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НЕЧЕТКО-МНОЖЕСТВЕННЫЙ ПОДХОД ДЛЯ ОЦЕНКИ РИСКА ИНВЕСТИЦИОННЫХ ПРОЕКТОВ
Соседская А.В.
Кубанский государственный университет, студентка экономического факультета г. Краснодар, Российская Федерация Шмалько С.П. Кубанский государственный университет, к.п.н., доцент кафедры «Информационных образовательных технологий» г. Краснодар, Российская Федерация
FUZZY LOGIC APPROACH FOR RISK ASSESSMENT OF
INVESTMENT PROJECTS
Sosedskaya A. V.
Kuban State University, student of economic faculty Shmalko S.P.
Kuban State University, Ph. D., associate Professor of the Department "Information educational technologies" Krasnodar city, Russian Federation
АННОТАЦИЯ
В данной статье рассматривается актуальный на данный момент метод оценки риска инвестиционных проектов, который основан на применении теории нечётких множеств. Нечёткие множества являются одним из наиболее популярных инструментов оценки всевозможных сценариев развития инвестиционных проектов.
ABSTRACT
This article discusses a currently relevant method of risk estimate of investment projects, which is based on the theory of fuzzy sets. Fuzzy sets are one of the most popular tools to estimate different scenarios of the investments development.
Ключевые слова: теория нечёткой логики, нечёткие множества, оценка риска, степень риска инвестиционных проектов
Keywords: fuzzy logic, fuzzy sets, investments, risk estimate, risk degree of an investment project
In a modern reality of developing financial markets the fuzzy logic theory is a relatively new approach of conditions disclosure, in which there is uncertainty of the aims, further activities. It obstructs and excludes the using of the exact quantitative methods.
The distinguishing characteristic of using fuzzy logic method is the introduction of linguistic variables, which are subjective expressions that couldn't be described with a help of mathematical language. It is difficult to give an objective appraisal of them. For example, the measure of risk can be described using the set of modifiers such as: "very", "quite", "not",
"too" and etc. According to certain sources linguistic variables are used to name as "term set".
Using fuzzy logic is directly connected with membership functions, which help to figure out how an arbitrary element of the universal set belongs to the fuzzy set.
Membership function (X) is some mathematical function, which carries out the degree of belonging the set X to fuzzy set N. the larger the argument x corresponds to a fuzzy set N, the larger the value (X), so the closer the argument value to 1.
Constructing membership functions can be based on estimates of one or two experts (or expert community). It is allocated two main groups of methods for constructing membership functions of fuzzy sets according to expert estimates: direct and indirect methods [2].
Direct methods are caused by the experts who settle the rules to determine the values of the membership function nA (X)that characterize the element. Examples of methods are the direct assignment of the membership function by the graphs, tables or formulas. The obvious "drawback" of these methods is the subjectivity.
Indirect methods are characterized by the choice of function values, which satisfy conditions stated
before. Expert information is only starting material for further processing. This group of methods includes the construction of membership functions based on pair-wise comparisons, rank estimates using the statistical data and etc.
Fuzzy set theory as a separate mathematical section is based on premises of normality (altitude of fuzzy set equals to 1), convexity (all a-cross-sections - convex sets) and unimodality (a function has only one extremum on a given interval). They were first formulated by L. Zadeh and R. Bellman [4].
Types of membership functions: trapezoidal, triangular, Gauss's function (symmetric or bilateral), sigmoid.
Picture 1 - Triangular (left) and trapezoidal (right) membership functions
Picture 2 - Symmetric Gauss's function
One of the most commonly used in the practice is the triangular membership function of estimation of investment projects. A triangular number N is specified using three parameters: the minimum (a) modal (b) and maximum (c) values, A = (a, b, c), which
corresponds to the pessimistic, basic and optimistic scenarios.
