Report No. EC-164

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This report may not be published nor may it be quoted as representing the view of the Bank and its affiliated organizations. The,Y do not accept responsibility for its

accuracy or completeness.

INTERNATIONAL BANK FOR RECONSTRUCTION AND DEVELOPMENT

L.~TERNATIONAL DEVELOP}'lE~T ASSOCIATION

TECHNIQUES FOR

PROJECT APPRAISAL

UNDER UNCERTAINTY

Econdttric s Department Prepared by: Shlomo Reutlinger

August 21, 1968

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PREFACE

This study is concerned with the apprai~al of events which have uncertain outcomes~ This issue is generally recognized, but is usually not explicitly'considered in otherwise detailed cost-benefit analysis of investment projects. Application of contingency allowances and sensi­tivity analyses have been used as partial remedies. However, for the most part, project benefits are still est'.imated and reported in terms of one single outcome which does not take CLCCOunt of) or record, valuable infor­mation about the extent of uncertainty of project-related events.

This paper recommends that the best available judgments about the various factors underlyint the cost and benefit estimates of the project be recorded 1n terms of probability distributions and that these distribu­tions be aggregated in a mathematically correct manner to yield a probability distribution of the rate of return, or net present worth, of the project. This procedure in no way eliminates the problem of making judgments about events and relationships in the face of limited and incomplete information, nor does it suggest a uniqu~ and simple formula fc;>r choosing among projects or project strategies with v~~ying degrees of riskiness. However, this type of analysis would ensure and encourage that available information about events affecting the outcome of the project would be more fully utilized and correctly transformed into information about uncertain project results. Project-related decisions could be made more easily and more intelligently ~f returns on ~rojects were reported not in terms of a single rate, or a "'ide range of possible returns with undefined likelihoods 0 f occurrence, but in terms of a probability distribution.'

The present paper should be viewed primarily as providing a conceptual framework for further study into the scope and limitations of practical application of probability appraisal and pursuant project decisions. Several case studies are currently being investigated in the Bank by Louis Pou1iquen and Tariq Husain. The author has benefited from many discussions with them. The author also wishes to acknowledge helpful comments by Herman G. vap der Tak and Jan de Weille of the Sector and Project Group in the Economics Department and written comments on an earlier draft of this paper by Bank staff members: Messrs. B. Balassa, L. Goreux, A. Kundu, M. Schrenk, A. E. Tiemann, D. J. Wood of the Economics Department; and D. S. Ballantine, I. T. Friedgut, V. W. Hogg, P. O. Malone, H. o. Muth, M. Pa1ein, S. Y. Park, M. P1ceagl1, L. Pou1iquen, A. P. Pusar, V. Rajagopalan, S. Takahashi, V. Wouters of the Projects Department. However, the views expressed in this paper are those of the author, and he alone is responsible for them.

This staff study is part of continuing work in the Economics Department on problems of sector and project analysis. Other studies in this field which have been given circulation outside the Bank are listed on the back cover.

A. M. Kamarck Director

Economics Department

PREFACE

I.

II.

III.

f . .

TABLE OF CONTENTS

INTRODUCTION AND SUMHARY

ASSESSMENT OF UNCERTAIN EVENTS .. . . . . I if • • • •

1. 2. 3. 4. s.

Formulation of Anticipations • The Probabilistic Formulation Probabilistic Formulation Facilitates Aggregation Probabilistic Approach Utilizes Hore Information • Probabilistic Formulation can be Subjected to

Meaningful Empirical Te$t • .. .. • .. .. • .. • • .. 6. Estimation of Probability Distributions

PROBAB ILITY APPRAISAL OF PROJECT RETURNS miDER UNCERTAINTY • • .. • • • .. • .. . . . . 1. 2. 3.

4. s.

General Outline • • • • The Aggregation Problem •• ' ... Illustration of Alternative Procedures for . ...,

Aggregating Probab~lity Distributions Discussion of Alternative Estimating Procedures Conceptual Problems Related to Probability

Appraisal •• • • • • .. • •

Biased Estimate • Correlation • • •

Page No.

i

1

6

6 7 7 9

10 10

12

12 12

15 20

22

22 24

6. Specific Uncertainty Problems in Project Appraisal 28

7.

uncertainty About Annual Net Benefi.ts .'. Equal Annual Correlated and Uncorrelated

Benefits ....... \ • • .. .. • .. • • Annual Benefits Contain an Uncertain Trend "Life" of Project .. I_ • • .......

Estimating Probability bistribution of . ..

Annual Bene fi ts • .. '. • .. • .. • • . . . Summary and Conclusions on Aggregation of

Probability Distributions ........... . . . . . ANNEX TO CHAPTER III

A HYPOTHETICAL EXAMPLE . . . . . . . . . The Hodel • • • • .. .... ..... Probability Distribution of the Inputs "Conventional" Appraisal of Present Value

of Benefits • .. .. • • ~ .. .. .. .. .. • • e

Appraisal by Probability Analysis • • • .. . ..

28

29 33 38

38

40

43

43 44

45 46

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PROJECT DECISIONS UNDER UNCERTAINTY 0 • • i , • • •

1. 2. 3. 4. 5.

Introduction • • • • • • • • • • • • 1 i • • •

A Hypothetical Example of the Deci.ion Problem Project Appraisal and Utility Theor1 • • Public Investment Decisions • • i • • • •

Selected Decision Problems • • • • ; • • • • •

. , . . . . • • •

· . II •

· . Value of Information Theory • Alternative Project Strategies Time Related Problems • • • • • • • • •

Page No.

48

48 48 50 54 57

57 59 S9

APPENDIX A: Tables 1 and 2 • • • • · . . . • 64/65

APPENDIX B: REVIEW OF SOME BASIC CONCEPTS AND RULES FROM PROBABILITY CALCULUS AND SPECIFI(, APP1.lCATION

REFERENCES

Tables.

111-1

111-2

111-3

111-4

111-5

111-6

111-7

111-8

• • • • • $ • ... • • • • • • • • • • • • • • • • •

Probability Distributions of Revenue (X) and I~.lves tment Cos t (Y) • • • • • • • • • • • •

Prob~bility Distributions of Present Value (R)

· . . · . . . • •

Cumulative Probability Distribution of R • • • • • • •

Ratio of Coefficient of Variation of Preaent Value (CR) to Coeffic~ent of Variation of Succes8ively

UQcorrelated Annual Benefits, (CB), (for a stream

of equal me~ 8nn~l benefits) ~ • • • • • • · . Hypothetical Data for' Calculating a Variance of

the Present Value when ~enefit8 from Successive Years are Uncorrelated or Perfectly Correlated

Elasticity of Variance of Present Valu~ of a Stream of Benefits with Respect to V ":'iance of Successively Uncorre1ated Benefits (when

· . .

Bt • B + e t ). · • • • • • • • • • • • • • • · . . . . Elasticity of the Mean Present Value (a) with

Respect to Mean Growth Rate (b) and Mean Initial Benefits (ti) •••••••••••

Elastic.iti/as of the Variance of Present Value V(R) -with Re;spect to V(b), V(B ) and V(e) •

I 0

· . . . . • • · . .

67

72

15

16

18

31

32

34

36

37

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Figures

111-1 Probability Distribution of Present Value 19

111-2 Illustration of Bias when Yield is Calcdlated as a Function of Average Rainfall • • • • • • 25

III A-I Simulated Probability Distribution of Present Value 47 ,\

III A-2 Simulated Probability Distribution of Internal Rate of Return • • • • • • • • 47

Probability Distributions of Present Value 58

IV-2 Probability Distributions of Present Value for Alternative Project Strategies • • • • " 60

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I. INTRODUCTION AND SUMMARY

1. The primlry purpose of this study is to present a feasible method for evaluating the'riskiness of investment projec~s. A second objective 1s to show how quantitative evaluations of the ri~;kines8 of projects might be used in various decision problems. Throughout, the emphasis is on methodology and problems of measurement, not on description of various kinds of uncertainty problems, nor is much attention paid to theories which have no immediate applicability to project appraisal. Uncertainty is everywhere, as anyone knows; hence, a gener~l descriptive study of un­certainty is unnecessary and the specific sources of uncertainty must be identified for each specific case. In general, however, the uncertainty conditions relevant for this study are those unique to a particular proj­ect, and not to "global" uncertainties which affect the outcomes of all projects within'a cot.mtry.

2. This study does not recommend a specific "best" attitude for a public investment authority or an international lending agency with re­spect to undertaking projects with uncertain outcomes. To do so, would be as presumptuOus as to advise a government on the income distribution or composition of goods and services it should promote. The pursuit of economic development is clearly inconsistent with a policy of avoiding all risks (there simply are no worthwhile projects' ,without risks). At the other extreme, most people would agree that a project. which has a rea­sonably high probability of turning out badly should not be undertaken if that outcome would mean a considerable deterioration of the present economic well-being of a country. However, between those two extreme choices lie many alternatives whose desirability would deperld on the subjective. preferences or aversions to risk of the decision makers and their constit­uents ..

3. This study deals at length with the question how to evaluate and present.in sununary form a measure of the relative riskiness of projects, on the general asst.lIlption that a "good" judgment of risk is an important ingredient for reaching a "best" decision. For all practical purposes, decisions involving choi~es among uncertain economic returns from invest­ment have one thing in common: they ask for judgments about the likelihood of the measure of returns used in the evaluation. FoT. some the most relevant measure of returns is "the most likely one" (the mode). Others use exclu­siv~ly a conservative estimate, that is one which has a "high chance of p"eitl,g ~xceeded" ,~nd still others wish to consider an entire set of returns, andi'their respe.ctive likelihoods. Or as Marshak has stated it very elo­quently:

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n •••• Instead of assuming an individual !who thinks he knows the future events, we assume an individual who thinks he knows the probabilities of future events. We may call this situation the situation of a game of chance, and consider it as a better although still in complete approximation to reality than the usual assumption that people believe themselves to be prophets". 1/

T. Marshak, "Molley and the Theory of Assets", Econometrica, 6:311-325, (1938) •

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4. No attempt is made in this study to present a comprehensive review of all decision theories dealing with uncertainty. SUch comprehensive reviews are available elsewhere. 1/ They are useful to students and research workers but 11lore often than not they leave the pract~tioner's problem unre­solved. Only a small set of uncertainty hypotheses and decision criteria are presented in this paper. 2/ They reflect t in the judgment of the author, the most useful and generally correct approaches to a large number of prob­lems arising in project appraisal.

5.: The commonly accepted procedure tn project evaluation calls for I'

the calculation of the return from each prol,j ect and for criteria by which to choose from among different projects on the basis of the estimated re­turns. 3/ The essence of the uncerta~nty problem is simply that many of the variables affecting the outcome of a particular plan of action are not controllable by the planner or decision maker. 4/ Hence, project evaluation which t~kes due account of uncertainty involves (a) judgments about the likelihood of occurrence of the non-controllable variables, (b) calculation of a whole set ot possible outcomes or returns for each ~roject, and (c) cl:iteria for choosing among projects on the basis of sets of pos­sible ret~rns from each'project.

1/

See, for example: K. J. Arrow, "Alternative Approaches to the Theory of Choice in Risk-Taking Situations", Econometrica, 19:404-437, (1951); M. Friedman and L. T. Savage, "The Utility Analysis of Choice InvoJ:ving RiskY', Journal of Political Economy, LVI, (August 1948); R. Dorfman, "Basic Economi~ and Technologic Concepts", Chapter in A. ~as', et aI, Design of Water Resource Systems, Harvard University Press, 1962; rr~ E. Farrar, The Investment Decision Under Uncertainty, Prentice-Hall, 1962.

The point of view taken in this study parallels most nearly the way F. Modigliani and K. J, Cohen have stated it: " •••• ~xobably the best avail­able tool at this stage is the' so-called 'expect~#-utility' theory •••• starting from certain basic postulates of ratio~~l behaviour this theory shows; that the information available to the ag~nt concerning an un­certain event can pe represented by a 'sub j ecti ve' probabili,ty distri­bution and that there exists a (cardinal) utility function such that the agent acts as tqough he were endeavoring to maximise the ~xpected value of his utility •• ,." ("The Significance and Uses of Ex Ante Data", Chapter IV in Expectations, Uncertainty and Business Behaviour, edited by M. J. Bowman.

The criteria are for the most part derived from a deterministic model which assumes that the exact returns are'known.

I

Such non-controllable variables migh't be prices, incomes, population, size of labor force and climate. While governments may have some (eontrol over s9me of ~hese variables, they are not likely to be interested or to succeed in contrplling them completely.

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6. Chapter II assesses briefly the nature of uncertainty and the kind of judgments which are basic ingredients for the decision making pro­c~ss. Particular attention is paid to the notion that. the uncertaility which is relevant for most decisions is best characte~ized in t~rms of a decision agent's subjective beliefs about· the likelihoods of o'ccurrence of various outcomes of the uncertain event. Such probability distributions' may be based, of course, on "more" or "les's" evidence and in this sense might be labeled 'lRore" or "less" subjective. While for any given event it may be desirable to marshall more evidence, if this is possible and not too costly, it is postulated that for reachini a decision it makes little difference whether an event is "known" in terms of a more subjec­tive or a more 'objective probability distribution. It would be a sad mis­take to subscribe to a decision theory which fails to consider variables, or recommends to discount information on variables, simply because their outcomes or probability distributions olE outcotnes are not known with cer­tainty. Errors of omission might be more important than errors of commis­sion. What matters is only whether an event has important consequences for a decision, and not how "objective" or "subjective" the estimate or probability distribution of its outcome is. !/

7. Among the various characterizations of uncertainty advocated by different theories, the probabilistic approach has been singled out primarily because this lends ,itself best to an appraisal of the possible outcomes of a proj ect which 1'13 affected by uncertainties from many different sources. It is shown how probability judgments about many basic variables and parameters affecting the final outcome of a project can be aggregated into an estimate' of the probability distribution of that final outcome. The advantages of "building up" such an estimate ~~re many: (a) it is gener­ally easier to formulate judgments about the outcomes of basic events than about the outcomes from a project be~ause such events are frequently recur­ring, whereas projects are usually unique in some respect, (b) the outcomes of events, such as rainfall, production functions or prices are likely to be eva1,~ted with less emotional bias and more factual evidence than ,8 project's overall benefits, (c) judgments about the outcomes of various "simple" events utilize the experience of many experts who should be in the best position to know, and, finally, (d) analytical insights into the desirability of restructuring a project can be gained from knowledge of the specific contribution of each underlying factor to the probability distribution of the project's final outcome. .

