TILL CASH MANAGEMENT MODEL by GORDON ARTHUR S I C K B.Sc., U n i v e r s i t y of Calgary, M.Sc, U n i v e r s i t y o f T o r o n t o , 1971 1972 A T H E S I S SUBMITTED IN P A R T I A L FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION v in FACULTY OF COMMERCE, & BUSINESS ADMINISTRATION We a c c e p t t h i s thesis to the required as conforming standard THE U N I V E R S I T Y OF B R I T I S H COLUMBIA December, © Gordon A r t h u r 1976 Sick, 1976 In p r e s e n t i n g t h i s thesis an advanced degree at the L i b r a r y s h a l l I f u r t h e r agree in p a r t i a l fulfilment of the U n i v e r s i t y of B r i t i s h make i t freely available that permission for the requirements f o r Columbia, I agree r e f e r e n c e and f o r e x t e n s i v e copying o f this that study. thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s of representatives. this thesis It i s understood that copying or p u b l i c a t i o n f o r f i n a n c i a l gain shall not be allowed without my written permission. Department of The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook Vancouver, V6T 1W5 Place Canada Columbia 4 ABSTRACT This thesis develops a model f o r the management of t i l l cash (currency and coin) of a branch f o r a Vancouver area c r e d i t union. model i s developed i n two p a r t s . The 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, e x h i b i t i n g both a growth trend and high autocorrelations at large lags. In the second model, a growth trend f o r real weekly cash flows i s f i r s t estimated, incorporating an asymptotic capacity c o n s t r a i n t . The real cash flow trend i s converted to a nominal trend and used as the weight in a l i n e a r weighted l e a s t squares model f o r d a i l y cash flows, i n which the explanatory v a r i a b l e s are dummy variables to i n d i c a t e days of the week, months of the year, incidence of pay days, e t c . model i s also The consistency of the r e s u l t i n g forecast 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 d e l i v e r y lag times f o r orders and other assumptions are inappropriate in t h i s t i l l cash management s e t t i n g . To relax the steady state assumptions a general dynamic programming framework i s ii that developed f o r the cash management model that allows f o r e i t h e r penalty costs f o r cash-outs (cash shortages) or a chance c o n s t r a i n t i n v o l v i n g the p r o b a b i l i t y of a cash-out. Because of n o n - s t a t i o n a r i t y of the cash flows the dynamic program cannot be solved d i r e c t l y i s obtained using a simulation technique. s but an approximate s o l u t i o n The. r e s u l t i n g algorithm i s tested on h i s t o r i c a l data and the r e s u l t s are discussed iii briefly. TABLE OF CONTENTS ABSTRACT ii LIST OF TABLES vi LIST OF FIGURES vii ACKNOWLEDGMENTS viii Chapter 1 2 3 INTRODUCTION 1 1.1 The Nature of the Problem 1 1.2 Overview 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 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 iv Chapter 4 5 Page 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 P o l i c i e s . . . . 65 4.6 Summary 69 A STOCHASTIC PROGRAMMING MODEL TO ORDER TILL CASH 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 A p p l i c a t i o n of the Model 86 5.7 Summary 90 Appendix: The Dynamic Program Corresponding to the Objective Function (5.4.2) 6 72 91 CONCLUSIONS. 93 LIST OF REFERENCES , v 95 LIST OF TABLES Table- Page 1 Estimated Regression C o e f f i c i e n t s f o r D a i l y 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 vii ACKNOWLEDGMENTS The author would Like to thank,,the o f f i c e r s of the c r e d i t union f o r the data and valuable discussions, Professor Robert White f o r advisory assistance and Professor William Ziemba f o r technical assistance. Any errors or omissions are the r e s p o n s i b i l i t y of the author alone. The research was performed while the author was in r e c e i p t of a Samuel Bronfman Award and a U n i v e r s i t y of B r i t i s h Columbia Graduate Fellowship. viii Chapter 1 N ITRODUCTO IN 1.1 The Nature of the Problem The purpose of t h i s study i s to develop a model to order t i l l cash (currency and coin) f o r a Vancouver area c r e d i t union branch, with the o b j e c t i v e of minimizing costs, while maintaining adequate balances to meet the members' demand. The demand f o r cash i s not d e t e r m i n i s t i c , and part of the problem i s to determine the s i z e of precautionary cash balances which w i l l reduce t h e . r i s k of a cash-out to an appropriate l e v e l desired by management. A cash-out occurs i f the balance of cash on hand f a l l s below some l e v e l which management regards as the p r a c t i c a l minimum level f o r smooth operation. This l e v e l 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 c o i n . The costs that are to be minimized are the f i x e d costs of placing a cash order and the opportunity costs of i d l e cash balances. If a p o l i c y i s taken to make large orders, few orders are required and f i x e d order costs are low, but cash balances are high, creating large opportunity costs. Conversely, i f orders are small and frequent, f i x e d order costs are high and opportunity costs are low. Minimization of t o t a l costs e n t a i l s 1 2 f i n d i n g an optimal t r a d e - o f f between f i x e d order costs and v a r i a b l e opportunity costs. Thus, the problem i s e s s e n t i a l l y one of constrained o p t i m i z a t i o n : minimize costs subject to some constraint involving the r i s k of a cashout. In t h i s sense, i t i s a standard inventory problem. Inventory theory i s generally oriented towards solving a n a l y t i c a l l y t r a c t a b l e problems which do not always f i t real world s i t u a t i o n s very w e l l . of t h i s project require an operational model. The terms of reference 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 r a r e l y f i t p e r f e c t l y ) , but how well w i l l the model perform when i t s s p e c i f i c a t i o n s are v i o l a t e d . O p e r a t i o n a l i t y 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 c r e d i t union branch has o n - l i n e 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 o n s to the algorithm, but to minimize development costs, the algorithm should be rather s t r a i g h t f o r w a r d Operational costs r e l a t e mainly to the personnel time required to generate d a i l y 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 t h i s operational environment that the cash order model i s developed. 1.2 Overview Development of the model requires the s o l u t i o n of two major sub- problems: estimation of the demand f o r cash and construction of the actual order algorithm. 3 Chapter 2 discusses the t h e o r e t i c a l aspects of forecasting the demand f o r cash. F i r s t , various methods of adaptive time series f o r e c a s t i n g are considered, culminating in a discussion of Box-Jenkins time series techniques. Then, the problems of developing a l i n e a r regression model f o r forecasting cash demand are considered. In Chapter 3, the r e s u l t s of both Box-Jenkins estimation and weighted l e a s t squares estimation are presented. although preferable The Box-Jenkins method, in the sense that i t i s an adaptive forecasting technique, i s found to be i n a p p l i c a b l e because the demand f o r cash i s non-stationary. The n o n - s t a t i o n a r i t y a r i s e s p a r t l y because of the general growth in demand l e v e l and p a r t l y because f a c t o r s of v a s t l y d i f f e r e n t orders of seasonality are present. In order to use weighted l e a s t squares, a forecasting model f o r the general trend l e v e l in cash demand i s f i t t e d by l e a s t squares, and then d a i l y cash flows, deflated by the general trend l e v e l , are regressed on dummy variables r e l a t i n g to months of the year, days of the week, paydays, e t c . Tests of the p r e d i c t i v e a b i l i t y of the model are also discussed in t h 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 t u r e are discussed. models are based on steady state assumptions demand f o r cash. Some of these such as s t a t i o n a r i t y of the The appropriateness of these assumptions is discussed, and f o r purposes of comparison, a dynamic programming formulation of the problem i s presented. In Chapter 5, an approximate s o l u t i o n to the dynamic programming problem i s developed. It e s s e n t i a l l y y i e l d s a v a r i a b l e control 4 l i m i t model. Cash i s ordered when the cash balance is inadequate f o r transactionary and precautionary requirements, which vary according to the expected demand during the order d e l i v e r y lag period. The order s i z e i s selected to minimize expected costs per day during the l i f e of the order, as computed by s i m u l a t i o n . 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 f o r other c r e d i t union branches. Chapter 2 FORECASTING THE DEMAND FOR CASH: THEORY 2.1 Introduction Forecasting i s an inductive reasoning process whereby h i s t o r i c r e l a t i o n s h i p s between a v a r i a b l e of i n t e r e s t (the dependent v a r i a b l e ) and explanatory variables (the independent v a r i a b l e s ) are analyzed, i n order to predict (perhaps with e r r o r ) future values of the dependent variable. A general explanatory model i s of the form *t where y f x t = fCx ...- x l t s n t ) + u (2.1.1) t i s the dependent v a r i a b l e at time t (e.g. cash f l o w ) , i s a function whose form i s determined a priori but which may have e m p i r i c a l l y estimable parameters, l t ' * * * ' X t are the explanatory variables at time t and (e.g., lagged observations of y, , or dummy variables f o r days of the week), u^ i s the forecast e r r o r , the d i s t r i b u t i o n of which may be important to the f o r e c a s t e r . We may develop the model so that E ( u ) = 0, so t h a t , given (x^,'»»,x t the forecast f o r y i s 5 ), 6 y = f(.x --.,x ). r 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 c a l values of any given dependent v a r i a b l e . Rarely do a l l of these models agree in t h e 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 c a l data, because the h i s t o r i c a l r e l a t i o n s h i p s may not continue to hold in the f u t u r e . It is possible to f i t a model on an e a r l y subset of h i s t o r i c a l data and t e s t i t s performance on a l a t e r subset (as i s done l a t e r in t h 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 r e l a t i o n s h i p s . Thus, i f inappropriate treatment of, or omission of, some explanatory v a r i a b l e 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 g n i f i c a n t in the future. In the present a p p l i c a t i o n , f o r 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 p r e d i c t well on h i s t o r i c data. w i l l increase without bound over time, some w i l l Some of the estimates increase to a f i n i t e asymptotic l i m i t , and some w i l l u l t i m a t e l y decrease a f t e r a c e r t a i n time. Selection of an appropriate model cannot be made o b j e c t i v e l y with h i s t o r i c a l data only, but must involve some subjective judgment of the forecaster. W i t h i n . t h i s framework, the theories of several p o t e n t i a l f o r e casting models f o r cash demand at the c r e d i t union branchliare analyzed. In Section 2.2, cash demand i s regarded in a time series framework whereby f u t u r e " cash flows are estimated in terms of the series of h i s t o r i c a l flows only. Such models are frequently adaptive i n the sense that c e r t a i n 7 types of f o r e c a s t i n g errors tend to be corrected over time. placed on Box-Jenkins time s e r i e s models. Emphasis is I t i s pointed out t h a t , i f the cash demands have s i g n i f i c a n t 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 f o r h i s t o r i c a 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 c o n s t r a i n t . Then cash demands are d e f l a t e d by t h i s trend level in a weighted l e a s t squares l i n e a r model which uses as explanatory v a r i a b l e s dummy v a r i a b l e s i n d i c a t i n g the incidence of various days of the week, months of the year, paydays and holidays. This model uses a wider range of explanatory v a r i a b l e s than the Box-Jenkins model and, as a r e s u l t , may provide v i a b l e 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 v a r i a b l e s in the model (2.1.1) are lagged observations of the independent v a r i a b l e (y t -j , y t 2'***')- Many c l a s s i c a l f o r e c a s t i n g models such as a r i t h m e t i c moving averages and exponentially weighted moving averages have t h i s form. An e x c e l l e n t discussion of such models i s given in Wheelwright and Makridakis [1973]. They are adaptive in the s t r i c t sense t h a t , i f there i s an increase or decrease in the mean level of the v a r i a b l e s y^, a r i t h m e t i c 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 l e v e l changes again. If the time series f l u c t u a t e s r a p i d l y over time, sophi- s t i c a t e d weighting schemes are required to reduce the response time i n the adaptation process and thus reduce forecast erros. C l a s s i c a l theory says l i t t l e about methods of s e l e c t i n g optimal values f o r the time period f o r moving average or the weight f o r an exponentially weighted average.;. A l s o , these techniques f a i l to provide a n y d e s c r i p t i o n of the d i s t r i b u t i o n of the random e r r o r term u^. in ( 2 . 1 . 1 ) . For the problem at hand, some d e s c r i p t i o n of the e r r o r term u^ i s valuable in analyzing the r i s k of a cash-out. A l s o , cash flows f l u c t u a t e s i g n i f i c a n t l y from day to day, week to week and month to month because of predictable v a r i a t i o n s in demand. A naive weighting scheme in a weighted average forecast would be highly inaccurate because of the frequent r e 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 f o r q u a n t i f i c a t i o n 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 i n g l e v a r i a b l e y t (such as cash f l o w ) . One i s the i n f i n i t e autoregressive (AR) form: y where t y^ i s the t = 6 + u t + <h y t - 1 + <j> y _ 2 t + 2 ••• observation i n the s e r i e s , 6 i s a constant, {u^} i s a sequence of independent i d e n t i c a l l y dist r i b u t e d random shocks (unobserved), (2.2.1) 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 t h a t , 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 t e r n a t i v e 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 = 6 + u - t 6: t V ] - 6 2 U t _ . 