Present a case study comes to confirm theoretical suppositions formulated by the authors, which asserts that in commercial banks we can determine internal cession price using mathematical methods – like the offered by matrix system of linear programming.internal cession price, profit centers, lineal programming problem

Grigore LUPULESCU

Ioan NISTOR

Ion DUBREU

English

Research Papers in Economics

Year VII, No.8/2008                                                                                                   37 
 
 
DETERMINATION OF THE INTERNAL CESSION PRICE 
IN THE COMMERCIAL BANKS THROUGH 
MATHEMATICAL METHODS – A CASE STUDY 
 
Prof. Ioan NISTOR, PhD 
”Babeş-Bolyai” University, Cluj-Napoca 
Lect. Grigore LUPULESCU, PhD Student  
”TITU MAIORESCU” University, Bucharest 
Ion DUBREU - informatics specialist 
Commercial Bank - Carpatica, Tg-Jiu
 
The internal cession price of a 
commercial bank represents, in our view, 
the transfer rate at which the resources 
of the bank are transferred from one 
responsibility center to another. The price 
is merely conventional and it is 
determined based on certain 
management rules and not on the rules 
confirmed by the state of the market and 
having no impact on the general results 
of that particular commercial bank. 
Here are several arguments in 
favor of the usefulness of the 
determination of the internal cession 
price in a commercial bank: 
 those in charge feel more 
motivated. If internal cession prices are 
determined, managers of the 
responsibility centers can evaluate (their 
own) performance; 
 operation of the bank is 
improved. The process is based on the 
idea that personal interest of the 
manager is conditioned by the results of 
his work and the convergence between 
the two will lead to the maximization of 
the efficiency of the responsibility center 
he manages. For it might happen that 
decisions of the manager of a certain 
responsibility center are benefic for that 
particular center, but have negative 
effects on the bank in general; 
 
 profitability can be evaluated. 
The internal cession price allows the 
profitability of a responsibility center to be 
evaluated, at a certain extent. However 
determination of the internal cession 
price contains also some subjective 
elements and as a consequence 
evaluation of the profitability of the 
centers based on these prices can be a 
permanent source of conflict. 
In our view, another reason why 
the internal cession price has to be 
determined in a commercial bank is the 
marginal resource (the last resource 
acquired or sold in order to balance 
investment resources) which is defined 
as an excess from the point-of-view of 
the releasing responsibility center and a 
deficit from that of the receiving 
responsibility center. 
Within a responsibility center of a 
bank the above-defined marginal 
resource can usually be acquired only by 
redistribution of resources among the 
responsibility centers of the bank or from 
the responsibility center of the Treasury 
Management Department (the 
department which deals with external 
acquiring or investing the deficit, 
respectively the excess of resources of 
the bank in general, a department which 
is also called the treasury pool). 
Dealing with the marginal resource 
compels us to introduce another two 
terms: 
  marginal cost, the excess cost 
created by acquiring an additional 
resource. It is an addition to the class of 
total expenses expended due to the 38                                                                         Finance – Challenges of the Future 
 
increase of a certain volume of 
resources; 
  marginal income, the excess 
income gained from investing an 
additional resource. 
In view of the above-mentioned the 
following association of terms can be 
performed regarding the responsibility 
centers of a bank. 
  Deficit resource center – marginal 
resource to be acquired – marginal cost; 
  Excess resource center – marginal 
resource to be sold – marginal income; 
Relevant literature suggests 
several views regarding the 
determination of the internal cession 
price, one of these being the usage of 
mathematical models for such a purpose. 
The case study hereby is based on such 
a view. 
In the case study hereby the 
budgeted and actual excess and deficit of 
resources of 6 (six) profit centers of a 
commercial bank at the end of a calendar 
month will be presented using 
hypothetical data due to practical 
considerations.  
Each of the 6 (six) profit centers 
represents a typical position in which any 
other profit center of the same bank can 
find itself. 
1.  The excess and deficit 
achieved by the 6 (six) profit centers 
compared to the budgeted values are 
presented in Table 1. 
 
