% $Id: PA-charts.Rnw 1950 2012-05-20 10:19:00Z braverock $

%% no need for  \DeclareGraphicsExtensions{.pdf,.eps}

\documentclass[12pt,letterpaper,english]{article}
\usepackage{times}
\usepackage[T1]{fontenc}
\IfFileExists{url.sty}{\usepackage{url}}
                      {\newcommand{\url}{\texttt}}

\usepackage{babel}
\usepackage{noweb}
\usepackage{Rd}

\usepackage{Sweave}
\SweaveOpts{engine=R,eps=FALSE}
%\VignetteIndexEntry{PerformanceAnalytics Charts and Tables Reference}
%\VignetteDepends{PerformanceAnalytics}
%\VignetteKeywords{returns, performance, risk, benchmark, portfolio}
%\VignettePackage{PerformanceAnalytics}



%%%%%%%%%%%%%%%%%%%%%% End Setup Lines 

\begin{document}

%\author{Peter Carl\\Guidance Capital Management \And Brian G. Peterson\\Diamond Consulting}
% \plainauthor{Peter Carl, Brian Peterson}
\author {Peter Carl \& Brian G. Peterson}

\title{\pkg{PerformanceAnalytics} Charts and Tables Overview}
% \plaintitle{PerformanceAnalytics Charts and Tables Overview}

% \keywords{returns, performance, risk, benchmark, portfolio}




\makeatletter
\makeatother
\maketitle

\begin{abstract}
  This vignette gives a brief overview of (some of) the graphics and display wrapper
  functionality contained in \pkg{PerformanceAnalytics} including most of the
  charts and tables . For a more complete overview of the package's functionality
  and extensibility see the manual pages.  We develop the examples using data
  for six (hypothetical) managers, a peer index, and an asset class index.
\end{abstract}

\tableofcontents

\section{Introduction}

\code{\link{PerformanceAnalytics}} is a library of functions
designed for evaluating the performance and risk characteristics of
financial assets or funds. In particular, we have focused on functions
that have appeared in the academic literature over the past several
years, but had no functional equivalent in \R.

Our goal for \code{\link{PerformanceAnalytics}} is to make it
simple for someone to ask and answer questions about performance and
risk as part of a broader investment decision-making process. There
is no magic bullet here -- there won't be one \emph{right} answer
delivered in these metrics and charts. Investments must be made in
context of investment objectives. But what this library aspires to
do is \emph{help} the decision-maker accrete evidence organized to
answer a specific question that is pertinent to the decision at hand.
Our hope is that using such tools to uncover information and ask better
questions will, in turn, create a more informed investor and help
them ask better quality decisions.

This vignette provides a demonstration of some of the capabilities
of \code{\link{PerformanceAnalytics}}. We focus on the graphs and tables, but
comment on some other metrics along the way. These examples are not
intended to be complete, but they should provide an indication of
the kinds of analysis that can be done. Other examples are available
in the help pages of the functions described in the main page of
\code{\link{PerformanceAnalytics}}.


\section{Set up PerformanceAnalytics}

These examples assume the reader has basic knowledge of \R ~and
understands how to install \R, read and manipulate data, and create
basic calculations. For further assistance, please see \code{\link{http://cran.r-project.org/doc/manuals/R-intro.pdf}}
and other available materials at \code{\link{http://cran.r-project.org}}.
This section will begin with installation, discuss the example data
set, and provide an overview of charts attributes that will be used
in the examples that follow.


\subsection{Install PerformanceAnalytics}

As of version 0.9.4, \code{\link{PerformanceAnalytics}} is available
via CRAN, but you will need version 0.9.5 or later to replicate the examples
that follow.  \R~ users with connectivity can simply type:
\code{install.packages("PerformanceAnalytics")}.  Or, if you
already have an earlier version installed, see \code{update.packages}.
A number of packages are required, including \code{Hmisc}, \code{zoo},
and the various Rmetrics packages such as \code{fBasics}, \code{fCalendar},
and \code{fExtremes}. After installing \code{\link{PerformanceAnalytics}},
load it into your active \R~ session using \code{library("PerformanceAnalytics")}.