8. The "subjective" definiti6n of probability implies at once that the process of estimation is both an art and a science. The quality of judgm,~nts involved in estimation will vary with the nature of the variables

'"~I

1/ Following the terminology suggested by Frank Knight,-r events whose proba­bili~y distributions can be objectively1;tnown are siometimes labeled as Itri~~:~", and subjectively concei'ved dis tributions are called "uncertainties". The 'pl\)int of view taken in this s tudy ~s that this is not a meaningful c1ass~\fication, both because there are only more or less "objective" estim~\~es and because the extent of objectivity does not necessarily' alter their interpretation in terms of decisions.

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and tq.)e~ appraiser's expertise in interpolating and dtrapolating related observations and experience. Quite generally, desitahle prerequisites for good judgments are (a) knowledge of past outcomes of the event (experience and data), (b) knowledge of basic relationshi'l)s which could explain why the outcomes of the event might have varied in the past and how they might vary in the future (a model), and (c) sound procedur~s for interp~eting the interaction of model and data (st~tistics). To the extent that formal' theory, subject matter expertise and analytical tools can assist in the estimation process, they are assumed to be known to project appraisers and are not elaborated in this study.

9. Chapter III discusses'how to aggregate probability distributions of relevant factors and parameters into a probability distribution of the economic returns for a project. The problems arising when uncertain estimates ~f the various factors are combined have been for the most part neglected or inadequately treated by conventional appraisal methods, al­though for these aggregation problems at least it is possible to prescribe a uniquely correct methodology. The factors one chooses to consider in any economic appraisal of a project, and the prediction of their outcomes and estimations of how :they interrelate are always a matter fo·r subjective judgments within the liinits described by relevant theory and subject matter expertise. The organization's control over these judgments does n~~ extend far beyond its capacity for hiring able engineers, agronomists, economists, etc., who will come up with the best possible judgments consistent with the state of the arts. Contrariwise, the aggregation procedures theJllSe1ves caD. and should be exactly prescribed in order to transmit as far as possible and as correctly as possible all the information at;ld judgments made on each of the relevant factors affecting the costs and benefits from a project.

Ii

10. Probability appr;;tisal or,:.-isk analysis as discussed in 'Chapter III does nothing more than to suggest 'that the proper probability calculus be used in aggregating probability judgments about the many events influencing the f~nal outcome of a pro~ect. Just as it is generally accepted that 2 + 2 is 4 and not. 5, so there are logical, though lti1.9S generally known, rules for aggregating probability distributions oJ uncertain events. A major reason why these rules have not been more wid~1y used is the complex­ity and multitude of calculations which are required in their application. However, present day availability of hjgh speed computers makes their application not only desirable but definitely feasible. The only exception

I to this recommendation is the case where even the most, pessimistic estimates for all of the vari~bles and parameters affe~ting a project's net benefits result in a sati~factory measure of the return. In this case a probability appraisal might:still satisfy sODle intellectual curiosity but would be redund;ant for the overall project decision. Even in this

!

case, however, one :could find it useful to do probability appraisals if the objective is to investigate alternative ways of implementing the project.

11. The primary purpose of' Chapter III is to illustrate, wi,th some highly stylized and hypothetical streama of costs and benefits, why appli­catio'n of the probability calculus to aggregation is important and to show, the sensitivity' of the present value of a stream of net benefits to proba­bility distributions and correlations of various basic ev~nts. The

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')j I;

method of apprdximat,ion by a simulated sample is briefly described, and recommended for estimation of probability distributions of rates of returns from actual projects. The simplicity of calculati.on and the adaptability of this method to any type of model and conceivable set of probability dis­tributions make this a preferable method, provided' that the resulting distribution is, approximated by an adequate sample. 11 Only under very restrictive assumptions about the model and the distributions would,it be practical to calculate means and standard d~via:tions of an aggre'sated variable by using mathematical methods for aggregation. Mathematical aggregation of probability distributions may be useful also for partial analysis.

12. The final crucial phase of project appraisal is, of course, the ranking of alternative projects, or of courses of action to be taken in a given project. Unfortunately, in contrast to the material on aggrega­tion procedures, bVt not unlike the subjec~ of deriving basic probability judgments on t simpl~! events, the choice among alternative courses of action subject to uncertain outcomes involves a large element of subjective judgment. At least there is a very large number of decisions which could not be classified in any objective way as either "good" or "bad", or "right" or "wrong" (in an .!. priori sense) no m.~tter how one defines ut.ility or preferences. In addition, decisions involving public projects raise questions about the distribution of benefits and costs, and many prefer­ences with respect to risk are to be taken into account. Some of these decision probl~ms are discussed in Chapter IV. While theory can not suggest a unique general solution to these pro.bl~s, it is neverthe-less quite apparent that decision makers do wish tu know the likelihoods of outcomes from alternative courses of action in order to reach more satis­factory decisions. Hence, project appraisals are better if tlr~y provide tilis information. Furthermore, there ar~ certain limited activities of concarn in project appraisal, such as galthering of additional information or, s:trategies involving sequential decisJlons, which can be bes,t appraised in a'probabilistic decision framework. Chapter IV presents a brief dis­cussion of th~ al)plication of probabilistic information to such decision problems .'-

11 Probability appraisal by simulation is being applied to several Bank projects by Messrs. Louis Pouliquen and Tariq Husain. These case .s·tudies will illustrate problems in making explicit probabi­lity judgments and particular methodological aspects of estimatiot~\, by simulation.

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II. ASSESSMENT OF UNCERTAIN EVENTS

1. .Millions of dollars have been invested in t'roject A in anticipa-tion of great benefits to the country. But, on hindsight, the benefits have been disappointing or even inadequate to cover costs. Elsewhere, Project B has had far better results than anticipated during its planning stage. Should projects like Project A have never been undertaken. and the Project B kind of investments have been expanded? This in a nut-shell, is the problem which arouses interest in the study of decisions under uncertainty. Clearly, the success of one project . ,and the failure of another is no evidence· that a wrong decision has been made. They merely give rise to two kinds of ques­tions: have the realized outcomes been anticipated or are they a complete· surprise and, given "correct" anticipation, was the decision a "correct" one, i.e., an a £!iori optimal decision? 1/

1. Formulation of Anticipations

2. First it is wo,rthwhile to ask then what is a "correct" anticipation and how the several formulations of anticipations under uncertail\ty qualify by the "correctness" criterion. Does "correct" mean that an anticipation must be confirmed, by the realization? Certainly not. 1.1 It is almost axiomatic that under uncertainty, no anticipation could be expected to be·correct in this sense. Instead, a "correct" anticipation could be defined as one which is not refuted by a realization. Applying this criterion it. is evident that a single valued anticipation of an outcome can hardly qualify. Certainly, an anticipation whi~h stretches over the entire range of possible realizations will be a "correct" anticipation.

3. Unfortunat~ly "correct" anticipations in this objective sense I,lre not necessarily a satisfactory formulation for reaching decisions and it'is after all primarily for the purpose of making choices that anticipations are formulated. Correct anticipations i.n this sense are neither unique nor necessarily very helpful for reaching decisions.

4. Consider for instance a statement of anticipation whereby an outcome is said to be highl.Y likely within a given range and extremely unlJkely outside this range. This anticipation is as "correct" as one whicq assigns no likelihoods at all to possible outcomes i'h the sense that no realizationscoitld Irefute the stated anticipations:. Similarly, correct is a statement which assigns numerical values to the relative likelihoods of various Dutcomes. The choice between alternative formula­tions of "correct" anti,cipations must be sought then on other grounds.

!I The "optimal" decision problem is discussed i~ Chapter IV.

11 Note that the anticipation in which we are here interested is not. a whole set of outcomes, but a single outcome. If the outcomes of an event. are observable many times over and if the deciaion pertatns to ~he .entire set of outcomes, an anticipation of a frequency distribution of outcomes might be nearly"'''correct lt in t1'\e sense that an anticipation can be expec~ed to be approxinJately realized (if the number of observations on'which the anticipation is based and the number of realizations is large epough).

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,: "/

5. Some seek to distinguish between good and bad formulations of the nature of uncertainty on the basis of relative objectivity in the deriva­tion of the estimate. Clear~y; a statement of anticipation which defines possible outcomes in terms of specific relative likelihoods is less universally acceptable than one which does not distinguish between llk~li­h,oods. Similarly, the likeli11;pods of outcomes of an eventwhlch can be o;bserved many times over, are l{iess disputable than the likelihoods of a nOil­l:ecurring. event. It is not clear, however, how refevant an objective /., formulation of.anticipations is for analyzing how investors do or e~eh ought to act. 1/ /'~

2. The Prob8bilisti~ Formulation

6. Most decision theories adopt a particular formulation of anticipa-tions on the basis of how closely it is thought to correspond with the way decision makers actually think about uncertain ou~comes i~ relation to their decisions. 1/ ·There is pretty much agreement that the li~elihoods of out­comes do concern decision makers and that it makes lit,tIe difference for a decision whether' these likelihoods are judgments based on mere hunches or on an enormous ,amount of frequency evidence. Furthermore, since the likelihoods of outcomes and, to a more limited extent, even the full range of outcomes generally cannot be objectively determined, it is now commonly accepted that "uncertainty of the con§equences, which is controlling for beq1i;viour is basi­cally that existing in the mind of the chooser", 1/ that is, ~:he evaluation of risk is subjective.

3. Probabilistic Formulation Facilitates Aggr~gation

7. A good portion of this paper is devoted to ari<~xpositio~·; of how investors and project appraisers might go about formulat:t·q.g their expecta­tions about the outcomes from a given investment, say the ~ate of return or

1/ Game-theoretic decision models are primarily justified on the basis of their reliance on the objective formulation of anticipations. But then again it is difficult to see why the objective fCrMl1:lation of

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the uncertainty condition should be important when the ch~~ce criteria or the choice from among many decision models is a subjective matter.

R. Me Aldeman, "Criteria for Capital Investment", Operational Research Quarterly, March 1965 summarizes the ongoing debate between objectivists and subjectivists pretty well: "As there are so many subjec'tive elements in the choic'e of criterion to use, there seem to be no valid grounds for objecting to subjective elements within the, crj.terion. Certainly, it seems that wheneyer a criterion cQntaining subjective elements is pro­posed there will be cries that it is 'not objective. Likewise, however, if a criterion that claims objectivity is proposed, there are cries that it does not take into account the decision maker's subjective valuation of pay-offs involved, nor his subjective beliefs.

1/" Marshak, "Alternative Approaches to the Theory of Choice in Risk-Taking Situations", Econometrica, Vol. 19, No.4, October 1951, pp. 404-417 •.

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the discounted present value of net bElnefits. Any estimate of such an out­come from an investment usually needs to be developed from information about the effects of many variables (cost items, production, prices, etc.) and

,their values. This is in essence what an investment appraisal is all about. Similarly, of course, the various outcomes from an investment under uncer­tainty conditions arise also from the wide range of values which relevant non-controllable variables and parameters of the relevant relationships take on as a result of uncertainty. Now, it may be satisfactory (and "objectively" more correct than formulation of another anticipation) to say that production, prices and various cost items will fall into specified range~. But, to "build" up an estimate of, say, the. rate of return, from wide ranges of the relevant variables ·and parameters without regard to likelihoods and particularly the likelihoods of compensating events would lead to quite unaccep'table results.

8. Surely it is difficult to see that there should be much chance f~r all the worst, or the best, of the anticipations to appear in combination, no matter how difficult it is to specify the basic probability distributions. The only way, so far, to handle this aggregation problem is to use the pro­bability calculus. It is one thing to believe that one event has as good a chance to turn out favorably as unfavorably, and another matter tq believe that there is as much of a ,chance that luck or misfortune will hold out simul­taneously for a large number of independent events as there is a chance for some turning out favorably and others unfavorably. If it is thought, for instance, that X, Y and Z are independent events (that is their outcomes are in no way correlated), then the probability calculus tell us that the pro­bability of encountering a combination of the most unfavorable outcomes of all three events is the product of the probabilities of the most unfavorable outcome of each event. 1/

9. To illustrate the aggregation problem we might .consider a simple game. A person is given the opportunity to purcbase a ticket at the cost of $1.00 which entitles him to a $10.00 reward if the last digit of the first nuulber in the telephone directory turns out to be an even nUmber. He gets no reward if it turns out to be an uneven number. It is conceiv~ble that a conservative person should refuse the offer without even investigating the probabilities of the outcomes. It is, however, difficult to conceive of a person who would be conservative enough not to investig~te the probab~11ties of an even and odd number, if the same game were to·consist of ~he sum . of the returns for each of the readings of 10 numbers similarly se1~cted. For the game involving the selection of a single number he may simply assume that the probability of getting an even numbered last digit is'not greater -that' 90 percent. This might be sufficient informat:ion for a conservative person. He is not int~rested in the game since even a 10 percent chance at loosing $1.00 is too much of a gamble for him. If. the probability of getting an even number in the last digit were 90 percent, however, the chance of not

~/ !f the most unfavorabte outcomes are Xl' Yl and Zl and the respect~ve

iprobabilitie~ are P1 / P2 and P3 then :'P'(X1YlZl)·1JI Pl.P2· PJ.

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recovering the cost of the ticket would be only one in ten billion (and con­versely the chances for a gain are almost certain), a result which might he enough to change even the mind of a most conservative person. Without a probability approach, the aggregation effect would be virtually unassail­able. We could metely say that the "payoff" in the second case would be between zero and $100.00.

