Method for Determining the Risk Profile of Investors Based on the Relationship of Two Stock Investing Problems Текст научной статьи по специальности «Экономика и бизнес»

ORIGINAL PAPER

(CO ]

doi: 10.26794/2587-5671-2024-28-4-136-143 udc 336.767,519.853(045) jel g11, c61

Method for Determining the Risk Profile of Investors Based on the Relationship of Two stock Investing Problems

V. A. Gorelika, T. V. Zolotovab

a Federal Research Center "Informatics and Control" RAS, Moscow, Russia;

Pedagogical State University, Moscow, Russia; b Financial University, Moscow, Russia

abstract

The subject of research in this paper is the investor's risk profile as a characteristic of his behavior in the stock market. The purpose of the study is to assess the investor's risk profile in the form of a risk ratio in a model with a linear convolution of expected return and variance. A financial consultant can use this information to create a portfolio of financial instruments that corresponds with an investor's acceptance of risk. This makes the study relevant because it addresses the problem of minimizing potential risks in the management of an investment portfolio, which is related to the investor's attitude toward risk. The scientific novelty lies in the development of a mathematical approach to solving the problem of determining the risk profile based on the relationship between the solutions of two problems of choosing an investment portfolio, expressed as conditions on the parameters under which the solutions of these problems exist and coincide. Wherein, mathematical programming methods were used, as well as the Python programming language. As a result, the risk coefficient is expressed in terms of the model parameter with a constraint on profitability; a classification of the risk profile according to the acceptable value of the risk coefficient is proposed; the method is implemented as a set of programs and demonstrated on the example of the Russian stock market. The conclusion is made about the possibilities of trust managers using this approach when making decisions on choosing the best portfolio. Keywords: risk coefficient; risk profile; expected return; criteria convolution; investment portfolio

For citation: Gorelik V.A., Zolotova T.V. Method for determining the risk profile of investors based on the relationship of two stock investing problems. Finance: Theory and Practice. 2024;28(4):136-143. DOI: 10.26794/2587-5671-2024-284-136-143

ОРИГИНАЛЬНАЯ СТАТЬЯ

Метод определения риск-профиля инвестора на основе взаимосвязи двух задач фондового инвестирования

В.Д. Горелик3, Т. В. Золотова"

а Федеральный исследовательский центр «Информатика и управление» РАН, Москва, Россия;

Московский педагогический государственный университет, Москва, Россия;

ь Финансовый университет, Москва, Россия

BY 4.0

АННОТАЦИЯ

Предметом исследования в данной работе является профиль риска инвестора как характеристики его поведения на фондовом рынке. Цель исследования - дать оценку риск-профиля инвестора в виде коэффициента риска в модели с линейной сверткой ожидаемой доходности и дисперсии. Актуальность такого исследования, с одной стороны, обусловлена проблемой снижения возможных рисков при управлении инвестиционным портфелем, которая, в свою очередь, связана с отношением самого инвестора к риску, а с другой стороны - дает возможность финансовому консультанту создать портфель финансовых инструментов в соответствии с уровнем согласия инвестора на риск. Научная новизна заключается в разработке математического подхода к решению проблемы определения риск-профиля на основе взаимосвязи решений двух задач выбора портфеля инвестиций, выраженной в виде условий на параметры, при которых решения этих задач существуют и совпадают. При этом использовались методы математического программирования, а также язык программирования Python. В результате коэффициент риска выражен через параметр модели с ограничением на доходность, предложена классификация риск-профиля по допустимому

© Gorelik V.A., Zolotova T.V., 2024

значению коэффициента риска; метод реализован в виде комплекса программ и продемонстрирован на примере российского фондового рынка. Сделан вывод о возможностях применения данного подхода доверительным управляющим при принятии решений о выборе наилучшего портфеля.