<<LoadLibrary,echo=F,results=hide>>=
library('PerformanceAnalytics')
#source("PerformanceAnalytics/R/charts.PerformanceSummary.R")
@


\subsection{Load and review data}

%
\begin{figure}

\caption{First Lines of the managers Object }

\label{fig:First-Lines-of}

<<LoadData,echo=T>>=
data(managers)
#managers=read.csv("/home/peter/dev/R/managers.csv",row.names=1)
head(managers)
@
\end{figure}


First we load the data used in all of the examples that follow. As
you can see in Figure \ref{fig:First-Lines-of}, \code{managers}
is a data frame that contains columns of monthly returns for six hypothetical
asset managers (HAM1 through HAM6), the EDHEC Long-Short Equity hedge
fund index, the S\&P 500 total returns, and total return series for
the US Treasury 10-year bond and 3-month bill. Monthly returns for
all series end in December 2006 and begin at different periods starting
from January 1996. Similar data could be constructed using \code{mymanagers=read.csv("/path/to/file/mymanagers.csv",
row.names=1)}, where the first column contains dates in the YYYY-MM-DD
format.

A quick sidenote: this library is applicable to return (rather than
price) data, and has been tested mostly on a monthly scale. Many library
functions will work with regular data at different scales (e.g., daily,
weekly, etc.) or irregular return data as well. See function \code{\link{CalculateReturns}}
for calculating returns from prices, and be aware that the \code{zoo}
library's \code{\link{aggregate}} function has methods for \code{tseries}
and \code{zoo} timeseries data classes to rationally coerce irregular
data into regular data of the correct periodicity.

With the data object in hand, we group together columns of interest
make the examples easier to follow. As shown in Figure \ref{fig:Assess-and-Organize},
we first confirm the dimensions of the object and the overall length
of the timeseries.

%
\begin{figure}

\caption{Assess and Organize the Data}

\label{fig:Assess-and-Organize}

\begin{center}

<<CalcDataDimensions,echo=T>>=
dim(managers)
managers.length = dim(managers)[1]
colnames(managers)
manager.col = 1
peers.cols = c(2,3,4,5,6)
indexes.cols = c(7,8)
Rf.col = 10
#factors.cols = NA
trailing12.rows = ((managers.length - 11):managers.length)
trailing12.rows
trailing36.rows = ((managers.length - 35):managers.length)
trailing60.rows = ((managers.length - 59):managers.length)
#assume contiguous NAs - this may not be the way to do it na.contiguous()?
frInception.rows = (length(managers[,1]) - length(managers[,1][!is.na(managers[,1])]) + 1):length(managers[,1])

@

\end{center}
\end{figure}


Next, we group the columns together. The first column, HAM1, contains
the subject of our analysis in the examples below. Columns two through
six contain returns of other managers using a similar investing style
that might be considered substitutes -- a {}``peer group,'' of sorts.
Columns seven and eight contain key indexes. The EDHEC Long/Short
Equity index is a {}``peer index,'' the S\&P 500 Total Return index
is an {}``asset-class index''. We combine those into a set of {}``indexes''
for later analysis. Column nine we skip for the time being; column
10 we use as the risk free rate for each month.

Then we do the same thing for the rows of interest. We calculate the
row numbers that represent different trailing periods and keep those
handy for doing comparative analyses.


\subsection{Create charts and tables for presentation}

Graphs and charts help to organize information visually. Our goals
in creating these functions were to simplify the process of creating
well-formatted charts that are frequently used for portfolio analysis
and to create print-quality graphics that may be used in documents
for broader consumption. \R's graphics capabilities are substantial,
but the simplicity of the output of \R's default graphics functions
such as \code{\link{plot}} does not always compare well against
graphics delivered with commercial asset or portfolio analysis software
from places such as MorningStar or PerTrac.


\subsubsection*{Color Palettes}

We have set up some specific color palattes designed to create readable
line and bar graphs with specific objectives. We use this approach
(rather than generating them on the fly) for two reasons: first, there
are fewer dependencies on libraries that don't need to be called dynamically;
and second, to guarantee the color used for the n-th column of data.
Figure \ref{fig:Examples-of-Color} shows some examples of the different
palates.