4. Probabilistic Approach Utilizes More Iriformation

10. The appraisal and evaluation of a project is usually a collective effort of a team of experts. A good project appraisal draws on the knowledge of many disciplines such as engineering, agronomy, hydrology, statistics, economics, sociology, and politics. It is a major contention underlying the appraisal procedures suggested in this monograph that a good apprai.sal should attempt to distinguish between what each discipline and, if embodied within different individuals, what each expert contributes to the final appraisal and evaluation of a project. It is not uncommon that particularly under uncertainty conditions these contributions get confounded ,beyond recogni­tion. It is then not uncommon for the agronomist .. )r the engineer to "contribute" a production function which reflects his assessment of the political and social (!onditions or to "discount" the parameters by what he believes ought to be his government's attitudes towards particular outcomes. Conversely, and particularly if unaware that the engineer has already "colored" his estimate of technical coefficients, the person in charge of assessing a set of possible benefits from a project may "adjust" the techni­~a1 coefficients to what he believes (and is in a best position to know) are "realistic" levels. There are, of course, many legitimate interaction effects which make it desirable and necessary for the different exper .... :s to collaborate in preparing projections. However, beyond these, projections of a particular event should as nearly as possible reflect what the apprai­ser believes to be the possible outcome of that event under explicitly stated conditions. This is in fact facilitated by the probabilistic approach.

11. When an engineer is required to summarize a projection of a parti-cular event, say, the water yield supplied by a given size reservoir, in terms of one unique number, he must throwaway a great deal of his knowledge about this event. Knowing that the unique estimate supplied by him will form the basis and the only basis for a unique estimate of a benefit-cost measure, he will be tempted to give an estimate which he believes to reflect the decision maker's preference or aversion towards risk. He may give a most conservative estimate - one which he knows has a high probability to be exceeded~ he may give what he believes to be the most likely outcome, or otherwise.

12. Conversely, the final decision maker, who must consider the riski­ness of various projects in choosing between them, is in no less a difficult position. Deprived of knowledge of the likelihoods of r~alizing outcomes 6f some technical events other than those reported, be must estimate technical information ~V'bich the engineer is in a better position to estimate and might have actually estimated but which due to faulty appraisal procedures has not been recorded.

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5. Probabilistic Formulation can be Subjected to Meaningful Empirical Test

13. In our search for a good formulatioA of a atatement of antici-pation about an uncertain event we have in essence rejected those formula­tions which either are practically always refuted <the single valued estimate) or are never refuted (the r~nge without probabilities) by the actual outcome of the event. It is noteworthy that a valuable attribute of the probability formulation is the fact that, while with it an anticipation cannot be . refuted by a single outcome of an event, it can so be refuted at least in a probability sense by observing several outcomes of an event. In a sense then one could say that in a world of uncertainty the only correct and use­ful knowledge and information is one which is reported in probability terms and can be refuted in these term3.

6. Estimation of Probability Distributions

14. Few cut-and-dried rules can be given for actually estimating the probability distributions of basic events or parameters used in a cost­benefit analysis. Very often it might mean nothing more than stating explicitly the information experts have been using all along in making their projections. For instance, if annual rainfall is one of the uncertain vari­ables, a frequency distribution derived from past observations may be available and used if meteorologists think that this is the best estimate of the probability distribution of future rainfall. Frequently it is thought that a better estimate of the probability distribution could be obtained by fitting the frequency data to a known cu~e.

15. Estimates of parameters such as price and income elas.ticities or production coefficients are often derived by formal stat~stical analyses of data. Although not $lways reported, these analyses can supply estimates of the probability distributions of the parameters as well as single valued estimates.

16. If a formal statistical "best" estimate of the distribution of an event is either not available or not relevant, the expert might proceed by first projecting the lim.tts of the range of possible outcomes of the event on the basis of historic.l or other comparable data and his experience with the event under similar circumstances. Subsequently, t~is range is ~ub­div1.ded in1:O 2 to 5 subranges. Then these subranges cau be first tanked on the basis of ''more'' or "less l ' likely. Subsequently, relative magnitudes eould be assigned to these likelihoods, such that the sum of the weights· add up to unity. Alternatively, it may be easier sometimes to. ask for tQe limits of the range which encompasses the actual outcome of a certain eVQut with a specified probability_

17. Depending on the variable involved and on how one wishes to use th~ probability information, it may be desirable to specify a continuous distri­bution or one whi~h isspeci,.fied for discrete values of a variable .. : Often one may be satisfied with e~timatingthe range which encompasses all or almOst all likely outcomf!S and then to assume on 'the basis of prior knowl­edge that the variable i~ distributed as one of several',knGWD theoretical

- 11 -

I

probability distribut:ions. If a normal distribution is hypothesized for instance, it is sufficient to ask for what the "expert" believes to be the mean ~r the mode -and ~he limits of the range which would have a rare chance of being exceeded. If a Beta distribution is hypothesized, the mean and standard deviation can ,be estimated by asking the expert for a pessimistic (p) most likely (m) and optimistic (0) prediction. 11

18. The main point to be stressed with regards to the assessment of probability distributions of basic events and parameters affecting the re.turns of a project is that it is desirable to avoid "coloring" these pt'O­

bability judgments by risk preference or risk aversion considerations. The e~ltimated probability distribution should as nearl:>' as possible reflect what the appraiser believes to be- the possible outcomes of a particular event and

;\ their respec,tive likelihoods. A project may be rejected because it may have a small chance of failure, regardless of a high probability of success, but this does not at all imply that it is appropriate to neglect reporting of probabilities for highly favorable outcomes of basic events (such ~s physi­cal yields and prices). The reason for this and the inappropriateness of appraising only limited aspects of the p'rohability distributions will be e~plained later on.

19. Estimation of probability distributions is simply a way of expli-citly stating as best as we know how what we do know about the outcome of a p~rticular event. However, in spite of the subjective and limited fore­sight that a probability distribution can portray, it is difficult to see " how ignoring whatever little is or can be known about it, should result in a more useful appraisal except by mere chance.

1/ The mean is then (p + 0 + 4m)/6 and the standard deviation is (0 - p)/6. For /8 good discussion of estimating probability distributions in the con­text of investment appraisal, see B. Wagle, HA Statistical Analysis of Risk in Capital Investment Projects", Operations Research ,Quarter"!y', Vol. 18, No.1.

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III. PROBABILITY APPRAISAL OF PROJECT RETURNS UNDER UNCERTAINTY

1. General Outline

1. 'Project appraisal in general involves an evaluation of how:' certain simple events interact to produce a final outcome. Assuming certainty about the state of the simple events and the relationship between then and a final outcome, the appraisal of an investment consists essentially if identifying the events most relevant to the final outcome of a given course of action, such as the outputs A, B, C .••• , the required inputs D, C, E •••• , and the corresponding prices. Subsequently, logical (mathematically correct)proce­dures such as addition and multiplication are used to calculate the economic returns of the project.

2. An adequate appraisal of projects involving uncertainty requires judgments, just the same as under certainty, of the kind of events relevant to the outcome from a given course of .action. But instead of presenting exact estimates of the relevant events·, an appraiser must form a judgment of the likelihoods of various states of the same events. He must then use the probability calculus to derive meaningful aggregations of the inter­actions of the simple events. This!\ chapter primarily deals with reasonably correct aggregation procedures for ~eriving a probability distribution of a cos t-benefit measure used in proj eCi~ appraisal. Concern here is with the logical steps to be taken in aggregating probability beliefs of investment appraisers about various relatively "simple" events into probability dis­tributions of the total net benefits from an investment. Correct aggregation procedures do not, of course, in any way substitute for "good" judgments in . the choice of relevant variables and their estimated projected probability distributions. They are merely a means for assuring that "good" judgments are preserved in the process of aggregation.

2. The Aggregation Problem,

3. It will do here to review briefly the benefit-cost calculations used most commonly in investment appraisals, when uncertainty is not ex­plicitly taken into consideration. This review and somewhat formal and' precise way of stating thei commonly used procedures will assist us, how­ever, in learning the modifications needed in any analysis which explicitly takes i~ccount of uncertainty about the outcomes of specified elements in the analysis. The basic benefit-cost formula is:

(3.1) R .: ""-. . -t L (1 + r) B .. t ""

t m 0, 1, •••• , n

where "R is the total net benefit from an investment" discounted to the present time, B

t is the annual !l£.t benefit and r is the marginal cost

of capital. 1/ ..

1/ A variant of this flonnula :f.s the internal'rate of return calculation, in which case r is calcula.ted from formula, (3.1) Vatting R equal zero. For the exposition intended in this chapter, ft'mal<es no difference whether R or r is the final variable of interest. '

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4. The annual benefits and costs are, of course, derived from knowl-edge about certain other variablese In the crude~,t form of analysis, the appraiser would have to conside't' various changes in the production of goods and servIces"their respective values and the quantities and costs of the resources needed as a result of the investment. In a more compre­hensive appraisal, further explicitly stated relationships may be used to estimate the' net berie~its. Prices, for instanJce, may be related to projected per capita incomes, population, imports or exports. In case pf an irrigation project, additional outputs may be a function of moisture deficiency or· rainfall, and the number of producing units affected by the investment may ,depend on the available amount of water (which in turn depends on rainfall) and the fanners' responsiveness to adopt irrigation.

I

5c In a very general way, the total net benefits from an investment can be said to be a function of some exogenous variables and parameters which describe the quantitative relationship between variables. Exogenous variables are variables which an analyst chooses no~ to explain in any formal way by other functional relationships because, either the state of the variable has only a small effect on the variable of interest or he finds it too difficult, time .consuming and cos tly to carry out further analysis or else, because the variables which explain are as difficult to forecast as the variable to be explained. For instance, the analyst may choose not to explain the price of fertilizer because this variable has only a small effect on the net benefits for a certain irrigation project~ On the other hand, he may not choose to study the prices of products in any formal way because price forecasting is a costly and time-consuming activity. Finally, he may not study the functional relationship between yields and rainfall because he can not forecast rainfall in the future any better than he can directly forecast yields.

6. The problem which concerns us in this chapter is how to aggre-gate probability distributions of exog'enous variables and parameters. For an oversimplified illustration, conuider a simple project whose costs and benefits are fully realized in two years, such that the present value R is,

(6.1) R • + -1

where a • (1 + r) ,r is the opportunity cost of capital, B1 is the net return (positive or negative) in the first year and B2 is the net return in the second year. Asswne furthermore 'that Bl is the sum of two costs which in turn are the products of physical inputs and their unit costs, Yl and Y and C1 and C2 respectively. B2 is ~he s~m of net revenues derived trom two sources which in turn are the proGucts of physical out­puts and their per unit prices, Xl and X2 and VI and V2 respectively, i.e.,

(6.2) •

and

(6.3) •

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Furthermore, physical output X2 is known to be a quadratic function of a

certain input Z. The parameters of this functional relationship are also random variables subj ect to probability dis tributions, i. e. ,

(6.4)

Then, by substituting equations (6.2), (6.3) and (6.4) into (6.1)e the present value can be seen to be a function of the exogenous vat'iables el ,e2, Y1 , Y2, Pl' P2' Xl and Z and the parameters a, eo' e1 and e2:

2 2 (6.5) R - a(el Y1 + C2 YZ) + a (Vl Xl + V2 eo + Vl e1 Z + Vl et Z )

7. One procedure for· deriving the ptobabil1ty distributions of R is to recompute equation (6,S) for each possible combination of the outcomes of the "basic" variables,and furthermore, to calculate the probability of each combination. Assuming that the ~robability distribution of each variable is stated in terms of 4 possible outcomes, even a crude and simple

11 analysis as described here would amount to (4) • 4.000,000 calculations, 11, being the number of tlbasic" variables. In the analysis of an actual project with benefits stretching out over many years~ the number of vari­ables would be much higher, and in spite of possible shortcuts and even higher calculating speeds of e1ect~onic computers, it is difficult to see that this procedure has any great merit.. Recall that in addition to cal .... culating the returns, the computer would need to calculate also the product of all the probabilities for each combination and then to reaggregate the returns and their probabilities into a distribution.

8. A second procf!dure which is certainly feasible if; to estimate · the probability distribution of R on the basis of a aimula-ted sample. 11 All possible otitcomes of the variables affecting the returns from a proJ­ect and their probabilities al"e fed into a computer. The computer is then instructed to select at random one outcome of each of the variables, allo,,,ing for realistic restrictions for interdependencies in the variables. Given the selected outcomes of all the variables, the corresponding' ~eturns on the investment are calculated. This process is ,repeated until a!arge enough sample is obtained for a close approximation to the' actual proba­bility distribution of the returns (R). This procedure requires absolutely no new mathematical sk1.lls on the part of project appraisers, They merely must supply estimates of probability distributions. There are already available computer programs which (a) select at random values from these dis tributions, (b) calcula~'e the present value of internal Tate of '['eturn or any other measure of project benefits and (c) after repeating the same process a desired number of times compute a frequency distribution of the measure of benefits. In practice, the size of ~he sample is detennined by

!I This method is sometimes referred to as stochastic simulation. A remarkably lucid exposition of this method is g1.ven by D. B. Hertz" "Risk Analysis in Capital~ Expenditure Decisions". Harvard Business Review, January/Febru~ry,; 1964.

- 15 -(/ t

trial and error. The sample is considered ,large enough when the frequency distribution does.rlot change much when the sample size is further increased.

9. A third procedure is to apply the probab/ility calculus directly to the calculation of c'ertain characteristics of the probability distribu­tion of R (or any other measure of aggregated benefits). This procedure is based on the application of one of the most important con~epts involving probability distributions, namely, that of mathematical expectations. In

. the next ~wo sections we will discuss the limitations as well as the attrac­tive features of the two practically feasible ,aggI;egation pfocedure:3 - the simulation method and the mathematical method - by way of illustrations.

3. Illustration of Alternative Procedures for Aggregating Probability Distributions

10. At this point a very simple illustration of what has been dis-, cussed so far should be useful. While it is usually not feasible to calculate the exact probability distribution of an aggregate measure (the first procedure), a very simple case is presented here in which an exact distribution can be easily calculated. Subsequently, the two approxima­tion (but feasible) procedures will be used and the results compared with the "true" dist'ribution.

11. The object is to know the probability distribution of a present value of net revenue (R), based on knowledge of the probability distribu­tions of an initial investment cost (Y) and a revenue (X), discounted by a factor of 0.5 (say the revenue is received 10 years later and the dis­count rate is 7%), i.e.,

(Present Value) • (.5) (Gross Revenue) - (Invesoment Cost) or in symbols

(11.1) R • (.5)(X) - y

The assumed probability.distribution of X and Yare given in Table III-I.

