#' @title Decompose portfolio VaR into individual factor contributions
#'
#' @description Compute the factor contributions to Value-at-Risk (VaR) of
#' portfolio returns based on Euler's theorem, given the fitted factor model.
#' The partial derivative of VaR w.r.t. factor beta is computed as the expected
#' factor return given portfolio return is equal to its VaR and approximated by a
#' kernel estimator. Option to choose between non-parametric and Normal.
#'
#' @importFrom stats quantile residuals cov resid qnorm
#' @importFrom xts as.xts
#' @importFrom zoo as.Date index
#'
#' @details The factor model for a portfolio's return at time \code{t} has the
#' form \cr \cr \code{R(t) = beta'f(t) + e(t) = beta.star'f.star(t)} \cr \cr
#' where, \code{beta.star=(beta,sig.e)} and \code{f.star(t)=[f(t)',z(t)]'}. By
#' Euler's theorem, the VaR of the asset's return is given by:
#' \cr \cr \code{VaR.fm = sum(cVaR_k) = sum(beta.star_k*mVaR_k)} \cr \cr
#' where, summation is across the \code{K} factors and the residual,
#' \code{cVaR} and \code{mVaR} are the component and marginal
#' contributions to \code{VaR} respectively. The marginal contribution to VaR
#' is defined as the expectation of \code{F.star}, conditional on the loss
#' being equal to \code{portVaR}. This is approximated as described in
#' Epperlein & Smillie (2006); a triangular smoothing kernel is used here.
#'
#' @param object fit object of class \code{tsfm}, or \code{ffm}.
#' @param weights a vector of weights of the assets in the portfolio. Default is NULL,
#' in which case an equal weights will be used.
#' @param factor.cov optional user specified factor covariance matrix with
#' named columns; defaults to the sample covariance matrix.
#' @param p tail probability for calculation. Default is 0.05.
#' @param type one of "np" (non-parametric) or "normal" for calculating VaR.
#' Default is "np".
#' @param invert a logical variable to choose if change VaR to positive number, default
#' is False
#' @param use an optional character string giving a method for computing factor
#' covariances in the presence of missing values. This must be (an
#' abbreviation of) one of the strings "everything", "all.obs",
#' "complete.obs", "na.or.complete", or "pairwise.complete.obs". Default is
#' "pairwise.complete.obs".
#' @param ... other optional arguments passed to \code{\link[stats]{quantile}} and
#' optional arguments passed to \code{\link[stats]{cov}}
#'
#' @return A list containing
#' \item{portVaR}{factor model VaR of portfolio return.}
#' \item{n.exceed}{number of observations beyond VaR.}
#' \item{idx.exceed}{a numeric vector of index values of exceedances.}
#' \item{mVaR}{length-(K + 1) vector of marginal contributions to VaR.}
#' \item{cVaR}{length-(K + 1) vector of component contributions to VaR.}
#' \item{pcVaR}{length-(K + 1) vector of percentage component contributions to VaR.}
#' Where, K is the number of factors.
#'
#' @author Douglas Martin, Lingjie Yi
#'
#' @seealso \code{\link{fitTsfm}}, \code{\link{fitFfm}}
#' for the different factor model fitting functions.
#'
#' \code{\link{portSdDecomp}} for factor model Sd decomposition.
#' \code{\link{portEsDecomp}} for factor model ES decomposition.
