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FEDERAL RESERVE BANK OF RICHMOND MONTHLY REVIEW Linear Programming: A Nczu Approach To Bank Portfolio Management Corporate Financing and Liquidity;: 1968-1972 I\ V o lu m e 58 N u m b e r 11 i NOVEMBER 19 7 2 The M o n t h l y R e v i e w is produced by the Research Department of the Federal Reserve Bank of Richmond. Subscriptions are available to the public without charge. Address inquiries to Bank and Public Relations, F ed eral Reserve Bank of Richmond, P. 0 . B ox 27622, Richmond, I "iryinia 23261. Articles may be reproduced if source is given. Please provide the Bank’s Research Department with a copy of any publication in which an article is used. LINEAR PROGRAMMING: A New Approach to Bank Portfolio Management Perhaps the most important and most difficult problem facing any commercial bank’s senior manage ment on a continuing basis is asset portfolio manage ment. Portfolio decisions made at any given time directly affect a bank’s current income and profits. Moreover, current decisions may significantly influ ence income and profit flows in future periods. What makes asset selection difficult is that alternative courses of action invariably present trade-offs be tween profits, liquidity, and risk. Evaluating and weighing these factors is an inherently complex task. The problem has been compounded during recent years by the pressure on commercial banks to main tain adequate profits in the face of increased compe tition for funds both from nonbank financial institu tions and from various money market instruments. As a result of this increased pressure, the com mercial banking industry has begun to seek more sophisticated approaches to portfolio management. Management scientists are assisting the industry by devising improved decision techniques that can be understood and effectively employed by bankers.1 One technique receiving considerable attention is lin ear programming. Linear programming is a basic analytical procedure, or “ model,” employed exten sively in management science and operations research. Although the theory underlying the technique in volves advanced mathematics, the model’s structure is straightforward and can be understood by manage ment personnel having only minimal training in mathematics. The purpose of this article is to de scribe the technique in a nonmathematical manner and to indicate how it can be used in the bank port folio management process. Section I outlines two currently popular approaches to asset management and points out some of their principal deficiencies. Section II describes the linear programming model and uses a highly simplified numerical example to indicate the model’s applicability to bank portfolio decisions. Section III discusses how banks might 1 Two management scientists, Kalman J. Cohen and Frederick S. Hammer, have been instrumental in this effort. Their published work in this area, on which the present article draws extensively, is listed in the accompanying references. employ the model in practice and attempts to suggest the model’s proper role in the overall portfolio deci sion process. Section IV summarizes the technique’s advantages in banking applications and points out some of its limitations. I. CURRENT APPROACHES The typical bank’s balance sheet lists a variety of assets and liabilities. Liabilities, such as demand and savings deposits, are sources of bank funds. Assets, such as business loans, consumer installment loans, and government securities, are uses of bank funds. The essence of the asset management prob lem is the need to achieve a proper balance between ( 1 ) income, ( 2 ) adequate liquidity to meet such contingencies as unanticipated loan demand and de posit withdrawals, and (3 ) the risk of default. The problem arises because assets carrying relatively high yields, such as consumer installment loans, are gen erally less liquid and riskier than assets having rela tively low yields, such as short-term government securities. The “ Pooled-Funds” Approach D u rin g the early postw ar period, funds w ere generally available to banks in ample supply at low cost. C onse quently, m ost banks follow ed what has been termed a “ pooled-funds” approach in deciding how to allocate funds among competing assets. Under the pooled-funds concept, a bank begins its asset selection procedure by arbitrarily defining a fixed liquidity standard, usually some target ratio of re serves and secondary reserve assets to total deposits. Using this standard, the bank then allocates each dollar it attracts, from whatever source, in identical proportions among various categories of assets. A principal deficiency of this procedure is its failure to take into account variations in liquidity needs that arise from variations in the structure of liability and loan accounts .2 - The “structure” of an individual bank’s liabilities refers to the proportionate allocation of total funds among various liability categories such as demand deposits, savings deposits, and certifi cates of deposit. Similarly, the structure of a bank’s loan accounts refers to the allocation of total loans among various classes of loans. FEDERAL RESERVE B A N K OF R IC H M O N D 3 The “Asset Allocation” Technique T h e pooledfunds approach served most banks reasonably well during the late 1940’s and early 1950’s when funds were relatively plentiful and the majority of bank liabilities were noninterest-bearing demand deposits. Since that time, the financial environment in which banks operate has changed dramatically. Nonbank financial institutions, particularly savings and loan associations and mutual savings banks, began to compete vigorously with individual commercial banks for deposits during the 1950’s. In addition, cor porate treasurers, motivated by sharp increases in the yields of such money market instruments as Treasury bills and high-grade commercial paper, be gan to trim their working balances held in commer cial bank demand deposits to bare minimums. The banking industry has responded to these deposit drains by developing new sources of funds, notably negotiable certificates of