Ключевые слова: коэффициент риска; риск-профиль; ожидаемая доходность; свертка критериев; инвестиционный портфель

Для цитирования: Gorelik V.A., Zolotova T.V. Method for determining the risk profile of investors based on the relationship of two stock investing problems. Финансы: теория и практика. 2024;28(4):136-143. DOI: 10.26794/2587-5671-202428-4-136-143

INTRODUCTION

The development of the Russian economy largely depends on the stock market, which plays an important role in the redistribution of financial resources. The economic situation in the Russian Federation contributes to an increase in investment volumes in the securities of Russian enterprises. Economic progress is closely related to the results of investment activity. Attracting investors is one of the key issues related to the Russian stock market. Investing in shares of Russian companies carries a high risk of losing investment.

The main function of the financial market is associated with the transformation of risks. The market mechanism analyzes many types of risks and forms the so-called risk premium. It is important to note that even under conditions of market equilibrium, when all risks are fairly assessed, securities will not be equally attractive to all investors. Factors influencing investor preferences include their financial condition, individual attitude to risk, composition of assets and liabilities, current market conditions and much more. It is important to note that attempts to completely avoid risk result in a portfolio return approaching the risk-free rate, which may not be in the interest of the investor. Identifying the specific types of risk, that need to be addressed allows for a controlled increase in portfolio performance.

To achieve positive results in investment activities, investors form portfolios of securities that reduce the risk of losses and maximize profits [1-3]. To reduce the risks associated with managing investment portfolios, various equity portfolio management strategies are used. One approach to making such investment decisions is to determine the investor's risk profile. The risk profile of an investor determines the level of a person's willingness to take the risk associated with the loss of an investment. Each investor has a different attitude to market volatility or risk, and this attitude depends, for example, on factors such as available funds, age, etc. Risk profiling allows both the investor and the financial advisor to create a portfolio of financial instruments that corresponds to the risk investor profile [4]. The investor's trustee can take

proactive or reactive measures to minimize and, in certain cases, even prevent potential losses after the risk profile is determined.

The investor's risk profile is divided into three types: conservative (low risk propensity), moderate (moderate risk propensity), and aggressive (the greatest willingness to withstand market volatility) [5, 6]. In normal practice, testing is carried out to determine the risk profile [7, 8]. A selection of different data collection instruments was used in [9]: several structured online questionnaires, designed to provide an understanding of risk profiles and personalities, and a software package for simulating investments, used to track investors decisions when managing a portfolio. The article [10] proposes the experimental method, with the application of structured questionnaires, and computer simulation of investments with Expecon software utilizing data on real financial instruments that are available on the market. The articles [11-13] are devoted to the development of platforms for robotic consultants for risk analysis and investor profiling. An empirical method for studying investor risk tolerance is discussed in [14].

This paper proposes to determine the investor's risk profile using quantitative methods, i.e., the problem of finding the optimal portfolio is solved, taking into account the individual attitude of the investor to risk, expressed as a risk coefficient. Thus, the main role of the manager is to determine the goals, restrictions, the choice of appropriate types of securities, the selection of acceptable returns and risks, as well as the formulation of the optimization problem.

A portfolio manager must deal with multicriteria tasks and the problem of lack of information when choosing methods for assessing future results. The choice of an approach to multicriteria, uncertainty and risk leads to a formulated mathematical programming problem that can be solved using available optimization methods [15-19], and using machine-learning methods [20]. When choosing a portfolio, an investor is primarily interested in the expected return and the risk of loss. Really, the investor is interested in profitability, but one cannot determine it. Therefore,

when optimizing a portfolio, this uncertainty is formalized by averaging real return values for the previous period and introducing a risk assessment as a deviation from the average. Accordingly, in this paper, the accounting for these two indicators is formalized in the form of two optimization problems. In the first problem, the linear convolution of the mathematical expectation and the return variance is maximized. In the second problem, the variance is minimized, and the mathematical expectation of the return must be equal to the given value. Conditions on the parameters under which the solutions of these problems exist and coincide are given.

To implement this analytical approach, a software package was developed using the Python programming language version 3.8.5 and the Google Colaboratory environment. It was tested on a practical example with real data from the Russian stock market.

