# Portfolio Implementation Risk Management Using Evolutionary Multiobjective Optimization
## 2 Materials and Methods
### Experimental Setup  
Given that solving a single optimization problem would not be enough to evaluate the approach, a sliding window is used. This way, instead of using all the data at once to estimate a single vector of future returns and a variance-covariance matrix, we define several subsets of returns for consecutive dates. This creates a number of partially overlapping single-period optimization problems that include a wide range of market conditions that are tackled one by one.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Code** \\ \hline Frank Russell 1000 Growth & FRUS1 GR \\ Frank Russell 1000 Value & FRUS1 VA \\ Frank Russell 2000 Growth & FRUS2 GR \\ Frank Russell 2000 Value & FRUS2 VA \\ S\&P GSCI Commodity Total Return & GSCITOT \\ MSCI EAFE & MSEAFFL \\ Barclays Corporate Intermediate & 595758 \\ Barclays Intermediate US GVT/CREDIT & Y04946 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Indices used in the experimental work. S\&P GSCI Commodity Total Return is the Standard & Poor’s former Goldman Sachs Commodity Index and MSCI EAFE is the Morgan Stanley Capital International large and mid-cap securities index for Europe, Australasia and Far East.  
More specifically, we start by defining a 10-year window consisting of the \(n=120\) consecutive months between January 2000, \(t_{1}\), and December 2009, \(t_{n}\). This data is subsequently used to estimate the parameters required to find the efficient frontier for January 2010, \(t_{n+1}\). Once this first problem is solved, we generate a second one that considers \(t_{2}\) to \(t_{n+1}\) to estimate the vector of returns and the variance-covariance matrix for February 2010, \(t_{n+1}\). It is worth noting that the approach is agnostic to the parameter estimation method, as these are part of the problem specification, not the solution strategy. For convenience, we used the standard approach of forecasting returns using the sample average, but the algorithm would be the same if we used Black-Litterman, a perfect forecast, or any other alternative.  
The previous process is repeated 60 times to generate 60 different optimization problems, finding the efficient frontiers for the January 2010 to December 2014 period. That is, we define 60 different single-period problem instances using the sliding window. For each of them, we search for the robust efficient frontier relying on training samples of 10 years worth of data. The estimated robust solutions for the following unobserved time period are subsequently assessed.  
For each sliding window, 20 independent runs are performed with a population size of \(\mu=100\), running for \(70\times\mu\) function evaluations. The further parameter settings are as follows. The real representation of the decision variables is used. The variation operator involves the SBX crossover and polynomial mutation applied with the probability of \(p_{c}=0.9\) and \(p_{m}=1/n\) (where \(n\) is the number of decision variables), respectively. The crossover and mutation distribution indexes are \(\eta_{c}=20\) and \(\eta_{m}=20\), respectively.  
Regarding the constraints of the optimization problem, in addition to requiring full investments, and long-only positions, the cardinality of the portfolios was limited by investing in a minimum of two indexes, and a maximum of 6. The relative weight of the investment in every index considered in the asset allocations was also required to be between 10% and 90%.