R/attribution.R
In R-Finance/PortfolioAttribution: Performance attribution tools used for identifying sources of portfolio return and risk.
#' performs sector-based single-level attribution
#'
#' Performs sector-based single-level attribution analysis. Portfolio
#' performance measured relative to a benchmark gives an indication of the
#' value-added by the portfolio. Equipped with weights and returns of portfolio
#' segments, we can dissect the value-added into useful components. This
#' function is based on the sector-based approach to the attribution. The
#' workhorse is the Brinson model that explains the arithmetic difference
#' between portfolio and benchmark returns. That is it breaks down the
#' arithmetic excess returns at one level. If returns and weights are available
#' at the lowest level (e.g. for individual instruments), the aggregation up to
#' the chosen level from the hierarchy can be done using
#' \code{\link{Return.level}} function. The attribution effects can be computed
#' for several periods. The multi-period summary is obtained using one of
#' linking methods: Carino, Menchero, GRAP, Frongello or Davies Laker. It also
#' allows to break down the geometric excess returns, which link naturally over
#' time. Finally, it annualizes arithmetic and geometric excess returns
#' similarly to the portfolio and/or benchmark returns annualization.
#'
#' The arithmetic excess returns are decomposed into the sum of allocation,
#' selection and interaction effects across \eqn{n} sectors:
#' \deqn{R_{p}-R_{b}=\sum^{n}_{i=1}\left(A_{i}+S_{i}+I_{i}\right)}
#' The arithmetic attribution effects for the category i are computed
#' as suggested in the Brinson, Hood and Beebower (1986):
#' Allocation effect
#' \deqn{A_{i}=(w_{pi}-w_{bi})\times R_{bi}}{Ai = (wpi - wbi) * Rbi}
#' Selection effect
#' \deqn{S_{i}=w_{pi}\times(R_{pi}-R_{bi})}{Si = wpi * (Rpi - Rbi)}
#' Interaction effect
#' \deqn{I_{i}=(w_{pi}-w_{bi})
#' \times(R_{pi}-R_{bi})}{Ii = (wpi - wbi) * Rpi - Rbi}
#' \eqn{R_{p}}{Rp} - total portfolio returns,
#' \eqn{R_{b}}{Rb} - total benchmark returns,
#' \eqn{w_{pi}}{wpi} - weights of the category \eqn{i} in the portfolio,
#' \eqn{w_{bi}}{wbi} - weights of the category \eqn{i} in the benchmark,
#' \eqn{R_{pi}}{Rpi} - returns of the portfolio category \eqn{i},
#' \eqn{R_{bi}}{Rbi} - returns of the benchmark category \eqn{i}.
#' If Brinson and Fachler (1985) is selected the allocation effect differs:
#' \deqn{A_{i}=(w_{pi}-w_{bi})
#' \times (R_{bi} - R_{b})}{Ai = (wpi - wbi) * (Rbi - Rb)}
#' Depending on goals we can give priority to the allocation or to
#' the selection effects. If the priority is given to the sector allocation
#' the interaction term will be combined with the security selection effect
#' (top-down approach). If the priority is given to the security selection,
#' the interaction term will be combined with the asset-allocation effect
#' (bottom-up approach).
#' Usually we have more than one period. In that case individual arithmetic
#' attribution effects should be adjusted using linking methods. Adjusted
#' arithmetic attribution effects can be summed up over time to provide the
#' multi-period summary:
#' \deqn{R_{p}-R_{b}=\sum^{T}_{t=1}\left(A_{t}'+S_{t}'+I_{t}'\right)}
#' where \eqn{T} is the number of periods and prime stands for the adjustment.
#' The geometric attribution effects do not suffer from the linking problem.
#' Moreover we don't have the interaction term. For more details about the
#' geometric attribution see the documentation to
#' \code{\link{Attribution.geometric}}. Finally, arithmetic annualized excess
#' returns are computed as the arithmetic difference between annualised
#' portfolio and benchmark returns:
#' \deqn{AAER=r_{a}-b_{a}}{AAER = ra - ba} the geometric annualized excess
#' returns are computed as the geometric difference between annualized
#' portfolio and benchmark returns:
#' \deqn{GAER=\frac{1+r_{a}}{1+b_{a}}-1}{GAER = (1 + ra) / (1 + ba) - 1}
#' In the case of multi-currency portfolio, the currency return, currency
#' surprise and forward premium should be specified. The multi-currency
#' arithmetic attribution is handled following Ankrim and Hensel (1992).