THEORETICAL BASIS OF THE METHOD

The models under consideration are based on the assumption that there is a set of assets, which is described by the vector 7 = (r1,.,7i,., ~rn), where r — the expected return of the i -th financial instrument, and the covariance matrix V = (o,,.) .

V ij 'nx.n

The investor's strategy consists in the distribution of funds between assets and is described by the vector x = (x1v..,xi,...,xn) , where xt — share of funds invested in the i -th financial instrument.

The formulation of the first problem of determining the optimal portfolio:

max

X

[rx-a( xVx)], xe = 1,

(1)

where a > 0 — weighting factor that determines the investor's attitude to risk (risk coefficient), e = (1,..., 1).

The optimal composition of the portfolio x and the corresponding value of the Lagrange multiplier X are found from the system of linear algebraic equations:

r -2aVx* = À*e, x*e = 1.

(2)

ч \ h ( b Л 1 x (a) = —+ g—h a ^ a j

2a

(3)

The formulation of the second problem of determining the optimal portfolio:

min xVx, rx = rp, xe = 1,

(4)

where rp — the expected return of the portfolio, given by the investor.

*

Optimal portfolio composition x and the corresponding values of the Lagrange multipliers X* and X2 are found from the system of linear algebraic equations:

2Vx * = A*r + X2e, rx * =rp, x *e = 1. (5)

*

Find x from the first vector equation of the system (5):

x * = 2 (x* g+^2h).

(6)

Substituting (6) into the second and third equations of system (5), we obtain a system for finding X* and X2. It is shown that if all ri are distinct, then ca - b2 > 0. Then we have the values of the Lagrange multipliers

x * 2 (arp - b) X * 2 (c - brp) X =-^—, At =-

ca - b

ca - b

It is proved that problem (4) has a solution if the following conditions are satisfied

J b b2 li -J b b2 I .

max < —,— ¡4, < min < —,— !• > 1

arp ac

If the risk coefficient a =

arp ac

ca - b 2 (arp - b)

(7)

> 0 , then the

solutions of problems (1) and (4) coincide.

To classify the investor's risk profile according to the type of conservative, moderate, aggressive it is convenient to map range values [0, ^ ) of the coefficient a into the segment [0, 1] using the function

Assume that the covariance matrix V is nondegenerate and introduce the notation for scalar

quantities a = eV~le , b = rV~le , c = rV~lr and

vectors h = V~le and g = V~xr . Solving system (2), we obtain the composition of the optimal portfolio

P(a) =1 - J

1

+ a

(8)

Let's connect the investor's risk profile with the value of the coefficient p, namely,

• aggressive, if pe[0,0.25],

• moderate, if pe(0.25,0.75],

• conservative, if pe(0.75,1].

Note that this classification of the risk profile does not exclude the possibility of its adjustment by financial consultants.

2008-01-09 0,018400 0-011200 0.02» 0-005600 0-051200 0.080600 0023100 ■0,017100 -0.001500 -0.033400

2008-01-10 0.019013 0.007665 ■0.000822 ■0.006177 0003624 0.005102 0.046834 0.012568 0.018495 O.OM&22

2006-01-11 0.091167 0.017167 0.009960 ■0.012609 0.000000 0.045685 0.000829 ■0.D8I310 0.011801 0.000000

2008-01-14- -0.031385 0.02327G 0.021886 -0.015717 -0.009524 0.029029 0.(98079 0.002776 9.002865 -0.006566

2008-01-15 ■0,003352 0,019377 -0,009513 -0,006487 0.000000 ■0,009388 0,018979 ■0.009018 9,006762 -0.015702

W 2022*12*2! 0.004952 0-007(45 ■0-000125 0.005933 0.00419$ ■0001853 0-02650? 0021821 0-010M?

