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 modeldeveloped in this work.

Introduction – Monetary policy – commercial banks’ objectives

Profit is made from different revenues that can be put into two main categories:

1. interest2. 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!

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

Mathematical model

1. Indexes i - type of indebtedness j - commercial bank l - type of investment p - marginal reserve requirements

percentage t - time period of indebtedness

(macro period) - time period of investments

(micro period) St

Mathematical model

2. Parameters op - reserve requirements percentage dlt - minimal demand for credit glt - maximal supply of credit ol - the number of repayments of credit

instalments bi - the number of repayments of

indebtedness instalments kit - interest rate of indebtedness mjlt - interest rate of investment xjil0 - bank’s indebtedness at the beginning

of the observed period Wjl0 - 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 - 1, if the percentage of marginal/special

reserve requirements is p; 0 otherwise vjilpt - 1, if bank’s indebtedness is bigger then

repayment related to the previousindebtedness; 0 otherwise

xjilpt , wjl ≥ 0; 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 previouscredit

Qjipt - bank’s debt

Rjlt - bank’s credit

Mathematical model

Expression for notes:

j,i,l,p,t (a)

j,l,t (b)

j,l,t (c)

j,i,p,t (d)

j,l,t (e)

, )(1 1

)1,max(0

t

btjilpilpjil

ijiplt

i

xzxb

y

,

tS

jljlt wW

, )(1 1

)1,max(0

t

otjljl

ljlt

l

WWo

U

, ))((3

1 10

l

t

jilpjilpjiljipt yxxQ

, )(1

0

t

jljljljlt UWWR

Mathematical model

Model:

with constraints:

i,l,t (1)

j

with constraints :

tj

ljltjlt

zUW

,2,1

)(min

,1p

ilptz

t ,

,,,)(max

pijiptit

ljltjlt

piltQkRm

Mathematical model

Model - constraints:

j,l,t (2)

j,i,l,p,t (3)

j,i,l,p,t (4)

j,i,l,p,t (5)

0

1

111,,

1,,0

))))(1(

)()100

1()(100

((

jl

t

jl

t

jl

t

pijilpjilpjilpt

t

pijilpjilpilpjilpt

ijiljlt

WWUyxv

yxzp

vxop

opW

, ilptjilpt Mzx

, )(1

jilpt

t

jilpjilp Mvyx

, )1()(1

jilpt

t

jilpjilp vMxy

Mathematical model

Model - constraints:

j,i,l,p,t(6)

j,i,l,p,t (7)

xjilpt , wjl ≥ 0; zilpt , vjilpt {0,1}, j,i,l,p,t (8)

, jltjlt dW

, jltjlt gW

Mathematical model - difference between models

t

jljl

jl

jl

t

jl

t

jl

t

pijilpjilpjilpt

i

t

pijilpjilpilpjilptjiljlt

WW

W

WWUyxv

yxzp

vxop

opW

20

0

1

1

0

1

111,,

1,,0

0,100

1max5.0

))(1(

)(100

100

6,040

100

0

1

111,,

1,,0

))(1(

)(100

100

6,040

100

jl

t

jl

t

jl

t

pijilpjilpjilpt

i

t

pijilpjilpilpjilptjiljlt

WWUyxv

yxzp

vxop

opW

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

jlptz

jilptv

Real situation: j=34, i=2, l=3, p=70, t=12

171136 0-1 variables

and constraint (2) is cubic

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%

HeuristicRead model parameters

Choose initial bank for solving

Choose initial marginal/ special reserve

requirement for solving

Solve relaxed linear problem

Has the marginal/special reserve

requirement been found for all kinds

of loans?

Yes

No

Choose next marginal/ special reserve requirement for

solving

Have all the banks been

considered?

Yes

No

Choose the next bank

for solving

Print out the calculatedvalues for the highest,

lowest and mean marginal / special reserve

requirementfor the banking system

Stop

Start

What does it mean “Choose the next value for MRR”?

neighbourhood of ±5% for a closer look,

neighbourhood of ±1%

Heuristic

What does it mean “Is the MRR found?” jump!

Wjlt - the total amount of bank’s credit in the macro period

jltjltjlt gWd

*jltjlt gWMRRMRR

*jltjlt dWMRRMRR

Ratio of interest rate on housing loans

and MRR under break-even point

10

20

30

40

50

60

70

80

4,12 4,93 5,17 5,33 5,81 6,27 6,77,12 7,43 7,56 7,58 7,86 8,28 8,33 8,63 8,8

9,15 9,46 9,68 9,92

interest rate (%)

MRR (%)

Computational results – model without CCBB

Ratio of interest rate on other loans to households

and MRR under break-even point

10

20

30

40

50

60

70

80

7,62 7,99 8,29 8,54 8,64 8,92 9,17 9,48 9,7510,0

810,2

410,4

810,5

310,7

711

,1611

,3711

,9312,6

913,2

114,8

516,0

6

interest rate (%)

MRR (%)

Computational results – model without CCBB

Ratio of interest rate on loans to enterprises

and MRR under break-even point

10

20

30

40

50

60

70

80

interest rate (%)

MRR (%)

Computational results – model without CCBB

10

20

30

40

50

60

70

80

4 6 8 10 12 14 16 18interest rate (%)

MRR (%)

housing loans

other loans tohouseholds

loans toenterprises

Computational results – model without CCBB

Ratio of interest rate on housing loans and MRR under break-even point

10

20

30

40

50

60

70

80

4,124,93

5,175,33

5,816,27 6,7

7,127,43

7,567,58

7,868,28

8,338,63 8,8

9,159,46

9,68

interest rate (%)

MRR (%)

Computational results – model with CCBB

Ratio of interest rate on other loans to households

and MRR under break-even point

10

20

30

40

50

60

70

80

7,62 7,99 8,29 8,54 8,64 8,92 9,17 9,48 9,7510,0

810,2

410,4

810,5

310,7

711

,1611

,3711

,9312,6

913,2

114,8

516,0

6

interest rate (%)

MRR (%)

Computational results – model with CCBB

Ratio of interest rate on loans to enterprises

and MRR under break-even point

10

20

30

40

50

60

70

80

5,63

6,14

6,61

6,87

7,39

7,49

7,56

7,76

8,09

8,31

8,58

8,74

9,05

9,34

9,53

9,85

10,1

10,3

10,5

10,7 11

11,3

11,8

12,2

12,7

13,3

15,2

interest rate (%)

MRR (%)

Computational results – model with CCBB

Computational results – model with CCBB

10

20

30

40

50

60

70

80

4 6 8 10 12 14 16 18interest rate (%)

MRR (%)

housing loans

other loans tohouseholds

loans toenterprises

Computational results - comparison of models

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

Housing loans

Other loans to households

Enterprises loans

min 18,00 57,00 42,00

max 65,00 80,00 78,00

avg 44,62 67,71 62,29

model without CCBB model with CCBB

Housing loans

Other loans to households

Enterprises loans

min 18,00 44,00 42,00

max 65,00 76,00 78,00

avg 44,23 60,65 61,97

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

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 increases in order to obtain the same profit (0-1 variable becomes 0)

interest rate increases, MRR changes (0-1 variable changes)

jilptv

jlptz

Further research