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Till cash management model Sick, Gordon Arthur 1976

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TILL CASH MANAGEMENT MODEL by GORDON ARTHUR SICK B . Sc . , U n i v e r s i t y o f C a l g a r y , 1971 M.Sc, U n i v e r s i t y o f T o r o n t o , 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION v i n FACULTY OF COMMERCE, & BUSINESS ADMINISTRATION We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December, 1976 © Gordon A r t h u r S i c k , 1976 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my written permiss ion. Department of 4 The Univers i ty of B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 ABSTRACT This thesis develops a model for the management of t i l l cash (currency and coin) of a branch for a Vancouver area cred i t union. The model i s developed in two parts. F i r s t , a model i s estimated to forecast cash demand and then a cash order algorithm i s developed. Two s t a t i s t i c a l models are developed to estimate cash demand. The f i r s t employs Box-Jenkins time series techniques. This model f a i l s because the cash flow data are non-stationary, exh ib i t ing both a growth trend and high autocorrelations at large lags. In the second model, a growth trend for real weekly cash flows i s f i r s t estimated, incorporating an asymptotic capacity constra int. The real cash flow trend i s converted to a nominal trend and used as the weight in a l i nea r weighted least squares model for da i l y cash flows, in which the explanatory variables are dummy variables to indicate days of the week, months of the year, incidence of pay days, etc. The consistency of the resu l t ing forecast model i s also discussed. To develop a cash order algorithm, steady state models are f i r s t considered. These models are generally based on stationary cash demand, constant del ivery lag times for orders and other assumptions that are inappropriate in th i s t i l l cash management set t ing. To relax the steady state assumptions a general dynamic programming framework i s i i developed f o r the cash management model that allows fo r e i ther penalty costs f o r cash-outs (cash shortages) or a chance constra int involv ing the p robab i l i t y of a cash-out. Because of non-stat ionar i ty of the cash flows the dynamic program cannot be solved d i r e c t l y s but an approximate so lut ion i s obtained using a simulation technique. The. resu l t ing algorithm i s tested on h i s t o r i c a l data and the resu l t s are discussed b r i e f l y . i i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES vi LIST OF FIGURES v i i ACKNOWLEDGMENTS v i i i Chapter 1 INTRODUCTION 1 1.1 The Nature of the Problem 1 1.2 Overview 2 2 FORECASTING THE DEMAND FOR CASH: THEORY 5 2.1 Introduction 5 2.2 Forecasting Cash Demand as a Time Series 7 2.3 Forecasting Demand by Regression 13 2.4 Summary 25 3 FORECASTING THE DEMAND FOR CASH: APPLICATION 29 3.1 Introduction 29 3.2 The Data 29 3.3 Estimation by Box-Jenkins Techniques 32 3.4 Estimation by Regression Techniques 35 3.5 Summary 46 i v Chapter Page 4 THE CASH MANAGEMENT PROBLEM IN GENERAL 48 4.1 Introduction 48 4.2 The Problem and Its Environment 49 4.3 Steady State Models 51 4.4 Dynamic Programming Formulation 58 4.5 Operational Approximations to Optimal Po l i c ie s . . . . 65 4.6 Summary 69 5 A STOCHASTIC PROGRAMMING MODEL TO ORDER TILL CASH 72 5.1 Introduction 72 5.2 Preliminary Considerations 73 5.3 The Chance Constraint 75 5.4 The Objective Function 78 5.5 The Cash Order Algorithm 81 5.6 Appl icat ion of the Model 86 5.7 Summary 90 Appendix: The Dynamic Program Corresponding to the Objective Function (5.4.2) 91 6 CONCLUSIONS. 93 LIST OF REFERENCES , 95 v LIST OF TABLES Table- Page 1 Estimated Regression Coeff ic ients for Daily Demand 38 2 Cash Delivery Order Lags 87 3 Performance of the Cash Order Model and of Management 87 vi LIST OF FIGURES Figure Page 1 Growth of Real Cash Demand 20 2 The Cash Order Algorithm 82 3 The Minimum Cash Order Subroutine 83 v i i ACKNOWLEDGMENTS The author would Like to thank,,the o f f i ce r s of the c red i t union for the data and valuable discussions, Professor Robert White for advisory assistance and Professor Will iam Ziemba for technical assistance. Any errors or omissions are the re spons ib i l i t y of the author alone. The research was performed while the author was in receipt of a Samuel Bronfman Award and a Univers ity of B r i t i s h Columbia Graduate Fellowship. v i i i Chapter 1 INTRODUCTION 1.1 The Nature of the Problem The purpose of th i s study i s to develop a model to order t i l l cash (currency and coin) for a Vancouver area cred i t union branch, with the objective of minimizing costs, while maintaining adequate balances to meet the members' demand. The demand for cash i s not determin i s t ic , and part of the problem i s to determine the s ize of precautionary cash balances which w i l l reduce the.r i sk of a cash-out to an appropriate level desired by management. A cash-out occurs i f the balance of cash on hand f a l l s below some level which management regards as the pract ica l minimum level for smooth operation. This level may be zero, but could, more meaningfully, be about $10,000, since a smaller level would not allow a l l t e l l e r s to have a reasonable supply of a l l denominations of currency and coin. The costs that are to be minimized are the f ixed costs of placing a cash order and the opportunity costs of i d l e cash balances. If a po l icy i s taken to make large orders, few orders are required and f ixed order costs are low, but cash balances are high, creating large opportunity costs. Conversely, i f orders are small and frequent, f ixed order costs are high and opportunity costs are low. Minimization of tota l costs enta i l s 1 2 f inding an optimal trade-off between f ixed order costs and var iable opportunity costs. Thus, the problem i s e s sent ia l l y one of constrained optimizat ion: minimize costs subject to some constraint involving the r i s k of a cash-out. In th i s sense, i t i s a standard inventory problem. Inventory theory i s generally oriented towards solving ana l y t i c a l l y t ractable problems which do not always f i t real world s i tuat ions very we l l . The terms of reference of th i s project require an operational model. Thus, in deciding whether or not a given inventory model i s appropriate, the question i s not whether the real problem f i t s the assumptions of the model (real problems rare ly f i t pe r fec t l y ) , but how well w i l l the model perform when i t s spec i f icat ions are v io la ted. Operational ity of the cash order model also means that the costs of development, day to day operation and maintenance of the model should be reasonable. Since the cred i t union branch has on- l ine computer f a c i l i t i e s , there are no major computational r e s t r i c t i on s to the algorithm, but to min-imize development costs, the algorithm should be rather stra ightforward-Operational costs re late mainly to the personnel time required to generate da i ly input. Maintenance costs are the costs of re-estimating any parameters of the model i f the structure of the problem changes over time. It i s in th i s operational environment that the cash order model i s developed. 1.2 Overview Development of the model requires the solut ion of two major sub-problems: estimation of the demand for cash and construction of the actual order algorithm. 3 Chapter 2 discusses the theoret ica l aspects of forecasting the demand for cash. F i r s t , various methods of adaptive time series forecast ing are considered, culminating in a discussion of Box-Jenkins time series techniques. Then, the problems of developing a l inear regression model for forecasting cash demand are considered. In Chapter 3, the resu lts of both Box-Jenkins estimation and weighted least squares estimation are presented. The Box-Jenkins method, although preferable in the sense that i t i s an adaptive forecasting technique, i s found to be inappl icable because the demand for cash i s non-stationary. The non-stat ionar ity arises part ly because of the general growth in demand level and part ly because factors of vast ly d i f fe rent orders of seasonality are present. In order to use weighted least squares, a forecasting model for the general trend level in cash demand is f i t t e d by least squares, and then da i l y cash flows, deflated by the general trend l e v e l , are regressed on dummy variables re la t ing to months of the year, days of the week, paydays, etc. Tests of the predict ive a b i l i t y of the model are also discussed in th i s chapter. Chapter 4 i s a discussion of the general aspects of the cash management problem. Several models that have appeared in the inventory theory and cash management l i t e r a tu re are discussed. Some of these models are based on steady state assumptions such as s ta t i ona r i t y of the demand for cash. The appropriateness of these assumptions i s discussed, and for purposes of comparison, a dynamic programming formulation of the problem i s presented. In Chapter 5, an approximate solut ion to the dynamic pro-gramming problem i s developed. It e s sent ia l l y y ie lds a var iable control 4 l i m i t model. Cash i s ordered when the cash balance is inadequate for transactionary and precautionary requirements, which vary according to the expected demand during the order del ivery lag period. The order s ize is selected to minimize expected costs per day during the l i f e of the order, as computed by s imulation. The chapter also discusses tests of the model on h i s t o r i c data. In Chapter 6, a few concluding remarks are made about the extent to which the techniques developed here can be generalized to the cash management problem for other c red i t union branches. Chapter 2 FORECASTING THE DEMAND FOR CASH: THEORY 2.1 Introduction Forecasting is an inductive reasoning process whereby h i s t o r i c re lat ionships between a variable of interest (the dependent var iable) and explanatory variables (the independent var iables) are analyzed, in order to predict (perhaps with error) future values of the dependent var iable. A general explanatory model i s of the form * t = f C x l t . . . - s x n t ) + u t (2 .1.1) where y t i s the dependent var iable at time t (e.g. cash f low), f i s a function whose form i s determined a priori but which may have empir ica l ly estimable parameters, x l t ' * * * ' X t are the explanatory variables at time t (e.g., lagged observations of y, , or dummy variables for days of the week), and u^ i s the forecast error , the d i s t r i bu t i on of which may be important to the forecaster. We may develop the model so that E(u t ) = 0, so that, given (x^, '»»,x ), the forecast for y i s 5 6 y = f ( . x r - - . , x n ) . . (2,1.2) There are generally many independent variables and many more models which may be used to explain h i s t o r i ca l values of any given depen-dent var iab le. Rarely do a l l of these models agree in the i r estimates, and a forecaster must choose which model i s most appropriate. He cannot simply choose the model which best f i t s h i s t o r i ca l data, because the h i s to r i ca l re lat ionships may not continue to hold in the future. It i s possible to f i t a model on an early subset of h i s t o r i ca l data and test i t s performance on a l a te r subset (as i s done l a te r in th i s paper), but there i s s t i l l the outstanding question of s t a b i l i t y of re lat ionsh ips. Thus, i f inappropriate treatment of, or omission of, some explanatory var iable causes l i t t l e harm to the performance of the model on h i s t o r i c data, there i s no assurance that the same type of error w i l l be i n s i gn i f i can t in the future. In the present app l icat ion, for example, cash flows are increasing over time. Many forecasting methods that have increasing estimates over time can be f i t t e d to predict well on h i s t o r i c data. Some of the estimates w i l l increase without bound over time, some w i l l increase to a f i n i t e asymptotic l i m i t , and some w i l l u l t imately decrease after a certain time. Selection of an appropriate model cannot be made object ive ly with h i s t o r i ca l data only, but must involve some subjective judgment of the forecaster. Within.this framework, the theories of several potential fore-casting models fo r cash demand at the cred i t union branchliare analyzed. In Section 2.2, cash demand i s regarded in a time series framework whereby future" cash flows are estimated in terms of the series of h i s t o r i ca l flows only. Such models are frequently adaptive in the sense that certa in 7 types of forecasting errors tend to be corrected over time. Emphasis i s placed on Box-Jenkins time series models. I t i s pointed out that, i f the cash demands have s i gn i f i cant autocorrelations at very long lags, such as seasonal spans of one month or one year, i t may not be possible to f i t a parsimonious. Box-Jenkins model to cash demands. In Section 2.3, a two-part model i s developed for h i s t o r i ca l cash demand. F i r s t a general trend over time i s f i t t e d to real cash demands which incorporates the a -priori notion that the branch i s growing towards an asymptotic capacity constraint. Then cash demands are deflated by th i s trend level in a weighted least squares l inear model which uses as explanatory variables dummy variables ind icat ing the incidence of various days of the week, months of the year, paydays and holidays. This model uses a wider range of explanatory variables than the Box-Jenkins model and, as a re su l t , may provide v iable estimates even when the Box-Jenkins model f a i l s . 2.2 Forecasting Cash Demand as a Time Series In a time series model, the only independent variables in the model (2.1.1) are lagged observations of the independent var iable (y t -j , y t 2 ' * * * ' ) - Many c l a s s i ca l forecasting models such as arithmetic moving averages and exponentially weighted moving averages have th i s form. An excel lent discussion of such models i s given in Wheelwright and Makridakis [1973]. They are adaptive in the s t r i c t sense that, i f there is an increase or decrease in the mean level of the variables y^, ar ithmetic and exponentially weighted moving average forecasts increase or decrease accordingly and converge over time to the new l e v e l . Thus absolute 8 forecast errors decrease in an adaptive manner, unless the general level changes again. If the time series f luctuates rapid ly over time, sophi-st icated weighting schemes are required to reduce the response time in the adaptation process and thus reduce forecast erros. C lass ica l theory says l i t t l e about methods of se lect ing optimal values for the time period for moving average or the weight for an exponentially weighted average.;. Also, these techniques f a i l to provide anydesc r ip t ion of the d i s t r i bu t i on of the random error term u^ . in ( 2 . 1 . 1 ) . For the problem at hand, some descr ipt ion of the error term u^ i s valuable in analyzing the r i sk of a cash-out. Also, cash flows f luctuate s i g n i f i c an t l y from day to day, week to week and month to month because of predictable var iat ions in demand. A naive weighting scheme in a weighted average forecast would be highly inaccurate because of the frequent re-versals in cash demand. For example, an exponential scheme would forecast high demand in January simply because there was high demand in December. The need for quant i f icat ion of forecast errors and a more sophisticated weighting scheme suggests the use of Box-Jenkins techniques. Many authors have considered two general models of a time series of observations on a s ingle var iable y t (such as cash f low). One i s the i n f i n i t e autoregressive (AR) form: y t = 6 + u t + <h y t - 1 + <j>2 y t _ 2 + ••• (2 .2 . 