R/CAPM.beta.R
In braverock/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{SFM.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{SFM.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
#' @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
#' @param \dots Other parameters like max.it or bb specific to lmrobdetMM regression.
#' @param digits (Optional): Number of digits to round the results to. Defaults to 3.
#' @param benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
#' @param method (Optional): string representing linear regression model, "LS" for Least Squares
#' and "Robust" for robust. Defaults to "LS
#' @param family (Optional):
#' If method == "Robust":
#' This is a string specifying the name of the family of loss function
#' to be used (current valid options are "bisquare", "opt" and "mopt").
#' Incomplete entries will be matched to the current valid options.
#' Defaults to "mopt".
#' Else: the parameter is ignored
#' @param warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#'
#' @author Dhairya Jain, Peter Carl
#' @seealso \code{\link{BetaCoVariance}} \code{\link{SFM.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)
#' SFM.beta(managers[, "HAM1"], managers[, "SP500 TR"], Rf = managers[, "US 3m TR"])
#' SFM.beta(managers[,1:3], managers[,8:10], Rf=.035/12)
#' SFM.beta(managers[,1], managers[,8:10], Rf=.035/12, benchmarkCols=FALSE)
#'
#' betas <- SFM.beta(managers[,1:6],
#' managers[,8:10],
#' Rf=.035/12, method="Robust",
#' family="opt", bb=0.25, max.it=200, digits=4)
#' betas["HAM1", ]
#' betas[, "Beta : SP500 TR"]
#'
#' SFM.beta.bull(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"])
#' SFM.beta.bull(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' method="Robust")
#'
#' SFM.beta.bear(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"])
#' SFM.beta.bear(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' method="Robust")
#'
#' TimingRatio(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"])
#' TimingRatio(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' method="Robust", family="mopt")
#'
#' chart.Regression(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' fit="conditional",
#' main="Conditional Beta")
#'
#'
#' @rdname SFM.beta
#' @export SFM.beta CAPM.beta
SFM.beta <- CAPM.beta <- function (Ra, Rb, Rf = 0, ..., digits=3, benchmarkCols=T, method="LS", family="mopt", warning=T)
{ # @author Peter Carl, Dhairya Jain
# 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.
# digits (Optional): Number of digits to round the results to. Defaults to 3.
# benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
# method (Optional): string representing linear regression model, "LS" for Least Squares
# and "Robust" for robust. Defaults to "LS
# family (Optional):
# If method == "Robust":
# This is a string specifying the name of the family of loss function
# to be used (current valid options are "bisquare", "opt" and "mopt").
# Incomplete entries will be matched to the current valid options.
# Defaults to "mopt".
# Else: the parameter is ignored
# warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#
# Output:
# Market Beta
# FUNCTION:
# .coefficients fails if method is "Both"
if (warning && method == "Both"){
warning("Using 'Both' while using SFM.beta will lead to ill-formatted output");
}
# Get the NCOL and colnames from Ra, and Rb
Ra.ncols <- NCOL(Ra);
Rb.ncols <- NCOL(Rb);
Ra.colnames <- colnames(Ra);
Rb.colnames <- colnames(Rb)
# Get the excess returns of Ra, Rb over Rf
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
# Get the result matrix
result.all <- getResults(xRa=xRa, xRb=xRb,
Ra.ncols=Ra.ncols, Rb.ncols=Rb.ncols,
method = method, family = family, ...);
# Process the results and return them
return (processResults(result.all, "beta", Ra.ncols, Rb.ncols,
Ra.colnames, Rb.colnames, method, "Beta",
digits, benchmarkCols))
}
#' @rdname SFM.beta
#' @export SFM.beta.bull CAPM.beta.bull
SFM.beta.bull <- CAPM.beta.bull <-
function (Ra, Rb, Rf = 0, ..., digits=3, benchmarkCols=T, method="LS", family="mopt")
{ # @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.
# digits (Optional): Number of digits to round the results to. Defaults to 3.
# benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
# method (Optional): string representing linear regression model, "LS" for Least Squares
# and "Robust" for robust. Defaults to "LS
# family (Optional):
# If method == "Robust":
# This is a string specifying the name of the family of loss function
# to be used (current valid options are "bisquare", "opt" and "mopt").
# Incomplete entries will be matched to the current valid options.
# Defaults to "mopt".
# Else: the parameter is ignored
# warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#
# Output:
# Bull market beta
# FUNCTION:
if (!is.null(list(...)$subset)){
stop("Can't pass subset in the optional parameters")
}
#
return (SFM.beta(Ra, Rb, Rf, subset="Bull", digits=digits, benchmarkCols=benchmarkCols, method=method, family=family, ...))