3743 2022-12-27 >0.001665 0.016125 0.012026 0.004613 0.005551 ■0.000380 0.006767 0.008655 ■0.007449 ■0.006139

3744 2022-12-20 -0.007506 ■0.020498 -0.013714 ■0.003227 -0.003276 ■0.000760 -0.011702 -0.000985 ■0.002768 -0.030&82

3745 2022-12-29 0.000000 -0.000675 0.006736 0.001619 -0.002769 0,012367 -0.004633 0.012672 9,010394 0.008194

3746 2022-12-30 0,003361 0,017789 -0,002148 0,011810 0.021867 0,0133« -0,005431 0.016129 9,001348 -0.013245

3747 rows* 11 columns

Fig. 1. Content of Dataframe Profit_DF

Source: Compiled by the author.

tools for implementing the proposed method

The developed software package selects the period and the initial list of shares, data processing, entering the defining parameters and checking the conditions, calculating the risk profile of the investor and the composition of the optimal portfolio (the programs were written with the participation of a student at the Financial University A. V. Karasev).

For the specific implementation of the practical part, stock quotes of ten companies of the Russian stock market were selected:

• AFK Sistema (AFKS)

• Gazprom (GAZP)

• Lukoil (LKOH)

• NLMK (NLMK)

• NOVATEK (NVTK)

• Pole (PLZL)

• Rosneft (ROSN)

• Sberbank (SBER)

• VTB (VTBR)

• Severstal (CHMF)

As the data under consideration, we will take the daily closing prices of the corresponding shares for the period from 2008 to 2022, inclusive. The choice of this period is determined by the completeness of data on closing prices for the selected shares.

Export of quotes of company shares was made from the site "Investing.com".1 On the main page of "Investing. com" in the "Quotes" tab, select "Shares", "Russia", then

1 Investing.com. URL: https://ru.investing.com/equities/russia (accessed on 05.02.2023).

select the type of shares, for example, shares of VTB (VTBR). Then you need to open the tabs "Overview", "Past data" and select "Time period" — "Day". Set the upload boundaries from "01/01/2008" to "12/31/2022" and sort the data in ascending order for the convenience of further manipulations, click on "Download data".

A unified data frame profit_df containing the date and returns for the specified date for the corresponding tickers is shown in Fig. 1.

Further, the possibility of limiting the considered period is implemented by entering the start and end dates from the keyboard, as well as selecting certain securities from the specified list.

Then, in the "Select stocks" block, we will enter the tickers of the stocks we are interested in, from which we want to make an investment portfolio.

A program has been developed that calculates the vector of mathematical expectations of returns mean_ vec and the covariance matrix of returns cov_matrix for selected stocks for a limited period (Fig. 2).

For the selected set of stocks, we find the vector of mathematical expectations of returns and the covariance matrix (Fig. 3).

We use the inverse covariance matrix for calculations (Fig. 4).

Fig. 5 shows the fragment program for calculating mathematical expressions from conditions (7).

Fig. 6 shows a program that allows, at a given level of profitability, to check the conditions that ensure the equivalence of problems (1) and (4). We introduce the required value of the mathematical expectation of the daily return of the portfolio rp and check the conditions (7).

[25] mean_vec = profit_df.iloc[:, l:].mean() mean_vec

[27] covmatrix = profitdf,iloc[:, l:].cov() cov_matrix

Fig. 2. Formation of Mathematical Expectations of stock Returns and Covariance Matrix

Source: Compiled by the author.

vtbr дагр nlmk plil sber

vtbr -0.000526 vtbr 0,000590 0.000342 0.000252 0.000178 0.000445

gazp 0.000516 gazp 0,000342 0.000655 0.000215 0.000182 0.00036B

nlmk 0 .000038

plzl 0.000698 nlmk 0,000252 0.000215 0.000427 0.000158 0.000264

sber -0.000042 plzl 0.000178 0.000182 0.000158 0.000560 0.000184

dtype: float64 sber 0,000445 0.000368 0.000264 0.000104 0.000621

Fig. 3. Calculation of Mathematical Expectations of stock Returns and Covariance Matrix

Source: Author calculations.

