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R/CAPM.beta.R
In PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis
#' calculate single factor model (CAPM) beta
#'
#' The single factor model or CAPM Beta is the beta of an asset to the variance
#' and covariance of an initial portfolio. Used to determine diversification potential.
#'
#' This function uses a linear intercept model to achieve the same results as
#' the symbolic model used by \code{\link{BetaCoVariance}}
#'
#' \deqn{\beta_{a,b}=\frac{CoV_{a,b}}{\sigma_{b}}=\frac{\sum((R_{a}-\bar{R_{a}})(R_{b}-\bar{R_{b}}))}{\sum(R_{b}-\bar{R_{b}})^{2}}}{beta
#' = cov(Ra,Rb)/var(R)}
#'
#' Ruppert(2004) reports that this equation will give the estimated slope of
#' the linear regression of \eqn{R_{a}}{Ra} on \eqn{R_{b}}{Rb} and that this
#' slope can be used to determine the risk premium or excess expected return
#' (see Eq. 7.9 and 7.10, p. 230-231).
#'
#' Two other functions apply the same notion of best fit to positive and
#' negative market returns, separately. The \code{CAPM.beta.bull} is a
#' regression for only positive market returns, which can be used to understand
#' the behavior of the asset or portfolio in positive or 'bull' markets.
#' Alternatively, \code{CAPM.beta.bear} provides the calculation on negative
#' market returns.
#'
#' The \code{TimingRatio} may help assess whether the manager is a good timer
#' of asset allocation decisions. The ratio, which is calculated as
#' \deqn{TimingRatio =\frac{\beta^{+}}{\beta^{-}}}{Timing Ratio = beta+/beta-}
#' is best when greater than one in a rising market and less than one in a
#' falling market.
#'
#' While the classical CAPM has been almost completely discredited by the
#' literature, it is an example of a simple single factor model,
#' comparing an asset to any arbitrary benchmark.
#'
#' @aliases CAPM.beta CAPM.beta.bull CAPM.beta.bear TimingRatio SFM.beta
#' @param Ra an xts, vector, matrix, data frame, timeSeries or zoo object of
#' asset returns
#' @param Rb return vector of the benchmark asset
#' @param Rf risk free rate, in same period as your returns
#' @author Peter Carl
#' @seealso \code{\link{BetaCoVariance}} \code{\link{CAPM.alpha}}
#' \code{\link{CAPM.utils}}
#' @references Sharpe, W.F. Capital Asset Prices: A theory of market
#' equilibrium under conditions of risk. \emph{Journal of finance}, vol 19,
#' 1964, 425-442. \cr Ruppert, David. \emph{Statistics and Finance, an
#' Introduction}. Springer. 2004. \cr Bacon, Carl. \emph{Practical portfolio
#' performance measurement and attribution}. Wiley. 2004. \cr
###keywords ts multivariate distribution models
#' @examples
#'
#' data(managers)
#' CAPM.alpha(managers[,1,drop=FALSE],
#' managers[,8,drop=FALSE],
#' Rf=.035/12)
#' CAPM.alpha(managers[,1,drop=FALSE],
#' managers[,8,drop=FALSE],
#' Rf = managers[,10,drop=FALSE])
#' CAPM.alpha(managers[,1:6],
#' managers[,8,drop=FALSE],
#' Rf=.035/12)
#' CAPM.alpha(managers[,1:6],
#' managers[,8,drop=FALSE],
#' Rf = managers[,10,drop=FALSE])
#' CAPM.alpha(managers[,1:6],
#' managers[,8:7,drop=FALSE],
#' Rf=.035/12)
#' CAPM.alpha(managers[,1:6],
#' managers[,8:7,drop=FALSE],
#' Rf = managers[,10,drop=FALSE])
#' CAPM.beta(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' CAPM.beta.bull(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' CAPM.beta.bear(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' TimingRatio(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' chart.Regression(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE],
#' fit="conditional",
#' main="Conditional Beta")
#'
#' @rdname CAPM.beta
#' @export CAPM.beta SFM.beta
CAPM.beta <- SFM.beta <- function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a CAPM beta.
# Inputs:
# Ra: vector of returns for the asset being tested
# Rb: vector of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a vector
# of the same length as x and y.
# Output:
#
# FUNCTION:
Ra = checkData(Ra)
Rb = checkData(Rb)
if(!is.null(dim(Rf)))
Rf = checkData(Rf)
Ra.ncols = NCOL(Ra)
Rb.ncols = NCOL(Rb)
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
pairs = expand.grid(1:Ra.ncols, 1:Rb.ncols)
result = apply(pairs, 1, FUN = function(n, xRa, xRb)
.beta(xRa[,n[1]], xRb[,n[2]]), xRa = xRa, xRb = xRb)
if(length(result) ==1)
return(result)
else {
dim(result) = c(Ra.ncols, Rb.ncols)
colnames(result) = paste("Beta:", colnames(Rb))
rownames(result) = colnames(Ra)
return(t(result))
}
}
#' @rdname CAPM.beta
#' @export
CAPM.beta.bull <-
function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a conditional CAPM beta for up markets.
