The paper presents three economic-mathematical models for the formation of the company's investment program based on: (1) principle of guaranteed net present value; (2) principle of maximizing of the average expected net present value under predetermined upper estimation of its dispersion; (3) principle of maximizing the average expected net present value under predetermined upper estimation of the probability of its inaccessibility. Proposed solutions of the problems allow us to give a system estimation of the enterprise investment attractiveness which can be used in selecting an e ective investment portfolio based on risk appetite of decision makers.
Currently, there are many models of the choice of investment policy. The best known model of nancial markets is the model of Markowitz-Tobin for portfolio management [6, 10]. This model allows maximization of the expected result with acceptable risk. The main element of control is a periodic portfolio diversi cation.
To implement the strategic objectives of the enterprise also requires e ective management of its investment activities. The goal of this management is the most e cient implementation of possible projects bringing the maximum nancial result with minimal risk under limited investment resources and the uncertainty of their volumes [2, 3, 9, 12, 13, 16, 17]. In this case the investments are long term, and the investment program is essentially a strategic plan for development of the company [3, 13, 16, 17]. This plan is calculated for certain time, and includes a list of various projects which are listed with the detail volumes of nancial investments.
The aim of the paper is to show mathematical models determining the optimal investment program for the enterprise, i. e. nding the order of implementation of many independent projects. There are some known heuristic algorithms [3, 14] for building an investment program which provides the system of author's preferences. Analyses of the set of e ective investment programs, each of which: (i) maximizes the expected net present value (NPV) under a certain level of risk of impossibility to execute the program, (ii) ensures minimal risk of impossibility of the program for a certain value of the expected NPV, { are more reasonable way for selection of the optimal enterprise investment program. As a risk measure, either variance of NPV by analogy with the approach of Markowitz-Tobin [6, 10] at formation of a portfolio of securities [4, 7], or the probability of inaccessibility of the desired mean NPV [5, 8] can be used.
Construction of the e cient investment programs set (i. e. Pareto set by the criteria space of \risk { NPV") allows to select an e ective investment program taking into account the risk appetite of decision makers. The paper presents mathematical models of building a set of e cient investment programs.
The article consists of four sections, conclusion and bibliography (17 references). Designations and the basic relations are introduced in section 2. A mathematical model implementing the principle of guaranteed payo is presented in section 3. Search e cient portfolios under uncertainty and risk [10] are discussed in the section 4. Two more mathematical models based on di erent de nitions of the concept of \risk": (i) variance of NPV, (ii) unattainability probability of expected NPV are proposed. Model example of a problem and the numerical solution for a variety of options for building the investment program of the enterprise are considered in section 5. Conclusion summaries the study. 2
General formulation of the problem Main criterion for forming an optimal investment program of the enterprise is NPV of the investment program [1, 11]. Let { P = fp1; p2; : : : ; png be set of n of investment projects that can be included to the investment program; { L = fl1; l2; : : : ; lng be set of durations of implementation of investment projects (i. e. accounting period); { m be planning horizon (the number of billing periods); { R = fr0; r1; : : : ; rm 1g be xed nancial resources or funding the company's investment program at billing periods.
Each of the investment projects pj , j = 1; 2; : : : ; n can be characterized by two parameters: { value Cjs of the net presents value of the project pj that started during s-th period; { need volume Ijsi to nance the investment project pj launched at the period s over current period i.
Indicators of income and expenditure are predictable values. They depend on a number of factors. Therefore it is advisable to consider Cjs and Ijst as random variables. We receive interval estimations of the net present value [Cjs; Cjs], needs [Ijst; Ijst], and nancial resources of the enterprise [ri; ri] based on a retrospective analysis for each project pj , j = 1; 2; : : : ; n, and for all settlement periods i, s = 1; 2; : : : m.
Let us introduce the boolean variables xjs = (1; beginning of the project pj is period s;
0; beginning of the project pj is not period s:
Realizable subset of projects of the set P is a subset of projects that can be nanced within the available nancial resources for all settlement periods i; s = 1; 2; : : : m. NPV of the investment program is the sum of discounted net income of projects included into the investment program [11, 13].
