cvar/0000755000176200001440000000000014330710446011207 5ustar liggesuserscvar/NAMESPACE0000644000176200001440000000055514330676704012443 0ustar liggesusers# Generated by roxygen2: do not edit by hand S3method(ES,default) S3method(ES,numeric) S3method(VaR,default) S3method(VaR,numeric) S3method(predict,garch1c1) export(ES) export(GarchModel) export(VaR) export(VaR_cdf) export(VaR_qf) export(sim_garch1c1) importFrom(Rdpack,reprompt) importFrom(gbutils,cdf2quantile) importFrom(stats,quantile) importFrom(stats,runif) cvar/README.md0000644000176200001440000000617114210660660012472 0ustar liggesusers # cvar [![CRANStatusBadge](https://www.r-pkg.org/badges/version/cvar)](https://cran.r-project.org/package=cvar) [![R-CMD-check](https://github.com/GeoBosh/cvar/workflows/R-CMD-check/badge.svg)](https://github.com/GeoBosh/cvar/actions) [![codecov](https://codecov.io/gh/GeoBosh/cvar/branch/master/graph/badge.svg?token=2SW9HKG71Y)](https://app.codecov.io/gh/GeoBosh/cvar) ## Overview Compute expected shortfall (ES) and Value at Risk (VaR) from a quantile function, distribution function, random number generator or probability density function. ES is also known as Conditional Value at Risk (CVaR). Virtually any continuous distribution can be specified. The functions are vectorised over the arguments. The computations are done directly from the definitions, see e.g. Acerbi and Tasche (2002). Some support for GARCH models is provided, as well. ## Installing cvar The [latest stable version](https://cran.r-project.org/package=cvar) is on CRAN. install.packages("cvar") The vignette [Guide_cvar]( https://CRAN.R-project.org/package=cvar/vignettes/Guide_cvar.pdf) shipping with the package gives illustrative examples (can also be opened from R with `vignette("Guide_cvar", package = "cvar")`). You can install the [development version](https://github.com/GeoBosh/cvar) of `cvar` from Github: library(devtools) install_github("GeoBosh/cvar") ## Overview Package `cvar` is a small `R` package with, essentially two functions — `ES` for computing the expected shortfall and `VaR` for Value at Risk. The user specifies the distribution by supplying one of the functions that define a continuous distribution—currently this can be a quantile function (qf), cumulative distribution function (cdf) or probability density function (pdf). Virtually any continuous distribution can be specified. The functions are vectorised over the parameters of the distributions, making bulk computations more convenient, for example for forecasting or model evaluation. The name of this package, "cvar", comes from *Conditional Value at Risk* (CVaR), which is an alternative term for expected shortfall. We chose to use the standard names `ES` and `VaR`, despite the possibility for name clashes with same named functions in other packages, rather than invent possibly difficult to remember alternatives. Just call the functions as `cvar::ES` and `cvar::VaR` if necessary. Locations-scale transformations can be specified separately from the other distribution parameters. This is useful when such parameters are not provided directly by the distribution at hand. The use of these parameters often leads to more efficient computations and better numerical accuracy even if the distribution has its own parameters for this purpose. Some of the examples for `VaR` and `ES` illustrate this for the Gaussian distribution. Since VaR is a quantile, functions computing it for a given distribution are convenience functions. `VaR` exported by `cvar` could be attractive in certain workflows because of its vectorised distribution parameters, the location-scale transformation and the possibility to compute it from cdf's when quantile functions are not available. cvar/man/0000755000176200001440000000000013501140665011761 5ustar liggesuserscvar/man/GarchModel.Rd0000644000176200001440000000317014330275321014255 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/garch.R \name{GarchModel} \alias{GarchModel} \title{Specify a GARCH model} \usage{ GarchModel(model = list(), ..., model.class = NULL) } \arguments{ \item{model}{a GARCH model or a list.} \item{...}{named arguments specifying the GARCH model.} \item{model.class}{a class for the result. By default \code{GarchModel()} decides the class of the result.} } \value{ an object from suitable GARCH-type class } \description{ Specify a GARCH model. } \details{ Argument \code{model} can be the result of a previous call to \code{GarchModel}. Arguments in \code{"..."} overwrite current components of \code{model}. \code{GarchModel} guarantees that code using it will continue to work transparently for the user even if the internal represedtation of GARCH models in this package is changed or additional functionality is added. } \examples{ ## GARCH(1,1) with Gaussian innovations mo1a <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5) mo1b <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, cond.dist = "norm") ## equivalently, the parameters can be given as a list p1 <- list(omega = 1, alpha = 0.3, beta = 0.5) mo1a_alt <- GarchModel(p1) mo1b_alt <- GarchModel(p1, cond.dist = "norm") stopifnot(identical(mo1a, mo1a_alt), identical(mo1b, mo1b_alt)) ## additional arguments modify values already in 'model' mo_alt <- GarchModel(p1, beta = 0.4) ## set also initial values mo2 <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, esp0 = - 1.5, h0 = 4.96) ## GARCH(1,1) with standardised-t_5 mot <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, cond.dist = list("std", nu = 5)) } cvar/man/ES.Rd0000644000176200001440000001307314330676704012574 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/VaR.R \name{ES} \alias{ES} \alias{ES.default} \alias{ES.numeric} \title{Compute expected shortfall (ES) of distributions} \usage{ ES(dist, p_loss = 0.05, ...) \method{ES}{default}( dist, p_loss = 0.05, dist.type = "qf", qf, ..., intercept = 0, slope = 1, control = list(), x ) \method{ES}{numeric}( dist, p_loss = 0.05, dist.type = "qf", qf, ..., intercept = 0, slope = 1, control = list(), x ) } \arguments{ \item{dist}{specifies the distribution whose ES is computed, usually a function or a name of a function computing quantiles, cdf, pdf, or a random number generator, see Details.} \item{p_loss}{level, default is 0.05.} \item{...}{passed on to \code{dist}.} \item{dist.type}{a character string specifying what is computed by \code{dist}, such as "qf" or "cdf".} \item{qf}{quantile function, only used if \code{dist.type = "pdf"}.} \item{intercept, slope}{compute ES for the linear transformation \code{intercept + slope*X}, where \code{X} has distribution specified by \code{dist}, see Details.} \item{control}{additional control parameters for the numerical integration routine.} \item{x}{deprecated and will soon be removed. \code{x} was renamed to \code{p_loss}, please use the latter.} } \value{ a numeric vector } \description{ Compute the expected shortfall for a distribution. } \details{ \code{ES} computes the expected shortfall for distributions specified by the arguments. \code{dist} is typically a function (or the name of one). What \code{dist} computes is determined by \code{dist.type}, whose default setting is \code{"qf"} (the quantile function). Other possible settings of \code{dist.type} include \code{"cdf"} and \code{"pdf"}. Additional arguments for \code{dist} can be given with the \code{"..."} arguments. Argument \code{dist} can also be a numeric vector. In that case the ES is computed, effectively, for the empirical cumulative distribution function (ecdf) of the vector. The ecdf is not created explicitly and the \code{\link[stats]{quantile}} function is used instead for the computation of VaR. Arguments in \code{"..."} are passed eventually to \code{quantile()} and can be used, for example, to select a non-defult method for the computation of quantiles. Except for the exceptions discussed below, a function computing VaR for the specified distribution is constructed and the expected shortfall is computed by numerically integrating it. The numerical integration can be fine-tuned with argument \code{control}, which should be a named list, see \code{\link{integrate}} for the available options. If \code{dist.type} is \code{"pdf"}, VaR is not computed, Instead, the partial expectation of the lower tail is computed by numerical integration of \code{x * pdf(x)}. Currently the quantile function is required anyway, via argument \code{qf}, to compute the upper limit of the integral. So, this case is mainly for testing and comparison purposes. A bunch of expected shortfalls is computed if argument \code{x} or any of the arguments in \code{"..."} are of length greater than one. They are recycled to equal length, if necessary, using the normal \R recycling rules. \code{intercept} and \code{slope} can be used to compute the expected shortfall for the location-scale transformation \code{Y = intercept + slope * X}, where the distribution of \code{X} is as specified by the other parameters and \code{Y} is the variable of interest. The expected shortfall of \code{X} is calculated and then transformed to that of \code{Y}. Note that the distribution of \code{X} doesn't need to be standardised, although it typically will. The \code{intercept} and the \code{slope} can be vectors. Using them may be particularly useful for cheap calculations in, for example, forecasting, where the predictive distributions are often from the same family, but with different location and scale parameters. Conceptually, the described treatment of \code{intercept} and \code{slope} is equivalent to recycling them along with the other arguments, but more efficiently. The names, \code{intercept} and \code{slope}, for the location and scale parameters were chosen for their expressiveness and to minimise the possibility for a clash with parameters of \code{dist} (e.g., the Gamma distribution has parameter \code{scale}). } \examples{ ES(qnorm) ## Gaussian ES(qnorm, dist.type = "qf") ES(pnorm, dist.type = "cdf") ## t-dist ES(qt, dist.type = "qf", df = 4) ES(pt, dist.type = "cdf", df = 4) ES(pnorm, 0.95, dist.type = "cdf") ES(qnorm, 0.95, dist.type = "qf") ## - VaRES::esnormal(0.95, 0, 1) ## - PerformanceAnalytics::ETL(p=0.05, method = "gaussian", mu = 0, ## sigma = 1, weights = 1) # same cvar::ES(pnorm, dist.type = "cdf") cvar::ES(qnorm, dist.type = "qf") cvar::ES(pnorm, 0.05, dist.type = "cdf") cvar::ES(qnorm, 0.05, dist.type = "qf") ## this uses "pdf" cvar::ES(dnorm, 0.05, dist.type = "pdf", qf = qnorm) ## this gives warning (it does more than simply computing ES): ## PerformanceAnalytics::ETL(p=0.95, method = "gaussian", mu = 0, sigma = 1, weights = 1) ## run this if VaRRES is present \dontrun{ x <- seq(0.01, 0.99, length = 100) y <- sapply(x, function(p) cvar::ES(qnorm, p, dist.type = "qf")) yS <- sapply(x, function(p) - VaRES::esnormal(p)) plot(x, y) lines(x, yS, col = "blue") } } \seealso{ \code{\link{VaR}} for VaR, \code{\link[=predict.garch1c1]{predict}} for examples with fitted models } cvar/man/VaR.Rd0000644000176200001440000001532014330675543012752 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/VaR.R \name{VaR} \alias{VaR} \alias{VaR_qf} \alias{VaR_cdf} \alias{VaR.default} \alias{VaR.numeric} \title{Compute Value-at-Risk (VaR)} \usage{ VaR(dist, p_loss = 0.05, ...) VaR_qf( dist, p_loss = 0.05, ..., intercept = 0, slope = 1, tol = .Machine$double.eps^0.5, x ) VaR_cdf( dist, p_loss = 0.05, ..., intercept = 0, slope = 1, tol = .Machine$double.eps^0.5, x ) \method{VaR}{default}( dist, p_loss = 0.05, dist.type = "qf", ..., intercept = 0, slope = 1, tol = .Machine$double.eps^0.5, x ) \method{VaR}{numeric}(dist, p_loss = 0.05, ..., intercept = 0, slope = 1, x) } \arguments{ \item{dist}{specifies the distribution whose ES is computed, usually a function or a name of a function computing quantiles, cdf, pdf, or a random number generator, see Details.} \item{p_loss}{level, default is 0.05.} \item{...}{passed on to \code{dist}.} \item{intercept, slope}{compute VaR for the linear transformation \code{intercept + slope*X}, where \code{X} has distribution specified by \code{dist}, see Details.} \item{tol}{tollerance} \item{x}{deprecated and will soon be removed. \code{x} was renamed to \code{p_loss}, please use the latter.} \item{dist.type}{a character string specifying what is computed by \code{dist}, such as "qf" or "cdf".} } \description{ \code{VaR} computes the Value-at-Risk of the distribution specified by the arguments. The meaning of the parameters is the same as in \code{\link{ES}}, including the recycling rules. } \details{ \code{VaR} is S3 generic. The meaning of the parameters for its default method is the same as in \code{\link{ES}}, including the recycling rules. \code{VaR_qf} and \code{VaR_cdf} are streamlined, non-generic, variants for the common case when the \code{"..."} parameters are scalar. The parameters \code{x}, \code{intercept}, and \code{slope} can be vectors, as for \code{VaR}. Argument \code{dist} can also be a numeric vector. In that case the ES is computed, effectively, for the empirical cumulative distribution function (ecdf) of the vector. The ecdf is not created explicitly and the \code{\link[stats]{quantile}} function is used instead for the computation of VaR. Arguments in \code{"..."