diff --git a/R/SFM.fit.models.R b/R/SFM.fit.models.R index 5c860082..99184b9d 100644 --- a/R/SFM.fit.models.R +++ b/R/SFM.fit.models.R @@ -68,7 +68,8 @@ SFM.fit.models <- function(Ra, Rb, Rf=0, family = "mopt", # option is that program will ask you to choose a plot # plots: Boolean to output plots after the function. Defaults to TRUE # Output: - # Optional Graphs comparing models, and returns a fitted object + # Graphs comparing models & a fitted object is returned, If plots = TRUE is used + # Only a fitted object is returned, If plots = FALSE is used # FUNCTION: @@ -115,8 +116,13 @@ SFM.fit.models <- function(Ra, Rb, Rf=0, family = "mopt", else { plot(fmLSrob, which.plots = which.plots) } + + # Return the fitted object, but make it invisible + return (invisible(fmLSrob)); } - # Return the fitted object to the user - return(fmLSrob); + else{ + # Return the fitted object to the user + return(fmLSrob); + } } \ No newline at end of file diff --git a/inst/PA-Robust-Betas.tex b/inst/PA-Robust-Betas.tex new file mode 100644 index 00000000..54c3ebb4 --- /dev/null +++ b/inst/PA-Robust-Betas.tex @@ -0,0 +1,786 @@ +%% LyX 2.3.7 created this file. For more info, see http://www.lyx.org/. +%% Do not edit unless you really know what you are doing. +\documentclass[11pt]{extarticle}\usepackage[]{graphicx}\usepackage[]{xcolor} +% maxwidth is the original width if it is less than linewidth +% otherwise use linewidth (to make sure the graphics do not exceed the margin) +\makeatletter +\def\maxwidth{ % + \ifdim\Gin@nat@width>\linewidth + \linewidth + \else + \Gin@nat@width + \fi +} +\makeatother + +\definecolor{fgcolor}{rgb}{0.345, 0.345, 0.345} +\newcommand{\hlnum}[1]{\textcolor[rgb]{0.686,0.059,0.569}{#1}}% +\newcommand{\hlstr}[1]{\textcolor[rgb]{0.192,0.494,0.8}{#1}}% +\newcommand{\hlcom}[1]{\textcolor[rgb]{0.678,0.584,0.686}{\textit{#1}}}% +\newcommand{\hlopt}[1]{\textcolor[rgb]{0,0,0}{#1}}% +\newcommand{\hlstd}[1]{\textcolor[rgb]{0.345,0.345,0.345}{#1}}% +\newcommand{\hlkwa}[1]{\textcolor[rgb]{0.161,0.373,0.58}{\textbf{#1}}}% +\newcommand{\hlkwb}[1]{\textcolor[rgb]{0.69,0.353,0.396}{#1}}% +\newcommand{\hlkwc}[1]{\textcolor[rgb]{0.333,0.667,0.333}{#1}}% +\newcommand{\hlkwd}[1]{\textcolor[rgb]{0.737,0.353,0.396}{\textbf{#1}}}% +\let\hlipl\hlkwb + +\usepackage{framed} +\makeatletter +\newenvironment{kframe}{% + \def\at@end@of@kframe{}% + \ifinner\ifhmode% + \def\at@end@of@kframe{\end{minipage}}% + \begin{minipage}{\columnwidth}% + \fi\fi% + \def\FrameCommand##1{\hskip\@totalleftmargin \hskip-\fboxsep + \colorbox{shadecolor}{##1}\hskip-\fboxsep + % There is no \\@totalrightmargin, so: + \hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}% + \MakeFramed {\advance\hsize-\width + \@totalleftmargin\z@ \linewidth\hsize + \@setminipage}}% + {\par\unskip\endMakeFramed% + \at@end@of@kframe} +\makeatother + +\definecolor{shadecolor}{rgb}{.97, .97, .97} +\definecolor{messagecolor}{rgb}{0, 0, 0} +\definecolor{warningcolor}{rgb}{1, 0, 1} +\definecolor{errorcolor}{rgb}{1, 0, 0} +\newenvironment{knitrout}{}{} % an empty environment to be redefined in TeX + +\usepackage{alltt} +\usepackage[T1]{fontenc} +\usepackage[latin9]{inputenc} +\usepackage{geometry} +\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in} +\setlength{\parskip}{\medskipamount} +\setlength{\parindent}{0pt} +\usepackage{color} +\usepackage{float} +\usepackage{url} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{graphicx} +\usepackage{setspace} +\onehalfspacing +\usepackage[unicode=true,pdfusetitle, + bookmarks=true,bookmarksnumbered=true,bookmarksopen=false, + breaklinks=false,pdfborder={0 0 1},backref=false,colorlinks=false] + {hyperref} + +\makeatletter +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. +%%%%%%%%%%%%%%%%%%%% book.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% +% sample root file for the chapters of your "monograph" +% +% Use this file as a template for your own input. +% +%%%%%%%%%%%%%%%% Springer-Verlag %%%%%%%%%%%%%%%%%%%%%%%%%% + + +% RECOMMENDED %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +% choose options for [] as required from the list +% in the Reference Guide + + +\usepackage[bottom]{footmisc}% places footnotes at page bottom + +\usepackage{hyperref} +\def\UrlBreaks{\do\/\do-} + +\usepackage[style=authoryear,natbib=true,firstinits=true,backend=biber]{biblatex} + +\setlength{\bibitemsep}{0.5\baselineskip} +\DeclareNameAlias{sortname}{last-first} + +\addbibresource{robregJAM.bib} +\renewcommand\cite{\citet} + +%\renewcommand*{\nameyeardelim}{\addspace} +%\renewbibmacro{in:}{} + +\usepackage{txfonts} +\usepackage{upgreek} +\usepackage{booktabs} + +\hypersetup{pdfstartview={XYZ null null 1.00}} + +\makeatother +\IfFileExists{upquote.sty}{\usepackage{upquote}}{} +\begin{document} + + +\title{Robust Betas for PerformanceAnalytics} +\maketitle +\begin{center} +R. Douglas Martin and Dhairya Jain +\par\end{center} + +\tableofcontents{} + +\section{Introduction} + +A