# Inputs:
# Ra: time series of returns for the asset being tested
# Rb: time series of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a time series
# of the same length as x and y.
# Output:
# Bear market beta
# FUNCTION:
Ra = checkData(Ra)
Rb = checkData(Rb)
if(!is.null(dim(Rf)))
Rf = checkData(Rf)
Ra.ncols = NCOL(Ra)
Rb.ncols = NCOL(Rb)
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
pairs = expand.grid(1:Ra.ncols, 1:Rb.ncols)
# .beta fails if subset contains no positive values, all(xRb <= 0) is true
if (all(xRb <= 0)) {
message("Function CAPM.beta.bull: Cannot perform lm. No positive Rb values (no bull periods).")
return(NA)
}
result = apply(pairs, 1, FUN = function(n, xRa, xRb)
.beta(xRa[,n[1]], xRb[,n[2]], xRb[,n[2]] > 0), xRa = xRa, xRb = xRb)
if(length(result) ==1)
return(result)
else {
dim(result) = c(Ra.ncols, Rb.ncols)
colnames(result) = paste("Bull Beta:", colnames(Rb))
rownames(result) = colnames(Ra)
return(t(result))
}
}
#' @rdname CAPM.beta
#' @export
CAPM.beta.bear <-
function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a conditional CAPM beta for down markets
# Inputs:
# Ra: time series of returns for the asset being tested
# Rb: time series of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a time series
# of the same length as Ra and Rb.
# Output:
# Bear market beta
# FUNCTION:
Ra = checkData(Ra)
Rb = checkData(Rb)
if(!is.null(dim(Rf)))
Rf = checkData(Rf)
Ra.ncols = NCOL(Ra)
Rb.ncols = NCOL(Rb)
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
pairs = expand.grid(1:Ra.ncols, 1:Rb.ncols)
# .beta fails if subset contains no negative values, all(xRb >= 0) is true
if (all(xRb >= 0)) {
message("Function CAPM.beta.bear: Cannot perform lm. No negative Rb values (no bear periods).")
return(NA)
}
result = apply(pairs, 1, FUN = function(n, xRa, xRb)
.beta(xRa[,n[1]], xRb[,n[2]], xRb[,n[2]] < 0), xRa = xRa, xRb = xRb)
if(length(result) ==1)
return(result)
else {
dim(result) = c(Ra.ncols, Rb.ncols)
colnames(result) = paste("Bear Beta:", colnames(Rb))
rownames(result) = colnames(Ra)
return(t(result))
}
}
#' @rdname CAPM.beta
#' @export
TimingRatio <-
function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This function calculates the ratio of the two conditional CAPM betas (up and down).
beta.bull = CAPM.beta.bull(Ra, Rb, Rf = Rf)
beta.bear = CAPM.beta.bear(Ra, Rb, Rf = Rf)
result = beta.bull/beta.bear
if(length(result) ==1)
return(result)
else {
names = colnames(Rb)
rownames(result) = paste("Timing Ratio:", names)
return(result)
}
}
.beta <- function (xRa, xRb, subset) {
# subset is assumed to be a logical vector
if(missing(subset))
subset <- TRUE
# check columns
if(NCOL(xRa)!=1L || NCOL(xRb)!=1L || NCOL(subset)!=1L)
stop("all arguments must have only one column")
# merge, drop NA
merged <- as.data.frame(na.omit(cbind(xRa, xRb, subset)))
# return NA if no non-NA values
if(NROW(merged)==0)
return(NA)
# add column names and convert subset back to logical
colnames(merged) <- c("xRa","xRb","subset")
merged$subset <- as.logical(merged$subset)
# calculate beta
model.lm = lm(xRa ~ xRb, data=merged, subset=merged$subset)
beta = coef(model.lm)[[2]]
beta
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# Copyright (c) 2004-2020 Peter Carl and Brian G. Peterson
#
# This R package is distributed under the terms of the GNU Public License (GPL)
# for full details see the file COPYING
#
# $Id$
#
###############################################################################
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#' calculate single factor model (CAPM) beta
#'
#' The single factor model or CAPM Beta is the beta of an asset to the variance
#' and covariance of an initial portfolio. Used to determine diversification potential.
