CC BY
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UDC 517.958:52/59
Vestnik SibGAU Vol. 17, No. 1, P. 79-83
A MATHEMATICAL MODEL OF OIL PRICE ASSESSMENT
K. V. Safonov
Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: safonovkv@rambler.ru
The article deals with the development of the mathematical model of oil price assessment. The methodological foundation is determined for developing the mathematical model: two axioms stating the unique properties of oil as a commodity. The first one claims that oil is a commodity being determined at auctions and is not related to its value as a measure of abstract labor invested, the second axiom states that the markdown in oil price will not cause the increase in the demand for it, as the demand is determined only by the economy state of the demander. Among the factors of oil pricing an imbalance of oil supply and demand in the world market is chosen to be the dominant factor. The mathematical model is represented in two models. The first one assumes that for any excess of supply over demand, the price of oil tends to zero, i. e. for a sufficiently large number of auctions it becomes lower than any predefined level. The second theorem states that in the case of the excess of demand over supply oil price tends to infinity (a finite number of sessions exceeds any predefined level) in case of the dominance of imbalance. The most likely forecast resulting from the hypothesis that the developed mathematical model is correct is the trend of the price decrease reaching its extremely low level and a further transition into a long-term period characterized by the price increase trend.
Keywords: the price of oil, a mathematical model, the unique properties of the oil trade, the dominant factor in pricing, supply and demand imbalance.
Вестник СибГАУ Том 17, № 1. С. 79-83
О МАТЕМАТИЧЕСКОЙ МОДЕЛИ ФОРМИРОВАНИЯ ЦЕНЫ НА НЕФТЬ
К. В. Сафонов
Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнёва Российская Федерация, 660037, г. Красноярск, просп. им. газ. «Красноярский рабочий», 31
E-mail: safonovkv@rambler.ru
Разработана математическая модель формирования цены на нефть. Для построения математической модели определены методологические основания: две аксиомы, устанавливающие специфические свойства нефти как товара. Первая утверждает, что нефть является товаром, цена которого определяется на аукционах и не связана с его стоимостью как мерой вложенного абстрактного труда, вторая аксиома утверждает, что снижение цены на нефть не вызывает повышения спроса на неё, поскольку спрос определяется лишь состоянием экономики покупателя. Среди факторов ценообразования нефти выбран первостепенный, доминирующий фактор - дисбаланс предложения и спроса на нефть на мировом рынке. Математическая модель представлена в двух теоремах. Первая утверждает, что при любом превышении предложения над спросом цена нефти стремится к нулю, т. е. за достаточно большое количество аукционов становится ниже любого заданного уровня. Вторая теорема утверждает, что при условии превышения спроса над предложением цена нефти стремится к бесконечности (за конечное число сессий превысит любой заданный уровень) при условии доминирования дисбаланса. Наиболее вероятным прогнозом, вытекающим из гипотезы о том, что построенная математическая модель верна, является тренд на снижение цены до предельного, экстремально низкого значения, а затем переход к длительному периоду, который характеризуется трендом роста цены.
Ключевые слова: цена на нефть, математическая модель, уникальные товарные свойства нефти, доминирующий фактор ценообразования, дисбаланс предложения и спроса.
Introduction. In the middle of 2014 the process of a markdown in oil price commenced. It became in the focus of public attention as the value of oil prices for the economies of oil-producing countries is very large [1-10]. When estimating, experts talk about a "fair" price level at which the markdown may stop. Initially the levels of $90, $80 and $60 per Brent oil barrel were mentioned, gradually $50 and $40 were spoken of, currently a possibility of $30 and even lower prices are predicted.
The forecasts of upcoming price increase or at least its pegging given by media are positively received by the society. Meanwhile, the markdown is running on, which actualizes the search of a reasonable and, finally, well-thought-out model of oil price forming describing the current state of affairs from the middle of 2014 to the present day. The feasibility of such models will allow to hope that the forecasting subsequent upon these models is likely to come true.