2 ... where ^ t ^ ' ^ ' ^- t^ and 0 i , 8 , ••• are the moving average weights, u a r e (2.2.2) before a s 2 The AR and MA forms are equivalent in the sense t h a t , given one, the other can be found, although i t may not be a convergent s e r i e s . For f o r e c a s t i n g 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> .y _ = fi + u p t p 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 p a r t s , respectively). Introducing the backshift operator B where By = t y B^y = y _ l , and then d e f i n i n g the AR polynomial t < (j>(B) = 1 - <h B - <J> B 2 2 - ••• - <J) B (2.2.4) - ••• - 0 B (2.2.5) P p and the MA polynomial 0(B) = 1 - e a B - 6 t = 6 + e-(B) u 2 B 2 q q we can rewrite (2.2.3) as *(B) y (2.2.6) t A d e s i r a b l e property of (2.2.6) i s s t a t i o n a r i t y , which means that the form of the stochastic process does not vary over time (in p a r t i c u l a r , the s c a l i n g and l o c a t i o n of the v a r i a t i o n should be constant). S t a t i o n a r i t y i s equivalent to having the zeros of the AR polynomial outside the u n i t c i r c l e . If there i s a zero s t r i c t l y i n s i d e the u n i t c i r c l e , the s e r i e s e x h i b i t s explosive behaviour. Zeros on the u n i t c i r c l e r e s u l t in mild n o n - s t a t i o n a r i t y and correspond to f a c t o r s in <j)(B) of the form (1 + B ) s or (1 - B ) f o r s = 1 , 2 , 3 , « " . The form 1 - B s corresponds to taking the d i f f e r e n c e between the current observation and the observation at the (seasonal) lag s. Powers of (1 - B ) correspond to taking d i f f e r e n c e s 11 of d i f f e r e n c e s . One can remove the d i f f e r e n c e factors from the AR part to rewrite (2.2.6), with obvious changes in n o t a t i o n , as ti c D 4>(B) (1 - B) ,(1 - B ) Q S In p r a c t i c e , i f the observed series y t y t =• 6 + 6(B) u (2.2. t e x h i b i t s n o n - s t a t i o n a r i t y (such as growth over t i m e ) , the forecaster should t r y various forms of d i f f e r encing before attempting to solve f o r <J>, 6 and 6. Analogous to s t a t i o n a r 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 p a r t . By recursive s u b s t i t u t i o n 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 s t i p u l a t e s that t h i s i n f i n i t e series s h a l 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 f o r forecasting purposes, since one does not know what appropriate values of e a r l y , unobserved data to use. I n v e r t i b i l i t y i s equivalent to the zeros of the MA polynomial l y i n g 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 t a t i o n a r i t y requirements, an ARMA process i s uniquely determined by i t s a u t o c o r r e l a t i o n s . Thus one can look at the estimated autocorrelations of the series (along with p a r t i a l autoc o r r e l a t i o n s and inverse autocorrelations which are r e l a t e d to them) to i d e n t i f y an appropriate model of the form (2.2.7) f o r an observed time series. The parsimony concept requires s e l e c t i o n of a model where d, D, p and q are as small as possible. This presents a p o t e n t i a l problem 12 f o r the a p p l i c a t i o n of Box-Jenkins techniques to forecast cash demand, f o r one would expect a l o t of information content in high order autocorrelations. For d a i l y cash demands there should be high p o s i t i v e auto- c o r r e l a t i o n s 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 y e a r ) . 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 autocorrel a t i o n at that l a g . Moreover, some weeks only have 4 days and t h i s w i l l upset the basic a u t o c o r r e l a t i o n structure at 5 day lags. A l l of t h i s suggests i t may be hard to f i t a parsimonious model to d a i l y data. The problems f o r d a i l y cash flow estimation are mitigated somewhat by considering weekly cash flows. reasonable (but s t i l l A long 260 day log i s then a more long) 52 week l a g . The problem of holidays reducing some weeks to 4 days does not a f f e c t weekly lag structures as s e r i o u s l y as i t does d a i l y lag s t r u c t u r e s . The actual estimation of (2.2.7) i s accomplished by minimizing the sum of squared residuals (^u^ ), i t e r a t i v e l y by the Marquardt algorithm. 2 Under the hypothesis that the u^'s are:independent N(0,a), t h i s y i e l d s maximum l i k e l i h o o d estimates of 0, <p and 6. Unbiased k-day ahead forecasts that minimize mean square pred i c t i o n e r r o r 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 subs t i t u t e d f o r the random shocks (and the k unrealized random shocks are set to 0). The forecast errors are normally d i s t r i b u t e d i f the random 13 shocks 6 _-|) 2 are, and the k-day forecast error variance i s (1 + 9 o^ 2 where the 0-'s 2 : + ••• + 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 s t r a i g h t f o r w a r d . The main problem to be faced when applying Box-Jenkins i s the question of whether an i n v e 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 s e r i e s . It was argued t h a t , in order to capture most of the explainable v a r i a t i o n i n 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 a d d i t i o n a l variables i s to use a regression model: y = xB + u f where • y = • • • x = Oi "N • , • E(u u ) = 1 0 2 • • • x - u = • (2.3.1) 1 N U N 0 'aj 14 x^. i s a k x 1 vector of explanatory variables at time t , i s a random e r r o r , with Eu = 0, and 3 i s a k x 1 vector of regression c o e f f i c i e n t s . Appropriate explanatory variables x ^ include dummy variables which take the value 1 f o r the various days of the week, months of the year, pay days and holidays, and are zero otherwise. ( H»"*'Xfct^' X w e ma ^ nave ' f° r e x a m That i s , i f x// = Pl > e 1 i f day t i s a Tuesday x lt = * 0 otherwise. Another component that must be considered f o r x^ i s the level of i n t e r e s t rates, since c l a s s i c a l economic models regard i t as a determinant of demand for money. There are several problems with introducing i n t e r e s t rates into the model. F i r s t , over the short three year period of a v a i l a b l e data, i n t e r e s t rates are highly c o l l i n e a r with time, so i t would be d i f f i c u l t to d i s t i n g u i s h growth e f f e c t s from i n t e r e s t rate e f f e c t s . d i f f i c u l t to decide which i n t e r e s t rate series to use. Second, i t i s C e r t a i n l y the i n t e r e s t rate most l i k e l y to influence a c r e d i t union member's demand for cash i s the i n t e r e s t 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 e f f e c t s (over a three year period) would be spurious. It i s d i f f i c u l t to argue that other i n t e r e s t rate s e r i e s , such as commercial paper r a t e s , have any observable e f f e c t on demand f o r currency. T h i r d l y , i f some macro-economic model were a v a i l a b l e to express demand f o r money in terms of i n t e r e s t 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 n e c e s s a r i l y true that demand f o r currency is as strongly affected by general economic conditions as the demand f o r other forms of money. Thus, f o r operational s i m p l i c i t y , i n t e r e s t 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 t h i s i s to have some explanatory v a r i a b l e that increases over time (such as t ) . This would only model an a d d i t i v e growth e f f e c t and f a i l to model any increase in the e f f e c t s of the dummy explanatory v a r i a b l e s 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 f o r example also should increase over time. To model t h i s , one could allow polynomial time trending of the c o e f f i c i e n t s . y t = x ' t Bo + x ' That i s , consider the model B i t + ••• + x " B t t p t + u p 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 c o e f f i c i e n t s which are m u l t i p l i e d 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 e c t , each c o e f f i c i e n t i n B i s allowed to f o l l o w a trend over time, where B = Bo + B i t + B_t 2 + ••• + 3 t . p P This model has the l i n e a r form h = w t + u, 16 where f(x , tx u ] t : -.tPxu) w, (xkt, txkts \t)' Quadratic time trending, f o r example, t r i p l e s the number of c o e f f i c i e n t s to be estimated, which would y i e l d low s i g n i f i c a n c e l e v e l s f o r 3 i f there are a large number of explanatory v a r i a b l e s . for e x t r a p o l a t i v e f o r e c a s t s . Moreover, polynomials are poor That i s , polynomials may have maxima or minima f o r t > T which simply r e s u l t from the need to f i t high order d e r i v a t i v e s in the range 1 < t < T, and do not i n any way r e f l e c t any p r e d i c t i v e power of the polynomial model. (for s c a l a r 3 o , For example, f i t t i n g the simple trend model 3i,.32) y = 3o + S i t + 3 t t 2 2 + u t to weekly cash flows y^ resulted i n a polynomial that predicted a peak i n demand i n August 1976 ( j u s t s l i g h t l y beyond the data range)! C l e a r l y such growth models are inappropriate. As the f i n a l choice to model the growth, l e t us simultaneously consider a r e l a t e d problem, that of h e t e r o s c e d a s t i c i t y . I n i t i a l regressions indicated that o ^ , the variance of u^ increases over time and, indeed, i s 2 proportional to d time t . 2 t , the square of the general level of cash demand, at That i s , o^ 2 d 2 t . Without loss of g e n e r a l i t y , we may take d to be the general l e v e l of weekly demand. t This suggests using a m u l t i p l i c a t i v e growth model to simultaneously model growth and eliminate h e t e r o s c e d a s t i c i t y . In the l i n e a r model (2.3.1), weighted l e a s t squares i s required to eliminate 17 heteroscedasticity. In t h i s case the appropriate weights are 1/d^- The mode.l i s then y 1 t -± = x ' t 6 + u (t=l,..-,N) t (2.3.2) or, W = Z3 + u (2.3.3) where Wi w = • • • • -VJd u/di Zi' ,z = — • • • • u = • • — 1 • • 1 k, and x N J E(U) = 0 cov(U) = q 2 I This can be rewritten as y t = d (z 'B + u ) which i s a m u l t i p l i c a t i v e form. t t t (2.3.4) Note that w^ = y^/dj. i s the cash flow at time t expressed as a proportion of the general trend l e v e l 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 e x h i b i t no growth over time (that i s , 18 as 0-1 dummy v a r i a b l e s ) . The d^ provide f o r growth of y t and eliminate h e t e r o s c e d a s t i c i t y at the same time. Other possible sets of weights {d > t which in some way i n d i c a t e the s i z e of the branch may be considered, such the number of members or the t o t a l amount of deposits. Here i t i s again important to consider whether h i s t o r i c a l r e l a t i o n s h i p s between the " s i z e " measures and demand w i l l i n t o the f u t u r e . continue If care i s not taken to exclude i n a c t i v e members from a count of members, changes in the proportion of a c t i v e members on the r o l l s w i l l d i s t o r t the r e l a t i o n s h i p between cash demand and number of members. The branch opened a new b u i l d i n g in May 1973, and i n i t i a l l y experienced rapid growth f o r some time afterwards, but the rate of growth has decreased since then. In the e a r l y part of the period, most members were new and a c t i v e , but a f t e r 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 i n d i c a t i v e of s i z e , but t h i s i s not l i k e l y to be a stable p r e d i c t o r 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, e t c . that may be in vogue from time to time. That i s , at various times members may change t h e i r t o t a l deposits without changing t h e i r demand f o r cash. Thus, the weight d^ i s chosen to be proportional to the general trend l e v e l 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 f o r 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 r y to forecast i n f l a t i o n , but t h i s i s not necessary since Consumer P r i c e 19 Index figures are a v a i l a b l e monthly and l i n e a r e x t r a p o l a t i o n beyond the l a t e s t two months of unavailable data provides an e x c e l l e n t forecast of the CPI level (although, of course, e x t r a p o l a t i o n i s a poor p r e d i c t o r of changes in CPI l e v e l s ) . nominal cash demand. It i s necessary, however, to forecast l e v e l s of Thus, l e t trend l e v e l of real weekly cash demand at time t , CP index at time t CP index f o r January 1976 ' trend l e v e l of nominal weekly cash demand at time t , expressed i n January 1976 d o 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 i n real demand d^. The real weekly cash flows generally increased over the period f o r which data were a v a i l a b l e , but the growth rate tended to decrease over the period. Indeed, regressions to f i t piecewise l i n e a r trends in time showed a monotonic increasing but concave cash trend. This i s s i m i l a r to the general a c t i v i t y l e v e l s of other branch operations because rapid growth followed the opening of a new b u i l d i n g in May 1973, but by 1976, f u r t h e r growth was dampened by the general physical constraints of the b u i l d i n g ( e . g . , t e l l e r wicket space i s now l i m i t e d and long line-ups at the t e l l e r s ' wickets act to discourage members from using the branch cash s e r v i c e s ) . Thus, growth in real cash demand appears to be approaching a capacity cons t r a i n t , as in Figure 1, and t h i s should be incorporated i n the forecast model. 20 d t capacity c o n s t r a i n t 1973 1974 Figure 1. 1975 1976 Growth of Real Cash Demand. A simple way of modelling such an asymptotic constraint i s by a s u i t a b l y scaled and located rectangular hyperbola, namely -k c - t + t. where (2. c i s the capacity c o n s t r a i n t , k i s a p o s i t i v e s c a l i n g constant, and to i s a r e l o c a t i o n parameter f o r time (t + t f o r a l l t in the model) 0 > 0 Johnston [1972, p. 52] suggests such r e c i p r o c a l transformations with the artificial restriction t 0 = 0. This i s a l i n e a r f u n c t i o n of can be f i t t e d by l e a s t squares. t That i s , l e t _ nominal weekly cash flow at time t p. For a f i x e d value of t 0 1 t + t and ( 21 k t + t + e 0 (2.3.6) t where, f o r e = 0 E(e) = 0 and E(ee') = Z f o r some a > 0. As in the regression G = a (2.3.3) the variance of e^ i s pro- portional to the square of the trend l e v e l in cash demand. Since E(e) = 0, the l e a s t squares estimators c and £ are unbiased, but i n e f f i c i e n t under the h e t e r o s c e d a s t i c i t y , which merely places more weight on l a t e r observations. This i s the most desirable departure from homoscedasticity, in t h i s case. To s e l e c t the time l o c a t i o n parameter t , i t s u f f i c e s to estimate (2.3.6) 0 f o r various values of t 0 and choose t 0 to minimize the sum of squared residuals. 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^ i n the weighted l e a s t squares model (2.3.3) to obtain the approximate generalized l e a s t squares estimate, 3 = ( X E~ 1 1 X ) " di X ' 1 Z u Y = • W = t lo Z'W • • • • where 1 yi/di 0 2 ( Z ' Z ) " djiJ ly /d j N N For a r e a l i z a t i o n of the dummy explanatory variables zj., we have an estimate of nominal cash demand at time t of h = C - where d t t \ + t r (2.3.8) y^ = cash flow forecast f o r time t , p .t d t = CPI at time t CPI f o r January 1976 = c k t + t 0 ' • = forecast trend level of real weekly cash flow at time t , z_ = vector of explanatory dummy variables f o r time t. 3 = estimated regression c o e f f i c i e n 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 (2.3.9) te The f o l l o w i n g proposition gives consistency r e s u l t s f o r the forecasts and variance of forecast e r r o r s . Proposition: The p r e d i c t i o n s y t and y^ are both consistent p r e d i c t i o n s of y^ and a consistent estimator of the variance of y^ i s d where s 2 s [ l + z^(Z'Z)" 2 2 (2.3.10) z ] 1 t 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 + [1 (N + t o f ' J to)" 1 V Nx2 di Then = (V'V)" 1 V and the covariance matrix of (V'V)" 1 V E V(V'V)" 1 1 s (2.3.11) 24 By i n t e r p r e t i n g enlargement of the sample s i z e N to mean the observation of a sequence of independent but s t o c h a s t i c a l l y i d e n t i c a l c r e d i t union branches over the same time period, the regressor matrix V i s "constant in repeated samples" (Theil [1971, pp. 3 6 4 - 3 6 5 ] ) , so that N ' M V V ) and N " 1 V E g V both converge to p o s i t i v e d e f i n i t e matrices. We can rewrite ( 2 . 3 . 1 1 ) as N [ N ( V V ) ' _ 1 (N 1 V _ 1 Z V) N ( V ' V ) ] and t h i s - 1 g tends to 0 as N -> °°. Thus, the variances of c and k tend to 0 f o r large N, 3 e s t a b l i s h i n g the consistency of these estimators. Thus, the elements of K 0 ) 2 are consistent estimators of those of. I = E(U'U). U A l s o , the elements of these matrices are bounded (in p r o b a b i l i t y ) since the trend l e v e l s d^ are bounded by the capacity c o n s t r a i n t c. Define the p o s i t i v e d e f i n i t e matrix = l i m N" a Z'Z = plim N" X' E~ X. Then by the consistency and N-*» N-*» boundedness, we also have lim N ( X ' E~ X) = Q . Then p l i m ( N X' E" , X,— . Q 1 1 2 1 y U _1 N" 1 X 1 E" X) = Q 1 - = 0. ments e s t a b l i s h that plim N and _1 1 The same boundedness and consistency argu- _ i plim N" N-*» 1 1 X'fZ' U (E 1 1 _ 1 - E" ) U = 0 1 - E" ) U = 0. 