time series Single Factor Model (SFM) has the general form + +\begin{equation} +r_{t}=\alpha+\beta f_{t}+\epsilon_{t}\,t=1,2,\cdots,T\label{eq: singleFactorModel} +\end{equation} + +where $r_{t}$ is a time series of asset returns, such as those for +a stock, an ETF or a hedge fund, $f_{t}$ is a factor return, $\epsilon_{t}$ +is the error term, and $\alpha$ and $\beta$ are unknown intercept +and slope coefficients that need to be estimated based on observed +asset and factor returns. The factor return $f_{t}$ is typically +either: (1) a market proxy $f_{M,t}$, such as the CRSP value-weighted +market index $f_{M,t}=f_{CRSP,t}$ in the context of the Capital Asset +Pricing Model (CAPM), or (2) an active manager's index benchmark such +as the S\&P500, Russell 1000, Russell 2000, and Russell 3000, among +others. + +The SFM errors for arbitrary intercept $a$ and slope $b$ values +are defined as: + +\begin{equation} +\epsilon_{t}(a,b)=r_{t}-a-bf_{t},\,t=1,2,\cdots,T.\label{eq: errors} +\end{equation} + +For estimates $a=\hat{\alpha}$ and $b=\hat{\beta}$, the above errors +become the fitted model \emph{residuals} + +\begin{equation} +\hat{\epsilon}_{t}=\epsilon_{t}(\hat{\alpha},\hat{\beta})=r_{t}-\hat{\alpha}-\hat{\beta}f_{t}\label{eq: residuals} +\end{equation} + +and one has the fit-plus-residuals representation of the asset return: + +\begin{equation} +r_{t}=\hat{\alpha}-\hat{\beta}f_{t}+\hat{\epsilon}_{t}.\label{eq: fitPlusResiduals} +\end{equation} + +The unknown coefficients $\alpha$ and $\beta$ are currently almost +universally estimated using the method of least squares (LS), which +is computed by minimizing the sum of the squared errors $\sum_{t=1}^{T}\epsilon_{t}^{2}(a,b)$ +with respect to $a$ and $b$. Unfortunately, both asset returns and +factor returns often contain outliers which adversely influence the +LS intercept and slope estimates, and one needs a robust alternative +which is computed as a complement to LS, or a replacement for LS, +depending on the context. + +In this Vignette we describe and illustrate the use of a highly robust +estimator of SFM slope and intercept parameters. This estimator called +the \emph{mOpt} estimator (the Robust estimator), and it has an intuitive +weighted-least-squares (WLS) interpretation that it minimizes the +weighted sum of squared regression residuals $\sum_{t=1}^{T}w_{t}(a,b)\epsilon_{t}^{2}(a,b)$, +where $w_{t}(a,b)$ is a special data-dependent weight function described +in Section \ref{sec: mOpEstimatorDetails}. + +The \emph{mOpt} estimator was recently introduced for time series +factor models in \citet{MartinXiaTS2022}, who used it to study the +performance of mOpt relative to LS in estimating CAPM betas for the +cross-section of liquid U.S. stocks from 1963 to 2018, and for fitting +multifactor time series models such as the Fama-French 3 factor model.\footnote{This paper appeared in the March 2022 issue of The Journal of Asset +Management, and is available in opens source form at \url{https://link.springer.com/content/pdf/10.1057/s41260-022-00258-0.pdf}.} The study revealed extensive adverse influence of outliers on the +LS betas. In particular, it was shown that the LS and mOpt betas differ +in absolute values by at least 0.3 for roughly 26\% of microcap stocks, +14\% of smallcaps and 7\% of bigcaps, and by at least 0.5 for roughly +12\% of microcap stocks, 5\% of smallcaps and 2\% of bigcaps. + +Section 2 introduces the Robust SFM fitting functions in the PerformanceAnalytics +R package, and illustrates their use for package \texttt{managers} +data set. Section 3 provides some mathematical details for the mOpt +Robust estimator, and Section 4 provides concluding comments. + +\section{The PerformanceAnalytics Robust Betas Functions} + +PerformanceAnalytics package contains the following two main functions +for computing LS and mOpt (Robust) SFM model fits: +\begin{itemize} +\item \texttt{chart.SFM} +\item \texttt{SFM.fit.models} +\end{itemize} +The \texttt{chart.SFM} function computes both LS and Robust alphas +and betas and overlays the corresponding straight line fits to a scatter +plot of asset returns versus benchmark or market proxy returns. The +\texttt{SFM.fit.models} function also computes both LS and Robust +SFM fits, in order to provide: (a) a comparative tabular display of +the LS and Robust alphas, betas, and related statistics, and (b) an +optional display of any subset of 10 different comparative graphical +displays of the LS and Robust SFM fits. + +In addition the following function allows the user to compute either +mOpt or LS robust SFM fits, with mOpt the default, for any combination +of one or more sets of asset returns and one or more benchmarks: +\begin{itemize} +\item \texttt{SFM.coefficients} +\end{itemize} +In order to use these functions, an R user needs to first install +the current version of PerformanceAnalytics from CRAN (\url{https://cran.r-project.org/web/packages/PerformanceAnalytics/index.html}), +and load it with: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{library}\hlstd{(PerformanceAnalytics)} +\end{alltt} +\end{kframe} +\end{knitrout} + + + +\subsection{The \texttt{managers} Data Set} + +The following examples will use the \texttt{xts} time series data +set \texttt{managers} included with PerformanceAnalytics, so we first +load this data set and determine its class, dimensions, and names +with the code: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{data}\hlstd{(managers)} +\hlkwd{class}\hlstd{(managers)} +\end{alltt} +\begin{verbatim} +## [1] "xts" "zoo" +\end{verbatim} +\begin{alltt} +\hlkwd{dim}\hlstd{(managers)} +\end{alltt} +\begin{verbatim} +## [1] 132 10 +\end{verbatim} +\begin{alltt} +\hlkwd{names}\hlstd{(managers)} +\end{alltt} +\begin{verbatim} +## [1] "HAM1" "HAM2" "HAM3" "HAM4" "HAM5" +## [6] "HAM6" "EDHEC LS EQ" "SP500 TR" "US 10Y TR" "US 3m TR" +\end{verbatim} +\end{kframe} +\end{knitrout} + +The results show that \texttt{managers} is an \texttt{xts} data object +consisting of 10 time series for 132 months. Now we replace the last +4 names of \texttt{managers} with shorter convenient names with no +spaces: + + + +Next we use the function \texttt{tsPlotMP} from the PCRA package to +create the time series plots shown in Figure \ref{fig: mgrsTseries}. + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} + +\hlstd{PCRA}\hlopt{::}\hlkwd{tsPlotMP}\hlstd{(managers,} \hlkwc{scaleType} \hlstd{=} \hlstr{"same"}\hlstd{,} \hlkwc{axis.cex} \hlstd{=} \hlnum{0.5}\hlstd{,} \hlkwc{stripText.cex} \hlstd{=} \hlnum{0.5}\hlstd{)} +\end{alltt} +\begin{verbatim} +\end{verbatim} +\end{kframe} +\end{knitrout} + +\begin{figure}[H] +\begin{centering} +\includegraphics[width=1\textwidth]{Plots/mgrsTseries.png} +\par\end{centering} +\caption{Time Series of managers Data} + +\label{fig: mgrsTseries} +\end{figure} + +The figure shows that some of the times series data begins in January +1996, but some series begin later, and all series continue until December +2008. It will be convenient for the calculations below to use the +maximum time window of the managers data such that none of the time +series have missing data. This maximum length window is easily determine +using the \texttt{na.omit} and \texttt{range} functions as follows: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{range}\hlstd{(}\hlkwd{index}\hlstd{(}\hlkwd{na.omit}\hlstd{(managers)))} +\end{alltt} +\begin{verbatim} +## [1] "2001-09-30" "2006-12-31" +\end{verbatim} +\end{kframe} +\end{knitrout} + +In order to avoid using last four months fraction of the year 2001, +we use the following code line to delete those four months and rename +the result: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlstd{mgrs} \hlkwb{<-} \hlstd{managers[}\hlstr{"2002/"}\hlstd{]} \hlcom{# 5 full years from 2002 through 2006} +\end{alltt} +\end{kframe} +\end{knitrout} + +\subsection{Use of \texttt{chart.SFM}} + +The function \texttt{chart.SFM} computes LS and mOpt robust alphas +and betas, and makes a graphical display of the resulting LS and mOpt +straight line fits, superimposed on the scatter plot of asset returns +and factor returns. The arguments of \texttt{chart.SFM} are viewed +with: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{args}\hlstd{(chart.SFM)} +\end{alltt} +\begin{verbatim} +## function (Ra, Rb, Rf = 0, main = NULL, ylim = NULL, xlim = NULL, +## family = "mopt", xlab = NULL, ylab = NULL, legend.loc = "topleft", +## makePct = FALSE) +## NULL +\end{verbatim} +\end{kframe} +\end{knitrout} + +NOTE: With the default NULL optional arguments for the \texttt{xlim} +and \texttt{ylim} axes limits, \texttt{chart.SFM} uses sensible data +dependent values, as will be seen in resulting plot in the figure +below. You can obtain more information about the arguments of \texttt{chart.SFM} +by using the \texttt{help()} function: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{help}\hlstd{(chart.SFM)} +\end{alltt} +\end{kframe} +\end{knitrout} + +Figure \ref{fig: mOptLSforHAM6} shows the result of using \texttt{chart.SFM} +for the HAM6 returns and the S\&P500 as the benchmark. + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{chart.SFM}\hlstd{(mgrs}\hlopt{$}\hlstd{HAM6, mgrs}\hlopt{$}\hlstd{SP500, mgrs}\hlopt{$}\hlstd{RF, }\hlkwc{makePct} \hlstd{=} \hlnum{TRUE}\hlstd{)} +\end{alltt} +\begin{verbatim} +\end{verbatim} +\end{kframe} +\end{knitrout} + +\begin{figure}[H] +\begin{centering} +\includegraphics[width=0.67\textwidth]{Plots/mOptLSforHAM6.png} +\par\end{centering} +\caption{Robust mOpt Beta and LS Beta for HAM6} + +\label{fig: mOptLSforHAM6} +\end{figure} + +You can use a simple \texttt{for} loop to plot the mOpt and LS fits +for any subset of the HAM1 through HAM6 funds and LSEQ, for example +you do so for HAM3 through HAM6 with the code