Needless to say, the public are aware of the range of factors currently encouraging the markdown. The following factors are unanimously claimed to work in such a way: excess of oil supply over demand on the world market, the change in values of the US dollar due to the actions of The Federal Reserve System of the USA, different steps taken by OPEC members, current oil reserves information, the number of drilling rigs in the USA, weather forecasts, etc.
Nevertheless, the lack of holistic comprehension of oil price forming process model in the media is acknowledged. In fact, under the conditions of progressing markdown factors, it is desirable to understand what the reasons, frequently mentioned by the experts, for upcoming price increase to any "fair" level are.
At least judging by the information presented in the media, it makes sense to speak about the fact that the contribution of each markdown factor is not differentiated enough; its importance is not fairly appreciated.
As a result, the majority of people interested in oil price as a factor affecting social policy of oil-producing countries expect positive course of events in the nearest future and the issue of excess of oil supply over demand seems to them quite solvable. Many people, for example, think that if oil supply exceeds demand by 5 %, the price will soon consequently reduce by 5 % and, as a result, a new "fair" price making 95 % of the previous one will be set which will eliminate the problem.
Of course, it is not so. Anticipating, we should note that 5 % excess of supply over demand will lead to longtime sequential multiple-stage markdown. As a result, at least theoretically, it may make 5 % of the staring price, so the reduction may make 95 % - the price will be decreased by 20 times! This is, as we will ensure, the mechanism of oil price forming.
Coming back to the feasibility of forecasting, we state that one of the ways of adequate understanding of current state of affairs is to develop a mathematical model of oil price forming.
The mathematical model should reasonably describe the markdown since the middle of 2014, explain its causes and forecast further price forming. Naturally, the model actuality should be viewed approximately, as a trend.
The aim of this article is developing and analysis of the mathematical model of oil price forming.
Developing the mathematical model of oil price forming. It should be noted that mathematical model method is a general-purpose and powerful instrument of human perception, especially efficacious in the cases when experiment conduction is impossible. Mathematical model method allows to study a phenomenon basing on the development and analysis of its mathematical model -an approximate description in mathematical parlance [11].
Mathematical model development implies a certain simplification of a complex phenomenon, its particular idealization basing on the identification of major factors which characterize the evolvement of the phenomenon and on neglecting minor, less important factors [12-15].
Neglecting minor factors allows developing the mathematical model basing on major dominant factors and hereafter it is sufficient to study the model thoroughly. As a result, the model response is determined approximately, as a trend fixed by major factors; the trend presumes inconspicuous movements caused by the influence of minor factors.
Mathematical modeling normally includes the following range of stages:
- Development of the mathematical model of the phenomenon based on its simplification, identification of dominant factors which cause this phenomenon;
- Exploration of the model by means of mathematical methods, gaining the information which describes the further evolvement of the phenomenon;
- Comparing of the theoretically forecast model response with what will actually occur;
- Specification and correcting of the mathematical model in case there is a difference between theoretical and practical aspects;
- Exploration of the specified model, description of its response;
- Repeated correcting of the mathematical model basing on the comparison of theory and practice etc.
It should be realized that mathematical model development is to be relied on solid methodological foundation, well-proven facts and information. In our case such a foundation can be represented by the unique properties of oil as a commodity which make it different from many other commodities. It seems reasonable to associate oil price behavior with its specific commodity properties as many other commodity prices, unlike oil price, do not show sustained drop.
What are the unique properties of such a commodity as oil, or, to be more precise, the specific properties of its price forming which can be used as a methodological foundation of the prospective mathematical model?
In fact, these commodity properties of oil are well-known, however, it is essential to see their true value. We suggest using these properties as the following prime postulates which act as the axioms describing oil price forming principles.
The following two statements are suggested as such axioms.
Axiom 1. Oil is a commodity which price is determined at auction sellings on several fixed venues and is not directly related to its cost as a measure of abstract
labor invested and to the market price of other commodities which, unlike oil price, are determined by millions of market transactions.
