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Current article carries on the portfolio optimization process focusing on increasing expected return of the portfolio. Consequently, the optimized portfolio is one of higher risk to yield higher returns. As a result there are several portfolios with varying risk and return ratio. Investors are free to choose based on the degree of risk tolerance.

Suppose that an investor is ready to accept a higher risk level to increase the expected return of the portfolio. Let the maximum acceptable standard deviation of the return of the portfolio be 2.5%. We will carry out the optimization procedure of weight coefficients for searching for the maximum return of the portfolio with an additional restriction on the standard deviation (it should not exceed 2.5%).

This leads us to the following structure of P3: MCD stocks again got the greatest weight (61.63%). Then DIS (21.74%), HON (14.17%), IBM (1.61%) and HD (0.84%) were ranged in weight descending order. KO stocks got zero weight and are not included in portfolio P3.

Diagram 2: P3 Portfolio Structure |

As a result, portfolio P3, created to maximize the expected return, so that it does not exceed the standard deviation level of 2.55, has the expected monthly return of 0.65% and the Sharpe ratio of 0.26.

Table 4: Realized returns, standard deviations and Sharpe ratios of P1, P2 and P3 portfolio |

P1 | Р2 | Р3 | |

Mean Return | 0.49% | 0.24% | 0.40% |

StDev | 4.25% | 5.73% | 4.51% |

Sharpe Ratio | 0.11 | 0.04 | 0.09 |

The platform allows building a chart, reflecting the dynamics of portfolio P3 by means of PCI tools. The chart includes five assets with corresponding weights in the base part, and Dow Jones Industrial Average index in the quoted part.

Chart 3: Portfolio, maximizing return in relation to the index Dow Jones |

Here we see a growing structure as in the previous case. The growth during the whole observed period came to be more considerable than in case of P2, but the fluctuation during the crisis looks more significant – a direct result of the increase of the allowable risk.

To make the picture complete, let us also build a portfolio for a risk adverse investor that seeks to maximally lower the risks through portfolio diversification. To do that, we resort to the optimization procedure of the weights of the available assets, looking for the minimum value of the standard deviation. Portfolio P4 is characterized by the following set of weights:

Diagram 3: P4 Portfolio Structure |

The diagram shows that this time the largest weight coefficient fell on KO stocks (21.35%), after all, it was the asset for which we got the smallest standard deviation from input data. Afterwards, HON (20.77%), MCD (18.13%), DIS (17.11%), HD (13.89%) and IBM (8.75%) were ranged in descending weight order. This portfolio is characterized by expected return of 0.45%, standard deviation- 1.60% and the Sharpe ratio - 0.28.

Table 5: Realized returns, standard deviations and Sharpe ratios of P1, P2, P3 and P4 portfolios |

P1 | Р2 | Р3 | Р3 | |

Mean Return | 0.46% | 0.52% | 0.65% | 0.45% |

StDev | 1.74% | 1.72% | 2.50% | 1.60% |

Sharpe Ratio | 0.26 | 0.30 | 0.26 | 0.28 |

PCI toolset will again help in building the chart of the portfolio and estimating its behavior in relation to Dow Jones Industrial Average index within the specified time.

Chart 4: Portfolio, minimizing standard deviation in relation to the Index Dow Jones |

It is remarkable how stable the behavior of portfolio P4 is. Though the absolute growth during the whole period is lower than in case of previously formed portfolios - comparatively narrow range of fluctuations that reflects lower volatility has its advantages.

If we plot all four portfolios (Р1, Р2, Р3, Р4) in the coordinates of risk and return, we can confirm only that portfolio P2 (maximum Sharpe ratio) is preferred to the "random" portfolio P1, since its position (in the higher left sector relative to P1) indicates higher return and lower standard deviation.

Chart 5: Risk-return profiles of Portfolios p1, p2, p3, p4 |

The choice between portfolios P2, P3 and P4 depends on investor’s individual preferences and restrictions. On the one hand, if an investor is ready to accept relatively high risk level, his choice will probably shift to portfolio P3 with the highest expected return (among the options considered). On the other hand if an investor seeks to minimize risks, he may choose portfolio P4, which has the lowest standard deviation (among the options considered). We would call portfolio P2, a balanced portfolio, having the best return per unit of risk.

Charts, built through GeWorko method and PCI tools, are a visual confirmation of effective application of the principles of contemporary portfolio theory, alongside with vast number of opportunities. Based on quantitative estimations of risk, return and covariance of various assets, we have found a variety of "successful" portfolios, which have been systematically outperforming the market over recent years and in the meantime have come along numerous investment opportunities. The present analysis is based on only seven assets (6 stocks + index) of the same class of financial instruments. The potential gains from diversification can be considerably increased, if we add other asset classes into the analysis that have lower or negative correlation with each other. Therefore, there are endless opportunities that You obtain in case of applying GeWorko method for market analysis and trading.

See the beginning of the article "Portfolio Structure Optimization through GeWorko Method (Part 1)"