Table 1. The excess and deficit achieved by the studied profit centers at the end 
of a calculation period 
lei or thousands of lei or millions of lei 
Budgeted results  Actual results  Disparity  Profit 
centers  Excess Deficit  Excess Deficit  Excess  Deficit 
0  1  2  3  4  5 = 3-1 or 3+2 6 = 4-2 or 4+1 
PC1 +100 -  +150  -  +50  (E1)  - 
PC2 +125 -  +70  -  -55  (E2)  - 
PC3  +50  -  -  45 (95)   95 (D3) 
PC4 -  110 -  100 -  -10  (D4) 
PC5 -  70  -  95  -  25  (D5) 
PC6  -  95  +20 (115)  -  +115 (E6)  - 
Total   275  275  240  240     
 
2.  Formation of the coefficients' 
matrix (A). This will be formed 
automatically by the computer program 
and it will be communicated to the local 
managers at the beginning of the 
evaluated period so that each manager 
would be aware of the internal cession 
price he will be allowed according to the 
results, i.e. the excess or deficit achieved 
compared to the budgeted values, at the 
very beginning of the process.  
This coefficients’ matrix may reflect 
the general policy of the bank; possible 
positive or negative deviations (not 
dependent on the local management 
policy) compared to the parameters 
determined at the beginning of the 
process will be reflected in the profit 
center of the General Management 
Department (Treasury Management 
Department – treasury pool), which 
connects the external environment of the 
bank and its internal profit centers. The 
Treasury Management Department is 
directly controlled by the general 
management of the bank and in our view, 
its results will reflect only its own 
performance (the general management’s 
performance) without reflecting 
performances of local managements. 
In our view, performances of local 
management should not be affected by 
factors outside their own activity. The 
bank’s general policy is formulated by the 
general management, but it is carried out 
by the local management. Year VII, No.8/2008                                                                                                   39 
Each profit center (PC) will be 
assigned the following coefficients, 
according to its position in disparities 
regarding excess (E) and deficit (D), as 
follows: 
Disp. E, D         +E       -D      -E      +D 
(thousands, millions etc.) 
  0-20    5 5 5 1 
  21-40    4 4 5 1 
  41-60    3 3 5 1 
  61-80    2 2 5 1
  81-100  1 1 5 1 
One variant for formation of the 
coefficients' matrix is the one presented 
in the following logical scheme (logical 
scheme no. 1). 
The actual method of determining 
the coefficients is applied according to 
the general policy of the bank, at the 
beginning of the financial year; therefore 
the coefficients determined at the 
beginning of a financial year cannot be 
changed during this interval. 
Based on the conditions 
established above the coefficients' matrix 
(A) is as follows: 
 
CP1 CP2  CP3  CP4 CP5 CP6 
A=    























1 0 0 0 5 3
0 1 5 5 0 0
0 5 5 5 0 0
0 5 5 5 0 0
1 0 0 0 5 3
1 0 0 0 1 3
 
D4 
Significance of the coefficients: 
e.g. +3 in the first row and first column 
signifies that three units of the excess 
achieved in PC1 will be assigned to PC1; 
+1 in the first row and last column 
(column 6) signifies that one unit of the 
excess achieved by PC1 will be assigned 
to PC6. The coefficients’ matrix (A) will 
be formulated automatically by the 
computer program created for this 
purpose so that all classification errors 
regarding PCs will be excluded, they will 
be classified according to the Excess or 
Deficit achieved compared to the 
budgeted values. 
3. Restrictions in the system are as 
follows: 
X1, X2, X3, X4, X5 ≥ 0 AND 30 ≤ 
X6 ≥ 40 
4.  The function of the problem is 
maximization of the profit obtained by the 
means of the excess and deficit 
achieved by each PC and can be 
formulated as follows: 
Fx = 150X1 + 70X2 - 45X3 - 
100X4 - 95X5 + 20X6               maximum 
5.  The column of the absolute 
terms (B) is made up of the excess or 
deficit budgeted by each PC, as follows: 
  100; 
  125; 
    50; 
  110; 
    70; 
E1      95; 
E2  6.  The absolute term d = 0, the 
term in which we will be able to find (after 
solving the mathematical model using the 
simple reduction algorithm) the maximum 
of the function proposed by us (i.e. the 
maximum of the profit obtained by the 
means of the excess and deficit of 
resources); 
D3 
D5 
E6 
The mathematical model of the 
problem of lineal programming will have 
the following formulation: 
 
