# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 3 Results  
The results in terms of the three performance metrics are summarized in Table 2. This table presents the mean and median values over the 60 scenarios and 20 runs per scenario for both the non-robust and the robust versions of the algorithm. The latter is studied with different values of the dispersion parameter \(\epsilon\). The presented results clearly show connection between the choice of the algorithm and the quality of the output. The robustness-based SMS-EMOA systematically provides better results than the base version with respect to the considered metrics. The statistical significance of all the differences between the median values for the non-robust version of the algorithm versus the robust one was tested with the Mann-Whitney test, and they were significant at the 1% level. As expected, the quality of obtained solutions with respect to robustness is highly related to the dispersion parameter. As the value of \(\epsilon\) is decreased, the algorithm provides solutions with a higher robustness to both estimation and implementation risks measured by the corresponding metrics. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**SMS-EMOA**} & \multicolumn{2}{c}{**Implem. Risk**} & \multicolumn{2}{c}{**Stability**} & \multicolumn{2}{c}{**Extr. Risk**} \\ \hline  & **Version** & \(\epsilon\) & **Mean** & **Diff. (\%)** & **Mean** & **Diff. (\%)** & **Mean** & **Diff. (\%)** \\ \hline \multirow{8}{*}{Mean} & _Standard_ & & 0.0770 & & 68.56 & & 2984.71 & \\  & & 0.99 & 0.0356 & \(-\)53.8\% & 53.44 & \(-\)22.0\% & 2289.04 & \(-\)23.3\% \\  & & 0.9 & 0.0148 & \(-\)80.7\% & 50.14 & \(-\)26.9\% & 2124.96 & \(-\)28.8\% \\  & & 0.6 & 0.0069 & \(-\)91.1\% & 47.55 & \(-\)30.6\% & 1996.60 & \(-\)33.1\% \\  & & 0.4 & 0.0043 & \(-\)94.5\% & 45.07 & \(-\)34.3\% & 1862.10 & \(-\)37.6\% \\  & & 0.2 & 0.0021 & \(-\)97.3\% & 43.77 & \(-\)36.2\% & 1788.66 & \(-\)40.1\% \\  & & 0.1 & 0.0015 & \(-\)98.0\% & 43.40 & \(-\)36.7\% & 1769.18 & \(-\)40.7\% \\ \hline \multirow{8}{*}{Median} & _Standard_ & & 0.0769 & & 39.30 & & 4276.12 & \\  & & 0.9 & 0.0327 & \(-\)57.5\% & 32.25 & \(-\)17.9\% & 3000.91 & \(-\)29.8\% \\ \cline{1-1}  & & 0.9 & 0.0144 & \(-\)81.3\% & 30.40 & \(-\)22.6\% & 2918.74 & \(-\)31.7\% \\ \cline{1-1}  & & 0.6 & 0.0062 & \(-\)92.0\% & 29.29 & \(-\)25.5\% & 2705.15 & \(-\)36.7\% \\ \cline{1-1}  & & 0.4 & 0.0036 & \(-\)95.4\% & 27.27 & \(-\)30.6\% & 2550.84 & \(-\)40.3\% \\ \cline{1-1}  & & 0.2 & 0.0016 & \(-\)97.9\% & 26.39 & \(-\)32.8\% & 2448.58 & \(-\)42.7\% \\ \cline{1-1}  & & 0.1 & 0.0010 & \(-\)98.7\% & 26.12 & \(-\)33.5\% & 2442.56 & \(-\)42.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results for baseline and robustness-based S-metric selection evolutionary multiobjective optimization algorithms (SMS-EMOA). Differences in percentage vs. standard version. It is an interesting observation, as the definition of a robustness measure used in the fitness assignment procedure only relies on the notion of implementation risk. The calculation of fitness does not explicitly involve a measure for estimation risk. This way, the experimental results suggest that there is some sort of connection between the two risks, as solutions more tolerant to implementation risk appear to be more tolerant to estimation risk as well. The advantage of the proposed approach lies in its ability to address both risks. Given that experiments involved 60 optimization problems with 20 independent runs for each problem, Figure 1 illustrates, in the form of boxplots, the connection between the dispersion parameter \(\epsilon\) and the distribution of the results regarding performance metrics. The presented plots show how the dispersion across metrics tends to grow for larger values of the parameter \(\epsilon\).