URN:ISBN 951-683-726-3 <URL:http://lipas.uwasa.fi/~ts/cesb/cesb.html>

Veikko Jääskeläinen & Timo Salmi & Jaakko Hara

An Intertemporal Linear Programming Model with Deterministic Loan-Deposit Relationships for Optimal Credit Expansion Strategy in a Bank

CONTENTS

Abstract

1 Introduction

1.1 Review of relevant research on bank portfolio models

1.2 The present problem

1.3 Motivation for using loan-deposit relationships

2 Exposition of the model

2.1 Preliminaries

2.2 Description of the fictitious planning situation

2.3 Objective function

2.4 Loan-deposit relationships

2.5 Funds availability constraints

2.6 Statutory, market, and management policy constraints

3 Solution and analysis

3.1 Output

3.2 Discussion on the solution

3.3 Further remarks

4 Conclusion

References

Footnotes

*Helsinki School of Economics, Finland
**University of Vaasa, Finland
***Union Back of Finland

An Intertemporal Linear Programming Model with Deterministic Loan-Deposit Relationships for Optimal Credit Expansion Strategy in a Bank

This paper presents an intertemporal linear programming model for exploring optimal credit expansion strategies of a commercial bank in the framework of dynamic balance sheet management assuming that it is both technically feasible and economically relevant to establish a linear relationship between the bank's credit expansion and the deposits received by the bank induced by the credit expansion process.

The inclusion of the relationship between the credit expansion and the deposits induced thereby in the inter temporal model leads to optimal solutions which run counter to intuitive reasoning. The optimal solution may, e.g., exclude the purchase of investment securities in favor of loans to be granted even in the case where the nominal yield on securities is higher than the yield on loans. The optimal solution may also contain a variable representing the utilization of a source of funds, e.g., funds obtained from the central bank, which implies the payment of a rate of interest on these funds higher than any yield obtained on the bank's portfolio of loans and securities. Since the objective function of the model is the maximization of the difference between the total yield on the securities and loans portfolio and the total interest on the various deposits and other liabilities that the bank obtains, it would be hard to arrive at these results by intuitive reasoning. The explanation for the results obtained is the dynamic relationship between the loans granted and deposits received. This relationship seems to be particularly relevant in Finnish banking due to high market shares for the largest commercial banks.

Veikko Jääskeläinen & Timo Salmi & Jaako Hara (1976/1996), An Intertemporal Linear Programming Model with Deterministic Loan-Deposit Relationships for Optimal Credit Expansion Strategy in a Bank, published on the World Wide Web as http://www.uwasa.fi/~ts/cesb/cesb.html at the University of Vaasa, Finland.

KEYWORDS: banking, loan management, quantitative budgeting, linear programming

1.1 REVIEW OF RELEVANT RESEARCH ON BANK PORTFOLIO MODELS

Implementing mathematical programming models for planning the structure and magnitude of portfolios of bank assets and liabilities has gained notable acceptance at least in U.S. commercial banks.¹ Examples of reported applications are given by [2], [11], [14], and [30]. This is due to the increasing utilization of computers and the development of quantitative planning techniques. In the present paper we consider a deterministic intertemporal linear programming model for finding an optimal credit expansion strategy under typically Finnish conditions.

Cohen [12, p. 294] and [13, pp. 10-11] established a well-founded classification of portfolio approaches to dynamic balance sheet management into two major categories: (l) asset allocation models, and (2) intertemporal linear programming models. Asset allocation modelling, which is the more widely used, but the less satisfactory of the two, determines rules indicating which specific source of funds may be converted into which particular uses of funds [12, p. 294]. An example of an asset allocation model is given by [11].