The essence of axiom 1 seems to be well-known. We pay attention to the fact that it underlines the weak connection of a forming oil price with its actual cost which is determined by average production expenses. Thus, the possibility of a considerable gap between the price and the cost, both downward and upward, is postulated.
Axiom 2. Oil price markdown does not trigger increase in demand, as it is determined by the economy state of a demander; increase in demand is possible only at a low price for oil reservation purposes, however, the extra demand exists only provided that oil price is low and reduces as soon as the price rises.
The content of Axiom 2 is also well-known. We will note that this property of oil differs from many other commodities, cheapening of which leads to expansion of consumption.
The emphasis is laid on the fact that these statements are empiric basing on the experience of human activity related to oil extraction and marketing, they have been proved through practice over decades, they are axioms from the logical point of view and require no further proof.
The aggregate of these two simple axioms is almost sufficient for understanding the main trends in price forming. The missing link for developing the mathematical model of oil price forming is, as it was mentioned before, the identification of major and minor price-forming factors.
In our opinion, the major foremost factor of oil price change is the imbalance of supply and demand; the value of this factors significantly prevails over all the other ones, making it dominant.
In case market oil supply exceeds demand, this very factor becomes the basic one leading to the markdown; in case supply is lower than demand, the main trend shows price increase.
Relying on Axioms 1 and 2 and considering the imbalance of supply and demand to be dominant, we obtain the mathematical model of oil price forming by two basic theorems.
The first one corresponds to the current state of affairs of exceeding supply over demand, namely, the following theorem describing the main trend in price forming is fair:
Theorem 1. If oil market supply exceeds demand for any arbitrarily small value A1 > 0, realized by market participants, in the process of repeated auction selling oil price tends to zero, i. e. in finite number of steps it becomes lower than any predefined level.
In case demand exceeds supply, price forming is described by the following theorem.
Theorem 2. If oil market demand exceeds supply for any arbitrarily small value A2 > 0, realized by market participants, in the process of repeated auction selling oil price tends to infinity, i. e. in finite number of steps it will exceed any predefined level becoming nonproportionally high in relation to the cost of other commodities.
Naturally, these theorems are provable. The proof of theorem 1. As supply exceeds demand, the auction
participants are forced to agree on the price lowering by a certain value, for example, by $0.5 in order to sell their oil volumes. According to axiom 1, both demanders and suppliers, determining oil price at the next session of auction selling, will see that, according to Axiom 2, the demand does not increase, which still means A1 > 0, leading suppliers to the necessity of reducing the price by $0.5. As a result of the iterative process, in finite number of steps the oil price will arbitrarily tend to zero.
The prove to theorem 2. If supply is lower than demand, the demanders at auction selling agree on a certain price increase, say, at $0.5 in order to purchase required amounts of oil. In compliance with axiom 1, both demanders and suppliers, determining oil price at the next session of auction selling, will see that the demand does not reduce, which still means A2 > 0, leading demanders to the necessity of agreeing on a $0.5 price increase again. As a result of repeating auctions, in a finite number of trading sessions oil price may exceed, at least theoretically, any predefined level.
We should underline the fact that, firstly, oil price is determined by the results of auctions that is why these results depend mainly on supply and demand balance. For example, if milk price were determined in the same way, it would differ from the one we see in shops, it could take on surprising values.
Secondly, the theorems say about the supply and demand im/balance value "realized by market participants". We imply that market participants realize and weigh all the aggregate of factors affecting the price including oil reserves information, the number of drilling rigs in the USA, Chinese economy state information, geopolitical events, weather forecasts, etc. Evaluation of excess supply over demand is conducted with regard to all the minor factors and definitely includes emotional component.
Analysis of the developed mathematical model. We
will show what conclusions can be drawn when analyzing the developed mathematical model. Relating to the current state of events, the price forming mechanism described by theorem 1 results in the following statements.
Firstly, it is not correct to speak about, as it occurs sometimes, oil price slump; the price lowers quite smoothly, without leaps, which proves once again that the main reason is quite explainable and is the excess of supply over demand.
