Calculates Expected Shortfall(ES) (also known as) Conditional Value at Risk(CVaR) or Expected Tail Loss (ETL) for univariate, component, and marginal cases using a variety of analytical methods.
ETL(
R = NULL,
p = 0.95,
...,
method = c("modified", "gaussian", "historical"),
clean = c("none", "boudt", "geltner", "locScaleRob"),
portfolio_method = c("single", "component"),
weights = NULL,
mu = NULL,
sigma = NULL,
m3 = NULL,
m4 = NULL,
invert = TRUE,
operational = TRUE,
SE = FALSE,
SE.control = NULL
)a vector, matrix, data frame, timeSeries or zoo object of asset returns
confidence level for calculation, default p=.95
any other passthru parameters
one of "modified","gaussian","historical", see Details.
method for data cleaning through Return.clean.
Current options are "none", "boudt", "geltner", or "locScaleRob".
one of "single","component","marginal" defining whether to do univariate, component, or marginal calc, see Details.
portfolio weighting vector, default NULL, see Details
If univariate, mu is the mean of the series. Otherwise mu is the vector of means of the return series, default NULL, see Details
If univariate, sigma is the variance of the series. Otherwise sigma is the covariance matrix of the return series, default NULL, see Details
If univariate, m3 is the skewness of the series. Otherwise m3 is the coskewness matrix (or vector with unique coskewness values) of the returns series, default NULL, see Details
If univariate, m4 is the excess kurtosis of the series. Otherwise m4 is the cokurtosis matrix (or vector with unique cokurtosis values) of the return series, default NULL, see Details
TRUE/FALSE whether to invert the VaR measure, see Details.
TRUE/FALSE, default TRUE, see Details.
TRUE/FALSE whether to ouput the standard errors of the estimates of the risk measures, default FALSE.
Control parameters for the computation of standard errors. Should be done using the RPESE.control function.
This function provides several estimation methods for the Expected Shortfall (ES) (also called Expected Tail Loss (ETL) or Conditional Value at Risk (CVaR)) of a return series and the Component ES (ETL/CVaR) of a portfolio.
At a preset probability level denoted \(c\), which typically is between 1 and 5 per cent, the ES of a return series is the negative value of the expected value of the return when the return is less than its \(c\)-quantile. Unlike value-at-risk, conditional value-at-risk has all the properties a risk measure should have to be coherent and is a convex function of the portfolio weights (Pflug, 2000). With a sufficiently large data set, you may choose to estimate ES with the sample average of all returns that are below the \(c\) empirical quantile. More efficient estimates of VaR are obtained if a (correct) assumption is made on the return distribution, such as the normal distribution. If your return series is skewed and/or has excess kurtosis, Cornish-Fisher estimates of ES can be more appropriate. For the ES of a portfolio, it is also of interest to decompose total portfolio ES into the risk contributions of each of the portfolio components. For the above mentioned ES estimators, such a decomposition is possible in a financially meaningful way.
The ES at a probability level \(p\) (e.g. 95%) is the negative value of
the expected value of the return when the return is less than its
\(c=1-p\) quantile. In a set of returns for which sufficently long history
exists, the per-period ES can be estimated by the negative value of the
sample average of all returns below the quantile. This method is also
sometimes called “historical ES”, as it is by definition ex
post analysis of the return distribution, and may be accessed with
method="historical".
When you don't have a sufficiently long set of returns to use non-parametric or historical ES, or wish to more closely model an ideal distribution, it is common to us a parmetric estimate based on the distribution. Parametric ES does a better job of accounting for the tails of the distribution by more precisely estimating shape of the distribution tails of the risk quantile. The most common estimate is a normal (or Gaussian) distribution \(R\sim N(\mu,\sigma)\) for the return series. In this case, estimation of ES requires the mean return \(\bar{R}\), the return distribution and the variance of the returns \(\sigma\). In the most common case, parametric VaR is thus calculated by
$$\sigma=variance(R)$$
$$ES=-\bar{R} + \sqrt{\sigma} \cdot \frac{1}{c}\phi(z_{c}) $$
where \(z_{c}\) is the \(c\)-quantile of the standard normal
distribution. Represented in R by qnorm(c), and may be accessed with
method="gaussian". The function \(\phi\) is the Gaussian
density function.
The limitations of Gaussian ES are well covered in the literature, since most financial return series are non-normal. Boudt, Peterson and Croux (2008) provide a modified ES calculation that takes the higher moments of non-normal distributions (skewness, kurtosis) into account through the use of a Cornish-Fisher expansion, and collapses to standard (traditional) Gaussian ES if the return stream follows a standard distribution. More precisely, for a loss probability \(c\), modified ES is defined as the negative of the expected value of all returns below the \(c\) Cornish-Fisher quantile and where the expectation is computed under the second order Edgeworth expansion of the true distribution function.
Modified expected shortfall should always be larger than modified Value at Risk. Due to estimation problems, this might not always be the case. Set Operational = TRUE to replace modified ES with modified VaR in the (exceptional) case where the modified ES is smaller than modified VaR.