Intertemporal linear programming models for bank dynamic balance sheet management determine (given as inputs e.g. forecasts of loan demand, deposit levels, yields and costs of various alternatives over a several-year planning horizon) the sequence of period-by-period balance sheets which will maximize the bank's net return subject to constraints on the bank's maximum exposure to risk, minimum supply of liquidity, and a host of other relevant considerations [12, p. 249]. The pioneering work was presented by Chambers and Charnes [6] in 1961 and later by Cohen and Hammer [14] in 1967, who extended the model to cover the relevant set of factors encountered in a real-life application at the Bankers Trust Company. Cohen [12, p. 314] describes this paper as follows: "It presents an intertemporal linear programming model whose decision variables relate to assets, liabilities, and capital accounts. The model incorporates constraints pertaining to risk, funds availability, management policy, and market restrictions. Intertemporal effects and the dynamics of loan-related feedback mechanisms are considered, and the relative merits of alternative criterion functions are discussed." Other examples of deterministic linear programming models are given by [3], [16], [24], [30], and [32, Ch. 5-6].

The basic approach and the problem of bank dynamic balance sheet management have been extended in various directions. One major direction has been dictated by the desire to explicitly incorporate risk and uncertainty in the banking models.² The chance-constrained approach to models designated primarily for bank balance sheet management has been developed e.g. in [8] and [9] (the latter presenting the principles only). Linear programming under uncertainty (or stochastic programming with recourse) has been applied e.g. in [2], [4], [16] (the last presenting the principles only), [17], [19] (which is rather a bond portfolio model under uncertainty), and [33]. Quadratic programming approach establishing risk-return trade-offs, in the spirit of Markowitz [29] or by other relevant techniques have also been attempted. These models relate, however, solely to decisions on the assets-side of the balance sheet, as e.g. the one period model in [21], and the intertemporal model in [10]. Furthermore, extensions have been developed, for example, to consider decisions on optimal currency arbitrage in multinational banks in a two-stage linear programming framework [34] and [22] (which is a minor extension of the former), and decisions on opening and closing bank branch offices in a long-range goal programming frame work [25].

1.2 THE PRESENT PROBLEM

In the current paper we present an intertemporal linear programming approach for finding an optimal credit expansion strategy in a bank in the framework of dynamic balance sheet management. For reasons which will be discussed in the next section, in building the model it is assumed that it is both possible and sound to assess linear relationships between the expansions in the loans made by the bank and the increases in incoming deposits. For expository purposes the model will be deliberately oversimplified.³ For instance, a high level of aggregation is consequently used and choices between different maturities are omitted.⁴ Nevertheless, the model roughly simulates a decision-making situation likely to occur in Finnish banking.

1.3 MOTIVATION FOR USING LOAN-DEPOSIT RELATIONSHIPS

It has been indicated e.g. in [14, p. 158] in 1967 that bankers, quite rationally, often make loans at contract rates which are less than the market rates of interest obtainable on alternative investment options. Furthermore, it can also be optimal for a bank to borrow at a higher rate of interest than the rate on the investment option with the highest yield, even when liquidity considerations do not force this action. Lassila [28, pp. 81-86] demonstrated in 1966 using a marginalistic approach that borrowing at high interest rates at the Central Bank⁵ in order to grant lower-interest loans can be consistent with straightforward short-term profit maximization for the bank. These phenomena simply result from the fact that the deposits received depend on the loans made, and, consequently, by borrowing at the Central Bank for credit expansion the bank generates sufficient additional deposits (which it invests in proper options) to make the borrowing profitable in spite of the high interest rate.

It has been observed, for instance, by Rossi [31, p. 142] in 1955 that the higher the market share of a bank, the greater will also be the total credit expansion resulting from a marginal unit of exogenous funds and the multiplier effect of induced deposits.⁶ Since the market share of the largest Finnish commercial banks tends to be high,⁷ incorporating a pertinent relationship in their strategic models seems particularly relevant.

Cohen and Hammer [14, p. 158] made three alternative suggestions for the incorporation of the discussed feature into intertemporal linear programming models for bank dynamic balance sheet management: (1) loan making is made a predetermined constant, (2) imputed yield rates on the loans are applied, or (3) loan related feedback mechanisms are incorporated in the intertemporal constraints in order to reflect changes in the market share as the result of the bank's relative performance in meeting its loan demand. Thore [33, pp. 126-127] presented a technique for linking uncertain future changes in deposits to drawing rights created on loans granted by using a linear relationship in his model, which is basically an asset allocation model only. As the discussion this far indicates, a modification of Cohen's and Hammer's third alternative will be adopted along the lines indicated by Thore. It has to be assumed that the difficult problem of estimating the relevant parameters of loan-deposit functions can be satisfactorily solved in actual practice, since the first two of Cohen's and Hammer's alternatives are not plausible especially in the Finnish banking environment.⁸

As will be seen, the model to be presented is a deterministic intertemporal linear programming model. The development of the deposits is, however, fundamentally a stochastic process with an underlying trend component which depends partially on the loan making policy adopted by the bank. This paper will be concluded by a suggestion of further research in this direction.

