Selected Papers l 23

MathematicalModels forFinancialManagement

By MERTON H. MILLERand DANIEL ORR

GRADUATE SCHOOL OF BUSINESS

UNIVERSITY OF CHICAGO

M ERTON H. MILLER, the Edward Eagle Brown Pro-fessor of Finance and Economics in the GraduateSchool of Business, is an outstanding scholar in theapplication of quantitative analysis to a wide varietyof management problems, particularly in finance.He co-developed the celebrated Modigliani-Millertheorems on financial policy and the cost of capital.Before joining the Graduate School of Business fac-ulty, Professor Miller had been an economist withthe U.S. Treasury Department and with the Boardof Governors of the Federal Reserve System, a lec-turer at the London School of Economics, and afaculty member of Carnegie Institute of Technol-ogy‘s Graduate School of Industrial Administration.He received the A.B. degree magna cum laude fromHaruard and the Ph.D. from Johns Hopkins. Herecently was awarded a National Science Founda-tion Senior Post-Doctoral Fellowship for a year ofstudy and research at the Center for Operations Re-search and Econometrics at the Universite de Lou-vain, Belgium.

DANIEL ORR, now on the faculty of the Universityof California at La Jolla, was Assistant Professor ofMathematical Economics at the Graduate School ofBusiness of the University of Chicago until 1966.His work in the mathematical analysis of businessproblems has appeared in such periodicals as TheJournal of Business, Economica, Quarterly Journalof Economics, and American Economic Review. Heis co-author with Merton H. Miller of a series ofpublished papers on money supply and demand,and is author of Models of the Maximizing Firm,to be published by Prentice-Hall, Inc. He receivedthe A.B. degree from Oberlin, and the M.A. andPh.D. from Princeton. He has been on the facultiesof Princeton, Amherst, and the University of Cali-fornia at San Diego. He was an operations analystfor Procter and Gamble Co. and has done consult-ing work for Mathematica, Inc., Arthur D. Little,Inc., and The RAND Corporation.

THIS PAPER was presented at the Conference on Fi-nancial Research and Its Implications for Manage-ment, held at Stanford University June 9-11, 1966.The underlying study and report were done withthe assistance of two students of the GraduateSchool of Business of The University of Chicago-ROBERT CRAMER, now with the United CaliforniaBank, and PETER KUYPER, now with the RichfieldOil Company.

Mathematical Modelsfor Financial ManagementONE STREAM of current research in finance in-volves the extension to the field of finance ofthe methods and approaches that have cometo be called “operations research” or “manage-ment science.” Researchers working alongthese lines try to develop mathematical rep-resentations or “models” of typical decision-making problems in finance and, where theyare given the opportunity to do so, to test andapply these models in actual decision settings.At the moment, this stream of research is stilla relatively small one-really only a trickle ascompared to the flood of material pouring outon the subjects of capital budgeting, or valua-tion. But it is a stream that can be expected togrow rapidly in the years ahead with the im-provement in mathematical and computertechnology and especially with the increase inthe number of people who are being taught touse the tools effectively and creatively.

Rather than attempting any broad surveyof work to date, this paper will present a singleexample of this type of research, describingboth the development of the mathematicalmodel and its application in a specific firm.Such an example can convey more graphicallyand more convincingly than any amount ofpreaching many of the important implicationsfor management of this kind of research.

The Cash Balance as an Inventory

The particular financial problem involvedin our example is that of managing a cashbalance in conjunction with a portfolio ofshort term securities. And the particularmathematical model that will be used is a type

of inventory model that might be called a“control-limit” model.

