daily cash forecasting with multiplicative models of cash flow patterns

Daily Cash Forecasting with Multiplicative Models of Cash Flow PatternsAuthor(s): Bernell K. Stone and Tom W. MillerSource: Financial Management, Vol. 16, No. 4 (Winter, 1987), pp. 45-54Published by: Wiley on behalf of the Financial Management Association InternationalStable URL: http://www.jstor.org/stable/3666108 .

Accessed: 12/06/2014 22:19

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

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

.

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

http://www.jstor.org

This content downloaded from 195.34.79.228 on Thu, 12 Jun 2014 22:19:51 PMAll use subject to JSTOR Terms and Conditions

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

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

http://www.jstor.org/stable/3666108?origin=JSTOR-pdf

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


Cash Flow Applications

Daily Cash Forecasting with

Multiplicative Models of

Cash Flow Patterns

Bernell K. Stone and Tom W. Miller

Bernell K. Stone is the Harold F. Silver Professor of Finance at

Brigham Young University. Tom W. Miller is an Associate Professor of Finance at Emory University.

0 Daily cash forecasting is a problem of considerable practical importance. Good daily cash flow forecasts can improve net yield on short-term investments, reduce the cost of commercial paper issuance and other short-term borrowing, reduce idle cash balances, and in general facilitate cash management. In a survey conducted by the National Corporate Cash Manage- ment Association concerning topics that required both further education and improved practice, the member- ship ranked daily cash forecasting as the number one priority in both areas. Because of its importance, many companies undertake serious efforts to forecast daily cash flows. While there have been a few successful systems, most companies have achieved, at best, mod- erate success.

This article briefly structures the daily cash flow forecasting problem and examines reasons why so many company efforts at daily cash flow forecasting have failed. It shows that resolving a time varying mix

of seasonals (time dependent patterns) is the central statistical issue of most daily cash flow forecasting - a problem that standard statistical treatments of simul- taneous seasonals has generally neglected. Next, this article examines in depth a family of easy-to-estimate log-linear models for treating day-of-week and day-of- month interaction.

I. Explanations for Forecast Failures When a statistician or systems analyst is assigned

the task of developing a daily forecast of the net cash flow, the logical response is to obtain a history of net daily flows and fit one of the standard statistical time series models to the data. However, the application of standard statistical forecasting models to the net daily cash flow of a large firm virtually guarantees failure of the forecast effort.

There are four common reasons for the failure of most efforts at daily cash forecasting. Three reasons

45



46 FINANCIAL MANAGEMENT/WINTER 1987

pertain to proper problem structuring and the fourth involves issues of proper statistical modeling.

Reason 1: Major Flow Separation. Major flows are defined as all large flows that are not periodic and not generally estimatible from past history, e.g., taxes, dividends, sinking fund payments, lease payments, and major receipts such as project completion payments, contract sales, and asset disposition proceeds.

If standard extrapolative, periodicity, or auto-re- gressive techniques were applied to the net daily cash flows without removing the major flows, the standard techniques would fail, since the major flows constitute large flows on the days they occur, are a large proportion of the total flow, and cannot be modeled as either a trend or periodic function. For instance, income tax payments typically occur at most six times per year, do not exhibit a regular periodicity, and have amounts that bear little relation to past cash flows including even past tax payments.

While generally not predictable via standard statistical techniques, the major flows are easy to handle in daily forecasting since most are known in both timing and amount 30 to 90 days before their occurrence. In fact, major known-in-advance payments constitute 50% to 70% of the total payments in most companies; therefore, simply identifying and properly scheduling major payments can usually ensure 50 + % accuracy for cash outflows with no formal statistical forecasting system. The cash flow to which statistical techniques pertain are the nonmajor flows.

Reason 2: Component Identification. Even after all major flows have been removed from the cash flow stream, it is almost always necessary to divide the remaining nonmajor flow into components. At a mini- mum, nonmajor inflows and outflows should be sepa- rated, but generally even more detail is needed. It is often necessary to track and forecast dozens or even hundreds of different nonmajor cashflow streams. This is because different cashflow streams have different statistical properties, especially apparent in day-of- month and day-of-week patterns. A time varying mix of different patterns means no statistical stability, and therefore, very limited ability either to estimate forecast parameters from past history or to use them for forecasting.