2.1 PRELIMINARIES

Given the planning situation it is our aim to assess for a simulated Finnish bank the optimal strategy for credit expansion⁹ and borrowing at the Central Bank.¹⁰ This is done by determining the optimal sequence of the bank's future balance sheet positions. For this purpose we construct a scaled-down two-period linear programming model which is to maximize undiscounted net operating earnings.¹¹ The planning horizon is restricted to two periods (years, this time) for expository convenience. This is sufficient, since extending the number of periods in the model should be straightforward.¹² Simplifications are also made in other respects in this paper. A very high level of aggregation is used, certain options open to a bank are left out, choices between different maturities are omitted, the set of the model-constraints is kept at a minimum, and a deterministic approach is adopted. For two obvious reasons these simplifications need not, however, be as severe as they may sound. First, we are involved with a strategic problem, which allows a rather crude approach. Second, and even more important, the model is easily extended along the lines which can be conveniently traced back to earlier literature, e.g. [3], [11], [14], and [32].

2.2 DESCRIPTION OF THE FICTITIOUS PLANNING SITUATION

The highly aggregated initial balance sheet is given below.

   Assets  (0 000 000 Fmk)    Liabilities  (0 000 000 Fmk)
   Cash                50     Time deposits       380
   Loans              650     Demand deposits     150
   Investments        250     Total deposits          530
   Other assets        20     Borrowing at the C.B.   120
                              Stockholders' equity    300
                              Other liabilities        20
                      ---                             ---
                      970                             970

   Cash          0%     Deposits                6%
   Loans        10%     Borrowing at the C.B.  12%
   Investments  11%

Define the following direct decision variables:

• LOAN(t) = loans granted during period t.
• INVE(t) = investments made during period t.
• CEBA(t) = increase¹³ in the borrowing at the Central Bank during period t.
• CLCA(t) = Closing cash at the end of period t.

• DEPO(t) = increase in time¹⁴ deposits during period t.¹⁵

The objective is to maximize the undiscounted sum of net operating earnings for the bank before taxes over the two-period planning horizon. The justification for using a planning horizon of two periods only was already discussed in Section 2.1. Using the sum of net operating earnings as a criterion is adopted as the most obvious, but not necessarily the only choice.¹⁶ Not discounting the pertinent net operating earnings is a feature which could be amended when the model is extended to include more periods.¹⁷ Omitting taxation from consideration is also a simplification made for convenience, since it can be justifiably argued that taxation can affect the nature of the optimal solutions in banking models of the kind under observation [12, p 301], [16, p. 43].

Practitioners often argue that profit maximization is not an operational criterion in bank planning, and contend that maximization of a bank's deposits at the end of the planning horizon should be used instead.¹⁸ Their main argument is that short-term profit maximization is not consistent with the banks' long-term objectives. For example, in their opinion, short-term profit maximization gives no motivation for credit expansion, nor for borrowing at the Central Bank for purposes other than retaining liquidity. Our numerical example shows that at least this argument is not valid.

In constructing the objective function, and the subsequent funds availability constraints, the "Cohen and Hammer [14] flow assumptions" rather than the "Chambers and Charnes [6] flow assumptions" are adopted. Thus the flows occur at constant rates during each period.

Consider first the yield on loans. It is assumed that all the loans granted are amortized in a span of five years. This simplifying assumption means that 20 per cent of the principal of the loans mature each year at a constant rate. Consequently the total interest yield on the loans granted will be

   0.1 {(650)+0.8 650+LOAN1)}/2
   + 0.1 {(0.8 650+LOAN1)+(0.6 650+0.8LOAN1+LOAN2)}/2
   = 104 + 0.14LOAN1 + 0.05LOAN2.

