A STOCHASTIC APPROACH FOR EVALUATING SHORT=TERM FINANCING- ALTERNATIVES UNDER commons OF RISK . Thesis for the Degree of ‘Ph. D; MICHIGAN STATE UNIVERSITY JOHN WALTON ELLE-S 1971 yv-E‘F a 0-7639 [BRA‘RY Michigan Stsw University This is to certify that the thesis entitled ' H": {’1 ' "’T‘l'rfi ‘ "n“u‘wfi ' fl" *1“ "1"? 1.- ‘r’1" “a « —U JLLa-aL» .A. , A .., --‘ a. -N--\JL1 a v-9 ._J ..-.LJU—-‘., —$‘ a r' ,; ' "‘1" ,' " 7.1.... ' ' '17“ ‘ ‘V " 7 -‘ ‘t ‘ m‘t’t‘r‘“r‘ .~.11V&b.~—._ M“- d. - L‘o-‘L‘u —LI all A¢‘~~4L“‘AL——o-I’»—Ia -.. - _ . ,1. —- --"1--.' -,.l I 5.1 l-- a .. VLIqrd‘» vvaad-_..._ U‘!._.‘ V... ...... ~ . ‘1 .1 1" “t n w n ' "‘ U v._‘ .1 .o— n.— has been accepted towards fulfillment of the requirements for Ldegreein 2 Major professor ABSTRACT A STOCHASTIC APPROACH FOR EVALUATING SHORT-TERM FINANCING ALTERNATIVES UNDER CONDITIONS OF RISK BY John Walton Ellis At least some facet of the cash management problem is faced by every manufacturing firm. In the decision process, management must decide upon an optimal minimum cash balance and a policy for handling cash shortages. A model to aid management in selecting an optimal financing strategy from a given set of alternatives is presented in this study. The objective is the maximization of the utility of the unrestricted ending cash balance at the end of a specific planning period. A sequential, multi-period approach that considers risk is incorporated in the simulation model. The net cash flow values that serve as input to the model are generated by a simulation of the stochastic processes of the input variables which make up the net cash flow. The condition of stochastic dominance is utilized to evaluate John Walton Ellis the cumulative probability distributions which were generated from the model output. In summary, two quantitative techniques, simulation and the condition of stochastic dominance have been integrated and applied to analyzing and selecting an optimal financing strategy from a given set of alternatives. A STOCHASTIC APPROACH FOR EVALUATING SHORT-TERM FINANCING ALTERNATIVES UNDER CONDITIONS OF RISK BY John Walton Ellis A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Accounting and Financial Administration 1971 ACKNOWLEDGMENTS In their capacity as members of the dissertation committee the author is grateful to Dr. Myles Delano, Dr. Gardner Jones, and Dr. Ronald Marshall for their helpful criticism. The author also wishes to express his sincere appreciation to Dr. James Don Edwards, Chairman of the Accounting and Finance Department, for arranging financial support throughout the author's course of study. In addition, the author gratefully acknowledges use of the Michigan State computing facilities, made possible through support, in part, from the National Science Foundation. Finally, the author gives special thanks to his wife, Susan, for editing and typing the preliminary manuscript. Without her patience and encouragement, this work could not have been completed. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . LIST OF TABLES o o o o O O 0 O O 0 LIST OF FIGURES . . . . . . . . . Chapter I. II. II. INTRODUCTION . . . . . . . . Statement of the Problem . . Plan of the Study . . . . . Certainty, Risk and Uncertainty Organization of Remainder of the SURVEY OF THE RELATED LITERATURE-- METHODS AND MODELS . . . . . 0 Introduction . . . . . Deterministic Approaches . . Stochastic Approaches . . . Thesis A DECISION-THEORY STATEMENT OF THE PROBLEM . . Objective . . . . Single- Stage Decision Problem (DP) Under Certainty . . . Model of the Single- Stage Problem Under Risk . . . . . . Multi- -Stage Decision Problem Under Risk The N-Stage Decision Problem . The Decision Problem to be Analyzed in this Study . . . . . The Model in This Study as An Analysis of a Subset, A9, of the Strategies 61 in 1 . iii Page ii vi . viii UlubNH |-’ 23 23 23 27 29 31 32 35 Page Assumptions . . . . . . . . . . . 36 Probabilistic Dynamic Programming as a Possible Approach to Solving the General Stochastic Multi-Stage Model . 38 Simulation 0 O O C O O O O O O O 41 IV. MODEL DESCRIPTION . . . . . . . . . . 43 Introduction . . . . . . . . . . 43 Me thOdo logy O O O O O O O O O O O 4 6 The Transformation Function . . . . . 47 The Short-Term Financing Alternatives . . 51 Computer Evaluation of the Financing Alternatives . . . . . . 53 Model Input Data . . . . . . . . . 63 Interest Rates on the Financing Alternatives . . . . . . 63 The Minimum Required Cash Balance . . . 76 V. PRESENTATION AND EVALUATION OF RESULTS . . . 77 The Net Cash Flow Simulation . . . . . 78 Constraints . . . . . . . . . . . 79 Stockout Penalties . . . . . . . . 81 Interest Rates . . . . . . . . . . 83 The Minimum Required Cash Balance . . . 83 The Computer Programs Involved in the MOde 1 O O I O O O O O O 8 5 Model Output . . . . . . . . . . 85 Stochastic Dominance . . . . . . . . 86 Evaluation at a Minimum Required Cash Balance of $3 Million . . . . . . 88 Evaluation at a Minimum Required Cash 'Balance of $4.5 Million . . . 91 Evaluation at a Minimum Required Cash Balance of $6 Million . . . . . . 98 Evaluation When the Minimum Required Cash Balance is a Stochastic Variable . . . . . . . . . . 100 Summary . . . . . . . . . 107 The Results of the Model in Terms of the Short-Term Financing Problem . . 109 iv Page VI. SUMMARY AND CONCLUSIONS . . . . . . . . 114 Summary . . . . . . . . . . . . 114 Conclusions . . . . . . . . . . . 116 Suggestions for Further Research . . . . 116 Contributions of the Study . . . . . . 118 BIBLIOG'MPHY O O O I O O O O O O O O O O 119 APPENDICES O O O O O O O O O O O O O O O 125 A. Computer Subroutines for Probability Distributions O O I C O O O O O O O 126 B. Net Cash Flow Simulation Program and Input Data . . . . . . . . . . . . 129 C. Net Cash Flow Output . . . . . . . . . 145 D. Simulation of the Financing Alternatives-— Computer Program and Input Data . . . . . 150 E. Output from the Simulation of the Financing Alternatives . . . . . . . . 166 G. Tables of Data for the Cumulative Probability Distributions of the Financing Alternatives . . . . . . . . 171 LIST OF TABLES Table Page 1. Net cash flow simulation variables . . . . . 71 2. Data for the cumulative probability distri- butions of the financing alternatives at a minimum required cash balance of $3 million and a stockout penalty of 1 percent . 172 3. Data for the cumulative probability distri— butions of the financing alternatives at a minimum required cash balance of $4.5 million and a stockout penalty of 1 percent . . . . . . . . . . . . . 173 4. Data for the cumulative probability distri— butions of the financing alternatives at a minimum required cash balance of $4.5 million and a stockout penalty of 5 percent . . . . . . . . . . . . . 174 5. Data for graphs of the comparative effects of stockout penalties for Type 2 at a minimum required cash balance of $4.5 million . . . . . . . . . . . . . 175 6. Data for graphs of the comparative effects of stockout penalties for Type 4 at a minimum required cash balance of $4.5 million . . . . . . . . . . . . . 176 7. Data for the cumulative probability distri— butions of the financing alternatives at a minimum required cash balance of $6 million and a stockout penalty of 1 percent . 177 8. Data for the cumulative probability distri— butions of the financing alternatives at a minimum required cash balance of $6 million and a stockout penalty of 5 percent . 178 vi Table Page 9. Data for graphs of the comparative effects of stockout penalties for Type 2 at a minimum required cash balance of $6.0 million . . . . . . . . . . . . . 179 10. Data for graphs of the comparative effects of stockout penalties for Type 4 at a minimum required cash balance of $6.0 million . . . . . . . . . . . . . 180 11. Data for the cumulative probability distri— butions of the financing alternatives at a minimum required cash balance of $4-$5 million and a stockout penalty of 1 percent . 181 12. Data for the cumulative probability distri— butions of the financing alternatives at a minimum required cash balance of $4-$5 million and a stockout penalty of 5 percent . 182 13. Data for the cumulative probability distri- butions of the financing alternatives at a minimum required cash balance of $5.5— $6.5 million and a stockout penalty of 1 percent . . . . . . . . . . . . . 183 vii Figure 11. 12. 13. LIST OF FIGURES Diagram of the model . . . . . . . . A general flow chart of the financing alternatives simulation . . . . . . Line of credit . . . . . . . . . . Term loan . . . . . . . . . . . . Commercial paper . . . . . . . . . Accounts receivable loan . . . . . . . General diagram of the net cash flow . . . Net cash flow variables--basic outline . . Net cash flow variables--detai1ed outline . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $3 million and a stockout penalty of 1 percent . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $4.5 million and a stockout penalty of 1 percent . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $4.5 million and a stockout penalty of 5 percent . . . . Comparative cumulative probability distribu- tions for Type 2 at stockout penalties of 0, l and 5 percent with a minimum required cash balance of $4.5 million . . . . . viii Page 45 50 54 56 60 62 70 72 73 89 92 95 96 Figure 14. 15. 16. 16. 18. 19. 20. 21. Comparative cumulative probability distribu- tions for Type 4 at stockout penalties of 0, 1 and 5 percent with a minimum required cash balance of $4.5 million . . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $6 million and a stockout penalty of 1 percent . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $6 million and a stockout penalty of 5 percent . . . . Comparative cumulative probability distribu- tions for Type 2 at stockout penalties of 0, l and 5 percent with a minimum required cash balance of $6 million . . . . . Comparative cumulative probability distribu— tions for Type 4 at stockout penalties of 0, l and 5 percent with a minimum required cash balance of $6 million . . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $4-$5 million and a stockout penalty of 1 percent . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $4-$5 million and a stockout penalty of 5 percent . . . . The cumulative probability distributions for the financing alternatives at a minimum required cash balance of $5.5-$6.5 million and a stockout penalty of 1 percent . . ix Page 97 99 101 102 103 105 106 108 CHAPTER I INTRODUCTION Statement of the Problem Cash management and short-term financial planning have become increasingly complex and important in the past decade. As the cost of borrowing and yields on marketable securities have increased, firms have begun to recognize the advantages of economizing on cash holdings, speeding up cash inflows, controlling cash outflows, and investing excess cash.1 The rapid expansion of business activities has required increasing quantities of working capital. At least some facet of the cash balance problem is faced by every manufacturing firm. Only recently have firms become cognizant of the importance of economizing cash holdings, however. The need for cash stems from a lack of balance between cash inflows and outflows, and the difficulty of accurately predicting some of these flows. An adequate amount of cash must be maintained to perform regular transactions and to meet unexpected cash require- ments. Management must decide on an Optimal minimum cash 1Yair E. Orgler, Cash Management—-Methods and JModels (Belmont, California: Wadsworth Publishing Company, Inc., 1970), p. 25. balance and a policy for handling cash shortages. It is the intent of this study to develop a model that is capable of assisting management in selecting an optimal short-term financing alternative under conditions of risk, when a net cash drain is anticipated. Plan of the Study The model presented in this study is designed to assist management in determining an Optimal short-term financing strategy. The criterion for decision will be the maximization of the utility of the unrestricted ending cash balance1 at the end of a specific planning period. The financing requirement of a firm in a given planning horizon is determined by the initial cash balance and the projected net cash flow. The cash balance is affected by the stochastic nature of the net cash flow for each period. The cash balance may be decreased by a number of methods including investing surplus cash in marketable securities and/or paying off short-term debt. Conversely, in situations demanding more cash, the cash balance may be increased by reducing the firm's holdings of 1The unrestricted ending cash balance is defined as the ending cash balance less that portion of cash that is restricted to the future payments of principal and interest specific to each financing alternative, plus the amount of interest and principal due from invested surplus cash. marketable securities and/or undertaking some short-term financing. The cash demands must be satisfied at a reasonable cost, and a minimal level of cash must be maintained. It will be assumed in this study that management specifies the minimum required cash balance for each period. The financing alternatives will be evaluated considering this minimum required balance. A model will be introduced to evaluate four short- term financing strategies--line of credit, term loan, commercial paper, and accounts receivable loan. Input data to the model will include the following information for each period: (1) the distribution of the interest rates of the alternative financing strategies; (2) the net cash flow distribution; (3) the minimum required cash balance; and (4) constraints on the financing alternatives. The net cash flow data and the data for the cumulative probability distributions of the unrestricted ending cash balances for each financing alternative will be generated by Monte Carlo simulation. The latter cumulative proba- bility distributions will be compared by the condition of stochastic dominancel to determine an optimal alternative. 1See page 36, Chapter V for a discussion of the condition of stochastic dominance. Certainty, Risk and Uncertainty Models currently being utilized can be divided into two basic types: (1) those that are restricted to variables known with certainty--deterministic models; and (2) those that allow probabilistic values for certain variables, thus incorporating risk--stochastic models. Certainty is defined as the condition that exists when all parameters are completely known and the variability associated with these parameters is zero. Thus, each action is known and leads invariably to a Specific outcome. The classical distinction between risk and uncertainty was preposed by F. H. Knight (Risk, Uncertainty and Profit, Houghton-Mifflin Company, 1921).1 Accordingly, risk is defined as the situation in which the outcomes are not completely known but objective probabilities can be associated with possible outcomes. Uncertainty refers to situations for which probabilistic results cannot even be predicted in probabilistic terms. The distinction between risk and uncertainty in an organizational context cannot be so clearly delineated, however. Decision-makers generally have some feelings about the probabilities of future events. These feelings will affect judgments made by management.2 lAlfred Rappaport, "Sensitivity Analysis in Decision- Making," The Accounting Review, LXII (July, 1967), 441. 21bid. Thus, although these terms have been used interchangeably in the literature, the term risk will be used in this study. Organization of Remainder ofIthe Thesis The cash management problem has been established as significant and worthy of further study. A model has been outlined in this chapter which represents an approach to aid management in solving some facets of this problem. Chapter II presents and discusses some models in the literature related to the cash management problem. The deterministic and stochastic approaches to solving cash management problems are delineated. The models are discussed in terms of their relevance to the development of problem-solving techniques and to the model in this study. Chapter III is devoted to the development of the model in decision-theory framework. The single-stage decision problem under certainty is extended to include several stages and incorporate risk. The model in this study is presented as a multi-stage decision problem, and the assumptions of the model are outlined. Finally, the simulation approach to the model is discussed. Chapter IV develops the model itself. The financing alternatives to be evaluated are discussed in detail, and the method of computer analysis presented. The input data to the model are defined. Chapter V consists of the presentation and evalua- tion of the results. The inputs to the model and the model output are presented. The cumulative probability distri- butions for each financing alternative at the specified minimum required cash balances, are graphed. The distribu- tions are then compared using the condition of stochastic dominance. An optimal financing alternative is then selected for each minimum required cash balance. A final chapter consists of the summary and conclusions. CHAPTER II SURVEY OF THE RELATED LITERATURE-- METHODS AND MODELS Introduction In recent years mathematical programming techniques and computers have been applied to the solution of various problems in finance. In this chapter some general models discussed in the literature will be reviewed. The primary intent of this discussion is to indicate references that relate to cash management problems. Several capital investment decision-making models will also be presented since the approaches may be applicable to short-term financing problems. Two basic approaches are used to solve cash manage- ment problems—-the deterministic approach in which risk is not considered, and the stochastic approach which incorpo- rates risk. The deterministic method is based upon the assumption that the values of the variables and parameters in any specific problem are known with certainty. The objective of these models is to maximize or minimize some deterministically defined function. The assumption of complete information, essential to these models, has been thought to impair their validity, since variability of conditions is an observable fact. PrOponents of the approach contend, however, that no existing method is capable of adequately incorporating this variability, and, in the attempt to do so, often introduce more error than they resolve. The stochastic approach is becoming more widely accepted; h0wever, as methods are develOped which more closely approximate real world conditions. The stochastic approach allows some of the variables and parameters in any specific problem to be defined as random. The random values are described by some proba- bility distribution. The degree to which risk is incorporated varies with the specific model under considera- tion. The stochastic approach allows management to make decisions with regard to some of the variable input data. Their personal experience may be invaluable in enabling the model to reflect the specific conditions of the firm under study. The development of the deterministic and stochastic models which relate to the cash management problem in this study will be described in this chapter. Deterministicggpproaches Inventory Models.--The cash management problem is very similar to the problem Of inventory management.1 Baumol2 formulated one Of the first models to solve cash balance problems by utilizing a deterministic approach incorporating the classical economic lot size model Of inventory management. His objective was to minimize the total cost Of the borrowing transactions over some planning horizon. The model determined how much cash, C*, is obtained, how often this cash is to be obtained, L*, and the average cash balance, 5*, as follows: 1!: [me 2b Trans fer C . L ' . a: = With an average cash balance of. C 21' where: b fixed cost for borrowing or withdrawal 1 Opportunity interest rate per day m constant dollar expenditure per day L = number Of days in a period Graphically, the model can be expressed as follows: 1Edwin J. Elton, and Martin J. Gruber, "Dynamic Programming Applications in Finance," The Journal Of Finance, XXVI, No. 2 (May, 1971), 499. 