(2 rânduri libere, 11p)


Studies and Scientific Researches. Economics Edition, No 20, 2014  http://sceco.ub.ro 

 
GAME THEORY: MINIMISING THE COST OF CAPITAL VS. 

MAXIMISING THE RETURN OF INVESTORS  
  

Brindusa Tudose  
 Petre Andrei University of Iasi 
  brindusatudose@gmail.com 

 
 
Abstract  
 The application of game theory to financial transactions focuses on two categories of 
stakeholders: users of financing (firms) and providers of financing (investors). The core of 
game theory consists in the strategy that a partner is able to build starting from the possible 
decisions of the other partner (each party having opposing interests). In fact, we deal here with 
a cooperative game in which both opponents seek to maximise their own chances of winning. 
The article aims to highlight the manner in which mathematical game theory is transposed in 
the field of corporate finance by balancing the firm’s objectives (maximising market value by 
minimising the cost of raising capital) and the investors’ objectives (maximising returns on 
investments). The intended novelty of this paper lies in developing a model for optimising a 
firm’s financial structure and assessing it in terms of investors’ interests. 
  
Keywords  
 financial structure; optimisation criteria; cost minimisation; profit maximisation 
  
JEL Classification  
G32; C70  
 
 
 
Introduction 
The issue of firms’ financing has been and continues to be one of the most hotly 
debated topics for a variety of analysts/researchers, not necessarily finance experts 
(for instance mathematicians, psychologists, economists). Their works have focused 
on the optimal mix of the two sources of financing (internal and external), in order to 
secure the success of major objectives undertaken by management; hence, the 
financing decision – one of the key decisions in financial management – emerges as a 
genuine challenge. Most of the theories formulated so far have built analytical 
frameworks for grounding, understanding and implementing firms’ capital structure 
policies. However, the theories developed have certain limitations, which have posed 
challenges for further research and experienced varied and wide-ranging debates, with 
key concepts re-emerging in novel forms (Tudose, 2013). 
Inaugurated by Emile Borel (1921) but comprehensively laid out in only 1944 by 
John von Neuman (a mathematician) and Oskar Morgenstern (an economist), the 
mathematical theory of games opened up a highly diverse research field. Lately, we 
have observed increasing interest by finance researchers in reassessing theories of 
finance against the background of game theory. 
With reference to the above, it must be emphasised that certain mathematicians have 
argued that “mathematical game theory has a broad applicability in the field of 
financial transactions, yet raises problems related to building specific functions and 
deliver concrete solutions for various models and classes of determinist or 
probabilistic financial transactions” (Purcaru, 1998).  
The main objective undertaken in this paper is to show how the mathematical game 
theory is transposed in the field of corporate finance by balancing the firm’s 

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objectives (maximising market value by minimising the cost of raising capital) and 
the investors’ objectives (maximising returns on investments). When a firm posts 
minimum cost of capital it is essential to ascertain not only the context that enabled it 
to achieve this objective but also the extent to which investors were able to achieve 
their goals too. 
  
 
 Landmarks in specific research 
The adequacy of capital structure represents a major decision for any firm; this is 
because the decision is founded not only on the need to maximise shareholder returns, 
but also on the need to ensure the firm’s capacity to cope with its competitive 
environment. The views on the optimal financial structure have varied over time; 
going beyond the irrelevance theory, it has been recognised that there is a link 
between capital structure and firm value. 
The idea proposed 10 years ago, arguing that “there is no universal theory of the debt-
equity choice, and no reason to expect one” (Myers, 2001), reoriented research to the 
level of empirical analyses. 
As far as the firm is concerned, its financial structure is designed based on certain 
criteria (Toma & Alexandru, 2003) as follows: the firm’s rate of return; the cost of 
raising capital; maximum debt carrying capacity and the flexibility of the financial 
structure; risk; resource allocation; firm’s objectives. Additionally, other criteria may 
be considered (Tudose, 2006) including: the size of the firm; the weakening of 
shareholders’ controlling power (in case of new share issues); technology risk (which 
may be offset by leasing equipment); adjusting repayment methods to the firms’ 
financial constraints (certain reimbursement methods may be more convenient even if 
costlier); loan granting procedures (depending on the urgency of the financing 
needed, a more expensive alternative may be accepted that provides faster access to 
funding); loan guarantees being demanded (firms that are unable to deliver the 
required guarantees must opt for other types of lending, higher priced, but less 
stringent in terms of collaterals required). 
One of the domains of optimization applicability is represented by the financial 
structure. As the structural elements are variable, they can be combined in such a 
manner in which the maximization of the firm’s value will be attained.  
In other words, the optimization of the financial structure can lead to an improvement 
of the organization and of firms function. By accomplishing this objective favourable 
premises will be created for both shareholders and creditors; this will have a positive 
impact on the future development of the firms. 
However, when dealing with the optimization issue, in terms of economic theory, 
must not be adopted a fixed, rigid attitude. As long as the maximization and the 
minimization are matters of dynamics, the ratio between the useful output and the 
superior quality input will define the premises of optimization. 
When an firm records its maximum level of efficiency, and thus the minimum cost 
level of the capital (the optimum level), it is compulsory to express the environmental 
circumstances – exterior and interior environment, financial or of any other nature – 
that allowed the accomplishment of this objective. 
 