Reason 3: Information System Support. Tracking dozens and usually hundreds of nonmajor cash flow streams as well as hundreds of individual major flows requires a computer-based information- control system able to focus on the various cash flow

streams. Developing a daily cash forecasting system is as much a problem of designing and managing an information-control system as it is a problem of statistical forecasting. Most companies do not have such an information-control system in place. Thus, they often lack the detailed data necessary to treat the relevant components.

Reason 4: Pattern Resolution. For several reasons, daily cash flows always have a day-of-week pattern and generally have a day-of-month pattern as well. Many cash flow events are tied to monthly or weekly events, e.g.: (1) a monthly, biweekly, or weekly pay- roll, (2) credit terms tied to day-of-month, such as EOM (end-of-month) due dates, and (3) internal processing procedures and policies such as preparing certain payment streams on certain days of the week. The nature of the mail system and check processing system also create strong day-of-week patterns because of weekends, even when the underlying cash flow gen- eration process has no day-of-week pattern. For instance, mail delivery means a greater mail volume on Monday than other days of the week. Likewise, check processing often means a weekend distortion with efforts to expedite processing on Friday and a cumulative volume build-up on Monday that often spills over to Tuesday because of processing capacity limitations.

The day-of-week pattern is not a periodic subcycle of the day-of-month cycle since there are 28 possible combinations of day-of-week and length of month. Conventional statistical techniques assume a stable mix, estimate the combined effect, and then project the combined effect into the future. Using a misspecified model assuming a stable mixture means low predictive power as well as low explanatory power at the parameter estimation stage. In daily cash forecasting, it is necessary to explicitly model the day-of-month and day-of-week patterns, measure model parameters (i.e., resolve the individual patterns from actual cash flow data involving a mix of the two effects), and then construct a forecast by using the appropriate day-of- week and day-of-month effects for a particular forecast period.

II. Approaches to Daily Cashflow Forecasting

Most daily forecasting techniques can be placed in three generic categories: (1) statistical extrapolation (e.g., moving averages, exponential smoothing), (2) scheduling, and (3) distribution.

Scheduling is mapping information that precedes



STONE AND MILLER/MULTIPLICATIVE MODELS OF CASH FLOW 47

a cashflow into its cashflow implications. For instance, receiving and approving an invoice for payment with terms of net 30 implies a future date for check mailing given a company's payment policy, e.g., 30 days after the invoice date if paid in accord with credit terms. Scheduling is an information system-based approach to daily forecasting.

Distribution is the mapping of a total amount into the daily cash flow implications on the basis of an assumed distribution by day of the total. For instance, if $300,000 in checks were written on Thursday, then the predicted cash withdrawal at the disbursing bank on the following Tuesday through Monday from these checks might be $30,000, $60,000, $120,000, $60,000, and $30,000 respectively.

A very common use of the distribution approach in practice is to use data from the company's month-by- month cash budget to obtain monthly totals for various cash flow streams, e.g., the cash flow arising from nonmajor vendor disbursements in the specialty metals division or the expected cash flow from collections of credit sales of industrial fans on terms of net 60.

A. The Interacting Seasonal Problem Purely extrapolative techniques such as moving

averages or exponential smoothing generally do not work well, because daily cash flows usually have both a day-of-week and a day-of-month pattern along with any annual seasonal that may be present in a cash flow stream. The usual ways of jointly modeling trends and patterns (seasonals) also meet trouble due to the exis- tence of both day-of-month and day-of-week patterns.

Treating day-of-month and day-of-week effects to- gether can result in a case of a time varying mix of interacting seasonals. Hence, it is much more complex than the standard problem of a stable mix of two seasonals treated in the forecasting literature [e.g., 3, 4, 7, 9, 11]. The reason is that the day-of-month and day- of-week mix varies over time, since a month is not an integer multiple of a week (except for February when it is not a leap year). Excluding February, there are 14 possible combinations of day-of-month and day-of- week interactions (7 possible days for start-of-month and 2 possible lengths, namely 30 and 31 calendar days). February presents another 14 possible combinations (7 possible days for start of month and 2 possible lengths, namely 28 and 29). Thus, there are 28 possible day-of-week, day-of-month interactions.