#'
#' This function uses a linear intercept model to achieve the same results as
#' the symbolic model used by \code{\link{BetaCoVariance}}
#'
#' \deqn{\beta_{a,b}=\frac{CoV_{a,b}}{\sigma_{b}}=\frac{\sum((R_{a}-\bar{R_{a}})(R_{b}-\bar{R_{b}}))}{\sum(R_{b}-\bar{R_{b}})^{2}}}{beta
#' = cov(Ra,Rb)/var(R)}
#'
#' Ruppert(2004) reports that this equation will give the estimated slope of
#' the linear regression of \eqn{R_{a}}{Ra} on \eqn{R_{b}}{Rb} and that this
#' slope can be used to determine the risk premium or excess expected return
#' (see Eq. 7.9 and 7.10, p. 230-231).
#'
#' Two other functions apply the same notion of best fit to positive and
#' negative market returns, separately. The \code{CAPM.beta.bull} is a
#' regression for only positive market returns, which can be used to understand
#' the behavior of the asset or portfolio in positive or 'bull' markets.
#' Alternatively, \code{CAPM.beta.bear} provides the calculation on negative
#' market returns.
#'
#' The \code{TimingRatio} may help assess whether the manager is a good timer
#' of asset allocation decisions. The ratio, which is calculated as
#' \deqn{TimingRatio =\frac{\beta^{+}}{\beta^{-}}}{Timing Ratio = beta+/beta-}
#' is best when greater than one in a rising market and less than one in a
#' falling market.
#'
#' While the classical CAPM has been almost completely discredited by the
#' literature, it is an example of a simple single factor model,
#' comparing an asset to any arbitrary benchmark.
#'
#' @aliases CAPM.beta CAPM.beta.bull CAPM.beta.bear TimingRatio SFM.beta
#' @param Ra an xts, vector, matrix, data frame, timeSeries or zoo object of
#' asset returns
#' @param Rb return vector of the benchmark asset
#' @param Rf risk free rate, in same period as your returns
#' @author Peter Carl
#' @seealso \code{\link{BetaCoVariance}} \code{\link{CAPM.alpha}}
#' \code{\link{CAPM.utils}}
#' @references Sharpe, W.F. Capital Asset Prices: A theory of market
#' equilibrium under conditions of risk. \emph{Journal of finance}, vol 19,
#' 1964, 425-442. \cr Ruppert, David. \emph{Statistics and Finance, an
#' Introduction}. Springer. 2004. \cr Bacon, Carl. \emph{Practical portfolio
#' performance measurement and attribution}. Wiley. 2004. \cr
###keywords ts multivariate distribution models
#' @examples
#'
#' data(managers)
#' CAPM.alpha(managers[,1,drop=FALSE],
#' managers[,8,drop=FALSE],
#' Rf=.035/12)
#' CAPM.alpha(managers[,1,drop=FALSE],
#' managers[,8,drop=FALSE],
#' Rf = managers[,10,drop=FALSE])
#' CAPM.alpha(managers[,1:6],
#' managers[,8,drop=FALSE],
#' Rf=.035/12)
#' CAPM.alpha(managers[,1:6],
#' managers[,8,drop=FALSE],
#' Rf = managers[,10,drop=FALSE])
#' CAPM.alpha(managers[,1:6],
#' managers[,8:7,drop=FALSE],
#' Rf=.035/12)
#' CAPM.alpha(managers[,1:6],
#' managers[,8:7,drop=FALSE],
#' Rf = managers[,10,drop=FALSE])
#' CAPM.beta(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' CAPM.beta.bull(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' CAPM.beta.bear(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' TimingRatio(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE])
#' chart.Regression(managers[, "HAM2", drop=FALSE],
#' managers[, "SP500 TR", drop=FALSE],
#' Rf = managers[, "US 3m TR", drop=FALSE],
#' fit="conditional",
#' main="Conditional Beta")
#'
#' @rdname CAPM.beta
#' @export CAPM.beta SFM.beta
CAPM.beta <- SFM.beta <- function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a CAPM beta.
# Inputs:
# Ra: vector of returns for the asset being tested
# Rb: vector of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a vector
# of the same length as x and y.
# Output:
#
# FUNCTION:
Ra = checkData(Ra)
Rb = checkData(Rb)
if(!is.null(dim(Rf)))
Rf = checkData(Rf)
Ra.ncols = NCOL(Ra)
Rb.ncols = NCOL(Rb)
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
pairs = expand.grid(1:Ra.ncols, 1:Rb.ncols)
result = apply(pairs, 1, FUN = function(n, xRa, xRb)
.beta(xRa[,n[1]], xRb[,n[2]]), xRa = xRa, xRb = xRb)
if(length(result) ==1)
return(result)
else {
dim(result) = c(Ra.ncols, Rb.ncols)
colnames(result) = paste("Beta:", colnames(Rb))
rownames(result) = colnames(Ra)
return(t(result))
}
}
#' @rdname CAPM.beta
#' @export
CAPM.beta.bull <-
function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a conditional CAPM beta for up markets.