The first category of colorsets are designed to provide focus to the
data graphed as the first element, and include \code{redfocus},
\code{bluefocus}, and \code{greenfocus}. These palettes are
best used when there is an important data set for the viewer to focus
on. The other data provide some context, so they are graphed in diminishing
values of gray. These were generated with \code{RColorBrewer},
using the 8 level "grays" palette and replacing
the darkest gray with the focus color. To coordinate these colorsets
with the equal-weighted colors below, replace the highlight color
with the first color of the equal weighted palette from below. This
will coordinate charts with different purposes.

The second category is sets of equal-weighted colors. These colorsets
are useful for when all of the data should be observed and distinguishable
on a line graph. The different numbers in the name indicate the number
of colors generated (six colors is probably the maximum for a readable
linegraph, but we provide as many as twelve). Examples include \code{rainbow12equal}
through \code{rainbow4equal}, in steps of two. These colorsets
were generated with \code{rainbow(12, s = 0.6, v = 0.75)}. The
\code{rich12equal} and other corresponding colorsets were generated
with with package \code{gplots} function \code{rich.colors(12)}.
Similarly \code{tim12equal} and similar colorsets were generated
with with package \code{fields} function \code{tim.colors(12)},
a function said to emulate the Matlab colorset. The \code{dark8equal},
\code{dark6equal}, \code{set8equal}, and \code{set6equal}
colorsets were created with package \code{RColorBrewer}, e.g.,
\code{brewer.pal(8,"Dark2")}. A third
category is a set of monochrome colorsets, including \code{greenmono}
, \code{bluemono}, \code{redmono}, and \code{gray8mono} and
\code{gray6mono}.

To see what these lists contain, just type the name.

<<ShowColors,echo=T>>=
tim12equal
@

These are just lists of strings that contain the RGB codes of each
color. You can easily create your own if you have a particular palate
that you like to use. Alternatively, you can use the \R~ default colors.
For more information, see:

\code{\link{HTTP://research.stowers-institute.org/efg/R/Color/Chart/index.htm}}


%
\begin{figure}

\caption{Examples of Color Palettes}

\label{fig:Examples-of-Color}

\begin{center}

<<ShowPalettes,echo=F,fig=T>>=
layout(rbind(c(1,2),c(3,4)))
chart.CumReturns(managers[,c(manager.col,peers.cols)],
lwd=2,legend.loc="topleft", main="default")
chart.CumReturns(managers[,c(manager.col,peers.cols)],lwd=2,
legend.loc="topleft",colorset=redfocus, main = "redfocus")
chart.CumReturns(managers[,c(manager.col,peers.cols)],lwd=2,
legend.loc="topleft",colorset=rainbow6equal, main="rainbow6equal")
chart.CumReturns(managers[,c(manager.col,peers.cols)],lwd=2,
legend.loc="topleft",colorset=tim6equal, main = "tim6equal")
@

\end{center}
\end{figure}



\subsubsection*{Symbols}

Similarly, there are a few sets of grouped symbols for scatter charts . These
include \code{opensymbols}, \code{closedsymbols}, \code{fillsymbols},
\code{linesymbols}, and \code{allsymbols}.


\subsubsection*{Legend locations}

In the single charts the legend can be moved around on the plot. There
are nine locations that can be specified by keyword: "bottomright",
"bottom", "bottomleft", "left",
"topleft", "top", "topright",
"right" and "center". This places
the legend on the inside of the plot frame at the given location.
Further information can be found in \code{\link{xy.coord}}.
Most compound charts have fixed legend locations.


\subsubsection*{Other Parameters}

We have tried to leave access to all of the extensive list of parameters
available in \R's traditional graphics. For more information, see
\code{\link{plot.default}} and \code{\link{par}}. In the
example above, we passed \code{\link{lwd}} the value of 2, which
affects the line width. We might also alter the line type or other
parameter.