Since the implementation of the investment project pj may begin no later than at the period m lj , then the following condition holds. Conditions for realization of the investment program may be written in the form NPV of the whole investment program is equal to
m lj Cj = X Cjsxjs:
s=0
C =
n n m lj
X Cj = X X Cjsxjs:
j=1 j=1 s=0
(
Application cautious strategy aimed at maximizing the guaranteed NPV is reduced to the solution of the problem
n m lj C(x) = X X here x = fxjs : j = 1; 2; : : : ; n; s = 0; 1; 2; : : : m the constraints lj g, admissible set B satis es n i X X Ijsixjs j=1 s=0 ri; i = 0; 1; 2; : : : ; m
1;
Building e ective investment programs under risk It is possible to look for optimal investment program of the enterprise in set of e ective investment programs (i. e. Pareto set in the space of criteria \risk { NPV"). Intelligent decision support systems allow to select the most suitable investment program based on the identi ed system decision-makers preferences. 4.1
NPV dispersion as risk measure Let us use the expectation of NPV of the investment program as a measure of income, and its variance as the risk measure.
Assuming that the net present value of each project pj 2 P is uniformly distributed in the interval [Cjs; Cjs] for all s = 1; 2; : : : ; lj we nd the expectation of net present value
8 n m lj
EfC(x)g = E <X X Cjsxjs
9 n m lj
= = X X (xjs EfCjsg) =
:j=1 s=0
;
We nd the variance of the net present value given the nature of the Boolean
variables x and independence between the net present value of the various projects
8 n m lj
DfC(x)g = D <X X Cjsxjs
9 n m lj
= = X X (xjs DfCjsg) =
:j=1 s=0
;
where x = fxjs : j = 1; 2; : : : ; n; s = 0; 1; 2; : : : m ljg, admissible set D
satis es the constraints (
Tasks (10) and (11), as well as the task of (
Probability of given NPV inaccessibility as risk measure Let C be a predetermined level NPV. Probability of achieving a given level be P fC(x) > Cg = P 8<Xn mXlj Cjsxjs > C=9 : :j=1 s=0 ; Let us introduce events
m lj Ej : X Cjsxjs = yj; j = 1; 2; : : : ; n; s=0 n X yj > C j=1 consist of the fact that the income from the project pj will not be less yj. These events are used by methods of reduction problems with probabilistic criteria to a deterministic view [15].
It follows from (8) and (9) that Thus, the construction of an e cient investment program at risk is reduced to problems
n m lj EfC(x)g = X X j=1 s=0
Cjs + Cjs xjs
2 n m lj (Cjs DfC(x)g = X X j=1 s=0
Cjs)2 12
! xjs !
max ! x2D: DfC(x)g d
; min ! x2D: EfC(x)g e ; P fEjg = m lj m lj X xjsP fCjs > yjg = X s=0 s=0 yj Further instead of maximizing the probability of Pf g we consider the problem of maximizing its logarithm. Due to the monotony of the logarithmic function !
: ! : (10) (11) (12) the optimal solutions to both problems are the same. We have
n 0m lj ln P fC(x) > Cg = X ln @ X j=1 s=0 This equation determines the probability of investment program setpoint NPV inaccessibility, and later this probability is used as a measure of risk.