   0.11 {(250)+(250+INVE1)}/2
   + 0.11 {(250+INVE1)+(250+INVE1+INVE2)}/2
   = 55 + 0.165INVE1 + 0.055INVE2.

We have

   0.06 {(530)+(530+DEP01)}/2
   + 0.06 {(530+DEP01)+(530+DEPO1+DEP02)}/2
   = 63.6 + 0.09DEPO1 + 0.03DEP02.

   0.12 {(120)+(120+CEBA1)}/2
   + 0.12 {(120+CEBA1)+(120+CEBA1+CEBA2)}/2
   = 28.8 + 0.18CEBA1 + 0.06CEBA2

   max NOE = 0.14LOAN1 + 0.165INVE1 - 0.09DEPO1 - 0.18CEBA1
             + 0.05LOAN2 + 0.055INVE2 - 0.03DEP02 - 0.06CEBA2
             + 1.60.

The objective function is to be maximized subject to the constraints and equations to be considered next.

2.4 LOAN-DEPOSIT RELATIONSHIPS

For the reasons already discussed in detail in Section 1.3, we assume that it is both technically feasible and economically sound to use the relationships

   DEPO(t) = f(LOAN(t),LOAN(t-1),....,LOAN(1)); t=1,...,T.

                       t-1
   DEPO(t) = a(tau) +  sum b(t,t-tau)LOAN(t-tau); t=1,...,T.
                      tau=0

In the numerical example it is assumed that the relevant equations are

   DEPO1 = 50 + 0.4LOAN1
   DEP02 = 50 + 0.2LOAN1 + 0.4LOAN2.

The general linear relationships were given above in the most general form. In order to carry out an actual estimation of the required loan-deposit relationships, simplifying assumptions have to be made to reduce the number of the parameters to be estimated. Koyck suggested an approach for the treatment of distributed lags in investment analysis.²¹ This approach has in fact been utilized by Palda in the analogous problem of estimating the carryover effects of marketing efforts in the well-known Lydia Pinkham Case. 13 Since the estimation problem of the case resembles our own, it seems that the approach might be plausible also for our purposes. On the basis of Finnish banking experience our simulated estimates seem to be reasonable enough.

2.5 FUNDS AVAILABILITY CONSTRAINTS

It has to be required individually for each period that the uses of funds do not exceed the sources of funds. Technically this requirement can be constructed in two equivalent ways. The first is to utilize inter temporal linkages by noticing that the ending balance for a period is at the same time the beginning balance of the next. The second way, which is used in this paper, is to construct the funds availability constraints cumulatively. The flow assumptions were already given in Section 2.3, where the objective function was presented. We require for the first period that

   incremental     incremental
   loan making     investments     interest  on  deposits
   LOAN1        +  INVE1       +   0.06x530  +  0.03DEPO1

     interest on  borrowing      fixed exp.  closing cash
   + 0.12x120  +  0.06CEBA1      + 30     +     CLCA1  .le.

   init. cash amortization       interest   on   loans
   50     +    0.2x650   +  0.10(650+0.8 650)/2  + 0.05LOAN1


                                 incremental    incremental
   interest  on  investments     deposits       borrowing
   + 0.11x250  +  0.055INVE1  +  DEPO1      +     CEBA1

In order to facilitate the computer solution these terms can be rearranged to give

   0.95LOAN1  +  0.945INVE1  -  0.97DEPO1  -  0.94CEBA1
   + CLCA1  .le.  189.80.

   incremental        incremental
   loans              investments        i n t e r e s t
   LOAN1 + LOAN2  +  INVE1 + INVE2  +  0.06x530 + 0.06x530

               o n      d e p o s i t s         i n
   + 0.03DEPO1 + 0.06DEPO1 + 0.03DEP02  +  0.12x120

   t e r e s t    o n    b o r r o w i n g
   + 0.12x120 + 0.06CEBA1 + 0.12CEBA1 + 0.06CEBA2

                     closing      initial
       fixed exp.    cash         cash         a m
   +   30  +  35  +  CLCA2  .le.  50       +   0.2x650

     o r t i z a t i o n         i n t e r e s t
   + 0.2 650  + 0.2LOAN1   +   0.10(650+0.8 650)/2

                             on
   + 0.10(0.8 650+0.6 650)/2 + 0.05LOAN1 + O.1O(LOAN1

           l o a n s            i n t e r e s t
   +0.8LOAN1)/2 + 0.05LOAN2 + 0.11x250 + 0.11x250

           o n    i n v e s t m e n t s     incremental
   + 0.055INVE1 + O.11INVE1 + 0.055INVE2   +  DEPO1

                      incremental
   deposits           borrowing
   + DEP02     +      CEBA1 + CEBA2.

