#' calculates total attribution effects using GRAP smoothing
#'
#' Calculates total attribution effects over multiple periods using
#' GRAP linking method. Used internally by the \code{\link{Attribution}}
#' function. Arithmetic attribution effects do not naturally link over time.
#' This function uses GRAP smoothing algorithm to adjust attribution effects
#' so that they can be summed up over multiple periods
#' Attribution effect are multiplied by the adjustment factor
#' \deqn{A_{t}' = A_{t} \times G_{t}}{At' = At * Gt} where
#' \deqn{G_{t}=\prod^{t-1}_{i=1}(1+R_{pi})\times\prod^{n}_{t+1}(1+R_{bi})}
#' \eqn{A_{t}'}{At'} - adjusted attribution effects at period \eqn{t},
#' \eqn{A_{t}}{At} - unadjusted attribution effects at period \eqn{t},
#' \eqn{R_{pi}}{Rpi} - portfolio returns at period \eqn{i},
#' \eqn{R_{bi}}{Rbi} - benchmark returns at period \eqn{i},
#' \eqn{R_{p}}{Rp} - total portfolio returns,
#' \eqn{R_{b}}{Rb} - total benchmark returns,
#' \eqn{n} - number of periods
#' The total arithmetic excess returns can be explained in terms of the sum
#' of adjusted attribution effects:
#' \deqn{R_{p} - R_{b} = \sum^{n}_{t=1}\left(Allocation_{t}+Selection_{t}+
#' Interaction_{t}\right)}
#'
#' @aliases Grap
#' @param rp xts of portfolio returns
#' @param rb xts of benchmark returns
#' @param attributions xts with attribution effects
#' @param adjusted TRUE/FALSE, whether to show original or smoothed attribution
#' effects for each period
#' @return returns a data frame with original attribution effects and total
#' attribution effects over multiple periods
#' @author Andrii Babii
#' @seealso \code{\link{Attribution}} \cr \code{\link{Menchero}} \cr
#' \code{\link{Carino}} \cr \code{\link{Frongello}} \cr
#' \code{\link{Attribution.geometric}}
#' @references Bacon, C. \emph{Practical Portfolio Performance Measurement
#' and Attribution}. Wiley. 2004. p. 196-199 \cr GRAP (Groupe de Recherche en
#' Attribution de Performance) (1997) \emph{Synthese des modeles d'attribution
#' de performance}. Paris, Mars.\cr
#' @keywords arithmetic attribution, GRAP linking
#' @examples
#'
#' data(attrib)
#' Grap(rp = attrib.returns[, 21], rb = attrib.returns[, 22], attributions = attrib.allocation, adjusted = FALSE)
#'
#' @export
Grap <-
function(rp, rb, attributions, adjusted)
{ # @author Andrii Babii
# DESCRIPTION:
# Function to provide multi-period summary of attribution effects using
# GRAP linking. Used internally by the Attribution function
# Inputs:
# rp xts of portfolio returns
# rb xts of benchmark returns
# attributions attribution effects (allocation, selection, interaction)
# Outputs:
# This function returns the data.frame with original attribution effects
# and total attribution effects over multiple periods
# FUNCTION:
G = rp
T = nrow(rp)
G[1] = prod(1 + rb[2:T]) #GRAP factor for the first period
if (T == 2){
G[2] = (1 + rp[1])
}
if (T > 2){
G[T] = prod(1 + rp[1:(T - 1)]) #GRAP factor for the last period
}
if (T > 3){
for(i in 2:(T - 1)){
r = 1 + rp[1:(i-1)]
b = 1 + rb[(i+1):T]
G[i] = apply(r, 2, prod) * apply(b, 2, prod)
}
}
g = matrix(rep(G, ncol(attributions)), nrow(attributions),
ncol(attributions), byrow = FALSE)
adj = attributions * g
total = colSums(adj)
if (adjusted == FALSE){
attributions = rbind(as.data.frame(attributions), total)
} else{
attributions = rbind(as.data.frame(adj), total)
}
rownames(attributions)[nrow(attributions)] = "Total"
return(attributions)
}
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