[Returnanalytics-commits] r2212 - in pkg/PerformanceAnalytics: . R man
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Fri Jul 27 18:01:01 CEST 2012
Author: matthieu_lestel
Date: 2012-07-27 18:01:01 +0200 (Fri, 27 Jul 2012)
New Revision: 2212
Added:
pkg/PerformanceAnalytics/R/FamaBeta.R
pkg/PerformanceAnalytics/R/Selectivity.R
pkg/PerformanceAnalytics/man/FamaBeta.Rd
pkg/PerformanceAnalytics/man/Selectivity.Rd
Modified:
pkg/PerformanceAnalytics/NAMESPACE
pkg/PerformanceAnalytics/R/AppraisalRatio.R
pkg/PerformanceAnalytics/R/ProspectRatio.R
pkg/PerformanceAnalytics/man/AppraisalRatio.Rd
pkg/PerformanceAnalytics/man/DownsideDeviation.Rd
pkg/PerformanceAnalytics/man/ProspectRatio.Rd
Log:
modified jensen's alpha, alternative modified jensen's alpha, selectivity and fama beta with examples and documentation
Modified: pkg/PerformanceAnalytics/NAMESPACE
===================================================================
--- pkg/PerformanceAnalytics/NAMESPACE 2012-07-27 15:24:37 UTC (rev 2211)
+++ pkg/PerformanceAnalytics/NAMESPACE 2012-07-27 16:01:01 UTC (rev 2212)
@@ -83,6 +83,7 @@
Return.relative,
sd.annualized,
sd.multiperiod,
+ Selectivity,
SemiDeviation,
SemiVariance,
SharpeRatio,
Modified: pkg/PerformanceAnalytics/R/AppraisalRatio.R
===================================================================
--- pkg/PerformanceAnalytics/R/AppraisalRatio.R 2012-07-27 15:24:37 UTC (rev 2211)
+++ pkg/PerformanceAnalytics/R/AppraisalRatio.R 2012-07-27 16:01:01 UTC (rev 2212)
@@ -1,28 +1,41 @@
#' Appraisal ratio of the return distribution
#'
-#' Appraisal ratio is the Jensen's alpha adjusted for systemeatic risk. The numerator
+#' Appraisal ratio is the Jensen's alpha adjusted for specific risk. The numerator
#' is divided by specific risk instead of total risk.
#'
-#' \deqn{Appraisal ratio = \frac{alpha}{\sigma_{\epsilon}}}{Appraisal ratio = Jensen's alpha / specific risk}
+#' Modified Jensen's alpha is Jensen's alpha divided by beta.
#'
-#' where \eqn{alpha} is the Jensen's alpha and \eqn{\sigma_{epsilon}} is the specific risk.
+#' Alternative Jensen's alpha is Jensen's alpha divided by systematic risk.
#'
+#' \deqn{Appraisal ratio = \frac{\alpha}{\sigma_{\epsilon}}}{Appraisal ratio = Jensen's alpha / specific risk}
+#'
+#' \deqn{Modified Jensen's alpha = \frac{\alpha}{\beta}}{Modified Jensen's alpha = Jensen's alpha / beta}
+#'
+#' \deqn{Alternative Jensen's alpha = \frac{\alpha}{\sigma_S}}{Alternative Jensen's alpha = Jensen's alpha / systematic risk}
+#'
+#' where \eqn{alpha} is the Jensen's alpha, \eqn{\sigma_{epsilon}} is the specific risk,
+#' \eqn{\sigma_S} is the systematic risk.
+#'
#' @aliases AppraisalRatio
#' @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 period number of periods in a year monthly scale = 12, quarterly = 4)
+#' @param method is one of "appraisal" to calculate appraisal ratio, "modified" to
+#' calculate modified Jensen's alpha or "alternative" to calculate alternative
+#' Jensen's alpha.