1 u Hence by T h e i l ' s Theorem 8.4 [1971, p. 3 3 9 ] , plim /N ( i - 6) = 0, s N-*» 2 is a consistent estimator of a , and the matrix s ( Z ' Z ) converges in p r o b a b i l i t y 2 to the covariance matrix of 3. 2 In p a r t i c u 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 [l + z|(Z'Z) 2 /\ Since y t /\ = - 1 z ] t . s\ and d t i s a consistent estimator of d , formula (2.3.10) t must be the asymptotic variance of y^. follows from that of (3, c and k. The consistency of y^ and y^ This establishes the p r o p o s i t i o n . We may conjecture that the asymptotic variance (2.3.10) of y^ i s also asymptotically the variance of y^. At any r a t e , that variance w i l l be a meaningful p r a c t i c a l approximation to the variance of y^. Thus, we have generalized l e a s t squares estimates f o r cash demand, along with asymptotic r e s u l t s about the estimators and t h e i r variances. This completes our analysis of the regression model. 2.4 Summary In t h 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 l e a s t squares model which uses dummy regressors to i n d i c a t e 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 differences: y •"• u - y t t = y = .6 + u t - 1 t " t-i " y t 6+ 9 - e V 6 + ( l - eB) u = l Vl t S u b s t i t u t i n g the second equation r e c u r s i v e l y into the f i r s t to eliminate u _-|, _ 2 > * ' * y i e l d s : successively u t h t-l y = t = u =r + U t + " 6 6 ( y t-l " h-Z + (1 + 6)6 + (1 - 6 ) y _ t t ~ 1 6 + 9 u 9y _ + t t-2 J - 0 2 u _ 2 t 2 • • • = u Since E u + (l+e+e +e +..-)6 + (l-0)(y _ 2 t 3 t 1 + 0y _ t + 0 y _3+---) 2 2 t = 0 a l i n e a r unbiased forecast of y , given _-]> ^ t - 2 ' " * * v t y t = (l+0+0 +---)6 + ( 1 - 6 ) ( y _ 2 t t 1 + 0y _ t i s t + 9 y _ 2 2 t 3 + ••• ) Under mild r e s t r i c t i o n s on the y ' s , t h 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 . A l s o , note that f o r 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 c a l forecasting models. 2 The Consumer P r i c e Index to be used i s the S t a t i s t i c s Canada seasonally unadjusted monthly CPI f o r Vancouver. One may argue that a seasonally adjusted index would be b e t t e r , since i t would not d i s t o r t some <~~ of the seasonal v a r i a t i o n s in cash f l o w that should be estimated in the ' model,' but such a f i g u r e i s not a v a i l a b l e . More importantly, one might argue that the CPI bundle of goods includes items that are paid f o r by cheque as well as cash. None of the indices a v a i l a b l e r e f l e c t cash payments better than the general CPI, however. 27 °The i n t e r p r e t a t i o n of V as being constant i n repeated samples only i n d i c a t e s that trend estimates are consistent f o r time points i n or s h o r t l y a f t e r the 3 year data range, N. To t e s t the accuracy of e x t r a p o l a t i v e forecasts the model w i l l be f i t t e d i n Chapter 3 on the f i r s t two years of data and tested on the t h i r d year. This i s a useful-adjunct to the-consistency r e s u l t s presented here which are only of an i n t e r p o l a t i v e nature.A l t e r n a t i v e l y , one can i n t e r p r e t sample enlargement as the a d d i t i o n of more observations f o r the same c r e d i t union as time proceeds. To consider t h i s type of consistency, f i r s t note that N •1 N I (to+t)' t=l V'V N -2 I (to+t) t=l I (to+t)' t=l (V'V) Let nh2 T i n -1 m 2j (m i 2 2 The 1 m2 2 N-MzUo+t)- ) E(t +t)- 1 0 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 Tfj" I ( t + t ) 1 The square root of the second term i s t i c a l l y l i k e - -j-N •+ » , m 2 2 dt t=l 1 / 0 and hence (V'V) However, m l a In 7N t +N to+1 ^ c n -> 0 as N -> °°. 0 D e n a v e s asympto- Therefore, as f 0 =— ^^t) E(t +t) w n 0 0 as N -> 0 0 since the - (Z(to+t)- )" - 2 1 0 numerator and f i r s t term i n the denominator have f i n i t e l i m 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 . Similarly 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 i n the denominator tends to zero and the second term ro oi -1 tends to ». Thus (V'V) as N •> f o r some a > 0. 0 a 00 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 t e - v a l u e d matrix, as N -> °° whose diagonal e n t r i e s are non-zero. - v Hence as N -*- °°, the matrix (V'V)" '0 0 0] a 1 for a 1 > 0. V'Z V ' ( V V ) " 1 tends to a matrix of the form Thus, the estimator of the capacity constraint c i s c o n s i s t e n t , but the estimator k, which describes how q u i c k l y the capacity i s approached, i s not consistent. This means that f o r projections of growth into the d i s t a n t f u t u r e , the forecast of ultimate capacity i s "accurate" ( c o n s i s t e n t ) . The projected rate of growth to that capacity may be over or under estimated, but forecast errors induced by t h i s are bounded since the projection of capacity i s " a c c u r a t e . " Chapter 3 ' FORECASTING THE DEMAND FOR CASH: APPLICATION 3.1 Introduction In t h 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 s p e c i a l problems associated with them. 3.3 presents the r e s u l t s of Box-Jenkins a n a l y s i s . Section It w i l l be found there that attempts to estimate a s t a t i o n a r y , i n v e r t i b l e 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 l e a s t squares l i n e a r model with dummy variables to forecast cash demand. This r e s u l t s i n an estimable model, the forecasting a b i l i t y of which i s studied by f i t t i n g the model on the f i r s t two years of data and t e s t ing the r e s u l t i n g t h i r d year forecasts f o r bias and other sources of e r r o r . 3.2 The Data Data on cash balances and cash orders f o r the c r e d i t union branch were analyzed f o r the period May 1973 to A p r i l 1976. This period com- menced with the opening of a new b u i l d i n g , and e a r l i e r data are l i k e l y to be s t r u c t u r a l l y d i f f e r e n t at least i n terms of the general l e v e l of cash flow, i f not also i n the pattern of v a r i a t i o n s i n the flows. 29 30 Closing t i l l cash balances were obtained f o r the 768 days from General Ledger accounts and information on cash d e l i v e r i e s was obtained from d e l i v e r y invoices prepared by the B r i t i s h Columbia Central C r e d i t Union, through which cash orders were placed. t i o n of d a i l y cash flows. This allowed the computa- In a d d i t i o n , during December 1974 several deposits were s o l i c i t e d from a large l o c a l merchant to cover cash shortages. These were treated in the same way as regular cash d e l i v e r i e s f o r the c a l c u l a t i o n of d a i l y cash flows. Of the r e s u l t i n g 767 d a i l y cash flows, a l l but 36 were negative (outflows), because the branch dealt p r i m a r i l y with consumer accounts rather than commercial accounts. C o l l e c t i o n of the data was a c t u a l l y 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 s e t t i n g debits and c r e d i t s to the General Ledger were not made on the same day. A l t e r n a t i v e l y , i f the head t e l l e r was absent f o r several days and did not post the changes i n the treasury cash balance,^ while other t e l l e r s withdrew money from the treasury (and posted t h e i r c r e d i t s immediately) the same type of discrepancy would occur. These discrepancies resulted in a p o s i t i v e (negative) flow one day followed by a negative ( p o s i t i v e ) flow of equal magnitude a few days l a t e r . The methods of making General Ledger e n t r i e s varied from time to time and f o r the middle h a l f of the data period, each t e l l e r ' s c l o s i n g 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 t h i s provided enough d e t a i l , i n conjunction with the balance in the t e l l e r ' s interchanges account (which normally i s z e r o ) , to i n f e r the c o r r e c t adjustment f o r these discrepancies. Such adjustments could not be made, of course, f o r 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 e n t r i e s were inadvertently posted to the wrong account and corrected a few days later. This also would r e s u l t i n an p o s i t i v e (negative) flow one day, followed a few days l a t e r by a negative ( p o s i t i v e ) flow of equal magnitude. L i t t l e could be done to eliminate t h i s type of discrepancy. A t h i r d type of discrepancy could have resulted from cash d e l i v e r y 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 f o r other branches were found i n c o r r e c t l y f i l e d with those of the branch involved in t h i s study. This type of discrepancy would r e s u l t in a p o s i t i v e flow one day, which, unlike 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 ( i n d i c a t i n g days of the week, months of the year, pay days and h o l i d a y s ) , 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 c o e f f i c i e n t 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 e d with the branch studied. 32 3.3 Estimation by Box-Jenkins Techniques When the theory of the Box-Jenkins technique was presented i n Section 2.2 i t was pointed out t h a t , with s i g n i f i c a n t a u t o c o r r e l a t i o n information at short and long lags, a time series model f o r d a i l y cash flows might be hard to f i t and that aggregation to weekly cash flows would present fewer problems f o r estimation. f i t t e d , neither can a d a i l y model. If a weekly model cannot be Thus, the i n i t i a l r e s u l t s presented here are f o r 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 o n a r i t y . Consecutive and 4 ^ order seasonal d i f f e r e n c i n g were considered on both raw and logged data. Note th that 4 order seasonal d i f f e r e n c i n g corresponds approximately to a seasonal lag of one month. The a u t o c o r r e l a t i o n s and p a r t i a l a u t o c o r r e l a t i o n s of these differenced series suggested a 4 ^ process. r order moving average These models are of the form 0-B )y S where 0 t = 6 + (l-OiB-ezB 2 - e B 3 3 - 0 ^ - 0 B ) u 5 5 t 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 l e a s t squares algorithm with i n i t i a l estimates obtained by solving a system of 1inear Yule-Walker type equations i n v o l v i n g the inverse 2 autocorrelations and the MA parameters {Q-}. This y i e l d e d the form 33 (l-B' )y + R 2 t = .382 = (1 - .117 B + .250 B (1.84) - .266 B 2 (4.25) 3 (-4.65) - .698 B ) u 4 t (5.22) + 2479 (5.05) (3.3.1) The t - s t a t i s t i c s (147 degrees of freedom) f o r each parameter are given in parentheses, and a l l except that f o r 0^ are to t e s t the hypothesis the parameter equals 0. e 4 The t - s t a t i s t i c f o r G 4 that i s to t e s t the hypothesis = i. At f i r s t sight the "model (3.3.1) appears to be quite reasonable in that a l l c o e f f i c i e n t s but 8 i are s i g n i f i c a n t l y d i f f e r e n t from zero. However, to check the i n v e r t i b i 1 i t y of the process (which i s e s s e n t i a l to f o r e c a s t i n g ) , 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 f o l l o w s : 0(B) = - (B + 1)(B - ,1)(B - i ) (B + i ) = 1 - B * 1 (3.3.3) The only major differences between (3.3.2) and (3.3.3) are the constant c o e f f i c i e n t and the f i r s t f a c t o r . polynomial i n (3.3.1). But (3.3.3) i s simply the d i f f e r e n c i n g S u b s t i t u t i n g the approximation (3.3.3) into (3.3.1) and c a n c e l l i n g the f a c t o r 1 - B y K t to eliminate the o v e r - d i f f e r e n c i n g y i e l d s = u t + 6'. 34 This implies that y t i s e s s e n t i a l l y white noise. Thus, the autocorrelations created by d i f f e r e n c i n g constituted almost a l l of the autocorrelations a v a i l a b l e f o r estimating (3.3.1) and induced an MA polynomial that i s v i r t u a l l y i d e n t i c a l to the AR d i f f e r e n c i n g polynomial. Of course, the t - s t a t i s t i c s of (3.3.1) i n d i c a t e that the MA polynomial i s not exactly (1 - B *), but the only d i f f e r e n c e between the polynomials i s in the f a c t o r 1 (B + 1.34). A l l the other f a c t o r s of 1 - B * cancel e x a c t l y . 1 When a l l of the other p l a u s i b l e 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 t h e i r roots e s s e n t i a 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 - B *), which 1 indicates o v e r - d i f f e r e n c i n g , even though d i f f e r e n c i n g i s required to achieve s t a t i o n a r i t y . Thus, from a Box-Jenkins viewpoint, there is. not enough informat i o n in the autocorrelations of weekly cash flows to overcome the white noise, and the time s e r i e s 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 e r r o r i s eliminated and the remaining series may be estimable. Since the real Requirement i s to have a model f o r d a i l y rather than weekly cash flows, t h i s l a s t hypothesis was not tested f o r weekly cash flows but f o r d a i l y cash flows deflated by the trend. That i s , in the notation of (2.3.3) l e t w^ = y / d t d a i l y cash flow y^ deflated by the trend d^. order and seasonal t be the For a l l combinations of f i r s t (s = 5 days) d i f f e r e n c i n g , the series of autocorelations f a i l e d to die out at large lags, i n d i c a t i n g 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 model. Box-Jenkins 35 In summary, the Box-Jenkins estimation f a i l e d because the s e r i e s of'Cash flows i s non-stationary. forms. The n o n - s t a t i o n a r i t y occurs in two One source of n o n - s t a t i o n a r i t y i s the trend. The other source of n o n - s t a t i o n a r i t y i s a seasonal d r i f t r e l a t e d to the autocorrelations at long lags. 3.4 Estimation by Regression Techniques Since the attempt at Box-Jenkins estimation of the demand f o r cash was a f a i l u r e we consider here a d i f f e r e n t c o l l e c t i o n of explanatory variables and use the model developed in Section 2.3, which employs a non-linear trend estimate m u l t i p l i e d by a weighted l e a s t squares l i n e a r model. The non-linear trend (2.3.6) of 156 real weekly cash flows, d^, was f i t t e d by l e a s t squares y i e l d i n g 170,000 - 28.9 x 1 0 ( t + 240)" 6 (26.32) (-9.00) (3.4.1) 1 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. variance over time. As expected, the residuals displayed increasing As a r e s u l t , the estimated t - s t a t i s t i c s and variance of the e r r o r term are a l l biased. The estimated c o e f f i c i e n t s , however, are unbiased and consistent. The asymptotic capacity level of real weekly cash demand that i s forecast by (3.4.1) i s $170,000 (expressed in January 1976 d o l l a r s ) . Sub- s t i t u t i n g t = 767 y i e l d s an estimated trend level f o r May 1976 of $141,000 36 or 83% of the asymptotic level of cash demand. This f i g u r e is thought to be reasonable i n l i g h t of the actual remaining p o t e n t i a l f o r growth at the branch. To t e s t the goodness of f i t f o r (3.4.1), a quadratic polynomial i n time was added to the regression and i t did not s i g n i f i c a n t l y increase R. 2 F i t t i n g the model f o r the f i r s t and l a s t halves of the data separately did not r e s u l t i n s i g n i f i c a n t l y d i f f e r e n t estimates f o r the regression c o e f f i c i e n t s c and k. (Again, we must emphasize that standard s i g n i f i c a n c e t e s t s are not meaningful in such an heteroscedastic s i t u a t i o n , and these statements of s i g n i f i c a n c e are h e u r i s t i c only.) Note that since p r e c i s e l y three years of data were used to estimate the trend, no bias w i l l be introduced as a r e s u l t of seasonal f l u c t u a t i o n s in demand. The trend estimates m u l t i p l i e d by the CPI l e v e l s p t were then used as weights d^ in the generalized l e a s t squares model (2.3.3) to estimate the f l u c t u a t i o n s in cash demand about the trend l i n e . pendent v a r i a b l e s The inde- were dummy v a r i a b l e s i n d i c a t i n g 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 v a r i a b l e s 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 e r r o r and i t s c o r r e c t i o n . In order to reduce any tendency of these errors to bias the estimated regression c o e f f i c i e n t s , adjustments to the data were made whenever there was a p a i r of r e s i d u a l s 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 r e s u l t i n g r e s i d u a l s were equal. Such adjustments were made in 8 cases, out of the 759 data observations that were used f o r the demand estimation. The f i n a l form of the demand equation (2.3.3), with k = 28 explanatory v a r i a b l e s , i s presented in Table 1. To avoid m u l t i - c o l l i n e a r i t y the regression was run with no dummy v a r i a b l e f o r January. Since the other winter months, November and February, had c o e f f i c i e n t s that were i n s i g n i f i c a n t l y d i f f e r e n t from zero, they were dropped from the model and hence equated to January. The r>.. regression c o e f f i c i e n t s f o r Tuesday and Wednesday mid-month and month-end pay dates were e s s e n t i a l l y i d e n t i c a l , so they were included as one dummy variable. A few comments about the regression c o e f f i c i e n t s are in order. Recall that the equation i s f o r d a i l y cash demand expressed as a proportion of the general weekly trend l e v e l of cash flow, and the explanatory variables are e i t h e r 0 or 1. Thus the i t h regression c o e f f i c i e n t indicates the e s t i - mated increase in cash demand (expressed as a proportion of the weekly trend l e v e l ) that i s caused by incidence of the event for-which the i v a r i a b l e i s an i n d i c a t o r f u n c t i o n . For example, an ordinary Tuesday in January that i s not on or p r i o r to a pay day w i l l have a estimated cash demand of 5.32% of the general weekly trend l e v e l . From the f i r s t f i v e c o e f f i c i e n t s we can see that the demand f o r cash r i s e s through the week, peaking on Saturday (even though Saturday has shorter hours of operation than the other days). 