#' Currency returns are decomposed into the sum of the currency surprise and
#' the forward premium: \deqn{R_{ci} = R_{cei} + R_{fpi}}{Rci = Rcei + Rfpi}
#' where
#' \deqn{R_{cei} = \frac{S_{i}^{t+1} - F_{i}^{t+1}}{S_{i}^{t}}}
#' \deqn{R_{fpi} = \frac{F_{i}^{t+1}}{S_{i}^{t}} - 1}
#' \eqn{S_{i}^{t}}{Sit} - spot rate for asset \eqn{i} at time \eqn{t}
#' \eqn{F_{i}^{t}}{Fit} - forward rate for asset \eqn{i} at time \eqn{t}.
#' Excess returns are decomposed into the sum of allocation, selection and
#' interaction effects as in the standard Brinson model:
#' \deqn{R_{p}-R_{b}=\sum^{n}_{i=1}\left(A_{i}+S_{i}+I_{i}\right)}
#' However the allocation effect is computed taking into account currency
#' effects:
#' \deqn{A_{i}=(w_{pi}-w_{bi})\times (R_{bi} - R_{ci} - R_{l})}{Ai =
#' (wpi - wbi) * (Rbi - Rci - Rl)}
#' Benchmark returns adjusted to the currency:
#' \deqn{R_{l} = \sum^{n}_{i=1}w_{bi}\times(R_{bi}-R_{ci})}
#' The contribution from the currency is analogous to asset allocation:
#' \deqn{C_{i} = (w_{pi} - w_{bi}) \times (R_{cei} - e) + (w_{pfi} - w_{bfi})
#' \times (R_{fi} - e)}
#' where \deqn{e = \sum^{n}_{i=1}w_{bi}\times R_{cei}}
#' The final term, forward premium, is also analogous to the asset allocation:
#' \deqn{R_{fi} = (w_{pi} - w_{bi}) \times (R_{fpi} - d)}{Rfi = (wpi - wbi) *
#' (Rfpi - d)}
#' where \deqn{d = \sum^{n}_{i=1}w_{bi}\times R_{fpi}}
#' and \eqn{R_{fpi}} - forward premium
#' In general if the intent is to estimate statistical parameters, the
#' arithmetic excess return is preferred. However, due to the linking
#' challenges, it may be preferable to use geometric excess return if the
#' intent is to link and annualize excess returns.
#'
#' @aliases Attribution
#' @param Rp T x n xts, data frame or matrix of portfolio returns
#' @param wp vector, xts, data frame or matrix of portfolio weights
#' @param Rb T x n xts, data frame or matrix of benchmark returns
#' @param wb vector, xts, data frame or matrix of benchmark weights
#' @param method Used to select the priority between allocation and selection
#' effects in arithmetic attribution. May be any of: \itemize{ \item none -
#' present allocation, selection and interaction effects independently,
#' \item top.down - the priority is given to the sector allocation. Interaction
#' term is combined with the security selection effect, \item bottom.up - the
#' priority is given to the security selection. Interaction term is combined
#' with the sector allocation effect}
#' By default "none" is selected
#' @param wpf vector, xts, data frame or matrix with portfolio weights of
#' currency forward contracts
#' @param wbf vector, xts, data frame or matrix with benchmark weights of
#' currency forward contracts
#' @param S (T+1) x n xts, data frame or matrix with spot rates. The first date
#' should coincide with the first date of portfolio returns
#' @param F (T+1) x n xts, data frame or matrix with forward rates. The first
#' date should coincide with the first date of portfolio returns
#' @param Rpl xts, data frame or matrix of portfolio returns in local currency
#' @param Rbl xts, data frame or matrix of benchmark returns in local currency
#' @param Rbh xts, data frame or matrix of benchmark returns hedged into the
#' base currency
#' @param bf TRUE for Brinson and Fachler and FALSE for Brinson, Hood and
#' Beebower arithmetic attribution. By default Brinson, Hood and Beebower
#' attribution is selected
#' @param linking Used to select the linking method to present the multi-period
#' summary of arithmetic attribution effects. May be any of:
#' \itemize{\item carino - logarithmic linking coefficient method
#' \item menchero - Menchero's smoothing algorithm
#' \item grap - linking approach developed by GRAP
#' \item frongello - Frongello's linking method
#' \item davies.laker - Davies and Laker's linking method}
#' By default Carino linking is selected
#' @param geometric TRUE/FALSE, whether to use geometric or arithmetic excess
#' returns for the attribution analysis. By default arithmetic is selected
#' @param adjusted TRUE/FALSE, whether to show original or smoothed attribution
#' effects for each period. By default unadjusted attribution effects are
#' returned
#' @return returns a list with the following components: excess returns with
#' annualized excess returns over all periods, attribution effects (allocation,
#' selection and interaction)
#' @author Andrii Babii
#' @seealso \code{\link{Attribution.levels}},
#' \code{\link{Attribution.geometric}}
#' @references Ankrim, E. and Hensel, C. \emph{Multi-currency performance
#' attribution}. Russell Research Commentary. November 2002 \cr Bacon, C.