} are passed eventually to \code{quantile()} and can be used, for example, to select a non-defult method for the computation of quantiles. In practice, we may need to compute VaR associated with data. The distribution comes from fitting a model. In the simplest case, we fit a distribution to the data, assuming that the sample is i.i.d. For example, a normal distribution \eqn{N(\mu, \sigma^2)} can be fitted using the sample mean and sample variance as estimates of the unknown parameters \eqn{\mu} and \eqn{\sigma^2}, see section \sQuote{Examples}. For other common distributions there are specialised functions to fit their parameters and if not, general optimisation routines can be used. More soffisticated models may be used, even time series models such as GARCH and mixture autoregressive models. } \note{ We use the traditional definition of VaR as the negated lower quantile. For example, if \eqn{X} are returns on an asset, VAR\eqn{{}_\alpha}{_a} = \eqn{-q_\alpha}{-q_a}, where \eqn{q_\alpha}{-q_a} is the lower \eqn{\alpha}{a} quantile of \eqn{X}. Equivalently, VAR\eqn{{}_\alpha}{_a} is equal to the lower \eqn{1-\alpha}{1-a} quantile of \eqn{-X}. } \examples{ cvar::VaR(qnorm, c(0.01, 0.05), dist.type = "qf") ## the following examples use these values, obtained by fitting a normal distribution to ## some data: muA <- 0.006408553 sigma2A <- 0.0004018977 ## with quantile function, giving the parameters directly in the call: res1 <- cvar::VaR(qnorm, 0.05, mean = muA, sd = sqrt(sigma2A)) res2 <- cvar::VaR(qnorm, 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) # 0, intercept/slope equivalent to mean/sd ## with quantile function, which already knows the parameters: my_qnorm <- function(p) qnorm(p, mean = muA, sd = sqrt(sigma2A)) res1_alt <- cvar::VaR(my_qnorm, 0.05) abs((res1_alt - res1)) ## with cdf the precision depends on solving an equation res1a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # 3.287939e-09 abs((res2a - res2)) # 5.331195e-11, intercept/slope better numerically ## as above, but increase the precision, this is probably excessive res1b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) # 6.938894e-18 # both within machine precision abs((res2b - res2)) # 1.040834e-16 ## relative precision is also good abs((res1b - res2)/res2) # 2.6119e-16 # both within machine precision abs((res2b - res2)/res2) # 3.91785e-15 ## an extended example with vector args, if "PerformanceAnalytics" is present if (requireNamespace("PerformanceAnalytics", quietly = TRUE)) withAutoprint({ data(edhec, package = "PerformanceAnalytics") mu <- apply(edhec, 2, mean) sigma2 <- apply(edhec, 2, var) musigma2 <- cbind(mu, sigma2) ## compute in 2 ways with cvar::VaR vAz1 <- cvar::VaR(qnorm, 0.05, mean = mu, sd = sqrt(sigma2)) vAz2 <- cvar::VaR(qnorm, 0.05, intercept = mu, slope = sqrt(sigma2)) vAz1a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2)) vAz2a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2)) vAz1b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2), tol = .Machine$double.eps^0.75) vAz2b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2), tol = .Machine$double.eps^0.75) ## analogous calc. with PerformanceAnalytics::VaR vPA <- apply(musigma2, 1, function(x) PerformanceAnalytics::VaR(p = .95, method = "gaussian", invert = FALSE, mu = x[1], sigma = x[2], weights = 1)) ## the results are numerically the same max(abs((vPA - vAz1))) # 5.551115e-17 max(abs((vPA - vAz2))) # "" max(abs((vPA - vAz1a))) # 3.287941e-09 max(abs((vPA - vAz2a))) # 1.465251e-10, intercept/slope better max(abs((vPA - vAz1b))) # 4.374869e-13 max(abs((vPA - vAz2b))) # 3.330669e-16 }) } \seealso{ \code{\link{ES}} for ES, \code{\link[=predict.garch1c1]{predict}} for examples with fitted models } cvar/man/cvar-package.Rd0000644000176200001440000001036014327473625014610 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/cvar-package.R \docType{package} \name{cvar-package} \alias{cvar-package} \alias{cvar} \title{Compute Conditional Value-at-Risk and Value-at-Risk} \description{ Compute expected shortfall (ES) and Value at Risk (VaR) from a quantile function, distribution function, random number generator or probability density function. ES is also known as Conditional Value at Risk (CVaR). Virtually any continuous distribution can be specified. The functions are vectorised over the arguments. Some support for GARCH models is provided, as well. } \details{ There is a huge number of functions for computations with distributions in core \R and in contributed packages. Pdf's, cdf's, quantile functions and random number generators are covered comprehensively. The coverage of expected shortfall is more patchy but a large collection of distributions, including functions for expected shortfall, is provided by \insertCite{VaRES2013;textual}{cvar}. \insertCite{PerformanceAnalytics2018;textual}{cvar} and \insertCite{actuarJSS2008;textual}{cvar} provide packages covering comprehensively various aspects of risk measurement, including some functions for expected shortfall. Package \pkg{cvar} is a small package with, essentially, two main functions --- \code{ES} for computing the expected shortfall and \code{VaR} for Value at Risk. The user specifies the distribution by supplying one of the functions that define a continuous distribution---currently this can be a quantile function (qf), cumulative distribution function (cdf) or probability density function (pdf). Virtually any continuous distribution can be specified. The functions are vectorised over the parameters of the distributions, making bulk computations more convenient, for example for forecasting or model evaluation. The name of this package, "cvar", comes from \emph{Conditional Value at Risk} (CVaR), which is an alternative term for expected shortfall. We chose to use the standard names \code{ES} and \code{VaR}, despite the possibility for name clashes with same named functions in other packages, rather than invent possibly difficult to remember alternatives. Just call the functions as \code{cvar::ES} and \code{cvar::VaR} if necessary. Locations-scale transformations can be specified separately from the other distribution parameters. This is useful when such parameters are not provided directly by the distribution at hand. The use of these parameters often leads to more efficient computations and better numerical accuracy even if the distribution has its own parameters for this purpose. Some of the examples for \code{VaR} and \code{ES} illustrate this for the Gaussian distribution. Since VaR is a quantile, functions computing it for a given distribution are convenience functions. \code{VaR} exported by \pkg{cvar} could be attractive in certain workflows because of its vectorised distribution parameters, the location-scale transformation, and the possibility to compute it from cdf's when quantile functions are not available. Some support for GARCH models is provided, as well. It is currently under development, see \code{\link{predict.garch1c1}} for current functionality. In practice, we may need to compute VaR associated with data. The distribution comes from fitting a model. In the simplest case, we fit a distribution to the data, assuming that the sample is i.i.d. For example, a normal distribution \eqn{N(\mu, \sigma^2)} can be fitted using the sample mean and sample variance as estimates of the unknown parameters \eqn{\mu} and \eqn{\sigma^2}, see section \sQuote{Examples}. For other common distributions there are specialised functions to fit their parameters and if not, general optimisation routines can be used. More soffisticated models may be used, even time series models such as GARCH and mixture autoregressive models. } \examples{ ## see the examples for ES(), VaR(), predict.garch1c1() } \references{ \insertAllCited{} } \seealso{ \code{\link{ES}}, \code{\link{VaR}} } \author{ Georgi N. Boshnakov } cvar/man/predict.garch1c1.Rd0000644000176200001440000001124414210645012015266 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/garch.R \name{predict.garch1c1} \alias{predict.garch1c1} \title{Prediction for GARCH(1,1) time series} \usage{ \method{predict}{garch1c1}(object, n.ahead = 1, Nsim = 1000, eps, sigmasq, seed = NULL, ...) } \arguments{ \item{object}{an object from class \code{"garch1c1"}.} \item{n.ahead}{maximum horizon (lead time) for prediction.} \item{Nsim}{number of Monte Carlo simulations for simulation based quantities.} \item{eps}{the time series to predict, only the last value is used.} \item{sigmasq}{the (squared) volatilities, only the last value is used.} \item{seed}{an integer, seed for the random number generator.} \item{...}{currently not used.} } \value{ an object from S3 class \code{"predict_garch1c1"} containing the following components: \item{eps}{point predictions (conditional expectations) of the time series (equal to zero for pure GARCH).} \item{h}{point predictions (conditional expectations)of the squared volatilities.} \item{model}{the model.} \item{call}{the call.} \item{pi_plugin}{Prediction intervals for the time series, based on plug-in distributions, see Details.} \item{pi_sim}{Simulation based prediction intervals for the time series, see Details.} \item{dist_sim}{simulation samples from the predictive distributions of the time series and the volatilties. } } \description{ Predict GARCH(1,1) time series. } \details{ Plug-in prediction intervals and predictive distributions are obtained by inserting the predicted volatility in the conditional densities. For predictions more than one lag ahead these are not the real predictive distributions but the prediction intervals are usually adequate. For simulation prediction intervals we generate a (large) number of continuations of the given time series. Prediction intervals can be based on sample quantiles. The generated samples are stored in the returned object and can be used for further exploration of the predictive distributions. \code{dist_sim$eps} contains the simulated future values of the time series and \code{dist_sim$h} the corresponding (squared) volatilities. Both are matrices whose \code{i}-th rows contain the predicted quantities for horizon \code{i}. The random seed at the start of the simulations is saved in the returned object. A speficific seed can be requested with argument \code{seed}. In that case the simulations are done with the specified seed and the old state of the random number generator is restored before the function returns. This setup is similar to \code{\link{sim_garch1c1}}. } \note{ This function is under development and may be changed. } \examples{ op <- options(digits = 4) ## set up a model and simulate a time series mo <- GarchModel(omega = 0.4, alpha = 0.3, beta = 0.5) a1 <- sim_garch1c1(mo, n = 1000, n.start = 100, seed = 20220305) ## predictions for T+1,...,T+5 (T = time of last value) ## Nsim is small to reduce the load on CRAN, usually Nsim is larger. a.pred <- predict(mo, n.ahead = 5, Nsim = 1000, eps = a1$eps, sigmasq = a1$h, seed = 1234) ## preditions for the time series a.pred$eps ## PI's for eps - plug-in and simulated a.pred$pi_plugin a.pred$pi_sim ## a DIY calculation of PI's using the simulated sample paths t(apply(a.pred$dist_sim$eps, 1, function(x) quantile(x, c(0.025, 0.975)))) ## further investigate the predictive distributions t(apply(a.pred$dist_sim$eps, 1, function(x) summary(x))) ## compare predictive densities for horizons 2 and 5: h2 <- a.pred$dist_sim$eps[2, ] h5 <- a.pred$dist_sim$eps[5, ] main <- "Predictive densities: horizons 2 (blue) and 5 (black)" plot(density(h5), main = main) lines(density(h2), col = "blue") ## predictions of sigma_t^2 a.pred$h ## plug-in predictions of sigma_t sqrt(a.pred$h) ## simulation predictive densities (PD's) of sigma_t for horizons 2 and 5: h2 <- sqrt(a.pred$dist_sim$h[2, ]) h5 <- sqrt(a.pred$dist_sim$h[5, ]) main <- "PD's of sigma_t for horizons 2 (blue) and 5 (black)" plot(density(h2), col = "blue", main = main) lines(density(h5)) ## VaR and ES for different horizons cbind(h = 1:5, VaR = apply(a.pred$dist_sim$eps, 1, function(x) VaR(x, c(0.05))), ES = apply(a.pred$dist_sim$eps, 1, function(x) ES(x, c(0.05))) ) ## fit a GARCH(1,1) model to exchange rate data and predict gmo1 <- fGarch::garchFit(formula = ~garch(1, 1), data = fGarch::dem2gbp, include.mean = FALSE, cond.dist = "norm", trace = FALSE) mocoef <- gmo1@fit$par mofitted <- GarchModel(omega = mocoef["omega"], alpha = mocoef["alpha1"], beta = mocoef["beta1"]) gmo1.pred <- predict(mofitted, n.ahead = 5, Nsim = 1000, eps = gmo1@data, sigmasq = gmo1@h.t, seed = 1234) gmo1.pred$pi_plugin gmo1.pred$pi_sim op <- options(op) # restore options(digits) } cvar/man/sim_garch1c1.Rd0000644000176200001440000000257113501140665014516 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/garch.R \name{sim_garch1c1} \alias{sim_garch1c1} \title{Simulate GARCH(1,1) time series} \usage{ sim_garch1c1(model, n, n.start = 0, seed = NULL) } \arguments{ \item{model}{a GARCH(1,1) model, an object obtained from \code{GarchModel}.} \item{n}{the length of the generated time series.} \item{n.start}{number of warm-up values, which are then dropped.} \item{seed}{an integer to use for setting the random number generator.} } \value{ a list with components: \item{eps}{the time series,} \item{h}{the (squared) volatilities,} \item{eta}{the standardised innovations,} \item{model}{the GARCH(1,1) model,} \item{.sim}{a list containing the parameters of the simulation,} \item{call}{the call.} } \description{ Simulate GARCH(1,1) time series. } \details{ The simulated time series is in component \code{eps} of the returned value. For exploration of algorithms and eestimation procedures, the volatilities and the standardised innovations are also returned. The random seed at the start of the simulations is saved in the returned object. A speficific seed can be requested with argument \code{seed}. In that case the simulations are done with the specified seed and the old state of the random number generator is restored before the function returns. } \note{ This function is under development and may be changed. } cvar/DESCRIPTION0000644000176200001440000000255514330710446012724 0ustar liggesusersPackage: cvar Type: Package Title: Compute Expected Shortfall and Value at Risk for Continuous Distributions Version: 0.5 Authors@R: person(given = c("Georgi", "N."), family = "Boshnakov", role = c("aut", "cre"), email = "georgi.boshnakov@manchester.ac.uk") Description: Compute expected shortfall (ES) and Value at Risk (VaR) from a quantile function, distribution function, random number generator or probability density function. ES is also known as Conditional Value at Risk (CVaR). Virtually any continuous distribution can be specified. The functions are vectorized over the arguments. The computations are done directly from the definitions, see e.g. Acerbi and Tasche (2002) . Some support for GARCH models is provided, as well. URL: https://geobosh.github.io/cvar/ (doc), https://github.com/GeoBosh/cvar (devel) BugReports: https://github.com/GeoBosh/cvar/issues Imports: gbutils, Rdpack (>= 0.8) RdMacros: Rdpack License: GPL (>= 2) Collate: VaR.R cvar-package.R garch.R RoxygenNote: 7.2.0 Suggests: testthat, fGarch, PerformanceAnalytics NeedsCompilation: no Packaged: 2022-11-03 09:30:19 UTC; georgi Author: Georgi N. Boshnakov [aut, cre] Maintainer: Georgi N. 