It may be a little startling at first to thinkof your firm’s cash balance as just another in-ventory-an inventory of dollars, so to speak-but is it really so farfetched? Consider, for ex-ample, some raw material item that your com-pany stocks and ask yourself why you keep somuch of it around or why you don’t simplyorder each day’s or each hour’s requirementon a hand-to-mouth basis. The answer is, ofcourse, that this would be a very wasteful pol-icy. The clerical and other costs involved inplacing orders for the material are not trivial;and there would be further costs incurred inthe form of production delays or interruptionsif materials were slow in arriving or if require-ments on any day should happen to be higherthan had been anticipated. Why, then, noteliminate these costs once and for all by plac-ing one big order for a mountain of the stuff?Here, of course, the answer would be thatthere are also costs connected with holding in-ventory. These would include not just thephysical costs connected with the storage spaceand handling, but also the cost of deteriora-tion, or of obsolescence, or of adverse pricefluctuations, and especially of the earningsforegone on the capital tied up in the inven-tory. The inventory management problem forany physical commodity is thus one of strikinga balance between these different kinds ofcosts; and the goal is to develop a policy inwhich orders will be placed on the average atjust the right frequency and in just the rightamounts so as to produce the smallest com-bined costs of ordering, of holding inventory,and of running out of stock.

Similarly with cash. If you want to add toor subtract from your inventory of cash bymaking a transfer to or from your portfolio ofsecurities, there is an order cost involved, part-ly in the form of internal clerical and decision-

3

making costs and partly in the form of broker-age fees, wire transfer costs and the like. In theother direction, if you try to cut down thesein-and-out costs by holding large cash balances,there is a substantial holding cost in the formof the interest loss on the funds tied up in thebalance. As for the costs connected with run-ning out of cash, these are perhaps too obviousto require discussion before a group of thiskind.

The Control-Limit Approach toCash Management

Accepting the inventory analogy as valid,what form of inventory management policywould be suitable for cash balances? Here,since the typical cash balance fluctuates upand down and, in part, unpredictably, itseemed to us that the most natural approachfor a wide variety of cases might be a “control-limit” policy.

How one particular kind of control-limitpolicy might work when applied to a cash bal-ance is illustrated in Figure 1. We say “oneparticular kind” since the control-limit ap-proach is quite flexible and many differentvariations can be used depending on the cir-cumstances. The one illustrated happens to bean especially simple one and one that can beshown to be appropriate whenever the inter-nal clerical and decision-making costs are themain costs involved in making portfolio trans-actions. It is also the form of policy actuallyused in the specific application to be describedlater.

The wiggly line that starts at the left at m,,traces out the hypothetical path of a cash bal-ance over time. As drawn, it first seems to fluc-tuate aimlessly until about day t,,, at whichpoint a rising trend appears to set in. Duringthis interval receipts are exceeding expendi-tures and the cash balance is building up. The

4

buildup is allowed to continue until the dayon which the cash balance first reaches orbreaks through the upper control limit of hdollars. At this point in time-day t, on thegraph-a portfolio purchase is made in anamount large enough to restore the cash bal-ance to the return point z. Once back at z, thecash balance is allowed to wander again. Nofurther purchases or sales are made until thebalance either breaks through the upper boundat h again or until it breaks through the lower

5

control limit, I, as at day ta. When the lowercontrol limit is reached, a sale of securitiesfrom the portfolio is signalled in an amountsuch that the balance is once again restored tothe return point z.

An Optimal Solution for aSimple Special Case

Given that this kind of control policy seemsreasonable-and we would argue that it is rea-sonable not only in dealing with some types ofcash management problems but in many otherkinds of settings where there is a substantialcost in managerial intervention to restore a“wandering” system to some desired state-thetask of the researcher then becomes that of ap-plying mathematical or numerical methods todetermine the optimal values of the limits. Byoptimal we mean values that provide the mostadvantageous trade-off between interest losson idle cash and the costs involved in transfersof cash to and from the portfolio. As it turnsout, there happen to be some simple, but im-portant special cases in which these optimalvalues can be derived in relatively straightfor-ward fashion and where the results can be ex-pressed in the form of a simple, compact form-ula. In particular, we have been able to obtainsuch a formula for the optimal values of thelimits for cases which meet the following con-ditions: (1) Where it is meaningful to talkabout both the cash balance and the portfolioas if they were each single homogeneous as-sets.1 (2) Where transfers between cash and theportfolio may take place at any time but onlyat a given “fixed” cost, i.e., a cost that is thesame regardless of the amount transferred, thedirection of the transfer or of the time since

1 We have also recently been able to develop approxi-mately optimal solutions for certain special kinds of“three-asset” models, i.e., models in which there are twokinds of securities (e.g., a line of credit and commercialpaper) in addition to cash.