2William J. Baumol, "The Transactions Demand for Cash: An Inventory Theoretic Approach,“ Quarterly Journal foEconomics, LXVI (November, 1952), 545—556. 10 4—L*—> The Baumol model held that the cost Of borrowing or with- drawal, the Opportunity interest rate,and disbursements over time were constant and known. Only the transactions demand for cash was actually considered. Despite these recognized limitations the model provided a definite contribution to the literature and was a basis for con- siderable further work in this area. Tobinl develOped a model similar to Baumol's. Tobin's model determined the Optimal average cash and bond holdings to maximize interest earnings net Of transactions costs. Conversely, as has been shown, Baumol was interested in minimizing the total cost Of the transactions irrespective Of the financial efficacy of investing in bonds as Opposed to holding cash. Linear Programming.--An article by Charnes, COOper and Miller2 presented the first application Of linear lJames Tobin, "The Interest-Elasticity of Trans- actions Demand for Cash," The Review of Economics and §tatistics, XXXVIII, NO. 3 (August, 1956),”241-247. 2A. Charnes, W. W. COOper, and M. H. Miller, "Application Of Linear Programming to Financial Budgeting and Costing Of Funds," The Journal Of Business, XXXII, NO. 1 (January, 1959), 20—46. 11 programming tO finance. Their approach considered the Optimal allocation Of funds within the firm which included the Operating environment and the existing organization Of financial and physical resources. The Objective was the maximization Of profits, subject to capacity, buying and selling constraints, and a liquidity requirement. Charnes, Cooper and Miller extended the basic approach to include lending and borrowing. The use of linear programming was a significant contribution to the solution Of finance problems. The technique presented in this article found later application in solving short-term cash management problems even though their Specific model could not be used in this regard. An extension Of the model by Charnes 23.31} was developed by Ijiri, Levy, and Lyon.l It covers marketable securities transactions and is limited to a single-period approach. The model was further amplified in later work by Charnes, COOper, and Ijiri.2 Modifications and extensions Of the models using linear programming led to eXperimentation in its application. Although the specific emphases Of the models may have differed, the basic techniques, which led lY. Ijiri, F. K. Levy, and R. C. Lyon, "A Linear Programming Model for Budgeting and Financial Planning," Journal Of AccountingyResearch, I, NO. 2 (Autumn, 1963),, 198-212. 2A. Charnes, W. W. COOper and Y. Ijiri, "Break- Even Budgeting and Programming to Goals," Journal Of Agcounting_Research, I, NO. 1 (Spring, 1963), 16-43. 12 to the develOpment Of increasingly comprehensive approaches, were established. Robichek, Teichroew and Jones1 developed a multiple- period linear programming model for Optimizing short-term financing decisions. Their discussion was concerned with the decision Of how much and when money should be acquired from specific sources Of short—term funds to minimize cost to the firm. The model made the following assumptions: (1) the minimum cash balance required at all times was Specified; (2) the cash inflows and/or outflows were known with certainty; and (3) the costs and limitations Of all Of the financial alternatives were known. With these assumptions and given cash balance requirements, the model was developed. The Robichek, Teichroew and Jones (RTJ) model is particularly significant with regard tO the model in this study. The basic approaches are similar except that the RTJ model utilizes linear programming and thus solves the problem of selecting an Optimal financing alternative, simultaneously. The model in this study is probabilistic and considers the selection Of an Optimal financing alternative, sequentially. Similarities Of the two models include the consideration Of multiple periods, the —¥ 1A. A. Robichek, D. Teichroew, and J. M. Jones, "Optimal Short-Term Financing Decision," Management Science, XII, NO. 1 (September, 1965), 1-36. 13 assumption Of known constraints on the financing alterna- tives, and the certainty with which the minimum cash balance is known. Orgler §t_§l.l develOped a linear programming model differing only slightly from the RTJ approach. D. E. Peterson2 presented a model, also utilizing linear pro- gramming, but which was more comprehensive in that it incorporated the following sources Of short-term financing: accounts payable, line Of credit, accounts receivable and sale Of commercial paper.3 The discussion Of selecting an Optimal financing method from specific alternatives in this article assumed that the needs Of a firm were known with complete certainty. The certainty constraints necessary in any linear programming approach are its major short-coming. Linear programming provides a useful and valid tool for problem-solving as long as its limitations are recognized. 1Orgler, pp. 70-118. 2D. E. Peterson, A Quantitative Framework for Financial Management (Homewood, Illinois: Richard D. Irwin, Inc., 1969), Chapter 7. 3The model in this study, although stochastic in nature, Offers four similar financing alternatives. 14 Dynamic Programming.--Maol presented an approach for managing the cash balance during a finite planning horizon through the use Of a two-step systematic search procedure using dynamic programming. The cash balance decision was viewed as a problem involving a sequence Of interrelated decisions. These decisions included whether to retain cash holdings in the form Of securities or in the form of cash, and which quantitites Of securities should be held or sold by the firm. The dynamic programming approach is one of the valid methods for solving cash management problems sequentially. The importance Of the inter-temporal relationships lends credibility to dynamic programming (or recursive Optimization) as a technique of solving these problems. This approach will be discussed more fully in the section on probabilistic dynamic programming. Stochastic Approaches Many Of the techniques utilized in deterministic models have been extended to allow them to incorporate risk. A gradual refinement Of the models has taken place, as computer techniques have been increasingly applied and accepted for use in solving business problems. Computers have made feasible the solution to heretofore complex 1James C. T. Mao, Quantitative Analysis of Financial Decisions (Toronto: The Macmillan Company, Collier- MacmiIIan Canada, Ltd., 1969), pp. 505-514. 15 mathematical problems, particularly problems involved in the probabilistic dynamic programming and simulation approaches. Various approaches are discussed in the following section which assisted the author in selecting the components Of the model presented in this study. InventoryyMOdels.--Miller and Orr1 extended the Baumol model to allow the net cash flows to be stochastic. The model was based on the assumption that the cash balance varies irregularly and unpredictably over time. The random behavior of the cash flows was generated in a sequence Of independent Bernoulli trials. The model attempted to minimize the eXpected cost per day Of managing the firm's cash balance over a finite planning horizon. The cash balance was allowed to fluctuate freely within predetermined maximum, h, and minimum, 0, values. When either Of these values was reached, a transfer Of funds or liquidation of assets was effected to achieve an "Optimal" level, 2. A fixed transfer cost independent of the amount Of funds transferred was assumed. Optimal values Of z and h were found as follows: 1Merton H. Miller and Daniel Orr, "A Model of the Demand for Money by Firms,“ Quarterly Journal of Economics, LXXX (August, 1966), 413—435. 16 3ym2t 4v h* = 32* where: y cost per transfer 8 II constant expenditure in dollars per day <3 II daily rate Of interest earned on the portfolio t = number Of Operating cash transactions per day The model is expressed graphically as follows: Cash (S) h z . A , 0 %Time Linear Programming.—-Stochastic linear programming models in which either coefficients Of the Objective function or technical coefficients Of the constraints are treated as random variables have been employed under conditions Of risk. Since the origination Of this linear programming 17 application with the works Of Tintnerl and Babbar,2 there have been numerous theoretical extensions. Frequently the linear stochastic model is transformed into a consistent non-linear programming model which is solved by non-linear techniques resulting in an Optimal feasible solution to the original model. Application Of stochastic linear program- ming has been limited because Of the computational burdens involved.3 Chance-Constrained Programming.--Naslund4 incorpo- rated risk into his model through the use Of chance- constrained programming. The model involved allocating a limited fund between several investment projects (classi- fied as independent) over a finite horizon. Borrowing and lending were added as decision variables. Risk was intro- duced into the model through probabilistic constraints, one for each year, reflecting the fraction Of the time that the firm's original financial plan was allowed to be violated. 1G. Tintner, "Stochastic Linear Programming with Applications to Agricultural Economics," Second Symposium in Linear Programming, Washington (Washington, D.C.: NationaI Bureau Of Standards, 1955). 2M. M. Babbar, "Distributions of Solution of a Set Of Linear Equations with Applications to Linear Programming," Journal Of the Statistical Association, L (1955), 155-164. 3Rappaport, p. 443. 4Bertil Naslund, "A Model of Capital.Budgeting Under Risk," The Journal of Business, XXXIX, NO. 2 (April, 1966), 257—271. 18 The decision-maker had tO be able and willing to specify the probability level at which the constraints must be met. These probabilistic constraints were transformed into deterministic equivalents by inverting the probability con- straints and replacing the technical coefficients by their means and variances. The variables were assumed to be normally distributed. The Objective function Of Naslund's model was the maximization Of the expected value Of the firm at the end of the planning period. Chance-constrained programming is significant as a conceptually sound approach Of considering risk. However, the constraints introduced by defining all probability distributions as normal and the inability to handle multi- period, interrelated data sequentially discount its application to the model in this study.1 Dynamic Programming.--Eppen and Fama2 coauthored three related articles concerned with the cash balance problem. Initially, a linear-programming model was developed to indicate the properties Of Optimal Operating policies for stochastic cash-balance and simple dynamic portfolio problems. The Objective Of the model was to L l . . . For a further discu331on Of chance-constrained programming see A. Charnes and W. W. Cooper, "Chance- Constrained Programming," Management Science, October, 1959. 2Gary D. Eppen, and Eugene F. Fama, "Solutions for Cash-Balance and Simple Dynamic-Portfolio Problems," Journal 9§_Business, XLI, NO. 1 (January, 1968), 94-112. 19 minimize the discounted expected costs over some infinite horizon. A later article1 included a special adaptation Of this model using a dynamic programming approach for establishing general properties Of cash balance problems. In their latest work, Eppen and Fama2 discussed the determination Of an Optimal Operating policy for a three asset stochastic cash balance problem. They retained the dynamic programming approach used in the above mentioned models, however. In contrast to these models, which con- sidered holding cash in one Of two forms--cash or earning assets, the model prOposed in this article divided the non-cash holdings into two earning assets categories-- "stocks" and "bonds." "Stocks" assets comprised the major portion Of the firm's earnings, and were assumed to have higher expected returns per period than "bonds." "Stocks" also had higher transactions costs. Thus, it cOuld pay the firm to hold some "bonds" as protection against fluctuations in the cash account. The article discussed this last point in some detail. 1Gary D. Eppen, and Eugene F. Fama, "Cash Balance and Simple Dynamic Portfolio Problems with PrOportional Costs," International Economic Review, X, NO. 2 (June, 1969), 2Gary D. Eppen, and Eugene F. Fama, "Three Asset Cash Balance and Dynamic Portfolio Problems," Management Science, XVII, NO. 5 (January, 1971), 311-319. 20 Eppen and Fama presented a relatively sophisticated and realistic general approach to the cash management problem through utilization of the inventory method. Their approach is a recent, advanced and comprehensive presenta- tion of the recursive relationships involved in the probabilistic dynamic programming solution to the cash management problem. The relationships were "solved" to develop a set of rules or policies specific to the conditions which exist in any particular situation. The use of probabilistic dynamic programming in the above articles by Eppen and Fama led the author to explore this approach for possible use in his study. Their approach was theoretical, however. Practical application of probabilistic dynamic programming to the multi-stage, sequential cash management problem which incorporates risk will be possible only when a practical computer approach for solving dynamic programming problems is developed. Simulation was subsequently selected. Several models which utilize simulation will be presented in the following section. Simulation.--Donaldsonl developed a probabilistic approach to the evaluation of corporate debt capacity. The purpose involved determining the probability that interest _‘ lGordon Donaldson, Corporate Debt Capacity (Boston: Harvard Business School, 196I), Chapter 7 and Appendix B. 21 on a given amount of debt could force a company into cash insolvency should a severe recession occur. The approach incorporated some random variables and some constants to determine the probability of an ending cash balance Of less than zero. Certain cash outflows were deemed mandatory. The level of sales and average collection period on receivables were assumed to be statistically independent with a known combined probability distribution. Utilizing the model, management could ascertain the probability distribution of its ending cash balance during a recession. Maol adOpted Donaldson's model and utilized it in a computer simulation eXperiment. By a simulation of the cash flows, the probability of cash insolvency was determined. The method involved the use of probability distributions of sales, resulting in the generation of a random sales level. A collection period was then deter— mined for that level Of sales and the ending cash balance was calculated. As a result of the simulation, a probability distribution of the ending cash balance was produced and the risk of insolvency determined. Mao also develOped a model to determine the pattern of financing to minimize the net interest cost in view Of an anticipated net cash drain in the subsequent twelve month period. Mao introduced uncertainty into the model by randomizing the sales variable. All other variables E 1Mao, pp. 566-577. 22 were assumed to be known with certainty. Three alternative financing strategies were studied in the stated time horizon resulting in a frequency distribution Of the net interest cost for each financing strategy. The probability distributions, when compared, would assist management in deciding upon the preferred financing strategy for their individual firm. The basic model presented by Mac was considered in the preparation Of the model in this study. The author elected to allow more of the variables to be defined as probabilistic. These variables were selected to approxi- mate real world conditions. Further, whereas Mao did not elaborate on a recommended method for comparing the financing alternatives, this author suggests such a technique. The Objective functions of the models also differ, but the basic approach remains similar. Problems resulting from the significant degree of risk in the economic environment and the increasing complexity involved in financial policy decisions demand SOphisticated methods of analysis. Simulation, as a technique of incorporating risk, has become increasingly appropriate. CHAPTER III A DECISION-THEORY STATEMENT OF THE PROBLEM Objective The Objective of this study is to solve the multi- stage decision problem of determining an optimal short- term financing strategy under conditions of risk by Optimizing the decision-maker's utility function. The decision problem is presented in this study in terms of a multi-stage decision model (DM). Initially, in order to explain and clarify the relationships involved in the multi-stage model, a single-stage deterministic model will be presented.1 It will be subsequently amplified to include several stages and incorporate risk. Single-Stage Decision Problem (DP) Under Certainty Any situation where a decision is needed can be considered as a decision problem. This problem can be stated in terms of a decision model (DM). The model is composed Of a decision structure and a search problem (SP). 1The method of Carr and Howe [Charles R. Carr, and Charles W. Howe, Quantitative Decision Procedures in Manage- ment and Economics (New York: McGraw-Hill Book Company, 1964] was utiIized in this discussion (pp. 38-40). 23 24 The decision structure1 of this model can be expressed schematically as follows: where: C NCF (Ct,NCFt,r ti .(.. T(Ct,NCF d t'rti' ti) t+1 ll [—4 ti) t d V II U(C — U[T(Ct,NCF t+1 t'rti' ti’l d W(Ct,NCF ) t'rti' ti set of input variables to stage t describing the relevant aspects of the decision-maker's environment prior to making a decision at point t.2 For example: cash balance at the beginning Of stage t and interest on invested surplus cash that will be received at the beginning of stage t. the financing alternative under consideration. net cash flow in stage t, i.e., for the period Of time from time point t to time point t+1 at which the next decision is to be made. 1 Although t, defining the number of periods, is actually superfluous in the single-stage model, the author elected to include it to familiarize the reader with the notation to be used in the subsequent multi-stage model descriptions. 2 For the single-stage model t=l. It should be noted that stage t is a period Of time and that the decision is to be made at the beginning of stage t, i.e., at point t. 25 rti = interest rate per appropriate length of time under alternative i for stage t. dti = the amount of the minimum required cash balance Specified at the beginning Of stage t for the end of stage t. T = transformation function U = utility function W = payoff function Decision Structure.--The decision structure is composed of a state space, 9 an action space, P a t' t' transformation function, T, a utility function, U, and a payoff function, W, where: State space Qt-= {wt : (Ct' rti' NCFt)} Under certainty, Q is a single-element set containing the non-controllable values involved in the particular decision situation. Action space Ptl = {d is R Pt represents the set Of all feasible actions dtieth (specifying minimum required cash balance values) and the particular decision situation. The action space states the feasible actions for the particular decision situation. As the stage progresses, dti will influence the amount the firm borrows or reinvests. 