Table III-I: Probability D.istributions of Revenue (X) and Investment Cost (Y)

-----X (Revenue) Y (Investment Cost)

Value Probability Value Probability

20 .10 8 .20 22 •. 20 10 .60 25 .40 12 .20 28 .20 ·30 .10

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12. The "true" distribution of the Present Value is derived by cal-culating. R for each possible combination of X and Y, and the probability .. of each comb ina tion to occur. In this case there are 15 possible comb ina- '" { tions. Assuming that the distribution of X and Yare independent (i.e., ' .. that" the probabilities of getting a p~rticular value of X are in no' way' affected by what value of Y has oCC;frred or vice versa), the pro,bability of any particular combination of X ana Y is the product of the probabilities of the respective values of X and Y. For instance, the probability of X' having a value of 20 and Y a value of 8 is (.10)(.20) - .02. The true probability distribution of R based on the assumed probability distributions of X and Y given above are presented in the second column of Table III-2.

Table III-2: Probability Distributions of Present Value (R) :

Present Value (R)

-2 -1 o 0.5 1 2 2.5 3 4 4.5 5 6 7

Mean: Variance:

Pro b a b iIi tie s

"True l1

Distribution

.02

.04

.06

.08

.12

.06

.24

.06

.12

.08

.06

.04

.02

2.50 3.75

Simulated Sample

(50 obs.)

.06 o

.04

.06

.08

.06

.30

.02

.14

.10

.04

.10 o

2.77 3.82

Simulated Sample

(100 obs.)

.03

.03

.05

.07

.06

.08

.21

.03

.15

.13

.03

.10

.03

2.94 4.24

13.' To estimate the probability distribution of R by simulation,. it I

is necessary to draw at random a large number of X and Y values from thet.r respective probabi1i'ty distributions and to compute a value of R for each set of X and Y values drawn. The frequency distribution of R when a large enough sample is used will tend to approximate the "true" probability distribution of R. (In the case used for illustration, there would be

\

no point, of course,:.to I'IJse the simulated sampling method, since the total flwnber of possible combinations is only 15, while a "large enough" sample would require a l1!.in~um of, say, 100 "observations" on R.)

14. To illustrate the method of simulation by a sample, we have used the last digits of tele~hon~ numbers in a. directory as a r~lndomization

•• J

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device. 11 Samples of size 50 and 100 were chosen by drawing the appro­priate number of observation for each X and Y and pairing them at random. To obtain the values of X, for instance, we let the last digit (0) represent an X'value of 20, (1) and (2) a value of 22, (3) (4) (5) and (6) a value of 25, (7) and (8) a value of 28 and (9) a value of 30. To obtain. a series of Y numbers, we let a las t digit of (0) and: (1) represent a. Y value of 8, (2), (3), ,(4), (5), (6) and (7) a value of 10 and (8) and (9) a value of 12. ~The probability distributions of R corresponding with 50 and 100 pair:~ of randomly selected X and Y values from their respective probabi­lity distributions are presented in Columns 3 and 4 of Table 111-2.

• • I

15. The third pro~edure consists of c;a1cu1ating the mean and the variance of the present value (R) and to interpret the results in terms of a normal distri,bution. This is, of course, only an approximation pro­cedure, since we know already that in our case, the "true" distribution is a discrete distribution (i.e., the variables of interest take on only dis­rlrete values) and the probabilities do not follow an exact pattern as would l:.e expected from a normal distribution. To begin with, however, let us see how to calculate the mean and the variance of R and how to interpret these in terms of a normal distribution.

16. Denote the means of X and Y by X and Y respectively, and their variances by VeX) and V(Y). In our case, (from the basic data presented in Table III-I):

X =- L (prob i) (Xi)

:a (.10) (20) + (.20) (22) + (.40) (25) + (.20) (28) + (.10) (30)'" II

Y = [, (prob i) (Yi )

- (.20) (8) + (.60) (10) + (.20) (12) • 10

VeX) • l: (prob i) (Xi - X)2

(.10) (_5)2 + (.20) (_3)2 + .20 (3)2 + .10 (5)2 • 8.6

and

V(Y) .. L (prob i) (Yi - Y) 2

.20 (_2)2 + .20 (2)2 • 1.6

17,. Given these data, it is a simple matter to calculate the mean, i, and the variance, V(R), of the present value as follows:

i = (.5) (X) - Y • (.5) (25) - 10 = 2.5

!I There exist many random selection computer "packages" which select at random. values from various kinds of probability distributions.

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and

VCR) = (.5)2 VeX) + V(Y)

= (.25) (8.6) + (1.6) :II: 3.75

assuming that X and Yare not correlatedo 11 $~\

i/ 18. To determine the probability that R is less than any value; Ri' one computes the ratio

S = (Ri

- R)I I VCR)

and looks up the probability in a table of the standard normal distribution. The cumulative distribution based on the assmnption that R approximates a normal distribution is g'iven in the last column of Table 111-3, and is presented graphically in Figure III-I. 1/ For comparisons the cumulative probabilities from the "true" probability distribution and the simulated samples are also presented in Table I:~II-3 and Figure 111-1. .

Table 1I1-3: Cumulative Probability Distribution of R

Cumulative Probabilities, Prob (R < Ri

) Present

Sample Sample Approximation Value I'True" 50 100 by Normal'

Ri Distribution Observa- Observ(t- Distribution tions tions

-2.0 0.02 0.06 0.03 0.01 -1.0 0.06 0.06 0.06 0.04 0 0.12 0.10 0.11 0.10 0.5 0.20 0.16 0.18 0.15 1.0 0.32 0.24 0.24 0.22 2.0 0.38 0.30 0.32 0.40 2.5 0.62 0.60 0.53 0.50 3.0 0.68 0.62 0.56 0.60 4.0 0.80 0.76 0.71 0.78 4.5 0.88 0.80 0.84 0.85 s.o 0.94 0.90 0.87 0.90 6.0 0.98 1.00 0.97 0.96 7.0 1.00 1.00 1.00 0.99

1/ See Appendix B for the mathematical deriv~tion of the formulae. Note that the mean and variance calculated by these form~~lae are the "true" mean and variance of R.

1,.1

'1-../ A cumulative distribution shows the probabilities that the event will be less than a stated value.

• ~ • .t

Probability

.8

.3

.2

..

Figure III-l~

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i " ! !'

3

I I i f

f : I ; 1 ..

s

, I I

tIl I I I I

II: I i ; t I I , ; , !

'·"·;.r'''':f1!F'' t .,,: t·.-t'

-t' ~ t'" . ,t!·· I

I"! • I'

• t I ,

: I ".I I j "I

I I + I : r ,t I •

.6 7 'Present value

Probahili tIl DLstrlbutions of Preelant Value \.

- 20 -

4. Discussion of Alternative Estimating Procedures

19. The brief illustration and comparison of results of the two estimation procedures - the simulation method and the mathematical method -suffices to point up at least, in principle, the possible advantageous .. features and the shortcoming of both methods.

20. For simulation the project appraiser needs no knowledge of the probability calculus whatsoever. There is no chance of making any error in the calculations. All the appraiser needs to do is to present the compu­ter with a model and the constant values or probability distributions of the relevant parameters and variables and the computer (with a programr.l¢~) can grind out an estimated probability distribution of the desired aggre­gate measure. Furthermore, this method requires no assumptions with respect to the relevant distJ:ibutions, since the calculated sample gives directly an es tima te of the '(~tue" dis tribution, whatever its shape. '

21. The primary disadvantage of the simulation method is its complete reliance on the availability of a computer. Furthermore, any "run" is highly specific to the postulated inputs. If any variations in the assump­tions or in the project itself are to be: investigated, a new computer "run" is necessary. Frequently, an appraisal :~eam may wish to pursue considera­tlon of alternatives, while st,ill in thei' field, based on the results of a previous analysis. With simulation thisrmay not be feasible. Another unresolved issue.is the optimum sample s!tize. However, since in most cases very little computing time on a large cotnputer ,.,ill be required, the prac­tical solution might be to choose a relatively large sample, or to devise

'a sampling procedure by stages with a statistical test to determine whether additional observations should be calculated.

22. The mathematical method is only useful if one wishes to consider a relatively simple model consisting of aggregation of only a few major uncertain variables. In this case, the method is cheap, requiring.little more than pencil and paper and a desk calculator. Furthermore, once the mean and the variance have been calculated for one set of parameters it i~' easy to estimate the effects of changes in any of the para~eters or pro! ~ bability distributions of the important variables on the mfan and varia1ce of the desired aggregate measure. '

23. The mathematical method does require some minimal knowledge of the probability calculus; however, this is no major prob1eme An invest­me1?~t appraiser usually kno,.,s how to calculate means and variances and' can be easily familiarized Jolith a fely basic rules needed for deriving the mean and variance of an aggregative measure. 1/ The real problem is to d~te~ine how useful it j.s to know the mean and the variance if one does not know the e>:act shape of the proqability distribution of the aggregate .mea~ure.

!I Some of these basic rules are given in Appendix B.

:

- 21 -

24. There are several ways of "sweeping, the distribution problem under the table". None, however, ate completely satisfactory. There are, for instance, some decision-makers, or so it is assumed in much of the literature on risk appraisal, whose objective function is such that they require to know only the mean and variance. This aspect of the problem is further discussed in the foll9wing chapter. At least very superficially, however, it may be seriously que~tioned whether decision making under uncer­tainty can be generally reduced to a maximization of a weighted function of a mean and variance of some measure of income. On the other hand, particu­larly when one wishes to consider various alternative ways of designing a given project and when one, therefore, essentially deals with the same kind of probability distribution, it may be quite sufficient to hav'e information on the means and variances of alternative designs.

25. Then there are, of course, quite a number of projects for which the aggregate measure of the net benefits would be approximately charac­terized by a normal distribution, which is completely specified once the mean and variance are kno~nl. Say, for instance, that the present value is simply the sum of a string of discotmted annual net benefits, each of which are assmned to follow a nc)rmal distribution in this case, the p:cesent value l\f'ould indeed bJ~ a normal distribution as well. 1/ Furthermore, even if the annual net benefits were fLot normally distributed, the present value dis­tribution would still be aLpproximately normal, if a large number of annual benefits with approximately equal weights were to be summed (i.e., if the interest rate were relatively low). 11 There is no assurance, however, tha~ the interpretation of the calculated mean and variance of a present value of an internal rate of return in tenns of a normal probability distribution is a reasonable procedure in all cases. It is first of all an empirical question on how much the true distribution deviates from normality and also it matters on how sensitive the decision criteria are.

26. The data presented in Table 111-3 illustrates the problem fairly well. The true distributlon of the aggregate measure R ,is certainly not a normal distribution, yet the cumulative probabilities estimated by using the mathematical expectations method and interpret.ing the mean and variance in terms of a norma];~d4.stribtltion do not differ much from the "true" cumula­tive probabilities. CertllLinly, the distribution derived from a sample of 100 does not give bettE!r estimates (though a sufficiently large sample would have had at least a high probability of doing better).

27. In sun~ry, as aL matter of general practice the simulation method is the preferable method ~rhenever a complete probability appraisal is de­sired. With computers bec~oming increasingly accessible and appropriate programs more generally available, the simulation method is likely to be aC,tually less costly both in terms of manpower and mathematical skills required than the mathematical method. In addition and quite importantly,

11 The internal rate of return, however, would not be exactly normally distributed.

Y This is shown by the Central Limit Theorem.

- 22 -

simplation is likely .to give a better estimate of the true distri,~ution than can be expected from assuming a normal di$tribution. The mathema­tical method is likely to prove useful, however, if partial analyses of the impact of uncertainty in a few selected variables is desired and quick approximate answers are needed.

28. For the rema.i_nder of this chapter only the mathematical appraisal method is used in ord~r: (a) to show how, in general, the results from a probability appraisal may differ from the results obtained by a conven­tionally practiced project appraisal, and (b) in order to further illus­trate aome essential rules from the probability calculus. The simulation procedure is not very well suited to derive generalizations and is simple and straightforward enough not to require further illustrations.

5. Conceptual Problems Related to Probability Appraisal

29. Throughout the discussion so far we have assumed that the reason for making a probability appraisal is that decision makers are somehow inter­e~~ed in knowing not only a single-valued measure of a project's outcome, sucnCii~"the one most likely or an average one, but wish to know also other possible outcomes and their respective probabilities. Under the heading of "Biased Estimates" below we explain why probability appr.aisal is desir­able even if the decision maker were to be satisfied with ~erely knowing, a single valued estimate. Another problem discussed in this section is the problem of estimating the probability distribution of the present value or the internal rate of return for stochastic variables, some of whic~ are correlated.

Biased Estimates

30. Even if the decision maker were interested only in a single point estimate and not in the entire probability distribution, it w~uld be desir­able to do a probability appraisal in some cases in 'order to avo'id consistent errors of estimation.

31. One example is the practice of aggregating most likely values (modes) of various variables. To illustrate. the folly of this method of aggregation, ~on~ider first a stmple ~ase wnere one is interested in es tima ting the mos t likely revenue. from 'forecas ts 0 f price and s.ales. Say, the market ('Analyst predicts a 60~' chance that the price will be $10 and a 40% chance that the price will be oply $5. Sales ~r~ given a 60% chance to be 100 units and a 40% chance to be 50 units. The most likely revenue calculated from the most likely price and most likely sales is obviously $1,000. However, a probab~lity appraisal ~ould have clearly shown that this is not the correct ~stimate of the most likely revenue. Asswning that pr.ice and sales are not correlated, the true probability distribution of revenue 1s as follow~:

- 23 -

Price Sales Probability Revenue

10 10 .36 1,000 10 5 2' . ... 500

5 10 .24 500 5 5 .16 250

Clearly, the most likely revenue is $500 (\vith a B't"obability of .48) and not $1,000 (with a probability of .36). If the respective probabilities would have been 70 and 30%, the most likely revenue would have been the product of the modes of price and sales. HO'vever, in general, when an aggregate measure is the (\l7eighted) sum of many different variables or products of variables, the simple aggregation of modes will not give a good estimate of the true mode of the aggregate measure.

32. The same reasoning, of course, applies when one is intcrest(ld in getting a "conservative ll estimate, \.:herc stich an estimate is defined as an outcome which has a large chance of being e1{cceded. 1f one ,,,ere to aggregate such conservative estimates for different prices and sales, etc., the result would generally be a rate of teturn ~ith an undefined extent ()f· "conservativeness". In fact, to follow such a rule-of-thumb would certainly lead to non-comparable rates of returns estimates for different projects in terms of the degree of "conservatism" implied.

