A Linear Programming Model for Commercial Bank Liquidity Management

Wiley and Financial Management Association International are collaborating with JSTOR to digitize, preserve and extend access to Financial Management.

http://www.jstor.org

A Linear Programming Model for Commercial Bank Liquidity Management Author(s): Bruce D. Fielitz and Thomas A. Loeffler Source: Financial Management, Vol. 8, No. 3 (Autumn, 1979), pp. 41-50Published by: on behalf of the Wiley Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3665037Accessed: 16-03-2015 20:34 UTC

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of contentin a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship.For more information about JSTOR, please contact [email protected].

This content downloaded from 193.226.34.226 on Mon, 16 Mar 2015 20:34:03 UTCAll use subject to JSTOR Terms and Conditions

http://www.jstor.org

http://www.jstor.org/action/showPublisher?publisherCode=black

http://www.jstor.org/action/showPublisher?publisherCode=fma

http://www.jstor.org/stable/3665037

http://www.jstor.org/page/info/about/policies/terms.jsp


A Linear Programming Model

for Commercial Bank Liquidity Management

Bruce D. Fielitz and Thomas A. Loeffler

Bruce D. Fielitz is Research Professor of Finance at Georgia State University. Thomas A. Loeffler is a Financial Analyst in the Office of the Comptroller of the Currency.

Introduction

The primary purpose of commercial bank liquidity management is to support other banking functions by maintaining reserves to meet unanticipated deposit withdrawals and an inventory of near cash funds to satisfy potential credit demands. Because of the substantial volume and the frequent turnover of assets and liabilities, liquidity management requires close attention to money market portfolio management as well as to the support of deposit and credit activities. The numerous constraints on such activity suggest that liquidity management may best be accomplished using mathematical programming.

This paper describes the development of a practical and usable mathematical programming model for liquidity management in a medium to large commercial bank. Although there are differences in the models, the motivation for the application discussed here comes from Komar [7].

The bank has total assets of approximately $800 million and an investment portfolio (securities and

federal funds sold) of approximately $200 million. Since the bank has had no prior experience with mathematical programming techniques, the model must be simple in order to insure that management understands and accepts it. While a simple model may abstract and oversimplify some real world complex- ities, the model discussed here can still improve liquidity management.

The model is designed to help the manager make decisions by providing a comprehensive framework within which to evaluate alternative strategies.

Development of the Model

Since the mid-1960s, there have been dramatic fluctuations in interest rates and in the availability of funds. Especially during periods of disintermediation, commercial banks have felt the need to manage liquidity more effectively.

The application of a mathematical programming

© 1979 Financial Management Association 41



FINANCIAL MANAGEMENT/AUTUMN 1979

model is in part a response to such changes in the nature of financial markets. The application is further motivated by a recognition that, even while liquidity is viewed as supporting other banking functions, it is possible to realize significant returns by actively managing liquidity for profit.

Framework of Model Development

Liquidity management is characterized by the need to achieve a formal objective in a constrained environment. Such achievement is complicated by the substantial amount of data that must be processed.

As the initial step in development of the model, senior bank management met with the project team to specify the objective of liquidity management and the operating constraints. The group included the manager responsible for liquidity management and managers representing the credit and deposit functions whose areas of responsibility depend on the performance of the liquidity management function.

After this meeting, the project team had to translate the objective and constraints into a quantitative model. This was particularly sensitive for model development, because the project team needed to quantify accurately the verbal descriptions of the function provided by the senior managers, none of whom had a management science background.

In order to select a single objective function, as required in a simplified mathematical programming formulation of the model, it was necessary to determine which criterion for liquidity management - support or profit - was more appropriate. The participants agreed that the support function should take precedence. Such a ranking of criteria can be readily accomplished in mathematical programming by desig- nating the support function as a series of constraints that must be satisfied before the profitability criterion is maximized. Thus, the model may be interpreted both conceptually and technically as maximizing profitability within the constraint of first supporting other bank functions.

There are, of course, other constraints that limit the ability of the liquidity manager to maximize profits. Accounting and regulatory restrictions, as well as risk and return preferences of the bank managers, con- strain the solution. Each restriction must be carefully delineated and quantified so that the resulting model closely parallels the activities of the liquidity manager.

Any model of the bank liquidity management function must be able to accommodate and process large amounts of data. The mathematical programming model described here facilitates systematic analysis of

the vast amount of information that confronts the liquidity manager daily. Yields on alternative liquid assets and liabilities, cash flows, and liquidity needs are analyzed systematically in the model.