1Although it is possible that the action space, P could be different for each financing alternative, 1, inferring that Pti should be used, Pt will be used to simplify this discussion. t! 2R is defined as the set of real numbers. 26 Transformation function T T: (Qt X Pt) + Ct expresses the consequences of the non- -controllable value wte.Qt and the decision dti€:Pt. T(wt,dt = Ct+1€'Ct+1 where Ct+1 is the set of all possible conse- quences Of the interaction of the state space at and the action space Pt, and Ct+1 is a single consequence. The transformation function, T(Ct, NCFt, rti rdti) = Ct+lr transforms the inputs to stage t and the l decision in stage t into the output state. Utility function U U: Ct+1 + R is a mapping such that (U(Ct+1) is the utility Of a consequence Of ct+1e:Ct+1 to the decision-maker. Alternatively, U(Ct+1) E U[T(Ct, NCFt, rtir dti)] measures the utility of the transformation Of the input state to the output state. Payoff function W W: (Qt X Pt) + R eXpresses the result of the interaction of the state space and the action space in terms of a payoff to the decision- maker. The payoff function, W(Ct, NCFt, rti, ), is defined in terms of utility as %llows: W: (Qt X Pt) + R, such that U(Ct+1) :U[T(Ct, NCFt, rti, dti)] E W(Ct, NCFt, rti' dti)° SP.-- The search problem is to Optimize the utility function, U, over all of the decisions, d in the action ti' Space, Pt' Thus: For a given i, i=l,...,4 p Eax U[T(Ct, N Ft, rti’ dti)]. ti 1The output state becomes the input state for the subsequent period t+l. 27 The optimal1 decision for each i is denoted by d*ti and the tr ti’ d*ti)]’ Then the t’ rti' d*ti)] for each i are compared so as to: Optimum utility as U[T(Ct, NCF r U[T(Ct, NCF max U[T(Ct, NCF i d* .)]. rti' t1 t! Result.--The SP solution gives the following results: 1. The Optimal decision, d*ti*' with respect to the utility function, U. t+1) : d*ti*)]' of the utility 2. The optimal value, U(C U[T(Ct, NCF r t' ti' function with respect to all possible actions in the action space, Pt' Model Of the Single-Stage PrObIem Under Risk The single-stage deterministic model will now be modified to incorporate risk. The net cash flow and the interest rates will be assumed to be unknown prior to making the decision in stage t, and defined as random (stochastic) variables. The model of this decision structure is represented sohematicalLy in the following diagram: lAsterisks denote Optimal values. S Q T(C1'NCFl’rli’dli) = C2 U(CZ) ' U[T(Cl, NCFl, rli' (111)] d .) NCF 11' 11 W(C r 1' 1' Decision Structure.--The state space is modified to indicate that the net cash flow, and the interest rates are unknown. The action Space remains the same SE,--The SP is now concerned with the expected value Of the utility function. Mathematically, these modifications can be incorporated to result in the following relationships: For a given 1, i=l,...,4 Eax E[U[T(Cl, NCFl, rli' dli)]|1]. 1i The Optimal decision for each i is denoted by d*1i and the Optimal expected utility as E[U[T(C1, NCFl, rli’ d*li)]|i]. T * ' hen the E[U[T(Cl, NCFl, rli' d li)]|1] for each i are Conmared so as to max E[U[T(Cl, NCFl, rli’ d*1i)]li]. 1 .Result.-7The result is the Optimal decision, d*li*’ eund the Optimum expected utility, E[U[T(C1p NCFl: rli' d*li*)]]' for that decision structure. 29 Multi-Sta e Decision Problem Unaer Risg The model is now expanded to include more than one stage. The resulting model will incorporate random (stochastic) variables and multiple stages. The decision structure is represented schematically for two stages as follows: NCFI '11 ‘11 "CF: ‘21 d21 4‘ § § 4 6 L Ci‘ “1 t_1 r(ci'mpi'ru'du’ ' c2 t_2 "cz'uc’2"2i"21’ ' c3; . . U(cz) — "(7(61'NC’1"11'd11’1 U(C3) a UlT(C2.NCPz.r2i.dzi)l 1 W(c1,Ncr1.rli.d1i) - "(Cz'NC52"zi'd21) t {C }; C = multiple-element set containing the nos-controllable values indicating the state of the system for stage t. at is known immediately prior to making the decision in stage t. where: 9 C1 = the input state to stage one,where Cl 5 91 C2 = the output from stage one which serves as the input state to stage two, where C2 2 02 C3 = the output from stage two that will serve as the input state to stage three, where C3 5 03. Decision Structure.--The state space is expanded to include the variables in stage two as well as those in stage 30 one. The action space is similarly expanded to include both stages. The utility function also incorporates both stages. The payoff function for each stage is related to only the variables within the stage to which it applies. SP,--The SP is to maximize the expected utility for the final stage in the multi-stage problem. The utility of the final stage is expressed as U(C3) E U[T(C2, NCFZ, r , d Since the intermediate value C2 = 21 21)]° T(C1, NCFl, r dli)’ the latter expression can be entered li' into the preceding expression in place of C2: U(C3) 2 U[T[T(Cl, NCF , dli), NCFZ, r 1' r11 21' 21]]- In general, T* may be used to denote a composite of multiple transformation functions. In the preceding two- stage problem, this composite transformation function is described as follows: 'I: T [C1, NCFl, rli’ dli' NCF2, r21, d2i] T[T(Cl, NCFl, rli' dli)' NCFZ, rZi’ dzi]. For a given i, i=1,...,4 max E[U[T*(C NCFl, r i'dZi a 1' 11' dli' NCFz' r21' d21)1|1]' 1 31 Then: max E[U[T*(Cl, NCF 1 'k * ' 1' rli' dli’ NCFz' rZi’ dzi’llll' Result.--The result is the Optimal set Of decisions, d*i*' where d*i* = (d*li*’ d*2i*)’ and the Optimum expected utility of the decision structure, * * * E[U[T (cl, NCFl, rli, d 11*, NCFZ, r21, d 21*)]]° The N-Stage Decision Problem The preceding description of the two-stage problem may be eXpanded to include any N number of stages: Let NCF = (NCF1,...,NCFN) t Y o RN _ z N di — (ali,...,dNi) 6 P1 x P2 x...x PN _ P c R _ . N ri -' (rli'ooo'rNi) 8 ZQ" R The transformation function can also be restated as: T: (01 X Y X Z X P) + Q where N+l T(Cl' NCF, ri, di) = C for all N+1 E Qn+1 (C1, NCF, ri, di) E (91 X Y X Z X P) The solution to the N-stage problem is the set Of decisions, d*i*, where d*i* = (d*li*""'d*Ni*)’ and the Optimum eXpected utility Of the decision structure, 32 E[U[T*(Cl, NCF, ri, d*i*)]]' where T* is the composite of the N individual transformation functions. The Decision Problem to be AnaIyzed in This Study The objective of the model in this study is to evaluate short-term financing under conditions of risk. The criterion for decision is the expected utility of the unrestricted ending cash balance Of the firm at the end of the final stage in the planning period.1 It is assumed that the minimum required cash balance has been specified. Given this required minimum, the model determines that financing alternative which maximizes the expected utility. Hence, the decision for this study is a subset of the total decision problem Of determining the Optimal minimum required balance level and Optimal financing alternative. This total multi- stage decision problem is discussed in the following section. Subsequently, the decision problem in this study will be presented in terms of that total problem. The multi-stage probabilistic decision structure is represented schematically as follows: 1The unrestricted ending cash balance is defined as the ending cash balance less that portion of cash that is restricted to the future payments Of principal and interest specific to each financing alternative, plus the amount Of interest and principal due from invested surplus cash. 33 r(c1, ncrl, rli, 61) a c2 T(CN_1, NCFN_1, rN_1i 6N_1) . cN NCFI 1:11 61 ! NCFN rNi 6 N 1 J. i ' , i '. l C c Q ' C 1 l ' J- J N+1 6 t ‘ W(Cl' “cpl' ‘11: 51) w(cN, NCFN, rNi, 5N) where the following definitions hold for t=l,...,N: Ct = {Ctli’ Ct2i' Ct3i' Ct4i} Ctli = cash balance at the beginning of stage t under financing alternative i. CtZi = amount Of interest and principal payable at the beginning Of stage t under financing alternative 1. Ct3i = amount of interest and principal receivable from invested surplus cash at the beginning Of stage t under financing alternative i. Ct4i = maximum amount available for borrowing during stage t under financing alternative i. NCFt = net cash flow random variable for stage t. rti = interest rate random variable for stage t under financing alternative 1. 6ti = a strategy function such that: ) E d..e:P 0' NCFl'ooo'd t]. 6ti(cl' d1; t-li’ NCFt-l \ \f’ _/ h t ti 34 This 6 i is a strategy function for etermining the value of the minimum required cash balance in stage t for financing alternative 1. Thus, all essential prior history of the process, hti: is known immediately before the decision is made. T = transformation function (2'. II utility function W = payoff function and let: NCF E (NCF1,...,NCFN) ri E (rli,...,rN) 61 E {611""'6Ni} where 61(hi,..., ht""' hN) E (dli""’dti""'dNi) 5 di and, ht 2 (c1, dli' NCF1,...,dt_li, NCFt_1) Decision Structure.-- State space at = {Ct} Action space1 A = {Si} Utility function U, where U(C ) E N+l N’ rNi' 6Ni” 5 U[T*(Cl, NCF, ri, 61)]. U[T(CN, NCF ear- For a given i, i=1,...,4 )Ii] = E[U(T*(Cl, NCF, ri, aillli]. mix E[U(CN+1 i 1The action Space is now a set Of strategies, A, as Opposed to the earlier definition as a set of actions, P. 35 The Optimal strategy for a given alternative i is denoted by 6*i. Then: max {max {E[U(CN+1)Ii]}} 1 6i = max E[U[T*(C NCF r. 6*.)]|i]. i 1' ' 1' 1 Result.--The result is the Optimal strategy function, 5*i*' Where 6*i*(hl’ooo,h hN) E d*i*l and d*i* = t’...’ , and the Optimum expected utility of the 5*1.)11. (d ,ooo'd* * . . ) 11* N1* decision structure, E[U[T*(Cl, NCF, ri, The Model in This Study as An Analysis Of‘a SuBSet, Ag, of the Strategies g1 In A In this study, certain minimum required cash balance strategy functions, 6?, were evaluated. These 63 are elements of a set of minimum required cash balance strategy functions, Ag, where A? = {63}1 and A9 is a subset Of A. The evaluation procedure utilized in this study consisted of evaluating the financing alternatives for a set Of strategy functions which resulted from the selected set of minimum required cash balance values.2 1The value Of a strategy function is a set of minimum required cash balances. For example: {($3,ooo,000),...,($3,ooo,000)}. This indicates that the minimum required cash balance is $3,000,000 for each period. 2See page 83, Chapter V for a discussion of the minimum required cash balances selected for evaluation and presentation in this study. 36 The output state of the final stage was determined for each financing alternative. Then the alternatives were compared and an Optimal alternative selected. This process was repeated for each set of minimum required cash balances considered in the study. In notational form the following procedure was followed for each set of minimum required cash balances: For each i, i=1,...,4, determine E[U(C [Ag]. n+1) Then: g mix E[U(CN+1)IAi]. Result.--The result is the Optimal financing alternative 1* and the Optimum utility E[U(CN+1)|AE*]. The results may be subOptimal since: E[U(C IA?*] 1 E[U(CN-t-lHAiW] i the expected value that N+l) would result from an optimal combination of financing alternatives and minimum required cash balance strategies the expected value that would result if the firm were allowed to utilize combinations Of financing alternatives or change alternatives from period to period. Assumptions l. The beginning balances of Ctli' CtZi’ Ct3i’ C ., are known with certainty by management. t41 2. Management specifies the minimum amount of cash that the firm desires to hold in each stage. [A 37 Borrowing is effected at a given annual rate of interest, different for the different types of borrowing. The interest rates are probabilistically defined, and management will decide the type of distribution which best describes the interest rates for each stage. Borrowing constraints will be impOsed, Specific to the type borrowing. Borrowing shall be accomplished as needed during the stage in which the need arises, the amounts of which will follow from the specification of the minimum required cash balance. Funds acquired through borrowing are repayable, including interest owed, at the beginning of the stage due unless otherwise stated. All surplus cash will be invested in an earning asset at the beginning of the stage available. Management will project the type of distribution for the interest rate on reinvested cash. All invested surplus cash will be available, plus interest, at the beginning of the stage following the stage in which it was invested. 38 10. The model considers the same financing alternative for all thirteen periods. The alternatives are handled separately and no combinations are allowed. The periods are examined sequentially. ProbabiliStic Dynamic Programming; as a Possible Approach to Solving; the General Stochastic Multi- Stage ModeI The general model, presented in the preceding section, is defined as stochastic in nature in that it contains variables that are described as random. These variables are expressed in the form Of probability distributions. The effect of these random variables makes the solution to a multi-stage problem appreciably more difficult when the variables are considered simultaneously. Thus, methods which allow for sequential solution are preferable to those that do not. Two methods have been generally discussed in the literature for use in solving this type Of problem--probabilistic dynamic programming and simulation. Probabilistic dynamic programming, an Optimization technique, is worthy of discussion even though, without a practical computer approach for solving complex dynamic programming problems, it is not suitable for routine use. —__ 1George L. Nemhauser, Introduction to Dynamic Pro rammin (New York: John WiIey and SOns, Inc., 1966), pp. - o 39 The stochastic, multi-stage model consists Of an N-stage stochastic system with random variables that affect the utility function and transformation function at each stage. The following definitions are made: h t input state tO stage t (history Of the process to stage t) 6ti = decision strategies in stage t NCFt = net cash flow } random variables in stage t r . = interest rates t1 . Ut = Ut(ht' NCFt, rti’ dti) - ut111ty function for stage t ht-l = Tt(ht, NCFt, rti' Gti) - transformation function for stage t The subscript t refers to that stage where there are t stages left in the process. NCF and r are assumed to t ti be independently distributed with probability distributions Pt(NCFt) and Pt(rti)‘ Probabilistic dynamic programming also assumes that the total utility (RN) for the N stage problem is equal to the sum of the utility of each stage. RN(CN'OOO’C1' NCFN'ooopNCFJ-p rNi’oooprli' 6Ni'ooo'61i) N = Z- Ut(ct' NCFt, rti' Gti) t-l ft(Ct) = the maximum expected utility as a function of the input state Ct 4O ft(ct) = max 2 z Pt(NCFt) - Pt(rti)[Ut(Ct, NCFt, rti, sti) + ft_1(Tt(Ct, NCFt, rti' dti))], where t=2,...,N and fl(Cl) = U1(C1, NCFl, rli’ 611), where t=l. Probabilistic dynamic programming presents a feasible approach to the stochastic multi-stage problem. Some probabilistically defined variables in this study exhibit interdependence between stages, however. This interdependence does not necessarily discount solution of the model by probabilistic dynamic programming. Rather, the interdependence of the net cash flows between periods necessitates the redefinition of the probability distribu- tions, Pt(NCFt), as conditional probabilities, denotes the net cash flows Pt(NCF | NCFt_l), where NCF t t-l prior to period t. Since additional state variables must be added to account for the interdependence, the problem becomes more difficult and time-consuming to solve.1 Similarly, the financing alternatives affect more than one stage and the additional state variables necessary again complicate solution. Until a practical computer approach for solving 1Each additional state variable expands the recursive relationships and causes the problem to become more complex. 41 large-scale probabilistic dynamic programming models is developed, the problems remain computationally complex . l and expens1ve. Simulation Simulation provides another method for solving the multi-stage probabilistic problem. Simulation is defined as ". . . a numerical technique for conducting experiments with certain types of mathematical models which describe the behavior Of a complex system on a digital computer over extended periods of time."2 AS in probabilistic dynamic programming, simulation allows the multi-stage stochastic process to reveal itself sequentially. A solution may be achieved by maximizing the final stage utility function. Since the net cash flow of the model in question involves random variables as well as interdependence of those variables (lag structures), analytical solutions are difficult. Naylor notes that even though an analytical solution may exist for a particular model, it may be less costly in terms of analyst and computer time to run a simulation. That is, the additional information gained from an analytical solution may not be sufficient to justify 1Orgler, p. 41. 2Thomas H. Naylor, Computer Simulation Experiments with Models of Economic Systems (New York: John Wiley and Sons, Inc., 1971), p. 2. . 42 the search time involved.1 In this particular model, the net cash flow determination provided a major point Of discussion. The author explored various techniques and elected to run a simulation. The simulation inputs were Obtained from real-world data as discussed in Chapter IV. 1Ibid., p. 5. CHAPTER IV MODEL DESCRIPTION Introduction The model to be presented in this study is designed 'to assist management in the evaluation of alternative short-term financing strategies. The criterion for 1anning horizon2 is determined by the initial cash balance Ennd the projected net cash flows. Given a need to effect Iiinancing, four alternative strategies are evaluated by ‘tflhe model, and an Optimal strategy selected. Input to tflhe model for each period consists of the constraints on tile financing alternatives, the distribution of the iJIterest rate for each financing alternative, the net cash 1See p. 32, Chapter III for a definition Of the L‘lnrestricted ending cash balance. 2The planning horizon in this study will consist of thirteen periods (or stages) with each period being four weeks in length. 43 44 flow distribution, and the minimum cash balance specified by management. The effect on the cash balance for each period specific to the individual financing alternative is calculated utilizing a simulation program. The cash balance is affected, for example, by any principal and interest payments made in the period, by the investment of surplus cash in marketable securities or by the con- version of marketable securities into cash. At the end of the thirteenth period the unrestricted ending cash balance for each financing alternative is determined. The unrestricted ending cash balance must be calculated for many different values of the variables associated with I each financing alternative because of the probabilistic nature Of the cash flows and interest rates. Ultimately, two-hundred values of the unrestricted ending cash balances for each financing alternative are generated. These two- hundred values are punched on cards by the computer. The cards are then sorted and a cumulative probability distribution drawn for each alternative. The four result- ing distributions are then compared through the use Of stochastic dominance, and an Optimal short-term financing alternative selected. A general diagram of the model can be seen in Figure l. The following definitions correspond to the diagram: 45 NBTCF 1 r11 RBQHIN1 NETCF13 r131 REQMIN13 i 1 Jr 4 J. .L .L t (2 C C. 1 1 .—____._. o o o o o o __.___._13. 