: 656.615:004.272

Arsenyev Yu.N., Davidova T.Y.


The territorial branch of the ARDIFE in the town of Tula

Tula State Teachers Training University named after Leo Tolstoy


Consideration of the models of optimization of investicial and financial objects with an illustration then on concrete examples are considered, optimization approaches to the utility value and assessed value of investmentinto the intellectual capital of the region with the multitude of features and alternatives.

Under certain enumeration of development alternatives of the intellectual capital and economic performances stipulated by them, may emerge situations of risk (scenario probabilities of process developments are acquainted), uncertainty (probability values are impossible to obtain) and obscurity (assertions which cant unambiguously be assigned to valid or fallible). By means of methods of information collection, processing, analysis and predication it is possible to decrease the degree of expectation unreliability, disbenefit of their consequences. Constructing models and applying methods of investment estimation it is possible to optimize values of objective functions taking into account changes of production parameters and realization of enterprise production or intellectual capital. Alternatives of changes of intellectual capital and enterprise development may correspond to various values of objective functions. In cases of undominated alternatives the choice of the best alternative is advisably to conduct on the basis of the acquainted criteria [1].

Utility value of alternative solutions risk is mapped by the function of risk priority, transforming miscellaneous investors risk attitude: positive (convergence to risk), negative (fear of risk) or neutral (indifference to risk). Under negative risk attitude the investor usually chooses under common mathematical expectation that alternative which has smaller standard deviation. As this takes place a part of information is lost, motivating the compilation of detecting the function of risk priority. Replacing mathematical expectation and moments of risk of the objective functions (CK) with expected utility (profit) it is possible to correlate them with objectives, expectation of degree achievement and accounting investors risk attitude. Bernullis criterion is used to estimate the profit of a compromise of possible investment with the search for the maximum of moral expectation for every alternative , determining utility function of unreliable results (expectations), for example CK. In this process the reliable result (equivalent) is found which has similar profit as two unreliable results and probabilities values of the occurrence of an event. Utility function in the risk situation evaluates investors risk attitude: under indifferent risk attitude reliable equivalent corresponds to the value of the result to be expected, under positive (negative) risk attitude the value of reliable equivalent is higher (lower) than the value of the result to be expected.

Methods of optimization of investment-finance include a multitude of economic, mathematical and applied models. In action widely known models of taking unit solutions under single objective function (multitude of static and dynamic models) and programmed universal solutions (classical, technological, compensatory and uncompensatory, MADM-methods , MAUT-methods, LIMAP-methods and MODM-methods with programmed multiple-lens solution of matters of vector maximum).

In conditions of expectation equivocation the most well-known methods of taking investment-financial decisions are methods of correction; sensibility analysis; simulative and sensitive risk; solution trees.

Correction method specifies change in the range of assumed data of the reckoning of investment (mathematical expectation value is often replaced by discounts or risk premiums). When using cost of capital value for determining utility value and risk assessment they raise interest rate or the rate of current payouts, reduce the rate of current payments or working lifespan. This guarantees that objective function will reach the maximum with high probability. The disadvantage of this method consists in subjectivity of determination and calculation of corrections in the integrated rather than differentiated given data accounting of unreliability of the expectations and their consequences.

Sensibility analysis method affords investor decision-making to take into account the link between the assumed data and the values of objective functions of alternatives, to answer the questions, relating changes of values objective function under given variation of input quantities as well as bandwidths where its possible to change the values of input quantities under given worst value or the limit of values of the objective function. Sensibility analysis is connected with the formation of the decision-making model and the determination of its data, flavor and number of studied input quantities, valuation of their influence to the end result in one or more periods of planned interval of time, collation of choice alternatives and their valuations of relative utility value. At determining relative utility value of investment, focused on expansion of production they usually evaluate: the complex of parameters (critical values of purchase costs, sale price of goods, overall production and volume of sales, fixed costs or variable costs, rates of payouts and expenses, interest rate, working lifespan, liquidation revenues). When analyzing the sensibility it is necessary to: determine the values of input quantities deviations from the assumed data; estimate the alternative values of input quantities; calculate objective function values on their basis. Planning production turnout with the overall production, equal to the volume of sales, market entity chooses from the multitude of investment alternatives labeled by the set of assumed data and calculates the cost of capital just as CK = -A0 + ((pcrit - avt)xt - Aft)q-t + Lq-t, where t time index; T maintenance end; A0 purchase payments; pt selling price in the point of time t; avt payments per unit in the point of time t, determined by the sales(production) quantities; xt - sales(production) quantity in the point of time t; Aft payments in the point of time t unaffected by the sales or production quantities; L - liquidation revenue; q-t - discount coefficient in the point of time t (q = 1 + i).

Having applied computer modeling CK its possible to find the critical value of the upper and lower limits of the profitability of the best alternative, its deviation and also its probability basis for evaluation of profit or risk of the alternative. Critical selling price is determined as pcrit = [A0 + (avx + Af)q-t + Lq-t] / xq-t].

The deviation of the given combination of values from the values of the critical function testifies to the dependence of the utility value from the values of input quantities and their compromise. If the evaluations of the investment alternatives relative profit are necessary than its more convenient to consider them mutually. Then its possible to determine critical values combinations for every input quantity.

By comparison a range of investment projects its possible to find a critical value CK whereby investment has common value of objective function with the fixation of spectral band utility value and values of input quantities. With the aid of critical values or variations of input quantities values its possible to trace the influence of input data on the relative profit of investment objects under ascendancy of analytical estimations number [1].