line: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwa{for}\hlstd{(k} \hlkwa{in} \hlnum{3}\hlopt{:}\hlnum{6}\hlstd{)\{} + \hlkwd{chart.SFM}\hlstd{(mgrs[,k], mgrs}\hlopt{$}\hlstd{SP500, mgrs}\hlopt{$}\hlstd{RF, }\hlkwc{makePct} \hlstd{= T,} + \hlkwc{main} \hlstd{=} \hlkwd{names}\hlstd{(mgrs[,k]))} +\hlstd{\}} +\end{alltt} +\end{kframe} +\end{knitrout} + +\begin{figure}[H] +\begin{centering} +\includegraphics[width=0.5\textwidth]{Plots/HAM3.png}\includegraphics[width=0.5\textwidth]{Plots/HAM4.png} +\par\end{centering} +\begin{centering} +\includegraphics[width=0.5\textwidth]{Plots/HAM5.png}\includegraphics[width=0.5\textwidth]{Plots/HAM6.png} +\par\end{centering} +\caption{Robust mOpt Beta and LS Beta for HAM3 - HAM6} + +\label{fig: mOptLSforHAM3-6} +\end{figure} + + +\subsection{Use of \texttt{SFM.fit.models}} + +The function \texttt{SFM.fit.models} has two main capabilities. The +first is to compute both LS and Robust alpha and beta coefficient +estimates, along with their standard errors and t-statistics, and +display them in easy-to-compare table form. The second is to make +side-by-side graphical displays of Robust and LS model fitting results. +First, we find out what the arguments of \texttt{SFM.fit.models} are +with: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{args}\hlstd{(SFM.fit.models)} +\end{alltt} +\begin{verbatim} +## function (Ra, Rb, Rf = 0, family = "mopt", which.plots = NULL, +## plots = TRUE) +## NULL +\end{verbatim} +\end{kframe} +\end{knitrout} + +Use the first code line below to compute the Robust and LS coefficients +with no graphics, and save the result of the fitted models object +fitHAM6, displays. Then use the next 3 code lines to see what the +class of fitHAM6 is, display the LS and Robust alpha and beta coefficient +estimates, and display a complete comparative LS and Robust statistics +summary: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlstd{fitHAM6} \hlkwb{<-} \hlkwd{SFM.fit.models}\hlstd{(mgrs}\hlopt{$}\hlstd{HAM6, mgrs}\hlopt{$}\hlstd{SP500, }\hlkwc{Rf} \hlstd{= mgrs}\hlopt{$}\hlstd{RF, } + \hlkwc{plots} \hlstd{=} \hlnum{FALSE}\hlstd{)} +\hlkwd{class}\hlstd{(fitHAM6)} +\end{alltt} +\begin{verbatim} +## [1] "lmfm" "fit.models" +\end{verbatim} +\begin{alltt} +\hlkwd{round}\hlstd{(}\hlkwd{coef}\hlstd{(fitHAM6),}\hlnum{3}\hlstd{)} +\end{alltt} +\begin{verbatim} +## (Intercept) Beta +## LSFit 0.006 0.325 +## RobFit 0.007 0.548 +\end{verbatim} +\begin{alltt} +\hlkwd{summary}\hlstd{(fitHAM6)} +\end{alltt} +\begin{verbatim} +## +## Calls: +## LSFit: lm(formula = Alpha ~ Beta, data = merged, subset = subset) +## RobFit: RobStatTM::lmrobdetMM(formula = Alpha ~ Beta, data = merged, +## subset = subset, control = RobStatTM::lmrobdet.control(family = family)) +## +## Residual Statistics: +## Min 1Q Median 3Q Max +## LSFit: -0.04504 -0.01249 0.003436 0.01277 0.04062 +## RobFit: -0.06474 -0.01175 0.003893 0.01185 0.06478 +## +## Coefficients: +## Estimate Std. Error t value Pr(>|t|) +## (Intercept): LSFit: 0.006331 0.002632 2.405 0.0194 * +## RobFit: 0.006753 0.002392 2.824 0.0065 ** +## +## Beta: LSFit: 0.325048 0.073839 4.402 4.67e-05 *** +## RobFit: 0.547858 0.080665 6.792 6.55e-09 *** +## --- +## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 +## +## Residual Scale Estimates: +## LSFit: 0.02028 on 58 degrees of freedom +## RobFit: 0.01943 on 58 degrees of freedom +## +## Multiple R-squared: +## LSFit: 0.2504 +## RobFit: 0.2364 +\end{verbatim} +\end{kframe} +\end{knitrout} + +To make a wide variety of plots that compare Robust and LS SFM model +fitting results use + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{SFM.fit.models}\hlstd{(mgrs}\hlopt{$}\hlstd{HAM6,mgrs}\hlopt{$}\hlstd{SP500,}\hlkwc{Rf} \hlstd{= mgrs}\hlopt{$}\hlstd{RF)} +\end{alltt} +}\end{kframe} +\end{knitrout} + +which results in the following output in the Console: + +\texttt{\textcolor{blue}{Make plot selections (or 0 to exit): }} + +\texttt{\textcolor{blue}{1: Normal QQ Plot of Residuals}} + +\texttt{\textcolor{blue}{2: Kernel Density Estimate of Residuals}} + +\texttt{\textcolor{blue}{3: Residuals vs. Mahalanobis Distance}} + +\texttt{\textcolor{blue}{4: Residuals vs. Fitted Values}} + +\texttt{\textcolor{blue}{5: Sqrt Residuals vs. Fitted Values}} + +\texttt{\textcolor{blue}{6: Response vs. Fitted Values}} + +\texttt{\textcolor{blue}{7: Residuals vs. Index (Time)}} + +\texttt{\textcolor{blue}{8: Overlaid Normal QQ Plot of Residuals}} + +\texttt{\textcolor{blue}{9: Overlaid Kernel Density Estimate of Residuals}} + +\texttt{\textcolor{blue}{10: Scatter Plot with Overlaid Fit(s)}} + +\texttt{\textcolor{blue}{Selection:}} + +The first time you try this, we suggest that you enter each of the +choices 1 through 10 after Selection, and after each selection you +with see the corresponding plot type, then enter 0 to exit from the +graphics display menu. If you just want a particular subset of graphical +displays, e.g., types 2 and 7, just enter 2 to see the first plot +and then enter 7 after Selection, followed