The usual assumption in modeling interacting seasonals is that one pattern is a perfect and recurring subcycle of another [e.g., 4, 7, 9]. A time varying mix

of seasonals greatly complicates the forecast task for both estimation and prediction. Rather than simply measuring jointly a historical seasonal interaction and then extrapolating this joint effect, as is done in the usual statistical treatments of seasonality, it is necessary to first estimate the underlying day-of-month and day-of-week patterns, i.e., to model and resolve the seasonals into separate effects, and then to construct from the estimated seasonals the combined effect appropriate for a particular day-of-month and day-of-week.

B. Past Research Stone and Wood [18] recognize the problem of re-

solving day-of-month and day-of-week interactions. They use an additive model of the interaction. Miller and Stone [10] develop a general class of day-of- month, day-of-week interaction models that include the additive model as a very special case.

This paper seeks to structure the task of resolving (separately estimating) a time varying mix of day-of- month and day-of-week effects and then recombining them into an appropriate prediction. The concern here is to develop alternative ways to model the day-of- month and day-of-week interaction that can also be implemented in practical daily cash forecasting situations.

III. The Distribution of Monthly Totals In the distribution approach to daily cash flow fore-

casting, a total flow is spread over the relevant days on the basis of the proportion of the total predicted for each day. In this paper, we shall limit the treatment to the distribution of a monthly total. The techniques developed here can be adapted to other total distribution situations such as those described by Anvari [1] and by Stone and Miller [16].

A set of proportions represents the daily distribution of the total. These models can incorporate both day-of- month and day-of-week effects if they are appropriate- ly reflected in the proportions. If t is a day-of-month index and w is the day-of-week index and p,, is the proportion of the monthly total occurring on day t when day t occurs on weekday w, the cash flow prediction equation is:

CFtw =

Ptw(TOTAL). (1)

The sum of the proportions is generally required to equal one for any given month.




Exhibit 1. Actual and Forecasted Cash Flows for the Additive Model

Millions of $

1.790 S1.790Actual -

1.580 Forecasted

1.370 /

1.160

0.950

0.740

0.530

0.320

0.110

- .100 --, -*: -- : 1 2 3 4 5 6 7 8 9 10 1112131415161718192021

IV. Additive Versus Multiplicative Structures

Stone and Wood [18] use an additive model to char- acterize the cash flow pattern as a combination of day- of-month and day-of-week effects, namely:

tw = at + bw. (2)

The variable a, is the proportion of the cash flow that would occur on workday t in the absence of any day- of-week effect. The set {a,} thus characterizes the monthly pattern, while the set

{bw} (with w equal to one on Monday and five on Friday) measures day-of-week effects.

Equation (2) does not provide for interaction between the day-of-month and day-of-week effects. In particular, the day-of-week effect is the same at high levels and low levels of the cash flow. However, the day-of-week effect often arises from factors such as weekend shifts in mail times, company processing patterns, and payment system clearing of deposited checks. Thus, the day-of-week adjustment should generally reflect variation in the level of the basic monthly flow.

Day of the Month V. The Need for a Multiplicative Model

Awareness of the need for a multiplicative model arose from efforts to use the additive model. For some product lines, such as fabricated steel, credit terms such as EOM (due at end of the month) or 2/20M n 20NM (a 2% discount if paid by the next 20th of the month or else payment in full by the 20th of the following month) lead to cash flows that are concentrated in a particular part of the month. In such cases, the {a,} characterizing the day-of-month proportions were found to be large during part of the month and near zero during the rest of the month.

Exhibit 1 illustrates this problem for an extreme situation in which the day-of-week adjustment was not only negative (reflecting below average cash flows on that day of the week) but was also bigger in magnitude than the near-zero estimates for the day-of-month flow on some workdays.' Exhibit 1 shows that the additive

'A description of the data used for this illustration is provided in the Appendix. Estimates of the day-of-month and day-of-week parameters for the additive and multiplicative specifications employed in this study are presented in Exhibit 5.