vtbr gazp nlmk plzl sber

vtbr 1.0 vtbr 4037.797394 -609.049459 -647.816870 -181.060099 -2202.751357

gazp 1.0 gazp -609.049459 2475.669965 -275.337073 -255.346884 -838.026319

nlmk 1.0

plzl 1.0 nlmk -647.816870 -275.337073 3474,592059 -450.628332 -714.762674

sber 1.0 plzl -181.060099 -255.346884 -450,628332 2105.606688 -152.918728

dtype: float64 sber -2202,751357 -838.026319 -714,762674 -152.918728 4034.890055

Fig. 4. Calculation Inverse Covariance Matrix

Source: Author calculations.

ritiol1 (meanjec f inv.ravjatm ! t.Tj I (rj * le i invjovjatm i >.!))

ratio! = Ile Î inv_COv_Mttii S №âii_vet.T) " 2) I ((Mn_v& ( imcorjiitrix I mesn_ve:,T) * (e S inv_rav_»uix I e,I|) Fig. 5. Matrix operations

Source: Author calculations.

For a given level of return rp = 0.0005 got the value of the risk coefficient a = 5.4799.

The program for calculating optimal portfolios uses formulas (3) and (6), as well as numerical optimization methods of the CVXPY library. Fig. 7 shows the result of calculating the optimal portfolios for the selected set of financial instruments.

As can be seen from Fig. 5, in this example, there is a negative value of one of the components of the portfolio composition vector. A negative value of the share, as you know, means a short sale. If it is required to build a portfolio without short sales, then the process of finding a solution in problems (1), (4) is reduced to enumeration of square submatrices of the original covariance matrix.

condition * (max(ratiol, ratio2) < 1) or (min(ratiol, ratio2) > 1) print(condition)

True

terml ■ mean_vec É inv_cov_matrix Î mean_vec.T

term2 = e 0 inv_cov_matrix 8 e.T

term3 ■ (e # inv_cov_matrix @ mean_vec.T) ** 2

alpha = (terml » termi - term3) / (2 » (tj * termî - (n»an_vec Î inv_cov_matrix •> e.T))]

True alpha ■ 5,47935274407859 Fig. 6. Checking conditions and calculating the risk profile for a given level of return

Source: Author calculations.

alpha: 5.47985274407B59

Vtbr -0-122 357

gazp 0 >264364

nlmk 0.372643

plZl 0-413066

sber 0.072284 dtype: float<S4

r_p - 0.0005

vtbr -0.122357

gazp 0,264364

nlmfc 0.372643

plzl 0.413066

aber 0.072284

dtype: float64

Fig. 7. Finding optimal portfolios

Source: Author calculations.

When using tools, one can build a portfolio without short sales, for example, by adding a non-negativity condition on the variables and using a numerical method for convex programming problems (1), (4).

For the found value a = 5.4799 , we get the value P = 0.8457 by formula (8). This means that, in accordance with the above classification, the constructed portfolio is suitable for a conservative investor.

conclusion

Thus, the investor's risk profile, which is a certain characteristic of the investor's behavior in the stock market, helps the trustee to choose the right investment strategy. At the same time, for each type of investor, the return on his portfolio is related to, for example, the inflation rate or the rate on deposits.

The purpose of a conservative investor is to protect against inflation and preserve his capital. It is reasonable for him to get paid for investing 70% of the money at the rate of return on deposits or the rate at which inflation is indicated.

The moderate investor is focused on stable accumulation. He only invests 50% in bonds and the

rest in stocks. The return on investment should be well above the rate of inflation.

An aggressive investor is prepared to accept chances in order to generate high profits. He tries to maximize return on investment by investing about 80% of the assets in highly volatile financial instruments.

If, when making decisions on choosing the best portfolio, convolution of sum-type criteria with a risk coefficient for dispersion is used, then the problem of setting the risk coefficient (investor's risk profile) arises. In this paper, a mathematical approach to solving such a problem is developed.

