STATISTICAL TRANSFORMATION OF PROBABILISTIC INFORMATION by MOON HOE LEE B. E., University of Malaya, I960 C. Eng., London, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Faculty Commerce and Business Administration We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1967 In presenting this thesis i n partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for ex- tensive copying of this thesis for scholarly purposes may be granted by the Head of ray Department or by his representatives. It i s understood that copying or publication of this thesis for financ i a l gain shall not be allowed without my written permission. Faculty of Commerce & Business Administration The University of British Columbia Vancouver 8, Canada Date August 1, 196? iii ABSTRACT This research study had shown various probable r a t i o n a l methods of quantifying subjective information i n a p r o b a b i l i t y d i s t r i b u t i o n with p a r t i c u l a r reference to the evaluation of economic projects by computer simulation. Computer simulation to give a l l the possible outcomes of a c a p i t a l project using th-3 Monte Carlo technique (method of s t a t i s t i c a l t r i a l s ) provides a strong p r a c t i c a l appeal f o r the evaluation of a r i s k y project. However, a p r a c t i c a l problem i n the a p p l i c a t i o n of computer simulation to the evaluation of c a p i t a l expenditures i s the numerical q u a n t i f i c a t i o n of uncertainty i n the input variables i n a p r o b a b i l i t y distribution. One serious shortcoming i n the use of subjective p r o b a b i l i - t i e s i s that subjective p r o b a b i l i t y d i s t r i b u t i o n s are not i n a reproduc i b l e or mathematical form. They do not, therefore, allow f o r v a l i d a t i o n of t h e i r general s u i t a b i l i t y i n p a r t i c u l a r cases to characterize input variables by independent means. At the same time the p r a c t i c a l d e r i v a t i o n of subjective p r o b a b i l i t y d i s t r i b u t i o n s i s by no means considered an easy or exact task. The present study was an attempt to suggest a s i m p l i - f i c a t i o n to the probloui of deriving a p r o b a b i l i t y d i s t r i b u t i o n by the usual method of d i r e c t l i s t i n g of subjective p r o b a b i l i t i e s . The study examined the possible a p p l i c a b i l i t y of four theoret i c a l p r o b a b i l i t y d i s t r i b u t i o n s (lognormal, Weibull, normal and t r i a n g u l a r ) to the evaluation of c a p i t a l projects by computer simulation. Both theory iv and procedures were developed for employing the four theoretical probability distributions to quantify the probability of occurrence of input variables i n a simulation model. The procedure established for f i t t i n g the lognormal probability function to three-level estimates of probabilistic information was the principal contribution from this study to research i n the search for improved techniques for the analysis of risky projects. A priori considerations for studying the lognormal function were discussed. Procedures were also shown on how to apply the triangular probability function and the normal approximation to simulate the outcomes of a capital project. The technique of f i t t i n g the Weibull probability function to three-level estimates of forecasts was adopted from a paper by William D. Lamb. The four theoretical probability functions wore applied to a case problem which was analyzed using subjective probabilities by David B. Hertz and reported i n the Harvard Business Review % The proposal considered was a $10/-million extension to a chemical processing plant for a mediumsized industrial chemical producer. The investigations of the present study disclosed that the lognormal function showed considerable promise as a suitable probability distribution to quantify the uncertainties surrounding project variables. The normal distribution was also found to hold promise of being an appropriate distribution to use i n simulation studies. The Weibull probability function did not show up too favourably by the results obtained when i t was applied to the case problem under study. The triangular probability V function was found to be either an inexact or unsuitable approximation to use i n simulation studies as shown by the results obtained on this case problem. Secondary investigations were conducted to test the sensitivity of Monte Carlo simulation outputs to ( l ) number of s t a t i s t i c a l t r i a l s ; (2) assumptions made on t a i l probabilities and (3) errors i n the threelevel estimates. TABLE OF CONTENTS P CHAPTER I. INTRODUCTION A G E # 1 The Research Proposal Terminology 3 * 3 J u s t i f i c a t i o n of the Problem of Study 4 Reasons f o r Adopting T h e o r e t i c a l P r o b a b i l i t y Distributions , 8 Rationale f o r Computer Simulation...... f II. 13 Research Methodology. 14 D e l i m i t a t i o n s of t h i s Study 17 Organization of Subsequent Chapters 18 REVIEW OF LITERATURE ON CAPITAL BUDGETING 20 C r i t e r i a of P r o f i t a b i l i t y 20 I n t e r n a l Rate of Return 21 Present Worth 23 Cost of C a p i t a l . . . . . . . . 2 3 Conventional Risk C r i t e r i a 24 A n a l y t i c a l P r o b a b i l i s t i c Models 25 P o r t f o l i o and Programming Models 28 C a p i t a l Budgeting under C a p i t a l Rationing. 28 vii CHAPTER III. PAGE STATISTICAL THEORY 30 Transformation of Skew D i s t r i b u t i o n s 30 Two-Parameter Lognormal D i s t r i b u t i o n . 31 Estimation of the Parameters f o r the TwoParameter Lognormal D i s t n b u t i o n Three-Parameter Lognormal D i s t r i b u t i o n . 34 36 Estimation of the Parameters f o r the ThreeParameter Lognormal D i s t r i b u t i o n 38 Algorithm f o r Estimating M, S and t 40 Weibull Distribution 41 Estimation of the W e i b u l l Parameters 44 Algorithm f o r Estimating a, b and c....... IV. 44 Triangular D i s t r i b u t i o n 46 Normal Approximation. 48 PROGRAMME PROCEDURES 50 Source of Data 50 Determination of Incremental Cash Flows 51 Determination of I n t e r n a l Rates of Return 52 Treatment of Simulated Data 53 Generation of Pseudorandom and Normally D i s t r i b u t e d Pseudorandom Numbers 54 viii CHAPTER V. PAGE 56 ANALYSES OF RESULTS AND FINDINGS Parameter Estimation................ 56 Main Simulation Studies 59 Sensitivity Analyses 67 Sensitivity to Number of Statistical Trials 68 Sensitivity to Tail Probabilities 6& Sensitivity to Errors i n the Three-Level Estimates VI. 81 SUMMARY AND CONCLUSIONS 98 Summary. 98 Conclusions 101 Future Research. 102 BIBLIOGRAPHY. 104 APPENDIX A. Values of Parameters for Input Variables f o r ThreeParameter Lognormal Distribution (.01 Tail Probabilities ).....« B. 110 Values of Parameters and Statistics for Input Variables for Two-Parameter Lognormal Distribu- tion (. 01 Tail Probabilities ) C. Ill Values of Parameters for Input Variables for Weibull Distribution (.01 Tail Probabilities) 112 ix APPENDIX D. PAGE Values of Parameters for Input Variables for Normal Distribution (.01 Tail Probabilities) E. 113 Frequency Distribution of Internal Rates of Return for 36OO Monte Carlo Trials (Lognormal Probabilities)...... F. 114 Frequency Distribution of Internal Rates of Return f o r 36OO Monte Carlo Trials (Weibull Probabilities) G. 115 Frequency Distribution of Internal Rates of Return for 36OO Monte Carlo Trials (Normal Probabilities )........ H. 116 Frequency Distribution of Internal Rates of Return for 3599 Monte Carlo Trials (Triangular Probabilities) I. 117 Frequency Distribution of Internal Rates of Return for 36OO Monte Carlo Trials (Empirical Probabilities ) J. Computer Programme for Finding the Three Parameters of the Lognormal Distribution. K. 119 Computer Programme for Finding the Three Parameters of the Weibull Distribution L. 118 122 Computer Programme for Monte Carlo Simulation, Using Lognormal Input Probabilities 125 X APPENDIX M. PAGE Computer Programme f o r Monte C a r l o S i m u l a t i o n , Using Weibull Input P r o b a b i l i t i e s N, Computer Programme f o r Monte C a r l o S i m u l a t i o n , Using'Normal I n p u t P r o b a b i l i t i e s 0. 143 Computer Programme f o r Monte C a r l o S i m u l a t i o n , Using Triangular Input P r o b a b i l i t i e s P. 134 152 S u b r o u t i n e s f o r R i s k A n a l y s i s ( f r o m Mr. D a v i d B. H e r t z ) l6l xi LIST OF TABLES TABLE I. II. PAGE Basic Input Data. Influence of Input Probabilities on Statistical Measures of Internal Rates of Return ($) HI. 16 63 Tests of Hypotheses that Simulated Frequency Distributions of Internal Rates of Return Are Normal Distributions IV. 64 Effect of Variations of Number of Monte Carlo Trials on Statistical Measures of Internal Rates of Return (#) (Lognormal Probabilities) V. 77 Effect of Variations of Number of Monte Carlo Trials on Statistical Measures of Internal Rates of Return ($) (Weibull Probabilities) VI. 78 Effect of Variations of Number of Monte Carlo Trials on Statistical Measures of Internal Rates of Return ($) (Normal Probabilities) VII. 79 Effect of Variations of Number of Monte Carlo Trials on Statistical Measures of Internal Rates of Return (#) (Triangular Probabilities) VIII. 80 Effect of Variations of Tail Probabilities on Statistical Measures of Internal Rates of Return ($) (Lognormal Probabilities)... 86 xii TABLE IX. PAGE Effect of Variations of T a l l Probabilities on Statistical Measures of Internal Rates of Return <S) (Weibull Probabilities).... X. 8? Effect of a Ten Per Cent Change to Input Data on Statistical Measures of Internal Rates of Return (#) (Lognormal Probabilities) XI. 95 Effect of a Ten Per Cent Change to Input Data on Statistical Measures of Internal Rates of Return ($) (Weibull Probabilities) XII. 96 Effect of a Ten Per Cent Change to Input Data on Statistical Measures of Internal Rates of Return ($) (Normal Probabilities) 97 xiii LIST OF FIGURES FIGURE 1. PAGE Two-Parameter Lognormal P r o b a b i l i t y Density Function 2. 33 Three-Parameter Lognormal P r o b a b i l i t y Density Function 37 c 3. W e i b u l l P r o b a b i l i t y Density Function 41 4. T r i a n g u l a r P r o b a b i l i t y Density Function 46 5. Normal P r o b a b i l i t y Density Function. 48 6. P r o b a b i l i t y D i s t r i b u t i o n of Simulated Rates of Return f o r 3600 T r i a l s 7. 6l Complementary Cumulative P r o b a b i l i t y D i s t r i b u t i o n of Simulated Rates of Return f o r 36OO T r i a l s 8. 62 E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on P r o b a b i l i t y D i s t r i b u t i o n of Returns (Lognormal P r o b a b i l i t i e s ) 9. 69 E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on Complementary Cumulative Distribution of ftetums (Lognormal P r o b a b i l i t i e s ) 10. 70 E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on P r o b a b i l i t y D i s t r i b u t i o n of Returns 71 (Weibull P r o b a b i l i t i e s ) 11. E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on Complementary Cumulative of Returns (Weibull P r o b a b i l i t i e s ) . Distribution xiv FIGURE 12. PAGE E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on P r o b a b i l i t y D i s t r i b u t i o n of Returns (Normal P r o b a b i l i t i e s ) 13. 73 E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on Complementary Cumulative D i s t r i b u t i o n of Returns (Normal P r o b a b i l i t i e s ) 14. 74 E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on P r o b a b i l i t y D i s t r i b u t i o n of Returns (Triangular P r o b a b i l i t i e s ) 15. 75 E f f e c t of V a r i a t i o n s of Number of Monte Carlo T r i a l s on Complementary Cumulative D i s t r i b u t i o n of Returns (Triangular P r o b a b i l i t i e s ) . . 16. 76 E f f e c t of V a r i a t i o n s of T a i l P r o b a b i l i t i e s on P r o b a b i l i t y D i s t r i b u t i o n of Returns w i t h 200 Monte Carlo T r i a l s (Lognormal P r o b a b i l i t i e s ) . . . . 17. 82 E f f e c t of V a r i a t i o n s of T a i l P r o b a b i l i t i e s on Complementary Cumulative D i s t r i b u t i o n of Returns w i t h 200 Monte Carlo T r i a l s (Lognormal Pr o b a b i l i t i es) 18. 83 E f f e c t of V a r i a t i o n s of T a i l P r o b a b i l i t i e s on P r o b a b i l i t y D i s t r i b u t i o n of Returns w i t h 200 Monte Carlo T r i a l s (Weibull P r o b a b i l i t i e s ) . . . 19. 84 E f f e c t of V a r i a t i o n s of T a i l P r o b a b i l i t i e s on Complementary Cumulative D i s t r i b u t i o n of Returns w i t h 200 Monte Carlo T r i a l s (Weibull P r o b a b i l i t i e s ) . 85 XV FIGURE 20. PAGE Effect of a Ten Per Cent Change to Input Data on Probability Distribution of Returns with 200 Monte Carlo Trials (Lognormal Probabilities).... 21. 89 Effect of a Ten Per Cent Change to Input Data on Complementary Cumulative Distribution of Returns with 200 Monte Carlo Trials (Lognormal Probabilities) 22. 90 Effect of a Ten Per Cent Change to Input Data on Probability Distribution of Returns with 200 Monte Carlo Trials (Weibull Probabilities) 23. 91 Effect of a Ten Per Cent Change to Input Data on Complementary Cumulative Distribution of Returns with 200 Monte Carlo Trials (Weibull Probabilities).. 24. 92 Effect of a Ten Per Cent Change to Input Data on Probability Distribution of Returns with 200 Monte Carlo Trials (Normal Probabilities) 25. 93 Effect of a Ten Per Cent Change to Input Data on Complementary Cumulative Distribution of Returns with 200 Monte Carlo Trials (Normal Probabilities) 94 xvi ACKNOWLEDGMENT Mr. William D. Lamb, of General Electric Company, Louisville, Kentucky, U.S.A., very kindly sent me a copy of his paper "A Technique for Subjective Probability Assignment i n Risk Analysis Problems", read at the Institute of Management Sciences, American Meeting, i n Boston i n April 1967. Similar appreciative acknowledgment i s due to Mr. David B. Herts of Mckinsey & Company, Inc., New York who sent me a number of Fortran II subroutines for risk analysis and to Professor Frederick S. H i l l i e r of Stanford University, California, who sent me a copy of his draft monograph, "The Evaluation of Risky Interrelated Investments." Mr. A.G. Fowler, of the Computing Centre, University of British Columbia, was kind enough to shift the Fortran II subroutines into Fortran IV. Generous programming assistance was provided, by the staff at the Computing Centre, University of British Columbia. Of special help was the sugges- tion from Mr. A.G. Fowler which provided the rationale for the programme on sorting of simulated data into a frequency distribution and the test programme for finding the real root closest to one of a polynomial prepared by Mr. W.J. Coulthard. CHAPTER I INTRODUCTION The research reported h e r e i n was d i r e c t e d a t s o l v i n g a s p e c i f i c .. problem i n the a p p l i c a t i o n of computer s i m u l a t i o n t o the evaluation of capital projects. The problem i s the d i f f i c u l t y encountered i n p r a c t i c e of l i s t i n g s u b j e c t i v e p r o b a b i l i t i e s over the range of an i n p u t v a r i a b l e , recognizing the unusual n o t i o n of p r o b a b i l i t y f o r e c a s t e r s o r analysts have t o contend w i t h . One serious shortcoming of the use o f s u b j e c t i v e p r o b a b i l i t y d i s t r i b u t i o n s i s t h a t they have no reproducible or mathemat i c a l forms. They do not, t h e r e f o r e , a l l o w f o r v a l i d a t i o n o f t h e i r general s u i t a b i l i t y t o c h a r a c t e r i z e economic v a r i a b l e s by independent means % This problem i s probably one of the reasons why the a p p l i c a t i o n of computer s i m u l a t i o n t o the evaluation of c a p i t a l p r o j e c t s has not been as w i d e l y and as r e a d i l y accepted i n p r a c t i c e as i t would be expected., since the technique was f i r s t suggested by Hess and QvtigLey'- i n I963 and l a t e r made known t o a wide readership by Hertz i n 1964. The present study explored the p o s s i b i l i t i e s of employing t h e o r e t i c a l p r o b a b i l i t y d i s t r i b u t i o n s t o f a c i l i t a t e the s p e c i f i c a t i o n of a p r o b a b i l i t y d i s t r i b u t i o n f o r s u b j e c t i v e data. The s t a t i s t i c a l p r i n c i - ples developed i n t h i s t h e s i s are a p p l i c a b l e i n general t o the •^Sidney W. Hess and Harry A. Quigley, "Analysis of Risk i n Investments Using Monte Carlo Techniques, "Chemical Engineering Symposium Series 42: S t a t i s t i c s and Numerical Methods i n Chemical Engineering (New York: American I n s t i t u t e of Chemical Engineering, 1963)1 55-63 2 David B. Hertz, "Risk Analysis i n C a p i t a l Investment," Harvard Business Review* 42:1 (January-February, 1964), 95-106. 2 transformation of any probabilistic information into a probability distribution. Though the focus of the thesis was on the application of the s t a t i s t i c a l principles to the evaluation of capital projects, an important area of interest to most industrial organizations, these principles could conceivably be applied to quantify the uncertainty of subjective data i n security portfolio analysis, for instance. The evaluation of a capital project often requires the making of decisions i n the face of uncertain information or future events. Typical projects i n long range planning characterized by economic uncertainty include new product development, f a c i l i t i e s expansion and research and development programmes. mous suras of cash outlays. be high. The major capital investments involve enorThe dollar penalty for a wrong decision can Thus, the capital expenditure decision i s one often fraught with risks. The uncertainty and complexity inherent i n the capital project evaluation process make i t rather desirable to analyze risk quantitatively. A probability distribution i s a means of portraying the uncertainties of the economic variables entering into the analysis and of quantifying risk i n the measures of profitability. Computer simulation enables one to deal readily with the complexities typical of large economic undertakings. 3 I. THE RESEARCH PROPOSAL I t was the objective of this study ( l ) to develop the theory and the procedures for f i t t i n g theoretical probability distributions (lognormal, normal and triangular) to three-level estimates of probab i l i s t i c information - the pessimistic, the most l i k e l y and the optimistic estimates; (2) to present the theory and procedures developed by Larnb-^ for f i t t i n g the Weibull probability distribution to three-level estimates of probabilistic information; and (3) to investigate the behaviour of the distribution of internal rates of return obtained by computer simulation for a proposed capital investment using the lognormal, Weibull, normal and triangular probability functions, each i n turn, to quantify the probability of occurrence of economic variables which enter the simulation model as input factors. II. TERMINOLOGY Computer simulation i s a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical models that describe the behaviour of a business or economic system (or some component thereof) over extended periods of real time. The Monte Carlo technique i s a simulation sampling procedure i n which a stochastic variate i s obtained by drawing randomly from the probability 3 William D. Lamb, "A Technique for Subjective Probability Assignment i n Risk Analysis Problems," (paper presented to the Institute of Management Sciences, American Meeting, Boston, Massachusetts, April 5-7. 1967) ihomas H. Naylor and ethers, Computer Simulation Techniques (New York: John Wiley & Sons, Inc., 1966), p.3. 4 distribution of a variable. A Stochastic variable i s a real-valued function defined over a sample space associated with the outcome of a conceptual chance experiment.-* A stochastic variate i s a sample value of a stochastic variable^. A f r a c t i l e or quantile of order j i n a set of values i s the value which i s associated with the cumulative relative frequency value of j , where j 7 i s a fraction between 0 and 1 ' . An algorithm i s a complete and unambiguous set of numerical procedures g leading to the solution of a mathematical problem . III. JUSTIFICATION OF THE PROBLEM OF STUDY The traditional approach to the evaluation of a capital project has usually been made on a deterministic basis. The practice i s to combine a l l the favourable single-valued forecasts into a single measure of investment worth and to use this measure of economic desirability as a single criterion for accepting or rejecting a proposed project. ^Naylor and others, op. c i t . , p.43. 6 Ibid. ' A . Hald, Statistical Theory With Engineering (New York: John Wiley & Sons, Inc., 1 9 5 2 ) , pp. 6 0 - 6 7 . %.D. Conte, Elementary Numerical Analysis McGraw-Hill Book Company, 1 9 6 5 ) , p. i x . Applications (New York: More recently, some writers' have advocated probabilistic analysis which aims to introduce aspects of risk and uncertainty explicitly into the methodology of evaluation. One promising probabilistic model i s the Monte Carlo technique, which simulates the possible outcomes of a project by a sampling procedure. The method may be applied using a hand calculator ^ or a computer. 1 Hertz showed that simulation might yield an expected rate of 11 return (14.6 f>) very different from the figure by combining a l l the best estimates taken. (25.2$) 12 Others and could, then, lead to a different decision to be had shown that when uncertainty concerning revenues, costs and project l i f e existed, the internal rate of return method was so sensitive to errors of estimation i n the variables as to limit severely the value of information provided by the expected rate of return alone. However, to apply the Monte Carlo simulation approach i t i s f i r s t necessary to be able to specify a probability distribution for each of the variables affecting an investment decision. The assignment of ^Hertz, op. c i t . ; Frederick S. H i l l i e r , "The Derivation of Probabilistic Information for the Evaluation of Risky Investments," Management Science, 9: 3 (April, 1963), 443-457; Richard F. Hespos and Paul A. Strassmann, "Stochastic Decision Trees for the Analysis of Investment Decisions," Management Science, 11:10 (August, 19&5), B 252- B 254. ^Hess and Quigley, op.cit.; Leon Warren Woodiield, "An Experiment i n Application of the Monte Carlo Method for Simulating Capital Budgeting Decisions Under Uncertainty" (unpublished Doctor of Business Administration's dissertation, Michigan State University, 19&5)t pp.18-19. 1{ ^Hertz, op.cit., p.103. M a r t i n B. Solomon, Jr., "Uncertainty and Its Effect on Capital Investment Analysis," Management Science, 12: 8 (April, 1966), B 334-B 339. 6 subjective p r o b a b i l i t i e s i s presently advocated. Advocators of the Monte Carlo technique have thus f a r assumed away or minimized the diffi- c u l t y of d e r i v i n g such e m p i r i c a l (subjective) p r o b a b i l i t y d i s t r i b u t i o n s Lamb pointed out, "one c r i t i c a l problem i n using p r o b a b i l i s t i c models i s the numerical encoding of uncertainty i n a probability distribution." 14 The use of a subjective p r o b a b i l i t y d i s t r i b u t i o n i s not without i t s disadvantages. Most f o r e c a s t s , f o r example, sales f o r e c a s t s , commonly not the e f f o r t of any i n d i v i d u a l or group. are I t w i l l then be a very d i f f i c u l t and an i n e x a c t task to assign p r o b a b i l i t y values to such composite estimates. Wagle mentioned a f i n d i n g by Dutton (of Esso) i n h i s experience w i t h obtaining s u b j e c t i v e p r o b a b i l i t i e s , "A d i s t u r b i n g f a c t was the noticeable tendency f o r estimators to spend l e s s time on the more uncertain v a r i a b l e s . " 1 5 More r e c e n t l y , Pessemier*^ suggested the a p p l i c a t i o n of the two- parameter lognormal and the two-parameter Weibull p r o b a b i l i t y d i s t r i b u t i o n s to sales analysis and t e s t market d e c i s i o n s . He advocated f i t t i n g the two-parameter lognormal and W e i b u l l functions to two-level estimates of sales forecasts - the p e s s i m i s t i c and the o p t i m i s t i c estimates. severe shortcoming of the use of the two-parameter p r o b a b i l i t y l^Hertz, o p . c i t . , pp. pp. B252-254. 14 99-101; A function Hespos and Strassraann, o p . c i t . , Lamb, op. c i t . , abstract. 15B. Wagle, "A S t a t i s t i c a l Analysis of Risk i n C a p i t a l Investment P r o j e c t s , " Operational Research Quarterly, 1 8 : 1 (March, 1 9 6 7 ) , 19. l^Edgar A. Pessemier, New Product D e c i s i o n s : An A n a l y t i c a l Approach (New York: McGraw-Hill Book Company, 1966), pp. 141-210. 