# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 2 Materials and Methods
### Portfolio Optimization Problem  
[PERSON] suggested that the aggregate behavior of the components should be analyzed according to two dimensions: risk and return.  
Given the linear nature of portfolio structure, the first of them is just the weighted average of the expected return for the assets. Defining risk, however, is more complicated. [PERSON] suggested modeling this component through the variance or the standard deviation of returns, bearing in mind that the pattern of returns for the assets included in portfolios could be such that the movements could be negatively correlated, hence providing diversification benefits.  
There are different ways to characterize the risk, such as semivariance and expected shortfall. In this work, the risk is determined by the standard deviation of the returns \(\sigma_{p}\), which is the most prevalent way in the literature.  
The problem, therefore, can be considered as multiobjective with two conflicting objectives formally defined as follows.  
Maximize total expected return:  
\[E(R_{p})=\Sigma_{i=1}^{n}w_{i}\mu_{ri}, \tag{1}\]  
Minimize portfolio risk:\[\sigma=\sqrt{\Sigma_{i=1}^{n}\Sigma_{j=1}^{n}w_{i}w_{j}\sigma_{ij}}, \tag{2}\]  
subject to:  
\[\Sigma_{i=1}^{n}w_{i}=1, \tag{3}\] \[0\leq w_{i}\leq 1\quad\forall i\in\{1,\ldots,n\}, \tag{4}\]  
where \(n\) is the number of investment alternatives, \(\mu_{ri}\) is the expected return of the asset \(i\), \(\sigma_{ij}\) is the covariance between assets \(i\) and \(j\), and \(w_{i}\) are the relative weights of the components of the portfolio that represent the decision variables.  
The first constraint, modeled in Equation (3), requires full investment whereas the second, given in Equation (4), prevents the investor from shorting. In addition to that, we will also consider two real world constraints:  
Cardinality: there is a maximum \(C_{\max}\) and minimum \(C_{\min}\) number of assets in which it is possible to invest simultaneously:  
\[C_{\min}\leq\Sigma_{i=1}^{n}c_{i}\leq C_{\max}, \tag{5}\] \[c_{i}\in\left\{0,1\right\},i=1,\ldots,n, \tag{6}\]  
where \(c_{i}\) is a binary variable. If asset \(i\) is held in the portfolio (\(w_{i}\
eq 0\)) the \(c_{i}=1\). Otherwise, \(c_{i}=0\).  
Investment limits: in case an investment alternative is present in the portfolio, its relative weight \(w_{i}>0\) should be in the interval \([l_{i},\,u_{i}]\), where:  
\[0.0\leq l_{i}\leq w_{i}\leq u_{i}\leq 1.0. \tag{7}\]  
The addition of these constraints, especially the cardinality one, increases the complexity of the problem very significantly. In addition to these, investors might be subject others imposed by regulators or internal processes. In this paper, we do not intend to consider all the possible constraints. Given that the set that might be relevant would depend on the specific circumstances of the investor, we just added a few that complicate the basic problem enough to justify the use of heuristics instead of standard quadratic programming.  
As we mentioned before, given the multiobjective nature of the problem, the solution is not a single portfolio but a set of them. They define a curve in the risk-return space that represents the dominating portfolios as, for every risk level, no portfolio offers a higher expected return.  
It is worth noting that there are several well-known alternatives to characterize portfolio risk. In addition to variance/standard deviation, one could consider value-at-risk, risk of loss, shortfall risk, etc. The algorithm introduced in this paper could work with any of them just as easily. However, we will stick to the most popular characterization for the experimental work.