38 Table 1 Estimated Regression C o e f f i c i e n t s f o r Daily Demand Estimated Coefficient Standard Error of Dummy V a r i a b l e z.^ i s a 0-1 Indicator .0532 .0120 ** Tuesday Function f o r : .0844 .0114"** Wednesday .1139 .0123*** Thursday .2358 .0123'** Friday .2848 .0121' .0178 .0138 ** .0375 .0144*** .0233 .0139* .0349 .0140*** Saturday March Apri 1 May June .0530 .0139*** .0193 .0136 .0210 .0140 September .0291 .0137* October .1249 .0148*** .0931 .0198*** .1285 .0440* July August December Tuesday or Wednesday and Pay Day Thursday and Ktd-Mbntlr Pay.Day .1817 .0439*** .0707 .0261 Friday and Mid-Month Pay Day .1345 . 0263** Friday and Month-End Pay Day .0936 .0260*** Saturday and Mid-Month Pay Day .0820 .0254*** Saturday and Month-End Pay Day Thursday .and'Month-End./Pay Day CONTINUED 39 Table 1 (Continued) Coefficient Subscript i R 2 •Estimated Coefficient Standard Error of 3 3 22 -.0207 .0212 23 -.0505 .0194 ** 24 -.0377 .0190 25 .0481 .0165 ** 26 .0630 ,0171 ** 27 .0546 ,0197 28 .1336 ,0260** = .5468 Dummy Variable i s a 0-1 Indicator Function f o r : Thursday and Pay Day Occurs E a r l i e r in Week Friday and Pay Day Occurs E a r l i e r in Week Saturday and Pay Day Occurs E a 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, J u l y , August, September and December only) 759 Observations Durbin Watson S t a t i s t i c = 2.08 Standard Error of the Regression-s .0950 S i g n i f i c a n t at 10% l e v e l ** S i g n i f i c a n t NOTE: at 1% l e v e l Mid-month and month-end pay days occur on the 15th or month-end, r e s p e c t i v e l y , unless that day i s a holiday or on a weekend, i n which case the pay day i s s h i f t e d ahead. Monday pay days are recorded on Tuesday. A Saturday pay day i s a day a f t e r a Friday pay day. 40 Because of Christmas, December has by f a r the largest cash demand, but the other winter months have l i g h t demand. Demand in A p r i l i s high, l i k e l y because the branch i s located near a u n i v e r s i t y where cash demand would increase as students and f a c u l t y prepare f o r vacations. As to be expected, there i s a sharp increase in the demand f o r 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 a n t i c i p a t e d that on Fridays and Saturdays f o l l o w i n g a pay day there would be an increased demand f o r cash since people would wait u n t i l these convenient days.to cash t h e i r cheques. Hoever, the r e s u l t s i n d i c a t e that people generally cash t h e i r cheques r i g h t on a pay day or perhaps even cash post-dated cheques before pay day, since there i s a s i g n i f i c a n t increase in demand on the days preceding a pay day. As a r e s u l t , there i s a c t u a l l y a s i g n i f i c a n t decrease in demand on Fridays and Saturdays f o l l o w i n g a pay day. As to be expected, holidays p r e c i p i t a t e d a large increase in . the demand f o r cash on the day p r i o r to the holiday. This e f f e c t was not observed f o r Easter, Thanksgiving or Remembrance day holidays, however, so these holidays were excluded from consideration by only considering holidays in May, June, J u l y , August, September and December; The Durbin-Watson s t a t i s t i c shows no s i g n i f i c a n t a u t o c o r r e l a t i o n of the e r r o r s . Furthermore, a Box-Jenkins i d e n t i f i c a t i o n of the r e s i d u a l s was performed and, with only one exception, a l l the autocorrelations and p a r t i a l autocorrelations out to 60 day lags were w i t h i n a 95% confidence i n t e r v a l about 0. In p a r t i c u l a r , f o r the 5 day (1 week) l a g , t h e . autocorrelation was .01 and the p a r t i a l a u t o c o r r e l a t i o n 41 was 0. At the 5 day (2 week) lag the autocorrelation and p a r t i a l auto- c o r r e l a t i o n were both -.'02. This indicates' that there i s no observable e f f e c t on cash demand as a r e s u 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 g n i f i c a n t a u t o c o r r e l a t i o n structure i n d i c a t e s that the f i r s t two types of data discrepancies e i t h e r are i n s i g n i f i c a n t or occur at random lags. There was no evidence of h e t e r o s c e d a s t i c i t y , except f o r 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 t e s t was performed on the n u l l hypothesis that the estimated c o e f f i c i e n t s were unchanged from one period to the next. The n u l l hypothesis was barely rejected at the 5% l e v e l , but not at the 1% l e v e 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 f o r each c o e f f i c i e n t . Polynomial time trend- ing i s l i k e l y the best"way of modelling a trend in the c o e f f i c i e n t s , but as discussed e a r l i e r , polynomial projections can be very u n r e l i a b l e . Plotting the residuals of the regression against a cumulative normal d i s t r i b u t i o n revealed-a f a t - t a i l e d d i s t r i b u t i o n , somewhat skewed to the r i g h t . A f t e r pooling the residuals into 20 classes in the +3.5 standard error range and performing a chi-square goodness of f i t t e s t , the normality hypothesis Kolmogorov-Smirnov had to be rejected at the 1% l e v e l . Similarly a goodness of f i t t e s t implied r e j e c t i o n of the normality 42 hypothesis at the 1% l e v e l . This i s l a r g e l y a r e s u l t of the data errors that arose from the bookkeeping discrepancies. The same chi-square and Kolmogorov-Smirnov t e s t s implied r e j e c - t i o n (at the 1% l e v e l ) of the hypotheses that the r e s i d u a l s were d i s t r i buted as Poisson or binomial. In order to t e s t 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 c o e f f i c i e n t s of the model, using exactly the same methods as discussed f o r the f u l l set of data. This model was used to forecast nominal cash flows in the t h i r d year. It was assumed that CPI data i s a v a i l a b l e with a 50 working day l a g , so in forecasting the cash demand in the t h i r d year, instead of using the actual CPI, the CPI was estimated by a l i n e a r p r o j e c t i o n of the CPI trend over the 250 working days p r i o r to the l a t e s t a v a i l a b l e CPI figures. Thus, i f p^ i s the published CPI l e v e l f o r the month which includes day t , the CPI estimate i s : n p - n fi• + f t - 5 0 " t-300) tt " t i - 5ou 0 5 I — b Pt-300 ~~ >•J V. 1 p 1 P _ p - This CPI estimate i s used in the usual manner to forecast cash demand in the f i n a l year. Theil [1966, pp. 26-36] has suggested several techniques f o r analyzing the p r e d i c t i o n errors of a forecast method. Let p r e d i c t i o n f o r day t ( t = l , ' * * , n ) r e a l i z e d value f o r day t (t=l,• • • ,~n) . 43 T h e n t h e mean square error of prediction MSEP = - - E(y\ 9 = £ i y , Letting y j : = 1 z y t is y t and i z(y and r - - — - t y)(y - following decomposition MSEP = (y - u u 2 m + (s = $ p A s proportion and for a poor u and is is of MSEP t h a t forecast the of variation something the due it over which S E s 2 2(1 + - r) (3.4.5) ss , A p S A ^ ^ - r ' ) s proportions, p S a n d A of unequal the and u from b i a s should variance to 2 A MSEP results a good f o r e c a s t , s ) MSEP c inequality - p ( p,- U termed s ' M U These are t holds: y) Let >-et y ) — s the - t in the be s m a l l , the u series, co-variation forecaster + u m has + u s c general is and of the = 1. forecast m is also forecasts control the level, p r o p o r t i o n due should little u to be m i n i m i z e d , and realizations, (except, 44 perhaps, to use more explanatory v a r i a b l e s in his f o r e c a s t s ) . For the forecasts of cash flows in the l a s t year, MSEP = 1.729 x 10 u m = .0045 u S = .0628 u c = .9327 8 These r e s u l t s i n d i c a t e that v i r t u a l l y no e r r o r arose because of bias in the general trend l e v e l estimates, and l i t t l e e r r o r r e s u l t s from i n c o r r e c t l y estimating the l e v e l of v a r i a t i o n . On the other hand, 93% of the error i s due to d i f f e r e n t c o v a r i a t i o n ( e s s e n t i a l l y , noise). These r e s u l t s suggest that the model performance could not have been improved without the use of more explanatory v a r i a b l e s . Another breakdown of the errors i s : MSEP = (y - y ) u + (s 2 + u +.u - rs ) 2 f t (s„ - r s ) - " P ' MSEP" R r b f + (1 - r ) s 2 f t 2 f t t MSEP U Then u p = I. . If one regresses y = t a ± by + e t t (t=l,"«,n) 45 then perfect forecasts correspond to a = e^ = 0 and b = 1. Clearly, e t f 0 in general, so t h i s i s an i r r e d u c i b l e component of forecast e r r o r . The proportion of MSEP due to t h i s component i s u ° . On the other hand, i f a d i f f e r s g r e a t l y from zero, there i s substantial bias in the forecasts m and u i s large. If the regression slope b d i f f e r s g r e a t l y from 1, u R m R i s large. For a good forecast u and u are small. We have already seen m R that u i s s m a l l , and u = .0367 which i s also small. The mean absolute e r r o r of p r e d i c t i o n was $9084 while the mean error of p r e d i c t i o n i s only $883. This means that on average the model over-estimated the general trend l e v e l in the l a 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 e a r l y from the f i r s t two years to the l a s t year. For t h i s model, the MSEP i s 3.000 x 10 . 8 As an analogue to T h e i l ' s second U - s t a t i s t i c (Theil [1966, p. 28]), we may consider the r a t i o of the MSEP of the f u l l model to the MSEP of the benchmark. smaller the r a t i o , the better the f o r e c a s t . ) (The This r a t i o i s .58, i n d i c a t i n g that the f u l l model does indeed outperform naive e x t r a p o l a t i o n (which, of course, has a r a 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 R 2 in the regressions), which a f f e c t s both the model and the benchmark, prevents the r a t i o from approaching zero. In summary the model performs quite well in t e s t s of p r e d i c t i v e a b i l i t y and in p a r t i c u l a r , the estimation of the trend curve appears to be good. 46 3.5 Summary In t h i s chapter, we have studied the r e s u l t s of the estimation of the two main models of cash demand. The Box-Jenkins time series tech- nique was found to be i n a p p l i c a b l e due to n o n - s t a t i o n a r i t y of the cash flow data due to a growth trend and the presence of s i g n i f i c a n t autocorrel a t i o n information at very long lags. A s a t i s f a c t o r y m u l t i p l i c a t i v e model for cash demand was obtained i n two parts. F i r s t , a trend f o r real cash demand that incorporated a capacity constraint was estimated. Then d a i l y cash flows deflated by t h i s trend were regressed on dummy variables i n d i c a t i n g days of the week, months of the year, paydays and holidays. The residuals of t h i s were too f a t - t a i l e d to be normally d i s t r i b u t e d . regression This resulted in part from bookkeeping discrepancies which created pairs of reversed residuals of large magnitude. The second ( m u l t i p l i c a t i v e ) 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 p r e d i c t i o n errors that r e sulted in the t h i r d year. Most of the mean squared p r e d i c t i o n e r r o r was a r e s u l t of random noise, and very l i t t l e was due to bias or i n c o r r e c t l y estimated s c a l i n g of the v a r i a t i o n in cash flows. In short, the model i s good f o r p r e d i c t i o n purposes. The two-part m u l t i p l i c a t i v e model w i l l be used in the cash order algorithm to predict cash demand and provide a 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 the p r e d i c t i o n e r r o r s , which w i l l be used in i n the determination of precautionary cash buffer s i z e s . 47 FOOTNOTES TO CHAPTER 3 The treasury i s 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 i n d i v i d u a l t e l l e r s as needed. Inverse autocorrelations were developed by Cleveland [1972] and are dual to the autocorrelations i n the sense that reversing the roles of the AR and MA parameters also reverses the roles of the autoc o r r e l a t i o n s and inverse a u t o c o r r e l a t i o n s . Chapter 4 THE CASH MANAGEMENT PROBLEM IN GENERAL 4.1 Introduction In t h i s chapter we s h a l l discuss the general considerations i n - volving the management of t i l l Section 4.2 discusses cash. some of the aspects that make t i l l cash management at the branch involved in t h i s study d i f f e r e n t from other cash management problems and inventory problems in general. Important points raised there include the n o n - s t a t i o n a r i t y of cash demand and the v a r i a b l e lag times f o r d e l i v e r y of cash orders. Section 4.3 presents some cash management models* that-have been b u 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 - O r r and Eppen-Fama. Models that give exact solutions for t h e i r optimal parameters are generally steady s t a t e . Section 4.4 casts the cash management problem as a general dynamic programming model which does not require any steady s t a t e assumption. Optimal solutions are hard to c a l c u l a t e from t h i s model, but the model i s examined in Section 4.5 to f i n d which approximations can be made to s i m p l i f y computation with a minimal loss of o p t i m a l i t y . 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 t h i s s e c t i o n , the cash management problem i s defined and the general context of the problem i s presented. The o b j e c t i v e of cash management i s to s e l e c t a p o l i c y 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 earning cash. This problem may be c a s t . i n the form of a recourse problem or a chance constrained problem. In the recourse problem, the o b j e c t i v e func- t i o n also includes a penalty f o r cash-outs (or cash shortages). In the chance constrained problem, the o b j e c t i v e function only includes order costs and opportunity c o s t s , but the o b j e c t i v e i s minimized subject to the c o n s t r a i n t of holding the p r o b a b i l i t y of a cash-out below some predetermined l e v e l d . A more s p e c i f i c discussion of the recourse and chance constrained formulations w i l l ' b e ' p r e s e n t e d in Section 4.4. The most important features of the problem are as f o l l o w s : 1. The demand f o r cash i s a non-stationary s t o c h a s t i c process f o r which forecasts and the d i s t r i b u t i o n of residual error are a v a i l a b l e (by the analysis of Chapter 3). 