Fuzzy evaluation of the value of shares of the issuer companies on the stock market using the example of Exxon Mobil

Dorokhov Evgenii Vladimirovich ORCID: 0000-0001-7869-4530 PhD in Economics Postgraduate Student, Lomonosov Moscow State University, Faculty of Economics 119991, Russia, Moscow, Leninskie Gory str., 1, p. 46 e.v.dorokhov@mail.ru Other publications by this author

Introduction

In modern financial and economic conditions caused by the consequences of the global economic crisis and the COVID-19 pandemic, as a result of which there is significant volatility in stock market quotations, the problem of determining the most reliable valuation of the shares of issuing companies for stock market participants (FR), stock investors, owners and purchasers of companies is relevant. The valuation of shares of issuing companies has two aspects: on the one hand, the owners of shares are entitled to dividends, which, as a rule, are determined by objective financial and economic indicators of the company; on the other hand, the uncertainty and irrationality of the formation of the market price of shares in the process of trading on the stock market are subjective.

There are various methods for evaluating shares of issuing companies. The main methods using cash flow discounting procedures to assess the value of company shares are analyzed in ^[1-3]. In the article ^[1], the methodology of the income approach is used to evaluate a stock asset. In ^[2], a number of methods are considered: comparative valuation based on investment multipliers; dividend and income discounting of cash flows; option pricing using Black–Scholes models and the binomial model. In the study ^[3], profit and loss statements of companies are used to implement discounting of cash flows. Shiller developed a procedure for smoothing cyclical fluctuations in company revenues (net profit) ^[4-5] by averaging them and adjusting for inflation over a period of time when calculating the investment multiplier P/E (P is the market value of a share, E is earnings per share). This procedure allows for a more accurate assessment of the value of company shares, unlike the classical scheme. In the study ^[4], based on the concept of multiplier volatility of a stock, a formula is proposed for estimating the value of shares of companies in the oil and gas industry. Methods of modeling financial systems and stock asset management based on probabilistic and fuzzy multiple descriptions of these uncertain processes are considered in studies ^[6-8]. The systematization of the works of Russian and foreign authors on methods for calculating the value of VaR (Value at risk), taking into account current trends, is considered in the study ^[9]. Based on the VaR method, the article ^[10] analyzes the stock quotes of enterprises in the real sector of the Republic of Uzbekistan. The market price is estimated using the Markowitz model and the CAPM (Capital Asset Pricing Model) model Sharpe. The evaluation of information efficiency and clustering of forecasting volatility in financial markets is carried out in ^[11-14]. With the help of various modifications of the ARCH and GARCH models, widely used for financial data analysis, stock market volatility forecasting models have been developed. The peculiarities of the influence of external factors on the stock assets of Russia are considered in ^[15-16]. The article ^[15] shows that the volume of GDP and the cost of oil have a dominant influence on them. The need for state regulation on the stock market in an unfavorable geopolitical situation is justified in ^[16]. Research ^[17-18] is devoted to the presentation of methods for forecasting and modeling prices on the stock market, which are based on the use of software platforms with elements of neural networks and artificial intelligence. Using the example of shares of American high-tech companies Facebook, Google and Nasdaq, the article ^[19] examines the processes of manipulation of stock asset prices. The signs of manipulation are formulated and a method for their identification is proposed.

The given review of the current state of the problem under study indicates that there are only methods using specialized criteria, and there is no universal method for evaluating shares of issuing companies. Many of the above methods are highly specialized and are used mainly by stock analytical companies. The most common and easy to apply in practice are methods of comparative valuation using investment multipliers, as well as discounting cash flows. This article proposes a methodology for evaluating shares of issuing companies, the subject of which is the calculation of a pair of parameters that determine investment attractiveness: forecast profitability (projected cash flow generated) and the riskiness of the object of assessment.

Fuzzy methodology for estimating the value of shares of issuing companies

The market prices of the shares of the issuing companies are formed in the process of exchange trading based on the combined expert assessments of the participants of the FR. The market price of the issuing company's shares can be divided into the following components:

· a fundamental component reflecting the objective economic and financial performance of the company;

· the component associated with various types of risks (strategic, financial, environmental, technological, operational, personnel, legal, reputational, sectoral, economic, political);

· an emotional component that characterizes the emotional level of market participants in the process of exchange trading and depends on both internal and external factors;

· a manipulated component reflecting the impact of manipulators on the market price of a stock in order to make a profit.