}
#' @rdname SFM.beta
#' @export SFM.beta.bear CAPM.beta.bear
SFM.beta.bear <- CAPM.beta.bear <-
function (Ra, Rb, Rf = 0, ..., digits=3, benchmarkCols=T, method="LS", family="mopt")
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a conditional CAPM beta for down markets
# Inputs:
# Ra: time series of returns for the asset being tested
# Rb: time series of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a time series
# of the same length as Ra and Rb.
# digits (Optional): Number of digits to round the results to. Defaults to 3.
# benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
# method (Optional): string representing linear regression model, "LS" for Least Squares
# and "Robust" for robust. Defaults to "LS
# family (Optional):
# If method == "Robust":
# This is a string specifying the name of the family of loss function
# to be used (current valid options are "bisquare", "opt" and "mopt").
# Incomplete entries will be matched to the current valid options.
# Defaults to "mopt".
# Else: the parameter is ignored
# warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#
# Output:
# Bear market beta
# FUNCTION:
if (!is.null(list(...)$subset)){
stop("Can't pass subset in the optional parameters")
}
return (SFM.beta(Ra, Rb, Rf, subset="Bear", digits=digits, benchmarkCols=benchmarkCols, method=method, family=family, ...))
}
#' @rdname SFM.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 = SFM.beta.bull(Ra, Rb, Rf = Rf, ...)
beta.bear = SFM.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)
}
}
###############################################################################
# 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
#
# $Id$
#
###############################################################################
braverock/PerformanceAnalytics documentation built on Feb. 16, 2024, 5:37 a.m.
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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{SFM.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{SFM.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
#' @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
#' @param \dots Other parameters like max.it or bb specific to lmrobdetMM regression.
#' @param digits (Optional): Number of digits to round the results to. Defaults to 3.
#' @param benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
#' @param method (Optional): string representing linear regression model, "LS" for Least Squares
#' and "Robust" for robust. Defaults to "LS
#' @param family (Optional):
#' If method == "Robust":
#' This is a string specifying the name of the family of loss function
#' to be used (current valid options are "bisquare", "opt" and "mopt").
#' Incomplete entries will be matched to the current valid options.
#' Defaults to "mopt".
#' Else: the parameter is ignored
#' @param warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#'
#' @author Dhairya Jain, Peter Carl
#' @seealso \code{\link{BetaCoVariance}} \code{\link{SFM.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)
#' SFM.beta(managers[, "HAM1"], managers[, "SP500 TR"], Rf = managers[, "US 3m TR"])
#' SFM.beta(managers[,1:3], managers[,8:10], Rf=.035/12)
#' SFM.beta(managers[,1], managers[,8:10], Rf=.035/12, benchmarkCols=FALSE)
#'
#' betas <- SFM.beta(managers[,1:6],
#' managers[,8:10],
#' Rf=.035/12, method="Robust",
#' family="opt", bb=0.25, max.it=200, digits=4)
#' betas["HAM1", ]
#' betas[, "Beta : SP500 TR"]
#'
#' SFM.beta.bull(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"])
#' SFM.beta.bull(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' method="Robust")
#'
#' SFM.beta.bear(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"])
#' SFM.beta.bear(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' method="Robust")
#'
#' TimingRatio(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"])
#' TimingRatio(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' method="Robust", family="mopt")
#'
#' chart.Regression(managers[, "HAM2"],
#' managers[, "SP500 TR"],
#' Rf = managers[, "US 3m TR"],
#' fit="conditional",
#' main="Conditional Beta")
#'
#'
#' @rdname SFM.beta
#' @export SFM.beta CAPM.beta
SFM.beta <- CAPM.beta <- function (Ra, Rb, Rf = 0, ..., digits=3, benchmarkCols=T, method="LS", family="mopt", warning=T)