2. The d i s t r i b u t i o n of forecast errors i s skewed to the r i g h t and too f a t - t a i l e d to be normal, nor i s the d i s t r i b u t i o n Poisson or binomial. 3. Cash flows are generally net outflows (only 30 net inflows i n 767 d a i l y observations).so that v i r t u a l l y no loss of o p t i m a l i t y i s involved by considering only non-negative transfers or orders. 50 4. The lag time between placement and r e c e i p t of orders i s two to four working days (depending on the day of the week when the order i s placed), and the length of t h i s lag i s of the same order of magnitude as the expected optimal time between re-order points. 5. Cash can be ordered d a i l y , but the i n t e r a c t i o n of v a r i a b l e d e l i v e r y 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 o n a l to order i f the order w i l l a r r i v e no e a r l i e r than w i l l an order placed a day or two l a t e r when more information on cash balances i s available. 6. There are two assets: term, l i q u i d i n t e r e s t earning assets. non-interest earning cash and shortThe extension to more assets of varying terms to maturity i s beyond the scope of t h i s study, since i t would e n t a i l a comprehensive asset and l i a b i l i t y management problem. The c r e d i t union o f f e r s i n t e r e s t on the minimum d a i l y balance f o r some accounts, so i t i s reasonable to suppose that the i n t e r e s t earning asset i s j u s t a deposit in such an account, thereby avoiding the need to consider more than two assets. The opportunity cost of cash i s the d a i l y y i e l d i on the i n t e r e s t earning asset. 7. Cash orders are only a v a i l a b l e f o r use the day a f t e r they are received, since cash parcels are counted a f t e r hours. Opportunity costs of cash accrue as of the actual d e l i v e r y 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 d e l i v e r i e s , are not operational then. in the demand process every weekend. That i s , there i s a d i s c o n t i n u i t y 51 9. The cost of placing a cash order Q is the piecewise l i n e a r function a + bQ if Q > 0 0 if Q = 0 F(Q) 10. A cash-out occurs when the cash balance f a l l s below a pre- s p e c i f i e d balance x > 0. It may be that x > 0, r e f l e c t i n g the f a c t that the branch i s e f f e c t i v e 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 c o i n . 11. Emergency cash can be obtained on short notice by s o l i c i t i n g deposits from c e r t a i n l o c a l merchants, however there are c e r t a i n i n t a n g i b l e costs associated with the r i s k of transporting the cash without an armoured car. The cash order models presented in t h i s chapter and the next should be evaluated on the basis of how well they w i l l perform in t h i s environment. 4.3 Steady State Models In t h i s s e c t i o n , 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 p o l i c y can be cast in terms of parameters that are constant over time. The steady state assumptions are generally v i o l a t e d by the f o l l o w i n g features of the problem: n o n - s t a t i o n a r i t y of the demand f o r cash, v a r i a b l e d e l i v e r y lags f o r cash orders, the i r r a t i o n a l i t y 52 of ordering cash on c e r t a i n days of the week and the d i s c o n t i n u i t y of the cash demand process on weekends. Asserting that the steady s t a t e assumptions are generally v i o l a t e d by various features of the problem requires two q u a l i f i c a t i o n 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 K a r l i n f o r non-stationary demand, i s discussed, where s t a t i o n 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 s t a t e s . (The demand states are defined so that cash demand i s s t a t i o n a r y w i t h i n 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 i o l a t e d , and then the most appropriate way to measure the extent to which the assumptions are v i o l a t e d i s by the amount of increased cost caused by the r e s u l t i n g sub-optimality. We s h a l l evaluate the steady state models i n t h i s framework. One of the best-known cash management models i s by Baumol [1952]. He assumes a non-stochastic, constant demand f o r cash over time, and his model can r e a d i l y be modified to allow f o r a d e l i v e r y lag period f o r orders, since the cash requirement during the d e l i v e r y period i s assumed to be known with c e r t a i n t y . Let y be the d a i l y demand f o r cash, b the brokerage fee or f i x e d order cost, i the d a i l y opportunity i n t e r e s t cost of i d l e cash balances, and Q* the optimal order s i z e . can v e r i f y that Q* = /(2b y / i ) . By elementary c a l c u l u s , one This i l l u s t r a t e s the c l a s s i c a l t r a d e - o f f 53 between o r d e r c o s t s and h o l d i n g ( i n t e r e s t ) c o s t s i n i n v e n t o r y t h e o r y : if ( f i x e d ) o r d e r c o s t s a r e high r e l a t i v e t o t h e i n t e r e s t c o s t , l a r g e o r d e r s s h o u l d 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 o r d e r c o s t s a r e r e l a t i v e l y low, small o r d e r s s h o u l d be made, but more o f t e n . control l i m i t p o l i c y o f simple demand f o r c a s h . d u r i n g point. This i s a ( s , S ) form, where t h e t r i p l e v e l s i s the the d e l i v e r y l a g p e r i o d and S = Q* i s t h e r e t u r n That i s , when t h e cash balance f a l l s below s, o r d e r an amount S. T h i s g i v e s a pure t r a n s a c t i o n s demand f o r cash. I f t h e r e i s a need f o r p r e c a u t i o n a r y b a l a n c e s , one s i m p l y has t o add t h e s i z e o f t h e p r e c a u t i o n a r y b u f f e r t o s and S. Archer [1966] d i s c u s s e s some o f t h e important c o n s i d e r a t i o n s i n t h e d e t e r m i n a t i o n o f p r e c a u t i o n a r y cash b a l a n c e s . In e f f e c t , he p o i n t s out t h a t i f y^ i s t h e s t o c h a s t i c demand f o r cash d u r i n g t h e d e l i v e r y l a g p e r i o d and management d e s i r e s t o r i s k a cash-out o n l y w i t h p r o b a b i l i t y a, then s s h o u l d be t h e 1 - a p o i n t o f t h e c u m u l a t i v e 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 t h e r e - o r d e r p o i n t s i n c l u d e s p r o v i s i o n f o r both p r e c a u t i o n a r y and t r a n s a c t i o n s demand f o r cash d u r i n g the d e l i v e r y l a g p e r i o d . F o r example, i f t h e d a i l y demands f o r cash a r e independent and i d e n t i c a l l y N.(y, a ) and t h e o r d e r l a g p e r i o d i s n days, and N, 2 distributed i s the o n e - t a i l e d 1 - a p o i n t o f t h e s t a n d a r d 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, h a ) and s = ny + N-j_ Vn o so t h a t ny i s t h e t r a n s a c t i o n s demand and 2 a N-|_ /n a i s t h e p r e c a u t i o n a r y demand f o r c a s h . a S i n c e t h e Baumol model was developed i n the c o n t e x t o f a c o n s t a n t demand f o r c a s h , i t i s i n c o n s i s t e n t t o add p r e c a u t i o n a r y 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 t h a t the s t o c h a s t i c demand a l s o makes t h e 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. ing Their model i s based on the f o l l o w - assumptions: 1. F o r 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 d a y , 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. T h e s e B e r n o u l l i trials are t h e o n l y exogenous changes t h a t can o c c u r in t h e cash 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 instrument y i e l d i n g d a i l y i n t e r e s t a t the rate i . 3. T r a n s f e r s b e t w e e n t h e a s s e t a c c o u n t s may...be made instantaneously in either direction, f o r a fixed c o s t b. 4. The c a s h b a l a n c e must n o t 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 t h e 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 t h i s case the optimal values of z and h are given by 3 bm t 1/3 4 i 2 z = h = 3z They also show that m t i s the d a i l y variance of cash flows (c ), 2 2 so t h a t , in t h i s n o - d r i f t case, the optimal control parameters can be evaluated in terms of the known q u a n t i t i e s b, i and a , 2 are needed about the parameters m and t . and no s p e c i f i c assumptions 55 Since there i s a pronounced downward d r i f t i n cash balances i n our case, we have 0 < p < ^. In t h i s case, M i l l e r and Orr present com- p l i c a t e d formulae f o r 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 . Implementa- 2 t i o n 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 b u t i o n of cash flows. For small values of p, such as in our case, t h i s can be approximated by a Poisson d i s t r i b u t i o n , so t h a t , in p r i n c i p l e , such a model could be made operat i o n a l , by f i t t i n g a Poisson d i s t r i b u t i o n 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 - O r r model in our problem r e s u l t s from the f a c t that the demand f o r t i l l s t a t i o n a r y , so that the parameters m and p vary over time. cash i s non- They point out that i f the seasonality of demand i s 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 t y p i c a l period between order and reorder, i t i s not c l e a r that the modified model w i l l approach optimal behaviour. Indeed, Orr [1970] makes t h i s s p e c i f i c caveat: . . . i f adjustment t r a n s f e r s are made a t t h r e e day i n t e r v a l s on average, w h i l e t h e p e r i o d of pronounced d r a f t i s a month o r more, t h e r e i s no l a r g e problem . . . i f . . . cash f l o w s a r e c h a r a c t e r i z e d by extreme d r i f t o v e r s h o r t p e r i o d s , then t h e steady s t a t e mode of a n a l y s i s may be i n a p p r o p r i a t e : t r a n s i e n t e f f e c t s may dominate t h e l o n g e r - t e r m movements t h a t a r e w e l l - h a n d l e d in t h e s t e a d y - s t a t e approach.! In Section 4 . 4 . i t w i l l become apparent that there are r e a l l y two problems that a r i s e from the short-term n o n - s t a t i o n a r i t y of cash flows. F i r s t , there i s the problem that Orr mentions of sub-optimality a r i s i n g 56 from the f a c t that a myopic order p o l i c y improperly accounts f o r the costs that occur a f t e r m and p are changed and leaves t r a n s i e n t e f f e c t s when m and p are changed. The second problem arises because s e l e c t i o n of the parameters m and p requires some d e s c r i p t i o n of the demand process that w i l l be in e f f e c t u n t i l the next order a r i s e s . But the time of the second order depends on the s i z e of the f i r s t order, which in turn depends on the values of m and p used f o r the f i r s t order. This i s a c i r c u l a r problem, f o r which there i s no obvious s o l u t i o n . Another major v i o l a t i o n of the M i l l e r - O r r axioms i s the f a c t t h a t , not only i s there a major order d e l i v e r y lag time (almost as long as the l i f e of a t y p i c a l order), but the length of t h 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 - O r r algorithm to take account of t h i s or the f a c t that i t i s i r r a t i o n a l to order cash on some days of the week, e i t h e r . These problems make i t highly u n l i k e l y that an operational version of the M i l l e r - O r r model can be applied to the t i l l cash management problem. There are two other minor v i o l a t i o n s of the M i l l e r - O r 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 a r i s e s from the d i s c o n t i n u i t i e s of the problem on weekends: cash demand stops, while opportunity costs continue. The second a r i s e s from the question of whether the cash demand process i s r e a l l y B e r n o u l l i a n . For s u f f i c i e n t l y small d i v i s i o n s of the day, the B e r n o u l l i process approximates various continuous processes f o r various values of m and p, however, so t h i s i s not l i k e l y to be a serious problem. In a l a t e r paper, M i l l e r and Orr [1968] provide a rather substantial defence f o r the approximation of cash demand by a B e r n o u 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 i z e of cash balances, t r a n s f e r costs are l i n e a r functions of the amount t r a n s f e r r e d , cash d e l i v e r i e s are instantaneous and that the sequence of cash balances forms a d i s c r e t e Markovian s t o c h a s t i c process. They use a simple f o r (u, U; D,d) inventory p o l i c y 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 s e s above d, and no action i s taken i f the balance i s between u and d. They set up a l i n e a 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 f o r 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 i b u t e d . ) He assumes proportional penalty and holding costs, and f i x e d t r a n s f e r costs. He solves f o r the optimal parameters of a (u, U; D,d) inventory p o l i c y by minimizing Tim N " 1 (EC.,) where C^ i s the t o t a l stochastic inventory cost incurred out to a horizon N. A l l of these papers y i e l d s p e c i f i c solutions f o r inventory control parameters, but only by r e l y i n g on the s t a t i o n a r i t y of the underl y i n g cash demand process. The n o n - s t a t i o n a r i t y of the demand process prevents these models from being applied in p r e c i s e l y the same way i t prevents the M i l l e r - O r r model from being applied in our problem. Similarly the varying d e l i v e r y lag times and d i s c o n t i n u i t i e s undermine the steady state assumptions of a l l models discussed in t h i s chapter. 58 The p r i c e paid to obtain a simple steady state s o l u t i o n appears to be too high in terms of the r e s u l t i n g sub-optimality of decisions. In the next s e c t i o n , 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 i b u t e s : the solutions are optimal, but too complicated to compute. 4.4 Dynamic Programming Formulation Cash management problems are r e a l l y j u s t special inventory theory problems. The two main features that d i s t i n g u i s h 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 g e n e r a l i t y wi11 also allow us to r e l a x the assumption in the previous section that the problem has a steady state nature. The general s o l u t i o n to inventory theory problems i s often cast in a dynamic programming framework. A general discussion of dynamic programming i s given, f o r example, by Ziemba [1975] and s p e c i f i c formulat i o n s 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 t h e r a recourse model (with a penalty function f o r cash-outs) or as a chance constrained model (constraining the r i s k of a cash-out), and the f o l l o w i n g discussion derives both. To be consistent with the s t a t i s t i c a l portion of the paper we s h a l l enumerate working days, not calendar days. Thus i f t i s a Saturday, 59 t+1 i s a Tuesday. A l s o , to remain c o n s i s t e n t i w i t h the r e s t of the paper, we s h a l l enumerate time i n 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^.(x ) the opportunity cost of the opening cash balance x^ > 0 on day t t , . P( f-) x D e the- penalty cost f o r a cash-out of s i z e x^ < 0, and y^ be the s t o c h a s t i c cash flow on day t with cumulative d i s t r i b u t i o n function <p^(y^). Define the cost of holding the opening cash balance x^.