## 6 Discussion of Results
### CEF-ARPO with Uncertain Inputs  
Key to our research is a widely held assumption--in line with most of the existing literature--that the returns vector and the variance and covariance matrix are well-know and deterministic; i.e., they are known with certainty. Whilst this assumption is analytically and computationally convenient, its practical appeal may be limited. In order to account for this practical restraint, and to test if our algorithm is robust to small changes in the returns vector and the variance-covariance matrix, we use the data set of Hang Seng stock market to perform computational experiments that involve some degree of uncertainty in the first and second moments. We consider three different scenarios. In Scenario 1 (S1), a random disturbance term distributed with a normal distribution with mean 0 and standard deviation 0.00015 is added to each return of the original instance. Under this scenario, an original return of 0.00350406 (which is the mean return of the instance) now takes on values within the interval of 0.00350406 \(\pm\)2\(\cdot\)0.00015=(0.00320406,0.00380406) with the probability of 95.45%, which can be deemed a realistic interval. In Scenario 2 (S2), a random disturbance term distributed with a normal distribution with mean 0 and standard deviation 0.000025 is added to each element of the covariance matrix. Under this scenario, the original covariance of 0.00113094 now takes on values within the interval of (0.00108094,0.00118094) with the probability of 95.45%. In Scenario 3 (S3), Scenarios 1 and 2 are jointly simulated. The results are compared against Scenario 0 (S0), which depicts the CEF-ARPO with \(k_{min}=1\), \(k_{max}=10\), and portfolio weights confined between 0.01 and 1. Table 7 reports the portfolio variance and the APL loss relative to S0 of the three scenarios, for 10 initial and 10 last portfolio returns on the CEF-ARPO. Table 8 summarizes the results of ARPO for different scenarios generated from Instance 1 (Hang Seng), in which problem cardinality constrains are modified. The average APLs for the three scenarios are 3.324%, \(-\)0.109% and 3.274%, respectively (See also Table 9, where the average APL, average computational times and the number of infeasible rates of return are summarized). Thus our algorithm delivers a reasonable-quality solution also in an environment that allows for inaccurate inputs. However, these results should be interpreted with caution. Importantly, it should be recognized that S1-S3 amount to different optimization problems. A relative larger APL for S1 may imply that it is more costly--in terms of the portfolio variance--to attain a given portfolio return when individual asset returns are randomized than when they are not. The induced random variation in individual asset returns effectively boosts the corresponding elements of the variance and covariance matrix. This leads to a larger overall portfolio variance for a given portfolio return. As a result, portfolios on the higher end of the efficient frontier become more costly and even infeasible (See also Table 9). In S2, where individual variances and covariances are randomized, the resulting CEF-ARPO remains very similar to the CEF-ARPO, as manifested by the average APL. Indeed, in contrast to S1, adding a normally distributed random disturbance term to variances and covariances between individual assets does not appear to alter the optimal solution. For portfolio on the lower end of the efficient frontier, the APL takes on low positive values; however, for more rewarding and riskier portfolios with feasible returns, the solution is better than for the CEF-ARPO, suggesting further diversification opportunities for investors in high-yield portfolios. Finally, S3 shows combined effects of S1 and S2. Next, we carry out a stability analysis of the ARPO algorithm to ratify our main findings. The stability analysis consists of six scenarios (numbered successively) as follows, which are evaluated against S0. * Scenario 4 (S4): the minimum number of assets in the portfolio is increased from 1 to 2.