7 i s i t s failure to incorporate the expected value or most l i k e l y estimate, which represents the forecaster's best judgment, into the analysis, Lamb*^ lately experimented with the application of the three-parameter Weibull probability distribution to Monte Carlo analysis. He proposed the f i t t i n g of the three-parameter Weibull probability function to threelevel estimates of a variable. His method i s a practical one and shows promise of being an adequate general approach to f i t t i n g theoretical probability distributions to probabilistic information. The present study was an attempt to pursue further along similar lines of research initiated by Lamb and Pessemier. In a complementary direction, linear and integer programming procedures^ and a sequential selection model^ have been developed for the selection of a portfolio of projects from among a number of alternatives. These portfolio models require as input data some measures of cen- t r a l tendency and dispersion of returns (usually the expected return and variance). The Monte Carlo analysis i s able to generate the statistical measures required by the use of programming models. If i t i s preferred to use analytical procedures to evaluate a project instead of simulation, the s t a t i s t i c a l principles developed for the transformation of probabilistic information into the lognormal or the Weibull probability function are also able to provide the s t a t i s t i c a l measures required by the use of ^Lamb, op. c i t . 1 ft Joel Cord, "A Method of Allocating Funds to Investment Projects When Returns Are Subject to Uncertainty, "Management Science, 10:2 (January, 19^4), 335-3^1; H. Martin Weingartner, "Capital Budgeting of Interrelated Projects: Survey and Synthesis," Management Science, 12:7 (Marck,1966),485-516. 19 G. David Quirin, The Capital Expenditure Decision (Homewood: Richard D. Irwin, Inc., 1967), pp. 223-248. 8 programming models. IV. REASONS FOR ADOPTING THEORETICAL PROBABILITY DISTRIBUTIONS The existing available methods for assigning probability d i s t r i - butions have not proved to be really satisfactory or easy to apply. The possible methods of constructing probability distributions are (1) direct derivation of subjective probabilities; (2) estimate of mean and variance; and (3) use of observed frequencies based on historical data. The f i r s t method consists of asking the forecaster(s) a series of questions concerning the chances or likelihoods a value of a variable w i l l or w i l l not be exceeded.. The answers to these questions are plotted as a cumulative relative frequency curve for that variable. appears to be a rational and practical one. The technique However, i t is not a method without i t s drawbacks or an easy one to apply and a search for a more objective method of obtaining probability distributions i s perhaps indicated. The second method of the use of an estimated mean and variance to convey a probability distribution i s rather unsatisfactory because the 20 method leads to "bell-shaped thinking." The two statistics of mean and variance are sufficient to determine a normal distribution but not a skew distribution. The use of observed frequencies based on historical data i s a Lamb, op. c i t . , p.2. 20 9 valid method i f these frequencies are relevant for predicting future events. The scope for using observed frequency distributions i n project planning i s naturally limited. A possible fourth method i s the use of theoretical probability distributions. I t i s known that many real processes approximate certain standard theoretical probability distributions. The lognormal probability distribution has been selected for a close examination of i t s applicability to computer simulation of economic projects as there i s a priori basis for believing i t to be a valid d i s t r i bution to describe random processes or changes affecting business data. If a variable x i s affected by many random causes, each of which produces a small effect proportional to x i t s e l f , the logarithm of x i s approximately normal-and x has a skew lognormal distribution. * 2 Naylor explained the circumstances for the use of the lognormal distribution: The lognormal distribution i s frequently used to describe random processes that represent the product of several small independent events. This property of the lognormal distribution i s known as the "law of proportionate effects" and provides the basis on which we can decide to assume that a lognormal distribution describes a particular random v a r i a b l e . 22 A more formal definition of the law of proportionate effect was provided by Aitchison and Brown: The law of proportionate effect then i s : A variate subject to a process of change i s said to obey the law of proportionate effect i f the change i n the variate at any step of the process Two J.F. Kenney and E.S. Keeping, Mathematics of Statistics, Part (Princeton: D. Van Nostrand Company, Inc., 1956), p. 122. 77 Naylor and others, p.99. 10 i s a random proportion of the previous value of the variate. ^ 2 Holt recorded a p r i o r i considerations f o r employing the lognormal f u n c t i o n as a p r o b a b i l i t y d i s t r i b u t i o n to describe f o r e c a s t errors thus: There are both general a p r i o r i considerations and reasons of mathematical convenience to recommend t h i s p a r t i c u l a r type of d i s t r i b u t i o n . As f o r the a p r i o r i considerations, i t i s not i m p l a u s i b l e to suppose t h a t sales to any one customer i n a period of time might be determined by the product rather than sum of a great many different random f a c t o r s . We can imagine, f o r i n s t a n c e , t h a t the orders placed by a company f o r a product are determined by the previous-period usage of the product m u l t i p l i e d by an adjustment f a c t o r which r e f l e c t s a f o r e c a s t of f u t u r e usage, m u l t i p l i e d by a scrap-loss f a c t o r , m u l t i p l i e d by a f a c t o r r e f l e c t i n g someone's d e s i r e to increase inventory holdi n g , and so on. The logarithm of s a l e s , I n S, would then be the sum of independent,, random f a c t o r s . As the number of these random f a c t o r s i n c r e a s e s , the d i s t r i b u t i o n of I n S approaches the normal d i s t r i b u t i o n according t o the c e n t r a l l i m i t theorem. Consequently, the sales r a t e , S, would be lognormally d i s t r i buted. To be sure, i f there were a very l a r g e number of customers and no c o r r e l a t i o n between the r e s p e c t i v e s i z e of order placed by each, then aggregate sales would s t i l l approach normality, but the orders are i n f a c t l i k e l y to be c o r r e l a t e d . I t would, however, not be surprising to f i n d t h a t the lognormal d i s t r i b u t i o n might f i t b e t t e r where the number of orders received per period i s not very l a r g e , as w i l l tend to happen w i t h wholesale data of the type analyzed here, than when dealing w i t h a l a r g e number of small orders, characteri s t i c of a r e t a i l o p e r a t i o n . ^ 2 Several w r i t e r s have recorded t h e i r observations on the t e s t of f i t of the lognormal f u n c t i o n to business or economic data. Holt ^ 2 3j. A i t c h i s o n and J.A.C. Brown, The Lognormal D i s t r i b u t i o n (Cambridge: U n i v e r s i t y of Cambridge Press, 1957). P.22. 2 24 Charles C. Holt and others, Planning Production, I n v e n t o r i e s , and Work Force (Englewood C l i f f s : P r e n t i c e - H a l l , I n c . , I960), p. 283. 5lbid, p.284. 2 11 found the lognormal f i t for sales data for electrical motor parts and cooking utensils to be good and concluded, "On the whole the evidence we have analyzed suggests that the lognormal distribution holds considerable promise as a useful approximation to the distribution of sales for many products, and this conclusion i s supported by other investigators. 26 " The FAO conducted a forecast for the world demand for paper using the lognormal distribution. ? 2 Aitchison and Brown'' devoted a whole monograph 1 of the lognormal distribution i n economics. to the uses The areas where lognormal theory wore applicable, they cited, included economics and sociology, physical and industrial processes, income statistics and consumers' behaviour. They noted: Expenditures on particular commodities, or the prices paid per unit of a commodity by individual families, are often approximately lognormal; . . . . . . Evidence of lognormality i n price statistics (the distribution of price changes over time for a large number of commodities) led Davies i n 1946 to advocate the use of the geometric mean i n index numbers. " It i s not inconceivable that other project variables l i k e production costs, investment outlays and project l i f e are subject to random causes that produce small changes proportional to the values of these variables. For instance, production costs may fluctuate as a result of considerable overtime to meet rush orders; as a result of 26 See, e.g. Martin J. Beckmann and F. Bobkoski, "Airline Demand: An Analysis of Some Frequency Distributions," Naval Research Logistics Quarterly, 5:1 (March, 1958), 43-51. ?Food and Agriculture Organization, World Demand for Paper to 1975. Rome, I 9 6 0 . 2 28 2 A i t c h i s o n and Brown, op.cit., pp. 100-105; 121-140. 9 l b i d , p. 102. 12 unforeseen labour disputes; as a result of normalizing work schedules; as a result of random variations i n overheads and as a result of other contingent expenses. Each of these factors may be expected to produce a random change proportional to an expected normal level of production costs. Another important consideration i n studying the lognormal distribution i s that i t i s a flexible distribution, basically skew but can take on an asymptotically normal form. A widely used authoritative text i n statistics stated: Experience shows that only a comparatively small number of the distributions met with i n practical l i f e can be described by the normal distribution. Distributions influenced by economic, psychological and biological iactors are generally . skew.3° Aitchison and Brown reported, "In economic data skew frequency curves are the rule rather than the exception. This isby no means an original observation, as we have made clear i n our introductory chapter."31 An important factor computationally i s that the lognormal distribution i s a f a i r l y convenient distribution to work with mathematically and the distribution i s amenable to computer simulations. 32 Weibull-'^ introduced a very convenient distribution computationally, which i s essentially an extension of the exponential distribu- 30Hald, op. cit., p. 159. 31 Aitchison and Brown, op.cit., p. x v i i J 3 Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability," Journal of Applied Mechanics (September, 1951)» 293-297. 2 13 tion. Lamb l i s t e d three advantages of the Weibull distribution: " ( 1 ) ' f i t s ' any form of information (symmetric or skewed i n either direction), ( 2 ) mathematical ease i n f i t t i n g the distribution, and ( 3 ) simplicity, hence economy, i n using the distribution i n a computer simulation program."33 Weibull demonstrated the distribution to f i t the physical 34 characteristics of a wide range of classes of materials. 35 Others have found the distribution useful i n equipment r e l i a b i l i t y studies. The normal distribution was experimented with because i t i s one of the better known and probably most frequently used distributions. The triangular distribution was V. RATIONALE FOR considered because of i t s simple features, COMPUTER SIMULATION Variabilities of forecasts of key factors like, demand, selling prices, investment outlays, production costs and project l i f e show up i n the future. The Monte Carlo analysis i s a simulation technique which can provide a s t a t i s t i c a l distribution of rates of return from a l l possible combinations of the values of the key factors resulting from fluctuations due to random causes. Complex interrelationships among the factors l i k e the price/ demand relationship can be maintained through the provision of several 33 Lamb, op.cit., abstract. 34 Weibull, op. c i t . -'-'John H.K. Kao, "A New Life-Quality Measure for Electron Tubes," Transactions, Institute of Radio Engineers, Professional Group on Reliabil i t y and Quality Control, 7 (April, 1956). 1-11. 14 sets of distributions for these factors: "Thus, i f price determines the total market, we f i r s t select from a probability distribution the price for the specific computer run and then use for the total market a probability distribution that i s logically related to the price selected." 36 The analysis of large economic projects usually involves many complex factors and interactions among them. Economic projects relate to future expectations and recourse to past performance of similar projects i s seldom available. Complex capital budgeting problems l i k e these are very amenable to computer simulation, akin to laboratory experiments. VI. RESEARCH METHODOLOGY After the problem had been formulated, attention was given to the derivation of equations and algorithms for f i t t i n g the lognormal, triangular, normal and Weibull distributions to three-level estimates of probabilistic information. Though the concept of f i t t i n g a probab i l i t y distribution to three-level estimates parallelled that of Lamb , 37 the derivation of the equations for determining the three parameters of the lognormal distribution differed from Lamb's derivation of the parameters for the Weibull distribution because of the different mathematical forms of the two distributions. The idea of f i t t i n g the triangular distribution to three-level estimates came as a suggestion from ^Hertz, op. c i t . , p. 101. 37 -"Lamb, op. c i t . 15 Professor John Swirles of the Faculty of Commerce and Business Administration, University of British Columbia. Developing the equations for the triangular distribution, however, was the investigator's own work. The equations for f i t t i n g the Weibull distribution to three-level estimates were adopted from Lamb's work. The formulae for the statis- t i c a l measures of mean, mode, median and variance for the lognormal and the Weibull distributions were collected together from various research publications, Data for the evaluation of an actual proposal, a $10/-million extension to a chemical processing plant for a medium-sized industrial chemical producer, was conveniently taken from Hertz's article, shown i n table I. The reasons for adopting Hertz's published data were ( l ) to provide a possible comparison of the output statistics using subjective and theoretical probabilities and (2) i t was f e l t that r e a l i s t i c data l i k e these were d i f f i c u l t to obtain i f a separate case study was attempted. Programme procedures, detailed i n chapter IV, were then evolved. The next phase of the work entailed the preparation of the Fortran programmes for the determination of the parameters for the l o g normal and Weibull distributions for the input data and the four Fortran programmes for the main Monte Carlo analysis each for one of the four theoretical probability distributions under study. TABLE I BASIC INPUT DATA Input Variables Most l i k e l y estimates 2^0,000 Market size (tons) ^10 Selling prices ($ / ton) Market growth rate (p.a.) 0.03 Share of market (p.a.) 0.12 Total investment required (million $ ) 9.5 10 Useful l i f e of f a c i l i t i e s (yr.) Residual value (million $) U.5 Variable costs ($ / ton) U35 Fixed costs (thousands $) 300 Pessimistic estimates 100,000 385 0 0.03 7 5 3.5 370 250 Optimistic estimates 3U0,000 575 0.06 0.17 10.5 15 5 5U5 375 NOTE: Pessimistic and optimistic estimates represent approximately 1% to 99% probabilities, i.e., there i s only a 1 i n a 100 chance that the value actually achieved w i l l be less than the pessimistic estimate or greater than the optimistic estimate. Source: David B. Hertz, "Risk Analysis i n Capital Investment", Harvard Business Review, U2 : 1 (January-February, I 9 6 U ) , 95-106 17 Subroutines for risk analysis written i n Fortran II were kindly furnished on request by Mr. David B. Hertz of McKinsey & Company, Inc., New York. however. The investigator did not use these computer subroutines, The investigator, unfortunately, did not have the main programme used by Mr. Hertz i n his article as this would have provided the programme procedures adopted by Mr. Hertz for combining the nine input factors and similar procedures could be adopted by the present investigations to achieve comparability of methods. A methodical approach to programme checkout was taken. Each computer routine of a main programme was tested for accuracy by independent runs. The routines were then assembled into a main programme. Each main programme was then test run to provide sample printouts of computations done by the computer to establish that i t would perform as intended. The internal rate of return figures printed out were checked by another independent (library) programme for finding the roots of a polynomial. Main simulation runs with 3600 s t a t i s t i c a l t r i a l s were carried out. Secondary simulation runs with 200 and 100 s t a t i s t i c a l t r i a l s were carried out to analyze the sensitivity of output statistics to various assumptions made. Results were then analyzed. VII. DELIMITATIONS OF THIS STUDY The programme procedures for calculating the internal rates of return did not include tax and depreciation considerations as these were considered non-essential refinements to the conduct of the present research. Considerations for tax and depreciation methods could be 18 written into the computer programmes without d i f f i c u l t y for a pract i c a l adaptation of the programmes. The internal rate of return was selected as a profitability index i n order to achieve comparability of method with that adopted by Hertz. Other criteria of profitability l i k e benefit-cost ratio and net present value could be substituted. There are other aspects of the capital budgeting problem under uncertainty which have net been specially taken into account i n the research undertaken. A brief reference i s alluded to some of these other aspects and brief mention i s also made of alternative approaches to probabilistic analysis and programming models i n the review of literature on capital budgeting. VIII. ORGANIZATION OF SUBSEQUENT CHAPTERS Chapter I I gives an overview of some of the c r i t i c a l viewpoints of the literature on capital budgeting. I t starts with a short critique of the two most widely advocated criteria of measuring the desirability of a project - internal rate of return and present worth. A critical review of some of the methods of incorporating risk into the decision criterion follows. Alternative probabilistic models for the evaluation of individual projects are briefly examined. A survey of the literature on the development of programming models for the selection of interrelated projects i s indicated. Brief mention i s also made of the special situations relating to budgeting under capital rationing. 19 Chapter III gives the s t a t i s t i c a l principles and derivation of formulae relevant to the present study. Chapter IV gives an outline of the procedures for computing cash flows and internal rates of return and for sorting simulated data into a frequency distribution used i n the computer programmes, and an explanation of the computer library programmes foi* the generation of uniformly distributed and normally distributed random numbers. The f i f t h chapter deals with the analysis of computer output statistics and the report of findings. The f i n a l chapter gives a summary of the major developments of the study and the conclusions drawn, and indications of possible areas for future research connected with this study. CHAPTER II REVIEW OF LITERATURE ON CAPITAL BUDGETING The subject of capital budgeting i s a topic of interest to many groups - business and public administrators, economists, accountants and engineers. Much has been written i n various publications of business, economic, accounting, engineering and scientific bodies on the subject but only a brief examination of the more topical viewpoints of the literature w i l l be reviewed here. I. CRITERIA OF PROFITABILITY A criterion of profitability which summarizes the economic desirability of a proposed project into a single index serves the purpose of providing a common basis for comparing alternatives. There i s general agreement among financial writers that incremental or net cash flows should be used to compute the measure of investment worth. 1 There i s broad agreement that an estimate of the cost of capital should be used as a yardstick of comparison. There i s less precise concensus of opinion i n the theoretical literature as to the correct measure of profitability to use. The two commonly discussed criteria that have some claims of theoretical basis are (1) internal rate of return and (2) present worth. The question of whether the reinvestment of the stream of net benefits generated from a project i s relevant to the For a definitive treatment of common concepts used i n capital budgeting, see Quirin, op. c i t . 21 evaluation process has not been clearly answered i n the literature. Internal rate of return. The internal rate of return i s , by definition, the rate of discount which equates the present value of net cash benefits to the present value of cash outlays. have been levied against i t . Two criticisms F i r s t l y , due to the existence of multiple internal rates of return, this measure does not always give a unique decision. Secondly, as the method i s purely a discounting process, i t appears d i f f i c u l t i e s may arisa as to the implicit assumptions of the reinvestment rates when projects are ranked by their internal rates of return. 2 Bernhard was one of the f i r s t to examine analytically the assumptions underlying the mechanics of computing the internal rate of return. He was of the opinion that, since the internal rate of return method did not give unambiguous answers, the method was not i n general a meaningful procedure for comparing projects. H© also stated that the internal rate of return and present value methods might lead to opposite rankings of two mutually exclusive proposals and that some writers had erroneously concluded that the two methods were completely equivalent. Teichroew, Robichek and Montalbano i n two stimulating articles^ in Management Science gave a valuable analytical treatment of the multiple internal rate of return problem and offered interpretative 2 Richard H. Bernhard, "Discount Methods for Expenditure Evaluation - A Clarification of Their Assumptions," The Journal of Indust r i a l Engineering (January-February, 1962), 19-26. 3Daniel Teichroew, Alexander A. Robichek and Michael Montalbano, "An A>xalysis of Criteria for Investment & Financing Decisions Under Certainty," Management Science, 12:3 (November, 1965)1 151-179; ."Mathematical Analysis of Rates of Return Under Certainty," Management Science, 11:3 (January, I965), 395-403. 22 s o l u t i o n s to a number of p o s s i b l e cases where m u l t i p l e r a t e s could occur. 4 Baldwin suggested a m o d i f i c a t i o n to the usual procedure f o r c a l c u l a t i n g the i n t e r n a l r a t e of r e t u r n . This was t o compound the stream of net b e n e f i t s a t the cost of c a p i t a l t o a terminal year, d i s count the stream of c a p i t a l outlays a t the same cost of c a p i t a l t o the i n i t i a l year and f i n d the r a t e of discount which equated the sum of the compounded net b e n e f i t s t o the sum of the discounted c a p i t a l outlays. This idea of Baldwin of adding a compounding process t o the discounting one had not appeared t o have been taken up anywhere else i n the l i t e r a l t u r e , except a recent d o c t o r a l d i s s e r t a t i o n ^ a t the U n i v e r s i t y of C a l i f o r n i a , Los Angeles, had devoted i t s e l f wholly to the concept. One d i f f i c u l t y w i t h Baldwin's method i s t h a t when p r o j e c t s w i t h unequal l i v e s are ranked, any choice of a t e r m i n a l year w i l l i n v o l v e e x p l i c i t assumptions. A s l i g h t l y d i f f e r e n t m o d i f i c a t i o n was suggested by Pessemier, which was t o discount the stream of c a p i t a l outlays a t the cost of c a p i t a l and to f i n d the discount r a t e which equated the stream of net b e n e f i t s t o the sum of the discounted stream of c a p i t a l outlays. To deal with the problem of unequal economic l i v e s Pessemier suggested a weighted average of the i n t e r n a l r a t e of r e t u r n computed as above and L Robert H. Baldwin, "How to Assess Investment Proposals," Harvard Business Review, J? (Hay-June, 1959), 98-104. 5 G u i l f o r d C a r l i l e Babcock, "Growth to Future Value as a Measure of Investment Worth," (unpublished Ph.D. d i s s e r t a t i o n , U n i v e r s i t y of C a l i f o r n i a , Los Angeles, 1966) 23 "the normal rate of return anticipated on capital invested i n the interval from the end of the proposal's l i f e until the end of th9 maximum ecnnomic l i f e being considered for competing alternatives."^ Present worth. There are two variants of this measure - benefit-cost ratio and net present value. Benefit-cost ratio i s the ratio of the stream of net benefits discounted at the cost of capital to the stream of capital outlays discounted at the cost of capital. Net present value i s the difference of the two discounted streams, instead of the ratio. Since the computation involves only a discounting process, i t appears that the reinvestment potential of the stream of net benefits generated, from the time they become available to the terminal year, have not been considered. Again, the same conceptual problem of unequal economic lives applies to this measure. Cost of capital. The definition of the cost of capital has for long ranained one of the most intractable problems of financial theory. Financial writers l i k e Solomon and Quirin had suggested a weighted average of the costs of equity and debt instruments, using the current market values of equity and debt instruments as weights, as a workable solution for the present.7 Pessemier, op. c i t . , p. 79. Quirin, op. c i t . , p. 142. 