# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 2 Materials and Methods
### Evaluation Metrics  
Quantitative metrics for evaluating the outcomes produced by different runs of stochastic algorithms provides a basis for their comparison and statistical inference. Given that a portfolio optimization is a multiobjective problem, one could tempt applying quality indicators, such as hypervolume or spread [40], widely used in the field of evolutionary multiobjective optimization. The use of such indicators would be inappropriate in the context of the present study, whose main concern is to obtain reliable portfolios with respect to different definitions of robustness. Moreover, even for a well-converged approximation to the efficient front, more diverse solutions in the objective space do not necessarily mean the better results, as a preference information is considered to focus on the most relevant regions.  
This study investigates the robustness of portfolios to implementation and estimation risks. The latter is estimated by a herein suggested metric, whereas the former is evaluated using metrics introduced in [24]. The description of these metrics is provided below.  
* **Implementation Risk** This metric estimates the robustness of a given portfolio structure taking into consideration the changes in risk and return due to deviations in an initially adopted investment strategy. The portfolio robustness is estimated based on the ratio between deviations in risk-return values and portfolio weights. This metric is calculated as shown in Equation (10). Smaller values of this metric correspond to more robust portfolios.
* **Stability** This metric measures the sensitivity of the portfolios included in the efficient frontier to a number of potential scenarios where asset returns and their variance-covariance matrix deviate from the expected values. The stability of a portfolio is measured by averaging the Mahalanobisdistances between the expected pair of risk-return in the efficient frontier and the expected risk-return in \(S\) different scenarios. The metric is calculated as follows: \[ST=\frac{1}{S}\sum_{i=1}^{S}\frac{1}{N}\sum_{p=1}^{N}d_{M}(x_{p},x_{pi})^{2},\] (13) where \(d_{M}(x_{p},x_{pi})\) is the Mahalanobis distance between \((E(R_{np}),\sigma_{np}^{2})\) and \((E(R_{npi}),\sigma_{npi}^{2})\), the risk and return for portfolio \(p\) in the period \(t_{n}\) computed using the parameters for the scenario \(i\). A set of 500 likely alternative scenarios is generated from historical data by using a nonparametric bootstrap. The higher the average dispersion, the less reliable the expected behavior of the current solution.
* **Extreme Risk** This metric evaluates the sensitivity of the solutions to worst-case scenarios. It is closely related to the measure of Stability. The difference is that, instead of considering the average for all \(S\) scenarios, a small subset is only taken into account. Specifically, the computation uses 1% of most problematic scenarios. These scenarios are identified by resampling historical returns 500 times, computing the average returns and the variance-covariance matrix for each of them, and including in the computation five scenarios for which the behavior of the portfolios in the solution differ more from the expected ones. The higher the metric, the higher the risk.