- on day t as H (x ) t if x t >0 ,p(x ) if x t < 0 t C (x ) t t t 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 t e r a weekend, i n order to f u l l y r e f l e c t the opportunity cost incurred over the weekend. Let g (Q > x^) be the expected discounted cost as of time t t t of current and future order p o l i c y i f the current (opening) cash balance i s x^ and an order of s i z e Q i s placed at time t (temporarily assuming t a zero d e l i v e r y lag) and an optimal p o l i c y i s followed f o r a l l subsequent orders. Then we have a dynamic program i n the f o l l o w i n g recursive form: g ( Q , x ) = F(Q ) + Y t t t mm t -oo 'Qt+1>0 W x t - y + Q) t d * ( y ) t (4.4.1) 60 That i s , the expected c o s t a t time, t i s d i s c o u n t e d expected optimal the sum o f the o r d e r c o s t , c o s t at time t+1, h o l d i n g c o s t of the opening balance at time The optimal Q* t > 0 such the and the d i s c o u n t e d expected t+1. i n v e n t o r y p o l i c y a t time t i s to s e l e c t an o r d e r that 9 (Q , t t * ) t = min g ( Q » f t t x ) . t ^t However, the r e c u r s i v e r e l a t i o n ( 4 . 4 . 1 ) does not s p e c i f y any values g value f o r some time T. u n l e s s we g i v e t i t a starting minimize expected c o s t s for a r b i t r a r i l y c o n v e r g e n c e ) , or we may minimize t o t a l h o r i z o n T. _ 1 We may e i t h e r horizons expected c o s t s (with y < 1 to 9i(Qi*» g (Q , x T x ) = 0. T can be computed r e c u r s i v e l y by means o f approximately, the long run average c o s t of an optimal ensure out to some f i x e d In the l a t t e r case one may s e t y = 1 (no d i s c o u n t i n g ) i n i t i a l i z e a t the h o r i z o n T by s e t t i n g T large of and Then ( 4 . 4 . 1 ) and cash is, management policy. In o r d e r to g e n e r a l i z e the problem to a l l o w f o r v a r y i n g l a g t i m e s , we may extend a f o r m u l a t i o n p r o v i d e d by Bellman et al. p. 87j by e n l a r g i n g orders. space to i n c l u d e h i s t o r i c a l D e f i n e the o r d e r h i s t r o y Q_ = (Q _-|> Q t the d e l i v e r y lag day t + L ( t ) The t o t a l the s t a t e - L(t) t t 2 delivery [1955, but u n r e c e i v e d ' D e f > 0 so t h a t an o r d e r p l a c e d on day t i s '' n e r e c e i v e d on 1, counted a f t e r hours and p l a c e d i n t o use on day t + L ( t ) . of p a s t . o r d e r s that w i l l m) t =- l f become a v a i l a b l e Q _j « t L ( t . d ) > j f o r use on day t is (4.4.2) 61 0 if itj 1 if i=k 6.. where In our a p p l i c a t i o n , the maximum d e l i v e r y 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 ) only f o r t = j + L ( j ) . We now require a convention regarding Lt the opening cash balance x^. be used on day t . That i s , We s h a l l say x^ includes the orders due to x t = t l ~ t 1 x y + ^^t^" ^ s n 0 t a t l 0 n ^ s convenient because holding costs apply to the c l o s i n g balance of day t-1 as well as the orders that are received and counted that night, f o r use the next day. We s h a l l also adopt the convention that t + L ( t ) = t + 1 + L ( t + 1). = 0 whenever That i s , we s h a l l never consider ordering cash on an " i r r a t i o n a l " day f o r which an order placed next day with more information w i l l a r r i v e j u s t as soon. L e t t i n g h (Q '> 0^., x^) be the present value of the expected t t order p o l i c y costs given the order h i s t o r y 0^ and cash balance x^, when the current order i s Q t and a l l subsequent orders are selected o p t i m a l l y , we have the f o l l o w i n g analogue of (4.4.1). . o o h ( Q ; 3.t» t t x t ) = F(Q ). t + Y JC t + 1 (x t - y + R(Q t + 1 )) d$ (y) t - o o + ^min Y -» •h t + 1 (Q t + 1 ;i t + 1 ,x t - y + R(Q_ ))} d«> (y) t+1 t >0 (4.4.3) 62 Note that Q t on the r i g h t hand side of (4.4.3) i s w e l l - d e f i n e d given and 0^ on the l e f t s i d e . As before, set h T (Q-j;; Q-p x ) = 0 f o r some T large T, in order to r e c u r s i v e l y compute and minimize (4.4.3) f o r 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 t o l e r a t e cash-outs with a p r o b a b i l i t y a (0 < a < 1). To determine the day f o r which the cash-out constraint i s to be a p p l i c a b l e note that the cash order i s not a v a i l a b l e before t + L ( t ) , so i t i s senseless to set a c o n s t r a i n t regarding the cashout r i s k on any e a r l i e r day. A l s o , f o r an order on day t , one cannot be content with constraining the r i s k 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 ) , ; w h i c h i s the e a r l i e s t time an order placed on day t + 1 can be used. This can occur when the d e l i v e r y lag times vary so that L ( t ) + 1 = L(t + 1 ) . Hence, a more r e a l i s t i c c o n s t r a i n t is to constrain the r i s k 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 a v a i l a b l e . To do t h i s , we should perform the minimization in (4.4.3) so that f o r t=l,2,3,••• Q t > 0 and Prob x. - y, + 1 1 t*L(t+l) ^ I (-y, + R(Q.)) > x > j= +i J - J - 1 - a. J t If dp (y •) ~ N ( E ( y . , a - ) , and the y . are independent, t h i s has the deteri m i n i s t i c form 2 (4.4. 63 t+L(t+1) E(y ) • t "I E(y ) - R(a,) d + / a t j a t+L(t+l) N, < X. - x t 1-a - (4.4.5) j=t+l where N n 1-a i s the 1-a point of the standard normal d i s t r i b u t i o n . ' Note that the term in square brackets i s the expected cumulative cash flow ( a f t e r adjusting f o r the r e c e i p t 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 f o r cash. The other term on the l e f t i s the buffer of cash required to hold the p r o b a b i l i t y of a cash-out below a, and hence i s the precautionary demand f o r cash. By increasing the s i z e of the current order Q^, the term in square brackets can be decreased to s a t i s f y the i n e q u a l i t y , 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 v e as e a r l y as an order placed on day t . l a s t case, i t i s i r r a t i o n a l to order money and we set Q t In t h i s = 0. This model accommodates a l l the features of the problem mentioned in Section 4.2. In p a r t i c u l a r , i t incorporates n o n - s t a t i o n a r i t y of the cash flows because Q t depends on t . i s incorporated in L ( t ) and R ^ ) . The v a r i a b l e d e l i v e r y time f o r orders The d i s c o n t i n u i t y of the model on weekends (demand stops, opportunity costs continue) i s incorporated in the f a c t that the costs H and depend on t . t The model has, however, one major flaw. It i s f a r too compli- cated to compute a l l of the recursive formulae (4.4.3). F(Q ) t The order cost i s easy to evaluate (in our case i t i s a l i n e a r function f o r Q^. > 0 and i s zero f o r = 0). The second term in (4.4.3) i s the expected t o t a l holding and penalty cost E y t (C, + 1 (x. - y^ + R(Q. -|))) of the stochastic t+ opening cash balance on day t+1, and i s r e a d i l y computable by numerical i n t e g r a t i o n , or by simulation. The t h i r d term i s the discounted expected 64 value of optimal future order p o l i c y given the order h i s t o r y 0^). on day t (and order The integrand has no t r a c t a b l e representation because i t i s i t s e l f a s o l u t i o n to an optimization problem, and the character of t h i s recursive optimization problem changes s i g n i f i c a n t l y with t , due to the n o n - s t a t i o n a r i t y of y^, v a r i a b l e d e l i v e r y lags and d i s c o n t i n u i t i e s on weekends. Since there i s no steady state c h a r a c t e r i z a t i o n of the recursive optimization problem, i t must be re-solved at each i t e r a t i o n . If numerical methods are used, the number of c a l c u l a t i o n s required grows exponentially with the number of i t e r a t i o n s of (4.4.3), so that the problem i s v i r t u a l l y i n s o l u b l e f o r large values of T. To appreciate the l e v e l of complexity introduced j u s t by nons t a t i o n a r i t y , we may consider a model by Iglehart and K a r l i n [1962] f o r a non-stationary inventory process. They assumed k demand states with Markov t r a n s i t i o n p r o b a b i l i t i e s between s t a t e s . density <pj{y). Each state has demand In our a p p l i c a t i o n , we could l e t the states i be the days of the year (k = 365), in which case the p r o b a b i l i t y t r a n s i t i o n matrix would have a 1 corresponding to the t r a n s i t i o n 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 pro3 portional order costs only) they show that the optimal p o l i c y i s chara c t e r i z e d by k c r i t i c a l numbers x. such t h a t , i f the inventory f a l l s below x.j in state i , one should order up to the level x^. The s o l u t i o n requires solving up to k! transcendental equations and (k-1) renewal; equations. Even f o r k = 20 s t a t e s , t h 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 p o l i c y . However, i t can be used as a benchmark for evaluation of operational approximations. w i l l be discussed i n the next s e c t i o n . This 65 4.5 Operational Approximations to Optimal P o 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 evaluat i o n of the l a s t term which i s the present value of the expected cost of future optimal order p o l i c y given that i s ordered on day t . An opera- t i o n s o l u t i o n of the problem requires some sort of approximation f o r t h 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 h e u r i s t i c order p o l i c y and simulate the operation of the model with various parameter assignments, s e l e c t i n g those parameter values which minimize the long run costs. The problem i s to determine what sort of h e u r i s t i c p o l i c y to use, bearing in mind that a more r e a l i s t i c p o l i c y has more parameters, while the number of computer runs required to f i n d an optimal parameterization grows exponentially with the number of parameters. one could simulate the use of a simple (s,S) For example order p o l i c y to f i n d which values of the parameters s and S order p o l i c y to f i n d 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 f o r an order a r r i v i n g before a Friday pay day in December w i l l c e r t a i n l y not be the same as the optimal values of the parameters f o r an order a r r i v i n g on a Tuesday in January with the next pay day two weeks away. would be better to use a family {(s^, S^):i Thus i t e 1} of order p o 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 t h i s dependence i s h e u r i s t i c . Furthermore, f o r a comprehensive range of states I, t e s t i n g the model for N d i f f e r e n t 66 values runs of each of which would The the costs will of the (s., be p r o h i b i t i v e l y second type the dynamic the best may s e t next program policy the the order out the d a i l y order to all next optimal rate of the that fixed order costs order at all.) 2 it simulation of costs the y to the assumption beyond t h e of the this at next best we s h a l l present point p o l i c y ;.• expected cost the that re-order order notion, average cast the per day time t=l. of We 1. probability at distributions x + time ic 1 > 1. on t h e e v e n t That is, ; condition ic = ••• = and 0/^ =0 often occurs Then r e c u r s i v e * that f (In 0. the optimal substitution of the presence order (4.4.3), is to place noting of no that * F(Q ) = ••• = F(Q 2 per Q the order occurs ic on t h e e v e n t policy horizon T starting discount ' computer involves To f o r m a l i z e terms N' expensive. same as in require approximation point. (4.4.3) Condition that of f u t u r e optimal be a p p r o x i m a t e l y before ) would ) = 0, y i e l d s the following formula for long run costs day: MQi; Q X l l5 ) = A ( T , Ch) + E(min h T + ] (4, ) where A(T, C -OO -00 J + l ( X l + Q'I)-= ( " i y F(Q.f)'+ + R ( ^i + 1 ) ) ) D < M Y I ) D < } ) T ( Y T ) 67 E(min h T+1 ) = T + r V r ^T+T Xa + -co T I i=l (-Yi + R ( Q i + 1 ))) d<h(yi) d<J) (y ) T T This gives expected costs out to.the horizon T f o r the order Q as the t sum of expected costs out to x (in A ( T , Q ) ) and the expected costs from x T+1 to T (in E(min h ^ - j ) ) . Note that T i s a function of the cash flows y , so that conditioning on T means that the d i s t r i b u t i o n functions t ^ ( y ) are conditioned on x . t t For s i m p l i c i t y t h i s dependence i s 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 a r i s e by this. flows. A The f i r s t term, A ( x , Q i ) , can be evaluated by simulating the cash The second term, E(min h -j)» i s much more d i f f i c u l t to evaluate. T+ reasonable approximation f o r the second term i s Qi i s the optimal order at time t = 1. ~r- h (Qi; x Q_i, X i ) , where This i s the expected cost f o r the whole period (given x) m u l t i p l i e d by the f a c t o r (T-x)/T to adjust f o r the shorter time to the horizon. That i s , we approximate the expected d a i l y cost of optimal p o l i c y over the period x+1 to T with the expected d a i l y cost of optimal p o l i c y over the whole period 1 to T. Now, suppose * Qi i s close to Q i , 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 hi(Qi; Q i , xi) = A(x, Q ) + x T-x T h i ( Q i ; Q.!, x j or T Mth; Xi) = T A(T,...QI) (4.5.2) 68 Thus, the average cost per day to the horizon T i s approximately x - 1 A(x, (h c o n d i t i o n a l 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 v a r i a b l e x to obtain the f o l l o w i n g approximation to the expected average cost per day objective f u n c t i o n : E (xT 1 A(x, Q J ) (4.5 . A simulation technique f o r evaluating the o b j e c t i v e f u n c t i o n , for a given value of Q l s i s presented i n Section * can be used to s e l e c t Q . 5.6. A g r i d search * Note that as Q approaches Q i n the g r i d search, the q u a l i t y of the approximation (4.5.2) improves because * hi (Qi; O^i, x i ) approaches hj. (Qi; Q_i, x ) . I f the model i s cast i n a chance x c o n s t r a i n t framework, the g r i d search should be performed only over values of Qi that s a 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 assuming that the s t r u c t u r e of demand a f t e r the placing of the next order i s approximately the same as the s t r u c t u r e before the next order i s placed. This i s e s s e n t i a l l y a steady state assumption, so i t i s i n s t r u c t i v e to consider how a steady state model l i k e that of M i l l e r - O r r would f i t into t h 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 i n place of E (X t _ 1 A(x, Q i ) ) . However, by examining ( 4 . 5 . 1 ) and the d e f i n i t i o n of A(x, Qi) we see that i t depends on the d i s t r i b u t i o n 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 i b u t i o n s i n order to evaluate the steady state 69 parameters. But such a summary requires knowledge of x. That i s , i f a re-order i s required before a holiday or some other day of heavy demand, the summary should not forecast such heavy cash requirements as i f the r e order i s to be received a f t e r the holiday. In order to evaluate x f o r the M i l l e r - O r r model, we would have to s e l e c t Qiand then perform some sort of simulation of the cash demand to determine the next order p o i n t . But the value f o r Q 2 depends on the summary parameters derived from x. Thus, some sort of i t e r a t i o n back and f o r t h between values of x and values of * Qi would be required to evaluate E ( X T A ( X , Q )). evaluate t h i s expectation d i r e c t l y by simulation. It seems simpler to In t h 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 - O r r (or any other) steady state s o l u t i o n , 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 s e l e c t i n g M i l l e r - O r r parameters by some sort of i t e r a t i o n process. 4.6 Summary In Section 4.2 we examined several of the d i s t i n g u i s h i n g features of the t i l l cash management problem, and foremost amongst these were the n o n - s t a t i o n a r i t y of the data, the v a r i a b l e d e l i v e r y lag times f o r orders and the d i s c o n t i n u i t y of the cash flow/opportunity cost r e l a t i o n ship on weekends. In Section 4.3 several steady state models f o r 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 s e r i o u s l y v i o l a t e d to y i e l d good solutions in t h i s content. 