Behavioral Implications

Liquidity management involves a number of specialists making independent decisions. The function may actually be performed by several individuals; one, for example, concentrating solely on CDs, another concerned with Treasury securities, and still another managing the municipal portfolio. Typically, a single individual who is responsible for the overall performance of the liquidity management function coordinates these separate functions.

A model designed to improve the efficiency of liquidity management must be able to accommodate such an environment. Almost by definition, a mathematical programming model focuses on the coordination of these activities via an interrelated set of objectives and constraints. In this particular application, use of the model not only improved coordination and cooperation among the managers, but also, for the first time, made available quantitative guidelines and objectives.

Technical Implications To implement the mathematical programming

model for liquidity management, it was necessary not only to create a computer-based model, but also to consider the interface between the model and the bank's accounting records and data bases.

First came analysis of the relative merits of batch processing and time sharing. As will be demonstrated later, the aggregation of variables favors the use of time sharing. Furthermore, the necessity for speed to make decisions in response to changing yield structures, security availability, and unexpected cash flows requires that the model be readily accessible to the user, without delays in the conversion of raw data to usable results. A conversational, time sharing program was therefore developed to facilitate data entry and to process quickly the data and solution output in usable form. (A partial example of the program's conversational format is included as an Appendix.)

To expedite the solution process, automation of data entry, processing, and retrieval is important. There was no obstacle here, because the bank already had highly-sophisticated investment portfolio data processing procedures. Without the ability to interface with bank accounting records and data bases, a li-

42



FIELITZ AND LOEFFLER/BANK LIQUIDITY MANAGEMENT

quidity management model based upon a management science approach would be very difficult to implement.

The Model The liquidity management model requires input

information reflecting money and capital market supply and demand conditions (i.e., yields, maturity, and availability of money and capital market instruments), tax rates, anticipated credit demands and deposit withdrawals, and other factors affecting commercial bank liquidity. The output provided by the model indicates the amount to be held of each type of liquid asset and liability and the highest net earnings (lowest net cost) consistent with the constraints.

The liquidity variables are assumed to be con- tinuous. Given the size of the bank's investment portfolio, this assumption appears justified and is consistent with the bank's pre-model experience. To insure that the continuity assumption is not compromised, constraints are included in the model reflecting feasible attainment levels for the liquidity variables.

The relationships among variables are assumed to be linear. While at the extreme, some of the relationships are nonlinear or, more likely, piecewise linear with step function risk premium yield structures, the model includes constraints which are designed to preserve the linearity assumption. Specifically, upper and lower limits on asset acquisition or sale are formulated so that the viability of the assumed constant risk premium yield structures is maintained.

Together, the continuity and linearity assumptions allow the use of linear programming.

The Variables The following asset categories are considered in the

model as decision variables: 1) Treasury securities, 2) Agency securities, 3) Municipal securities, 4) Project notes, 5) Federal funds, 6) Certificates of Deposit, 7) Repurchase Agreements (and reverse repurchase agreements), and 8) Federal Reserve Discount Win- dow borrowings. Other variables such as bankers' acceptances and Eurodollars can easily be incor- porated into the model if desired. Each of the variables is defined appropriately as the purchase or sale of a liquid asset (say, a Treasury bill) or as the issuance of a liquidity liability (a certificate of deposit), depending on whether the instrument is seen as a source or use of funds. In most cases, the variables are further delineated by maturity, type of issue, and other characteristics. Exhibits 1 and 2 present the variables categorized as sources or uses of funds.

Exhibit 1. Decision Variables Representing Uses of Funds*

Variable Instrument Type

xlj (= 1 * ., m,) Treasury Securities Asset x2j (j = . , m2) Agency Securities Asset X3j (= 1 * . m3) Municipal Securities Asset X4j (j= , ... m4) Project Notes Asset Xsj 0= 1 * * * m,) CD- Bank Asset x6j (j= 1, . , m6) Reverse Repurchase

Agreements Asset X7 Federal Funds Sold Asset

*Uses of funds are defined solely in terms of the purchase of liquid assets. Commercial banks cannot readily reduce their liquidity liabilities prior to maturity and, at maturity, reductions of liquidity liabilities are treated automatically by the program as exogenous adjustments to the constraints.