   0.66LOAN1 + 0.95LOAN2 + 0.835INVE1 + 0.945INVE2
   - O.91DEPO1 - 0.97DEP02 - 0.82CEBA1 - 0.94 CEBA2
   + CLCA2 .le. 311.60.

In the above constraints DEPO1 and DEP02 are defined by the loan- deposit relationships given in Section 2.4.

The borrowing potential at the Central Bank is limited by the lending policy of the Central Bank.²² The upper limit on the total increase in the borrowing at the Central Bank is assumed to be 40 monetary units for the entire planning horizon. As a result, our optimal strategy must also satisfy the constraint

   CEBA1 + CEBA2 .le. 40.

The Finnish bank law imposes minimum reserve requirements, which the bank is obliged to meet (contrary to the Federal Reserve Board examiners' criteria which are not rigid legal restrictions). These requirements are 20% of the liabilities payable on demand plus 5% of certain longer term liabilities less certain liquid assets. In our simplified model this gives rise to two statutory constraints:

   closing       demand       time deposits
   cash          deposits

   CLCA1  .ge.   0.20x150  +  0.05(380 + DEPO1)

   CLCA2  .ge.   0.20x150  +  0.05(380 + DEPO1 + DEP02)

Market constraints reflect the externally generated economic and institutional realities of the market place. In a way the loan- deposit relationships are such constraints. No other market constraints are included in this paper. (Loan demand constraints, which might seem relevant at first sight, are not needed in our model, because of the chronic Finnish situation of excess loan demand.)

Management policy constraints can be used to express the level of activities acceptable to bank's management. They refer to factors outside the model and beyond the planning horizon.²³ It is conventional to include requirements on certain balance-sheet- ratios to retain liquidity and to reduce risk. The shadow prices of the pertinent constraints are important indicators in evaluating the impact of the management's policies. We exclude, however, management policy constraints, since the two statutory constraints presented earlier are quite sufficient for our expository purposes.

Finally, it has to be required, as is usual, that all the variables of the model be non-negative.

3.1 OUTPUT

When the model is solved we obtain the following optimal values of the variables:

   LOAN1 = 385.56  LOAN2 = 573.00  incremental (new) loans
   INVE1 =   0.00  INVE2 =   0.00  incremental investments
   CEBA1 =  40.00  CEBA2 =   0.00  incremental borrowing
   CLCA1 =  59.21  CLCA2 =  77.00  closing cash
   DEPO1 = 204.22  DEP02 = 356.31  incremental deposits
              NOE = 47.96          objective function

                                   period 1     Period 2
   Loan-deposit equations          0.154        0.065
   Funds availability constraints  0.154        0.065
   Borrowing constraint                   0.018
   Reserve requirements            0.107        0.030

                         Initial   Period 1   Period 2
Assets (0 000000 Fmk)
 Cash                      50.00      59.21       77.00
 Loans                    650.00     905.56     1271.49
 Investments              250.00     250.00      250.00
 Other assets              20.00      20.00       20.00
                         -------    -------     -------
                          970.00    1234.77     1618.49

Liabilities (0 000000 Fmk)
 Total deposits           530.00     734.22     1090.53
 Borrowing at the C.B.    120.00     160.00      160.00
 Stockholders' equity     300.00     300.00      300.00
 Retained earnings           -        20.45       47.96
 Other liabilities         20.00      20.00       20.00
                         -------    -------     -------
                          970.00    1234.77     1618.49

The three consecutive balance sheets indicate the optimal two-year strategy for credit expansion, and borrowing at the Central Bank. (As is recalled only two periods were included for expository convenience). We see clearly that it is optimal for our simulated bank to expand loans in the current profit-maximizing solution. This loan expansion and borrowing strategy is optimal even though the interest rate on loans (10%) is less than the rate on investments (11%). It is also observed that it is profitable to resort to borrowing at the Central Bank although the relevant interest rate (12%) is higher than the rate of the best option available for the use of the funds (11%). The solution of the numerical example utilizes the entire borrowing capacity already during the first period in order to attain the maximum impact through allocating these funds to credit expansion. No new investments are made by the model solution, because of the more advantageous alternative of credit expansion, which is fully utilized in the absence of loan demand (and management policy) constraints.

