known, the relationships (1)(4) allow to determine the resulting corrected CornishFisher distribution $\mathcal{cCF}_{\mu_s, \sigma_s, \kappa_s, \gamma_s}$.
 Otherwise, a robust iterative numerical method is used to tentatively^{23} compute the skewness and excess kurtosis parameters $\kappa$ and $\gamma$
 If this computation is successful, the increasing rearrangement procedure of Chernozhukov et al.^{18} is applied to the resulting corrected CornishFisher distribution $\mathcal{cCF}_{\mu_s, \sigma_s, \kappa_s, \gamma_s}$ in order to transform it into a valid distribution
 Otherwise, an error is raised
Example of usage  Computing the modified ValueatRisk of Bitcoin
Bitcoin is an example of asset exhibiting strong nonnormal characteristics^{24}, for which the standard measures of ValueatRisk like Gaussian ValueatRisk or modified ValueatRisk would be inaccurate.
But what about modified ValueatRisk based on the corrected CornishFisher expansion?
In order to investigate the accuracy of this measure, that I will call corrected CornishFisher ValueatRisk (cCFVaR), Figure 11 compares, over the period 20 August 2011  06 April 2023, the empirical distribution of Bitcoin daily returns^{14} to the corrected CornishFisher distribution $\mathcal{cCF}_{\mu_s, \sigma_s, \kappa_s, \gamma_s}$ with actual moments:
 $\mu_s \approx 0.001863$, the sample mean of Bitcoin returns over the considered period
 $\sigma_s \approx 0.047369$, the sample standard deviation of Bitcoin returns over the considered period
 $\kappa_s \approx 1.368879$, the sample skewness of Bitcoin returns over the considered period
 $\gamma_s \approx 24.594523$, the sample excess kurtosis of Bitcoin returns over the considered period
It seems that the corrected CornishFisher distribution does a pretty good job in approximating the empirical return distribution of Bitcoin, except in the right tail though.
Figure 12 and Figure 13 compare the left tail of these two distributions.
There figures confirm that the corrected CornishFisher distribution accurately approximates the empirical return distribution of Bitcoin down to a confidence level of $\approx 95\%$, but no lower.
This can also be confirmed numerically, with a comparison between historical ValueatRisk and corrected CornishFisher ValueatRisk at different confidence levels:
Confidence level $\alpha$  $\text{HVaR}_{\alpha}$  $\text{cCFVaR}_{\alpha}$ 

95%  6.90%  6.86% 
97.5%  9.53%  10.63% 
99%  13.36%  16.51% 
99.5%  15.92%  21.56% 
99.9%  27.04%  35.08% 
All in all, this example empirically demonstrates that modified ValueatRisk, when corrected following Maillard^{6} results, works well for highly nonnormal distributions with not too small tail probabilities.
Conclusion
The goal of this post was to highlight that accuracy issues reported by practitioneers with modified ValueatRisk have been understood since more than ten years, but that, as AmedeeManesme et al.^{4} put it:
this point […] does not seem to have received sufficient attention
If you are such a practitioneer, I hope that this post will encourage you to double check how modified ValueatRisk is computed by your internal risk management software.
Waiting for an answer from your (puzzled) IT teams, feel free to connect with me on LinkedIn or follow me on Twitter.
–

See Zangari, P. (1996). A VaR methodology for portfolios that include options. RiskMetrics Monitor First Quarter, 4–12. ↩ ↩^{2}

See Martin, R. Douglas and Arora, Rohit, Inefficiency of Modified VaR and ES. ↩ ↩^{2} ↩^{3} ↩^{4}

For example, European financial regulators require to use mVaR in order to compute the Summary Risk Indicator (SRI), i.e. the risk score, of Packaged Retail Investment and Insurance Products (PRIIPs) starting 1st January 2023, c.f. regulatory Technical Standards on the content and presentation of the KIDs for PRIIPs. ↩