Exhibit 2. Decision Variables Representing Sources of Funds*

Variable Instrument Type

yk (k= 1, . . , ni) Treasury Securities Asset y2k (k= 1, ..., n) Agency Securities Asset y3k (k= 1, , n3) Municipal Securities Asset Y4k (k= 1, . . , n4) Project Notes Asset Y5k (k= 1, .., n) CD-Bank Asset y6k (k= 1, . .., n6) Repurchase Agreements Liability Y7 Federal Funds Bought Liability y8k (k= , .. ., n8) CD - Public Money Liability y9k (k= 1 .. , n9) CD - Money Market Liability Yio Discount Window Liability

Borrowings

*Sources of funds are defined as sales of liquid assets and purchases of liquidity liabilities.

In both Exhibits 1 and 2, the xij and Yik values represent dollar quantities. In the case of variables representing the purchase of a liquid asset (use of funds) or the sale of a liquidity liability (source of funds), the quantity reflects current market value. In the case of variables representing the sale of a liquid asset (source of funds), the quantity reflects book value.'

The first subscript on variables x and y refers to the type of variable as defined in the exhibits. In Exhibit 1, the subscript j indicates specific issues or options available for purchase under each general category of the variable. For example, x,j refers to the purchase of a certificate of deposit from the jth bank, while X3j

'Defining the sale of a liquid asset at book value facilitates the calculation of the associated objective function coefficient (discussed in the following section) and the formulation of the securities' gain (loss) constraint, Equation (14).

43




refers to the purchase of a certain municipal with given characteristics reflecting risk, maturity, coupon rate, etc. In Exhibit 2, the subscript indicates a specific asset available for sale or a specific liquidity liability. In both exhibits, the mi and n, values refer to the total number of issues or options available in each category.

The Objective Function

The objective function of the model is stated in terms of maximizing the after-tax profit generated from management of the liquidity variables. The objective function has the general form:

Maxz = Z. bijxj- - 2 b,yikY. (1) ij i k

Since the model described here is single-period, the coefficients of the objective function, bi(j or k) must describe the relative attractiveness of purchasing new liquid assets, selling some assets from inventory, or acquiring new liquidity liabilities, without knowing how long the instruments will be retained or what future interest rate levels will be. Thus, the current, after-tax yields to maturity (YTMs) of assets and liabilities and (using the terminology of Komar [7]) the closely-related market alternative yields (MAYs) are used as coefficients in the objective function, because they are directly comparable and because they require only current information in their calculation. The model uses management interpretation of the future direction of interest rates (as contained in the maturity structure constraints) to select those assets and liabilities that generate the highest return (lowest cost) within the constraints imposed by future term structure considerations.

The bij and bik values in the objective function vary by sign and by method of computation. For all use-of- funds variables defined in Exhibit 1, the bij values are after-tax yields to maturity. Adjustments are made to reflect differential tax treatment depending on whether the current price is at a discount, equal to, or at a premium in relation to the face value.

For the source-of-funds variables reflecting the issuance of a liability, the bik values are computed similarly. For the source-of-funds variables in Exhibit 2 that reflect the sale of assets, however, the bik values are market alternative yields or yields forgone. Again, adjustments are made depending on whether book value is at a discount, equal to, or at a premium in relation to the face value. All yield calculations are made using Fisher's [5] algorithm for finding exact rates of return. Yields are stated as annual compound rates.

The signs on the bi(j or k) values are as follows. Uses of funds (assets purchased) represent positive contributions to profit (yield received) and, therefore, the bij values are positive. Sources of funds (the sale of assets or the issuance of liabilities) represent negative contributions to profit because of yield forgone or ex- penses incurred. Therefore, the appropriate sign for the bik values is negative, as shown in Equation (1).

The yields computed for the source and use of funds variables are used as the coefficients in the objective function of the linear programming model. The model compares the coefficients economically, and the algorithm selects that combination of liquid assets and liquidity liabilities that maximizes the after-tax profit to the bank consistent with the primary function of liquidity management.2 The Constraints

Liquidity management in a commercial bank is constrained by the external environment and by the in- ternal risk and return preferences of bank management: institutional constraints and management constraints.

Institutional Constraints. Institutional constraints are those restrictions imposed for accounting, legal, regulatory, or market reasons. (Such restrictions are institutional in the sense that they apply to all commercial banks, or at least to all commercial banks in a particular state or of a particular type.3) They include 1) activity level constraints, 2) pledging constraints, and 3) the cash flow constraint.

