Financial efficiency of investment project is being evaluated in this paper. It is showed that the net present value function is constant and that quota value is equal to c0 when i converges to the infinite. Optimal rates i are analyzed in certain cases and everything is illustrated through examples.investment, efficiency, project, investor, money flow, interest rate

Dominika Crnjac

Miljenko Crnjac

English

Research Papers in Economics

Miljenko Crnjac • Dominika Crnjac 368
INVESTMENT PROJECT EFFICIENCY EVALUATION
Miljenko Crnjac, Professor
Faculty of Economics
Dr.sc. Dominika Crnjac
Faculty of Electrical Engineering Osijek
Summary
Financial efﬁ  ciency of investment project is being evaluated in this paper. It is 
showed that the net present value function is constant and that quota value is equal to 
c0 when i converges to the inﬁ  nite. Optimal rates i are analyzed in certain cases and 
everything is illustrated through examples.
Keywords: investment, efﬁ  ciency, project, investor, money ﬂ  ow, interest rate.
1. Introduction
Given that project justiﬁ  cation appraisal is carried out by measuring investments 
and the effects of these investments, and given that these effects can generally be 
different, we can observe them in different conditions and from different points of 
view.
In this paper we will concentrate on ﬁ  nancial efﬁ  ciency of investment project, 
which should answer the question how justiﬁ  ed it is to invest ﬁ  nancial assets into a 
certain project from the viewpoint of the company making the investment.
The project duration period is T(n). In time T(n) investments and expected 
investment effects are observed. For many business projects it is not possible to 
determine exact money ﬂ  ow, therefore statistical theory is used.
Assuming that ct is money difference between money income and money outlay 
in any time interval during the project.
ct = money income in moment t – money outlay in moment t.
If  money  ﬂ  ow  is  constant,  then  money  ﬂ  ow  net  rate  can  be  expressed  as 
follows:
) ( ) ( ) ( 2 1 t c t c t c − = ,
where c1(t) is income rate, and c2(t) is outlay rate of money assets in the project 
moment t.INVESTMENT PROJECT EFFICIENCY EVALUATION 369
If the difference between income and outlay is larger then zero, then we talk 
about positive money ﬂ  ow, and if the difference is less then zero, then we talk about 
negative money ﬂ  ow.
For a particular investment project, let us assume that we know T(n) (economic 
duration of a project) and net effects ct.
It is not hard to conclude that the investment project value at its beginning is in 
a discrete case:
(0) 1 1 2
0 2 1 ( )
1 (1 ) (1 ) (1 )
n n
n n
c c c c
NVSV i c
i i i i
−
− = + + + + +
+ + + +
K ,
i.e.:
(0)
0
( ) (1 )
n
k
k
k
NVSV i c i
−
=
= + ∑
Equation 1
Where:
NVSV(0)(i)  net present value at the beginning of a project
i    interest rate
T(n)  project duration (usually years)
ck     project net effect at the end of year k.
It is in investor’s best interest to measure investment proﬁ  tability with regard to 
other investments, as this will indicate whether it is advisable to enter the ﬁ  nancial 
business. The issue here is whether we can enrich the given capital better than 
creditors.
Similarly to the previous situation, at the time T when the project ﬁ  nishes, we 
have the following situation:
∑ ∫
− − + + +
t
T
t T t T
t d t i t c i c
0
) 1 ) ( ( ) 1 ( ,
thus the net present value dependent on interest rate i equals to:
0
( ) (1 ) ( )(1 )
T
T t T t
t
t
NVSV i c i c t i dt
− − = + + + ∑ ∫
Equation 2Miljenko Crnjac • Dominika Crnjac 370
If we presume that c(t) = 0, then we get Equation (1).
By substituting q = (1 + i) 1, we get ( )
t
t
t
NVSV i c q =∑ , i. e. (1).
Note 1.
