Path: blob/master/sandbox/riskbudgetpaper(superseded)/text paper/ExhibitsOct_2010.tex
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\documentclass[11pt]{article}1\usepackage{graphicx}2\usepackage{natbib}3\usepackage{lscape}4\renewcommand{\topfraction}{0.85}5\renewcommand{\textfraction}{0.1}6\oddsidemargin=0.5cm \topmargin=0cm \setlength{\textheight}{21cm}7\setlength{\textwidth}{15.5cm}8\renewcommand{\baselinestretch}{1.2}9\renewcommand{\figurename}{Exhibit}10\renewcommand{\tablename}{Exhibit}11\usepackage{amsmath}1213\begin{document}1415{\bf \large \noindent EXHIBITS}1617\setcounter{page}{19}181920\bigskip212223\begin{figure}[h]24%\begin{center}25\caption{Percentage CVaR contribution of asset 1 in function of its portfolio weight for a two-asset portfolio with asset returns that have a bivariate normal distribution with means $\mu_1$ and $\mu_2$, correlation $0.5$ and standard deviations $\sigma_1$ and $\sigma_2$, respectively. }26\includegraphics[width=6.5cm,angle=270]{sensitivity_rho50.eps}27%\end{center}28\end{figure}293031\begin{figure}[h]32%\begin{center}33\caption{Weight and CVaR allocation of bond-equity portfolios, together with the in-sample annualized mean and monthly 95\% CVaR over34the period January 1976-June 2010. }35\vspace{1cm}36\scalebox{0.87}{37\begin{tabular}{|lc cc c cc c cc | } \hline38& & \multicolumn{2}{c}{Weight allocation} & &\multicolumn{2}{c}{CVaR allocation} & & Ann. mean & 95\% CVaR \\39& & Bond & Equity & &Bond & Equity & & & \\ \hline40Equal-weight & &50\% &50\% & &3.47\% & 96.53\% & & 8.90\% & 4.87\% \\4160/40 weight & &60\% &40\% & & 13.79\% & 86.21\% & & 8.63\% & 4.03\%\\42Min CVaR & &96.86\% & 3.14\% & & 96.86\% & 3.14\% & & 7.63\% & 2.44\%\\43Min CVaR concentration & &77.01\% &22.91\% & & 50\% &50\% & & 8.17\% & 3.00\%\\4460/40 risk allocation & &81.23\% &18.77\% & & 60\% & 40\% & & 8.05\% & 2.80\%\\45\hline46\end{tabular}47}48%\end{center}49\end{figure}5051\vspace{-6cm}52%\begin{figure}[tb]53%\begin{center}54%\caption{Monthly statistics of real returns on total return indices of the Merrill Lynch US bond, S\&P500, MSCI EAFE and S\&P GSCI indices over the period January 1976 - December 2009. }55%\vspace{1cm}56%\begin{tabular}{|lrrrr| } \hline57%&US bond & S\&P 500 &EAFE& GSCI \\ \hline58%Mean (in \%) & 0.32 &0.52 &0.39& 0.10 \\59%StdDev (in \%)&1.86 &4.46 &4.98& 5.50 \\60%Skewness& 0.89& -0.74 &-0.78 &-1.03 \\61%Exc. Kurtosis & 8.75 &2.48 &1.82& 5.45 \\62%95\% CVaR (in \%) & 1.24 &12.55& 13.51 &20.66 \\63% \hline64%\end{tabular}65%\end{center}66%\end{figure}6768%\newpage6970\begin{figure}[tb]71%\begin{center}72\caption{Annualized mean returns versus the monthly portfolio CVaR and largest percentage CVaR contribution for the mean-StdDev, mean-CVaR and mean-CVaR concentration efficient73portfolios. The frontier is estimated using all January 1976-June 2010 monthly returns. }74\includegraphics[width=12cm,angle=270]{frontier_fourassets.eps}75%\end{center}76\end{figure}777879\begin{figure}[tb]80%\begin{center}81\caption{Weight and CVaR allocation of mean-StdDev, mean-CVaR and mean-CVaR concentration efficient82portfolios for various levels of annualized portfolio returns. The frontier is estimated using all January 1976-June 2010 monthly returns. }83\includegraphics[width=12cm,angle=270]{stackedweightsriskcont_efficientfrontier.eps}84%\end{center}85\end{figure}8687888990\newpage9192\begin{figure}[tb]93%\begin{center}94\caption{Stacked bar weight and CVaR contribution plots for the equal-weight, minimum CVaR and minimum CVaR concentration portfolios invested in the Merrill Lynch US bond, S\&P 500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly.}95\includegraphics[width=12cm,angle=270]{stackedweightsriskcont_benchmark.eps}96%\end{center}97\end{figure}9899\newpage100101102\begin{figure}[tb]103%\begin{center}104\caption{Monthly CVaR of the equal-weight and risk budget optimized portfolios invested in the Merrill Lynch US bond, S\&P 500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly. The shaded regions indicate a bear market regime. }105\includegraphics[width=12cm,angle=270]{portfolioCVaR.eps}106%\end{center}107\end{figure}108109\newpage110111\begin{figure}[tb]112%\begin{center}113\caption{Relative performance of the risk budget optimized portfolios versus the equal-weight portfolio invested in the Merrill Lynch US bond, S\&P 500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly. The shaded regions indicate a bear market regime. }114\includegraphics[width=12cm,angle=270]{RelPerf_EW.eps}115%\end{center}116\end{figure}117118\newpage119120\begin{figure}[tb]121%\begin{center}122123\caption{Summary statistics of monthly out-of-sample returns on investment strategies over the period January 1984 - June 2010. }124\vspace{1cm}\scalebox{0.85}{125\begin{tabular}{|lc c cccc c cc| } \hline126& & Equal & \multicolumn{4}{l}{Min CVaR} & & \multicolumn{2}{l|}{Min CVaR Concentration} \\127\cline{4-7} \cline{9-10}128& & Weight & & 40\% Position & 40\% CVaR & ERC & & & 40\% Position \\129& & & & Limit & Alloc Limit & & & & Limit \\ \hline130\multicolumn{10}{|l|}{\emph{Full period (in \%)}} \\131\multicolumn{2}{|l}{Ann. Mean} & 7.32 & 8.07 & 7.72 & 7.99 & 8.22 & & 8.23 & 7.63 \\132\multicolumn{2}{|l}{Ann. StdDev} & 10.42 & 4.53 & 8.77 & 5.35 & 5.78 & & 5.78 & 8.46 \\133\multicolumn{2}{|l}{Monthly Hist CVaR} & 7.42 & 2.34 & 6.15 & 2.95 & 3.35 & & 3.35 & 5.91 \\134%HI of Hist $C_{i}$CVaR & 0.06 & 0.21 & 0.09 & 0.16 & 0.12 & & 0.12 & 0.07 \\135\multicolumn{2}{|l}{Portfolio turnover} & 1.26 & 2.14 & 3.55 & 2.64 & 1.74 & & 1.74 & 1.51 \\136\multicolumn{10}{|l|}{\emph{Bear stock market (in \%)}} \\137\multicolumn{2}{|l}{Ann. Mean } & -24.36 & 6.31 & -17.29 & -0.61 & -3.81 & & -3.79 & -16.51 \\138\multicolumn{2}{|l}{Ann. StdDev} & 15.46 & 6.00 & 13.03 & 7.10 & 7.44 & & 7.44 & 12.58 \\139\multicolumn{2}{|l}{Monthly Hist CVaR} & 13.71 & 3.30 & 11.05 & 5.37 & 6.28 & & 6.27 & 10.80 \\140\multicolumn{10}{|l|}{\emph{Normal/Bull stock market (in \%)}} \\141\multicolumn{2}{|l}{Ann. Mean } & 13.37 & 8.40 & 12.50 & 9.63 & 10.52 & & 10.52 & 12.24 \\142\multicolumn{2}{|l}{Ann. StdDev } & 8.08 & 4.20 & 6.92 & 4.81 & 5.16 & & 5.16 & 6.67 \\143\multicolumn{2}{|l}{Monthly Hist CVaR} & 4.09 & 2.04 & 3.50 & 2.27 & 2.38 & & 2.38 & 3.38 \\144\hline \multicolumn{10}{|l|}{ \emph{Drawdowns higher than 10\%} } \\145\multicolumn{2}{|l}{Credit crisis$^{*}$} & 0.48 & 0.09 & 0.37 &0.13 & 0.15 & & 0.15 & 0.38 \\146\multicolumn{2}{|l}{Dot-com bubble burst$^{**}$}& 0.25 & & 0.19 & & & & & 0.17 \\147\multicolumn{2}{|l}{Asian-Russian crisis$^{***}$}& 0.12 & & & & & & & \\ \multicolumn{2}{|l}{ Black Monday$^{****}$} & 0.11 & & 0.12 & & & & &148\\149\hline \multicolumn{10}{|l|}{ \emph{Summary statistics on level and concentration of portfolio losses exceeding 10\%} } \\150$-w_t'r_t$ & \multicolumn{1}{l}{ median} & 0.16 & 0.11 & 0.13 &0.12 & 0.12 & & 0.12 & 0.12 \\151&\multicolumn{1}{l}{ max } & 0.22 & 0.17 & 0.21 &0.15 & 0.14 & & 0.14 & 0.19 \\152$\max_i\frac{(w_{(i)t}r_{(i)t})}{w_t'r_t}$ & \multicolumn{1}{l}{ median} & 0.43 &0.97 & 0.51 & 0.85 & 0.83 & & 0.83 & 0.58 \\153& \multicolumn{1}{l}{ max } & 0.45 & 1.02 & 0.74 &0.97 & 0.92 & & 0.92 & 0.66 \\ \hline154\end{tabular}155}156%\end{center}157{\scriptsize $^{*}$ May-Oct 2008 for the Min CVaR strategy, June 2008-Feb 2009 for all other styles. $^{**}$ Sep 2000-Sep 2002.158\newline $^{***}$ April-Aug 1998. $^{****}$ Sep-Nov 1987. }159\end{figure}160161162163164165\newpage166167\begin{figure}[tb]168%\begin{center}169\caption{Stacked bar weight and CVaR contribution plots for the constrained minimum CVaR portfolios invested in the Merrill Lynch US bond, S\&P500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly.}170\includegraphics[width=12cm,angle=270]{MinCVaR_alternatives.eps}171%\end{center}172\end{figure}173174175176177\end{document}178179180