Exhibit 2. Actual and Forecasted Cash Flows for the Multiplicative Model

Millions of $

1.790 Actual

1.580' Forecasted

1.370

1.160

0.950

0.740

'0.530 /

0.320

0.110

-.100 - - , - 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Day of the Month model predicted negative cash inflows on three workdays, namely, the first, sixteenth, and twenty-first workdays of the month. A negative cash flow from accounts receivable payments is clearly impossible! Further investigation shows that high cash flow times tended to have day-of-week adjustments with magnitudes that were too small, especially the peak flow occurring on workday nine. Conversely, low cash flow times (workdays 1, 2, 3, 16, 17, 18, 19, and 21) tended to have adjustment magnitudes that were too large.

This error structure was not easy to identify because some of the {b,} were positive and some were negative. Hence, there was not a clear series of either excessively high or excessively low errors but rather just greater error variance at the high and low cash flow points as indicated in Exhibits 1 and 3.

The problem here is that the least squares estimation of the additive model provides only an average day-of- week adjustment that cannot reflect variations in the estimated value of the day-of-month proportions. To solve this problem, it is clearly necessary to make the level of the day-of-week adjustment vary with the level of the basic monthly cycle, i.e., to allow for an interac-

tion between the day-of-month and day-of-week effects.

VI. A Multiplicative Model A proportional adjustment to the day-of-month ef-

fect can be modeled as a multiplicative scaling in which the proportion p,, is given by:

P,, = a*(1 + b*). (3)

Here, aT and b* are multiplicative analogues of at and b,. Thus, a* is the multiplicative estimate of the proportion of the monthly total that would occur on workday t in the absence of any day-of-week effect, and b* is the multiplicative day-of-week adjustment. For example, when b* is greater than zero, the adjustment is upward.

This multiplicative day-of-week adjustment varies with the level of the monthly cycle as intended. Note also that since the average value of at is about 0.05 (i.e., there are about twenty workdays per month), b* must be about twenty times as large as

bw to provide the

same average correction. Thus, b, and b* are not nu-




Exhibit 3. Forecast Errors for the Additive and Multiplicative Models

Millions of $

0.340 Additive

8I Multiplicative 0.280

0.220

0.160

0.100 '

,

0.040

- 020

-. 140

-.200 " - 1 2 3 4 5 6 7 8 9 1011 12131415161718192021

Day of the Month

merically comparable.

VII. Estimation of the Multiplicative Model

The additive model is obviously amenable to linear dummy variable estimation. Because the multiplicative model appears to mix the monthly and weekly patterns, it appears to require more complex nonlinear estimation. However, the apparent nonlinearity can be resolved by performing log-linear estimations. Taking logarithms of both sides of Equation (3) gives:

log(ptw) = log(a*) + log(1 + b*). (4)

This can be estimated using the dummy variable regression equation,

M W

log(ptw) = I A,m, +

Bwd + tw,, (5)

t=l w=l

where At and Bw

are regression coefficients, M is the

number of workdays in a month, W is the number of workdays in a week, m, is a day-of-month dummy variable that is one on workday t and is zero otherwise, d, is a day-of-week dummy variable that is one on weekday w and is zero otherwise, and

tw, is an error

term. Comparing (4) and (5), the transformation relat- ing the regression coefficients to the model parameters is:

A, = log(a*) (6)

B, = log(1 + b*). (7)

Solving Equations (6) and (7) gives estimates of the {at} and the {b*}.

VIII. Performance of the Multiplicative Model

Exhibit 2 shows the multiplicative forecast for the illustrative case in which the additive model broke down. The multiplicative forecast tracks the actual flow better than the additive. Exhibit 3 compares the




forecast errors. The key point is not that the multiplicative model performs better since this example was in- tentionally selected to illustrate an extreme case of additive model failure. Rather the concern is with the comparative error structure. The multiplicative forecasts are more consistent across the extreme intra- month variation in cash flow magnitude in contrast to the occurrence of relatively large errors for the additive model at both the high and low cash flow points. In this case, the additive and multiplicative models had simi- lar estimates for the day-of-month and day-of-week patterns although the statistical significance was much better for the multiplicative. Thus, the major reason for the superior performance of the multiplicative model in this case appears to be its ability to have the magnitude of day-of-week correction vary with the level of cash flow. In effect, it more accurately reflects the day-of-week, day-of-month structure than does the additive model in this case.

