AN INTERTEMPORAL LINEAR PROGRAMMING MODEL WITH DETERMINISTIC
LOAN-DEPOSIT RELATIONSHIPS FOR OPTIMAL CREDIT EXPANSION STRATEGY
IN A BANK


Veikko Jskelinen*, Timo Salmi** , Jaakko Hara***


Originally working paper 76 - 7            March 1976
European Institute for Advanced Studies in Management

Rewritten as WWW publication http://uwasa.fi/~ts/cesb/ May 1996
FTP version ftp://garbo.uwasa.fi/pc/research/tscesb10.zip
University of Vaasa, Finland

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

Presented at the E.I.A.S.M. Workshop on Money and Banking Theory,
February 3, 1978


                     ALL RIGHTS RESERVED



ABSTRACT

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.


1 INTRODUCTION

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 Bank_5/ 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 EXPOSITION OF THE MODEL

2.1 PRELIMINARIES

Given the planning situation it is our aim to assess for a simulated
Finnish bank the optimal strategy for credit expansion_9/ 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

The deterministic yield and interest rates are as follows for both
periods

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

Note that 6% < 10% < 11% < 12%.  Furthermore, we assume that there
is an excess demand for loans.

Define the following direct decision variables:

LOAN(t) = loans granted during period t.
INVE(t) = investments made during period t.
CEBA(t) = increase_13/ in the borrowing at the Central Bank during
          period t.
CLCA(t) = Closing cash at the end of period t.

In addition, define the variables

DEPO(t) = increase in time_14/ deposits during period t._15/


2.3 OBJECTIVE FUNCTION

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.

For investments we have

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

The interest cost of deposits is calculated on the average total
deposits. The interest is paid at the end of the pertinent year._19/

We have

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

The interest cost of borrowing at the Central Bank amounts to_20/

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

Finally, the fixed expenditures (paid in cash) are 30 monetary units
for the first and 35 units for the second year. By combining the
above terms we get the objective function

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

This tedious conversion process would naturally be computerized in a
real-life application. The conversion in this paper is given for the
benefit of the reader.

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.

In order to retain the linearity of our model we utilize linear
relationships. It is assumed that the difficult problem of
estimating the parameters can be satisfactorily solved. The general
linear relationships between deposits and loans are given by

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

In the above equations the a:s delineate the autonomic growth of
deposits and the b:s indicate the effect of the adopted credit
expansion strategy.

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

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

These equations are included as constraints in our intertemporal
linear programming model.

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  

   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    189.80.

For the second subperiod we have the following constraint.

   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       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.

When the terms are rearranged, we have

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

As stated earlier, such conversions should be computerized in actual
applications.

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  40.


2.6 STATUTORY, MARKET, AND MANAGEMENT POLICY CONSTRAINTS

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        0.20x150  +  0.05(380 + DEPO1)

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

Furthermore, according to the Finnish bank law, stockholders' equity
must be at least 4% of total liabilities less cash and certain other
asset categories. This requirement is not incorporated in our model,
since we know in advance that it would be redundant. It is
interesting to note that if the excluded requirement were
efficiently binding this would indicate an imminent need for issuing
new equity, which is a recurring event in Finnish banking.

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 SOLUTION AND ANALYSIS

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

The values of the shadow prices are:

                                   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

The initial and the projected balance sheets are given below.


                         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


3.2 DISCUSSION ON THE SOLUTION

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
of_24/ 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/


4 CONCLUSION

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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FOOTNOTES

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].