deposit, commercial paper issued through affiliates, and Eurodollar borrowings. While these innovations have permitted the banking industry to grow at an adequate rate, they have proved costly, resulting in increased pressure on bank profits. Therefore, a premium has been placed on efficient bank balance sheet management. The management tool developed to meet the need for more sophisticated portfolio management was the so-called Asset Allocation technique.3 The distin guishing feature of this procedure is that it takes explicit account of a bank’s liability structure in guiding asset choice. More specifically, the Asset Allocation approach recognizes that the velocity of various types of liabilities differs systematically from one liability category to another.4 The procedure specifies that funds obtained from liabilities with rapid turnover rates (such as demand deposits) should be invested relatively heavily in assets of short maturity, and, conversely, that funds obtained from low velocity liabilities (such as certificates of deposit) should be invested relatively heavily in long-term assets. In its most extreme form, the technique divides a bank into subsystems by liability classes: for example, a “ demand deposit bank,” a “ time deposit bank,” and a “ Eurodollars bank.” Funds flowing into each of these “ banks,” that is, funds obtained from each liability source, are then allocated proportionately among alternative assets using for mulas that reflect liability velocities. For example, the demand deposit formula might specify relatively high proportions of short-term government securities and short-term business loans, while the time deposit formula might specify a relatively high proportion of mortgages. Faced with an ever-widening range of diverse sources of funds, many bank portfolio managers have adopted the Asset Allocation approach because of its explicit attention to asset-liability linkages. But while the method represents an improvement over earlier procedures, it possesses several fundamental weaknesses.5 First, velocity is a poor guide to the liquidity requirements imposed by a given class of liabilities. A far more relevant consideration is ac count stability, that is, the net daily variation of an account’s total balance. It is widely recognized that no correlation necessarily exists between stability and velocity .6 Second, the technique is arbitrary and inflexible. It is arbitrary because no clearly-defined bank goal (such as some form of constrained profit maximization) guides the determination of the vari ous fund conversion formulas. It is inflexible be cause no systematic procedure is provided for altering the formulas in the face of changing external con ditions, such as shifts in particular asset yields. 3 The Asset Allocation or “ conversion of funds” procedure was originally devised by Harold E. Zarker. See Harold E. Zarker, Conversion of Commercial Bank Funds (Cambridge, Massachusetts: Bankers Publishing Company, 1942). 4 The velocity of a given liability account is the ratio of the dollar flow within that account during some specified time period to the average stock of dollars in the account during the same period. The reciprocal of velocity is then the length of time an average dollar remains in the account. 5 For a more extensive critique, see Kalman J. Cohen and Frederick S. Hammer, ed., Analytical Methods in Banking (Homewood, Illinois: Richard D. Irwin, Inc., 1966), pp. 45-53. 8 See George R. Morrison and Richard T. Selden, Time Deposit Growth and the Employment of Bank Funds (Chicago: Association of Reserve City Bankers, 1965), p. 12. 7 A comprehensive treatment of linear programming is contained in G. Hadley, Linear Programming (Reading, Massachusetts: AddisonWesley Publishing Company, Inc., 1962). 4 Third, by compartmentalizing a bank into various subsystems, the method diverts attention from the overall goals of the bank’s operations and fails to recognize important bank activities. interactions between various The linear programming approach described below avoids these difficulties. il. THE LINEAR PROGRAM M ING MODEL: AN EXAMPLE Linear programming is a general mathematical procedure for maximizing target variables subject to constraints." The linear programming model has been extensively applied in industrial production analysis, where the objective typically is to maximize profits by achieving the proper product mix within the constraints imposed by technical production pro cedures, resource availability, and resource costs. This section presents a simple numerical example designed to illustrate how the model can be used by bank portfolio managers. The example employs a set of graphs to assist readers unfamiliar with the M O N TH LY REVIEW, NOVEMBER 1972 model in grasping the essence of the technique’s sub stantive content. W hile graphs are a useful explan atory device, their employment restricts the scope of the illustration. Consequently, the example is a necessarily artificial and unrealistic representation of the actual portfolio decision process. Nonetheless, the illustration conveys the flavor of the technique and demonstrates its applicability to bank balance sheet decisions. Consider a hypothetical bank that holds two classes of liabilities, demand deposits (D D ) and time de posits ( T D ) , and that can choose between two classes of assets, loans ( L ) and securities ( S ) . Hence, the bank’s balance sheet takes the following fo r m : Assets L S Total Funds Constraint A s indicated above, the bank has $100 million to allocate between loans and securities. Consequently, the sum of its loan and securities balances cannot exceed $100 million. This constraint can be expressed mathematically a s : L + S < 100 million where the symbol < means “ less than or equal to .” 