24 II. CONVENTIONAL RISK CRITERIA "Treatment of risk," English wrote, "has bothered most users of traditional methods. It has been somewhat unsatisfictory to take account of risk by adjusting the interest rate, by foreshortening expected l i f e , by understating expected income, or the l i k e , a l l on an arbitrary basis. The seeming precision of the calculations which are used does not seem justified, i n relation to the crudity of such arbitrary approaches to risk."8 In a highly theoretical article, Lintner showed that "there can be no 'risk-discount' rate to be used i n computing present values to accept or reject individual projects. In particular, the 'cost of capital' as defined (for uncertainty) anywhere i n the literature i s not the appropriate rate to use i n these decisions even i f a l l new projects have the same 'risk' as existing assets."9 In their preface to the second edition of their popular book on capital budgeting, Bierman and Smidt wrote: We are not satisfied that the cost of capital, implicitly including a risk discount i n addition to a measure of time value preference, can be used to reject investments with yields less than the cost of capital. There may be situations when we want to accept investments with yields less than the cost of capital.* J.M. English, "Economic Comparison of Projects Incorporating a Utility Criterion i n the Rate of Return," The Engineering Economist, 10 (Winter, 1965), 1-14. ^John Lintner, "The Valuation of Risk Assets and the Selection of Risky Investments i n Stock Portfolios and Capital Budgets, "Review of Economics and Statistics, XLVII: 1 (February, 19&5). 15. Harold Bierman, Jr. and Seymour Smidt, The Capital Budgeting Decision: Economic Analysis & Financing of Investment Decisions, Second Edition (New York: Maci4illan Company, 1966), p. v i i . 25 A very commonly used measure, especially i n situations characterized by uncertainty, i s the payback method. I t i s a practical simpleminded device to cope with business uncertainties i n want of a better method but, unfortunately, i t does not stand up to theoretical rationa- . l.i.z. mg1 .1 III. ANALYTICAL PROBABILISTIC MODELS One alternative approach to computer simulation suggested i n the literature i s the analytical mathematical procedure advanced by Hillier.* In this paper he derived the probability distribution of the 2 two measures of profitability, the net present value and the internal rate of return. One rather restrictive assumption i n Hillier's method was the assumption that the net present value of a project was normally distributed. If the net present value could not be assumed normally distributed, the suggestion was offered that Tchebycheff's inequality might be used to make weak probability statements about the net present value of a project. In H i l l i e r ' s later work^ he pointed out that the Tchebycheff's inequality was usually very conservative. The derivation of the cumulative probability distribution of the internal rates of return was based on the normality assumption of net present value. Usee Quirin, op. c i t . , pp. 1 2 211-212. H i l l i e r , op. c i t . F r e d e r i c k S. H i l l i e r , "The Evaluation of Risky Interrelated Investments" (draft monograph , Stanford University), p. 2 8 . 26 H i l l i e r had acknowledged Seymour Kaplan and Richard Bernhard's observations that the derivation of the cumulative probability distribution of internal rates of return based on the normality of net present value assumption was valid only under certain restrictive 14 conditions. Horowitz, using a deterministic model to examine the validity of Hillier«s assumption of the normality of net present value, found that the assumption of normality of net present value required the assumption of a linear relationship between each economic variable and net present value and had this to conclude: Indeed, setting coincidence aside, as long as probab i l i t y assessments are made with regard to the individual parameters determining present value, normality i n the underlying distributions w i l l effect normality i n the distribution of present values only when the relationship between present value and each probabilistic parameter i s linear. Moreover, i f regression techniques are relied upon to estimate the parameter [and these techniques are indeed becoming more prevalent i n this connection], the parameter w i l l be normally distributed. Admitting probab i l i s t i c properties for more than a single parameter, thereby necessitating compounding probabilities, could influence the distribution of present values i n a variety of ways, but achieving normality i n the distribution would almost surely be the result of a most fortuitous and exceptional turn of events.15 Wagle pointed out: . . . the starting point i n Hillier's analysis are the means and variances of the various cash flows emanating from different sources. Now i n many situations, these may not " , "Supplement to The Derivation of Probabilistic Information for the Evaluation of Risky Investments," Management Science, 11: 3 (January, 1965), 485-487. 1M l^Ira Horowitz, "The Plant Investment Decision Revisited," The Journal of Industrial Engineering, XVII: 8 (August, 1966), 422. 27 be known directly. What may be available are the means, and variances of factors which make up each cash flow. Wagle sought to present an improvement over H i l l i e r ' s model, 17 using data drawn from Hertz's article ' and using PERT statistics to compute the means and standard deviations of the key variables of market size, selling prices, etc. In his model he employed matrix theory to analyze correlation effects and also considered the l i f e of the project to be probabilistic, an improvement over Hillier's use of a single estimate. I t i s , perhaps, not generally known that the PERT statistics stand on doubtful theoretical ground even as approximations to the Beta distribution. Several writers had questioned the validity of the PERT statistics to describe the Beta distribution. Grubbs, for example, after a s t a t i s t i c a l examination of the PERT statistics, concluded: In looking at some of the pertinent parts of PERT, we apologize for raising some questions on the soundness of the theoretical bases, although we think i t only f a i r to do so i n spite of the fact that the Polaris program was so successf u l . We merely ask the question: Is not further work needed to validate the PERT statistics? We think so, for sound theoretical bases are of fundamental importance to the progress of science and Operations Research.* 8 ^Wagle, op. c i t . , p.15 *?Hertz, °P» lt« c Frank E. Grubbs, "Attempts to Validate Certain PERT Statistics or 'Picking on PERT'," Operations Research. 10 (1962), 915.. l8 28 IV. PORTFOLIO In u t i l i t y ment AND PROGRAMMING a prize-vanning analysis MODELS paper, Adelson t o evaluate t h e impact 19 c o m b i n e d game t h e o r y a n d 7 of uncertainty upon plant invest- decisions. 20 Van tion Home employed of interrelated Bayesian theory t o i l l u s t r a t e the selec- projects. 21 Quirin on Markowitz A presented, a n o t a b l e s e q u e n t i a l selection model based theory. large number of articles on t h e development of programming 22 models have valuable had appeared survey, appeared V. critique CAPITAL BUDGETING most firms. UNDER rationing Rationing before CAPITAL of capital or external considerations have ing and t h e time Weingartner o f programming techniques that RATIONING or another exists may b e i n t e r n a l as a r e s u l t o f market t o be given to both dimension had given a 1966. i n one form sif-imposition situations journals. and synthesis i n the literature Capital i n i n scholastic the cost t o some as a extent consequence imperfections. of capital i n the decision of Special under process. 2 rationThree 3 19 y through R.K. Adelson, Decision 20 James Combinations "Criteria Theory," Van Home, of Risky f o rCapital Operational Research Investment: Quarterly "Capital-BudgetingDecisions Investments," Management An Approach (March, 19&5).19-49. Involving Science, 1 3 : 2(October, 1966), B 84 - 3 9 2 . 2 i Q u i r i n , op. c i t . , pp. 223-240 22 v/eingartner, 2 cit., 3 F o r o p .c i t . a discussion pp.175-197 and Bierman of the capital and Smidt, rationing op. c i t . , pp. case, see Quirin, I8I-I89. op. 29 problems i n c a p i t a l rationing r a i s e d by L o r i e and Savage have become c l a s s i c a l i n the f i n a n c i a l l i t e r a t u r e . ^ 2 24 James H . L o r i e and Leonard J . Savage, "Three Problems i n Rationing C a p i t a l , " reprinted i n James Van Home (ed.), Foundations f o r F i n a n c i a l Management (Homewood: Richard D. Irwin, Inc., 1 9 6 6 ) , pp. 2 9 5 - 3 0 9 . CHAPTER III STATISTICAL THEORY The s t a t i s t i c a l principles presented i n this chapter provide the theoretical bases for the mathematical encoding of subjective information i n a probability distribution and for the generation of stochast i c variates i n simulation studies. Part of the body of s t a t i s t i c a l theory given here was taken from various research publications relating to other studies and from one unpublished work by William D. Lamb. Some of the principles presented represented the investigator's own research. In particular, the technique for the application of the three-parameter lognormal distribution to computer simulation of the possible outcomes of economic projects was the investigator's independent work. The methods for applying the triangular distribution and the normal approximation to computer simulation studies were developed by the investigator. Some of the formulae published i n articles and monographs relating to other studies and restated here were derived from f i r s t principles by the investigator i n order to understand the background on the use of the formulae. I. TRANSFORMATION OF SKEtf DISTRIBUTIONS I t i s theoretically possible to find a normal function z of a variable x the distribution of which i s skew. 1 •^ald, op. c i t . , pp. 159-187. The skew distribution 31 may then be transformed into a normal distribution. The cumulative distribution function, P(x), of the skew distribution may then be written as P (x) = ( 1 / ( 2 T f ) ) l / 2 exp (1) ( - v / 2 ) dv 2 -oo =F(z) where z i s a function of x and v i s a variable Of integration. The f i r s t derivative of equation (1) gives the probability density function p (x) = ( 1 / ( 2 T T ) 1 / 2 ) exp ( - z / 2 2 ) dz (2) One practical form of z i s given by z = ( g (x) - m ) / s (3) where the transformation function g(x) i s normally distributed about a constant m with standard deviation s. The transformation function g(x) = In x i s used i n the derivation of the probability density function for the two-parameter lognormal distribution and the transformation function g(x) = In (x-t) i s used i n the derivation of the probab i l i t y density function for the three-parameter lognormal distribution. II. TWO-PARAMETER LOGNORMAL DISTRIBUTION A random variable x has a lognormal distribution i f the natural 2 logarithm of the random variable, In x, i s normally distributed with Although the distribution i s not so restricted, only natural logarithms w i l l be discussed. 32 mean m and standard deviation s-^. Using the transformation function g(x) = In x the cumulative distribution function i s defined by equation ( l ) as P (x) = F (z), (0<x<co) W where z = (In x - m) / s The f i r s t derivative of equation (4) with respect to x gives the probability density function (x) = ( l / ( s x (2TT) / )) exp (- (In x-m) 1 P 2 which looks l i k e figure 1. 3Aitchison and Brown, op. c i t . , pp. 7 - 9 . l/2 / 2 s 2 ) (5) 33 1.0 Variable x FIGURE 1 TWO-PARAMETER LOGNORMAL PROBABILITY DENSITY FUNCTION The variable x has a positively skew, unimodal, continuous distribution, which i s completely specified by the two parameters m and s. The variable cannot assume zero values since the transformation g(x) = In x i s not defined for x = 0. The lognormal fractiles of order j are obtained from x . • = exp ( m + z . s ) 3 3 where Zz i s the standard normal variate. lognormal fractiles are given by x ^ = exp ( m - 1.28 s) and x Q = exp ( m '+ 1.28 s ) (6) For example, the 0.1 and 0.9 Equation ( 6 ) provides the basis for generating the two-parameter lognormal variate i n simulation studies. The i moment about the origin i s given by th L = exp ( i m + 1/2 i ± 2 s ) (7) 2 The mean, mode, median and variance of the lognormal distribution are given by x = exp ( m + s / 2 ) (8) x (9) 2 = exp ( m - s c 2 ) x ^ = exp (m) s III. 2 x = exp ( 2 m + 2 s (10) 2 ) - exp ( 2 m + s 2 ) (ll) ESTIMATION OF THE PARAMTERS FOR THE WO-PikRAMETER LOGNORMAL DISTRIBUTION L Consider two fractiles x^ = exp ( m + z^ s ) and x. = exp ( m + z . s ) where x. < x.. ^Aitchison and Brown, op. c i t . , pp. 40-42. 35 Then, (In x ± - m)/z i = (In x. - 3 m) / - ' °J so that m = ( z.. In x ± - z ± In x.. ) / (z^ - z ) i (12) Since In x^ - z . s = In x. - s z. th en s = (In Xj - In J^) / (aj - (13) z ) ± The estimates for m and s for the 0.01 and 0.99 fractiles are given by m = ( Inx + In s = (In x -lnx )/2 (14) and j 0 1 ) / (15) 4.6526 Aitchison and Brown state that the maximum efficiency of the method of estimation by fractiles i s attained when the fractiles are symmetrically placed. Maximum efficiency of 81% i n estimating m tained with fractiles of order 0.2? and 0.73. i s ob- Maximum efficiency of 65$ 2 i n estimating s i s obtained with fractiles of order 0.07 and 0.93. The joint efficiency of 60$ i n estimating m and s i s obtained with fract i l e s of order 0.1 and 0.9. 36 IV. THREE - PARAMETER LOGNORMAL DISTRIBUTION Consider a random variable x such that a simple displacement of x, x' = x-t i s lognormally distributed.^ The range of x i s t<_x < co where the location parameter t defines the lower bound value for x. Assume that the natural logarithm of (x-t) i.e. In (x-t) i s normally distributed with mean M and standard deviation S. Using the transformation function g(x) = In (x-t) the cumulative distribution function i s given by P (x) = F (z), (t < x<°o) (16) where z = ( In (x-t) - M) / S The f i r s t derivative of equation (16) with respect to x gives the probability density function p(x) = ( 1/(S (x-t) (2TT) / ) ) exp; (- (ln(x-t)-M)/2S ) 1 2 ^Aitchison and Brown, op. c i t . , pp. 14-15. 2 (17) 37 1.0 c •P tf i—I (D « Variable x FIGURE 2 THREE-PARAMETER LOGNORMAL PROBABILITY DENSITY FUNCTION The density function i s roughly depicted i n figure 2. The variable x has a positively skew, unimodal, continuous distribution, which i s completely specified by the three parameters M, S and t. The two-parameter distribution i s the special case for which t = 0. The lognormal fractiles of order j are given by x. = exp ( M + z^S) + t J « where z. i s the standard normal variate. 3 (18) 38 Equation (18) provides the basis for generating the threeparameter lognormal variate i n simulation studies. The i ^ * moment about t are given by 1 L = exp ( i M + 1/2 i ± 2 S 2 ) (19) The mean, mode, median and variance of the distribution are given by (20) x = exp ( M + 1/2 S ) + t 2 X = exp ( M - S q )+ t 2 (21) (22) x ^ = exp (M) + t s 2 x = exp ( 2 M + 2 3 2 ) _ exp ( 2 M + S 2 ) (23) V. ESTIMATION OF THE PARAMETERS FOR THE THREE-PARAMETER LOGNORMAL DISTRIBUTION. Consider two fractiles x^ = exp ( M + z S ) + t i and x. = exp ( M + z S ) + t 3 3 i where x^ < X J Then (In (xi-t) -M)/ZJ, = ( In (x - t ) -M)/z. 3 3 so that M = (z. ln(x -t) - z 3 1 ± In (x -t) ) / (z.-z*)A 3 3 (24) 39 Since I n ( X i - t ) - z S = I n ( x - t ) -Z..S ± then S = (In ( x - t ) - In (x -t) ) 7 i J From equation . (z -z ) (25) ± J (25) t = (xj-Xi From equation exp ( (zy.li) S) ) / (1- exp ( ( z ^ - z ^ S ) ) (26) (21) M = In (x -t) + S (27) 2 0 E l i m i n a t i n g M from equations (24) and (27) gives S (z. 2 ln(x -t)-a i i I n (x - t ) ) / ^ . ^ ) - I n ( x - t ) c (28) For symmetrical f r a c t i l e s equations (26) and (28) reduce to t = 3^ - exp (2zS) (29) and S 2 = ( l n ^ t ) + where z. = - z = / z / i and 3 3 "1-i la{x t))/Z limir - In (x -t) Q (30) Algorithm f o r estimating M, S and t Assume that the most l i k e l y estimate XQ and two outboard values, x., the pessimistic estimate and x., the o p t i m i s t i c estimate, of a variable are known. The steps f o r the determination of M, S and t by the NewtonRaphson i t e r a t i v e procedure 1. are as follows: Obtain the values of the standard normal variates z^ and z . from a 3 table of the standard normal d i s t r i b u t i o n , corresponding to t h e i r cumul a t i v e p r o b a b i l i t y values of i and j . 2. Make an i n i t i a l guess of S, say = 0.1. 3. With the assumed value SQ_, compute t ^ from equation 4. L.H.S. of equation (28) = S-j . 5. Compute R.H.S. of.equation (28). 6. Define f 7. Select another value of S, say Sg = (26). 2 x = R.H.S. - L.H.S.-^ 1 0.2. 8. Repeat steps 3 to 6 and define f g = R.H.S.g-L.H.S.g 9. Use the Newton-Raphson method to s e l e c t a better approximation of S, thus k+1 S = k " k 3 f < k- k-l) / < k S S f - k-l> f 10. Continue i t e r a t i n g u n t i l f ^ i s made as small as desired, say 11. With the f i n a l values of S and t , compute M from equation (27). the equations (26) to (28) 0.0001. have been numerically solved f o r M, S and t . R a i s e r S. Kunz, Numerical Analysis (New York: McGraw-Hill Book Company, Inc., pp. 1957), Thus, 10-12. 41 VI. WEIBULL DISTRIBUTION Variable x -H FIGURE 3 WEIBULL PROBABILITY DENSITY FUNCTION The cumulative d i s t r i b u t i o n f u n c t i o n of the three-parameter 7 d i s t r i b x i t i o n named a f t e r W e i b u l l has the simple form P(x) = 1 - exp for c £ x g oo ( - ((x-c) / b ) ) and a a (3D o where a = shape parameter b = s c a l e parameter c = lower bound l o c a t i o n parameter. The complementary cumulative i s R(x) = exp ( - ( (x-c) / b ) W e i b u l l , op. c i t . a ) (32) 42 The f i r s t derivative with respect to x of equation (31) gives the probability density function p(x) = (a/b ) (x-c) a 41 exp. ( - ( (x-c) / b 1 a )) (33) The Weibull density function i s roughly depicted i n figure 3. The density function i s a continuous distribution and i s an extension of the ecponential distribution. I t i s unimodal when the shape parameter a i s greater than unity. When a=l, the Weibull distribution reduces to the exponential distribution. The density function i s approximately 8 symmetrical when the shape parameter i s about 3.5. For values of the shape parameter greater than 3.5» "the distribution i s negatively skewed i.e. "tailing off to the l e f t " and for smaller values, the distribution is positively skewed i.e. "tailing off to the right". The generation of a Weibull variate i s easily accomplished by q the inverse transformation method . 7 If r i s a uniformly distributed number on the interval 0 to 1, then r = P(x) or r = R(x) since (1 - r) i s also a uniform random number. The Weibull variate i s then given by the formula x = b ( - In r J / * + c 1 ^Lamb, op. c i t . , p.15. 9 Nayl or and others, op. c i t . , p. 70. (34) 43 The i^^moment about the o r i g i n i s 10 L =l^Lo ± i i c i _ k bk T (1 + k/a)/ ( k j ( i - k ) j ) (35) where T i s the gamma f u n c t i o n . 1 1 The mean and variance are obtained by s u b s t i t u t i n g i = 1 and 2 i n equation (35). Thus, (36) x = b T ( 1 + 1/a) + c s = b 2 2 T (1 + 2/a) - T 2 ( 1 + 1/a) (37) The median may be found by s e t t i n g r = 1/2 i n equation (34), namely x = b ( - I n l/Z) ^ 1 O + c (38) S e t t i n g the f i r s t d e r i v a t i v e of equation (33) w i t h respect t o x t o zero, the mode may be obtained, thus x Q = b ( (a-l) / a ) l / a + c (39) The two-parameter Weibull d i s t r i b u t i o n and i t s c h a r a c t e r i s t i c f u n c t i o n s are obtained by s e t t i n g c = 0. 10 Kao, op. c i t . , p. 10 ^ F o r tabulated values of gamma f u n c t i o n s , see Herbert B r i s t o l Daght, Tables of I n t e g r a l s and Other Mathematical Data, 4th edit i o n (New York: The MacMillan Company, 1961), p. 260. 44 VII. ESTIMATION OF THE WEIBULL PARAMETERS 12 Let = probability that x w i l l be less than the pessimistic estimate x^ ; P. = probability that x w i l l be greater than the optimistic estimate x.. *J Assume that x.. , x., P. , P^ and x , the most l i k e l y estimate of i j x J o a variably are known. Rearranging the terms of equation (39) gives b = ( a/ (a-l) ) l / (x -c) a (40) o Substituting the expression for b and the known values of the estimates into equation (32) gives 1 - P. = exp ( - ( (a-l)/a) ( ( x ^ c ) / (x^-c) ) ) a (41) and P J = exp (- ( (a-l)/a) ( (x .-c)/(x -c) ) ) 3 a o Algorithm for estimating a,b and c Solve equations (41) and (42) for a and c by the NewtonRaphson iterative procedure.* 1. 3 Make an i n i t i a l guess of a, say a^ = 2 . 12 See Lamb, op. c i t . , pp. 13-15. * Kunz, op. c i t . , pp. 10-12. 3 (42) 45 2. Solve equation (41) f o r c, i . e . c l = (*i - Vl> / - i (1 d } where = (a-L (-Ui ( 1 - P ) ) / - 1 ) ) ± 3. l / a l S u b s t i t u t e a^ and c^ i n t o equation (42) to o b t a i n the complementary- cumulative f u n c t i o n P f o r these two values. 1 4. Define f = P. - ? 5. S e l e c t another value of a, say a,-, = 1.5. 6. Repeat steps 2 to 4 and d e f i n e f 7. Use the Newton-Raphson technique to s e l e c t a b e t t e r approximation of ± ? = P. - P„ a i.e. ^+1 8. = \ ~ k f ( a k " k-l a Continue i t e r a t i n g u n t i l f ) 1 ( f k - f k _!) i s made as small as d e s i r e d , say 0.0001. With the f i n a l values of a and c, compute b from equation (40). the equations (40) to (42) have been numerically solved f o r a, b and c. Thus, 46 VIII. TRIANGULAR DISTRIBUTION 1.0 O a m b Variable x FIGURE 4 TRIANGULAR PROBABILITY DENSITY FUNCTION Assume that the pessimistic, most l i k e l y and optimistic estimates a, m and b of a variable are known. As the area under the probability density curve i n practical cases may be assumed to be unity, the height of the triangle (figure 4) i s given by h = 2/ (b-a) (43) Using similar triangles, the probability density functions may be derived and are given by the formulae p(x) = 2 (x-a) / ( (m-a) (b-a) ) (44) for a < x < m. and p(x) = 2 (b-x) / ( (b-m) (b-a) ) for m < x < b. (45) 47 The cumulative density functions are given by P(x) = (x-a) / ( (m-a) (b-a) ) (46) 2 for a < x < m and P(x) = 1 - (b-x) / ( (b-m) (b-a) ) (47) 2 fcr m < x < b If r i s a uniform random number on the interval 0 to 1 , the application of the inverse transformation method* leads to the genera4 tion of triangular variates by the following formulae x = for a Cr (m-a) (b-a) + a (48) x < ra and 1/2 x = b - ( (1-r) (b-m) (b-a) ) (49) for m<x < b The mean and variance of the distribution are given by = (a + m + b)/ 3 x s = ( (b-a) + (m-a) (m-b) ) / 18 2 2 14 Naylor and others, op. c i t . , p. 70. (50) (51) 48 IX. NORMAL APPROXIMATION a> -P H Variable x FIGURE 5 NORMAL PROBABILITY DENSITY FUNCTION Assume that the pessimistic, most l i k e l y and o p t i m i s t i c estimates a, m and b of a variable and the p r o b a b i l i t i e s r e l a t i n g to the pessimistic and o p t i m i s t i c estimates are known. The standard deviation may be estimated as follows: s = (a-b) / Z where Z i s the number of standard deviations ab. (52) corresponding to the range For example, with 0 . 0 1 l e f t and r i g h t t a i l p r o b a b i l i t i e s , z - 4 , 6 5 2 6 . With the mean and the standard deviation known, the normal v a r i a t e i s computed from the expression x = m + zs (53) where z i s a standard normal variate, which may be generated by a computer routine. CHAPTER IV PROGRAMME PROCEDURES This chapter describes the procedures used i n the computer programmes for computing incremental cash flows, internal rates of return, and for the sorting of simulated data into a frequency d i s t r i bution. A description of the statistics printed out by the main pro- grammes i s stated. The methods for the generation of random uniform numbers and random standard normally distributed numbers used i n the library subroutines available at the Computing Centre, University of British Columbia, are briefly outlined. I. SOURCE OF DATA The basic data used i n the simulation runs were drawn from Hertz's article i n the Harvard Business Review and i s shorn i n table I on page 16. The project under study was a proposal to consider a $10/-million extension to the existing f a c i l i t i e s of a chemical processing plant for a medium-sized industrial chemical producer. The basic data might be grouped under three categories: 1. Market analyses, which provided the estimates for the total market size, market growth rate, the firm's share of the market and selling prices. These factors were subject to uncertainty as they depended on future anticipations of consumer acceptance of the product. 51 2. Investment cost analyses, which provided the estimates for the total investment required, useful l i f e of f a c i l i t i e s and salvage value of the f a c i l i t i e s . These analyses took into account the service l i f e and operating cost characteristics expected i n the future, which were subject to error and uncertainty. 