See AmedeeManesme, CO., Barthelemy, F. & Maillard, D. Computation of the corrected Cornish–Fisher expansion using the response surface methodology: application to VaR and CVaR. Ann Oper Res 281, 423–453 (2019). ↩ ↩^{2} ↩^{3} ↩^{4}

See Stoyan V. Stoyanov, Svetlozar T. Rachev, Frank J. Fabozzi, Sensitivity of portfolio VaR and CVaR to portfolio return characteristics, Working paper. ↩ ↩^{2} ↩^{3}

See Maillard, Didier, A User’s Guide to the Cornish Fisher Expansion. ↩ ↩^{2} ↩^{3} ↩^{4} ↩^{5} ↩^{6} ↩^{7} ↩^{8} ↩^{9}

See Lamb, John D., Maura E. Monville, and KaiHong Tee. Making Cornish–fisher Fit for Risk Measurement, Journal of Risk, Volume 21, Number 5, Pages 5381. ↩ ↩^{2} ↩^{3} ↩^{4} ↩^{5} ↩^{6} ↩^{7}

Like 1% quantile or even less. ↩

See Jorion, P. (2007). Value at risk: The new benchmark for managing financial risk. New York, NY: McGrawHill. ↩

In this post, returns are assumed to be logarithmic returns. ↩

This is the case when the portfolio return cumulative distribution function is strictly increasing and continuous; otherwise, a similar formula is still valid, with $F_X^{1}$ the generalized inverse distribution function of $X$, but these subtleties  important in mathematical proofs and in numerical implementations  are out of scope of this post. ↩

See Boudt, Kris and Peterson, Brian G. and Croux, Christophe, Estimation and Decomposition of Downside Risk for Portfolios with NonNormal Returns (October 31, 2007). Journal of Risk, Vol. 11, No. 2, pp. 79103, 2008. ↩ ↩^{2}

Asset returns have a tendency to follow a distribution closer and closer to a Gaussian distribution the more the time period over which they are computed increases; this empirical property is called aggregational Gaussianity, c.f. Cont^{25}. ↩

The associated adjusted prices have been retrieved using Tiingo. ↩ ↩^{2}

This also means that it is possible to have $\text{mVaR}_{95\%} > \text{mVaR}_{99\%} $, which requires some funny arguments to be explained… ↩

See Barton, D.E., & Dennis, K.E. (1952). The conditions under which GramCharlier and Edgeworth curves are positive definite and unimodal. Biometrika, 39(34), 425–427. ↩

I will not enter into the mathematical details in this post, but it suffices to say that this procedure allows to correct the behavior of the CornishFisher expansion when used outside of its domain of validity thanks to a sorting operator. ↩

See Chernozhukov, V., FernandezVal, I. & Galichon, A. Rearranging Edgeworth–Cornish–Fisher expansions. Econ Theory 42, 419–435 (2010). ↩ ↩^{2}

In addition, Maillard^{6} mentions that when the skewness and excess kurtosis parameters are small enough, in a loose sense, they coincide with the actual skewness and excess kurtosis of the CornishFisher distribution, which perfectly explains the behavior of the modified ValueatRisk observed in practice with return distributions close to normal^{5}. ↩

Actually, the result of Lamb et al.^{7} is a little bit more generic: they establish that the system of nonlinear equations is invertible on a region which includes the domain of validity of the CornishFisher expansion. ↩

The domain of validity of the corrected CornishFisher expansion is the mathematical image, by the functions $f_1$ and $f_2$, of the domain of validity of the CornishFisher expansion. ↩

In the context of this blog post, the given distribution is a return distribution (asset, portfolio, strategy…). ↩

This tentative computation is theoretically justified by the results from Lamb et al.^{7}. ↩

See Joerg Osterrieder, The Statistics of Bitcoin and Cryptocurrencies, Proceedings of the 2017 International Conference on Economics, Finance and Statistics (ICEFS 2017). ↩

See R. Cont (2001) Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance, 1:2, 223236. ↩