13 H" (Simulate Two-Hundred Times) Determine Unrestricted I Ending Cash l Balance l I I (Simulation Complete) l y’ Two-Hundred Unrestricted Ending Cash Balance Figures for Each Financing Alternative Create Cumulative Distribution;— from Card Output for Each Financ1ng Alternative Evaluate Distributions and Selectl Optimal Financing Alternative ' Figure 1.--Diagram of the model. 46 i = the financing alternative being evaluated NETCF the net cash flow in period t t rti = the interest rate in period t for financing alternative i REQMINt = the minimum required cash balance for period t The following designations have been changed to correspond to the notation in the computer program: NETCFt = the net cash flow in period t. and replaces NCF in the preceding models.l REQMINt = 6ti(ht) E dti = the minimum required cash balance. REQMINt replaces 6t in the preceding models. Methodology The model considers the same financing alternative for all thirteen periods in the planning horizon. The periods are examined sequentially. The alternatives are handled separately and no combinations are allowed. For the first period, the input data are considered to determine whether excess cash exists to be invested, or whether the short-term financing alternative being evaluated should be implemented. If short-term financing is needed, the con- straints applied to that alternative are initiated. The output consists of the cash balance, interest and principal 1See Chapter III. 47 payable in future periods, any surplus cash receivable in future periods under that alternative and any interest on that invested surplus cash. The alternative i=1 is applied to each Of the thirteen periods. At the end Of the thirteenth period, the unrestricted ending cash balance under i=1 is determined. This process is repeated for all four financing alternatives (i=1,...,4). Four unrestricted ending cash balances, one for each alternative, result. If all values for the variables in the model were deterministic, the analysis would be complete at this point. The four values would be compared and an Optimal financing alternative selected. However, a number Of the variables are probabilistic and, as such, are defined by probability distributions. Therefore, the entire analysis must be repeated a number of times. In this model, the analysis is run two- hundred times for each financing alternative resulting in a probability distribution which is an empirical estimate of the theoretical distribution for each alternative. These distributions are then compared, and an Optimal alternative selected by stochastic dominance. The Transformation Function The transformation function is responsible for converting the input state of period t, Ct' and any 48 transactions in that period into the output state for period t, C The transformation function states the t+l‘ relationships between the input state, the net cash flow, the amount Of money borrowed or invested, the interest rate and the minimum required cash balance for the period to which it applies. Thus, the transformation function is stated as T(Ct, NETCF r ., REQMINt) = C t' t1 t+l° of the transformation function can be divided into four The actions steps, as follows: 1. The transformation function determines the potential cash balance1 from the input state for period t, C and the value of the net cash flow and minimum ti required cash balance2 at the beginning of period t. PCASBAL(t) = CASH(t-l) + NCF (t) [NCF(t) = NETCF(t) + INVINC(t) + INVSPC(t-l) - FINTRN(t)] where: NCF(t) = adjusted net cash flow for period t PCASBAL(t) = potential cash balance for period t prior to any short-term financing in period t NETCF(t) = net cash flow in period t, a random variable INVINC(t) = interest received at beginning of period t on surplus cash invested in the prior period t-l 1The potential cash balance for a period consists of the ending cash balance of the previous period plus the net cash flow for that period and any adjustments necessary as a result of prior financing arrangements. 2See p. 76 of this chapter for a discussion Of the minimum required cash balance. 49 INVSPC(t-l) = surplus cash invested in period t-l FINTRN(t) = payment of principal and interest at the beginning of period t CASH(t-l) = ending cash balance for period t-l Ct = input state consisting Of: beginning cash balance in period t; any interest on invested surplus cash that will be received at the beginning Of period t; invested surplus cash that will be converted back to cash at the begin- ning of period t; the amount of interest and principal payments at the beginning Of period t for the financing alternative under considera— tion; and the amount available for borrowing in period t under financing alternative 1. 2. The transformation function then determines the difference between the potential cash balance and the minimum required cash balance: DIF(t) = PCASBAL(t) - REQMIN(t) where: DIF(t) = difference between the potential cash balance and the minimum required cash balance REQMIN(t) = minimum required cash balance for period t. 3. The DIF(t) is then evaluated. If the DIF(t) is positive, the excess surplus cash is invested. This invested surplus is converted to cash at the beginning of the subsequent period. The interest on this invested cash is assumed to be received at the beginning of the subse- quent period.1 If the DIF(t) is negative, the short-term 1See Figure 2. 50 ® Initialize Alternative Financing Strategies 1. Constraints 2. Interest Rate Distributions (§§:E§Ep 1 for 1:1, 200 Input for each Period l,...,13 1. Net Cash Flow 2. Accounts Receivable l NCF(t) = NETCF(t) + INVINC(t) + INVSPC(t) - FINTRN(t) PCASBAL(t) = CASH(t-l) 4 NCF(t) DIF(t) = PCASBAL(t) - REQMIN(t) NO I /\ (Number of times simulation is run) (Number of financing alternatives) (Number of periods) JIF(t) 1 0 V Effect Short-Te Financing rm L Yes J [Invest Surplus Cash] A\\ fine Deep 3) f Unrestricted Ending Cash Balance is Determined for Financing Alternative I Type Unrestricted Ending Cash Balance for Four Financing Alternatives End Loop i) i) Punched on Cards_w End Loop 1 Figure 2.--A general flow chart of the financing alternatives simulation. 51 financing alternative under consideration is implemented. The amount to be borrowed is determined, as well as the payment schedule for the principal and interest. 4. The output state for period t, C is then t+l’ determined. It consists of the cash balance at the beginning Of the period t+l, the amount of surplus cash invested in period t and interest on that surplus cash (both Of which will be received at the beginning of the period t+l), and the amount of interest and principal resulting from previous Short-term financing payable at the start of period t+1. Four financing alternatives are included in the model. In the subsequent section, these short-term financing alternatives will be discussed. Further, the techniques Of evaluating these alternatives using the computer will be presented. The Short-Term Financing Alternatives Four financing alternatives are considered by the mode1--line of credit, term loan, commercial paper, and accounts receivable loan. Line Of Credit.--A line Of credit consists of an informal agreement between a commercial bank and the customer regarding the maximum amount Of unsecured credit the bank will allow the customer at any one time. The 52 bank requires that the customer maintain a cash balance at the bank, the amount of which is directly prOportional to either the amount of funds borrowed, or the amount of the commitment. Term Loan.--A term loan is a formal loan agreement effected at a commercial bank. Repayment of the loan and interest payments are made in regular periodic install- ments. The duration of the loan may be up to ten years but the minimum and maximum limits are arbitrary. For the purposes of this study, the loan will extend over six periods. Commercial Paper.--Commercial paper is issued by a firm in the form of unsecured short-term negotiable promissory notes which are sold in the money market. Unlike the prime rate or bank loan, commercial paper rates fluctuate considerably reflecting money market conditions. Issuance of commercial paper is generally a less expensive means of obtaining funds than a short-term loan from a commercial bank. However, only the most credit—worthy firms can issue it. Accounts Receivable Loan.--The accounts receivable constitute one of the most liquid assets of a firm. The commercial bank will allow a firm to borrow utilizing their accounts receivable as security for the loan. The 53 bank assesses the quality and size of the receivables to determine the percentage of the face value that can be borrowed. The accounts receivable loan is a more or less continuous financing arrangement with the maximum amount of the loan allowable varying with the amount and quality of the accounts receivable of the firm. Computer Evaluation of the Financing Klternatives Line of Credit.--The maximum amount that can be borrowed on a line of credit is by definition a determinis- tic value at any point in time. For the purposes of this study, that value is assumed to remain constant for each period in the planning horizon. The maximum amount that can be borrowed with this financing alternative, and the distribution of interest rates for each period form the input data to the model. The model then determines if the maximum amount that can be borrowed is sufficient to cover the financing needs of the firm. If it is suffi- cient, the amount needed is borrowed. It it is not sufficient, the maximum available is borrowed and the 1,2 remainder is classified as a stockout. The cash balance for each period then is equal to the potential cash balance 1A stockout indicates that a less than optimal quantity of cash can be obtained and results in an unrestricted ending cash balance that is below that achievable when the desired amount can be borrowed. 2See Figure 3. Amount of credit already used = maximum ? Yes ' Enough available to cover amount needed to borrow |No borrowing allowedj NO Amount—available is borrowed, and Amount needed remainder is con- is borrowed sidered a stock- out Interest payable next period is determined from amount borrowed Amount borrowed is specified as payable next period L Cash balance at the end of period is determined1 Figul‘e 3.--Line of credi'... 1CASH(t) = PCASBAL(t) + DIF(z) 2The flow chart is for each period. 55 plus the amount borrowed. At the end of the thirteenth period, the unrestricted ending cash balance is equal to the cash balance in the thirteenth period, plus interest on invested surplus cash not received, plus invested surplus cash, less principal and interest due the subsequent period. Term Loan.--Input data for the term loan financing alternative include: (1) the maximum and minimum amounts that can be borrowed each period; (2) the maximum amount that can be outstanding at any time; and (3) the distribu- tion of interest rates for each period.1 The model evaluates the amount needed in terms <1f the maximum and minimum constraints on the amount that can.be borrowed as follows: If the amount available is greater than the amount needed and: l. The amount needed is less than the minimum, then nothing can be borrowed and the needed amount is considered a stockout. 2. The amount needed is greater than the maximum, then the maximum is borrowed and the remainder is considered a stockout. 3. The amount needed is between the maximum and the minimum, then the amount needed is borrowed. 1See Figure 4. 56 mflfi wow 0 Eseaxme podum. can» mmoa can Ezeflcas coflumd cmcu umumoum mannaflm>m unzOE< mm» m c03ouuon swan Eseaxne aanuo>o mum Essacae cadumm zany mama manuaam>n uc9084 02 m Esawcaa vowuom annu mama .aoooc 9:5054 mm» m/ mannawm>m/// uc:080 ,, can» umummum ./ ©0600: uc502< OZ m wama nodumm can» mama can Edawcwe beauam can» “mucoum tarmac acacia OZ 57 .poHumm comm new ma uumno 30am one~ Auvaa + auqummm ucsofi¢ anewuom me mewsoHHOM on» mo comm cw band on Haw3 amououcA can Hmmwocwum may we nuxamumco vocasumumv we ammumuca can Homaocaua mo waspmnom ucmexmmom mosaaumumo A ma ohms or on ummu“.:« mo ucsofid TlIII ~.cm0a Sumanl.v muswwh usoxooum o wouocwmcoo ma umocumaou osoxuoum o cwumcwmcoo ma HGQCNMEQH ”CM I. UQ3OHHOQ mun $3OHHCD WH @03OHHM omBOHHm mca3ouuon oz Esefime nodumm v _pmvomc undead mcw3ouuon oz. can vascuuon ma Esfiaxne GOMuom 58 If the amount available is less than the amount needed and: l. The amount needed is less than the minimum, then nothing can be borrowed and the needed amount is considered a stockout. 2. The amount available is greater than the maximum, then the maximum is borrowed and the remainder is considered a stockout. 3. The amount available is between the maximum and the minimum, the amount available is borrowed and the remainder is considered a stockout. One-sixth of the principal and interest due is to be paid each period. The program determines the dollar amount to be paid as principal and interest. The cash balance for each period is equal to the potential cash balance plus the amount borrowed. At the end of the thirteenth period the unrestricted ending cash balance equals the cash balance plus the interest on invested cash not yet received, plus invested surplus cash, less principal and interest payments for the next six periods. Commercial Paper.--In theory, the quantity of commercial paper that can be issued by a firm has no static maximum. However, management generally sets a maximum issuance which is considered to be in the best interests of the financial well-being of the firm. This maximum 59 amount of commercial paper that is allowed to be outstand- ing at any one time, and the interest rate to be paid on that outstanding paper each period serve as input data for the model. The model then determines if the amount needed exceeds the maximum amount available with this alternative. If the amount needed is available it is borrowed. If the amount needed exceeds that available, what is available is borrowed and the remainder is considered a stockout. The model also determines the interest due on the amount borrowed. The interest must be paid in the period during which it was borrowed. Thus, the cash balance at the end of the period is equal to the potential cash balance, plus the amount borrowed, less any interest paid. At the end of the thirteenth period the unrestricted ending cash balance is equal to the cash balance at the end of the final period, plus invested surplus cash and any interest on those funds, less the amount of principal due for the next three periods.1 Accounts Receivable Loan.--The accounts receivable balance, the maximum amount in dollars that can be borrowed, and the maximum percentage of accounts receivable that can be used as collateral for the loan are entered as input to 1See Figure 5. Amount outstanding = maximum amount allowed ? Enough available to cover amount [No borrowing needed to borrow NO Amount available Amount needed is borrowed, and is borrowed remainder is considered a stockout Interest payable this period is determined from amount borrowed Amount borrowed is specified as payable three periods in the future Cash balance1 at the end of allowea] the period is determined. Figure 5.--Commercial paper.2 1The cash balance CASH(t) = PCASBAL(t) + DIF(t) - INTPAY(t). 2The flow chart is for each period. 61 the model. The interest rate to be applied each period is also entered. The model then determines if the maximum amount that may be borrowed on the accounts receivable has been exceeded. If it has, the amount needed is declared a stockout. If the maximum amount available has not been exceeded, the program considers the total dollar value of the accounts receivable and applies to that value the percentage upon which money can be borrowed. If the resultant value has been exceeded, the amount needed is considered a stockout. If the amount available has not been exceeded, the amount needed is borrowed assuming the full amount is available. If the full amount needed is not available, what can be borrowed, is, and the remainder is a stockout. Interest on the loan is to be paid at the beginning of the period subsequent to the one in which the loan was effected. The amount borrowed is repayable at the beginning of the next period. The cash balance for each period is equal to the potential cash balance plus the amount borrowed. The unrestricted ending cash balance at the end of the thirteenth period is equal to the invested surplus cash and interest on those funds, less principal and interest payable the subsequent period.1 1See Figure 6. Enough available to cover amount needed to borrow , [No borrowing allowed] No Yes Amount available is borrowed, and remainder is considered a stockout Amount needed is borrowed L l Interest payable next period is determined from amount borrowed Amount borrowed is specified as payable next period- Cash balance at the endfiOf the period is determined2 Figure 6.--Accounts receivable loan.3 1Maximum = minimum (specified minimum dollar amount or the percentage of the dollar amount of the accounts receivable). 2CASH(t) = PCASBAL(t) + DIF(t) 3The flow chart is for each period. 63 Model Input Data There are four basic categories of input data to the model. These include, for each period, the constraints on the financing alternatives, the interest rates or distributions specific to the financing alternatives, the net cash flow distribution and the minimum required cash balance. The constraints on the financing alternatives were presented in the preceding section. The other three categories will be discussed in detail in the following section. Interest Rates on the Financing Alternatives The interest rate for each financing alternative is entered into the model as input data for each period. The interest rates may be submitted either as deterministic values or as probabilistic values with normal or uniform distributions.l Management specifies the value that is used according to the certainty with which the value can be described at the point in time when the model is to be run. When the interest rate is a deterministic value, only that figure is necessary as input. However, when the interest rate is projected to be a probabilistic value with a normal distribution, the expected value of the lComputer subroutines for handling the determin- istically or probabilistically defined interest rates may be found in Appendix A. 64 interest rate and the standard deviation for each period must be entered. Similarly, when a uniform distribution is projected, the maximum and minimum interest rate values for each period must be included. The interest rates are subject to the constraints applied to the financing alternatives with regard to the number of periods for which the loan is made and the repayment schedule. Methods of Determining the Net Cash Flow.--In theory, there are a number of approaches which could be used to determine the random values of the net cash flow. Three methods will be briefly discussed in the following section: (1) time series analysis: (2) the net cash flow as some function of one or more random variables; and (3) simulation. A time series analysis of the historical net cash flow values could be made to estimate the stochastic process that generated those values.1 The stochastic process would then be applied to generate the input net cash flow values for the model. Another approach could consist of analytically developing the stochastic process of the net cash flow as some function of the random variable, sales, which itself is generated by a stochastic 1 Paul E. Pfeiffer, Conce ts of Probabilit Theor (New York: McGraw-Hill Book Company, I965), pp. $91-295. 