Let us introduce determinate variables zji, j = 1; 2; : : : ; n, i = 0; 1; : : : ; m 1 and let us consider events Aji = fIjsi zjig and fBi = Pjn=1 zji rig. Event Aji means the fact that at period i the resources required by the project pj started at any period s m lj do not exceed a value of zij. Event Bi means the fact that the resources required for all performed projects at the period i do not exceed value ri. Probabilities of the introduced events are
P fIjsi zji
P 8<Xn zji :j=1
9 ri= = ; ri ri ri :
Let us nd the probability of the conditions 3 realizability of the investment program considering the variables zji as xed. We have
8 n i P <ri(x) = X X Ijsixjs : j=1 s=0
9 ri= =
; = P 8<Xn zji :j=1
9 n m lj ri= Y X (xjsP fIjsi ; j=1 s=0 zjig) : (14) for any billing period i = 0; 1; 2; : : : ; m and taking into account 1 . Finding of equation (14) logarithm ln P fri(x) rig =
P fri(x) > rig ; ln P fIjsi 8 n ln P <X zji :j=1 9 = ri ;
8 n = P <X zji > ri 9 = = :j=1 ; Ijsi zji ; zji Equation (15) de nes the probability of exceeding of resources required for the calculation period i = 0; 1; 2; : : : ; m with the enterprise investment program.
Thus, if be tolerable risk unreachable investment program setpoint NPV, i be tolerable risk of exceeding the investment program of the enterprise, the resources required for the calculation period i = 0; 1; 2; : : : ; m then the problem of de ning the maximum of the expected income can be represented as follows (16) (17) (18) (19) (20) (21) (22) C(x; y; z) = n X yj ! mx;ya;xz j=1 m lj
X Cjsxjs = yj ; j = 1; 2; : : : ; n; zji Let us consider the application of the above mathematical models to the next task. To implement proposed n = 7 investment projects, all projects according to preliminary calculations are cost-e ective. The planning horizon of the investment program is m = 11 billing periods. The duration of projects j = 1; 2; : : : ; 7 is the same and amounts to lj = 8 billing periods, thus beginning of any project is possible only in the billing period s = 0; 1; 2; 3.
Table 1 contains the interval estimations of projects NPV depending on the time of s start implementation.
Table 2 contains interval estimations of the allowed amount of nancing the company's investment program for the calculation period i. correspond to di erent Pareto optimal investment programs having an average NPV equal to 3297 and 3782 respectively. Increasing the level of unreachable probability of possible expected value of NPV in stochastic models also leads to various investment programs with increasing average expected value of NPV.
Since the stochastic model (16){(22), in contrast to the model (10){(11), allows the risk of exceeding the investment program of the enterprise resources required for the calculation period, then the potential average expected NPV in the stochastic model are higher.
Images of all the projects in Table 4 for the coordinate systems \variance NPV" { \expected NPV" and \probability unreachable" { \expected NPV" are shown in Figure 1.
Investment programs built in the example are e ective (i. e. belong to the Pareto set in the space of criteria).
Considered models of optimal investment program with known distribution of funds for each period allow to shape the Pareto-optimal investment programs. Presented modi cation of this model which takes into account the uncertainty of nancial resource volumes to support investment projects.
Solutions of the respective tasks provide systematic assessment of investment attractiveness of the enterprise can be used by intelligent supporting systems for choice of e cient portfolio based on derivative criteria of performance: (i) the payback period of the investment, (ii) the rate of return on capital, (iii) the di erence between the amount of income and investment costs (non-recurring expenses) for the entire useful life of the investment project, (iv) reduced production costs, and (v) risk appetite of decision makers. look. St. Petersburg., 01{03 December". vol. 3, pp. 64{67. Publishing house \KultInformPress", St. Petersburg (2012) 8. Kozina, E.N.: Mathematical model of the investment program of the enterprise.
In: Proceedings of XII All-Russian Conference on Control (VSPU 2014) Moscow,
June 16{19, 2014. pp. 1250{1253. Institute of Control Sciences, Institute of Control
Sciences, Moscow (2014)
9. Mikhalina, L.M., Golovanov, E.B.: Investment activity variance of small businesses
in a corrupt environment. Bulletin of South Ural State University. Series:
Economics and Management 8(
Nauka, Novosibirsk (2001)
11. Panyukov, A.V.: Mathematical modeling of economic processes. URSS, Moscow
(2009)
12. Schlegel, D., Frank, F., Britzelmaier, B.: Investment decisions and capital
budgeting practices in german manufacturing companies. International Journal of
Business and Globalisation 16(