The market value of shares of issuing companies formed by stock market participants during the trading process, without taking into account the emotional and manipulated component, will be considered undisturbed. The undisturbed market value of shares of issuing companies is the price formed by stock market participants only on the basis of initial (historical) objective financial and economic indicators and forecast development scenarios, taking into account various types of risk.

Let's consider a fuzzy economic and mathematical model for describing the risks of the issuer's development, in which its forecast financial and economic indicators are modeled by fuzzy numbers ^[20]. Various forecast scenarios of the company's development are characterized by their own set of forecast financial and economic indicators (parameters). Fuzzy indicators of the forecast scenarios for the development of the issuing company are set on the basis of expert assessments, taking into account its historical financial and economic indicators and their current values obtained as a result of statistical observations. Let T be the forecast financial and economic indicators (profit, income, cash flows per share) generated by the company at the forecast stage t (usually a year). These indicators are presented as fuzzy numbers for the corresponding forecast scenario of the issuer's development. Based on these fuzzy indicators and in accordance with the procedures for calculating net discounted income ^[21], its total fuzzy value P for a certain forecast time period T is determined by the formula:

, (1)

where rt is the forecasted discount rate.

The resulting fuzzy value of P(T) according to formula (1) is interpreted as a forward–looking estimate of the shares of the issuing company, provided that at each time stage t during the entire time period T, the company will generate the value of the indicator T ₍profit, income, cash flows per share). If this condition is met, investments in this company at a cost of P(T) for stock market participants make practical financial sense.

The formation of the current (higher or lower) market price of the issuing company's shares by the participants of the FR depends on their confidence in the further duration of its positive or negative development.

Based on formula (1), the fuzzy value of the investment multiplier M(T), which is widely used by stock market participants to evaluate the value of company shares, is found from the expression:

. (2)

The practice of stock trading shows that the investment multiplier of the issuing company usually has approximately the same value for companies similar to this stock market. In order to smooth out significant changes in the E indicator during crisis events, it is proposed to use the moving average of its E _cp value for 6-8 previous time stages to evaluate the investment multiplier according to formula (2) ^[22].

In vague terms, the indicator M(T) is interpreted as follows: most participants of the FR in the process of forming the current market price of shares of the issuing company P(T) are sufficiently confident in their expectation that this company at each forecast time stage t of the forecast time period T will generate profit, income, cash flows per share at least the predicted value of Ẽ during the subsequent time stages belonging to the fuzzy set M(T).

The developed methodology for the fuzzy valuation of the shares of issuing companies includes:

· databases of historical quotations and financial and economic indicators (profit, income, cash flows per share) of the issuing company;

· forecast fuzzy scenarios for the development of the issuing company, formed on the basis of these databases, as well as expert assumptions about the future development of the internal and external economic situation;

· calculation using formulas (1, 2) of the fuzzy valuation of the shares of the issuing company and its investment multiplier.

This methodology is applicable to various issuing companies with sufficient liquidity and regularly published financial and economic reports, including for the relevant companies of the Russian stock market.

The methods of forecasting and modeling stock market prices in research ^{[17, 18]}, based on the use of software platforms with elements of neural networks and artificial intelligence, as well as the methods considered in ^[9-14], differ in a very narrow specialization and a large amount of processed data. The application of these techniques requires significant labor, as well as information and computing resources. These techniques are mainly used by specialized organizations that are engaged in analytical research of the stock market. Many methods (^{[6-8],[15-16]}) are descriptive in nature. Therefore, these methods are very difficult to apply in practice in a formalized manner to assess the value of shares of issuing companies. The presented fuzzy economic and mathematical methodology for estimating the value of shares of issuing companies includes procedures that are quite simple and common in practice. The proposed methodology makes it possible to assess the inherent degree of risk in the current quotations of shares of issuing companies. Fuzzy scenarios are an adequate description of the forecasted development of issuing companies in accordance with their historical and financial and economic indicators, as well as trends in the development of internal and external economic conditions.