#' @param \dots any other passthru parameters
#' @author Matthieu Lestel
#' @references Carl Bacon, \emph{Practical portfolio performance measurement
-#' and attribution}, second edition 2008 p.72
+#' and attribution}, second edition 2008 p.77
#'
#' @keywords ts multivariate distribution models
#' @examples
#'
#' data(portfolio_bacon)
-#' print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.0952
+#' print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="appraisal")) #expected -0.375
+#' print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="modified"))
+#' print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="alternative"))
#'
#' data(managers)
#' print(AppraisalRatio(managers['1996',1], managers['1996',8]))
@@ -31,8 +44,10 @@
#' @export
AppraisalRatio <-
-function (Ra, Rb, Rf = 0, period = 12, ...)
+function (Ra, Rb, Rf = 0, method = c("appraisal", "modified", "alternative"), ...)
{
+ method = method[1]
+
Ra = checkData(Ra, method="matrix")
Rb = checkData(Rb, method="matrix")
@@ -46,7 +61,15 @@
if (calcul) {
Period = Frequency(Ra)
- result = CAPM.jensenAlpha(Ra,Rb,Rf,Period)/SystematicRisk(Ra,Rb,Rf,Period)
+ switch(method,
+ appraisal = {
+ epsilon = Ra - Rb * CAPM.beta(Ra,Rb,Rf) - CAPM.alpha(Ra,Rb,Rf)
+ specifikRisk = sqrt(sum((epsilon - mean(epsilon))^2)/length(epsilon))*sqrt(Period)
+ result = CAPM.jensenAlpha(Ra,Rb,Rf,Period)/specifikRisk
+ },
+ modified = {result = CAPM.jensenAlpha(Ra,Rb,Rf,Period)/CAPM.beta(Ra,Rb,Rf)},
+ alternative = {result = CAPM.jensenAlpha(Ra,Rb,Rf,Period)/SystematicRisk(Ra,Rb,Rf, Period)}
+ ) # end switch
}
else {
result = NA
@@ -55,10 +78,14 @@
}
else {
Ra = checkData(Ra)
- result = apply(Ra, MARGIN = 2, AppraisalRatio, Rb = Rb, Rf = Rf, ...)
+ result = apply(Ra, MARGIN = 2, AppraisalRatio, Rb = Rb, Rf = Rf, method = method,...)
result<-t(result)
colnames(result) = colnames(Ra)
- rownames(result) = paste("Appraisal ratio (Risk free = ",Rf,")", sep="")
+ switch(method,
+ appraisal = {rownames(result) = paste("Appraisal ratio (Risk free = ",Rf,")", sep="")},
+ modified = {rownames(result) = paste("Modified Jensen's alpha (Risk free = ",Rf,")", sep="")},
+ alternative = {rownames(result) = paste("Alternative Jensen's alpha (Risk free = ",Rf,")", sep="")}
+ ) # end switch
return(result)
}
}
Added: pkg/PerformanceAnalytics/R/FamaBeta.R
===================================================================
--- pkg/PerformanceAnalytics/R/FamaBeta.R (rev 0)
+++ pkg/PerformanceAnalytics/R/FamaBeta.R 2012-07-27 16:01:01 UTC (rev 2212)
@@ -0,0 +1,65 @@
+#' Fama beta of the return distribution
+#'
+#' Fama beta is a beta used to calculate the loss of diversification. It is made so
+#' so that the systematic risk is equivalent to the total portfolio risk.
+#'
+#' \deqn{\beta_F = \frac{\sigma_P}{\sigma_M}}{Fama beta = portfolio standard deviation / benchmark standard deviation}
+#'
+#' where \eqn{\sigma_P} is the portfolio standard deviation and \eqn{\sigma_M} is the
+#' market risk
+#'
+#' @aliases FamaBeta
+#' @param Ra an xts, vector, matrix, data frame, timeSeries or zoo object of
+#' asset returns
+#' @param Rb return vector of the benchmark asset
+#' @param \dots any other passthru parameters
+#' @author Matthieu Lestel
+#' @references Carl Bacon, \emph{Practical portfolio performance measurement
+#' and attribution}, second edition 2008 p.78
+#'
+#' @keywords ts multivariate distribution models
+#' @examples
+#'
+#' data(portfolio_bacon)
+#' print(FamaBeta(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 1.03
+#'
+#' data(managers)
+#' print(FamaBeta(managers['1996',1], managers['1996',8]))
+#' print(FamaBeta(managers['1996',1:5], managers['1996',8]))
+#'
+#' @export
+FamaBeta <-
+function (Ra, Rb, ...)