As was stated earlier, the estimation of the parameters in the loan- deposit relationship is a difficult problem, which was ignored in the present version. Therefore it is imperative to test the sensitivity of the optimal solution to the changes in the parameters of the loan deposit relationships. A sensitivity analysis performed on these parameters indicated that the current optimal solution of our example is not unduly sensitive in this respect.

The opportunity costs given by the shadow prices are an essential part of the output in intertemporal linear programming models for bank balance sheet planning. They can be used to evaluate the incremental benefits of a marginal relaxation of any of the model's constraints. For example, the shadow prices of the funds availability constraints (0.154 and 0.065) indicate that the marginal value of one additional unit of cash available in the first period, and optimally utilized over the horizon, is 0.154 + 0.065 = 0.219. This cumulative figure corresponds to an average annual rate of²⁴ 10.4%. (In contemplating this figure remember the shortness of the planning horizon and the fact that taxation was excluded from the model.) The marginal value of an additional unit of cash available in the second period corresponds to an annual rate of 6.5%. Since the horizon is much closer in the latter case, the feedback effects are weakened.

3.3 FURTHER REMARKS

In this paper we have constructed in a simplified framework an intertemporal linear programming model for dynamic balance sheet management in order to assess an optimal credit expansion and borrowing strategy in a simulated bank. In actual applications model of the discussed kind have to be made much more detailed, as was briefly discussed in Section 2.1. Furthermore, it is a well-known fact that in applications of operations research techniques the technical model building is only a minor part of the problem solving process. The relevant external and internal data have to be gathered and processed, the solution obtained and processed, and the results implemented.²⁵

The current paper presented an intertemporal linear programming approach for a commercial bank for finding an optimal strategy of credit expansion, and borrowing at the Central Bank. The model included linear dynamic relationships between the credit expansion process and the deposits induced thereby. It was demonstrated that the nominal yield and cost rates of the optional uses and sources of funds are not sufficient as criteria for decision-making even if the objective is a straight forward maximization of the bank's net operating earnings.

The approach applied in this paper is deterministic and it is finally suggested that the approach should be extended to account for the stochastic nature of the generation of deposits together with their dependence on the credit expansion process. The yield and the cost rates could be made stochastic, too. This approach should be attempted in a multi-period framework. It is our intention to attempt this in a future paper along the lines developed in [17], [27], [33], and the current paper.

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[31] Rossi, Reino. Suomen luottojärjestelmä ja rahalaitosten luotonantokyky (= The Finnish Credit System and the Lending Capacity of the Banks). Bank of Finland Institute for Economic Research Publications Series B:17, Helsinki: K. F. Puromiehen Kirjapaino O.Y., 1955.

[32] Sunderland, Neil V. Bank Planning Models: Some quantitative methods applied to bank planning problems. Bankwirtschaftlige Forschungen Band 23. Institut fur Schweizerisches Bankwesen der Universitat Zurich. Institut fur Bankwirtschaft an der Hochschule St. Gallen. Berne/Stuttgart: Verlag Paul Haupt, 1974.

[33] Thore, Sten. "Programming Bank Reserves Under Uncertainty," The Swedish Journal of Economics, Vol. LXX, No. 3 (September 1968), 123-137.

[34] Thore, Sten. "A programming model for optimal currency arbitrage New York - London - Frankfurt," presented at the European meeting of the Econometric Society, Barcelona, September 1971.

¹ See [20, pp. 5-6] for a short review of the history of applying operations research techniques in banking problems. Nevertheless, it is still discouragingly common to approach financial decision problems in a piecemeal fashion, i.e. by considering one alternative at a time on the basis of its relative merits and dismerits by applying more or less elaborate rules of thumb.