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GARCH(1,1) with standardised-t_5 mo_t <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, cond.dist = list("std", nu = 5)) expect_identical(mo_t, GarchModel(mo1a, cond.dist = list("std", nu = 5))) }) test_that("garch1c1 related functions work ok", { ## try to deal with the misterious error on TravisCI and devtools::test() ## set.seed(123) a_mo <- GarchModel(omega = 0.4, alpha = 0.3, beta = 0.5) a <- sim_garch1c1(a_mo, n = 100, n.start = 100, seed = 1234) a_pred <- predict(a_mo, n.ahead = 5, Nsim = 100, eps = a$eps, sigmasq = a$h, seed = 1235) ## 2019-03-13 deal with change in RNG in R-devel (for 3.6.0), see email from Kurt Hornik in Org/ ## I couldn't make his suggestion work. ## TODO: maybe it would be better to just load 'a' from a saved version. if(getRversion() < "3.6.0"){ a_saved <- "a_before_6.0.RDS" a_pred_saved <- "a_pred_before_6.0.RDS" }else{ a_saved <- "a.RDS" a_pred_saved <- "a_pred.RDS" } expect_equal_to_reference(a, a_saved) expect_equal_to_reference(a_pred, a_pred_saved) ## as above but without 'seed' sim_garch1c1(a_mo, n = 100, n.start = 100) predict(a_mo, n.ahead = 5, Nsim = 100, eps = a$eps, sigmasq = a$h) expect_equal(.get_dist_elem(NULL, "r"), .dist$norm$r) ## for now expect_error(.get_dist_elem("unknowndist", "r")) expect_error(.get_dist_elem(list("unknowndist"), "r")) expect_error(.get_dist_elem(5, "r")) .get_dist_elem(list("norm"), "r") .get_dist_elem(list("norm", mean = 5, sd = 3), "r") .get_dist_elem(list("norm", mean = 5, sd = 3, n = 10), "r") .get_dist_elem("std", "r") .get_dist_elem(list("std", df = 5), "r") .get_dist_elem(list("std", df = 5, n = 10), "r") .get_dist_elem(list("ged"), "r") expect_equal(.get_dist_elem(NULL, "p"), .dist$norm[["p"]]) .get_dist_elem("norm", "d") .get_dist_elem("norm", "p") .get_dist_elem("norm", "q") .get_dist_elem("norm", "r") .get_dist_elem(list("norm"), "p") .get_dist_elem(list("norm", mean = 4, sd = 2), "p") ## for now expect_error(.get_dist_elem("unknowndist")) expect_error(.get_dist_elem(list("unknowndist"))) expect_error(.get_dist_elem(5)) }) cvar/tests/testthat/test-VaR.R0000644000176200001440000001107214330671763016012 0ustar liggesuserslibrary(cvar) context("VaR") test_that("VaR works ok", { muA <- 0.006408553 sigma2A <- 0.0004018977 ## with quantile function res1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) res2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) # 0, intercept/slope equivalent to mean/sd expect_error(cvar::VaR(dnorm, dist.type = "pdf", p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)), "Not ready") cvar::VaR_qf(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) ## with cdf the precision depends on solving an equation res1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # 3.287939e-09 abs((res2a - res2)) # 5.331195e-11, intercept/slope better numerically expect_equal(res1, VaR_qf(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A))) expect_equal(res2a, VaR_cdf(pnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A))) set.seed(1236) a.num <- rnorm(100) VaR(a.num) ## test the fix for issue #2 expect_true(ES(a.num) == ES(a.num, p_loss = 0.05)) expect_true(ES(a.num) != ES(a.num, p_loss = 0.01)) ## as above, but increase the precision, this is probably excessive res1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) expect_lt( abs((res1b - res2)), 1e-12) # 6.938894e-18 # both within machine precision expect_lt( abs((res2b - res2)), 1e-12) # 1.040834e-16 ## relative precision is also good abs((res1b - res2)/res2) # 2.6119e-16 # both within machine precision abs((res2b - res2)/res2) # 3.91785e-15 ## an extended example with vector args, if "PerformanceAnalytics" is present if(require("PerformanceAnalytics")){ data(edhec, package = "PerformanceAnalytics") mu <- apply(edhec, 2, mean) sigma2 <- apply(edhec, 2, var) musigma2 <- cbind(mu, sigma2) ## compute in 2 ways with cvar::VaR vAz1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = mu, sd = sqrt(sigma2)) vAz2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = mu, slope = sqrt(sigma2)) vAz1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2)) vAz2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2)) vAz1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2), tol = .Machine$double.eps^0.75) vAz2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2), tol = .Machine$double.eps^0.75) ## analogous calc. with PerformanceAnalytics::VaR vPA <- apply(musigma2, 1, function(x) PerformanceAnalytics::VaR(p = .95, method = "gaussian", invert = FALSE, mu = x[1], sigma = x[2], weights = 1)) ## the results are numerically the same max(abs((vPA - vAz1))) # 5.551115e-17 max(abs((vPA - vAz2))) # "" max(abs((vPA - vAz1a))) # 3.287941e-09 max(abs((vPA - vAz2a))) # 1.465251e-10, intercept/slope better max(abs((vPA - vAz1b))) # 4.374869e-13 max(abs((vPA - vAz2b))) # 3.330669e-16 } }) test_that("ES works ok", { expect_equal(ES(qnorm, dist.type = "qf"), ES(qnorm) ) ## Gaussian expect_equal(ES(qnorm, dist.type = "qf"), ES(pnorm, dist.type = "cdf") ) ## t-dist expect_equal(ES(qt, dist.type = "qf", df = 4), ES(pt, dist.type = "cdf", df = 4) ) expect_equal(ES(pnorm, p_loss = 0.95, dist.type = "cdf"), ES(qnorm, p_loss = 0.95, dist.type = "qf") ) ## - VaRES::esnormal(0.95, 0, 1) ## - PerformanceAnalytics::ETL(p=0.05, method = "gaussian", mu = 0, ## sigma = 1, weights = 1) # same ## this uses "pdf" expect_equal(ES(dnorm, p_loss = 0.05, dist.type = "pdf", qf = qnorm), ES(pnorm, p_loss = 0.05, dist.type = "cdf") ) ## 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P^VP _w-{]&;WzSEj镝R*DݘU)&k\q]h)0ds;TkdTufXuD ^6Fn[;U" &zt}1I!M.84v,@a+1;Pʖ"bF1łsͫR{ ~ߡC[6_tm_--C[W]#"py-e&J|*7A/瓟=e' ֳI o0ؼCmٮpJo3:> ݅ _6Q\uZAO$%;uk/Tw>͂IzfFl^LC|dÉ߹uƌm΅^*"F65;%I =$.ta "ݼ K?4[U0;9 X7zi_.Bp^]#T@r;ƣ- aJ">p=p]GɞAD& ֹ[>>U FW:ӧ oPwv Y C'_O&x <Zv/0Dg|1L=S p ƶ4\:sP.G8V]j5"K+c\$A>lX]7CCghoK}CߝMM󚖸ᭉEzx-8y-sKk59m_|쫍O$|q>{iCKA/*ō<=:D'߬-D9=վv R}՝էm8%Y BxzPc{:8;_"B\ 9t,mوˬN ɍO_ aOo&η::SBVّD a`_Z+4L3ܤiq3Vr cvar/tests/testthat.R0000644000176200001440000000006413630522362014334 0ustar liggesuserslibrary(testthat) library(cvar) test_check("cvar") cvar/vignettes/0000755000176200001440000000000014330705053013215 5ustar liggesuserscvar/vignettes/Guide_cvar.Rnw0000644000176200001440000003272614330671606015775 0ustar liggesusers% preamble taken from the vignette in package strucchange % \documentclass[nojss,article]{jss} \usepackage[T1]{fontenc} %\usepackage{a4wide} \usepackage[left=2cm,right=2cm,bottom=15mm]{geometry} \usepackage{graphicx,color,alltt} \usepackage[authoryear,round,longnamesfirst]{natbib} \usepackage{hyperref} \usepackage{amsmath, amsfonts} % \usepackage{Sweave} %\definecolor{Red}{rgb}{0.7,0,0} %\definecolor{Blue}{rgb}{0,0,0.8} %\definecolor{hellgrau}{rgb}{0.55,0.55,0.55} %\newcommand{\E}{\mbox{$\mathsf{E}$}} %\newcommand{\VAR}{\mbox{$\mathsf{VAR}$}} %\newcommand{\COV}{\mbox{$\mathsf{COV}$}} %\newcommand{\p}{\mbox{$\mathsf{P}$}} %\newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} %\newenvironment{smallexample}{\begin{alltt}\small}{\end{alltt}} %\setlength{\parskip}{0.5ex plus0.1ex minus0.1ex} %\setlength{\parindent}{0em} % %\bibpunct{(}{)}{;}{a}{}{,} % %\newcommand{\ui}{\underline{i}} %\newcommand{\oi}{\overline{\imath}} %\RequirePackage{color} %\definecolor{Red}{rgb}{0.5,0,0} %\definecolor{Blue}{rgb}{0,0,0.5} %\definecolor{hellgrau}{rgb}{0.55,0.55,0.55} % %\hypersetup{% %hyperindex,% %colorlinks,% %linktocpage,% %plainpages=false,% %linkcolor=Blue,% %citecolor=Blue,% %urlcolor=Red,% %pdfstartview=Fit,% %pdfview={XYZ null null null}% %} % SweaveOpts{engine=R,eps=FALSE} %\VignetteIndexEntry{Brief guide to R package cvar} %\VignetteDepends{cvar} %\VignetteKeywords{expected shortfall, ES, CVaR, VaR, value at risk} %\VignettePackage{cvar} <>= library(cvar) pd <- packageDescription("cvar") @ \newcommand{\VaR}[2][\alpha]{\text{VaR}_{#1}(#2)} \newcommand{\ES}[2][\alpha]{\text{ES}_{#1}(#2)} \title{Brief guide to R package \pkg{cvar}} \author{Georgi N. Boshnakov \\ University of Manchester } \Plainauthor{Georgi N. Boshnakov} \Address{ Georgi N. Boshnakov\\ School of Mathematics\\ The University of Manchester\\ Oxford Road, Manchester M13 9PL, UK\\ URL: \url{http://www.maths.manchester.ac.uk/~gb/} } %\date{} %\maketitle \Abstract{ Package \pkg{cvar} is a small package with, essentially, two functions --- \code{ES()} for computing the expected shortfall and \code{VaR()} for Value at Risk. The user specifies the distribution by supplying one of the functions that define a continuous distribution---currently this can be a quantile function (qf), cumulative distribution function (cdf) or probability density function (pdf). Virtually any continuous distribution can be specified. This vignette is part of package \pkg{cvar}, version~\Sexpr{pd$Version}. %$ } \Keywords{expected shortfall, ES, CVaR, VaR, value at risk} \Plainkeywords{expected shortfall, ES, CVaR, VaR, value at risk} \begin{document} \section{Introduction} \label{sec:introduction} There is a huge number of functions for computations with distributions in core R and in contributed packages. Pdf's, cdf's, quantile functions and random number generators are covered comprehensively. The coverage of expected shortfall is more patchy but a large collection of distributions, including functions for expected shortfall, is provided by \citet{VaRES2013}. \citet{PerformanceAnalytics2018} and \citet{actuarJSS2008} provide packages covering comprehensively various aspects of risk measurement, including some functions for expected shortfall. Package \pkg{cvar} is a small package with, essentially two functions --- \code{ES} for computing the expected shortfall and \code{VaR} for Value at Risk. The user specifies the distribution by supplying one of the functions that define a continuous distribution---currently this can be a quantile function (qf), cumulative distribution function (cdf) or probability density function (pdf). Virtually any continuous distribution can be specified. The computations are done directly from the definitions, see e.g. \citet{acerbi2002expected}. The functions are vectorised over the parameters of the distributions, making bulk computations more convenient, for example for forecasting or model evaluation. The name of this package, "cvar", comes from \emph{Conditional Value at Risk} (CVaR), which is an alternative term for expected shortfall. We chose to use the standard names \code{ES} and \code{VaR}, despite the possibility for name clashes with same named functions in other packages, rather than invent possibly difficult to remember alternatives. Just call the functions as \code{cvar::ES} and \code{cvar::VaR} if necessary. Locations-scale transformations can be specified separately from the other distribution parameters. This is useful when such parameters are not provided directly by the distribution at hand. The use of these parameters often leads to more efficient computations and better numerical accuracy even if the distribution has its own parameters for this purpose. Some of the examples for \code{VaR} and \code{ES} illustrate this for the Gaussian distribution. Since VaR is a quantile, functions computing it for a given distribution are convenience functions. \code{VaR} exported by \pkg{cvar} could be attractive in certain workflows because of its vectorised distribution parameters, the location-scale transformation and the possibility to compute it from cdf's when quantile functions are not available. \section{Value at Risk and Expected Shortfall} %To avoid confusion, we give the mathematical definitions here. We use the traditional definition of VaR as the negated lower quantile. More specifically, let $Y$ be the variable of interes, such as return on a financial asset. Suppose that it is modelled as a random variable with distribution function $F^{Y}(y)$. Then VaR is defined as% \footnote{Beware that a definition without negation is also used in both the literature and in the R packages.} \begin{equation*} \VaR{Y} = - q_{\alpha}^{Y} . \end{equation*} where $q_{\alpha}^{Y}$ is the lower $\alpha$-quantile of $Y$, i.e. we have \begin{equation*} \alpha = F(q_{\alpha}^{Y}) = P(Y \le q_{\alpha}^{Y}) . \end{equation*} Typical values for $\alpha$ are $0.05$ and $0.01$. Equivalently, we could write\footnote{This equation shows why some authors use the ``big numbers'', e.g. 0.95 and 0.99, in the notation for the VaR.