Secondly, there is no a priori certain fair price. Being launched due to certain conditions (excess of supply over demand), price reduction process cannot cease on its own accord while the condition is fulfilled. The cease can occur only when supply and demand are balanced and the price formed under such a condition can be considered fair.
Thirdly, the current market situation is characterized by a significant steady excess of supply over demand. As it becomes clear, there is no use expecting the price to stop as a result of imbalance elimination. As the result of decrease process the price may reach an extremely low level and stop just right after it because further decrease is impossible. In practice this level may be about $10 (for oil price it is a "physical" zero, further decrease bears
no sense). To some extent, it would be the fair price corresponding to the situation. Reaching this level may occur in a relatively short period of time, for example, within half-year.
When experts say that the fair price is $40 or $50, they surreptitiously compare the price with minimal production expenses, i. e. with the cost which, as we see, has nothing to do with oil price.
Fourthly, price decrease process can be controlled only by decreasing oil supply on the world market.
Fifthly, the most important conclusion is related to the answer to the key question: what will happen when the price reaches "zero level", but supply still exceeds demand?
Theorem 1 cannot be applied in this case as the price can no longer decrease. There is also no reason to forecast a small price bounce and further retrieval to the zero level. It seems that such a situation should be described within a framework of a special mathematical model which is beyond the scope of this article.
Anyway, logical and practical way out of the situation of reaching a rock bottom price level may be to acknowledge one of the scenarios: either supply stops exceeding demand, or supply excess ceases being a dominant factor, which has been mentioned above, and another factor becomes dominant.
In the first case, when supply reduces, the conditions of theorem 2 will start acting, according to which the price will start increasing unrestrictedly.
In the second case other dominant factors will appear. Basing on the hypothesis that this mathematical model is correct, market participants are very likely to change the behavior. Indeed, black marketers play an important role making profit on oil price changes. The situation in which they are not able to move down will lead to their necessity to gain profit from the realization of price increase sustained trend. Thus, in both cases the commence of unrestricted price increase process is most likely to happen. According to the developed theory and the standing practice it can be forecast that the maximum markdown and moving to increase period is possible in the period from six months to one year.
The increase period duration, as it can be assumed, will be sufficient enough for the price to reach a high level stage by stage.
Sixthly, the represented mathematical model allows to derive a range of conclusions, for example, that longtime pricing at the level of approximately $50 under the condition of imbalance is impossible for a long period of time, say, one year. At the same time it should be noted that the average price for this period can be quite high due to the fact that a significant markdown can be compensated by a considerable increase.
Conclusion. This article presents and works out the problem of developing the mathematical model of oil price forming.
The methodological foundation is determined for the development of such a mathematical model. The two axioms stating the unique properties of oil as a commodity are chosen as the basis. The first axiom says that oil is the commodity which price is determined at auctions on 2-3 world venues and is not directly related
to its cost as a measure of abstract labor invested and the prices of other commodities which are determined as a result of a great number of market transactions.
The weak connection of the forming oil price with its actual cost which is objectively determined by average market production expenses postulates the possibility of a considerable gap between the price and the cost, both downward and upward.
The second axiom states that the markdown in oil prices involves no demand increase for it, as the demand is determined by the economy state of a demander; market demand increase is possible at a lower price as a part of demanders can purchase oil for reservation purposes, however, extra demand exists only under condition of a low price and it vanishes if the price increases. This commodity property of oil is essential as it differs oil from the majority of commodities, cheapening of which leads to a significant increase of consumption.
These statements are empirical basing on the long-term experience of oil extracting and marketing and are axiomatic from the logical point of view.
Among the factors that determine oil price the imbalance of supply and demand on the world market is chosen as a major dominant factor.
Neglecting multiple minor factors allows to develop the mathematical model basing on the mentioned dominant factor and to analyze it. As a result, the mathematical model response is described approximately, presuming inconspicuous movements which are generated by the influence of minor factors.
The mathematical model is presented in two theorems. The essence of the first one is that oil price becomes lower than any predefined level in a considerable number of auctions under every excess of supply over demand.