#' \emph{Practical Portfolio Performance Measurement and Attribution}. Wiley.
#' 2004. Chapter 5, 6, 8 \cr Christopherson, Jon A., Carino, David R., Ferson,
#' Wayne E. \emph{Portfolio Performance Measurement and Benchmarking}.
#' McGraw-Hill. 2009. Chapter 18-19 \cr Brinson, G. and Fachler, N. (1985)
#' \emph{Measuring non-US equity portfolio performance}. Journal of Portfolio
#' Management. Spring. p. 73 -76. \cr Gary P. Brinson, L. Randolph Hood, and
#' Gilbert L. Beebower, \emph{Determinants of Portfolio Performance}. Financial
#' Analysts Journal. vol. 42, no. 4, July/August 1986, p. 39-44 \cr
#' Karnosky, D. and Singer, B. \emph{Global asset management and performance
#' attribution. The Research Foundation of the Institute of Chartered Financial
#' Analysts}. February 1994. \cr
#' @keywords attribution
#' @examples
#'
#' data(attrib)
#' Attribution(Rp = attrib.returns[, 1:10], wp = attrib.weights[1, ], Rb = attrib.returns[, 11:20],
#' wb = attrib.weights[2, ], method = "top.down", linking = "carino")
#'
#' @export
Attribution <-
function (Rp, wp, Rb, wb,
wpf = NA, wbf = NA, S = NA, F = NA, Rpl = NA, Rbl = NA, Rbh = NA,
bf = FALSE,
method = c("none", "top.down", "bottom.up"),
linking = c("carino",
"menchero",
"grap",
"frongello",
"davies.laker"),
geometric = FALSE, adjusted = FALSE)
{ # @author Andrii Babii
# DESCRIPTION:
# Function to perform the attribution analysis.
# Inputs:
# Rp T x n xts, data frame or matrix of portfolio returns
# wp vector, xts, data frame or matrix of portfolio weights
# Rb T x n xts, data frame or matrix of benchmark returns
# wb vector, xts, data frame or matrix of benchmark weights
# wpf vector, xts, data frame or matrix with portfolio weights of
# currency forward contracts
# wbf vector, xts, data frame or matrix with benchmark weights of
# currency forward contracts
# S (T+1) x n xts, data frame or matrix with spot rates
# F (T+1) x n xts, data frame or matrix with forward rates
# Rpl xts, data frame or matrix of portfolio returns in local currency
# Rbl xts, data frame or matrix of benchmark returns in local currency
# Rbh xts, data frame or matrix of benchmark returns hedged into the
# base currency
# Outputs:
# This function returns the attribution effects with multi-period summary
# and annualized excess returns
# FUNCTION:
# Transform data to the xts objects
Rb = checkData(Rb)
Rp = checkData(Rp)
WP = wp # Save original weights in order to avoid double conversion later
WB = wb
wp = Weight.transform(wp, Rp)
wb = Weight.transform(wb, Rb)
if (nrow(wp) < nrow(Rp)){ # Rebalancing occurs next day
Rp = Rp[2:nrow(Rp)]
Rb = Rb[2:nrow(Rb)]
}
if (ncol(Rb) == 1){
Rb = matrix(rep(coredata(Rb), ncol(Rp)), nrow(Rp), ncol(Rp))
}
if (ncol(Rb) != ncol(Rp)){
stop("Please use benchmark xts that has columns with benchmarks for each
asset or one common benchmark for all assets")
}
method = method[1]
linking = linking[1]
currency = !(is.null(dim(wpf)) & is.null(dim(wbf)) &
is.null(dim(S)) & is.null(dim(F)) &
is.null(dim(Rpl)) & is.null(dim(Rpl)) &
is.null(dim(Rpl)))
if (geometric == FALSE & linking != "davies.laker"){
# The function makes all computations for the arithmetic attribution
# case (except for Davies and Laker linking)
# Compute attribution effects (Brinson, Hood and Beebower model)
# If portfolio is single-currency
if (!currency){
Rc = 0
L = 0
} else{ # If multi-currency portfolio
S = checkData(S)
F = checkData(F)
wpf = Weight.transform(wpf, Rp)
wbf = Weight.transform(wbf, Rb)
Rc = lag(S, -1)[1:nrow(Rp), ] / S[1:nrow(Rp), ] - 1
Rd = lag(F, -1)[1:nrow(Rp), ] / S[1:nrow(Rp), ] - 1
Re = Rc - coredata(Rd)
Rl = Rb - coredata(Rc)
Rk = Rp - coredata(Rc)
Rfp = Re / (1 + Rd)
E = reclass(matrix(rep(rowSums(Re * coredata(wb)), ncol(Rb)), nrow(Rb),
ncol(Rb)), Rp)