# Inputs:
# Ra: time series of returns for the asset being tested
# Rb: time series of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a time series
# of the same length as x and y.
# Output:
# Bear market beta
# FUNCTION:
Ra = checkData(Ra)
Rb = checkData(Rb)
if(!is.null(dim(Rf)))
Rf = checkData(Rf)
Ra.ncols = NCOL(Ra)
Rb.ncols = NCOL(Rb)
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
pairs = expand.grid(1:Ra.ncols, 1:Rb.ncols)
# .beta fails if subset contains no positive values, all(xRb <= 0) is true
if (all(xRb <= 0)) {
message("Function CAPM.beta.bull: Cannot perform lm. No positive Rb values (no bull periods).")
return(NA)
}
result = apply(pairs, 1, FUN = function(n, xRa, xRb)
.beta(xRa[,n[1]], xRb[,n[2]], xRb[,n[2]] > 0), xRa = xRa, xRb = xRb)
if(length(result) ==1)
return(result)
else {
dim(result) = c(Ra.ncols, Rb.ncols)
colnames(result) = paste("Bull Beta:", colnames(Rb))
rownames(result) = colnames(Ra)
return(t(result))
}
}
#' @rdname CAPM.beta
#' @export
CAPM.beta.bear <-
function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a conditional CAPM beta for down markets
# Inputs:
# Ra: time series of returns for the asset being tested
# Rb: time series of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a time series
# of the same length as Ra and Rb.
# Output:
# Bear market beta
# FUNCTION:
Ra = checkData(Ra)
Rb = checkData(Rb)
if(!is.null(dim(Rf)))
Rf = checkData(Rf)
Ra.ncols = NCOL(Ra)
Rb.ncols = NCOL(Rb)
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
pairs = expand.grid(1:Ra.ncols, 1:Rb.ncols)
# .beta fails if subset contains no negative values, all(xRb >= 0) is true
if (all(xRb >= 0)) {
message("Function CAPM.beta.bear: Cannot perform lm. No negative Rb values (no bear periods).")
return(NA)
}
result = apply(pairs, 1, FUN = function(n, xRa, xRb)
.beta(xRa[,n[1]], xRb[,n[2]], xRb[,n[2]] < 0), xRa = xRa, xRb = xRb)
if(length(result) ==1)
return(result)
else {
dim(result) = c(Ra.ncols, Rb.ncols)
colnames(result) = paste("Bear Beta:", colnames(Rb))
rownames(result) = colnames(Ra)
return(t(result))
}
}
#' @rdname CAPM.beta
#' @export
TimingRatio <-
function (Ra, Rb, Rf = 0)
{ # @author Peter Carl
# DESCRIPTION:
# This function calculates the ratio of the two conditional CAPM betas (up and down).
beta.bull = CAPM.beta.bull(Ra, Rb, Rf = Rf)
beta.bear = CAPM.beta.bear(Ra, Rb, Rf = Rf)
result = beta.bull/beta.bear
if(length(result) ==1)
return(result)
else {
names = colnames(Rb)
rownames(result) = paste("Timing Ratio:", names)
return(result)
}
}
.beta <- function (xRa, xRb, subset) {
# subset is assumed to be a logical vector
if(missing(subset))
subset <- TRUE
# check columns
if(NCOL(xRa)!=1L || NCOL(xRb)!=1L || NCOL(subset)!=1L)
stop("all arguments must have only one column")
# merge, drop NA
merged <- as.data.frame(na.omit(cbind(xRa, xRb, subset)))
# return NA if no non-NA values
if(NROW(merged)==0)
return(NA)
# add column names and convert subset back to logical
colnames(merged) <- c("xRa","xRb","subset")
merged$subset <- as.logical(merged$subset)
# calculate beta
model.lm = lm(xRa ~ xRb, data=merged, subset=merged$subset)
beta = coef(model.lm)[[2]]
beta
}
###############################################################################
# R (http://r-project.org/) Econometrics for Performance and Risk Analysis
#
# Copyright (c) 2004-2020 Peter Carl and Brian G. Peterson
#
# This R package is distributed under the terms of the GNU Public License (GPL)
# for full details see the file COPYING
#
# $Id$
#
###############################################################################
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