Application of Constraint Logic Programming to ALM modeling

Computer Science in Economics and Management 4: 107-116, 1991. 107 ~) 1991 Kluwer Academic Publishers. Printed in the Netherlands.

Application of Constraint Logic Programming to Asset and Liability Management in Banks

J O H A N M. B R O E K AI Lab., Rotterdam School of Management, Erasmus University, P.O. Box 1738, 3000 DR, Rotterdam, The Netherlands, Email: [email protected].

and

H E N N I E A.M. D A N I E L S Institute for Language Technology and Artificial Intelligence, Tilburg University, P.O. Box 90153, 5000 LE, Tilburg, The Netherlands, Email: [email protected].

(Received: 18 September 1990)

Abstract. Constraint logic programming is a relatively new and promising paradigm. In this paper it is shown that this approach yields flexible tools to support financial decision making. As an example we present an asset and liability management model implemented in the CHIP language.

Key words. Asset and liability management, constraint logic programming, risk assessment.

I. Introduction

Banks, as financial intermediaries, perform the function of improving the efficiency of financial markets. These activities create profitable opportunities, but may also introduce risks. The structure of the banks balance sheet and the fluctuations in interest rates cause risk. Interest rate risk (IRR) is defined as the risk that the level of future net interest income (NII) will be affected by movements in interest rates [8]. Several developments in the financial world forced banks to manage interest rate risk [4].

• Increased competition led to lower margins. • Financial markets and the consumers of financial services are becoming more

sophisticated. • Advances in financial theory, analytical techniques, and processing technol-

ogy, made decision support more productive, and consequently more affordable.

These factors urged banks to allocate their resources more efficiently. This has led to the creation of complex and innovative instruments such as options, interest rate swaps, futures and mortgage-backed securities [7]. In addition to the creation of new investment instruments, a whole range of risk control methodologies emerged. These control methods and tools are the instruments of asset and liability management (ALM). ALM is the management of the structure of a

108 J.M. BROEK AND H.A.M. DANIELS

bank's balance sheet in such a way that interest-related earnings, e.g. net interest income, are maximized within the overall risk preferences of the bank's management [4]. ALM can be viewed as a process consisting of three phases, the gathering of exposure data, the analysis, and the decision making. These data can be derived from many decentralized accounting systems and have to be prepared for the analysis. The analysis phase consists of two parts. Firstly, one tries to assess the current risk positions using formal models. Secondly, one devises optimal strategies for the future. The results, presented as reports or business graphs, provide some quantitative rationale and support for management to make policy decisions.

In this paper we describe an asset and liability management simulation model and its implemention in the constraint logic programming language CHIP. The underlying theory of the model was developed in collaboration with the Amro Bank. It is one of the first practical applications of constraint logic programming in the financial domain; see also [1, 6].

The outline of this paper is as follows. In Section 2 we consider several ALM modeling approaches. In the third section the principles of constraint logic programming are briefly outlined. Furthermore, the implementation of a particu- lar ALM model in CHIP is explained. The results of the simulation are presented in Section 4.

2. Asset and Liability Modeling

2.1. OVERVIEW

The purpose of an ALM model is to offer a framework for understanding the bank's earnings and risk dynamics. The ALM model should,

• provide a realistic representation of current exposures to interest rate risk and uncertainty,

• translate these exposures to several risk control accounts, and • reveal feasible asset and liability choices that alter the current exposure of the

bank toward the desired exposure.

Although an ALM model is not an accounting system, it should be able to realistically reflect the actual timing, magnitude and accounting treatment of all expense and income flows. This type of dynamic modeling requires assumptions about managerial behavior, probable loan and deposit demands and the path taken by interest rates. In a robust modeling environment sophisticated relationships may be expressed to establish the relationships between an asset's or liability's growth and other factors. Such relationships may incorporate external economic data or internal policy.

Roughly, there are three generic types of asset and liability models. The first is the maturity gap approach which is currently used by most banks. The second is the simulation approach. The third and newest is the durationga p approach. Only

CONSTRAINT LOGIC PROGRAMMING FOR BANKS 109

some of the (dis)advantages are discussed. Toevs and Haney give an extensive comparison of these models in [8].

The maturity gap approach is easy to use and the results are easily interpreted. This method makes comparison with other banks possible. However, it also has some serious drawbacks. It does not consider interest rates, and therefore describes only a small part of the IRR. The maturity gap approach only describes IRR at one point in time.

The duration gap approach does consider interest rates. Instead of the book values of assets & liabilities it uses market values. It is difficult to determine the market value of non-marketable assets & liabilities. The duration gap approach is still new and its impact on ALM in European banks is still unknown.

Simulation models employ different analytical techniques than those used in maturity gap and duration gap models. Simulations produce their results in a forward-looking context, whereas the alternatives produce their results statically [8].

