=Paper=
{{Paper
|id=Vol-1726/paper-09
|storemode=property
|title=Forming of the Competitive Investment Programs for Enterprises
|pdfUrl=https://ceur-ws.org/Vol-1726/paper-09.pdf
|volume=Vol-1726
|authors=Anatoly V. Panyukov,Ekaterina N. Kozina
}}
==Forming of the Competitive Investment Programs for Enterprises==
https://ceur-ws.org/Vol-1726/paper-09.pdf
Forming of the Competitive Investment
Programs for Enterprises
Anatoly V. Panyukov and Ekaterina N. Kozina
South Ural State University, Chelyabinsk, Russia
kozinzen@susu.ru
paniukovav@susu.ru
Abstract. The paper presents three economic-mathematical models for
the formation of the company’s investment program based on: (1) prin-
ciple of guaranteed net present value; (2) principle of maximizing of the
average expected net present value under predetermined upper estima-
tion of its dispersion; (3) principle of maximizing the average expected
net present value under predetermined upper estimation of the proba-
bility 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 effective investment portfolio based on
risk appetite of decision makers.
Keywords: investment program, net present value, risk dispersion, prob-
ability, stochastic programming
1 Introduction
Currently, there are many models of the choice of investment policy. The best
known model of financial 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 diversification.
To implement the strategic objectives of the enterprise also requires effective
management of its investment activities. The goal of this management is the most
efficient implementation of possible projects bringing the maximum financial
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 financial
investments.
The aim of the paper is to show mathematical models determining the op-
timal investment program for the enterprise, i. e. finding the order of imple-
mentation of many independent projects. There are some known heuristic algo-
rithms [3, 14] for building an investment program which provides the system of
author’s preferences. Analyses of the set of effective 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 efficient investment programs set (i. e. Pareto set by the
criteria space of “risk – NPV”) allows to select an effective investment program
taking into account the risk appetite of decision makers. The paper presents
mathematical models of building a set of efficient investment programs.
The article consists of four sections, conclusion and bibliography (17 refer-
ences). Designations and the basic relations are introduced in section 2. A math-
ematical model implementing the principle of guaranteed payoff is presented
in section 3. Search efficient portfolios under uncertainty and risk [10] are dis-
cussed in the section 4. Two more mathematical models based on different defini-
tions of the concept of “risk”: (i) variance of NPV, (ii) unattainability probability
of expected NPV are proposed. Model example of a problem and the numeri-
cal 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 = {p1 , p2 , . . . , pn } be set of n of investment projects that can be included
to the investment program;
– L = {l1 , l2 , . . . , ln } be set of durations of implementation of investment
projects (i. e. accounting period);
– m be planning horizon (the number of billing periods);
– R = {r0 , r1 , . . . , rm−1 } be fixed financial 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 finance 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 [C js , C js ],
needs [I jst , I jst ], and financial 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
(
1, beginning of the project pj is period s,
xjs = (1)
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 financed within the available financial 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
m−lj
X
xjs ≤ 1, j = 1, 2, . . . , n (2)
s=0
holds. Conditions for realization of the investment program may be written in
the form
Xn X i
Ijsi xjs ≤ ri , i = 0, 1, 2, . . . , m − 1. (3)
j=1 s=0
If investment project pj may be included into the investment program then, in
view of (2), its NPV be
m−lj
X
Cj = Cjs xjs . (4)
s=0
NPV of the whole investment program is equal to
n
X n m−l
X Xj
C= Cj = Cjs xjs . (5)
j=1 j=1 s=0
3 The maximin strategy
Application cautious strategy aimed at maximizing the guaranteed NPV is re-
duced to the solution of the problem
n m−l
X Xj
C(x) = C js xjs → max, (6)
x∈D
j=1 s=0
here x = {xjs : j = 1, 2, . . . , n, s = 0, 1, 2, . . . m − lj }, admissible set B satisfies
the constraints
n X
X i
I jsi xjs ≤ ri , i = 0, 1, 2, . . . , m − 1; (7)
j=1 s=0
m−lj
X
xjs ≤ 1, j = 1, 2, . . . , n; (8)
s=0
xjs ∈ {0, 1}, j = 1, 2, . . . , n, s = 0, 1, .., m − lj . (9)
We reach the warranty of optimal value of the problem (6)–(9) due to the use
of lower bounds C js for NPV, and rjs for volumes of financing for periods, and
I jst for upper bounds for all financing needs.
The problem (6)–(9) is the boolean linear programming problem with a non-
negative matrix of conditions, so it can be solved by pseudo polynomial algorithm
based on dynamic programming.
4 Building effective investment programs under risk
It is possible to look for optimal investment program of the enterprise in set
of effective 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 identified 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 ∈ P is uniformly dis-
tributed in the interval [C js , C js ] for all s = 1, 2, . . . , lj we find the expectation
of net present value
Xn m−l
Xj n m−l
X Xj
E{C(x)} = E Cjs xjs = (xjs · E{Cjs }) =
j=1 s=0 j=1 s=0
n m−l
!