65 process.1 A similar technique consists of using the stochastic processes of more than one random variable. The net cash flow is then some function of a finite number of random variables. The stochastic process thus determined is then used to generate the input net cash flow data. Finally, the stochastic processes of the input variables may be sampled using simulation to generate the net cash flow values which can be used in the model. The analytically complex derivation of the stochastic process of the net cash flow is then eliminated. 1. Time series analysis.2 A stochastic process can be defined as a random phenomenon which arises through a process which is develOping in time and in a manner controlled by probabilistic laws.3 In time series analysis observations are made at a constant interval over time. The statistical theory of time series analysis attempts to determine the mechanism generating an observed time series. The stochastic process, {X(t), t e T}, can lIbid., pp. 164-170. 2For a further discussion of time series analysis see: E. J. Hannan, Time Series Analysis (London: Butler and Tanner, Ltd., 1967); Andrew H. Jazwinski, Stochastic Processes and Filtering Theory (New York: Academic Press, Inc., 1970Y: and The Research Section of the Royal Statistical Society, "Symposium on Stochastic Processes," The Journal of the Royal Statistical Society, Series B, XI, No. 2,51949 (reprint). 3Emanuel Parzen, Stochastic Processes (San Francisco: Holden-Day, Inc., 19627, p. 22. 66 be described by defining the joint probability law of the n random variables X(t1),...,X(tn) for all intergers n and all n points t1....,tn e T. This joint probability law can be expressed by the joint distribution function,1 F, for all real numbers xl,...,xn as follows: FX(tl),...,X(tn)(Xl""'xn)==Pr{x(tl):LX1""'X(tn)f-Xn} where F is a finite dimensional distribution x(t1) ,...,X(tn) of the process. The joint distribution function must be determined. Once the distribution of the stochastic process is estimated in the time series analysis, that distribution is utilized to generate the values of the net cash flow. The problem rapidly becomes involved and time- consuming to solve. 2. The net cash flow distribution as a function of one or more random variables. The stochastic process of the net cash flow can be determined if the distribution of some random variable, usually sales, and the function relating the net cash flow to this random variable are known. However, in this study, there are several random variables as well as some deterministic variables which enter into the calculation of the net cash flow. In order 1A distribution function indicates the cumulative probability of the random variable having a value 1 some real value. ' 67 to estimate the stochastic process of the net cash flow, the distributions of the random variables must be known. Further, the function relating these variables to the net cash flow must be known. The function which relates the random variables to the net cash flow is not a linear function nor are all the variables defined by the same type of distribution. The Central Limit Theoreml states that “under very general conditions, the sum (or average) of a large number of independent random variables is a random variable which is approximately normally distributed." However, in this study, where only a few random variables are considered and the variables are not independent, the Central Limit Theorem fails to aid in achieving a solution. An analytical solution becomes necessary which may be difficult to solve. Thus, the author has discounted the application of this approach of determining the net cash flow distribution since it appears that the problems involved belie practical use by the layman. An exploration of the two above approaches may well reveal valid methods for determining the stochastic process of the net cash flow. The approaches are both statistically complex, however, and as such are significant studies in themselves. They are perhaps more suitable leeiffer, p. 134. 68 for those pursuing expertise in mathematics or statistical analysis. 3. Simulation.1 Simulation is another approach that can be used to generate net cash flow data. 'The stochastic processes of the random input variables are sampled to generate the net cash flow values. The method eliminates the problems associated with deriving the stochastic process of the net cash flow. The variables, both deterministic and stochastic, are defined and the program written and tested. Management can then simply supply the input variable values and derive the net cash flow values. The author concedes that this approach may not be a panacea for all the problems involved in determining the net cash flow information necessary for the model. Never- theless, it is an entirely adequate approach. It has a distinct advantage over the other approaches in that it requires less complex input from the firm.2 Experience has shown that this is an important consideration for manage- ment. Further, while the importance of the net cash flow data should not be understated, a method which generates 1See Chapter III, page 41 for a further discussion of simulation. 2That is to say, that no analytical or time series analysis must be made before the model can be used. 69 data representing a reasonable approximation of the net cash flow and yet is not unduly complex, is entirely suf- ficient. Monte Carlo simulation will be used to generate values for the cash inflow and cash outflow variables and parameters for each period.1 There are two basic categories of variables.2 Some have their values determined from the values of other variables and the parameter value which relates them. Others are not specifically dependent on the values of any other variable. The variables will be defined as either probabilistic or deterministic. Parameter values, which relate one variable to another numerically, will be handled deterministically or stochastically depending upon whether risk is involved. Management decision will influence some of the variables when a statement of policy can affect the cash flow. Figures 8 and 9 present a detailed outline of the net cash flow variables and how they are defined. Probability Distribution Determination.--Three types of probability distributions are used in this study to define the probabilistic variables--normal, uniform and general. Each distribution is contained in a subroutine in the program and can be accessed according to the type of 1See Figure 7. 2See Table l. Parameter values 70 Deterministic variables SIMULATION PROGRAM L_ I i I I I (Simulate tonhundred times) I I I I I ‘W Net cash flow values for each period Probability distributions for stochastic variables Figure 7.--General diagram of the net cash flow. 71 __ D - Deterministic Variable S - Stochastic Variable P - Parameter *Combined in general and administrative costs. TABLE l.--The net cash flow simulation variables. Variables Type Determination Symbol TOTAL SALES S SALES Cash Receipts Cash Sales S P CS Collection of Accounts Receivable D P CAR New Debentures D ND Common Stock Issued D COMSI Sale of Fixed Assets D SFA Cash Disbursements Direct Wages S P DW Raw Materials 8 P RM Factory Overhead* FOH Selling Costs* SELL General and Administrative Costs D/S P GA Research and Development Costs* RD Interest Payments on Long-Term Debt D DIP Tax Payments D/S TXP Miscellaneous Costs D/S P MSCL Debenture Repayment D DRP Debenture Retirement Fund D DRF Capital Expenditures D/S CAP Dividends--Preferred D DIVP Dividends--Common D DIVC 72 Net Cash Flow for Each Period (T) A. Total sales (used as a basis for determining the values of some of the variables) (SALES) B. Cash inflows (CR) CR(T) = CS(T) + CAR(T) + ND(T) + COMSI(T) + SFA(T) U1J>OJNH C. Cash Cash sales (CS) Collection of accounts receivable (CAR) Issuance of new long-term debt (ND) Issuance of common or preferred stock (COMSI) Sale of fixed assets (SFA) outflows (CD) CD(T) = DW(T) + RM(T) + FOH(T) + SELL(T) + GA(T) + RD(T) + DIP(T) + TXP(T) + DRP(T) + DRF(T) + CAP(T) + DIVP(T) + DIVC(T) + MSCL(T) + SPMISC(T) \DmflmmwaH 10. 11. 12. 13. 14. Direct wages (DW) Raw material (RM) Factory overhead (FOH) Selling cost (SELL) General and administrative cost (GA) Research and development cost (RD) Interest on long-term debt (DIP) Tax payments (TXP) Repayment of long-term debt (DRP) Sinking fund for long-term debt (DRF) Capital expenditure (CAP) Dividends--common (DIVC) Dividends--preferred (DIVP) Miscellaneous (MSCL) and special miscellaneous (SPMISC) D. Net cash flow calculation (NETCF) NETCF(T) = CR(T) - CD(T) E. Additional input data Value of outstanding accounts receivable and timing of their collection during initial periods of the simulation. Figure 8.--Net cash flow variables--basic outline. '73 unmxauco codmmonman nmaonzu flannenouou umoo mngu uo cOnunOQ cmxnw any an auvcoom anon: mmqnumnumncnspo can Hmnmcou .m .no>o3oz .Uonounmcoo xaamsnn>nuCn on 0» Evan 30HHu ou nonunvoe on annmmo coo Eunmonm 0:9 .xnomoumo o>nuanumnanEUuIvcnIHonucoo on» on cuppa can omcnaeoo ono3 mumv omen» .xcumEoo anon on» :H .ucoEmoHu>wp can nonmmmon 0cm .umoo mandamm .pmogn0>o >nouumm .m.v.m mmqcm wo ucsOEm 0cm manEnu on» monounvcn ucwEmmmcmz UnumnCnanmuov "unannnm> mummmc voxwu no wamm .m xooum connouwnm no season we mocmsmmn xcm mo ucsOEm can oCnEnu on» woumonpc« ucweuomcmz OnumMCnEnmumu "mHQMnnm> xuoum cmnnmumnd no COEEoo uo mocmsmmu .v anon EnouuqcoH 30c uo guacammn >co mo ucsOEm can mansnu on» mwumonccH unwewmmcmx caumnannwuwv "mHQMnnm> anon EnmulmcoH 30: mo mocuzmmm .m .cOnnwm £000 mo mcnccwmmn on» an nmuu mm nouumaaoo on 0» moanm>nmumn :mmu mo unseen on» mmumHsoamu Emnmonm 0:8 ponnmd sumo cmuooaaou on on wanm>nooon mucsooom manpcmumuso mo omwucmunmd one .A mama unpmno sumo mo canuooaaoo muoHQEoo on >nmmmwomc vaHnmm mo nonfisc one .u uvmnouco on umsE dump menonHON 0:9 Unumncnsnmuov "wagonnm> UHDM>MQUOH m0c3000fl NO GOHUOOHHOU .N a uOHnmm cw mmamm unpmno on» mn Asymmu mnonz AABV mmamm Hmuoa 74 .ocaaaso voanmaonIImoHnMnno> 30am nmoo aoZII.m onsuah amnocom "cenasnnnamac hanannononm mamou msoocmHHoomnE Honuodm ona no macsosm ona mo monanannmnonm can macsoam .mcHEna ona mnoaco acoEoomcmEIIUnamaHanononm mamoo msoocmHHoUmns Honoomm ona no assoEd new qcnana ona moaooaocn acoEomocmEIIOnam«:nEnoaoo Onamnaanononm no unamHCaEnoaop uoanmano> mdoocoaaoooae Huauomm mammanco scammonmon nmsonna connenoaoc amou man» no conunom noxno on» an Anvomnz onon3 mmqnv xuoam oonnomonm mo acsOEo new manna ona moamoancn acoEommcmz UnamaCnEnoaoU «onnoanu> nonnowonmIImpcan>ao pawn muconn>nn xooam coeeou mo acsofio new manna ona moamonccn acoEomocoz Unamncaenoaop uoHnMnno> :oEEoounmvconn>ao Honocom "QOnasnnnamac maaannononm monsaaccodxo auanmmo ona mo macsoso ona mo monanannmnonm new mangoeo .ocnsna ona mnoaco acoEommcmEIIUaamaaanononm monaaancomxo Hmanmmo uo acsoao 0cm mCMEna oza moao0aosn acoEomocmEIIUnamneasnoaoo onamnaanmnonm no UnamnCaEnoaop ”oaanno> onsanccomxo Hmaammo macofi>mm noon manxcnm xco mo acacEd can manEna ona moaouncCa acoeommcoz Unamaannoaoo "oHnMnno> anon EnoanmcoH new vcsw meaxcnm anon EnoaumcoH uo acoE>odon xcm no acsosd can mansna ona moamoaucn acoEooocoz OaamnCnEnoaop "oanmanm> anon EnoanmcoH no acoeammom Honocom “c0nasnanamno hanannonOnm .macoE>om xoa ona mo macsoeo ona mo monaaannonond can mangoem .anEna ona mnoaco acoeomwcoEIIoaaonHanononm macosxmm xma mo aQDOEd can mcHEaa ona moaMOnucn acoEommcoEIIOnamaCnEnoaoo Onamnnanononm no UnamHCnEnoaop "oanoanm> manoeuum xua anon EnoanmcoH co amonoaca mo acsoEd new manEna ona moaounccw acoEomocoz unamncasnoaoc uoHnMano> anon Enoalmcod co amoumuCH .va .MH .NH .aa .oa 75 distribution that most closely describes each probabilistic variable. The normal distribution requires as input, the mean and standard deviation of the parameter or variable. The upper and lower bounds of the parameter or variable value must be submitted when the uniform distribution is to be used. The general distribution allows management to submit possible values for the variable in question and indicate for each value the probability of occurrence associated with it. Data for the Net Cash Flow.--The financial information necessary to simulate the net cash flow and for subsequent use in determining the unrestricted ending cash balances for the financing alternatives, was collected from real-world data. A Mid-west-based firm in the Motor Carrier Industry (Company Z) agreed to provide the data. The firm has an annual revenue of approximately $50 Inillion and extends its operations over a multi-state area. The basic financial planning structure for the firm ins divided into periods of four weeks, each period ending 011 a Saturday, with thirteen periods per year. Data for eeach period of the calendar years 1969 and 1970 were 1lected as well as data from the first four periods of 19771. The accounting information was then converted to a cash basis . 76 The Minimum Required Cash Balance The minimum required cash balance, REQMINt a 6 (h ) E d for each period is assumed in this study ti ti ti’ to be a deterministic value Specified by management. The amount of funds borrowed or reinvested, DIFt, is directly related to the interaction between this required minimum, and the values of the input state variables, Ct’ and the net cash flow, NETCFt.l As previously discussed,2 the results obtained may be suboptimal because only certain minimum required cash balance values are considered in the analysis. Ideally, the selection of an optimal financing alternative would require an analysis including all minimum required cash balances. The minimum required cash balance value Specified by management serves as an indicator of the feeling of management regarding the necessary balance between liquidity and profitability. A higher liquidity requirement indi- <:ates a more conservative attitude toward cash management. (Zonversely, a lower liquidity requirement may reveal more VVillingness on the part of management to risk a potential caish insolvency in order to gain higher profitability. 1See pages 48-51. 2See Chapter III, page 36. 3Higher profitability may be achieved by investing the :funds rather than holding them as cash. CHAPTER V PRESENTATION AND EVALUATION OF RESULTS In the preceding chapters, the short-term financing decision has been discussed and a model to assist manage- ment in making that decision has been proposed. The model is designed to evaluate four specific financing alterna- tives--line of credit, term loan, commercial paper and accounts receivable loan. Input data to the model for each period included: (1) the distribution of the interest rates of the financing alternatives; (2) the net cash flow distribution; (3) the minimum required cash balance; and (4) the constraints on the financing alternatives. Manage- Inent received as output from the model a set of cumulative enmdxical probability distributions, one for each financing Ethernative, which were evaluated to select the optimal financing alternative for any specified short-term financing decision. The application of the model to actual data will 1x3 presented and discussed in this chapter. The results will be evaluated using the condition of stochastic dmminance. Finally, the model itself will be evaluated in 77 78 terms of its applicability, effectiveness and practica— bility for use in the solution of the short-term financing problem. The input data will be presented in the following sections. The Net Cash Flow Simulation Input Data.--The probability distributions for the stochastic input variables1 to the net cash flow simulation were obtained through discussion with the management of the test company, Company Z. The distributions2 of the stochastic variables were defined as follows: 1. Revenue had a uniform distribution 2. Tax payments had a uniform distribution 3. Special miscellaneous costs had a general distribution 4. The breakdown of credit and cash revenue had a uniform distribution 5. The parameter relating wages to total revenue had a uniform distribution 6. The parameter relating raw materials to total revenue had a normal distribution 7. The parameter relating general and administrative costs to total revenue had a normal distribution 8. The parameter relating miscellaneous costs to total revenue had a uniform distribution. 1See pp. 73-74 of Chapter IV for a full discussion of both the stochastic and the deterministic input variables. . 2The probability distributions which are used in this study are discussed on pp. 69, 75 of Chapter IV. 79 All other input variables were deterministically defined. The Specific values for the input data to the net cash flow simulation are listed in Appendix B as they appear in the program. Output Data.--The output data consisted of the two-hundred net cash flow values for each of the thirteen periods in the planning horizon considered in this study. A cumulative net cash flow for each of the two-hundred simulations over the planning horizon was also given as printed output. In addition, the amount of the outstand- ing accounts receivable at the beginning of each of the thirteen periods was generated by this program. The latter data functioned as part of the input to the model in order to facilitate the evaluation of the accounts receivable loan alternative. An example of the printed net cash flow output is included in Appendix C. C_onstraints The financing alternatives will be designated by type in the discussion in this chapter as follows: Type 1 refers to Line of Credit Type 2 refers to Term Loan Type 3 refers to Commercial Paper Type 4 refers to Accounts Receivable Loan TYPe l was divided into two categories, 1A and 13, since 'flufi compensating balance required with a line of credit 80 may be determined in two ways. The method of calculating the compensating balance influenced the evaluation of the alternative. A number of constraints were imposed specific to each financing alternative. A line of credit is an informal agreement between a commercial bank and the customer regarding the maximum amount of unsecured credit the bank will allow the customer at any one time. For the purposes of this study, that maximum amount was $5 million. The bank requires that the customer maintain a compensating balance at the bank. This amount may be either directly proportional to the amount of the funds borrowed, Type 1A, or to the amount of the commitment, Type 18. In this study, 15 percent of the funds borrowed equaled the com- pensating balance for Type 1A. Type 13 had a required com— pensating balance which was equal to 15 percent of the $5 million commitment, or $750 thousand. A term loan is a formal loan agreement effected at ii commercial bank. Repayment of the loan and interest Payments are made in regular periodic installments. For thna purpose of this study the loan extended over six periods. Thf probability distributions. Stochastic dominance may be Iaseful for the analysis of a variety of decision problems ¥ lJosef Hadar, Mathematical Theory of Economic EBehavior (Reading, Massachusetts: AddISon4Wesley Publishing Clompany, 1971), pp. 261- 262; and Josef Hadar and William R. Piussell, "Rules for Ordering Uncertain Prospects, " The Pkmerican Economic Review, LIX (1969), 25- 34. 87 under risk. The condition of stochastic dominance makes possible a prediction about preference for probability distributions without requiring any knowledge of the utility function of the decision-maker. Thus, consider two uncertain prospects, X and Y, and X is larger than Y in the sense of first degree stochastic dominance (FSD). Then X is preferred to Y if it is desirable to maximize the expected utility. X is larger than Y in the sense of FSD when the cumulative probability of X being less than or equal to vi is less than or equal to that of Y for all unrestricted ending cash balance values, vi. This can be stated as follows: This condition is depicted in the following graph. The lower cumulative probability curve exhibits stochastic dominance as defined in the preceding section. 