Valuation of shares of Exxon Mobil Corporation based on unclear scenarios of its development

The testing of the developed methodology is carried out on the basis of model scenarios for the development of the largest company in the US oil and gas sector, Exxon Mobil Corporation (Exxon Mobil). The choice of Exxon Mobil to assess the undisturbed value of its shares is based on the fact that currently companies of the American Federal Republic are less exposed to various kinds of risks related to geopolitics than companies of the Russian stock market that are under sanctions. Historical statistics of the company's financial and economic data show that its average annual profit per share from 2012 to 2022 is US$ 5.15 ^[23]. Due to high energy prices in 2022, Exxon Mobil's earnings per share peaked at $13.26 during this period, which is more than 157% more than its average annual profit from 2012 to 2022 ^[24]. This indicator had a minimum value in the 2020 COVID-19 pandemic crisis (-5.25 USD).

The IMF predicts a global slowdown in the global economy in 2023-2024, which may negatively affect the cost of energy resources and, as a result, the financial and economic performance of Exxon Mobil ^[23]. According to the forecasts of the Fund for the development of the world economy in 2023-2024. and taking into account Exxon Mobil's earnings per share in the 1st and 2nd quarters of 2023. ($2.79 and $1.94, respectively) ^[25] it can be expected that the company's profit for 2023 it will be much less than the value of its profit for 2022, but more than its average profit from 2012 to 2022. Taking into account the prevailing economic conditions, as well as the development trend of Exxon Mobil in 2023, it is advisable to consider rather unfavorable scenarios for the company's development in order to assess the value of its shares. In this regard, and taking into account the beginning of the cycle of refinancing rate increases in the United States, the pessimistic and basic model scenarios for the development of Exxon Mobil are being considered.

The value of the investment multiplier M = P/E _cp as of the end of July 2023 is 20.82. As noted above, the indicator M is usually interpreted as follows: most FR participants in the process of forming the market price of shares of the issuing company P are confident that this company will generate profit, income, cash flows per share of at least the value of E _cp during the next M stages. Therefore, the forecast scenarios for the development of Exxon Mobil (basic and pessimistic) are further considered for 20 stages (years) from the end of 2022.

The baseline scenario assumes that in 2023, annual earnings per share are expected in the range of 8-10 USD (the trend at the beginning of the year will continue), then in the next 5 years, starting in 2024, the forecast profit is expected to be higher by about 1.5–2% per year. In the following years, annual earnings per share are projected to grow by 1-1.5% per year. It is assumed that the average annual discount rate in the next 3 years will increase from 5-6% in 2023 to 6-8% in 2025. In the following years, the rate will remain at the level of 6-8% per annum. The forecast for the discount rate is based on the stages of raising the refinancing rate in the United States after quantitative easing, which has been in effect since 2008, and inflation expectations. As of July 2023, the refinancing rate in the United States is 5.25–5.50% per annum ^[26].

Based on the generated forecast scenario for the development of Exxon Mobil, the forecast annual earnings per share are modeled using fuzzy triangular numbers (E01, E1, E02) ^[6]. The values E01, E1, E02 uniquely characterize the membership function of a triangular number, where E01 and E02 represent the left and right boundaries of the zero confidence level of a fuzzy triangular number, and the value E1 determines the value of its unit confidence. It is assumed that the fuzziness (blurriness) of the interval (E01, E02) of the predicted values of E1 of this parameter increases by 1.5% with each forecasted year. The forecast data of annual earnings per share based on the specified baseline forecast scenario for the development of Exxon Mobil are presented in Table 1.

Table 1. Forecast data in the form of fuzzy triangular profit numbers per share of Exxon Mobil (baseline scenario)

Source: compiled by the author based on the forecast scenario for the development of Exxon Mobil

The forecast data in Table 1 shows that the baseline scenario represents a fairly moderate annual growth in earnings per share of Exxon Mobil after the company's strong economic performance in the first half of 2022. (the possibility of recession is taken into account). Table 2 shows the data for calculating the valuation of the undisturbed value of Exxon Mobil shares according to formula (1) in accordance with the forecast baseline scenario.

