#' calculates total attribution effects using Frongello smoothing
#'
#' Calculates total attribution effects over multiple periods using
#' Frongello linking method. Used internally by the \code{\link{Attribution}}
#' function. Arithmetic attribution effects do not naturally link over time.
#' This function uses Frongello smoothing algorithm to adjust
#' attribution effects so that they can be summed up over multiple periods
#' Adjusted attribution effect at period t are:
#' \deqn{A_{t}' = A_{t}\times\prod^{t-1}_{i=1}(1+r_{pi})+R_{bt}
#' \times\sum^{t-1}_{i=1}A_{i}'}
#' \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{R_{p}}{Rp} - total portfolio returns,
#' \eqn{R_{b}}{Rb} - total benchmark returns,
#' \eqn{n} - number of periods
#'
#' @aliases Frongello
#' @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{Grap}} \cr \code{\link{Carino}} \cr
#' \code{\link{Attribution.geometric}}
#' @references Bacon, C. \emph{Practical Portfolio Performance Measurement and
#' Attribution}. Wiley. 2004. p. 199-201 \cr Frongello, A. (2002) \emph{Linking
#' single period attribution results}. Journal of Performance Measurement.
#' Spring, p. 10-22. \cr
#' @keywords arithmetic attribution, Frongello linking
#' @examples
#'
#' data(attrib)
#' Frongello(rp = attrib.returns[, 21], rb = attrib.returns[, 22], attributions = attrib.allocation, adjusted = FALSE)
#'
#' @export
Frongello <-
function(rp, rb, attributions, adjusted)
{ # @author Andrii Babii
# DESCRIPTION:
# Function to provide multi-period summary of attribution effects using
# Frongello linking. Used internally by the Attribution function
# Inputs:
# rp xts of portfolio returns
# rb xts of benchmark returns
# attributions attribution effects (e.g. allocation, selection, interaction)
# Outputs:
# This function returns the data.frame with original attribution effects
# and total attribution effects over multiple periods
# FUNCTION:
adj = attributions
if (nrow(rp) > 1){
adj[2, ] = coredata(adj[2, ]) * drop((1 + rp[1, 1])) + drop(rb[2, 1]) *
coredata(adj[1, ])
}
if (nrow(rp) > 2){
for(i in 3:nrow(rp)){
adj[i, ] = coredata(adj[i, ]) * drop(prod(1 + rp[1:(i-1), 1])) +
drop(rb[i, ]) * coredata(colSums(adj[1:(i-1), ]))
}
}
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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