There are two generic types of ALM simulation models: non-optimization models, and models that perform some kind of optimization. Models of the first type require an interest rate scenario, a starting balance, and an AL strategy as input. The output is an estimation of net interest income and other control information, such as liquidity and solvability ratios. Because these models are simple they can also be very big and produce relatively precise answers. By systematically changing parameters, such as interest rate scenarios and AL strategies, one can perform sensitivity analysis or 'what-if' analysis. This type of analysis is useful in assessing the model. It also helps the model-builder or model-user to understand more about the problem.

ALM optimization models can be subdivided into two classes:

• Models based on Markowitz's theory of portfolio selection. This theory assumes that managers utilize risk-averse utility functions. The value of an asset then depends not only on the expectation and average of its return but also on the covariance of its return with the return of all other existing and potential investments. These models concentrate on risk minimization.

• Models which maximize the future stream of profits subject to portfolio mix constraints.

In the remainder of this section we present an ALM model of the second kind that was developed at the Amro Bank (Amsterdam). Similar models are de- scribed in [5].

2.2. THE AMRO BANK MODEL

2.2.1. Parameters

Nj denotes the net interest income in period j, V 0. the volume of section i of the balance sheet in period j, and Rq the interest rate of section i in period j. The net


interest income Nj is defined as A L

Nj= E VijRij- E VijRij, i = 1 i = A + I

where i = 1 . . . . , A are assets and i = A + 1 . . . . , L are liabilities.

Sections of the balance sheet usually consists of subsections. The interest rate of a section is the weighted average of the interest rates of its subsections. We assume that the adjustment of the interest rates can be written as a convex combination of Rij_ 1 and the interest rate of the new production R~

Rii = (1 - ~i)R~j_ I + ~R~ .

This equation is known as the partial adjustment equation. The coefficient ~t can be estimated with linear regression. The interest rate of the new production is determined by the market mechanism, and can be estimated from interest indicator Rmj

R p = otiRmj -F •i'

where ~ and fli are coefficients. The changes in the volumes of different sections can be caused in two ways.

The bank can increase V~j by increasing its market-share Sii, and secondly Vq can increase because the total market Tij of section i increases (ceteris paribus). We assume that T~j are exogenous variables.

For some sections i, V~j has a lower bound (other than 0) because the redemption in one period is limited. Redemption in period j is a fraction of V~j_I and is denoted by /x i. Like the partial adjustment coefficient ~ , /~ i is assumed to be independent of time j. In the following section some of the important constraints of the problem are explained.

2.2.2. Constraints

The stability constraint prevents sharp decreases in NII. The NII in period j should not decrease more than a fraction s of NII in period j - 1.

Nj ~ sNj_I .

It is interesting to know the maximum value of s given a certain interest rate scenario because it can serve as a measure of risk.

Because most of the spreads on interest rates are positive the bank can always increase its NII by increasing the total amount of assets and liabilities. However , this may be expensive and result in a lower net income. The margin constraint states that the margin, i.e., NII divided by total assets, must always be more than a certain minimum m.

A

Nj>>-m E V~j, i = l

where i = 1 , . . . , A are assets.


The capital ratio is defined as equity divided by total liabilities. The capital ratio must always be equal to the capital ratio of the starting balance (i.e., the capital ratio in period 0).

The liquidity constraint should prevent liquidity traps. There are two sets of liquidity and solvency constraints. The Dutch Central Bank (DNB), supervises the liquidity and solvency positions of the banks. The bank itself also defines rules for liquidity and solvency. A liquidity constraint is of the form

A L

E riVl7 ~ E riVij, i = 1 i = A + I

where i = 1 , . . . , A are assets, i = A + 1 , . . . , L are liabilities, and 0~< r i~< 1. The solvency constraint is of the form,

A

Vej ~ E fiVij , i = 1

where V,j denotes the total amount of equity, i = 1 , . . . , A are assets, and 0 ~< f~ ~< 1. A liquidity constraint states that for every short term liability a certain amount of short term or marketable assets must be present. A solvency constraint states that the amount of equity must always be more than the summation of fractions of the asset volumes.

The balance sheet is always balanced. Therefore,

A L

Ev,,= E v,,, i = 1 i = A + I

where i = 1 , . . . , A are assets and i = A + 1 , . . . , L are liabilities.

2.2.3. Goal Functions

Clearly, the bank may consider different strategies or combinations of strategies. This leads to the formulation of different optimization problems in which different (combinations of) goal functions are considered. A few are mentioned below.

Following the 'greedy' strategy one tries to maximize N: in every period. First one maximizes N1, then N 2, and so on. This is an interesting strategy because to some extent it simulates the behavior of a bank without special A L regulations. Because in this case the bank does not anticipate or plan, the fluctuations in NII may be large and the stability is low. Clearly, this strategy does not lead to a maximum total net interest income.