X Xj C js + C js
= · xjs .
j=1 s=0
2
We find the variance of the net present value given the nature of the Boolean vari-
ables x and independence between the net present value of the various projects
Xn m−l
Xj n m−l
X Xj
D{C(x)} = D Cjs xjs = (xjs · D{Cjs }) =
j=1 s=0 j=1 s=0
n m−l
!
X Xj (C js − C js )2
= · xjs .
j=1 s=0
12
Thus, the construction of an efficient investment program at risk is reduced to
problems
n m−l
!
X Xj C js + C js
E{C(x)} = · xjs → max , (10)
j=1 s=0
2 x∈D: D{C(x)}≤d
n m−l
!
X Xj (C js − C js )2
D{C(x)} = · xjs → min , (11)
j=1 s=0
12 x∈D: E{C(x)}≥e
where x = {xjs : j = 1, 2, . . . , n, s = 0, 1, 2, . . . m − lj }, admissible set D
satisfies the constraints (7)–(9), d and e be levels of permissible dispersion and
the expectation respectively.
Tasks (10) and (11), as well as the task of (6)–(9), are the problems of Boolean
linear programming with non-negative conditions of the matrix, so they can be
resolved by pseudopolynomial algorithm based on dynamic programming.
4.2 Probability of given NPV inaccessibility as risk measure
Let C be a predetermined level NPV. Probability of achieving a given level be
X n m−l
Xj
P {C(x) > C} = P Cjs xjs > C . (12)
j=1 s=0
Let us introduce events
m−lj n
X X
Ej : Cjs xjs = yj , j = 1, 2, . . . , n, yj > C
s=0 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
m−lj m−lj
!
X X C js − yj
P {Ej } = xjs P {Cjs > yj } = xjs .
s=0 s=0
C js − C js
Hence, given the independence of the NPV value of the different projects, we
have,
n m−l
n
!
Y Y Xj C js − yj
P {C(x) > C} = P {Ej } = xjs .
j=1 j=1 s=0
C js − C js
Further instead of maximizing the probability of P{·} we consider the problem
of maximizing its logarithm. Due to the monotony of the logarithmic function
the optimal solutions to both problems are the same. We have
!
n m−lj
X X C js − y j
ln P {C(x) > C} = ln xjs =
j=1 s=0
C js − C js
n m−l n m−l
! !
Xj C js − yj Xj yj − C js
∼
X X
= xjs ln =− xjs .
j=1 s=0
C js − C js j=1 s=0
C js − C js
The second equality is a consequence of (8) and (9), and the last equality is a
consequence of approximate equality ln(1 − ξ) ∼
= −ξ.
On the other hand
ln P {C(x) > C} = ln [1 − P {C(x) < C}] ∼
= −P {C(x) < C} ,
consequently
n m−l
!
Xj yj − C js
P {C(x) < C} ∼
X
= xjs . (13)
j=1 s=0
C js − C js
This equation determines the probability of investment program setpoint NPV
inaccessibility, and later this probability is used as a measure of risk.
, n, i = 0, 1, . . . , m − 1
Let us introduce determinate variables zji , j = 1, 2, . . .P
n
and let us consider events Aji = {Ijsi ≤ zji } and {Bi = j=1 zji ≤ ri }. 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
Xn
zji − I jsi n r i − zji
j=1
X
P {Ijsi ≤ zji } = , P zji ≤ ri = .
I jsi − I jsi
j=1
ri − ri
Let us find the probability of the conditions 3 realizability of the investment
program considering the variables zji as fixed. We have
n X
X i
P r (x) = Ijsi xjs ≤ ri =
i
j=1 s=0
Xn Yn m−l
Xj
=P zji ≤ ri · (xjs P {Ijsi ≤ zji }) . (14)
j=1 j=1 s=0
for any billing period i = 0, 1, 2, . . . , m − 1 . Finding of equation (14) logarithm
and taking into account
ln P {ri (x) ≤ ri } ∼
= −P {ri (x) > ri } ,
I jsi − zji
ln P {Ijsi ≤ zji } ∼
= −P {Ijsi > zji } = − ,
I jsi − I jsi
Xn
X n X n zji − ri
zji ≤ ri ∼
j=1
ln P = −P zji > ri = − ,
j=1
j=1
ri − ri
as well as conditions (8) and (9) we have
Xn
zji − ri n m−l
!
j=1
X Xj I jsi − zji
P {ri (x) > ri } = + xjs ,
ri − ri j=1 s=0
I jsi − I jsi
i = 1, 2, . . . , m. (15)
Equation (15) defines the probability of exceeding of resources required for the
calculation period i = 0, 1, 2, . . . , m with the enterprise investment program.
Thus, if α be tolerable risk unreachable investment program setpoint NPV,
βi 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 defining the maximum of the expected income can be represented as follows
n
X
C(x, y, z) = yj → max (16)
x,y,z
j=1
m−lj
X
Cjs xjs = yj , j = 1, 2, . . . , n; (17)
s=0
n m−l
!