IX. Alternative Specifications of Multiplicative Interactions

The preceding case is the common one of a basic monthly cycle with a day-of-week disturbance. An- other situation is the reverse of this - a basic weekly cycle with a day-of-month disturbance. In an additive model, these two situations have the same structural specification. However, for a multiplicative formula- tion, an alternative model specification is required, namely,

Ptw = (1 + a )bw.

(8)

Here, b+ measures the weekly cycle and a' measures the day-of-month disturbance. In this case, we require lb+ = 1 and a = 0.

When the proportions are near zero, another problem arises. The logarithmic function is extremely sen- sitive to small changes in arguments near zero and is undefined for arguments equal to zero. The basic specification can be transformed to handle this problem. Instead of estimating

pw, (1 +

ptw) can be estimat-

ed. The logarithm of (1 + p-,) is defined even when the value of p,, equals zero. Moreover, it is stable for small values of ptw, i.e., those near zero. Therefore, when- ever some of the {ptw} are either zero or very near zero, all of the {p,} in Equations (3) and (8) can be replaced with {1 + p,}.

Exhibit 4 summarizes these specification alternatives and their estimation equations. It is stressed that there is not just one linearizable multiplicative model

but rather a set of at least three log-linear multiplicative specifications. They represent alternative ways of modeling day-of-month and day-of-week interactions that are proportional to the level of the basic flow. As Exhibit 4 shows, all of these models share the common feature of being linearizable by a logarithmic transformation. Likewise, all use past values of daily cash flow proportions to estimate the day-of-month and day-of-week coefficients. Thus, all the multiplicative alternatives provide complementary estimates of future p,, values from the estimated day-of-month and day-of-week coefficients.

X. Concluding Comments These log-linear multiplicative models of day-of-

month, day-of-week interactions are alternatives to the additive model developed by Stone and Wood [18]. When we say "alternatives," we do not mean to sug- gest that they are necessarily superior to or a replace- ment for the additive model in all situations. Rather, we view them as complements that allow proportional interactions. Unless there is a strong a priori reason for choosing a particular specification, a forecaster using dummy variable regression for implementing the distribution approach should consider both the additive and the alternative multiplicative models and estimate all the relevant specifications. The primary reason for this recommendation is that all these alternatives can be estimated from the same data and use the same estimation technique - linear dummy variable regression. A forecaster can test the performance of each forecast specification. Choice between the alternatives can be made on the basis of statistical fit and forecast performance.

The issue of additive versus multiplicative distur- bances highlights a potential problem - the possibility that the true weekly disturbance and the true monthly disturbance are mixtures of both additive and multiplicative components or even possibly more complex interactions. The concern here is not to present the de- finitive daily cash flow forecasting framework for the distribution approach and the many statistical complexities that arise with complex interaction models. Readers interested in more complex day-of-week and day-of-month interactions are referred to Miller and Stone [10] for both their specification and in-depth treatment of the associated estimation complexities. Readers are, however, warned that these more complex interaction models achieve their generality at the cost of many more parameters and much more difficult estimation procedures. Since available data is usually



Exhibit 4. A Summary of the Alternative Multiplicative Models

Basic Monthly Cycle with a Basic Weekly Cycle with a Weekly Disturbance Monthly Disturbance Relative Disturbance Model

Model Equation Ptw = at(1 + b*) Ptw = (1 + a)b+ 1 +

Ptw = (1 + a')(l + b)

Logarithmic Transformation log(ptw) = log(a*) + log(l + b*) log(ptw) = log(1 + a+) + log(b+) log(1 + pw)

= log(l + a') + log(l + b')

Parameter to Coefficient Transformation At= log(a*t) and Bw

= log(l + b*) At = log(l + at) and Bw

= log(b) A' = log(l + a) and B' = log(l + bw)