70 In Section 4.4, a dynamic programming model f o r cash management was presented. Although the dynamic programming s o l u t i o n i s v i r - t u a l l y impossible to compute, i t was pointed out that c e r t a i n 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 s e l e c t orders with the o b j e c t i v e 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 t h i s chapter the stochastic programming model to determine near-optimal t i l l cash orders i s developed. t i o n s are discussed in Section 5.2. Several technical considera- The main f i n d i n g 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 d e t e r m i n i s t i c equivalent of the chance c o n s t r a i n t . A method of evaluating the value of objective function by simulation f o r any order s i z e Q i s discussed in Section 5.4. The o b j e c t i v e function in Section 5.4 i s more complicated than the one presented in Chapter 4, and the dynamic program underlying t h i s v a r i a n t 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 c a l cash flow data in Section 5.6. This allows evaluation of the e f f e c t i v e 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 level. The model performance is also compared to the h i s t o r i c a 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 considered. These points w i l l be discussed i n t h i s s e c t i o n . 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 ) to encourage the holding of adequate cash balances that t w i l l reduce the r i s k of cash-outs. The chance constrained formulation eliminates the penalty function but adds a c o n s t r a i n t to the s e l e c t i o n of the Q ' s t which reduces the r i s k of a cash-out to some level set by management. The chance c o n s t r a i n t has a d e t e r m i n i s t i c 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 i b u t i o n s of the d a i l y cash flows since occur only f o r extreme cash flows. cash-outs In Chapter 3 i t was found that the r e s i d u a l s of the estimated cash flows formed a d i s t r i b u t i o n 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 o u t l i e r s could have simply been a r e s u 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 i b u t i o n s of the d a i l y cash demands. It i s not c l e a r what adjustment, i f any, could be made to the t a i l s of the d i s t r i b u t i o n s in the recourse model so that the estimated expected penalty cost of a given cash order p o l i c y w i l l be a reasonable approximation o f i t s true value. However, in Section 4.4 i t was shown that f o r normal forecast errors the chance c o n s t r a i n t 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. + « • ?+o'Z,. , N, where y . ~ N ( E ( y . ) , a . ) and the t t+L (t+1) 1-a •'j "'j J 2 y . are independent. 2 A m i s - s p e c i f i c a t i o n of the t a i l s of the d i s t r i b u t i o n s of the y . ' s w i l l m i s - s p e c i f y the r i s k l e v e l a, but the concept of a preJ cautionary buffer requirement expressed as a sum of money i s nevertheless something which management can r e l a t e to previous experience. Thus, a manager may be a l i t t l e uneasy about s p e c i f y i n g a r i s k of a cash-out of 1% ( e s p e c i a l l y i f he understands that the d i s t r i b u t i o n of the t a i l s of the forecast errors i s not known a c c u r a t e l y ) , but he i s more l i k e l y to feel at ease i f he s p e c i f i e s a precautionary buffer of, say, $60,000 when the order d e l i v e r y 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 f u n c t i o n . In Section 4.2 i t was suggested that an adequate form would be b + aQ if Q > 0 b, a > 0 t F(Q ) t =4 if Q t Since only non-negative orders are considered and the long run t o t a 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, f o r large T. Thus the c o n t r i b u t i o n to long run costs t=l t=l z z by the v a r i a b l e order cost term i s independent of order p o l i c y and without loss of g e n e r a l i t y we can assume a = 0, so that fb if Q t > 0 (5.2.1) F(Q ) t 0 if Q t = 0 75 We s h a l l also be more e x p l i c i t at t h i s point about the cost function of the opening cash balance C^.(x^.). This i s a piecewise l i n e a r function H (x ) = i x (D(t) - D(t-l)) t t t if x t > 0 if x t < 0 C (x ) t t [P(x ) = " p ' x t t where i i s the d a i l y i n t e r e s t rate of the i n t e r e s t earning asset, (5. D(t) i s the number of calendar days a f t e r day one, and p > 0 i s the penalty cost rate f o r negative balances (p = 0 f o r the chance constrained model). Note f o r 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 f o l l o w i n g a h o l l i d a y . 5.3 The Chance Constraint As discussed in the l a 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 b u t e d , i t was pointed out that we can model the chance c o n s t r a i n t as though the errors were normally d i s t r i b u t e d as long as we remember that the p r o b a b i l i s t i c c o n s t r a i n t only gives a r e l a t i v e , but not absolute, q u a n t i f i c a t i o n of the r i s k of a cashout. Thus, we s h a l l use the chance c o n s t r a i n t (4.4.5) which i s based on normal forecast e r r o r s . In Section 2.3, an estimate of the variance of the d a i l y cash flow estimate y. given in formula (2.3.10) i s 76 i where d t s [l + z^CZ'Z)" 2 2 t i s the forecast trend l e v e l , s regression, 2 1 z J (5.3.1) t i s the standard error of the i s the vector of dummy v a r i a b l e s f o r day t and s ( Z ' Z ) 2 i s the estimated covariance matrix of the regression c o e f f i c i e n t s |3. For the purposes of c a l c u l a t i n g the precautionary buffer s i z e in the chance c o n s t r a i n t , i t i s a l i t t l e d i f f i c u l t to c o n t i n u a l l y compute a quadratic form i n v o l v i n g the 28 x 28 covariance matrix. The dummy com- ponents of the vector z^ are described in Table 1 i n C h a p t e r 2 along with the regression c o e f f i c i e n t s and t h e i r standard e r r o r s . A c a r e f u l examina- t i o n of Table 1 indicates that at most 4 components of z^ are non-zero at any time: one each f o r the day of the week, month of the year, pay- day and holiday. A l s o , the largest standard e r r o r of the regression c o e f f i c i e n t s i s .044, and the standard errors are t y p i c a l l y .015 to .020. Consider the f o l l o w i n g type of day where the c o e f f i c i e n t s of the non-zero dummy v a r i a b l e s have the l a r g e s t possible standard e r r o r s : June with a month-end pay day, j u s t before a holiday. z 17t = z 28t = 1 a n c ' a 1 1 t ' i e o t n e r a Thursday in Then z ^ dummy v a r i a b l e s are zero. t = z^ = Then from the estimated covariance submatrix corresponding to the non-zero v a r i a b l e s , we have s 2 z.MZ'Z) - 1 z + = lO" 4 ,0026 (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 77 Now s = (.095) 2 2 = .0090. From (5.3.1) we see that the forecast variance is .0026 d ~ 2 + .0090 d j 2 = .0116 d 2 t . This i s an example where the f i r s t term i s larger than usual (using 4 c o e f f i c i e n t s with large standard e r r o r s ) , yet i t i s r e l a t i v e l y small compared to the t o t a l forecast variance. Since we are not using the forecast variances to compute absolute l e v e l s of cash-out r i s k , but only to compute approximate precautionary b u f f e r s , 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 f o r e casts y ^ ' s s 2 d 2 t . have a r e l a t i v e l y constant variance given approximately by Thus, the d e t e r m i n i s t i c equivalent of the chance constraint (4.4.5) involving the r i s k of a cash-out y where i + t+L(t+l) j I = t + l ~ (y J - becomes A J )) + /Llt+lT s d L N i-a < x i - x (5.3. i s the opening balance on day t , x i s the t r i p balance for a cash-out, y. i s the forecast of the cash flow f o r day j , R(0.) i s the t o t a l of orders to be received on day j , L(t) i s the number of days in the d e l i v e r y lag f o r an order placed on day t and Ny_ standard normal d i s t r i b u t i o n . a i s the l - a point of the This c o n s t r a i n t s p e c i f i e s that the prob- a b i l i t y of running out of cash j u s t before an order placed next day i s received s h a l l be less than a. (t+1) As before, we s h a l l only require the chance constraint to hold on days when i t i s r a t i o n a l to order cash (that i s , when t + L(t) < t + 1 + (L(t+1)). 78 F i n a l l y we observe t h a t , without loss of g e n e r a l i t y , we may set x = 0. If management f e e l s that i t has e s s e n t i a 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 p o l i c y . 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) i s not the only dynamic programming"formulation of the inventory problem - i t was presented there only because i t i s the simplest to understand. Hence the approximate o b j e c t i v e function (4.5.3), which i s E^T" 1 A ( T , Q )), (5.4.1) i s not the only o b j e c t i v e function that can be used in the problem. From the dynamic programming formulation (4.5.1), we see that x - 1 (A(x,Qi)) i s the average of the order cost F(Ch) and the holding costs Cj- -|( t -|) x + + of opening balances over the period s t a r t i n g on the working day when the current order i s placed (t=l) to the working day before the next order i s placed (t = x ) . However, the s e l e c t i o n of the order s i z e Qi a f f e c t s holding costs over a d i f f e r e n t period in time. If cash i s ordered on working day t , i t i s ready f o r use on day t + L ( t ) but i t a c t u a l l y a r r i v e s at 79 the branch on day t + |_(t) - 1 i n order t o be counted a f t e r hours f o r use on the next day. Thus the s e l e c t i o n of order s i z e Q a f f e c t s the : holding costs of the opening cash balances on days 1 + L ( l ) to ]) ( i+[_(1) ^ x + L(x+1), namely x •" ' '^T+L(-T+1 )^ T+L(T+1 X ^' F u r t n e r m o r e > we indexed the holding cost function by t t o account f o r the increased cost of holding cash balances over weekends, while T i s a count o f working days only, so i t i s more appropriate t o average the costs over the number of calendar days rather than working days. ing day t be D(t) calendar days a f t e r day one. Accordingly, l e t work- 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 v a r i a b l e ) , a more reasonable o b j e c t i v e function i s the expected average d a i l y cost (on a calendar day basis) of ordering and holding costs which are a f f e c t e d by the s i z e of the order Q i namely E ;T A 1 ( T , Ch D(T+L(T+1)) (5.4 where A'(T,QI) f o o T+L(T+1 ) . - U J I _ o o c n ' ) ' t+i = F(Q i ) + t+1 ( X i + l 2 +R ( V ^ - W+D } ) d (5 - 4 (For s i m p 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 l s o , that i t i s c o n d i t i o n a l on T . ) A dynamic program corresponding to t h i s o b j e c t i v e function i s given i n the Appendix at the 80 end of t h 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 t e r order w i l l a r r i v e on the same day. Suppose a simulation model generates a new sequence of cash flows {y^} given { y h t in each i t e r a t i o n . Then we can condition on {y^K there i s a well defined T noting that such that cash will--be ordered,-at T . + 1 Then we can use (5.4.3) to evaluate the c o n d i t i o n a l expectation f , A," ( T , Q I ) ) T|{y }[D(T+L(T+l))J _ A' (T Q) D(T*+L(T*+1.)) T T*+L(T*+1) F(Qi) + L ^ 2 t+1 C ( + X l t = l + L ( l ) _ I (-y^+RtQ,.))) (5.4.4) i=2 D(x* + L(x*+1)) Now, taking the expectation with respect to fA' (T, Qj) D(T+L(T+1)) = £ iy^h fA' (x, Q J iy } E t x|{y } t (5.4.5) D(T+L(T+1)) Thus (5.4.5) can be evaluated by c a l c a u l t i n g (5.4.4) f o r each i t e r a t i o n of a simulation of {y^} and taking the average of the r e s u l t i n g values which simply amounts to taking the expectation with respect to This can be done f o r any value of Q can be minimized by a g r i d search. l s iy^}- so that the o b j e c t i v e functions This i s the essence of the cash order algorithm to be presented in the next s e c t i o n . 81 5.5 The Cash Order Algorithm In t h i s section the formulae presented in the previous sections are used to develop an algorithm to order cash by s e l e c t i n g an order s i z e to minimize the expected average costs per day (as computed by a simulation model) subject to a constraint involving the r i s k of i n c u r r i n g 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, f o r which an order i s being considered. Since there are f i x e d 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 c o n s t r a i n t (5.3.2) i s v i o l a t e d with Q t = 0. Thus the minimum acceptable order on day t i s t+L(t+1) - A QMIN(t) = /LTt+TT s d. N, + I j=t L t+L(t)-1 y- + x - x. J £ j=t+l L R(Q ) (5.5.1) J where s i s the standard e r r o r of the regression in Table 1, d^. is the estimated weekly trend level in cash flows, N-j_ i s the a point of the a standard normal d i s t r i b u t i o n , y. i s the forecast cash flow on day j and R(Q.) i s the t o t a l of previously made orders that are due to a r r i v e on day j . Note that R ( ^ t + L ( )) t Thus we can omit the dates = Q a t n d R (Q.j) = 0 f o r t+L(t) < j < t+L (t+1). t+L.(t) + 1, • • •, t+L (t+1) in the l a s t summation of (5.5.1) and the date t+L(t) corresponds to QMIN(t). If Q min U ) > 0 an order i s placed, unless 1 + L(t+1) = L ( t ) , in which case an order placed on day t+1 would a r r i v e j u s t as soon as. an;' ,j 82 Start Current day i s ' r/ / * Yes K QMIN(1) f 0 V <J Stop N 1 + L(2) = L ( l ) y 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 _i - d t t - i * R E S ( ] + 7 5 8 * F R A N °) Yes QMIN(t) f 0 * .or 1 + L(t+1) = L ( t ) t = t+1 No = t-1 T |EXPC0ST(Q) EXPCOST(Q) + 1 BOUND B 0 U N D + 1 A'(T,Q) NSIM D ( T + L ( T + 1 )) NSTM * D(x"ilL(?l )) 2 • QBEST = Q N = N+l r ^ N > N S I M ) y ? s ><f EXPCOST(QBEST) < EXPCOST(Q) EXPCOST (QBEST )\No < BOUND Yes / Stop V Q* = QBEST Figure 2. The Cash Order Algorithm. \ J t+L(t+1) ~ QMIN(t) = /L(t+1) s d N _ + t + x - x - 1 t+L(t)-l I j=t+l L j ^ Figure 3. Return J a y\ R(Q ) J ^ The Minimum Cash Order Subroutine. 