} \begin{equation*} \VaR{Y} = q_{1-\alpha}^{-Y} . \end{equation*} The expected shortfall is the (partial) expectation of $\VaR{Y}$: \begin{equation*} \ES{Y} = \frac{1}{\alpha}\int_{0}^{\alpha}\VaR[\gamma]{Y} d\gamma . \end{equation*} Suppose now that $Y$ is obtained from another random variable, $X$, by a linear transformation: \begin{equation*} Y = a + bX \end{equation*} When this is the case, there is a simple relation between the VaR's and ES's of $Y$ and $X$: \begin{align*} \VaR{Y} &= -a + b \VaR{X} \\ \ES{Y} &= -a + b \ES{X} \end{align*} In practice, $X$ is often chosen to be standardised but this is not necessary. Note also that if a bunch of VaR's and ES's are needed, say for normally distributed variables, this can be computed very efficiently using this property. \section{VaR} \label{sec:var} Here we compute the VaR associated with a standard normal r.v. The three variants are equivalent since 0.05 and \code{"qf"} are default for the last two arguments. <<>>= cvar::VaR(qnorm, p_loss = 0.05, dist.type = "qf") cvar::VaR(qnorm, p_loss = 0.05) cvar::VaR(qnorm) @ \code{x} can be a vector: <<>>= cvar::VaR(qnorm, p_loss = c(0.01, 0.05)) @ Let's set some more realistic values for the parameters of the normal distribution. Suppose that the daily returns on some stock have sample mean $0.006408553$ and sample variance $0.0004018977$. Then $N(0.006408553, 0.0004018977)$ can be taken as normal distribution fitted to the data. <<>>= muA <- 0.006408553 sigma2A <- 0.0004018977 @ This computes VaR using the fitted normal distribution: <<>>= res1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) @ For the normal disribution we could also use the intercept-slope arguments, since the parameters are precisely the intercept and the slope: <<>>= res2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) # 0, intercept/slope equivalent to mean/sd @ If we compute VaR using the cdf, the intercept-slope method has some numerical advantage: <<>>= ## with cdf the precision depends on solving an equation res1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # 3.287939e-09 abs((res2a - res2)) # 5.331195e-11, intercept/slope better numerically @ Of course, it is almost always better to use the quantile function when it is available. We can use smaller tollerance to improve the precision (the value \verb+.Machine$double.eps^0.75+ = \Sexpr{.Machine$double.eps^0.75} is probably excessive): <<>>= ## as above, but increase the precision, this is probably excessive res1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) # 6.938894e-18 # both within machine precision abs((res2b - res2)) # 1.040834e-16 @ The relative precision is also good. <<>>= abs((res1b - res2)/res2) # 2.6119e-16 # both within machine precision abs((res2b - res2)/res2) # 3.91785e-15 @ For examples with vectorised distribution parameters we use data from \pkg{PerformanceAnalytics}. Compute means and variances of the variables in a data frame and store them in vectors: <<>>= ## if(require("PerformanceAnalytics")){ data(edhec, package = "PerformanceAnalytics") mu <- apply(edhec, 2, mean) sigma2 <- apply(edhec, 2, var) musigma2 <- cbind(mu, sigma2) @ We compute VaR using \code{PerformanceAnalytics::VaR}: <<>>= ## analogous calc. with PerformanceAnalytics::VaR vPA <- apply(musigma2, 1, function(x) PerformanceAnalytics::VaR(p = .95, method = "gaussian", invert = FALSE, mu = x[1], sigma = x[2], weights = 1)) @ The results below compare to the value, \code{vPA}, obtained here. The computations below are similar to the previous example but the distribution parameters are vectors. The first pair of results are numerically the same: <<>>= vAz1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = mu, sd = sqrt(sigma2)) vAz2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = mu, slope = sqrt(sigma2)) max(abs((vPA - vAz1))) # 5.551115e-17 max(abs((vPA - vAz2))) # "" @ Computing VaR from cdf shows some advantage for the location-scale method: <<>>= vAz1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2)) vAz2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2)) max(abs((vPA - vAz1a))) # 3.287941e-09 max(abs((vPA - vAz2a))) # 1.465251e-10, intercept/slope better @ The advantage remains for smaller tolerance: <<>>= vAz1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2), tol = .Machine$double.eps^0.75) vAz2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2), tol = .Machine$double.eps^0.75) max(abs((vPA - vAz1b))) # 4.374869e-13 max(abs((vPA - vAz2b))) # 3.330669e-16 @ \section{Expected shortfall} \label{sec:expected-shortfall} \code{ES} has essentially the same arguments as \code{VaR}. The examples below are obtained almost literally by replacing the calls to \code{VaR} with \code{ES()} ones. Here we compute the VaR associated with a standard normal r.v. The three variants are equivalent since 0.5 and \code{"qf"} are default for the last two arguments. <<>>= cvar::ES(qnorm, p_loss = 0.05, dist.type = "qf") cvar::ES(qnorm, p_loss = 0.05) cvar::ES(qnorm) @ \code{x} can be a vector: <<>>= cvar::ES(qnorm, p_loss = c(0.01, 0.05)) @ Let's set some more realistic values for the parameters of the normal distribution: <<>>= muA <- 0.006408553 sigma2A <- 0.0004018977 @ This demonstrates the use of \code{intercept} and \code{slope}. These arguments use the linear transformation property discussed in the theoretical section. For the normal disribution the parameters are precisely the intercept and the slope: <<>>= res1 <- cvar::ES(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) res2 <- cvar::ES(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) @ If we compute ES using the cdf, the intercept slope method has some numerical advantage: <<>>= ## with cdf the precision depends on solving an equation res1a <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # abs((res2a - res2)) # intercept/slope better numerically @ We can use smaller tollerance to improve the precision (the value \verb+.Machine$double.eps^0.75+ = \Sexpr{.Machine$double.eps^0.75} is probably excessive): <<>>= ## as above, but increase the precision, this is probably excessive res1b <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) abs((res2b - res2)) @ The relative precision is also good. <<>>= abs((res1b - res2)/res2) abs((res2b - res2)/res2) @ \newpage{} \bibliography{REFERENCES} \end{document} cvar/vignettes/REFERENCES.bib0000644000176200001440000000233113501140665015314 0ustar liggesusers@article{acerbi2002expected, title={Expected shortfall: a natural coherent alternative to value at risk}, author={Acerbi, Carlo and Tasche, Dirk}, journal={Economic notes}, volume={31}, number={2}, pages={379--388}, year={2002}, doi = {10.1111/1468-0300.00091}, url = {https://arxiv.org/pdf/cond-mat/0105191.pdf}, } @Manual{PerformanceAnalytics2018, title = {PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis}, author = {Brian G. Peterson and Peter Carl}, year = {2018}, note = {R package version 1.5.2}, url = {https://CRAN.R-project.org/package=PerformanceAnalytics}, } @Manual{VaRES2013, title = {VaRES: Computes value at risk and expected shortfall for over 100 parametric distributions}, author = {Saralees Nadarajah and Stephen Chan and Emmanuel Afuecheta}, year = {2013}, note = {R package version 1.0}, url = {https://CRAN.R-project.org/package=VaRES}, } @Article{actuarJSS2008, title = {actuar: An R Package for Actuarial Science}, author = {Christophe Dutang and Vincent Goulet and Mathieu Pigeon}, journal = {Journal of Statistical Software}, year = {2008}, volume = {25}, number = {7}, pages = {38}, url = {http://www.jstatsoft.org/v25/i07}, } cvar/vignettes/auto/0000755000176200001440000000000014210656663014176 5ustar liggesuserscvar/vignettes/auto/Guide_cvar.el0000644000176200001440000000240714210656663016573 0ustar liggesusers(TeX-add-style-hook "Guide_cvar" (lambda () (TeX-add-to-alist 'LaTeX-provided-class-options '(("jss" "nojss" "article"))) (TeX-add-to-alist 'LaTeX-provided-package-options '(("fontenc" "T1") ("geometry" "left=2cm" "right=2cm" "bottom=15mm") ("natbib" "authoryear" "round" "longnamesfirst"))) (add-to-list 'LaTeX-verbatim-environments-local "alltt") (add-to-list 'LaTeX-verbatim-macros-with-braces-local "path") (add-to-list 'LaTeX-verbatim-macros-with-braces-local "url") (add-to-list 'LaTeX-verbatim-macros-with-braces-local "nolinkurl") (add-to-list 'LaTeX-verbatim-macros-with-braces-local "hyperbaseurl") (add-to-list 'LaTeX-verbatim-macros-with-braces-local "hyperimage") (add-to-list 'LaTeX-verbatim-macros-with-braces-local "hyperref") (add-to-list 'LaTeX-verbatim-macros-with-delims-local "path") (TeX-run-style-hooks "latex2e" "jss" "jss10" "fontenc" "geometry" "graphicx" "color" "alltt" "natbib" "hyperref" "amsmath" "amsfonts") (TeX-add-symbols '("ES" ["argument"] 1) '("VaR" ["argument"] 1)) (LaTeX-add-labels "sec:introduction" "sec:var" "sec:expected-shortfall") (LaTeX-add-bibliographies "REFERENCES")) :latex) cvar/R/0000755000176200001440000000000014330677051011414 5ustar liggesuserscvar/R/garch.R0000644000176200001440000004277414330267202012632 0ustar liggesusers#' Specify a GARCH model #' #' Specify a GARCH model. #' #' Argument \code{model} can be the result of a previous call to \code{GarchModel}. #' Arguments in \code{"..."} overwrite current components of \code{model}. #' #' \code{GarchModel} guarantees that code using it will continue to work #' transparently for the user even if the internal represedtation of GARCH #' models in this package is changed or additional functionality is added. #' #' @param model a GARCH model or a list. #' @param ... named arguments specifying the GARCH model. #' @param model.class a class for the result. By default \code{GarchModel()} #' decides the class of the result. #' @return an object from suitable GARCH-type class #' #' @examples #' ## GARCH(1,1) with Gaussian innovations #' mo1a <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5) #' mo1b <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, cond.dist = "norm") #' #' ## equivalently, the parameters can be given as a list #' p1 <- list(omega = 1, alpha = 0.3, beta = 0.5) #' mo1a_alt <- GarchModel(p1) #' mo1b_alt <- GarchModel(p1, cond.dist = "norm") #' stopifnot(identical(mo1a, mo1a_alt), identical(mo1b, mo1b_alt)) #' #' ## additional arguments modify values already in 'model' #' mo_alt <- GarchModel(p1, beta = 0.4) #' #' ## set also initial values #' mo2 <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, esp0 = - 1.5, h0 = 4.96) #' #' ## GARCH(1,1) with standardised-t_5 #' mot <- GarchModel(omega = 1, alpha = 0.3, beta = 0.5, cond.dist = list("std", nu = 5)) #' #' @export GarchModel <- function(model = list(), ..., model.class = NULL){ ## TODO: check the correctness of the parameters ## alpha, beta, cond.dist, 2nd order stationary, initial value for eps_t^2, h_t ## 2019-03-15 TODO: handle fGARCH models ## if(inherits(model, "fGARCH")){ ## ## }else{ dots <- list(...) if(length(dots) > 0) model[names(dots)] <- dots class(model) <- "GarchModel0" if(is.null(model.class) && length(model$alpha) == 1 && length(model$beta) == 1) class(model) <- "garch1c1" # GARCH(1,1), 'c' stands for 'comma' ## } model } ## 2022-11-01 change .dist from 'list' to 'env', was: # .dist <- list( # norm = list(d = call("dnorm", x = NA, mean = 0, sd = 1), # p = call("pnorm", q = NA, mean = 0, sd = 1), # q = call("qnorm", p = NA, mean = 0, sd = 1), # r = call("rnorm", n = NA, mean = 0, sd = 1)), # std = list(d = call("dstd", x = NA, mean = 0, sd = 1, nu = NA), # p = call("pstd", q = NA, mean = 0, sd = 1, nu = NA), # q = call("qstd", p = NA, mean = 0, sd = 1, nu = NA), # r = call("rstd", n = NA, mean = 0, sd = 1, nu = NA)), # ged = list(d = call("dged", x = NA, mean = 0, sd = 1, nu = NA), # p = call("pged", q = NA, mean = 0, sd = 1, nu = NA), # q = call("qged", p = NA, mean = 0, sd = 1, nu = NA), # r = call("rged", n = NA, mean = 0, sd = 1, nu = NA)) # ) .dist <- new.env() ## N(0,1) is built-in .dist$norm <- list(d = call("dnorm", x = NA, mean = 0, sd = 1), p = call("pnorm", q = NA, mean = 0, sd = 1), q = call("qnorm", p = NA, mean = 0, sd = 1), r = call("rnorm", n = NA, mean = 0, sd = 1)) # generalise and prepare to make fGarch suggested; was: # # .dist$std <- list(d = call("dstd", x = NA, mean = 0, sd = 1, nu = NA), # p = call("pstd", q = NA, mean = 0, sd = 1, nu = NA), # q = call("qstd", p = NA, mean = 0, sd = 1, nu = NA), # r = call("rstd", n = NA, mean = 0, sd = 1, nu = NA)) # # .dist$ged <- list(d = call("dged", x = NA, mean = 0, sd = 1, nu = NA), # p = call("pged", q = NA, mean = 0, sd = 1, nu = NA), # q = call("qged", p = NA, mean = 0, sd = 1, nu = NA), # r = call("rged", n = NA, mean = 0, sd = 1, nu = NA)) .dist_list <- list( ## TODO: allow different packages here by letting 'package be vector or completely ## refactor all this. Only '.get_dist_elem()' should know the implementation. std = list(package = "fGarch", d = list(str2lang("fGarch::dstd"), x = NA, mean = 0, sd = 1, nu = NA), p = list(str2lang("fGarch::pstd"), q = NA, mean = 0, sd = 1, nu = NA), q = list(str2lang("fGarch::qstd"), p = NA, mean = 0, sd = 1, nu = NA), r = list(str2lang("fGarch::rstd"), n = NA, mean = 0, sd = 1, nu = NA)), ged = list(package = "fGarch", d = list(str2lang("fGarch::dged"), x = NA, mean = 0, sd = 1, nu = NA), p = list(str2lang("fGarch::pged"), q = NA, mean = 0, sd = 1, nu = NA), q = list(str2lang("fGarch::qged"), p = NA, mean = 0, sd = 1, nu = NA), r = list(str2lang("fGarch::rged"), n = NA, mean = 0, sd = 1, nu = NA)) ) .get_dist_elem <- function(dist, what){ if(is.list(dist)){ params <- dist[-1] dist <- dist[[1]] }else params <- list() if(is.null(dist)){ ## TODO: this probably should be handled by the caller ## normal distribution is default res <- .dist$norm[[what]] }else{ stopifnot(is.character(dist), length(dist) == 1) gen <- .dist[[dist]] if(!is.null(gen)) # already prepared res <- gen[[what]] else{ # not prepared yet gen_list <- .dist_list[[dist]] if(is.null(gen_list)) stop("unknown distribution specified.") if(!requireNamespace(gen_list$package)) stop("requested distribution property needs package ", gen_list$package, "\nplease install package '", gen_list$package, "' and try again") ## prepare ## .dist is env, so assignment is persistent .dist[[dist]] <- lapply(gen_list[-1], as.call) # dropping $package res <- .dist[[dist]][[what]] if(is.null(res)) stop("property '", "'what not available for '", dist, "'") } } if(length(params) > 0) # set parameters if specified res[names(params)] <- params res } #' Simulate GARCH(1,1) time series #' #' Simulate GARCH(1,1) time series. #' #' The simulated time series is in component \code{eps} of the returned value. #' For exploration of algorithms and eestimation procedures, the volatilities #' and the standardised innovations are also returned. #' #' The random seed at the start of the simulations is saved in the returned #' object. A speficific seed can be requested with argument \code{seed}. In #' that case the simulations are done with the specified seed and the old state #' of the random number generator is restored before the function returns. #' #' @param model a GARCH(1,1) model, an object obtained from \code{GarchModel}. #' @param n the length of the generated time series. #' @param n.start number of warm-up values, which are then dropped. #' @param seed an integer to use for setting the random number generator. #' #' @return a list with components: #' \item{eps}{the time series,} #' \item{h}{the (squared) volatilities,} #' \item{eta}{the standardised innovations,} #' \item{model}{the GARCH(1,1) model,} #' \item{.sim}{a list containing the parameters of the simulation,} #' \item{call}{the call.