L = reclass(matrix(rep(rowSums(Rl * coredata(wb)), ncol(Rb)), nrow(Rb),
ncol(Rb)), Rp)
D = reclass(matrix(rep(rowSums(Rd * coredata(wb)), ncol(Rb)), nrow(Rb),
ncol(Rb)), Rp)
# Contribution to currency
Cc = (wp - wb) * (Re - E) + (wpf - wbf) * (Rfp - E)
# Forward premium
Df = (wp - wb) * (Rd - D)
Cc = cbind(Cc, rowSums(Cc))
Df = cbind(Df, rowSums(Df))
colnames(Cc) = c(colnames(S), "Total")
colnames(Df) = colnames(Cc)
}
# Get total portfolio returns
if (is.vector(WP) & is.vector(WB)){
rp = Return.portfolio(Rp, WP, geometric = FALSE)
rb = Return.portfolio(Rb, WB, geometric = FALSE)
} else{
rp = Return.rebalancing(Rp, WP, geometric = FALSE)
rb = Return.rebalancing(Rb, WB, geometric = FALSE)
}
names(rp) = "Total"
names(rb) = "Total"
# Get individual attribution effects
if (bf == TRUE){ # Brinson and Fachler (1985) allocation effect
allocation = coredata(wp - wb) * (Rb - coredata(Rc) - coredata(L) -
rep(rb, ncol(Rb)))
} else{ # Brinson, Hood and Beebower (1986) allocation effect
allocation = coredata(wp - wb) * (Rb - coredata(Rc) - coredata(L))
}
selection = (Rp - coredata(Rb)) * wb
interaction = (wp - wb) * (Rp - coredata(Rb))
# Get total attribution effects
n = ncol(allocation) # number of segments
allocation = cbind(allocation, rowSums(allocation))
names(allocation)[n + 1] = "Total"
selection = cbind(selection, rowSums(selection))
names(selection)[n + 1] = "Total"
interaction = cbind(interaction, rowSums(interaction))
names(interaction)[n + 1] = "Total"
# Adjust attribution effects using one of linking methods if there are
# mutliple periods
if (nrow(allocation) > 1){
if (linking == "carino"){
allocation = Carino(rp, rb, allocation, adjusted)
selection = Carino(rp, rb, selection, adjusted)
interaction = Carino(rp, rb, interaction, adjusted)
}
if (linking == "menchero"){
allocation = Menchero(rp, rb, allocation, adjusted)
selection = Menchero(rp, rb, selection, adjusted)
interaction = Menchero(rp, rb, interaction, adjusted)
}
if (linking == "grap"){
allocation = Grap(rp, rb, allocation, adjusted)
selection = Grap(rp, rb, selection, adjusted)
interaction = Grap(rp, rb, interaction, adjusted)
}
if (linking == "frongello"){
allocation = Frongello(rp, rb, allocation, adjusted)
selection = Frongello(rp, rb, selection, adjusted)
interaction = Frongello(rp, rb, interaction, adjusted)
}
}
# Arithmetic excess returns + annualized arithmetic excess returns
excess.returns = rp - coredata(rb)
if (nrow(rp) > 1){
er = Return.annualized.excess(rp, rb, geometric = FALSE)
excess.returns = rbind(as.matrix(excess.returns), er)
}
colnames(excess.returns) = "Arithmetic"
# Select the appropriate result corresponding to the chosen method
result = list()
result[[1]] = excess.returns
result[[2]] = allocation
result[[3]] = selection
if (method == "top.down"){ # Top-down attribution
result[[3]] = result[[3]] + interaction
}
if (method == "bottom.up"){ # Bottom-up attribution
result[[2]] = result[[2]] + interaction
}
if (method == "none"){
result[[4]] = interaction
}
} else{ # The function takes output of the corresponding function
# (Attribution.geometric or DaviesLaker)
if (geometric == TRUE){
attrib = Attribution.geometric(Rp, WP, Rb, WB)
}
if (linking == "davies.laker"){
attrib = DaviesLaker(Rp, WP, Rb, WB)
}
result = attrib
}
# Label the output
if ((method == "none" & geometric == FALSE) | linking == "davies.laker"){
names(result) = c("Excess returns", "Allocation", "Selection",
"Interaction")
} else{
names(result) = c("Excess returns", "Allocation", "Selection")
}
# If multi-currency portfolio
if (currency){
result[[length(result) + 1]] = Cc
result[[length(result) + 1]] = Df
names(result)[(length(result)-1):length(result)] =
c("Currency management", "Forward Premium")
}
return(result)
}
R-Finance/PortfolioAttribution documentation built on May 8, 2019, 4:48 a.m.