8 Chart 1 depicts this restriction graphically. Any point on the graph represents some combination of loans and securities. For example, point X corre sponds to a loan balance of $60 million and a securi8 The opposite symbol > TOTAL FUNDS CONSTRAINT Securities ($ Millions) Liabilities DD TD Capital Accounts Assume that the rate of return on loans is 10 percent during some relevant decision period, but that no loan matures and no loan can be sold during the period. Assume further that securities yield 5 per cent during the period and can be liquidated at any time without the risk of capital loss. Total funds available to the bank are fixed at, say, $100 million, distributed as follow s: $45 million in demand deposit accounts, $45 million in time deposit accounts, and $10 million in capital and surplus. Finally, assume for illustrative simplicity that the bank incurs no costs in attracting and maintaining deposits. The bank would like to select an asset portfolio that maximizes its total return over the period. If this were all that were involved, the optimal asset selection decision would be obvious: channel all avail able funds into loans, the asset yielding the higher return. The bank recognizes, however, certain con straints upon its actions. In reality, the constraints are numerous. The present example will consider three. (1 ) C h a rt I means “greater than or equal to.” Loans ($ Millions) ties balance of $70 million. The diagonal line A A ' (the graphical representation of the equation L S = 100 million) is the locus of points at which loans and securities total $100 million. A t point Y , for example, the loan balance is $50 million, the securities balance is $50 million, and total assets are therefore $100 million. A t any point above and to the right of line A A ', such as X , total assets exceed $100 million. A t any point below and to the left of A A 7, such as Z, total assets are less than $100 million. The total funds constraint requires that the point representing the bank’s asset portfolio either fall on A A ' or within the shaded region below and to the left of A A '.9 Liquidity Constraint T h e bank recognizes that, because loans cannot be liquidated during the time period under consideration, some quantity of negoti able securities must be held to meet unanticipated deposit withdrawals. Therefore, the bank makes it a rule always to maintain some minimum ratio of securities to total assets. Assume that, with $45 million of demand deposits and $45 million of time deposits, the bank always maintains a securities balance equal to or greater than 25 percent of total 9 Strictly, with total funds equal to $100 million, the balance sheet identity requires that L + S equal exactly $100 million, that is, that the point representing the bank’s asset portfolio fall on line A A 1. For the purpose of illustrating the linear programming technique, it is helpful to treat the constraint as an inequality rather than an equality. This deviation will not affect the example’s solution. FEDERAL RESERVE B A N K OF R IC H M O N D 5 assets. The mathematical expression for this con straint i s : (2 ) S > .2 5 (L + S ), or, equivalently and more conveniently, a s : (3 ) S > .3 3 (L ). Constraint (3 ) is depicted graphically by Chart 2. It requires that the bank’s asset portfolio fall on line OB or at some point in the shaded region above the line. On the presumption that time deposits are gen erally more stable than demand deposits, the bank’s management varies its liquidity ratio inversely with shifts in the ratio of time to total deposits. Hence, an increase in the ratio would cause line OB to rotate downward, thereby enlarging the shaded area of ac ceptable portfolio. Conversely, a reduction in the ratio would rotate OB upward, diminishing the area of acceptable portfolios. The effects of such shifts will be considered below. Loan Balance Constraint Because the bank co n siders lending its most important activity, it imposes certain restrictions on its loan balance. Specifically, the bank attempts to satisfy all of the requests for loans submitted by its principal customers. Assume that the aggregate demand of these customers totals $30 million during the period. This constraint is depicted by Chart 3. The restriction requires the bank’s portfolio to fall on or to the right of line CC'. The mathematical statement of the constraint i s : (4 ) L > The Feasible Region T h e three constraints just outlined are all relevant when the bank’s management meets to allocate available funds between loans and securities. Chart 4 shows how the constraints taken as a group restrict the bank’s range of choice. Any asset portfolio represented by a point outside the shaded region E F G violates one or more of the con straints. Conversely, any portfolio represented by a point within or on one of the boundaries of this region satisfies all of the constraints. Therefore, the portfolio selected must lie within the region or on one of its boundaries. For this reason, the area is called the “ feasible region.” The Objective Function T h e reader will recall the assumption that loans yield 10 percent and se curities 5 percent during the relevant time period. Consequently, the bank’s total income during the period equals 10 percent of its loan balance plus 5 percent of its securities balance.10 Mathematically: (5 ) Income = .1 0 (L ) -f- ,0 5 (S ). 10 For simplicity, the possibility of loan default is ignored. Chart 3 Chart 2 LIQUIDITY CONSTRAINT Securities ($ Millions) 30 million. LOAN BALANCE CONSTRAINTS Securities ($ Millions) 100 80 60 40 20 20 40 60 Loans ($ Millions) 6 M O N TH LY REVIEW, NOVEMBER 1972 80 100 Expression 5 is called the objective function of the linear programming problem. Chart 5 depicts the “ family” of objective functions represented by equa tion 5. Each member of the family, that is, each of the parallel lines on the graph, corresponds to some unique income level. On the graph, the line closest to point O corresponds to income of $1 million, the middle line to income of $3 million, and the outer most line to income of $5 million .11 Hence, the bank’s income increases as the objective function shifts upward and to the right. The Optimal Asset Portfolio Chart 4 along with several members of equation 5’s family of objective functions. From what has been said, it should be clear that the bank can find its income-maximizing portfolio by pushing the objective function outward as far as possible