100‘ 50‘ (1650 4100 percentage of occurrences vi (in thousands) In this study, P(X i.Vi) i P(Y 1 vi) has been restated as: P(X > vi) : P(Y > vi). Thus, 88 the cumulative probability of X being greater than vi is greater than or equal to that of Y for all vi. This is the statement of stochastic dominance that will be used in this study. The format of the graphs used in the following sections to explain the results of the model is presented below: 4100 <1650 vi (in thousands) percentage of occurrences In the above graph, X exhibits the condition of first degree stochastic dominance over Y. The following sections will be devoted to an analysis and evaluation of the output of the model. Evaluation at a Minimum Required CaSh Balance of $3 Million The four financing alternatives were evaluated at three different minimum required cash balance levels. At a minimum required balance of $3 million, Type lA exhibited stochastic dominance over all other alternatives.1 Type 4 1See Figure 10. £39 SBOUBJJRDOO 30 18(4an anxaetnmna .HH onnme "mongom .acounoa a mo xaamcod asoxUOam a pop conanE mm wo oocoaon nmou ponnsvon ESEncHE w an mo>waocnoaam manUCMan ona non mconasnnnamnp >an~nnononm o>namH:ESU one .oa onsonm mnoaaoc wo mucomsona an moocmaon nmco qcnpco poaoanawonca D .l .1 T. 7. Z Z .C Z Z Z Z .C .C E CL C 9 I.— 8 6 0 T. z rt #3 Ca 9 L 8 6 0 TL 3 C. 0 O o 0 0 o 0 0 0 O O 0 O 0 0 0 0 O o o 0 0 0 o o O C 0 0 0 0 0 0 p e e P 2 - A-.. a ON .. IIOH om II Irma av 11 . L1 om om LT \ .. 1: mm Om. 11 . II Om 5.1.. Ilmm cm 1.] Itl 0‘ col. up. .yme oon-. .. «a. 1.0m and- Ixmm ONHL 11 CO omHI 11mm . v om>e.II- OVHI . 11 OF M omxsll cmal N omhal.l. Iwmh om>alll coal na 11 om .2“ meta. I--- 0nd: .1 mm coal 1. om coal 1v mm OONL H 00H saoualxnooo go nuaozod enraetnmng 90 had a cumulative probability distribution similar to and nearly as optimal as that of Type 1A. This degree of similarity can be expected since the only significant dif- ference between the two alternatives at this particular minimum required cash balance was the interest rate paid in each period. Type 4 was not affected by the constraint governing the maximum amount of funds available with this alternative since this amount of borrowing was not needed. Type 4 experienced a stockout only once in the two- hundred simulations at this level. Type 3 was dominated by Types 1A and 4 because of the requirement that the firm borrow the funds for three periods. Thus, interest had to be paid for the three periods. Alternatives 1B, 2, and 3 failed to exhibit stochastic dominance among themselves. These alternatives were clearly inferior to Types 1A and 4, however. In summary, the rankings of the alternatives at this minimum required cash balance can be explained by the low level of borrowing which eliminated some of the constraints on several of the alternatives and caused the interest rates and the duration of interest payments to be highly signifi- cant. Types 2 and 4 were evaluated both with and without stockout penalties. Neither the 1 percent nor the 5 percent stockout penalty caused any significant change in the cumulative probability distribution of Type 4 because 91 essentially no stockouts occurred. Type 2 experienced 102 stockouts in the simulation and was affected by the 1 percent stockout-to some degree. It was not the least desirable alternative consistently, even with the stockout penalty, however. When a 5 percent penalty was imposed, the desirability of this alternative was significantly reduced. Evaluation at a Minimum Required CaSH Balance of $475 MiIlion At a minimum cash balance of $4.5 million the rankings of the alternatives began to reflect the effects of the constraints. The data will first be evaluated con- sidering a 1 percent stockout penalty, and then re- evaluated when a 5 percent stockout penalty was imposed. Stockopt Penalty at l Percent1.--Type 3 and Type 1A exhibited stochastic dominance over the other alternatives. These two types had similar cumulative probability distributions.2 Type 3 did not exhibit stochastic dondnance over Type lA since, at three points on the curve,3 the cumulative probability distribution for Type 3 failed ¥ 1See Figure 11. 2The additional interest payment on the compensating {balance for Type 1A was nearly equal to the three-period Interest payment required for Type 3. . 3The stochastic dominance failed at the unrestricted ending cash balance levels of $2.3 million, $3.7 million and $3.65 million (see Table 3). 92 SBDUBJJUOOO JO Jaqmnu anxietnmng .HHH manna “ounsom .acoonod a mo >anmcom asoxooam m can COnHHnE m.cm Lo oucoamn nmoo ponnsvon agendas m an mo>naocnoaam unaccounm ona nOL mc0nasnnnamnp >anannmn0nd o>naoHsEzo one .HH onsmnm onwaaop mo mpcomsona cw moucoaon nmoo canpco poaonnamoncs b. T. T. T. Z Z Z Z Z Z Z Z Z Z S E r. E C C E E E E .7 .7 9 L 8 6 0 T. Z S .7 S 9 L 8 6 0 T. Z C .7 c. 9 L 8 6 0 T. S 0 0 0 0 O 0 0 0 0 0 O 0 0 O 0 0 0 0 0 0 0 0 O 0 O 0 0 0 0 0 0 0 0 0 0 O 0 O O O 0 0 0 O 0 O 0 0 0 0 0 p P _ p p n n p b h n p p p P h p n b [p p b I I S I 4... ON II IT OH on I- #2 0? IT 1.. ON om ll ATMN 00 L! 110M 2. .. in cm :1 110? ow +1 Irmv co? #8 OHHLY 11mm ONHIT Item can: +3 OQHLI #rOF om? \ \\.\ :2. 03 \ \.\\.\ a mane i! .f \I.I“\ \ \ M 03%.“. I 410m . \ I I 2.? \ \\ ~ 2.5!! I3 ~ ‘ m. WQX . l 2: \ \\\ a a L. \. fix 2 mane -.-- .eea . \\~\. 8;. mm 002 cc." saouazznooo JO quaozad anxiwtnmna 93 to be equal to or greater than that for Type lA at those points. It is possible that if more simulations were run, a stochastic dominance would be exhibited. It should be noted, however, that a large number of firms are unable to issue commercial paper. Type 3 would then be eliminated leaving 1A clearly dominant over the remaining alternatives. Firms which are able to utilize Type 3 often attempt to combine Type 3 and Type lA when borrowing.l The data shown in Figure 11 tend to justify this approach. Type 2 is the least desirable alternative since it is dominated by all of the other alternatives. The dura- tion of the term loan in this study was six periods. Interest must be paid for each of the six periods and the total interest paid was greater than that paid on the alternatives which have a single—period repayment schedule. The lower percent interest rate available with Type 2 was not sufficient to make it a more desirable alternative in this case. In addition, Type 2 experienced stockouts in each of the two-hundred simulations. The stockouts iJuzreased the divergence between the cumulative probability dis tributions . No stochastic dominance could be established between Types 1B and 4. Type 4 was less desirable than it .was at.the $3 million level because the alternative “ 1James C. Van Horne, Fundamentals of Financial Mana ement (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., I, pp. 235-236. 94 experienced 178 stockouts at the $4.5 million level. Thus, the stockout penalty was imposed and caused the alternative to become less attractive. Stockout Penalty at 5 Percentl.--The data for alternatives 1A, 1B, and 3 were identical to those discussed in the preceding section Since these alternatives did not experience stockouts. Type 2 was markedly affected by the 5 percent stockout penalty because stockouts occurred 200 times in the same number of simulations. Type 4 was also significantly affected by the stockout penalty although to a lesser degree than Type 2. Type 4 became noticeably less desirable in the regions of the graph where larger sums were being borrowed. The differences between Type 4 and Type 1B became less marked in the lower regions of the graph where less borrowing was effected. No stochastic dominance was exhibited by Type 1B over Type 4. Type 2 and the effects of the stockout penalties on that alternative are presented graphically in Figure 13. Stockout penalties of both 1 and 5 percent were imposed. Figure 14 presents Type 4 and the effects of the stockout penalties on the cumulative probability distributions of that alternative . 1See Figure 12. 95 .>H oanme "oonsom .acoonod m we >a~ocod asoxUOam m can confinne m.vm Lo oocoaon nmou ponnsqon ESEnCnE w an mo>naocnoaao manocm2am ona new mconasnnnamnp xanannonOnd o>naoaseno one .NH onsonm mnmnnop mo mpcomsona an moucoaon ammo manpcm poaonnamoncn h I I I z z Z Z z z Z z z Z CL CL E E E .V v 9 L 8 6 0 T. Z .L P r: o, L 8 Av 0 T. Z 8 6 0 I S 0 0 0 0 nv n. 0 nu O 0 0 0 n. 0 nu 0 0 0 0 O 0 0 0 0 0 0 0 0 no 0 0 0 0 n. 0 nu 0 0. 0 0 0 e _ r _ _ .I n e a n a a . _ a a . I] 1‘19. on. 11m \YHVW\.uu o? .\ w... #3 \. \;< N on r VIN? :2 I _\ m 0v .1. IION m e ome Ilmn 3 T. % ow. 1.0m u m on :mm m 1 cm. Iov O J o om . Irm¢ m n cos. on J 1 m OHH L o . .mm a s o~a.. 1.0m oma. Jvmw 00H; :1th v odxhnlll 03. m weanil. lime 8? N were III :8 m5 maxelll and: .44 OQ>BIIII I if mm 097... Irom cad: .x x mm com Hog SBOUSJJDDOO 30 qua313d BAIQPIUUIUD 965 .> onnme ”oonsom .COnHaaE m.vw uo oceanon nnoo tonnavon ESEnCnE o nan: acounod m new .H .0 Mo moaanocod asoxooam an N omha n0u wCOMasnwnaonp >aa~nnonond o>naoHsEsu o>naonodeoo .ma onsmam mnoHHOp mo unconsona aw noocoHon nmou manvco poaonnamonco OS9I> IOOLI iooet .moat aoooz 10012 .oozz Toocz -oo:z «005: +009: “OOLZ 4008: ..006z +~000£ «FOOI£ ~ooz2 oocc ..oovc I»oosc 1009: ~00L€ >008£ 006E —ooov 0019 F k\§kx m OH ma on ov om saouazznooo JO Jaqmnu anxqetnmna mm Sill on an IIII; . I we Call om mm om ..mm OOH saouazznooo JO abenuaoxad anxietnmna 97 .H> magma saouaxxnooo ;o Jaqmnu anxietnmno ”oonsom .COaHHnE m.vw wo oocoaon ammo ponnsqon ESEnCHE m can: acoonod m can .H .o mo monaaocom asoxooam as v od>a new chnasnnnaan >aaaanon0nd o>naoHsEso o>naonodeou .va onsoam mnoHHOp wo unconsona an moocoaon nmoo unnpco poaonnamonca .0 I T. T. z z z Z z Z 7» z Z Z E E CL CL CL CL S C... CL CL .7 v 9 l.— 8 6 O Tt Z (L V S 9 L 8 6 0 T. Z CL .7 S 9 L 8 6 o I S 0 0 0 0 o O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 o 0 0 0 0 0 0 0 o o 0 0 o 0 0 0 0 0 o 0 0 0 0 o 0 _ n _ _ L . _ w n n L n ,P p p _ . VIII_ _ _ _ _ OH ‘1 fl m om .. I 0H on .e I. ma ov .. : ow cm 11 % mm ow .. I on on I. 5 mm cm 1 1. cc ca I % mv OOH.I I: om OHHL 1T mm ONH 1 11 cm and. .rmm ova.. ..on omal Lyme owal atom 02 I If mm wm--|l oma. ma Item code C II no com. gvooa saouaJJnooo go abeauaolad aArieInmna 98 Evaluation at a Minimum Required Cash Balance of $6 Million Stockout Penalty at 1 Percentl.--At a minimum required cash balance of $6 million Type 3 exhibited stochastic dominance over all other alternatives. Type 3 dominated Type lA because the dollar value of the interest rate for Type 3, even for three periods, was less than the interest for Type 1A for one period because of the large compensating balance required with Type 1A.2 At this particular minimum required cash balance, every alternative except for Type 2 exhibited stochastic dominance over at least one other alternative. The alternatives ranked as follows: Type 3, Type IA, Type 18, Type 4, Type 2. When sufficient funds were borrowed the constraints on the alternatives were fully effective. Types 2 and 4 were clearly dominated by the other alternatives since both of these types eXperienced stockouts for every simulation. Although Type 1A dominated Type 13, the difference between the desirability of the two alternatives was dindnishing. At higher levels of borrowing, the actual challar values of the compensating balances converge. Thus, if a simulation were run for a minimum required cash 1See Figure 15. 2The difference between the two interest rates times the amount needed by the firm equals that amount of interest that is less than the interest rate of Type 1A times the amount of the compensating balance. 9S) Ionxnctnmna SODUOJJUOOO JO 10(1an oa cm on ov om cm or am oe coax: oaalt omalfl OMHII ovAJv omalv ooHII omH 11} nm\ omang com 059T> FOOLI ”0081 *006T ~000Z .HH> oanoe .acoonom a mo >aaocom asoxUOam M van cenaane mm wo oocoaon nmmo nonnsvon E:En¢ns m an mo>namcnoaao mcHUCMan ona new mconaonanamnp xanaanonOnm o>aamaoezu one mnoaaoo mo mvcmmsona an moocoaon :moo unnpco poaonnamonc: “OOIZ ~OOZZ "OOCZ ~00VZ "OOSZ ~009Z r00L: *008Z ~006Z “000$ "oonsom "OOLE A Y 1" ~008$ .mn onsone ”006$ m N we 4H odxp. om>e om>9 oa>9 ~000V I cofi saouelxnooo Jo abequaoxad anxqernmng 100 balance of $8 million, the curves of Types 1A and 1B would be approximately the same. Stockout Penalty at 5 Percent.--The data for alternatives 1A, 1B, and 3 presented in Figure 16 were identical to those plotted in Figure 15 and were included merely for comparative purposes. The 5 percent stockout penalty with a minimum required cash balance of $6 million significantly affected Types 2 and 4. Type 4 was less severely effected by the stockout penalty than Type 2 because the maximum amount that could be borrowed with Type 4 was greater than with Type 2. Thus, even though both alternatives experienced extensive stockouts, Type 2 was the less desirable alternative. Figures 17 and 18 present Types 2 and 4 and the changes effected by the stockout penalties at 1 and 5 per- cent. The significant impact of the stockout penalties at the $6 million level is demonstrated. Evaluation When the Minimum Raquired Cash Balance is a Stochastic Variable In an attempt to explore further the effects of the minimum required cash balance on the rankings of the financing alternatives, this value was allowed to be probabilistically defined. 101 .HHH> oanme "oonsom .acoonod m mo xaamcod asoxuoam m paw cenaane mm mo ouc0aon not» tonnsqon ESEncHE m ao mo>naocnoano SCHUCMCnu ona n0m mCOnaannnamnp xanaanoQOnd o>nuznzaou one .oa onsmnm mnoHHOp wo monomsona an moucoaon nmoo wanpco poaoanamonca D .l T. T. Z Z Z Z Z Z Z Z Z Z r... r... c: c... E F. E 6 c... cc .7 .7 9 L 8 6 0 I Z 6 .7 S 9 L 8 6 0 T Z c... .7 c. 9 L 8 6 O T. S 0 0 0 G 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0 0 C 0 0 C 0 0 0 O 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 C C 0 0 C 0 0 0 I a a _ tI a _ e _ a e . a In L . e _ _ e p F \I OH I." .‘I‘\ AT \\ om .e. \I ).v\ .\I.\ \ \\ on .I \l \ 1. CV IT \ \|I I." om 1. \ . cm I ..\ \ \\ : on I W \_ . x..\ I D \ m cm 1T . \ \\\ f n ‘ \ x 1 m. x. \ u. 2.. I \ \ \ e a _\ oo r \ a m T _ t\\ l. m. can .r: .\ I a . 1 . 0 ONHII+ \ \ i I) . s . o _ X\\ .«W OMH 11.. \a. If m 2: .rr \.\\ . IT a ._ «a m own 4 x 5 \ \.\\ 1:. \ can 1., \\ I . \\'.‘ chalk \. 2: L.\.\ om.” 1V CON saouaJJnooo go abenerJad anxietnmng 102 SBOUBJJDDOO JO Jaqumu QATQPIHWUD .xH oHnoB “oonsom .COAHHHE w» Lo oocoaon nmmo Donnsvon EDEHCHE o nan3 acoonom m vco .H .0 mo monanocod aaoxooam am ~.oa>e ace weenasnnnamnp xannnnmnOnd o>na6nsesu o>naonoaeoo .en onsonm mnmnnop mo mpcmmsona an moocmnon nmou manpco poaonnamonco > M u m m w. u u a w. n R a u m. M U. a a w. a re... a u a w n S O O 0 0 0 0 O o 0 o 0 0 0 0 0 o O 0 0 0 C 0 0 0 0 0 0 O 0 0 0 0 0 0 0 O O 0 0 o 0 0 o o 0 0 0 0 o 0 0 a a t 0 t i . a l a + .H-.-w. .w I” :”I w...” mwnanIIIe r IIhI It \. .\ . I. OH L1 \ o x 11m om t Ioa on I :ma 0? 11 \ ..v :ON 3 .. .\ emm \ S A. \. .bm \. Oh AV Nsn vmm cm 4 _ Irov om ._ .13 . con..n Iom _ e: . .. Lam oma.-~ loo _ Omar n \. Imp o: ._ as. 4 .\ om r Arm ATmh \. can \-\ +8 _ \\ on." ..._ \\ wm..|..l Imm . wHIIIII S: e \ \ o - tea 1 .\ omH.N\ :mm OONL tooa saouaJnooo go abeguaoxad enrgetnmwna 1.013 saouazznooc go Jaqmnu anrqetnmna .x manna CS9T> - OOLI a 0081 "oonsom - 006T - OOOZ .COHanE om uo oocmHmn nmmo ponnsvon EdEaCaE m can: acoonod m use .H .o «o monaamcod anonUCam am v od»e new chaasnnnamap xanannonOnm o>aaoazeoo o>namnodeoo mnoaaop Lo mpcmmsona an R OOIZ a OOZZ L 0072 » OOSZ + 009: moocoaon nmmu manpco poaonnamonca e OOLZ I oosz ~ COLE * 008$ .mn maaane ' 006E . 0007 OOT7 _ » ooos !* OOIE 0H ma ow mm on mm ov mv om mm ow mm on ms om mm 3 mm ooa saouaxlnooc go ebeiuaozad GAIQPInan 104 The $4-$5 Million Range with a 1 Percent Stockout Penalty1.--When a uniform distribution of $4-$5 million was selected and the results of the simulation were tabulated and graphed, it appeared that in allowing this one additional value to be stochastic, the cumulative probability distributions of the financing alternatives could no longer be evaluated using stochastic dominance. The most conclusive statement that could be made was that Type 2 was clearly the inferior alternative. Types 1A, 1B, 3 and 4 exhibited no significant dominance with regard to each other. The $4-$5 Million Range with a 5 Percent Stockout Penalty2.--When a 5 percent stockout penalty was imposed, Type 4 became a less desirable alternative. Type 4 still failed to become inferior to Types 1A, 1B, and 3 at all jpoints, however. Figures 19 and 20 can be compared with the deterministic graph at the $4.5 million level.3 In the deterministic graph alternative 3 had a cumulative probability distribution which ranked above the other alternatives. Although Type 3 did not exhibit true stxochastic dominance,4 it was close to being dominant over ‘ 1See Figure 19. 2See Figure 20. 3See Figure 11. 4Type 3 failed to exhibit stochastic dominance at tfllree points on the curve. .1()5 5831181111030 JO 13(1an GAIQPIGUUD .Hx mange "oonsom .acoonoa a we xaaocoa asoxUOam m can connane mmIem Lo ooconmn ammo ponnsoon EdEacHE m an mo>naocnoaam oCnUCMCnm ona new mconasnnnamnp xanannmnOnd o>namHoEsu one .mH onoqnm mnoHHOU mo mpcomsona an woocmHmn nmoo wonpco poaonnamonco > m ”u M a m u u u n u w a u "u Mm u n“ a n ”a n u M ”u w u u mu 6 n n n n n n w m n m. w m anIm _ m m t a. . Ir a \. - .I II III OH 1T . lim o~ I . -Ion I . p. IImH \ \ ow \\ n\ IION . Mn on ;u\ IImm ow Iron or Irmm om Irov om Ixmv ooa Ixom 0AA Ixmm oma Ixom oma 11mm Q .I. can a m an I m om>e_lI. 4 on own N omens I.I Le me on we? II 00H 4H 0“; III 110m on." It Ia. mm omHI I oo oaHI mm oo~ ooa seouazznooo go abequaoxad anxietnmna 106 SBOUBJJnDDO JO Jaqumu aA‘annuIng .HHx onnoe "oonsom .acoonoa m we >aaocom asoxooam o poo COnHHnE mmva uo oocmaon nmou ponnsvon ESEncHE m an mo>naocnoaam manucmcHu ona n0u macaaonnnamnp >anannmnond o>naoazeso one .om onsoam mnmaaop mo mpcumsona an mouconon nmou manpco poaonnamonc: w u m. a w. u a a u a w u. u a. m. u S o o 0 0 0 o 0 0 O 0 0 0 O o o C o O 0 0 0 0 o o o 0 0 o o o o k h h — — b b P p p p P h r oa II .f m ea .. IS on I- t3 ov II . % o~ .\\ om I1 \\ I. mm 8 I .\ . Ion \ \ 1" \ . 13mm on \ \\\\ .I \ \\_\. ATOV em I \ x..\ cm I IYmv ooa. Irom OHHI mm own . Ivoo omnI I.mw 3L. IE omHll 17 m5 owHI I.om on>9-ll- ohHI v Lvmm m oa>e.l|! omHI N od>eI.Il rem ma oaxaIlll omHI omaBIIII mm com ooa saouazznooo go abeguaozad anrgetnmno 107 Type 1A, and these two alternatives exhibited dominance over the other alternatives. In the stochastic graph not even this degree of dominance was exhibited. The relatively low level of borrowing caused ranking of the alternatives to be less distinct than they became when higher levels of borrowing were effected. When the minimum required cash balance was allowed to be stochastic, interpretation of the data became even more difficult and no advantage could be seen in this approach at this level. The $5.5-$6.5 Million Range.--A stochastic minimum required cash balance was also tested at a higher range to determine if stochastic dominance could be established in ranking the alternatives at another level. The results of this simulation, as seen in Figure 21, not only exhibited stochastic dominance between all of the alternatives, but presented a graph remarkably similar to l The that of the deterministic level of $6 million. constraints on the alternatives were fully effective at this level, and definitive rankings were more likely to toe achieved with the stochastic as well as the deterministic approach . Shammary The results of the simulations examining the four financing alternatives have been presented in the preceding 1See Figure 15. 108 antqetnmng SBDUBJJUODO JO Jaqmnu 0H cm on ov cm on Oh om om OOH omH ona omH 03 com .HHHx onnme "oonsom .acoonod A we >aamcod asexooam o nco cOaHHnE m.omIm.mm mo oucwaon nmoo ponnsvon ESEHCHE m an mo>naocnoaao ocaocoCau ona new mCOnaznnnamno >anannon0nm o>aaoaoeso one .HN onsmam muwHHOU MO mUCMm30£u Cw meCQHQD Cmmo DCAUCO UOUUHuUwOMCD > C... E CL CL CL S E v .V nnmmwnuunwwnmumnusvs9L860I S o o 0 0 0 o O o 0 o 0 0 0 0 0 0 0 O 0 o O 0 0 o o c o 0 0 o O 0 0 0 0 0 0 0 O 0 0 0 O 0 0 0 0 O 0 0 0 p I S III a PI _ . V. n _ p . e _ p t b _ b . .I‘LLIV {\QIIIIILOVIHJMIIIIIII\.J \I\\[.I\\\.