3. Operating and fixed cost analyses, which provided estimates for variable costs and overhead expenses, which were also subject to uncertainty. II. DETERMINATION OF INCREMENTAL CASH FLOWS It was recognized at the time of the investigation that the procedures adopted for calculating incremental cash flows might differ from the approach taken by Hertz. The exact procedure used by Hertz for combining the various input data to determine incremental cash flows was not known. In addition, Hertz had referred to the possibility of taking into consideration the correlation of input factors, for example, price/demand relationships. I t was not too obvious from the article whether he used more than one probability distribution curve for each factor i n the analysis of this project. The steps for the computation of the yearly incremental cash flows adopted by the investigator were as follows: 1. Generate randomly a value for the l i f e of the f a c i l i t i e s , which f a l l s within the estimated pessimistic/optimistic range. The l i f e of the project, so found, w i l l determine the number of yearly revenue and cost 52 figures to be used to arrive at an internal rate of return figure. 2. Generate randomly a value for the i n i t i a l total market size, market growth rate, investment outlay and residual value within the pessimistic/ optimistic range for each factor for each computation of the internal rate of return. 3. Compute the total size of the market for each year of the l i f e of the f a c i l i t i e s from the generated values of the i n i t i a l total market and the market growth rate. 4. Generate randomly the values of selling prices, share of market, variable costs and fixed costs within the pessimistic/optimistic range for each factor for each year of the l i f e of the f a c i l i t i e s . 5. Compute the total variable costs, the revenues and the incremental cash flows for each year of the l i f e of the f a c i l i t i e s . Total variable cost = variable cost per ton x market size x share of market Revenue = market size x share of market x selling price Incremental cash flow = revenue - total variable cost - fixed cost III. DETERMINATION OF INTERNAL RATES OF RETURN Compute the internal rate of return for the yearly incremental cash flows as determined i n the previous section by the formula I = C F ^ (1+r) + CF / (1+r) + .... 2 2 + (CF + S) / (1 + r ) n n 53 where I = investment outlay CF = incremental cash flow n = l i f e of f a c i l i t i e s S = residual value r = internal rate of return Repeat the processes of sections I I and I I I for as many trials as i s desired. A computer subroutine for calculating internal rates of return using an iterative procedure was specially designed. IV. TREATMENT OF SIMULATED DATA A computer routine was designed to sort the simulated internal rates of return into a frequency distribution, compute the probability, cumulative and complementary cumulative density functions and print out these distributions as tabular data. The procedure for sorting the internal rates of return was to place the generated values into bins of predetermined class intervals and a count of the number of values i n each bin was programmed. Lower and upper limits of the frequency d i s t r i bution were established and an error exit was provided for simulated internal rates of return which f e l l outside the established limits of the frequency distribution. A routine for computing the mean and the standard deviation of the frequency distribution of internal rates of return was also 54 programmed. The mean of the frequency distribution gives the expected internal rate of return. The standard deviation i s a measure of the degree of dispersion of the frequency distribution. The mode, which i s the internal rate of return that corresponds to the highest frequency of the distribution can be determined directly from the printout. The median, which i s the internal rate of return that corresponds to the 50$ value of the cumulative distribution can be read directly or by interpolation from the printout. A clock subroutine was called from various parts of the main programme to give an idea of the time taken by each computer routine. V. GENERATION OF PSEUDORANDOM AND NORMALLY DISTRIBUTED PSEUDORANDOM NUMBERS. A random number generator (RAND) and a normally distributed random number generator (RANDN) are available as library subroutines at the Computing Centre, University of British Columbia. The method 1 for the generation of pseudorandom numbers with a uniform distribution over the interval 0 to 1 i s R where R fi n+1 = ^ X ( 2 ? + X ) + I M S i s the last random number computed. The generator may be i n i t i a l i z e d with any random number having a maximum of nine decimal digits. ''"T.E. Hull and A.R. Dobell, "Mixed Congruential Random Number Generators for Binary Machines," Journal Association for -Computing Machinery, New York, 11:1 (January, 1964), 31-34. 55 The accuracy may be gauged from these typical results: Number of generations 5,000 10,000 20,000 Mean Standard deviation .494439 .497492 .496897 .286485 .288228 .290012 The theoretical standard deviation for a uniform distribution is 1/ (12) / or 0.288675. 1 2 The normally distributed pseudorandom number generator (RANDN) 2 makes the following evaluations on successive calls : Odd calls: = (-2 L O G (Ell?-/ RANDN 2 X (((E2) -(E3) )/((S2) +(E3) )) 2 2 2 2 Even calls: = (-2 L O G ( E l ^ RANDN 2 x , <2xE2xE3)/( (E2) +(E3) ) 2 2 wnere EL = RAND (0.), E2 = 2 x RAND (0.) -1, E3 = 2 x RAND (0.)-l and ((E2) + (E3) ) < 1 2 2 (RAND i s a function which generates uniformly distributed random numbers between 0 and 1). J.F. Hogg, "Programme Writeup for the Normally Distributed Random Number Generator," Computing Centre, University of British Columbia. CHAPTER V ANALYSES OF RESULTS AND FINDINGS This chapter gives an account of the experimental determination of the values of the parameters for the lognormal and Weibull functions and the processing and analyses of simulated data. The account includes interpretation of results and a report of the principal findings. I. PARAMETER ESTIMATION I n i t i a l guesses of 0 . 1 and 0 . 2 for parameter S (equation ( 2 8 ) of Chapter III) were f i r s t selected for use i n the computer programme (appendix J ) to f i t the lognormal function to the input data shown i n table I on page 1 6 . The average number of iterations to reduce f ^ to within 0 . 0 0 0 1 was between 5 and 9 . Typical values of the three parameters M, S, and t are shown i n appendix A. The computer time taken to find the lognormal parameters for the nine input variables was 1 2 seconds on the IBM 7 0 4 0 system. It was decided to vary the assumptions concerning the t a i l probabilities for the input data shown i n table I. Tail probabilities of the following combinations left: .01 .05 .01 .1 .1 .2 .3 right: .05 .01 .1 .01 .1 .2 .3 were experimented with. Successful determination of the lognormal parameters for the input data was obtained for the f i r s t , second and the fourth to seventh combinations of t a i l probabilities. For the third 57 combination of t a i l probabilities, the evaluation of a logarithm of a negative number was encountered i n the iterative procedure for finding the lognormal parameters for the input factors of variable costs and fixed costs. An error message recorded by the computer terminated the running of the programme. It was generally found that the estimation of lognormal parameters presented no d i f f i c u l t y when the t a i l probabilities were assumed equal. The lognormal parameters for the input variables l i s t e d i n table I, when they had been scaled up by 1 0 $ and down by 1 0 $ were obtained without d i f f i c u l t y . Fitting the two-parameter lognormal function to the pessimistic and the optimistic estimates of the nine input variables l i s t e d i n table I was tried. These parameter values and the associated statistics are shovm i n appendix B. I t was observed that a l l the measures of central tendency (mean, mode and median) for the nine input variables showed poor correspondence to the most l i k e l y (best) estimates for these variables shown i n table I. It was, therefore, decided not to make a main simu- lation run using the two-parameter lognormal function. I n i t i a l guesses of shape parameter values of 4 . 0 and 3 . 5 were f i r s t tried in the computer routine (appendix K) for finding the Weibull parameters for the nine input variables listed i n table I but using the l e f t and right t a i l probabilities of 0 . 0 5 . Lamb"*" stated, that he used 4 . 0 and 3 . 5 shape parameter values as i n i t i a l guesses. "Lamb, op. c i t . , p. 1 5 . However, i t 58 was discovered that with these selected guesses the iterative method did not converge when i t was applied to the input factors of variable costs and fixed costs as the shape parameter escalated to a high negative value. This difficulty disappeared when the i n i t i a l guesses were changed to 2.0 and 1.5, and these values were used throughout the remaining investigations for determining Weibull parameters. The Weibull parameters for the nine input factors l i s t e d i n table I were experimented with the following combinations of t a i l probabilities: left: .01 .05 .01 .1 .05 .1 .2 .3 right: .01 .05 .1 .01 .01 .1 .2 .3 A l l except the fourth and f i f t h combinations of t a i l probabilities presented no d i f f i c u l t y . When the fourth combination of t a i l pro- babilities was tried, the shape parameter for the market size variable escalated to a high negative number and the computer terminated the running of the programme for the remaining investigations. The f i f t h combination was then tried and a l l the nine input variables except the residual value variable gave normal answers. The shape parameter for the residual value variable escalated to a high negative number. The Weibull parameters for the nine input variables l i s t e d i n table I, when they had been adjusted upwards by 10$ and downwards by 10$ were also found. The average number of iterations to reduce f was between 6 and 11. k to within 0.0001 The computer time taken to find parameters for 59 nine input variables was about 18 seconds on the IBM 7C40 system. Typical values of the Weibull parameters are given i n appendix C. The investigations indicated that the determination of Weibull parameters generally presented no d i f f i c u l t y whan the t a i l probabilities were assumed equal. Lamb indicated that experience showed that i f the input variables were greatly skew i n one direction, the Newton-Raphson technique irould not converge on the Weibull parameters, as there wasa limit to the skewness of the Weibull function. He indicated, however, these occurrences would be quite rare. The standard deviations to the input data listed i n table I, using the normal approximation, were determined. These values are shown i n appendix D. II. MAIN SIMULATION STUDIES The main simulations comprised the computer runs for 3^00 dis- counted cash flow computations, using the lognormal, Weibull, normal and triangular probability functions to quantify the probability of occurrence of the input variables listed i n table I. The analyses of the results obtained from these simulations were designed (1) to compare the computer output performance of each input probability function with that of subjective probabilities as given by Hertz, and with one another and 60 (2) to test the hypothesis that the frequency distributions of internal rates of return obtained with as many as 36OO s t a t i s t i c a l t r i a l s might be assumed normal distributions. In discussing the findings, i t was borne i n mind that there was no a priori basis for believing that the results obtained by Hertz were in any way considered close to the "correct" solutions. However, i t was conceivable that the results obtained by Hertz provided r e a l i s t i c solutions since considerable expertize was employed to make the economic forecasts and to derive the associated subjective probabilities. In figure 6 are shown the probability density functions of simulated interna], rates of return for various input probability functions. In figure 7 are shown the corresponding complementary cumulative d i s t r i butions. In table II are tabulated the s t a t i s t i c a l measures of mean, mode, median and standard deviation of the probability distributions of simulated internal rates of return. 1. These observations may be made: I t i s evident that the probability distributions are sensitive to the assumptions of different input probability functions; 2. The normal and lognormal functions gave complementary cumulative curves i n surprisingly close agreement to that given by subjective input probability function (figure 7). The triangular probability function gave a complementary cumulative curve poor i n comparison to that of the empirical probability function, with the Weibull function giving a curve occupying an intermediate position. 3. Figure 6 cannot be used directly to compare the performance of theoretical input probability functions with the curve obtained by Hertz. LEGEND Empirical (Hertz) Lognormal Weibull -10 0 10 20 Anticipated Per Cent Return FIGURE 6 PROBABILITY DISTRIBUTION OF SIMULATED RATES OF RETURN FOR 36OO TRIALS Anticipated Per Cent Return FIGURE 7 COMPLEMENTARY CUMULATIVE PROBABILITY DISTRIBUTION OF SIMULATED RATES OF RETURN FOR 36OO TRIALS 63 TABLE II INFLUENCE OF INPUT PROBABILITIES ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) Type cf input probability function Statistic Empirical Lognormal Weibull Normal Mangular Mean 14.6 14.37 9.64 16.46 Mode 15.0 10.0 15.0 Median 5,15,25 16.76 11.5 7 13.4 3.73 2.5 1.1 Standard deviation not given 9.07 8.97 8.98 9.07 64 TABLE III TESTS OF HYPOTHESES THAT SIMULATED FREQUENCY DISTRIBUTIONS OF INTERNAL RATES OF RETURN ARE NORMAL DISTRIBUTIONS Input probability function Chi-square value Computed From table .05 significance level No. of degrees of freedom Accept/reject hypothesis Lognormal 51.68 16.92 9 Reject Weibull 27.40 15.51 8 Reject Normal 73.05 15.51 8 Reject Triangular 54.02 15.51 8 Reject 65 Hertz's result gave three modes at 5$» 15$ and 25$. input probability functions gave unimodal curves. to compare the unimodal curves with one another. normal functions gave curves i n f a i r l y close f i t . The theoretical However, i t i s possible The lognormal and. The triangular probability function gave a curve shifted to the l e f t with the curve using the Weibull function taking on an intermediate position. These curves have approximately equal dispersion as measured by their standard deviations given i n table II but with their mean positions shifted. The low value of the mean for the curve as returned by the triangular input probability function i s explained by the fact that a relatively larger number of negative internal rates of return were simulated. An important result i s that the lognormal input probability function gave an expected value of 14.37$ (mean of the distribution) i n good agreement to the expected value of 14.6$ obtained by Hertz. It should be explained that expected values (means) and standard deviations were directly calculated by the computer. These figures would be considered more accurate than the medians which were obtained by interpolation from the cumulative distributions or the modes which were grouped over a 5$ class interval. The normal input probability function gave an expected value of 16.46$ not far from the figure of 14.6$. The lognormal and normal input probability functions also returned a modal internal rate of return of 15$i 4. agreeing with the middle modal value of 15$ obtained by Hertz. The symmetrical curves obtained with theoretical probability functions (figure 6) suggested the chi-square tests for their normality. 66 Table III gave the results of the chi-square tests. The conclusion to be drawn from these tests i s that even with a relatively large number of t r i a l s the probability distribution curves are s t i l l not normal. I t took 15 ninutes computer time for a run of 36OO discounted cash flow computations on the IBM 7040 system for the main programmes using lognormal and Weibull input probabilities. I t took 12 minutes and 9 minutes computer time to make the similar number of calculations using normal and triangular input probabilities. On the faster IBM 7044 the computer times required to make the above calculations would be about 4/10 the times quoted above. Most of the computer time was taken up with the loop for random sampling of input variables and computing the internal rates of return. 3 1/2 seconds. The sorting process took only Hertz quoted i t took 2 minutes computer time to make 36OO discounted cash flow calculations for the same project using subjective probabilities. The observed differences i n computer times taken by the programmes used i n this thesis and the programme used by Hertz could be attributed to the more accurate but more complicated processes for generating random numbers, and to the evaluations of mathematical functions for the programmes using lognormal, Weibull, and triangular input probabilities. The computer times taken to make a run of 200 t r i a l s were about 1 minute on the IBM 7040 system and 1/2 minute on the IBM 7044 system. The cash flows generated i n the simulations were deflated by 10,000 to avoid the problem of excessive accumulator overflows with 2 3ee Samuel B. Richmond, Statistical Analysis, Second Edition (New York: The Ronald Press Company, 1964) p. 143 for the procedure of calculating the theoretical normal frequencies from sample data, 67 the computer routine for the determination of internal rates of return. III. SENSITIVITY ANALYSES Simulation runs were conducted to provide an insight into the sensitivity of output to (1) number of s t a t i s t i c a l t r i a l s ; (2) assump- tions made on t a i l probabilities and (3) errors made i n the three-level estimates. Hess and Quigley , using a single variable deterministic Monte 3 Carlo simulation model, found that 100 discounted cash flow computations gave a complamentary cumulative distribution curve very close to the theoretically correct solution. Hertz** used 3600 discounted cash flow calculations i n the nine variable problem. The problem of sample size has not been completely solved i n the currant literature. The f i r s t sensitivity investigation of the present study was intended to give sample results for a nine variable case. In the third sensitivity investigation a l l the input estimates shown i n table I were f i r s t scaled up by 10$ and then scaled down by 10$. The purpose was to assess the extent of the variation of the simulated data when estimating errors occurred i n the same direction. In any given practical situation i t would be expected that errors would occur i n estimating the confidence attached to pessimistic and optimist i c estimates, i . e . , i n t a i l probabilities, and i n estimating the values of the variables themselves. Some of these errors would produce cancelling 3 ^Hess and Quigley, op. c i t . , p. 59. 4 Hertz, op. c i t . , p. 101. 68 effects on the results obtained from a simulation. Sensitivity to number of statistical trials.Figures 8, 10, 12, and 14 show the effect of variations of the number of discounted cash flow computations on the probability distributions of simulated internal rates of return using lognormal, Weibull, normal and triangular probabil i t y functions to quantify the probability of occurrence of input variables. Figures 9, 11i 13 and 15 show the corresponding complementary cumulative distribution curves. Tables IV to VII show the variations of the number of discounted cash flow computations on various statistical measures, using the four input probability functions of lognormal, Weibull, normal and triangular. The general observation is that 100 and 200 s t a t i s t i c a l trials gave probability distributions i n f a i r l y close agreement to that given by 3600 s t a t i s t i c a l t r i a l s , except at the peaks and at the t a i l s and i n some instances at the sides of the distributions. Better f i t s are ob- tained with the complementary cumulative distribution curves. The small variations i n the statistical measures of central tendency and dispersion evident from tables IV to VII support the view that the curves in figures 8 to 15 have reasonably good f i t s . Sensitivity to t a i l probabilities. In figures 16 and 18 are plotted the probability distributions of simulated internal rates of return obtained from 200 statistical t r i a l s for various combinations of t a i l probabilities using the lognormal and Weibull functions to quant i f y the probability of occurrence of input variables. In figures 17 and 19 are plotted the corresponding complementary cumulative distribution curves. Tables VIII and IX show the effect of the variations of t a i l o.h LEGEND 3600 200 100 £ 0.3 +5 d Xi G) -P -Ti o •H 0 . 2 c .5 > •rt o o 0.1| ra o c CO o Anticipated Per Cent Return FIGURE 8 EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON PROBABILITY .DISTRIBUTION OF RETURNS (LOGNORMAL PROBABILITIES) Trials Trials Trials Anticipated Per Cent Return FIGURE 9 EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS (LOGNORMAL PROBABILITIES) -O O LEGEND 3600 Trials 200 T r i a l s 100 T r i a l s FIGURE 10 EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON PROBABILITY DISTRIBUTION OF RETURNS (WEIBULL PROBABILITIES) Chances of Achieving Indicated Return or Larger o ro o O • ON "T"— CO — , — o o I 9 o 1 O > S3 s-, H >-3 > o M H < S3 d O M to Ol P3 > »^ O M ir to 3 (-3 »-3 § M M 1 O 1^ o H* o Cd H 3 c+ § a H i H 1—1 a H d<D Q. C• DD OH"-5 CD O 3 t+ 1 S 3 ro W O ^ o so> S 3 H3 t-< o i-3 S 3 CO g CO o 25 11 H M O O O Q O t—3 *~3 *-3 H" H* O O H* JO pj (U W OJ to M M H LEGEND 3600 Trials 200 T r i a l s 100 T r i a l s Anticipated Per Cent Return FIGURE 12 EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON PROBABILITY DISTRIBUTION OF RETURNS •(NORMAL PROBABILITIES) -10 0 10 20 Anticipated Per Cent Return 30 FIGURE 13 EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS (NORMAL PROBABILITIES) h0 LEGEND Trials 2 0 0 Trials 1 0 0 Trials 3599 Anticipated Per Cent Return FIGURE l l ; EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON PROBABILITY DISTRIBUTION OF RETURNS (TRIANGULAR PROBABILITIES) EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS (TRIANGULAR PROBABILITIES) 77 TABLE IV EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (LOGNORMAL PROBABILITIES) Statistic Number of discounted cash f l o w c a l c u l a t i o n s 3600 200 100 Mean 14.37 14.48 Mode 15.0 11.5 9.07 15.47 15.0 12.6 Median Standard d e v i a t i o n 8.41 15.0 11.1 7.97 78 TABLE V EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (WEIBULL PROBABILITIES ) Statistic Number of discounted cash flow calculations 3600 Mean Mode 9.64 10.0 200 9.24 12.5 100 10.07 10.0 Median 7.0 6.86 7.2 Standard deviation 8.97 9.44 8.59 79 TABLE VI EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN (#) (NORMAL PROBABILITIES) Number of discounted cash f l o w c a l c u l a t i o n s Statistic 3600 200 100 Mean 16.46 15.89 16.28 Mode 15.0 20.0 10.0 Median 13.4 13.4 13.1 .. Standard d e v i a t i o n 8.98 8.51 8.29 80 TABLE VTI EFFECT OF VARIATIONS OF NUMBER OF MONTE CARLO TRIALS ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (TRIANGULAR PROBABILITIES) Statistic Mean Mode Median Standard deviation Number of discounted c a s h flow calculations 3599 200 100 3.73 2.5 1.1 9.07 3.32 0.0 1.0 8.58 3.63 0.0 1.02 8.29 81 probabilities of the lognormal and Weibull functions on various statist i c a l measures. A major shift of the probability distribution and complementary cumulative distribution curves i s evident when a change of symmetrical t a i l probabilities of 0.01 to 0.1 was made. Unsymmetrical t a i l probabilities of order between 0.01 to 0.1 produced intermediate curves. Errors of the order of 5$ to 10$ i n confidence attached to pessimistic and optimistic estimates, i.e., i n t a i l probabilities, would not be expected to be uncommon i n practice. The variations i n the mea- sures of central tendency shown i n tables VIII and IX support the view that the Monte Carlo outputs are sensitive to variations, within pract i c a l limits, i n t a i l probabilities. Sensitivity to errors i n the three-level estimates. Figures 20, 22 and 24 show the extent of the movement of the probability d i s t r i butions of simulated internal rates of return, assuming a l l the three estimates of each of the input variables shown i n table I are incorrect by 10$ i n the same direction, using the lognormal, Weibull and normal probability functions to quantify the probability of occurrence of the input variables. Figures 21, 23, and 25 show the corresponding comple- mentary cumulative curves. A simulation run using the triangular input probability function was not carried out for this sensitivity analysis, as the triangular probability function had shown to be a rather inexact approximation to use i n the earlier investigations. Tables X to XII show the effect of a 10$ error i n a l l the three-level estimates of a l l the input variables shown i n table I on various s t a t i s t i c a l measures, using lognormal, Weibull aid normal probabilities as input probability functions. Anticipated Per Cent Return FIGURE 16 EFFECT OF VARIATIONS OF TAIL PROBABILITIES ON PROBABILITY DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (LOGNORMAL PROBABILITIES) LEGEND RIGHT TAIL LEFT TAIL 0.01 0.05 0.01 0.10 0.01 0.01 0.01 0.05 0.10 0.10 ho US Anticipated Per Cent Return FIGURE 17 EFFECT OF VARIATIONS OF TAIL PROBABILITIES ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (LOGNORMAL PROBABILITIES) 00 Chances of Achieving Indicated Return o +79 • • » H fO VA> • fr* Chances of Achieving Indicated Return Or Larger 86 TABLE VIII EFFECT OF VARIATIONS OF TAIL PROBABILITIES ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (LOGNORMAL PROBABILITIES) Tail probabilities Mean Mode Median Standard deviation Left Right .01 .01 15.4? 