Table 2. Estimated estimates in the form of fuzzy triangular numbers of the undisturbed value of Exxon Mobil shares (baseline scenario)

Source: formed by the author based on the calculation of the projected baseline scenario for the development of Exxon Mobil

The data for calculating the estimate of the fuzzy value of the investment multiplier M according to formula (2) in the form of fuzzy triangular numbers in accordance with the forecast baseline scenario are presented in Table. 3. As an indicator of E in formula (2), its moving average value for the 8 previous time stages is selected.

Table 3. Estimated estimates in the form of fuzzy triangular numbers for the evaluation of the investment multiplier M of Exxon Mobil shares (baseline scenario)

Source: formed by the author based on the calculation of the projected baseline scenario for the development of Exxon Mobil

Figure 1 shows a graph of fuzzy estimates of the undisturbed value of Exxon Mobil shares (in US dollars), based on the data in Table 2, depending on the values of the forecast stages (baseline scenario).

Figure 1. Dynamics of parameters P01, P1, P02 characterizing fuzzy triangular numbers of estimates of the undisturbed value of Exxon Mobil shares (baseline scenario)

Source: Constructed by the author based on the data in Table 2

The pessimistic scenario for the development of Exxon Mobil is based on the assumption of a possible recurrence of the economic situation during the COVID-19 pandemic-related crisis of 2019-2020. It is assumed that in 2023 the annual profit per share will be approximately 7-8 USD, then in the next four years, starting in 2024, it will remain constant and equal to the average profit from 2012 to 2022 — 5.15 USD, and in subsequent years it is projected to grow by 1.5–2% per year. Table 4 shows the data for calculating the valuation of the undisturbed value of Exxon Mobil shares according to formula (1) in accordance with the forecast pessimistic scenario.

Table 4. Estimated estimates in the form of fuzzy triangular numbers of the undisturbed value of Exxon Mobil shares (pessimistic scenario)

Source: formed by the author based on the calculation of the forecast pessimistic scenario for the development of Exxon Mobil

The data for calculating the estimate of the fuzzy value of the investment multiplier M according to formula (2) in the form of fuzzy triangular numbers in accordance with the forecast pessimistic scenario are presented in Table. 5. As an indicator of E in formula (2), its moving average value for the 8 previous time stages is selected.

Table 5. Estimated estimates in the form of fuzzy triangular numbers for the evaluation of the investment multiplier M of Exxon Mobil shares (pessimistic scenario)

Source: formed by the author based on the calculation of the forecast pessimistic scenario for the development of Exxon Mobil

Figure 2 shows a graph of fuzzy estimates of the undisturbed value of Exxon Mobil shares (in US dollars), based on the data in Table 4, depending on the values of the forecast stages (pessimistic scenario).

Figure 2. Dynamics of parameters P01, P1, P02 characterizing fuzzy triangular numbers of estimates of the undisturbed value of Exxon Mobil shares (pessimistic scenario)

Source: Constructed by the author based on the data in Table 4

The extension of the interval (P01, P02) in Fig. 1 and Fig. 2 with an increase in annual values means that with an increase in the depth of forecasting, uncertainty in stock valuations increases.

The closing level of the market prices of Exxon Mobil shares at the end of 2022 amounted to $110.30 ^[27]. This value belongs to the fuzzy sets of triangular numbers from the 17th to the 20th forecast stages (Table 2, Fig. 1), which are vague estimates of the undisturbed value of Exxon Mobil shares, subject to the implementation of the basic forecast scenario. I.e., the market price of shares at the end of 2022, formed by the participants of the FR, shows that the presented model basic forecast scenario is quite likely for them. Calculations carried out on the basis of a model pessimistic forecast scenario for the development of Exxon Mobil demonstrate that this market price of its shares far exceeds the fuzzy estimates of the undisturbed value of its shares (Table 4, Fig. 2). Therefore, it can be argued that the participants of the FR as of the end of 2022 consider the implementation of such a pessimistic forecast scenario unlikely.