} #' #' @note This function is under development and may be changed. #' #' @export sim_garch1c1 <- function(model, n, n.start = 0, seed = NULL){ ## TODO: seed is not used currently garch1c1_fields <- c("omega", "alpha", "beta", "cond.dist", "eps0", "h0", "eps0sq") stopifnot(all(names(model) %in% garch1c1_fields)) omega <- model$omega alpha <- model$alpha beta <- model$beta rgen <- .get_dist_elem(model$cond.dist, "r") eps0 <- model$eps0 h0 <- model$h0 eps0sq <- model$eps0sq ## see Francq & Zakoian, p. 142, for alternative initialisations if(is.null(eps0sq)){ model$eps0sq <- eps0sq <- if(is.null(eps0)) omega / (1 - alpha - beta) else eps0^2 } if(is.null(h0)){ model$h0 <- h0 <- omega / (1 - alpha - beta) } N <- n + n.start h <- eps <- numeric(N) RNGstate <- .RNGstate(seed) ## 2020-03-04 was: if(!is.null(RNGstate$oldRNGstate)) # or !is.null(seed) ## (see comment at a similar command) if(!is.null(seed)) on.exit( if(is.null(RNGstate$RNGstate)){ ## TRUE id seed is NULL but also if .Random.seed was not set. "nothing to do" }else if(is.null(RNGstate$oldRNGstate)){ rm(".Random.seed", envir = .GlobalEnv) }else{ assign(".Random.seed", RNGstate$oldRNGstate, envir = .GlobalEnv) } ) rgen$n <- N # or: rgen[[2]] <- N eta <- eval(rgen) h[1] <- omega + alpha * eps0sq + beta * h0 for(i in 1:(N - 1)){ eps[i] <- sqrt(h[i]) * eta[i] h[i + 1] <- omega + alpha * eps[i]^2 + beta * h[i] } eps[N] <- sqrt(h[N]) * eta[N] ## TODO: return eta only conditionally? ## TODO: include the model in the returned value? ## TODO: return also the initialisation values (in a separate component)? list(eps = eps[(n.start + 1):N], h = h[(n.start + 1):N], eta = eta[(n.start + 1):N], model = model, .sim = list(seed = seed, rand.gen = rgen, n = n, n.start = n.start), call = match.call() ) } ## modelled after the beginning of `stats:::simulate.lm()` .RNGstate <- function(seed = NULL){ ## 2020-03-04 - changing the logic below ## ## if(!exists(".Random.seed", envir = .GlobalEnv, inherits = FALSE)) ## runif(1) oldRNGstate <- if(exists(".Random.seed", envir = .GlobalEnv)) get(".Random.seed", envir = .GlobalEnv) else NULL if (is.null(seed)) list(RNGstate = oldRNGstate) else{ ## This doesn't resolve the problem ## suppressWarnings(RNGversion("3.5.0")) # 2019-03-13 temporary, RNG changed in R-devel. ## # see email from Kurt Hornik in Org/ set.seed(seed) RNGstate <- structure(seed, kind = as.list(RNGkind())) ## on.exit(assign(".Random.seed", oldRNGstate, envir = .GlobalEnv)) list(RNGstate = RNGstate, oldRNGstate = oldRNGstate) } } #' Prediction for GARCH(1,1) time series #' #' Predict GARCH(1,1) time series. #' #' Plug-in prediction intervals and predictive distributions are obtained by #' inserting the predicted volatility in the conditional densities. For #' predictions more than one lag ahead these are not the real predictive #' distributions but the prediction intervals are usually adequate. #' #' For simulation prediction intervals we generate a (large) number of #' continuations of the given time series. Prediction intervals can be based on #' sample quantiles. The generated samples are stored in the returned object and #' can be used for further exploration of the predictive #' distributions. \code{dist_sim$eps} contains the simulated future values of #' the time series and \code{dist_sim$h} the corresponding (squared) #' volatilities. Both are matrices whose \code{i}-th rows contain the predicted #' quantities for horizon \code{i}. #' #' The random seed at the start of the simulations is saved in the returned #' object. A speficific seed can be requested with argument \code{seed}. In #' that case the simulations are done with the specified seed and the old state #' of the random number generator is restored before the function returns. #' This setup is similar to \code{\link{sim_garch1c1}}. #' #' @param object an object from class \code{"garch1c1"}. #' @param n.ahead maximum horizon (lead time) for prediction. #' @param Nsim number of Monte Carlo simulations for simulation based quantities. #' @param eps the time series to predict, only the last value is used. #' @param sigmasq the (squared) volatilities, only the last value is used. #' @param seed an integer, seed for the random number generator. #' @param ... currently not used. #' #' @return an object from S3 class \code{"predict_garch1c1"} containing #' the following components: #' \item{eps}{point predictions (conditional expectations) of the time series (equal #' to zero for pure GARCH).} #' \item{h}{point predictions (conditional expectations)of the squared volatilities.} #' \item{model}{the model.} #' \item{call}{the call.} #' \item{pi_plugin}{Prediction intervals for the time series, based on plug-in #' distributions, see Details.} #' \item{pi_sim}{Simulation based prediction intervals for the time series, see Details.} #' \item{dist_sim}{simulation samples from the predictive distributions of the time #' series and the volatilties. } #' #' @examples #' op <- options(digits = 4) #' #' ## set up a model and simulate a time series #' mo <- GarchModel(omega = 0.4, alpha = 0.3, beta = 0.5) #' a1 <- sim_garch1c1(mo, n = 1000, n.start = 100, seed = 20220305) #' #' ## predictions for T+1,...,T+5 (T = time of last value) #' ## Nsim is small to reduce the load on CRAN, usually Nsim is larger. #' a.pred <- predict(mo, n.ahead = 5, Nsim = 1000, eps = a1$eps, #' sigmasq = a1$h, seed = 1234) #' #' ## preditions for the time series #' a.pred$eps #' #' ## PI's for eps - plug-in and simulated #' a.pred$pi_plugin #' a.pred$pi_sim #' #' ## a DIY calculation of PI's using the simulated sample paths #' t(apply(a.pred$dist_sim$eps, 1, function(x) quantile(x, c(0.025, 0.975)))) #' #' ## further investigate the predictive distributions #' t(apply(a.pred$dist_sim$eps, 1, function(x) summary(x))) #' #' ## compare predictive densities for horizons 2 and 5: #' h2 <- a.pred$dist_sim$eps[2, ] #' h5 <- a.pred$dist_sim$eps[5, ] #' main <- "Predictive densities: horizons 2 (blue) and 5 (black)" #' plot(density(h5), main = main) #' lines(density(h2), col = "blue") #' #' ## predictions of sigma_t^2 #' a.pred$h #' #' ## plug-in predictions of sigma_t #' sqrt(a.pred$h) #' #' ## simulation predictive densities (PD's) of sigma_t for horizons 2 and 5: #' h2 <- sqrt(a.pred$dist_sim$h[2, ]) #' h5 <- sqrt(a.pred$dist_sim$h[5, ]) #' main <- "PD's of sigma_t for horizons 2 (blue) and 5 (black)" #' plot(density(h2), col = "blue", main = main) #' lines(density(h5)) #' #' ## VaR and ES for different horizons #' cbind(h = 1:5, #' VaR = apply(a.pred$dist_sim$eps, 1, function(x) VaR(x, c(0.05))), #' ES = apply(a.pred$dist_sim$eps, 1, function(x) ES(x, c(0.05))) ) #' #' ## fit a GARCH(1,1) model to exchange rate data and predict #' gmo1 <- fGarch::garchFit(formula = ~garch(1, 1), data = fGarch::dem2gbp, #' include.mean = FALSE, cond.dist = "norm", trace = FALSE) #' mocoef <- gmo1@fit$par #' mofitted <- GarchModel(omega = mocoef["omega"], alpha = mocoef["alpha1"], #' beta = mocoef["beta1"]) #' gmo1.pred <- predict(mofitted, n.ahead = 5, Nsim = 1000, eps = gmo1@data, #' sigmasq = gmo1@h.t, seed = 1234) #' gmo1.pred$pi_plugin #' gmo1.pred$pi_sim #' #' op <- options(op) # restore options(digits) #' #' @note This function is under development and may be changed. #' #' @export predict.garch1c1 <- function(object, n.ahead = 1, Nsim = 1000, eps, sigmasq, seed = NULL, ...){ model <- object omega <- model$omega alpha <- model$alpha beta <- model$beta model$eps0 <- eps0 <- eps[length(eps)] model$h0 <- h0 <- sigmasq[length(sigmasq)] # TODO: compute if not present? ## TODO: add non-zero mean and possibly ARMA conditional mean pred_eps <- numeric(n.ahead) ## tsff slides, p. II.7.4/y1718 pred_h <- numeric(n.ahead) pred_h[1] <- omega + alpha * eps0^2 + beta * h0 if(n.ahead > 1) for(i in 2:n.ahead){ pred_h[i] <- omega + (alpha + beta) * pred_h[i - 1] } fq <- .get_dist_elem(model$cond.dist, "q") fq$sd <- sqrt(pred_h) fq[[2]] <- 0.025 lwr <- eval(fq) fq[[2]] <- 0.975 upr <- eval(fq) eps_pi <- cbind(lwr, upr) ## need local variants of withr::with_seed() withr::with_preserve_seed() ## but such are not available (as of 2018-09-28), so do it DIY: ## ## Memorise the state in any case but restore the old state only if 'seed' is not NULL. RNGstate <- .RNGstate(seed) ## 2020-03-04 was: ## ## if(!is.null(RNGstate$oldRNGstate)) # or !is.null(seed) ## ## But the check is not equivalent to is.null(seed)! ## Moreover, if RNGstate$oldRNGstate is NULL, .Random.seed should be removed, ## (so doing it now) if(!is.null(seed)) on.exit( if(is.null(RNGstate$RNGstate)){ ## TRUE id seed is NULL but also if .Random.seed was not set. "nothing to do" }else if(is.null(RNGstate$oldRNGstate)){ rm(".Random.seed", envir = .GlobalEnv) }else{ assign(".Random.seed", RNGstate$oldRNGstate, envir = .GlobalEnv) } ) h_sim <- eps_sim <- matrix(NA_real_, nrow = n.ahead, ncol = Nsim) for(i in 1:Nsim){ wrk <- sim_garch1c1(model = model, n = n.ahead, n.start = 0) eps_sim[ , i] <- wrk$eps h_sim[ , i] <- wrk$h } sim_pi <- t(apply(eps_sim, 1, function(x) quantile(x, c(0.025, 0.975)))) structure(list( eps = pred_eps, h = pred_h, model = model, call = match.call(), pi_plugin = eps_pi, pi_sim = sim_pi, dist_sim = list(eps = eps_sim, h = h_sim, RNGstate = RNGstate$RNGstate) ), class = "predict_garch1c1") } cvar/R/VaR.R0000644000176200001440000004416714330677051012243 0ustar liggesusers#' Compute Value-at-Risk (VaR) #' #' @description \code{VaR} computes the Value-at-Risk of the distribution specified by the #' arguments. The meaning of the parameters is the same as in \code{\link{ES}}, including #' the recycling rules. #' #' @note We use the traditional definition of VaR as the negated lower quantile. For example, #' if \eqn{X} are returns on an asset, VAR\eqn{{}_\alpha}{_a} = \eqn{-q_\alpha}{-q_a}, #' where \eqn{q_\alpha}{-q_a} is the lower \eqn{\alpha}{a} quantile of \eqn{X}. #' Equivalently, VAR\eqn{{}_\alpha}{_a} is equal to the lower \eqn{1-\alpha}{1-a} #' quantile of \eqn{-X}. #' #' @inheritParams ES #' #' @param intercept,slope compute VaR for the linear transformation \code{intercept + #' slope*X}, where \code{X} has distribution specified by \code{dist}, see Details. #' #' #' @details #' \code{VaR} is S3 generic. The meaning of the parameters for its default method is the #' same as in \code{\link{ES}}, including the recycling rules. #' #' \code{VaR_qf} and \code{VaR_cdf} are streamlined, non-generic, variants for the common #' case when the \code{"..."} parameters are scalar. The parameters \code{x}, #' \code{intercept}, and \code{slope} can be vectors, as for \code{VaR}. #' #' Argument \code{dist} can also be a numeric vector. In that case the ES is computed, #' effectively, for the empirical cumulative distribution function (ecdf) of the #' vector. The ecdf is not created explicitly and the \code{\link[stats]{quantile}} #' function is used instead for the computation of VaR. Arguments in \code{"..."