33. The problem of bia~ exists also if on~ is aggregating means of probability distributions of several elementary events to estimate a mean of the aggregate. Fortunately, there are likely to be many cases when such estimates are not biased, however. One such case is if the aggregate is a tunction linear in the uncertain variables. [Say, preserit value (R) is a sum of the discounted benefits (B

i) in sever,11 years. Then. the mea'n

of revenue (i) is the sum of the discounted me;an annual benefits (Hi)' i. e. ,

(33.1) R n- + B + a 2n-

2 + lilt 0 a 1 · ......... .

where a • (1 + r)-l and r is the discount rate.

34. But there are many cases when the dependent variable is a func-tion which is non-linear in the. independent variable. Consider, for i.nstance, that one wishes to estimate the mean yield (Y) of an agricultural crop based on one's knowledge between rainfall (W) and yield (Y) and the proba­bility distribution of rainfall. Assume, furthermore, that yield (Y) increases at a decreasing rate when rainfall (W) increases in a given range, such that

(34.1) Y = 10 + 6 W

Assume that \v has a 50% chance of being 1 and 50% chance of being 5. Corres~ondingly, the yield has equal chances of being 15.5 and 27.5 and the true mean yield is 2l.5~ If, however, instead of a complete proba­bility appraisa! we had calculated the mean yield by substituting the mean rainfall (W = 3) in equation (34.1) our estimate would have been 23.5 .. This overestimate is, of course, intuitively expected since a decreasing rate in yield additions implies that a loss in yield due to a less than average rainfall is not fully compensated by the gain in yield when rainfall is to the same extent more than average. The notion of bias can be seen very easily, graphically, at least in this simple example. In Figure 11I-2, Y is the "true" expected yield, whereas, f(W) is the estim~te '\vhich would be obtained by simply substituting fJ for W in the yield equation.

35. Since non-linear functions used in economic projections are frequently convex (increasing at a decreasing rate), the lj.kely bias from neglecting to do a proper probability analysis is likely to be an. over­estimation of benefits. For instance, if ~ve base an estimate of benefits from irrigation on average wate~ availability we are likely to overestimate the benefits if watf!r availability is highly variable and the additional returns to water beyond the average amount are small in comparison with the loss in revenue due to an amount of water less than the average. A likely source of a similar bias in project appraisal is the calculation of cost-benefits on the basis of an average "life" of the investment. The present value as a function of investment "life" certainly increases at a decreasing rate. The bias can be quite large, particularly, if the dis­CQunt rate is high and the average lnvestment "life" fair,-ly short. Several possible sources of such biases are discussed below. The point to be made is that it mny be desirable to do a complete probability appraisal even if the appraiser is only interested in a single-valued estimate and not in the entire probability dist1Cibution of some measure of a project's benefits.

Correlation

36. A major problem in appraising a project subject to stochastic events is correlation. Generally; the existence of correlation indicates incomplete model specification. Therefore, 1.£ significant correlations are suspected, the best way to avoid misleading predictions is to explicitly recognize further undet'lying systemat:i.c relationships among variables and to substitute uncorrelated variables. The problem of correlation and how to cope with it can be best illustrated by a few examples.

37. Consider that the objective is to estimate revenue (R) based on prior estimates of price (T) and sales (S), i.e.,

(37.1) R I: (T)(S)

Assume, furthermore, that the price is believed to have a 50-50 chance of being 1 or 3 and similarly sales have a 50-50 chance of being 200 or 400. The distribution, mean and variance of the revenue, assuming. no correlation, are as follows:

~ ': I j I

"1

~ I I .. I

...... i ·";·1 I., . , ~.; ~

'.;* I • ~"", t

I.;·'+·! : j , :. t

Figure 111-2:

I I

- 25 -

! f

i •

• , t

! I t I' i ~ t

I

• I I

• 1

, '

, , . I'-~ t; I:.

1-: I 1 . I

" I: l~ "

1 I! I. .. ~ 1 ~ . ..·t4 ..... Mri .. ---·,-,l-~~~ !.--; r I ,I ! l i' t . ., , ~ ,. ~ l' 1 ,. r' ~ u. II; , , iIi t, t ~ •

t t f If' •. ,.".

, f' j I ; ,: t ~ I r 1 i I' • !

~ • f t { •

,; :: I I- I I: i t·· 1

I ,. i .' . i • t .;; r·1 : I , • , I '

.-~ '~!' '~r;"I~:"'T: 'rr~ITrr; : i j t ; i; !;,; .'. •

I ~ ..' 1!" i : ' . It. t - , I I, r 1 ,I t :: i Lij .!

ill j! I :,' ii', ;,~, t·:r . · i t-! : . I ! I I l' -I t";

.~ --:-.+-1----........ - -Till:'...L~ 'r i : I : l I -j"l- i. - ! I·' · r 1,·1 ·1 -j- 1 . I '~'; .-j.... i j *:

l·i I I I : :1 ~. ,\ .. 1,,: t -• . I I I I I 1 • -1 ~ I., , I I I I I '1 I ~ I

, i II j t ! I I' : ': T I r.; ~ : I • , I.l!! . '-1 · I . 'I I I t I I

· I 'i: I I, i . I I II I J. I: • 1 I ...

... ! to' 4,.4- ..... . j--L..f-~ .. 1,_~ ......... 1_L.~ ..... "": · ", , , , J ' I I ! . . , 1 ~ : .• ;

, : ' ~ 'I : ~ .

:'~,: i ~','" ;.!': ,. I " ; ~.,~. , ' : ,I ...

I I ;1

!! 'I • I , I .~ I I

l . t ~ I j' 'I '( It,

Illustration ot Bias when Yield is Calculated as a 'Function ot Average R8!i118l1

j(

- 26 -

Probability Combination Revenue Mean and (Price,Sales) (R) Variance

.25 (1,200) 200 R - 600

.25 (1,400) 400 VCR) - 140,000

.25 (3,200) 600

.25 (3,400) 1,200

38. Contrariwise, if price and sales in the above example had been perfectly correlated however, i.e., for instance, if whenever sales are 200, price is 3, and whenever sales are 600. price is 1, the "true" distri­bution of revenue, the mean and variance would have been as follows:

Probability

.50

.50

Combination (Price,Sales)

1,400 3,200

Revenue (R)

400 600

Mean and Variance

R • 500 VCR) • 10,000

39. Clearly, whether or not price and sales are correlated makes quite a difference in how we should calculate the distribution of revenue. Both, mean and variance of the product of price and sal .. es are different, depending on the extent of correlation. The correlati(.n between price and sales could have been accounted for by specifying an additional equation describing the relationship between sales and price as follows:

(39.1) T == 5 .01 S

or if price and sales are not perfectly correlated as implied by the linear demand function (39.1), then by

(39.2) T • 5 .01 S + e

where e represents random effects· on pJ;'ice not correlated with sales. The mathematical equations for derivilng the mean and varia.nce of the revenue in both the correlated and. uncorrelated case are presented in Appendix B.

40. Another likely source of correlation is that two or more variables are related in a systematic way to a third variable. For illustration, consider that we wish to calculate the probability distribution of a total revenue on the basis of what we believe are the probability distributions' of revenue from two sources, R(l) and R(2), i.e.,

(40.1) R = _el) + R(2)

given the following data:

Probability

~50 .50

- 27 -

Revenue (1) R(l)

50 90

Probability

.50

.50

Revenue (2) R(2)

40 60

and disregarding the possibility of correlation~ the probability distri­bution of total revenue, its mean and variance are. sununarized below:

Probability Combination Total Mean and [R(l). R(2)] Revenue (R) Variance

.25 50, 40 90 R :. 120 .

.25 50, 60 110

.25 90, 40 130 VCR) • 500

.25 90, 60 150

t

41. If however, the uncertainty in both revenues had the same under-lying cause, say, both would be the higher of the given values if a protective tariff were to prevail and enactment of the legislation is given a 50-50 chance, then R(l) and R(2) are perff~ct1y correlated. Thfi corresponding probability distribution mean and variance of total revenue would in this case be as follows:

Probability

.50

.50

Combination [R(1),R(2)]

(SO, 40) (90, 60)

Revenue (R)

90 150

Mean and Variance

R = 120 VCR) • 900

42. Note that correlation does not affect the mean of a sum, however, the variance can be substantially altered by presence of correlation. Further specification of the model would have given an explicit equation for the close relationship of the respective revenues with the size of tariff (Z). From our data we can interpolate that these relationships may be as follows:

(42.1) R(l) • 50 + 4 Z

and

(42.2) R(2) • 40 + 2 Z , '

Adding th~se equations to the model we would substitute (42.1) and (42.2) into (40.1), i.e.,

... 28 -

(42.3) R • 90 + 6 Z

and estimate the probability distribution of total revenue, its mean and variance directly from this equation, as follows:

Probability

.50

.50

Tariff (Z)

o 1.0

Total Revenue

90 150

Mean and Variance

i - 120 VCR) • 900

It should be noted, of course, that the method of calculating all possible out.comes is not an operational procedure and is used here only to illustrate concepts. In the next. section we will show how to estimate the mean and variance of total revenue by using the mathematics of expectations. 1/

43. Correlation, then, is the problem of accounting for relationships between the included variables themselves, and between included variables and excluded variables and since in any practical appraisal only a few relationships can be explicitly stated, one could not possibly hope for the removal of all correlation. At best one can hope that the problem is recog­nized and understood and that one explicitly accounts for the mor~~ important relationships and makes judgments about their quantitative nature.

6. Specific Uncertainty Problems in Project Appraisal

44. The purpose of the ensuing discussion is twofold: to illustrate how to calculate the melln and variance of the present value and other sel­ected variables and to ~erive some generalizations for assessing projects subject to uncertainty. The operational uses of the mathematical estima­ti~n procedure are, of course limited to highly stylized appraisal models. Such, however, are not infrequently quite useful at least as preliminary exercises prior to a more complete appraisal.

Un·:ertainty About Annual Net Benefits -- 1

45. Let us begin by assuming that somehow the appraisal has proceeded to the point of having an estimate of a stream of annual benefits (positive or negative), that the annual benefits can be estimated in terms of pro­bability distributions_and that the discount rate is known. The expected (mean) present value (R) 1s then simpl)r a, weighted sum of the' expected. (mean) annual benefits, i.e.,

Y The mean a!!d varianc.~ can be readily ~a~culated from equation (42.3). Note that z • ~I and V(Z) "25. Then R • 90 + 6 (5) • 120 and

V(R) • (62)(25) • 900.

..

- 29 -

(45.1) i · - - 2-Bo + aBl + a B2 + ••••

/. . -1 where a • (1 +'r) and r is the discount rate. Note that for estimating the mean present value it matters not whether successive benefits are cor­related. The variance of R, VCR) depends, however very much on the extent of corre1a~ion between successive benefits. If successive benefits are not

" correlated the variance of the present value is

(45.2)

However, if the benefits are perfectly correlated,

(45.3) V(R)· [/V(Bo) + ~'L IV(Bl > + a 2 IV(B2

) + •••• + at IV(Bt

)+ •••• ] 2

Equal Annual Correlated and Uncorrelated Benefits

46. The above equations for deriving the mean and variance of the present value can be easily solved when at least a large number of succes­sive benefits have means and variances which are equal or follow a general trend •. Let us first assume that annual benefits consist of a known or unknown level of benefits, B, and positive or negative, but from year to year uncorrelated deviations from this level, et , such that,

(46.1) • B +

Then, if B is known and et is assumed to be zero on the average, Bt

• Band V(Bt) • V(e). S,ince we ass\UJled that the e t in successive years are not correlated the annual benefits are clearly not correlated as well. If B, the level of the annual benefits, is not known except in tennsof an average ~ and a variance V(B), but e t is zero in all years, Bt • Band V(Bt) a V(B).

In this case the annual benefits are perfectly correlated. We now proceed t~\analyze.the corresponding calculations of the mean and variance of the p::lt~'sent value of a stream of benefits in these two extreme cases.

47. Regardless of whether successive benefits are correlated the mean of the present value is

(47.1)

However, the variance of the present value depends on correlation. 1/ If the successive benefits are uncorrelated, i.e., if the uncertainty is due to year to year fluctuations,

(47.2) ~ 2t VCR) • ( ~ a ) V(et )

If the, level of benefits is the only source of uncertainty,

1/ .. f

In allt.the equations presented here it is assumed that V(et

) is the

same in all years, i.e., V(el ) • V(e2) • •••• • V(en)o

- 30 -

(47.3)

Derivation of these equations is discussed in Appendix B. Values of.(l: a2t ) for up to 30 years and 6% and 10% discount rates are presented in Appendix A, Table 1.

48. As should be expected, the variance in the present value"~s rela-tively much smaller when successive benefits are uncorrelated than w~en they are positively correlatedj; For instance, we may be uncertain whether~the annual benefits are plus or minus 30% of the mean. No correlation means that overestimates are likely to be compensated by underestimates. Pe}:'fect positive correlation of all successive benefit means, however, that the same forces, are at work, and if we had overestimated the benefits in the first year overestimation is to occur in all subsequent years.

49. To"compare varianc~~ it is frequently convenient to use the ratio C· V(B)/i, the so-called coefficient of variation. C is then a measure, of the relative variance and allows us to make some general statements about the impact of various kinds of uncertainty on the uncer­tainty implied for the overall returns for a project.

50. The coefficient of variation of the present value, C" when r

successive benefits are perfectly correlated simply equals the coefficient of variation of the annual benefits, eB, i.e.,

(50.1) V V(R) / R =[ L: ,at V V(B)]/[ Eat ii] or

=

However, if successive benefits are not correlated the coefficient of variation of the present value is

(50.2) =

Table 111-4 gives the ratio of the coefficients of variation of the present value relative to the coefficient of variation of the annual benefits for selected "life spans" of projects and discotmt rates. As is to be expected, the longer the life of the project and the lower the discount rate, the more "errors" of estimcltion in the true benefits compensate for each ot'iler. Qpite generally, Table;I11-4 shows that year to year variations in benefits, say, due 'to weather or other non-systematic erro~~ (in terms of persistence over time), are relatively quite unimportant in assessing the uncertainty about the present value (or rate of return). In general, the standard deviation of the present value derived from a give~ standard deviation of year to year correlatec;l annual benefits will be ap'proximately 3 to 4 times as large as when deriv~d from the same standard d~viat1_on of year to year uncorrelated benefits.