 
Methodology of the research  
 Approaching the issue from the firm’s perspective, we acknowledge that it is possible 
to determine the optimal state (based on the cost of raising capital from different 
sources) by using linear optimisation models. For these purposes, we will build a 
company finance structure optimisation model, based on the weighted average cost of 

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GAME THEORY: MINIMISING THE COST OF CAPITAL VS. MAXIMISING THE RETURN OF INVESTORS 

capital criterion. We will simultaneously explain the extent to which investors 
(financing providers) are able to achieve their objectives. 
The value of a financial transaction – V(I,F) – is not perceived in similar fashion by 
investors (I) and the firm (F); whereas investors aim to maximise the rate of return of 
their investment, the firm seeks to minimise the cost of raising capital. Because 
investors and the firm do not act independently, one might argue that both parties will 
focus on a strategy that will diminish (as much as possible) the effects of the other 
party’s strategy. A financial transaction (investment for investors, capital raising for 
firms) is optimal (or mutually advantageous, both for the firm and for the investors) if 
the strategies m 10 and m20 are in place to ensure that: 
 

)I,F(Vmaxmin)F,I(Vminmax 1m2m2m1m =  
 

Approaching the issue from the firm’s perspective, we acknowledge that it is possible 
to determine the optimal state (based on the cost of raising capital from different 
sources) by using linear optimisation models.  
In order to solve the linear programming problem, I will start from an admissible 
basic solution (forming a finite set) and will subsequently identify the solutions in 
order to minimise or maximise a particular function.  
  
 
Presentation of the model 
The milestones propose are: a) defining the target function (required to perform the 
optimisation); b) defining the system of equations/inequalities specific to the financial 
structure to be optimized, in terms of the cost of financing; c) defining the linear 
optimisation problem; d) solving the problem. 
 

 Defining the target function (required to perform the 
optimisation) 
 That can be achieved as follows: 
f(x) = weighted average cost of capital → minimum, or 
 

.mink
CC

C
k

CC
C

)x(f i
ip

i
p

ip

p
→

+
+

+
= ,  

 
where:  

Cp = equity capital,  
Ci = debt capital,  
kp = cost of equity capital,  
ki = real cost of debt capital (taking into account any tax shields). 

As linear programming models only allow linear (minimum or maximum) 
optimisation functions, we must redefine the optimisation function. Consequently, by 
using x 1 to denote the ratio )CC(C ipp +  and x 2 for the ratio )CC(C ipi + , the 
optimisation function becomes: 
 

.minkxkx)x(f i2p1 →+= , 
 

where the costs of the two types of capital are known, and x 1 and x 2 are the variables 
(the unknowns). 
 

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Tudose 

Defining the system of equations/inequalities specific to the 
financial structure to be optimized, in terms of the cost of financing 
a) Firms have access to two main types of financial resources, own capital or equity 
and borrowed capital or debt, respectively; as regards the level of accessibility, it 
should be noted that a firm can finance itself with 100% equity, but cannot finance 
themselves with 100% debt. The first equation of the system reflects the financial 
structure: 
 

1
CiCp

Ci
CiCp

Cp
=

+
+

+
 

 
According to the previously employed notation, the equation can be reformulated as 
follows:  

1xx 21 =+  
 

b) Own capital – after making an investment financed exclusively by own capital, the 
level of equity cannot exceed the sum of the initial own capital (Cp0) and the value of 
the investment (I). Consequently, the second component of the system will be an 
inequality: 

)IC(C 0pp +≤   
 or  

)IC(
)CC(

C
)CC( 0p

ip

p
ip +≤+

+   

 or  
)IC(x)CC( 0p1ip +≤+ . 

 
c) Borrowed capital – after the investment is made – cannot exceed the sum of the 
initial borrowed capital and the value of the investment (in case the investment was 
financed exclusively by borrowed capital). The third component of the system will 
also be an inequality: 

)IC(C 0ii +≤   
or 

)IC(
)CC(

C
)CC( 0i

ip

i
ip +≤+

+  

or  
)IC(x)CC( 0i2ip +≤+ . 

 
From the standpoint of the creditor/investor who seeks to maximise his profit, this 
transaction is only justified if it involves a premium aligned with the level of risk he is 
taking. Once loan is agreed, it is argued that both parties (the firm and the investors) 
have achieved their goals. 
d) Borrowed capital – after the investment is made – cannot exceed double the value 
of own capital (a prerequisite for ensuring the survival of the company and its 
financial balance). 
 