Sensibility analysis method directs statement of costs for a number of variants of investing production (production purchase) with regard for critical values of production and expenses, comparison of critical volumes of production and limits of alternative profitability and the choice of the best variant with regard for unreliability of expected production volumes.

Decision-making algorithm on the basis of the simple risk includes procedures: formation of decision-making model; determination of probability distribution of unreliable input quantities; account of the stochastic dependence between unreliable input quantities; calculation of probability distribution for the objective function; interpretation of the results. Under probability distribution of individual input values discontinuous or abrupt distribution is used. Abrupt distribution is usually applied under given type of distribution with the evaluation of its parameters (mathematical expectation, standard deviation under normal distribution, outermost and innermost limits at the triangular distribution). The determination of probability distribution is sophisticated in case of single investment. Probability correspondences between unreliable input quantities are considered in pair or multiple regression coefficient and correlation and probability distribution of input quantities.

Probability calculation of objective function is carried on analytical (distribution of its values is calculated from input quantities distribution) or simulative (multitude of accounts is fulfilled and using the method of random numbers its necessary to choose samples from the probability distribution of input quantities) method, then multitude of these accounts gives the distribution of the objective function values. The basis of risk evaluation is the distribution of calculated objective function values. Absolute values of separate class frequencies are transferred in relative values composing the base of probability distribution evaluation, distribution function or the risk profile of the objective function.

Let us assume that funds are invested into projects A and B (table 1) for environment enhancement and security assurance of manufacture functioning. Unreliable input quantities are prices, payouts, operation revenue, production and fulfillment volumes of A project. These quantities, excluding production and fulfillment volumes of A project, have triangular distribution (their distribution parameters modal (mod), minimal (min) and maximal (max) value limits are summarized in table 1). Its necessary to evaluate the investment risk into these projects. After having calculated capital cost of projects A and B and constructed distribution functions CK let us find optimum value of the objective value. Collocating and pattern of distribution function testifies to average values of the objective function and their dispersion. Values of mathematical expectation CK, its standard deviation and negative profit probability (p) of projects A and B duly comprised: M[CKA] = 10,11 rub.; SA = 10,045; P = 0,18; M[CKB] = 9,67 rub.; SB = 6,676; PB = 0,08. Investor accounting of the character of the distribution function and calculated indexes is important for the exposure of risk connected with investment and negative profit probability and it testifies about absence of the absolute advantage of the alternatives. Project A has higher values of mathematical expectation, standard deviation and negative profit probability than Project B and stochastic domination is missing here.

Sensibility (sensitivity) and investment decisions risk may be evaluated autonomously and in complex. Complex approach includes the analysis of: unreliable input quantities and their distribution probabilities; stochastic interdependencies between unreliable input quantities; input quantities considered to be reliable.

Under sensitive risk analysis objective function value is given and later with the help of imitation modeling because of random values of other unreliable input quantities we calculate the critical value of input quantity or the compromise of critical values from the input quantities range with the search of their distribution. In comparison to common analysis of sensibility sensitive risk analysis is distinguished by the fact that under systematic variation of input quantities their probability distribution varies and that gives a multitude of probability distributions of objective quantity and under compatible analysis we determine critical value distribution instead of critical value. Having applying for the data from table 1 sensitive risk analysis method with the value change of the selling prices triangular distribution (minimal, modal and maximal values) on 10% and 5% under the constancy of probability distribution of all the other values we obtain selling prices triangular distribution, values of their mathematical deviation and standard deviation on the basis of imitation modeling (table 2). Investment planning based on the solution tree method affords to find optimal decisions at the beginning of the planned period and successive instants of time accounting possible environmental conditions and their probabilities. Mathematical expectation CK value usually arises as objective quantity. As this takes place a set of decisions given maximum of mathematical expectation of capital cost is considered to be optimal.

In the large the solution tree method is oriented towards the evaluation of the flexible models under small number of unreliable quantities and their values diapason in a single monetary objective function with decision, containing only expected values of mathematical expectation CK without their possible deviations. The compromise of sensitivity analysis method and solution tree method gives additional possibilities of the best decision search under conditions of uncertainty but under more difficult computer calculations.

Table 1. Indexes of investment projects A and B (thous rub.)

Input alternative quantities

Investment indexes

project A

project B

Purchase payments A0



Interest rate i, %



Liquidation revenue L

Input quantities values

Min Mod Max Min Mod Max

0 20 50 0 12 30

Selling price, r, rub.

92 100 105

92 100 105

Payments for production unit:

Independent from production av, rub.

* dependent on production Af


45 50 60

15 16 17


45 50 60

11,5 12,5 13,5

Quantity of sales and production xt:

- in period t = 1

- in period t = 2

- in period t = 3

- in period t = 4

- in period t = 5


900 1000 1200

950 1050 1150

1000 1100 1200

950 1050 1150

900 1000 1100


800 8000 800

800 8000 800

800 8000 800

800 8000 800

800 8000 800

Table 2. Capital costs of project A realization (triangular distribution)

Imitation Circle

Triangular distribution parameter

Min Mod Max

Mathematical expectation CK

Standard deviation CK





- 28490






- 9374






















1. Arsenyev Yu.N., Davidova T.Yu. Hybrid intellect systems. Economics. Management. Education. - M.: High School, 2008. - 566 p.