Coefficient to Parameter Transformation a = exp(At) and b* = exp(Bw)

- 1 a+ = exp(At) - 1 and bt+ = exp(B) a' = exp(A') - 1 and b = exp(B') - 1

M W M W M W

Dummy Variable Regression Equation log(ptw) = Atmt + Bdw

+ t* log(ptw) = Atmt + Bd, + s log(1 + p) = Am + Bd, + '

t=l w l t=l w=l t=l w=l

These columns present three different multiplicative models for specifying day-of-month and day-of-week interactions. The fact that there are different estimators for day- of-month and day-of-week effects is stressed by use of *, +, and ' to denote the various parameters. In performing regressions, the dependent variable is obtained from the data by entering the proportion of the monthly total for all t and w. Holidays are not explicitly treated here. See Miller and Stone (10) for treatment of holiday issues and several alternative ways of incorporating them in the forecast.

U' gIr

z z 3> n PM

z m m z -,4

z -4 m

Imd

co l*4




limited to twelve to thirty-six months in most actual company forecast situations, practical use of the general interaction models usually requires structural as- sumptions to reduce the general interaction parameters to a reasonable number. Moreover, parameter estimation generally requires constrained, non-linear estimation with its many complexities.

The significant attribute of the purely multiplicative interaction models developed here is that the log-linear transformation provides an important class of interaction models that are linearizable with no more parameters than the purely additive (no-interaction) linear model. The dramatic reduction in parameters and thus in required data is the advantage of the multiplicative interaction models developed here over the general interaction specification in Miller and Stone [10]. References

1. M. Anvari, "Forecasting Daily Outflows from a Bank Ac- count," Omega (No. 3 1983), pp. 273-277.

2. K. Boyd and V. A. Mabert, "Two Stage Forecasting Ap- proach at Chemical Bank of New York for Check Process- ing," Journal of Bank Research (Summer 1977), pp. 101-107.

3. G. E. Box and G. M. Jenkins, Time Series Analysis, San Francisco, Holden Day, 1970.

4. R. G. Brown, Smoothing, Forecasting, and Prediction of Discrete Time Series, Englewood Cliffs, N.J., Prentice Hall, Inc., 1963.

5. G. W. Emery, "Some Empirical Evidence on the Properties of Daily Cash Flow," Financial Management (Spring 1981), pp. 21-28.

6. J. B. Guerard, Jr. and K. D. Lawrence, "Forecasting Daily Cash Collections," Lehigh University Working Paper, Pre- sented at the First Annual Working Capital Research Sym- posium (Montreal, July 1985).

7. C. C. Holt, F. Modigliani, J. F. Muth, and H. A. Simon, Planning, Production, Inventories, and Work Force, En- glewood Cliffs, N.J., Prentice Hall, Inc. 1960.

8. S. F. Maier, D. W. Robinson, and J. H. Vander Weide, "A Short-Term Disbursement Forecasting Model," Financial Management (Spring 1981), pp. 9-20.

9. J. O. 4cClain, "Dynamics of Exponential Smoothing with Trend and Seasonal Terms," Management Science (May 1974), pp. 1300-1304.

10. T. W. Miller and B. K. Stone, "Daily Cash Forecasting: Alternative Models and Techniques," Journal of Financial and Quantitative Analysis (September 1985), pp. 335-351.

11. R. H. Morris and C. R. Glassey, "The Dynamics and Statistics of Exponential Smoothing Operators," Oper- ations Research (July-August 1963), pp. 561-569.

12. D. C. Montgomery and L. A. Johnson, Forecasting and Time Series Analysis, New York, McGraw-Hill Book Co., 1976.

13. H. Rinne and R. A. Wood, "Daily Sales Measurement Control," Working Paper, Finance Department, Penn State University (November 1983).

14. B. K. Stone, "The Use of Forecasts and Smoothing in Control-Limit Models for Cash Management," Financial Management (Spring 1972), pp. 72-84.

15. B. K. Stone and T. W. Miller, "Daily Cash Forecasting: A Structuring Framework," Journal of Cash Management (October 1981), pp. 35-50.