+{
+ Ra = checkData(Ra, method="matrix")
+ Rb = checkData(Rb, method="matrix")
+
+ if (ncol(Ra)==1 || is.null(Ra) || is.vector(Ra)) {
+ calcul = FALSE
+ for (i in (1:length(Ra))) {
+ if (!is.na(Ra[i])) {
+ calcul = TRUE
+ }
+ }
+
+ if (calcul) {
+ n1 = length(Ra)
+ n2 = length(Rb)
+ Period1 = Frequency(Ra)
+ Period2 = Frequency(Rb)
+ result = sqrt(var(Ra)*(n1-1)/(n1))*sqrt(Period1) / (sqrt(var(Rb)*(n2-1)/n2)*sqrt(Period2))
+ }
+ else {
+ result = NA
+ }
+ return(result)
+ }
+ else {
+ Ra = checkData(Ra)
+ result = apply(Ra, MARGIN = 2, FamaBeta, Rb = Rb, ...)
+ result<-t(result)
+ colnames(result) = colnames(Ra)
+ rownames(result) = paste("Fama Beta ", sep="")
+ return(result)
+ }
+}
Modified: pkg/PerformanceAnalytics/R/ProspectRatio.R
===================================================================
--- pkg/PerformanceAnalytics/R/ProspectRatio.R 2012-07-27 15:24:37 UTC (rev 2211)
+++ pkg/PerformanceAnalytics/R/ProspectRatio.R 2012-07-27 16:01:01 UTC (rev 2212)
@@ -3,7 +3,7 @@
#' Prospect ratio is a ratio used to penalise loss since most people feel loss
#' greater than gain
#'
-#' \deqn{ProspectRatio(R) = \frac{frac{1}{n}*\sum^{n}_{i=1}(Max(r_i,0)+2.25*Min(r_i,0) - MAR)}{\sigma_D}}{ProspectRatio(R) = (1/n * sum(Max(ri,0) + 2.25 * Min(ri,0)) - MAR) / DownsideRisk}
+#' \deqn{ProspectRatio(R) = \frac{\frac{1}{n}*\sum^{n}_{i=1}(Max(r_i,0)+2.25*Min(r_i,0) - MAR)}{\sigma_D}}{ProspectRatio(R) = (1/n * sum(Max(ri,0) + 2.25 * Min(ri,0)) - MAR) / DownsideRisk}
#'
#' where \eqn{n} is the number of observations of the entire series, MAR is the minimum acceptable return and \eqn{\sigma_D} is the downside risk
#'
Added: pkg/PerformanceAnalytics/R/Selectivity.R
===================================================================
--- pkg/PerformanceAnalytics/R/Selectivity.R (rev 0)
+++ pkg/PerformanceAnalytics/R/Selectivity.R 2012-07-27 16:01:01 UTC (rev 2212)
@@ -0,0 +1,37 @@
+#' Selectivity of the return distribution
+#'
+#' Selectivity is the same as Jensen's alpha
+#'
+#' \deqn{Selectivity = r_p - r_f - \beta_p * (b - r_f)}{Selectivity = r_p - r_f - beta_p * (b - r_f)}
+#'
+#' where \eqn{r_f} is the risk free rate, \eqn{\beta_r} is the regression beta,
+#' \eqn{r_p} is the portfolio return and b is the benchmark return
+#'
+#' @aliases Selectivity
+#' @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 any other passthru parameters
+#' @author Matthieu Lestel
+#' @references Carl Bacon, \emph{Practical portfolio performance measurement
+#' and attribution}, second edition 2008 p.78
+#'
+#' @keywords ts multivariate distribution models
+#' @examples
+#'
+#' data(portfolio_bacon)
+#' print(Selectivity(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -1.41
+#'
+#' data(managers)
+#' print(Selectivity(managers['1996',1], managers['1996',8]))
+#' print(Selectivity(managers['1996',1:5], managers['1996',8]))
+#'
+#' @export
+
+Selectivity <-
+function (Ra, Rb, Rf = 0, ...)