1 ) where y^ is the t observation in the ser ies, 6 i s a constant, {u^} i s a sequence of independent i den t i c a l l y dis-tr ibuted random shocks (unobserved), 9 and {cj> }^ i s a sequence of AR weights. This i s the most general form of a weighted average scheme, and expresses the current observation as a weighted sum of e a r l i e r observations plus a random shock. Note that, in general, i t i s not an "average" since the weights <j>. need not sum to 1 and 6 need not be zero. A l te rna t i ve l y , one can express the current observation as a weighted sum of the sequence of e a r l i e r unobserved random shocks. This i s the moving average (MA) form: y t = 6 + u t - 6: V ] - 62 U t _ 2 . . . . (2.2.2) where ^ t ^ ' ^ ' ^- ut^ a r e a s before and 0 i , 8 2 , ••• are the moving average weights, The AR and MA forms are equivalent in the sense that, given one, the other can be found, although i t may not be a convergent ser ies . For forecasting purposes, the general AR and MA forms are impossible to estimate since they have an i n f i n i t e number of parameters. Box and Jenkins [1970] proposed that the forecaster should consider a f i n i t e combination of (2.2.1) and (2.2.2) which parsimoniously uses a f i n i t e number of parameters. This they c a l l an auto-regressive moving-average (ARMA) process. Its general form i s y t - <h y t _ ! d>p.yt_p = fi + u t - ei u ^ e q u t _ q (2.2.3) 10 It can be seen that (2.2.3) represents the current observation y t as the sum of a constant 6, a current random shock u^, and weighted sums of past observations and past shocks (the AR and MA parts, respec-t i v e l y ) . Introducing the backshift operator B where By t = y B^y = y t _ l < , and then defining the AR polynomial (j>(B) = 1 - <h B - <J>2 B 2 - ••• - <J)pBP (2.2.4) and the MA polynomial 0(B) = 1 - e a B - 6 2 B 2 - • • • - 0 q B q (2.2.5) we can rewrite (2.2.3) as *(B) y t = 6 + e-(B) u t (2.2.6) A desirable property of (2.2.6) is s t a t i ona r i t y , which means that the form of the stochastic process does not vary over time (in pa r t i cu l a r , the scal ing and locat ion of the var iat ion should be constant). S tat ionar i ty i s equivalent to having the zeros of the AR polynomial outside the unit c i r c l e . If there is a zero s t r i c t l y inside the unit c i r c l e , the series exhibits explosive behaviour. Zeros on the unit c i r c l e resu l t in mild non-stat ionar ity and correspond to factors in <j)(B) of the form (1 + B ) s s or (1 - B ) for s = 1 , 2 , 3 , « " . The form 1 - B corresponds to taking the difference between the current observation and the observation at the (seasonal) lag s. Powers of (1 - B ) correspond to taking differences 11 of d i f ferences. One can remove the difference factors from the AR part to rewrite (2.2.6), with obvious changes in notation, as ti c D 4>(B) (1 - B) Q,(1 - B S) y t =• 6 + 6(B) u t (2.2. In pract ice, i f the observed series y t exhibits non-stat ionar ity (such as growth over t ime), the forecaster should t ry various forms of d i f f e r -encing before attempting to solve for <J>, 6 and 6. Analogous to s ta t ionar i t y of the AR part i s i n v e r t i b i l i t y of the MA part. By recursive subst i tut ion of observations with e a r l i e r observa-tions in (2.2.6) or (2.2.7) we can represent the current observation as an i n f i n i t e AR series of past observations. The i n v e r t i b i l i t y requirement st ipulates that th i s i n f i n i t e series shal l converge - that i s , less and less weight i s placed on e a r l i e r and e a r l i e r observations to determine the current observation. This i s an important requirement for forecasting purposes, since one does not know what appropriate values of ear ly , un-observed data to use. I n v e r t i b i l i t y is equivalent to the zeros of the MA polynomial ly ing s t r i c t l y outside the unit c i r c l e . ' ' Given the i n v e r t i b i l i t y and s ta t ionar i t y requirements, an ARMA process i s uniquely determined by i t s autocorrelat ions. Thus one can look at the estimated autocorrelations of the series (along with pa r t i a l auto-correlat ions and inverse autocorrelations which are related to them) to ident i f y an appropriate model of the form (2.2.7) for an observed time ser ies . The parsimony concept requires select ion of a model where d, D, p and q are as small as possible. This presents a potential problem 12 for the appl icat ion of Box-Jenkins techniques to forecast cash demand, for one would expect a l o t of information content in high order auto-cor re lat ions . For da i l y cash demands there should be high pos i t ive auto-correlat ions at 5 day lags because of the correspondence of days of the week, and there should also be high autocorrelations at 260 day lags because of the correspondence of months of the year (there are approximately 260 working days per year). Since the number of days in a month i s not constant, the lags corresponding to semi-monthly pay days w i l l vary in a c y c l i c fashion with a very long period which w i l l create an important autocorre-la t ion at that lag. Moreover, some weeks only have 4 days and th i s w i l l upset the basic autocorrelation structure at 5 day lags. A l l of th i s suggests i t may be hard to f i t a parsimonious model to da i l y data. The problems for da i ly cash flow estimation are mitigated somewhat by considering weekly cash flows. A long 260 day log i s then a more reasonable (but s t i l l long) 52 week lag. The problem of holidays reducing some weeks to 4 days does not af fect weekly lag structures as ser iously as i t does da i l y lag structures. The actual estimation of (2.2.7) i s accomplished by minimizing the sum of squared residuals (^u^ 2), i t e r a t i v e l y by the Marquardt algorithm. Under the hypothesis that the u^'s are:independent N(0,a), th i s y ie lds maximum l i ke l i hood estimates of 0, <p and 6. Unbiased k-day ahead forecasts that minimize mean square pre-d i c t i on error can be developed from the random shock form (2.2.2) or the ARMA form (2.2.3) where h i s t o r i c a l one-period forecast errors are sub-s t i tuted for the random shocks (and the k unrealized random shocks are set to 0). The forecast errors are normally d i s t r ibuted i f the random 13 shocks are, and the k-day forecast error variance i s (1 + 9 : 2 + ••• + 62_-|) o ^ 2 where the 0- 's are given by the random shock form (2.2.2). If a Box-Jenkins model can be f i t t e d , the creation of forecasts and estimation of forecast error variances i s quite straightforward. The main problem to be faced when applying Box-Jenkins i s the question of whether an i n ve r t i b l e , stationary model can be f i t t e d . 2.3 Forecasting Demand by Regression In Section 2.2 i t was pointed out that forecasts (of demand) by time series techniques only use as explanatory variables the e a r l i e r observations of the same ser ies. It was argued that, in order to capture most of the explainable var iat ion in cash demand, autocorrelations at long lags would be important, and the model would not l i k e l y be parsimonious. Another approach i s to use more explanatory variables in the forecasting model (2.1.1). One way to incorporate addit ional variables is to use a regression model: y = xB + u (2.3.1) where • • f "N • Oi 2 0 y = • • -x = • • x 1 N u = • • UN , E(u u 1 ) = • 0 ' a j 14 x^ . i s a k x 1 vector of explanatory variables at time t , i s a random error , with Eu = 0, and 3 i s a k x 1 vector of regression coe f f i c i en t s . Appropriate explanatory variables x ^ include dummy variables which take the value 1 for the various days of the week, months of the year, pay days and holidays, and are zero otherwise. That i s , i f x// = ( X H » " * ' X f c t ^ ' w e m a^ n a v e' f ° r e x a m P l e > 1 i f day t i s a Tuesday x l t = * 0 otherwise. Another component that must be considered for x^ i s the level of interest rates, since c l a s s i ca l economic models regard i t as a determinant of demand for money. There are several problems with introducing interest rates into the model. F i r s t , over the short three year period of ava i lab le data, interest rates are highly co l l i nea r with time, so i t would be d i f f i c u l t to d i st inguish growth effects from interest rate e f fec t s . Second, i t i s d i f f i c u l t to decide which interest rate series to use. Certa in ly the interest rate most l i k e l y to influence a cred i t union member's demand for cash i s the interest rate offered on demand deposits. However, these rates are changed infrequently and any s t a t i s t i c a l l y estimable effects (over a three year period) would be spurious. It i s d i f f i c u l t to argue that other interest rate ser ies, such as commercial paper rates, have any observable e f fect on demand for currency. Th i rd ly , i f some macro-economic model were ava i lable to express demand for money in terms of interest rates, such a model would be oriented towards money in the form of demand deposits 15 and other major forms of cash, rather than currency. It i s not necessari ly true that demand for currency is as strongly affected by general economic conditions as the demand for other forms of money. Thus, for operational s imp l i c i t y , interest rates were excluded from the model. The model also must explain the h i s t o r i c growth in cash demand. One possible way to model th i s i s to have some explanatory var iable that increases over time (such as t ) . This would only model an addit ive growth ef fect and f a i l to model any increase in the effects of the dummy explana-tory variables over time. That i s , i f general cash demand increases over time, one would also expect that the extra cash demand on a payday for example also should increase over time. To model t h i s , one could allow polynomial time trending of the coe f f i c i en t s . That i s , consider the model y t = x t ' Bo + x t ' B i t + ••• + x t " B p t p + u t (t = 1,---,N), where the x^'s are vectors of, say, 0-1 dummy variabiles.and the B's are k x 1 vectors of regression coef f i c ient s which are mul t ip l ied by powers of t . The degree of the polynomial p i s t y p i c a l l y less than or equal to 2 (quadratic time trending). In e f f ec t , each coe f f i c i en t in B i s allowed to fol low a trend over time, where B = Bo + B i t + B_t 2 + ••• + 3 t p . P This model has the l inear form h = w t + u, 16 where w, f ( x u , t x ] t : (xkt, txkts -.tPxu) \t)' Quadratic time trending, for example, t r i p l e s the number of coef f i c ient s to be estimated, which would y i e l d low s igni f icance levels for 3 i f there are a large number of explanatory var iables. Moreover, polynomials are poor for extrapolat ive forecasts. That i s , polynomials may have maxima or minima for t > T which simply resu l t from the need to f i t high order derivat ives in the range 1 < t < T, and do not in any way r e f l e c t any predict ive power of the polynomial model. For example, f i t t i n g the simple trend model (for scalar 3o, 3 i , . 3 2 ) y t = 3o + S i t + 3 2 t 2 + u t to weekly cash flows y^ resulted in a polynomial that predicted a peak in demand in August 1976 (just s l i g h t l y beyond the data range)! C lear ly such growth models are inappropriate. As the f i n a l choice to model the growth, l e t us simultaneously consider a related problem, that of heteroscedast ic ity. I n i t i a l regressions indicated that o^ 2, the variance of u^ increases over time and, indeed, i s proportional to d t 2 , the square of the general level of cash demand, at time t . That i s , o^2 d t 2 . Without loss of general i ty, we may take d t to be the general level of weekly demand. This suggests using a mu l t i p l i c a t i ve growth model to simultaneously model growth and eliminate heteroscedast ic ity. In the l inear model (2.3.1), weighted least squares i s required to eliminate 17 heteroscedast ic ity. In th i s case the appropriate weights are 1/d^- The mode.l i s then y t 1 - ± = x t ' 6 + u t (t=l,..-,N) (2.3.2) or, W = Z3 + u (2.3.3) where W i Z i ' u / d i 1 w = • • -VdJ • • , z = • • 1 k, x N J — • • u = • • — • • and E(U) = 0 cov(U) = q 2 I This can be rewritten as y t = d t ( z t ' B + u t ) (2.3.4) which i s a mu l t i p l i c a t i ve form. Note that w^ = y^/dj. i s the cash flow at time t expressed as a proportion of the general trend level of weekly cash demand, so we can think of w^ as being normalized cash demand at time^t. The z. should be selected so as to exh ib i t no growth over time (that i s , 18 as 0-1 dummy var iab les ) . The d^ provide for growth of y t and el iminate heteroscedasticity at the same time. Other possible sets of weights {dt> which in some way indicate the s ize of the branch may be considered, such the number of members or the to ta l amount of deposits. Here i t i s again important to consider whether h i s t o r i c a l re lat ionships between the " s i z e " measures and demand w i l l continue into the future. If care i s not taken to exclude inact ive members from a count of members, changes in the proportion of act ive members on the r o l l s w i l l d i s to r t the re lat ionship between cash demand and number of members. The branch opened a new bui lding in May 1973, and i n i t i a l l y experienced rapid growth for some time afterwards, but the rate of growth has decreased since then. In the early part of the period, most members were new and act ive, but a f ter some time, many members, became•inactive, making member counts a poor measure of demand l e v e l . The magnitude of the l i a b i l i t i e s of the branch i s also ind icat i ve of s i ze , but th i s i s not l i k e l y to be a stable predictor of cash flow, since l i a b i l i t i e s can vary depending on the various retirement savings programs, home ownership savings programs, etc. that may be in vogue from time to time. That i s , at various times members may change the i r to ta l deposits without changing the i r demand for cash. Thus, the weight d^ is chosen to be proportional to the general trend level of cash flows at time t . In order to model the cash flow trend, l e t us examine the two main h i s t o r i c reasons for the increasing cash flow trend. These are the increase in nominal demand due to i n f l a t i o n and the increase in real demand due to growth of the branch. It would be f o l l y to t ry to forecast i n f l a t i o n , but th i s i s not necessary since Consumer Pr ice 19 Index f igures are avai lable monthly and l i near extrapolation beyond the l a tes t two months of unavailable data provides an excel lent forecast of the CPI level (although, of course, extrapolation i s a poor predictor of changes in CPI l e ve l s ) . It i s necessary, however, to forecast levels of nominal cash demand. Thus, l e t trend level of real weekly cash demand at time t , CP index at time t CP index for January 1976 ' trend level of nominal weekly cash demand at time t , expressed in January 1976 do l l a r s . Then d t = P t d t • (2.3.4) Estimation of the trend now reduces to estimation of the trend in real demand d^. The real weekly cash flows generally increased over the period for which data were ava i lab le , but the growth rate tended to decrease over the period. Indeed, regressions to f i t piecewise l inear trends in time showed a monotonic increasing but concave cash trend. This i s s imi la r to the general a c t i v i t y levels of other branch operations because rapid growth followed the opening of a new bui lding in May 1973, but by 1976, further growth was dampened by the general physical constraints of the bui lding (e.g., t e l l e r wicket space i s now l imited and long l ine-ups at the t e l l e r s ' wickets act to discourage members from using the branch cash serv ices) . Thus, growth in real cash demand appears to be approaching a capacity con-s t r a i n t , as in Figure 1, and th i s should be incorporated in the forecast model. 20 d t capacity constraint 1973 1974 1975 1976 Figure 1. Growth of Real Cash Demand. A simple way of modelling such an asymptotic constraint is by a su itably scaled and located rectangular hyperbola, namely c -- k t + t. (2. where c i s the capacity constra int, k i s a pos i t ive scal ing constant, and to i s a relocation parameter for time (t + t 0 > 0 for a l l t in the model) Johnston [1972, p. 52] suggests such reciprocal transformations with the 1 a r t i f i c i a l r e s t r i c t i o n t 0 = 0. This i s a l inear function of can be f i t t e d by least squares. That i s , l e t t + t ( and _ nominal weekly cash flow at time t t p. For a f ixed value of t 0 21 k + e (2.3.6) t + t 0 t where, for e = 0 E(e) = 0 and E(ee') = Z G = a for some a > 0. As in the regression (2.3.3) the variance of e^ i s pro-portional to the square of the trend level in cash demand. Since E(e) = 0, the least squares estimators c and £ are unbiased, but i n e f f i c i e n t under the heteroscedast ic ity, which merely places more weight on l a te r observations. This i s the most desirable departure from homoscedasticity, in th i s case. To select the time locat ion parameter t 0 , i t suf f ices to estimate (2.3.6) fo r various values of t 0 and choose t 0 to minimize the sum of squared res iduals. The estimated weekly real cash trend i s k t + t 0 . and the weekly nominal cash trend i s 22 These values of d t can be used in place of d^ in the weighted least squares model (2.3.3) to obtain the approximate generalized least squares estimate, 3 = ( X 1 E~ 1 X ) " 1 X ' Z u Y = ( Z ' Z ) " 1 Z'W where d i 2 0 y i / d i • • • W = • t • lo djiJ ly N /d N j For a rea l i za t i on of the dummy explanatory variables zj., we have an estimate of nominal cash demand at time t of h = d t \ C -t + t r (2.3.8) where y^ = cash flow forecast for time t , = CPI at time t  p.t CPI for January 1976 ' k d t = c • = forecast trend level of real weekly t + t 0 cash flow at time t , z_ = 3 = vector of explanatory dummy variables for time t. estimated regression coe f f i c i en t . Analogously, l e t y^ be the (unobtainable) estimate of y^, given perfect knowledge of the trend. That i s , l e t 23 d t W t z t e (2.3.9) The fol lowing proposition gives consistency results for the forecasts and variance of forecast errors. Proposit ion: The predictions y t and y^ are both consistent predictions of y^ and a consistent estimator of the variance of y^ i s d 2 s 2 [ l + z^ (Z ' Z ) " 1 z t] (2.3.10) where s 2 i s the residual variance obtained by estimating (2.3.3) with the estimated weights d^ instead of d^. PROOF: We have seen that c and k are unbiased. To see that they are consistent, l e t fl (1 + to)" 1 V Nx2 [1 (N + t o f ' J Then = (V'V)" 1 V d i and the covariance matrix of 1 s (V 'V ) " 1 V E V (V 'V ) " 1 (2.3.11) 24 By interpret ing enlargement of the sample s ize N to mean the observation of a sequence of independent but s tochast i ca l l y ident ica l c red i t union branches over the same time period, the regressor matrix V i s "constant in repeated samples" (Theil [1971, pp. 364 -365 ] ) , so that N ' M V V ) and N" 1 V E g V both converge to pos i t ive de f i n i te matrices. We can rewrite (2 .3 .11 ) as N _ 1 [ N ( V V ) ' 1 ( N _ 1 V Z g V) N (V ' V ) - 1 ] and th i s tends to 0 as N -> °°. Thus, the variances of c and k tend to 0 for large N, 3 establ i sh ing the consistency of these estimators. Thus, the elements of K 2 0 ) are consistent estimators of those of. I U = E(U'U). Also, the elements of these matrices are bounded (in probab i l i ty ) since the trend levels d^ are bounded by the capacity constraint c. Define the pos i t ive de f i n i te matrix Qy = l im N" 1 a 2 Z'Z = plim N" 1 X' E~ 1 X. Then by the consistency and N-*» N-*» U boundedness, we also have lim N _ 1 (X ' E~ 1 X) = Q . Then p l im(N _ 1 X' E"1, X,— . N" 1 X1 E"1 X) = Q - = 0. The same boundedness and consistency argu-ments establ i sh that plim N _ i X ' f Z ' 1 - E" 1) U = 0 and plim N" 1 U 1 ( E _ 1 - E" 1) U = 0. N-*» u Hence by The i l ' s Theorem 8.4 [1971, p. 339] , plim /N ( i - 6) = 0, s 2 i s a N-*» consistent estimator of a 2, and the matrix s 2 (Z 'Z ) converges in p robab i l i t y to the covariance matrix of 3. In pa r t i cu l a r , note that 3 i s consistent. Thus w^ = z| 3 i s a consistent estimator of w^, with variance given asymptotically by 25 s 2 [ l + z | ( Z ' Z ) - 1 z t ] . /\ /\ s\ Since y t = and d t i s a consistent estimator of d t , formula (2.3.10) must be the asymptotic variance of y^. The consistency of y^ and y^ follows from that of (3, c and k. This establishes the proposit ion. We may conjecture that the asymptotic variance (2.3.10) of y^ i s also asymptotically the variance of y^. At any rate, that variance w i l l be a meaningful pract ica l approximation to the variance of y^. Thus, we have generalized least squares estimates for cash demand, along with asymptotic results about the estimators and the i r variances. This completes our analysis of the regression model. 