1. Activity Level Constraints. The activity level constraints prevent the model from selling more of an asset than is held. These constraints, therefore, refer only to the assets representing sources of funds in Ex- hibit 2 and can be stated simply as:

(2) (3)

Ylk < Tyk k = 1, . . ., n Y2k < Ty2k k = 1, . . ., n2

* * *

Y5k < Tysk k = 1, ...,. n5.

2It should be noted that yield forgone from assets sold is associated with a decision-making criterion, as are the yields associated with issuance of liquidity liabilities or the acquisition of liquid assets. To calculate the actual, financial accounting value of the objective function (profit or loss), it is necessary to add to the objective function the revenues from assets held in the portfolio prior to the solution of the model. The program does this automatically. 3For instance, some state chartered banks that are not members of the Federal Reserve System may hold some of their reserves in the form of liquid assets. An institutional constraint would apply to this situation.

(6)

44




The variables are as defined in Exhibit 2 where the ni values are the total number of issues of each type of variable available for sale. The Tyik values represent the total amount (book value) of each variable held.

2. Pledging Constraints. Commercial banks are required to hold collateral for the acquisition of certain types of liabilities. Included among these liabilities are Federal Reserve discount window borrowings and, for this particular bank, public money (state and municipal time deposits). For the latter restriction, the constraint takes the form:

2X3j + 2X4j - zy3k -

2y4k -

2Y8k 2 J J k k k

(CD - Public Money) - Ty3k - Ty4k, (7)

where (CD - Public Money) is the amount of state and municipal time deposits held by the bank and not maturing during the decision horizon, and the other variables are as defined previously.

3. Cash Flow Constraint. Certain variables (sources of funds) contribute positively to cash flow, and certain variables (uses of funds) contribute negatively. Sources of cash flow include the sale of assets or the issuance of liabilities, the Yik. Uses of cash flow are represented by the purchase of assets, the xij. Following Komar [7], the cash flow constraint can be written as:

- 2 aij xij + : 2 aik Yik > C-M (8) ii ii 1J l J

where C is defined as exogenous cash flow (projected credit demand, deposit withdrawals, etc.) such that C>O represents a cash outflow and C<O represents a cash inflow. M is the net sum of maturing liquidity variables. The ai (j or k) values represent adjustments of book values to actual cash flow values. These adjustments are calculated:4

Current Price aijork) = Book Value

(Current Price - Book Value) (Tax Rate) Book Value

Reserve Requirement (%) ±

Accrued Interest (1 - Tax Rate) Book Value ( )

For each of the liquidity variables, the appropriate

4Although tax effects have no immediate, direct impact on cash flows (the impact is felt only when tax payments are actually made), the model explicitly considers tax effects on a period-to-period basis.

adjustments are made.5 Given the above description of the terms in Equa-

tion (8), let c equal the cash flow resulting from the purchase or sale of liquid assets or the issuance of liquidity liabilities. The model provides sufficient cash flow to meet obligations as long as c > C-M.

Management Constraints. Management constraints that reflect policy restrictions are peculiar to individual banks. Constraints are periodically reviewed and modified to reflect changes in risk/return preferences, projected yields, projected profits, and other policy-oriented factors. Management constraints include 1) portfolio composition constraints, 2) a liquidity capability constraint, 3) maturity constraints, and 4) a securities gain (loss) constraint.

1. Portfolio Composition Constraints. Portfolio composition constraints are restrictions preventing undue reliance on certain liquidity instruments. Such a constraint might limit the amount of funds that could be derived from a particular source, or might affect the distribution of funds among liquid assets. Each such constraint has the form:

Xij < Uij or Yik < Uik, (10)

where xij or yik represent the appropriate assets and liabilities, and each Ui (j or k) represents an upper bound on the value of that particular variable. This constraint can easily be supplemented by another constraint of the form:

xij 2 Lij or yik 2 Lik, (11) where Li (j or k) represents a lower bound beneath which the value of a variable must not fall.

Rather than restricting activity to some absolute value, percentage portfolio composition constraints may also be formulated. Such constraints may have the form:

ZX4j - 2Y4k + Ty4k J k

.X4j - 2Y4k + Ty4k + 2y3k + Ty3k J k k

P. (12)

In this illustration, project notes are limited to P percent of the total tax-exempt portfolio. Of course, many other variations of this basic formulation are possible.