#'
#' @examples
#' # Time Series Factor Model
#' data(managers)
#' fit.macro <- factorAnalytics::fitTsfm(asset.names=colnames(managers[,(1:6)]),
#' factor.names=colnames(managers[,(7:9)]),
#' rf.name=colnames(managers[,10]), data=managers)
#' decomp <- portVaRDecomp(fit.macro,invert = TRUE)
#' # get the factor contributions of risk
#' decomp$cVaR
#'
#' # random weights
#' wts = runif(6)
#' wts = wts/sum(wts)
#' names(wts) <- colnames(managers)[1:6]
#' portVaRDecomp(fit.macro, wts)
#'
#'
#' # Fundamental Factor Model
#' data("stocks145scores6")
#' dat = stocks145scores6
#' dat$DATE = as.yearmon(dat$DATE)
#' dat = dat[dat$DATE >=as.yearmon("2008-01-01") &
#' dat$DATE <= as.yearmon("2012-12-31"),]
#'
#' # Load long-only GMV weights for the return data
#' data("wtsStocks145GmvLo")
#' wtsStocks145GmvLo = round(wtsStocks145GmvLo,5)
#'
#' # fit a fundamental factor model
#' fit.cross <- fitFfm(data = dat,
#' exposure.vars = c("SECTOR","ROE","BP","MOM121","SIZE","VOL121",
#' "EP"),date.var = "DATE", ret.var = "RETURN", asset.var = "TICKER",
#' fit.method="WLS", z.score = "crossSection")
#'
#' decomp = portVaRDecomp(fit.cross)
#' # get the factor contributions of risk
#' decomp$cVaR
#' portVaRDecomp(fit.cross, weights = wtsStocks145GmvLo)
#'
#' @export
portVaRDecomp <- function(object, ...){
# check input object validity
if (!inherits(object, c("tsfm", "ffm"))) {
stop("Invalid argument: Object should be of class 'tsfm', or 'ffm'.")
}
UseMethod("portVaRDecomp")
}
#' @rdname portVaRDecomp
#' @method portVaRDecomp tsfm
#' @importFrom zoo index
#' @export
portVaRDecomp.tsfm <- function(object, weights = NULL, factor.cov, p=0.05, type=c("np","normal"),
invert = FALSE, use="pairwise.complete.obs", ...) {
# set default for type
type = type[1]
if (!(type %in% c("np","normal"))) {
stop("Invalid args: type must be 'np' or 'normal' ")
}
# get beta.star: 1 x (K+1)
beta <- object$beta
beta[is.na(beta)] <- 0
n.assets = nrow(beta)
asset.names <- object$asset.names
# check if there is weight input
if(is.null(weights)){
weights = rep(1/n.assets, n.assets)
}else{
# check if number of weight parameter matches
if(n.assets != length(weights)){
stop("Invalid argument: incorrect number of weights")
}
if(!is.null(names(weights))){
weights = weights[asset.names]
}else{
stop("Invalid argument: names of weights vector should match with asset names")
}
}
# get portfolio beta.star: 1 x (K+1)
beta.star <- as.matrix(cbind(weights %*% as.matrix(beta), sqrt(sum(weights^2 * object$resid.sd^2))))
colnames(beta.star)[dim(beta.star)[2]] <- "Residuals"
# factor returns and residuals data
factors.xts <- object$data[,object$factor.names]
resid.xts <- as.xts(t(t(residuals(object))/object$resid.sd) %*% weights)
zoo::index(resid.xts) <- as.Date(zoo::index(resid.xts))
if (type=="normal") {
# get cov(F): K x K
if (missing(factor.cov)) {
factor.cov = cov(as.matrix(factors.xts), use=use, ...)
} else {
if (!identical(dim(factor.cov), as.integer(c(ncol(factor), ncol(factor))))) {
stop("Dimensions of user specified factor covariance matrix are not
compatible with the number of factors in the fitTsfm object")
}
}
# get cov(F.star): (K+1) x (K+1)
K <- ncol(object$beta)
factor.star.cov <- diag(K+1)
factor.star.cov[1:K, 1:K] <- factor.cov
colnames(factor.star.cov) <- c(colnames(factor.cov),"Residuals")
rownames(factor.star.cov) <- c(colnames(factor.cov),"Residuals")
# factor expected returns
MU <- c(colMeans(factors.xts, na.rm=TRUE), 0)
names(MU) <- c(colnames(factor.cov),"Residuals")
# SIGMA*Beta to compute normal mVaR
SIGB <- beta.star %*% factor.star.cov
}
# initialize lists and matrices
K <- length(object$factor.names)
VaR.fm <- rep(NA, 1)
idx.exceed <- list()
n.exceed <- rep(NA, 1)
mVaR <- rep(NA, 1+K)
cVaR <- rep(NA, 1+K)
pcVaR <- rep(NA, 1+K)
names(mVaR)=names(cVaR)=names(pcVaR)=colnames(beta.star)
# return data for portfolio
match = colnames(object$data) %in% asset.names
R.xts <- object$data[,match]
R.xts <- R.xts * weights
R.xts = as.xts(rowSums(R.xts), order.by = zoo::index(R.xts))
names(R.xts) = 'RETURN'
if (type=="np") {
# get VaR for asset i
VaR.fm <- quantile(R.xts, probs=p, na.rm=TRUE, ...)
