# Optimization of the Structure of the Investment Portfolio of High-Tech Companies Based on the Minimax Criterion

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## Abstract

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## 1. Introduction

## 2. Literature Foundations

#### Minimax Approach, Model, Methodology

_{i}).

_{k}the rank of the k-th industry according to the number of the industry in the list (by priority). Similar to task (1), for industries, the task is set:

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

DSS | Decision Support System; |

SAD | computer-aided design system; |

m | number of industries; |

n | the number of companies participating in the analysis (from all selected industries for analysis); |

FL | Estimates of financial leverage, that is, the ratio of borrowed capital to equity expressed in fractions of a unit or as a percentage; |

IR | integral ranks of companies (from the best, equal to one) obtained during the implementation of the author’s algorithm; |

Ω | there are many restrictions on the structure of the portfolio (the sum of the shares is one); |

Ω^{η} | there are many restrictions on the structure of the portfolio (the sum of the shares is one and the required yield is fixed at the level of η_{p}, taking into account the returns of the assets included in the portfolio η_{i}, i = 1,…,n; |

$\tilde{\theta}$ | vector (with components that make up the investment shares, giving a total of one), for a problem with risk assessment in terms of financial leverage; |

$\widehat{\theta}$ | vector (with components that make up the investment shares, giving a total of one), for a problem with an assessment in terms of an integral rating of companies; |

θ | vector (with components that make up the investment shares, giving one in total), for the optimal portfolio structure, correction of the solution of the problem with risk assessment in terms of financial leverage by solving the problem for risk assessment in terms of integral rating, the solution of the second problem is the correction coefficients for optimizing the solution of the first problem; |

W_{i} | industry rank, for further optimization of a portfolio solution containing companies from several industries; |

$\stackrel{\u2323}{\theta}$ | vector (with components that make up the investment shares, giving a total of one), for a problem with an assessment in terms of an integral rating of industries; |

θθ | vector (with components that make up the investment shares, giving one in total), for the optimal portfolio structure, correction of the solution of the problem with risk assessment in terms of financial leverage by solving the problem for risk assessment in terms of integral rating, the solution of the second problem is the correction coefficients for optimizing the solution of the first problem, as well as taking into account the correction of the solution for each company taking into account the industry attribute (solving the problem of optimizing the rating of industry positions in the portfolio), if there is one industry, then the result coincides with the calculation; |

z, zz | intermediate indicators that do not have significant significance in terms of their interpretation. |

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Company, PJSC | Net Profit, Thousand Rubles | Equity Capital, Thousand Rubles | Revenue, Thousand Rubles | Assets, Thousand Rubles | Debt Capital, Thousand Rubles | Risk (Debt/Equity, Financial Leverage), Shares (%) | Profitability (Net Profit to Equity), Shares (%) |
---|---|---|---|---|---|---|---|

Gazprom | 651,124,114 | 11,334,679,889 | 4,758,711,459 | 15,916,355,497 | 4,581,675,608 | 40.40% | 5.70% |

Lukoil | 405,759,769 | 957,169,199 | 444,471,354 | 2,209,166,567 | 1,251,997,368 | 130.80% | 42.40% |

Surgutneftegaz | 105,478,643 | 4,303,834,579 | 1,555,622,592 | 4,553,686,428 | 249,851,849 | 5.80% | 2.50% |

Novatek | 237,224,510 | 718,557,978 | 528,544,385 | 899,787,613 | 181,229,635 | 25.20% | 33.00% |

Company | Net Profit, Thousand Rubles (Index D) | Equity Capital, Thousand Rubles (Index C) | Revenue, Thousand Rubles (Index B) | Assets, Thousand Rubles (Index A) | Place, 1—Leader (Including Revenue) | The Company’s Share in The Portfolio by Rating Position |
---|---|---|---|---|---|---|

Gazprom | 651,124,114 | 11,334,679,889 | 4,758,711,459 | 15,916,355,497 | 1 | 48% |

Lukoil | 405,759,769 | 957,169,199 | 444,471,354 | 2,209,166,567 | 3 | 16% |

Surgutneftegaz | 105,478,643 | 4,303,834,579 | 1,555,622,592 | 4,553,686,428 | 2 | 24% |

Novatek | 237,224,510 | 718,557,978 | 528,544,385 | 899,787,613 | 4 | 12% |

Company | Rating (C, A) | Rating (C, A, B) |
---|---|---|

PJSC Gazprom | 3 | 1 |

PJSC Lukoil | 4 | 3 |

PJSC Surgutneftegaz | 15 | 15 |

PJSC Novatek | 6 | 6 |

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**MDPI and ACS Style**

Borodin, A.; Tvaronavičienė, M.; Vygodchikova, I.; Panaedova, G.; Kulikov, A.
Optimization of the Structure of the Investment Portfolio of High-Tech Companies Based on the Minimax Criterion. *Energies* **2021**, *14*, 4647.
https://doi.org/10.3390/en14154647

**AMA Style**

Borodin A, Tvaronavičienė M, Vygodchikova I, Panaedova G, Kulikov A.
Optimization of the Structure of the Investment Portfolio of High-Tech Companies Based on the Minimax Criterion. *Energies*. 2021; 14(15):4647.
https://doi.org/10.3390/en14154647

**Chicago/Turabian Style**

Borodin, Alex, Manuela Tvaronavičienė, Irina Vygodchikova, Galina Panaedova, and Andrey Kulikov.
2021. "Optimization of the Structure of the Investment Portfolio of High-Tech Companies Based on the Minimax Criterion" *Energies* 14, no. 15: 4647.
https://doi.org/10.3390/en14154647