\section{Create Charts and Tables}

With that, we are ready to analyze the sample data set. This section
starts with a set of charts that provide a performance overview, some
tables for presenting the data and basic statistics, and then discusses
ways to compare distributions, relative performance, and downside
risk.


\subsection{Create performance charts}

%
\begin{figure}

\caption{Draw a Performance Summary Chart}

\label{fig:Draw-a-Performance}

\begin{center}

<<Graph1,echo=T,fig=T>>=
charts.PerformanceSummary(managers[,c(manager.col,indexes.cols)], colorset=rich6equal, lwd=2, ylog=TRUE)
@

\end{center}
\end{figure}


Figure \ref{fig:Draw-a-Performance} shows a three-panel performance
summary chart and the code used to generate it. The top chart is a
normal cumulative return or wealth index chart that shows the cumulative
returns through time for each column. For data with later starting
times, set the parameter \code{begin = "axis"},
which starts the wealth index of all columns at 1, or \code{begin =
"first"}, which
starts the wealth index of each column at the wealth index value attained
by the first column of data specified. The second of these settings,
which is the default, allows the reader to see how the two indexes
would compare had they started at the same time regardless of the
starting period. In addition, the y-axis can be set to a logarithmic
value so that growth can be compared over long periods. This chart
can be generated independently using \code{chart.CumReturns}.

The second chart shows the individual monthly returns overlaid with
a rolling measure of tail risk referred to as Cornish Fisher Value-at-Risk
(VaR) or Modified VaR. Alternative risk measures, including standard
deviation (specified as \code{StdDev}) and traditional
Value-at-Risk (\code{VaR}), can be specified
using the \code{method} parameter. Note that \code{StdDev} and
\code{VaR} are symmetric calculations, so a high and low measure
will be plotted. \code{ModifiedVaR}, on the other hand, is asymmetric
and only a lower bound is drawn. These risk calculations are made
on a rolling basis from inception, or can be calculated on a rolling
window by setting \code{width} to a value of the number of periods.
These calculations should help the reader to identify events or periods
when estimates of tail risk may have changed suddenly, or to help
evaluate whether the assumptions underlying the calculation seem to
hold . The risk calculations can be generated for all of the columns
provided by using the \code{all} parameter. When set to \code{TRUE},
the function calculates risk lines for each column given and may help
the reader assess relative risk levels through time. This chart can
be generated using \code{chart.BarVaR}.

The third chart in the series is a drawdown or underwater chart, which
shows the level of losses from the last value of peak equity attained.
Any time the cumulative returns dips below the maximum cumulative
returns, it's a drawdown. This chart helps the reader assess the synchronicity
of the loss periods and their comparative severity. As you might expect,
this chart can also be created using \code{chart.Drawdown}.


\subsection{Create a monthly returns table}

Summary statistics are then the necessary aggregation and reduction
of (potentially thousands) of periodic return numbers. Usually these
statistics are most palatable when organized into a table of related
statistics, assembled for a particular purpose. A common offering
of past returns organized by month and accumulated by calendar year
is usually presented as a table, such as in \code{\link{table.CalendarReturns}}.

%
\begin{figure}

\caption{Create a Table of Calendar Returns}

\label{fig:Calendar-Returns}

\begin{center}

<<CalendarReturns,echo=T>>=
t(table.CalendarReturns(managers[,c(manager.col,indexes.cols)]))
@

\end{center}
\end{figure}


Figure \ref{fig:Calendar-Returns} shows a table of returns formatted
with years in rows, months in columns, and a total column in the last
column. For additional columns, the annual returns will be appended
as columns. Adding benchmarks or peers alongside the annualized data
is helpful for comparing returns across calendar years. Because there
are a number of columns in the example, we make the output easier
to read by using the \code{t} function to transpose the resulting
data frame.


\subsection{Calculate monthly statistics}

Likewise, the monthly returns statistics table in Figure \ref{fig:Statistics-Table}
was created as a way to display a set of related measures together
for comparison across a set of instruments or funds. In this example,
we are looking at performance {}``from inception'' for most of the
managers, so careful consideration needs to be given to missing data
or unequal time series when interpreting the results. Most people
will prefer to see such statistics for common or similar periods.
Each of the individual functions can be called individually, as well.