84 order placed on day t , making i t i r r a t i o n a l to use less information and order on day t . At t = l , i f an order i s to be placed, the g r i d search s t a r t s with the minimal order Q = QMIN(l) and examines larger values, in i n c r e ments of QSTEP. This ensures that the chance c o n s t r a i n t i s s a t i s f i e d . For each order s i z e Q, the cash flows are simulated NSIM times. In each i t e r a t i o n the sequence { x i , X 2 , x , » » * } i s calcauTted by s e t t i n g 3 y -l = y = X t X t t-l + t-f d t-1 " R E S I D U A L y t-l R( + 0 + 7 5 8 * FRAND) V where RESIDUAL i s the array of 759 unsorted r e s i d u a l s from the regression in Table 1, Section 3.4 and FRAND i s a random number generator that gives independent uniform v a r i a b l e s on [0,1]. The sequence of cash flows i s continued u n 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 f o l l o w i n g expectation (conditioned on (y^)) by formula (5.4.4): F [A (x Q i ) ' T|{y HD(x+L(x+l)), 1 t This i s then averaged into the o b j e c t i v e function which i s the uncondit i o a l expectation EXPCOST = E x A' (T , Q i ) ) D(x+L(x+1) 85 Having computed the o b j e c t i v e function EXPCOST(Q) a f t e r NSIM i t e r a t i o n s of the simulation, the o b j e c t i v e i s compared to that f o r the l a s t best order (QBEST), and i f b e t t e r , QBEST i s updated and the object i v e i s computed f o r the next larger order, Q + QSTEP. If the order Q i s not better than QBEST, then QBEST i s a l o c a l minimum of the o b j e c t i v e f u n c t i o n . {y^} For a given cash flow sequence 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, a s . w e l l , which causes an o v e r a l l decrease in the o b j e c t i v e function. Thus l o c a l minima need not be global minima. However to obtain a lower bound f o r the o b j e c t i v e function suppose that f o r a given (y^)* increasing the order from Q to Q causes an increase in T * of one working 1 A (T*+1 0') Then d ( t 4 I + L ( T * + 2 ) ) A 1 day- - 1 Ix* 0) 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 i s unchanged by the increase in order s i z e . This lower bound i s computed at each i t e r a t i o n 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 l o b a l l y optimal i f EXPCOST(QBEST) < BOUND. algorithm. This provides a stopping r u l e f o r the Note that f o r very large orders, the t o t a l opportunity costs grow q u a d r a t i c a l l y with order s i z e (since both T and the average balance grow asymptotically with Q) while the denominator increases only l i n e a r l y with Q. Thus, a f i n i t e order i s optimal and the stopping r u l e w i l l the algorithm in a f i n i t e number of steps. stop 86 The algorithm e s s e n t i a l l y provides a v a r i a b l e (s,S) l i m i t model. control The decision to order i s made by evaluation of the chance constraint and the decision of how much to order i s determined by the simulation and g r i d search. Both decisions depend on the cash flow f o r e - casts and d e l i v e r y lag times so that the parameters s and S vary over time. 5.6 A p p l i c a t i o n of the Model In t h i s section we s h a l l study the performance of the model with various values f o r 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 . cash.flow y^ was replaced by ^ 0 , 0 0 0 regarded as the occurrence of a y z l r o p o r That i s , the s i m p l i c i t y , a cash-out was cash balance so that x = 0. All tests of the model were performed with NSIM = 100 i t e r a t i o n s of the cash flow simulation f o r each order s i z e and orders were selected in steps of QSTEP = $5000 over the minimum order. The d e l i v e r y lag period L ( t ) f o r an order placed on day t varies according to the day of the week because the order i s received from a bank which operates on a Monday to Friday work week in contrast to the Tuesday to Saturday work week of the c r e d i t union. Orders are placed i n the vault on the day of a d e l i v e r y and counted a f t e r hours, for use the next day. A l i s t of the d e l i v e r y 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) Thursday Friday 3 Wednesday Friday Saturday 3 Thursday Tuesday Wednesday 4 Friday Tuesday Wednesday 3 Saturday Tuesday Wednesday 2 Tuesday Table 3 Performance of the Cash Order Model and of Management Fixed Order Cost $b Opportunity Cost of Cash 365 x i % Risk Level , (Normal Errors) a Probability of a CashOut in Simulation Number of Orders Placed Number of Actual Cash-Outs Average Cost Per Year ($) Management: 7 - - 153 2 15,700 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 25 Model: 88 i t i s i r r a t i o n a l to order cash on Thursday or Friday, since a Saturday order a r r i v e s on the same day. Table 3 o u t l i n e s 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 f o l l o w s : Fixed order costs b include the cost of armoured car d e l i v e r y and the bank charge f o r preparing United States and Canadian coin and currency p a r c e l s . These t o t a l $18.50., but other f i x e d costs, such as the cost of the t e l l e r s ' time to count the cash, make $25 and $40 f i x e d costs more r e a l i s t i c . The model was tested under both cost assumptions. The c r e d i t union o f f e r s 7% annual i n t e r e s t on the minimum d a i l y balance in one type of savings account, and t h i s may be taken as the opportunity cost of cash. A l t e r n a t i v e 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 t h i s opportunity cost as w e l l . The model was tested with three r i s k l e v e l s a of a cash-out: a = .00J , a = .01, a = .025. These r i s k l e v e l s are,only v a l i d under the assumption that the errors are normally d i s t r i b u t e d . Since the f o r e - 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 t h i s l e v e l 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 t o t a l number of i t e r a t i o n s (where the t o t a l number of i t e r a t i o n s = NSIM x number of orders placed) i s a good i n d i c a t i o n 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 r o b a b i l i t y of a Cash-Out in Simulation." In a d d i t i o n a count of the t o t a l number of cash-outs that would have 89 occurred with the actual cash flows, when the optimal order p o l i c y i s followed, i s provided. Referring to Table 3, we may f i r s t compare performance of management with the model, using f i x e d 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, i n c u r r i n g costs of $15,700 per year. On two occasions (or f o r 1.3% of the orders placed), deposits were s o l i c i t e d from a l o c a l merchant to cover.cash shortages ($2600 and $20,800). In contrast, with a = .025 the model y i e l d e d only a 1% r i s k of a cash-out in simulated performance and no cash-outs i n p r a c t i c e , while t o t a l . c o s 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 year). less than those of management ($13,900/ 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 r o b a b i l i t i e s i s to note that the "insurance premium" associated with reducing the p r o b a b i l i t y of a cash-out from .010 to .008 i s $11,600-$ll,100 = $500 per year, while the "insurance premium" f o r reducing the cash-out r i s k from .010 to .002 i s $2800 per year. When the f i x e d order costs and opportunity costs of holding cash were varied with a = .001 , the simulated cash-out p r o b a b i l i t y was always between .001 and .002 and the model responded i n a predictable manner in terms of the number of orders placed. That i s , increased 90 f i x e d costs caused fewer orders to be placed and increased opportunity costs caused more orders to be placed. In summary, the model provides more protection against cash-outs while i n c u r r i n g smaller operating costs than management's h i s t o r i c a l performance. A l s o , the model readjusts i t s long run p o l i c y in the correct d i r e c t i o n as the r a t i o of f i x e d order costs to opportunity costs i s varied. 5.7 Summary In t h i s chapter, the algorithm f o r ordering t i l l developed. cash was In Section 5.2 i t was asserted that a chance constrained model would be more appropriate than a recourse model, and the exact forms of the order and holding cost functions were e s t a b l i s h e d . The c a l c u l a t i o n of the d e t e r m i n i s t i c equivalent of the chance constraint was discussed in Section 5.3. In Section 5.4, i t was pointed out that a somewhat more complicated o b j e c t i v e function than that proposed in Section 4.5 would y i e l d a c l o s e r approximation to t r u l y optimal decisions and an operational method of evaluating the o b j e c t i v e function by simulation was presented. The actual cash order algorithm f o r minimizing the o b j e c t i v e function by means of a g r i d search was presented in Section 5.5. The performance of the cash order model was compared to the cash order performance of management in Section 5.6. It was established there that implentation would y i e l d lower costs and/or greater protection against the r i s k of cash-outs. 91 APPENDIX TO CHAPTER 5 THE DYNAMIC PROGRAM CORRESPONDING TO THE OBJECTIVE FUNCTION (5.4.2) Let V "t$ < ti < t r a t i o n a l to order cash. order placed a f t e r t < t3 < ••• be the days on which i t i s 2 This i s a l i s t of those days t such that an cannot a r r i v e at the same time as an order placed l a t e r ( i . e . t + L(t ) > t + L(t )'whenever t > t ). • Let n 2 (5.A.1 ) C.(x.) t=t + L ( t ) n n Z Z This i s the sum of the holding costs of opening cash balances which would be a f f e c t e d by the s i z e of the order Q , but not by the s i z e of the next S possible order Q, . Then l e t V l \ ( Q n t 1 n ••• oo Q n r min + l -00 >0 _ - -00 h n+1 L n (Q ; Oj. n+1 n+1 t L } . . . dc|, d* n L J x C = F(Q. ) + n + Qt ' t n n L ( n+1 , x. n+1 L } n+1 ) dcj). z n ••• dcj). n+1 , (5.A.2) n+l Note that the minimizations are performed subject to the c o n s t r a i n t (5.3.2) 92 and - for t n < t < t n + ] + L(t n + 1 ), x t = x + t _ I n t n (- y._ + R (£..)). } + This i s a dynamic program where we i t e r a t e from n = N to n = 0 and the horizon i s T = t... N Now i f we condition on the time T + 1 = t next order, recursive s u b s t i t u t i o n of (5.A.2) f o r h' , ' * * , h ! to t f o l l o w i n g analogue of (4.5.1) where t = 1: of the n+1 y i e l d s the n 0 hl(Qi; Qi, x j = A ' ( T , Q J + E(min h. ) , ^n+l where A 1 i s defined in (5.4.3). (5.A.3) Again we approximate the expectation in (5.A.3) by hi, adjusted now f o r the d i f f e r e n t 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 n+l t J S u b s t i t u t i n g into (5.A.3) y i e l d s the f o l l o w i n g approximation to the average r J T d a i l y cost: hi DTfT = QJ _ A ' ( , -QJ +L(t ; )-1)- D(T + L( +L)) A'(T D(t N + 1 T n i Computing the expectation over T of the above expression y i e l d s (5.4.2). Chapter 6 CONCLUSIONS At t h 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 t h i s p r o j e c t . From a s t a t i s t i c a l viewpoint i t i s perhaps s u r p r i s i n g that so much random v a r i a t i o n - i s present in the recorded d a i l y cash flows. If, as suspected, a large portion of the v a r i a t i o n 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 d a i l y basis. Despite the f a c t that the cash demand at the c r e d 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 r e s u l t of n o n - s t a t i o n a r i t y induced by the trend towards increasing cash demand as well as e f f e c t i v e n o n - s t a t i o n a r i t y a r i s i n g from the long periods of the cycles (up to one year long) and the even greater l e a s t common m u l t i p l e of these periods. The estimation of the demand f o r 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 s p e c i f i c case at hand and cannot be r e a d i l y generalized to other branches. I t e n t a i l e d f i t t i n g a non-linear 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 f o r other 93 94 branches. The subsequent estimation of demand v a r i a t i o n about the trend was performed by regression techniques that can r e a d i l y 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 distinguished from standard inventory theory problems as i s the c l a s s i c a l cash management problem, in which the lag time f o r transactions i s zero and the s t o c h a s t i c cash balance can d r i f t up or down. with t i l l However, the main problems associated cash management appear to be the n o n - s t a t i o n a r i t y of demand and other features that make i t hard to develop a steady state s o l u t i o n . approximate dynamic programming s o l u t i o n to the t i l l An cash management problem at a p o t e n t i a l order point was obtained by conditioning on the time of the next order and replacing the expected costs of optimal behaviour a f t e r that time with the average cost p r i o r 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 t h i r d or some l a t e r order, in order to incorporate more i t e r a t i o n s i n the dynamic program before invoking an approximation. Unfortunately the number of c a l c u l a t i o n s increases exponentially with the number of dynamic programming i t e r a t i o n s . The major t r a d e - o f f between t h e o r e t i c a l correctness and operat i o n a l 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 a c t u a l l y out-perform management and can r e a d i l y be implemented with e x i s t i n g o n - l i n e 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). Journal "A Model f o r the Determination of Cash Balances," of Financial and Quantitative Analysis, 1_, 1-14. Baumol, W.J. (1952). "The Transactions Demand f o r Cash: An Inventory Theoretic Approach," The Quarterly Journal of Economics, LXVI, 545-556. Bellman, R., I. Glicksberg and 0. Gross. (1955). Inventory Equation," Management Science, Box, G.E.P. and G.M. Jenkins. (1970). Time Series and Control, Hoi den-Day, New York. "On the Optimal 83-104. 2, Analysis: Forecasting Cleveland, W.S. (1972). "Inverse Autocorrelations of a Time Series and Their A p p l i c a t i o n s , " Technometrics, XIV, 277-298. Constantinides, G.M. (1976). " S t o c h a s t i c 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 i q u i d i t y : A M u l t i - P e r i o d Stochastic Model," Journal of Financial and Quantitative Analysis, 329-343. Eppen, G.D. and E. Fama. (1968). "Solutions f o r Cash Balance and Simple Dynamic P o r t f o l i o Problems," Journal of Business, 4j_, 94-112. : . (1969). "Cash Balance and Simple P o r t f o l i o Problems with Proportional Costs," International Economic Review, JJ), 119-133. Iglehart, D. and S. K a r l i n . (1962). "Optimal P o l i c y f o r Dynamic Inventory Processes with Nonstationary Stochastic Demands," i n Studies in Applied Probability and Management Science (K.J. Arrow, S. K a r l i n and H. Scarf, eds.), Chapter V I I I , Stanford U n i v e r s i t y Press, Stanford, C a l i f o r n i a . 9& 96 Johnston, J . K a r l i n , S. (1972). Econometric Methods, 2nd e d . , McGraw-Hill, New York. (1958). "Steady State S o l u t i o n s , " in Studies in the Mathematical Theory of Inventory and Production (K.J. Arrow, S. K a r l i n and H. Scarf, eds.), Chapter V I I I , Stanford U n i v e r s i t y Press, Stanford, C a l i f o r n i a . M i l l e r , M.H. and D. Orr. (1966). "A Model of the Demand f o r Money by Firms," The Quarterly Journal of Economics, LXXX, 413-435. . (1968). "The Demand f o r Money by Firms: Extensions of A n a l y t i c R e s u l t s , " Journal of Finance, 23> 735-759. Neave, E.H. (1970). "The Stochastic Cash Balance Problem with Fixed Costs f o r Increases and Decreases," Management Science, 16, 472-490. Nelson, C R . (1973). Applied Time Series Analysis casting, Hoi den-Day, San Francisco. Orr, D. for Managerial Fore- (1970). Cash Management and the Demand for Money, Praeger Publishers, New York. Stone, B.K. (1972). "The Use of Forecasts and Smoothing in C o n t r o l - L i m i t Models f o r Cash Management," Financial Management, Spring, 72-84. T h e i l , H. (1966). Applied . (1971). Principles Economic Forecasting, of Econometrics, North-Holland, Amsterdam. Wiley, New York. Ziemba, W.T. (1975). "Dynamic Programming," i n Stochastic Optimization Models in Finance (W.T. Ziemba and R.G. Vickson, e d s . ) , Academic Press, New York, 43-56.
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Till cash management model Sick, Gordon Arthur 1976
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Title | Till cash management model |
Creator |
Sick, Gordon Arthur |
Date Issued | 1976 |
Description | This thesis develops a model for the management of till cash (currency and coin) of a branch for a Vancouver area credit union. The model is developed in two parts. First, a model is estimated to forecast cash demand and then a cash order algorithm is developed. Two statistical models are developed to estimate cash demand. The first employs Box-Jenkins time series techniques. This model fails because the cash flow data are non-stationary, exhibiting both a growth trend and high autocorrelations at large lags. In the second model, a growth trend for real weekly cash flows is first estimated, incorporating an asymptotic capacity constraint. The real cash flow trend is converted to a nominal trend and used as the weight in a linear weighted least squares model for daily 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 resulting forecast model is also discussed. To develop a cash order algorithm, steady state models are first considered. These models are generally based on stationary cash demand, constant delivery lag times for orders and other assumptions that are inappropriate in this till cash management setting. To relax the steady state assumptions a general dynamic programming framework is developed for the cash management model that allows for either penalty costs for cash-outs (cash shortages) or a chance constraint involving the probability of a cash-out. Because of non-stationarity of the cash flows the dynamic program cannot be solved directly, but an approximate solution is obtained using a simulation technique. The resulting algorithm is tested on historical data and the results are discussed briefly. |
Subject |
Cash flow - Mathematical models Banks and banking, Cooperative - British Columbia - Vancouver |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0094015 |
URI | http://hdl.handle.net/2429/20320 |
Degree |
Master of Science in Business - MScB |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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