{ # @author Peter Carl, Dhairya Jain
# 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.
# digits (Optional): Number of digits to round the results to. Defaults to 3.
# benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
# method (Optional): string representing linear regression model, "LS" for Least Squares
# and "Robust" for robust. Defaults to "LS
# family (Optional):
# If method == "Robust":
# This is a string specifying the name of the family of loss function
# to be used (current valid options are "bisquare", "opt" and "mopt").
# Incomplete entries will be matched to the current valid options.
# Defaults to "mopt".
# Else: the parameter is ignored
# warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#
# Output:
# Market Beta
# FUNCTION:
# .coefficients fails if method is "Both"
if (warning && method == "Both"){
warning("Using 'Both' while using SFM.beta will lead to ill-formatted output");
}
# Get the NCOL and colnames from Ra, and Rb
Ra.ncols <- NCOL(Ra);
Rb.ncols <- NCOL(Rb);
Ra.colnames <- colnames(Ra);
Rb.colnames <- colnames(Rb)
# Get the excess returns of Ra, Rb over Rf
xRa = Return.excess(Ra, Rf)
xRb = Return.excess(Rb, Rf)
# Get the result matrix
result.all <- getResults(xRa=xRa, xRb=xRb,
Ra.ncols=Ra.ncols, Rb.ncols=Rb.ncols,
method = method, family = family, ...);
# Process the results and return them
return (processResults(result.all, "beta", Ra.ncols, Rb.ncols,
Ra.colnames, Rb.colnames, method, "Beta",
digits, benchmarkCols))
}
#' @rdname SFM.beta
#' @export SFM.beta.bull CAPM.beta.bull
SFM.beta.bull <- CAPM.beta.bull <-
function (Ra, Rb, Rf = 0, ..., digits=3, benchmarkCols=T, method="LS", family="mopt")
{ # @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.
# digits (Optional): Number of digits to round the results to. Defaults to 3.
# benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
# method (Optional): string representing linear regression model, "LS" for Least Squares
# and "Robust" for robust. Defaults to "LS
# family (Optional):
# If method == "Robust":
# This is a string specifying the name of the family of loss function
# to be used (current valid options are "bisquare", "opt" and "mopt").
# Incomplete entries will be matched to the current valid options.
# Defaults to "mopt".
# Else: the parameter is ignored
# warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#
# Output:
# Bull market beta
# FUNCTION:
if (!is.null(list(...)$subset)){
stop("Can't pass subset in the optional parameters")
}
#
return (SFM.beta(Ra, Rb, Rf, subset="Bull", digits=digits, benchmarkCols=benchmarkCols, method=method, family=family, ...))
}
#' @rdname SFM.beta
#' @export SFM.beta.bear CAPM.beta.bear
SFM.beta.bear <- CAPM.beta.bear <-
function (Ra, Rb, Rf = 0, ..., digits=3, benchmarkCols=T, method="LS", family="mopt")
{ # @author Peter Carl
# DESCRIPTION:
# This is a wrapper for calculating a conditional CAPM beta for down markets
# Inputs:
# Ra: time series of returns for the asset being tested
# Rb: time series of returns for the benchmark the asset is being gauged against
# Rf: risk free rate in the same periodicity as the returns. May be a time series
# of the same length as Ra and Rb.
# digits (Optional): Number of digits to round the results to. Defaults to 3.
# benchmarkCols (Optional): Boolean to show the benchmarks as columns. Defaults to TRUE.
# method (Optional): string representing linear regression model, "LS" for Least Squares
# and "Robust" for robust. Defaults to "LS
# family (Optional):
# If method == "Robust":
# This is a string specifying the name of the family of loss function
# to be used (current valid options are "bisquare", "opt" and "mopt").
# Incomplete entries will be matched to the current valid options.
# Defaults to "mopt".
# Else: the parameter is ignored
# warning (Optional): Boolean to show warnings or not. Defaults to TRUE.
#
# Output:
# Bear market beta
# FUNCTION:
if (!is.null(list(...)$subset)){
stop("Can't pass subset in the optional parameters")
}
return (SFM.beta(Ra, Rb, Rf, subset="Bear", digits=digits, benchmarkCols=benchmarkCols, method=method, family=family, ...))
}
#' @rdname SFM.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 = SFM.beta.bull(Ra, Rb, Rf = Rf, ...)
beta.bear = SFM.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)
}
}
###############################################################################
# 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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