%
\begin{figure}

\caption{Create a Table of Statistics}

\label{fig:Statistics-Table}

\begin{center}

<<MonthlyReturnStats,echo=T>>=
table.Stats(managers[,c(manager.col,peers.cols)])
@

\end{center}
\end{figure}


When we started this project, we debated whether or not such tables
would be broadly useful or not. No reader is likely to think that
we captured the precise statistics to help their decision. We merely
offer these as a starting point for creating your own. Add, subtract,
do whatever seems useful to you. If you think that your work may be
useful to others, please consider sharing it so that we may include
it in a future version of this library.


\subsection{Compare distributions}

For distributional analysis, a few graphics may be useful. The result
of \code{\link{chart.Boxplot}}, shown in Figure \ref{fig:Create-a-Boxplot},
is an example of a graphic that is difficult to create in Excel and
is under-utilized as a result. A boxplot of returns is, however, a
very useful way to observe the shape of large collections of asset
returns in a manner that makes them easy to compare to one another.

%
\begin{figure}

\caption{Create a Boxplot}

\label{fig:Create-a-Boxplot}

\begin{center}

<<Graph10,echo=T,fig=T>>=
chart.Boxplot(managers[ trailing36.rows, c(manager.col, peers.cols, indexes.cols)], main = "Trailing 36-Month Returns")
@

\end{center}
\end{figure}


It is often valuable when evaluating an investment to know whether
the instrument that you are examining follows a normal distribution.
One of the first methods to determine how close the asset is to a
normal or log-normal distribution is to visually look at your data.
Both \code{\link{chart.QQPlot}} and \code{\link{chart.Histogram}}
will quickly give you a feel for whether or not you are looking at
a normally distributed return history. Figure \ref{fig:Create-a-Histogram}
shows a histogram generated for HAM1 with different display options.

%
\begin{figure}

\caption{Create a Histogram of Returns}

\label{fig:Create-a-Histogram}

\begin{center}

<<Graph13,echo=T,fig=T>>=
layout(rbind(c(1,2),c(3,4)))
chart.Histogram(managers[,1,drop=F], main = "Plain", methods = NULL)
chart.Histogram(managers[,1,drop=F], main = "Density", breaks=40,
methods = c("add.density", "add.normal"))
chart.Histogram(managers[,1,drop=F], main = "Skew and Kurt", methods = c
("add.centered", "add.rug"))
chart.Histogram(managers[,1,drop=F], main = "Risk Measures", methods = c
("add.risk"))
@

\end{center}
\end{figure}


Look back at the results generated by \code{\link{table.Stats}}.
Differences between \code{\link{var}} and \code{\link{SemiVariance}}
will help you identify \code{\link{[}fBasics]{skewness}} in
the returns. Skewness measures the degree of asymmetry in the return
distribution. Positive skewness indicates that more of the returns
are positive, negative skewness indicates that more of the returns
are negative. An investor should in most cases prefer a positively
skewed asset to a similar (style, industry, region) asset that has
a negative skewness. Kurtosis measures the concentration of the returns
in any given part of the distribution (as you should see visually
in a histogram). The \code{\link{[}fBasics]{kurtosis}} function
will by default return what is referred to as \dQuote{excess kurtosis},
where zero is a normal distribution, other methods of calculating
kurtosis than \code{method="excess"} will set
the normal distribution at a value of 3. In general a rational investor
should prefer an asset with a low to negative excess kurtosis, as
this will indicate more predictable returns. If you find yourself
needing to analyze the distribution of complex or non-smooth asset
distributions, the \code{nortest} package has several advanced
statistical tests for analyzing the normality of a distribution.




\subsection{Show relative return and risk}

Returns and risk may be annualized as a way to simplify comparison
over longer time periods. Although it requires a bit of estimating,
such aggregation is popular because it offers a reference point for
easy comparison, such as in Figure \ref{fig:Show-Relative-Risk}.
Examples are in \code{\link{Return.annualized}}, \code{\link{StdDev.annualized}},
and \code{\link{SharpeRatio.annualized}}.

