Theoretical bases of a fuzzy pay-off method for real option valuation in energy sector

According to A. Damodaran's classification reserve opportunities of the company, and also industrial tests and scientific researches belong to a separate type of real options, namely growth option. Besides, A. Damodaran refers strategic merges and absorption to this type of real options. Characteristic of an option of growth is that it, in fact, presents the prerequisite or a link to chains of the interconnected projects, opening new opportunities for growth in future [Damodaran, A., 2012, p. 46].

Real options "for" projects are connected with an assessment of investment opportunities while real options "in" projects create administrative flexibility directly.

Classical examples of real options "for" projects is the assessment of oil fields, mines, pharmaceutical research projects where in general it is necessary to estimate result of such projects and to make the decision on expediency of investment into these projects. Strengthening the base and columns on a multilevel parking can be an example of real options "in" projects.

Practical use of real options "in" the project didn't gain distribution and development of this theory is only at the initial stage now. Though many theoretical and empirical researches are devoted to real options "for" the project as well as in the case of real options "in" the project their practical distribution is still complicated.

First, it is connected with complexity of understanding of real options by the managers of the companies. Respectively, a psychological factor takes place when, without understanding the tool or seeing complexity in its studying, the manager refuses to use the real option, claiming that it isn't applicable specifically for its firm.

Second, M. Amram and N. Kulatilaka see complexity that the qualitative analysis providing numerical values corresponding to real situation is expensive, also the management often follows the developed strategy, and it also results in need of rather deep studying of real options and formations, so-called, option thinking [A. Damodaran, 2012]. The option thinking allows to use real options for active company management and decision-making process.

Third, the option thinking of managers means formation of the trained organizations by them. A. V. Bukhvalov argues about it in his article "Real options in management: classification and appendices" [Bukhvalov, A.V., 2013]. This rather essential restriction in use of the real options as the control systems developed by world practice, has taken root recently in Russia.

Fourth, the usage of real options limits the difficult mathematical apparatus which is initially developed for the financial markets and respectively there is a discussion concerning objectivity of results of calculations. According to certain authors, there is a fear that estimates that real options can be overestimated, especially in emerging markets where information support for decision-making is lower, than in the developed ones, when the experience of forecasting the market tendencies is worse. Real options are used at adoption of strategically important decisions therefore the slightest miscalculations can lead to serious financial losses.

However active development of the real options theory proves relevance of this tool in development and estimation of the strategy and promotes gradual overcoming of these restrictions. The alliance of financial, strategic and own vision of managers makes real options to be a valuable tool in formation of flexibility of the firm in the conditions of the uncertain environment.

In the modern finance theory the traditional methods and approaches applied to an assessment of efficiency of investment projects often show the limitation as in the majority they are intended for the companies functioning in stable spheres of business. Such methods don't include administrative flexibility, risk and uncertainty of the project in calculations, therefore, they ignore possibility of stage-by-stage planning and financing the project where there is an opportunity in the course of implementation of the project to change investment data in the external and internal environment.

In Russia there is rather rough process of creation of the new hi-tech companies, and the market differs by high degree of uncertainty.

Innovative enterprises are characterized by uncertainty of situations, risk of a decision-making process, plurality of available options of capital investments, limitation of financial resources for investment, by a large number of entrance indicators, and also existence of information which is poorly formalized and can't be considered as application of only quantitative methods.

Economic literature, as a rule, doesn't give definition of the "uncertainty" because of big abstractness of this concept. Uncertainty and risk are inseparably linked, but have rather essential difference. Uncertainty can't be excluded completely, especially while planning and forecasting.

In more general sense the concept can give such definition to uncertainty -insufficiency of data on conditions in which economic activity, low degree of anticipation of these conditions proceeds. Uncertainty is connected with risk of planning, decision-making, implementation of actions at all levels of economic system.

They consider risk as possibility of a deviation of the investment income (or a concrete conditional cash flow) from the expected size; the changes and the scale of fluctuations of the possible income (streams) is wider, the risk is higher.

More reasonable is approach of risk definition is neoclassical one, though consequences of risk are most often shown in the form of financial losses. The risk of losses is an alternative of obtaining the additional income.

We consider that various and numerous parameters weren't risk factors of certain investments (for example, a jump in prices of raw materials, increase in terms of construction of new shop, violation of the production technology, emergence in the market of the strong contender, etc.) – all of them finally are shown only in two aspects:

• -    the actual positive conditional cash flows will be less expected;

• -    the actual negative conditional cash flows will be more expected (on an absolute value).

Thus, uncertainty is the ineradicable quality of the market environment connected to the market conditions, and extent of its influence isn't determined. Then the risk is a measurable possibility of loss caused by need of decisionmaking in the course of interaction with the market environment possessing uncertainty.

The problem of the accounting of uncertainty and risk arises in investment calculations at determination of efficiency of investments when the investor is compelled to define for himself to what risk he is ready to go in order to receive a desirable result. Thus the solution of this two-criteria task is complicated by the fact that tolerance of investors to risk is individual.