2.4 Summary In th i s chapter we have studied two s t a t i s t i c a l models which may be used to forecast cash demand. The f i r s t was a Box-Jenkins time series model in which forecasts of future cash flows are weighted sums of previous cash flows. The second model was the product of a non-linear trend model, and a weighted least squares model which uses dummy regressors to indicate days of the week, months of the year, pay days and holidays. The s t a t i s t i c a l estimation of both models w i l l be discussed in the next chapter. 26 FOOTNOTES TO CHAPTER 2 V o i l l u s t r a t e , consider an MA process (q=l) o f - f i r s t successive di f ferences: y t - y t - 1 = .6 + u t - e V l = 6 + ( l - eB) u t •"• u t = y t " y t - i " 6 + 9 Vl Substituting the second equation recurs ively into the f i r s t to successively eliminate ut_-|, u t _2>* ' * y i e ld s : h = y t - l + U t + 6 " 6 ( y t - l " h-Z ~ 6 + 9 u t - 2 J = u t + (1 + 6)6 + (1 - 6 )y t _ 1 + 9 y t _ 2 - 0 2 u t _ 2 =r • • • = u t + ( l + e + e 2 + e 3 + . . - ) 6 + ( l - 0 ) ( y t _ 1 + 0y t _ 2 + 02yt_3+---) Since E u t = 0 a l inear unbiased forecast of y t , given vt_-]> ^ t - 2 ' " * * i s y t = (l+0+02+---)6 + (1-6)(y t _ 1 + 0y t _ 2 + 9 2 y t _ 3 + ••• ) Under mild r e s t r i c t i on s on the y ' s , th i s converges i f f | e | < 1, i . e . , i f f 1 - 6B has i t s zero outside the unit c i r c l e . Also, note that for 6 = 0 , y. i s simply an exponentially weighted moving average of e a r l i e r observations, one of the c l a s s i ca l forecasting models. 2 The Consumer Price Index to be used i s the S t a t i s t i c s Canada seasonally unadjusted monthly CPI for Vancouver. One may argue that a seasonally adjusted index would be better, since i t would not d i s to r t some <~~ of the seasonal var iat ions in cash f low that should be estimated in the ' model,' but such a f igure i s not ava i lab le. More importantly, one might argue that the CPI bundle of goods includes items that are paid for by cheque as well as cash. None of the indices ava i lable r e f l e c t cash payments better than the general CPI, however. 27 °The interpretat ion of V as being constant in repeated samples only indicates that trend estimates are consistent for time points in or short ly a f te r the 3 year data range, N. To test the accuracy of extrapolat ive forecasts the model w i l l be f i t t e d in Chapter 3 on the f i r s t two years of data and tested on the th i rd year. This i s a useful-adjunct to the-consistency resu lts presented here which are only of an interpo lat i ve nature.-A l te rna t i ve l y , one can interpret sample enlargement as the addit ion of more observations for the same cred i t union as time proceeds. To consider th i s type of consistency, f i r s t note that N V'V N I (to+t)' t=l I (to+t)' t=l N I (to+t) t=l •1 -2 Let (V'V) -1 T i n n h 2 (m 2i m 2 2 j The m22 1 E(t 0+t)- N - M z U o+t)- 1) The f i r s t term in the denominator converges to a f i n i t e l i m i t as N °°. 1 N 1 The square root of the second term i s Tfj" I (t 0 +t) w n ^ c n D e n a v e s asympto-t i c a l l y l i k e - -j--t=l dt 7N In t0+N to+1 -> 0 as N -> °°. Therefore, as 1 N •+ » , m 2 2 / 0 and hence (V'V) f 0 However, m l a = — ^^t) E ( t 0 + t ) - 2 - (Z(to+t)- 1)" 0 as N -> 0 0 since the numerator and f i r s t term in the denominator have f i n i t e l im i t s and the second term in the denominator has an i n f i n i t e l i m i t . S im i la r l y m 1 2 = m 2 1 = z ( t o + t ) - 2 / ^ t o ; l ) - i _ £ ( t o + t ) - i - - as N -since the f i r s t term in the denominator tends to zero and the second term ro oi as N •> 0 0 for some a > 0. tends to ». Thus (V'V) -1 0 a 28 The elements of. V'EgV each behave asymptotically l i k e those of V'V since the d^ 2 diagonal elements of Z g are bounded. Hence V'E V ( V ' V ) - 1 tends to a f i n i te -va lued matrix, as N -> °° whose diagonal entr ies are non-zero. - v Hence as N -*- °°, the matrix (V 'V)" V'Z V ' (VV) " 1 tends to a matrix of the form '0 0] 0 a 1 for a1 > 0. Thus, the estimator of the capacity constraint c is consistent, but the estimator k, which describes how quickly the capacity is approached, i s not consistent. This means that for projections of growth into the distant future, the forecast of ultimate capacity i s "accurate" (consistent). The projected rate of growth to that capacity may be over or under estimated, but forecast errors induced by th i s are bounded since the projection of capacity is "accurate." Chapter 3 ' FORECASTING THE DEMAND FOR CASH: APPLICATION 3.1 Introduction In th i s chapter, the two cash demand models developed in Chapter 2 are estimated s t a t i s t i c a l l y . : , The next .section i s devoted to a discussion of the data and some of the special problems associated with them. Section 3.3 presents the results of Box-Jenkins analys is. It w i l l be found there that attempts to estimate a stat ionary, i nve r t ib le time series model f a i l . Section 3.4 presents the estimation of a cash flow trend, which i s used in a weighted least squares l inear model with dummy variables to forecast cash demand. This results in an estimable model, the forecasting a b i l i t y of which is studied by f i t t i n g the model on the f i r s t two years of data and te s t -ing the resu l t ing th i rd year forecasts for bias and other sources of error. 3.2 The Data Data on cash balances and cash orders for the cred i t union branch were analyzed for the period May 1973 to Apr i l 1976. This period com-menced with the opening of a new bui ld ing, and ea r l i e r data are l i k e l y to be s t ruc tu ra l l y d i f fe rent at least in terms of the general level of cash flow, i f not also in the pattern of var iat ions in the flows. 29 30 Closing t i l l cash balances were obtained for the 768 days from General Ledger accounts and information on cash de l i ver ies was obtained from del ivery invoices prepared by the B r i t i s h Columbia Central Credit Union, through which cash orders were placed. This allowed the computa-t i on of da i l y cash flows. In add i t ion, during December 1974 several deposits were s o l i c i t e d from a large local merchant to cover cash shortages. These were treated in the same way as regular cash de l i ver ies for the c a l -culat ion of da i l y cash flows. Of the resu l t ing 767 da i l y cash flows, a l l but 36 were negative (outflows), because the branch dealt pr imar i ly with consumer accounts rather than commercial accounts. Co l lect ion of the data was actua l ly quite complicated because maintenance of the General Ledger accounts was oriented towards ensuring correct figures at month end and many discrepancies crept into the accounts during the month. Three general types of discrepancies were observed, only some of which could be corrected. The f i r s t type of discrepancy arose when two t e l l e r s exchanged cash between themselves and o f f se t t ing debits and credits to the General Ledger were not made on the same day. A l te rna t i ve l y , i f the head t e l l e r was absent for several days and did not post the changes in the treasury cash balance,^ while other t e l l e r s withdrew money from the treasury (and posted the i r credits immediately) the same type of discrepancy would occur. These discrepancies resulted in a pos i t i ve (negative) flow one day followed by a negative (posit ive) flow of equal magnitude a few days l a te r . The methods of making General Ledger entries varied from time to time and for the middle half of the data period, each t e l l e r ' s c los ing balance was posted d i r e c t l y to the General Ledger (rather than as an aggregate with a l l 31 other t e l l e r s ) . Frequently th i s provided enough d e t a i l , in conjunction with the balance in the t e l l e r ' s interchanges account (which normally is zero), to i n fe r the correct adjustment for these discrepancies. Such adjustments could not be made, of course, for the time periods when a l l t e l l e r s were posted to the General Ledger together. The second type of discrepancy occurred when journal entr ies were inadvertently posted to the wrong account and corrected a few days l a te r . This also would resu l t in an pos i t ive (negative) flow one day, followed a few days l a te r by a negative (posit ive) flow of equal magnitude. L i t t l e could be done to eliminate th i s type of discrepancy. A th i rd type of discrepancy could have resulted from cash del ivery invoices being f i l e d under the wrong branch. Indirect evidence of the p o s s i b i l i t y of such discrepancies was provided by the observation that • invoices for other branches were found incor rect l y f i l e d with those of the branch involved in th i s study. This type of discrepancy would resu l t in a pos it ive flow one day, which, unl ike the f i r s t two types of d i s -crepancies, would not be followed by a compensating negative flow a few days l a t e r . If the f i r s t two types of discrepancies occur randomly and independently of the dummy regressor variables ( indicat ing days of the week, months of the year, pay days and hol idays), then no bias w i l l be introduced into the estimates 3 and y of Section 2.3, although the estimated residual variance s 2 w i l l be biased upwards. The second type of discrepancy w i l l bias some or a l l of the regression coef f i c ient s (for demand) downwards, although the bias i s not l i k e l y to be serious since only 3 orders were found m i s - f i l ed with the branch studied. 32 3.3 Estimation by Box-Jenkins Techniques When the theory of the Box-Jenkins technique was presented in Section 2.2 i t was pointed out that, with s i gn i f i c an t autocorrelat ion information at short and long lags, a time series model for da i l y cash flows might be hard to f i t and that aggregation to weekly cash flows would present fewer problems for estimation. If a weekly model cannot be f i t t e d , neither can a da i l y model. Thus, the i n i t i a l resu l t s presented here are for weekly cash flow y^. ^ Since the cash flows, generally increased over time, f i r s t d i f -ferences were required to achieve s t a t i ona r i t y . Consecutive and 4 ^ order seasonal d i f ferencing were considered on both raw and logged data. Note t h that 4 order seasonal d i f ferencing corresponds approximately to a seasonal lag of one month. The autocorrelations and pa r t i a l autocorrelations of these differenced series suggested a 4 ^ 0 r order moving average process. These models are of the form 0 - B S ) y t = 6 + ( l-OiB-ezB 2 - e 3 B 3 - 0 ^ - 0 5 B 5 ) u t where s = 1 or 4, y^ ..= weekly cash flow or logged weekly cash flow, and 04 j 0 but perhaps 0 5 = 0. For raw weekly cash flows with s = 4 and 0 5 = 0, the model was f i t t e d by the Marquardt least squares algorithm with i n i t i a l estimates obtained by solving a system of 1inear Yule-Walker type equations involving the inverse 2 autocorrelations and the MA parameters {Q-}. This y ie lded the form 33 ( l - B ' + ) y t = (1 - .117 B + .250 B 2 - .266 B 3 - .698 B 4) u t + 2479 R2 = .382 (1.84) (4.25) (-4.65) (5.22) (5.05) (3.3.1) The t - s t a t i s t i c s (147 degrees of freedom) for each parameter are given in parentheses, and a l l except that for 0^ are to test the hypothesis that the parameter equals 0. The t - s t a t i s t i c for G 4 i s to test the hypothesis e 4 = i . At f i r s t sight the "model (3.3.1) appears to be quite reasonable in that a l l coef f i c ient s but 8 i are s i g n i f i c an t l y d i f fe rent from zero. However, to check the i n ve r t i b i 1 i t y of the process (which is essential to forecast ing), one must examine the factored from of 9(B): 0(B) = -.698(B + 1.34)(B - 1.05)(B + .05 - 1.01i)(B + .05 + 1.01i) (3.3.2) Note that the l a s t three roots of 0(B) are v i r t u a l l y on the unit c i r c l e , so that forecasts based on (3.3.1) w i l l be unstable. Moreover, we can approximate 0(B) as fol lows: 0(B) = - (B + 1)(B - ,1)(B - i ) (B + i ) = 1 - B1* (3.3.3) The only major differences between (3.3.2) and (3.3.3) are the constant coe f f i c i en t and the f i r s t factor . But (3.3.3) i s simply the di f ferencing polynomial in (3.3.1). Substituting the approximation (3.3.3) into (3.3.1) and cancel l ing the factor 1 - BK to el iminate the over-differencing y ie lds y t = u t + 6'. 34 This implies that y t i s e s sent ia l l y white noise. Thus, the autocorrelations created by d i f ferencing constituted almost a l l of the autocorrelations ava i lab le for estimating (3.3.1) and induced an MA polynomial that i s v i r t u a l l y ident ica l to the AR di f ferencing polynomial. Of course, the t - s t a t i s t i c s of (3.3.1) indicate that the MA polynomial i s not exactly (1 - B1*), but the only difference between the polynomials i s in the factor (B + 1.34). A l l the other factors of 1 - B1* cancel exactly. When a l l of the other plaus ible weekly models that were mentioned e a r l i e r were f i t t e d exactly the same phenomena happened: The factored MA polynomials had a l l of the i r roots e s sent ia l l y on the unit c i r c l e and the factors included those of the AR polynomial (1 - B) or (1 - B1*), which indicates over-di f ferencing, even though dif ferencing i s required to achieve s ta t i ona r i t y . Thus, from a Box-Jenkins viewpoint, there is. not enough informa-t ion in the autocorrelations of weekly cash flows to overcome the white noise, and the time series model i s not estimable. One might conjecture that i f the trend i s eliminated from cash demand, as with the trend model discussed in Section 2.3, a major source of the random error i s eliminated and the remaining series may be estimable. Since the real Requirement i s to have a model for da i l y rather than weekly cash flows, th i s las t hypothesis was not tested for weekly cash flows but for da i l y cash flows deflated by the trend. That i s , in the notation of (2.3.3) l e t w^ = y t / d t be the da i l y cash flow y^ deflated by the trend d^. For a l l combinations of f i r s t order and seasonal (s = 5 days) d i f ferenc ing, the series of autocorelations f a i l e d to die out at large lags, ind icat ing that there was s t i l l too much information at long lags to allow the f i t t i n g of a parsimonious Box-Jenkins model. 