The procedure for determining the relationship between the coefficient a and the expected level of return can help financial advisers in making informed decisions on compiling investment portfolios for clients with different risk profiles.

The technical implementation of the proposed method makes it possible to automate the process of determining the investor's risk profile. The use of the Python 3.8.5 programming language and the user-friendly Google Colaboratory environment, which does not require the installation of additional software, allow multiple users to work together.

references / СПИСОК ЛИТЕРАТУРЫ

1. Bacon C. R. Practical portfolio performance measurement and attribution. New York, NY: John Wiley & Sons; 2023. 529 p.

2. Gorelik V. A., Zolotova T. V. Criteria for evaluation and optimality of risk in complex organizational systems. Moscow: Dorodnicyn Computing Centre of RAS; 2009. 162 p. (In Russ.).

Горелик В. А., Золотова Т. В. Критерии оценки оптимальности риска в сложных организационных системах. М.: ВЦ РАН; 2009. 162 с.

3. Kalayci C. B., Ertenlice O., Akbay M. A. A comprehensive review of deterministic models and applications for mean-variance portfolio optimization. Expert Systems with Applications. 2019;125:345-368. DOI: 10.1016/j. eswa.2019.02.011

4. Dekanova K. V. Evaluation of opportunities to increase the potential for attracting funds from private investors through their rational interaction with an independent financial (investment) adviser. Sibirskaya finansovaya shkola = Siberian Financial School. 2022;(2):77-87. (In Russ.). DOI: 10.34020/1993-4386-20222-77-87

Деканова К. В. Оценка возможностей повышения потенциала привлечения средств частных инвесторов за счет рационального их взаимодействия с независимым финансовым (инвестиционным) советником. Сибирская финансовая школа. 2022;(2):77-87. DOI: 10.34020/1993-4386-2022-2-77-87

5. Kislitsyna L. V., Shnitova G. A., Makhonina O. V., Safonova V. A. Methodology for categorizing investors for the purposes of forming their investment portfolio. Zhurnal prikladnykh issledovanii = Journal of Applied Research. 2021;(6-7):667-672. (In Russ.). DOI: 10.47576/2712-7516_2021_6_7_667

Кислицына Л. В., Шнитова Г. А., Махонина О. В., Сафонова В. А. Методика категоризации инвесторов для целей формирования их инвестиционного портфеля. Журнал прикладных исследований. 2021;(6-7):667-672. DOI: 10.47576/2712-7516_2021_6_7_667

6. Zhiluk D. A., Skorokhod A. Yu. Categorization of private investors: New risks and opportunities. Izvestiya Sankt-Peterburgskogo gosudarstvennogo ekonomicheskogo universiteta. 2020;(4):42-47. (In Russ.).

Жилюк Д. А., Скороход А. Ю. Категоризация частных инвесторов: новые риски и возможности. Известия Санкт-Петербургского государственного экономического университета. 2020;(4):42-47.

7. Kachalov A. A. Determining level of risk tolerance of individual investor. Finansovyi biznes = Financial Business. 2021;(11):56-58. (In Russ.).

Качалов А. А. Особенности процесса определения риск-профиля частного инвестора. Финансовый бизнес. 2021;(11):56-58.

8. Nguyen L., Gallery G., Newton C. The joint influence of financial risk perception and risk tolerance on individual investment decision-making. Accounting & Finance. 2019;59(S 1):747-771. DOI: 10.1111/acfi.12295

9. De Bortoli D., da Costa N., Jr., Goulart M., Campara J. Personality traits and investor profile analysis: A behavioral finance study. PloS One. 2019;14(3): e0214062. DOI: 10.1371/journal.pone.0214062

10. Goulart M., da Costa N. C.A., Jr., Andrade E. B., Santos A. A.P. Hedging against embarrassment. Journal of Economic Behavior & Organization. 2015;116:310-318. DOI: 10.1016/j.jebo.2015.04.014