15 12.6 8.41 .01 .05 11.17 10 7.8 10.93 .05 .01 11.77 10 8.8 9.16 .1 .01 11.17 10 8.8 10.06 .1 .1 3.18 5 1.3 16.76 87 TABLE IX EFFECT OF VARIATIONS OF TAIL PROBABILITIES ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (WEIBULL PROBABILITIES) Tail probabilities Left Right .01 .01 .1 .01 .1 .1 Mean Mode Median Standard deviation 9.24 6.86 9.44 6.84 12.5 5.0 3.7 2.54 5.0 0.35 11.91 10.90 88 One good measure of the percentage shift of the curves (using data with 10$ error) from the curves (using original data) i s the percentage movement of the expected (mean), internal rates of return. The lognormal input probability function produced a percentage change i n the expected value of 32.3$ probability function are -49.5$; the figures for the Weibull input 69.5$ and -42.8$ and tiat for the normal input and probability function are 40.5$ and -42$. These figures may also be compared with the percentage changes i n the expected internal rates of return calculated by combining a l l the best estimates of the input variables. The deterministic expected internal rate of return for original data to be found i n this way i s internal rate of return)-' and for data and for data revised 10$ 25.2$ revised 10$ 20.88$ (Hertz downwards i s gave for this expected 14.54$, giving upwards i s 27.3$ percentage changes of 30.7$ and -30.3$ The dispersions of the probability distributions of internal rates of return, as measured by their standard deviations, i n the overestimation and underestimation cases, remain practically of the same order as i n the case with unrevised data. -'The investigator could not see how Hertz's figure was obtained. LEGEND O r i g i n a l input data A l l input data increased by 10£ A l l input data decreased by 10$ Anticipated Per Cent Return FIGURE 20 EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON PROBABILITY DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (LOGNORMAL PROBABILITIES) Anticipated Per Cent Return FIGURE 21 EFFECT OF A T M PER CENT CHANGE TO INPUT DATA ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (LOGNORMAL PROBABILITIES) LEGEND Original input data A l l input data increased by 10$ A l l input data decreased by 10% A Anticipated Per Cent Return FIGURE 22 E F F E C T OF A T E N PER CENT CHANGE TO INPUT DATA ON P R O B A B I L I T Y D I S T R I B U T I O N OF RETURNS WITH 200 MONTE CARLO T R I A L S (WEIBULL P R O B A B I L I T I E S ) Anticipated Per Cent Return FIGURE 23 EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (WEIBULL PROBABILITIES) vo ^ LEGEND Original input data A l l input data increased by 10% A l l input data decreased by Anticipated Per Cent Return FIGURE 2k EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON PROBABILITY DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (NORMAL PROBABILITIES) EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON COMPLEMENTARY CUMULATIVE DISTRIBUTION OF RETURNS WITH 200 MONTE CARLO TRIALS (NORMAL PROBABILITIES) M3 95 TABLE X EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (LOGNORMAL PROBABILITIES) Errors i n three-level estimates of input variables Statistic Original data 10$ overestimation 15.47 20.35 10.0 15.0 25.0 Median 5.9 12.6 17.7 Standard deviation 7.92 Mean Mode 10$ underestimation 7.82 8.41 9.94 96 TABLE XI EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (WEIBULL PROBABILITIES) Statistic Mean Mode Median Standard deviation Errors i n three-level estimates of input variables. Original 10$ under10$ overdata estimation estimation 5.33 10.0 3.1 8.12 9.24 12.5 6.86 9.44 15.65 15.0 12.5 11.31 97 TABLE X I I EFFECT OF A TEN PER CENT CHANGE TO INPUT DATA ON STATISTICAL MEASURES OF INTERNAL RATES OF RETURN ($) (NORMAL PROBABILITIES) Statistic Mean Mode Median Standard d e v i a t i o n Errors i n t h r e e - l e v e l estimates of input variables. 10$ underOriginal 10$ overestimation data estimation 9.22 10.0 6.7 7.90 22.36 15.0 15.89 20.0 13.4 8.51 . 18.9 9.53 CHAPTER VI SUMMARY AND CONCLUSIONS This chapter summarizes the more important developments of the whole study and indicates the contributions of the research study. Some problems arising from the study are mentioned as areas for possible future research. I. SUMMARY The application of the computer to "simulate 11 an economic pro- ject provides a statistical distribution of possible outcomes. yields a quantitative measure of risk. This The simulation can be programmed to provide other statistical measures of central tendency and dispersion of value to linear or integer programming models and other portfolio selection models. One c r i t i c a l problem i n the use of the Monte Carlo simulation technique (method of s t a t i s t i c a l t r i a l s ) i s the numerical encoding of uncertainty of input variables i n a probability distribution. The derivation of subjective probability distributions i s by no means considered an easy or exact task. One serious shortcoming of the use of subjective probability distributions i s that they have no reproducible or mathematical forms. They do not, therefore, allow for valida- tion of their general suitability i n particular cases to characterize input variables by independent means. The present study proposed a simp3.ification to the problem.of deriving a probability distribution by the usual method pf direct l i s t i n g of subjective probabilities. The study examined the possible applicability of four theoretical probability 99 distributions (lognormal, Weibull, normal and triangular) to the evaluation of capital projects by computer simulation. Both theory and nume- r i c a l procedures were developed for employing these theoretical probab i l i t y distributions to quantify the probability of occurrence of input variables i n a simulation model. In particular, the procedure established for f i t t i n g the lognormal probability function to three-level estimates of input variables was the investigator's independent work. Procedures were also developed by the investigator for the generation of stochastic variates with the triangular probability distribution and with the normal approximation. The procedure for f i t t i n g the Weibull probability function to three-level estimates of input variables was adopted from a paper by William D. Lamb. The techniques developed for employing the four theoretical probability functions i n project simulation studies were applied to a case study problem which was analyzed and reported by David B. Hertz in the Harvard Business Review. The proposal considered was a $ 1 0 / - million extension to a chemical processing plant for a medium-sized industrial chemical producer. The experimental results obtained from the simulations using the lognormal and normal probability functions compared favourably with the results using subjective probabilities reported by Hertz. The log- normal function returned an expected internal rate of return (mean of distribution of simulated internal rates of return) of 14.37$ while the normal approximation gave a value of 16.46$. internal rate of return of 14.6$. Hertz reported an expected The complementary cumulative d i s t r i - bution curves using the lognormal and normal functions gave surprisingly 100 good f i t s to the curve obtained by Hertz. The experimental results returned by the Weibull function compared less favourably. The poor simulation results given by the triangular probability function possibly indicated that the use of this probability function was probably too inaccurate an approximation to represent the uncertainties i n this case problan. It should be pointed out that there was no a priori basis for assuming that the simulation res-alts obtained by Hertz using subjective probabilities gave the "correct" solutions. However, i t was con- ceivable that the results obtained by Hertz provided r e a l i s t i c solutions, since considerable expertize was employed to make the economic forecasts and to derive the associated subjective probabilities. Part of the differences i n the f i t s of the distributions could be due to the possible differences i n the procedures for combining the input factors and i n the accuracy of the methods for generating random numbers used by the investigator and by Hertz. Secondary investigations were carried out to test the sensitivity of computer simulation outputs to (1) number of s t a t i s t i c a l t r i a l s ; (2) assumptions made on t a i l probabilities and (3) errors i n the threelevel estimates. It was found that for a nine variable problem 1 0 0 and 2 0 0 discounted cash flow computations gave results f a i r l y close to that with 3^00 discounted cash flow computations. Tail probabilities were found to affect significantly the Monte Carlo outputs. An idea of how errors occurring i n the three-level estimates would affect the outputs were shown. 101 II. CONCLUSIONS This research project had shown various probable rational methods of quantifying subjective information i n a probability d i s t r i bution with particular reference to the evaluation of economic projects by computer simulation. The application of the lognormal probability func- tion to this type of analysis was the single principal contribution of this research study to the search for improved techniques i n the analysis of risky projects. There were a priori considerations for studying this probability function. Some investigators had shown that the lognormal function provided a good f i t to historical data on sales and prices of commodities. These are probably two of the variables subject to the greatest uncertainty, which enter into the analysis of most economic projects. A fundamental property of the lognormal function is that i t obeys the law of proportionate effects, i.e., the lognormal distribution describes random processes which produce small effects (variations) proportional to the previous value of the variable. I t is not an im- plausible supposition that other project variables, other than sales and commodity prices, like production costs, investment outlays and physical or economic lives of f a c i l i t i e s are subject to random causes that produce changes proportional to the values of the variables (as small percentage changes.) The lognormal distribution is mathematically tractable and amenable to simulation. The investigations of the present study disclosed that the lognormal function showed considerable promise as a suitable probability distribution to quantify the uncertainties surrounding project variables. 102 The normal d i s t r i b u t i o n was also found t o hold promise of being an approp r i a t e d i s t r i b u t i o n t o use i n s i m u l a t i o n s t u d i e s . The Weibull probabi- l i t y f u n c t i o n d i d not show up too favourably by the r e s u l t s obtained when i t was applied t o the case problem under study. The t r i a n g u l a r proba- b i l i t y f u n c t i o n was found t o be e i t h e r an inexact or unsuitable approxi- mation t o use i n s i m u l a t i o n studies as shown by the r e s u l t s obtained on t h i s case problem. III. FUTURE RESEARCH There i s need f o r f u r t h e r research t o v a l i d a t e t h e o r e t i c a l p r o b a b i l i t y d i s t r i b u t i o n s against h i s t o r i c a l data on s a l e s , p r i c e s , production c o s t s , investment outlays and p h y s i c a l or economic l i v e s of f a c i lities. There i s also scope t o examine other t h e o r e t i c - u p r o b a b i l i t y d i s t r i b u t i o n s f o r p o s s i b l e mathematical t r a c t a b i l i t y t o use i n s i m u l a t i o n studies with p a r t i c u l a r reference t o p r o j e c t evaluation. One such pro- b a b i l i t y d i s t r i b u t i o n of p o t e n t i a l value that comes t o mind i s the gamma probability function. A u s e f u l i n v e s t i g a t i o n w i l l be t o experiment w i t h the f i t t i n g of the lognormal and t h e Weibull p r o b a b i l i t y functions t o d i f f e r e n t sets of data, using unequal t a i l p r o b a b i l i t i e s . H i l l i e r considered s e r i a l c o r r e l a t i o n i n sales ( o r revenues) as an important f a c t o r i n h i s a n a l y t i c a l p r o b a b i l i s t i c model. One p o s s i b l e area of research w i l l then be on how t o account f o r s e r i a l c o r r e l a t i o n i n sales and p o s s i b l y other v a r i a b l e s i n Monte Carlo s i m u l a t i o n . The f i t t i n g techniques developed i n t h i s t h e s i s o f f e r p o t e n t i a l 103 applications to stock portfolio selection models. One obvious area for future research is to f i t the lognormal and the Weibull probability distributions to stock price series to examine their usefulness to describe stock price changes. BIBLIOGRAPHY 105 BIBLIOGRAPHY A. BOOKS A i t c h i s o n , J . and J.A.C. Brown. The Lognormal D i s t r i b u t i o n . Cambridge: Cambridge U n i v e r s i t y P r e s s , 1957. Bierman, Harold J r . and Seymour Smidt. The C a p i t a l Budgeting Decision: Economic Analysis and Financing of Investment Decisions. Second e d i t i o n . New York: The Macmillan Company, 1966. Conte, S.D. Elementary Numerical A n a l y s i s . New York: McGraw-Hill Book Company, I n c . , 19&5. Dwight, Herbert B r i s t o l . Tables of I n t e g r a l s and Other Mathematical Data. Fourth e d i t i o n New York: The Macmillan Company, 1961. c Hald, A. S t a t i s t i c a l Theory With Engineering A p p l i c a t i o n s . John Wiley & Sons, Inc., 1952. New York: H o l t , Charles C. and others. Planning Production, I n v e n t o r i e s , and Work Force. Englewood C l i f f s : P r e n t i c e - H a l l , Inc., i960. Kenney, J.F. and E.S. Keeping. Mathematics of S t a t i s t i c s . P a r t Two. Second e d i t i o n . Princeton: D. Van Nostrand Company, I n c . , 1956*. Kunz, Kaiser S. Inc., 1957. Numerical A n a l y s i s . New York: McGraw-Hill Book Company, Markowitz, Harry M. P o r t f o l i o S e l e c t i o n : E f f i c i e n t D i v e r s i f i c a t i o n of Investments. New York: John Wiley & Sons, Inc., 1959. McMillan, Claude and Richard F. Gonzalez. Systems A n a l y s i s : A Computer Approach t o Decision Models. Homewood: Richard D. I r w i n , I n c . , I965. Naylor, Thomas H. and others. Computer S i m u l a t i o n Techniques. .John Wiley & Sons, Inc., 19&6T New York: Pessemier, Edgar A, New Product Decisions: An A n a l y t i c a l Approach. York: McGraw-Hill Book Company, Inc., V)6Z~. New Q u i r i n , G. David. The C a p i t a l Expenditure Decision. Homewood: Richard D. I r w i n , Inc., 1 9 6 ? . " Richmond, Samuel B. S t a t i s t i c a l A n a l y s i s . Second e d i t i o n . Ronald Press Company, 191357" Shreider, Yu. A. (ed.). 1966. New York: The The Monte Carlo Method. Oxford: Pergamon Press, Van Horne, James. Foundations f o r F i n a n c i a l Management. Homewood: Richard D. I r w i n , Inc., 196*6*1 106 B. PERIODICALS Adelson, R.M."Criteria f o r Capital Investment: An Approach through Decision Theory," Operational Research Quarterly (March, 1965), 19-49. Baldwin, Robert H. "How to Assess Investment Proposals," Business Review, 37 (May-June, 1959). 98-104. Harvard Beckmann, Martin J . and F. Bobkoski. " A i r l i n e Demand: An Analysis of Some Frequency D i s t r i b u t i o n s , " Naval Research L o g i s t i c s Quarterly,. 5:1 (March, 1958), 43-51. Bernhard, Richard H. "Discount Methods, f o r Expenditure Evaluation - A C l a r i f i c a t i o n of Their Assumptions," Journal of I n d u s t r i a l Engineering (January - February, 1962), 19-26. Cord, J o e l . "A Method f o r A l l o c a t i n g Funds to Investment Projects When Returns Are Subject to Uncertainty," Management Science, 10:2 (January, 1964), 335-341. English, J.M. "Economic Comparison of Projects Incorporating a U t i l i t y C r i t e r i o n i n the Rate of Return," The Engineering Economist, 10 (Winter, 1965), 1-14. Green, P.E. "Risk Attitudes and Chemical Investment Decisions," Chemical Engineering Progress, 59:1 (January, 1963). 35=40. Grnibbs, Frank E„ "Attempts to Validate Certain PERT S t a t i s t i c s or •Picking on PERT'," Operations Research, 10 (1962), 912-915. Hertz, David B. "Risk Analysis i n Capital Investment," Review, 42:1 (January - February, 1964), 95-106. Harvard Business Hespos, Richard F. and Paul A. Strassmann. "Stochastic Decision Trees for the Analysis of Investment Decisions," Management Science, 11:10 (August, 1965), B244-3259. H i l l i e r , Frederick S. "The Derivation of P r o b a b i l i s t i c Information f o r the Evaluation of Risky Investments," Management Science, 9:3 ( A p r i l , 1963), 443-457. . • "Supplement toihe Derivation of P r o b a b i l i s t i c Information f o r the Evaluation of fiisky Investments ," Management Science, 11:3 (January, 1965), 485-437. , and David V. Heebink. "Evaluating Risky Capital Investments," C a l i f o r n i a Management Review (Winter, 1965), 71-80. Horowitz, I r a . "The Plant Investment Decision R e v i s i t e d , " Journal of I n d u s t r i a l Engineering (August, 1966), 416-422. 107 Hull, T.E. and A.R. Dobell. "Mixed Congruential Random Number Generators for Binary Machines," Journal, Association for Computing Machinery, New York, 11:1 (January, 19o¥), 31-40. Kao, John H.K. "A New Life-Quality. Measure for Electron Tubes," Transactions, Institute of Radio Engineers, Professional Group on Reliability and Quality Control, 7 (April, 1956), 1-11. Lintner, John. "The Valuation of Risk Assets and the Selection of Risky Investments i n Stock Portfolios and Capital Budgets," Review of Economics and Statistics, XLVII: 1 (February, 1965). 13-37. MacCrimmon, Kenneth R. and Charles A. Ryavec. "An Analytical Study of the PERT Assumptions," Operations Research, 12 (1964), 16-38. Robichek, Alexander A. and Stewart C. Myers. "Conceptual Problems i n the Use of Risk-Adjusted Discount Rates," Journal of Finance, XXI (December, 1966), 727-730. Solomon, Martin B. Jr. "Uncertainty and Its Effect on Capital Investment Analysis," Management Science, 12:8 (April, 1966), B334-B339. Teichroew, Daniel, Alexander A. Robichek and. Michael Montalbano. "Mathematical Analysis of Rates of Return Under Certainty," Management Science, 11:3 (January, I965). 395-403. . "An Analysis of Criteria for Investment and. Financing Decisions Under Certainty," Management Science, 12:3 (November, 1965), 151-179. Van Home, James. "Capital Budgeting Decisions Involving Combinations of Risky Investments," Management Science, 13:2 (October, 1966), B 84-B 92. Wagle, 3. "A Statistical Analysis of Risk i n Capital Investment Projects," Operational Research Quarterly (March, I 9 6 7 ) , 13-33. Weibull, Waloddi. "A Statistical Distribution Function of Wide Applicability," Journal of Applied Mechanics (September, 1951), 293-297. Weingartner, H. Martin. "Capital Budgeting of Interrelated Projects: Survey and Synthesis," Management Science, 12:7 (March, 1966), 485-516. C. OTHER PUBLICATIONS Food and Agriculture Organization, United Nations. World Demand for Paper to 1975. Rome, i960. 108 Hess, Sidney W. and Harry A. Quigley. "Analysis of Risk i n Investments Using Monte Carlo Techniques," Chemical Engineering Symposium Series 42: Statistics and Numerical Methods in"" Chemical Engineering (New York: American Institute of Chemical Engineering, 1963), pp.55-63. H i l l i e r , Frederick S. The Evaluation of Risky Interrelated Investments. Technical Report No. 73. Contract Nonr-225 (53) (NR - 642-002) with the Office of Naval Research, 1964. Norton, John H. "The Role of Subjective Probability i n Evaluating New Product Ventures," Chemical Engineering Progress Symposium Series , 42: Statistics and Numerical Methods i n Chemical Engineering (New York: American Institute of Chemical Engineering, 1963)»PP. 49-54. D. DISSERTATIONS Babcock, Guilford Carlile. "Growth to Future Value as a Measure of Investment Worth." Unpublished Ph. D. dissertation, The University of California, Los Angeles, 1966. Woodfield, Leon Warren. "An Experiment i n Application of the Monte Carlo Method for Simulating Capital Budgeting Decisions Under Uncertainty." Unpublished Doctor of Business Administration's dissertation, Michigan State University, I965. E. UNPUBLISHED PAPERS H i l l i e r , Frederick S. "The Evaluation of Risky Interrelated Investments." Unpublished Research Draft Monograph, Stanford University, August I966. Hogg, J.F. "Programme Writeup for the Normally Distributed Random Number Generator." Computing Centre, The University of British Columbia. Lamb, William D. "A Technique for Subjective Probability, Assignment i n Risk Analysis Problems." Paper read at the Institute of Management. Sciences, American Meeting, Boston, Massachusetts, April 5-7, 1967. APPENDICES APPENDIX A VALUES OF PARAMETERS FOR INPUT VARIABLES FOR THREE-PARAMETER LOGNORMAL DISTRIBUTION (.01 TAIL PROBABILITIES) Input Variables Market Size Selling Prices Market growth rate Share of market Total investment Useful l i f e Residual value Variable costs Fixed costs Mu Sigma Location 22.72087008 12.64226064 1.1264811*2 9.36861288 24.07318944 6.2U4278UO 21.87094176 5.40789616 12.24798448 0.69972047 lO"* 0.13201797 10-3 0.41806588 lO" 0.25686586 lO"* 0.26396054 lO"** 0.41732054 lO" 0.10233075 lO" 0.16450233 0.12696467 X X X 2 X X X X 2 3 . -0.73711252 -0.30884994 -0.30547297 -Ooll7l4734 -0.28491062 -0.50504848 -0.31463622 0.21779640 0.94774523 X X X X X X 10 106 10 10 10* 10 10 10 10* 10 U 3 10 X X 5 X APPENDIX B VALUES OF PARAMETERS & STATISTICS FOR INPUT VARIABLES FOR TWO-PARAMETER LOGNORMAL DISTRIBUTION (.01 TAIL PROBABILITIES) Input variables Parameters Mu Sigma Market size 12.12U81 0,263030 Mode Median 172,065 184,391 Variance Mean 190,881 10 O.26ICO x 10 6.15381 0.08622 U67 471 U72 1,664 Market growth rate -8.314H6 2.36472 0 0.000245 0.004 0.0043 Share of market -2.63926 0.37282 0.062 0.071 0.077 O.OOO87 Total investment 15.96415 0.08715 8,508,3U9 8,573,214 8,605,831 0.565 x i o 2.15874 0.23613 8.66 8.91 4.55 Residual value 15.24661 0.07666 4,158,787 4,183,300 4,195,610 o„io4 x 10 Variable costs 6.10714 0.08324 446 451 1,U12 12.63195 0.08715 Selling prices Useful l i f e Fixed costs 8.19 303,870 449 306,186 307,351 1 2 0.720 x 10 9 12 APPENDIX C VALUES OF PARAMETERS FOR INPUT VARIABLES FOR WEIBULL DISTRIBUTION (.01 TAIL PROBABILITIES) Input variables Shape Scale Location Market size 6.5410143 312,504 -54,677 Selling prices 8.6)i38869 Market growth rate 3.4909639 0.04686741 -0.01254839 Share of market 7.52787U5 0.20525403 -0.08140436 Total investment 20.976092 Useful l i f e 3.4909638 Residual value 9.4724828 Variable costs 2.3838524 Fixed costs 2.5655055 313.63139 12,847,080 7.8112344 2,681,085 99.873293 75,939 200.79796 -3,317,197 2.9086019 1,850,308 355.4994 237,360 APPENDIX D VALUES OF PARAMETERS FOR INPUT VARIABLES FOR NORMAL DISTRIBUTION (.01 TAIL PROBABILITIES) Input variables Mean Standard deviation Market size 2^0,000 510 0.03 0.12 9,500,000 10 4,500,000 435 " 300,000 51,584 U0.837 0.013 0.030 752,268 Selling prices Market growth rate Share of market Total investment Useful l i f e Residual value Variable costs Fixed costs 2.149 322,400 37.613 26,867 P APPENDIX E FREQUENCY DISTRIBUTION OF INTERNAL RATES OF RETURN FOR 36OO MONTE CARLO TRIALS (LOGNORMAL PROBABILITIES) Frequency distribution Range of internal rates of return {%) -32.5 -27.5 -22.5 -17.5 -12.5 - 7.5 - 2.5 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 to -27.5 -22.5 -17.5 -12.5 - 7.5 - 2.5 Probability density function Complementary cumulative density function 1 0,000278 2 2 0.000556 0.000556 0.001111 0.999722 0.999167 0.998611 0.997500 0.005556 0.991941; 4 20 61 203 22.5 27.5 32.5 37.5 42.5 47.5 589 0.016944 0.056389 0.133056 0.218889 0.223889 0.163611 380 168 0.046667 52.5 5 57.5 1 2.5 7.5 12.5 17.5 479 788 806 0.105556 58 0.016111 23 10 0.006389 0.002778 0.001389 0.000278 0.975000 O.9186II 0.785556 0.566667 0.342778 0.179167 0.073611 0.026944 0.010833 O.OOUhhh O.OOI667 0.000278 0.000000 APPENDIX F FREQUENCY DISTRIBUTION OF INTERNAL RATES OF RETURN FOR 36OO KONTE CARLO TRIALS (WEIBULL PROBABILITIES) Frequency distribution Range of i n t e r n a l rates of return {%) -27.5 -22.5 -17.5 -12.5 - 7.5 - 2.5 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 to -22.5 -17.5 -12.5 - 7.5 - 2.5 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 1*7.5 2 8 20 66 177 462 • 718 864 648 352 176 73 27 6 1 Probability density function Complementary cumulative density function 0.000556 0.002222 0.005556 0.018333 0.049167 0.128333 0.199444 0.2l;0000 0.180000 0.097778 0.048889 0.020278 0.007500 0.001667 0.000278 0.999li4ii 0.997222 0.991667 0.973333 0.924167 0.795833 0.596389 0.356389 O.176389 O.O786II 0.029722 0.009444 0.001944 0.000278 0.000000 APPENDH G FREQUENCY DISTRIBUTION OF INTERNAL RATES OF RETURN FOR 36OO MONTE CARLO TRIALS (NORMAL PROBABILITIES) Range of internal rates of return {%) -27.5 -22.5 -17.5 -12.5 - 7.5 - 2.5 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 1*2.5 47.5 52.5 to -22.5 -17.5 -12.5 - 7.5 - 2.5 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 Frequency distribution Probability density function 1 0 3 7 30 130 350 708 834 694 438 246 100 39 16 3 1 0.000278 0.000000 0.000833 0.001944 0.008333 0.036111 0.097222 0.196667 0.231667 0.192778 0.121667 0.068333 0.027778 0.010833 0.004444 0.000833 0.000278 Complementary cumulative density • function 0.999722 0.999722 0.998889 0.996944 0.988611 0.952500 0.855278 0.658611 0.426944 0.234167 0.112500 0.044167 O.OI6389 0.005556 0.001111 0.000278 0.000000 APPENDIX H FREQUENCY DISTRIBUTION OF INTERNAL RATES OF RETURN FOR 3 5 9 9 MONTE CARLO TRIALS (TRIANGULAR PROBABILITIES) Range of internal rates of return {%) -1*2.5 -37.5 -32.5 -27.5 -22.5 -17.5 -12.5 - 7.5 - 2.5 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 to Frequency distribution Probability density function Complementary cumulative density function O.000834 0.000000 0.999166 0 5 0.001389 0.997777 9 0.002501 0.995276 31 0.008614 O.986663 -12.5 81 0.022506 0.964157 7.5 - 2.5 2.5 7.5 12.5 189 0.052515 0.911642 458 0.127258 0.784385 842 0.233954 0.550431 843 588 0.234232 0.316199 0.163379 0.152820 17.5 329 0.091414 0.061406 22.5 27.5 32.5 37.5 42.5 142 0.039455 0.021951 57 0.015838 0.006113 17 3 0.004724 O.OOI389 0.000834 0.000556 2 0.000556 0.000000 -37.5 -32.5 -27.5 -22.5 -17.5 - 3 0.999166 118 APPENDIX I FREQUENCY DISTRIBUTION OF INTERNAL RATES OF RETURN FOR 36OO MONTE CARLO TRIALS (EMPIRICAL PROBABILITIES) Internal rates of return ($) 0 5 10 15 20 25 30 Frequency distribution* Probability density function Complementary cumulative density function 126 572 194 778 389 1090 0.035 0.159 0.054 0.965 0.806 0.752 0.538 0.430 0.126 0.000 451 0.214 0.108 0.304 0.126 *The figures in this column were computed basing on the figures i n the last column tabulated i n Hertz's article. Source: David B. Hertz, "Risk Analysis i n Capital Investment," Harvard Business Review, 42:1 (January - February, 1964), 95-106. APPENDIX J COMPUTER PROGRAMME FOR FINDING THE THREE PARAMETERS OF THE LOGNORMAL DISTRIBUTION $JOB 19097 MOON HOE L E E $ FORT RAN C •' C " ' F I N D T H E P A R A M E T E R S OF LOGNORMAL P R O B A B I L I T Y D E N S ITY" F U N C T I O N C BY THE NEWTON-RAPHSON ITERATIVE PROCEDURE. C NOTATION MU A N D S I G M A A R E T H E MEAN A N D S T A N D A R D D E V I A T I O N O F T H E C U N D E R L Y I N G NORMAL. T = LOCATION PARAMETER "" " " ' C MU A N D S I G M A MAY B E C O N S I D E R E D A S S C A L E A N D S H A P E P A R A M E T E R S OF C LOGNORMAL P.D.F. " C " " " X L = PESSIMISTICESTIMATE XM C ESTIMATE C P L = LOWER BOUND P R O B A B I L I T Y C "' C DL AND DU ARE CORRESPONDING THE" LOWER AND TO PU. PL AND = MODAL PU UPPER = ESTIMATE UPPER BOUND QUANT I L E S OF .. ' R xTT = " O P T I M I S T I C " P R O B A B I L_I_TY_ N(0,1) " C _ "3 0 0 " „ _____ _ CONTINUE " " " ~ " " "" READ ( 5 * 1 ) X L » XM » XU IF ( X L . E Q . ( - 1 . 