**6. Conclusions**

Our study shows that is possible to create robust portfolios where the risk is minimized, and the return is maximized. Theory behind is [2] which study focus on 'efficient frontier of optimal investment', while advocating a diversified portfolio to reduce risk. To perform it, six portfolio models are proposed, and its formation are made by a solver, where the selected solving method is the GRG Nonlinear engine for linear solver problems. Then we compare results with the benchmark (S&P 500), a linear regression model (available for request) and other "popular" portfolios (already known by many investors – also, only available by request) as well.

Results show that the GRG Nonlinear engine is powerful, providing excess returns to all six models. We design models for three types of investors: conservative, moderate and aggressive. For a conservative investor, portfolio 1 fits the best followed by portfolios 6 and 3. Portfolio 1 shows a strong sharp ratio equal to 4, presenting though, very low volatility but with lower returns when comparing to portfolios 6 and 3. None of this mentioned portfolios show a negative year during the period of 2000 to 2018. Portfolio 3 presents a high performance (14,15% annually). For an aggressive investor portfolio 2 is the best choice because it maximizes the overall return. It is the most volatile portfolio, but it may generate an average income superior to 15% annually. For a moderate investor, fan is wider but still we would exclude the portfolio 1 because it will not generate to much return and portfolio 2 may be a little volatile. Even that, portfolio 2 only presents three negatives years which is still better than the benchmark (3 versus 7).

We went further in our research and we figure out that GRG is robust, and its returns exceeds the other models mentioned in the first paragraph of this section (linear and "classical" portfolios).

Our contribution for this study is to provide a wider variety of portfolios that can be easily used by institutional and private investors and considering that nowadays there are plenty ETFs or funds available in the market is easy to everyone to apply one of the proposed models. Also, it is proved that it is possible to design very efficient portfolios, increasing returns and at the same time, lowering the risk. The results enforce the MPT from [2, 3].

As it happens in all models, there are, of course, some limitations as well. First, we may not guarantee that portfolios constitutions (1–6) will present the same results in the future because we are relaying in past returns and we would need, at least one more decade to understand if, for example, "good" decades present similar behavior between themselves. Another limitation found, the lack of the real VIX tracker (ETF/ETN). Available ETFs of VIX are a mix of mid-short term that do not reflect the actual index. Note that VIX plays an important role in GRG models.