by 0 to Exit. Alternatively, +use of the command + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{SFM.fit.models}\hlstd{(mgrs}\hlopt{$}\hlstd{HAM6, mgrs}\hlopt{$}\hlstd{SP500, }\hlkwc{Rf} \hlstd{= mgrs}\hlopt{$}\hlstd{RF, }\hlkwc{which.plots} \hlstd{=} \hlkwd{c}\hlstd{(}\hlnum{2}\hlstd{,}\hlnum{7}\hlstd{))}\end{alltt} +\begin{verbatim} +\end{verbatim} +\end{kframe} +\end{knitrout} +results in the following in the Console: + +\texttt{\textcolor{blue}{Hit to see next plot:}} + +Then pressing Enter results in display of the type 2 plot in the top +of Figure \ref{fig: plotTypes 2 and 7}, and pressing Enter again +results in display of the bottom 7 plot in Figure \ref{fig: plotTypes 2 and 7}, +and then the above line in the Console disappears. Try it out. + +\begin{figure}[H] +\begin{centering} +\includegraphics[width=0.6\textwidth]{Plots/kernelDensityEstimates.png} +\par\end{centering} +\begin{centering} +\includegraphics[width=0.6\textwidth]{Plots/residualsVsTime.png} +\par\end{centering} +\caption{The Top Plot is Type 2 and the Bottom Plot is Type 7} + +\label{fig: plotTypes 2 and 7} +\end{figure} + + +\subsection{Use of SFM.coefficients} + +The function \texttt{SFM.coefficients} was designed to support computing +either Robust mOpt (the default) or LS multiple single factor model +(SFM) for one or more assets and one or more benchmarks. Here are +the arguments of \texttt{SFM.coefficients}: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{args}\hlstd{(SFM.coefficients)} +\end{alltt} +\begin{verbatim} +## function (Ra, Rb, Rf = 0, subset = TRUE, ..., method = "Robust", +## family = "mopt", digits = 3, benchmarkCols = T, Model = F, +## warning = T) +## NULL +\end{verbatim} +\end{kframe} +\end{knitrout} + +Here we use the first four managers HAM1, HAM2, HAM3, HAM4 as the +assets, and the SP500 and Bond10Yr as the benchmarks: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlstd{funds} \hlkwb{<-} \hlstd{mgrs[, }\hlkwd{c}\hlstd{(}\hlstr{"HAM1"}\hlstd{, }\hlstr{"HAM2"}\hlstd{, }\hlstr{"HAM3"}\hlstd{, }\hlstr{"HAM4"}\hlstd{)]} +\hlstd{benchmarks} \hlkwb{<-} \hlstd{mgrs[, }\hlkwd{c}\hlstd{(}\hlstr{"SP500"}\hlstd{, }\hlstr{"Bond10Yr"}\hlstd{)]} +\end{alltt} +\end{kframe} +\end{knitrout} + +Now we make Robust and LS fits of the four mangers to the two benchmarks, +and examine the class of the resulting \texttt{fit.Rob} (\texttt{fit.LS} +has the same matrix class). + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlstd{(fit.ROB} \hlkwb{<-} \hlkwd{SFM.coefficients}\hlstd{(funds, benchmarks,} \hlkwc{method} \hlstd{=}\hlstr{"Robust"}\hlstd{))} +\end{alltt} +\begin{verbatim} +## Alpha : SP500 Alpha : Bond10Yr Beta : SP500 Beta : Bond10Yr +## HAM1 0.006 0.012 0.595 -0.334 +## HAM2 0.001 0.002 0.184 -0.275 +## HAM3 0.003 0.008 0.586 -0.331 +## HAM4 0.003 0.017 1.190 -0.286 +\end{verbatim} +\begin{alltt} +\hlstd{(fit.LS} \hlkwb{<-} \hlkwd{SFM.coefficients}\hlstd{(funds, benchmarks,} \hlkwc{method} \hlstd{=}\hlstr{"LS"}\hlstd{))} +\end{alltt} +\begin{verbatim} +## Alpha : SP500 Alpha : Bond10Yr Beta : SP500 Beta : Bond10Yr +## HAM1 0.006 0.011 0.599 -0.426 +## HAM2 0.002 0.004 0.216 -0.215 +## HAM3 0.002 0.007 0.555 -0.378 +## HAM4 0.008 0.015 0.923 -0.429 +\end{verbatim} +\begin{alltt} +\hlkwd{class}\hlstd{(fit.ROB)} +\end{alltt} +\begin{verbatim} +## [1] "matrix" "array" +\end{verbatim} +\begin{alltt} +\hlkwd{summary}\hlstd{(fit.ROB)} +\end{alltt} +\begin{verbatim} +## Alpha : SP500 Alpha : Bond10Yr Beta : SP500 Beta : Bond10Yr +## Min. :0.00100 Min. :0.00200 Min. :0.1840 Min. :-0.3340 +## 1st Qu.:0.00250 1st Qu.:0.00650 1st Qu.:0.4855 1st Qu.:-0.3317 +## Median :0.00300 Median :0.01000 Median :0.5905 Median :-0.3085 +## Mean :0.00325 Mean :0.00975 Mean :0.6388 Mean :-0.3065 +## 3rd Qu.:0.00375 3rd Qu.:0.01325 3rd Qu.:0.7438 3rd Qu.:-0.2833 +## Max. :0.00600 Max. :0.01700 Max. :1.1900 Max. :-0.2750 +\end{verbatim} +\end{kframe} +\end{knitrout} + +We note that since \texttt{method = ``Robust''} is the default, +that argument may be omitted in the first code line above. + +By default, the benchmark Alpha and Beta results are in columns, and +those of the assets are in rows. This is because portfolio managers +often have many assets in their portfolio and only a few benchmarks. +Note that \texttt{fit.Rob} and \texttt{fit.LS} are R matrix objects, +and the results are printed with the default digits = 3. You can get +the robust fit results displayed with benchmarks in rows, and 6 significant +digits with the code line: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{SFM.coefficients}\hlstd{(funds, benchmarks,} \hlkwc{benchmarkCols} \hlstd{=} \hlnum{FALSE}\hlstd{,} \hlkwc{digits} \hlstd{=} \hlnum{6}\hlstd{)} +\end{alltt} +\begin{verbatim} +## HAM1 HAM2 HAM3 HAM4 +## Alpha : SP500 0.005602 0.001172 0.003275 0.002534 +## Alpha : Bond10Yr 0.011842 0.002482 0.007719 0.017016 +## Beta : SP500 0.594613 0.183761 0.586422 1.189505 +## Beta : Bond10Yr -0.334156 -0.274526 -0.330747 -0.285967 +\end{verbatim} +\end{kframe} +\end{knitrout} + +You can use the function \texttt{SFM.alpha} if you only want alpha +estimates, and use \texttt{SFM.beta} if you only want beta estimates. +For example the following gives robust alphas + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{SFM.alpha}\hlstd{(funds, benchmarks,} \hlkwc{digits} \hlstd{=} \hlnum{4}\hlstd{)} +\end{alltt} +\begin{verbatim} +## Alpha : SP500 Alpha : Bond10Yr +## HAM1 0.0058 0.0110 +## HAM2 0.0023 0.0044 +## HAM3 0.0024 0.0071 +## HAM4 0.0078 0.0148 +\end{verbatim} +\end{kframe} +\end{knitrout} + +and the following gives LS betas: + +\begin{knitrout} +\definecolor{shadecolor}{rgb}{0.969, 0.969, 0.969}\color{fgcolor}\begin{kframe} +\begin{alltt} +\hlkwd{SFM.alpha}\hlstd{(funds, benchmarks,} \hlkwc{method} \hlstd{=} \hlstr{"LS"}\hlstd{,} \hlkwc{digits} \hlstd{=} \hlnum{2}\hlstd{)} +\end{alltt} +\begin{verbatim} +## Alpha : SP500 Alpha : Bond10Yr +## HAM1 0.01 0.01 +## HAM2 0.00 0.00 +## HAM3 0.00 0.01 +## HAM4 0.01 0.01 +\end{verbatim} +\end{kframe} +\end{knitrout} + +\section{The mOpt Robust SFM Fit Mathematical Details\label{sec: mOpEstimatorDetails}} + +The SFM model \ref{eq: singleFactorModel} can be written in the following +form: + +\begin{align*} +r_{t} & =\mathbf{\tilde{f}}_{t}^{\prime}\boldsymbol{\theta}+s\epsilon_{t},\;t=1,2,\cdots,T +\end{align*} + +where $\mathbf{\tilde{f}}_{t}^{\prime}=(1,f_{t})$, $\boldsymbol{\theta}=(\alpha,\beta)^{\prime}$, +and $\epsilon_{t}$ is a standardized error term that is scaled by +the scale parameter $s$. The robust estimate $\hat{\boldsymbol{\theta}}=(\hat{\alpha},\hat{\beta})$ +is a solution of the weighted least squares (WLS) estimating equation + +\begin{equation} +\sum_{i=1}^{T}w_{t}\mathbf{\tilde{f}}_{t}\left(r_{t}-\mathbf{\tilde{f}}_{t}^{\prime}\hat{\boldsymbol{\theta}}\right)=0\label{eq: regMestWLSEqn} +\end{equation} + +where the $w_{t}$ are the data--dependent weights + +\begin{equation} +w_{t}=w_{t}(\hat{\boldsymbol{\theta}};\hat{s})=w_{mOpt}\left(\frac{r_{t}-\mathbf{\tilde{f}}_{t}^{\prime}\hat{\boldsymbol{\theta}}}{\hat{s}}\right)\label{eq: regMestWeights} +\end{equation} + +and the shape of the weight function $w_{mOpt}(t)$ is shown in Figure +\ref{fig: mOptWtFunction}. + +\begin{figure}[H] +\begin{centering} +\includegraphics[width=0.7\textwidth]{Plots/mOptWt.png} +\par\end{centering} +\caption{The mOpt weight function ($c=3.00$)} + +\label{fig: mOptWtFunction} +\end{figure} + +The mOpt weight function gives a weight of 1 to all sufficiently small +robustly scaled residuals $\hat{\epsilon}_{t}=(r_{t}-\mathbf{\tilde{f}}_{t}^{\prime}\hat{\boldsymbol{\theta}})/\hat{s}$, +and smoothly transitions to zero weight for robustly scaled residuals +whose absolute is greater than 3.00. All asset and factor returns +pairs $(r_{t},\mathbf{\tilde{f}}_{t})$ whose scaled residuals have +absolute values larger than 3.0 are are \emph{rejected} by the by +the $\hat{\boldsymbol{\theta}}$ estimator. For normally distributed +data and true parameter values, the probability that such a pair is +rejected is only 0.27\%, and the estimator is essentially equivalent +to the LS estimator. + +The weights $w_{t}=w_{t}(\hat{\boldsymbol{\theta}};\hat{s})$ depend +on the values of $\hat{\boldsymbol{\theta}}$, $r_{t}$, and $\mathbf{\tilde{f}}_{t}$. +Consequently. the WLS equation (\ref{eq: regMestWLSEqn}) is a nonlinear +function of the data $(r_{t},f_{t}),t=1,\cdots,T$, and $\hat{\boldsymbol{\theta}}$ +must be computed with some type of iterative nonlinear algorithm. +It is quite convenient that the estimate $\hat{\boldsymbol{\theta}}$ +may be expressed in the nonlinear weighted least squares (WLS) mathematical +form + +\begin{equation} +\hat{\boldsymbol{\theta}}=\left(\sum_{i=1}^{T}w_{t}(\hat{\boldsymbol{\theta}};\hat{s})\mathbf{\tilde{f}}_{t}\mathbf{\tilde{f}}_{t}^{\prime}\right)^{-1}\left(\sum_{i=1}^{T}w_{t}(\hat{\boldsymbol{\theta}};\hat{s})\mathbf{\tilde{f}}_{t}r_{t}\right)\label{eq: robregWLSversion-1} +\end{equation} + +which lends itself to the iterated weighted least squares (IRWLS) +algorithm: + +\begin{equation} +\hat{\boldsymbol{\theta}}^{k+1}=\left(\sum_{i=1}^{T}w_{t}(\hat{\boldsymbol{\theta}}^{k};\hat{s})\mathbf{\tilde{f}}_{t}\mathbf{\tilde{f}}_{t}^{\prime}\right)^{-1}\left(\sum_{i=1}^{T}w_{t}(\hat{\boldsymbol{\theta}}^{k};\hat{s})\mathbf{\tilde{f}}_{t}r_{t}\right),\,k=0,1,2,\cdots.\label{eq: regMestIRWLS-1} +\end{equation} + +The mOpt Robust estimator is computed with the above IRWLS algorithm, +using a highly robust but inefficient initial estimate $\hat{\boldsymbol{\theta}}^{0}$. +For further details, see \citet{MartinXiaTS2022}. + +\printbibliography +\end{document} diff --git a/inst/robregJAM.bib b/inst/robregJAM.bib new file mode 100644 index 00000000..1bbc7973 --- /dev/null +++ b/inst/robregJAM.bib @@ -0,0 +1,1751 @@ +% Encoding: UTF-8 + +@InCollection{MartinClarkGreen:2010, + author = {Martin, R. Douglas and Clark, Andrew and Green, Christopher G.