0 ) _ ) _ S T O P 1 FORMAT ( 3F20.8 ) ' READ ( 5 * 2 ) PL» PU 2 FORMAT (8F10.4) _ . • READ ( 5 » 7 ) "DL'»'"6U". 7 FORMAT (8F12.6) WRITE ( 6 » 3 ) X L » XM » X U _ '~3~~ " ' F O R M A T <1 H 0 » 24H~ P E S S I M T S T I C E S T I M A T E = » F 2 0 . 8 *2 0 X » 2 3 H M O S T L I K E L Y E 1 S T I M A T E = * F 2 0 . 8 •// 2 3 H O P T I M I S T I C E S T I M A T E = ,F20.8) WRITE ( 6 * 6 ) PL» PU _ _ _ _ __ " 6 " F O R M A T ' ( 1 H 0 » 2 7 H L O W E R B O U N D P R O B' ABl' OTY"~=~" • F 1 0 . 4 7 l O X »~26 HUP P E R ~ BOU N~ " ID PROBABILITY = .F10.4) WRITE ( 6»4 ) _ _' . 4 5 FORMAT (1H0»5H L O O P • 7 X • 5 H S I G M A » 1 2 X» 1 8 H L 0 C A T I ON __ ; ; PARAMETER »13X»3HLHS 1»13X*3HRHS»8X»7HRHS-LHS ! W R I T E ( 6 * 5 ) _ _' ~" F O R M A T ( IX ) ™ " ~ " ~ " ~ " " " "~ " ~ ~ LOOP = 0 SD1 = 0.1 _ _ _ T l = ( X U -' ( X L * E X P ( ( D U - D L T* S D 1 ) ) ) / " ( 1 . 0 - E X F ' { ( D 1 J - D L T * " S D 1 " ) ) " " XLHS = SD1 ** 2 RHS = ( ( D U * A L 0 G ( X L - T 1 )-DL*ALO_G( XU-T1J ) / ( D U - D L ) ) - A L O G ( X M - T l ) ^ F l = RHS - XLHS LOOP = LOOP + 1 WRITE ( 6 » 2 0 ) LOOP. S D 1 . T l , XLHS, RHS, F l ; g 20 _ _ FORMAT ( I X . 14', F 1 6 . 8 » G20.8' , 3 F 2 0 . 8 ) IF ( A B S ( F l ) .LE. 0.0001) GO TO 1 0 0 _ SD2 =0.2 5 0 '~ "CONTINUE T2 = (XU - ( X L * E X P ( ( D U - D D * S 0 2 ) ) ) / <1.0-EXP ( ( D U - D L ) * S D 2 ) ) X L H S = SD2 ** 2 _ _ _ _ . R H S = ( ( D U * A L O G ( X L - T 2 ) - D L * A L C G ( X U - T 2 ) > /" ( D U - D L ) ) - A L 0 G ( X M - T 2 j F 2 = RHS - X L H S LOOP = LOOP + 1 _ . _ W R I T E ( 6 , 2 0 ) LOOP* S D 2 * T2» X L H S , RHS» F 2 ~ IF (ABS(F2) .LE. 0.0001) GO TO 2 0 0 IF ( L O O P . G T . 3 0 ) GO T O _ 2 0 5 'SDNEW = S D 2 - F 2 * ( S 0 2 - S D 1 ! 7 ( F 2 - F 1 ) ~~ SD1 = SD2 Fl = F2 "SD2 = SDNEW * ~ " ~ " . " ' ~ ' G O TO 5 0 100 CONTINUE _ . _ " XMU = A L O G ( X M - T 1 f + X L H S W R I T E ( 6 , 2 5 ) L O O P * XMU, S D 1 » Tl 25 F O R M A T ( _ H 0 » 1 8 H N O . I T E R A T I O N S = , I A » / 7 2 X » 5 H M U =_ * F 2 0 . 8 , 2X_, 8 H S I G M A 1 = , G 2 0 . 8» 2 X , 2 1 H L O C A T I O N ' P A R A M E T E R = • G 2 0 V 8 T " ~ "~ GO TO 3 0 0 200 CONTINUE _ • XMU = A L O G ( X M - T 2 ) + X L H S W R I T E ( 6 , 2 5 ) L O O P * XMU» S D 2 » T2 GO TO 3 0 0 205 CONTINUE ~ ' ' '"' " " " " W R I T E (6»30) 30 F O R M A T ( 1 H 0 • 4 0 H P R O G R A M NOT C O N V E R G I N G - T E R M I N A T E D . J "XMU = A L 0 G ( X M - T 2 ) + X L H S W R I T E ( 6 » 2 5 ) LOOP * XMU, SD2» T2 • GO TO 3 0 0 _ • ____ END' ' '. ' ~ ~" "" ' ~ " SENTRY - APPENDIX K COMPUTER PROGRAMME FOR FINDING THE THREE PARAMETERS OF THE WEIBULL DISTRIBUTION $JOB 19097 MOON HOE L E E ^FORTRAN C _ _ , C F I N D THE P A R A M E T E R S OF WEI B U L L " P R O B A B I L I T Y D E N S I T Y F U N C T I O N C BY THE NEWTON-RAPHSON I T E R A T I V E P R O C E D U R E . C_ NOTATION A = SHAPE PARAMETER B = SCALE_ PARAMETER C C = LOCATION " P A R A M E T E R " " " " " " C XL = P E S S I M I S T I C E S T I M A T E XM = MODE XU = O P T I M I S T I C ESTIMATE = LOWER BOUND P R O B A B I L I T Y PU = UPPER BOUND P R O B A B I L I T Y c c~ 300 P _ L ""' " READ ( 5 » 1 ) XL» XM» XU I F ( XL • E Q . ( - 1 . 0 ) ) STOP _ _ READ ( 5 , 2 ) PL.PU 1 FORMAT (4F20.8) 2 FORMAT ( 8 F 1 0 . 4 ) WRITE ( 6 , 3 ) XL»XM,XU 3 FORMAT ( 1 H 0 . 2 4 H P E S S I M I S T I C E S T I M A T E = » F 2 0 . 8 * 2 X , 7 H M 0 D E • ,F20.8,2 ^ 1 X , 2 2 H 0 P T I M I S T I C ESTIMATE = . F 2 Q . 8 ) """WRITE ( 6 , 4 ) " PL,PU 4 FORMAT ( 1 H 0 , 2 7 H LOWER BOUND P R O B A B I L I T Y » , F 1 0 • 4 , 1 0 X , 2 6 H U P P E R BOUN ID PROBABILITY = , F 1 0 . 4 ) " " WRITE ( 6 , 5 ) " " ~ " ' " " " • 5 FORMAT (IX) _ LOOP = 0 '_ Al = 2.0 Dl'=((A1/(A1-1.0)) * (-ALOG(l.O-PL)))** (l.O/AU Cl = (XL-XM*D1) / ( 1 . 0 - D 1 ) _ _ _^ ARG1 = ( ( A l - 1 . 0 J / A l ) * ( ( X U - C 1 ) / (XM-ClT) * * Al" PU1 = E X P ( - A R G l ) F l = PU - P U 1 LOOP = LOOP + 1 WRITE ( 6 , 2 0 ) LOOP, A l » C l , PU1 20 FORMAT ( I X , 14 , G 2 0 . 8 , 2 X , G 2 0 . 8 • 2 X , _ F 1 2 . 8 ) _ IF (ABS(Fl) . L E . 0 . 0 0 0 1 ) " GO'TO 100 A2 = 1 . 5 50 CONTINUE _ _ _ __ D2 = ( ( A 2 / ( A 2 - 1 . 0 ) ) * (-ALOG(l.O-PL)))** (1.0/A?) C2 = ( X L - X M * D 2 ) / (1.0-D2) ARG2 = ( ( A 2 - 1 . 0 ) / A 2 ) * ( ( X U - C 2 ) / _( X M - C 2 ) ) A2__ _ PU2 = EXP ( - A R G 2 ) " " ~ " " " " " " " " " ' " " " " " " " " F 2 = PU - PU2 LOOP = LOOP + 1 5 100 WRITE ( 6 , 2 0 ) LOOP7 " A T , ' " C 2 T " P D T IF (ABS(F2) .LE. 0.0001) GO TO 2 0 0 I F (LOOP • G T . 3 0 ) GO TO_ 2 0 5 ANEW = '"AY~- F2"*~TA12^ /* ( F ' 2 - F T ) A l = A2 F l = F2 _ __ A 2 = ANEW """" " ™ " ~~" ' GO TO 5 0 CONTINUE _ B = : • " ~ " ~ Cl _ _ " ' . _ ~" __ ( A l 7""( A l - l T O ) f * » T I T O / A l T * • ( X M - C 1 )'' " WRITE ( 6 » 2 5 ) L00P,A1»B»C1 FORMAT ( 1 H 0 . 1 8 H N O . I T E R A T I O N S = _ » I 4 . / / I X »_18HSHAPE P A R A M E T E R ^__»G2 1 0 . 8 » 2 X , 1 8 H S C A L E PARAMETER = , G 2 0 . 8 , 2 X • 2 1 H L O C A T I O N P A R A M E T E R = • G 2 0 2.8 ) GO TO 3 0 0 ___ 2 0 0 C O N T I NUE " " " " " - - - — B = (A2 / (A2-1.0)) ** (1.0/A2) * (XM-C2) WRITE ( 6 , 2 5 ) L00P»A2,B,C2 GO TO 3 0 0 " ' ~ " 205 CONTINUE WRITE ( 6 * 3 0 ) " 3 0 F O R M A T ( 1 H 0 , 4 0 H PROGRAM NOT C O N V E R G I N G - " T E R M I N A T ED "T" B = (A2 / (A2-1.0)) ** (1.0/A2) * (XM-C2) WRITE ( 6 , 2 5 ) L 0 0 P » _ A 2 » _____ C2 GO TO 3 0 0 END SENTRY 25 3 i 8 6 R TT '" Zi APPENDIX. L COMPUTER PROGRAMME FOR MONTE CARLO SIMULATION, USING LOGNORMAL INPUT PROBABILITIES $JOB 19097 $ PAGE ST I ME SIBFTC'MAIN MOON HOE L E E 25 20 . c F I N D D I S T R I B U T I O N OF R A T E S 0F_ R E T U R N BY MONTE C A R L O S I M U L A T I O N • U S I N G LOGNORMAL F U N C T I O N TO Q U A N T I F Y P R O B A B I L I T Y O F O C C U R R E N C E OF I N P U T V A R I A B L E S . U - P R E F I X = S C A L E PARAMETER__MU D - P R E F I X = SHAPE"PARAMETER SIGMA T - P R E F I X = LOWER BOUND L O C A T I O N P A R A M E T E R P L = LOWER BOUND P R O B A B I L I T Y _ P U = U P P E R BOUND P R O B A B I L I T Y ~ •" D I M E N S I O N XMK T Y R ( 5 0 ) . P R I C E ( 5 0 ) . S H A R E ( 5 0 ) » V C < 5 0 ) . T O T V C ( 5 0 ) , F C ( 5 0 ) * 1REVNUE(50) >CF(50) »ROR(5000) DIMENSION I B I N ( 1 0 0 > » B I N L I M ( 1 0 0 ) . P D F ( 1 0 0 ) »CDF(100),COMCDF(100) REAL MMKT•OMKT•MPRICE*OPRICE•MGRATE»OGRATE»MSHARE*OSHARE*MINV,OIN 1V*MLIFE»0LIFE*MSALV»0SALV»MVC»0VC*MFC»0FC 1 FORMAT ( 15 ) 2 FORMAT (8F10.4) 4 FORMAT ( F 2 0 . 8 . 5 X . 2 E 1 5 . 8 ) 5""""FORMAT C3F20.8) 10 FORMAT ( 3 9 H N O . OF S T A T I S T I C A L T R I A L S T E R M I N A T E D = * I 4 ) 50 FORMAT ( I H ) _ ' 60 FORMAT ( 1 H 0 ) 70 FORMAT ( 1 H 1 ) 1100 FORMAT ( 4 0 X . 5 2 H C A P I T A L P R O J E C T _ E V A L U A T I O N BY MONTE C A R L O _ S I M U L A T 1 0 IN) " ~ " ' ~ 1200 FORMAT ( 4 1 X , 5 0 H $ 1 0 M I L L I O N E X T E N S I O N TO C H E M I C A L P R O C E S S I N G P L A N T ) 1300 FORMAT ( 4 0 X » 5 3 H S T A T I ST I C A L D I S T R I B U T I O N S OF I N T E R N A L R A T E S OF R E T U 1 RN ) 1400 FORMAT ( 5 4 X . 2 5 H 1 ON A BEFORE TAX B A S I S ) ) 1500 FORMAT ( I X • 26HRANGE OF I R R I N P E R C E N T A G E • 2 X . 2 2 H F R E Q U E N C Y D I S T R I BUT 11 ON » 2 X , 2 4 H PROBAB I L I T Y D I STR I BUT I ON~» 2X , 22HCUMUL AT I V E PROB • D I S t . V 1 X 2 »3 OHCOMPLEMENTARY C U M . P R O B . D I S T . ) 1600 FORMAT ( 1 X , F 8 . 2 . 3 X , 2 H T O • 3 X * F 8 . 2 » 1 2 X • I 4 » 1 6 X • F 1 2 • 6 • 1 4 X • F 1 2 . 6 * 1 5 X , F l 2 ' 1 . 6 ) " " "~ " " " " " 1700 FORMAT ( 2 5 X * 9H T O T A L = . 2 X » I 5»6X•6HSUM =»4X,F12.6) 1800 FORMAT ( 6 7 H I N T E R N A L R A T E S OF RETURN O U T S I D E RANGE OF FREQUENCY D I 1STRIBUTION - ) 1900 FORMAT (1X.6G20.8) 1950 FORMAT ( 6 7 H NUMBER OF EXTREME V A L U E S OF I N T E R N A L R A T E OF R E T U R N D I C_ C C C C " " C C C S 1SREGARDED = ,14) FORMAT • ( 8 X . 1 5 H I N P U T V A R I A B L E S , 1 8 X . 5 2 H P A R A M E T E R S OF LOGNORMAL PROBA 1 B I L I T Y D E N S I T Y F U N C T I ON , 16 X , 13HPROBAB I L I T I ES ) _ 2 200 FORMAT ( 4 4 X , 5 H S H A P E , 1 5 X , 5 H S C A L E , 1 4 X , 8 H L 0 C A T I O N , 1 3 X , i 1 H L 0 W E R BOUND* 1 2 X , 1 1 H U P P E R BOUND) 2250 FORMAT ( 4 3 X , 7 H ( S I G M A ) , 1 4 X , 4 H ( M U ) ) 2300 FORMAT ( 9 X 11HMARKET S I Z E » 1 2 X , G 2 0 . 8 , 2 X , 6 2 0 . 8 , 2 X » G 2 0 . 8 • 1 O X , F 1 0 . 4 , 2 X 1,F10.4) 2400 FORMAT <9X 14HSELLING P R I C E S , 9 X , G 2 0 . 8 , 2 X » G 2 0 . 8 , 2 X , G 2 0 . 8 , 1 0 X , F 1 0 . 4 » 12X,F10.4) 2550 FORMAT <9X 18HMARKET GROWTH R A T E » 5 X , G 2 0 . 8 , 2 X , G 2 0 . 8 » 2 X , G 2 0 . 8 , 1 O X , F 1 10.4,2X*F10 4) _ _ _ "2600" FORMAT (9X 1,2X,F10.4) 15HSHARE OF MARKET,8X,G20•8,2X,G20.8,2X,G26.8,1 OX,F16.4 2 700 FORMAT (9X 14,2X,F10.4) 16HT0TAL INVESTMENT,7X,G2 0.8,2X,G20.8,2X,G20.8,10X,F10. FORMAT (9X 18HLIFE OF FACILITIES,5X,G20.8,2X,G20.8,2X,G20.8,10X,F1 2800 10.4»2X•F10 4 ) "2 900 FORMAT (9X 1 3HSALVAGE VALUE,1 OX,G20.8,2X,G20.8,2X•G20.8,10X,F10.4, 12X,F10.4> 3 000 FORMAT (9X 14HVARI ABLE COSTS,1 OX,G20.8,2X,G20.8,2X,G20.8,1 O X , F l 0 . 4 1,2X,F10.4) COSTS,12X,G20.8,2X,G20.8,2X,G20.8,10X,F10.4,2X 3100 FORMAT (9X 11HFIXED 1 » F10. 4 ) 5 000 FORMAT (8X»15HINPUT VAR I ABLES , 14 X , 21H PES S IMIST'l C ESTIMATES, 12X.21H 1M0ST LIKELY EST I MATES,12X,21H0PTIMI ST IC ESTIMATES) FORMAT (9X 11HMARKET SIZE,16X,G20.8,11X,G20.8,11X,G20.8 ) 5010 5020 FORMAT (9X 14HSELLING PRICFS,13X ,G20.8 » 1 I X , G 2 0 . 8 , 1 1 X , G 2 0 . 8 J T 5030 FORMAT (9X 18HMARKET GROWTH RATE,9X,G20.8,11X,G20.8,11X,G20 . 8 ) 5040 FORMAT (9X 15HSHARE OF MARKET,12X,G20.8,1IX,G20.8,11X,G20.8 ) 5050 FORMAT (9X 16HT0TAL INVESTMENT,11X,G20.8,1IX,G20.8,1IX,G20.8 ) 5060 FORMAT (9X 18HLIFE OF FACILITIES,9X,G20.8,11X,G20.8,11X,G20.8) 5070 FORMAT (9X 13HSALVAGE VALUE,14X,G20.8,11X,G20.8,11X,G20.8) 5080 FORMAT (9X 1- HVARIABLECOSTS,13X,G20.8,11X,G20.8,11X,"G20.8)' 5 090 FORMAT (9X 11HFIXED COSTS,16X,G20.8,11X,G20.8,11X,G20.8) START CLOCKIO.) ' "6 000 ~ CONTINUE READ (5,1) N IF (N .EQ. (-1)) STOP READ IN LOGNORMAL PARAMETERS 'FORINPUT VARIABLES'. " READ (5*4) U M K T . D M K T » T M K T » U P R I C E » D P R ICE»TPR I C E » U G R A T E » D G R A T E » T G R A T 1E,USHARE,DSHARE,TSHARE,UINV,DINV,TINV,ULIFE,DLIFE,TLIFE,USALV,DSAL s 2100 8 6 01 TT Zi + 10 9 i— ro ->3 1 2 V * T S A L V . UVC»DVC*TVC,UFC,DFC»TFC R E A D I N T H R E E - L E V E L E5 T I MATES FOR I N P U T V A R I A B L E S READ (5* 5 ) PMKT__MMKT ,OMKT »PPR I CE_»_MPR I C E « O P R I C E » PGRAT E__MG_R A T E • OGR AT 1 E . P S H A R E • MSH ARE »OSHARE , P I N V , V I N V , 0 I N V • P L I F E » M L I F E~» OL I F E • P S A L V • MS A L 2 V • O S A L V » PVC»MVC»OVC,PFC»MFC»OFC READ ( 5 » 5 ) RORMIN • RORMAX» RANGE PL»PU READ ( 5 . 2) W R I T E (6 7 0 ) W R I T E (6 5 0 0 0 ) • ~ W R I T E ( 6 6 0 )". " ~ W R I T E ( 6 5 0 1 0 ) PMKT»MMKT,OMKT WRI T E (6 6 0 ) W R I T E (6 5 0 2 0 ) " P P R I C E T M P ' R I C E , O P R I C E WRITE ( 6 60 ) W R I T E ( 6 5 0 3 0 ) PGRATE*MGRATE»OGRATE WRITE < 6 6 0 ) W R I T E ( 6 5 0 4 0 ) PSHARE.MSHARE*OSHARE WRI T E ( 6 6 0 ) WRITE (6 5 0 5 0 ) P I N V . M I N V . O I N V WRITE ( 6 60 ) W R I T E (6 5 0 6 0 ) P L I F E » M L I F E » 0 L I F E " W R I T E (6 6 0 ) W R I T E (6 5 0 7 0 ) P S A L V » M S A L V * O S A L V WRITE ( 6 60 ) W R I T E ( 6 5 0 8 0 ) PVC »MVC»OVC WRITE (6 60 ) WRITE (6 5 0 9 0 ) PFC.MFC.OFC WRITE ( 6 70) WRITE ( 6 2100) WRITE (6 60 ) WRITE WRITE WRITE "WRITE WRITE WRITE "WRITE WRITE WRITE WRITE WRITE WRITE 01 . "IT zi ( 6 2200) ( 6 60 ) ( 6 2250) : ( 6 60) ( 6 2300) DMKT*UMKT*TMKT*PL»PU ( 6 60 ) "(6 2 4 0 0 ) " D P R I C E » U P R I C E » T P R I C E * P L » P U " ( 6 60 ) ( 6 2 5 5 0 ) DGRATE*UGRATE»TGRATE •PLj-PU (6 6 0 ) " " ~ ( 6 2 6 0 0 ) DSHARE.USHARE,TSHARE»PL*PU ( 6 60 ) co 450 1505 1510 1520 1530 1540 C u 2 00 210 C 1 5 50 1560 WRITE ( 6 , 2 7 0 0 ) " D I N V , U I N V , T I N V , P L ,PU WRITE ( 6 , 6 0 ) WRI TE ( 6 , 2 8 0 0 1 D L I F E , U L I F E , T L I F E , P L , P U WRITE ( 6 , 6 0 ) ~ WRITE ( 6 , 2 9 0 0 ) D S A L V , U S A L V , T S A L V , P L » P U WRITE ( 6 , 6 0 ) WRITE ( 6 , 3 0 0 0 ) D V C U V C T V C P C P U WRI TE ( 6 , 6 0 ) WRITE ( 6 , 3 1 0 0 ) D F O U F C » T F C , P L » P U WRITE ( 6 , 7 0 ) R = RANDN ( 0 . 4 9 2 8 7 5 3 6 ) K = 0 L = 0 TIME1 = CLOCK(START) / 6 0 . RANDOM S A M P L I N G OF I N P U T V A R I A B L E S DO 4 5 0 M = 1 , N ROR ( M ) = 1 0 0 0 0 . 0 CONTINUE DO 5 0 0 M = 1 V N X L I F E = EXP ( U L I F E + RANDN ( 0 . 0 ) * D L I F E ) + T L I F E I F ( X L I F E . L T . P L I F E _ . 0 R . X L I FE__.GT._0_L I F E [ _ G 0 TO 1 5 0 5 NCF'= X L I F E " XMKT = EXP ( UMKT + RANDN ( 0 . 0 ) * DMKT ) + TMKT I F ( XMKT . L T . PMKT ._0 R . XMKT . GT«__0MKT ) G0_ TO 1 5 1 0 G R A T E = E X P ( UGRATE + RANDN ( 0 . 0 ) * DGRATF ) + f G R A T E I F ( GRATE . L T . PGRATE . O R . GRATE . G T . OGRATE ) GO TO 1 5 2 0 X I N V = EXP ( U I N V + RANDN ( 0 . 0 ) * D I N V )_ + T I N V I F ( X I N V . L T . P I N V . O R . X I N V . G T . ' 0 1 NV ) GO TO' 1 5 3 0 S A L V = EXP ( U S A L V + RANDN ( 0 . 0 ) * D S A L V ) + T S A L V I F ( S A L V . L T . P S A L V . O R . S A L V . G T . O S A L V ) GO TO 1 5 4 0 COMPUTE MARKET S I Z E F O R " E A C H ' Y E A R XMKTYR ( 1 ) = XMKT I F ( NCF . L T , , 2 ) GO TO 2 1 0 DO 200 I = 2 , N C F ' XMKTYR ( I ) = XMKT YR ( I - 1 ) * ( 1 . 0 + GRATE ) CONTINUE CONTINUE COMPUTE C A S H FLOWS FOR EACH YEAR DO 300 I = 1 , NCF P R I C E ( I ) = EXP ( U P R I C E + R A N D N ( 0 . 0 ) * DPR I C F ) + T PR I C E ' " I F ( P R I C E ( I ) . L T . PPRICE . O R . P R I C E ( I ) . G T . O P R I C E ) GO TO 1 5 5 0 SHARE ( I ) = EXP ( USHARE + RANDN ( 0 . 0 ) * DSHARE ) + T S H A R E 9 I 8 6 01 TT Zl A ro 1570 1580 300 C 2000 2 020 6 00 700 500 12 C C C "C" c c ~c I F ( S H A R E ( 1 f V L T V PSHARE . O R . S H A R E ? I ) . G T . OSHARE ) GO VC(I) = EXP (UVC + R A N D N ( 0 • 0 ) * DVC ) + TVC I F ( V C ( I ) _ . L T . P V C_ . 0 R . VC (J )___G T . OVC ) GO TO 1 5 7 0 T O T V C ( I ) = VC ( T ) * X M K T Y R " ( I ) * SHARE ( I ) FC ( I ) = EXP < UFC + RANDN ( 0 . 0 ) * DFC ) + T F C I F ( FC ( I ) . I T . PFC . O R . FC ( I ) „ „ . G T . OFC ) GO _T0 1 5 8 0 REVNUE(I) = X M K T Y R ( I ) * SHARE( I ) * PRICE(I) CF ( I ) = REVNUE ( I ) - TOTVC ( I ) - FC (I) CONTINUE _ . COMPUTE I N T E R N A L R A T E S OF RETURN I T E M = NCF + 1 DO 2 0 0 0 I J = 1 • NCF I = ITEM IJ 1 1 = 1 + 1 CF ( I I ) = CF ( I ) _ _ _ " CONTINUE" CF ( 1 ) = XINV CF ( I T E M ) = CF ( I T E M ) + S A L V DO 2 0 2 0 I = l ' » ITEM CF ( I ) = CF ( I ) / 10000.0 CONTINUE _ _ C A L L POLY ( I T E M »CF » ROOT* LOOP) I F ( L O O P . G T . 5 0 ) GO TO 6 0 0 GO TO 7 0 0 CONTINUE K = K + 1 GO TO 5 0 0 CONTINUE ROR ( M ) 1 00.0 = ( ROOT - 1 . 0 ) CONTINUE T I ME 2 = C L O C K ( S T A R T ) / 60. WRITE (6*60) WRITE ( 6 * 1 0 ) K SORTING OF INTERNAL R0 RMIN = STARTING RORMAX = E N D I N G RANGE = BIN SIZE WRITE WRITE (6,60) ( 6 , 1800) RATES OF RETURN V A L U E OF FREQ UENCY V A L U E OF FREQUENCY OR C L A S S INTERVAL INTO A FREQUENCY D l S T R I BUT 10N DISTRIBUTION TO 1560 71 _ DISTRIBUTION. H o c C W R I T E "(6750) COMPUTE FREQUENCY DO 750 " IBIN 750. " : DISTRIBUTION. c 1 = 1 , 100__ ( I ) =0 . " 800 8 - - - CONTINUE 0! DO 800 M = 1 » N • I F ( ROR (M) • L T • (RORMIN) JOR. ROR (M) . G E . (RORMAX) ) I = ( R O R ( M ) + (-1.0) * ( R O R M I N ) J / R A N G E + 1.0 900 IBIN ( I ) = GO T O 800 CONTINUE IF IBIN ( I ) + 1 GO TO 900 ~ " _ " " ' • ( R O R ( M ) . N E . 10000.0 ) L = L + 1 . N E . 10000.6 ) W R I T E (6.1900) IF ( ROR ( M ) CONTINUE WRITE (6,60) WRITE ROR (M) _ {6»1950)"T ~ C COMPUTE P R O B A B I L I T Y D E N S I T Y F U N C T I O N AND C L A S S LIMITS. NBIN = ((RORMAX) + (-1.0) * ( R O R M I N ) ) / R A N G E BINLIM (1) = RORMIN DO 1000 1=1. NBIN _ l-0AT ( I BIN ( I ) ) / FLOAT ( N - K - L ) ' " B I N L I M (1 + 1) = B I N L I M TO +" R A N G E 1000 CONTINUE C _ C O M P U T E C U M U L A T I V E D E N S I T Y F U N C T I O N _AND_ C O M P L E M E N T A R Y _ C CUMULATIVE DENSITY FUNCTION. " " "' C D F (1) = P D F (1) C O M C D F (1) = 1.0 - CDF Q ) P D F ( n D O 1050 CDF ( I ) = p 1 = = 2 CDF * NBIN ( I - 1) + COMCDF ( I ) = 1.0 - CDF_ 1050" CONTINUE I TOTAL = 0 S U M = 0.0 "" D O 2500 I = 1 • " N B I N I TOTAL = I TOTAL + I B I N SUM = SUM + PDF ( I ) "2 500" CONTINUE T I ME3 = C L O C K ( S T A R T ) / WRITE (6,70) " " " ~ W R I T E (6,1100) WRITE (6,60) WRITE s (6,1200) PDF ( I ) U) ( I ) 60. H VO • TT" zi WRITE (6,60) WRITE (6»1300) W R I T E . (.6.60) WRITE WRITE (6.1400) (6.60) WRITE ( 6 , 1500) _ _ WRITE (6.50) WRITE ( 6 . 1 6 0 0 ) ( B I N L I M ( I ) » B I N L I M ( 1 + 1) » I B I N ( I ) , P D F ( I ) » C D F ( I ) . C O M C D F • I = 1 » NBIN)_ WRITE (6.60) ' " "~ " WRITE ( 6 . 1 7 0 0 ) IT O T A L • SUM COMPUTE M E A N A N D S T A N D A R D I S E V I A T I 0K__._ SUM = 0 . 0 ----- C DO 4 0 0 0 M = 1 . N IF (ROR(M) .LT. (RORMINJ .0R_. R O R ( M ) _ . G E . ( R 0 R M A X ) ""IF ( R 0 R ( M ) . N E • 1 0 0 0 0 . 0 F ' S U M ~ = 'SUM'""* "RFRT'M'T" 4000 CONTINUE A M E A N = S U M / F L O A T _ _ ( N__K-L_) SUM2 = 0.0 " " ~ DO 4 0 1 0 M = 1 . N IF ( R O R ( M ) . L T . f R O R M I N ) _ _ . 0 R . _ R 0 R (M2__._GE_. J R O R M A X ) IF ( R O R ( M ) •NE. 1 0 0 0 0 . 0 ) SUM2 = SUM2 CONTINUE STD = SORT ( S U M 2 _/ F L O A T ( N - K - L ) ) WRITE (6,60) " " 4010 WRITE (6,3200) AMEAN FORMAT ( 3 6 H MEAN OF I N T E R N A L 3200 I T ) WRITE 3300 150 151 152 153 " " " " +" ("RORTMT RATES_OF_RETURN " " = ) ^G 0 ) GO T0__4000 TO 4 0 1 0 -~AMEANT'**~2 •F16.8,2X.8HPER " " " " CEN ; (6.60) WRITE ( 6.3300 ) STD F O R M A T ( 5 0 H S T A N D A R D D E V I A T I O N "OF I N T E R N A L R A T E S 18.2X.8HPER CENT) TIME4 = CLOCK (START) / 6 0 . WRITE (6.70) " " ~ " ~ ~ WRITE (6,150) TIME1 FORMAT (1H0,9H TIME1 = .F12.4) WRITE (6.1.51) TIME 2 " " FORMAT (1H0.9H TIME2 = .F12.4) WRITE ( 6 . 1 5 2 ) T I ME 3 FORMAT ( 1 HO .9 H T I M E 3 " ~ " = . F 1 2 . 4 ) " ~ " WRITE (6.153) TIME4 FORMAT (1H0.9H TIME4 = ,F12.4) OF RETURN _ ~"~ = ,F16. _ s GO TO 6 0 0 0 END $ I B F T C POLY C~ ~ ~ C F I N D THE R E A L " 9 i . ' ROOT " CLOSEST . 8 "6 Ol • TO 1 OF A POLYNOMIAL C _ _ _ _ _ _ _ S U B R O U T I N E POLY ( N C F » CF * ROOT • LOOP j D I M E N S I O N CF ( 5 0 ) LOOP =. 0 ._ SOLTN = 1.5 5 CONTINUE PX = CF ( 1 ) PDX = F L O A T "(NCF - 1 ) * P.X DO 200 1 = 2 . NCF PX = CF ( I ) + PX * _ S 0 L T N " ' " P D X = F L O A T ( N C F - I ) * CF ( I ) + PDX 200 CONTINUE PDX = PDX / SOLTN PX = S O L T N - PX / PDX LOOP = LOOP + 1 I F ( A B S ( PX - S O L T N ) . L E . 0 . 0 0 0 1 ) I F ( LOOP . G T . 5 0 j G O TO 4 0 0 S O L T N = PX GO TO 5 "300 CONTINUE ROOT = PX RETURN 400 CONTINUE RETURN END _ ; SENTRY OF DEGREE NCF-1. ___ :__ i r Zl + _ j . * SOLTN GO TO • 300 APPENDIX M COMPUTER PROGRAMME FOR MONTE CARLO SIMULATION, USING WEIBULL INPUT PROBABILITIES SJOB SPAGE ST.I ME $ I 8FTC C 19097 ._ c c c c G 9 LEE L 8 MAIN 5 01 FIND C MOON H O E 25 20 D I STR I B U J I O N _OF RATES OF . R E T U R N BY MONTE CARLO SIMULATION US I NG W E I B U L L F U N C T I O N TO Q U A N T I F Y P R O B A B I L I TY O F "OCCURRENCE""""'"" " OF I N P U T V A R I A B L E S . A - P R E F I X = SHAPE_ PARAMETER B - P R E F I X = SCALE PARAMETER' C - P R E F I X = LOWER BOUND L O C A T I O N PARAMETER PU UPPER BOUND P R O B A B I L I T Y PL = LOWER BOUND P R O B A B I L I T Y D I M E N S I O N XMKTYR ( 5 0 ).» PR I CE ( 5 0 ) » S H A R E ! 50 ) . V C < 5 0 ) . T O T V C ( 5 0 ) . F C ( 5 0 ) . R E V N U E t 5 0 ) »CF ( 50 ) »ROR( 5 0 0 0 ) _ (100).COMCDF(100) " D I M E N I BSI NI ( 1O0 0N) » "B I "N L I M ( "1 0"0 ) . P "D F '( 1 0 0 ") . C D F""" MMKT.CMKT.MPRICE.OPRICE.MGRATE.OGRATE.MSHARE.OSHARE.MINV.OIN REAL I V . M L I FE »OL I F E . M S A L V . O S A L V . M V C . OVC » MFC . OFC FORMAT ( I 5 ) 1 FORMAT ( 8 F 1 0 . 4 ) 2 FORMAT ( 3 F 2 0 . 8 ) •• . '' 5____ " F O R M A T OF S T A T I S T I C A L TRIALS TERMINATED 14 ) < 3 9 H ' N O . 10 FORMAT ( I H ) 50 FORMAT ( 1 H 0 ) 60 FORMAT ( 1H1 ) 70 FORMAT ( 4 0 X , 5 2 H C A P I T A L P R O J E C T E V A L U A T I O N BY MONTE CARLO S I M U L A T I O 1100 IN ) FORMAT ( 4 1 X . 5 0 H S 1 0 M I L L I O N E X T E N S I O N TO C H E M I C A L P R O C E S S I N G P L A N T ) 1200 FORMAT ( 4 0 X . 5 3 H S T A T I S T I C A L D I S T R I B U T I O N S OF I N T E R N A L R A T E S OF RETU 1300 1 RN ) 1 40 Cf FORMAT ( 5 4 X , 2 5 H ( ON A BEFORE TAX B A S I S ) ) FORMAT ( I X » 26HRANGE OF I R R I N P E R C E N T A G E , 2 X • 2 2 H F R E Q U E N C Y D I S T R I BUT 1500 1 I O N . 2 X . 2 4 H P R O B A B I L I T Y D I S T R I B U T I O N • 2 X . 2 2 H C U M U L A T I V E PROB. DIST..IX 2 .30HCOMPLEMENTARY CUM. PROB. D l ' S T . ) '""'"' FORMAT (1X.F8.2.3X,2HT0»3X,F8.2,12X.I4,16X.F12.6,14X.F12.6.15X.F12 1600 1.6) _ 1700' FORMAT ( 2 5 X » 9HT OTA L '= . 2 X » I 5 . 6X , 6 HSU M ' =". 4X , F "l 2 . 6 ) " 1800 FORMAT ( 6 7 H I N T E R N A L R A T E S OF RETURN O U T S I D E RANGE OF FREQUENCY D I 1STRIBUTION -) 1900 FORMAT ( 1 X . 6 G 2 0 . 8 ) " 1950 FORMAT ( 6 7 H NUMBER OF EXTREME V A L U E S OF I N T E R N A L RATE OF R E T U R N D I 1SREGARDED =,14) F O R M A T ( 8 X » 1 5 H I N P U T VARIABLES•19X•50HPARAMETERS OF WE I BULL PROBABIL 1ITY DENSITY FUNCTI ON,17X.13HPR0BABILITIES> FORMAT (44X, 5HSHAPE » 1 5 X » 5HSCAL. E^14X^8 HLOCATI 0N_, 1_3X»_1 1HL0WER_ BOUND » 2 200 12X» 11HUPPER BOUND) "~ ~ ~ " ~" • '~~" ' FORMAT (9X, 11HMARKET S I Z E » 1 2 X • G 2 0 . 8 • 2 X , G 2 0 . 8 , 2 X * G 2 0 . 8 , 1 0 X , F 1 0 • 4 • 2 X 2300 1•F10.4) FORMAT (9X 14HSELLING P R I C E S * 9 X » G 2 0 . 8 , 2 X • G 2 0 . 8 . 2 X • G 2 0 . 8 , 1 OX,F10.4» "2400 12X.F10.4) -2 5 50, FORMAT (9X 18HMARKET GROWTH _RJkJJ^5X »G20^_8^2X ,G20 . 8, 2X • G20_. 8 » 1 OXj F l 10.4,2X»F10 FORMAT <9X 15HSHARE OF M A R K E T , 8 X . G 2 0 . 8 , 2 X » G 2 0 . 8 • 2 X • G 2 0 . 8 • 1 O X • F l 0 • 4 2600 1,2X,F10.4) FORMAT (9X 16HT0TAL INVESTMENT,7X,G20.8» 2X,G20.8,2X.G20.8,10X,F10. "2 700 14,2X,F10.4 FORMAT (9X 18HLIFE OF F A C I L I T I E S , 5X,G20. 8» 2_X»G20. 8 , 2 X , G20. 8 ,1 OX . F1 2800 10.4,2X,F10 FORMAT (9X 13HSALVAGE VALUE,1 OX,G20.8,2X,G20.8,2X,G20.8,10X,Fl0.4, 2900 12X»F10.4) 3 000 "FORMAT " (9X 14HVARI ABLE COSTS,1 O X , G 2 0 . 8 , 2 X , G 2 0 . 8 » 2 X , G 2 0 . 8 , 1 O X , F l 0 . 4 1»2X,F10.4) FORMAT (-9X 11HFIXED C O S T S , 1 2 X »_G 2 0 . 8 , 2 X , G 2 0 . 8 , 2 X , G 2 0 . 8 , 1 0 X , F 1 0 . 4 , 2 X 3100 "l » F 1 0 . 4 ) " " " ' ' " " " ~ 5000 FORMAT (8X,15HINPUT VARIABLES,14X,21HPESSIMISTIC EST I MATES,12X , 2 1H 1M0ST LIKELY EST I MATES,12X,2IHOPT IMI ST IC ESTIMATES) 5010 FORMAT <9X 11HMARKET S I Z E » 1 6 X » G 2 0 . 8 » 1 1 X , G 2 0 . 8 » 1 1 X , G 2 0.8) 5020 FORMAT (9X 14HSELLING PR ICES,13X,G20.8,11X•G20 8,11X,G20.8) 5030 FORMAT (9X 18HMARKET GROWTH RATE,9X,G20.8,11X,G20.8,11X•G20.8 ) 5040 FORMAT (9X 15HSHARE OF" MARKET,12X,G26•8,11X,G20.8,11X•G20.8 )""" INVESTMENT,11X,G20.8,11X,G20.8»1IX,G20.8) 5050 FORMAT (9X 16HTOTAL 5060 FORMAT (9X 18HLIFE OF^FACI LIT IES,9X,G20. 8,11X,G20.8, 11X,G20^8_) 5070 FORMAT (9X 13HSALVAGE VALUE » 1 4 X , G 2 0 . 8 , 1 1 X , G 2 0 . 8 , 1 1 X , G 20.8) 5080 FORMAT (9X 14HVARI ABLE COSTS,13X,G20.8,11X,G20.8,11X,G20.8 ) COSTS,16X,G20.8,11X,G20.8,11X,G20.8) 5 090 FORMAT (9X 11HFIXED "" S T A R T = ' C L O C K ! 