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\begin{document}
{\bf \large \noindent EXHIBITS}
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\bigskip
\begin{figure}[h]
\begin{center}
\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. }
\includegraphics[width=9cm,angle=270]{sensitivity_rho50.eps}
\end{center}
\end{figure}
\newpage
\begin{figure}[tb]
\begin{center}
\caption{Weight and CVaR allocation of bond-equity portfolios, together with the in-sample annualized monthly mean and monthly 95\% CVaR over
the period January 1976-June 2010. }
\vspace{1cm}
\scalebox{0.87}{
\begin{tabular}{|lc cc c cc c cc | } \hline
& & \multicolumn{2}{c}{Weight allocation} & &\multicolumn{2}{c}{CVaR allocation} & & Ann. mean & 95\% CVaR \\
& & Bond & Equity & &Bond & Equity & & & \\ \hline
Equal-weight & &50\% &50\% & &3.47\% & 96.53\% & & 8.90\% & 4.87\% \\
60/40 weight & &40\% &60\% & & 0\% & 100\% & & 9.17\% & 5.82\%\\
Min CVaR & &96.86\% & 3.14\% & & 96.86\% & 3.14\% & & 7.63\% & 2.44\%\\
Min CVaR concentration & &77.01\% &22.91\% & & 50\% &50\% & & 8.17\% & 3.00\%\\
60/40 risk allocation & &72.90\% &27.10\% & & 40\% & 60\% & & 8.28\% & 3.18\%\\
\hline
\end{tabular}
}
\end{center}
\end{figure}
\vspace{-6cm}
%\begin{figure}[tb]
%\begin{center}
%\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. }
%\vspace{1cm}
%\begin{tabular}{|lrrrr| } \hline
%&US bond & S\&P 500 &EAFE& GSCI \\ \hline
%Mean (in \%) & 0.32 &0.52 &0.39& 0.10 \\
%StdDev (in \%)&1.86 &4.46 &4.98& 5.50 \\
%Skewness& 0.89& -0.74 &-0.78 &-1.03 \\
%Exc. Kurtosis & 8.75 &2.48 &1.82& 5.45 \\
%95\% CVaR (in \%) & 1.24 &12.55& 13.51 &20.66 \\
% \hline
%\end{tabular}
%\end{center}
%\end{figure}
%\newpage
\begin{figure}[tb]
\begin{center}
\caption{Efficient frontier, weight and CVaR allocation of mean/CVaR concentration efficient portfolios. The frontier is estimated using all January 1976-June 2010 monthly returns. }
\includegraphics[width=12cm,angle=270]{frontier_fourassets.eps}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\caption{Efficient frontier, weight and CVaR allocation of mean/CVaR concentration efficient portfolios. The frontier is estimated using all January 1976-June 2010 monthly returns. }
\includegraphics[width=12cm,angle=270]{stackedweightsriskcont_efficientfrontier.eps}
\end{center}
\end{figure}
\newpage
\begin{figure}[tb]
\begin{center}
\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\&P500, MSCI EAFE and S\&P GSCI indices. The portfolios are rebalanced quarterly.}
\includegraphics[width=12cm,angle=270]{stackedweightsriskcont_benchmark.eps}
\end{center}
\end{figure}
\newpage
\begin{figure}[tb]
\begin{center}
\caption{Monthly CVaR of the 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. }
\includegraphics[width=12cm,angle=270]{portfolioCVaR.eps}
\end{center}
\end{figure}
\newpage
\begin{figure}[tb]
\begin{center}
\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. }
\includegraphics[width=12cm,angle=270]{RelPerf_EW.eps}
\end{center}
\end{figure}
\newpage
\begin{figure}[tb]
\begin{center}
\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.}
\includegraphics[width=12cm,angle=270]{MinCVaR_alternatives.eps}
\end{center}
\end{figure}
\newpage
\begin{figure}[tb]
\begin{center}
\caption{Summary statistics of monthly out-of-sample returns on investment strategies over the period January 1984 - June 2010. }
\vspace{1cm}\scalebox{0.85}{
\begin{tabular}{|l c cccc c cc| } \hline
& Equal & \multicolumn{4}{l}{Min CVaR} & & \multicolumn{2}{l|}{Min CVaR Concentration} \\
\cline{3-6} \cline{8-9}
& Weight & & 40\% Position & 40\% CVaR Alloc & ERC & & & 40\% Position \\
& & & Limit & Limit & & & & Limit \\ \hline
\multicolumn{9}{|l|}{\emph{Full period (in \%)}} \\
Ann. Mean & 7.32 & 8.07 & 7.74 & 7.99 & 8.22 & & 8.23 & 7.63 \\
StdDev & 3.01 & 1.31 & 2.53 & 1.54 & 1.67 & & 1.67 & 2.44 \\
Hist CVaR & 7.42 & 2.34 & 6.15 & 2.95 & 3.35 & & 3.35 & 5.91 \\
HI of Hist $C_{i}$CVaR & 0.06 & 0.21 & 0.09 & 0.16 & 0.12 & & 0.12 & 0.07 \\
Portfolio turnover & 1.26 & 2.14 & 3.55 & 2.64 & 1.74 & & 1.74 & 1.51 \\
\multicolumn{9}{|l|}{\emph{Bear stock market (in \%)}} \\
Ann. Mean & -24.36 & 6.31 & -17.25 & -0.66 & -3.81 & & -3.79 & -16.52 \\
StdDev & 4.46 & 1.73 & 3.76 & 2.05 & 2.15 & & 2.15 & 3.63 \\
Hist CVaR & 13.71 & 3.30 & 11.04 & 5.36 & 6.28 & & 6.27 & 10.80 \\
\multicolumn{9}{|l|}{\emph{Normal/Bull stock market (in \%)}} \\
Ann. Mean & 13.37 & 8.40 & 12.51 & 9.64 & 10.52 & & 10.52 & 12.24 \\
StdDev & 2.33 & 1.21 & 2.00 & 1.39 & 1.49 & & 1.49 & 1.93 \\
Hist CVaR & 4.09 & 2.04 & 3.50 & 2.26 & 2.38 & & 2.37 & 3.38 \\
\hline \multicolumn{9}{|l|}{ \emph{Drawdowns higher than 10\%} } \\
% \multicolumn{9}{|l|}{\emph{Credit crisis} } \\
Credit crisis$^{*}$ & 0.48 & 0.09 & 0.37 &0.13 & 0.15 & & 0.15 & 0.38 \\
% \multicolumn{9}{|l|}{\emph{Dot-com bubble burst}} \\
Dot-com bubble burst$^{**}$& 0.25 & & 0.19 & & & & & 0.17 \\
% \multicolumn{9}{|l|}{\emph{Asian-Russian crisis}} \\
Asian-Russian crisis$^{***}$ & 0.12 & & & & & & & \\ % \multicolumn{9}{|l|}{\emph{Black Monday}} \\
Black Monday$^{****}$ & 0.11 & & 0.12 & & & & & \\
\hline
\end{tabular}
}
\end{center}
{\scriptsize $^{*}$ May-Oct 2008 for the Min CVaR strategy, June 2008-Feb 2009 for all other styles. $^{**}$ Sep 2000-Sep 2002.
\newline $^{***}$ April-Aug 1998. $^{****}$ Sep-Nov 1987. }
\end{figure}
\end{document}