35 In summary, the Box-Jenkins estimation f a i l e d because the series of 'Cash flows i s non-stationary. The non-stat ionar ity occurs in two forms. One source of non-stat ionar ity is the trend. The other source of non-stat ionar ity i s a seasonal d r i f t related to the autocorrelations at long lags. 3.4 Estimation by Regression Techniques Since the attempt at Box-Jenkins estimation of the demand for cash was a f a i l u r e we consider here a d i f fe rent co l l e c t i on of explanatory variables and use the model developed in Section 2.3, which employs a non-linear trend estimate mul t ip l ied by a weighted least squares l i near model. The non-linear trend (2.3.6) of 156 real weekly cash flows, d^, was f i t t e d by least squares y ie ld ing 170,000 - 28.9 x 10 6 ( t + 240)" 1 (3.4.1) (26.32) (-9.00) s 2 = 9.9 x 10 8 where t i s the number of working days from May 1, 1973 and t - s t a t i s t i e s are given in parentheses. As expected, the residuals displayed increasing variance over time. As a re su l t , the estimated t - s t a t i s t i c s and variance of the error term are a l l biased. The estimated coe f f i c i en t s , however, are unbiased and consistent. The asymptotic capacity level of real weekly cash demand that i s forecast by (3.4.1) is $170,000 (expressed in January 1976 do l l a r s ) . Sub-s t i t u t i n g t = 767 y ie lds an estimated trend level for May 1976 of $141,000 36 or 83% of the asymptotic level of cash demand. This f igure is thought to be reasonable in l i gh t of the actual remaining potential for growth at the branch. To test the goodness of f i t for (3.4.1), a quadratic polynomial in time was added to the regression and i t did not s i g n i f i c an t l y increase R2. F i t t i n g the model for the f i r s t and las t halves of the data separately did not resu l t in s i g n i f i c an t l y d i f fe rent estimates for the regression coef f i c ient s c and k. (Again, we must emphasize that standard s ign i f icance tests are not meaningful in such an heteroscedastic s i tua t i on , and these statements of s ign i f icance are heur i s t i c only.) Note that since precise ly three years of data were used to estimate the trend, no bias w i l l be introduced as a resu l t of seasonal f luctuat ions in demand. The trend estimates mul t ip l ied by the CPI levels p t were then used as weights d^ in the generalized least squares model (2.3.3) to estimate the f luctuat ions in cash demand about the trend l i n e . The inde-pendent variables were dummy variables ind icat ing the incidence or proximity of days of the week, months of the year, semi-monthly and monthly pay days and holidays. An i n i t i a l regression was run using a l l of these variables as regressors and the residuals were examined for evidence of the f i r s t two types of data discrepancies discussed in Section 3.2. Such discrepancies manifest themselves as reversed pairs of residuals of large magnitude, separated by only a few days, and correspond to a bookkeeping error and i t s correct ion. In order to reduce any tendency of these errors to bias the estimated regression coe f f i c i en t s , adjustments to the data were made whenever there was a pair of residuals with opposite 37 signs, each having a magnitude in excess of 3 standard errors of the regression. The adjustment was made so that the sum of the cash flows was unchanged and the resu l t ing residuals were equal. Such adjustments were made in 8 cases, out of the 759 data observations that were used for the demand estimation. The f i n a l form of the demand equation (2.3.3), with k = 28 explanatory var iables, i s presented in Table 1. To avoid mult i -co l l i n e a r i t y the regression was run with no dummy var iable for January. Since the other winter months, November and February, had coef f i c ient s that were i n s i g n i f i c an t l y d i f fe rent from zero, they were dropped from the model and hence equated to January. The r>.. regression coe f f i c ient s for Tuesday and Wednesday mid-month and month-end pay dates were es sent ia l l y i d e n t i c a l , so they were included as one dummy var iable. A few comments about the regression coe f f i c ient s are in order. Recall that the equation i s for da i l y cash demand expressed as a proportion of the general weekly trend level of cash flow, and the explanatory variables are e i ther 0 or 1. Thus the i t h regression coe f f i c i en t indicates the e s t i -mated increase in cash demand (expressed as a proportion of the weekly trend leve l ) that i s caused by incidence of the event for-which the i variable i s an indicator function. For example, an ordinary Tuesday in January that i s not on or pr ior to a pay day w i l l have a estimated cash demand of 5.32% of the general weekly trend l e ve l . From the f i r s t f i ve coe f f i c ient s we can see that the demand for cash r i ses through the week, peaking on Saturday (even though Saturday has shorter hours of operation than the other days). 38 Table 1 Estimated Regression Coeff ic ients for Daily Demand Estimated Coeff ic ient Standard Error of Dummy Variable z.^ i s a 0-1 Indicator Function for : .0532 .0844 .1139 .2358 .2848 .0178 .0375 .0233 .0349 .0530 .0193 .0210 .0291 .1249 .0931 .1285 .1817 .0707 .1345 .0936 .0820 .0120 .0114" .0123* .0123' . 0 1 2 1 ' .0138 .0144* .0139* .0140* .0139* .0136 .0140 .0137* .0148* .0198* .0440* .0439* .0261 . 0263 .0260* .0254* ** ** * ** ** ** * * * ** ** ** * * Tuesday Wednesday Thursday Friday Saturday March Apri 1 May June Ju ly August September October December Tuesday or Wednesday and Pay Day Thursday and Ktd-Mbntlr Pay.Day Thursday .and'Month-End./Pay Day Friday and Mid-Month Pay Day Friday and Month-End Pay Day Saturday and Mid-Month Pay Day Saturday and Month-End Pay Day CONTINUED 39 Table 1 (Continued) Coef f ic ient Subscript i •Estimated Coeff ic ient 3 Standard Error of 3 Dummy Variable i s a 0-1 Indicator Function f o r : 22 23 24 25 26 27 28 -.0207 -.0505 -.0377 .0481 .0630 .0546 .1336 .0212 .0194 .0190 ** .0165 ** ,0171 ** ,0197 ,0260 ** Thursday and Pay Day Occurs Ea r l i e r in Week Friday and Pay Day Occurs Ea r l i e r in Week Saturday and Pay Day Occurs Ea r l i e r in Week Tuesday and Pay Day Occurs Later in Week Wednesday and Pay Day Occurs Later in Week Thursday and Pay Day Occurs Later in Week Holiday Occurs Next Day (May, June, Ju ly , August, September and December only) R2 = .5468 759 Observations Durbin Watson S t a t i s t i c = 2.08 Standard Error of the Regression-s S ign i f i cant at 10% level .0950 ** S ign i f i cant at 1% level NOTE: Mid-month and month-end pay days occur on the 15th or month-end, respect ively, unless that day i s a holiday or on a weekend, in which case the pay day i s sh i fted ahead. Monday pay days are recorded on Tuesday. A Saturday pay day i s a day a f te r a Friday pay day. 40 Because of Christmas, December has by far the largest cash demand, but the other winter months have l i gh t demand. Demand in Apr i l i s high, l i k e l y because the branch is located near a univers i ty where cash demand would increase as students and facu l ty prepare for vacations. As to be expected, there i s a sharp increase in the demand for cash on monthly and semi-monthly pay dates. For Thursday and Friday pay dates, the cash demand was much heavier at month-end than at mid-month, probably because many people are paid only at the end of the month. A priori i t was ant ic ipated that on Fridays and Saturdays fol lowing a pay day there would be an increased demand for cash since people would wait un t i l these convenient days.to cash the i r cheques. Hoever, the resu lts indicate that people generally cash the i r cheques r ight on a pay day or perhaps even cash post-dated cheques before pay day, since there i s a s i gn i f i cant increase in demand on the days preceding a pay day. As a re su l t , there i s actua l ly a s i gn i f i cant decrease in demand on Fridays and Saturdays fol lowing a pay day. As to be expected, holidays precip itated a large increase in . the demand for cash on the day pr ior to the holiday. This e f fect was not observed for Easter, Thanksgiving or Remembrance day holidays, however, so these holidays were excluded from consideration by only considering holidays in May, June, Ju ly , August, September and December; The Durbin-Watson s t a t i s t i c shows no s i gn i f i cant autocorrelat ion of the errors. Furthermore, a Box-Jenkins i d en t i f i c a t i o n of the residuals was performed and, with only one exception, a l l the autocorrelations and pa r t i a l autocorrelations out to 60 day lags were within a 95% confidence interva l about 0. In pa r t i cu la r , for the 5 day (1 week) lag, the . autocorrelation was .01 and the pa r t i a l autocorrelation 41 was 0. At the 5 day (2 week) lag the autocorrelation and pa r t i a l auto-cor re la t ion were both -.'02. This indicates ' that there i s no observable e f fect on cash demand as a resu l t of bi-weekly pay periods (as opposed to the semi-monthly pay periods used in the regression). Furthermore, the lack of any s i gn i f i can t autocorrelat ion structure indicates that the f i r s t two types of data discrepancies e ither are i n s i gn i f i can t or occur at random lags. There was no evidence of heteroscedast ic ity, except for an increased variance of the residuals around Christmas. A plot of squared residuals against squared estimates, however, indicated there was no general reason to believe that residual variance increases with the general level of cash demand. The data period was divided into two equal sub-periods and a Chow test was performed on the nul l hypothesis that the estimated coef f i c ient s were unchanged from one period to the next. The nul l hypothesis was barely rejected at the 5% l e v e l , but not at the 1% leve l . - On the whole, i t appears best to use the estimates based on a l l of the data and avoid any attempts to forecast a trend for each coe f f i c i en t . Polynomial time trend-ing i s l i k e l y the best"way of modelling a trend in the coe f f i c i en t s , but as discussed e a r l i e r , polynomial projections can be very unre l iab le. P lot t ing the residuals of the regression against a cumulative normal d i s t r i bu t i on revealed-a f a t - t a i l e d d i s t r i bu t i on , somewhat skewed to the r i ght . After pooling the residuals into 20 classes in the +3.5 standard error range and performing a chi-square goodness of f i t te s t , the normality hypothesis had to be rejected at the 1% l e ve l . S im i la r l y a Kolmogorov-Smirnov goodness of f i t test implied reject ion of the normality 42 hypothesis at the 1% l e v e l . This i s largely a resu l t of the data errors that arose from the bookkeeping discrepancies. The same chi-square and Kolmogorov-Smirnov tests implied re jec-t ion (at the 1% leve l ) of the hypotheses that the residuals were d i s t r i -buted as Poisson or binomial. In order to test the forecasting a b i l i t y of the model, the f i r s t two years of data were used to re-estimate the coef f i c ient s of the model, using exactly the same methods as discussed for the f u l l set of data. This model was used to forecast nominal cash flows in the th i rd year. It was assumed that CPI data i s ava i lab le with a 50 working day lag, so in forecasting the cash demand in the th i rd year, instead of using the actual CPI, the CPI was estimated by a l i near projection of the CPI trend over the 250 working days pr ior to the la tes t ava i lab le CPI f igures. Thus, i f p^ i s the published CPI level for the month which includes day t , the CPI estimate i s : n - n fi• + 1 f P t -50 " pt-300) p t " p t - 50 1 5 _ — b - ~ ~ t i ou I Pt-300 >• V. J This CPI estimate i s used in the usual manner to forecast cash demand in the f i na l year. Theil [1966, pp. 26-36] has suggested several techniques for analyzing the predict ion errors of a forecast method. Let predict ion for day t ( t= l , ' * * ,n) real ized value for day t (t=l,• • • ,~n) . 43 Then the mean square e r r o r o f p r e d i c t i o n i s MSEP = - E(y\ - y j : L e t t i n g 9 = £ i y t , y = 1 z y t and i z ( y t - y ) ( y t - y t ) and r - - — - - — s p s A the f o l l o w i n g d e c o m p o s i t i o n h o l d s : MSEP = (y - y ) 2 + ( s p - s A ) 2 + 2(1 - r ) s s A , (3.4.5) L e t u m = $ ' >-et u M S E p s ( s p , - S A ^  U MSEP ' a n d c 2 ^ - r ) s p S A  U MSEP These a r e termed i n e q u a l i t y p r o p o r t i o n s , and u m + u s + u c = 1. u m i s t he p r o p o r t i o n o f MSEP t h a t r e s u l t s f rom b i a s i n the g e n e r a l f o r e c a s t l e v e l , and f o r a good f o r e c a s t , i t s h o u l d be s m a l l , u i s t he p r o p o r t i o n due t o a poor f o r e c a s t o f the v a r i a n c e o f t he s e r i e s , and s h o u l d a l s o be m i n i m i z e d , u i s t he v a r i a t i o n due to unequal c o - v a r i a t i o n o f f o r e c a s t s and r e a l i z a t i o n s , and i s someth ing o v e r wh ich the f o r e c a s t e r has l i t t l e c o n t r o l ( e x c e p t , 44 perhaps, to use more explanatory variables in his forecasts). For the forecasts of cash flows in the l a s t year, MSEP = 1.729 x 10 8 u m = .0045 u S = .0628 u c = .9327 These results indicate that v i r t u a l l y no error arose because of bias in the general trend level estimates, and l i t t l e error resu lts from incor rect l y estimating the level of va r i a t ion . On the other hand, 93% of the error i s due to d i f fe rent covariat ion (es sent ia l l y , noise). These results suggest that the model performance could not have been improved without the use of more explanatory var iables. Another breakdown of the errors i s : MSEP = (y - y ) 2 + ( s p - r s f t ) 2 + (1 - r ) s f t 2 u R - " P ' r b f t (s„ - r s f t ) 2 MSEP" U MSEP Then u + u +.u = I. . If one regresses y t = a ± by t + e t ( t = l , " « , n ) 45 then perfect forecasts correspond to a = e^ = 0 and b = 1. C lear ly , e t f 0 in general, so th i s i s an i r reduc ib le component of forecast error. The proportion of MSEP due to th i s component i s u°. On the other hand, i f a d i f f e r s greatly from zero, there i s substantial bias in the forecasts m R and u i s large. If the regression slope b d i f f e r s greatly from 1, u m R i s large. For a good forecast u and u are small. We have already seen m R that u is smal l , and u = .0367 which i s also small. The mean absolute error of predict ion was $9084 while the mean error of predict ion i s only $883. This means that on average the model over-estimated the general trend level in the la s t year by less than $1000/day. As a benchmark we may consider a naive forecast model that simply extrapolates nominal cash flows l i n ea r l y from the f i r s t two years to the la s t year. For th i s model, the MSEP i s 3.000 x 10 8 . As an analogue to The i l ' s second U - s t a t i s t i c (Theil [1966, p. 28]), we may consider the ra t i o of the MSEP of the f u l l model to the MSEP of the benchmark. (The smaller the r a t i o , the better the forecast.) This r a t i o i s .58, ind icat ing that the f u l l model does indeed outperform naive extrapolation (which, of course, has a ra t i o of 1). One might have expected the r a t i o to be smaller, but the high level of white noise ( i . e . low R2 in the regressions), which affects both the model and the benchmark, prevents the ra t io from approach-ing zero. In summary the model performs quite well in tests of predict ive a b i l i t y and in pa r t i cu l a r , the estimation of the trend curve appears to be good. 46 3.5 Summary In th i s chapter, we have studied the resu lts of the estimation of the two main models of cash demand. The Box-Jenkins time series tech-nique was found to be inappl icable due to non-stat ionar ity of the cash flow data due to a growth trend and the presence of s i gn i f i cant autocorre-l a t i on information at very long lags. A sat i s factory mu l t i p l i c a t i ve model for cash demand was obtained in two parts. F i r s t , a trend for real cash demand that incorporated a capacity constraint was estimated. Then da i l y cash flows deflated by th i s trend were regressed on dummy variables ind icat ing days of the week, months of the year, paydays and holidays. The residuals of th i s regression were too f a t - t a i l e d to be normally d i s t r ibuted. This resulted in part from bookkeeping discrepancies which created pairs of reversed residuals of large magnitude. The second (mul t ip l i cat i ve ) model was tested by f i t t i n g i t on the f i r s t two years of data and studying the predict ion errors that re-sulted in the th i rd year. Most of the mean squared predict ion error was a resu l t of random noise, and very l i t t l e was due to bias or incor rect ly estimated scal ing of the var iat ion in cash flows. In short, the model i s good for predict ion purposes. The two-part mu l t i p l i ca t i ve model w i l l be used in the cash order algorithm to predict cash demand and provide a probab i l i t y d i s t r i bu t i on for the predict ion errors, which w i l l be used in in the determination of precautionary cash buffer s izes. 47 FOOTNOTES TO CHAPTER 3 The treasury is a balance of vault cash maintained by the head t e l l e r from which the t e l l e r s withdraw cash as they need i t . Cash parcels would go f i r s t to the treasury and then to indiv idual t e l l e r s as needed. Inverse autocorrelations were developed by Cleveland [1972] and are dual to the autocorrelations in the sense that reversing the roles of the AR and MA parameters also reverses the roles of the auto-corre lat ions and inverse autocorrelat ions. Chapter 4 THE CASH MANAGEMENT PROBLEM IN GENERAL 4.1 Introduction In t h i s chapter we shal l discuss the general considerations i n -volving the management of t i l l cash. Section 4.2 discusses some of the aspects that make t i l l cash management at the branch involved in th i s study d i f fe rent from other cash management problems and inventory problems in general. Important points raised there include the non-stat ionar ity of cash demand and the var iable lag times for del ivery of cash orders. Section 4.3 presents some cash management models* that-have been bu i l t on a steady-state framework, in which the basic structure of the problem does not change over time. Steady state models include those by Baumol, M i l l e r -Or r and Eppen-Fama. Models that give exact solutions for the i r optimal parameters are generally steady state. Section 4.4 casts the cash management problem as a general dynamic programming model which does not require any steady state assump-t i on . Optimal solutions are hard to calculate from th i s model, but the model is examined in Section 4.5 to f ind which approximations can be made to s impl i fy computation with a minimal loss of opt imal i ty . One of these approximations i s used in Chapter 5 to develop a cash order algorithm. 