0. ) CONTINUE 6000 READ ( 5,1 ) N _ I F ( N • E Q • ( - 1 ) ) STOP READ I N W E I B U L L PARAMETERS FOR I N P U T V A R I A B L E S . READ (5,5) A M K T , B M K T , C M K T , A P R I C E , B P R I C E , C P R I C E » A G R A T E , B G R A T E , C G R A T 1E,ASHARE,BSHARE,CSHARE,A I N V , B I N V , C I N V , A L I F E , B L I F E , C L I F E , A S A L V • B S A L 2V,CSALV,AVC,BVC,CVC,AFC,BFC,CFC READ I N T H R E E - L E V E L E S T I M A T E S FOR I N P U T V A R I A B L E S 2100 e READ < 5 » 5 T PMKT »MMKT » O M K T , P P R I C E • M P R I C E , O P R I C E • P G R A T E , M G R A T E V O G R A T I E • PSHARE , M S H A R E , O S H A R E • P I N V , M I N V » 0 1 N V , P L I F E » M L I F E , 0 L I F E , P S A L V • M S A L 2 V , O S A L V * PVC »MVC»OVC » P F C » M F C » O F C READ . 5 , 5) RORMIN*RORMAX.RANGE READ ( 5 , 2) PL.PU WRITE (6 70) WRITE (6 5 0 0 0 ) WRITE ( 6 60 ) W R I T E ( 6 5 0 1 0 ) P M K T . M M K T * OMKT "WRITE ( 6 60 ) WRITE ( 6 5 0 2 0 ) PPRICE.MPRICE.OPRICE WRITE (6 6 0 ) " " W R I T E ( 6 5 0 3 0 f" P GR AT E . MG R A T E » 0 G R A TE W R I T E (6 6 0 ) WRI TE ( 6 5 0 4 0 ) P S H A R E . M S H A R E • O S H A R E "WRITE ( 6 60) PINV,MINV,OINV WRITE ( 6 5 0 5 0 ) _ _ WRITE (6 6 0 ) W R I T E (6 5 0 6 0 ) P L I F E » M L I F E , 0 L I F E W R I T E (6 6 0 ) WRITE ( 6 5 0 7 0 ) PSALV,MSALV,OSALV "WRITE ( 6 6 0) W R I T E (6 5 0 8 0 ) PVCMVCOVC WRITE ( 6 6 0 ) W R I T E (6 5 0 9 0 ) " " PFC'VMFC ,~6"FC ~ WRITE ( 6 7 0 ) WRITE ( 6 2 1 0 0 ) WRITE ( 6 60 ) WRITE ( 6 2 2 0 0 ) W R I T E (6 6 0 ) _ W R I T E ( 6 2 3 0 0 ) A M K T , B M K T • C M K T • P L »PU WRITE ( 6 6 0 ) WRITE ( 6 2 4 0 0 ) APRICE»BPRICE,CPRICE,PL,PU " W R I T E (6 6 0 ) """ WRITE ( 6 2 5 5 0 ) AGRATE.8GRATE.CGRATE.PL.PU WRITE ( 6 60 ) WRITE ( 6 2 6 0 0 ) ASHARE•BSHARE,CSHARE•PL•PU WRITE ( 6 60 ) W R I T E (6 2 7 0 0 ) AINV,BINV,CINV,PL»PU WRITE (6 6 0 ) " " ' WRITE ( 6 2 8 0 0 ) ALIFE,BLIFE,CLIFE,PL,PU WRITE ( 6 60 ) : s 9 8 6 Cl Ti" 21 A r-> s 450 150 5 I 5 10 .15 20 1530 1 540 C 200 2 10 C 15 50 1560 "1570 WRITE ( 6 . 2 9 0 0 ) A S A L V , B S A L V , C S A L V , P L . P U WRITE ( 6 , 6 0 ) W R I T E ( 6 . 3 0 0 0 1 AVC » B V C . C V C . P L . P U WRITE ( 6 , 6 0 ) WRI TE ( 6 . 3 1 0 0 ) A F C B F C . C F C P L . P U WRITE ( 6 . 7 0 ) Y = RAND ( 0.51496382 ) K = 0 L = 0 ' • _ TIME1 = CLOCK(START) / 6 0 . RANDOM S A M P L I N G OF I N P U T V A R I A B L E S DO 4 5 0 M = 1 . N ROR ( M ) = 1 0 0 0 0 . 0 CONTINUE DO 5 0 0 M = 1 » N __ . ALIFE) + CLIFE XLIFE = BLIFE * (-ALOG(RAND(6.0))> ** (1.0 / IF ( XLIFE . L T . PLIFE .OR. XLIFE . G T . O L I F E ) GO TO 1 5 0 5 NCF = X L I F E XMKT = BMKT * ( - A L O G ( R A N D ( 0 . 0 > ) > * * ( 1 . 6 / A M K T ) + CMKT I F ( XMKT . L T . PMKT . O R . XMKT . G T . OMKT ) GO TO 1 5 1 0 GRATE = BGRATE * ( - A L O G ( RAND ( 0 • 0 )_)_) _ • 0 / A G R A T E ) + CGRATE_ 1 5 2 0 " I F ( GRATE . L T . PGRATE • OR • GRATE . G T . O G R A T E ) G O TO BINV * (-ALOGtRAND(O.O))) * * ( 1 . 0 / AINV) + CINV X I NV . G T . _ O I _ N V ) GO TO 1 5 3 0 I F ( XINV . L T. P I N V . O R . X I N V ( 1 . 0 / ASALV) + CSALV S A L V = B S A L V * " ( - A L 0 G ( R A ND ( 0 . 0 )") ) * * I F ! S A L V . L T . P S A L V . O R . S A L V . G T . O S A L V ) GO TO 1 5 4 0 COMPUTE MARKET S I Z E FOR EACH YEAR XMKTYR ( 1 ) = XMKT I F ( NCF . L T , 2 ) GO TO 2 1 0 DO 200 I = 2 , NCF ( 1 . 0 + GRATE ) XMKTYR ( I ) = XMKTYR"" ( I - 1 ) CONTINUE CONTINUE COMPUTE CASH FLOWS FOR EACH YEAR DO -300 I = 1 . NCF PRICE ( I ) = B P R I C E * ( -ALOG(_RAND( 0 . 0 M ) * * ( 1 . 0 / _ A P R J C E ) _ + C P R I C E I F ( P R I C E ( I ) ' . L T . P P R I C E . 0 R • P R I C E ( 17 . G T . O P R I C E > GO TO 15 5 0 ' SHARE ( I ) = BSHARE * (-ALOG(RAND(0.0))) ( 1 . 0 / A S H A R E ) + CSHARE I F ( S H A R E ( I ) . L T . PSHARE . O R . S H A R E ( I ) . G T . OSHARE ) GO TO 1 5 6 0 VC ( I ) = BVC * ( - A L O G ( R AND ( 0 • 0)7 ) " " " * * ( l" 0 / A V C ) + CVC I F ( VC ( I ) . L T . PVC . O R . VC ( I ) . G T . OVC ) GO TO 1 5 7 0 TOTVC ( I ) = VC ( I ) * XMKTYR ( I ) * SHARE (I) 9 L 8 6 01 TT H s 1580 300 C FC ( I ) = BFC * ( - A L O G t R A N D ( O . O ) >) ( 1 . 0 / AFC ) + C F C I F ( FC ( I ) . L T . PFC . O R . FC ( I ) • G T . OFC ) GO TO 1 5 8 0 REVNUE(I) = XMKTYR(I) * SHARE(I) * PRICE( I ) CF ( I J = REVNUE ( I ) - TOTVC ( I ) - FC ( I ) CONTINUE COMPUTE I N T E R N A L R A T E S _ 0 F _ R E T U R N I T E M = NCF + 1 " DO 2 0 0 0 I J = I V NCF I = ITEM - I J _ ; I 1 = 1 + 1 " CF ( I I ) = CF ( I ) CONTINUE CF ( 1 ) = - X I N V CF ( I T E M ) = CF ( I T E M ) + S A L V DO 2 0 2 0 I = 1 » I T E M 9 L 3 6 Ot IT : 2000 CF 2020 ( I ) = CF ( 1 ) 7 ' 1 0 0 0 0 . 0 CONTINUE CALL POLY ( I T E M . C F . R O O T ^ L O O P l I F (LOOP . G T . 5 0 ) GO TO 6 0 0 GO TO 7 0 0 „PPP.__. 700 500 C C c c c" c c CONTINUE K = K + 1 GO TO 5 0 0 CONTINUE ( ROOT 1.0 ) * ROR ( M ) CONTINUE TIME2 = CLOCK(START) / 60 WRITE (6,60) WRITE ( 6 , 1 0 ) K SORTING OF I N T E R N A L RATES VALUE 100.0 OF RETURN OF FREQUENCY INTO A FREQUENCY RORMIN = STARTING RORMAX RANGE - E N D I N G V A L U E " O F " FREQUENCY "~D 1ST R I BUT I ON = B I N S I Z E OR C L A S S I N T E R V A L WRITE ( 6 , 6 0 ) " WRITE (6,1800) WRITE (6,50) COMPUTE FREQUENCY D I S T R I B U T I O N . " DO 7 5 0 1 = 1 , . 1 0 0 IBIN ( I ) = 0 "DISTRIBUTION. DISTRIBUTION " ~ ~ H 750 CONTINUE DO 8 0 0 M = 1 * N I F ( ROR (M> . L T . (RORMIN)_ .0R_. _ ROR ( M_)__ .GE< ( R0RMA_O ) GO TO 9 0 0 I = ( ROR (M) + ( - i . 0) * ( RORMI N )")'" / RANGE -"1.6"" IBIN ( I ) = I B I N ( I ) + 1 GO TO 8 0 0 _ 900 "CONTINUE " " ~ " I F ( ROR (M) L = L + 1 NE« 1 0 0 0 0 . 0 I F ( ROR (M) ,NE. _ 1_0 0 0 0 . 0 WRITE ( 6 * 1 9 0 0 ) ROR ( M ) CONTINUE " "" 800 WRITE ( 6 * 6 0 ) WRITE ( 6 * 1 9 5 0 ) L _ C " COMPUTE P R O B A B I L I T Y DENSITY FUNCTION AND C L A S S L I M I T S . N B I N = ((RORMAX) + (-1.0) * ( R O R M I N ) ) / RANGE B I N L I M ( 1 ) = RORMIN ___ DO 1 0 0 0 1 = 1 * N B I N PDF ( I ) = FLOAT ( I B I N ( I ) ) / FLOAT ( N - K - L ) B I N L I M (1 + 1) = B I N L I M ( I ) + RANGE 1000 CONTINUE " C COMPUTE CUMULATIVE DENSITY FUNCTION AND COMPLEMENTARY C C U M U L A T I V E DENSITY F U N C T I O N . CDF ( 1 ) = PDF <1)' COMCDF ( 1 ) = 1.0 - CDF ( 1 ) DO 1 0 5 0 1 = 2 * N B I N CDF ( I ) " = CDF ( I - 1) + PDF ( I ) COMCDF ' I ) = 1.0 - CDF ( I ) 1050 CONTINUE __ I TOTAL = 0 ' ' SUM = 0.0 _ DO 2 5 0 0 1 = 1 * N B I N I TOTAL = I TOTAL + I B I N ( I f SUM = SUM + PDF ( I ) 2500 CONTINUE " T I M E 3 = CLOCK ( S T A R T ) / 6 0 . " ""' " ~ " ~' WRITE ( 6 . 7 0 ) WRITE ( 6 * 1 1 0 0 ) _ _ WRITE ( 6 * 6 0 ) WRITE ( 6 * 1 2 0 0 ) WRITE ( 6 , 6 0 ) _ __ WRITE ( 6 , 1 3 0 0 ) ~" WRITE ( 6 , 6 0 ) WRITE ( 6 , 1 4 0 0 ) L 8 5 Oi TT" Zi i O WRITE ( 6 , 6 0 ) WRITE ( 6 , 1 5 0 0 ) __ _ WRITE ( 6 , 5 0 ) WRITE ( 6 , 1 6 0 0 ) T B T N L T M T I f , BI NL IM ( I+ 1 ) , I B I N ( I ) , PDF ( I ) , C D F ( I ) » C O M C D F 1(I) , I = 1 , NBIN ) WRITE ( 6 , 6 0 ) . WRITE ( 6 , 1 7 0 0 ) I T O T A L , SUM C COMPUTE MEAN AND STANDARD D E V I A T I O N . SUM = 0 . 0 ' _ DO 4 0 0 0 M = 1 , - - - - " I F (ROR(M) . L T . (RORMIN) . O R . ROR(M) . G E . (RORMAX) ) GO TO 4 0 0 0 I F ( ROR(M) . N E . 1 0 0 0 0 , 0 ) SUM = SUM + ROR(M) 4000'"CONTINUE AMEAN = SUM / FLOAT (N-K-L) SUM2 = 0 . 0 _ DO 4 0 1 0 M = 1 , N I F (ROR(M) . L T . (RORMIN) . O R . ROR(M) . G E . (RORMAX) ) GO TO 4 0 1 0 I F (ROR(M) . N E . 1 0 0 0 0 • 0 ) SUM2_f SUM2_ + ( ROR ( M ) AMEAN )_** 2 4010 CONTINUE STD = SORT (SUM2 / FLOAT (N-K-L)) WRITE ( 6 , 6 0 ) __ _ " W R I T E ( 6 , 3 2 0 0 ) AMEAN ~ ~ * "" " " 3200 FORMAT <36H MEAN OF INTERNAL RATES OF K t T y R N = » F I 6 . 8 . 2 X . » 8 H P E R CEN IT) ""WRITE ( 6 , 6 0 j WRITE ( 6 * 3 3 0 0 ) STD 3300 FORMAT ( 5 0 H STANDARD D E V I A T I O N 0F_ INTERNAL_RATES_ OF RETURN ,F16. 1 8 , 2 X , 8 H P E R CENT) " " " " " " " " " " " ' " T I M E 4 = CLOCK ( S T A R T ) / 6 0 . _ WRITE ( 6 , 7 0 ) " " W R I T E ( 6 , 1 5 0 ) TIME1 150 FORMAT (1H0.9H T I M E 1 = , F 1 2 . 4 ) WRITE ( 6 , 1 5 1 ) TIME2 _ ____ _ _ _ _ _ _ _ 1 5 1 " FORMAT ( 1 H 0 , 9 H T I M E 2 = , F 1 2 . 4 ) " " "" WRITE ( 6 , 1 5 2 ) T I M E 3 152 FORMAT ( 1 H 0 , 9 H T I M E 3 = » F 1 2 . 4 ) WRITE ( 6 , 1 5 3 ) T I M E 4 153 FORMAT ( 1 H 0 , 9 H T I M E 4 = , F 1 2 . 4 ) GO TO 6 0 0 0 " " V END " " S I B F T C POLY C N ± FIND THE REAL ROOT CLOSEST TO 1 OF A POLYNOMIAL S U B R O U T I N E POLY ( NCF • CF »ROOT_ LOOP_)_ D I M E N S I O N CF ( 5 0 ) ~ LOOP = 0 SOLTN = 1 . 5 _ CONTINUE " " -- - • - — PX = CF ( 1 ) PDX = F L O A T ( N C F - _ 1 _ J DO 2 00 I = 2 • NCF PX = CF ( I ) + PX * S O L T N PDX = F L O A T ' ( N C F _ - _ I ) _ * CF__(_I_)_+ JPDX _*_SC_L_TN CONTINUE " ' PDX = PDX / SOLTN PX = S O L T N - PX / PDX LOOP = LOOP + 1 I F ( A B S ( PX - S O L T N ) . L E . 0 . 0 0 0 1 ) GO TO 3 0 0 I F ( LOOP . G T . 5 0 ) GO TO 4QQ S O L T N = PX GO TO 5 CONTINUE ROOT = P X RETURN CONTINUE RETURN END : OF DEGREE _ - NCF-1. \ \ APPENDIX N COMPUTER PROGRAMME FOR MONTE CARLO SIMULATION, USING NORMAL INPUT PROBABILITIES i S 19097 MOON HOE LEE $ JOB 25 $ PAGE 20 $ T I ME S I B F T C MAI N C F I N D D I S T R I B U T I O N OF R A T E S ^ F R E T U R N _ B Y _ M O N T E _CARLO S I M U L A T I O N C U SING THE.NORMAL D I S T R I B U T I O N TO Q U A N T I F Y PROBAB!LITY'0* ~c OCCURRENCE OF I N P U T V A R I A B L E S . C M - P R E F I X = MOST L I K E L Y ESTIMATE D - P R E F I X = STANDARD D E V I A T I O N C_ "c c c 1 2 4 5 i d 50 60 70 1100 1200 1300 1400 1500 1600 "1700 1800 1900 1950 PL = LOWER BOUND PROBABILITY PU UPPER BOUND PROBABILITY DIMENSION X M K T Y R ( 5 0 ) , P R I C E ! 5 0 ) » S H A R E ( 5 0 ) » V C ( 5 0 ) »TOTVC<50 5 , F C ( 5 0 ) , 1REVNUE(50)»CF(50)»ROR(5000) DIMENSION IBIN(100).BINLIM(100),PDF(100)»CDF(100).COMCDF(100) REAL MMKT.MPRICE,MGRATE,MSHARE,MINV.MLIFE.MSALV.MVC»MFC REAL O M K T » O P R I C E » O G R A T E » O S H A R E » O I N V » O L I F E , O S A L V , O V C , 0 F C FORMAT ( 1 5 ) FORMAT ( 8 F 1 0 . 4 ) FORMAT ( 2 F 2 0 . 8 ) FORMAT ( 3 F 2 0 . 8 ) FORMAT ( 3 9 H N O . OF S T A T I S T I C A L T R I A L S T E R M I N A T E D =,14) FORMAT ( I H ) __ FORMAT ( 1 H 0 ) * FORMAT" ( 1 H 1 ) "" " " FORMAT ( 4 0 X , 5 2 H C A P I T A L PROJECT E V A L U A T I O N BY MONTE C A R L O S I M U L A T I O IN ) FORMAT ( 4 1 X , 5 0 H $ 1 0 " M I L L I O N " EXT EN'S I O N ' TO"' CHEM I CA L P R O C E S S I NG" P LA NT ) FORMAT ( 4 0 X , 5 3 H S T A T I S T I C A L D I S T R I B U T I O N S OF I N T E R N A L R A T E S OF R E T U 1RN ) FORMAT ( 5 4 X , 2 5 H ( ON A BEFORE TAX B A S I S )) FORMAT ( 1 X , 2 6 H R A N G E OF IRR I N P E R C E N T A G E , 2 X , 2 2 H F R E Q U E N C Y D I S T R I BUT DIST.,IX 1 I 0 N , 2 X , 2 4 H P R 0 B A B I L I T Y D l S T R I BUT I O N . 2 X » 2 2 H C U M U L A T I V E PROB 2 , 3 0HCOMPLEMENTARY C U M . P R O B . D I S T . ) " ' " " FORMAT (1X,F8.2,3X,2HT0,3X,F8.2,12X,14,16X,F12.6,14X,F12.6,15X,F12 1.6) . _ . FORMAT ( 2 5 X , 9 H T O T A L = ' » 2X , 1 5 , 6 X , 6 H S U M = ,4X , F 1 2 . 6 ) FORMAT ( 6 7 H I N T E R N A L R A T E S OF RETURN O U T S I D E RANGE OF FREQUENCY D l 1STRIBUTION -) FORMAT ( 1 X , 6 G 2 0 . " 8 ) " " " " " " " FORMAT ( 6 7 H NUMBER OF EXTREME V A L U E S OF I N T E R N A L RATE OF RETURN D l 1SREGARDED =,14) 9 L 8 6 01 TT" Zl 2100 2 200 2300 2 400 2550 2 600 2700 2800 "2 9 0 0 3000 3100 5000 50 i d ' 5020 5030 5040 5050 5060 5070 5080 5 090 8000 FORMAT ( 8 X » 1 5 H I N P U T V A R I A B L E S » 2 0 X * 4 9 H P A R A M E T E R S OF NORMAL P R O B A B I L 1 I T Y D E N S I T Y FUNCTI O N » 1 7 X , 1 3 H P R O B A B I L I T I E S ) FORMAT ( 4 7 X , 2 0 H M O S T L I K E L Y EST I M A T E , 4 X , 1 8 H S T A N D A R D DEV I A T I O N , 1 3 X * 1 1 I H LOWER BOUND » 2 X V 1 1 HUP PER BOUND) " " ' " FORMAT ( 9 X 11HMARKET SIZE•24X,G20.8.1X•G20.8,16X,F10•4•2X,F10.4) FORMAT ( 9 X 1 4 H S E L L I N G PR I C E S * 2 1 X » G 2 0 . 8 , 1 X » G 2 0 . 8 * 1 6 X , F 1 0 . 4 , 2 X , F l 0 • 4 1) FORMAT ( 9 X 18HMARKET GROWTH R A T E • 17X * G 2 0 . 8 , 1X • G 2 0 . 8 •' 1 6X » F l 0 • 4 , 2 X * F 110.4) FORMAT <9X 1 5 H S H A R E OF MARKET,20X•G20.8,1X*G20•8,16X,F10.4»2X,Fl0. 14 ! FORMAT (9X 16HTOTAL I N V E S T M E N T * 1 9 X * G 2 0 . 8 » 1 X * G 2 0 . 8 . 1 6 X , F 1 0 . 4 , 2 X * F 1 0 1.4) FACILITIES»17X»G20.8*1X,G20.8*16X*F10.4,2X,F FORMAT <9X 1 8 H L I F E OF 110.4) 13HSALVAGE VALUE•22X,G20.8•1X,G20.8,16X•F10.4•2X•Fl0.4) FORMAT(9X 1 4 H V A R I A B L E COSTS»21X,G20.8,1X•G20.8»16X,F10.4,2X,F10.4 FORMAT (9X 1) COSTS,24X,G20.8»IX,G20.8»16X,F10.4*2X*F10.4) FORMAT (9X 11HFIXED FORMAT ( 8 X 1 5 H I N P U T V A R I A B L E S * 1 4 X , 2 1 H P E S S I M I S T I C EST I M A T E S • 1 2 X • 2 1 H 1MOST L I K E L Y E S T I M A T E S » 1 2 X , 2 1 H O P T I M I S T I C _ _ E S T I M A T E S ) ) " FORMAT * ( 9 X 11HMARK ET S I Z E , 1 6 X , G 2 0 . 8Tl I X » G 2 C 8 • 1 1 X » G 2 0 . 8 PRICES,13X,G20.8,11X,G20.8,11X*G20.8) FORMAT (9X 14HSELLING FORMAT ( 9 X 18HMARKET GROWTH R A T E , 9 X , 6 2 0 . 8 * 1 1 X • G 2 0 . 8 • 1 1 X • G 2 0 .J3 ) "" FORMAT <9X 1 5 H S H A R E O F " M A R K E T , 1 2 X , G 2 0 . 8 , 1 1 X » G 2 0 . 8 * 1 I X , G 2 0 . 8 ) FORMAT (9X 16HT0TAL I N V E S T M E N T * 1 1 X , G 2 0 . 8 , 1 1 X , G 2 0 . 8 * 1 I X , G 2 0 . 8 ) FORMAT ( 9 X 1 8 H L I F E OF F A C I L I T I E S , 9 X » 6 2 0 . 8 • 1 1 X • G 2 0 . 8 , 1 1 X * G 2 0 . 8 ) VALUE,14X»G20.8,11X,G20.8»11X,G20.8)" FORMAT (9X 13HSALVAGE FORMAT (9X 14HVARIABLE C O S T S , 1 3 X » 6 2 0 . 8 * 1 1 X , G 2 0 . 8 » 1 I X » 6 2 0 . 8 ) FORMAT _ ( 9 X 1 1 HF I_XED C O S T S * 16X , G 2 0 . 8 » 11X , G 2 0 . 8 » 1 1 X » G 2 0 . 8 ) _ S T A R T = CLOCK ( 0 . 6 " ) " CONTINUE . READ ( 5 * 1 ) N I F (N . E Q . ( - 1 ) ) STOP" READ I N NORMAL PARAMETERS FOR I N P U T V A R I A B L E S . R EAD ( 5 * 4 ) MMK T , DMKT , MPR I C E , J 5 P R I C E , MG R A T E * DGR A TE_» MSH ARE , D S H A R E * M I N 1V * D I N V * ML I F E » D L I F E »MSALV »DSAL V »MVC » D V C • M F C » DFC READ I N T H R E E - L E V E L E S T I M A T E S FOR I N P U T V A R I A B L E S READ ( 5 * 5 ) PMKT,MMKT,OMKT,PPRICE.MPRICE»OPRICE*PGRATE,MGRATE*OGRAT 1 E » P S H A R E > M S H A R E » 0 S H A R E • P I N V , M I N V • 0 1 N V » P L I F E * ML I F E • O L I F E • P S A L V , M S A L 2V,OSALV*PVC»MVC*OVC,PFC,MFC»OFC READ ( 5 * 5 ) R O R M I N * RORMAX * . R A N G E READ ( WRITE WR I T E WRITE WRITE WRITE WRITE WRI TE WRI TE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRI TE WRITE WRITE WRITE WRITE WRIT E WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE 5 . 2) P L . PU ( 6,70 ) ( 6, 5 0 0 0 ) ( 6, 6 0 ) (6 , 5 0 1 0 ) PMKT »MMKT,OMKT ( 6.60 ) ( 6 .5020) PPRICE.MPRICE.OPRICE (6 , 6 0 ) ( 6 . 5 0 3 0 ) P G R A T E . M G R A T E * OGRATE (6 , 6 0 ) ( 6• 5 0 4 0 ) P S H A R E ^ M S H A R E ^ O S H A R E (6 , 6 0 ) ( 6 • 5050)" PINV»MINV»OINV ( 6.60 ) (6 • 5 0 6 0 ) P L I F E . M L I F E t O L l F E (6 , 6 0 ) ( 6 • 5 0 7 0 ) PSALV^MSALV^OSALV ( 6,60 ) ( 6 . 5 0 8 0 ) PVOMVC^OVC ( 6.60 ) ( 6 , 5 0 9 0 ) PFC»MFC»OFC ( 6• 70) ( 6,2100) ( 6• 6 0 ) ( 6•2200) ( 6 • 60 ) ( 6 • 2 3 0 0 ) MMKT,DMKT,PL^PU ( 6• 6 0 ) ( 6 •2400) MPRICE^DPRICE^PLtPU ( 6 • 60 ) ( 6 • 2 5 5 0 ) M G R A T E • D G R A T E • P L * PU ( 6. 6 0 ) ( 6 • 2 6 0 0 ) MSHARE^DSHARE^PL^PU (6 . 6 0 ) " " ( 6. 2 7 0 0 ) M I N V t D I N V . » P L » P U ( 6. 6 0 ) ( 6. 2 8 0 0 ) M L I F E • D L I F E • P L V P U (6 . 6 0 ) [ 6, 2 9 0 0 ) M S A L V , D S A L V P L • P U ( 6• 6 0 ) ( 6 • 3 0 0 0 ) MVC»DVC » P L » P U (6 . 6 0 ) f WRITE WRITE R = (6,3100) (6,70) RANDN MFCDFC ,PL,PU 9 L ( 0.49287536 ) H L = 0 TIME1 = CLOCK (START) / _ 6 0 . RANDOM S A M P L I N G OF INPUT V A R I A B L E S C" . ~ DO 4 5 0 M = 1 , N ROR (M) = 10000.0 4 50 """CONTINUE " DO 5 0 0 M = 1 , N 1505 X L I F E = ML I F E + RANDN (0.0) » DLIFE IF ( X L I F E ' . L T . P L I F E . O R . X L I F E .GT. O L I F E NCF = XLIFE 1 5 1 0 _ XMKT = MMKT + R A N D N ( 0 . 0 ) * DMKT ~ ~ I F ( X M K T . L T . P M K T "".OR". X M K T .GT." O M K T ) G O 1520 G R A T E = MGRATE + RANDN ( 0 . 0 ) * DGRATE " ' ) GO TO " TO 1505 1510 __ I F ( G R A T E . L T . P G R A T E . O R . G R A T E . G T . O G R A T E ) G O T O 1 5 2 0 X I N V = MINV + RANDN ( 0 . 0 ) * DINV IF ( X I N V . L T . P I N V .OR. X I N V .GT. O I N V ) GO TO 1 5 3 0 1540 _ S A L V = MSALV + RANDN ( 0 . 0 ) *_DSA_LV_ _• IF ( SALV . L T . P S A L V • OR • S A L V .G"T.'"0SALV T G O to"~1540 C COMPUTE MARKET S I Z E FOR EACH YEAR XMKTYR ( 1 )_ = X M K T _ . 1 530 IF ( N C F . L T . 2 ) GO TO 2 1 0 DO 200 1 = 2 , NCF XMKTYR ( I ) = XMKTYR ( I - J _ _ ) _*'_( _1 • 0__+ GRATE J_ 2 00 210 C CONTI NUE " " ' CONTINUE C O M P U T E C A S H FLOWS FOR E A C H Y E A R DO 300 1 = 1, NCF 1550 PRICE ( I ) = MPRICE + RANDN ( 0 . 0 ) * D P R I C E IF ( P R I C E ( I ) . L T . P P R I C E .OR. P R I C E . I > • GT.__0PR I C E 1 5 6 O"' S H A R E ( I ) = M S H A R E + RANDN (0.0) *DSHARE 1570_ IF ( SHARE( I) . L T . PSHARE VC ( I ) = I F ye ( TOTVC 1580 ( I) ( I ) = FC ( I ) = IF ( FC REVNUE _____ _ CF (i) MVC MFC + RANDN .LT. PVC = .OR. VC SHARE(I) DVC_ (I) VC ( I ) * XMKTYR + RANDN (0.0) * DFC .OR. (I) ( I ) .LT. PFC ( I ) = .OR. (0.0)* XMKTYR REVNUEL (ij ( I ) * FC .GT. ( I )* SHARE .GT. OSHARE _ _ _ _ _ _ _ OVC SHARE ) GO JL_GO__0_ 1 5 ) GO .GT. -_J_ojyc_iij_-_Fc_rL) 1560 _ TO 15 7 0 "" "" ( I ) £ _ ( I ) * TO 50 OFC ) GO PRICE TO _ _ 1 5 8 0 • " _ " _° " ( I ) _______: : CONTINUE COMPUTE I N T E R N A L R A T E S OF RETURN I T E M =. NCF + 1 DO 2 0 00 I J = 1 » NCF I = ITEM - I J II = 1 + 1 CF ( I I ) = CF ( I ) CONTINUE CF ( 1 ) . = . - X I N V _ CF ( I T E M ) = CF ( I T E M ) + S A L V DO 2 0 2 0 I = 1 • ITEM CF ( I ) = CF ( I ) / 1 0 0 0 0 . 0 CONTINUE C A L L POLY ( I T E M . C F . ROOT ,LOOP) I F (LOOP .GT. 5 0 ) GO T0__600 GO TO 7 0 0 ~ " CONTINUE K = K + 1 _ GO TO 5 0 0 CONTINUE ROR (M) = ( R 0 0 1 _ _ _ - _ l . 0 _ l *_1_00_0 CONTINUE TIME2 = CLOCK(START) / 6 0 . WRITE ( 6 , 6 0 ) •_ WRITE ( 6 . 1 0 ) ~ K " S O R T I N G OF I N T E R N A L R A T E S OF RETURN RORMIN = S T A R T I N G V A L U E OF FREQUENCY RORMAX = ENDING V A L U E OF FREQUENCY RANGE" = B I N S I Z E OR C L A S S I N T E R V A L _ __ •__ " - - - - - INTO A FREQUENCY DISTRIBUTION DISTRIBUTION ~ - DISTRIBUTION. _ : r WRITE ( 6 , 7 0 ) _ _ _ _ _ _ _' _ _ WRITE ( 6 . 1 8 0 0 ) " " " " ' " " " ~ "" WRITE ( 6 . 5 0 ) COMPUTE FREQUENCY D I S T R I B U T I O N . _ DO 7 5 0 I = 1 . 1 0 0 " " " ~ "~ ~ ' ~ ~ " " " " "" " "" IBIN (I ) = 0 CONTINUE _ • _ DO 8 0 0 M ="1 . N " ' " I F ( ROR (M) . L T . (RORMIN) .OR. ROR (M) .GE- ( R O R M A X ) ) GO TO 9 0 0 I = ( R O R (M) + (-1.0) * ( R O R M I N ) ) / RANGE + 1.0 ~ ~ " " " • |_, S I BIN (I) = IBIN ( I) + 1 GO TO 800 900 CONTINUE _ IF < ROR (M) .NE. 10000.0 ) L = L + 1 IF ( ROR (M> .NE. 10000.0 ) WRITE (6,1900) ROR (M) 800 CONTINUE _ _ WRITE ( 6 . 6 0 ) ' " ' " WRITE (6, 1950) L C_ _ COMPUTE PROBABILITY DENSITY FUNCTION AND CLASS_ LIMITS. " N B I N = ( (RORMAX) + (-1.0) *" "("RORMIN ) ) 7 "RANGE BlNLIM (1) = RORMIN DO 1000 1 = 1 , NBIN _ " P D F (I.) = FLOAT (IBIN (I )j / FLOAT (N-K-L) BINLIM Cl+1) = BINLIM (I) + RANGE 1000 CONTINUE _ "C COMPUTE CUMULAT I VE DEN S I TY FUNCTION AND COMPLEMENTARY"" C CUMULATIVE DENSITY FUNCTION. CDF (1) = PDF (1) _ COMCDF (1 )"= 1.0 - CDF "( 1) DO 1050 1 = 2 , NBIN CDF (I) = CDF ( I - l ) +_PDF_m " C O M C D F ( I )" =1.0 - CDF (I ) 1050 CONTINUE I TOTAL = 0 • SUM = 0.0 DO 2500 1 = 1 , NBIN I TOTAL. = I TOTAL + I B I N _ m SUM = SUM + PDF ( I ) ~ '" 2500 CONTINUE TIME3 = CLOCK ( START ) / 60. " WRITE (6,70 ) WRITE (6,1100) WRITE (6,60) "WRITE ( 6* 1200-) WRITE (6,60) WRITE (6, 1300) "WRITE (6,60 j WRITE (6,1400) WRITE (6,60) ._ WRITE (6, 1 5 0 0 ) " ~ " ' - -- ~ WRITE (6,50) WRITE (6, 1600 ) (BINLIM(I)»BINLIM(1 + 1) ,IBIN(I) ,PDF(I),CDF(I),COMCDF H C _ 4000 4010 3200 3300 150 _ 151 152 153 1(1) . I = 1 . NBIN) "" " " ~~ ' " ~ ~ ~ WRITE (6.60) W R I T E ( 6 , 1 7 0 0 L _ I T O T A L *_S_U_ COMPUTE MEAN AND STANDARD D E V I A T I O N • SUM = 0 . 0 DO 4 0 0 0 M = 1 • N • _ __ I F ( R O R ( M ) . L T . ( R O R M I N ) . O R . R O R ( M ) . G E . [ R O R M A X ) ) GO TO ^ 0 0 0 I F ( R O R ( M ) . N E . 1 0 0 0 0 . 0 ) SUM = SUM + R O R ( M ) ^CONTINUE AMEAN = SUM" / ' F L O A T T N - K - U SUM2 = 0 . 0 DO 4 0 1 0 M = 1 _» N _ I F ( R O R ( M ) . L T . ( R O R M I N ) • O R • R O R ( M ) . G E . (RORMAX) T GO TO 4 0 1 0 I F ( R O R ( M ) . N E . 1 0 0 0 0 . 0 ) SUM2 = SUM2 + ( R O R ( M ) - A M E A N ) * * 2 CONTINUE _ __ _ STD = SORT ( SUM2 7 F L O A T ( N - K - L ) ) ' ' " ~ WRITE ( 6 * 7 0 ) . WRITE ( 6 . 6 0 ) __ W R I T E ( 6 . 3 2 0 0 ) AMEAN " - FORMAT ( 3 6 H MEAN OF I N T E R N A L R A T E S OF RETURN = . F 1 6 . 8 . 2 X . 8HPER CEN IT) _ WRITE ( 6 . 6 0 ) ' " " " W R I T E ( 6 . 3 3 0 0 ) STD FORMAT ( 5 0 H STANDARD J__E_V I A J _ 0_N__0 F INT_ERN_AIL_ R__TJ_S_0_F i^ET_JR_N_=_ ^„ 18 . 2 X . 8 H P E R C E N T ) T I ME 4 = CLOCK ( S T A R T ) / 6 0 . WRITE(6,70) _ _ _ _ _ WRITE ( 6 . 1 5 0 ) TIM E l ~ " " " • -- - - - — FORMAT ( 1 H 0 . 9 H T I M E 1 = . F 1 2 . 4 ) WRITE ( 6 . 1 5 1 ) TIME 2 [ " F O R M A T ( 1 H 0 . 9 H T I M E 2 '= . F 1 2 . 4 ) " " ' " ' ~ -----W R I T E ( 6 . 1 5 2 ) T I ME 3 FORMAT ( 1 H 0 . 9 H T I M E 3 = » F 1 _ 2 . 4 ) _ _ _ _ _ ' WRITE ( 6 . 1 5 3 ) TIM E 4 """ " FORMAT ( 1 H 0 . 9 H T I M E 4 = . F 1 2 . 4 ) GO TO 8 0 0 0 _ END " "" " ' •' ' " - - - - ----- S I B F T C POLY C C F I N D THE r ' : __ ' . _____ _ • ^ «g REAL SUBROUTINE ROOT POLY CLOSEST TO 1 OF A " P O C Y N O M I A L O F (NCF.CF.ROOT,LOOP) DEGREE _ NCF-1. DIMENSION CF (50) LOOP = 0 __ SOLTN = 1.5 5~ CONTINUE PX » CF (1) PDX = FLOAT (NCF - 1 ) » PX DO 2 00 I =" 2 "'• NCF PX = CF ( I ) + PX * SOLTN __PDX = FLOAT (NCF_-_ I) * CF ( I ) + PDX » SOLTN 2 001~ "CONTINUE PDX = PDX / SOLTN PX = SOLTN - PX t_ PDX LOOP = LOOP + 1 IF ( ABS( PX - SOLTN ) .LE. 0.0001 ) GO TO 300 IF ( LOOP .GT. 50 ) GO TO 400 SOLTN = PX GO TO 5 _3P0_ CONTINUE ~ROOT"= PX RETURN 400 _ CONTINUE _ "'""' RETURN END SENTRY APPENDIX. 0 COMPUTER PROGRAMME FOR MONTE CARLO SIMULATION, USING TRIANGULAR INPUT PROBABILITIES 19097 MOON HOE L E E $JOB 25 SPAGE 20 $T I ME M A I N $ IBFTC C F I N D D I S T R I B U T I O N O F _ R A T E S _ O F RETURN BY MONTE CARLO S I M U L A T I O N _C U S I N G THE TRI ANGULAR D I S T R I B U T I O N TO Q U A N T I F Y P R O B A B I L I T Y OF C O C C U R R E N C E OF I N P U T V A R I A B L E S . C P-PREFIX = PESSIMISTIC ESTIMATE M-PREFIX C_ MOST L I K E L Y E S T I M A T E O-PREFIX = OPTIMISTIC ESTIMATE C F - P R E F I X = C U M U L A T I V E D E N S I T Y F U N C T I O N AT MOST L I K E L Y E S T I M A T E C C " D I M E N S I O N X M K T Y R ( 5 0 ) .PR ICE< 5 0 ) » S H A R E ( 5 0 ) , V C ( 5 0 ) , T O T V C ( 5 0 ) »FC ( 5 0 ) • 1REVNUE(50)»CF(50),ROR(5000) DIMENSION I B I N ( 1 0 0 ) . B I N L I M ( 1 0 0 ) .PDF(100) .CDF(IOC),C0MCDF(100 ) ""REAL MMKT • OMKT .MPRICE »OPR I C E'» MGR ATE > OGRATE, M SHARE .OSHARE » M I N V . O l N 1V.MLIFE.0LIFE.MSALV.0SALV.MVC.OVCMFC.OFC FORMAT ( 1 5 ) FORMAT ( 3 F 2 0 . 8 j 5 FORMAT ( 3 9 H NO. OF S T A T I S T I C A L T R I A L S T E R M I N A T E D =,14) 10 FORMAT ( I H ) 50 ~ FORMAT ( 1 H 0 ) 60 FORMAT ( 1 H 1 ) 70 FORMAT ( 4 0 X . 5 2 H C A P I T A L P R O J E C T E V A L U A T I O N BY MONTE CARLO S I M U L A T I O 1100 "IN ) FORMAT ( 4 1 X . 5 0 H S 1 0 M I L L I O N E X T E N S I O N TO C H E M I C A L P R O C E S S I N G P L A N T ) 1200 FORMAT ( 4 0 X . 5 3 H S T A T I S T I C A L D I S T R I B U T I O N S OF I N T E R N A L R A T E S OF RETU 1300 1RN) FORMAT ( 5 4 X , 2 5 H ( ON A B E F O R E TAX B A S I S ) ) 1400 FORMAT ( 1 X , 2 6 H R A N G E O F I R R I N P E R C E N T A G E , 2 X , 2 2 H F R E Q U E N C Y D I S T R I BUT 1500 H O N , 2 X , 2 4 H P R 0 B A B I L I T Y D I S T R I BUT I O N , 2 X • 2 2 H C U M U L A T I V E PROB •. D I S T . , I X 2,30HCOMPLEMENTARY CUM. PROB. D I S T . ) FORMAT ( 1 X , F 8 . 2 , 3 X , 2 H T 0 , 3 X , F 8 . 2 , 1 2 X , I4,16X,_Fl2.6,14X,F12.6»15X,Fl2 1600 1.6) . " ' ' " " " " 1700 FORMAT ( 2 5 X . 9 H T 0 T A L " = , 2X • I 5,6X,6HSUM =,4X,F12.6) 1800 FORMAT .(67H I N T E R N A L R A T E S OF RETURN O U T S I D E RANGE OF FREQUENCY D l 1 S T R I B U T I 0 N -) " " FORMAT ( 1 X , 6 G 2 0 . 8 ) 1900 FORMAT ( 6 7 H NUMBER OF E X T R E M E _ V A L U E S OF I N T E R N A L RATE OF RETURN D l 1950 1SREGARDED =,I 4 ) FORMAT ( 8 X , 1 5 H I N P U T V A R I A B L E S » 2 7 X , 6 2 H D A T A FOR ENCODING OF U N C E R T A I 2100 1NTY WITH T R I A N G U L A R D I S T R I B U T I O N S ) v C! 21 H 2200 2 300 2400 2550 2600 2700 2800 2900 3000 3100 FORMAT ( 3 7 X » 2 1 H P E S S I M I S T I C ESTI MATES.12X•21HM0ST LIKELY ESTIMATES, 112X.21HOPTIMISTIC ESTIMATES) F O R M A T ( 9 X , 1 1 H M A R K E T S I Z F , 1 6 X • G 2 0 _8 , 1 1 X , _ G 2 0 . 8 , 1 1 X , G 2 0 . 8 ) FORMAT (9X , 1 4 H S E L L I N G P R I C E S , 1 3 X , G 2 0 . 8 » 1 1 X , G 2 0 . 8 , 1 1 X , G 2 0 • 8 ) ( 9 X » 1 8 H M A R K E T GROWTH R A T E * 9 X , G 2 0 . 8 » 1 1 X » G 2 0 . 8 * 1 1 X » G 2 0 . 8 ) FORMAT ( 9 X , 1 5 H S H A R E OF MARKET,12X»G20.8,1IX,G20.8»11X,G20.8!_ FORMAT ( 9 X , 1 6 H T 0 T A L I N V E S T MENT , 1 1 X , G 2 0 . 8 , 1 1 X , G ? 0 . 8 , 1 1 X , G 2 0 . 8 ) FORMAT ( 9 X » 1 8 H L I F E O F F A CILITIES,9X,G20.8»11X,G20.8,1IX,G20.8) FORMAT ( 9 X , 1 3 H S A L V A G E V A L U E • 14_X , G20_. 8 * 1 I X * G 2 0 . 8 » 1 1 X , G 2 0 . 8 ) FORMAT ( 9 X , 1 4 H V A R I A B L E C O S T S , 1 3 X , G 2 6 . 8 , 1 iY7G2 6 . 8 " , T 1 X 7 G 2 6 7 8 ) FORMAT ( 9 X , 1 1 H F I X E D C O S T S , 1 6 X ,G20.8,11X,G20.8,11X,G2 0.8) FORMAT _ _ _ _ _ _ _ START = CLOCK ( 0 . 0 ) N ~ ~ ' " ' "' ' ~ ~ READ (5*1) R E A D I N T H R E E - L E V E L E S T I M A T E S FOR I N P U T V A R I A B L E S R E A D ( 5 »5 ) P M K T , M M K T , O M K T ,_PPR I C E * M P R I C E , O P R I C E ,_PGR AT E , MGR A T E , OGR AT_ ' L E * P S H A R E * M S H A R E * O S H A R E , P I N V 7 M I N VT6'L" N V 7 P"C"F F E * M L I F E 7 O L I F F » P S A L V » M S A L 2V,OSALV,PVC»MVC,OVC*PFC»MFC,OFC READ ( 5 * 5 ) R O R M I N , RORMAX* RANGE "WRITE ( 6 70 ) WRITE ( 6 2100 ) WRITE ( 6 60 ) WRITE ( 6 2 200) WRITE ( 6 60 ) PMKT.MMKT»OMKT WRITE (6 2 3 0 0 ) W RITE ( 6 60 ) WRITE ( 6 2400 ) P P R I C E * M P R I C E * O P R I C E WRITE ( 6 60 ) WRITE ( 6 2 5 5 0) PGRATE»MGRATE*OGRATE WRITE ( 6 60 ) WRITE ( 6 2 600) PSHARE»MSHARE*OSHARE WRITE (6 60 ) PINV»MINV,OINV WRITE ( 6 2700) WRITE (6 60 ) WRITE ( 6 2 8 0 0 ) PL I F E » M L I F E » O L I F E WRITE ( 6 60 ) WRITE ( 6 2900) PSALV,MSALV,OSALV WRITE ( 6 6 0 ) " WRITE ( 6 3000) PVC*MVC*OVC WRITE ( 6 60 ) WRITE ( 6 3100) PFC *MFC,OFC WRITE ( 6 70 ) ) R = RAND ( 0 . 