The second theory claims that oil price will exceed any predefined level in a finite number of trading sessions under the condition of exceeding demand over supply.
The developed mathematical model, as it must be, provides a certain simplification, idealization of oil price forming process. Nevertheless, it is simple, it has a clear reasoning and is practically confirmed: it is the only model to adequately describe the current state of events on the oil market.
The most likely forecast resulting from the developed mathematical model is the decrease of the price reaching its extremely low level and a further transition into a long-term multiple-stage period of its increase.
Of course, if the oil supply will be limited and become lower than its demand, price markdown will cease and, according to theorem 2, the price will be increasing.
References
1. Neft' podorozhala na 3 % v posledniy den' 2015 goda [Oil rose by 3% in the last day of 2015] (In Russ.). Available at: http://www.vestifinance.ru/articles/66055 (accessed 02.01.2016).
2. SMI: stoimost' barrelya nefti prevysit v 2016 godu 50 dollarov [Media: the price of a barrel of oil will exceed $50 in 2016] (In Russ.). Available at: http: // www. banki.ru/news/lenta/?id=8564375&r1=rss&r2=common& r3=news (accessed 02.01.2016).
3. Barrel' Brent opustilsya do $36,6 [A barrel of Brent fell to $36,6 ] (In Russ.). Avaliable at: http://www.vedomosti.ru/business/news/2015/12/31/6229 99-barrel-brent-opustilsya-do-366/ (accessed 02.01.2016).
4. Brent i WTI poteryali v tsene posle dannykh o roste zapasov v SShA [Brent and WTI lost value after data on growth of stocks in the United States] (In Russ.). Available at: http://www.vedomosti.ru/economics/news/ 2015/12/30/622986-brent-wti (accessed 02.01. 2016).
5. Mirovye tseny na neft' torguyutsya raznonaprav-lenno [Global oil prices traded mixed] (In Russ.). Available at: http://www.gazeta.ru/business/news/2015/ 12/31/n_8080025. shtml (accessed 02.01.2016).
6. Tseny na neft' uskorili padenie: zapasy rastut, spros snizhaetsya, prognozy ne obnadezhivayut [Oil prices accelerated their decline: stocks are rising, demand is slowing, the forecasts are not encouraging] (In Russ.). Available at: http://www.newsru.com/finance/30dec2015/ oilafternoon.html (accessed 02.01.2016).
7. Retsessiya, urezanie raskhodov, revolyutsiya? Kak nizkie tseny na neft' skazyvayutsya na neftyanykh derzhavakh [The recession, cuts in expenditures, the revolution? How low oil prices affect oil powers] (In Russ.). Available at: http://inopressa.ru/article/ 30Dec2015/guardian/oil.html (accessed 02.01.2016).
8. Esli tseny na neft' sorta Brent dostignut $30 za barrel', para dollar/rubl' protestiruet uroven' 78 [If the price of Brent crude oil reached $30 per barrel, the dollar/ruble will test the level of 78] (In Russ.). Available at: http://www.finam.ru/analysis/marketnews90CFF (accessed 02.01.2016).
9. Siriya, Turtsiya, OPEK. Samye krutye momenty padeniya i rosta tseny nefti v 2015 g [Syria, Turkey, OPEC. The coolest moments of the rise and fall of oil prices in 2015] (In Russ.). Available at: URL: http://neftegaz.ru/news/view/144907 (accessed 02.01.2016).
10. Sberbank CIB zayavil o nepredskazuemosti ekonomicheskoy dinamiki pri nefti nizhe $25 [Sberbank CIB announced the unpredictability of the economic dynamics in oil below $25] (In Russ.). Available at: http://www.vedomosti.ru/finance/news/2015/12/30/ 622949-sberbank-cib (accessed 02.01.2016).
11. Matematicheskaya entsiklopediya. [Mathematical encyclopedia]. Moscow, Sovetskaya entsiklopediya Publ., 1982, Vol. 3, 574 p. (In Russ.).
