A Mathematical Model and Decision Support
System for Determination of the Values of the
Marginal Reserve Requirement as Instrument of
Monetary Policy
Darko Pongrac
Outline






Introduction
Mathematical model
Heuristic
Computational results
Conclusions
Future research
Introduction – situation in Croatia

Croatian National Bank (CNB) - aims:



price stability
supporting economic growth
Commercial banks – aims:

making profit
Introduction – situation in Croatia
Commercial banks



foreign ownership
indebtedness abroad with low interest rate
giving loans in Croatia with high interest rate
easy profit!
Introduction – situation in Croatia

commercial banks’ debt abroad increase the Croatian external
debt

Croatian external debt reached the level which is in the economic
theory considered as upper accepted level for external debt of a
country
Croatian National Bank (CNB):

uses the available instruments and
measures to control the external debt
Introduction – Monetary policy

according to our open economy, there exists high possibility for
transmitting inflation from abroad (for example: increasing of energy
prices on the world market has big influence on domestic prices)

transmitting inflation from abroad and high level of foreign debt can
effect high disturbance in country economy

special attention is focused on the external debt growth

historically low level of interest rates on the world capital markets
Introduction – Monetary policy

CNB has a limited number of available measures and
instruments for influencing commercial banks behaviour


slow measures
fast measures
Introduction – Monetary policy
Slow measures




Reserve requirements
Marginal reserve requirements (MRR)
Special reserve requirements (SRR)
Compulsory central bank bills (CCBB)
These measures were used in the model
developed in this work.
Introduction – Monetary policy

– commercial banks’
objectives
Profit is made from different revenues that can be put into two main
categories:
1.
2.
interest
fees

Interest and fee revenues connected to the credit activities are
shown through the effective interest rate.

Revenues from credit activities are a significant part of commercial
banks’ revenues.
Introduction – mathematical programming

mathematical programming is in high expansion with
evolution of the computers

specially expanded in last twenty years

we know difference between single level and multilevel
mathematical programming

Bialas and Karwan described, in 1982., multilevel
programming problem which includes n level
Mathematical model
Bilevel programming model

CNB – leader: minimize the increase in household’s consumption
(loans to households)

commercial banks – followers: maximize their profits
Conflict!
Mathematical model
CNB (leader):

controls the percentage of marginal reserve requirements (MRR)

controls the percentage of special reserve requirements (SRR)

regulate conditions on the purchase of the compulsory CNB bills
Mathematical model
Commercial banks’ loans are
divided in three main categories:



housing loans
loans to households
loans to enterprises
Mathematical model
1.
Indexes

i
j
l
p

t -

 -



-
type of indebtedness
commercial bank
type of investment
marginal reserve requirements
percentage
time period of indebtedness
(macro period)
time period of investments
(micro period)  St
Mathematical model
2.
Parameters


op
dlt
glt
ol
-

bi
-



kit mjlt xjil0 -

Wjl0 -


reserve requirements percentage
minimal demand for credit
maximal supply of credit
the number of repayments of credit
instalments
the number of repayments of
indebtedness instalments
interest rate of indebtedness
interest rate of investment
bank’s indebtedness at the beginning
of the observed period
bank’s credit at the beginning of the
observed period
Mathematical model
3.
Variables

xjilpt -
the amount of bank’s debt in the
observed period

wjl -
the amount of bank’s credit in the
observed period


zilpt -
vjilpt -
xjilpt , wjl ≥ 0;
1, if the percentage of marginal/special
reserve requirements is p; 0 otherwise
1, if bank’s indebtedness is bigger then
repayment related to the previous
indebtedness; 0 otherwise
zilpt , vjilpt  {0,1}
Mathematical model
4.
Notes

yjilpt -
the amount that the bank repays for
previous indebtedness

Wjlt -
the total amount of bank’s credit in
the macro period

Ujlt -
the total amount of clients’
repayments related to previous
credit

Qjipt -
bank’s debt

Rjlt -
bank’s credit
Mathematical model

Expression for notes:


t 1
1
y jiplt  ( x jil0   zilp x jilp ) ,
bi
  max( t bi ,1)
 j,i,l,p,t
(a)
W jlt   w jl ,
 j,l,t
(b)
t 1
1
U jlt  (W jl 0  W jl ) ,
ol
  max( t ol ,1)
 j,l,t
(c)
 St