.. '

Table 111-4:

Number

of

" Years

10 20 30

- 31 -

Ratio of Coefficient of Variation of Present . Va~ue (CR) to Coefficient ~f Variation of Suc-

ces'sively Uncorrelated Annual Benefits, (CB) ,

(for a stream of equal mean annual benefits)

CR / CB Discount Rate

6% f lOi! J 20%

.32 .33 .34

.24 .25 .31

.20 .23

51. A co~fficient of variation as large as 0.5, for instance, in the annual benefi.ts deriving from year to year uncorrelated "errors" about an assumed estimated mean level of benefits would result in a coefficient of variation of approximately only 0.15. That is, a 95% confidence statement (approximately plus and minus 2 standard deviations, assuming a normal distribution), such as, that the annual benefits might be fluctuating with-in a range of a to 200, would reduce to a simi1ariy defined confidence state­ment that the present value of the stream of benefits will be within the r

range of 700 to 1,300. Whereas, if successive annual benefits were perfectly correlated, the corresponding confidence statement with respect to the present

. value of the stream of benefits would have given a range of 0 to 2,000.

52. For another illustration l'.lf the effect of correlation consider the data in Table 111-5. The benefits in this case are accruing over an 8-year period. The discount rate is assumed to be 8%. If successive values of the benefits are not correlated the variance of the present value is 1,187 and the standard deviation 34. Uncorre1ated benefits would be an appropriate assumption t if say, the source of the variability of outcome is a result of year to year changes in climate or prices which in turn are not correlated. If, however, contrariwise, the uncertainty is for instance the direct consequence of not knowing consumer reaction to a new product, the benefit can be assumed to be almost perfectly correlated. That is, in each of the 8 years benefits may be high or low, but if consumers respond unfavorably the benefits will be low in every year and if the new product is favorably received, the benefits will be high every year. Assuming per­fect correlation, the same assumptions about the variances of the benefits in each year will result in a variance of the present value of 9,149 or a standard deviation of 96.

"

\\

- 32 -

Table'111-5: HI~othetica1 Data for Calculating a Variance of the Present Value when Benefits from Suc-cessive Years are Uncorre1ated or 2erfect1I Correlated

t 2t (t) a a V(Bt ) J V(Bt ) VCR)

(1) (2) (3) (4) (5)

1 .93 .86 225 15 Uncorrelated 2 .86 .74 400 20 VCR) - 1,187 3 .79 .63 225 15 St. dev. a 34 4 .74 .54 200 14 Correlated 5 .68 .46 256 16 6 .63 .40 289 17 VCR) - 9,149 7 .58 .34 324 18 St.dev. - 96 8 .54 .29 361 19

53. That uncorrelated fluctuations in annual benefits are likely to be unimportant for the uncertainty about the present value (or rate of return on the project) is particularly apparent if one considers the simul­taneous contribution of ' the variances from successively correlated and uncorrelated benefits on the,variance of the present value of a stream of

\1 •

benefits, i.e., if the vari2nce is an additive function of the variance in the level of benefits and the variance due to year to year fluctuations such that

(53.1)

Correspondingly, the variance of the present value of such a stream of benefits is

(53.2)

54. For measuring the sensitivity of the variance of present value to variances from different sources it is useful to calculate elasticities, i.e., the percentage ch~nge in the variance of the present value to a one percent change in the variance of a contributing factor. 1/ When the factors are additive and not correlated, these elasticities add up to unity.

1/ The elasticity of VCR) with respect to V(et ) is as follows:

E - 1 V(e t ) <t at 2) .

!ill. 1 + La

2t V(et )

These elastici,ties for various interest rates and relative sizes of V(et ) are presented in Table 111-6. The .. e1asticity of VCR) with

respect to V(B} is: 1 - E V(et)·

- 33 -

i Table III-6 gives e1asticit,ies of the coefficient of variation of the present value with respect to the coefficient of v'ariation ofl~'he SUCCf!S­

sively uncorrelated benefits. Clearly, unless the project "life" is very short and the year to year fluctuations are very large relative to the un­certainty about the level of beo.efitel' the distribution of the present value is quite insensitive to the projected year to year uncorre1ated variations in benefits. Fpr an. illustration, consider an irrigation project whose "life" is ~O years and the discotmt rate is 10%. The average expected annual return is not known with any certainty but is estimated to be a normal dis­t'ribution with a \ mean of 100 and a standard deviation of 10. Furthermore, no matter what average level of returns will actually occur, there are ex­pected year to year fluctuations. If the standard deviation of these fluctuations is also 10, the total variance of the present value turns out to be about 30,800 (the mean of the distribution of present value is 850 and the standard deviation is 175). Neglecting the year to year uncorre­lated errors the variance would have been 28,960 2.nd the standard deviation 170. Hence, the uncorre1ated year to year variations "contribute" only 6% to the variance of the present value. (Note tqat the elasticity with respect to these fluctuations in Table III-6 is .06.)

Annual Benefits Contain an Uncertain Trend

55. < One case of strong serial correlation in connection with variances of annual benefits, is, of course, the presence of trend, i.e., where the uncertainty ,~bout benefits derived in a particular year arises primarily fr,?m not knowing the trend. Benefits may increase over time because a structure must be constructed to an excessive capacity which will be only increasingly utilized with time, or in general, because of an inc~ease in demand over time for the services generated by the investment. Benefits may decline because of prpgressive physical or technological obsolescence of the invest­ment. For'il1ustration, let us imagine th~t an annual benefit (Bt ) can be represented by the following equation:

(55.1) = B + bt + o

where B is the benefit in the initial year, b is the annual rate of change o

and, t is the number of years since the initial benefits have occured and e

t represents many random effects which can make an annual b0nefit deviate

from its normal trend. \\, ;

56.'\1 Given the growth-of-benefits equation (55.!) the formula for calculating/the mean and variance of present value (R and V(R», based on knowledge "~r.1B and V(B ), band V(b) and V(e) (e is assumed to be zero)

".' 0 0 are as follows:

(56.1) R - ( I: at) B + ( L: tat) b 0

and 2 2 . +(L (56.2) V(R) K ( L at) V(Bo) tat) L 2t .~ . V(b), + ( . j a ) . '

'\

V(~t)

Table I1I-6:

Duration of Benefits

(years)

10

20

30

10

20

30

IG

20

3C

- 34 -

Elastic:i ty of Variance of PreSent Value' of a Stream of Benefits with Respect to Variance of Successively Uncorrelated Benefits (when Bt = B + et)

Dis C 0 u n t Rat e

6% 10% 20%

(1) V(et) = V(B)

.093 .097 .098

.053 .060 .088

.ol~c .051

(2) V(E:1t) • 2 V(B)

.171 .177 .18c

.10C .1lh .163

.076 .097

(3} V(et ) = 3 V(B)

.236 .2LL /.' .2LS

.lL3 .162 .225

.111 .1)8

."

'. f , .

-.

- 35 -

" 57. On ~he basis of the above equations, it is first interesting to note the relative ,sensitivity of R to B , the initial level of benefits and

o to b, the growth rate. Here again ue use the elasticity concept, i.e., the percentage change in the mean present value in response to the percentage change in the mean growth rate, etc.. Obviously, this "sensitivity" depends on the relative ·size of the growth rate, the discount rate and the "life" of the project. Table lIi-7 gives the elasticity with respect to changes in b, the growth rate, for two levels of growth rates. The high rate corresponds with a doubling of the initial benefits in 10 years (i.e., b c io/lO) and the low rate implies a doubling of the initial benefits by

the 20th year (i.e., b = B /20). For instance, given a project "life" o of 10 years, ,a 10'01 growth rate and a 20% discount rate, a 10 percent change in the growt~ rate would change the present value only 1.7 percent, whereas with a project "life" of 30 years, a high growth rate and 3. 6% discount rate the present value would change by 5.3 percent. Conversely, in the first case a 10 percent change in the initial level of benefits, B , the present

o value would change by 8.3 percent and in the second case by only 4.7 percent. ('1'he two elasticities always add up to unity.)

58. Table 111-8 illustrates the order of magnitude of the relative sensitivity of the variance of present value with respect to var.iances in b, Band e.l/ Throughout the assumption is that the relative vari.ances

o -of B , Band e are the same, in particular~ it has been assumed that the

o coefficient of variations of Band bare 0.25. That is if B is 10 and o 0

b is' 1, the respective standard deviation is 2.5 and .25, and the stand­ard deviation of e t is assumed to be 2.5 as well. The high growth rate

11 The elasticities of the present value with Bo' band e, EV(Bo)t EV(b)' and Ev(e

t) are

•

respl!ct to the va,y;:lances 0 f as follows:

1

~ ~ L a2t l V(e t )

+ ( L

1 +

EV(b) •

1 +

and

E V(e t ) •

1 +

( L

( L ( E

~ L

t 2 a )

t 2 a )

tat )2

t 2 a)

( L: a 2t) ~

2 • V(B ) ( L at) V(B )

0 0

1

V(B ) ( L a2t) V(e~ 0 +

V(B) ( L: tat )2 V(b)

1

L 2 V~B l ~ tat}

w !ill. + V(et ) ( E a 2t) V(.~)

• I

/1

Project "tife"

10 years

20 years

30 years

le years

20 lears

30 lears

- 36 -

Table 111-7: Elasticity of the Mean Present Value (R) With Respect to Mean Growth Rate (b) and

Mean lni tit,1 Benefits (B )/1

Elasticity Discount Rate With

I Resnect

to: 6% 10%

1. High Grovlth Rate

b .33 .32

Bo .67 .68

b .~6 .Ld

Bo .5L .57

b .53 .L8

Bo .L7 .52

2. Low GrovJth Ra.tE

b .20 ·.19

Bo .8c .81

b .)0 .27

Eo ·70 .73

b .36 .32

Bo .6L .68

20%

.29

.71

.35

.65

.37

.63

.17

.83

.21

.79

.23

.77

The elasticity of R with respect ~o b • :1

(~ at) Bo 1 + . -( L: tat) b

,

,

Projec:'t "1if€t II

10 years

20 years

10 years

20·years

. "

- 37 -

Table 111-8: Elastici ties of the V~riance of Present V::Jlue ~with Respect to V(b), V(Bo) and V(e).

Elasticity of VCR)

Discount Rate

yd. th respect

6~~ to'; 10% 2C%

._------

1. High Rate of Growth

V(b) .19 .17 .13

V(B ) 0

.7L .75 .78

V(et ) .07 .08 .C9

V(b) .Ll .3L .21

V(B ) 0

.57 .61 .76

V(et) .02 .05 .C3

2. Loyr Rate of Orouth

V(b) .05 cr: • ;J

.CIII

V(Bo) .86 .B6 .57

V(et) .09 .C9 .09

V(b) .15 .12 .06

V(Bo) .81 .83 .85

V(et) .0LI .05 .09

.~----- .. ---- ,-..,-... j ,---....... -- ...... - .. - ... ~ ~~- .,

- 38 -

assumes a doubling of benefits every 10 years, the 10w growth rate a doubling of benefits every 20 years. The elasticities given in Table 111-8 illustrate that the variance is most sensitive to the variance of B •

o This is, of cotlirse, to be expected since an error in the level of benefits would have a c(:)nstant effect on the present value each year, whereas, an error in the g!~owth rate would have littl~ effect in the early years and a large effect. in subsequent years when due to discounting for time it matters much less. Particularly noteworthy is the relative lack of sensi­tivity of present value estimates to year to year uncorrelated errors. This means, for instance, that for the purpose of predicting the present value for a fertilizer project, one can predict quite precisely if one knows the yield resp!)n~e to fert:ilizer under average weather conditions and it· matters little that in some years the benefits will be much higher than-in others due to uncorrelated fluctuations in climatic conditions.

'~ife" of Proj ect

59. One source.of uncertainty may be the "life" oJ a project, say, because of silting of a reservoir it is not known whether its usefulness will be terminated after 10, 20 or 30 years. Quite generally, of course, concern about uncertainty from this source becomes less important when the discount rate is high and the earlies t date of termi'nation is· quite a long time in the future. If the discount rate is 10 percent, for instance, the present value of benefits is hardly affected by whether the "life" of the project is 30 or 50 years. .1

60. If the uncertainty about project "life" is high, the project "life" is relatively short and the discount rate fairly high, the usual practice of using the median "life" of the project in calculating benefits can,. in fact, be quite mislea.ding in terms of what is realistically' expec­ted to happen. To illustrate the problem, consider a project which re­sults in equal annual benefits of 1,000 dollars. The "life" of the project is assumed to be anywhere fr.om 10 to 30 years, any number of years within this range believed to be equally likely. If one would then calculate the gross benefits, assuming a 20 year "life" of project, the, present valu~ of the benefits is 8,514 dollars:" While the chances of a longer life and consequently higher benefits andi:a shorter life and lower benefits are 50-50, the expected gains would not be offset, however,'by the expected losses. In fact, the "true" expected present value, i.e.', R, is 8,253 dollars. Furthermore, the chance of realizing a present value which is 1,000 dollars less than the one projected based on the median "life" of the project is about 30 percent, whereas, the chance. of getting 1,000 dollars more is zero.

Estimating Probability Distributions of Annual Benefits

61. Attention in this section focuses on selected general problems in estimating the probability distribution of the b~nefits in each year by aggregating probability distributions of various 'factors. Besides being the, building blocks for e.stimating present value of ,the project, the probability distributions of the annual benl~fi ts may be of j,nteres t as such. A high

\

- 39 -

return in one year may not compensate fully for a low return in another year for many reasons although the average return is satisfactory. It is often assumed that unstable benefits from year to year are less desirable than a stream of b~nefits of equal annual magnitudes, because the cost of borrowing is higher than the rate of return on savings or because of diminishing marginal utility of income. It is not at all unlikely, however, that .a nation'may prefer a project which yields uilstnble returns over the years. COtlsider for instance a nation which must de(!ide between two projects which over the"years would produce an equal amount of food per dollar in­vested. It woula be quite in the national interest to prefer a project with a highly fluctuating fooa output, if this nation could count on getting food grants (or soft loans) in years when its domestic food supply is insufficient to ~eed the population, and if in food surplus years the surpluses can be effectively used for generating investment and economic growth.