12. Samarskiy A. A., Mikhaylov A. P. Matemati-cheskoe modelirovanie: Idei. Metody. Primery. [Mathematical modeling: Ideas. Methods. Examples]. Moscow, Fizmatgiz Publ., 2001, 320 p. (In Russ.).
13. Abramov A. P., Ivanilov Yu. P. Fizika i matematicheskaya ekonomika [Physics and mathematical Economics]. Moscow, Znanie Publ., 1991, 32 p. (In Russ.).
14. Lotov A. V. Vvedenie v ekonomiko-matematicheskoe modelirovanie [Introduction to economic-mathematical modeling]. Moscow, Nauka Publ., 1984, 392 p. (In Russ.).
15. Kantorovich L. V., Gorstko A. B. Optimal'nye resheniya v ekonomike [Optimal Solutions in Economics]. Moscow, Nauka Publ., 1979, 229 p. (In Russ.).
Библиографические ссылки
1. Нефть подорожала на 3 % в последний день 2015 года [Электронный ресурс]. URL: http://www. vestifinance.ru/articles/66055 (дата обращения: 02.01. 2016).
2. СМИ: стоимость барреля нефти превысит в 2016 году 50 долларов [Электронный ресурс]. URL: http: // www.banki.ru/news/lenta/?id=8564375&r1=rss& r2=common&r3=news (дата обращения: 02.01.2016).
3. Баррель Brent опустился до $36,6 [Электронный ресурс]. URL: http://www.vedomosti.ru/business/ news/2015/12/31/622999-barrel-brent-opustilsya-do-366 (дата обращения: 02.01.2016).
4. Brent и WTI потеряли в цене после данных о росте запасов в США [Электронный ресурс]. URL: http://www.vedomosti.ru/economics/news/2015/12/30/62 2986-brent-wti (дата обращения: 02.01.2016).
5. Мировые цены на нефть торгуются разнонаправленно [Электронный ресурс]. URL: http://www. gazeta.ru/business/news/2015/12/31/n_8080025.html (дата обращения: 02.01.2016).
6. Цены на нефть ускорили падение: запасы растут, спрос снижается, прогнозы не обнадеживают [Электронный ресурс]. URL: http://www.newsru.com/ finance/30dec2015/oilafternoon.html (дата обращения: 02.01.2016).
7. Рецессия, урезание расходов, революция? Как низкие цены на нефть сказываются на нефтяных державах [Электронный ресурс]. URL: http://inopressa.ru/ article/30Dec2015/guardian/oil.html (дата обращения: 02.01.2016).
8. Если цены на нефть сорта Brent достигнут $30 за баррель, пара доллар/рубль протестирует уровень 78 [Электронный ресурс]. URL: http://www.finam.ru/ analysis/marketnews90CFF (дата обращения: 02.01.2016).
9. Сирия, Турция, ОПЕК. Самые крутые моменты падения и роста цены нефти в 2015 г. [Электронный ресурс]. URL: http://neftegaz.ru/news/view/144907 (дата обращения: 02.01.2016).
10. Sberbank CIB заявил о непредсказуемости экономической динамики при нефти ниже $25 [Электронный ресурс]. URL: http://www.vedomosti.ru/ finance/news/2015/12/30/622949-sberbank-cib (дата обращения: 02.01.2016).
11. Математическая энциклопедия. Т. 3. М. : Советская энциклопедия. 1982. С. 574.
12. Самарский А. А., Михайлов А. П. Математическое моделирование: Идеи. Методы. Примеры. 2-е изд. М. : Физматгиз, 2001. 320 с.
13. Абрамов А. П., Иванилов Ю. П. Физика и математическая экономика. М. : Знание, 1991. 32 с.
14. Лотов А. В. Введение в экономико-математическое моделирование. М. : Наука, 1984. 392 с.
15. Канторович Л. В., Горстко А. Б. Оптимальные решения в экономике. М. : Наука, 1979. 229 с.
© Safonov K. V., 2016