+{
+ Period = Frequency(Ra)
+ CAPM.jensenAlpha(Ra,Rb,Rf,Period)
+}
Modified: pkg/PerformanceAnalytics/man/AppraisalRatio.Rd
===================================================================
--- pkg/PerformanceAnalytics/man/AppraisalRatio.Rd 2012-07-27 15:24:37 UTC (rev 2211)
+++ pkg/PerformanceAnalytics/man/AppraisalRatio.Rd 2012-07-27 16:01:01 UTC (rev 2212)
@@ -2,7 +2,9 @@
\alias{AppraisalRatio}
\title{Appraisal ratio of the return distribution}
\usage{
- AppraisalRatio(Ra, Rb, Rf = 0, period = 12, ...)
+ AppraisalRatio(Ra, Rb, Rf = 0,
+ method = c("appraisal", "modified", "alternative"),
+ ...)
}
\arguments{
\item{Ra}{an xts, vector, matrix, data frame, timeSeries
@@ -12,27 +14,46 @@
\item{Rf}{risk free rate, in same period as your returns}
- \item{period}{number of periods in a year monthly scale =
- 12, quarterly = 4)}
+ \item{method}{is one of "appraisal" to calculate
+ appraisal ratio, "modified" to calculate modified
+ Jensen's alpha or "alternative" to calculate alternative
+ Jensen's alpha.}
\item{\dots}{any other passthru parameters}
}
\description{
Appraisal ratio is the Jensen's alpha adjusted for
- systemeatic risk. The numerator is divided by specific
- risk instead of total risk.
+ specific risk. The numerator is divided by specific risk
+ instead of total risk.
}
\details{
+ Modified Jensen's alpha is Jensen's alpha divided by
+ beta.
+
+ Alternative Jensen's alpha is Jensen's alpha divided by
+ systematic risk.
+
\deqn{Appraisal ratio =
- \frac{alpha}{\sigma_{\epsilon}}}{Appraisal ratio =
+ \frac{\alpha}{\sigma_{\epsilon}}}{Appraisal ratio =
Jensen's alpha / specific risk}
- where \eqn{alpha} is the Jensen's alpha and
- \eqn{\sigma_{epsilon}} is the specific risk.
+ \deqn{Modified Jensen's alpha =
+ \frac{\alpha}{\beta}}{Modified Jensen's alpha = Jensen's
+ alpha / beta}
+
+ \deqn{Alternative Jensen's alpha =
+ \frac{\alpha}{\sigma_S}}{Alternative Jensen's alpha =
+ Jensen's alpha / systematic risk}
+
+ where \eqn{alpha} is the Jensen's alpha,
+ \eqn{\sigma_{epsilon}} is the specific risk,
+ \eqn{\sigma_S} is the systematic risk.
}
\examples{
data(portfolio_bacon)
-print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.0952
+print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="appraisal")) #expected -0.375
+print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="modified"))
+print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="alternative"))
data(managers)
print(AppraisalRatio(managers['1996',1], managers['1996',8]))
@@ -43,7 +64,7 @@
}
\references{
Carl Bacon, \emph{Practical portfolio performance
- measurement and attribution}, second edition 2008 p.72
+ measurement and attribution}, second edition 2008 p.77
}
\keyword{distribution}
\keyword{models}
Modified: pkg/PerformanceAnalytics/man/DownsideDeviation.Rd
===================================================================
--- pkg/PerformanceAnalytics/man/DownsideDeviation.Rd 2012-07-27 15:24:37 UTC (rev 2211)
+++ pkg/PerformanceAnalytics/man/DownsideDeviation.Rd 2012-07-27 16:01:01 UTC (rev 2212)
@@ -3,9 +3,6 @@
\alias{DownsidePotential}
\alias{SemiDeviation}
\alias{SemiVariance}
-\alias{UpsideDeviation,}
-\alias{UpsidePotential}
-\alias{UpsideVariance,}