, 62. Estimation of the annual benefits involves a careful accounting for many variables and relationships between variables. It is not possible to give any general rules for deciding how many variables and relationships should be explicitly considered. A probability appraisal may suggest sometimes consideration of additional variables and relationships than would be otherwise explicitly considered because it is easier sometimes to assess the probability distribution of a simple thiln that of a more compl~x event. However, generally it is sufficient to estimate the pro­bability distributions of the variables and parameters which one would consider implicitly without probability appraisal.

63. The major problem in estimating the aggregated probability dis-tribution of the annual benefits is once again correlation. Say, the. annual operating costs are estimated by summing costs of various input:s. The uncertainty about each cost item arises from not knowing exactly (a) what quantity of the input will be needed and (b) the price of the input. Both the quantities and the prices will often be partially cor­related. Only a careful examination of the data and a thorough under­standing of the underlying relationships can lead to realistic appraisal of the approximate extent of correlation.

64. It is not possible to give rules for estimating correlations, just as it 1.s impossible to gi.ve general rules for estimating probability distributions. The point to be made here is simply to suggest that the estimate.d and reported cost-benefit measure of a project should adequately reflect what the project appraisers believe to be their best judgment about the correlation between probability distributions as'wel1 as the nature of the pr.obability distributions themselves. Otherwise, a careful probability analysis may be· self-defeating, in fact be misleadi.ng. That is, if a pro­bability distribution of a complicated event is to be estimated by aggre­gating probability distributions, all correlations must be explicitly considered. Othe.rwise, the gains from explicit considerations of many probability distributions may be offset by having neglected correlations which would be implicitly taken into account by a less disaggregated

-analysis.

- 40 -

65. One often sums a large number of costs and benefits to derive the total net benefit. If one assumes no correlation when in fact the different costs and revenues are perfectly correlated. the true variance may be as much as n times the estimated variance when n benefits are sunnned up; i.e., when 2 benefits are a'ggregated the true variance may be 2 times the estimated variance when 10 revenues and costs are aggregated the true variance may be 10 times the estimated variance. When many correlations are negative, the true variance may be zero or much'l~ss than the estimated variance. 1/

7. Summary and Conclusions on Aggregation of Proba~i1ity Distributions

66. (a) Judgments about the likelihoods of various outcomes

1/

of the elementary events affecting the costs and benefits from a project can be combined without great difficulty into an estimate of the approxi­mate probability distribution of the present value or the internal rate of return.

(b) There are available b~3ically two approaches for aggregating probability distributions into an estimate of the probability distribution of returns from a project: The simulation of a sample of c;ut­comes based on randomly selected sets of observations from the "input" probability distributions witn the aid of a high-speed computer and the mathematical approach of aggregating mathematical expectations into an estimate of the mean and variance of a project's returns.

(c) For most operational purposes the simulation procedure is, the only feasible method at this time. The ~imulation procedure req~ires no prior assumptions about the nature of ,the aggre­gated probability distribution and no mathematica~

Assuming B == BI + B2 + ... + B n

and B2 • a2 BI , .... , B - a B1 n n

then the true variance of B is

a )2 :

V(B) :a (1 + a2 + + V(~~) n

whereas, if it was asslDIled that [BI~ B21, •••• [B1 , Bn1 are not

correlated the va~iance of U would have been.cal~ulated as

V(B) == (1 + + .... Note that V(B)/V(B) is at maximum when a i • 1 •

- 41 -

skills on the part of project appraisers. The major disadvantage of the procedure is the fact that resulting estimates are specific to one set of "input" distributions.

I

Cd),. f A maj'or advantage of the mathematical approach elaborated in this chapter is that the mean and variance (or standard deviation) calculations

(e)

(f)

(g)

(h)

can be made quickly for relatively simple projects when.only a crude estimate of a project's returns based on a few crucial variables is needed or feasible. The method is useful for deriving gene­ralizations about the sensitivity of estimates to various elementary events.

I

Ye~r to year uncorre:.ated fluctuations are of re-lative minor importance in terms of the probability distribution of overall project benefits. Instead, one should concentrate on making sure that uncertainty about the persistent level of costs and benefits is properly taken into account.

Generally, the problem of estimating the extent of correlation between the various probability

. distributions of the elementary events is likely to prove the most difficult aspect in application of the procedure. When one estimates directly the probability distribution of-a complicated event (like the r~te of return) from observing past data or a combination of various judgments, correlations are implicitly reflected. If such a distribution is estimated, however, from synthesizing the effects of various contributing factors, correlations must be made explicit and taken into account. Other­wise, the gains from disaggregation must be seriously questioned •

. Properly done, explicit consideration of the pro­bability distributions of various events is not only important for estimating the probabilities of other than the average or most likely returns. The usual practice of aggregating averages or most probable values of a series of events may in fact not yield a correct estimate of the average or most likely returns of a project as intended.

Just as any quantification of economic relationship is generally only a question of more or less rather than yes or no, so the optimum extent of quantifi­cation of the uncertain elements in project appraisal

- 42 -

should differ from case to case. In .ddition"to the usual considerations for dete~irtlHg th~ '~ptimum amount of study underlying a project lppraisal; the importance of quantifying uncertainty would also depend on the specific decision context. If the project. is one of many unrelated projects it may be less important to quantify uncertainty than when the project is very large relative to a country's inves­ment program.

· .

".

- 43 -

ANNEX TO CHAPTER. III

A HYPOTHETICAL EXAMPLE

1. To illustrate the data requirements, the procedure and the re­sUlting information which is involved is a quantitative appraisal of the uncertainty of returns, consider the following hypothetical case. The project may be thought of as an irrigation project where the present value of the returns is a function of the cost of establishing a facility and n an~u~l equal net returns thereafter. The annual net returns consist of reven'ues obta.ined from the increased production of three crops. Both, the number ,of acres and the yield per acre in crop 1 largely depend on wages farmers can earn in an alternative employment. The price of commodity 2 is assumed to be negatively correlated with the output of crop 2 by the project. Prices of co~odities 1 and 3 are not affected by the project's output. First, the model for calculating present value of benefits is presented. The same model would be used whether one does a conventional appraisal or a probability appraisal. It is merely a systematic, explicit statement of how one might ao about estim~ting the outcome of a variable based on knowledge about cer­i:ain parameter;; and other variables. Next the probability distributions of parameters' which cannot be predicted to have a specific value with certainty are present~d. Finally, the results of a conventional appraisal and a pro­bability appraisal are presented arid contrasted.

The Model

2.

. .

(1) Acres in production of crop I (A) are a function of wage in an alternative employment (W).

(2) Yield per acre of crop I (Y) is a function of wage

(3)

(4)

in alternative employment (W) •

Production of crop 1 (Xl)

is acres (A) times yield (Y) and a random effect (el ).

Gross revenue from crop 1(51)

is price (Zl) is times pro-

duction (Xl).

(5) Price of crop 2 (Z2) is a

function of slope coeffi­cient (b), production of crop 2 (Xi) and a random

effect (e2).

(A) = 10 (W)

(Y) • 10 2(W)

• (A) (Y) + (el

)

•

- 44 -

(6) Gross revenue from crop 2 (S2) is price (Z2) times pro-

duction (X2).

(7) Gross revenue from crop 3 (53)

is price (Z3) times pro-

duction (X3).

(8) Annual net benefit (B) is a

function of gross revenues

(SI' 82 and 83) minus annual

cost.

(9) The sum of the discounting

factors ( L, at) is a functio~ of the "life" of the invest­

ment (n).

(10) Present value (R) is a function

of the initial investment (Bl ),

the sum of the discount factors

at ( l: at) and the annual

benefits (B). 1/ Probability Distribution of the "Inputs"

'J,.

3. The 10 "inputs" needed for es tima ting the model 'are: wage (W), prices (Zl) and (Z3i" production (X2) and (X3), and th~a initial i.nvestment

cost (B,.), the "life" of the investment (n) a price-output coefficient (b) .1.' •

and the two random effects (el ) and (e2). Let us now assume that. the". proba-

bility distributions of these "inputs" are believed to be as follows:

!I For internal rate of return calculations, the equation is 0 m -(B1) +

(1 + r)-t (B) and r is the vari~ble to be derived.

- 45 -

"Inputs tl Outcome Probability "Inputs" Outcome Probability

1 .30 1,600 .25 2 .40 Bl ($) 2,000 .50 3 .30 2,400 .25

W ($)

2.4 .33 5 .33 3.0 .33 n (years) 10 .33 3.6 .33 15 .33

3.5 .20 .06 .30 -5.0 .60 b .10 .40 6.5 .20 .14 .30

30 .33 -10 .30 50 .33 e l (ton) 0 .40 70 .33 +10 .30

14 .33 0.4 .30 20 .33 e2 ($) 0 .40 26 .33 0.4 .30

X.l (ton)

"Conventional" Appraisal of Present Value of Benefits

5. The "conventional" procedure for .esti.mating the present value of the benefits (or alternatively the internal rate of return) is to use "best' estimates" of the "inputs" and to calculate in sequence each of the 10 equations in the "model". Alternatively, to be on the "safe" side, one may use "conservative" estimates for the inputs in the cost-benefit calculations.

6. The conventional appraisal procedure would have resulted in an estimate of the present value of the discounted benefits of $845 (or a rate of return of about 16 percent) assuming the means of the probability distributions of the "inputs" represent their best estimates. Assuming "conservative" estim~\tes for each of the inputs are represented by 10 percent decreases in "inputs" Zl' Z3' X2 ' X3 and n, 10 percent increases

in W, Band b, -5 for el and - .2 for e2 , the present value would have

been estimated as $ - 114 (or a rate of return of about 6.5 percent). Next'we see how these estimates compare with estimates obtained by proba­bility analysis.

. n

- 46 -

Appraisal by Probability Analysis

7. The same model and "input" distributions were used to derive,~ simulated distribution of present value and the internal rate of return. 11 The dots in Figures III A-I and III A-2 represent relative frequencies. of present value and internal rate of return estimates, respectively, based on 200 random sets of drawing from the "input" distributions. The resulting distributions are, of course, specific to the particular sample of 200 sets, of drawings, though it is generally ass\IIled that with large enough samples,' chances are high that the sample distribution nearly approximates the "true" distribution. Noteworthy is the fact that the expected (mean) presented value is $508 andi:'~?9nsiderably less than estimated by the "conventional" analysis. ~/ 'Similarly, the mean internal rate of return is 12.4 percent as compared with almost 16 percent by conventional,analysis. It is, of course, also easy to give "conser"ative" estimates by merely glancing at the figu~es. For instance, chances of getting less than a present value of zero, or correspondingly, a rate of return of 8% is about 26%. Using a full probability appraisal it is clearly possible to derive "conservative estimates which have a well-defined interpretation and can be evaluated in terms of alternatives. Countrariwise, the "conventional" con­servative estimate is not a clearly defined entity in terms of the final benefit-cost measure.

!I The calculations were performed on a computer by courtesy of McKinsey E, Company, Inc., utilizing their general program for risk analy,;is.

I

II The exact mean present value solving the model by applying mathema~ tical. expectations, is $574. The difference in the mean present value

I when calc~lated "conventionally" and by proba~ility appraisal is entirely due to non-linear functional relations.

"

- 48 -

IV. PROJECT DECISIONS UNDER UNCERTAtNTY.

1. Introduction

1. After having shown how to ipcorporate uncertainty into the appraisal of projects and portions of project, we .hould consider how and to what extent such appraisals might be useful for the evaluation of public projects, i.e., for decisions. As indicated ear11.er, outcomes of decisions which are affected by the interaction of a great many noncontrol­lable variables are most adequately described in terms of probability dis­tribution and sometimes quite adequately by means and variances. 1/ For this reason, and because the objective of this paper is to focus primarily on the appraisal of large scale publicly financed projects, only decision theories which relate to choices between probability distributions o~ out­comes (specifically, means and variances) are discussed here.

2. Quite apart from the decision problems encountered in the general evaluation of projects, there are certain decisions which can be only in­telligently evaluated if one explicitly studies probab~lity distributions. These are problems of sequential decisions and certain timing aspects of investments and the whole subject of the value of gathering addit1.onal information. These subjects will be also briefly discussed below.

3. The first reason for making such appraisals as indicated in the previous chapter, is of course, quite intuitively, that decision makers want to know the most likely or average outcome as well as the e~tent of the riskiness of a project and a probability distribution or the mean and variance are convenient indices. The fact that decision makers ask for this kind of information is sufficient evidence that decisions are affected by it. If this information is not supplied by those who are in a position to know best, second-rate information may be substituted. On the other hand, if instead of supplying means and variances, the appraiser himself adjusts the estimate of the outcome of a project by some ~ncertainty factor, the organization is not unlikely to reach inves~ent decisions which are inconsistent and misguided in terms of higher level policy objectives.

4. Any theory of economic choice involves the following basic ingre-dients: (1) the decision agent's preference function, (2) his basic endowment and (3) alternative courses of action with specified consequence,. The interaction between these three aspects of a decision problem can be . best illustrated by a simple hypothetical example. •

2. A Hypothetical Example of the De~ision Problem

5. Consider an individual is told that be has been selected to receive a prize, and that it is his prerogative to choose between several options.

~I The characterization of probability distributions in terms of means and variances will be seen to be sufficient for s~~ types of distributions, specificallywhe~ a normal distribution is approximated and regardless of normality, if the decision maker wi~hes to maximi~e expected utility and the utility function of income is quadratic.

..

- 49 -

Assume furthermore, that the same individual has zero wealth and that it is two days before he will'receive his next monthly salary. By option (A) he will receive 10 dollars at once. Options (B) and (C) will each consist of two possible outcomes, to be decided upon by the tossing of a fair coin. By option (B) he would have a 50-50 chance of getting 5 or 15 dollars, r~spectively. Option (C) would vield him 0 or 20 dollars, each with a 50-50 chance. The need for survival w~uld pretty much rule out the latter option. The recipient's' choice between options (A) and (B) would depend on his subjective preference. If he likes to gamble, or expressed in another way, 'if his marginal utility is an increasing function of income, he will choose. option (B).