%
\begin{figure}

\caption{Show Relative Risk and Return}

\label{fig:Show-Relative-Risk}

\begin{center}

<<Graph3,echo=T,fig=T>>=
chart.RiskReturnScatter(managers[trailing36.rows,1:8], Rf=.03/12, main = "Trailing 36-Month Performance", colorset=c("red", rep("black",5), "orange", "green"))
@

\end{center}
\end{figure}


\code{\link{chart.Scatter}} is a utility scatter chart with
some additional attributes that are used in \code{\link{chart.RiskReturnScatter}}.
Different risk parameters may be used. The parameter \code{method}
may be any of \code{modVaR}, \code{VaR},
or \code{StdDev}.

Additional information can be overlaid, as well. If \code{add.sharpe}
is set to a value, say \code{c(1,2,3)}, the function overlays Sharpe
ratio line that indicates Sharpe ratio levels of one through three.
Lines are drawn with a y-intercept of the risk free rate (\code{rf})
and the slope of the appropriate Sharpe ratio level. Lines should
be removed where not appropriate (e.g., \code{sharpe.ratio = NULL}).
With a large number of assets (or columns), the names may get in the
way. To remove them, set \code{add.names = NULL}. A box plot may
be added to the margins to help identify the relative performance
quartile by setting \code{add.boxplots = TRUE}.


\subsection{Examine performance consistency}

Rolling performance is typically used as a way to assess stability
of a return stream. Although perhaps it doesn't get much credence
in the financial literature because of it's roots in digital signal
processing, many practitioners find rolling performance to be a useful
way to examine and segment performance and risk periods. See \code{\link{chart.RollingPerformance}},
which is a way to display different metrics over rolling time periods.

Figure \ref{fig:Examine-Rolling-Performance} shows three panels,
the first for rolling returns, the second for rolling standard deviation,
and the third for rolling Sharpe ratio. These three panels each call
\code{\link{chart.RollingPerformance}} with a different \code{FUN}
argument, allowing any function to be viewed over a rolling window.

%
\begin{figure}

\caption{Examine Rolling Performance}

\label{fig:Examine-Rolling-Performance}

\begin{center}

<<Graph5,echo=T,fig=T>>=
charts.RollingPerformance(managers[, c(manager.col, peers.cols, indexes.cols)], Rf=.03/12, colorset = c("red", rep("darkgray",5), "orange", "green"), lwd = 2)
@

\end{center}
\end{figure}



\subsection{Display relative performance}

The function \code{\link{chart.RelativePerformance}} shows the ratio of the
cumulative performance for two assets at each point in time and
makes periods of under- or out-performance easy to see. The value
of the chart is less important than the slope of the line. If the
slope is positive, the first asset (numerator) is outperforming the second, and
vice versa. Figure \ref{fig:Examine-Relative-Performance} shows the
returns of the manager in question relative to each member of the
peer group and the peer group index.

%
\begin{figure}

\caption{Examine Relative Performance of Assets}

\label{fig:Examine-Relative-Performance}

\begin{center}

<<Graph6,echo=T,fig=T>>=
chart.RelativePerformance(managers[ , manager.col, drop = FALSE], managers[ , c(peers.cols, 7)], colorset = tim8equal[-1], lwd = 2, legend.loc = "topleft")
@

\end{center}
\end{figure}


Looking at the data another way, we use the same chart to assess the
peers individually against the asset class index. Figure \ref{fig:Examine-Performance-Relative2}
shows the peer group members' returns relative to the S\&P 500. Several
questions might arise: Who beats the S\&P? Can they do it consistently?
Are there cycles when they under-perform or outperform as a group?
Which manager has outperformed the most?

%
\begin{figure}

\caption{Examine Performance Relative to a Benchmark}

\label{fig:Examine-Performance-Relative2}

\begin{center}

<<Graph6a,echo=T,fig=T>>=
chart.RelativePerformance(managers[ , c(manager.col, peers.cols) ], managers[, 8, drop=F], colorset = rainbow8equal, lwd = 2, legend.loc = "topleft")
@

\end{center}
\end{figure}



\subsection{Measure relative performance to a benchmark}

Identifying and using a benchmark can help us assess and explain how well we are
meeting our investment objectives, in terms of a widely held substitute.
A benchmark can help us explain how the portfolios are managed,
assess the risk taken and the return desired, and check that
the objectives were respected. Benchmarks are used to get better control of the
investment management process and to suggest ways to improve selection.