Instead of representing stability as a constraint we can formulate it as a goal. In that case the goal is to maximize the minimum stability s. Another strategy is to maximize the minimum margin m.

It is of course also possible to try combinations of these goals. This can be done by taking weighted goal functions or by achieving goals in different orders. Example: maximize-the-margin followed by maximize-the-stability does not give the same results as maximize-the-stability followed by maximize-the-margin. Alter-

112 EM. BROEK AND H.A.M. DANIELS

nating goals and constraints is particularly simple in CHIP, as will be illustrated in the next section.

3. Implementation in CHIP

CHIP (Constraint Handling in Prolog) is a language developed at ECRC in Munich. It is a new language combining the declarative aspects of logic programming which the efficiency of constraint solving techniques. It provides three computation domains: finite domain restricted terms, Boolean terms and linear rational terms. For each of them it uses specialized constraint solving techniques: consistency techniques for finite domains, equation solving in Boolean algebra for Booleans and a symbolic simplex-like algorithm for rationals. A great number of problems occurring in different areas, such as planning, scheduling, and design have been solved using CHIP [3].

In this paragraph we will show that CHIP is a suitable tool for modeling financial problems. In the case at hand only the part for solving constraints on linear rational terms is of importance. Given a set of constraints on linear rational terms CHIP can compute whether or not these constraints have solutions or are unsatisfiable.

The meta-predicate r m a x / 1 maximizes the value of its argument, which is constrained by other clauses. Optimization problems can be formulated as constraint satisfaction problems elegantly, by using the r m a x / 1 meta-predicate. This is illustrated in the following clause representing greedy optimization

/* [Niil[Rest] is a list of Nlls: [Niil, Nii2,...,Niin] */

greedy_maximization( [NiillRest ] ) :- rmax(Niil ), gre edy_max imi z a t ion(Res t ).

This example clearly shows the declarative nature of the language. Most of the implementation is straightforward Prolog programming. Some of

the essential parts of the implementation are listed below. The top level of the program reads:

top(Scenario, Time, Total):- format_s tart ing_balance (Nii ,Volumes ,Rates), generate_cons t raint s ( Scenario, O, Time, Total, Ni i,

Volumes, Rates, Results), rmax(Total ), output_resul t s (Result s ).

The four elements in the body of the clause t o p / 3 correspond to the four parts of the program: information retrieval from the (Prolog) database, creation of ALM constraints, maximization of total net interest income and finally presenta-


tion of the results as business graphics. A set of constraints of the same type can be created in a recursive loop. For example, the following clauses,

t o p : - c r e a t e _ c o n s t r a i n t s ( 0 , 2 0 , 1 0 ) .

create_constraints(Time,Time,_). create_constraints(Time0,Time2,VO):- TimeO < Time2, Time1 is Time0 + I, Vl = < l . l * V 0 , c r e a t e _ c o n s t r a i n t s ( T i m e l , T i m e 2 , V l ) .

create the constraints:

V0<j<21, Vj~<I.IVj_ 1 , V 0= i0.

Several versions of the ALM model have been implemented in CHIP. The programs are short, modular and easy to read. In the next paragraph some results are presented.

4. Results

In the prototype ALM model six assets are five liabilities (including equity) are considered. One simulation step is equal to three months. The following table lists the balance sheet items.

Item number 1 2 3 4 5 6

Name Short term loans Current account (overdraft) Consumer loans Mortgage loans Long term loans Securities and private placements (marketable)

7 8

9 10 11

Short term time deposits Current account Savings Long term funding Equity

Because little is known about future interest rates, one has to consider several interest rate scenarios. An interest scenario describes the time-dependent values of the interest rate indicators. In this model we use the following indicators; AIBOR: Amsterdam Inter Bank Offered Rates (3 months); K: the yield on government bonds with time to maturity 5-8 years; PD: the banking rate or


discount rate; PDplus: the banking rate with a special premium; AK2: the average of AIBOR and K. By comparing the results of different scenarios the bank tries to assess the present income and risk positions. The outcomes of different simulations form the basic data to formulate strategies to alter the current exposure. As mentioned in Section 2, there exist a multitude of different goals which are interesting to consider. In general, one strives to study cases in which specific combinations of goals are stressed. In the examples discussed in some detail below the interest rate scenario changes from a normal stable pattern to a so-called inverse term structure of interest rates. In such cases following different strategies results in significant differences in future volumes of balance sheet items and NII.

