## 6 Discussion of Results
### CEF-ARPO with Certain Inputs
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**[PERSON] et al. [41]**} & \multicolumn{2}{c}{**[PERSON] et al. [40]**} & \multicolumn{2}{c}{**[PERSON] [39]**} & \multicolumn{2}{c}{**ARPO**} \\ \cline{2-10}  & \multicolumn{2}{c}{**FD + QP**} & \multicolumn{2}{c}{**SD + QP**} & \multicolumn{2}{c}{**GA + QP**} & \multicolumn{2}{c}{**TS**} & \multicolumn{2}{c}{**LS + QP**} \\ \cline{2-10}  & **APL** & **T(s)** & **APL** & **T(s)** & **APL** & **T(s)** & **APL** & **T(Std) (s)** \\ \hline S\&P 100 & 4.77157 & 11.2 & 4.69371 & 18.8 & 4.69507 & 1163.7 & 4.69507 & 713 & 4.65095 & 15.5 (8.42) \\ NIKKE1225 & 0.24176 & 25.3 & 0.20219 & 45.9 & 0.20198 & 1465.8 & 0.20198 & 1603 & 0.20189 & 5.0 (17.20) \\ \hline \hline \end{tabular}  
* Table 1 summarizes the results of ARPO for the CEF, including the minimum average percentage loss (APL) and computational time in seconds (T(s)). APL is calculated as in [41]. We also report the standard deviation of computational time for ARPO. The performance indicators of ARPO are also compared against its competitors, including [39; 40; 41].  
\end{table} 
Table 1: \(Cont\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Curve Position** & **Required Return** & **UEF-Variance** & **ARPO-Variance** & **APL** & **Time (s)** \\ \hline
20 & 0.002861137 & 0.0006424068 & 0.0006424114 & 0.0007160572 & 4.212 \\
40 & 0.002941981 & 0.0006428092 & 0.0006429074 & 0.0152766949 & 2.74 \\
60 & 0.003022827 & 0.000  
Table 3 presents the UEF and the CEF-ARPO for the DAX 100 portfolio for 10 initial points and 10 last values of returns. Figure 1: UEF and CEF-ARPO. **(a)** Panel A—Hang Seng Stock Market (Hong Kong); **(b)** Panel B—DAX 100 Stock Market (Germany); **(c)** Panel C—FTSE 100 Stock Market (UK); **(d)** Panel D—S&P 100 Stock Market (US); **(e)** Panel E—NIKKEI Stock Market (Japan). Notes: Figure 1 illustrates the UEF and the CEF-ARPO for the five instances (Panel A—Hang Seng, Panel B—DAX 100, Panel C—FTSE 100, Panel D—S&P 100 and Panel E—NIKKEI 225). The UEF is represented with a black solid line, whereas the CEF-ARPO is represented with a dash red line. Portfolio variance is depicted on the horizontal axis, whereas the required rate of return is depicted on the vertical axis. The minimum average percentage loss is 0.00399% for the Hang Seng portfolio, 2.45403% for the DAX 100 portfolio, 1.88340% for the FTSE100 portfolio, 4.65095% for the S&P 100 portfolio and 0.20189% for the NIKKEI 225 portfolio. Table 4 presents the UEF and the CEF-ARPO for the FTSE 100 portfolio for 10 initial points and 10 last values of returns. Table 4 summarizes the UEF and the CEF-ARPO for the DAX 100 portfolio for 10 initial points and 10 last values of returns. In column 2, values of the required rate of return are provided. In column 3, values of the UEF solution (portfolio variance) are provided. In column 4, values of the CEF-ARPO solution (portfolio variance) are provided. In column 5, values of the minimum average percentage loss are reported. In column 6, computational times are reported. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Curve Position** & **Required Return** & **UEF-Variance** & **ARPO-Variance** & **APL** & **Time (s)** \\ \hline  
\hline \end{tabular} Notes: Table 4 summarizes the UEF and the CEF-ARPO for the FTSE 100 portfolio for 10 initial points and 10 last values of returns. In column 2, values of the required rate of return are provided. In column 3, values of the UEF solution (portfolio variance) are provided. In column 4, values of the CEF-ARPO solution (portfolio variance) are provided. In column 5, values of the minimum average percentage loss are reported. In column 6, computational times are reported.  
\end{table} 
Table 3: DAX 100 Stock Market (Germany).