without going beyond the point where some part of the function lies within the feasible region. Clearly, the income-maxi mizing objective function in this case is line N N '. This line barely touches the feasible region at point G. Any line to the right of N N ', such as P P ', lies entirely outside of the feasible region. Lines to the left of N N ', such as M M ', may contain points within A ll of the elements the feasible region but correspond to income levels relevant to the bank’s portfolio decision have now less than that represented by N N '. The solution to been developed. the problem is given by point G. The bank can The linear programming problem is summarized by the following mathematical state maximize its income, while observing all constraints, ment : by choosing the combination of loans and securities ( 6) Maximize income = .1 0 (L ) + .05( S ) Subject to: L + S < S > L > represented by point G : that is, by allocating $75 million to loans and $25 million to securities.12 This portfolio would yield $8.75 million of income during the period. The linear programming model has pro 100 million .33 ( L ) 30 million. vided the bank an objective procedure for deter mining its optimal portfolio. The model has taken by Chart 6 , which reproduces the feasible region of explicit and simultaneous account of the various fac tors assumed relevant to the decision. 11 The reader can easily confirm that any point on one of these lines represents a portfolio that yields the designated income. 12 For simplicity, the solution values are rounded to the nearest million. C h a rt 4 C h a rt 5 The solution to the problem is depicted graphically THE OBJECTIVE FUNCTION THE FEASIBLE REGION S e c u ritie s ($ M illio n s ) S e c u ritie s ($ M illio n s ) L oans ($ M illio n s ) FEDERAL RESERVE B AN K OF R IC H M O N D 7 Analytical Uses of the Model T he linear p ro gramming model can perform a number of useful analytical tasks for the bank in addition to suggesting reasonable approximations to income-maximizing portfolios. In particular, the model can specify how the bank’s optimal portfolio changes when one of the constraints changes. Through analysis of this sort, the bank can determine the costs, in terms of fore gone income, of the various constraints under which it operates. Knowledge of these costs, in turn, can assist the bank in such diverse tasks as deciding how much interest to pay depositors, determining the rate of return on capital, and deciding whether to borrow or lend in the Federal funds market. A simple ex tension of the above example will serve to illustrate. It will be recalled that the bank’s deposits total $90 m illion: $45 million of demand deposits and $45 million of time deposits. Imagine that the bank gain access to an additional $10 million of time deposits. These additional time deposits affect two of the con straints in problem ( 6 ). First, the total funds con time deposits and $45 million of demand deposits, management considers a 20 percent liquidity ratio constraint adequate. Under these conditions, the re straint is eased t o : problem that results from the easing of the total (7 ) striction becom es: (8 ) S > .20( L + S ). or: (9 ) S > .25( L ) . W ith these modifications, the mathematical state ment of the bank’s problem is changed from ( 6 ) to: (10 ) Maximize income = .1 0 (L ) -f- ,0 5 (S ) Subject t o : L + S < S > L > 110 million .25( L ) 30 million. Chart 7 depicts the altered situation graphically. E FG is the feasible region of the preceding problem. E 'F 'G ' is the extended feasible region of the new funds and liquidity constraints attendant upon the L -f- S < 110 million. $10 million increase in time deposits. Point G' Second, it will be recalled that, by assumption, the represents the solution to the new problem, with the bank’s management varies the minimum ratio of se objective function in position Q Q '. curities to total assets inversely with the ratio of time G', the bank’s new income-maximizing portfolio con to total deposits. tains $88 million of loans and $22 million of securi- Assume that, with $55 million of C h a rt 7 C h a rt 6 THE OPTIMAL ASSET PORTFOLIO S e c u ritie s ($ M illio n s ) SOLUTION TO PROBLEM (10) S e c u ritie s ($ M illio n s ) L oans ($ M illio n s ) 8 FRASER Digitized for As indicated by MO N TH LY REVIEW, NOVEMBER 1972 ties. Since yields have not changed, the bank’s in come is now $9.9 million. The solutions to problems ( 6 ) and (1 0 ) can assist the bank in determining how much to pay depositors for the $10 million increment of time deposits. Com paring incomes in the two cases, it is clear that the additional deposits produce $1.15 million of addi tional income ($9.9 million — $8.75 m illion), or $.115 per additional time deposit dollar. Conse quently, the bank can afford to pay up to a rate of 11.5 percent for each additional time deposit dollar .13 A t first glance, management might consider it ridicu lous to contemplate incurring additional deposit costs at a rate that exceeds the available return on either loans or securities. The reason it is profitable to do so is that the additional time deposits have both a direct and a secondary effect on the bank’s income. The direct effect in this case is the additional income resulting from the investment of the extra funds. The secondary effect is the additional income gener ated by the reallocation of the bank’s original $100 million of funds to a higher proportion of loans made possible by the eased liquidity constraint. The linear programming technique takes account of such sec ondary effects automatically. This illustration demon strates the potential usefulness of the comprehensive decision framework that characterizes the model .14 used in the example.) T o exploit the model fully, a bank should define as many asset decision variables as there are assets of significantly different yield, liquid ity, and risk in its portfolio. The model is capable of handling any number of decision variables. Problems containing more than two or three variables cannot be solved using graphs. Several standardized solu tion procedures (known as