X Xj yj − C js
xjs ≤ α; (18)
j=1 s=0
C js − C js
Xn
zji − ri n m−l
!
j=1
X Xj I jsi − zji
+ xjs ≤ βi , i = 1, 2, . . . , m; (19)
ri − ri j=1 s=0
I jsi − I jsi
xjs zji ≤ I jsi , j = 1, 2, . . . , n, s = 0, 1, 2, . . . , m − lj , i = 1, 2, . . . , m; (20)
m−lj
X
xjs ≤ 1, j = 1, 2, . . . , n; (21)
s=0
xjs ∈ {0, 1}, j = 1, 2, . . . , n, s = 0, 1, .., m − lj . (22)
5 Example
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-effective. The planning horizon of the invest-
ment 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 1. NPV of the projects
Project s=0 s=1 s=2 s=3
j C j0 C j0 C j1 C j1 C j2 C j2 C j3 C j3
1 655 850 585 780 522 717 466 661
2 246 441 220 415 196 391 175 370
3 164 359 146 341 131 326 117 312
4 383 578 342 537 305 500 272 433
5 334 529 298 493 266 461 237 964
6 972 1167 867 1063 774 970 691 887
7 414 609 369 565 330 525 294 490
Table 2 contains interval estimations of the allowed amount of financing the
company’s investment program for the calculation period i.
Table 2. Estimates of acceptable amounts of funding
i 0 1 2 3 4 5 6 7 8 9 10
ri 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800 1800
ri 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000
Table 3 contains interval estimations Ijsi costs of project j = 1, 2, . . . , 7 at
calculation periods i = 0, 1, . . . , 10 under start period s.
All valid risks βi of exceeded the resources required for the calculation period
i = 0, 1, 2, . . . , m of investment program of the company are equal to the risk
tolerance α to get the unreachable investment program setpoint NPV.
Results of the solution received by using MS Excel are presented in Table 4.
Columns from the second to the eighth of Table 4 correspond to the seven
best programs of investment companies that satisfy given constraints on the type
and amount of risk. This columns contain billing period number s : xjs = 1
of implementation beginning for each project j = 1, 1, . . . , n included in the
investment program.
Guaranteed NPV equals to 2644 and its variance is equal to 1058 with max-
imin strategy optimal investment program. The highest expected NPV for the
variance of D0 = 1058 is equal to 2916 and implements the investment program,
which differs from the maximin. Levels of allowable variance 1.5D0 and 3D0
Table 3. Costs of project j = 1, 2, . . . , 7 at periods i = 0, 1, . . . , 10, s = 0, 1, 2, 3
(thousands of rubles)
i=s i=1+s i=2+s i=3+s i=4+s j =5+s i=6+s i=7+s
j
I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi I jsi
1 100 120 100 119 140 150 120 132 88 100 72 80 50 58 40 50
2 300 320 300 303 300 400 80 100 80 100 80 100 90 100 90 100
3 88 99 120 144 130 155 88 100 75 80 59 60 58 60 55 60
4 400 500 680 700 199 215 140 150 140 150 140 150 140 150 140 150
5 530 600 530 600 530 600 70 80 70 80 70 80 70 80 70 80
6 680 850 680 850 390 500 390 500 350 400 180 200 180 200 180 200
7 480 500 380 385 330 380 320 350 300 330 300 330 300 330 300 330
Table 4. Competitive investment programs
Project Maximin NPV dispersion Probability of accessibility
j s : xjs = 1 s : xjs = 1 NPV: s : xjs = 1
3D0 = 3174 1.5D0 = 2116 D0 = 1058 α = 0.1 α = 0.05 α = 0.03
1 0 0 1 0 0 1 1
2 − 0 3 3 − 0 0
3 3 1 0 0 0 − −
4 − 3 − − 2 2 3
5 3 − 0 1 0 − −
6 0 0 1 2 0 0 0
7 0 0 3 0 0 0 3
Project 1 2 3 4 5 6 7
NPV 2644 3782 3297 2916 4044 3588 2944
D D0 = 1058 3174 2116 1058 3372 2253 1103
α 0 0.104 0.051 0.031 0.096 0.049 0.026
correspond to different Pareto optimal investment programs having an average
NPV equal to 3297 and 3782 respectively. Increasing the level of α 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 effective (i. e. belong to the
Pareto set in the space of criteria).
Fig. 1. Samples in different spaces criteria projects
Conclusion
Considered models of optimal investment program with known distribution of
funds for each period allow to shape the Pareto-optimal investment programs.
Presented modification of this model which takes into account the uncertainty
of financial 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 efficient portfolio based on derivative criteria of performance: (i) the
payback period of the investment, (ii) the rate of return on capital, (iii) the
difference between the amount of income and investment costs (non-recurring
expenses) for the entire useful life of the investment project, (iv) reduced pro-
duction costs, and (v) risk appetite of decision makers.
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