\title{downside risk (deviation, variance) of the return distribution}
\usage{
DownsideDeviation(R, MAR = 0,
@@ -14,9 +11,6 @@
SemiDeviation(R)
SemiVariance(R)
-
- DownsideDeviation(R, MAR = 0,
- method = c("full", "subset"), ..., potential = FALSE)
}
\arguments{
\item{R}{an xts, vector, matrix, data frame, timeSeries
@@ -34,25 +28,6 @@
\item{potential}{if TRUE, calculate downside potential
instead, default FALSE}
-
- \item{R}{an xts, vector, matrix, data frame, timeSeries
- or zoo object of asset returns}
-
- \item{MAR}{Minimum Acceptable Return, in the same
- periodicity as your returns}
-
- \item{method}{one of "full" or "subset", indicating
- whether to use the length of the full series or the
- length of the subset of the series below the MAR as the
- denominator, defaults to "subset"}
-
- \item{author}{one of "Bacon", "Sortino", indicating
- whether to}
-
- \item{\dots}{any other passthru parameters}
-
- \item{potential}{if TRUE, calculate downside potential
- instead, default FALSE}
}
\description{
Downside deviation, semideviation, and semivariance are
@@ -145,29 +120,9 @@
SemiDeviation(managers[,1:6])
SemiVariance (managers[,1,drop=FALSE])
SemiVariance (managers[,1:6]) #calculated using method="subset"
-#with data used in Bacon 2008
-
-portfolio_return <- c(0.3,2.6,1.1,-1.0,1.5,2.5,1.6,6.7,-1.4,4.0,-0.5,8.1,4.0,-3.7,
--6.1,1.7,-4.9,-2.2,7.0,5.8,-6.5,2.4,-0.5,-0.9)
-MAR = 0.5
-DownsideDeviation(portfolio_return, MAR) #expected 2.55
-DownsidePotential(portfolio_return, MAR) #expected 1.37
-
-#with data of managers
-
-data(managers)
-apply(managers[,1:6], 2, sd, na.rm=TRUE)
-DownsideDeviation(managers[,1:6]) # MAR 0\%
-DownsideDeviation(managers[,1:6], MAR = .04/12) #MAR 4\%
-SemiDeviation(managers[,1,drop=FALSE])
-SemiDeviation(managers[,1:6])
-SemiVariance (managers[,1,drop=FALSE])
-SemiVariance (managers[,1:6]) #calculated using method="subset"
}
\author{
Peter Carl, Brian G. Peterson, Matthieu Lestel
-
- Peter Carl, Brian G. Peterson, Matthieu Lestel
}
\references{
Sortino, F. and Price, L. Performance Measurement in a
@@ -186,23 +141,6 @@
especially end note 10
\url{http://en.wikipedia.org/wiki/Semivariance}
-
- Sortino, F. and Price, L. Performance Measurement in a
- Downside Risk Framework. \emph{Journal of Investing}.
- Fall 1994, 59-65. \cr Carl Bacon, \emph{Practical
- portfolio performance measurement and attribution},
- second edition 2008
-
- Plantinga, A., van der Meer, R. and Sortino, F. The
- Impact of Downside Risk on Risk-Adjusted Performance of
- Mutual Funds in the Euronext Markets. July 19, 2001.
- Available at SSRN: \url{http://ssrn.com/abstract=277352}
- \cr
-
- \url{http://www.sortino.com/htm/performance.htm} see
- especially end note 10
-
- \url{http://en.wikipedia.org/wiki/Semivariance}
}
\keyword{distribution}
\keyword{models}
Added: pkg/PerformanceAnalytics/man/FamaBeta.Rd
===================================================================
--- pkg/PerformanceAnalytics/man/FamaBeta.Rd (rev 0)
+++ pkg/PerformanceAnalytics/man/FamaBeta.Rd 2012-07-27 16:01:01 UTC (rev 2212)
@@ -0,0 +1,47 @@
+\name{FamaBeta}
+\alias{FamaBeta}
+\title{Fama beta of the return distribution}
+\usage{
+ FamaBeta(Ra, Rb, ...)