 
3X1 – X2 +0X3 +0X4 + 0X5 + 1X6 ≥ 100 budgeted excess to be maximized; 
3X1 + 5X2 + 0X3 +0X4 +0X5 +1X6 ≥ 125 budgeted excess to be maximized; 
0X1 + 0X2 + 5X3 + 5X4 - 5X5 +0X6 ≥ 50 budgeted excess to be maximized; 
0X1 +0X2 + 5X3 + 5X4 + 5X5 + 0X6 ≤ 110 budgeted deficit to be minimized; 
0X1 + 0X2 + 5X3 – 5X4 + 1X5 + 0X6 ≤ 70 budgeted deficit to be minimized; 
3X1 – 5X2 + 0X3 + 0X4 + 0X5 + 1X6 ≤ 95 budgeted deficit to be minimized; 
0X1 + 0X2 + 0X3 + 0X4 + 0X5 + X6 ≥ 30 
0X1 + 0X2 + 0X3 + 0X4 + 0X5 + X6 ≤ 40 40                                                                         Finance – Challenges of the Future 
Scheme no. 1. Logical scheme of formation of coefficients’ matrix A related to the 
above-defined problem
 
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 Year VII, No.8/2008                                                                                                   41 
In accordance with the laws of lineal programming, in order that the excess or deficit of 
resources can be allotted in round numbers the inequalities above have to be 
transformed into equalities and thus the problem will be the following: 
3X1 – X2 +0X3 +0X4 + 0X5 + 1X6 – X7 = 100 
3X1 + 5X2 + 0X3 +0X4 +0X5 +1X6 – X8 = 125 
0X1 + 0X2 + 5X3 + 5X4 - 5X5 +0X6 – X9 = 50 
0X1 +0X2 + 5X3 + 5X4 + 5X5 + 0X6 + X10 = 110 
0X1 + 0X2 + 5X3 – 5X4 + 1X5 + 0X6 + X11 = 70 
3X1 – 5X2 + 0X3 + 0X4 + 0X5 + 1X6 + X12 = 95 
0X1 + 0X2 + 0X3 + 0X4 + 0X5 + X6 - X13 = 30 
0X1 + 0X2 + 0X3 + 0X4 + 0X5 + X6 + X14 = 40 
X7 … X14 - are variables of compensation with irrelevant value. 
The matrix of the lineal programming problem will be as follows: 
CP1   CP2   CP3     CP4    CP5   CP6           




























  



 

 
0 0 0 0 0 0 0 0 0 20 95 100 45 70 150
40 1 0 0 0 0 0 0 0 1 0 0 0 0 0
30 0 1 0 0 0 0 0 0 1 0 0 0 0 0
95 0 0 1 0 0 0 0 0 1 0 0 0 5 3
70 0 0 0 1 0 0 0 0 0 1 5 5 0 0
110 0 0 0 0 1 0 0 0 0 5 5 5 0 0
50 0 0 0 0 0 1 0 0 0 5 5 5 0 0
125 0 0 0 0 0 0 1 0 1 0 0 0 5 3
100 0 0 0 0 0 0 0 1 1 0 0 0 1 3
        