11. Abraham F., Schmukler S. L., Tessada J. Robo-advisors: Investing through machines. Research and Policy Briefs. 2019;(21). URL: https://documents1.worldbank.org/curated/en/275041551196836758/pdf/Robo-Advisors-Investing-through-Machines.pdf

12. Capponi A., Ólafsson S., Zariphopoulou T. Personalized robo-advising: Enhancing investment through client interaction. Management Science. 2022;68(4):2485-2512. DOI: 10.1287/mnsc.2021.4014

13. Fisch C., Masiak C., Vismara S., Block J. Motives and profiles of ICO investors. Journal of Business Research. 2021:125:564-576. DOI: 10.1016/j.jbusres.2019.07.036

14. Mubaraq M. R., Anshori M., Trihatmoko H. The influence of financial knowledge and risk tolerance on investment decision making. JurnalEkonomiBisnis dan Kewirausahaan. 2021;10(2):140-153. DOI: 10.26418/ jebik.v10i2.47089

15. Ao M., Yingying L., Zheng X. Approaching mean-variance efficiency for large portfolios. The Review of Financial Studies. 2019;32(7):2890-2919. DOI: 10.1093/rfs/hhy105

16. García F., González-Bueno J.A, Oliver J. Mean-variance investment strategy applied in emerging financial markets: Evidence from the Colombian stock market. Intellectual Economics. 2015;9(1):22-29. DOI: 10.1016/j. intele.2015.09.003

17. Guo X., Chan R. H., Wong W.-K., Zhu L. Mean-variance, mean-VaR, and mean-CVaR models for portfolio selection with background risk. Risk Management. 2019;21(2):73-98. DOI: 10.1057/s41283-018-0043-2

18. Harman R., Prus M. Computing optimal experimental designs with respect to a compound Bayes risk criterion. Statistics & Probability Letters. 2018;137:135-141. DOI: 10.1016/j.spl.2018.01.017

19. Radner R. Decision and choice: Bounded rationality. In: Wright J. D., ed. International encyclopedia of the social & behavioral sciences. 2nd ed. Oxford: Elsevier Science; 2015:879-885.

20. Chen W., Zhang H., Mehlawat M. K., Jia L. Mean-variance portfolio optimization using machine learning-based stock price prediction. Applied Soft Computing. 2021;100:106943. DOI: 10.1016/j.asoc.2020.106943

about the authors / информация об авторах

Victor A. Gorelik — Dr. Sci. (Phys. and Math.), Leading Researcher of the Department of Simulation Systems and Operations Research, Federal Research Center "Informatics and Control" RAS, Moscow, Russia; Prof., Chair of the Theoretical Informatics and Discrete Mathematics, Moscow Pedagogical State University, Moscow, Russia

Виктор Александрович Горелик — доктор физико-математических наук, ведущий научный сотрудник отдела имитационных систем и исследования операций, Федеральный исследовательский центр «Информатика и управление» РАН, Москва, Россия; профессор кафедры теоретической информатики и дискретной математики, Московский педагогический государственный университет, Москва, Россия http://orcid.org/0000-0003-2435-0796 gorelik@ccas.ru

Tatyana V. Zolotova — Dr. Sci. (Phys. and Math.), Assoc. Prof., Prof. of the Chair of Modeling and System Analysis, Financial University, Moscow, Russia

Татьяна Валерьяновна Золотова — доктор физико-математических наук, доцент, профессор кафедры моделирования и системного анализа, Финансовый университет, Москва, Россия

http://orcid.org/0000-0001-5185-0687 Correspondent author / Автор для корреспонденции: tzolotova@fa.ru

Conflicts of Interest Statement: The authors have no conflicts of interest to declare. Конфликт интересов: авторы заявляют об отсутствии конфликта интересов.

The article was submitted on 21.08.2023; revised on 05.12.2023 and accepted for publication on 27.12.2023. The authors read and approved the final version of the manuscript.

Статья поступила в редакцию 21.08.2023; после рецензирования 05.12.2023; принята к публикации 27.12.2023.

Авторы прочитали и одобрили окончательный вариант рукописи.