48 49 4.2 The Problem and Its Environment In th i s sect ion, the cash management problem i s defined and the general context of the problem i s presented. The objective of cash management i s to select a pol icy that w i l l minimize the present value of the expected costs of maintaining an i n -ventory of cash. These costs include the costs of ordering cash, as well as the opportunity costs of assets held in the form of non-interest earn-ing cash. This problem may be cas t . in the form of a recourse problem or a chance constrained problem. In the recourse problem, the objective func-t ion also includes a penalty for cash-outs (or cash shortages). In the chance constrained problem, the objective function only includes order costs and opportunity costs, but the objective i s minimized subject to the constraint of holding the probab i l i ty of a cash-out below some pre-determined level d . A more spec i f i c discussion of the recourse and chance constrained formulations w i l l ' be 'p resented in Section 4.4. The most important features of the problem are as fol lows: 1. The demand for cash i s a non-stationary stochastic process for which forecasts and the d i s t r i bu t i on of residual error are ava i lable (by the analysis of Chapter 3). 2. The d i s t r i bu t i on of forecast errors i s skewed to the r ight and too f a t - t a i l e d to be normal, nor is the d i s t r i bu t i on Poisson or binomial. 3. Cash flows are generally net outflows (only 30 net inflows in 767 da i l y observations).so that v i r t u a l l y no loss of opt imal i ty i s involved by considering only non-negative transfers or orders. 50 4. The lag time between placement and receipt of orders is two to four working days (depending on the day of the week when the order i s placed), and the length of th i s lag is of the same order of magnitude as the expected optimal time between re-order points. 5. Cash can be ordered da i l y , but the interact ion of var iable del ivery lags makes ordering cash on some days i r r a t i o n a l . That i s , i t i s i r r a t i ona l to order i f the order w i l l a r r ive no ea r l i e r than w i l l an order placed a day or two l a te r when more information on cash balances i s ava i lab le . 6. There are two assets: non-interest earning cash and short-term, l i qu i d interest earning assets. The extension to more assets of varying terms to maturity is beyond the scope of th i s study, since i t would enta i l a comprehensive asset and l i a b i l i t y management problem. The c red i t union offers interest on the minimum da i l y balance for some accounts, so i t i s reasonable to suppose that the interest earning asset i s jus t a deposit in such an account, thereby avoiding the need to consider more than two assets. The opportunity cost of cash is the da i l y y i e l d i on the interest earning asset. 7. Cash orders are only ava i lable for use the day a f ter they are received, since cash parcels are counted a f te r hours. Opportunity costs of cash accrue as of the actual del ivery time. 8. Opportunity costs of cash are incurred over weekends even though other parts of the model, such as cash demand and the a b i l i t y to take de l i ve r i e s , are not operational then. That i s , there i s a d i scont inu i ty in the demand process every weekend. 51 9. The cost of placing a cash order Q is the piecewise l inear function F(Q) a + bQ i f Q > 0 0 i f Q = 0 10. A cash-out occurs when the cash balance f a l l s below a pre-speci f ied balance x > 0. It may be that x > 0, r e f l ec t i ng the fact that the branch i s e f f e c t i ve l y out of cash i f not a l l t e l l e r s have an adequate supply of a l l denominations of currency and coin. 11. Emergency cash can be obtained on short notice by s o l i c i t i n g deposits from certa in local merchants, however there are certa in intangible costs associated with the r i sk of transporting the cash without an armoured car. The cash order models presented in th i s chapter and the next should be evaluated on the basis of how well they w i l l perform in th i s environment. 4.3 Steady State Models In th i s sect ion, we w i l l review some of the cash management models which assume a steady unchanging inventory problem over time. For these models, the optimal order pol icy can be cast in terms of parameters that are constant over time. The steady state assumptions are generally v io lated by the fol lowing features of the problem: non-stat ionar ity of the demand for cash, var iable del ivery lags for cash orders, the i r r a t i o n a l i t y 52 of ordering cash on certa in days of the week and the d i scont inu i ty of the cash demand process on weekends. Asserting that the steady state assumptions are generally v io lated by various features of the problem requires two qua l i f i c a t i on s . F i r s t , i t i s often possible to enlarge the state space of the problem so that steady state assumptions do hold on the enlarged state space. For example, in Section 4.4. a model, which was developed by Iglehart and Kar l in for non-stationary demand, i s discussed, where s ta t i on -a r i t y i s induced by enlarging the state space from the set of cash demands to the Cartesian product of the set of cash demands and demand states. (The demand states are defined so that cash demand is stationary within them.) Such refinements generally leave solutions that are too complicated to be computed. Second, we should observe that a steady state model may be f i t t e d even when i t s assumptions are v io la ted, and then the most appropriate way to measure the extent to which the assumptions are v io lated is by the amount of increased cost caused by the resu l t ing sub-optimality. We shal l evaluate the steady state models in th i s framework. One of the best-known cash management models i s by Baumol [1952]. He assumes a non-stochastic, constant demand for cash over time, and his model can readi ly be modified to allow for a del ivery lag period for orders, since the cash requirement during the del ivery period i s assumed to be known with cer ta inty . Let y be the da i l y demand for cash, b the brokerage fee or f ixed order cost, i the da i l y opportunity interest cost of i d l e cash balances, and Q* the optimal order s i ze. By elementary ca lcu lus, one can ver i f y that Q* = /(2b y / i ) . This i l l u s t r a t e s the c l a s s i ca l trade-off 53 between order costs and holding ( i n t e r e s t ) costs i n inventory theory: i f ( f i x e d ) order costs are high r e l a t i v e to the i n t e r e s t c o s t , large orders should be made, but r e l a t i v e l y i n f r e q u e n t l y ; i f order costs are r e l a t i v e l y low, small orders should be made, but more o f t e n . This i s a c o n t r o l l i m i t p o l i c y of simple (s,S) form, where the t r i p l e v e l s i s the demand f o r cash.during the d e l i v e r y l a g period and S = Q* i s the r e t u r n p o i n t . That i s , when the cash balance f a l l s below s, order an amount S. This gives a pure t r a n s a c t i o n s demand f o r cash. I f there i s a need f o r precautionary balances, one simply has to add the s i z e of the precautionary b u f f e r to s and S. Archer [1966] discusses some of the important c o n s i d e r a t i o n s i n the determination of precautionary cash balances. In e f f e c t , he points out that i f y^ i s the s t o c h a s t i c demand f o r cash during the d e l i v e r y l a g period and management d e s i r e s to r i s k a cash-out only with p r o b a b i l i t y a, then s should be the 1 - a point of the cumulative p r o b a b i l i t y d i s t r i b u t i o n f o r y^. Then the re-order point s includes p r o v i s i o n f o r both precautionary and t r a n s a c t i o n s demand f o r cash during the d e l i v e r y lag per i o d . For example, i f the d a i l y demands f o r cash are independent and i d e n t i c a l l y d i s t r i b u t e d N.(y, a 2 ) and the order lag period i s n days, and N, i s the o n e - t a i l e d 1 - a point of the standard normal d i s t r i b u t i o n , then y^ i s d i s t r i b u t e d N(ny, ha 2) and s = ny + N-j_ a Vn o so that ny i s the t r a n s a c t i o n s demand and N-|_a /n a i s the precautionary demand f o r cash. Since the Baumol model was developed i n the context of a constant demand f o r cash, i t i s i n c o n s i s t e n t to add precautionary requirements to the c o n t r o l l i m i t , while d i s r e g a r d i n g the f a c t that the s t o c h a s t i c demand al s o makes the o b j e c t i v e f u n c t i o n s t o c h a s t i c . 54 M i l l e r and Orr [1966] have developed a steady state model which assumes purely stochastic cash flows. Their model is based on the fol low-ing assumptions: 1. For some s m a l l f r a c t i o n l / t o f t h e w o r k i n g day, t h e c a s h b a l a n c e w i l l i n c r e a s e by +m d o l l a r s w i t h p r o b a b i l i t y p and d e c r e a s e by -m d o l l a r s w i t h p r o b a b i l i t y 1-p. These B e r n o u l l i t r i a l s a r e t h e o n l y exogenous changes t h a t can o c c u r i n t h e c a s h b a l a n c e . 2. There i s a two a s s e t e n v i r o n m e n t : c a s h and an i n s t r u m e n t y i e l d i n g d a i l y i n t e r e s t a t t h e r a t e i . 3. T r a n s f e r s between t h e a s s e t a c c o u n t s may...be made i n s t a n t a n e o u s l y i n e i t h e r d i r e c t i o n , f o r a f i x e d c o s t b. 4. The c a s h b a l a n c e must not f a l l below z e r o . They appeal to arguments by Karl in [1958] that an appropriate objective function w i l l be of simple form, which in the i r case i s an (h,z) control l i m i t model where the cash balance i s returned to the point z whenever i t exceeds h or f a l l s below 0. If p = the stochastic process of cumulative cash flows has no d r i f t because inflows and outflows are equally probable. In th i s case the optimal values of z and h are given by z = 3 bm2t 4 i h = 3z 1/3 They also show that m 2t i s the da i l y variance of cash flows (c 2), so that, in th i s no -d r i f t case, the optimal control parameters can be evaluated in terms of the known quantit ies b, i and a 2 , and no spec i f i c assumptions are needed about the parameters m and t . 55 Since there i s a pronounced downward d r i f t in cash balances in our case, we have 0 < p < ^. In th i s case, M i l l e r and Orr present com-pl icated formulae for the optimal values of h and z which s p e c i f i c a l l y depend on m and t , and cannot be summarized in terms of a 2 . Implementa-t ion of such a model then requires evaluation of the parameters p, m and t , which together describe a binomial d i s t r i bu t i on of cash flows. For small values of p, such as in our case, th i s can be approximated by a Poisson d i s t r i bu t i on , so that, in p r i n c i p l e , such a model could be made opera-t i o n a l , by f i t t i n g a Poisson d i s t r i bu t i on to the observed cash flows. A more serious d i f f i c u l t y in implementing the M i l l e r -Or r model in our problem results from the fact that the demand for t i l l cash i s non-stationary, so that the parameters m and p vary over time. They point out that i f the seasonality of demand is of a long term nature only, m and p can be varied seasonally to obtain a good approximation to optimal behaviour. But i f the period of the seasonal process i s approximately the same length as a typ ica l period between order and reorder, i t i s not c lear that the modified model w i l l approach optimal behaviour. Indeed, Orr [1970] makes th i s spec i f i c caveat: . . . i f adjustment t r a n s f e r s are made at three day i n t e r v a l s on average, wh i l e the per iod of pronounced d r a f t i s a month o r more, there is no large problem . . . i f . . . cash f lows are c h a r a c t e r i z e d by extreme d r i f t over short pe r iods , then the steady s t a t e mode of a n a l y s i s may be i napp rop r i a te : t r a n s i e n t e f f e c t s may dominate the longer-term movements t ha t are we l l -hand led in the s teady - s t a te approach.! In Section 4 .4 . i t w i l l become apparent that there are r ea l l y two problems that ar i se from the short-term non-stat ionar ity of cash flows. F i r s t , there i s the problem that Orr mentions of sub-optimality a r i s ing 56 from the fact that a myopic order pol icy improperly accounts for the costs that occur a f te r m and p are changed and leaves transient ef fects when m and p are changed. The second problem arises because select ion of the parameters m and p requires some descr ipt ion of the demand process that w i l l be in e f fect unt i l the next order ar i ses . But the time of the second order depends on the s ize of the f i r s t order, which in turn depends on the values of m and p used for the f i r s t order. This i s a c i r c u l a r problem, for which there i s no obvious so lut ion. Another major v i o l a t i on of the M i l l e r -O r r axioms i s the fact that, not only i s there a major order del ivery lag time (almost as long as the l i f e of a typ ica l order), but the length of th i s lag changes according to the day of the week. There i s no obvious way to adjust the M i l l e r -Or r algorithm to take account of th i s or the fact that i t i s i r r a t i ona l to order cash on some days of the week, e i ther . These problems make i t highly un l ike ly that an operational version of the M i l l e r -Or r model can be applied to the t i l l cash management problem. There are two other minor v io lat ions of the M i l l e r -Or r assumptions, which do not undermine the basic f e a s i b i l i t y of the model. The f i r s t ar ises from the d i scont inu i t ie s of the problem on weekends: cash demand stops, while opportunity costs continue. The second arises from the question of whether the cash demand process i s r e a l l y Bernoull ian. For s u f f i c i e n t l y small d iv i s ions of the day, the Bernoull i process approximates various continuous processes for various values of m and p, however, so th i s i s not l i k e l y to be a serious problem. In a l a te r paper, M i l l e r and Orr [1968] provide a rather substantial defence for the approximation of cash demand by a Bernou i l l i process. 57 Many other authors have considered the cash management problem in a steady state form. Eppen and Fama [1968] assume that holding and penalty costs are proportional to the s ize of cash balances, transfer costs are l inear functions of the amount transferred, cash de l i ver ie s are instantaneous and that the sequence of cash balances forms a d iscrete Markovian stochastic process. They use a simple for (u, U; D,d) inventory pol icy whereby the cash balance i s moved up to U i f i t f a l l s below u, down to D i f i t r i ses above d, and no action i s taken i f the balance i s between u and d. They set up a l i nea r program to minimize the present value of expected costs over an i n f i n i t e horizon. Constantinides [1976] models the cash management problem in a continuous time framework with a Wiener process for demand. (A Wiener process has stationary independent increments, and at each point in time the cash balance i s normally d i s t r ibuted. ) He assumes proportional penalty and holding costs, and f ixed transfer costs. He solves for the optimal parameters of a (u, U; D,d) inventory pol icy by minimizing Tim N" 1 (EC.,) where C^ i s the tota l stochastic inventory cost incurred out to a horizon N. A l l of these papers y i e l d spec i f i c solutions for inventory control parameters, but only by re ly ing on the s ta t i ona r i t y of the under-ly ing cash demand process. The non-stat ionar ity of the demand process prevents these models from being applied in precisely the same way i t prevents the M i l l e r -O r r model from being applied in our problem. S imi la r l y the varying del ivery lag times and d i scont inu i t ie s undermine the steady state assumptions of a l l models discussed in th i s chapter. 58 The pr ice paid to obtain a simple steady state solution appears to be too high in terms of the resu l t ing sub-optimality of decisions. In the next sect ion, the problem w i l l be cast in a more general dynamic programming framework, which has a complementary set of a t t r ibutes : the solutions are optimal, but too complicated to compute. 4.4 Dynamic Programming Formulation Cash management problems are r ea l l y ju s t special inventory theory problems. The two main features that d i s t inguish most cash management problems from ordinary inventory problems are the assumption of zero lead time for transactions and the p o s s i b i l i t y of increases in cash balances as well as decreases. We have seen that our problem d i f f e r s from the standard cash management problem in these two respects, so perhaps i t i s best to consider the t i l l cash management problem as a more general inventory theory problem. This general ity wi11 also allow us to relax the assumption in the previous section that the problem has a steady state nature. The general solution to inventory theory problems i s often cast in a dynamic programming framework. A general discussion of dynamic programming i s given, for example, by Ziemba [1975] and spec i f i c formula-t ions for cash management are given by Eppen and Fama [1969], Daellenbach and Archer [1969] and many other authors. The model can be formulated as e i ther a recourse model (with a penalty function for cash-outs) or as a chance constrained model (constraining the r i sk of a cash-out), and the fol lowing discussion derives both. To be consistent with the s t a t i s t i c a l portion of the paper we shal l enumerate working days, not calendar days. Thus i f t i s a Saturday, 59 t+1 i s a Tuesday. Also, to remain consistentiwith the rest of the paper, we shal l enumerate time in a forward rather than backward fashion. Let x^ . be the opening cash balance On working day t , y be the one-day discount rate (0 < y < 1), F(Q) be the cost of ordering $Q > 0, H^.