3

t
Q jipt   ( x jil0   ( x jilp  y jilp )) ,
 j,i,p,t
(d)
 j,l,t
(e)
 1
l 1
t

R jlt  W jl 0   (W jl  U jl ) ,
 1
Mathematical model

Model:
min
z
 (W
jlt
 U jlt )
l 1, 2
j ,t
with constraints:
z
ilpt
 i,l,t
 1,
(1)
p
max  (  m jlt R jlt   k itQ jipt )
t ,l ,i , p
t
l
with constraints :
i, p
j
Mathematical model

Model - constraints:
t
100  op
p
W jlt  ((
)(  x jil0   v jilpt(1 
) zilp ( x jilp  y jilp )
op
100
i
i , p ,  1
t

 (1  v
i , p , 1
jilpt
t
t 1
 1
 1
)( x jilp  y jilp )))  U jl  W jl  W jl 0
x jilpt  Mz ilpt ,
t
(x


1
jilp
 y jilp )  Mv jilpt ,
t
(y


1
jilp
 j,l,t (2)
 x jilp )  M (1  v jilpt ) ,
 j,i,l,p,t
(3)
 j,i,l,p,t
(4)
 j,i,l,p,t
(5)
Mathematical model

Model - constraints:
W jlt  d jlt ,
 j,i,l,p,t
W jlt  g jlt ,
 j,i,l,p,t
xjilpt , wjl ≥ 0; zilpt , vjilpt  {0,1},
(6)
(7)
 j,i,l,p,t
(8)
Mathematical model - difference between models
t
  100  op 
100  p
W jlt   
zilp ( x jilp  y jilp )
  x jil0   v jilpt
  40  0,6op  i
100
i , p , 1


t

 (1  v
i , p , 1
W jlt
jilpt
)( x jilp
t
t 1

 y jilp )    U jl  W jl  W jl 0

 1
   1
t
  100  op 
100  p


  x jil0   v jilpt
 
zilp ( x jilp  y jilp )
  40  0,6op  i
100
i , p , 1



t
 (1  v
i , p , 1
jilpt
)( x jilp
t
t 1

 y jilp )    U jl  W jl  W jl 0

 1
   1


  1
 


W


  jl

t
 1 



 0.5  max   1

,0  W jl 0 
 W
100  
 2
jl
0









 



Heuristic

NP-hard problem (Ben-Ayed, Blair, 1989)

heuristic

nonlinear constraint (2) was relaxed in the way that the
binary variable vijlpt is fixed to 1 in all observed points
(the real situation), and the second binary variable zilpt is
fixed to 1 for a chosen value of marginal reserve
requirements in each observed period
Heuristic

z jlpt

v jilpt

Real situation: j=34, i=2, l=3, p=70, t=12
171136
and constraint (2) is cubic
0-1 variables
Heuristic

interest rates for banks’ debt are fixed to the chosen
values (euribor+1%,that is 4.5%), interest rates for
banks’ loans are known

all banks have the same conditions for indebtedness

we observe only macro periods

we observe the neighbourhood of ±5% of the chosen
marginal reserve requirements

for a closer look at the changes in banks’ behaviour the
neighbourhood changes to ±1%
Start
Heuristic
Read model
parameters
Choose initial bank for
solving
Choose initial marginal/
special reserve
requirement for solving



What does it mean
“Choose the next value
for MRR”?
neighbourhood of ±5%
for a closer look,
neighbourhood of ±1%
Solve relaxed linear
problem
Has the marginal/
special reserve
requirement been found
for all kinds
of loans?
Choose next
marginal/ special
reserve requirement
for solving
No
Choose the next
bank
for solving
Yes
Have all
the banks been
considered?
Yes
Print out the calculated
values for the highest,
lowest and mean marginal
/ special reserve
requirement
for the banking system
Stop
No
Heuristic
What does it mean “Is the MRR found?”

jump!