Modern Portfolio Theory is a collection of tools and techniques
by which a risk-averse investor may construct an optimal portfolio.
It encompasses the Capital Asset Pricing Model (CAPM), the efficient
market hypothesis, and all forms of quantitative portfolio construction
and optimization. CAPM provides a justification for passive or index
investing by positing that assets that are not on the efficient frontier
will either rise or lower in price until they are on the efficient
frontier of the market portfolio.

%
\begin{figure}

\caption{Create a Table of CAPM-Related Measures}

\label{fig:CAPM-Table}

<<tableCAPM,echo=T>>=
table.CAPM(managers[trailing36.rows, c(manager.col, peers.cols)],
managers[ trailing36.rows, 8, drop=FALSE],
Rf = managers[ trailing36.rows, Rf.col, drop=FALSE])
@
\end{figure}


The performance premium provided by an investment over a passive strategy
(the benchmark) is provided by \code{\link{ActivePremium}},
which is the investment's annualized return minus the benchmark's
annualized return. A closely related measure is the \code{\link{TrackingError}},
which measures the unexplained portion of the investment's performance
relative to a benchmark. The \code{\link{InformationRatio}}
of an Investment in a MPT or CAPM framework is the \code{ActivePremium}
divided by the \code{TrackingError}. \code{InformationRatio}
may be used to rank investments in a relative fashion.

The code in Figure \ref{fig:CAPM-Table} creates a table of CAPM-related
statistics that we can review and compare across managers. Note that
we focus on the trailing-36 month period. There are, in addition to
those listed, a wide variety of other CAPM-related metrics available.
The \code{\link{CAPM.RiskPremium}} on an investment is the measure
of how much the asset's performance differs from the risk free rate.
Negative Risk Premium generally indicates that the investment is a
bad investment, and the money should be allocated to the risk free
asset or to a different asset with a higher risk premium. \code{\link{CAPM.alpha}}
is the degree to which the assets returns are not due to the return
that could be captured from the market. Conversely, \code{\link{CAPM.beta}}
describes the portions of the returns of the asset that could be directly
attributed to the returns of a passive investment in the benchmark
asset. The Capital Market Line \code{\link{CAPM.CML}} relates
the excess expected return on an efficient market portfolio to its
risk (represented in CAPM by \code{\link{StdDev}}). The slope
of the CML, \code{\link{CAPM.CML.slope}}, is the Sharpe Ratio
for the market portfolio. The Security Market Line is constructed
by calculating the line of \code{\link{CAPM.RiskPremium}} over
\code{\link{CAPM.beta}}. For the benchmark asset this will be
1 over the risk premium of the benchmark asset. The slope of the SML,
primarily for plotting purposes, is given by \code{\link{CAPM.SML.slope}}.
CAPM is a market equilibrium model or a general equilibrium theory
of the relation of prices to risk, but it is usually applied to partial
equilibrium portfolios which can create (sometimes serious) problems
in valuation.

%<<Graph7,echo=T,fig=T>>=
%#source("chart.MultiScatter.R")
%#chart.MultiScatter(Ra=managers[,1,drop=F],Rb=managers[,8,drop=F],linear.fit=T,Rf=managers[,10],excess.returns=T)
%@

In a similar fashion to the rolling performance we displayed earlier,
we can look at the stability of a linear model's parameters through
time. Figure \ref{fig:Rolling-Regression} shows a three panel chart
for the alpha, beta, and r-squared measures through time across a
rolling window. Each chart calls \code{\link{chart.RollingRegression}}
with a different \code{method} parameter.