The results of the optimization are presented in the form of business graphics. The graph in the lower left corner displays the interest rate scenario. The part of the graph in the upper left corner shows the time-dependent behaviour of the NIL. In the seven small boxes on the right two curves are depicted. The dotted lines are the time-dependent average interest rates of the balance sheet (BS) items. The minimum and maximum values are written on the right side of the boxes. The solid lines display the time-dependent volumes of the BS-items which maximize the goal function(s). The minimum and maximum values are written down on the left side of the boxes. The BS-items that are displayed are 1, 2, 4, 5,

#~712

t l t4

j : m c T ~ u t c m : ,

l i - | ~ , t I o . ~ ' M 4

i • 1 • t

M.J~. • ~ m . te

iiii iii I ii ii

'ot&t Ni t : , 6.26659 ] HInif~m I l t s b t l t t y : , 8.99796]

k Hinlm~m Ir~rgJn :p B . ~ | S ] eurGps3= m'elmd~lp "scmu~-io 3 ~ s t b " Rest o~ EOnd Pro f t t : , -8 ] wJrG~t3: ~ "lK:lllnarto 4 m~ltb"

murQpi3: ~ ° m r t o 5 m(stb"

: : 8.26859 Hare? ( ; ) [~ eurqpa3: w a e r ~ " m r ~ o S m~mt~"

Fig. 1. Maximization of stability and total NII.


i s

t

:Tot~T Ntt ; , S.33507 ] : b l l n l n l t ~ stab411ty :* O.g] ~Htn~mt/o margt~ :s 8 . ~ 3 3 8 ] [Ro~t of Bond I~ofl~. :~ -8 ]

: ; 6.33587 P~oro? ( ; ) [~

Fig. 2.

1 H - h i " * * J - m a

• , U 0

...... ii iii . . . . . . . . . . . . "

'* "~ ; 'a k "~

n - r e . •

T o ~ t , ~

Maximization of margin and total NII.

6, 7, and 10. BS-items 3, 8, 9, and 11 are not displayed. The small window beneath the graphics window provides additional information: minimum stability, minimum margin, the profit generated by selling securities which is accounted to periods after the last simulation step ('rest of bond profit'), and total NII.

Figure 1 shows the outcome of maximizing the minimum stability first, and maximizing total NII thereafter. In Figure 2 the results of first maximizing the minimum margin, and then maximizing total Nil are depicted. Focussing on mortgage loans (BS-item 4) and long term loans (BS-item 5) one notices an interesting difference. This can be explained as follows. In the first case the bank's agents reduce the production of mortgage loans and long term loans. This is done to decrease the mismatch between the terms of the assets and the liabilities. In the second case, on the other hand, the bank increases the volumes of the BS-items 4 and 5 because it is rigorously maximizing margin. This example clearly illustrates the tradeoff between risk and income [2].

5. Conclusions

Among others we have shown that ALM decision models can be implemented in constraint logic programming languages in an elegant and flexible way. The constraints state relations between objects directly in the intended domain of


discourse. The development times and the programs are short, due to the declarative nature of these languages. In CHIP it is possible to express constraints of different types in the same declarative way. Furthermore the constraint solvers are fully embedded in a Prolog-like language. This makes the programs easy to read.

Recent research on decision support systems and computer assisted decision making stresses the need for integration of databases, modeling tools and graphics. The constraint logic programming paradigm may serve as an overall framework for this integration.

The exploitation of these ideas in financial modeling has just started. Un- doubtedly numerous other applications will be implemented in the near future, as the proliferation of tools like CHIP in business organizations is growing.

Acknowledgements

We express our gratitude to the ALM department of the Amro Bank for their cooperation in the development of the ALM model. We also thank the logic programming group of ECRC for providing support on CHIP. This research was supported (in part) by the Netherlands Organisation for Scientific Research (NWO).

References

1. Berthier, F., Managing underlying assumptions of a financial planning model in CHIP, Technical Report TR-LP-39, ECRC, 1988.

2. Broek, J.M., Applications of constraint logic programming to asset and liability management. Internal Memo 2, ITK, 1989.

3. Dincbas, M., van Hentenryck, P., Simonis, H., Aggoun, A., Graf, T. and Berthier, F. The constraint logic programming language CHIP in Proceedings of the International Conference on Fifth Generation Computer Systems, Tokyo, 1988.

4. Huizer, M.C., The ALM Function: Development and Strategy, in J. Wilson (ed.), Managing Bank Assets and Liabilities, Euromoney Publications, 1988.

5. Kusy, M,I., and Ziemba, W.T., A bank asset and liability management model, Oper. Res., 34, 1986.

6. Lassez, C., McAloon, K. and Yap, R., Constraint logic programming and option trading, IEEE EXPERT, 1987.

7. R.B. Platt (ed.), Controlling Interest Rate Risk, Wiley, New York, 1986. 8. Toevs, A. and Haney, W., Measuring and managing interest rate risk: a guide to asset/liability

models used in banks and thrifts, in R.B. Platt (ed.), Controlling Interest Rate Risk, Wiley, New York, 1986.

Documents

Application of Constraint Logic Programming to ALM modeling