#' calculates total attribution effects using Menchero smoothing
#'
#' Calculates total attribution effects over multiple periods using
#' Menchero linking method. Used internally by the \code{\link{Attribution}}
#' function. Arithmetic attribution effects do not naturally link over time.
#' This function uses Menchero 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 (M +a_{t})}{At' = At * (M + at)}
#' where \deqn{M=\frac{\frac{1}{n}(R_{p}-R_{b})}{(1+R_{p})^{\frac{1}{n}}-
#' (1+R_{b})^{\frac{1}{n}}}}
#' \deqn{a_{t} = \left(\frac{R_{p}-R_{b}-M\times\sum^{n}_{t=1}(R_{pt}-
#' R_{bt})}{\sum^{n}_{t=1}(R_{pt}-R_{bt})^{2}}\right)\times(R_{pt}-R_{bt})}
#' In case if portfolio and benchmark returns are equal the limit of the
#' above value is used:
#' \deqn{M = (1 + r_{p})^\frac{n-1}{n}}{M = (1 + rp)^((n - 1) / n)}
#' \eqn{A_{t}'}{At'} - adjusted attribution effects at period \eqn{t},
#' \eqn{A_{t}}{At} - unadjusted attribution effects at period \eqn{t},
#' \eqn{R_{pt}}{Rpt} - portfolio returns at period \eqn{t},
#' \eqn{R_{bt}}{Rbt} - benchmark returns at period \eqn{t},
#' \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 Menchero
#' @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{Carino}} \cr
#' \code{\link{Grap}} \cr \code{\link{Frongello}} \cr
#' \code{\link{Attribution.geometric}}
#' @references Bacon, C. \emph{Practical Portfolio Performance Measurement and
#' Attribution}. Wiley. 2004. p. 194-196 \cr Menchero, J. (2000) \emph{An
#' optimized approach to linking attribution effects over time}. Journal of
#' Performance Measurement. Fall. p. 36-42 \cr
#' @keywords arithmetic attribution, Menchero linking
#' @examples
#'
#' data(attrib)
#' Menchero(rp = attrib.returns[, 21], rb = attrib.returns[, 22], attributions = attrib.allocation, adjusted = FALSE)
#'
#' @export
Menchero <-
function(rp, rb, attributions, adjusted)
{ # @author Andrii Babii
# DESCRIPTION:
# Function to provide multi-period summary of attribution effects using
# Menchero 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:
rpc = prod(1 + rp) - 1 # Cumulative returns
rbc = prod(1 + rb) - 1
T = nrow(rp)
if (rpc == rbc){
M = (1 + rbc)^((T - 1) / T)
at = 0
} else{
M = ((rpc - rbc) / T) / ((1 + rpc)^(1 / T) - (1 + rbc)^(1 / T))
at = (rpc - rbc - M * sum(rp - rb)) * (rp - coredata(rb)) / sum((rp -
coredata(rb))^2)
}
m = matrix(rep(M + at, ncol(attributions)), nrow(attributions),
ncol(attributions), byrow = FALSE)
adj = attributions * m
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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