(xt) the opportunity cost of the opening cash balance x^ > 0 on day t , . P( xf-) D e the- penalty cost fo r a cash-out of s ize x^ < 0, and y^ be the stochastic cash flow on day t with cumulative d i s t r i bu t i on function <p^(y^). Define the cost of holding the opening cash balance x .^-on day t as C t ( x t ) H t ( x t ) i f x t > 0 ,p(x t) i f x t < 0 Note that the opportunity cost of the cash balance depends on t to the extent that i t i s higher on the day a f te r a weekend, in order to f u l l y r e f l e c t the opportunity cost incurred over the weekend. Let g t(Q t> x^) be the expected discounted cost as of time t of current and future order pol icy i f the current (opening) cash balance i s x^ and an order of s ize Q t i s placed at time t (temporarily assuming a zero del ivery lag) and an optimal pol icy i s followed for a l l subsequent orders. Then we have a dynamic program in the fol lowing recursive form: g t (Q t , x t ) = F(Q t) + Y mm - o o 'Qt+1>0 W x t - y + Q t) d * t ( y ) (4.4.1) 60 That i s , the expected cost at time, t i s the sum of the order cos t , the discounted expected optimal cost at time t+1, and the discounted expected holding cost of the opening balance at time t+1. The optimal inventory po l i c y at time t i s to se lec t an order Qt* > 0 such that 9 t ( Q t , * t ) f = min g t ( Q t » x t ) . ^t However, the recurs ive r e l a t i o n (4.4.1) does not spec i fy any values of g t unless we give i t a s t a r t i ng value fo r some time T. We may e i t he r minimize expected costs f o r a r b i t r a r i l y large horizons (with y < 1 to ensure convergence), or we may minimize to ta l expected costs out to some f i xed horizon T. In the l a t t e r case one may set y = 1 (no discount ing) and i n i t i a l i z e at the horizon T by se t t ing g x (Q T , x T ) = 0. Then T _ 1 9 i (Q i * » can be computed recu r s i ve l y by means of (4.4.1) and i s , approximately, the long run average cost of an optimal cash management p o l i c y . In order to genera l ize the problem to al low for varying de l i ve r y lag t imes, we may extend a formulat ion provided by Bellman et al. [1955, p. 87j by enlarging the state space to include h i s t o r i c a l but unreceived orders. Define the order h i s t roy Q_t = (Qt_-|> Q t 2 ' D e f ' ' n e the de l i ve ry lag L ( t ) > 0 so that an order placed on day t i s received on day t + L ( t ) - 1, counted a f t e r hours and placed into use on day t + L ( t ) . The to ta l of past.orders that w i l l become ava i l ab le fo r use on day t i s mt) =- lf Q t_j « L ( t . d ) > j (4.4.2) 61 where 6.. 0 i f i t j 1 i f i = k In our app l i cat ion, the maximum del ivery time i s 4 days so 0 < L(t) < 4 and the sum (4.4.2) has only 4 terms. Note that an order placed at time j enters R(0 L t) only for t = j + L ( j ) . We now require a convention regarding the opening cash balance x^. We shal l say x^ includes the orders due to be used on day t . That i s , x t = x t l ~ y t 1 + ^ ^ t ^ " ^ s n 0 t a t l 0 n ^ s convenient because holding costs apply to the closing balance of day t-1 as well as the orders that are received and counted that night, for use the next day. We shal l also adopt the convention that = 0 whenever t + L(t) = t + 1 + L(t + 1). That i s , we shal l never consider ordering cash on an " i r r a t i o n a l " day for which an order placed next day with more information w i l l a r r ive just as soon. Lett ing ht(Qt'> 0 .^, x^) be the present value of the expected order pol icy costs given the order history 0^ and cash balance x^, when the current order i s Q t and a l l subsequent orders are selected optimal ly, we have the fol lowing analogue of (4.4.1). . o o h t (Q t ; 3.t» x t ) = F(Q t). + Y J C t + 1 ( x t - y + R (Q t + 1 ) ) d$t(y) - o o ^ m i n • h t + 1 ( Q t + 1 ; i t + 1 , x t - y + R(Q_ t + 1))} d«>t(y) (4.4.3) + Y -» >0 62 Note that on the r ight hand side of (4.4.3) i s well-defined given Q t and 0^ on the l e f t s ide. As before, set h T(Q-j;; Q-p x T ) = 0 for some large T, in order to recurs ively compute and minimize (4.4.3) for any given time t . To formulate the model as a chance constrained model set the penalty cost P(x^) = 0 and define a cash-out to be the occurrence of a cash balance below x > 0. Suppose management w i l l to lerate cash-outs with a probab i l i t y a (0 < a < 1). To determine the day for which the cash-out constraint i s to be appl icable note that the cash order i s not avai lable before t + L ( t ) , so i t i s senseless to set a constraint regarding the cash-out r i s k on any e a r l i e r day. Also, for an order on day t , one cannot be content with constraining the r i sk of a cash-out on day t + L(t) since that w i l l not provide protection against a cash-out on the next day i f t + L(t) + 1 < t + 1+ L(t + l ) , ;which i s the ea r l i e s t time an order placed on day t + 1 can be used. This can occur when the del ivery lag times vary so that L(t) + 1 = L(t +1 ) . Hence, a more r e a l i s t i c constraint i s to constrain the r i sk of a cash-out on day t + L(t + 1), which i s the l a s t day before an order placed at t + 1 w i l l be ava i lab le . To do t h i s , we should perform the minimization in (4.4.3) so that for t=l,2,3,••• Q t > 0 and Prob t*L(t+l) ^ x. - y, + I (-y, + R(Q.)) > x > 1 - a . (4.4. 1 1 j= t+i J J - J -If dpi (y •) ~ N(E (y. , a - 2 ) , and the y. are independent, th i s has the deter-min i s t i c form 63 E(y t ) • t+L(t+1) "I j = t + l E(y d) - R(a,) + / a t j a t+L(t+l) N, < X. - x (4.4.5) 1-a - t where Nn i s the 1-a point of the standard normal d i s t r i bu t i on . Note 1-a ' that the term in square brackets i s the expected cumulative cash flow (after adjusting for the receipt of past and current orders) between t and the day before an order placed at t+1 w i l l be received, so i t i s the net transactions demand for cash. The other term on the l e f t i s the buffer of cash required to hold the probab i l i ty of a cash-out below a, and hence i s the precautionary demand for cash. By increasing the s ize of the current order Q^, the term in square brackets can be decreased to sa t i s f y the inequal i ty , unless t + L(t) > t + L(t+1) in which case an order placed on day t+1 w i l l a r r i ve as ear ly as an order placed on day t . In th i s l a s t case, i t i s i r r a t i ona l to order money and we set Q t = 0. This model accommodates a l l the features of the problem mentioned in Section 4.2. In pa r t i cu la r , i t incorporates non-stat ionar ity of the cash flows because Q t depends on t . The var iable del ivery time for orders i s incorporated in L(t) and R ^ ) . The d i scont inu i ty of the model on weekends (demand stops, opportunity costs continue) is incorporated in the fact that the costs H t and depend on t . The model has, however, one major flaw. It i s far too compli-cated to compute a l l of the recursive formulae (4.4.3). The order cost F(Q t) i s easy to evaluate (in our case i t i s a l i near function for Q^ . > 0 and i s zero for = 0). The second term in (4.4.3) is the expected tota l holding and penalty cost E (C, + 1 (x. - y^ + R(Q.t+-|))) of the stochastic y t opening cash balance on day t+1, and is read i ly computable by numerical integrat ion, or by simulation. The th i rd term i s the discounted expected 64 value of optimal future order pol icy given the order on day t (and order h istory 0^). The integrand has no t ractable representation because i t i s i t s e l f a solut ion to an optimization problem, and the character of th i s recursive optimization problem changes s i g n i f i c an t l y with t , due to the non-stat ionar ity of y^, var iable del ivery lags and d i scont inu i t ie s on weekends. Since there is no steady state character izat ion of the recursive optimization problem, i t must be re-solved at each i t e r a t i on . If numerical methods are used, the number of ca lculat ions required grows exponentially with the number of i terat ions of (4.4.3), so that the problem i s v i r t u a l l y insoluble for large values of T. To appreciate the level of complexity introduced ju s t by non-s ta t i ona r i t y , we may consider a model by Iglehart and Kar l in [1962] for a non-stationary inventory process. They assumed k demand states with Markov t rans i t i on p robab i l i t i e s between states. Each state has demand density <pj{y). In our app l i cat ion , we could l e t the states i be the days of the year (k = 365), in which case the probab i l i ty t rans i t i on matrix would have a 1 corresponding to the t rans i t i on from day t-1 to t and zeros elsewhere. Under some r e s t r i c t i v e conditions ( including y. > 0 and pro-3 portional order costs only) they show that the optimal pol icy i s char-acterized by k c r i t i c a l numbers x. such that, i f the inventory f a l l s below x.j in state i , one should order up to the level x^. The solut ion requires solving up to k! transcendental equations and (k-1) renewal; equations. Even for k = 20 states, th i s i s v i r t u a l l y impossible. For operational purposes, the dynamic programming problem (4.4.3) cannot be solved to y i e l d optimal inventory po l icy. However, i t can be used as a benchmark for evaluation of operational approximations. This w i l l be discussed in the next section. 65 4.5 Operational Approximations to Optimal Po l i c i e s It was pointed out in the l a s t section that the main stumbling block to solving the dynamic programming problem (4.4.3) was the evalua-t ion of the l a s t term which i s the present value of the expected cost of future optimal order pol icy given that i s ordered on day t . An opera-t ion solut ion of the problem requires some sort of approximation for th i s term. There are two main types of approximation that can be done. The f i r s t type of approximation i s to avoid performing the minimization in (4.4.3), and instead parameterize an heur i s t i c order pol icy and simulate the operation of the model with various parameter assignments, select ing those parameter values which minimize the long run costs. The problem i s to determine what sort of heur i s t ic pol icy to use, bearing in mind that a more r e a l i s t i c po l icy has more parameters, while the number of computer runs required to f ind an optimal parameteri-zation grows exponentially with the number of parameters. For example one could simulate the use of a simple (s,S) order pol icy to f ind which values of the parameters s and S order pol icy to f ind which values of the i parameters s and S y i e l d the least long run cost. However, the problem i s not a steady problem and the optimal values of the parameters for an order a r r i v ing before a Friday pay day in December w i l l ce r ta i n l y not be the same as the optimal values of the parameters for an order a r r i v ing on a Tuesday in January with the next pay day two weeks away. Thus i t would be better to use a family {(s^, S^):i e 1} of order po l i c i e s where the state index i depends on the sequence of expected demands over the next few days. The form of th i s dependence i s heur i s t i c . Furthermore, for a comprehensive range of states I, test ing the model for N d i f fe rent 66 v a l u e s o f each o f the ( s . , ) would r e q u i r e N' ' computer s i m u l a t i o n runs wh ich would be p r o h i b i t i v e l y e x p e n s i v e . The second t ype o f a p p r o x i m a t i o n i n v o l v e s the a s sumpt ion t h a t t h e c o s t s o f f u t u r e o p t i m a l o r d e r p o l i c y beyond the next r e - o r d e r p o i n t w i l l be a p p r o x i m a t e l y the same as t he c o s t s o f t he b e s t o r d e r p o l i c y ;.• b e f o r e the next o r d e r p o i n t . To f o r m a l i z e t h i s n o t i o n , we s h a l l c a s t the dynamic program ( 4 . 4 . 3 ) i n terms o f the average e x p e c t e d c o s t per day o f t he b e s t p o l i c y ou t t o h o r i z o n T s t a r t i n g a t t he p r e s e n t t ime t = l . We may s e t t h e d a i l y d i s c o u n t r a t e y t o 1. C o n d i t i o n a l l o f t he p r o b a b i l i t y d i s t r i b u t i o n s on t h e e v e n t ; t h a t t h e next o p t i m a l o r d e r o c c u r s a t t ime x + 1 > 1. T h a t i s , c o n d i t i o n ic ic ic on the even t t h a t Q 2 = ••• = =0 and 0/^ f 0. ( In t he p r e s e n c e o f f i x e d o r d e r c o s t s i t o f t e n o c c u r s t h a t t he o p t i m a l o r d e r i s t o p l a c e no o r d e r a t a l l . ) Then r e c u r s i v e s u b s t i t u t i o n o f ( 4 . 4 . 3 ) , n o t i n g t h a t * * F ( Q 2 ) = ••• = F(Q ) = 0, y i e l d s t he f o l l o w i n g f o r m u l a f o r l ong run c o s t s pe r day: M Q i ; Q l 5 X l ) = A ( T , Ch) + E(min h T + ] ) (4, where A ( T , Q'I ) -= F ( Q . f ) ' + -OO -00 C J + l ( X l + ( " y i + R ( ^ i + 1 ) ) ) D < M Y I ) D < } ) T ( Y T ) 67 E(min h T+1 ) = - c o T + r V r ^ T + T Xa + T i = l I ( - Y i + R ( Q i + 1 ) ) ) d < h ( y i ) d<J ) T ( y T ) This gives expected costs out to.the horizon T for the order Q t as the sum of expected costs out to x ( in A ( T , Q x ) ) and the expected costs from T+1 to T (in E(min h^-j ) ) . Note that T i s a function of the cash flows y t , so that conditioning on T means that the d i s t r i bu t i on functions ^ t ( y t ) are conditioned on x . For s imp l i c i t y th i s dependence is not indicated e x p l i c i t l y in the notation of (4.5.1), since no confusion w i l l ar i se by t h i s . The f i r s t term, A ( x , Q i ) , can be evaluated by simulating the cash flows. The second term, E(min h T +-j)» is much more d i f f i c u l t to evaluate. A reasonable approximation for the second term i s ~r- h x ( Q i ; Q_i, X i ) , where Qi i s the optimal order at time t = 1. This i s the expected cost for the whole period (given x) mu l t ip l ied by the factor (T-x)/T to adjust for the shorter time to the horizon. That i s , we approximate the expected da i l y cost of optimal pol icy over the period x+1 to T with the expected da i l y cost of optimal po l icy over the whole period 1 to T. Now, suppose * Qi i s close to Qi, so..that we can use h i (Q i ; Q_i, X i ) in place of * hi(Qr, Qj, x j . Then we can approximate (4.5.1) as h i ( Q i ; Q i , x i ) = A ( x , Q x ) + T-x h i ( Q i ; Q.!, x j T or T Mth; Xi) = T A ( T , . . . Q I ) (4.5.2) 6 8 Thus, the average cost per day to the horizon T is approximately x - 1 A(x, (h condit ional on the next re-order occurring x days from now, where Qi i s a near-optimal order. Now compute the expectation with respect to the random variable x to obtain the fol lowing approximation to the expected average cost per day objective function: E T(x- 1 A(x, Q J ) . ( 4 . 5 A simulation technique fo r evaluating the objective funct ion, for a given value of Q l s i s presented in Section 5 . 6 . A gr id search * * can be used to select Q . Note that as Q approaches Q in the gr id search, the qua l i ty of the approximation ( 4 . 5 . 2 ) improves because * hi (Qi; O i^, x i ) approaches hj. (Qi; Q_i, x x ) . If the model i s cast in a chance constraint framework, the gr id search should be performed only over values of Qi that sa t i s f y the chance constraint ( 4 . 4 . 4 ) or ( 4 . 4 . 5 ) . Note that the approximation of E(min h -j) i s made by assum-ing that the structure of demand af ter the placing of the next order i s approximately the same as the structure before the next order i s placed. This i s e s sent ia l l y a steady state assumption, so i t i s ins t ruct ive to consider how a steady state model l i k e that of M i l l e r -Or r would f i t into th i s framework. The M i l l e r -O r r model gives optimal expected costs per day (assuming steady state parameters) which could be used in place of E t ( X _ 1 A(x, Q i ) ) . However, by examining ( 4 . 5 . 1 ) and the de f i n i t i on of A(x, Qi) we see that i t depends on the d i s t r i bu t i on of cash demands out to the reorder point x+1. In order to use M i l l e r -O r r , we would require a summary of these d i s t r ibut ions in order to evaluate the steady state 69 parameters. But such a summary requires knowledge of x. That i s , i f a re-order is required before a holiday or some other day of heavy demand, the summary should not forecast such heavy cash requirements as i f the re-order i s to be received after the holiday. In order to evaluate x for the M i l l e r -O r r model, we would have to select Qiand then perform some sort of simulation of the cash demand to determine the next order point. But the value for Q2 depends on the summary parameters derived from x. Thus, some sort of i t e ra t i on back and forth between values of x and values of * Qi would be required to evaluate E T ( X A ( X , Q ) ) . It seems simpler to evaluate th i s expectation d i r e c t l y by simulation. In th i s way, we would be making a steady state approximation l i k e that required to invoke a M i l l e r -Or r (or any other) steady state so lut ion, but we are choosing the steady state parameters in a more meaningful and simpler way (by simply conditioning on x) than one could i f select ing M i l l e r -Or r parameters by some sort of i t e ra t i on process. 