Portfolio composition constraints allow recognition

5For example, consider agk the coefficient associated with the kth CD-Money Market instrument issued. When (CDs - Money Market) are issued, cash available is slightly reduced on the CD because of reserve requirements. ThUs, agk = 1-0- reserve requirement (%) +0 < 1, where the exact value of the coefficient depends on the actual reserve requirement.

45




of the relationship between the amount of funds secured from a particular source and the yield cost necessary to secure those funds. The linear programming model assumes the relationships are linear; in practice, however, many of these relationships are nonlinear or, at best, piecewise linear. In the case of federal funds, for example, any bank is theoretically confronted with an upper dollar limit on the amount of funds it can buy at a fixed rate. Beyond this limit, a further increase in federal funds carries with it a higher yield cost because of the associated higher risk premium. At the point where lenders demand a higher rate on the bank's federal funds liabilities, the original objective function coefficient no longer reflects the true cost to the bank of this source of funds. The es- tablishment of appropriate upper and lower bounds for portfolio composition can be used to preserve the assumed linear structure.

2. Liquidity Capability Constraint. To insure that the bank maintains adequate liquid assets for projected and unanticipated deposit withdrawals or credit demands, a liquidity capability constraint is included in the model. Several formulations of this constraint are possible. For example, Komar [7] suggests that changes in net liquid assets be matched dollar-for- dollar with changes in deposits. This is certainly feasible, but perhaps overly-conservative, since only a relatively small portion of each dollar of new deposits is actually committed to liquidity considerations.

In the model described here, liquidity capability is related to the assets of the bank. The constraint provides that net liquid assets should be no less than a given percentage of total assets. Management has the ability to change the amount of liquidity, based upon relevant changes in market conditions, simply by varying the proportion of assets devoted to liquidity. Obviously, the shadow price of this constraint is useful in making decisions regarding the correct amount of liquidity.

3. Maturity Constraints. To further reflect the subjective evaluation of future economic conditions and to compensate partially for the absence of a multiperiod framework, the liquidity manager may manipulate the right-hand-side values of the maturity constraints. These constraints take the form:

assets may achieve. Constraints may also be formulated that provide an absolute limit (as opposed to an average limit) for all assets in a given category. It should be apparent from the formulation of Equation (13) that a relationship exists between the maturity constraints and the term structure of interest rates. Given sufficient detail in the definition of the maturity ranges for the liquidity variables, it is possible to accommodate the manager's views of future interest rate levels relative to current levels. By manipulating the right-hand-side values, the manager may either lengthen or shorten the average maturity or absolute maturity of his portfolio. This allows him to inject a further element of his risk/return preference into the framework of the model.

4. Securities' Gain (Loss) Constraint. The securities' gain (loss) constraint is one of the general form:

Zliyi < R, (14)

where the li are gain (loss) coefficients, the y, represent sales of assets, and R is the amount of gains or losses that may be taken during the decision period. This constraint is useful in at least two ways. First, to the extent that bank managers and investors consider earnings after securities gains or losses to be important, the formulation of this constraint can contribute significantly to management of the bank's actual, after-tax earnings. Second, the linear programming model adjusts the variables in the optimal solution according to differences in objective function coefficients within a very narrow margin. Thus, to the extent that gains or losses are significant, the constraint represented in Equation (14) provides a technical im- pediment to the model making minor swaps.

Modifications

Many banks are unfamiliar with mathematical programming techniques and management science methods. To reduce the complexity of the initial application of linear programming (thereby reducing any institutional reluctance that might prevent successful implementation), a modified version of the model may be necessary.

(13) MINi < Z mij xij < MAXi, J

where the mij are maturity coefficients, the xij liquid assets, and MINi and MAXi are the minimum and maximum average maturity that the ith category of

Aggregation of Variables

The model we have described consists of several hundred variables and a similar number of constraints. For a model of more manageable propor- tions, the bank is now using an aggregated version. For example, instead of separately considering all

46




possible municipal securities that provide a source of liquidity to the bank (x3j, j = 1, . . . m,), the model groups these securities into maturity categories, cal- culating the total amount in each category. Similar aggregations and computations are also made for the other sources and uses variables. In the present structure of the model, this reduces the number of variables from several hundred to between 30 and 40.