} are #' passed eventually to \code{quantile()} and can be used, for example, to select a #' non-defult method for the computation of quantiles. #' #' In practice, we may need to compute VaR associated with data. The distribution comes #' from fitting a model. In the simplest case, we fit a distribution to the data, #' assuming that the sample is i.i.d. For example, a normal distribution \eqn{N(\mu, #' \sigma^2)} can be fitted using the sample mean and sample variance as estimates of the #' unknown parameters \eqn{\mu} and \eqn{\sigma^2}, see section \sQuote{Examples}. For #' other common distributions there are specialised functions to fit their parameters and #' if not, general optimisation routines can be used. More soffisticated models may be #' used, even time series models such as GARCH and mixture autoregressive models. #' #' #' @param tol tollerance #' #' @seealso \code{\link{ES}} for ES, #' #' \code{\link[=predict.garch1c1]{predict}} for examples with fitted models #' #' @examples #' cvar::VaR(qnorm, c(0.01, 0.05), dist.type = "qf") #' #' ## the following examples use these values, obtained by fitting a normal distribution to #' ## some data: #' muA <- 0.006408553 #' sigma2A <- 0.0004018977 #' #' ## with quantile function, giving the parameters directly in the call: #' res1 <- cvar::VaR(qnorm, 0.05, mean = muA, sd = sqrt(sigma2A)) #' res2 <- cvar::VaR(qnorm, 0.05, intercept = muA, slope = sqrt(sigma2A)) #' abs((res2 - res1)) # 0, intercept/slope equivalent to mean/sd #' #' ## with quantile function, which already knows the parameters: #' my_qnorm <- function(p) qnorm(p, mean = muA, sd = sqrt(sigma2A)) #' res1_alt <- cvar::VaR(my_qnorm, 0.05) #' abs((res1_alt - res1)) #' #' ## with cdf the precision depends on solving an equation #' res1a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) #' res2a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) #' abs((res1a - res2)) # 3.287939e-09 #' abs((res2a - res2)) # 5.331195e-11, intercept/slope better numerically #' #' ## as above, but increase the precision, this is probably excessive #' res1b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", #' mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) #' res2b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", #' intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) #' abs((res1b - res2)) # 6.938894e-18 # both within machine precision #' abs((res2b - res2)) # 1.040834e-16 #' #' ## relative precision is also good #' abs((res1b - res2)/res2) # 2.6119e-16 # both within machine precision #' abs((res2b - res2)/res2) # 3.91785e-15 #' #' #' ## an extended example with vector args, if "PerformanceAnalytics" is present #' if (requireNamespace("PerformanceAnalytics", quietly = TRUE)) withAutoprint({ #' data(edhec, package = "PerformanceAnalytics") #' mu <- apply(edhec, 2, mean) #' sigma2 <- apply(edhec, 2, var) #' musigma2 <- cbind(mu, sigma2) #' #' ## compute in 2 ways with cvar::VaR #' vAz1 <- cvar::VaR(qnorm, 0.05, mean = mu, sd = sqrt(sigma2)) #' vAz2 <- cvar::VaR(qnorm, 0.05, intercept = mu, slope = sqrt(sigma2)) #' #' vAz1a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", #' mean = mu, sd = sqrt(sigma2)) #' vAz2a <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", #' intercept = mu, slope = sqrt(sigma2)) #' #' vAz1b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", #' mean = mu, sd = sqrt(sigma2), #' tol = .Machine$double.eps^0.75) #' vAz2b <- cvar::VaR(pnorm, 0.05, dist.type = "cdf", #' intercept = mu, slope = sqrt(sigma2), #' tol = .Machine$double.eps^0.75) #' #' ## analogous calc. with PerformanceAnalytics::VaR #' vPA <- apply(musigma2, 1, function(x) #' PerformanceAnalytics::VaR(p = .95, method = "gaussian", invert = FALSE, #' mu = x[1], sigma = x[2], weights = 1)) #' ## the results are numerically the same #' max(abs((vPA - vAz1))) # 5.551115e-17 #' max(abs((vPA - vAz2))) # "" #' #' max(abs((vPA - vAz1a))) # 3.287941e-09 #' max(abs((vPA - vAz2a))) # 1.465251e-10, intercept/slope better #' #' max(abs((vPA - vAz1b))) # 4.374869e-13 #' max(abs((vPA - vAz2b))) # 3.330669e-16 #' }) #' #' @export VaR <- function(dist, p_loss = 0.05, ...){ UseMethod("VaR") } warn_about_x <- function(s){ warning("argument 'x' has been renamed to 'p_loss'. 'x' still works with this version of 'cvar' but will stop doing so in the next release.\n Please change your scripts suitably.\n It most cases there is no need to name this argument when calling ", s) } #' @name VaR #' #' @export VaR_qf <- function(dist, p_loss = 0.05, ..., intercept = 0, slope = 1, tol = .Machine$double.eps^0.5, x){ if(!missing(x)){ p_loss <- x warn_about_x("VaR_qf") } dist <- match.fun(dist) ## assumes "..." are scalar res <- if(length(p_loss) > 1) - sapply(p_loss, dist, ...) else - dist(p_loss, ...) - intercept + slope * res } #' @name VaR #' #' @export VaR_cdf <- function(dist, p_loss = 0.05, ..., intercept = 0, slope = 1, tol = .Machine$double.eps^0.5, x){ if(!missing(x)){ p_loss <- x warn_about_x("VaR_cdf") } dist <- match.fun(dist) ## assumes "..." are scalar res <- if(length(p_loss) > 1) - sapply(p_loss, cdf2quantile, ..., MoreArgs = list(cdf = dist, tol = tol)) else - cdf2quantile(p_loss, dist, tol = tol, ...) # TODO: interval? - intercept + slope * res } #' @name VaR #' #' @export VaR.default <- function(dist, p_loss = 0.05, dist.type = "qf", ..., intercept = 0, slope = 1, tol = .Machine$double.eps^0.5, x){ if(!missing(x)){ p_loss <- x warn_about_x("VaR") } dist <- match.fun(dist) ## TODO: add "rand" to dist.type ## TODO: include 'dist' in the condition and mapply? # fu <- switch(dist.type, # qf = function(y) - dist(y, ...), # cdf = function(y) - cdf2quantile(y, dist, tol = tol, ...), # pdf = { # stop("Not ready yet. Please supply quantile function or cdf") # }, # stop('argument dist.type should be one of "qf", "cdf" or "pdf"') # ) # sapply(p_loss, fu) res <- switch(dist.type, qf = - mapply(dist, p_loss, ...), cdf = - mapply(cdf2quantile, p_loss, ..., MoreArgs = list(cdf = dist, tol = tol)), pdf = { stop("Not ready yet. Please supply quantile function or cdf") }, stop('argument dist.type should be one of "qf", "cdf" or "pdf"') ) - intercept + slope * res } #' @name VaR #' #' @export VaR.numeric <- function(dist, p_loss = 0.05, ..., intercept = 0, slope = 1, x){ if(!missing(x)){ p_loss <- x warn_about_x("VaR") } ## dist is raw data here res <- - quantile(dist, p_loss, ...) - intercept + slope * res } ## VaR <- function(dist, p_loss = 0.05, dist.type = "qf", ..., ## intercept = 0, slope = 1, tol = .Machine$double.eps^0.5){ ## dist <- match.fun(dist) ## ## TODO: add "rand" to dist.type ## ## TODO: include 'dist' in the condition and mapply()? ## if(length(p_loss) == 1 && all(sapply(list(...), length) < 2)) ## res <- switch(dist.type, ## qf = - dist(p_loss, ...), ## cdf = - cdf2quantile(p_loss, dist, tol = tol, ...), # TODO: interval? ## pdf = { ## stop("Not ready yet. Please supply quantile function or cdf") ## }, ## stop('argument dist.type should be one of "qf", "cdf" or "pdf"') ## ) ## else{ ## # fu <- switch(dist.type, ## # qf = function(y) - dist(y, ...), ## # cdf = function(y) - cdf2quantile(y, dist, tol = tol, ...), ## # pdf = { ## # stop("Not ready yet. Please supply quantile function or cdf") ## # }, ## # stop('argument dist.type should be one of "qf", "cdf" or "pdf"') ## # ) ## # sapply(p_loss, fu) ## res <- - switch(dist.type, ## qf = mapply(dist, p_loss, ...), ## cdf = mapply(cdf2quantile, p_loss, ..., MoreArgs = list(cdf = dist, tol = tol)), ## pdf = { ## stop("Not ready yet. Please supply quantile function or cdf") ## }, ## stop('argument dist.type should be one of "qf", "cdf" or "pdf"') ## ) ## ## } ## - intercept + slope * res ## } #' @title Compute expected shortfall (ES) of distributions #' #' @description #' Compute the expected shortfall for a distribution. #' #' @details \code{ES} computes the expected shortfall for distributions specified by the #' arguments. \code{dist} is typically a function (or the name of one). What \code{dist} #' computes is determined by \code{dist.type}, whose default setting is \code{"qf"} (the #' quantile function). Other possible settings of \code{dist.type} include \code{"cdf"} #' and \code{"pdf"}. Additional arguments for \code{dist} can be given with the #' \code{"..."} arguments. #' #' Argument \code{dist} can also be a numeric vector. In that case the ES is computed, #' effectively, for the empirical cumulative distribution function (ecdf) of the #' vector. The ecdf is not created explicitly and the \code{\link[stats]{quantile}} #' function is used instead for the computation of VaR. Arguments in \code{"..."} are #' passed eventually to \code{quantile()} and can be used, for example, to select a #' non-defult method for the computation of quantiles. #' #' Except for the exceptions discussed below, a function computing VaR for the specified #' distribution is constructed and the expected shortfall is computed by numerically #' integrating it. The numerical integration can be fine-tuned with argument #' \code{control}, which should be a named list, see \code{\link{integrate}} for the #' available options. #' #' If \code{dist.type} is \code{"pdf"}, VaR is not computed, Instead, the partial #' expectation of the lower tail is computed by numerical integration of \code{x * #' pdf(x)}. Currently the quantile function is required anyway, via argument \code{qf}, #' to compute the upper limit of the integral. So, this case is mainly for testing and #' comparison purposes. #' #' #' A bunch of expected shortfalls is computed if argument \code{x} or any of the #' arguments in \code{"..."} are of length greater than one. They are recycled to equal #' length, if necessary, using the normal \R recycling rules. #' #' \code{intercept} and \code{slope} can be used to compute the expected shortfall for #' the location-scale transformation \code{Y = intercept + slope * X}, where the #' distribution of \code{X} is as specified by the other parameters and \code{Y} is the #' variable of interest. The expected shortfall of \code{X} is calculated and then #' transformed to that of \code{Y}. Note that the distribution of \code{X} doesn't need #' to be standardised, although it typically will. #' #' The \code{intercept} and the \code{slope} can be vectors. Using them may be #' particularly useful for cheap calculations in, for example, forecasting, where the #' predictive distributions are often from the same family, but with different location #' and scale parameters. Conceptually, the described treatment of \code{intercept} and #' \code{slope} is equivalent to recycling them along with the other arguments, but more #' efficiently. #' #' The names, \code{intercept} and \code{slope}, for the location and scale parameters #' were chosen for their expressiveness and to minimise the possibility for a clash with #' parameters of \code{dist} (e.g., the Gamma distribution has parameter \code{scale}). #' #' #' @param dist specifies the distribution whose ES is computed, usually a function or a name #' of a function computing quantiles, cdf, pdf, or a random number generator, see #' Details. #' #' @param p_loss level, default is 0.05. #' #' @param dist.type a character string specifying what is computed by \code{dist}, such as #' "qf" or "cdf". #' #' @param qf quantile function, only used if \code{dist.type = "pdf"}. #' #' @param intercept,slope compute ES for the linear transformation \code{intercept + #' slope*X}, where \code{X} has distribution specified by \code{dist}, see Details. #' #' @param control additional control parameters for the numerical integration routine. #' #' @param ... passed on to \code{dist}. #' #' @param x deprecated and will soon be removed. \code{x} was renamed to \code{p_loss}, #' please use the latter. #' #' #' @return a numeric vector #' #' @seealso \code{\link{VaR}} for VaR, #' #' \code{\link[=predict.garch1c1]{predict}} for examples with fitted models #' #' @examples #' ES(qnorm) #' #' ## Gaussian #' ES(qnorm, dist.type = "qf") #' ES(pnorm, dist.type = "cdf") #' #' ## t-dist #' ES(qt, dist.type = "qf", df = 4) #' ES(pt, dist.type = "cdf", df = 4) #' #' ES(pnorm, 0.95, dist.type = "cdf") #' ES(qnorm, 0.95, dist.type = "qf") #' ## - VaRES::esnormal(0.95, 0, 1) #' ## - PerformanceAnalytics::ETL(p=0.05, method = "gaussian", mu = 0, #' ## sigma = 1, weights = 1) # same #' #' cvar::ES(pnorm, dist.type = "cdf") #' cvar::ES(qnorm, dist.type = "qf") #' cvar::ES(pnorm, 0.05, dist.type = "cdf") #' cvar::ES(qnorm, 0.05, dist.type = "qf") #' #' ## this uses "pdf" #' cvar::ES(dnorm, 0.05, dist.type = "pdf", qf = qnorm) #' #' #' ## this gives warning (it does more than simply computing ES): #' ## PerformanceAnalytics::ETL(p=0.95, method = "gaussian", mu = 0, sigma = 1, weights = 1) #' #' ## run this if VaRRES is present #' \dontrun{ #' x <- seq(0.01, 0.99, length = 100) #' y <- sapply(x, function(p) cvar::ES(qnorm, p, dist.type = "qf")) #' yS <- sapply(x, function(p) - VaRES::esnormal(p)) #' plot(x, y) #' lines(x, yS, col = "blue") #' } #' #' @export ES <- function(dist, p_loss = 0.05, ...){ UseMethod("ES") } CVaR <- AVaR <- ES # synonyms for ES, currently not exported #' @name ES #' #' @export ES.default <- function(dist, p_loss = 0.05, dist.type = "qf", qf, ..., intercept = 0, slope = 1, control = list(), x){ if(!missing(x)){ p_loss <- x warn_about_x("ES") } dist <- match.fun(dist) if(length(p_loss) == 1 && all(sapply(list(...), length) < 2)){ if(dist.type == "pdf"){ fu <- function(p) - p * dist(p, ...) / p_loss iargs <- list(f = fu, lower = -Inf, upper = qf(p_loss)) # stop.on.error = FALSE,? }else{ fu <- function(p) VaR(dist = dist, p_loss = p, dist.type = dist.type, ...) / p_loss iargs <- list(f = fu, lower = 0, upper = p_loss) # stop.on.error = FALSE,? } iargs[names(control)] <- control res <- do.call("integrate", iargs)$value }else{ ## TODO: this is lazy if(dist.type == "pdf"){ if(is.list(qf)) res <- mapply(ES, MoreArgs = list(dist = dist, dist.type = dist.type, control = control), p_loss = p_loss, qf = qf, ...) else res <- mapply(ES, MoreArgs = list(dist = dist, dist.type = dist.type, control = control, qf = qf), p_loss = p_loss, ...) }else{ res <- mapply(ES, MoreArgs = list(dist = dist, dist.type = dist.type, control = control), p_loss = p_loss, ...) } } -intercept + slope * res } #' @name ES #' #' @export ES.numeric <- function(dist, p_loss = 0.05, dist.type = "qf", qf, ..., intercept = 0, slope = 1, control = list(), x){ if(!missing(x)){ p_loss <- x warn_about_x("ES") } v <- VaR.numeric(dist, p_loss = p_loss, ..., intercept = intercept, slope = slope) bad <- dist[dist <= - v] res <- - mean(bad) res } cvar/R/cvar-package.R0000644000176200001440000001115114330275453014062 0ustar liggesusers#' @author Georgi N. Boshnakov #' #' @importFrom stats quantile runif #' @importFrom gbutils cdf2quantile #' @importFrom Rdpack reprompt #' #' #' #' @docType package #' @name cvar-package #' @aliases cvar #' #' @title Compute Conditional Value-at-Risk and Value-at-Risk #' #' @description Compute expected shortfall (ES) and Value at Risk (VaR) #' from a quantile function, distribution function, random number #' generator or probability density function. ES is also known as #' Conditional Value at Risk (CVaR). Virtually any continuous #' distribution can be specified. The functions are vectorised over #' the arguments. Some support for GARCH models is provided, as well. #' #' @details #' #' There is a huge number of functions for computations with #' distributions in core \R and in contributed packages. Pdf's, #' cdf's, quantile functions and random number generators are #' covered comprehensively. The coverage of expected shortfall is #' more patchy but a large collection of distributions, including #' functions for expected shortfall, is