}, + title = {Robust Portfolio Construction}, + pages = {337--382}, + crossref = {Guerard:2010book}, +} +################################################################### +# +# Bibliography for FF92 extension papers +# Christopher G. 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Accessed March 2015 via Wharton Research Data Services.}, + title = {{CRSP/Compustat Merged Database}}, + year = {2015}, + key = {CCM:2015}, + publisher = {Graduate School of Business, The University of Chicago 2001-2004}, +} + +@Misc{PalaciosVora:2011, + author = {Palacios, Luis and Vora, Premal}, + note = {A {SAS} program to replicate the \cite{FamaFrench:1993} {SMB} and {HML} factors. WRDS Research Applications. Original version April 2009, Updated August 2011. Available from \url{http://wrds-web.wharton.upenn.edu/wrds/research/applications/risk/fama-french/index.cfm}. Accessed April 2014.}, + title = {{SAS} Program \texttt{fama\_french\_factors\_replication.sas}}, + year = {2011}, +} + +@Misc{COMP, + author = {Compustat}, + note = {Used with permission. All rights reserved. Accessed March 2015 via Wharton Research Data Services.}, + title = {{Standard \& Poor's Compustat\textregistered{} Data}}, + year = {2015}, + key = {COMPUSTAT}, +} + +@Manual{package:robust, + title = {robust: Robust Library}, + author = {Wang, Jiahui and Zamar, Ruben and Marazzi, Alfio and Yohai, Victor J. and Salibian-Barrera, Matias and Maronna, Ricardo and Zivot, Eric and Rocke, David and Martin, Doug and Maechler, Martin and Konis, Kjell}, + note = {R package version 0.4-16}, + year = {2014}, + xurl = {https://CRAN.R-project.org/package=robust}, +} + +@Manual{package:robustbase, + title = {robustbase: Basic Robust Statistics}, + author = {Rousseeuw, Peter and Croux, Christophe and Todorov, Valentin and Ruckstuhl, Andreas and Salibian-Barrera, Matias and Verbeke, Tobias and Koller, Manuel and Maechler, Martin}, + note = {R package version 0.92-6}, + year = {2016}, + xurl = {https://CRAN.R-project.org/package=robustbase}, +} + +@Manual{package:mmregress, + title = {mmregress}, + author = {Verardi, Vincenzo and Croux, Christophe}, + note = {Stata package}, + year = {2009}, +} + +@Unpublished{Berk:1995b, + author = {Berk, Johnathan B.}, + note = {Working paper, University of Washington}, + title = {An Empirical Reexamination of the Relation between Firm Size and Return}, + year = {1995}, +} + +@article{ GarzaGomezEtAl:2001, + author="Garza-G{\'o}mez, Xavier and Hodoshima, Jiro and Kunimura, Michio", + title="Reexamining the Robustness of the Market Value of Equity", + journal="Asia-Pacific Financial Markets", + year="2001", + volume="8", + number="2", + pages="61--85", + issn="1573-6946", +} + +@Article{Sharpe:1964, + author = {Sharpe, William F.}, + journal = {Journal of Finance}, + title = {Capital asset prices: A Theory of Market Equilibrium under Conditions of Risk}, + year = {1964}, + number = {3}, + pages = {425--442}, + volume = {19}, +} + +@Article{Lintner:1965, + author = {Lintner, John}, + journal = {The Review of Economics and Statistics}, + title = {The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets}, + year = {1965}, + month = feb, + number = {1}, + pages = {13--37}, + volume = {47}, +} + +@Article{HorowitzEtAl:2000a, + author = {Horowitz, Joel L. and Loughran, Tim and Savin, N. E.}, + journal = {Journal of Empirical Finance}, + title = {Three analyses of the firm size premium}, + year = {2000}, + issn = {0927-5398}, + number = {2}, + pages = {143--153}, + volume = {7}, + keywords = {Size premia, Asset pricing, Spline regressions}, +} + +@Article{MartinYohaiZamar:1989, + author = {Martin, R. D. and Yohai, V. J. and Zamar, R. H.}, + journal = {The Annals of Statistics}, + title = {Min-max bias robust regression}, + year = {1989}, + pages = {1608--1630}, + volume = {17}, +} + +@Article{Mossin:1966, + author = {Mossin, J.}, + journal = {Econometrica}, + title = {Equilibrium in capital asset markets}, + year = {1966}, + number = {4}, + pages = {768--783}, + volume = {34}, +} + +@Article{vanDijk:2011, + author = {van Dijk, Mathijs A.}, + journal = {Journal of Banking \& Finance}, + title = {Is size dead? A review of the size effect in equity returns}, + year = {2011}, + issn = {0378-4266}, + number = {12}, + pages = {3263--3274}, + volume = {35}, + keywords = {Size effect, Cross-section of equity returns, CAPM, Anomalies}, +} + +@Article{HorowitzEtAl:2000b, + author = {Horowitz, Joel L. and Loughran, Tim and Savin, N. E.}, + journal = {Research in Economics}, + title = {The disappearing size effect}, + year = {2000}, + issn = {1090-9443}, + number = {1}, + pages = {83--100}, + volume = {54}, + keywords = {Size effect, January anomaly, risk factors.}, +} + +@Unpublished{Crain:2011, + author = {Crain, Michael A.