|...\.. I: c\v\\.\ \\.. 4m IIO‘.\IHI..II\I.\. O \ ...\ I \\\\.T\\ IVOH \x K.\ I. II\\\:I\ I 3 \ . IN .\ \f\\ w. \- ~\ \\ 1v ON \. .\ vex . . \1 .mm A \ \ ~\\ \ \ \ 4T. 0‘ .. s\ tom \ .\ é .. .\ \ \\ :3 O\ \\ \. +1 \e\. \\ me 1T \\\ \ JT om I: ‘ \.\\ 17mm 1 .\ . \ VI 0 I4 % o\\\\.\\.. OW \ e \\ v, \ \\ \\ I mm o y \x . \ .\ ¢ \ l \ . Sr 4 05 ..o \\ § i \ \.. ll \ \ x me .Q \. \\\h\ 4. 1.\A\x X I 0‘ \\ ‘1.‘ flow . \ .V iv \ \. \\ x mm ; xv kYN \\\ A .\ ea : mm I 00H SBDUBJJHDDO 30 QUSDJBd BAIQPInan 109 section. The net cash flow data and the constraints on the alternatives remained constant while different levels of the minimum required cash balance and stockout penalties were tested. The alternatives were evaluated using stochastic dominance to select an optimal alternative specific to a certain minimum required cash balance. The stockout penalties affected the rankings of the alterna- tives but in no case changed the optimal alternative. The following table presents a summary of the results. Minimum Required Cash Balance Optimal Alternative Deterministic $3 million Type lA $4.5 million Type 3/Type lA $6 million Type 3 Stochastic $4-$5 million None $5.5-$6.5 million Type 3 The Results of the Model in Terms of the Short-Term Financing Problem Introduction.--The short-term financing problem is complex because of the difficulty of forecasting cash needs. This difficulty arises from the dynamic nature 110 of the individual firm, the industry of which it is a part, and the economy as a whole. The model presented in this study represents an attempt to facilitate manage- ment in making that short-term financing decision. The results of the application of the model to actual data have been presented in the preceding section. The primary objective of this section was to demonstrate the flexibility of the model with regard to input data, to evaluate the effectiveness of the model at various levels of borrowing, and to assess, insofar as possible, the practicability and applicability of the model to the short-term financing problem. In addition, the condition of stochastic dominance was explored as a technique for evaluating the results of the model simulations. Evaluation of the Model.--The results of the application of the model to actual data suggest that the model may be of significant value to management in the role for which it was designed. The model allows manage- ment to evaluate financial information necessary for the short-term financing decision, rapidly and relatively inexpensively. The model quantifies risk which is incorporated into the model through probabilistically defined net caSh flow variables and interest rates. Risk is an important consideration since variability of conditions is an observable fact and not recognizing it as such and failing to include it would impair the 111 validity of the model. Further, subjective judgments of management can be considered in the model with regard to some of the variable input data. Management may utilize the model in the decision- making process without having to define the utility function when stochastic dominance is exhibited. This capability alleviates the problem of deciding upon an apprOpriate utility function. The model is flexible, facilitating the changes necessary to allow reasonable approximation of conditions specific to the firm employing it. Further, management may vary the input data to the model, or to the net cash flow simulation, to evaluate the results of potential future actions being considered. The model indicates, if the condition of stochastic dominance is exhibited, that an Optimal financing alternative exists, and which alternative is Optimal. If there is no stochastic dominance, management will need to specify a utility function in order to maximize the expected value of that function. The results of this study suggest, however, that even if clear stochastic dominance is not demonstrated between the alternatives, those alternatives that are inferior are quite definitely established. Attributes of the Model.--The model considers the sequential, multiple-period nature of the cash 112 management problem. The incorporation of risk into the model allows it to be relatively realistic, and the simulation approach moderates its computational com- plexity. In contrast to other studies, the model in this study attempts to broaden its application to con- sider the options of holding, borrowing, and reinvesting cash. Further, this thesis presents an exploration of the applicability of the condition of stochastic dominance to the evaluation of short-term financing alternatives. This technique was successful in this study particularly at the higher levels of borrowing. A full understanding of the results of this study cannot be achieved without recognizing the limitations of the model. Four specific financing alternatives are studied. Whereas other alternatives could be added to the model, reprogramming would be required. Further, some firms may want to evaluate combinations of some of the alternatives. The model is not designed to consider combinations. It is also possible that some of the assumptions of the model may impair its ability to simulate real- Inorld conditions.1 It is recognized, however, that with Inearly every model, certain assumptions must be made. It 1See pages 36 and 37, Chapter III for a list of the assumptions of the model. 113 can only be hoped that those necessary assumptions do not measurably distort the results. One further limitation exists which may, to some extent, inhibit full evaluation of the model. As with any heuristic device, the only true test of its effectiveness and predictive powers is a long-term study of the effects of the model recommendations assuming that these recommendations were completely implemented by a test firm or firms. The difficulty of such a study in terms of time and COOperation of a test firm is enormous, and constitutes a significant study in itself. The author in this study elected to utilize real-world data for net cash flow input to the model, and to attempt to select realistic values and distributions for the remaining required input data. The care exercised in the selection of the input data suggests that the results represent a relatively accurate reflection of these data. The author Inaintains that the model can be evaluated and reasonable lpredictions about its effectiveness made even though a true test for validity has not been conducted. In summary, the model in this study has been applied to test data and the results presented. The model has been evaluated in terms of the results, the advantages ‘asfisociated with its use, and its limitations. The model appears to present a useful and realistic approach to the eValuation of short—term financing alternatives. CHAPTER VI SUMMARY AND CONCLUSIONS Summary Chapter I established the importance of the short- term financing problem, and the need for economizing cash holdings. A model was proposed which was designed to assist management in evaluating four short-term financing alternatives. The basic cash management decision process was discussed and some important terms defined. In Chapter ZEI, models in the literature which related to the cash management problem were presented and briefly discussed. The major differences between the deterministic and the stochastic approaches were outlined. .Methods, including those utilizing linear programming, dynamic programming and simulation, were summarized. The significance of each method in terms of the model in this study was elucidated. Chapters III and IV were primarily devoted to a iflnorough explanation of the model. In Chapter III the mCKiel was described in decision-theory framework. The Siligle-stage decision problem under certainty was extended ‘UD incorporate risk and include multiple stages. The 114 115 reader was guided through each step of the develOpment of the final multi-stage decision problem under risk specific to the model in this study. The assumptions of the model were also presented. Finally, the author discussed alternative methods of solving the model and indicated the rationale for utilizing simulation. Chapter IV presented a further discussion of the model. The short- term financing alternatives to be evaluated were defined and the method of computer analysis of each alternative was discussed. Finally, the input data to the model were presented in detail. The results of the application of the model to actual data were presented in Chapter V. Input data in terms of the net cash flow simulation, the constraints on the financing alternatives, the interest rates and the minimum required cash balance were specifically analyzed. The computer programs were briefly eXplained, and the Imodel output stated. The condition of stochastic dominance was defined, and its role in the evaluation of “the financing alternatives delineated. The results of 'the model were discussed and graphs of the resultant cnimulative probability distributions for the financing alternatives included. The model was then discussed in terms of its applicability to the short-term financing PITDblem and its usefulness to management. 116 Conclusions The objective Of the model in this study was to select an Optimal financing strategy from a given set of alternatives. The model included the capability of assessing the Options of holding, borrowing or reinvesting cash. The condition of stochastic dominance was employed successfully to evaluate the cumulative probability distributions of the financing alternatives. The results of the application Of the model to actual data suggested its potential value to management. The model was able to produce results that could be interpreted in terms of an optimal financing alternative. The model was flexible in terms of input data, and relatively inexpensive to run. Risk was incorporated into the model and a sequential, multiple-period approach was used. The model was designed to approximate real— ‘world conditions without becoming unduly complex compu- 'tationally. Finally, the model programs could be .implemented by a firm with a minimum amount of effort and computer knowledge. l§gggestions for Further Research In the process of developing the model, the author erncountered the problem of determining net cash flow VEtlues to use as input to the model. Several approaches 'UD this problem are mentioned in Chapter IV. It would 117 appear that a thorough exploration Of the problem would be a valuable contribution to the literature. An interesting related problem also involves the net cash flow. It may be possible that the stochastic process that describes the unrestricted ending cash balances for each alternative could be analytically derived from the stochastic process Of the net cash flow, assuming that the latter process were known. In such a study, the distribution of the interest rates and the constraints must also be known and/or defined. If the stochastic process of the net cash flow was not known and could not be defined, an alternative approach would be the utilization Of the stochastic processes Of the variables of which the net cash flow is composed. If such a technique were successful, computer facilities for solving a model would be unnecessary. This would allow very small businesses to benefit from the approach suggested in this study. Another aspect of the study which provides a fertile area for further research is the empirical verifi- INDB=3 FOR PARAMETER THETA 134 READ¢IN1¢1008IIND INDTHETAEE- -. _ _,.. A._ .8“ _- ._- w» __ PEADI1N10103O)THETA10THETA2 IFIIND oGTo 2)INDTHETA83 FOR PARAMETER DELTA READIINIOIO BIIND INDDELTA=2 READLIN111030)DELTA1ODELTAZ .-u. .I “_ ”n _ IFCIND oGT. EIINDDFLTAIB C.. . KK IS NUMBER OF PERIODS THAT CREDIT SALES TAKES TO BE COMPLETELY C COLLECTED KK = 4 READ (INIOIOO?)(CRSPERIJ)vJ=19KK) -nC OUTSTANDINQVCREDIT SALES READ (INIOIOOBI(CRSOLDIJIOJ=IOKK) OLDOUT<1I=O DO 370 J=10KK OLDOUTII): OLDOUT(1) + CRSOLD¢JI 37m CONTINUE . -- - ___-_ ---_.D.O_-jl7_1 - .J:—'.'2.1KK 7 _ OLDOUTIJ): OLDOUT(J-1) - CRSOLD(J-1) 7-371 CONTINUE 1 FORMAT (8H SALES 018 08H CS(T): 918 09H CRSIT): 018 o 1 12H ALPHA a 0F50309H ACTSRV: .18) 2 FORMAT (10H COLCRT: 918 09H CRS(J)= 918 912H CRSPERII)= oF5.3) 3 EQRMATKIBH,CR(T)= 018 00H CAR(T)= 018 08H NDII)=m9I80 D _.~_..— 1 12H COMSI(T)= olaooH SFA(T)8 0I8) 4 FORMAT ( 8H WAGES: 018.7H A = .F503915H RAW MATERIAL: .189 1 7H 8 = 0F503) 8 FORMAT I ‘ 8H GAITI= 0180 1 11H THFTA = cFS.3010H MSCL(T)= oIBolIH DELTA a oFSoB) “wwa_u5MEORMATu(9H DIPIT): .1816H TXP: mlBoéH DRP=.lLauléH.DRFFWQIBQ 1 6H CAP: 018.7H DIVP: 91807H DIVC: .1899H SPMISC: .18) 16 FORMATtlHOoZOH START OF PERIOD T: .13) ¢O FORMATCIPH CRSPERIJI= 04F503) 1000.FORMAT(918/418) 1On1 FORMAT(13F5¢3) -10n2.fQEflAIML9F503) 10n4 FORMAT (I8) IOnS FORMAT (438) tons FORMATIIZ) 1007 FORMAT(12918) 10mg FORMATIIZ) ..... .12QQ.EQRMAI(121“. 1010 FORMATIIROFSQBI 135 1011 FORMAT IIOF604/IOF604/IOF604/1OF604/IOF604/2F604) -unmLQLZHEQBMAT.(QIB) 1013 FORMAT I180F604) 103D FORMATI2F503) I400 FORMAT (17H DEBUG INPUT DATA) 14a: FORMAT (1H 011(1X0IBI) 14m: FORMAT IIH 04(3X0F503)) C_U_LAQB_EORMAT IIH.05(3XOF604)) 1411 FORMAT IIH 06(3X0FS.3)) I41? FORMAT(1HO06H MAXI 0I806H MAX2 01808H MAXZPA 01808H MINZPA 0180 I 6H MAX3 01806H MAX4 0I807H PER4= 0F604) I413 FORMAT (IHO0IOH CASHBAL OIB) 1414 FORMATI1H001?H CRSOLDIT) 0418) L415 FORMATIZX07IIO02XOI30IHA) 1416 FORMAT(2X061100IPX01301H8) 1417 FORMATt1H 01301X0I419) 141R FORMAT12X071IO0PX0130IHC) 1410 FORMAT(?X06110012X0I30IHD) 149“ FORMAT(IH 01301X01319) -4 ”1525-593MATIIH1954X915H NET CASH FLOW ) 1457 FORMATI1H1060XOPIH ACCOUNTS RECEIVABLE ) I4?R FORMATIIH 04H NO 013(0H PERIOD )OIIH CUMULATIVE) IARO FORMATIIH 04H NO OIBIRH PERIOD )) I43“ EORMATISH 04X02H I07X02H 207X02H 307X02H 407X0?H 507X02H 60 1 7XOPH 707XOPH 807x02H 907XOEHIO07X02HIIO ”‘wa 2m“ “m _ __ 7X02HIP07X02H1307X04H NCF) 1431 FORMAT(5H 04X02H IO7X02H 207X02H 307X02H 407X02H 507X02H 60 1 7X02H 707X0PH B07X02H 907X02H1007X02H110 2 7X02HIE07X02H13) WRITE IOUT20IAOO) WRITE(OUT20PO)(CRSPER(J)0J=IOKK) Oflpmu-uDD-1502.J = 10 N 0 (ID. 0K1 WRITE (OUT201401)NDIJ)0COMSIIJ)0SFA(J)0 DIPIJ)0TXRI(J)0DRP(J)0 I DRFIJ)0CAP1(J)0DIVPIJ)0DIVC(J)0FDGA(J) 1Sn9 CONTINUE WRITEIOUT2014OS)AIOA20BIOB2 WRITEIOUT20I4II)ALPHA10ALPHA20THETA10THETAP0DELTA10DELTA? _WWRIIE (OUT201414)(CRSOLDIJ)0J=10KK) SIMULATION DO LOOP FOR NUMBER OF TIMES RUN DO 1720 ITIS = 10200 DETERMINE CASH FLOW FOR PERIOD T DO IBOO T = 1013 [BASIC EQUATIONS_OF CASH INFLONS __ I - _ CP (T) = CS (T) + CAR (T) + ND (T) + COMSI (T) + SFA (T) 136 c CASH SALES AND CREDIT SALES IN PERIOD T 2QOQUCONIINUEI IFIINDS oGTo I)GO TO ?001 SALES=SI(T) GO TO 9010 ROOI IFIINDS - 3>2O030200502007 POn3 SS1=S1(T) _Eu-- SSZESZITI. I CALL UNIFRMIISSI 0352 0SALES) GO TO 2010 RnnR SS!=SI(T) SS?=S2IT) CALL NORMALIISSI 0592 0SALES) C - -MLEQ ..-TQ P.” 1.0. POO? II=NS(T) DO P008 121011 IVALII)=SB(T0I) PCNT(1)=PS (T01) 9OnR CONTINUE E,HMW”CALL‘GFNERAL(1VAL0PCNT0II 0SALES) RO1O CONTINUE C DETERMINE VALUE OF PARAMETER ALPHA IFIIMDALPHA .FO. 2ICALL UNIFRMRIALPHAI 0ALPHA2 0ALPHA) IFIIMDALPHA .EO. 3)CALL NORMALRIALPHAI 0ALPHA2 0ALPHA) cs (T): ALPHA * SALES U_LCRS (T) = SALES * (I - ALPHA) 7 CALCULATE ACCOUNTS RECEIVABLE AT START OF PERIOD T AND , COLLECTION OF CREDIT SALES IN PERIOD T0 KK 15 MAX NO. PERIODS BACK ACTSRV(I) = OLDOUTII) ACTSRVIT + 1)=ACTSRV(T) + CRS(T) - CAR(T) ~ .. .-.-.C.OL.CRT = 0 IF (T 0LE0 KK)POII02025 a011 IEIT .EO. 1)?01?0?OIS 90:? COLCRT=CRSOLDI1I ACTSRVIT)= OLDOUTII) GO TO POSS .3015_CQLCRT=CRSOLD¢T) L=T - 1 DO 2017 J=10L IaL - J + I COLCRT: CRSIJ)*CRSPER(I) + COLCRT IFIITIS .LE. E .OR. ITIS .E0. 50 .OR. ITIS .EO. 100 ) LEL-JwRITEIOUT202ICOLCRT0CRS(J)0CR5PERII) 90!? CONTINUE 00000 137 GO TO P035 “m2fl35.COLCRT=OI C ()O(1()n§ 901“ P01? nggflgrg P CS (T) + CAP (T) + NO (T) + COMSI nkwvxn¢3 91o: K = T - KK L: T - 1 DO 9030 J : K. L I = L - J + 1 COLCRT = cRS (J) * cRSPER (I) + COLCPT 1wRITE(OUT2.2)COLCRT0CRSoCRSPEQ‘1’ CONTINUE CAR (T) = COLCRT “15(ITIS OLE. S 0090 ITIS CEO. 50 (OR. ITIS 0E0. 100 ACTRDV(T + 1) =ACTSRV(T) + CRS(T) - CAR(T) CASH PFCFIPTS RASIC EQUATIONS OE CASH OUTFLOWS DFTFDMINE VALUF OF CAPITAL rxDENOITUQFS IF(INDCAP oGTo 1’60 TO ?105 CAD = CAPIIT) GO TO 211E II=NCAP(T) .00 P119 (=1!!! IVAL(I)=CAP3(TOI) CD (T) = ON (T) + RM (T) + EOH (T) + SELL (T) + GA RD (T) + DIP (T) + TXP (T) + ORR (T) + ORE CAP (T) + DIVP (T) + DIVC (TI DIRECT WAGFS _IE}LNQANQF002)CALL UNIERMRIAI 9A? oAI IE(INDA .EO.3)CALL NORMALRcA1 .A2 .A) Ow (T) a A * SALES PAW MATERIAL IE(INDE .EO. RICALL UNIERMR(BI 9P2 .5) IE(INDP .EO. SICALL NORMALRISI (P? .9) _ EM (I) =.R * SALES GFNFQAL AND ADMINISTRATIVE IE(INDTHETA .EO. PICALL UNIERMR(THETA1 .THETAz IEIINDTHETA .EO. 3ICALL NORMALR(THETAI oTHETAa GA (T) a FDGA (T) + THETA * SALES MISCELLANEOUS IELINDDELTA .EO. 2)CALL UNIERMRCALL NORMALP(DELTAI oDELTAz MSCLIT) = MISCIT) + DELTA * SALES OVERHEAD FOH (T) a PETA * SALES SELLING SFLL (T) a FDSFLL (T) + GAMMA * SALES ) (T) + SPA (T) (T) + (T) + oTHETA) OTHETA) ODELTA) oDELTA) 138 pCNT(I)=PCAP(TOI) -2111-CQNIINUF CALL GFNFRAL(YVALODCNTOII .CAp) C DETERMINF TAX PAYMFNTS 911R (P(INDTYP oGT. 1)GO TO 21?: TXP = TXP)(T) GO TO ?)35 -L3125wIIQNTXP(T) DO 2130 (31.11 IVAL(I)= TXP3(T.I) PCNT(I)= pTXp(TQI) ?11n CONTINUF CALL GENFRAL(IVAL.DCNT.II.TXP) z.nh .QEIEQMINE SPECIAL MISC VALUFS P11: [P(INDMISC .GT. 1)GO TO P145 SDMISC = SPM19C1(T) GO TO 2155 2146 II:NMISC(T) DO 2130 181.!) “IVAL(I)=SPMISC3(T.I) PCNT(I)= pMI§C(TO!) Plfin CONTINUE CALL GENERAL(IVALOPCNT.11.SDMISC) PIER CONTINUE 0 C CAQH DISBURSEMFNTS CD_(T) 2 ON (T) + PM (T) 1 DIP (T) + TXP + DRP (T) ? CAP + DIVP (T) + DIVC (T) + MSCL(T) (P(ITIS OLE. R .090 ITIQ OECO 5O .09. ITIS .500 100 )179901700 1709 CONTINUE wRITE (OUT2.16)T + GA (T) + DRE + SPMISC WRITE (OUT2.1) SALES. CS (T). CQS (T). ALPHA .ACTSRV(T) wQITF (QUT2.3) CR (T). CAR (T). ND (T). COMSI (T). wQITE (OUT2.4) Dw (T). A . RM (T). B WRITE(OUT2.5) GA(T).THETA .MSCL( WRITE (OUT2.6) DIP (T). TXP . DRP (T). DRF (T). 1 DIVP (T). DIVC (T) . SPMISC CJTQQMCONTINUE C NET CASH FLOW CALCULATIONS NETCF (T) = C9 (T) - CD (T) XNCTCF(ITIS.T) = NFTCF(T) XACTSRV(ITIS.T)= ACTSRV(T) IROO CONTINUF CI- -PUNCH OUT NET CASH FLOW RFSULTS WDITF(6?.141§)((NETCF(J).J=1.7)OIT19) SPA (T) T).DFLTA CAP (T) + + 139 WRITF(62.1416)((NETC=(J).J=8.13).ITIS) .WRITE(62.1418)((ACTSRV(J).J=197).ITI3) RZDO 9R1”) WQITE(62.1419)((ACTSQV(J).J=R.I3)OITIS) WRITE CUT NETCASH FLOW RESULTS AT END OF EACH ITIS FOR 13 PERIODS AND CUMULATIVE NET CASH FLOW FOR TOTAL NETCF( Id) :0 DETERMINE CUMULATIVF MET CASH FLOW .00 P200 (=19)? NFTCF(I4)= NETCF(IA) + NETCF(I) CONTINUE XNFTCF(ITIS.)4)= NETCF(14) IF(ITIS .LE. 5 00R. ITIS .EQ. 50 .39. ITIS .50. 100 )2210.1720 CONTINUF “WRITEIOUT2.I417)ITIS.(NETCF(J).J=I.14) 179q 1791 1799 WRITF(OUT2.14?O) ITIS.(ACTSRV(J).J=IOI3) CONTINUE wRITE OUT NETCASH FLOW AS ARRAY FOR ALL SIMULATIONS wRITE(OUT2.I4?6) wRITEIOUT2.14?S) WRITE(OUT2.I4BO) DO 17?) I=I.POO WRITE(OUT2.14)7)I.(XNETCF(I.J).J=I.I4) CONT I NUF wRITE OUT ACTSRV ARRAY FOR ALL SIMULATION WRITF(OUT?.I4?7) WRITE(OUTP.I429) wRITE(OUT2.1431) DO 17?2 1:19700 WRITF(OUT2.1420)I.(XACTSRV(I.J).J=1.13) CONTINUE FNO SUDROUTINE UNIERMI(A.R.X) TYRE INTEGER A.R.X P: QANF(-)) x = A + (B-A) * P RETURN END SURROUTINE UNIFRMR(A.P.X) TYPF REAL A. B. X R: RANE(-I) X = A + (R-A) * R RETURN END SUSROUTINE NORMALI(EX.STDX.X) TYRE INTEGER FX.STDX.X 140 911M8000 DO 4 ISM)? C! = RANF("I) ASSUM=SUM + R X=STDX *(SUM- 6.) + EX RETURN END ."SUBRQUTINE NORMALR(EX.STDX.X) TYRF REAL EX.STDX. X SUM=O.n DO 4 I=I.1? R = RANFI-I) a 9UM=SUM + R ”_~.wwx:STDX4i(SUM- 6.) + EX RETURN END SURROUTINE GFNFRAL (A.R.N.X) DIMENSION P(S).CR(5).A(5) TYPE REAL R.CR ..“MUN_TYPF.INTEGER X.N .A C DETERMINE CUMULATIVE DISTRIBUTION CP(I)= 9(1) IF(N .GT. 1)GO TO 20 C DETERMINISTIC VALUE X: A(I) H.,GO IO 2°C 9“ DO I I: 2.N CD(I)= CD(I-I) + P(I) 1 CONTINUE R = RANF(-1) C pICKS POINT ON THE CUMULATIVE DISTRIBUTION . 