5 1 4 9 6 3 8 2 9 L •i 6 01 i'T z\ I I K = 0 L = 0 T I M E 1 = CLOCK 9 (START) / 6 0 . ; ' INPUf VARIABLES ' ~ ~ FMKT = (MMKT - PMKT) / (OMKT - PMKT) F P R I C E = ( MPR I C E _ - _ PPR I C E ) / ( O P R I C E_____P_PR I C E ) F G R A T E = (MGRATE - PGR ATE) / ( OGR ATE - PGRATE5 F S H A R E = (MSHARE - P S H A R E ) / (OSHARE - P S H A R E ) F I N V = ( M I N V - P I N V ) / [QINjy ____PJNV__ ~ " F L I F E ="' ( M L I F E -" P L I FE ) /'""( OL I F E -" P L I F E ) F S A L V = (MSALV - P S A L V ) / ( O S A L V - P S A L V ) FVC = (MVC - P V C ) / (OVC - P V C ) •_ F F C = (MFC - P F C ) / (OFC - P C )" DO 4 5 0 M = 1 • N • __ ROR (M) = 1 0 0 0 0 . 0_ . ; 450 CONTINUE DO 5 0 0 M = 1 , N R = RAND ( 0 . 0 ) _ _ I F "( R .GT. F L I F E )" GO TO 4 0 0 0 X L I F E = SQRT( (ML I F E - P L I F E ) * ( O L I F E - P L I F E > * R ) + P L I F E GO TO 4 0 1 0 4000 XLIFE = OLIFE - SQRT((OLIFE-MLIFE) * (OLIFE-PLIFE) * (1.0-R)) 4010 CONTINUE _ NCF = X L I F E R = RAND ( 0 . 0 ) I F ( R .GT. FMKT ) GO TO 4 0 2 0 = SQRT ( (MMKT-PMKT ) .* (OMKT-PMKT) * R ) + PMKT GO TO 4030" " 4020 XMKT = OMKT - S Q R T ( ( O M K T - M M K T ) * (OMKT-PMKT' * ( 1 . 0 - R ) ) 4030 CONTINUE R = RAND ( 0 . 0 ) I F ( R .GT. FGRATE ! GO TO 4 0 4 0 GRATE = SQRT ( (MGRATE-PGRATE__*_ (_0_GR_ATE-PGRAT_E)__j_ R_>_+ PGRAJfE . GO TO 4 0 5 0 " ' " 4040 GRATE = OGRATE - SQRT ( (OGRATE-MGRATE) * ( O G R A T E - P G R A T E ) * ( 1 . 0 - R ) ) 4050 CONTINUE R = RAND ( 0 . 0 ) I F ( R .GT. F I N V ) GO TO 4 0 6 0 X I N V = S Q R T ( ( M I N V - P I N V ) * ( O I N V - P I N V ) * R )_+ P I N V _ _ _ GO TO 4 0 7 0 " " " " " " ' ~ " . 4060 X I N V = O I N V - SQR T ( ( 01 NV-M IN V ) * . ( O I N V - P I N V ) * ( 1 . 0 - R ) ) 4070 CONTINUE "C / _s " R A N D O M " SAMIPLTNG"'OF " ~ s 01 f I w C X M K ' ' T ; ; f .1 : \_{. R = RAND ( 0 . 0 ) I F ( R .GT. F S A L V ) GO TO 4 0 8 0 S A L V = SQRT ( ( M S A L V - P S A L V ) * ( OSA L V - P S ALV ) » R ) + P S A L V GO TO 4 0 9 0 ~ " ' " . " 4080 S A L V = O S A L V - SQRT ( ( O S A L V - M S A L V ) * ( O S A L V - P S A L V) * ( 1 . 0 - R ) ) 4090__ C O N T I N U E . _ C COMPUTE MARKET S I Z E FOR EACH YEAR XMKTYR ( 1 ) = XMKT I F ( NCF . L T . 2 ) GO TO 210_ DO 200 I = 2 . NCF XMKTYR ( I ) = XMKTYR ( I - 1 ) * ( 1.0 + GRATE ) 200 CONTINUE _____ _ ; 210 CONTINUE C COMPUTE C A S H FLOWS FOR EACH YEAR DO 300 I = 1» NCF _SJ. ~ R = R AND ( 0 . 0 ) I F ( R .GT. F P R I C E ) GO TO 4 1 0 0 PR I CE ( I ) = SQRT ( [MPR I_C E-_P PRJ^C E ) * ( OPR I C E - P P R I C E ) *R ) + P P R I C E GO TO 4 1 1 0 4100 P R I C E ( I ) = OPRICE - S Q R T ( ( O P R I C E - M P R I C E ) * ( O P R I C E - P P R I C E ) * ( 1 . 0 - R ) ) 4110 CONTINUE R = RAND ( 0.0 ) I F ( R .GT. F S H A R E ) GO TO 4 1 2 0 _ SHARE ( I ) = SQRT ( ( M S H A R E - P S H A R E ) * (OS H A R E - P SH A R E ) »R ) PS HA R E GO TO 4 1 3 0 4120 S H A R E ( I ) = OSHARE - S O R T ( ( O S H A R E - M S H A R E ) * ( O S H A R E - P S H A R E ) * ( 1 . 0 - R ) ) 4130 CONTINUE . __ __ R = RAND ( 0 . 0 ) " ' " ' " ~ ~~ ' I F ( R .GT. FVC ) GO TO 4 1 4 0 ._ VC ( I ) = SQRT ( ( M V C - P V C ) * ( O V C - P V C ) » R) + PVC ' G O T O 4150 4140 VC ( I ) = OVC - S Q R T ( ( O V C - M V C ) * ( O V C - P V C ) * ( 1 . 0 - R ) ) 4150 CONTINUE _____ _ TOTVC ( I ) = VC ( I ) * X M K T Y R ( T V * S H A R E ""( I 7 " ~ ~" " " " R = RAND ( 0 . 0 ) I F ( R .GT. FFC ) GO TO 4 1 6 0 _ _ ; FC ( I ) = SQRT ( ( M F C - P F C ) * ( O F C - P F C ) * R ) + P F C ~ " " " " " GO TO 4 1 7 0 4160 F C H ) = OFC - SQRT ( ( O F C - M F C ) * [ O F C - P F C ) * ( 1 . 0 - R ) ) _ _ 4170 CONTINUE " REVNUE ( I ) = XMKTYR ( I ) * S H A R E ( I ) * P R I C E ( I ) CF ( I ) = REVNUE ( I ) - TOTVC ( I ) - FC ( I ) s s , L _ 6 Ol_ TT" Z\ | 1 • " H ~." " " _ V> 300 C 2000 2020 600 700 5 00 CONTINUE COMPUTE I N T E R N A L R A T E S OF RETURN I TEM, = NCF + _1 DO 2 0 0 0 I J = 1 • NCF I = ITEM - I J II = 1 + 1 CF ( I I ) = CF ( I ) CONTINUE CF ( 1 ) = - X I NV _ CF ( I T E M ) = CF ( I T E M ) + S A L V DO 2 0 2 0 1 = 1 * ITEM CF ( I ) = C F _ ( _ ) _ / 1 0 0 0 0 . 0 _ _ _ CONTINUE" C A L L POLY ( I T E M , C F » R O O T , L O O P ) I F (LOOP GT. 5 0 ) GC TO 6 0 0 GO TO 7 0 0 CONTINUE K = K + 1 GO TO 5 0 0 CONTINUE ROR (M) = ( ROOT - 1.0 ) * 1 0 0 . 0 C O N T I N U E ~ ~ TI ME 2 = C L O C K ( S T A R T ) / 6 0 . WRITE ( 6 * 6 0 ) WRITE (6,10)K 8 6 Oi TT Zl. t I 1 C c c c c "c c 750 S O R T I N G OF I N T E R N A L R A T E S OF RETURN INTO A FREQUENCY DISTRIBUTION. RORMIN = S T A R T I N G V A L U E OF FREQUENCY D I S T R I B U T I O N RORMAX = ENDING _ V A L U E O F _F_REQUEN_CY_DJ_STR_I BUTIj0N_ RANG E = " B I N S I Z E OR " C L ASS I N T E R V A L ' WRITE ( 6 . 7 0 ) _ WRITE (6» 1 8 0 0 j WRITE ( 6 , 5 0 ) COMPUTE FREQUENCY D I ST RJ BUT K ) N ._ DO 75 0 I = 1 '» 1 0 0 IBIN ( I ) = 0 CONTINUE DO 8 0 0 M = 1 . N I F ( ROR (M) , L T . (RORMIN) .OR. ROR (M) .GE. I = ( ROR (M) + (-1.0) * ( R O R M I N ) ) / RANGE + ( R O R M A X ) ) GO 1.0 TO 900 ft IBIN (I ) = I BIN" ( I T + 1 GO TO 8 0 0 CONTINUE 900 800 I F " ( R O R "TM") ' . N E T T O O O O . O ) IF ) WRITE ( ROR (M) . N E . 10000.0 L = L + l (6.1900) ROR ( M ) 01 CONTINUE WRITE (6*60) Zi WRITE ( 6 . 1 9 5 0 ) L C _ COMPUTE P R O B A B I L I T Y DENS I TY_ F U N C T I ON_ AND C L A S S L J j M I T S . _____ N B I N = ( ( R O R M A X ) + 7-1.0) * "(RORMIN)) / RANGE B I N L I M ( 1 ) = RORMIN DO 1 0 0 0 I = 1 » NBIN _ PDF ( I ) = FLOAT ( I B I N ( I ) " ) / FLOAT { N - K - L ) B I N L I M ( 1 + 1 ) = B I N L I M ( I ) + RANGE 1000 CONTINUE _______ _ "C C O M P U T E C U M U L A T I V E D E N S I T Y F U N C T I O N AND'COMPLEMENTARY C CUMULATIVE DENSITY FUNCTION. CDF ( 1 ) = PDF ( 1) _ COMCDF ( 1 ) = 1 . 0 - CDF ( 1) DO 1 0 5 0 1 = 2 . N B I N CDF ( I ) = CDF ( I - 1) + P D F _ _ J ) COMCDF ( I ) ' = 1.0 - CDF CiT" 1050 CONTINUE I TOTAL = 0 ___ _____ SUM = 0 . 0 " ' DO 2 5 0 0 1 = 1 , N B I N I TOTAL = I TOTAL + I B I N _J_> _ SUM = SUM' + PDF ( I ) ~ ~ " - - ™ ' 2500 CONTINUE _ T I M E 3 = C L O C K ( S T A R T ) J J>0* WRITE ( 6 . 7 0 ) " " ' WRITE ( 6 . 1 1 0 0 ) WRITE ( 6 , 6 0 ) _____ . WRITE ( 6 . 1 2 0 0 ) " ' " " ' ' " " " WRITE ( 6 . 6 0 ) WRITE ( 6 . 1 3 0 0 ) _ _ _ _ WRITE ( 6 . 6 0 ) " "" ' """ " ~ ' " " WRITE ( 6 . 1 4 0 0 ) WRITE ( 6 . 6 0 ) _ WRITE ( 6 . 1 5 0 0 ) ' " - - •-- — — - ---WRITE ( 6 . 5 0 ) WRITE ( 6 . 1 6 0 0 ) ( B I N L I M U ) , B I N L I M ! 1 + 1 ) , I B I N ( I ) , P D F ( I ) , C D F ( I ) , C O M C D F 7 - ~ '" _ - H Oo 1(1) • I = 1 • NBIN) WRITE ( 6 . 6 0 ) W R I T E ( 6 . 1 7 0 0 ) I TOTAL • SUM C COMPUTE MEAN AND STANDARD D E V I A T I O N . SUM = 0.0 __DO 4 1 8 0 M = 1 » N _ I F ( R O R ( M ) . L T . ( R O R M I N ) .OR. ROR(M) .GE. (RORMAX j ) GO TO 4 1 8 0 I F ( R O R ( M ) .NE. 1 0 0 0 0 . 0 ) SUM = SUM + ROR(M) 4180 CONTINUE _ _ _ AMEAN = SUM /""FLOAT " T N - K - L ) " SUM2 = 0.0 _ DO 4 1 9 0 M = 1 » N _ _ I F ( R O R ( M ) . L T . ( R O R M I N ) .OR. ROR(M) .GE. (RORMAX) ) GO TO 4 1 9 0 I F ( R O R ( M ) .NE. 1 0 0 0 0 . 0 ) SUM2 - SUM2 + ( R O R ( M ) - AMEAN) * * 2 41?0_ CONTINUE _ ; STD = SQRT (SUM2 / FLOAT ( N - K - L ) ) WRITE ( 6 , 7 0 ) WRITE ( 6 . 6 0 ) WRITE ( 6 . 3 2 0 0 ) AM EAN 3200 FORMAT ( 3 6 H MEAN OF I N T E R N A L RATES OF RETURN = . F 1 6 . 8 . 2 X . 8 H P E R CEN 1T ) " W R I TE ( 6 . 6 0 7 W R I T E ( 6 . 3 3 0 0 ) STD 3300 FORMAT ( 5 0 H STANDARD D E V I A T I O N OF I N T E R N A L R A T E S OF RETURN = , F 1 6 . 1 8 . 2 X . 8 H P E R CENT) ~ "" T I M E 4 = CLOCK ( S T A R T ) / 6 0 . WRITE(6.70) WRITE ( 6 . 1 5 0 ) TIME 1 •• — •-• 150 FORMAT ( 1 H 0 . 9 H T I M E 1 = . F 1 2 . 4 ) WRITE ( 6 . 1 5 1 ) TIME2 "151 FORMAT ( 1H0.9H T I M E 2 = . F 1 2 . 4 ) W R I T E ( 6 . 1 5 2 ) TI ME 3 152 FORMAT ( 1 H 0 . 9 H T I ME3_ =_• F l 2_. 4 [ "WRITE ( 6 . 1 5 3 ) " T i M E 4 " " "~ ' ~ 153 FORMAT ( 1 H 0 . 9 H T I M E 4 = . F 1 2 . 4 ) STOP S I B F T C POLY C__ _ _ _ _ _ _ C ' F I N D THE R E A L ROOT "CLOSEST" TO T'OF" A" P O L Y N O M I A L OF D E G R E E C S U B R O U T I N E POLY ( N C F . C F • R O O T , L O O P ) vo ' NCF-1. e t• • D I MEN S I O N 200 300 400 s s CF ( 5 0 ) LOOP = 0 S O L T N = 1.5 CONTI NUE PX = CF ( 1 ) PDX = F L O A T (NCF - 1 ) _ * _ P X I = 2 » N C F DO 2 00 CF ( I ) + PX * SOLTN PX = F L O A T (NCF I ) * CF ( I ) + PDX * SOLTN PDX = NUE CONTI PDX = PDX / SOLTN PX = S O L T N - PX / PDX LOOP = LOOP + 1 I F ( A B S ( P X - S O L T N ) . L E . 0 . 0 0 0 1 ) GO TO 3 0 0 I F ( LOOP . GT. 50 ) 30 TO 4 0 0 S O L T N = PX GO TO 5 CONTI NUE ROOT = PX" RETUR N CONTI NUE RETUR N END / ? , 6 0! TT" A SENTRY H CN O APPENDIX P SUBROUTINES FOR RISK ANALYSIS (FROM MR. DAVID B. HERTZ) SIBFTC INPROB SUBROUTINE INPPOB(NVAR»FX,FY! INPR0020 C "c c c C C c c c 100 .200. 210 90 T H I S S U B R O U T I N E READS I N A NUMBER ( N V A R ) OF P R O B A B I L I T Y DENSITY F U N C T I O N S . W R I T E S THEM OUT FOR V E R I F I CAT I ON•NORMAL I Z E S THEM SO THAT THE AREA UNDER THE CURVE__IS UN I T Y • NUMER I C A L L Y INTEGRATES T H E M . A N D RETURNS T H E C U M U L A T I V E D I S T R I B U T I O N F U N C T I O N S TO THE C A L L I N G PROGRAM. 01 z\ CHANGES TO INPROB TO ALLOW FOR F I X E D V A R I A B L E S W I T H 1 0 0 P . C . PROB. AND TO CHANGE NAME F O R M A T . DIMENSION F I X V A L ( 3 0 ) INPR0030 DIMENSION F X U ) . F Y ( l ) » NAMEX(4)» N A M E Y ( 4 ) . A R E A ( 8 ) » X ( 8 ) , Y ( 8 ) INPR0040 COMMON F I X V A L INPR0100 WRITE ( 6 . 6 0 2 ) DO 1 1 3 I V A R = 1 . N V A R ~ " ' ~~ ~ I N P R O il'6" READ ( 5 . 6 0 0 ) (NAMEX(K).K=1,3)»I.<X(J),J=1.8>,(NAMEY(K).K=1,3).II,( 1Y(J),J=1»6) INPR0130 IF ( I - I I ) 1 0 0 . 2 0 0 . 100 I NPR0140 INPR0160 WRITE (6.603) IMPR0.170_ IF ( Y ( 1 ) - l 0 0 . 0 ) 1 0 1 . 2 1 0 . 1 0 1 FIXVAL(IVAR)=X(1) INPR0180 WRITE ( 6 . 9 0 ) (NAMEX(K) »K=1 , 3 ) . I » X ( 1 ) . ( N A M E Y ( K ) , K = 1 , 3 ) » I . Y ( 1 ) INPR0200 FORMAT ( 1 H 0 . 2 A 6 . A 2 . I 2 . F 1 0 . 3 . / 1 H . 2 A 6 . A 2 . I 2 . F 1 0 . 3 ) __ " G O T O 113 " ~ ~ ' " " " ' .p WRITE ( 6 , 6 0 1 ) ( N A M E X ( K ) . K = 1 . 3 ) » I . ( X ( J ) » J = 1 , 8 ) . ( N A M E Y ( K ) , K = 1 , 3 ) , I I » I N P R 0 2 40 1 <Y( J ) , J = 1 . 8 ) _ INPR0250 C A L C U L A T E MODE OF D I S T R I B U T I O N ~ INPR0260 M0DE=1 J N P R O 270_ DO 1 0 3 N = 2 . 8 INPR0 280 IF (Y(MODE)-Y(N)) 102.103.103 INPR0290 MODE = N I NPR0300 CONTINUE ; ' I NPR0310 XMODE=XMObE N U M E R I C A L L Y I N T E G R A T E AND C A L C U L A T E C E N T R O I D AREA(1)=0.0 I NPR0_3 20_ CENT=0.0 I NPR0330 DO 1 0 6 K = 2 . 8 INPR0340 DELX = A M A X 1 ( X ( K ) - X ( K - l ) , 0 . 0 ) _. _ I NPRO3_50_ DELY = Y ( K ) - Y ( K - l ) " " INPR0370 CENT=CENT+DELX*(X(K)+X(K-l))*0.5MAMIN1(Y(K),Y(K~1)) INPR0380 IF (DELY) 104.106.105 I 101 C j? tl.' 10. 9 3 ' 1 6 5 A 3 102 103 N P 0 ? ; ? n ON 104 CENT=CENT-DELY*DELX*0.5*(X(K-1.+DELX/3.0) INPR0^90 GO TO 106 INPR0400 1 0 5 _ C E N T = C E N T + D E L Y * D E L X * 0 . 5* ( X ( K-_l ) + 0_»_6666'6 6_7#DE_LX ) JNPR0410 106 AR E A ( K ) = AR E A ( K - 1 ) + A M A X 1 ( ( X ( K ) - X ( K - 1 ) ) * ( Y ( K ) + Y ( K - 1 ) )*0.5»6_0 ) " fNPR04?6~ C CONVERT D I S T R I B U T I O N TO C U M U L A T I V E DO 109 K = l ,8 __ ._ _ _ _ INPR0430 L = 8*(I-1) +K ~ ~ INPR0440 I F (AR E A ( 8 ) ) 1 0 7 . 1 0 7 , 1 0 8 INPR0450 107 FY(L)=0.0 ._ _ _ I N P R 0 4 6 0 _____ GO TO. 1 0 9 " " " . •" INPR0470 108 FY(L)=AREA(K)/AREA(8) . 10_9 FX(L)=X(K) INPR0490 ' C C A L C U L A T E MEAN AND MEDIAN OF X ~ IF ( A R E A . 8 ) ) 110,110,111 INPR0500 110_ XMEAN = 0.0 ___• INPR0510 GO TO 112 "' " " INPRo'5 2 0" 111 XMEAN=CENT/AREA(8) INPR0530 1 1 2 _ XMED=DEV1 ( FX ,FY , I j 0 . 5 ) I NPR0540 C W R I T E CUT C U M U L A T I V E D I S T R I BUT I ON,MEAN,MED I AN,MODE JB=8*(I-1)+1 INPR0550 JE = JB + 7 _ INPR0560 ~ W R I T E ( 6 , 6 0 4 ) ( FY ( J ) , j = J B , J E ) , XMEAN , XMED , XMODE " " ' "l NPR0580'" 113 CONTINUE INPR0590 _ RETURN ' JNPR0600 _ " 6 0 0 " " F O R M A T ( 2 A 6 , A 2 , 12 » 8 F 7 • 2 / 2 A 6 , A 2 , 12 , 8 F 7 . 2 ) " " I NPR0610 601 FORMAT ( 1 H 0 2 A 6 , A 2 , I 2 , 8 F 1 0 . 3 / 1 H - 2 A 6 » A 2 , I 2 , 8 F l 0 . ? / ) 602 FORMAT ( 1 H 1 / / 5 O X , 2 3 H F R E Q U E N C Y DI S T R I BUT I O N S / 4 5 X , 3 1 H A S S O C I A T E D W I T H I N P R 0 6 3 0 " 1 INPUT V A R I A B L E S / / / . "" ' '""" '" " " " * " " " " T N P R 0 6 46" ' 603 FORMAT ( 4 5 H 0 I N P U T CARDS MAY BE OUT OF ORDER OR I N C O R R E C T ) ^_ 604 FORMAT (25HOCUMULAT IVE_ D I S T R I B U T I ON 3 F 1 0 . 3 / 1 3 H Q MEAN OF X = F 1 0 . 4 , 9 f I X , 1 3 H M E D I A N OF X = F 10 • 4 , 9 X , 11H M 0 D E OF X = F 1 0 . 4 / / ) """" ' INPR0670 •jEND INPR0680 !2_ S I B F T C TALLY _ _ !l " ' " S U B R 0 U T I NE TALLY"'""rCi'STN0", VAL0E~* W I Df FT, S f ART ,M ,N , S0M , S0MSQ ) "~ TALLOOTO 10 C 3 _C E X P L A N A T I O N 0F__AR_GUMENT_S 8 c " ? C 6_ C s " ~ C ' 4 C 3 C " " " L I S T N O = L I S T NUMBER. V A L U E = V A L U E OF ITEM TO BE T A L L I E D . _ __ _ M = NUMBER OF I N T E R V A L S IN E A C H H T S T 0 G R " A M V " US U A L L Y"'r6"0T"THE" F I R S T D I M E N S I O N OF N I N THE C A L L I N G PROGRAM. WIDTH = WIDTH OF EACH I N T E R V A L . w "' c c c c" c c START = S T A R T I N G P O I N T OF LOWEST I N T E R V A L . N = AN OUTPUT VECTOR SHOWING THE NUMBER OF T A L L I E S IN EACH INTERVAL. SUM = THE C U M U L A T I V E SUM OF THE T A L L I E S . SUMSQ = THE C U M U L A T I V E SUM OF THE SQUARED T A L L I E S . N* SUM» SUMSQ MUST BE D I M E N S I O N E D IN T HF C A L L I N G PROGRAM. DIMENSION N(1 ) »SUM(1),SUMSQ(1) SUM ( L I ST NO ) =SUM ( L I STNO ) +VALUE SUMSQ(LISTNO)=SUMSQ(LI STNO)+VALUE**2 T A L L Y CHANGES LL=100*(LISTNO-1) _ IF (VALUE-START) l O l V l O l • 1 0 2 101 N(LL+1)=N(LL+1)+l RETURN _ E M =M IF (VALUE-START-EM*WIDTH) 104»103»103 LM= L L + M _____ ___ _J?_ N(LM)=N(LM)+1 RETURN 104 NUM=(VALUE-START)/WIDTH NUM = NUM+LL + 1 " " ~ " N(NUM)=N(NUM)+l RETURN _ '"END ' SIBFTC HISTO S U B R O U T I N E H I S T O ( L I ST NO »I TER • WI DTH» STA RT • M » N »_SUM_, SUMS_OJ_ ~ " D I M E N S I O N N(1)* EN(104) » P(104)•SUM(1)•SUMSQfl) EQUIVALENCE (EN,P) WRITE (6»150) L I S T N O » M » W I D T H » S T A R T » I T E R ~XT=ITER CUMSUM=0.0 DO 2 0 1 J= 1 *M . C ' H I S T O CHANGES FOR TIDEWATER JL=((LISTNO-1)*100)+J EN(J!=N(JL) _ 201 P(J) = EN(J)/XT AVG=SUM(LISTNO)/XT STDEV=SQRT(SUMSQ(LISTNO)/XT-AVG**2) WRITE ( 6 , 1 5 2 ) AVG • STDEV ZZ=START DO 2 0 2 J = 1 » M , 5 9 L . 3 or . T T TALL0030 TALL0050 TALL0060 TALL0070_ TALL0086 TALL0090 TALL0100 TALL0110 TALL0120 TALL0130 TALL0140 TALL0150 TALL0160_ "TALL 0 1 7 0 TALL0180 TALL0190 "f A L L 02 00" HIST0020_ "HIST0030 "HIST0050 HIST0070 H I S TO "080" HIST0090 HIST0100 HIST0110 HIST0120 HISTO130 HIST0140 HIST0150 "HIST0170" HIST0180 HIST0190 H ON JJ=J+4 HISTO?00 SUM5=SIGMA(P•J•JJ) HIST0210 CUMSUM=CUMSUM+SUM5 _ l A WRITE ( 6 , 1 5 1 . ZZ,(P(K>»K=J»JJ)•SUM5•CUMSUM 202 ZZ=ZZ+5.0*WIDTH HIST0250 RETURN _ „ _ _ •_ _ HIST0260 150 FORMAT" ( 1 8 H 1 H I S T O G R A M NUMBER I 3 / / 7 X , 1 4 H C 0 N S I S T I N G ^ F I 4 , 2 9 H I N T E R V H I S T 0 2 7 0 1 A L S . EACH I N T E R V A L I S F 1 0 . 3 . 3 8 H U N I T S W I D E . F I R S T I N T E R V A L S T A R T S H I S T 0 2 8 0 2 AT F 1 0 . 3 / / 7 X , 2 2 H H I S T 0 G R A M I S BASED ON I 5 ,J1 3H 0BS_ER_VATJ QNS ) _ _ HIST0290 1 5 1 " F O R M A T ( 4 H 0 X = F 1 0 • 3 ,6X , 5F1 2 • 5 , 2 F 2 0 • 5 ) i ~ ~ 152 FORMAT ( 2 9 H 0 MEAN OF H I S T O G R A M = F 1 0 . 3 , 1 0 X • 2 1 H S T A N D A R D D E V I __'_ 1 A T I 0 N = F 10 • 3 / / / / 3 0 X , 4 3HPR0P0R TI ON OF O B S E R V A T I O N S I N EACH I N T E R V A H I S T 0 3 20 2 L / / 1 5 H F I R S T I N T E R V A L , 1 5 X , 3 8 H ( E A C H L I N E C O N S I S T S OF F I V E INTERVALSHISTO330 3 ) , 2 6 X , 4 H F I V E / 3 H OF F I V E , 8 4 X , 8 H I N T E R V A L , 1 1 X , 1 O H C U M U L A T I V E / 1 O H BEGINHIST0340 4.S AT, 1 6 X » 3 H * l * , 9 X , 3 H * 2 * » 9 X , 3 H * 3 * » 9 X » 3 H * 4 * » _ 9 X , 3 H * 5 * » 1 7 X , 5 H T O T A L » l > X H l S T 0 ^ 5 0 5 ,5HT0TA.L ) •' " " " " " ~ ' ' ' " " ~ " ~ " " END HIST0370 S I B F T C RAND . S U B R O U T I N E RAND (URN) ~ " ~ ' " RAND0020 32 IF ( I N I T - 1 ) 4,20,4 RAND0030 4 I NIT= 1 _ _ _• RAND0040 RN = 0."0 " ' ; " " " '" ''""*' " " RAND0050 NBASE=URN RAND0060 20 CALL GRN(NBASE,RN) RAND0070 URN=ABS(AM0D(RN,1000.0))/1000.0 " " " RAND0080 RETURN RAND0090 END RAND0100 FA P ENTRY 00000 GRN 00000 0 5 0 0 00 4 0 0 0 0 1 GRN CLA 1 ,4 00001 0 6 2 1 00 0 0 0 0 0 6 STA GRN + 6 00002 0 6 2 1 00 0 0 0 0 1 0 STA GRN+8 00003 0 6 2 1 00 0 0 0 0 1 1 GRN+9 STA 0 5 0 0 00 4 0 0 0 0 2 00004 CLA 2,4 00005 0 6 2 1 00 0 0 0 0 1 6 STA GRN1 ## 0 5 6 0 00 0 0 0 0 0 0 00006 LDO 0 2 0 0 00 0 0 0 0 2 0 " ~ " 00007 MP Y BASE #* STO 0 0 0 1 0 - 0 6 0 0 00 0 .0 0 0 0 0 ## 00011 0 5 0 0 00 0 0 0 0 0 0 CLA 0 7 7 1 00 0 0 0 0 2 2 18 00012 ARS FLZ 0 0 0 1 3 - 0 5 0 1 00 0 0 0 0 2 1 ORA 0 3 0 0 00 0 0 0 0 2 1 00014 FLZ FAD H e S T n ? 0 00015 00016 00017 00020 00021 00022 0 2 4 1 00 0 0 0 0 2 2 - 0 6 0 0 00 0 0 0 0 0 0 0 0 2 0 00 4 0 0 0 0 3 +343277244615 + 233 0 0 0 0 0 0 0 0 0 + 210406111564 FDP STO TR A OCT OCT DEC ' END GRN1 BASE r-H DIV DIV #* 3,4 343277244615 2^000000000 131.072 S I B F T C DEV DEV00020 FUNCTION DEV.I) C CHANGES TO DEV - P U T T I N G F X , F Y I N COMMON AND ALLOWING FOR F I X . V A R . C C T H I S F O R T R A N F U N C T I O N S E L E C T S A RANDOM D E V I A T E FROM THE I / T H c" FREQUENCY D I S T R I B U T I O N ( C U M U L A T I V E ) S P E C I F I E D BY FX AND F Y . c c DI MENS ION TITLE(12)»SUM(10),SUMSQ(10) , F X ( 9 , 3 0 ) , F Y ( 8 , 3 0 ) , F I X V A L ( 3 0 ) D F V 0 0 0 3 0 COMMON F I X V A L , FX , FY DEV00040 DEV00070 I F ( F I X V A L f i n _3,4,3 DEV=F I X V A L ("I ) "" " DEV00080 RETURN DEV00090 M = 8 * I-6 _ DEV00100 N = 8* I "~ DEV00110 DEV00120 C A L L RAND ( Y ) 5 DEV00130 I F (FY(M-1*1J-Y_) 1 5 , 5 0 , 5 0 10 DEV00140 DO 3 5 J=M,N 15" DEV00150 IF ( F Y ( J . l ) - Y ) 35,25,25 20 D E V _ F X ( J 1 » 1 ) + ( F X ( J , 1 1 ~ £ X ( J l * 1 ) ) * ( Y FY (_J-1 »_1 __/_( F Y (_J_, 1 ) F Y _ J 1 t_l ) DE VOO1 6 0 25 ""DEV00170" 1) ' " " " DEV00180 30 RETURN D EV00190 35 CONTINUE D FV00200 40 DEV=FX(N,1) D E V00210 45 RETURN DEV002.20 50 DEV = F X ( M - 1 , 1 ) 'DEV00 2 30 55 RETURN DEV00240 END SIBFTC DEVI DEV10020 F U N C T I ON D E V 1 ( F X , F Y • I , Y ) DEV10030 DIMENSION F X ( 1 ) , F Y ( 1 ) DEV10050 M = 8 * I - 6 ._ _ DEV10060" ~ " N = 8*I DEV10070 10 I F ( F Y ( M - l ) - Y ) 15,50,50 DEV10080 15 DO 35 J=M,N ' H 0\ ON IF ( F Y ( J ) - Y ) 35.25.25 DEV1=FX(J-l)+(FX(J)-FX(J-l))*(Y-FY(J-l))/(FY(J)-FY(J-l)) RETURN CONTINUE DEV1=FX(N) RETURN DEV1=FX(M-1) RETURN END _ $ IBFTC CALROI SUBROUTINE CALROI (CASH•NYRS•ROI) DEV10090 DEV10T00 DEV10110 DEVI 0120 DEV10130 DEV10140 DEV10150" DEV10160 DEV10170 20 25 30 35 40 45 50 55 C C C "C" c c c c c "c 697 6 98 "699" 7 00 701 702 703 CALR0020 T H I S S U B R O U T I N E C A L C U L A T E S THE R A T E OF RETURN ( R O D , OR DISCOUNT" R A T E , WHICH C A U S E S A STREAM OF ANNUAL CASH FLOWS ( C A S H ) • OVER A S P E C I F I E D NUMBER OF Y E A R S _ ( N Y R S ) . TO HAVE A NET P R E S E N T VALUE OF Z E R O . A C C U R A C Y I S to" THE NEAREST TENTH OF A PERCENT". E X P L A N A T I O N OF NOTES NOTE = 1 GOOD S O L U T I O N NOTE = 2 S O L U T I O N BELOW LOWER BOUND NOTE = 3 S O L U T I O N ABOVE UPPER BOUND NOTE = 4 F A I L U R E TO CONVERGE • P R O B A B L Y AN ERROR DIMENSION C A S H ( l ) TOPROI=2.0 BOTROI=-0.3 INDEX=-1 CALL CALNPV (TOPROI.CASH,NYRS»TOPNPV) C A L L CALNPV (BOTROI»CASH,NYRS•BOTNPV)" IF (TOPNPV*BOTNPV) 700.700.697 IF (TOPNPV) 6 9 8 , 6 9 8 , 6 9 9 ROI=100.0*BOTROI " NOTE=2 RETURN _?„i _ _/_ ROI=100.0*TOPR01r "*"" 7 - 3 NOTE=3 RETURN '_ ' DO 7 0 9 K = l ,99 " IF (INDEX) 701.710.702 ROI = ( B O T R O I + T O P R O D / 2 . 0 GO TO 7 0 3 ~ ' ~ ROI=BOTROI + (TOPROI-BOTROI)*BOTNPV/(BOTNPV-TOPNPV) CALL CALNPV (ROI,CASH,NYRS•ENPV) CALR0030 CALR0050 CALR0060 CALR0070 CALR0080 CALR0090 CALR0100 CALR0110 "CALROI20 CALR0130 CALR0140 "CALR0150 CALR0160 CALR0170 "CALR0180 CALR0190 CALR0200 "CALR0210 CALR0220 CALR0230 IF (ENPV) 704.708.705 TCPROI=ROI T0PNPV=ENPV GO TO 7 0 6 705 BOTROI=R0I BOTNPV=ENPV 706 IIFF ( T O P R O I - B O T R O I - . O O l ) 7 0 7 . 7 0 7 , 7 0 9 707 ROI=0.5*(TOPROI+BOTROI) 708 ROI=100.0*ROI _ NOTE=l ' RETURN 709 INDEX = -1*INDEX__ 710N0TE=4 RETURN END S I B F T C CALNPV S U B R O U T I N E CALNPV (RATE,CASH,NYRS•ENPV) C T H I S S U B R O U T I N E C A L C U L A T E S THE NET PRESENT V A L U E ( E N P V ) OF A C S P E C I F I E D STREAM OF ANNUAL CASH FLOWS ( C A S H ) , O C C U R R I N G OVER A C NUMBER OF YEARS ( N Y R S ) , D I S C O U N T E D AT A S P E C I F I E D R A T E ( R A T E ) . C C DIMENSION C A S H ( l ) ENPV=0.0 DISC=1.0 DO 5 0 0 J = l ,NYRS DISC=DISC/(1.0+RATE) ENPV=ENPV+DISC*CASH(J) 500 RETURN END S I B F T C SIGMA" F U N C T I O N SIGMA (X»M,N) C "THIS FORTRAN "FUNCT ION "CALCULATES THE SUM "OF THE TERMS IN A S E R I E S c" D E S I G N A T E D AS X ( I ) , WHERE I V A R I E S FROM M THRU N. c 704 c 10 DIMENSION X ( 1 ) SIGMA=0.0 DO 10 I=M»N SIGMA=SIGMA+X(I) RETURN END CALR0240 CALR0250 CALR0260 CALR0270 CALR0280 CALR0290 CALR0300 CALR0310 CALR0320 "CALR0330 CALR0340 CALR0350 CALR0360 CALR0370 C A L R 0 38 0 CALN0020 CALN0030 CALN0050 CALN0060 CALN0070 CALN0080 CALN0090 CALN0100 CALN0110 SIGM0020 SIGM0030 SIGM0050 SIGM0060 SIGM0070 SIGM008 0
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Statistical transformation of probabilistic information Lee, Moon Hoe 1967
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Title | Statistical transformation of probabilistic information |
Creator |
Lee, Moon Hoe |
Publisher | University of British Columbia |
Date Issued | 1967 |
Description | This research study had shown various probable rational methods of quantifying subjective information in a probability distribution with particular reference to the evaluation of economic projects by computer simulation. Computer simulation to give all the possible outcomes of a capital project using the Monte Carlo technique (method of statistical trials) provides a strong practical appeal for the evaluation of a risky project. However, a practical problem in the application of computer simulation to the evaluation of capital expenditures is the numerical quantification of uncertainty in the input variables in a probability distribution. One serious shortcoming in the use of subjective probabilities is that subjective probability distributions are not in a reproducible or mathematical form. They do not, therefore, allow for validation of their general suitability in particular cases to characterize input variables by independent means. At the same time the practical derivation of subjective probability distributions is by no means considered an easy or exact task. The present study was an attempt to suggest a simplification to the problem of deriving a probability distribution by the usual method of direct listing of subjective probabilities. The study examined the possible applicability of four theoretical probability distributions (lognormal, Weibull, normal and triangular) to the evaluation of capital projects by computer simulation. Both theory and procedures were developed for employing the four theoretical probability distributions to quantify the probability of occurrence of input variables in a simulation model. The procedure established for fitting the lognormal probability function to three-level estimates of probabilistic information was the principal contribution from this study to research in the search for improved techniques for the analysis of risky projects. A priori considerations for studying the lognormal function were discussed. Procedures were also shown on how to apply the triangular probability function and the normal approximation to simulate the outcomes of a capital project. The technique of fitting the Weibull probability function to three-level estimates of forecasts was adopted from a paper by William D. Lamb. The four theoretical probability functions wore applied to a case problem which was analyzed using subjective probabilities by David B. Hertz and reported in the Harvard Business Review. The proposal considered was a $10/-million extension to a chemical processing plant for a medium-sized industrial chemical producer. The investigations of the present study disclosed that the log-normal function showed considerable promise as a suitable probability distribution to quantify the uncertainties surrounding project variables. The normal distribution was also found to hold promise of being an appropriate distribution to use in simulation studies. The Weibull probability function did not show up too favourably by the results obtained when it was applied to the case problem under study. The triangular probability function was found to be either an inexact or unsuitable approximation to use in simulation studies as shown by the results obtained on this case problem. Secondary investigations were conducted to test the sensitivity of Monte Carlo simulation outputs to (l) number of statistical trials; (2) assumptions made on tail probabilities and (3) errors in the three-level estimates. |
Subject |
Probability Distribution (Probability theory) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-07-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0102417 |
URI | http://hdl.handle.net/2429/36260 |
Degree |
Master of Science in Business - MScB |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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