}, + note = {Published October 29, 2011. Available at SSRN: \url{https://ssrn.com/abstract=1710076}}, + title = {A Literature Review of the Size Effect}, + year = {2011}, +} + +@Article{Black:1972, + author = {Black, Fischer}, + journal = {Journal of Business}, + title = {Capital market equilibrium with restricted borrowing}, + year = {1972}, + pages = {444--454}, + volume = {45}, +} + +@InCollection{BlackJensenScholes:1972, + author = {Black, Fischer and Jensen, M. C. and Scholes, M.}, + booktitle = {Studies in the Theory of Capital Markets}, + publisher = {Praeger}, + title = {The capital asset pricing model: some empirical tests}, + year = {1972}, + address = {New York}, + editor = {M.~C. Jensen}, +} + +@Article{Banz:1981, + author = {Banz, R. W.}, + journal = {Journal of Financial Economics}, + title = {Anomalies in relationships between securities' yields and yield surrogates}, + year = {1981}, + pages = {103--126}, + volume = {6}, +} + +@Article{Reinganum:1981, + author = {Reinganum, M. R.}, + journal = {Journal of Financial Economics}, + title = {Misspecification of Capital Asset Pricing: Empirical Anomalies Based on Earnings Yields and Market Values}, + year = {1981}, + issn = {0304-405X}, + month = mar, +} + +@Article{Keim:1983, + author = {Keim, D. B.}, + journal = {Journal of Financial Economics}, + title = {Size-related anomalies and stock return seasonality: further empirical evidence}, + year = {1983}, + month = jun, + pages = {13--32}, + volume = {12}, +} + +@Article{Stattman:1980, + author = {Stattman, D.}, + journal = {The Chicago MBA: A Journal of Selected Papers}, + title = {Book values and stock returns}, + year = {1980}, + pages = {25--45}, +} + +@Article{RosenbergReidLanstein:1985, + author = {Rosenberg, Barr and Reid, Kenneth and Lanstein, Ronald}, + journal = {Journal of Portfolio Management}, + title = {Persuasive evidence of market inefficiency}, + year = {1985}, + pages = {9--11}, + volume = {11}, +} + +@Article{FamaFrench:1993, + author = {Fama, Eugene F. and French, Kenneth R.}, + journal = {Journal of Financial Economics}, + title = {Common risk factors in the returns on stocks and bonds}, + year = {1993}, + pages = {3--56}, + volume = {33}, +} + +@Article{Dichev:1998, + author = {Dichev, Ilia D.}, + journal = {The Journal of Finance}, + title = {Is the Risk of Bankruptcy a Systematic Risk?}, + year = {1998}, + issn = {00221082, 15406261}, + number = {3}, + pages = {1131--1147}, + volume = {53}, + publisher = {[American Finance Association, Wiley]}, + url = {http://www.jstor.org/stable/117389}, +} + +@Article{Amihud:2002, + author = {Amihud, Y.}, + journal = {Journal of Financial Markets}, + title = {Illiquidity and stock returns: cross-section and time-series effects}, + year = {2002}, + pages = {31--56}, + volume = {5}, +} + +@Article{DavisFamaFrench:2000, + author = {Davis, James and Fama, Eugene F. and French, Kenneth R.}, + journal = {Journal of Finance}, + title = {Characteristics, Covariances, and Average Returns: 1929 to 1997}, + year = {2000}, + month = feb, + number = {1}, + pages = {389--406}, + volume = {55}, +} + +@Article{FamaFrench:1996, + author = {Fama, Eugene F. and French, Kenneth R.}, + journal = {The Journal of Finance}, + title = {Multifactor Explanations of Asset Pricing Anomalies}, + year = {1996}, + number = {1}, + pages = {55--84}, + volume = {51}, + copyright = {Copyright � 1996 American Finance Association}, + jstor_articletype = {research-article}, + jstor_formatteddate = {Mar., 1996}, + language = {English}, + myissn = {00221082}, + publisher = {Wiley for the American Finance Association}, +} + +@Article{FamaFrench:2008, + author = {Fama, Eugene F. and French, Kenneth R.}, + journal = {The Journal of Finance}, + title = {Dissecting Anomalies}, + year = {2008}, + number = {4}, + pages = {1653--1678}, + volume = {63}, + copyright = {Copyright � 2008 American Finance Association}, + language = {English}, + myissn = {00221082}, + publisher = {Wiley for the American Finance Association}, +} + +@TechReport{ErdoganGoldfarbIyengar:2004, + author = {Erdogan, E. and Goldfarb, D. and Iyengar, G.}, + institution = {Operations Research Center, Columbia University}, + title = {Robust Portfolio Management}, + year = {2004}, + note = {CORC Technical Report TR-2004-11}, +} + +@Article{GoldfarbIyengar:2003, + author = {Goldfarb, D. and Iyengar, G.}, + journal = {Mathematics of Operations Research}, + title = {Robust Portfolio Selction Problems}, + year = {1993}, + month = feb, + pages = {1--38}, +} + +@Unpublished{CeriaStubbs:2006a, + author = {Ceria, Sebastian and Stubbs, Robert A.}, + note = {Axioma Research Paper No. 003. 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Douglas and Clark, Andrew and Green, Christopher G.}, + title = {Robust Portfolio Construction}, + pages = {337--382}, + crossref = {Guerard:2010book}, +} +################################################################### +# +# Bibliography for FF92 extension papers +# Christopher G. 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