Mathematically, the triangular form of the membership function can be described as (1):
A = (А, q) = (a +ax (b - a), c +ax (b - c))
(1)
where at any a membership function juA (X) possesses values (2):
p = a tax (b - a), and q = c tax (b - c) (2)
Operations that can be performed on sets of fuzzy logic: addition (3), multiplication (4), division (5):
А 14 = В = (p, q),
где p = pit p2, q = q + q2 (3)
A x 4 = B = (p, q), где p = px x p2, q = ql x q2 (4)
A 1A = B = (P, 4), где P = PJ Ъ,
q = qj P2 (5)
if A, A are positive, and
P = Pi / P2, Ъ = Ъ / Ъ, when A1 is negative.
Consider an investment project (will be based on integral risk estimate of Voronov Kirill and Maximov Oleg - V&M), where NPV can be reduced to a triangular number (6):
NPV = (NPV1, NPV, NPV2) (6)
where NPV is a Net Present Value in positive
scenario, NPV - in negative scenario , NPV -expected NPV.
The project is profitable on the condition that (7): NPV> G (7)
V & m * н
where
Consider an investment project with the conditions that:
a) the project will be realized within 4 years,
T=4;
b) the size of initial investment is I = 5 million;
c) net cash flow (planned): in the range from CFmin = 0 to CFmax = 4 million;
If average values are: CFav = 2 млн.,
Rav = 15%, то NPVav = 0,71.
Thus, the triangular number of the project under consideration has such form
where G - the effectiveness criterion (usually equals to 0)
By setting the extreme values of net present income, membership function can be described (8, 9, 10):
(8)
(10)
d) the discount rate can range from 10% to 20% per annum;
e) residual (liquidation) value of the project equals to zero (G=0).
Using fuzzy logic method for risk estimate could be produced such results (11 h 12) [1]:
(11)
(12)
NPV = (-5; 0,71; 7,68). Its graphical interpretation is presented in the third picture (3).
As NPVmm < G = 0 < NPV , so attitude to previously mentioned formula: ax = 0,88, R = 0,394, V & M * = 0,28.
R x
1 +
1 -a1
a
ln (l -ax)
1 -(l - R )x
1 +
1 -a1
a
ln (1 -a1 )
G < NPVm
NPVm;„ < G < NPV
NPV < G < NPVm NPV < G
R =
G - NPVmi„
NPV - NPV
T ' m nv ' m
1,
G - NPVmi„
ax=\
NPV - NPVmm NPV - G
NPVmax - NPV 0,
G < NPVmax NPV < G
G < NPVmi„
NPV - < G < NPV
NPV < G < NPVm NPV < G
0
1
0
,mi, 0 0 0 0
NPVmm = -5 +-r +-- +-r- +-t = -5
min (1 + 0,2)1 (1 + 0,2)2 (1 + 0,2)3 (1 + 0,2)4
4 4 4 4
NPVmax = -5 +-r +-- +-г- +-г = 7,68
max (1 + 0,1)1 (1 + 0,1)2 (1 + 0,1)3 (1 + 0,1)4
Picture 3 - Graphical interpretation of the investment project
After receiving this information, depending on the additional parameters of the project and their own preferences, the Risk Manager can establish a risk scale using a certain gradation.
For example, attitude to table number 1, risk of project under consideration could be judged as average degree risk. Maybe the project will be carried out with restrictions on the avoidance of force majeure.
Table 1
The risk degree and possible solution
V&M Risk degree Possible solution
0 - 0,1 very low exactly carrying out the project
0,1 - 0,17 low carrying out with reserve
0,17 - 0,35 average carrying out with restrictions
0,35 - 0,45 high project revision
> 0,45 too high project abandonment
In real economic conditions, the method of fuzzy sets, which is based on both objective and subjective factors of various business processes helps to understand and to assess the capabilities of the business projects. Fuzzy logic approach is not generally used as the only possible but used in combination with various methods of analysis and consideration of uncertainties in conjunction with methods of expert evaluations and quantitative methods, based on mathematical statistics [6].
References
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