E2 
D3 
D4 
       D5 
E6 
 
Applying the method of the simple 
reduction algorithm on the matrix of the 
linear programming problem using the 
GIM-2008 computer program, the 3rd 
(third) step of the simple reduction 
algorithm shows the linear programming 
problem to have a solution in the 
aggregate of the positive numbers, which 
means the restriction system is 
compatible and the 6th (sixth) step offers 
the following results: 
a). Fx (the maximum of the function) = 
6392,000; 
b). The values of the unknowns are as 
follows:   X1 = 26,667; 
  X2  =  3,000; 
  X3  =  14,400; 
    X4 = 1,600;  
  X5 = 6,000; 
  X6  =  30,000. 
If the 3rd (third) step had not 
allowed for a solution, we could have 
concluded that the lineal programming 
problem does not allow for a solution in 
the aggregate of the positive numbers. In 
this case the restriction system would 
have proven incompatible. 
7. Analysis of the results obtained 
compared to the initial data 
Comparing the excess and deficit 
of the studied profit centers and having in 
view the results obtained by solving the 
lineal programming problem defined 
above the data summarized in Table 13 
have been obtained. 
 
 
 42                                                                         Finance – Challenges of the Future 
Table no. 2. The excess and deficit achieved by the studied profit centers 
compared to the solutions obtained by solving the lineal programming problem 
defined above 
 
Budgeted results  Actual results  Disparity  Profit 
centers 
(PC) 
Excess 
(E) 
Deficit 
(D) 
Excess 
(E) 
Deficit 
(D) 
Excess 
(E) 
Deficit (D) 
Value of the 
unknowns. 
0  1  2  3  4  5 = 3-1 
or 3+2 
6 = 4-2 or 
4+1 
7 
PC1 +100  -  +150  -  +50  (E1)  -  26,667 income 
PC2 +125  -  +70 -  -55  (E2)  -  3,000 income 
PC3  +50  -  -  45 (95)    95 (D3)  14,400 cost 
PC4 -  110  -  100  -  -10  (D4)  1,600 cost 
PC5 -  70  -  95 -  25  (D5) 6,000 cost 
PC6 -  95  +20 
(115) 
- +115 
(E6) 
-  30,000 income 
Total   275  275  240  240       
Maximum of the function (the profit achieved from the excess and deficit of 
the resources of the PC) 
6392,000 
Comparing the initial data with the 
results obtained by solving the 
mathematical model the following can be 
concluded: 
1)  the internal cession price 
obtained is different both for the profit 
centers achieving an excess of resources 
and for those achieving a deficit, 
according to the discrepancy between 
the actual and the budgeted 
achievements; 
2) profit center no. 1 (PC1) who 
exceeded its budgeted resources with 
50% will receive a cession price equaling 
its income, more than PC2 (who did not 
achieve its budgeted resources), but less 
than PC6 (who exceeded  its budgeted 
resources although it budgeted for a 
deficit); 
3) profit center no. 3 (PC3) will 
receive the highest cession price 
equaling costs, since it budgeted an 
excess of resources and achieved 
instead a deficit, while PC4 will receive 
the lowest internal cession price equaling 
costs, because it achieved an even lower 
deficit than it budgeted. PC5 achieved a 
higher internal cession price equaling 
costs than PC4, since it did not achieve a 
lower deficit than it budgeted and thus 
caused additional costs to the bank, but 
still its internal cession price is much 
lower than that of PC3 who budgeted an 
excess of resources and achieved 
instead a deficit; 
4)  the internal cession price 
determined with the above method helps 
the bank to redistribute the effect in a 
more correct way, according to the 
performance of each unit and stimulates 
the profit centers to work for achieving 
the budgeted parameters or to exceed 
them or to achieve a lower deficit than 
budgeted - all natural aspirations for any 
management of any bank. 
 
 
 
 
REFERENCES 
Nistor E. Ioan, 
Lupulescu Grigore, 
Dubreu Ion 
Mathematical approaches concerning the determination of the internal 
cession price in the commercial banks, Competitiveness and 
European Integration,  Alma Mater Publishing House, Cluj-Napoca,   
2007 
 

Determination of the internal cession price in the commercial banks through mathematical methods - A case study

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