The number of constraints in the model is also reduced, primarily through the elimination of most of the activity level constraints. Specifically, instead of one activity level constraint for each security in the portfolio, the model uses one such constraint for each group of securities. Thus, depending on the number of managerial and institutional constraints specified and the number of maturity aggregations defined, the model normally consists of between 25 and 40 constraints.

The appropriate objective function coefficients for the reduced constraint and variable matrix are determined in the following manner. The market alternative yields used to represent a given maturity aggregation and category are selected to reflect the maximum yield forgone by the bank. That is, the security with the highest market alternative yield from a particular maturity category of securities represents its category. The rationale for this selection is that, if a sale at this most attractive yield is indicated by the model, then all the securities from that category are sale candidates. On the other hand, should a sale not be suggested, then the ranging of the objective function coefficients allows the bank to determine whether or not lower yielding securities result in a modification of the solution.

Similarly, yield to maturity coefficients are based on the yields of securities from a particular category which have a maturity considered representative of that category. For example, Treasury bills might be divided into two maturity categories, 0-180 days and 181-360 days. The representative yields to maturity may be derived from Treasury bills having 90- and 270-day maturities, respectively.

All rates used are based on "asked" or "bid" prices, as appropriate, such that the yield to the bank is ad- justed for dealer commissions.

Single-Period Decision Horizon The current version of the model is specified in a

single-period framework. Multiperiod models neces- sitate inclusion of explicit forecasts of interest rate levels and cash flow conditions. Incorrect forecasts of these values often result in "optimal" decisions that,

in practice, are much worse than non-optimal ones [7]. Furthermore, substantial empirical evidence [4, 9, 10, 11, 12] suggests that money and capital markets are "efficient," and that interest rate changes cannot be forecast with the precision required by a short-term liquidity management model.

The importance of intertemporal considerations is recognized by including in the model a series of maturity constraints. These constraints permit managers to enter their subjective assessments of the future direction of interest rates. Thus, the maturity of any portion or all of the investment portfolio can be lengthened or shortened to be consistent with future expectations regarding the term structure of interest rates.

Furthermore, because of the ease with which this model may be accessed, different average and/or absolute maturity limit assumptions can be readily tested. Such simulations enable the user to capture some of the power of multiperiod models without their disadvantages. If the model is run frequently, the penalties for this somewhat myopic approach to liquidity management are likely to be small, because adjustments reflecting current market conditions and anticipated changes in market direction can be made quickly.6

Implementation At this particular bank, liquidity management is the

responsibility of the Investment Management Divi- sion. This division includes an investment manager who determines portfolio composition and trading strategies for long-term securities, a money manager who determines liquidity management policies, and analysts assigned to each of them. The division is coordinated by a senior vice president for investments and money management who reports directly to the chief operating officer of the bank.

At least once a week, the liquidity management staff meets to discuss and coordinate plans and policies for the coming week, the next four weeks, and the current quarter. Bankwide plans and policies are determined quarterly at the asset/liability senior management committee meeting. These plans and policies are designed to guide the operations of the investments division as well as the other divisions of the bank.

At the beginning of each week, the senior vice president for investments, the investment manager, and the money manager meet to review and revise, as ap- 'The authors would consider the framework presented in [1] to be an appropriate extension without undue loss of simplicity.

47




propriate, the constraints of the model. This typically involves analysis of the current liquidity position of the bank, a forecast of potential deposit withdrawals, a forecast of loan demand, and a review of expected interest rate trends. If the forecasted data are especially volatile at that time, alternative projections are generated to reflect other possible analyses. This weekly meeting provides the most critical inputs to the liquidity management model. It is here where the managers specify the parameters of the model. If more than one alternative is to be analyzed, each set of parameters is listed separately.

An analyst then enters the data through the model's interactive computer program. (A portion of the program is reproduced in the appendix.) It is designed to make data entry simple. The program is conversational and is written in familiar terms.7 The program automatically provides calculations of yields and constraints, organizing the data as required by the linear programming algorithm. It should be em- phasized that acceptance of the model has been sub- stantially enhanced because its language is familiar to the managers and analysts. One side benefit of the program is that trainees who are assigned to enter the data become acclimated to the language and concepts of investments much more rapidly than they did before.

When the analyst has completed data entry for one alternative, a solution is generated immediately at the terminal. The output is in a format that readily per- mits the managers to interpret the solution. The output includes the type of security classified by maturity group, if applicable, and the amount to be bought, sold, or issued for each security and/or group; the value of the objective function; marginal values; ranging of the objective function; and ranging of the constraints. When alternative possibilities are specified, data are entered separately for each alternative, and solutions are successively generated.