algorithms) exist, how ever, for solving large problems .15 In addition to handling as many decision variables as necessary, the linear programming model can ac commodate as many constraints as bank managers consider relevant to the portfolio decision process. Specifically, detailed and realistic sets of liquidity constraints can be built into the model reflecting lia bility and capital structures, cash flow patterns, sea sonal fluctuations in loan demand, and miscellaneous restrictions imposed by management on the basis of experience .10 A variety of other constraints are con ceivable, taking account of such operating factors as legal reserve requirements, corresponding balances, and the use of certain assets as collateral to support government deposits. Dynamic Considerations tration was static. The Section II illus That is, the bank’s decision proc ess was cast in terms of a single time period. Actual portfolio management is anything but static, and no III. APPLYING THE MODEL IN PRACTICE rational portfolio manager can confine his attention The example presented in the preceding section myopically to the present. For example, current has conveyed something of the flavor of the linear portfolios should provide adequate liquidity to ac programming technique. commodate anticipated future loan demand. This section builds on the As a example to describe more fully how the model might second example, loan decisions in the current period be applied to portfolio management in practice. may affect deposit levels in future periods. The One of section concludes with a few remarks regarding actual use of the technique at one large commercial the distinct advantages of the linear programming bank. linkages explicitly. Decision Variables and Constraints T he example developed above considered only two decision vari ables : that is, only two variables over which the bank had direct control. These were the bank’s loan and securities balances. In reality, of course, bank bal ance sheets break assets down into far more detailed categories. (They also show a much wider variety of liabilities than the twofold deposit classification 13 It should be emphasized that this conclusion applies only to addi tional time deposits, not to deposits already held. A bank could pay a higher rate for additional deposits by, for example, issuing a new certificate of deposit. u In actual linear programming applications, questions of the sort just discussed are analyzed in a more sophisticated manner, using the so-called “ dual” linear program. For an elementary treatment of duality in linear programming, see William J. Baumol, Economic Theory and Operations Analysis, 2nd ed. (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1965), pp. 103-28. framework is its capacity to treat such inter-temporal In portfolio decision applications, the model can be designed in such a way that it takes account of anticipated future conditions and generates optimal portfolios for several future periods as well as for the current period. The reader should not infer that management would, at some point, use the model to suggest desirable portfolios for, say, the next five quarters, and then slavishly follow the prescrip tions for each quarter as time passes. Obviously, the model should be updated and solved again as fore 15 The most widely used algorithm is the so-called “simplex” method. See Baumol, op. cit., pp. 82-97. 18 In their pioneering application of the linear programming method to bank portfolio management, Chambers and Charnes developed a detailed system of capital adequacy-liquidity constraints using some of the bank examination criteria employed by the Federal Reserve System. See D. Chambers and A. Charnes, “ Inter-Temporal Analysis and Optimization of Bank Portfolios,” Management Science, 7 (July 1961), 393-410. FEDERAL RESERVE B AN K OF R IC HM O ND 9 casts are superseded by knowledge of actual events. Rather, the value of explicit attention to the future lies in the resulting clarification of the factors rele vant to current decisions. Bank Goals It was assumed in the illustration that the banks objective was to maximize gross in come during the single time period considered. O b viously, actual banks must define more refined ob jectives. First, deposit interest and other operating expenses have to be considered. In the terminology of the model, the variable maximized should be net income in some form. The model can easily meet this requirement by treating bank expenses as negative increments in the objective function. Second, if, as suggested earlier, a multiperiod time framework is employed, management must select a means of dis counting future income to present value equivalents. A number of alternative procedures are available, any of which can be explicitly incorporated in the model.17 The model cannot itself select an objective; however, the model forces management to define some oper ating goal. Moreover, the model is structured in such a way that each specific portfolio decision has a definite quantitative effect on the goal variable and can be evaluated on this basis. Use of the Model at Bankers Trust Company During the 1960’s, a group of management scientists developed a linear programming model at Bankers Trust Company in New York to assist that bank’s management in reaching portfolio decisions .18 The model is quite detailed. It employs a multiperiod decision framework, a large number of balance sheet categories as decision variables, and numerous con straints of the type described above. Perhaps the most interesting aspect of the Bankers Trust experiment is the role played by the model in the overall decision process. The model has not served in any sense as a substitute for the judgment of management. Rather, its principal function has been to clarify the consequences of alternative deci sions. An excellent example is provided by manage ment’s use of the model to analyze liquidity ratio constraints. When the consulting team initially formulated the model, they included no constraint on the ratio of government securities to total assets. The bank’s executive management was troubled by this omission. 