Mathematical Model for Determination of Marginal Reserve Requirement Values in Monetary Policy

1. A Mathematical Model and Decision Support System for Determination of the Values of the Marginal Reserve Requirement as Instrument of Monetary Policy Darko Pongrac

2. Outline • Introduction • Mathematical model • Heuristic • Computational results • Conclusions • Future research

3. Introduction – situation in Croatia • Croatian National Bank (CNB) - aims: • price stability • supporting economic growth • Commercial banks – aims: • making profit

4. Introduction – situation in Croatia Commercial banks • foreign ownership • indebtedness abroad with low interest rate • giving loans in Croatia with high interest rate easy profit!

5. 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

6. 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

7. Introduction – Monetary policy • CNB has a limited number of available measures and instruments for influencing commercial banks behaviour • slow measures • fast measures

8. 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.

9. Introduction – Monetary policy– commercial banks’objectives • Profit is made from different revenues that can be put into two main categories: • 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.

10. 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 includesn level

11. Mathematical model Bilevel programming model • CNB – leader: minimize the increase in household’s consumption (loans to households) • commercial banks – followers: maximize their profits Conflict!

12. 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

13. Mathematical model Commercial banks’ loans are divided in three main categories: • housing loans • loans to households • loans to enterprises

14. Mathematical model • 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

15. Mathematical model • 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

16. Mathematical model • 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 previous indebtedness; 0 otherwise xjilpt ,wjl≥ 0; zilpt ,vjilpt  {0,1}

17. Mathematical model • Notes • yjilpt - the amount that the bank repays for previous indebtedness • Wjlt - the total amount of bank’s credit in themacro period • Ujlt - the total amount of clients’ repaymentsrelated to previous credit • Qjipt - bank’s debt • Rjlt - bank’s credit

18. 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)

19. Mathematical model • Model: with constraints: i,l,t (1) j with constraints :

20. 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)

21. 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)

22. Mathematical model - difference between models

23. Heuristic • NP-hard problem (Ben-Ayed, Blair, 1989) • heuristic • nonlinear constraint (2) was relaxed in the way that the binary variablevijlptis fixed to 1 in all observed points (the real situation), and the second binary variable zilptis fixed to 1 for a chosen value of marginal reserve requirements in each observed period

24. Heuristic • Real situation: j=34, i=2, l=3, p=70, t=12 171136 0-1 variables and constraint (2) is cubic

25. 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%

26. Start Read model parameters Choose initial bank for solving Choose initial marginal/ special reserve requirement for solving Solve relaxed linear problem Choose next marginal/ special reserve requirement for solving Has the marginal/ special reserve requirement been found for all kinds of loans? No Choose the next bank for solving Yes Have all the banks been considered? No Yes Print out the calculated values for the highest, lowest and mean marginal / special reserve requirement for the banking system Stop Heuristic • What does it mean “Choose the next value for MRR”? • neighbourhood of ±5% • for a closer look, neighbourhood of ±1%

27. Heuristic What does it mean “Is the MRR found?” • jump! • Wjlt - the total amount of bank’s credit in the macro period

28. Computational results – model without CCBB

29. Computational results – model without CCBB

30. Computational results – model without CCBB

31. Computational results – model without CCBB

32. Computational results – model with CCBB

33. Computational results – model with CCBB

34. Computational results – model with CCBB

35. Computational results – model with CCBB

36. Computational results - comparisonof 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 model without CCBB model with CCBB

37. 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

38. 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

39. 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 increases in order to obtain the same profit (0-1 variable becomes 0) • interest rate increases, MRR changes(0-1 variable changes)