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#' performs sector-based single-level attribution
#'
#' Performs sector-based single-level attribution analysis. Portfolio
#' performance measured relative to a benchmark gives an indication of the
#' value-added by the portfolio. Equipped with weights and returns of portfolio
#' segments, we can dissect the value-added into useful components. This
#' function is based on the sector-based approach to the attribution. The
#' workhorse is the Brinson model that explains the arithmetic difference
#' between portfolio and benchmark returns. That is it breaks down the
#' arithmetic excess returns at one level. If returns and weights are available
#' at the lowest level (e.g. for individual instruments), the aggregation up to
#' the chosen level from the hierarchy can be done using
#' \code{\link{Return.level}} function. The attribution effects can be computed
#' for several periods. The multi-period summary is obtained using one of
#' linking methods: Carino, Menchero, GRAP, Frongello or Davies Laker. It also
#' allows to break down the geometric excess returns, which link naturally over
#' time. Finally, it annualizes arithmetic and geometric excess returns
#' similarly to the portfolio and/or benchmark returns annualization.
#'
#' The arithmetic excess returns are decomposed into the sum of allocation,
#' selection and interaction effects across \eqn{n} sectors:
#' \deqn{R_{p}-R_{b}=\sum^{n}_{i=1}\left(A_{i}+S_{i}+I_{i}\right)}
#' The arithmetic attribution effects for the category i are computed
#' as suggested in the Brinson, Hood and Beebower (1986):
#' Allocation effect
#' \deqn{A_{i}=(w_{pi}-w_{bi})\times R_{bi}}{Ai = (wpi - wbi) * Rbi}
#' Selection effect
#' \deqn{S_{i}=w_{pi}\times(R_{pi}-R_{bi})}{Si = wpi * (Rpi - Rbi)}
#' Interaction effect
#' \deqn{I_{i}=(w_{pi}-w_{bi})
#' \times(R_{pi}-R_{bi})}{Ii = (wpi - wbi) * Rpi - Rbi}
#' \eqn{R_{p}}{Rp} - total portfolio returns,
#' \eqn{R_{b}}{Rb} - total benchmark returns,
#' \eqn{w_{pi}}{wpi} - weights of the category \eqn{i} in the portfolio,
#' \eqn{w_{bi}}{wbi} - weights of the category \eqn{i} in the benchmark,
#' \eqn{R_{pi}}{Rpi} - returns of the portfolio category \eqn{i},
#' \eqn{R_{bi}}{Rbi} - returns of the benchmark category \eqn{i}.
#' If Brinson and Fachler (1985) is selected the allocation effect differs:
#' \deqn{A_{i}=(w_{pi}-w_{bi})
#' \times (R_{bi} - R_{b})}{Ai = (wpi - wbi) * (Rbi - Rb)}
#' Depending on goals we can give priority to the allocation or to
#' the selection effects. If the priority is given to the sector allocation
#' the interaction term will be combined with the security selection effect
#' (top-down approach). If the priority is given to the security selection,
#' the interaction term will be combined with the asset-allocation effect
#' (bottom-up approach).
#' Usually we have more than one period. In that case individual arithmetic
#' attribution effects should be adjusted using linking methods. Adjusted
#' arithmetic attribution effects can be summed up over time to provide the
#' multi-period summary:
#' \deqn{R_{p}-R_{b}=\sum^{T}_{t=1}\left(A_{t}'+S_{t}'+I_{t}'\right)}
#' where \eqn{T} is the number of periods and prime stands for the adjustment.
#' The geometric attribution effects do not suffer from the linking problem.
#' Moreover we don't have the interaction term. For more details about the
#' geometric attribution see the documentation to
#' \code{\link{Attribution.geometric}}. Finally, arithmetic annualized excess
#' returns are computed as the arithmetic difference between annualised
#' portfolio and benchmark returns:
#' \deqn{AAER=r_{a}-b_{a}}{AAER = ra - ba} the geometric annualized excess
#' returns are computed as the geometric difference between annualized
#' portfolio and benchmark returns:
#' \deqn{GAER=\frac{1+r_{a}}{1+b_{a}}-1}{GAER = (1 + ra) / (1 + ba) - 1}
#' In the case of multi-currency portfolio, the currency return, currency
#' surprise and forward premium should be specified. The multi-currency
#' arithmetic attribution is handled following Ankrim and Hensel (1992).