# get F.star data object
factor.star <- merge(factors.xts, resid.xts)
colnames(factor.star)[dim(factor.star)[2]] <- "Residuals"
# epsilon is apprx. using Silverman's rule of thumb (bandwidth selection)
# the constant 2.575 corresponds to a triangular kernel
eps <- 2.575*sd(R.xts, na.rm =TRUE) * (nrow(R.xts)^(-1/5))
# compute marginal VaR as expected value of factor returns, such that the
# asset return was incident in the triangular kernel region peaked at the
# VaR value and bandwidth = epsilon.
k.weight <- as.vector(1 - abs(R.xts - VaR.fm) / eps)
k.weight[k.weight<0] <- 0
mVaR <- colMeans(factor.star*k.weight, na.rm =TRUE)
} else if (type=="normal") {
# get VaR for asset i
VaR.fm <- drop(beta.star %*% MU + sqrt(beta.star %*% factor.star.cov %*% t(beta.star))*qnorm(p))
# compute marginal VaR
mVaR <- drop(MU + SIGB * qnorm(p)/sd(R.xts, na.rm=TRUE))
}
# index of VaR exceedances
idx.exceed <- which(R.xts <= VaR.fm)
# number of VaR exceedances
n.exceed <- length(idx.exceed)
# correction factor to ensure that sum(cVaR) = asset VaR
cf <- as.numeric( VaR.fm / sum(mVaR*beta.star, na.rm=TRUE) )
# compute marginal, component and percentage contributions to VaR
# each of these have dimensions: N x (K+1)
mVaR <- drop(cf * mVaR)
cVaR <- drop(mVaR * beta.star)
pcVaR <- drop(100* cVaR / VaR.fm)
if(invert){
VaR.fm <- -VaR.fm
}
fm.VaR.decomp <- list(portVaR=VaR.fm, n.exceed=n.exceed, idx.exceed=idx.exceed,
mVaR=mVaR, cVaR=cVaR, pcVaR=pcVaR)
return(fm.VaR.decomp)
}
#' @rdname portVaRDecomp
#' @method portVaRDecomp ffm
#' @importFrom zoo index
#' @export
portVaRDecomp.ffm <- function(object, weights = NULL, factor.cov, p=0.05, type=c("np","normal"),
invert = FALSE , ...) {
# set default for type
type = type[1]
if (!(type %in% c("np","normal"))) {
stop("Invalid args: type must be 'np' or 'normal' ")
}
beta <- object$beta
beta[is.na(beta)] <- 0
n.assets = nrow(beta)
asset.names <- unique(object$data[[object$asset.var]])
# check if there is weight input
if(is.null(weights)){
weights = rep(1/n.assets, n.assets)
}else{
# check if number of weight parameter matches
if(n.assets != length(weights)){
stop("Invalid argument: incorrect number of weights")
}
if(!is.null(names(weights))){
weights = weights[asset.names]
}else{
stop("Invalid argument: names of weights vector should match with asset names")
}
}
# get portfolio beta.star: 1 x (K+1)
beta.star <- as.matrix(cbind(weights %*% beta, sqrt(sum(weights^2 * object$resid.var))))
colnames(beta.star)[dim(beta.star)[2]] <- "Residuals"
# factor returns and residuals data
factors.xts <- object$factor.returns
resid.xts <- as.xts( t(t(residuals(object))/sqrt(object$resid.var)) %*% weights)
zoo::index(resid.xts) <- as.Date(zoo::index(resid.xts))
if (type=="normal") {
# get cov(F): K x K
if (missing(factor.cov)) {