6

the previous transfer. 2 (3) Where such trans-fers may be regarded as taking place instanta-neously, that is, where the “lead time” in-volved in portfolio transfers is short enough tobe ignored. (4) Where the lower limit on thecash balance is determined outside the model,presumably as the result of negotiations be-tween the bank and the firm as to what thefirm’s required minimum balance is to be; and(5) Where the fluctuations in the cash balanceare entirely random. There may perhaps be atrend or “drift” as it is called in this kind ofanalysis; but aside from this kind of simplesystematic component, the day-to-day changesin the cash balance are completely unpredict-able.

As for the specific formula that constitutesthe solution under these assumptions, there islittle point in discussing it any length here.The complete derivations and other detailscan be found in a recently-published article.3It might, perhaps, just be worth noting herethat the solution defines the limits in termsof the fixed transfer cost, the daily rate of in-terest on the portfolio and the variability ofdaily changes in the cash balance (exclusive ofchanges related to the portfolio). As would beexpected, the higher the transfer cost andgreater the variability the wider the spreadbetween the upper and the lower limits; andthe higher the rate of interest, the lower thespread. There are, however, some surprises.In particular, for the “no-drift” case, it turnsout that despite the fact that the cash balance

2 Simple solutions also have been developed for thecase in which the cost is not fixed but proportional tothe amount transferred. More complicated, mixed casesinvolving both a fixed and a proportional componenthave been analyzed by our colleagues G. Eppen andE. Fama who have developed a very flexible methodof obtaining numerical values for the limits under awide variety of circumstances.

3 ”Model of the Demand for Money by Firms,” byM. H. Miller and D. Orr, Quarterly Journal of Eco-nomics (forthcoming).

7

is equally likely to go up or down, and it’sequally costly to buy or to sell securities, theoptimal return point z-the point at which theaverage long-run costs of operating the systemare lowest-does not lie midway between theupper and lower limits. Instead, it lies sub-stantially below the midpoint. To be precise,it lies at one-third of the way between the low-er and upper bounds and it stays at the one-third point regardless of the numerical valuesthat are assigned to the transfer costs or to thedaily rate of interest that can be earned on theportfolio. As these values are changed, thewhole system expands or contracts, but the re-lation between the parts remains the same.

A Test Application of the Simple Model

Your initial reaction is likely to be that thismodel and the assumptions on which it wasbased are much too special and restrictive tohave any important applicability to real-worldproblems. In management science, however, asin science generally, it is rash to pass judgmenton the range of applicability of a model solelyon the basis of the assumptions that underlieit. Mathematical models often turn out to besurprisingly “robust” and insensitive to errorsin the assumptions. The only safe way to de-termine how well or how poorly a model worksis to try it out and see.

In obtaining the basis for this kind of test ofthe model we were extremely fortunate in hav-ing the active collaboration of Mr. D. B.Romans, Assistant Treasurer of the UnionTank Car Company. 4 Mr. Romans had seenan earlier version of our original paper andwas struck by the similarity between the modeland his own policies in putting his firm’s idle

4 We have also benefited greatly from discussions ofcash management problems and practices with severalofficers of the Harris Trust and Savings Bank of Chi-cago. We hope that they will benefit too from thischance to see how the problem looks from the otherside of the account.

cash to work. The systematic investment ofidle cash in short term, money market securi-ties was a relatively new program for his com-pany-one that he had instituted only about ayear previously. The interest earnings for thatyear were quite large, not only in relation tothe costs involved but to the total budget ofthe treasurer’s department. Now that the year’sexperience had been accumulated, he wantedto go back over the record, to study it in detailand to see whether any changes in practicemight be suggested that would make the oper-ation even more profitable. He felt, and weagreed, that the model might be extremelyhelpful in this kind of evaluation. If the modeldid seem to behave sensibly when applied tothe company’s past cash flow then it might beused to provide an objective standard or“bogey” against which past performance couldbe measured.