#' Currency returns are decomposed into the sum of the currency surprise and
#' the forward premium: \deqn{R_{ci} = R_{cei} + R_{fpi}}{Rci = Rcei + Rfpi}
#' where
#' \deqn{R_{cei} = \frac{S_{i}^{t+1} - F_{i}^{t+1}}{S_{i}^{t}}}
#' \deqn{R_{fpi} = \frac{F_{i}^{t+1}}{S_{i}^{t}} - 1}
#' \eqn{S_{i}^{t}}{Sit} - spot rate for asset \eqn{i} at time \eqn{t}
#' \eqn{F_{i}^{t}}{Fit} - forward rate for asset \eqn{i} at time \eqn{t}.
#' Excess returns are decomposed into the sum of allocation, selection and
#' interaction effects as in the standard Brinson model:
#' \deqn{R_{p}-R_{b}=\sum^{n}_{i=1}\left(A_{i}+S_{i}+I_{i}\right)}
#' However the allocation effect is computed taking into account currency
#' effects:
#' \deqn{A_{i}=(w_{pi}-w_{bi})\times (R_{bi} - R_{ci} - R_{l})}{Ai =
#' (wpi - wbi) * (Rbi - Rci - Rl)}
#' Benchmark returns adjusted to the currency:
#' \deqn{R_{l} = \sum^{n}_{i=1}w_{bi}\times(R_{bi}-R_{ci})}
#' The contribution from the currency is analogous to asset allocation:
#' \deqn{C_{i} = (w_{pi} - w_{bi}) \times (R_{cei} - e) + (w_{pfi} - w_{bfi})
#' \times (R_{fi} - e)}
#' where \deqn{e = \sum^{n}_{i=1}w_{bi}\times R_{cei}}
#' The final term, forward premium, is also analogous to the asset allocation:
#' \deqn{R_{fi} = (w_{pi} - w_{bi}) \times (R_{fpi} - d)}{Rfi = (wpi - wbi) *
#' (Rfpi - d)}
#' where \deqn{d = \sum^{n}_{i=1}w_{bi}\times R_{fpi}}
#' and \eqn{R_{fpi}} - forward premium
#' In general if the intent is to estimate statistical parameters, the
#' arithmetic excess return is preferred. However, due to the linking
#' challenges, it may be preferable to use geometric excess return if the
#' intent is to link and annualize excess returns.
#'
#' @aliases Attribution
#' @param Rp T x n xts, data frame or matrix of portfolio returns
#' @param wp vector, xts, data frame or matrix of portfolio weights
#' @param Rb T x n xts, data frame or matrix of benchmark returns
#' @param wb vector, xts, data frame or matrix of benchmark weights
#' @param method Used to select the priority between allocation and selection
#' effects in arithmetic attribution. May be any of: \itemize{ \item none -
#' present allocation, selection and interaction effects independently,
#' \item top.down - the priority is given to the sector allocation. Interaction
#' term is combined with the security selection effect, \item bottom.up - the
#' priority is given to the security selection. Interaction term is combined
#' with the sector allocation effect}
#' By default "none" is selected
#' @param wpf vector, xts, data frame or matrix with portfolio weights of
#' currency forward contracts
#' @param wbf vector, xts, data frame or matrix with benchmark weights of
#' currency forward contracts
#' @param S (T+1) x n xts, data frame or matrix with spot rates. The first date
#' should coincide with the first date of portfolio returns
#' @param F (T+1) x n xts, data frame or matrix with forward rates. The first
#' date should coincide with the first date of portfolio returns
#' @param Rpl xts, data frame or matrix of portfolio returns in local currency
#' @param Rbl xts, data frame or matrix of benchmark returns in local currency
#' @param Rbh xts, data frame or matrix of benchmark returns hedged into the
#' base currency
#' @param bf TRUE for Brinson and Fachler and FALSE for Brinson, Hood and
#' Beebower arithmetic attribution. By default Brinson, Hood and Beebower
#' attribution is selected
#' @param linking Used to select the linking method to present the multi-period
#' summary of arithmetic attribution effects. May be any of:
#' \itemize{\item carino - logarithmic linking coefficient method
#' \item menchero - Menchero's smoothing algorithm
#' \item grap - linking approach developed by GRAP
#' \item frongello - Frongello's linking method
#' \item davies.laker - Davies and Laker's linking method}
#' By default Carino linking is selected
#' @param geometric TRUE/FALSE, whether to use geometric or arithmetic excess
#' returns for the attribution analysis. By default arithmetic is selected
#' @param adjusted TRUE/FALSE, whether to show original or smoothed attribution
#' effects for each period. By default unadjusted attribution effects are
#' returned
#' @return returns a list with the following components: excess returns with
#' annualized excess returns over all periods, attribution effects (allocation,
#' selection and interaction)
#' @author Andrii Babii
#' @seealso \code{\link{Attribution.levels}},
#' \code{\link{Attribution.geometric}}
#' @references Ankrim, E. and Hensel, C. \emph{Multi-currency performance
#' attribution}. Russell Research Commentary. November 2002 \cr Bacon, C.