When a terminal session with the model is completed, the managers analyze the output. Solutions are first reviewed in terms of whether it is feasible to implement them. In some cases, managers may have specified constraints that generate impractical solutions. Also, the complex interactions among the constraints may cause infeasible or impractical solutions. Further, a manager may object to a particular solution because it is not the one that was an-

7After the analyst has gained sufficient experience in entering the data, the program has available an option that suppresses most of the descriptive material and prompts the user with only a mnemonic abbreviation.

ticipated or desired. The investment committee is careful not to reject or accept a solution without ex- amining the results. For example, if, by specifying a particular set of parameters, a manager is able to generate a "forced" solution, comparisons of the value of the objective function from the "forced" solution with values from alternative solutions lead to a careful appraisal of profits forgone to obtain the particular "forced" solution.

When a strategy is both feasible and practical, it is included among those considered for implementation. Each solution, based upon different possibilities, is in turn subjected to this analytical process, and all feasible and practical strategies are retained for final con- sideration.

Upon the completion of screening, strategies are compared in terms of the likelihood of the occurrence of alternative happenings, and in terms of the sensitivity of the solutions to slight deviations from the postulated environment. Although the solutions can readily be ranked according to the values of the objective function, the managers may not be equally com- fortable with each of the strategies. Subjective risk assessments affect the final choice. The strategy selected represents a tradeoff between a quantitative measure of expected return and a qualitative risk assessment.

The strategy is then reviewed in terms of the sensitivity of the objective function coefficients and the right-hand-side values of the constraints. At this stage the solution may be found to be extremely sensitive to changes in projected yields, cash flows, or other exogenous inputs. If so, the strategy may be amended or rejected and the process returned to the prior step. If the sensitivity is acceptable, that is, if the solution is not dramatically changed as a result of slight changes in the parameters, the strategy is implemented.

The solution is not meant to be a rigid prescription for action. On the contrary, it is considered to be a strategic plan to which many tactical modifications may be necessary. The actual solution of the model may not be the most important result of this process. The residual benefits from this analytical process - improvements in communication, coordination, and, very often, education - may be more important than the technical features of the model itself.

Conclusion

The benefits of constructing and implementing such a model are not easily measured. Improved communication and coordination in managing liquidity

48




and better education of the staff are not readily quan- tifiable. Nor, for that matter, can the profits generated based upon decisions aided by the model be readily compared to profits generated based upon the unaided decisions of the staff. Independent, parallel performance cannot be measured because of the interaction of the participants with the model prior to strategy selection. Clearly, it is impossible for managers to both participate in the process and to select a strategy, unaided by the model, that would be comparable. Also, comparison on the basis of an ex post simulation of the model is not relevant because the managers' decisions are influenced by model recommendations.

There are, however, indications besides its con- tinued use that the introduction of the model has been successful. Strategy selection sessions have become a regular and important part of the liquidity management process. Analysts have been able to assume responsibility for decision-making much more rapidly since the model's introduction. Finally, the increase in emphasis on profitability has caused the Investment Management Division to be classified in the bank's accounting system as a profit center rather than simply as a source of funds to support other banking functions.

References 1. Parvis Aghili, Robert- H. Cramer, and Howard E.

Thompson, "Small Bank Balance Sheet Management: Applying Two-Stage Programming Models," Journal of Banking Research (Winter 1975), pp. 246-256.

2. Kalman J. Cohen and Frederick S. Hammer, Analytical Methods in Banking, Homewood, Ill., Richard D. Irwin, Inc., Part II, "Asset Management," 1966.

3. Kalman J. Cohen and Frederick S. Hammer, "Linear Programming and Optimal Bank Asset Management Decisions," Journal of Finance (May 1967), pp. 147-168.

4. Eugene F. Fama, "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance (May 1970), pp. 383-417.

5. Lawrence Fisher, "An Algorithm for Finding Exact Rates of Return," Journal of Business (January 1966, Supplement), pp. 111-118.