17 For a comparative discussion of these alternatives, see Kalman J. Cohen and Frederick S. Hammer, “ Linear Programming and Opti mal Bank Asset Management Decisions,” The Journal of Finance, 22 (May 1967), 159-62. IS Kalman J. Cohen served as a principal consultant in the Bankers Trust project. The following remarks summarize his description of the model and its application. See Cohen, “ Dynamic Balance Sheet Management: A Management Science Approach,” Journal of Bank Research, 2 (Winter 1972), 11-18. 10FRASER Digitized for They feared possibly adverse consequences in the market for the bank’s stock should the Bankers Trust balance sheet show a much lower ratio than the bal ance sheets of other large New Y ork banks. In formed of this criticism, the consulting team reformu lated the model to include a minimum ratio of 16 percent. Subsequently, the scientists used the model to specify the effects on profits of small reductions in the ratio. The model indicated that quite small re ductions could increase profits significantly. Man agement wras unaware of this sensitivity. On the basis of this information, a more flexible policy was adopted. This experience demonstrates the kind of informa tive dialogue that can develop between a bank’s exec utives and a team of management scientists using a relatively sophisticated linear programming model. It is precisely in such interchanges that the model’s value to management lies. IV. CONCLUSIONS This article has described the linear programming technique and has indicated how it can be applied to bank balance sheet management decisions. A few cautionary remarks and a brief summing up are now in order. Although the linear programming model is a powerful analytical tool, it is in no sense an automatic procedure for generating optimal portfolio decisions. The complex and continually changing conditions faced by banks cannot be fully specified by a set of equations. It is unlikely that any bank will ever know, precisely and definitively, its optimal portfolio at a point in time. A t best, techniques such as linear programming can only suggest a range within which the “ best” portfolio is likely to fall. Nor is the model a substitute for the judgment of experienced portfolio managers. W hile it is unneces sary for executives to understand in detail the mathe matical theory underlying the model or its computa tional procedures, management must be directly in volved in the construction and application of any operational model. Specifically, management must define the objectives of the bank’s operations so that the model can reflect these objectives. Further, man agement must specify the constraints it considers rele vant to asset selection decisions in order that these constraints can be built into the model. Finally, management must determine the specific questions that the model is used to analyze. In short, the model does not reduce the need for managerial judgment. On the contrary, it challenges that judgment in a very comprehensive manner. MO N TH LY REVIEW, NOVEMBER 1972 W ith due attention to the proper role of the model in the decision process, it seems clear that the linear programming approach has several distinct advan tages over many alternative asset management tools, on bank profits, and (3 ) how portfolios should be adjusted when economic and financial conditions change. The application of linear programming to asset such as the Asset Allocation method described earlier. management appears to be one of the more important First, the structure of the model forces a bank’s recent developments in banking .19 management to establish a definite operational ob find the costs of constructing and operating linear jective and provides a convenient framework for con programming models prohibitive. sidering factors relevant to portfolio choice. Second, becomes widespread among larger banks, however, the model simultaneously determines each element of small banks may find themselves exposed to the pro a bank’s optimal portfolio, given the particular goals cedure through the portfolio management services and constraints specified by management. provided by correspondents. Because Small banks may If the technique Consequently, all bank of its simultaneous approach, the model automatically ers should be aware of the technique and its impli takes account of trade-offs between decisions with cations. respect to one element of the portfolio and decisions Alfred Broaddus with respect to another element of the portfolio. Third, the model provides a convenient tool for evalu ating ( 1 ) the comparative consequences of alterna tive decisions, ( 2 ) the effect of alternative constraints 19 In this regard, it should be pointed out that linear programming is only one, and by no means the most advanced, of the modern quantitative models currently being employed in private industry. It is quite possible that in the future one or several of the other techniques may prove more useful in banking applications than linear programming. REFERENCES I. General Treatments of Linear Programming Two excellent and relatively nontechnical introduc tions to linear programming a re: Baumol, W illiam J. Economic Theory and O per ations A nalysis. 2nd ed. Englewood C liffs, New Jersey: Prentice-Hall, Inc., 1965, pp. 70128. Dorfm an, Robert. “ Mathematical or ‘ Linear’ Pro gram m ing.” Am erican Economic Review , 43 (December 19 53 ), 797-825. Advanced treatments of the technique a re: Gass, Saul. Linear P rogram m ing: M ethods and Applications. New Y o rk : M cGraw-H ill Book Company, 1958. Hadley, G. Linear Programming. Reading, M assa chusetts: Addison-W esley Publishing Company, Inc., 1962. II. Applications of Linear Programming to Bank Portfolio Management Chambers, D. and A . Charnes. “ Inter-Tem poral A nalysis and Optimization of Bank Portfolios.” Management Science, 7 (July 1 9 61 ), 393-410. Cohen, Kalman J. “ Dynamic Balance Sheet M an agem ent: A Management Science Approach.” Journal o f Bank Research, 2 (W inter 1 9 7 2 ), 9-19. Cohen, Kalm an J. and Frederick S. Hammer. “ Linear Program m ing and Optimal Bank Asset Management Decisions.” The Journal o f F i nance, 22 (M ay 1 9 6 7 ), 147-65. Cohen, Kalman J., Frederick S. Hammer, and Howard M. Schneider. “ Harnessing Computers for Bank A sset Managem ent.” The Bankers Magazine, 150 (Summer 1 9 6 7 ), 72-80. FEDERAL RESERVE B A N K OF R IC H M O N D 11 CORPORATE FINANCING AND LIQUIDITY 1968-1972 The financial positions of most U. S. corporations have undergone some significant structural changes sources of funds at various times between 1968 and since 1968. Over this period, considerable fluctu corporate liquidity positions, as measured by the ratio ations have occurred in the total quantity of credit of financial assets1 to short-term liabilities, showed a market funds obtained, as well as in the composition net decline between 1966 and 1972. 