%
\begin{figure}

\caption{Create a Rolling Regression}

\label{fig:Rolling-Regression}

\begin{center}

<<Graph8,echo=T,fig=T>>=
#source("PerformanceAnalytics/R/Return.excess.R")
charts.RollingRegression(managers[, c(manager.col, peers.cols), drop = FALSE], managers[, 8, drop = FALSE], Rf = .03/12, colorset = redfocus, lwd = 2)
@

\end{center}
\end{figure}


Likewise, we can assess whether the correlation between two time series
is constant through time. Figure \ref{fig:Rolling-Correlation} shows
the rolling 12-month correlation between each of the peer group and
the S\&P500. To look at the relationships over time and take a snapshot
of the statistical relevance of the measure, use \code{\link{table.Correlation}},
as shown in Figure \ref{fig:Calculate-Correlations}.

%
\begin{figure}

\caption{Chart the Rolling Correlation}

\label{fig:Rolling-Correlation}

\begin{center}

<<Graph9,echo=T,fig=T>>=
chart.RollingCorrelation(managers[,c(manager.col, peers.cols)], managers[, 8, drop = FALSE], colorset = tim8equal, lwd = 2, main = "12-Month Rolling Correlation")
@

\end{center}
\end{figure}


%
\begin{figure}

\caption{Calculate Correlations}

\label{fig:Calculate-Correlations}

<<tableCorr,echo=T>>=
table.Correlation(managers[, c(manager.col, peers.cols)], managers[, 8,
drop = F], legend.loc = "lowerleft")
@
\end{figure}



\subsection{Calculate Downside Risk}

Many assets, including hedge funds, commodities, options, and even
most common stocks over a sufficiently long period, do not follow
a normal distribution. For such common but non-normally distributed
assets, a more sophisticated approach than standard deviation/volatility
is required to adequately model the risk.

Markowitz, in his Nobel acceptance speech and in several papers, proposed
that \code{\link{SemiVariance}} would be a better measure of
risk than variance.
This measure is also called \code{\link{SemiDeviation}}. The
more general case of downside deviation is implemented in the function
\code{\link{DownsideDeviation}}, as proposed by Sortino and Price (1994),
where the minimum acceptable return (MAR) is a parameter to the function.
It is interesting to note that variance and mean return can produce
a smoothly elliptical efficient frontier for portfolio optimization
utilizing \code{\link{[}quadprog]{solve.QP}} or \code{\link{[}tseries]{portfolio.optim}}
or \code{\link{[}fPortfolio]{MarkowitzPortfolio}}. Use of semivariance
or many other risk measures will not necessarily create a smooth ellipse,
causing significant additional difficulties for the portfolio manager
trying to build an optimal portfolio. We'll leave a more complete
treatment and implementation of portfolio optimization techniques
for another time.

%
\begin{figure}

\caption{Create a Table of Downside Statistics}

\label{fig:Downside-Table}

\begin{center}

<<tableDownside,echo=T>>=
table.DownsideRisk(managers[,1:6],Rf=.03/12)
@

\end{center}
\end{figure}


Another very widely used downside risk measures is analysis of drawdowns,
or loss from peak value achieved. The simplest method is to check
the \code{\link{maxDrawdown}}, as this will tell you the worst
cumulative loss ever sustained by the asset. If you want to look at
all the drawdowns, you can use \code{\link{table.Drawdowns}}
to find and sort them in order from worst/major to smallest/minor,
as shown in Figure \ref{fig:Drawdowns-Table}.

%
\begin{figure}

\caption{Create a Table of Sorted Drawdowns}

\label{fig:Drawdowns-Table}

<<tableDrawdowns,echo=T>>=
table.Drawdowns(managers[,1,drop=F])
@
\end{figure}


The \code{\link{UpDownRatios}} function may give you some insight
into the impacts of the skewness and kurtosis of the returns, and
letting you know how length and magnitude of up or down moves compare
to each other. Or, as mentioned above, you can also plot drawdowns
with \code{\link{chart.Drawdown}}.




\section{Conclusion}

With that short overview of a few of the capabilities provided by
\code{PerformanceAnalytics},
we hope that the accompanying package and documentation will partially
fill a hole in the tools available to a financial engineer or analyst.
If you think there's an important gap or possible improvement to be
made, please don't hesitate to contact us.
\end{document}