4.6 Summary In Section 4.2 we examined several of the dist inguishing features of the t i l l cash management problem, and foremost amongst these were the non-stat ionar ity of the data, the var iable del ivery lag times for orders and the d i scont inu i ty of the cash flow/opportunity cost r e l a t i on -ship on weekends. In Section 4.3 several steady state models for cash management were considered, including those of Baumol, M i l l e r and Orr, and Eppen and Fama. It was pointed out that the steady state assumptions of these models were too ser iously v io lated to y i e l d good solutions in th i s content. 70 In Section 4.4, a dynamic programming model for cash manage-ment was presented. Although the dynamic programming solut ion i s v i r -t ua l l y impossible to compute, i t was pointed out that certa in approximations allow the model to be presented as one of minimizing expected average costs per day over the l i f e of the current order. This i s a steady state approximation to the problem. In the next chapter an algorithm, based on the material in Section 4.4, i s presented to se lect orders with the objective of minimizing expected costs per day out to the f i r s t re-order point. 71 FOOTNOTES TO CHAPTER 4 Daniel Orr [1970, p. 82]. Chapter 5 A STOCHASTIC PROGRAMMING MODEL TO ORDER TILL CASH 5.1 Introduction In th i s chapter the stochastic programming model to determine near-optimal t i l l cash orders i s developed. Several technical considera-tions are discussed in Section 5.2. The main f inding there i s the con-tention that the chance constrained model i s more appropriate than the recourse model. Section 5.3. develops an approximation to the determinist ic equivalent of the chance constra int. A method of evaluating the value of objective function by simulation for any order s ize Q i s discussed in Section 5.4. The objective function in Section 5.4 i s more complicated than the one presented in Chapter 4, and the dynamic program underlying th i s variant of the objective function i s presented in the Appendix. The actual cash order algorithm i s presented in Section 5.5 and i s tested on the three years of h i s t o r i ca l cash flow data in Section 5.6. This allows evaluation of the e f fect i ve insurance premium, in terms of annual operating costs, that i s required to constrain the r i s k of a cash-out to any given l e v e l . The model performance is also compared to the h i s t o r i ca l per-formance by management over the same data period. 72 73 5.2 Preliminary Considerations In order to develop a model based on the discussion in the previous chapter, two technical decisions have to be made. One must choose between recourse and chance constrained stochastic optimization and the functional form of the order costs and holding costs must be con-sidered. These points w i l l be discussed in th i s section. As discussed in Sections 4.2 and 4.4 of the previous chapter, the recourse formulation of the dynamic programming model uses the penalty function P(x t ) to encourage the holding of adequate cash balances that w i l l reduce the r i sk of cash-outs. The chance constrained formulation eliminates the penalty function but adds a constraint to the se lect ion of the Q t ' s which reduces the r i sk of a cash-out to some level set by management. The chance constraint has a determinist ic equivalent such as (4.4.4) or (4.4.5). Both formulations require a rather precise knowledge of the t a i l s of the d i s t r ibut ions of the da i l y cash flows since cash-outs occur only for extreme cash flows. In Chapter 3 i t was found that the residuals of the estimated cash flows formed a d i s t r i bu t i on that was too f a t - t a i l e d to be normal and i t was also argued that a l o t of the ou t l i e r s could have simply been a resu l t of bookkeeping discrepancies rather than actual cash flows. Thus, we do not have a very good knowledge of the t a i l s of d i s t r ibut ions of the da i l y cash demands. It i s not c lear what adjustment, i f any, could be made to the t a i l s of the d i s t r ibut ions in the recourse model so that the estimated expected penalty cost of a given cash order pol icy w i l l be a reasonable approximation of i t s true value. However, in Section 4.4 i t was shown that for normal forecast errors the chance constraint can be expressed as a sum of net transactionary and net 74 precautionary requirements. The precautionary requirements are in the form of the buffer /a. 2+«• ?+o'Z,. , N, where y. ~ N(E (y.), a. 2) and the t t+L (t+1) 1-a •'j " ' j J y. are independent. A mis - spec i f icat ion of the t a i l s of the d i s t r ibut ions of the y . ' s w i l l mis-specify the r i s k level a, but the concept of a pre-J cautionary buffer requirement expressed as a sum of money i s nevertheless something which management can re late to previous experience. Thus, a manager may be a l i t t l e uneasy about specifying a r i s k of a cash-out of 1% (especia l ly i f he understands that the d i s t r i bu t i on of the t a i l s of the forecast errors i s not known accurately) , but he is more l i k e l y to feel at ease i f he speci f ies a precautionary buffer of, say, $60,000 when the order del ivery lag i s 3 days. For these reasons, the problem w i l l be formulated in a chance-constrained framework, rather than as a recourse model. Now we may consider the form of order cost function. In Section 4.2 i t was suggested that an adequate form would be F(Q t) =4 b + aQ t i f Q > 0 b, a > 0 i f Q t Since only non-negative orders are considered and the long run to ta l of the cash orders w i l l equal the long run net cash outflow we must have T T I aQ. = a I y, for large T. Thus the contr ibut ion to long run costs t=l z t=l z by the var iable order cost term is independent of order po l icy and without loss of general ity we can assume a = 0, so that fb i f Q t > 0 F(Q t) (5.2.1) 0 i f Q t = 0 75 We shal l also be more e x p l i c i t at th i s point about the cost function of the opening cash balance C^.(x^.). This i s a piecewise l i near function H t ( x t ) = i x t (D( t ) - D ( t - l ) ) i f x t > 0 [P(x t) ="p'x t i f x t < 0 (5. C t ( x t ) where i i s the da i l y interest rate of the interest earning asset, D(t) i s the number of calendar days a f ter day one, and p > 0 is the penalty cost rate for negative balances (p = 0 for the chance constrained model). Note for example that D(t) - D ( t - l ) = 3 i f day t i s Tuesday (to take account of weekend cash holdings) and D(t) - D ( t - l ) = 1 i f day t i s any other working day of the week not fol lowing a ho l l iday. 5.3 The Chance Constraint As discussed in the la s t section i t i s most appropriate to formulate the problem in a chance-constrained framework. Although the forecast errors are too f a t - t a i l e d to be normally d i s t r i bu ted , i t was pointed out that we can model the chance constraint as though the errors were normally d i s t r ibuted as long as we remember that the p robab i l i s t i c constraint only gives a r e l a t i v e , but not absolute, quant i f i cat ion of the r i sk of a cashout. Thus, we shal l use the chance constraint (4.4.5) which i s based on normal forecast errors. In Section 2.3, an estimate of the variance of the da i l y cash flow estimate y. given in formula (2.3.10) i s 76 i t 2 s 2 [ l + z ^ C Z ' Z ) " 1 z t J (5.3.1) where d t i s the forecast trend l e v e l , s 2 i s the standard error of the regression, i s the vector of dummy variables for day t and s 2 (Z ' Z ) i s the estimated covariance matrix of the regression coef f i c ient s |3. For the purposes of ca lcu lat ing the precautionary buffer s ize in the chance constra int, i t i s a l i t t l e d i f f i c u l t to cont inual ly compute a quadratic form involving the 28 x 28 covariance matrix. The dummy com-ponents of the vector z^ are described in Table 1 inChapter 2 along with the regression coe f f i c ient s and the i r standard errors. A careful examina-t ion of Table 1 indicates that at most 4 components of z^ are non-zero at any time: one each for the day of the week, month of the year, pay-day and holiday. Also, the largest standard error of the regression coef f i c ient s is .044, and the standard errors are t y p i c a l l y .015 to .020. Consider the fol lowing type of day where the coef f i c ient s of the non-zero dummy variables have the largest possible standard errors: a Thursday in June with a month-end pay day, ju s t before a holiday. Then z ^ t = z^ = z 17t = z 28t = 1 a n c ' a 1 1 t ' i e o t n e r dummy variables are zero. Then from the estimated covariance submatrix corresponding to the non-zero var iab les , we have s 2 z . M Z ' Z ) - 1 z + = l O " 4 (1,1,1,1) 1.51 -.50 -1.20 .18' T -.50 1.95 .22 -.37 1 -1.20 .22 19.31 -.11 1 .18 -.37 -.11 6.74, X ,0026 77 Now s 2 = (.095) 2 = .0090. From (5.3.1) we see that the forecast variance i s .0026 d~ 2 + .0090 d j 2 = .0116 d t 2 . This i s an example where the f i r s t term i s larger than usual (using 4 coe f f i c ient s with large standard er ror s ) , yet i t i s r e l a t i v e l y small com-pared to the to ta l forecast variance. Since we are not using the forecast variances to compute absolute levels of cash-out r i s k , but only to compute approximate precautionary buffers, i t i s reasonable to neglect the f i r s t term which i s complicated to compute and instead suppose that the fore-casts y^ 's have a r e l a t i v e l y constant variance given approximately by s 2 d t 2 . Thus, the determinist ic equivalent of the chance constraint (4.4.5) involving the r i sk of a cash-out becomes t + L ( t + l ) ~ A y + I (y - )) + /Llt+lT s d N < x - x (5.3. i j = t + l J J L i -a i where i s the opening balance on day t , x is the t r i p balance for a cash-out, y. i s the forecast of the cash flow for day j , R(0.) i s the to ta l of orders to be received on day j , L(t) i s the number of days in the del ivery lag for an order placed on day t and Ny_ a i s the l - a point of the standard normal d i s t r i bu t i on . This constraint spec i f ies that the prob-a b i l i t y of running out of cash just before an order placed next day (t+1) i s received shal l be less than a. As before, we shal l only require the chance constraint to hold on days when i t i s rat ional to order cash (that i s , when t + L(t) < t + 1 + (L(t+1)). 78 F ina l l y we observe that, without loss of general i ty, we may set x = 0. If management feels that i t has e s sent ia l l y run out of cash when the cash balance i s x = $5000, say, then the $5000 i s not a manageable item. Any holding costs associated with x cannot be increased or de-creased by any cash management po l i cy . Thus, when reporting cash balances x^ we may suppose that x has already been deducted, so that a cashout occurs when x^ < 0. 5.4 The Objective Function The dynamic programming formulation of equations (4.4.3) and (4.5.1) is not the only dynamic programming"formulation of the inventory problem - i t was presented there only because i t i s the simplest to under-stand. Hence the approximate objective function (4.5.3), which i s E ^ T " 1 A ( T , Q ) ) , (5.4.1) is not the only objective function that can be used in the problem. From the dynamic programming formulation (4.5.1), we see that x - 1 ( A ( x , Q i ) ) i s the average of the order cost F(Ch) and the holding costs Cj- +-|( xt +-|) of opening balances over the period start ing on the working day when the current order is placed (t=l) to the working day before the next order i s placed (t = x ) . However, the se lect ion of the order s ize Qi af fects holding costs over a d i f fe rent period in time. If cash i s ordered on working day t , i t i s ready for use on day t + L(t ) but i t actua l ly arr ives at 79 the branch on day t + |_(t) - 1 in order to be counted af ter hours fo r use on the next day. Thus the select ion of order s ize Q : a f fects the holding costs of the opening cash balances on days 1 + L ( l ) to x + L(x+1), namely ]) (xi+[_(1) ^  •" ' '^T+L(-T+1 )^ X T+L(T+1 ^' F u r t n e r m o r e > we indexed the holding cost function by t to account for the increased cost of holding cash balances over weekends, while T i s a count of work-ing days only, so i t i s more appropriate to average the costs over the number of calendar days rather than working days. Accordingly, l e t work-ing day t be D(t) calendar days a f ter day one. For example i f day 1 i s a Saturday, D(2) = 3. Since the f i r s t order i s placed on day 1 and the next order i s placed on day T+1 (a random var iab le ) , a more reasonable objective function i s the expected average da i l y cost (on a calendar day basis) of ordering and holding costs which are affected by the s ize of the order Qi namely E ; T A1 ( T , Ch D ( T + L ( T + 1 ) ) (5.4 where A ' ( T , Q I ) = F ( Q i ) + f o o T + L ( T + 1 ) t+1 . I n ) c t + i ( X i + l2 + R ( V } ) ^ - d W + D ( 5- 4 - U J _ o o ' ' (For s imp l i c i t y of notation we have replaced the d i f f e r e n t i a l d<j).j(y.j) with dcj).. Recall a lso, that i t i s conditional on T . ) A dynamic program corresponding to th i s objective function i s given in the Appendix at the 80 end of th i s chapter. Again the reader i s reminded that we set Q^ . = 0 whenever t + L(t) = t + 1 + L(t+1). That i s , we do not order cash i f a l a te r order w i l l a r r i ve on the same day. Suppose a simulation model generates a new sequence of cash flows {y^} in each i t e r a t i on . Then we can condition on {y^K noting that given { y t h there i s a well defined T such that cash will--be ordered,-at T . + 1 Then we can use (5.4.3) to evaluate the condit ional expectation f A " ( T , Q I ) ) , , _ A' (T Q2) T | { y T } [ D ( T + L ( T + l ) ) J D(T*+L(T*+1.)) T*+L(T*+1) t+1 F(Qi) + L ^ C ( X l + I (-y^+RtQ,.))) t = l + L ( l ) _ i=2 D(x* + L(x*+1)) (5.4.4) Now, taking the expectation with respect to iy^h fA' (T, Qj) D(T+L(T+1)) = £iyt} Ex|{y t} fA' (x, Q J D(T+L(T+1)) (5.4.5) Thus (5.4.5) can be evaluated by ca lcau l t ing (5.4.4) fo r each i t e r a t i on of a simulation of {y^} and taking the average of the resu l t ing values which simply amounts to taking the expectation with respect to iy^}-This can be done for any value of Q l s so that the objective functions can be minimized by a gr id search. This i s the essence of the cash order algorithm to be presented in the next section. 81 5.5 The Cash Order Algorithm In th i s section the formulae presented in the previous sections are used to develop an algorithm to order cash by select ing an order s ize to minimize the expected average costs per day (as computed by a simulation model) subject to a constraint involving the r i sk of incurr ing a cash-out. A flow-chart of the order algorithm is presented in Figure 2. The algorithm i s to be applied each morning with the opening cash balance as input. As before, the days are re-numbered so that t=l i s the current day, for which an order i s being considered. Since there are f ixed costs of ordering cash and opportunity costs of holding excess cash balances, i t i s always best to postpone making an order on day t unless the chance constraint (5.3.2) i s v io lated with Q t = 0. Thus the minimum acceptable order on day t i s A t + L ( t + 1 ) - t+L(t ) - 1 QMIN(t) = /LTt+TT s d. N, + I y- + x - x. - £ R(Q ) (5.5.1) L j=t J L j=t+l J where s i s the standard error of the regression in Table 1, d^ . is the estimated weekly trend level in cash flows, N-j_a i s the a point of the standard normal d i s t r i bu t i on , y. i s the forecast cash flow on day j and R(Q.) i s the tota l of previously made orders that are due to arr ive on day j . Note that R ( ^ t + L ( t ) ) = Q t a n d R(Q.j) = 0 for t+L(t) < j < t+L (t+1). Thus we can omit the dates t+L.(t) + 1, • • •, t+L (t+1) in the la s t summation of (5.5.1) and the date t+L(t) corresponds to QMIN(t). If Q m i nU) > 0 an order is placed, unless 1 + L(t+1) = L ( t ) , in which case an order placed on day t+1 would ar r ive just as soon as. an;' , j-82 Start Current day is ' r-/ * K Yes / QMIN(1) f 0 N V 1 + L(2) = L ( l ) y <J Stop No Q = QMIN(l) QBEST = Q N = 1 EXPCOST(Q) = 0 BOUND = 0 Q = Q + QSTEP t = 2 x t = x t - 1 + R ^ ) - y t _ i - d t - i * R E S ( ] + 7 5 8 * F R A N ° ) QMIN(t) f 0 .or 1 + L(t+1) = L(t) Yes * t = t+1 No T = t-1 |EXPC0ST(Q) BOUND E X P C O S T ( Q ) + 1 1 A ' ( T , Q ) N S I M D(T+L(T+1 )) B 0 U N D + NSTM * D ( x " i l L ( ? l 2 ) ) N = N+l r ^ N > N S I M ) y ? s ><f EXPCOST(QBEST) < EXPCOST(Q) • QBEST = Q EXPCOST (QBEST )\No < BOUND Yes / Stop \ V Q* = QBEST J Figure 2. The Cash Order Algorithm. t+L(t+1) ~ QMIN(t) = /L(t+1) s d t N 1_ a + J y\ t+L(t)- l + x - x - I R(Q ) L j=t+l J j -^ Return ^ Figure 3. The Minimum Cash Order Subroutine. 