Since mathematical modeling of business de-cisions is still quite new, and since few peopleoutside the production area have had muchdirect connection with it, it is perhaps worthemphasizing that at no time was it intendedor contemplated that a model should be devel-oped to do the actual on-line decision-making.The purpose of the study was to be evaluationby the treasurer of his own operation. This isa valuable but unglamorous use of modelsthat tends to be overlooked amidst all thehoopla of the Sunday supplement variety sur-rounding the subject of automated manage-ment. An important point that must be keptin mind about mathematical models is thatthey are not intended to replace management-though like any other technological improve-ment they sometimes have that effect-butthat they provide managers with new tools ortechniques to be used in conjunction withother managerial techniques (including goodjudgment) for improving over-ah performance.

9

The Setting of the Operation

Since our objective was to compare themodel’s decisions over some trial period withthose of the Assistant Treasurer, the first stepwas to examine carefully the setting in whichhe actually operated and to see how closelyor how poorly the circumstances matched theassumptions of the model. As would be ex-pected, the results were mixed. On the onehand, there were some respects in which theassumptions fit quite well. The AssistantTreasurer did behave, for example, as if hewere in fact controlling only a single-centralcash balance. Note the phrase “as if,” becauseas a matter of fact the firm does have manyseparate balances in many banks. For pur-poses of cash management, however, the As-sistant Treasurer works with one single bal-ance representing the free funds that he canmarshal throughout the system without re-gard to the particular banks they happen tobe in at the moment (or where the funds de-rived from a portfolio liquidation must ulti-mately be routed).

It was also clear that there were substantial“order costs” involved in making portfoliotransfers. In the case of a portfolio purchase,for example, some of the main cost compo-nents include: (a) making two or more long-distance phone calls plus fifteen minutesto a half-hour of the Assistant Treasurer’stime, (b) typing up and carefully checking anauthorization letter with four copies, (c) carry-ing the original of the letter to be signed bythe Treasurer, and (d) carrying the copies tothe controller’s office where special accountsare opened, the entries are posted and furtherchecks of the arithmetic made. It is hard toestablish a precise dollar figure for these costs,but at least the approximate order of magni-tude for a complete round trip is probablysomewhere between $20 and $50. That this

10

is not a trivial amount of money in the pres-ent context becomes clear when you remem-ber that interest earnings at the then prevail-ing level of interest rates were running atabout $10 per day per hundred thousanddollars in the portfolio and that his averagesize of portfolio purchase during the testperiod was about $400,000.

Not surprisingly, we also found that therewas a considerable amount of randomness orunpredictability in the daily cash flow. Infact, the Assistant Treasurer did not evenattempt to forecast or project flows more thana day or two ahead except for certain largerecurring outflows such as tax payments, divi-dend payments, sinking fund deposits, trans-fers to subsidiaries and the like; and evenhere the forecasts were made more with aview to deciding the appropriate maturitiesto hold in the portfolio than as part of thecash balance control per se. As for the “drift”or trend, analysis of the cash flow over the 9-month test period showed no evidence of anysignificant drift in either direction.