By setting portfolio_method="component" you may calculate the ES
contribution of each element of the portfolio. The return from the function in
this case will be a list with three components: the univariate portfolio ES,
the scalar contribution of each component to the portfolio ES (these will sum
to the portfolio ES), and a percentage risk contribution (which will sum to
100%).
Both the numerical and percentage component contributions to ES may contain both positive and negative contributions. A negative contribution to Component ES indicates a portfolio risk diversifier. Increasing the position weight will reduce overall portoflio ES.
If a weighting vector is not passed in via weights, the function will
assume an equal weighted (neutral) portfolio.
Multiple risk decomposition approaches have been suggested in the literature. A naive approach is to set the risk contribution equal to the stand-alone risk. This approach is overly simplistic and neglects important diversification effects of the units being exposed differently to the underlying risk factors. An alternative approach is to measure the ES contribution as the weight of the position in the portfolio times the partial derivative of the portfolio ES with respect to the component weight. $$C_i \mbox{ES} = w_i \frac{ \partial \mbox{ES} }{\partial w_i}.$$ Because the portfolio ES is linear in position size, we have that by Euler's theorem the portfolio VaR is the sum of these risk contributions. Scaillet (2002) shows that for ES, this mathematical decomposition of portfolio risk has a financial meaning. It equals the negative value of the asset's expected contribution to the portfolio return when the portfolio return is less or equal to the negative portfolio VaR:
$$C_i \mbox{ES} = = -E\left[ w_i r_{i} | r_{p} \leq - \mbox{VaR}\right]$$
For the decomposition of Gaussian ES, the estimated mean and covariance matrix are needed. For the decomposition of modified ES, also estimates of the coskewness and cokurtosis matrices are needed. If \(r\) denotes the \(Nx1\) return vector and \(mu\) is the mean vector, then the \(N \times N^2\) co-skewness matrix is $$ m3 = E\left[ (r - \mu)(r - \mu)' \otimes (r - \mu)'\right]$$
The \(N \times N^3\) co-kurtosis matrix is
$$ m_{4} = E\left[ (r - \mu)(r - \mu)' \otimes (r - \mu)'\otimes (r - \mu)' \right] $$
where \(\otimes\) stands for the Kronecker product. The matrices can
be estimated through the functions skewness.MM and kurtosis.MM.
More efficient estimators were proposed by Martellini and Ziemann (2007) and
will be implemented in the future.
As discussed among others in Cont, Deguest and Scandolo (2007), it is
important that the estimation of the ES measure is robust to single outliers.
This is especially the case for modified VaR and its decomposition, since
they use higher order moments. By default, the portfolio moments are
estimated by their sample counterparts. If clean="boudt" then the
\(1-p\) most extreme observations are winsorized if they are detected as
being outliers. For more information, see Boudt, Peterson and Croux (2008)
and Return.clean. If your data consist of returns for highly
illiquid assets, then clean="geltner" may be more appropriate to
reduce distortion caused by autocorrelation, see Return.Geltner
for details.
Brian G. Peterson and Kris Boudt
Boudt, Kris, Peterson, Brian, and Christophe Croux. 2008. Estimation and decomposition of downside risk for portfolios with non-normal returns. 2008. The Journal of Risk, vol. 11, 79-103.
Cont, Rama, Deguest, Romain and Giacomo Scandolo. Robustness and sensitivity analysis of risk measurement procedures. Financial Engineering Report No. 2007-06, Columbia University Center for Financial Engineering.
Laurent Favre and Jose-Antonio Galeano. Mean-Modified Value-at-Risk Optimization with Hedge Funds. Journal of Alternative Investment, Fall 2002, v 5.
Martellini, L. and Ziemann, V., 2010. Improved estimates of higher-order comoments and implications for portfolio selection. Review of Financial Studies, 23(4):1467-1502.
Pflug, G. Ch. Some remarks on the value-at-risk and the conditional value-at-risk. In S. Uryasev, ed., Probabilistic Constrained Optimization: Methodology and Applications, Dordrecht: Kluwer, 2000, 272-281.
Scaillet, Olivier. Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance, 2002, vol. 14, 74-86.
VaR
SharpeRatio.modified
chart.VaRSensitivity
Return.clean
# \donttest{
data(edhec)
# first do normal ES calc
ES(edhec, p=.95, method="historical")
# now use Gaussian
ES(edhec, p=.95, method="gaussian")
# now use modified Cornish Fisher calc to take non-normal distribution into account
ES(edhec, p=.95, method="modified")
# now use p=.99
ES(edhec, p=.99)
# or the equivalent alpha=.01
ES(edhec, p=.01)
# CRAN (questionably(ahem) requires these methods to not run if you don't have Suggests loaded)
if(requireNamespace("robustbase", quietly = TRUE)){
ES(edhec, clean="boudt")
# add Component ES for the equal weighted portfolio
ES(edhec, clean="boudt", portfolio_method="component")
} # end CRAN workaround
# } # end CRAN Windows check
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