Cp2Ci ≤  
or 

)CC(
Cp

)CC(2
)CC(

C
)CC(

ip
ip

ip

i
ip +

+≤
+

+   

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GAME THEORY: MINIMISING THE COST OF CAPITAL VS. MAXIMISING THE RETURN OF INVESTORS 

or 
 )CC(x2x)CC( ip12ip +≤+ . 

 
The inequality can be tested by factoring in the known amount of the projected 
investment; therefore it is not indispensable in defining the system of 
equations/inequalities. 
 

Defining the linear optimisation problem 
Defining linear optimization problem involves aggregating of the previous 
relationships shown: 
 













≥

+≤+

+≤+
=+

0x,x

)IC(x)CC(

)IC(x)CC(
1xx

21

0i2ip

0p1ip

21

 

 
.minkxkx)x(f i2p1 →+=  

 
Solving the problem 

By incorporating in the model only two variables (x1 and x2), the system may be 
solved either graphically or by using the primary simplex algorithm or based on the 
two-phase optimisation method (just as to the matrix of the equation system one may 
or may not add elementary transformations to construct a unit matrix) (Reitano, 
2010). 
 
 
Limits of the model 
Minimising the cost of the firm’s equity capital must not be viewed in absolute terms, 
must not represent an objective in itself; rather this goal must be aligned with a 
broader focus on improving the financial structure and maximising the firm s value. 
The financial structure of a company which does not pay dividends (zero cost of 
equity) cannot serve as a reference for optimisation. A firm which pays a high number 
of dividends does not necessarily have an unfavourable financial structure (Onofrei, 
2009). The concerns regarding increasing the rate of return on equity do not derail the 
company from the path of optimising its financial structure. Due to their higher return, 
a company’s shares will be more attractive to investors and the increased share price 
will have “have a positive effect on the market value of the company” (Vernimmen, 
2009). 
Nor can the cost of debt capital be considered as an absolute reference. Each creditor 
requires a target rate of return in exchange for providing access to his financing 
source. Considering that investors (creditors) have several options for investing their 
capital, they will opt for the investment type offering a risk premium, the rate of 
return in excess of a risk-free return rate. 
The model we developed is based on a single-criterion optimisation. In practice, the 
cost of capital is not the sole determining factor in financial decisions. Building a 
model that should enable a multi-criterion optimisation of the firm’s financial 
structure will constitute a future focus of research for us. 
 
 
 

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Tudose 

Conclusions  
The defined model reinforces the claim that the mathematical game theory has an 
extensive area of application in the field of financial transactions, at the same time 
overcoming the issues that may arise in constructing specific functions; the concrete 
solutions for the function f(x): the weighted average cost of capital → minim will 
enable the determination of solutions for the optimisation function of the type 
max m1 min m2 V(I,F) = min m2 max m1V(F,I). The equilibrium state of the game is 
defined by the finite set of strategies identified for each actor.  
The purely financial premises that constitute the base for achieving an optimum 
capital structure are: knowing the cost of the financing sources and the fluctuation of 
the interest rate; knowing the typology of the enterprises because, depending on their 
dimension, the organisations have access to different sources of financing; knowing 
the financing needs of the enterprises; knowing the weight of participation for 
different financing sources at the enterprise level; knowing the financial environment 
at a macroeconomic level and of the financial-monetary policies – fiscal, monetary, 
credit, income – promoted in a given length of time; we also include here the degree 
of development and functionality of the capital markets because through them the 
enterprises can attract resources corresponding to various time horizons; knowing the 
behaviour of the parties that participate on the financial market and their willingness 
to take risks; knowing the psychological implications of indebtedness, because the 
financial structure of an enterprise will be greatly influenced by the attitude of the 
financial decision-makers regarding the possibility of falling into debt. 
  
 
References  
 Myers, S. (2001). Capital structure, The Journal of Economic Perspectives, 15(2),  

p. 81-102. 
Onofrei, M. (2009), Guvernanța financiară corporativă, București,  Editura Wolters 

Kluwer. 
Purcaru, I. (1998), Matematici financiare, București, Editura Economică. 
Reitano, R. (2010), Introduction to Quantitative Finance, Cambridge, MIT Press.  
Toma, M., Alexandru F. (2003), Finanțe și gestiune financiară de întreprindere, 

București, Editura Economică. 
Tudose, M. B. (2006), Gestiunea capitalurilor întreprinderii. Metode de optimizare, 

București, Editura Economică. 
Tudose, M. B. (2013), ”Real effects of the firm financing and risk management”, in 

Abordări conceptuale și metodologice ale proceselor economice, București, 
Editura Academiei Române. 

Vernimmen, P. coord. (2009), Finance d’entreprise, 7ème édition, Paris, Dalloz. 
 

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