As opposed to these similarities between theassumptions of the model and the reality ofthe firm’s operation there were very definitelya number of respects in which the fit was muchless comfortable. The model assumes, forexample, that when the lower bound on cashis hit or breached, there will be an immediatesale of securities out of the portfolio to makeup the cash deficiency. The Assistant Treas-urer, however, followed a policy of buyingonly non-marketable securities and holdingthem to maturity primarily because he wantedto try his new cash management programwithout requiring any change in the com-pany’s standard accounting procedures. Hence,if a large net cash drain occurred unexpected-ly on a day on which he had no maturingsecurity he simply let his cash balance dropbelow his normal minimum which he and

11

his banks regarded as an “average minimum”rather than as the strict minimum contem-plated by the model.

A discrepancy between the model and real-ity that was more disturbing appeared whenwe constructed the frequency distribution ofdaily cash changes by size of change over theg-month sample period of 189 working days.The distribution of these daily changes isshown in graphic form in Figure 2. The logicof the model requires that this distributionbe at least approximately of a form that stat-isticians refer to as “normal” or “Gaussian.”A hasty glance at the figure might lead oneto conclude that this requirement is met.Closer study reveals, however, that not onlyis the distribution not normal, but it almost

1 2

seems to be a member of a particularly ill-behaved class of “fat-tailed” distributions thathave come to be called “Paretian distribu-tions.“5 In these distributions-which may befamiliar to those of you who have been fol-lowing the debate about random walks in thestock markets-large changes occur much morefrequently than in the case of the normaldistribution. In fact, the frequency of largechanges is so much greater that we were quiteuncertain as to whether the model would be-have in even roughly sensible fashion orwhether it would simply find itself being whip-sawed to death by the violent swings throughthe control range. As indicated earlier, how-ever, there is only one way to tell; and that’sby trying it out and seeing what happens.

The Test of the Model Against the Data

To get a close basis for comparison with theAssistant Treasurer’s actual decisions it wasdecided to run the model under various alter-native assumptions about the true value ofthe transfer cost. That is, we would start witha conservatively high value of say $90 pertransfer; compute the optimal upper limit hand return point z; run the model against theactual data and tabulate its portfolio pur-chases and sales. If, as expected, the modelmade fewer transfers than the Assistant Treas-urer, then we would go back; use a lowervalue for the costs; recompute the new optimal

5 We say “almost seems to be” because despite theconspicuously fat tails, the distributions as computedcumulatively month by month remain roughly similarwith no tendency for the tails to get fatter and fatterover time as in a true Paretian process. The Paretian-like tails are mainly the reflection of such large, but rela-tively controllable and definitely size-limited items asdividends, taxes, transfers to and from subsidiaries andthe like.

6 Random Walks in Stock-Market Prices, Selected Pa-pers No. 16, Graduate School of Business, University ofChicago.

13

limits and so on until we had finally forcedthe model to make approximately the samenumber of transfers over the sample intervalas the Assistant Treasurer himself. Then, as-suming the model was behaving sensibly, wecould compare and contrast their patterns ofportfolio decisions over the interval as wellas get at least some rough idea of what figurefor the cost of a transfer the Assistant Treas-urer was implicitly using in his own operation.

The only difficulty encountered in imple-menting this straightforward kind of test wasin the matter of deciding precisely how manytransfers the Assistant Treasurer should beregarded as having made. Because of his policyof holding only non-marketable issues, hisportfolio tended to be of quite short averageduration. Hence there were inevitably days onwhich he had a maturity that proved to be tooearly. If he had merely rolled these issues overthere would have been no problem; we wouldsimply have washed that transaction out andnot counted either the maturity or the rein-vestment as a transfer. But it is clearly notalways efficient just to roll over the maturingissue. Given that a purchase must be madeanyway on that day, it would be wise to pickup any additional cash that also happenedto be lying around, even if the amount in-volved would not have been large enough byitself to have justified incurring a transfercost. Accordingly, we decided not to countany transfers on rollover days unless the As-sistant Treaurer indicated that the balancewas so large even without the maturing issuethat he would almost certainly have boughtanyway (in which case he would be chargedwith the purchase, but not the sale). Similar-ly with the case of net sale days. If there wasa larger maturity on a given day than wasactually needed to meet the cash drain and ifsome small part of the excess proceeds wererolled over, then he was charged with a sale,

14

but not a purchase. By this criterion we wereable to agree on a figure of 112 total trans-actions by the Assistant Treasurer during the189 test days of which 58 were purchases and54 were sales (maturities).