1972. Largely as a result of these financing policies, New financing, net of repayments, Total credit market funds obtained by corporations by nonfinancial corporate business was as low as $33 closely reflect the overall pace of economic activity billion in 1968 and 1970 but rose to nearly $63 billion and conditions in the credit markets. in 1969 and 1972, according to Federal Reserve Flow 1969, the great business expansion of the 1960’s was of these funds. of Funds data shown in Chart 1. In 1968 and Also evident from Chart 1 are the marked shifts between permanent (i.e., equity and long-term debt) and short-term 1 Financial assets as defined here are composed of the following: demand deposits and currency, time deposits, U . S. Government securities, open-market paper, state and local obligations, consumer credit, and trade credit. C h a rt I COMPOSITION OF CORPORATE FINANCING SEASONALLY ADJUSTED A N N U A L RATES B illio n s 80 Short-Term Sources (Bank Loans, C om m e rcial P aper, T rade D ebt, O th e r) 70 □ Long-Term Sources (Stocks, Bonds, M o rtg a g e s ) 60 : 50 1 40 30 20 10 - II III 1968 IV 1969 1970 Source: Federal Reserve F lo w o f Funds—N o n fin a n c ia l C o rp o ra te Business. 2 Digitized for1 FRASER M O N TH LY REVIEW, NOVEMBER 1972 1971 1972 still underway. Demand for goods and services was rose to historically high levels in 1969. Even though high, and corporations were expanding working capi short-term rates were higher than long-term rates in tal and plant and equipment. Although a large some cases, many corporations concentrated their portion of the necessary funds came from internal borrowing in the short-term sector of the market. In sources, many corporations also obtained funds in the The composition of the this manner, many firms held down the length of their commitment to high interest costs. funds borrowed in these two markets began to change In mid-1970, when economic activity had slowed dramatically by the third quarter of 1968, when cor and credit conditions had eased, the balance of cor money and capital markets. porate treasurers increasingly relied on various short porate borrowing shifted back toward longer-term term forms of financing— such as bank loans, trade sources of funds. The impetus for this shift in finan debt, and commercial paper. cing policy was provided when the Penn Central This trend, which con tinued through the second quarter of spurred by a combination of forces. 1970, was The heavy de filed for bankruptcy in June 1970 and defaulted on its commercial paper. Immediately, lenders became mand for funds by these corporations and other eco very cautious about making short-term, unsecured nomic units to finance the prevailing economic ex loans. pansion again was putting strong upward pressure corporate treasurers paid them off with proceeds from on interest rates in 1968 after the brief downturn sales of stocks and bonds. This shift back to long experienced in 1967. Further increases in interest term sources of funds continued during the second rates occurred in 1969 when the Federal Reserve half of 1970 and all of 1971, as shown in Chart 1. was tightening credit in order to stem the rising tide The widespread slowdown in economic activity ex of inflation. perienced during 1970 is reflected by the small quan- Both short-term and long-term rates Thus, when short-term obligations matured, C h a rt 2 PERMANENT SOURCES OF FUNDS SEASONALLY ADJUSTED A N N U A L RATES $ B illio n s 70 60 C o rp o ra te Bonds a n d M o rtg a g e s n Stocks 50 40 30 20 10 -10 I II III IV 1968 1969 1970 1971 1972 Source: Federal Reserve F lo w o f Funds—N o n fin a n c ia l C o rp o ra te Business. FEDERAL RESERVE B AN K OF R IC H M O N D 13 tity of credit market funds obtained by corporations years. at that time. W ith the improvement in economic incurred during 1970 and 1971 required additional Also, the massive amounts of long-term debt activity since late 1971, total financing has increased; equity funds to achieve a more desirable capital and the mix between short-term and long-term funds structure. Although some debt financing lowers the appears to have assumed a more traditional pattern. overall cost of capital to a firm, too much debt in the W ith only a few exceptions, corporations obtained capital structure increases financial risk, which tends a steadily increasing quantity of funds from the capi to raise the cost of capital. tal markets between 1968 and 1972, as shown in Between 1968 and 1972, short-term sources of Chart 2. Equity funds, in the form of either common funds were used extensively in expansionary periods or preferred stocks, experienced net decreases in 1968 but sparingly in recessionary periods. but were used much more frequently in 1971 and 1972. improved and operations expanded for most firms A s business During 1968, the merger movement was still strong, during the latter 1960’s, increased working capital which tended to reduce the net amount of stock was needed. outstanding. Deteriorating stock prices in 1969 made least expensive (over the long run) source of finan stock sales expensive and stock repurchases com cing, corporations assumed large quantities of short paratively attractive for most firms. term debt. A s prices in the stock market rebounded in 1970, sales of equity instruments also increased. Since