## 6 Discussion of Results
### CEF-ARPO with Certain Inputs
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Curve Position** & **Required Return** & **UEF-Variance** & **ARPO-Variance** & **APL** & **Time (s)** \\ \hline
20 & 0.002175078 & 0.0001368925 & 0.0001481318 & 0.0821031101 & 1.5320 \\
40 & 0.002252039 & 0.0001370119 & 0.0001483250 & 0.0825702001 & 0.2180 \\
60 & 0.002329001 & 0.0001372175 & 0.0001484472 & 0.0818386868 & 0.3910 \\
80  
Panel C of Figure 1 depicts the CEF for the FTSE 100 (United Kingdom) stock market. It indicates that for the FTSE 100 stock index -similarly to the DAX 100 stock index- the CEF departs from the UEF at the lower end of expected return. As the rate of expected return increases, the CEF converges to the UEF. Noteworthy, portfolios featuring an expected return at the lower end of the CEF tend to be riskier (i.e., with higher portfolio variance) than portfolios on the UEF. At the higher end of the CEF that features rewarding but risky portfolios, the expected rate of return can be achieved with fewer assets. Panel D of Figure 1 depicts the CEF for the S&P 100 (United States) stock market. It indicates that for the S&P 100 stock index -as with the DAX 100 and FTSE 100 stock indices- the CEF departs from the UEF at the lower end of expected return. As the rate of expected return increases, the CEF becomes visually indistinguishable from the UEF. Portfolio investments with an expected return at the lower end of the CEF involve relatively more risk than portfolios with the same expected return located on the UEF. Finally, Panel E of Figure 1 depicts the CEF for the NIKKEI (United States) stock market. It indicates that for the NIKKEI stock index, the relation between the CEF and the UEF follows a pattern similar to the Hang Seng stock index. Specifically, although the CEF departs from the UEF at the lower end of expected return, the difference is visually very small. As the rate of expected return increases, the CEF gradually approaches the UEF. At the higher end of the CEF that includes portfolios with high expected return and high risk, the APL approaches to zero. To evaluate differences between the UEF and the CEF-ARPO, we also provide the portfolio weights for Instance 3 (FTSE 100), where the required rate of return is 0.0041572635, which is an approximately central value within the overall of returns. This instance was executed twice with the same seed, where the maximum time of execution was 20 s. The UEF considered all assets with weights ranging from 0 to 1 inclusively. The CEF was constrained to the minimum of 1 and the maximum of 10 assets, with portfolio weights ranging from 0.01 to 1. The minimum values of the portfolio variance were 2.3872556507357437E-4 (the UEF) and 2.5098945345432527E-4 (CEF-ARPO). The percentage loss is 5.137%. The UEF selected 26 assets (the remaining assets were allocated zero weight), whereas the CEF portfolio selected 10 assets, the upper bound of the cardinality constraint. The 10 assets are the subset of assets selected in the UEF portfolio, where assets indexed with 61, 36 and 52 carry the largest weight in the CEF portfolio. Figure 2 presents portfolio weights.The horizontal axis indexes assets, whereas the vertical axis measures the weight of an asset. Figure 2: Portfolio Weights.