The Results of the Test

When we commenced the trial-and-error-process of matching the total number oftransactions by the model with those of theAssistant Treasurer our hope was that themodel might be able to achieve an averagedaily cash balance no more than say 20 to 30per cent above the Assistant Treasurer’s aver-age. We felt that if we could get that closeand if the model did behave sensibly, thenthere was a very real prospect of being ableto use the model as a bogey against which tomeasure and evaluate actual performance. Asit turned out, however, we found that at 112transactions, the model not only came closebut actually did better-producing an averagedaily cash balance about 40 per cent lowerthan that of the Assistant Treasurer ($160,000for the model as compared with about $275,-000). Or, looking at it from the other side,if we matched the average daily cash holdingsat $275,000, the model was able to reach thislevel with only about 80 transactions or aboutone-third less than the 112 actually required.

It can be argued, of course, that this sortof comparison is unfairly loaded in favor ofthe model, not only because it was appliedon a hindsight basis, but because the transfercosts would actually have been higher for themodel than the simple matching of total num-bers of transfers would seem to suggest. TheAssistant Treasurer, it will be recalled, neverreally sold a security; he merely let it run off.Hence the model would have had to incuradditional costs on at least those sales that didnot occur on the easily forecastable, large-out-flow days. Check of the numbers involved

15

showed, however, that the model would stillhave dominated in terms of net interest minustransfer costs over the sample period even ifthese extra costs of liquidation were includedon every sale. And, of course, that is muchtoo extreme an adjustment. Many of the ac-tual sale days of the model coincided with thelarge out-flow days and appropriately matur-ing securities could have been purchased tohit these dates. In fact, the post-mortemshowed that about half the model’s sales tookplace on days when the Assistant Treasureralso “sold” and nearly 80 per cent occurredeither on the same day or within one dayeither way of a day on which he scheduled amaturity.

Furthermore, the model too is operatingunder some handicaps in the comparison. Atno time, for example, did the model everviolate the minimum cash balance marked onthe Assistant Treasurer’s work-sheets, whereasno less than 10 per cent of his total dollar daysinvested were represented by the cash defi-ciencies on the days in which he let his bal-ance dip temporarily below the minimum.In addition, the model did not receive in-structions to change its policies before week-ends and holidays. The Assistant Treasurer,on the other hand, always knew when it wasFriday and was thus able to sock away addi-tional amounts on which he could get twoextra days’ interest.

All in all then the comparison would seemto be basically a fair one; and it is a tributeto the Assistant Treasurer’s personal and pro-fessional character that he never became ego-involved in the comparison or wasted timealibiing. He was concerned about one thingand one thing only: how to do an even betterjob.

The Comparison of Operating Policies

With this question in mind, we then went

16

on to make a detailed comparison of the ac-tual decisions with those called for by themodel. The complete record of these com-parisons is, of course, too long and too special-ized to be spelled out at length here, butthere are at least a few simple contrasts thatcan be presented to illustrate the sorts ofthings that turned up.

P”RCHMES wJo omitt.d,

FIGURE 2aIS - lm14 -13 -13 -11 -10 -9-o- - 51) P”RC”*sES7-6-5--4-J-2-

Figures 2a and 2b, for example, show thefrequency distributions of portfolio purchasesby size of purchase for the model and for theAssistant Treasurer. Notice that even thoughwe have forced the total number of transfersto match, the model makes somewhat fewerpurchases (54 as against 58) and does so inconsiderably larger average size (about $600,-000 as compared with only $440,000). Thedifference in operating policy is particularly