#' \emph{Practical Portfolio Performance Measurement and Attribution}. Wiley.
#' 2004. Chapter 5, 6, 8 \cr Christopherson, Jon A., Carino, David R., Ferson,
#' Wayne E. \emph{Portfolio Performance Measurement and Benchmarking}.
#' McGraw-Hill. 2009. Chapter 18-19 \cr Brinson, G. and Fachler, N. (1985)
#' \emph{Measuring non-US equity portfolio performance}. Journal of Portfolio
#' Management. Spring. p. 73 -76. \cr Gary P. Brinson, L. Randolph Hood, and
#' Gilbert L. Beebower, \emph{Determinants of Portfolio Performance}. Financial
#' Analysts Journal. vol. 42, no. 4, July/August 1986, p. 39-44 \cr
#' Karnosky, D. and Singer, B. \emph{Global asset management and performance
#' attribution. The Research Foundation of the Institute of Chartered Financial
#' Analysts}. February 1994. \cr
#' @keywords attribution
#' @examples
#'
#' data(attrib)
#' Attribution(Rp = attrib.returns[, 1:10], wp = attrib.weights[1, ], Rb = attrib.returns[, 11:20],
#' wb = attrib.weights[2, ], method = "top.down", linking = "carino")
#'
#' @export
Attribution <-
function (Rp, wp, Rb, wb,
wpf = NA, wbf = NA, S = NA, F = NA, Rpl = NA, Rbl = NA, Rbh = NA,
bf = FALSE,
method = c("none", "top.down", "bottom.up"),
linking = c("carino",
"menchero",
"grap",
"frongello",
"davies.laker"),
geometric = FALSE, adjusted = FALSE)
{ # @author Andrii Babii
# DESCRIPTION:
# Function to perform the attribution analysis.
# Inputs:
# Rp T x n xts, data frame or matrix of portfolio returns
# wp vector, xts, data frame or matrix of portfolio weights
# Rb T x n xts, data frame or matrix of benchmark returns
# wb vector, xts, data frame or matrix of benchmark weights
# wpf vector, xts, data frame or matrix with portfolio weights of
# currency forward contracts
# wbf vector, xts, data frame or matrix with benchmark weights of
# currency forward contracts
# S (T+1) x n xts, data frame or matrix with spot rates
# F (T+1) x n xts, data frame or matrix with forward rates
# Rpl xts, data frame or matrix of portfolio returns in local currency
# Rbl xts, data frame or matrix of benchmark returns in local currency
# Rbh xts, data frame or matrix of benchmark returns hedged into the
# base currency
# Outputs:
# This function returns the attribution effects with multi-period summary
# and annualized excess returns
# FUNCTION:
# Transform data to the xts objects
Rb = checkData(Rb)
Rp = checkData(Rp)
WP = wp # Save original weights in order to avoid double conversion later
WB = wb
wp = Weight.transform(wp, Rp)
wb = Weight.transform(wb, Rb)
if (nrow(wp) < nrow(Rp)){ # Rebalancing occurs next day
Rp = Rp[2:nrow(Rp)]
Rb = Rb[2:nrow(Rb)]
}
if (ncol(Rb) == 1){
Rb = matrix(rep(coredata(Rb), ncol(Rp)), nrow(Rp), ncol(Rp))
}
if (ncol(Rb) != ncol(Rp)){
stop("Please use benchmark xts that has columns with benchmarks for each
asset or one common benchmark for all assets")
}
method = method[1]
linking = linking[1]
currency = !(is.null(dim(wpf)) & is.null(dim(wbf)) &
is.null(dim(S)) & is.null(dim(F)) &
is.null(dim(Rpl)) & is.null(dim(Rpl)) &
is.null(dim(Rpl)))
if (geometric == FALSE & linking != "davies.laker"){
# The function makes all computations for the arithmetic attribution
# case (except for Davies and Laker linking)
# Compute attribution effects (Brinson, Hood and Beebower model)
# If portfolio is single-currency
if (!currency){
Rc = 0
L = 0
} else{ # If multi-currency portfolio
S = checkData(S)
F = checkData(F)
wpf = Weight.transform(wpf, Rp)
wbf = Weight.transform(wbf, Rb)
Rc = lag(S, -1)[1:nrow(Rp), ] / S[1:nrow(Rp), ] - 1
Rd = lag(F, -1)[1:nrow(Rp), ] / S[1:nrow(Rp), ] - 1
Re = Rc - coredata(Rd)
Rl = Rb - coredata(Rc)
Rk = Rp - coredata(Rc)
Rfp = Re / (1 + Rd)
E = reclass(matrix(rep(rowSums(Re * coredata(wb)), ncol(Rb)), nrow(Rb),
ncol(Rb)), Rp)