1F (R .LE. CP(1))10.100 1') X =A(1) GO TO 200 Inn I=I 101 I = I + I IF(CP(I) .LE. IoOO)GO TO 10% X =.A(I'I) UDITE(6I.I)I 1 FORMAT(24H CR(I) GT 1.00 ERROR I= .14) GO TO ?OO IDS IF(R .LE. CP(I))110010I 11D X = A(I) LHWZQQHQONTINUE RFTURN FND 'DUN.3IOO3QOO SALES DISTRIBUTION IS UNIFORM 6598890 6508800 6694570 6684570 6684570 6541770 6541770 6541770 5171 09 OQQRRQO 6.704;")? “06573 1 0 7815197 57!) CARTIAL EXPENDITURFE HAR GENERAL 01 T=I IOOOOO 11QDOO IEODOO 0’1 T=P .1OQOOD. lqfifififi. I=OOOO O” T=3 InODOO IROOOO .159999” 0? 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Havana. oomnooa nomnomp “novomu wwfiwmvu cache. can“. mnuwnn. muonmvu vacno novvmw onwsmm oonnvo. «nmvumu oomvvvu «canoe: «cwnmn. nannvnu oaonvfiu mnvvnss mnfinmm. n~nnvmu oxvvn nomsmn- Hmvnasu canov vnvvomu o aomama :34; xm “I4081EORMAT (1H .IO(3X.F6.41) 1411 FORMAT (1H .6(3X.FS.3)) 1412 FCRMAT(1HO.6H MAXI .18.6H MAX2 .18.8H MAXZRA .18.8H MINBPA .18. 1 6H MAXE .IR.6H MAXa .IR.7H PER4= .Féoa) 1413 FORMAT (1HO=1OH CASHRAL. .IP) 141A FORMAT(1HO.19H CRSOLD(T) .4191 1415 FORMAT(PX.7IIO) 1410 F0RMAT(?X.611O) 14:3" FORP~5AT(1H1.??X.IRH NET INTEREST COST.11X. I 33H UNRESTRICTED ENDING CASH BALANCE) 14p1 FORMAT( 1H .3X.?(5X.6HTYRE-1.E’SX.6HTYPE-2.S .61-ITYRE-3.E‘»X.6HTYRE-4)1 Inga FORMAT(1H .3X.Q(3X.IR)) 1493 FOQMAT11H1.IAX.18H NET INTFRFST COST-12X. 1 33H UNRESTRICTED ENDING CASH BALANCE. 14P4 10¢: 1496 14:“ 14:0 n ? 1 156 18X. OHSTOCK OUT) FORMAT(1H.02X03(4X06HTYPF-104XoéHTYPE-ao4xo6HTYPE-3o4xo6HTYPF-4)) F00MAT¢4RX.¢7H INCLUDING COQT OF STOCKOUT) FOQmAT(1H 912(7X919)) FOQMAT(n¢?X.In)> FOQMAT(?X06'I99?XQIR)I TEST INITIALIZATION OF INTFDFST DATFS no 1:07 J=19N wDITF(OUT201409) INTI!(J)oINT12(J)oINT21(J)cINT??(J)oINT31(J)9 INT32(J)c INT41(J)o INT4?(J)vINTSPC1(J)oINTSPC2(J) Iqfi'f CONTINUF HITS 94mm 9407 n URITF (0UT291412) MAXIOMAXPqNAXZPAcMINRPAQMAX39MAX4QPER4 WRITE (OUT?01413) CASHBAL START OF PROCTSSING CASH FLOW THDOUGH FINANCING ALTEQNATIVES HO 7'7?q ITIS=10POO DPATH1N101419HNF’TCFCJ)0J=1‘|7) ”VAD(IN191410)(METCF(J)9J=QO13) FVAU(IN1o1418)(ACTSRV(J)oJ=197) DCAD(1N101410)(ACTSRV(J)0J=9¢13) NFTINTA(1)= 0 NFTINTA(2)= O NFTINTA(3)= 0 NFTINTA(4)= 0 DO 3405 J=1913 19:01: DFOMIN1(J) IPEO?=DFOMIN?(J) CALL UNIFRMI(tQFO1oIDF07oIDFC) DFONINIJI=IRFQ CONTINUE “DO 2407 J=1913 xspC1 = INTSPC1(J) xspcz = INTSDC2(J) IFofo O TOTPAID = C INTPAY3=0 - - DO 1701 T=1011 C RFPAYMENT PLUS INTERFST AT REGINNING OF PERIOD T FINTRN (T) = STFRP (T) + INTPAY (T) IF(T oFOo 1)?42002430 949° NCF(T)=NETCF(T) + INVINC(T) PCASBAL (T) = CASHBAL + NCF (T) .- .M_GO TO 2501 . 941” NCF (T) = NETCF (T) + INVINC (T) + INVSPC (T-l) - FINTPN (T) —‘ O .4 Z n n C POTENTIAL CASH BALANCE PRIOR TO FINANCING ACTIVITY FOR PERIOD T PCASRAL (T) = CASH (T-1) + NCF (T) C DIFFERENCE BETWEEN POTENTIAL CASH BALANCE AND REQUIRED MINIMUM IF(ITYPF .Ed. 1)STF1= STFl -5F1(T-1) -IFJITYPF_pFOo 2)9TF?= STFZ - SF2RP(T) IF(ITYPE .50. 3 .AND. T .GT. 3)STF3= STFB -SF3(T-3) . 1F 158 9A1? CONT I NUF C - _ DECISION FOR PERIOD T IF ( DIF (T) oGTo 0)490104920 C INVEST SURPLUS CASH 49“) INVSPCI (T) = DIF (T) * (1 - TCSPC) INVINc1 (T+1) = 01? (T) * INTSPC(T) IF‘ITIS OLE. 3 0090 ITIE 0700 SO COR. ITIQ 0E0. 100 ) IWRITE(OUT2c10)INVINC1(T+1)oDIF(T)oINTspc(T)qINVSPC1(T) GO TO 8099 497“ IF ( DIF (T) oEQo 0)4921¢4930 C NO ACTION TAKFN (L991 INVSPCI (T) :0 INVINCI (T+1) = O ,GO_IOH5999 C DIP(T) IS LES-S THAN ZERO C “1UST UEF SHORT TERM FINANCING C NFED TO ROOROW D I F ( T) aggn DIP(T): -DIF(T) GO TO (R000951OOoRPOOoS3fiO)oITYPE C TYRE 1 - LINE OF CREDIT “On" IF (MAX1 0E0. STFI) FOlloEOFO C TOTAL DOLLAR "=7AXINUI-"T ALREADY USED Up S‘."1 =1 INDICATES STOCKOUT {-7011 91"]. = 1 ETKOUTI = DIF (T) IE( ITIC- OLE. 3) ".‘IRITE(OUT?O3I )STKOUTI c .“PQSSIBLF STOCKOUT COQT GO To 5999 c How DO AMOUNT NEEDED AND AWOUNT AVAILABLE COMPARE 'TCpO AVAILI = MAXI " STFI C§O¢1 IF(ICOMB OCT. EEOMIN(T)) C I DIP(T) = DIF-(T) + (ICC'MD - REQMIN(T)) Cw » INCLUDE COMPENSATING QALANCF OF 15 DER CENT OF AMOUNT NEEDED (7;qu DIF(T) = DIF(T) '3" 101") C COMPENSATING DALANCE IS 15 DFRCENT OE LINE OF CREDIT C DIP(T) = DIP(T) + ICOMR qo¢1 DIP(T) = DIP(T) * 1.1: IF (AVAILl .LT. DIP(T))EOPPQFUBO c . NOT FNOUGH AVAILABLE G09? SFI (T) = AVAIL1 9W1 = 1 FUTKOUTI = DIF (T) - SFI (T) IF(ITIS OLE. 3) ‘.:‘IRITE(OUT2031)STKOUTI C POSSIRLF STOCKOUT COST -GO TO 5031 C ENOUGH AVAILARLE 159 C CALCULATIONS OF INTEDFST ANO QEDAYMENT .uECSCLSF11(I) = DIF (T) :Cq1 CONTINUF X11 = INT11(T) X1? = INTI?(T) IF (INDINTI ofOo 2) CALL UNIFRMRIXII) X12) XINT1) IF (INDINTI .r0. 3) CALL NORMALR(X110 x12. X1NT1) _“§fil(T) = SFI(T)/(1 — TXCI) M _WLW INTPAY (T+1) = SFI (T) * XINTI + INTPAY (T+1) STFPP (T+1) = STFQP (T+1) + SFI (T) RTFI = STFI + SF) (T) IF(ITI§ OLEO 3 OOQO ITIC OEQO 50 OORO ITIQ OEOO 100 ) IWRITF(OUT?04O)SF1(T)OINTPAY(T+1)99TF1OXINT1 n.. -_GQ 10 5999 C TYPF ? - TFDM LOAN 510“ CONTINU? IF‘ITIS OLEO 3 OORO ITIS OEQO 50 OOQO ITIS OEOO 100 ) IWRITEIOUT?o4?)8TF29€FPRP(T) IF(NAXP oLEO GTF2)GIOIOSIPO MC _ .TQTAL DOLLAR vAXIMUM ALREADY USED up r:IOI SwI = I STKOUTI = DIF (T) IF(ITI$ OLFO 3) WRIT=(OUT2031)STKOUT1 C POSSIBLE STOCKOUT COQT GO TO BQQQ .SIpQ AVAIL? = MAX? - STFR IF (DIF (T) OCFO AVAIL2)51?P05140 319? IF (AVAIL? 06:. MIszA )81?306101 L51:"? IF (AVAIL2 OLFO MAX2DA )512405130 C BORROW AMOUNT AVAILABLE “194 SF? (T) = AVAIL? ”SWJH=.1 STKOUTI = DIF (T) - SF? (T) IF(ITIS oLEo 3) WRITF(OUT2031)5TKOUT1 POSRIRLF STOCKOUT COST GO TO SIRO C GORDON PERIOD MAX 1 MUM :13“ SFE (T) = MAxaoA SW1 = 1 RTKOUTI - DIF (T) - SF? (T) IF(ITIS .LF. 3) WDITE(OUT2~31)STKOUTI POSSIBLF STOCKOUT COST GO To 5150 QIQOMIE_(‘DIF (T) OGEo MIN2PA )514105101 5141 IF (DIF (T) OLEO MAXPRA )514205130 0 fl P \' C “14‘ 160 nonnow AMOUNT NFFOFO _£F2 (T) = OIF (T) GO To FIFO OFTAILF OF TvnF 2 FINANCING CIDLIT REPAYTIIENT OF PRINCIF’AL AND INTEREST AMONG FUTURE DERIODS CONTINUF xaI = INT91(T) X22 = INT??(T) ” IF (INOINTg .F0. 2) CALL UMIFRMR(X21O xya. xINTg) IF (INDINTZ .FQ. 3) CALL NCQflALR(X210 x22. XINT2) qFP(T) = SF2(T)/(1 - Txce) STF” : STF? + FF? (T) INTOMY SF? (T) * XINT? QTQMY 15F? (T) INTONV = INTDHY / TLD GTOMY = STOMV / TLD IF(ITI§ .L:. 3 OORO ITIQ OTQO 50 OODO ITIQ OEQO 100 1‘.’!RITE(OUT? O41 ) INTDMVO ‘3TDPAYO’1TF?O RFTHT) 9XINT? K = T + TLP U-L= T T 1 DO 6160 J = Lo K Cf: 2QP(J)= STDP‘Y INT-DAY (J) = INTDMY + INT-DAY (J) QTFpD (J, = sTFFD (J) + STDMY CONTINUE IE(ITIS OLEO 3 OORO ITIQ OFQO '50 OOQO ITIS OEOO 100 IIWQITF(OHT2914F?)((IoINTPAY(I)) oI=LOK) 5.90" “$901 3201 R9¢n ERQG GO TO 509° TYDE 3 - COA”~1EQCIAI_ ”ADP—R IF (T OLEO 3)FQOBOF?OI CONTINUI' -IF(ITIS OLEO 3 OOQO ITIE OLOO 50 OORO ITIS oEQO 100 INDITF(OUT?050)STF3OSE3(T-3) IF(MAX3 CLEO STF3)5210052?O TOTAL. DOLLAR ’~'AXIMU3=‘. ALQEADV OUTSTANDING (”I = I “TKOUTI = DIF (T) IE(ITIE OLEO 3) ”DITE(OUTP031)RTKOUTI DOQQIRLE QTOCKOUT CORT CO T0 5997 AVAIL3 2 MAX",- - QTE3 IE ( DIF (T) .CTO AVAIL3)§2F‘1)O523D HO‘." DO [XII/"DUNT NIT-EDIT” ANT," Ab-"C’UNT AVAILAIBLF CONDAQE NOT FNOUCH AVAILA")LE CE? (T) = AVAIL? ) ) ) 161 cml = 1 HQTKQUTI : DIF (T) - €F3 (T) IF(ITIS oLEo 3) WQITF(OUT?IBI)STKOUT1 C POSSIRLF STOCKOUT COQT GO TO E?31 C ENOUGH AVAILAn-LF FPQO SFi (T) = DIF (T) “323.1 . CIDNTI NU? X3) = INT31(T) X32 = INT32(T) IF (INDINT3 oFOo 2) CALL UNIFPMR(X?19 X32) IF (INDINT3 .F0. 3) CALL NOQMALR14?) XIMTtL) IF (INOINTZ) 07(71- 3) CALL. N0727"\L_3-’.(X419 X490 XIHTA) cF4(T) = SF4(T)/(1 - TXCfl) ,INTPAY (T+1) = aFa (T) * xwaa + INTPAY (T+1) QTFQD (T+1) = STFQP (T+1) + QFA (T) ‘72TF4 = CRTFQ + 7F!) (T) IF(ITIQ OLE. 3 OOQO ITIQ .Fq. 50 .390 ITIS .330 100 ) 1‘”QITF(OUT?96?)SFZHT).7"'":.D.P("‘+2)oINTPI‘.Y(T+1)0XINTa CO TO 9009 _c __ “fiINANCIMG HAS PUVN FIN sHEQ to: DFQIOO T C Now MUST MAC)? ADJUSTMENTS T”) fDDZ’fDT‘J-Tf? F‘OD. DVTDIOD T+1 59o“ CONTINUE (P(SWI CF00 O) GO TO EOOO C‘TOCK‘ITISOITYDF) 2 Q.TOCK(ITISQITYDF{) + STi-(OUTI §W1=C IGOQTLCO¥TINUF IMVSDC (T) = INVODCI (T) INVINC (T+1) = INVINCI (T+1) CASH (T) = pcnvap (") + CPI (T) + SF? (T) + SF? (T) + SF4 (T) (W 1 - INTDAYE - IN‘.’.‘~DC (T) INTPAYB :O C n -TOTAL INCOWE TOO” IMTEHTFT “N SUDDLUO CASH TOTINC = TOTINC + IMVYNC (T) C TOTAL INTERFST PAID OM SHORT THDM EIMANCIHG TOTDAID = TOTOAIO + IMTDAY (T) (P(ITIS OLE. 3 0090 (TI: .90. 59 0330 (TI? .20. 100 ) IWQITE<0UTR~11)Tv CA¢H(T). INVIVC(T).INTDAY(T)oTOTINc.TOTPAIO 1791 CONTINU: NFT INTEREST CO$T FOD ITYDV INTNORC IS INTEWKST WCT DECEI”LD AT fimn OF LAOT PERIOD SPCNR IS INVESTFD SUPPLUC Crtd HGT :VCEIVLD BY END OF LAsT PERIOD INTNOPD IS INTEPFST MOT SAIO AT EVO OF LAST PERIOD FINNP IS FINANCING PPIFLICL” MO 0:12 AT inc 0? LAST PERIOD .IMTNOQC =_IMVINC(IA) [P(ITYPrT 0 F0. 1 .09. ITYDF oFOo 4))70901703 00000 {163 I7n9 CONTINUF ,..l[‘JTN(-‘D7>= INTOAY‘ )4) -. FINNP: STFQFH 14) GO TO 1710 17A? [P(ITYD: 0E0. I704 J=l3 + 1 ’< = 13 + 3 _ _._.-D_Q 1707 (um. FINNF’ = FINND + STF‘QD( I) 1707 CONTINUE GO TO 1710 170..“ (FHTYD'T ONE. J a 13 + 1 K "' 13 + TLP (no 1795 I=J9K I.'\.'Tr~'O.DD = INTMOD') + INTPAYH) FINNP: FINNP + STF‘PF—‘(H 1706 CONTINUF 1F(IT1§ .LF. 3 .OQ. _---_),l‘-'RIIE(OUT2~14972) ( ( I o INTPAH z ) )9 1:.) 0K) (P(ITIS .LE. 3 COR. ITIg .EC. 50 .OR. 1T1? .EQ. 1!‘.’QITE(OUTZII452) ((Iof-‘>TFRP(I))OI=JOK) 171A CCNTINUF TOTPAID - TCTINC NETTINT :3 P.,','-='TINT/\(ITYPE) = NF'TINT «Down - INTMODO - C.ASH14( ITYPE)= CASL‘H 13) + r’. “0 .9”. (TIC .EQ. ICC ) ”IFctTIs .LE. 3 .OQ. ITIa - IWRITE(OUT2915)NETINT0INTNOQC.SDCNDoIHTVOPDvFIHNPoCASH14((TYPE). 7’.)17’)4017OC‘T 2“) GO TO 1710 1717 0E0. 50 .09. I.” TNODC + (:1?th P [TYPE CASH/\(ITISOITYPF): CASHI‘HIT‘HDFI) NETA(ITISOITYDE)= NFTINTA(ITYPF) ,OINTNORC=O INTTKfiszo FINNP: O QDCNQ: O I7n’)(70thINLW' PUNCH OUT NFT INTEDEST COST .VWDITf-‘(éfi’c 14==OHNFTINTM I ) 9 1:1.4) (CARHIA(I)¢I=194) N?TINTA(1)=C NFTINTA!2)=O NF-‘TINTA(3) =0 NFTINTA(4)=O )0 COPJT IFJLJF— ‘.','_F2_I_Tff QLJT AQQAY or: NF-‘T INTFDF‘ST LJMQFTSTQICTEFD F‘NUING CA‘RH F‘IALMTCF (:ho P(FJD 164 WQITF(OHT?QI4?O) .MRITE(OUT291421) DO 773’1 I=1.?OO WPITE(OUT2914P?) ( (NET/H IOJ)9J=1'4) 9(CD‘F‘.HA( IoJ) 9J=1'4)) 71” CONTIFMFT DO 7735 I=IQPOO DO 7736 J=104 NETA(10J)= NyTA(IoJ) + .01 * STOCK(I¢J) CASHA(IOJ) = FASHA(IQJ) - .01'31‘5‘>TOC!<(IOJ) 716 CONTINUP 71: CONTINUF DO 7738 1:19?’C “DITF(6201450)(NETA(I0J)0J=104)9(CA9HA(IQJ)0J=104) 710 CONTINUG WDITF(OHT?014°?) WQITE(OUTPo149fi) WDITF(OUT7914?4) no 7730 I=lo?flo WRITF(OUT2q14?6) (NETA(I¢J)¢J=1¢4)0(CASHA(IoJ)oJ=194)g , 1(STOCK(IQJ)=J=19A) W10 CONTINUF END fiURDOUTINE UNIFPMQ(A9”!X) TYDF QEAL A. R. X (3?: QANF ( ‘1) .X ; A + (Q-A) * D OFTUPN FND QURQOUTINF NQDMALD(FX0CTDX0X) TYPF QFAL FXQQTDKO X QUM=O.” DO 4 I=1¢19 D = DANF(-1) 4 SUM=SUM + R X=STDX *(QUM- 6.) + FX QETUR’N END WSUBDOUTINF UNIFRmI(AoDcX) TYPF‘ INTFTGF-‘F-P A0530! P: 0AN¢(-1) X = A + (R-A) * D QCTUQN END ’UN139033000 '"O““”° ““0nnnn 4““nnnn ¢OOOOOO 4°OOOOO annnnfin annnnmn Anonnnn annnnnn REQMIM 165 0n0000 4000000 4000000 4000000 REQMIN foQQflD..--SC.QO°DQ0.5.0900??? 50.10000 :‘Tr‘fwm‘ 55““??? 5’7“???“ 56“.?"‘“'3 5"""55‘ C., 3595.315? dnnqnfi SOOOOOO 5000000 SCOOOOO REQMIN BOAOOO CASH BALANCE AT STAQT INOQPC IS UNIFOPM 35125 .09'7-2‘5 .0475 .047‘5 .0427; .032": .0475 .047?) .0475) .0525 .0732?) .0525_.O?i?511 0573‘; .0625 .0575 .0575 .0577.“- .O:25 .0577 .057?) .0575 .0625 0062.53 .0625 .062512 OQOQQD.115“O MAXI) PERI INDINT1 IS UNIFOQM 0625 .0625 .0575 .0575 .0525 .0525 .0575 .0575 .0575 .0625 .0625 .0625 .062511 0725 .0725 .0675 .0675 .0625 .0625 .0675 .0675 .0675 .0725 .0725 .0725 .072512 000000 1000000 50000 MAX2.MAX2@A.MIN29A INDINT? IS UNIFOQM ”fiqfiinQSO .060 .0600 .0550 .0600 .0600 .0600 .0600 .0650 00650 00550 99650?1_ 0773 .0775'.0725'.0725 .0675 .0675 .0725 .0725 .0725 .0775 .0775 .0775 .077522 OO°OOO MAX3 INDINT3 Is UNIFORM 0575 .0575 .0525 .0525 .0475 .0475 .0525 .0525 .0525 .0575 .0575 .0575 .05753 0675 .0675 .0625 .0625 .0575 .0575 .0625 .0625 .0625 .0675 .0675 .0675 .0675 ”002200.4500, MAX4.DEQ4 ’ INDINTa 15 UNIFOQM 0025 .0825 .0775 .0775 .0725 .0725 .0775 .0775 .0775 .0825 .0025 .0825 .002541 0690 .0925 .0575 .0075 .OHPS .0625 .0075 .0575 .0075 .0925 .0925 .0925 .092542 APPENDIX E OUTPUT FROM THE SIMULATION FINANCING ALTERNATIVES 166 OF THE Imfl $454 1' .mi; ' UNRESTRICTED ENDING CASH BALANCE TYPE'l 2781271 'm2538860 1892345 1779593 1817882 2237585 2149889 3657269 2397100 2536460 2024100 3415705 2348079 2545288 2292404 3173182 1475519 2564984 2261468 1895324 3382186 1292310 2530304 2772286 2651160 2805241 2638583 2502300 2121614 1952164 1784123 1932894 2903427 2537892 3316457 3268724 1858833 2053661 3191464 1776011 3083023 2939232 1875641 2660322 2717013 1575458 7725-2 2691574 ’2480480 1843516 1712207 1750921 2151734 2105024 3593518' 2325433 2447188 1965361 3255559 2275468 2454385 2218140 2844570 1404325 2515666 2187971 1868022 3319708 1273042 2440050 2689264 2583428 2730336 2577185 2455064 2064053 1914595 1717730 1848294 2814183 2455041 3251896 3187876 1774807 1998519 3122911 1712179 3011490 2847365 1835293 2596364 2638039 1518937 167 TYPE-3 2551124 .254024{ 1902913 1774878 1815532 2239539 2162510 ‘3667641 2409113 2264271 2043957 3424803 2343938 2563160 2310437 3180138 1496996 2569153 2278860 1902889 3396441 1317127 2525173 2780466 2657157 2817969 2649698 2503108 2139268 1961414 1784114 1946959 2909762 2541750 3316240 3278246 1862967 2050929 3200128 1740925 3089574 2940518 1883609 2658524 2723663 1570954 TYPE'4 2771012 1893314 1778126 1806458 2229895 2144701 3649160 2386876 2524416 2025214 3405964 2342231 2538088‘ 2285707 3162363 1480184 2557502 2257790 1983055 3371340 1315291 2520842 2757805 2643613 2795301 2636651 2500085 2120851 1957930 1781185 1929984‘ 2890897 2525061 3305295 3258939 1846427 '2531662‘ 2053852" 3182589 1773316 3071988 2925950 1877896 2645196‘ 2703803 1565818 2811192 3469112 2265571 2674525 2649830 1461442 2094547 2692337 3572491 1049947 2189244 2028896 2521429 3358196 2786528 2801962 2259241 2283206 1783169 2261729 3184619 2850830 1782205 2994983 2108792 1648413 2765259 1896008 2619423 2489976 2314578 2445655 1467090 2395830 2496069 2658907 2825809 2125735 3538885 2207979 2051818 2240917 3061947 2460371 2438851 3086173 2334199 3721536 2262082 2508491 2868451 2356127 2728945 2975463 2178012 2574165 2582153 1428340 2039861 2619296 3401885 1014455 2157371 1972051 2454501 3283548 2704277 2723653 2214380 2237677 1707039 2169781 3123860 2772990 1745309 2917626 2055375 1612257 2678916 1812975 2535561 2426847 2244552 2374026 1452593 2313864 2416635 2567169 2740571 2084283 3463618 2116662 1998442 2173641 2989233 2366247 2367871 3011766 2256439 3471116 2188383 2431209 2777985 2278198 168 2809966 3475817 2264468 2685301 2661171 1467993 2111001 2700397 3585035 1047824 2200108 2048107 2528681 3187235 2788887 2814396 2263704 2287823 1786611 2266442 3185764 2865310 1798756 2999395 2127669 1667170 2763919 1901768. 2628473 2503346 2317067 2453088 1477259 2400852 2498351 2650263 2836606 2130286 3540801 2207836 L916108 2248741 3070267 2470544 2444896 3083158 2343254 2729775 2275963 2521397 2883142 2361080 2797352 3456762 2256415 2664522 2644103 1473285 2097078 2678951 3558895 1063111’“ 2197912 2035118 2515630 3349004 2773820 2789905 2255021’ 2286723 1772768 2257827 3171516 2838732 1787156”. 2982407 2109676 1665819 2753489 1884605 2603877 " 2478033 2302358 2439071 1491436 2382929 2489041 2650302 2810552 2136713 3529949 2201408 2051377 2234542 3051492 2452133 2437626 3074651 2327608— 3712172 2258196 2496979 2858361 2341992 '1»; 1726172 3126530 2222428 1865833 2181416 1902198 3203204 2500306 2955217 2109446 2665469 1669621 2381651 2365531 1905458 2744318 3447729 2413689 3142076 2705748 2175848 2442821 2724564 2536355 1867259 4118733 2960604 2487764 2895319 2534203 2358488 2403942 2864246 2796350 2093554 2292931 2399959 2749202 1920085 2264171 3141486 2396743 3384239 1852506 1833677 3139738 3133326 2481474 2410583 2199286 2678430 2664421 1667563 3041551 2143083 1814247 2097302 1851141 2874939 2420954 2876410 2020428 2583188 1613138 2294252 2287890 1847623 2619718 3383249 2344585 3073552 2617521 2089858 2382029 267342 245715 1820344 4075754 2685865 2415647 2817498 2452129 2299143 2321581 2790055 2704578 2817932 2226065 2321442 2674285 1854206 2187650 3087814 2339083 2790063 1791110 1773536 3047149 3044857 2407197 2344151 2146254 2592778 2581169 169 1723314 3136212 2236261 1868878 1920095 1906783 3220193 2512043 2960815 2115054 2654948 1670669 2394083 2374472 1921451 2284698 3456464 2411113 3156711 2697765 2180621 2112994 2735836 2536118 1887161 4025757” 2961827 2500411 2907026 2540568 2373745 2409575 2880400 2800468 2905377 2300524 2398944 2762025 1918829 2270991 3150635 2405008 3255714 1870794 1827520 2879381 3134239 2496016 2421668 2208600 2688581 2672041 1726256 3110863 2210005 1856930 2173117 1897350 3190881' 2491581 2947003 20965591 2651999 1671421 2363441m 2356441 1900973 2729198 3434647 2405531 2695458 2170676 2433478'"_ 2718358m‘ '”” 2524794 1877601 4110603"“ 2946800 2475185 2880386 2521174 2360331 2391518pm"” 2852005 2782491 2882517 2279009 2389160 273747?"* 1923537 2255313 3132257’ 2392920 3374272 3127581MN"'“”" _...__— 1851135'"“'““— 1829464 3129672 3117500 2468180 2399895 2190334m 2659982 2651294 2745779 2681802 2679083 1830197 1925772 2963251 2520418 2618386 2362921 1058425 1398758 2525718 2992073 1857936 1742781 1884080 2717271 3418521 2797189 2559302 2127953 2230676 2195316 2161722 3109342 2510008 2490554 2951778 2722227 2503620 2657301 2596239 2639154 1674030 2375131 2765766 1981179 3198672 2169022 2197052 2757815 2138698 3157830 2471090 2963316 3230433 2649121 2069548 3076990 2191638 2662619' 2601073 2577609 1763888 1849638 2883716 2440764 2538411 2280224 1007918 1340494 2447225 2919023 1818741 1683635 1852797 2640104 3344659 2707728 2459494 2061270 2171730 2135763 2105031 3024482 2439505 2427305 2874320 2640808 2409500 2586804 2529934 2562939 1596913 2307177 2694495 1922047 3137424 2086166 2128381 2683327 2091998 3072079 2389509 2878879 3150018 2552672 2026169 2660644 2138522 170 2748728 2684021 2693441 1836225 1932169 2969398 2520250 2614620 2370511 1049672 1396864 2531456 3005222 1860046 1753703 1887438 2736800 3435287 2816131 2565191 2136812 1915956 2207340 2166404 3109202 2528781 2496105 2960746 2727550 2515192 2665993 2605033 2649287 1695296 2390223 2778113 1983679 3203106 2186983 2213256 2247304 2147485 3166120 2488025 2970189 3235508 2649713 2077919 2991648 2199352 2732573 2672133 2670458 1829868- 1921933 2953202 2505876" 2609684 2353316 1074436"" 1399907 2512565 2980419 1856991 1739085 1893334 2707585 3404917 2784267 2549802 2130930 2224923 """ 2185054 2157340 3099348" " 2501373 2473890 2938f28'" 2708073 2498236 2640632" 2582607 2625926 1671710 2372670 2756162 1975207 3190441 2167118 2187077 2744567 2145242 3144582 2461219 2948492 3217183 2641894 2070487 3064859 2186930 APPENDIX F TABLES OF DATA FOR THE CUMULATIVE PROBABILITY DISTRIBUTIONS OF THE FINANCING ALTERNATIVES 171 172 TABLE 2.--Data for the cumulative probability distributions of the financing alternatives at a minimum required cash balance of $3 million and a stockout penalty of 1 percent. URECB* Egllars Type l-A Type l-B Type 2 Type 3 Type 4 Thousands <1650 200 200 200 200 200 1650 193 192 191 193 193 1700 192 192 191 192 192 1750 192 190 189 192 192 1800 189 189 189 189 189 1850 187 185 185 187 187 1900 183 181 181 183 182 1950 181 173 175 179 180 2000 173 166 166 170 171 2050 164 163 163 161 164 2100 162 161 161 158 162 2150 160 158 157 156 160 2200 157 152 152 154 157 2250 151 147 145 144 151 2300 144 137 135 138 144 2350 136 130 125 130 136 1400 127 126 124 123 127 2450 124 120 115 118 124 2500 117 111 104 110 117 2550 109 106 100 97 108 2600 103 95 93 93 103 2650 87 84 80 81 87 2700 83 82 75 73 83 2750 79 72 67 66 79 2800 ’ 66 62 56 53 66 2850 58 56 50 46 58 2900 51 47 41 41 51 2950 45 43 38 38 45 3000 41 39 33 34 41 3050 0 0 0 0 0 * Unrestricted ending cash balances. 173 TABLE 3.--Data for the cumulative probability distributions of the financing alternatives at a minimum required cash balance of $4.5 million and a stockout penalty of 1 percent. URECB* Egllars Type l-A Type l-B Type 2 Type 3 Type 4 Thousands