ETL {PerformanceAnalytics} | R Documentation

## calculates Expected Shortfall(ES) (or Conditional Value-at-Risk(CVaR) for univariate and component, using a variety of analytical methods.

### Description

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.

### Usage

    ETL(R = NULL, p = 0.95, ..., method = c("modified", "gaussian",
      "historical"), clean = c("none", "boudt", "geltner"),
      portfolio_method = c("single", "component"), weights = NULL,
      mu = NULL, sigma = NULL, m3 = NULL, m4 = NULL, invert = TRUE,
      operational = TRUE)

### Arguments

`R` | a vector, matrix, data frame, timeSeries or zoo object of asset returns
`p` | confidence level for calculation, default p=.95
`...` | any other passthru parameters
`method` | one of "modified","gaussian","historical", see Details.
`clean` | method for data cleaning through `Return.clean`. Current options are "none", "boudt", or "geltner".
`portfolio_method` | one of "single","component","marginal" defining whether to do univariate, component, or marginal calc, see Details.
`weights` | portfolio weighting vector, default NULL, see Details
`mu` | If univariate, mu is the mean of the series. Otherwise mu is the vector of means of the return series, default NULL, see Details
`sigma` | If univariate, sigma is the variance of the series. Otherwise sigma is the covariance matrix of the return series, default NULL, see Details
`m3` | 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
`m4` | 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
`invert` | TRUE/FALSE whether to invert the VaR measure, see Details.
`operational` | TRUE/FALSE, default TRUE, see Details.

### Background

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.

### Univariate estimation of ES

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(μ,σ) 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 σ. In the most common case, parametric VaR is thus calculated by

sigma=var(R)

VaR= -mean(R) + sqrt(sigma)*dnorm(z_c)/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 dnorm 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.

### Note

The option to `invert` the ES measure should appease both academics and practitioners. The mathematical definition of ES as the negative value of extreme losses will (usually) produce a positive number. Practitioners will argue that ES denotes a loss, and should be internally consistent with the quantile (a negative number). For tables and charts, different preferences may apply for clarity and compactness. As such, we provide the option, and set the default to TRUE to keep the return consistent with prior versions of PerformanceAnalytics, but make no value judgement on which approach is preferable.

### Author(s)

Brian G. Peterson and Kris Boudt

### References

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.

### See Also

`VaR`
`SharpeRatio.modified`
`chart.VaRSensitivity`
`Return.clean`

### Examples

        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)

        # now with outliers squished
        ES(edhec, clean="boudt")

        # add Component ES for the equal weighted portfolio
        ES(edhec, clean="boudt", portfolio_method="component")

[Package PerformanceAnalytics version 1.5.3 Index]