If the project duration time is inﬁ  nite, accumulation is not deﬁ  ned, whereas 
net present value is deﬁ  ned by Equation (2). It is not hard to see that the function 
NVSV(i) is a continuous function of interest rate and that  0 lim ( )
i NVSV i c
→∞ =
1.
Assuming that the investment is with ﬁ  xed interest rate i1 , equation (2) leads to 
the conclusion that the project is proﬁ  table if and only if NVSV(i) > 0.
If  NVSV(i) is moving from positive to negative values for i and around i0, then 
the project is proﬁ  table under these conditions if and only if  i1 < i0.
Note 2.
Financial efﬁ  ciency evaluation of a project can be determined by mathematical 
analysis  of  a  function  graph  NVSV(i).  When  using  net  present  value  criteria, 
the problem is how to choose optimal interest rate i , in which money ﬂ  ows are 
discounted.
2.  Examples
Example 1
If there is one investment c0 at the beginning of the project duration, and if all 
effects are constant and greater than zero throughout the project, then net present 
value is:
(0)
0 0
1
( )
( 1)
n
n
r
NVSV i c c
r r
−
= +
− 2
,
where  r = 1+ i0, 
100
0
0
p
i =
The investment project is:
1  M. Crnjac, Matematika za ekonomiste, Ekonomski fakultet, Osijek (2000)
2  M. Crnjac, Matematika za ekonomiste, Ekonomski fakultet, Osijek (2000)INVESTMENT PROJECT EFFICIENCY EVALUATION 371
1)  efﬁ  cient if  NVSV(0)(i0)>0
2)  neutral if  NVSV(0)(i0)=0
3)  inefﬁ  cient if  NVSV(0)(i0)<0
Example 2
Internal proﬁ  tability rate ir is the rate at which net present value of investment 
project is equal to zero. So, NVSV(0)(ir)=0,  i.e.  ∑
=
− = +
n
k
k
r k i c
0
0 ) 1 ( .
Given  that  i0  is  the  proﬁ  tability  rate  at  which  common  enterprise  projects  are 
approved, it is not hard to conclude that the investment project is:
1.  efﬁ  cient for  ir > i0
2.  neutral for  ir = i0
3.  inefﬁ  cient for  ir < i0 
Using iterative methods it is not hard to calculate internal proﬁ  tability rate from the 
equation ∑
=
− = +
n
k
k
r k i c
0
0 ) 1 ( .
Example 3
Asset return time T(ip) is the time necessary to return investments. It can be 
calculated by the equation 
( )
(0)
0
( ) (1 ) 0
p T i
k
k p
k
NVSV t c i
−
=
= + = ∑ , where NVSV(0)(t) is
the function of time. If  t0 is the return time for similar projects, the investment 
project is:
1.  acceptable if  T(ip) < t0,
2.  neutral if  T(ip) = t0,
3.  unacceptable if  T(ip) > t0,
Using the permanency principle we can calculate:Miljenko Crnjac • Dominika Crnjac 372
(0)
0 (0) NVSV c =
1
(0)
0
(1) (1 )
k
k p
k
NVSV c i
−
=
= + ∑
2
(0)
0
(2) (1 )
k
k p
k
NVSV c i
−
=
= + ∑
3
(0)
0
(3) (1 )
k
k p
k
NVSV c i
−
=
= + ∑
:
(0)
0
( ) (1 )
t
k
k p
k
NVSV t c i
−
=
= + ∑
until we get  NVSV(0)(t)<0  and  NVSV(0)(t+1)≥0 thus having return time T(ip) from 
the open interval (t, t+1).
By using linear interpolation3 we have:
(0)
(0) (0)
( )
( )
( ) ( 1)
p
NVSV t
T i t
NVSV t NVSV t
= +
− +
References:
1)   J. Bedeković i koautori; Planiranje investicijskih projekata, Ekonomski institut, 
Zagreb (1993)
2)    R. Scitovski, R. Galić i M. Šilac Benšić; Numerička matematika, vjerojatnost i 
statistika, Elektrotehnički fakultet, Osijek (1993)
3)   B. Novak, Odlučivanje u ﬁ  nancijskom upravljanju, Ekonomski fakultet, Osijek 
(2000)
4)   M. Crnjac, Matematika za ekonomiste, Ekonomski fakultet, Osijek (2000)
3    R. Scitovski, R. Galić i M. Šilac Benšić; Numerička matematika, vjerojatnost i statistika,  
Elektrotehnički fakultet, Osijek (1993)

Bedeković i koautori; Planiranje investicijskih projekata, Ekonomski institut,

Investment Project Efficiency Evaluation

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Investment Project Efficiency Evaluation

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