late 1970, corporations, as part of an overall effort to improve liquidity positions, have issued new stock at a faster rate than at any time during the last 12 14 Seeking the most readily available and After demand slackened and interest rates receded, the short-term debt was paid off with newly acquired permanent funds. The single largest source of short-term funds used during the expansionary periods was trade debt, or funds supplied by selling (primarily larger) firms to M O N TH LY REVIEW, NOVEMBER 1972 buying (primarily smaller) accounts payable. firms in the form of Unable to obtain other types of financing during tight money periods, many small firms are forced to use trade debt even though it is often extremely expensive. Since early 1972, business activity has accelerated, causing corporations to use short-term funds to fi nance rising accounts receivable and inventory. If the economy continues to expand as expected in the coming months, short-term funds will surely remain an important source of financing. These shifts in the quantity and composition of financing by corporations have significantly altered the relationship between their holdings of financial assets and short-term liabilities between 1966 and 1972. This measure, as shown in Chart 4, is some times referred to as the “ acid-test” ratio and is used to evaluate corporate liquidity. W ith the acceler ation of economic activity in 1966, and the rapid assumption of short-term debt by many corporations, the acid-test ratio fell sharply, suggesting reduced liquidity. It continued to do so until m id-1967, when corporations turned their short-term debt into long term debt. W ith the acceleration of economic activity in 1968, liquidity was reduced even further. The ratio hit a trough in m id-1970, about the same time that the Penn Central filed for bankruptcy. The resulting turmoil in the commercial paper market made both borrowers and lenders acutely aware of the impaired position of corporate liquidity. A l though corporations began repairing their weakened liquidity positions in 1970, it was not until 1971 that actual improvements were discernible. Corporate liquidity positions will probably not re turn to their 1966 levels in the near future. Con tinued economic expansion in 1973 will most likely result once again in the extensive use of short-term debt, which tends to decrease liquidity. Philip H. Davidson FEDERAL RESERVE BAN K OF R IC H M O N D 15 PUBLICATIONS OF THE FEDERAL RESERVE BANK OF RICH M O N D PERIODICALS AND SERIALS ANNUAL R E P O R T A review of the B an k ’s operations during the year along with a feature article discussing significant econom ic topics. Distributed annually in February. 1972. F IF T H D I S T R I C T F IG U R E S A compilation of economic statistics, including data on resources, in com e, em ploym ent, agriculture, m ining, business and trade, utilities, and finance. Figures on Fifth District States and Standard M etropolitan Statistical A reas are compared w ith data for the United States. Distributed biennially. 1972. M O N T H L Y R E V IE W Contains articles covering Fifth District financial and business developm ents and topics of national and international significance. Distributed m onthly. 1972. B U S IN E S S F O R E C A S T S A reference file of representative Distributed annually in February. 1972. business forecasts for the com ing year. SPECIAL STUDIES COM E W IT H M E T O T H E F. O. M. C . ! A 28-page pam phlet describing in laym an’s term s the ac tivities of the Federal Open M arket Com m ittee. T h e text w as originally prepared as an address. 1967. THE FED ER AL RESERVE T O D A Y A n 18-page booklet explaining Reserve System , the service functions, and m onetary policy. 1971. the structure of the Federal TH E FED ER AL RESERVE A T W O R K of the Federal R eserve System . A booklet discussing the structure, objectives, and functions 35 pages. 1971. IN S ID E T H E F E D E R A L R E S E R V E B A N K O F R IC H M O N D on a tour of the Federal R eserve Bank of Richm ond. functions w ith liberal use of pictures. 1971. T h is pocket-size booklet takes you It includes a brief description of the service IN S T R U M E N T S O F T H E M O N E Y M A R K E T T h is booklet, in addition to describing a num ber of short-term highly liquid instruments, also pictures in general terms the institutional arrangem ents of the markets in w hich these instruments are traded. T h e booklet begins w ith a general review of the m oney market, follow ed by a fairly detailed description of ten m oney market instruments. E m phasis throughout is on the interrelatedness of the various sectors com prising the m oney market. 96 pages. 1970. K E Y S F O R B U S IN E S S F O R E C A S T IN G A booklet containing broad statistical measures that have gained widespread recognition as key business indicators. R elates the behavior of these indicators to changes in the level of business. D escribes statistical techniques for distinguishing normal seasonal changes in business data from changes associated with cyclical m ovem en ts and underlying grow th trends. 24 pages. 1970. M E A S U R IN G P R IC E C H A N G E S : A S T U D Y O F T H E P R IC E I N D E X E S A 52 -page booklet on the nature of price indexes, written for use w ith courses in econom ics and statistics and for ref erence by econom ic analysts. T h e booklet reviews the behavior of prices from 1960 through 1970. T h is review is follow ed by a detailed discussion of the conceptual and statistical problem s associated with the design and construction of price indexes. T h e final section exam ines in detail the statistical characteristics of the Consum er Price Index, the W h o lesa le Price Index, and the G N P D eflator, and evaluates these indexes in relation to the applications that are com m on ly m ade of them. 1972. YOU AND YOUR M ONEY A 14-page, cartoon-style booklet dealing w ith the causes of inflation and deflation and som e of the available remedies. Suitable for high schools. 1954. 16 FRASER Digitized for M ONTHLY REVIEW, NOVEMBER 1972
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