6. Donald P. Jacobs et al., Financial Institutions, Homewood, Ill., Richard D. Irwin, 1966.

7. Robert I. Komar, "Developing a Liquidity Manage- ment Model," Journal of Bank Research (Spring 1971), pp. 38-53.

8. Robert W. Llewellyn, Linear Programming, New York, Holt, Rinehart and Winston, 1964.

9. Charles R. Nelson, Applied Time Series Analysis for Managerial Forecasting, San Francisco, Holden-Day, Inc., 1973.

10. Richard J. Rogalski, "Bond Yields: Trends or Random Walks?" Decision Sciences (October 1975), pp. 688-699.

11. Richard Roll, The Behavior of Interest Rates, New York, Basic Books, Inc., 1970.

12. V. Kerry Smith, and Richard G. Marcis, "A Time Series Analysis of Post-Accord Interest Rates," Journal of Finance (June 1972), pp. 589-605.

13. Harvey M. Wagner, Principles of Operations Research, Englewood Cliffs, N.J., Prentice-Hall, Inc., 1969.

EASTERN FINANCE ASSOCIATION CALL FOR PAPERS

The Eastern Finance Association will hold its annual meeting in Savannah, Georgia, April 17-19, 1980. There will be papers and discussions by academicians, by business professionals, and by govern- ment specialists on almost all aspects of domestic and international banking, finance, and investments. Plan to join us and participate.

Send a two-page abstract of your proposal on or before November 30, 1979, to Professor George C. Philippatos, Program Chairman, The Pennsylvania State University, Department of Finance, 701 Business Administration Building, University Park, PA 16802.

49



Appendix*

THIS PORTTION OF THE PROGRAI UTILIZES INFORMATION FROM BOTH THE USER AND DATA FILES 'ND COMPUTES THE NECESSARY INPUT DATA ON MANAGEiENT CONSTRAINTS PREPARATORY TO RUNNING THE LIQUIDITY MANAGEMENT LINEAR PROGRAMIMING MODEL.

CASH-FLOW CONSTRAINT.

THE CASH-FLOW CONSTRAINT WILL NOW BE CALCULATED. ONJE ADDITIONAL PIECE OF INFORMIATION IS NEEDED. IWHAT IS THE ESTIMATE OF THE TOTAL CASH-FLOWL C, REQUIR ED OF THE BANiK FOR THE COMING PERIOD EXCLUDING TRPADING VARIABLES AND MATURING LIQUIDITY VARIABLES? IF YOU ESTIMATE A NET CASH OUTFLOWJ ENTER C AS A POSITIVE NUMBER; IF YOU ESTIMATE A NET CASH INFLOW' FENTER C AS A NEGATIVE NUMBER (E.G., -1000000). (DO NOT ENTER COMMAS OR THE DOLLAR SIGN.)

ENTER C ?

-3· ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +

LIQUIDITY CAPABILITY-SIZE CONSTRAINT.

THE LIQUIDITY CAPABILITY-INVESTMENT PORTFOLIO SIZE CONJSTFAINT WILL NOT. BE COMiPUTED. THE INVESTIMENT PORTFOLIO (TREASURY, AGENCY.

UNIlCIPAL., PROJECT NOTES. F.H.A. 'S, CD'S, AND COMMER CIAL PAPER) iiUST NOT BE GREATER THAN' P PERCENT OF LOANS PLUS INVESTMENTS ( iNVESTMiENTS INCLUDE ALL ASSETS EXCEPT FED FUNJDS SOLD AND CASH- AN D- DUE- FROMI- EANKS S) .

I?HAT IS P, ENVTERED AS A DECIMAL BETWEEN 0 AND 1? ?.4

WHAT IS THE VALUE OF THE TOTAL LOAN PORTFOLIO? (DO NOT ENTER COMMAS OR THE DOLLAR SIGN.) ?

MUNICIPAL PORTFOLIO SIZE CONSTRAINT.

THE SIZE OF THE MUNICIPAL PORTFOLIO (EXCLUDING PROJECT NOTES AND F.H.A. 'S) WILL NOW BE CONSTRAINED. THE MUNICIPAL PORTFOLIO SHOULD NOT BE GREATER THAN M PERCENT OF TOTAL DEPOSITS. -WHAT IS IM ENTERED AS A DECIMAL BETWEEN 0 AND 1? ?.2

WHAT IS THE VALUE OF TOTAL DEPOSITS? (DO NOT ENTER COMM1AS OR THE DOLLAR SIGN.) ?

*Some responses have been deleted to protect the identity of the bank.



Documents

A Linear Programming Model for Commercial Bank Liquidity Management