84 order placed on day t , making i t i r r a t i ona l to use less information and order on day t. At t= l , i f an order i s to be placed, the gr id search starts with the minimal order Q = QMIN(l) and examines larger values, in incre-ments of QSTEP. This ensures that the chance constraint i s s a t i s f i e d . For each order s ize Q, the cash flows are simulated NSIM times. In each i t e ra t i on the sequence { x i , X 2 , x 3 , » » * } i s calcauTted by sett ing y t - l = y t - l + d t - f R E S I D U A L 0 + 7 5 8 * FRAND) X t = X t -1 " y t - l + R(V where RESIDUAL i s the array of 759 unsorted residuals from the regression in Table 1, Section 3.4 and FRAND i s a random number generator that gives independent uniform variables on [0,1]. The sequence of cash flows is continued un t i l a re-order i s required on day t - x + 1 (using the order c r i t e r i a discussed e a r l i e r ) . This allows computation of the fol lowing expectation (conditioned on (y^)) by formula (5.4.4): F [A1 (x Q i ) ' T|{y tHD(x+L(x+l)), This i s then averaged into the objective function which i s the uncondi-t i o a l expectation EXPCOST = E x A' (T , Q i ) ) D(x+L(x+1) 85 Having computed the object ive function EXPCOST(Q) a f te r NSIM i terat ions of the simulation, the objective is compared to that for the l a s t best order (QBEST), and i f better, QBEST i s updated and the objec-t i ve i s computed for the next larger order, Q + QSTEP. If the order Q is not better than QBEST, then QBEST is a local minimum of the objective function. For a given cash flow sequence {y^} the numerator of (5.4.4) increases s t r i c t l y with Q since larger orders cause larger cash balances with longer times to re-order. However, the denominator is integer-valued and i s only non-decreasing in Q. Thus, increasing Q s l i g h t l y may increase the numerator but not the denominator, although increasing Q by a greater amount may cause an increase in the denominator, as .wel l , which causes an overal l decrease in the objective function. Thus local minima need not be global minima. However to obtain a lower bound for the objective function suppose that for a given (y^)* increasing the order from Q to Q1 causes an increase in T * of one working A1 (T*+1 0 ' ) A1 Ix* 0) day- Then d ( t4 I + L (T *+2 ) ) - D(T*+1+L(T*+2)) ' T h a t i s » a t b e s t t h e denominator can be increased while the numerator is unchanged by the increase in order s i ze. This lower bound i s computed at each i t e ra t i on of the simulation and averaged into the expectation BOUND. Then i f the l a s t best order i s l o c a l l y optimal ( i . e . QBEST < Q) i t i s also g lobal ly optimal i f EXPCOST(QBEST) < BOUND. This provides a stopping rule for the algorithm. Note that for very large orders, the tota l opportunity costs grow quadrat ica l ly with order s ize (since both T and the average balance grow asymptotically with Q) while the denominator increases only l i nea r l y with Q. Thus, a f i n i t e order i s optimal and the stopping rule w i l l stop the algorithm in a f i n i t e number of steps. 86 The algorithm es sent ia l l y provides a var iable (s,S) control l i m i t model. The decision to order i s made by evaluation of the chance constraint and the decision of how much to order is determined by the simulation and gr id search. Both decisions depend on the cash flow fore-casts and del ivery lag times so that the parameters s and S vary over time. 5.6 Appl icat ion of the Model In th i s section we shal l study the performance of the model with various values for the cost and r i s k parameters, as well as compare the performance of the model to actual management performance. The tests were performed on a l l three years of data. In order to be able to take averages of the costs over the three years, a l l of the data were rescaled to a constant $140,000/week trend l e v e l . That i s , the cash.flow y^ was replaced by ^0,000 y p o r s imp l i c i t y , a cash-out was regarded as the occurrence of a z l r o cash balance so that x = 0. A l l tests of the model were performed with NSIM = 100 i te rat ions of the cash flow simulation for each order s ize and orders were selected in steps of QSTEP = $5000 over the minimum order. The del ivery lag period L(t) for an order placed on day t varies according to the day of the week because the order is received f r o m a bank which operates on a Monday to Friday work week in contrast to the Tuesday to Saturday work week of the cred i t union. Orders are placed in the vault on the day of a del ivery and counted af ter hours, for use the next day. A l i s t of the del ivery lags i s given in Table 2. Note that 87 Table 2 Cash Delivery Order Lags Order Placed Before Noon on Day t Order Delivered on Order Put to Use on Lag L(t ) Tuesday Thursday Friday 3 Wednesday Friday Saturday 3 Thursday Tuesday Wednesday 4 Friday Tuesday Wednesday 3 Saturday Tuesday Wednesday 2 Table 3 Performance of the Cash Order Model and of Management Fixed Order Cost $b Opportunity Cost of Cash 365 x i % Risk Level , (Normal Errors) a Probab i l i ty of a Cash-Out in Simulation Number of Orders Placed Number of Actual Cash-Outs Average Cost Per Year ($) Management: 25 7 - - 153 2 15,700 Model: 25 7 .025 .0103 192 0 11,100 25 7 .01 .0084 198 0 11,600 25 7 .001 .0019 192 0 13,900 25 10 .001 .0011 203 0 18,900 40 7 .001 .0011 159 0 15,200 40 10 .001 .0017 186 0 20,200 88 i t i s i r r a t i ona l to order cash on Thursday or Friday, since a Saturday order arr ives on the same day. Table 3 outl ines the performance of the model and of management, under various assignments of the cost and r i s k parameters. These parameter assigments are j u s t i f i e d as fo l lows: Fixed order costs b include the cost of armoured car del ivery and the bank charge for preparing United States and Canadian coin and currency parcels. These tota l $18.50., but other f ixed costs, such as the cost of the t e l l e r s ' time to count the cash, make $25 and $40 f ixed costs more r e a l i s t i c . The model was tested under both cost assumptions. The cred i t union offers 7% annual interest on the minimum da i l y balance in one type of savings account, and th i s may be taken as the opportunity cost of cash. A l te rna t i ve l y , aggressive investment of excess cash could y i e l d rates of up to 10% per year, so the model was tested with th i s opportunity cost as we l l . The model was tested with three r i s k levels a of a cash-out: a = .00J , a = .01, a = .025. These r i s k levels are,only v a l i d under the assumption that the errors are normally d i s t r ibuted. Since the fore-cast residuals were too f a t - t a i l e d to be normal, the cash-out r i s k may be higher or lower than a . To estimate th i s level of r i s k , a count of the number of cash-outs in the simulations corresponding to the optimal orders on each order date was kept. This number, divided by the tota l number of i te rat ions (where the to ta l number of i terat ions = NSIM x number of orders placed) i s a good ind icat ion of the r i s k of a cash-out and i s l i s t e d beside a under the heading "P robab i l i t y of a Cash-Out in Simulation." In addit ion a count of the tota l number of cash-outs that would have 89 occurred with the actual cash flows, when the optimal order pol icy i s fol lowed, i s provided. Referring to Table 3, we may f i r s t compare performance of manage-ment with the model, using f ixed costs of $25 per order and an opportunity cost of cash of 7% per year. In the 2 year and 11 month period from May 1973 to March 1976, management generally placed weekly orders, incurr ing costs of $15,700 per year. On two occasions (or for 1.3% of the orders placed), deposits were s o l i c i t e d from a local merchant to cover.cash shortages ($2600 and $20,800). In contrast, with a = .025 the model y ie lded only a 1% r i s k of a cash-out in simulated performance and no cash-outs in pract ice, while tota l .cos t s were only $ll,100/year. Decreasing a to .001 resulted in a 0.2% cash-out r i s k in simulated per-formance while the costs were s t i l l less than those of management ($13,900/ year). Over the 35 month period the model ordered cash about 200 times, compared to management's 153 orders, which suggests that management's commitment to weekly orders was a large source of the sub-optimality of i t s h i s t o r i c a l performance. Another way of regarding the simulated cash-out p robab i l i t i e s i s to note that the "insurance premium" associated with reducing the probab i l i ty of a cash-out from .010 to .008 i s $11,600-$ll,100 = $500 per year, while the "insurance premium" for reducing the cash-out r i s k from .010 to .002 i s $2800 per year. When the f ixed order costs and opportunity costs of holding cash were varied with a = .001 , the simulated cash-out probab i l i ty was always between .001 and .002 and the model responded in a predictable manner in terms of the number of orders placed. That i s , increased 90 f ixed costs caused fewer orders to be placed and increased opportunity costs caused more orders to be placed. In summary, the model provides more protection against cash-outs while incurr ing smaller operating costs than management's h i s t o r i ca l per-formance. Also, the model readjusts i t s long run pol icy in the correct d i rect ion as the ra t i o of f ixed order costs to opportunity costs i s varied. 5.7 Summary In th i s chapter, the algorithm for ordering t i l l cash was developed. In Section 5.2 i t was asserted that a chance constrained model would be more appropriate than a recourse model, and the exact forms of the order and holding cost functions were established. The ca lcu lat ion of the determinist ic equivalent of the chance constraint was discussed in Section 5.3. In Section 5.4, i t was pointed out that a somewhat more complicated objective function than that proposed in Section 4.5 would y i e l d a closer approximation to t r u l y optimal decisions and an operational method of evaluating the objective function by simulation was presented. The actual cash order algorithm for minimizing the objective function by means of a gr id search was presented in Section 5.5. The performance of the cash order model was compared to the cash order performance of management in Section 5.6. It was established there that implentation would y i e l d lower costs and/or greater protection against the r i sk of cash-outs. 91 APPENDIX TO CHAPTER 5 THE DYNAMIC PROGRAM CORRESPONDING TO THE OBJECTIVE FUNCTION (5.4.2) Let V "t$ < t i < t 2 < t3 < ••• be the days on which i t i s rat ional to order cash. This i s a l i s t of those days t such that an order placed a f te r t cannot ar r ive at the same time as an order placed l a te r ( i . e . t + L(t ) > t + L(t n) 'whenever t > t ). • Let 2 C.(x.) t=t +L(t ) Z Z n n (5.A.1 ) This i s the sum of the holding costs of opening cash balances which would be affected by the s ize of the order Q , but not by the s ize of the next S possible order Q, . Then l e t V l \ ( Q t 1 Qt ' x t } n n n n = F(Q. ) + Ln -00 -00 C d* . . . dc|, ( } n n n+1 n+1 + • • • J min h (Q t ; Oj. , x. ) dcj). ••• dcj). , Q+ >0 Ln+1 Ln+1 Ln+1 Ln+1 zn n+1 oo n r _ -ln+l (5.A.2) Note that the minimizations are performed subject to the constraint (5.3.2) 92 and - for t n < t < t n + ] + L ( t n + 1 ) , x t = x t + _ I (- y._} + R (£..)). n tn + This i s a dynamic program where we i te ra te from n = N to n = 0 and the horizon i s T = t... Now i f we condition on the time T + 1 = t of the N n+1 next order, recursive subst i tut ion of (5.A.2) for h' , ' * * , h ! y ie lds the to t n fol lowing analogue of (4.5.1) where t 0 = 1: hl(Qi; Qi, x j = A ' ( T , QJ + E(min h. ) , (5.A.3) ^n+l where A 1 i s defined in (5.4.3). Again we approximate the expectation in (5.A.3) by hi, adjusted now for the d i f fe rent number of calendar days i . ~ D(T)-D(t +L(t )-l) , associated with the two terms. That i s , E(min h. ) = —: rTrr~\ —~hi rn+l J t T J Subst itut ing into (5.A.3) y ie lds the fol lowing approximation to the average da i l y cost: hi = A ' ( T QJ _ A ' ( T , -QJ DTfT D ( t N + 1 + L ( t n ; i ) - 1 ) - D ( T + L ( + L ) ) Computing the expectation over T of the above expression y ie lds (5.4.2). Chapter 6 CONCLUSIONS At th i s point i t i s appropriate " t o *b r i e f l y review some of the general lessons that were learned by developing th i s project. From a s t a t i s t i c a l viewpoint i t i s perhaps surpr is ing that so much random va r i a t i on - i s present in the recorded da i l y cash flows. If, as suspected, a large portion of the var iat ion i s in the record keeping, i t i s because the records are oriented to being accurate on an aggregate monthly basis rather than a da i l y basis. Despite the fact that the cash demand at the cred i t union varies in a c y c l i c manner, Box-Jenkins time series techniques cannot be used to model the demand process. This i s a resu l t of non-stat ionar ity induced by the trend towards increasing cash demand as well as e f fec t i ve non-stat ionar ity a r i s ing from the long periods of the cycles (up to one year long) and the even greater least common mult ip le of these periods. The estimation of the demand for cash f i n a l l y had to be performed in a manner that was custom f i t t e d to the spec i f i c case at hand and cannot be readi ly generalized to other branches. It enta i led f i t t i n g a non-l inear trend to real cash demand that incorporated the notion of an asymptotic capacity l e v e l . Such a trend model may not be appropriate for other 93 94 branches. The subsequent estimation of demand var iat ion about the trend was performed by regression techniques that can readi ly be generalized to other branches, however. In terms of the development of a cash order algorithm, i t i s apparent that the t i l l cash management problem i s not so sharply dist inguished from standard inventory theory problems as i s the c l a s s i ca l cash management problem, in which the lag time for transactions i s zero and the stochastic cash balance can d r i f t up or down. However, the main problems associated with t i l l cash management appear to be the non-stat ionar ity of demand and other features that make i t hard to develop a steady state so lut ion. An approximate dynamic programming solut ion to the t i l l cash management problem at a potential order point was obtained by conditioning on the time of the next order and replacing the expected costs of optimal behaviour a f te r that time with the average cost pr ior to that time (rescaled because of the shorter time to the horizon). Greater accuracy, at much greater com-putational expense, might be obtained by conditioning on the time of the th i rd or some l a te r order, in order to incorporate more i te rat ions in the dynamic program before invoking an approximation. Unfortunately the number of ca lculat ions increases exponential ly with the number of dynamic programming i t e ra t i on s . The major trade-off between theoret ica l correctness and opera-t ional i t y occurs in the cash order algorithm, rather than the s t a t i s t i c a l estimation. Since the algorithm does actua l ly out-perform management and can readi ly be implemented with ex i s t ing on- l ine computer f a c i l i t i e s i t i s a success from an operational point of view. LIST OF REFERENCES Archer, S.H. (1966). "A Model for the Determination of Cash Balances," Journal of Financial and Quantitative Analysis, 1_, 1-14. Baumol, W.J. (1952). "The Transactions Demand for Cash: An Inventory Theoretic Approach," The Quarterly Journal of Economics, LXVI, 545-556. Bellman, R., I. Glicksberg and 0. Gross. (1955). "On the Optimal Inventory Equation," Management Science, 2, 83-104. Box, G.E.P. and G.M. Jenkins. (1970). Time Series Analysis: Forecasting and Control, Hoi den-Day, New York. Cleveland, W.S. (1972). "Inverse Autocorrelations of a Time Series and Their Appl icat ions, " Technometrics, XIV, 277-298. Constantinides, G.M. (1976). "Stochastic Cash Balance Management with Fixed and Proportional Transactions Costs," Management Science, 22, 1320-1331. Daellenbach, H.G. and S.H. Archer. (1969). "The Optimal Bank L iqu id i t y : A Mult i -Per iod Stochastic Model," Journal of Financial and Quantitative Analysis, 329-343. Eppen, G.D. and E. Fama. (1968). "Solutions for Cash Balance and Simple Dynamic Po r t fo l i o Problems," Journal of Business, 4j_, 94-112. : . (1969). "Cash Balance and Simple Po r t fo l i o Problems with Proportional Costs," International Economic Review, JJ), 119-133. Iglehart, D. and S. Kar l i n . (1962). "Optimal Po l icy for Dynamic Inven-tory Processes with Nonstationary Stochastic Demands," in Studies in Applied Probability and Management Science (K.J. Arrow, S. Kar l in and H. Scarf, eds.), Chapter VII I, Stanford Univers ity Press, Stanford, Ca l i f o r n i a . 9& 96 Johnston, J . (1972). Econometric Methods, 2nd ed., McGraw-Hill, New York. Ka r l i n , S. (1958). "Steady State Solut ions, " in Studies in the Mathe-matical Theory of Inventory and Production (K.J. Arrow, S. Kar l in and H. Scarf, eds.), Chapter VI I I, Stanford Univers ity Press, Stanford, Ca l i f o rn i a . M i l l e r , M.H. and D. Orr. (1966). "A Model of the Demand for Money by Firms," The Quarterly Journal of Economics, LXXX, 413-435. . (1968). "The Demand for Money by Firms: Exten-sions of Analyt ic Results," Journal of Finance, 23> 735-759. Neave, E.H. (1970). "The Stochastic Cash Balance Problem with Fixed Costs for Increases and Decreases," Management Science, 16, 472-490. Nelson, CR. (1973). Applied Time Series Analysis for Managerial Fore-casting, Hoi den-Day, San Francisco. Orr, D. (1970). Cash Management and the Demand for Money, Praeger Publishers, New York. Stone, B.K. (1972). "The Use of Forecasts and Smoothing in Control-Limit Models for Cash Management," Financial Management, Spring, 72-84. The i l , H. (1966). Applied Economic Forecasting, North-Holland, Amsterdam. . (1971). Principles of Econometrics, Wiley, New York. Ziemba, W.T. (1975). "Dynamic Programming," in Stochastic Optimization Models in Finance (W.T. Ziemba and R.G. Vickson, eds.), Academic Press, New York, 43-56. 

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