+}
+\arguments{
+ \item{Ra}{an xts, vector, matrix, data frame, timeSeries
+ or zoo object of asset returns}
+
+ \item{Rb}{return vector of the benchmark asset}
+
+ \item{\dots}{any other passthru parameters}
+}
+\description{
+ Fama beta is a beta used to calculate the loss of
+ diversification. It is made so so that the systematic
+ risk is equivalent to the total portfolio risk.
+}
+\details{
+ \deqn{\beta_F = \frac{\sigma_P}{\sigma_M}}{Fama beta =
+ portfolio standard deviation / benchmark standard
+ deviation}
+
+ where \eqn{\sigma_P} is the portfolio standard deviation
+ and \eqn{\sigma_M} is the market risk
+}
+\examples{
+data(portfolio_bacon)
+print(FamaBeta(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 1.03
+
+data(managers)
+print(FamaBeta(managers['1996',1], managers['1996',8]))
+print(FamaBeta(managers['1996',1:5], managers['1996',8]))
+}
+\author{
+ Matthieu Lestel
+}
+\references{
+ Carl Bacon, \emph{Practical portfolio performance
+ measurement and attribution}, second edition 2008 p.78
+}
+\keyword{distribution}
+\keyword{models}
+\keyword{multivariate}
+\keyword{ts}
+
Modified: pkg/PerformanceAnalytics/man/ProspectRatio.Rd
===================================================================
--- pkg/PerformanceAnalytics/man/ProspectRatio.Rd 2012-07-27 15:24:37 UTC (rev 2211)
+++ pkg/PerformanceAnalytics/man/ProspectRatio.Rd 2012-07-27 16:01:01 UTC (rev 2212)
@@ -18,7 +18,7 @@
}
\details{
\deqn{ProspectRatio(R) =
- \frac{frac{1}{n}*\sum^{n}_{i=1}(Max(r_i,0)+2.25*Min(r_i,0)
+ \frac{\frac{1}{n}*\sum^{n}_{i=1}(Max(r_i,0)+2.25*Min(r_i,0)
- MAR)}{\sigma_D}}{ProspectRatio(R) = (1/n *
sum(Max(ri,0) + 2.25 * Min(ri,0)) - MAR) / DownsideRisk}
Added: pkg/PerformanceAnalytics/man/Selectivity.Rd
===================================================================
--- pkg/PerformanceAnalytics/man/Selectivity.Rd (rev 0)
+++ pkg/PerformanceAnalytics/man/Selectivity.Rd 2012-07-27 16:01:01 UTC (rev 2212)
@@ -0,0 +1,50 @@
+\name{Selectivity}
+\alias{Selectivity}
+\title{Selectivity of the return distribution}
+\usage{
+ Selectivity(Ra, Rb, Rf = 0, period = 12, ...)
+}
+\arguments{
+ \item{Ra}{an xts, vector, matrix, data frame, timeSeries
+ or zoo object of asset returns}
+
+ \item{Rb}{return vector of the benchmark asset}
+
+ \item{Rf}{risk free rate, in same period as your returns}
+
+ \item{period}{number of periods in a year monthly scale =
+ 12, quarterly = 4)}
+
+ \item{\dots}{any other passthru parameters}
+}
+\description{
+ Selectivity is the same as Jensen's alpha
+}
+\details{
+ \deqn{Selectivity = r_p - r_f - \beta_p * (b -
+ r_f)}{Selectivity = r_p - r_f - beta_p * (b - r_f)}
+
+ where \eqn{r_f} is the risk free rate, \eqn{\beta_r} is
+ the regression beta, \eqn{r_p} is the portfolio return
+ and b is the benchmark return
+}
+\examples{
+data(portfolio_bacon)
+print(Selectivity(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -1.41
+
+data(managers)
+print(Selectivity(managers['1996',1], managers['1996',8]))
+print(Selectivity(managers['1996',1:5], managers['1996',8]))
+}
+\author{
+ Matthieu Lestel
+}
+\references{
+ Carl Bacon, \emph{Practical portfolio performance
+ measurement and attribution}, second edition 2008 p.78
+}
+\keyword{distribution}
+\keyword{models}
+\keyword{multivariate}
+\keyword{ts}
+
More information about the Returnanalytics-commits
mailing list