factor.cov <- object$factor.cov
} else {
if (!identical(dim(factor.cov), dim(object$factor.cov))) {
stop("Dimensions of user specified factor covariance matrix are not
compatible with the number of factors in the fitSfm object")
}
}
# get cov(F.star): (K+1) x (K+1)
K <- ncol(object$beta)
factor.star.cov <- diag(K+1)
factor.star.cov[1:K, 1:K] <- factor.cov
colnames(factor.star.cov) <- c(colnames(factor.cov),"Residuals")
rownames(factor.star.cov) <- c(colnames(factor.cov),"Residuals")
# factor expected returns
MU <- c(colMeans(factors.xts, na.rm=TRUE), 0)
names(MU) <- c(colnames(factor.cov),"Residuals")
# SIGMA*Beta to compute normal mVaR
SIGB <- beta.star %*% factor.star.cov
}
# initialize lists and matrices
K <- length(object$factor.names)
VaR.fm <- rep(NA, 1)
idx.exceed <- list()
n.exceed <- rep(NA, 1)
mVaR <- rep(NA, 1+K)
cVaR <- rep(NA, 1+K)
pcVaR <- rep(NA, 1+K)
names(mVaR)=names(cVaR)=names(pcVaR)=colnames(beta.star)
dat = object$data
# return data for portfolio
R.xts = tapply(dat[,object$ret.var], list(dat[,object$date.var], dat[,object$asset.var]), FUN = I)
R.xts <- R.xts * weights
R.xts = as.xts(rowSums(R.xts), order.by = object$time.periods)
names(R.xts) = 'RETURN'
if (type=="np") {
VaR.fm <- quantile(R.xts, probs=p, na.rm=TRUE, ...)
# get F.star data object
zoo::index(factors.xts) <- zoo::index(resid.xts)
factor.star <- merge(factors.xts, resid.xts)
colnames(factor.star)[dim(factor.star)[2]] <- "Residuals"
# epsilon is apprx. using Silverman's rule of thumb (bandwidth selection)
# the constant 2.575 corresponds to a triangular kernel
eps <- 2.575*sd(R.xts, na.rm =TRUE) * (nrow(R.xts)^(-1/5))
# compute marginal VaR as expected value of factor returns, such that the
# asset return was incident in the triangular kernel region peaked at the
# VaR value and bandwidth = epsilon.
k.weight <- as.vector(1 - abs(R.xts - VaR.fm) / eps)
k.weight[k.weight<0] <- 0
mVaR <- colMeans(factor.star*k.weight, na.rm =TRUE)
}
else if (type=="normal") {
VaR.fm <- drop(beta.star %*% MU + sqrt(beta.star %*% factor.star.cov %*% t(beta.star))*qnorm(p))
mVaR <- drop(MU + drop(SIGB) * qnorm(p)/sd(R.xts, na.rm=TRUE))
}
# index of VaR exceedances
idx.exceed <- which(R.xts <= VaR.fm)
# number of VaR exceedances
n.exceed <- length(idx.exceed)
# correction factor to ensure that sum(cVaR) = asset VaR
cf <- as.numeric( VaR.fm / sum(mVaR*beta.star, na.rm=TRUE) )
# compute marginal, component and percentage contributions to VaR
# each of these have dimensions: N x (K+1)
mVaR <- drop(cf * mVaR)
cVaR <- drop(mVaR * beta.star)
pcVaR <- drop(100* cVaR / VaR.fm)
if(invert){
VaR.fm <- -VaR.fm
}
fm.VaR.decomp <- list(portVaR=VaR.fm, n.exceed=n.exceed, idx.exceed=idx.exceed,
mVaR=mVaR, cVaR=cVaR, pcVaR=pcVaR)
return(fm.VaR.decomp)
}
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