L = reclass(matrix(rep(rowSums(Rl * coredata(wb)), ncol(Rb)), nrow(Rb),
ncol(Rb)), Rp)
D = reclass(matrix(rep(rowSums(Rd * coredata(wb)), ncol(Rb)), nrow(Rb),
ncol(Rb)), Rp)
# Contribution to currency
Cc = (wp - wb) * (Re - E) + (wpf - wbf) * (Rfp - E)
# Forward premium
Df = (wp - wb) * (Rd - D)
Cc = cbind(Cc, rowSums(Cc))
Df = cbind(Df, rowSums(Df))
colnames(Cc) = c(colnames(S), "Total")
colnames(Df) = colnames(Cc)
}
# Get total portfolio returns
if (is.vector(WP) & is.vector(WB)){
rp = Return.portfolio(Rp, WP, geometric = FALSE)
rb = Return.portfolio(Rb, WB, geometric = FALSE)
} else{
rp = Return.rebalancing(Rp, WP, geometric = FALSE)
rb = Return.rebalancing(Rb, WB, geometric = FALSE)
}
names(rp) = "Total"
names(rb) = "Total"
# Get individual attribution effects
if (bf == TRUE){ # Brinson and Fachler (1985) allocation effect
allocation = coredata(wp - wb) * (Rb - coredata(Rc) - coredata(L) -
rep(rb, ncol(Rb)))
} else{ # Brinson, Hood and Beebower (1986) allocation effect
allocation = coredata(wp - wb) * (Rb - coredata(Rc) - coredata(L))
}
selection = (Rp - coredata(Rb)) * wb
interaction = (wp - wb) * (Rp - coredata(Rb))
# Get total attribution effects
n = ncol(allocation) # number of segments
allocation = cbind(allocation, rowSums(allocation))
names(allocation)[n + 1] = "Total"
selection = cbind(selection, rowSums(selection))
names(selection)[n + 1] = "Total"
interaction = cbind(interaction, rowSums(interaction))
names(interaction)[n + 1] = "Total"
# Adjust attribution effects using one of linking methods if there are
# mutliple periods
if (nrow(allocation) > 1){
if (linking == "carino"){
allocation = Carino(rp, rb, allocation, adjusted)
selection = Carino(rp, rb, selection, adjusted)
interaction = Carino(rp, rb, interaction, adjusted)
}
if (linking == "menchero"){
allocation = Menchero(rp, rb, allocation, adjusted)
selection = Menchero(rp, rb, selection, adjusted)
interaction = Menchero(rp, rb, interaction, adjusted)
}
if (linking == "grap"){
allocation = Grap(rp, rb, allocation, adjusted)
selection = Grap(rp, rb, selection, adjusted)
interaction = Grap(rp, rb, interaction, adjusted)
}
if (linking == "frongello"){
allocation = Frongello(rp, rb, allocation, adjusted)
selection = Frongello(rp, rb, selection, adjusted)
interaction = Frongello(rp, rb, interaction, adjusted)
}
}
# Arithmetic excess returns + annualized arithmetic excess returns
excess.returns = rp - coredata(rb)
if (nrow(rp) > 1){
er = Return.annualized.excess(rp, rb, geometric = FALSE)
excess.returns = rbind(as.matrix(excess.returns), er)
}
colnames(excess.returns) = "Arithmetic"
# Select the appropriate result corresponding to the chosen method
result = list()
result[[1]] = excess.returns
result[[2]] = allocation
result[[3]] = selection
if (method == "top.down"){ # Top-down attribution
result[[3]] = result[[3]] + interaction
}
if (method == "bottom.up"){ # Bottom-up attribution
result[[2]] = result[[2]] + interaction
}
if (method == "none"){
result[[4]] = interaction
}
} else{ # The function takes output of the corresponding function
# (Attribution.geometric or DaviesLaker)
if (geometric == TRUE){
attrib = Attribution.geometric(Rp, WP, Rb, WB)
}
if (linking == "davies.laker"){
attrib = DaviesLaker(Rp, WP, Rb, WB)
}
result = attrib
}
# Label the output
if ((method == "none" & geometric == FALSE) | linking == "davies.laker"){
names(result) = c("Excess returns", "Allocation", "Selection",
"Interaction")
} else{
names(result) = c("Excess returns", "Allocation", "Selection")
}
# If multi-currency portfolio
if (currency){
result[[length(result) + 1]] = Cc
result[[length(result) + 1]] = Df
names(result)[(length(result)-1):length(result)] =
c("Currency management", "Forward Premium")
}
return(result)
}
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