# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 1 Introduction
We can also mention [21], which plays with modifications and extensions of common robust optimization techniques by using an hybrid heuristic as solver, or [22], that introduces a single/multiobjective inverse robust evolutionary approach based on non-probabilistic methods that tries to deal with uncertainty in parameters. Lastly, in [23], the authors explored the possibility of increasing the robustness of the solutions obtained by a multiobjective genetic algorithm, exposing the system to several markets. On the other hand, in [24; 25], the focus is more on the single-period portfolio optimization problem described above. In [26], a time-stamped resampling strategy is introduced along with a variant that extends the basic two-objective framework by explicitly adding a third robustness objective. A similar problem is tackled in [27] by a different strategy for a multiperiod setup. More specifically, the integration of an anticipatory learning method using Kalman Filters is proposed. However, the scenario considered in this work is different. When asset managers identify allocations of interest, the process of rebalancing existing portfolios or creating new ones is not always immediate. Once the decision regarding the target portfolio has taken, some considerations should be made before implementation. Is the market deep enough to meet the order at current price or is likely to move prices? Is it desirable to conceal temporarily the structure of the target portfolio? Similar questions might result in different trading strategies, some of which require building the desired position over time. During this period, especially in the later stages, the portfolio will be similar to the chosen one, but different. This deviation in the structure can result in properties in terms of risk and return that differ from the expected one. If the intermediate portfolio is especially sensitive, this can lead to undesired risk exposure. In this view, a portfolio would be considered robust to implementation risk if small deviations in its composition result in similar risk-return profiles. This desirable trait for portfolios is closely related to robust design optimization focusing on finding solutions that are reliable under given uncertainties. There are two closely related aspects: the feasibility and performance robustness. The robustness against performance deterioration is referred as to the performance robustness. It can be addressed either by optimizing the expectation and variance [28; 29] or by introducing additional constraints [28]. Different types of robustness measures for multiobjective optimization were investigated in [30]. A methodology for incorporating the robustness measure into fitness values of population members in MOEAs were developed in [31]. On the other hand, the robustness of solutions against failure is a subject of reliability-based optimization techniques. They usually address a stochastic optimization problem in which the constraints are converted into probabilistic constraints. The probability of a solution being feasible is determined by a reliability index. The solution reliability can be determined by either a Monte Carlo simulation that generates a number of samples and estimates the probability of failure or optimization-based procedures that estimate the distance from the solution to the constraint [32]. For good surveys on robust optimization, an interested reader is kindly referred to [33; 34]. This study follows the methodology developed in [31] and suggests a robustness-based S-metric selection evolutionary multiobjective optimization algorithm (R-SMS-EMOA) for portfolio optimization to account for inherit uncertainties. The approach also allows the user to express and incorporates a preference information in terms the desirable degree of solutions robustness. This information is used to adjust the optimization and to focus on the most preferred regions of the search space.