16. "Forecasting Disbursement Funding Require- ments: The Clearing Pattern Approach," Journal of Cash Management (October/November 1983), pp. 67-78.

17. - "Daily Cash Forecasting," Chapter 7, pp. 120-141 in Corporate Cash Management: Techniques and Analysis, F. J. Fabozzi and L. N. Masonson, editors, Homewood, IL: Dow Jones-Irwin, 1985.

18. B. K. Stone and R. A. Wood, "Daily Cash Forecasting: A Simple Method for Implementing the Distribution Ap- proach," Financial Management (Fall 1977), pp. 40-50.




Appendix The data for this illustration are based on daily cash

flows over an 18 month period from accounts receivable for the electric parts division of a major company. Most sales, approximately 90%, were made to regular customers on open accounts under master contracts. Each of these customers typically had multiple shipments from several plants and/or warehouses. Most of these customers received a single invoice for all of their shipments during the month with credit terms offering a 2% discount if paid by the 10th of the next month else full payment was due at the end of the next month. These attractive discount terms produced a strong day-of-month receipts pattern. Approximately 10% of the sales were not to regular customers and were a mixture of cash before shipment and varying terms such as net 10, net 20, and net 30. Before estimating the day-of-month and day-of-week parameters for the forecasting models, the daily cash flow data were adjusted in several ways. First, all government receipts, which were large, erratic cash flows, were removed and tracked separately. Second, all large accounts, defined as any customer having expected shipments of $100,000 or more per month, were also removed and tracked separately. These were viewed as major cash flows that were not amenable to statistical distribution. Third, a shift model of the type discussed in Stone and Wood [ 18] was used to correct for holiday effects. Thus, the net result was a time series of holiday-corrected, nonmajor cash flows by workday of the month from open accounts receivable with the same credit terms and with a strong day-of-month pattern plus another time series of cash flows with varying credit terms and with no particular day-of-month pattern. With these adjustments, the many receipts from approximately 4000 customers clearly fit the require- ments for statistically forecasting the cash flows via the distribution approach. Exhibit 5 provides estimates of the day-of-month and day-of-week parameters for the additive and multiplicative models used in this study.

The example presented in the paper is used to illustrate the problems and methodology. Neither the accu-

Exhibit 5. Estimates for the Additive and Multiplica- tive Models

Additive Model Multiplicative Model Pattern

Signifi- Signifi- Day of Estimate of cance Estimate of cance Month Parameter Level Parameter Level

1 .00632 .085 .00935 .001 2 .00951 .011 .00964 .001 3 .01992 .001 .01902 .001 4 .02577 .001 .02426 .001 5 .03707 .001 .05345 .001 6 .04242 .001 .04124 .001 7 .10207 .001 .09820 .001 8 .14078 .001 .14232 .001 9 .17119 .001 .17223 .001

10 .14732 .001 .14175 .001 11 .10353 .001 .09689 .001 12 .04901 .001 .04812 .001 13 .03828 .001 .03689 .001 14 .02687 .001 .02566 .001 15 .01983 .001 .01845 .001 16 .00668 .074 .00984 .001 17 .01017 .007 .01026 .001 18 .01063 .011 .00937 .001 19 .01109 .086 .00963 .001 20 .00999 .015 .00917 .001 21 .00011 .409 .00489 .001

Signifi- Signifi- Day of Estimate of cance Estimate of cance Week Parameter Level Parameter Level

Mon .01197 .001 .24744 .001 Wed - .00855 .003 - .19406 .001 Thu - .00422 .079 - .08409 .024 Fri .00251 .202 .12781 .004

One of the day-of-week parameters (Tue) was removed from the estimation equations to avoid a perfect collinearity problem. The day-of-week parameter for Tuesday was set equal to 0 for forecasting purposes. Miller and Stone [10] discuss the perfect collinearity problem and present alternative solutions.

racy nor the relative performance of the additive and multiplicative models is necessarily representative of all of the cash flow forecasting situations for most companies. In fact, the accuracy for this division's nonmajor cash flows using the multiplicative model was better than average for this company.



daily cash forecasting with multiplicative models of cash flow patterns

Documents