provided by #' \insertCite{VaRES2013;textual}{cvar}. #' \insertCite{PerformanceAnalytics2018;textual}{cvar} and #' \insertCite{actuarJSS2008;textual}{cvar} provide packages #' covering comprehensively various aspects of risk measurement, #' including some functions for expected shortfall. #' #' Package \pkg{cvar} is a small package with, essentially, two main #' functions --- \code{ES} for computing the expected shortfall #' and \code{VaR} for Value at Risk. The user specifies the #' distribution by supplying one of the functions that define a #' continuous distribution---currently this can be a quantile #' function (qf), cumulative distribution function (cdf) or #' probability density function (pdf). Virtually any continuous #' distribution can be specified. #' #' The functions are vectorised over the parameters of the #' distributions, making bulk computations more convenient, for #' example for forecasting or model evaluation. #' #' The name of this package, "cvar", comes from \emph{Conditional #' Value at Risk} (CVaR), which is an alternative term for #' expected shortfall. #' #' We chose to use the standard names \code{ES} and \code{VaR}, #' despite the possibility for name clashes with same named #' functions in other packages, rather than invent possibly #' difficult to remember alternatives. Just call the functions as #' \code{cvar::ES} and \code{cvar::VaR} if necessary. #' #' Locations-scale transformations can be specified separately #' from the other distribution parameters. This is useful when #' such parameters are not provided directly by the distribution #' at hand. The use of these parameters often leads to more #' efficient computations and better numerical accuracy even if #' the distribution has its own parameters for this purpose. Some #' of the examples for \code{VaR} and \code{ES} illustrate this #' for the Gaussian distribution. #' #' Since VaR is a quantile, functions computing it for a given #' distribution are convenience functions. \code{VaR} exported by #' \pkg{cvar} could be attractive in certain workflows because of #' its vectorised distribution parameters, the location-scale #' transformation, and the possibility to compute it from cdf's #' when quantile functions are not available. #' #' Some support for GARCH models is provided, as well. It is #' currently under development, see \code{\link{predict.garch1c1}} #' for current functionality. #' #' In practice, we may need to compute VaR associated with data. The distribution comes #' from fitting a model. In the simplest case, we fit a distribution to the data, #' assuming that the sample is i.i.d. For example, a normal distribution \eqn{N(\mu, #' \sigma^2)} can be fitted using the sample mean and sample variance as estimates of the #' unknown parameters \eqn{\mu} and \eqn{\sigma^2}, see section \sQuote{Examples}. For other #' common distributions there are specialised functions to fit their parameters and if #' not, general optimisation routines can be used. More soffisticated models may be used, #' even time series models such as GARCH and mixture autoregressive models. #' #' @references #' \insertAllCited{} #' #' @seealso #' \code{\link{ES}}, #' \code{\link{VaR}} #' #' @examples #' ## see the examples for ES(), VaR(), predict.garch1c1() #' NULL .onLoad <- function(lib, pkg){ Rdpack::Rdpack_bibstyles(package = pkg, authors = "LongNames") invisible(NULL) } cvar/NEWS.md0000644000176200001440000000447214330704765012323 0ustar liggesusers# cvar 0.5 (CRAN) * made `ES` generic (`VaR` was already generic). * moved `fGarch` from Imports to Suggests. * renamed argument `x` of `VaR` and `ES` to `p_loss`. `p_loss` seems more expressive and suggests that it relates to the losses, usually small numbers like `0.05`. Other suitable names like `alpha`, `p`, and `prob`, are commonly used as arguments to other functions that might be used as argument `dist` and make them mmore difficult to pass via the `...` arguments. For now, an warning is issued if `x` is used as a named argument in a call (e.g. `VaR(dist, x = 0.05)`) with the intend to turn that in an error in the next release of the package. This change should not be noticed by most users since it is much more natural not to name this argument and use something like `VaR(dist, 0.05)`. * moved `fGarch` from Imports to Suggests. # cvar 0.4.1 (CRAN) * when the input was numeric, `ES()` was not handling the level `x` properly (fixes issue #2, reported by Marius Bommert). * changed the JSS reference to use the new-style doi. * fixed a bug in the tests, in v0.4-0, that was causing faiure of the tests on travis, despite all checks on CRAN passing with OK. `devtools::test()` was failing too, but only on the first run in a session, details in the git commit. * set up GHA. # cvar 0.4-0 (CRAN) * fix tests to pass with the changed R random generator. * some new examples and minor documentation changes. # cvar 0.3-0 (CRAN) * now `\VignetteIndexEntry` in `Guide_cvar.Rnw` is plain text. * added experimental support for GARCH models - currently GARCH(1,1) (the API may change). * now the first argument of `VaR()` and `ES()` can be a numeric vector. This is useful, e.g., for computing VaR by simulation. * bugfix: in `VaR_cdf()` and `VaR_qf()`, the code for the `if/else` clauses had been wrongly swappped. # cvar 0.2-0 (CRAN) * suggest 'covr'. * setup for travisCI and Coveralls. * more tests. * corrected \VignetteIndexEntry in the vignette --- I used the vignette for Rdpack as a template but didn't change this entry, which resulted in the vignette appearing with a puzzling title on CRAN and other sites.! * added author@R in DESCRIPTION. * added experimental web site (docs/) # cvar 0.1-1 (CRAN) * added a doi to DESCRIPTION. # cvar 0.1-0 * first public release. cvar/MD50000644000176200001440000000302514330710446011517 0ustar liggesusers24eced12792c1b13caf5d9effbff02b1 *DESCRIPTION b0545f51806f58edf48ebda4ca295c50 *NAMESPACE 48cdda522be879807a2c579f5171c503 *NEWS.md 97a2fcafa7b92107143c144349aa82aa *R/VaR.R 5e895ee09b3f99bd12e7316851aa3192 *R/cvar-package.R a482cf2ed19a01118ed37639a6aef47e *R/garch.R 3fe35053c9c29f4e865ff8756136b42f *README.md ac4fbf496c435f90fafe07d19b5cf69b *build/partial.rdb 95e94bfb82e40a71d6cd33388b3acf9c *build/vignette.rds 2a87d8dae0626a5d81eeee24dd1f9a72 *inst/REFERENCES.bib c06deb911fae19138395ca4f79cc026b *inst/doc/Guide_cvar.R d714972828cc5fdcb5cfb0d8543c210e *inst/doc/Guide_cvar.Rnw 79436a37dfc24aeb9fff37dc8d252ae7 *inst/doc/Guide_cvar.pdf d869bfe71414189d4a07746103761dac *man/ES.Rd 9a27fb266a6e471b3d161ebaa3265284 *man/GarchModel.Rd 6378f3023b8242a12842b64627babf99 *man/VaR.Rd fd6a0b95012581d75acda800aa0bfd9a *man/cvar-package.Rd 6e604659f91a96169483194ecc101a2b *man/predict.garch1c1.Rd 41d5c1f841b5e53d6456c2c8c926e240 *man/sim_garch1c1.Rd 32e67ae25ca51c0a429ccd166a1bfef7 *tests/testthat.R 58140e1d8933e014abc7ab5d481b9111 *tests/testthat/a.RDS 17bfcf5140c800a7cf57b6bc07e96019 *tests/testthat/a_before_6.0.RDS cf640ab785180cfce3d4ea1524be4494 *tests/testthat/a_pred.RDS 6c50b97c3d91d256f30aa902eb813362 *tests/testthat/a_pred_before_6.0.RDS c7776039446c3b0bbeed1ad29910c3a2 *tests/testthat/test-VaR.R 2d588c9ad8807c0bf8954c6e98cbc8ed *tests/testthat/test-garch.R d714972828cc5fdcb5cfb0d8543c210e *vignettes/Guide_cvar.Rnw 905dd4a699da274ebc7f9f3f290fd70c *vignettes/REFERENCES.bib 14b361d4dbff27d5f2b5cf86782e6c84 *vignettes/auto/Guide_cvar.el cvar/inst/0000755000176200001440000000000014330705053012162 5ustar liggesuserscvar/inst/REFERENCES.bib0000644000176200001440000000231614127262070014264 0ustar liggesusers@article{acerbi2002expected, title={Expected shortfall: a natural coherent alternative to value at risk}, author={Acerbi, Carlo and Tasche, Dirk}, journal={Economic notes}, volume={31}, number={2}, pages={379--388}, year={2002}, doi = {10.1111/1468-0300.00091}, url = {https://arxiv.org/pdf/cond-mat/0105191.pdf}, } @Manual{PerformanceAnalytics2018, title = {PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis}, author = {Brian G. Peterson and Peter Carl}, year = {2018}, note = {R package version 1.5.2}, url = {https://CRAN.R-project.org/package=PerformanceAnalytics}, } @Manual{VaRES2013, title = {VaRES: Computes value at risk and expected shortfall for over 100 parametric distributions}, author = {Saralees Nadarajah and Stephen Chan and Emmanuel Afuecheta}, year = {2013}, note = {R package version 1.0}, url = {https://CRAN.R-project.org/package=VaRES}, } @Article{actuarJSS2008, title = {actuar: An R Package for Actuarial Science}, author = {Christophe Dutang and Vincent Goulet and Mathieu Pigeon}, journal = {Journal of Statistical Software}, year = {2008}, volume = {25}, number = {7}, pages = {38}, doi = {10.18637/jss.v025.i07}, } cvar/inst/doc/0000755000176200001440000000000014330705053012727 5ustar liggesuserscvar/inst/doc/Guide_cvar.Rnw0000644000176200001440000003272614330671606015507 0ustar liggesusers% preamble taken from the vignette in package strucchange % \documentclass[nojss,article]{jss} \usepackage[T1]{fontenc} %\usepackage{a4wide} \usepackage[left=2cm,right=2cm,bottom=15mm]{geometry} \usepackage{graphicx,color,alltt} \usepackage[authoryear,round,longnamesfirst]{natbib} \usepackage{hyperref} \usepackage{amsmath, amsfonts} % \usepackage{Sweave} %\definecolor{Red}{rgb}{0.7,0,0} %\definecolor{Blue}{rgb}{0,0,0.8} %\definecolor{hellgrau}{rgb}{0.55,0.55,0.55} %\newcommand{\E}{\mbox{$\mathsf{E}$}} %\newcommand{\VAR}{\mbox{$\mathsf{VAR}$}} %\newcommand{\COV}{\mbox{$\mathsf{COV}$}} %\newcommand{\p}{\mbox{$\mathsf{P}$}} %\newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} %\newenvironment{smallexample}{\begin{alltt}\small}{\end{alltt}} %\setlength{\parskip}{0.5ex plus0.1ex minus0.1ex} %\setlength{\parindent}{0em} % %\bibpunct{(}{)}{;}{a}{}{,} % %\newcommand{\ui}{\underline{i}} %\newcommand{\oi}{\overline{\imath}} %\RequirePackage{color} %\definecolor{Red}{rgb}{0.5,0,0} %\definecolor{Blue}{rgb}{0,0,0.5} %\definecolor{hellgrau}{rgb}{0.55,0.55,0.55} % %\hypersetup{% %hyperindex,% %colorlinks,% %linktocpage,% %plainpages=false,% %linkcolor=Blue,% %citecolor=Blue,% %urlcolor=Red,% %pdfstartview=Fit,% %pdfview={XYZ null null null}% %} % SweaveOpts{engine=R,eps=FALSE} %\VignetteIndexEntry{Brief guide to R package cvar} %\VignetteDepends{cvar} %\VignetteKeywords{expected shortfall, ES, CVaR, VaR, value at risk} %\VignettePackage{cvar} <>= library(cvar) pd <- packageDescription("cvar") @ \newcommand{\VaR}[2][\alpha]{\text{VaR}_{#1}(#2)} \newcommand{\ES}[2][\alpha]{\text{ES}_{#1}(#2)} \title{Brief guide to R package \pkg{cvar}} \author{Georgi N. Boshnakov \\ University of Manchester } \Plainauthor{Georgi N. Boshnakov} \Address{ Georgi N. Boshnakov\\ School of Mathematics\\ The University of Manchester\\ Oxford Road, Manchester M13 9PL, UK\\ URL: \url{http://www.maths.manchester.ac.uk/~gb/} } %\date{} %\maketitle \Abstract{ Package \pkg{cvar} is a small package with, essentially, two functions --- \code{ES()} for computing the expected shortfall and \code{VaR()} for Value at Risk. The user specifies the distribution by supplying one of the functions that define a continuous distribution---currently this can be a quantile function (qf), cumulative distribution function (cdf) or probability density function (pdf). Virtually any continuous distribution can be specified. This vignette is part of package \pkg{cvar}, version~\Sexpr{pd$Version}. %$ } \Keywords{expected shortfall, ES, CVaR, VaR, value at risk} \Plainkeywords{expected shortfall, ES, CVaR, VaR, value at risk} \begin{document} \section{Introduction} \label{sec:introduction} There is a huge number of functions for computations with distributions in core R and in contributed packages. Pdf's, cdf's, quantile functions and random number generators are covered comprehensively. The coverage of expected shortfall is more patchy but a large collection of distributions, including functions for expected shortfall, is provided by \citet{VaRES2013}. \citet{PerformanceAnalytics2018} and \citet{actuarJSS2008} provide packages covering comprehensively various aspects of risk measurement, including some functions for expected shortfall. Package \pkg{cvar} is a small package with, essentially two functions --- \code{ES} for computing the expected shortfall and \code{VaR} for Value at Risk. The user specifies the distribution by supplying one of the functions that define a continuous distribution---currently this can be a quantile function (qf), cumulative distribution function (cdf) or probability density function (pdf). Virtually any continuous distribution can be specified. The computations are done directly from the definitions, see e.g. \citet{acerbi2002expected}. The functions are vectorised over the parameters of the distributions, making bulk computations more convenient, for example for forecasting or model evaluation. The name of this package, "cvar", comes from \emph{Conditional Value at Risk} (CVaR), which is an alternative term for expected shortfall. We chose to use the standard names \code{ES} and \code{VaR}, despite the possibility for name clashes with same named functions in other packages, rather than invent possibly difficult to remember alternatives. Just call the functions as \code{cvar::ES} and \code{cvar::VaR} if necessary. Locations-scale transformations can be specified separately from the other distribution parameters. This is useful when such parameters are not provided directly by the distribution at hand. The use of these parameters often leads to more efficient computations and better numerical accuracy even if the distribution has its own parameters for this purpose. Some of the examples for \code{VaR} and \code{ES} illustrate this for the Gaussian distribution. Since VaR is a quantile, functions computing it for a given distribution are convenience functions. \code{VaR} exported by \pkg{cvar} could be attractive in certain workflows because of its vectorised distribution parameters, the location-scale transformation and the possibility to compute it from cdf's when quantile functions are not available. \section{Value at Risk and Expected Shortfall} %To avoid confusion, we give the mathematical definitions here. We use the traditional definition of VaR as the negated lower quantile. More specifically, let $Y$ be the variable of interes, such as return on a financial asset. Suppose that it is modelled as a random variable with distribution function $F^{Y}(y)$. Then VaR is defined as% \footnote{Beware that a definition without negation is also used in both the literature and in the R packages.