Wjlt -
the total amount of bank’s credit in the macro period
d jlt  W jlt  g jlt
MRR  MRR *  W jlt  d jlt
MRR  MRR  W jlt  g jlt
*
Computational results – model without CCBB
Ratio of interest rate on housing loans
and MRR under break-even point
MRR (%)
80
70
60
50
40
30
20
9,
92
9,
68
9,
46
9,
15
8,
8
8,
63
8,
33
8,
28
7,
86
7,
58
7,
56
7,
43
7,
12
6,
7
6,
27
5,
81
5,
33
5,
17
4,
93
4,
12
10
interest rate (%)
Computational results – model without CCBB
Ratio of interest rate on other loans to households
and MRR under break-even point
MRR (%)
80
70
60
50
40
30
20
7,
62
7,
99
8,
29
8,
54
8,
64
8,
92
9,
17
9,
48
9,
75
10
,0
8
10
,2
4
10
,4
8
10
,5
3
10
,7
7
11
,1
6
11
,3
7
11
,9
3
12
,6
9
13
,2
1
14
,8
5
16
,0
6
10
interest rate (%)
Computational results – model without CCBB
Ratio of interest rate on loans to enterprises
and MRR under break-even point
MRR (%)
80
70
60
50
40
30
20
5,
63
6,
27
6,
86
7,
39
7,
56
7,
62
8,
09
8,
33
8,
64
9,
05
9,
46
9,
75
10
,1
2
10
,4
7
10
,5
3
11
,0
3
11
,3
6
11
,9
3
12
,7
13
,9
3
16
,0
6
10
interest rate (%)
Computational results – model without CCBB
MRR (%)
80
housing loans
70
60
other loans to
households
50
40
loans to
enterprises
30
20
10
4
6
8
10
12
14
16
18
interest rate (%)
Computational results – model with CCBB
Ratio of interest rate on housing loans and MRR under break-even point
MRR (%)
80
70
60
50
40
30
20
9,
68
9,
46
9,
15
8,
8
8,
63
8,
33
8,
28
7,
86
7,
58
7,
56
7,
43
7,
12
6,
7
6,
27
5,
81
5,
33
5,
17
4,
93
4,
12
10
interest rate (%)
Computational results – model with CCBB
Ratio of interest rate on other loans to households
and MRR under break-even point
MRR (%)
80
70
60
50
40
30
20
7,
62
7,
99
8,
29
8,
54
8,
64
8,
92
9,
17
9,
48
9,
75
10
,0
8
10
,2
4
10
,4
8
10
,5
3
10
,7
7
11
,1
6
11
,3
7
11
,9
3
12
,6
9
13
,2
1
14
,8
5
16
,0
6
10
interest rate (%)
Computational results – model with CCBB
Ratio of interest rate on loans to enterprises
and MRR under break-even point
MRR (%)
80
70
60
50
40
30
15,2
13,3
12,7
12,2
11,8
11,3
11
10,7
10,5
10,3
10,1
9,85
9,53
9,34
9,05
8,74
8,58
8,31
8,09
7,76
7,56
7,49
7,39
6,87
6,61
6,14
10
5,63
20
interest rate (%)
Computational results – model with CCBB
MRR (%)
80
70
housing loans
60
50
other loans to
households
40
loans to
enterprises
30
20
10
4
6
8
10
12
14
16
18
interest rate (%)
Computational results model without CCBB
comparison of models
model with CCBB
Housing
Other loans to
households
Enterprises
loans
min
18,00
57,00
42,00
max
65,00
80,00
avg
44,62
67,71
Housing
loans
Other loans to
households
Enterprises
loans
min
18,00
44,00
42,00
78,00
max
65,00
76,00
78,00
62,29
avg
44,23
60,65
61,97

MRR in margin between 10 and 80% have effect on all banks and all types of their
loans

only 8 banks don’t have housing loans

higher effect on housing and enterprises loans, and lower effect on other loans to
households in first model

almost same effect on all type of loans in second model
Conclusion

according to our numerical analysis the rate of marginal reserve
requirements of 55% is an average rate on which banks stop
profiting on extending credits to the households, and that is exactly
the rate approved by the CNB’s decision

based on the results which set the marginal reserve requirements
rate of 40% as a rate which starts being unprofitable for banks to
extend households credits, we can see why formerly prescribed
marginal reserve requirement rates weren’t efficient in stopping the
external debt growth
Further research

looking into the possibility of introducing some new measures on
extending the credits to the households

looking for the possibility of introducing variable MRR which would
depend on foreign debt changes and the changes in the credits to
the households
-> heuristic based on tabu search
Further research
Heuristic based on tabu search:

trade off between decreasing the foreign loans’ increase (MRR
decreases) and increasing the interest rate (demand decreases)

the rule of searching the neighbourhood: commercial bank accepts
to decrease the foreign debt increase, and the interest rate
v jilpt variable
increases in order to obtain the same profit (0-1
becomes 0)
z jlpt
interest rate increases, MRR changes (0-1 variable
changes)