The fuzzy values of Exxon Mobil's investment multipliers in Tables 3 and 5 are equal to the ratio of the fuzzy estimate of the undisturbed value of the company's shares for each forecast stage of the corresponding scenario to the moving average of E _cp for the 8 previous time stages. The average value of this indicator for the four largest oil and gas companies in the United States (Exxon Mobil, Chevron, Conoco Phillips, EOG Resources) by the results of 2022 is 10.4 ^[28]. For the basic scenario, its value belongs to the fuzzy sets M(t) from the 18th to the 20th forecast stages, and for the pessimistic scenario – from the 16th to the 20th forecast stages. The results obtained show that these forecast scenarios for the development of Exxon Mobil within the framework of the presented methodology are adequate to the average value of the investment multipliers of the four largest oil and gas companies in the United States for 2022 and the market prices of Exxon Mobil shares at the end of 2022.

For the basic forecast scenario, the value of the company's market price at the end of 2022 belongs to the fuzzy sets of estimates of the undisturbed value of the company's shares from the 17th to the 20th forecast stages; the average value of the investment multipliers of the four largest oil and gas companies in the United States according to the results of 2022 belongs to the fuzzy sets M (t) from the 18th to the 20th forecast stages stages.

For a pessimistic forecast scenario, the value of the market price of the company's shares at the end of 2022 far exceeds the fuzzy estimates of the undisturbed value of its shares; the average value of the investment multipliers of the four largest US oil and gas companies according to the results of 2022 belongs to the fuzzy sets M (t) from the 16th to the 20th forecast stages.

Therefore, this basic forecast scenario is more consistent than the pessimistic one with the actual economic and financial condition of companies in the US oil and gas industry in 2022. The market prices of these companies, realized by the participants of the FR at the end of 2022, show that the presented model baseline forecast scenario will be more likely for them than the pessimistic one.

Thus, using the example of the presented basic and pessimistic model scenarios for the development of Exxon Mobil, FR participants have the opportunity to compare the current share prices of the company with their estimated fuzzy undisturbed estimates. And based on this comparison, taking into account the degree of risk of implementing these scenarios, make a decision whether it makes practical sense to invest at a given price in this company. Comparing the current values of the investment multipliers of Exxon Mobil shares with their estimated fuzzy estimates for the basic and pessimistic scenarios allows us to establish the adequacy of current market share prices for companies similar to this FR.

Conclusion

The study proposes a methodology for fuzzy valuation of shares, an integral part of which are databases of historical quotations and financial and economic indicators, as well as forecast fuzzy scenarios for the development of issuing companies. The presented methodology makes it possible, based on the specified database and the current financial and economic condition of the companies in question, to form various forecast scenarios for their evolution in accordance with the trends in the development of internal and external economic conditions. The forecast parameters of these scenarios are modeled by fuzzy numbers, the use of which makes it possible to most adequately reflect the uncertainty of the development of companies. The approbation of the developed methodology is carried out using the example of Exxon Mobil shares in the context of the implementation of basic and pessimistic forecast scenarios for the development of the economic situation, fuzzy estimates of the undisturbed value of its shares, as well as estimates of fuzzy investment indicators (multipliers), are determined. A comparison of the obtained fuzzy estimates of the value of shares and investment indicators in the context of the implementation of the basic and pessimistic model scenarios for the development of Exxon Mobil with their current values makes it possible to determine the inherent degree of risk in their current quotes. The presented method of fuzzy valuation of the shares of issuing companies may be in demand not only for stock market participants, owners and purchasers of companies, but also for ordinary investors.

The results of the fuzzy valuation of the shares of the issuing companies provide an opportunity:

· professional stock market participants and potential investors — to form the tactics and strategy of stock trading;

· owners and purchasers of companies — to determine the investment forecast assessment of issuing companies.

This technique is applicable to various issuing companies with sufficient liquidity, for which databases of their historical quotations and financial and economic indicators are available, including for the relevant companies of the Russian stock market.

With regard to the domestic stock market, the developed methodology allows organizations regulating the stock market, based on various scenarios for the development of Russian companies, to determine the forecast movement of quotations of these companies and the stock market as a whole. Knowledge of information about the development of the stock market as the main part of the financial sector of the country's economy can be used by appropriate organizations when implementing plans for the modernization of the country in accordance with Decree of the President of the Russian Federation dated 07/21/2020 No. 474 "On National Development Goals of the Russian Federation for the period up to 2030".