17

striking at the lower end of the size scale be-cause of the rigid rule built into the modelthat keeps it from ever buying in units small-er than h-z, which was about $250,000 whenthe model was set to produce 112 transfers. TheAssistant Treasurer, by contrast, made about13 purchases (or nearly 25 per cent of histotal purchases) in amounts smaller than thatsize including 5 in amounts of $100,000 or less.Even allowing for the fact that some of thesesmall transactions were for weekends, the totalimpression conveyed is one of an excessiveamount of small-lot purchasing activity. Thisimpression was further reinforced both by thevery low implicit transfer cost that was nec-essary to force the model to make 112 transfersas well as by the fact that more than 90 percent of the total interest earnings achievedby the model with 112 transfers could havebeen attained with only about 50 total trans-fers. Of these 50, moreover, only some 20were purchases and all were of fairly largesize.

Even more revealing are Figures 3a and 3bwhich show the distribution of the closingcash balance by size on days when no port-folio action was taken in either direction.Notice again that because of its rigid upperlimit the model never lets the cash balancego above h which in this case is about $400,-000. The Assistant Treasurer, however, seemsto be much less consistent in this respect, hav-ing foregone no less than 23 buying oppor-tunities of this amount or larger including3 of over a million dollars. When and whyso many opportunities were missed is still notentirely clear. Part of the trouble undoubtedlystems from the fact that the Assistant Treas-urer has many other responsibilities and can-not always count on being at his desk at thetime of day when the decision has to be made.And without actually interrupting to con-struct his worksheet, there is no way for him

18

OTHEP. CaIS (ow aid)FIGURE 3b

to determine whether an interruption of hisother work would really be profitable. Hope-fully, however, by making his limits more ex-plicit (in the spirit of the model) and by dele-gating to others much of the purely mechanicaltask of monitoring these limits, he will be ableto achieve in the future a significant reduc-tion in the size and frequency of these lostopportunities.

Conclusion

We have tried here to present a concreteexample of how mathematical methods canbe and are being applied to management prob-lems in the field of finance. The example hap-pens to be a particularly simple one. But itdoes at least serve to illustrate very neatly anumber of points about this kind of research

19

that senior financial managers would do wellto keep in mind.

First, it is important for financial manag-ers to disabuse themselves of the notion thatthere is something special or unique aboutfinancial problems. In particular, we haveseen that what is commonly regarded as apeculiarly financial problem-to wit, manag-ing the cash balance and a portfolio of liquidsecurities-turns out to be nothing more thanan inventory problem.

Second, mathematical models of decisionor control problems should not be thoughtof as something fundamentally different fromordinary management principles or tech-niques. They are merely more disciplined andsystematic ways of exploiting these principles.In particular, control-limit models of the kindwe have seen here-and remember that manyadditional variations are possible-are essen-tially extensions of the fundamental notion of“management by exception.”

Third, be careful not to prejudge mathe-matical models solely on the basis of the lackof literal realism in the assumptions under-lying them. To develop a workable model,simplifications-sometimes, extreme simplifi-cations-must be made. But, if it has beenproperly conceived, a simple model may stillperform extremely well. It is not a matter ofgetting something for nothing; rather that thegains made by doing a good job on the reallyessential parts of the problem are often morethan large enough to offset the errors intro-duced by the simplifications (errors, inciden-tally, that often cancel out).

Finally, remember that there is a trade-offbetween improving decision-procedures andimproving the information and forecasts usedin arriving at the decisions. In the presentinstance, for example, we saw a case in whicha model that assumed the cash flow to be com-pletely random was still able to do a very

successful job of decision-making. Nor is thisresult unique or exceptional. The slogan ev-erywhere today is “more, better and fasterinformation for management.” We suspect,however, that thanks to the computer, manyfirms may already be in the position of havingmore, better and faster information than theycan use effectively with present managementtechniques. There is likely to be as much ormore real pay-off in the years ahead in ration-alizing and improving decision proceduresthan there is in simply trying to get an evenbigger bang from the information explosion.