Traditionally for an assessment of change of random variables, development of hypotheses under laws of their distribution, and also the account and an assessment of correlation communications between these variables statistical information is used, as well as expert estimates, methods of imitating modeling, and analog methods. When using statistical and analytical methods experts face that market uncertainty has no statistical nature.

Use of analog methods doesn't give the necessary clearness of the obtained data, and in the analysis of unique innovative projects in general becomes impossible.

To investigate possible change of investments’ efficiency, it is necessary to describe properly the existing investment uncertainty of parts of future financial condition of the project, both regarding revenue and perspective expenses. According to authors, at an assessment of investments’ efficiency indicators of discounting taking into account risk and methodology of indistinct mathematics are the most acceptable.

For the first time in 1965 in work of the American mathematician Lofti Zade "Fuzzy sets" ("Indistinct sets") the concept of indistinct sets was submitted. Unlike the classical theory of sets where accessory of elements to a set is estimated in binary terms according to an accurate condition — an element either belongs, or isn't present to this set, the fuzzy logic allows to define intermediate values between standard estimations. By means of this mathematical apparatus the price variable can be considered as a variable with values " very low", "low", "average", "high", " very high". Thus, by means of this mathematical apparatus it was succeeded to approach the mechanism of computer processing and the analysis of data [Trigeorgis, L, 1996].

The theory of indistinct sets resolves the graduated assessment of the relation of belonging of elements to a set; this relation is described by means of function of accessory:

M a ⁽ x ) = x ^ [0,1]

X - is the universal set, and AaM is function of accessory or the certain mathematical function setting degree or confidence with which elements of some set of X belong to the set indistinct set of A. The more the argument x corresponds to an indistinct set of A, the more the value is, i.e. the values of argument are closer to 1.

The function ц. _a (x) accepts values in some linearly ordered great number of X. A great number of X is called as a set of accessories, X is frequently the piece (0,1).

Then the indistinct set A is defined as:

A = {( x , A ⁽ x )/ x G X )} A = {(x,A(x)|xeX)}

Basis of use of indistinct model is represented by indicators, input parameters in a look is indistinct - linguistic variables. Achievement of such result can be received at adjustment statistically or analytically obtained basic data at a size of risk of fluctuation of this parameter, set by an indistinct variable.

For determination of efficiency of the innovative project we use a discounting indicator taking risk into account. For definition of this indicator it is offered to use a well-known formula of calculation of the discounted cash flows, however for the accounting of risk of the innovative project it is necessary to recalculate cash flows at each stage taking probability of a conclusion of an innovative product to the following development stage into account.

The formula of calculation for rNPV in general looks as follows:

rNPV =

L

n t=0

CF i * P o (1 + r ) t * P

rNPV = ^tn_0 c F1*P°

( 1+r)t$ P i where NPV – the net specified value corrected on risk;

P_r — probability of a conclusion of technology in t timepoint;

— — probability of achievement of a cash flow of t CF in t timepoint.

Pi

The method of the discounted cash flows taking into account risk (rDCF) expands an assessment of a method of the discounted cash flows (DCF), partially includes real options, provides various scenarios of development of the project and represents a simple tree which includes risk of failure at each stage of innovative development.

For a task of an indistinct variable when determining components it is necessary to multiply them statistically or analytically receiving sizes on coefficient of risk of their deviation from the expected sizes which would be set by an indistinct and linguistic variable in the form of triangular indistinct numbers.

The model expression of the following is: parameter A is approximately equal and in range [ a_max a_min].

Adjustment of parameter is determined by a formula:

^ka = ^{[ k}a min ’ ^kA ’ ^ka max ^]

k a = [k a m i n^A , k a m ax ]

Values of parameters are defined with the risk expressed by indistinct number (5) - (7).

rNPV . = V ^й   CF mn t * P ⁰

min   ^t=1(1 + rmax) * P rNPV ■ = YN m 1 n CFmint P°

r^V™  ^= 1 (,+r^p!

rNPV

^N t =

CF * P avt 1 0

(1 + r ) * P

̅         *

(   ̅)*

rNPV max

V N CF^ t * P o

^ t =¹ (1 + r mn ) * P 1

*

rNPV =∑    (    )*

To calculate the cost of the innovative project we suggest to interpret function of accessory as density of distribution and to consider a population mean and dispersion of the corresponding random variable.

It is impossible to consider parameters of a random variable, operating with function of accessory of triangular indistinct number therefore we interpret this function as density of distribution of a random variable with the parameters corresponding to it - dispersion and population mean. In other words it is necessary to enter the complete parametrical description of behavior of a random variable at the normal law distribution of two parameters.

For determination of cost of a real option by rDCF method it is offered to use the following formula (8):

ROC =--- APos) ---* E ( A )

A( Pos ) + A ( Neg )

ROV=

(    )       _*

(     )    (     )

Possible average E(A) value for positive area of values for triangular indistinct number is determined in the following way.

Indistinct approach for the rDCF method gives additional opportunities to projects as it allows to consider interrelation between risk and opportunities.