² In their basically deterministic formulation Cohen and Hammer included risk indirectly by [14, pp. 150-151] constraining the capital adequacy ratio to an acceptable level, and by [12, p. 302] reruns of the model. As is relevantly stated in [10, p. 140], commercial banks operate under conditions of uncertainty - uncertainty about (1) future prices and return of the assets under investment consideration, and (2) future deposit levels. Relevant discussion can be found also in [13, pp. 13-14].

³ For the often recurring discussion on the justification of this procedure see e.g. [17, p. 45].

⁴ The model is constructed in a manner which allows the incorporation of the omitted details along the conventional lines presented by the relevant earlier papers.

⁵ By the "Central Bank" we mean in this paper the approximate Finnish counterpart of the Federal Reserve, in other words the Bank of Finland.

⁶ A reader who is interested in the general economic analysis of bank credit expansion is referred to [5] for further references.

⁷ The deposits at the two biggest Finnish banks amount to almost one fifth of the total deposits in Finnish banks. In the U.S.A. such a figure is uncommon even on a federal level. Furthermore, a much larger relative share of the total savings is deposited in banks in Finland than is in the U.S.A.

⁸ There is, in addition to the factors discussed earlier, an excess demand for loans in Finland which stresses the importance of intertemporal loan-deposit relationships.

⁹ As discussed, the oligopolistic situation of Finnish banking stresses the loan-deposit relationships and consequently the need for a credit expansion strategy.

¹⁰ The Finnish commercial banks normally operate at the upper limit of their borrowing possibilities from the Central Bank.

¹¹ We shall discuss the objective more fully in Section 2.3.

¹² For example, the following proposition [15, p. 75] could be useful. "While the number and lengths of the planning periods employed can be altered whenever desired, a typical example would be the use of five planning periods stretching over a four-year planning horizon. In this case, one might usefully regard the first two periods as each being three months long, the third period as six months long, the fourth period as one year long, and the fifth period as two years long."

¹³ The model excludes the possibility of reducing the level of borrowing at the Central Bank. This possibility could be easily added by redefining CEBA(t) as the change in the borrowing at the Central Bank during period t, and consequently setting CEBA(t) = CEBA(t)+ - CEBA(t)-. The same goes naturally for investments.

¹⁴ We assume, for convenience, that demand deposits remain unchanged through the horizon. Time deposits and demand deposits have been separated, because the Finnish bank law imposes different minimum reserve requirements on them.

¹⁵ DEPO(t) is defined as a variable, since in our model DEPO(t) = f(LOAN(t), LOAN(t-1),..., LOAN(1)).

¹⁶ For further relevant discussion on the choice of the objective in financial planning models see [14, pp. 159-162], [16, p. 44], and [23, pp. 678-679]. Reconsidering the question of the choice of the criterion actually might quite well be relevant.

¹⁷ It has even been demonstrated that not discounting the income stream is well-founded on certain grounds [7]. See also [1] for a critical discussion about the validity of discounting. Furthermore, this is a common approach in banking practice.

¹⁸ For a reasonably comprehensive Finnish review see [3].

¹⁹ This information is actually required to set up the funds availability constraints which are presented in a later section.

²⁰ A uniform rate of interest is assumed on the borrowing at the Central Bank. Here a rather strong simplification is made. The interest rate on this borrowing depends on the bank's total outstanding balance at the Central Bank. In Finland this interest rate increases at certain limits on the total, not the marginal borrowing. Thus the actual cost function is discontinuous at the limits. This fact makes the utilization of linear approximation inaccurate. A suitable scheme of reruns or a mixed integer linear programming approach might be needed.

²¹ For the references and a discussion see [26, pp. 128-129 and 137-139].

²² At the time of first writing this paper in 1976 this policy was particularly tight in Finland.

²³ [13, pp. 14-16] contains an instructive real-life case of the importance of applying these constraints.

²⁴ It is calculated by solving the following simple equation for p: (1+p/100)^2 = 1+0.219.

²⁵ For relevant further discussion the reader is referred to [11], [12, pp. 284-289], and [14, pp. 162-165].

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