} \begin{equation*} \VaR{Y} = - q_{\alpha}^{Y} . \end{equation*} where $q_{\alpha}^{Y}$ is the lower $\alpha$-quantile of $Y$, i.e. we have \begin{equation*} \alpha = F(q_{\alpha}^{Y}) = P(Y \le q_{\alpha}^{Y}) . \end{equation*} Typical values for $\alpha$ are $0.05$ and $0.01$. Equivalently, we could write\footnote{This equation shows why some authors use the ``big numbers'', e.g. 0.95 and 0.99, in the notation for the VaR.} \begin{equation*} \VaR{Y} = q_{1-\alpha}^{-Y} . \end{equation*} The expected shortfall is the (partial) expectation of $\VaR{Y}$: \begin{equation*} \ES{Y} = \frac{1}{\alpha}\int_{0}^{\alpha}\VaR[\gamma]{Y} d\gamma . \end{equation*} Suppose now that $Y$ is obtained from another random variable, $X$, by a linear transformation: \begin{equation*} Y = a + bX \end{equation*} When this is the case, there is a simple relation between the VaR's and ES's of $Y$ and $X$: \begin{align*} \VaR{Y} &= -a + b \VaR{X} \\ \ES{Y} &= -a + b \ES{X} \end{align*} In practice, $X$ is often chosen to be standardised but this is not necessary. Note also that if a bunch of VaR's and ES's are needed, say for normally distributed variables, this can be computed very efficiently using this property. \section{VaR} \label{sec:var} Here we compute the VaR associated with a standard normal r.v. The three variants are equivalent since 0.05 and \code{"qf"} are default for the last two arguments. <<>>= cvar::VaR(qnorm, p_loss = 0.05, dist.type = "qf") cvar::VaR(qnorm, p_loss = 0.05) cvar::VaR(qnorm) @ \code{x} can be a vector: <<>>= cvar::VaR(qnorm, p_loss = c(0.01, 0.05)) @ Let's set some more realistic values for the parameters of the normal distribution. Suppose that the daily returns on some stock have sample mean $0.006408553$ and sample variance $0.0004018977$. Then $N(0.006408553, 0.0004018977)$ can be taken as normal distribution fitted to the data. <<>>= muA <- 0.006408553 sigma2A <- 0.0004018977 @ This computes VaR using the fitted normal distribution: <<>>= res1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) @ For the normal disribution we could also use the intercept-slope arguments, since the parameters are precisely the intercept and the slope: <<>>= res2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) # 0, intercept/slope equivalent to mean/sd @ If we compute VaR using the cdf, the intercept-slope method has some numerical advantage: <<>>= ## with cdf the precision depends on solving an equation res1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # 3.287939e-09 abs((res2a - res2)) # 5.331195e-11, intercept/slope better numerically @ Of course, it is almost always better to use the quantile function when it is available. We can use smaller tollerance to improve the precision (the value \verb+.Machine$double.eps^0.75+ = \Sexpr{.Machine$double.eps^0.75} is probably excessive): <<>>= ## as above, but increase the precision, this is probably excessive res1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) # 6.938894e-18 # both within machine precision abs((res2b - res2)) # 1.040834e-16 @ The relative precision is also good. <<>>= abs((res1b - res2)/res2) # 2.6119e-16 # both within machine precision abs((res2b - res2)/res2) # 3.91785e-15 @ For examples with vectorised distribution parameters we use data from \pkg{PerformanceAnalytics}. Compute means and variances of the variables in a data frame and store them in vectors: <<>>= ## if(require("PerformanceAnalytics")){ data(edhec, package = "PerformanceAnalytics") mu <- apply(edhec, 2, mean) sigma2 <- apply(edhec, 2, var) musigma2 <- cbind(mu, sigma2) @ We compute VaR using \code{PerformanceAnalytics::VaR}: <<>>= ## analogous calc. with PerformanceAnalytics::VaR vPA <- apply(musigma2, 1, function(x) PerformanceAnalytics::VaR(p = .95, method = "gaussian", invert = FALSE, mu = x[1], sigma = x[2], weights = 1)) @ The results below compare to the value, \code{vPA}, obtained here. The computations below are similar to the previous example but the distribution parameters are vectors. The first pair of results are numerically the same: <<>>= vAz1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = mu, sd = sqrt(sigma2)) vAz2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = mu, slope = sqrt(sigma2)) max(abs((vPA - vAz1))) # 5.551115e-17 max(abs((vPA - vAz2))) # "" @ Computing VaR from cdf shows some advantage for the location-scale method: <<>>= vAz1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2)) vAz2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2)) max(abs((vPA - vAz1a))) # 3.287941e-09 max(abs((vPA - vAz2a))) # 1.465251e-10, intercept/slope better @ The advantage remains for smaller tolerance: <<>>= vAz1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2), tol = .Machine$double.eps^0.75) vAz2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2), tol = .Machine$double.eps^0.75) max(abs((vPA - vAz1b))) # 4.374869e-13 max(abs((vPA - vAz2b))) # 3.330669e-16 @ \section{Expected shortfall} \label{sec:expected-shortfall} \code{ES} has essentially the same arguments as \code{VaR}. The examples below are obtained almost literally by replacing the calls to \code{VaR} with \code{ES()} ones. Here we compute the VaR associated with a standard normal r.v. The three variants are equivalent since 0.5 and \code{"qf"} are default for the last two arguments. <<>>= cvar::ES(qnorm, p_loss = 0.05, dist.type = "qf") cvar::ES(qnorm, p_loss = 0.05) cvar::ES(qnorm) @ \code{x} can be a vector: <<>>= cvar::ES(qnorm, p_loss = c(0.01, 0.05)) @ Let's set some more realistic values for the parameters of the normal distribution: <<>>= muA <- 0.006408553 sigma2A <- 0.0004018977 @ This demonstrates the use of \code{intercept} and \code{slope}. These arguments use the linear transformation property discussed in the theoretical section. For the normal disribution the parameters are precisely the intercept and the slope: <<>>= res1 <- cvar::ES(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) res2 <- cvar::ES(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) @ If we compute ES using the cdf, the intercept slope method has some numerical advantage: <<>>= ## with cdf the precision depends on solving an equation res1a <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # abs((res2a - res2)) # intercept/slope better numerically @ We can use smaller tollerance to improve the precision (the value \verb+.Machine$double.eps^0.75+ = \Sexpr{.Machine$double.eps^0.75} is probably excessive): <<>>= ## as above, but increase the precision, this is probably excessive res1b <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) abs((res2b - res2)) @ The relative precision is also good. <<>>= abs((res1b - res2)/res2) abs((res2b - res2)/res2) @ \newpage{} \bibliography{REFERENCES} \end{document} cvar/inst/doc/Guide_cvar.R0000644000176200001440000001607314330705053015131 0ustar liggesusers### R code from vignette source 'Guide_cvar.Rnw' ################################################### ### code chunk number 1: Guide_cvar.Rnw:57-59 ################################################### library(cvar) pd <- packageDescription("cvar") ################################################### ### code chunk number 2: Guide_cvar.Rnw:205-208 ################################################### cvar::VaR(qnorm, p_loss = 0.05, dist.type = "qf") cvar::VaR(qnorm, p_loss = 0.05) cvar::VaR(qnorm) ################################################### ### code chunk number 3: Guide_cvar.Rnw:212-213 ################################################### cvar::VaR(qnorm, p_loss = c(0.01, 0.05)) ################################################### ### code chunk number 4: Guide_cvar.Rnw:220-222 ################################################### muA <- 0.006408553 sigma2A <- 0.0004018977 ################################################### ### code chunk number 5: Guide_cvar.Rnw:225-226 ################################################### res1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) ################################################### ### code chunk number 6: Guide_cvar.Rnw:231-233 ################################################### res2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) # 0, intercept/slope equivalent to mean/sd ################################################### ### code chunk number 7: Guide_cvar.Rnw:237-244 ################################################### ## with cdf the precision depends on solving an equation res1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # 3.287939e-09 abs((res2a - res2)) # 5.331195e-11, intercept/slope better numerically ################################################### ### code chunk number 8: Guide_cvar.Rnw:250-257 ################################################### ## as above, but increase the precision, this is probably excessive res1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) # 6.938894e-18 # both within machine precision abs((res2b - res2)) # 1.040834e-16 ################################################### ### code chunk number 9: Guide_cvar.Rnw:261-263 ################################################### abs((res1b - res2)/res2) # 2.6119e-16 # both within machine precision abs((res2b - res2)/res2) # 3.91785e-15 ################################################### ### code chunk number 10: Guide_cvar.Rnw:272-277 ################################################### ## if(require("PerformanceAnalytics")){ data(edhec, package = "PerformanceAnalytics") mu <- apply(edhec, 2, mean) sigma2 <- apply(edhec, 2, var) musigma2 <- cbind(mu, sigma2) ################################################### ### code chunk number 11: Guide_cvar.Rnw:281-285 ################################################### ## analogous calc. with PerformanceAnalytics::VaR vPA <- apply(musigma2, 1, function(x) PerformanceAnalytics::VaR(p = .95, method = "gaussian", invert = FALSE, mu = x[1], sigma = x[2], weights = 1)) ################################################### ### code chunk number 12: Guide_cvar.Rnw:291-295 ################################################### vAz1 <- cvar::VaR(qnorm, p_loss = 0.05, mean = mu, sd = sqrt(sigma2)) vAz2 <- cvar::VaR(qnorm, p_loss = 0.05, intercept = mu, slope = sqrt(sigma2)) max(abs((vPA - vAz1))) # 5.551115e-17 max(abs((vPA - vAz2))) # "" ################################################### ### code chunk number 13: Guide_cvar.Rnw:299-305 ################################################### vAz1a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2)) vAz2a <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2)) max(abs((vPA - vAz1a))) # 3.287941e-09 max(abs((vPA - vAz2a))) # 1.465251e-10, intercept/slope better ################################################### ### code chunk number 14: Guide_cvar.Rnw:309-317 ################################################### vAz1b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", mean = mu, sd = sqrt(sigma2), tol = .Machine$double.eps^0.75) vAz2b <- cvar::VaR(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = mu, slope = sqrt(sigma2), tol = .Machine$double.eps^0.75) max(abs((vPA - vAz1b))) # 4.374869e-13 max(abs((vPA - vAz2b))) # 3.330669e-16 ################################################### ### code chunk number 15: Guide_cvar.Rnw:329-332 ################################################### cvar::ES(qnorm, p_loss = 0.05, dist.type = "qf") cvar::ES(qnorm, p_loss = 0.05) cvar::ES(qnorm) ################################################### ### code chunk number 16: Guide_cvar.Rnw:336-337 ################################################### cvar::ES(qnorm, p_loss = c(0.01, 0.05)) ################################################### ### code chunk number 17: Guide_cvar.Rnw:341-343 ################################################### muA <- 0.006408553 sigma2A <- 0.0004018977 ################################################### ### code chunk number 18: Guide_cvar.Rnw:349-352 ################################################### res1 <- cvar::ES(qnorm, p_loss = 0.05, mean = muA, sd = sqrt(sigma2A)) res2 <- cvar::ES(qnorm, p_loss = 0.05, intercept = muA, slope = sqrt(sigma2A)) abs((res2 - res1)) ################################################### ### code chunk number 19: Guide_cvar.Rnw:356-363 ################################################### ## with cdf the precision depends on solving an equation res1a <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A)) res2a <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A)) abs((res1a - res2)) # abs((res2a - res2)) # intercept/slope better numerically ################################################### ### code chunk number 20: Guide_cvar.Rnw:368-375 ################################################### ## as above, but increase the precision, this is probably excessive res1b <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", mean = muA, sd = sqrt(sigma2A), tol = .Machine$double.eps^0.75) res2b <- cvar::ES(pnorm, p_loss = 0.05, dist.type = "cdf", intercept = muA, slope = sqrt(sigma2A), tol = .Machine$double.eps^0.75) abs((res1b - res2)) abs((res2b - res2)) ################################################### ### code chunk number 21: Guide_cvar.Rnw:379-381 ################################################### abs((res1b - 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