6. Consider now a second, somewhat different situation in l-1hich the same individual is told that his opti.ons are to receive (A) a prize of 1,000 dollars at once, (B) 990 dollars at once plus a 50-50 chance of 5 or 15 dollars and (C) 990 dollars at once plus a 50-50 chance of 0 or 20 dollars, respectively. Clearly, one distinguishing characteristic between the first and the second situation is the fact that the decision problem appears to be a trivial one in the latter case by comparison. Yet, note that in both instances, the course of action and their conse­quences are identical: 10 dollars for certain vs. 5 and 15, or a to 20 with a 50-50 chance respectively. The difference, comes, of course, in the endowment, i.e., the variability of outcomes relative to the total outcomes is small even for the option involving the largest gamble. Hence, it is quite conceivable that the same individual who prefers (A) or. (B) to (e) in the first case, will this time prefer option (C) to either (A) or (B).

7. In a third case, we might conceive of the same individual to be confronted with having to choose between (A) receipt of 1,000 dollars at once, (B) a 50-50 chance of receiving 500 uollars or 1,500 dollars, and (e) a 50-50 chance at receiving nothing or 2,000 dollars. The endo,~ent and the variability relative to the average of the respective outcomes in this situation are approximately the same as in the first cited case. However, it is not at all unlikely that the decision would not be the same as in the first case. In fact, we probably would not be too surprised if the same individual who would have chosen options (B) in the first case would pr~fer (A) in this case. Tbi,s, means that an individual's preference function is not a given datum at all but may vary with the magnitudes of the options.

8. Finally, we may conceive of a fourth situation in which the same individual is given the option of receiving 1,000 dollars at once, or alternatively to participate in a game of chance in which his payoff is to be determined by tossing a coin one hundred times., Under option (B) he would receive in each·toss either 6 or 16 dollars. Option (C) would pay him 0 to 24 dollars. Clearly, these uncertainty options are different from the ones described in the first three cases. The total payoffs are likely to approximate one hundred times the average payoff, i.e., 1»100 and 1,200 dollars for options (B) and (C), respectively. Since the total payoff:for option (B) can be expected to approximate a normal probability distribution with a standard deviation of about 50, we would have in fact a 98 percent confidence that the payoff from (B) would be greater than

','

- 50 -

1,000 dollars. Except for an extremely conservative individual, it is, therefore, quite clear that (B) would be preferred to (A) and'most likely (C) would be considered still more desirable than (B). 11 As a generali­zation of this case, it may be stated that if plans of action are,compared whose outcomes are the sum total of many small uncertain outcomes, the decision may be based on a comparison of the expected outcome from each plan, and the same optimization rules which apply to ordering courses of action with certain outcomes are relevant for this particular kind of uncertainty problem.

3¥ Project Appraisal and Utility Theory

9. To gain some perspective on the nature of economic decision problems under uncertainty it is worthwhile to distinguish first of all between the decision problem of an individual and of society in a de terminis tic l-lorld, and then to superimpose the problem 0 f uncertain consequences from alte~ative courses of action.

IOn Very generally, the objective of economic activity may be re-garded as one of maximizing utility or welfare. Utility need not be a measurable object in order to derive meaningful criterion for an indivi­dual's economic decisions. Instead, it suffices to assume .that utility is an increasing function of income to see that profit max~mization is consistent with utility maximization. In a world of certainty, planning for an individual would in fact involve no decision problem at all. The investment appraiser would examine the income (or returns) from alterna­tive course of action and, presumably, the one with the highest returns would be chosen.

11. Planning fO,.r society is more ambiguous, even in a deterministic world. At best, it can be said that given the same minimal asslDIlptions ab"ut utility (being an increasing function for each member of society), so(!iety l-lould be potentiallY. better off by maximizing income (Pareto Optimum). That is, if we abstract from possible interdependence of utility and if the gainers from a particular course of action were actually to com­pensate the losers, social utility would be maximized by maximizing overall

1/ Note that:

and

Similarly,

and

E(Bi

) ~ (.5) (6) + (.5) (16) c 11 2 2

V~r (Bi

) = .5 (6 - 11) +.5 (16 - 11) a 25

E(B) = 100 E(Bi

) ~ (100) (11) = 1,100

Var (B) := 100 Var (Bi ) • (110) (25) - 2,500

Standard dp-viation (B) a 50 E(C

t) = (.~) (0) + (.5) (24) - 12 . 2 2

Var (C.) = .5 (0 - 12) +.5 (24 - 12) ,'= 144 1

E(C) = (100) (12) = 1,200 Var (C) = 100 (144) = 14,400 Standard deviation (C) -= 120.

- 51 -

income. !.I In reality, compensation does not usuaIly take place. Hencf~, public investment decisi~ns usually require a bala:ncing of preferences for a certain inc~e distribution and maximizing overall income.

12. 'I In dealing with economic choices involving uncertain outcomes, it iS'necessary to know something more about an individual's utility func­tion. Presumably, it is possible to determine an f,~ndividual' s utility function with respect to uncertain incomes. To be sure, such a utility function measures relative utility and not absolute utility; that is, the scale on which utility is measured has no known origin. 1/ By this theory, the objective of the individual is to maximize expected utility. If an individual's utility function is linear, expected uti~ity is maximized whenever he chooses the course of action which maximizes his expected in­come. Othe~iise, when the utility function is curvilinear, it is neces­sary to know the probability distributions of income from alternative courses of action in order to ascertain which course maximizes expected utility. If the utility function can be approxi~ated (in the relevant range) by a quadratic function, expected utility is a function of the mean and variance of income, i.e., if

(12.1) u + all + 2

= a a21 0

then

(12.2) E(U) = a + al E(I) + u2 (V(l) + (E(I)2) 0

~here U is utility and I is income, and E(l) and V(I) are the mean and the variance of the probability distribution of income, respectively.

13. The risk averter or "conservative'· individual t\.'ould have a utility function tolhich increases at a decreasing rate, such as a < 0 and, hence, o the expected utility is a decreasing function of the variance (the larger the uncertain~y, the lower the expected utility). On the other hand, the risk taker or gambler's utility function increases at an increasing rate and, hence, his expected utility is an increasing function of the variance of income.

11 Conversely, if utility could be actually measured (not only ordinari.ly in terms of relative preferences) it would be possible to derive a social utility function. However, most economists would agree that the notion of measuring absolute utility of individuals, quite a part of the impossible task of aggregating, is quite obsolete.

lJ' This is the most widely accepted theory about man's behaviour in economic risk situations since von Neuman and Morgenstern published their Theory of Games and Economic Behaviour. For a very lucid and non-mathematical exposition of the natu~e of the theory and how to measure cardinal uti­lity functions the reader i~ refer~ed to Ralph Oa SwaIm's paper, "Utility Theot;'y":'" Insights into Rislt 'taking''', Harvard Business Review, Nov. -Dec. 1966. "

- 52 -

14. One inunediate consequence of the expected utility theory is, of course, that decision makers do need to or ought to know means. and variances (if not complete probability distributions) of returns from alternative in­vestments. Beyond this, however, whether the utility function of an indi'l1idu~ll can ever be adequately measured and whether the individual can be observed to behave consistently with that function so as to make this a predictive or' explanatory theory is a ques tion which is beyond the scope of this paper '. Furthermore, we also do not concern ourselves here with the question of which utility functions are or are not compatible with the stati~ and dynamic theory of a free enterprise system.

15. Before we attempt to summarize the implications of the appraisal and evaluation of proje\.ts subject to uncertainties in the context of public decisions, it is interesting to note, however, how the expected uti­lity approach has been evaluated in the context of large US corporations. In a recent study, R. o. Swalm 1/ estimated segments of utility functions by quizzing one hundred executives, of which a large number belonged to one corporation. The participating executives were asked to make choices (between courses of action which would lead to out~omes with different degrees of riskiness) in their capacities as corporate decision makers, not as private individuals dealing with their own funds. Furthermore, they were asked to tell what they would actually do and not what. they feel they ought !Q. do. Briefly, the most significant findings were:

(a) The measured utility functions reveal widely different attitudes towards risk taking (even for the executives making decisions for one and the same corporation).

(b) The sample revealed differences in utility functions ranging from one which would characterize an extremely conservative indi­vidual to one 'to/hich would be characteristic ' of a gambler.

(c) Nost respondents tended to be extremely con­servative when chances for negative returns were involved.

(d) The executives' utility functions seem to have been more closely related to the amounts with which the respondents were accustomed to deal with as individuals than to the fin­ancial position of the company.

From these major findings the author concludes that:

~I R. o. SwaIm, Ibid.

- 53 -

II Businessmen do not attempt to optimize the expected dollar outcome in risk situations involving what, to them, are large amounts •

••••• Cardinal utility theory offers a reasonable basis for judging the internal consistency of a series of decisions •••••

•••• The theory offers a relatively simple way of classifying many types of industrial decision makers. For example, a supervisor may learn that, in decisions involving significant risks, one man tends to be quite conservative, a second tends to be a gambler, and a third tends to be moderately conservative. If he is moderately conservative himself, he will be hap­pier delegating decisions to the third than to either of the other two •

••••• The action a junior executive recommends in a risk situation is a function of his own 'planning hori~on' ••••• rather than to the financial condition and position of his company ••••

~

• • • •• If the decision makers intervieweeL are at all representative of u.s. executives in general, our managers are surely not the takers of risk so often alluded to in the classical defense of the capitalistic system. Rather than seekj.ng risks, they shun them, consistently refusing to recommend risks that, from the overall company viewpoint, would almost surely be attrac­tive ••••• "

16. What, if any, are the implications of all this for project appraisal and evaluation of public decisions in general? Basically, the problems are simil~r. The only difference, an important one to be sure, is the additional difficulty of defining and measuring the utility function of society. Even ifsomeho,w wf.';' >'J·j\I'~d know how to make interpersonal utility comparisons, that 'is even \~t 'fr.re would know the weights to be used in aggregating utili­ties, there rel1Hd~ns the problem that cardinal utility as proposed by von Neumann and Morgenstern is a relative rather than absolute measure of utility and, hence, can not be additive. There is, therefore, no way of deriving the shape of society's utility function from the utility functions of individuals. One way of ascertaining a utility function which might re-

t flect at least the wishes of the controlling segment of the population (be it the majority or a minority) may be through trial and error and the political process. That is, the decision makers in government may learn from experience whether the public's disapproval of projects which have disappointed is more or less intensely offset by the pub1ic's applause for projects which have succeeded beyond expectations.

11'." . The slow disappearance of risk takers and its consequences for the growth potential of the capitalistic system may he a matter of concern with respect to govenunental as well as corporate decisions to the extent

- 54 -

that deci\~ions are made by lower echelon who might not be sufficiently ~e­warded for-unusually successful projects but are reprimanded for-.failures. In this context, it is worthwhile to remember, however, that whether an activity involving risk is undertaken or shelved depends not only:on the attitude of the decision maker but depends equally on whether or not 'the risks of many independent activities can be pooled. The best example is, of course, in the field of research and development activities. If ~n individual's payoff was to depend on whether he succeeds inventing a'process to copy nature's photosynthesis, even a gambling spirit might be discouraged from undertaking an activity with such a high chance of failure. Contrari­wise, a research laboratory manager with a much more conservative attitude towards risk might not hesitate to employ the same individual together with 1,000 other research workers assigned to other projects, because the chance of the whole oper.ation to fail becomes so much smaller.

18. The important thing here is that the riskiness of the project· should be evaluated at the top level (together with other projects), and that, to the extent that the decision process must be diffused, the lowe~ echelon decision makers should not be unduly reprimanded for failures and should not fail to be applauded and rewarded for successful outcomes. In any case, to appraise attitudes as well as the performance of an orga'oization, it would seem to be of vital importance to know the riskiness of various ,projects and also the specific sources of the risks in order to make at least: an informed judgment about the likely joint outcomes from a total investment program.

4. Public Investment Decisions

19. Fortunately for those having to advise on the selection of govern-ment financed projects (and the author of this paper) the kind of uncertainty decisions frequently encountered by governments are those described by the second and fourth case in the hypothetical example presented earlier. A government's investment plan consists frequently o~ many diverse projects with uncorrelated risks and as long as the objective is to maximize the sum of the benefltr, neither the risk preference function nor the ex~ct probability distributions of the outcomes from each project need to ~e known. As long as the utility is an increasing function of benefits, 'the government will want to choose the projects with the highest expected benefits. Iq these cases, compensating risks can be relied upon to assure that the e~­pected benefits will be approximately realized.

20. A government's investment plan may be also frequently viewed ana-logous with the case of a ~mall gamble offered to a well endowed individual. In this case, it matters not how large the variances of alternative projects are relative to their expected valuee., but rather it is the variance of al­ternative projects relative to the tQtal revenue of the nation which matters.­Since benefits from projects are likely to result in only a small net addi­tion' to the total revenue, even fairly large differences in the variances from various projects are not likely to be important considerations when viewed relatively to the total revenues, even if the risks are not com­pensated elsewh.~re in the economy.

- 55 -

21. Unfortunately, on more careful examination, the uncertainty issue involved in project appraisal is far from adequately represented by the two types of gambling situations described in the previous paragraphs. Let us then take up one by one the kind of project decis1.ons which must be appraised in terms of probability distributions of outcomes and preference functions of the decision agents, ot else, .at least require consideration of the prob­ability distributions of selected variables affect:ing the expected outcome of a project.

22.,. Large Specialized Projects or Aggregates of Smaller Projects whose Ou\:,comes are Highly Correlated. A plan which consists of a few large proj­ectsiS'not necessarily more uncertain in its consequences than a plan consisting of many small projects, except if the consequences of the large projects are fairly uncertain and the outcomes of the small projects are uncorrelated. The size of the project by itself is no indication of the gnticivated variability of outcomes. It is rather the degree of speciali-zation which is likely to increase its riskiness. A large infra-structure :I

project, for instance, whose benefits depend on many independent variables is more likely to 'realize the expected outcome (if properly estimated) than a highly specialized industry, whose benefits might depend almost entirely on an uncertain export market for a particular commodity. An infra-structure project may, of course, be affected by many variables which are highly cor-related, say the demand for different commodities which all depend in turn primarily on GNP, and its projected benefits would then be nearly as un-certain as the forecast of GNP. Viewed then in terms of the total benefits from an aggregation of public projects and assuming that at least when it comes to the evaluation of outcomes for an entire investment plan the pref-erence is to avert risk, caution must be exercised to select projects which not sO much have a small variance by themselves, but rather, whose outcomes are not highly correlated. In other words, one should seek not so much to avoid risky projects as to seek out compensating risks. A risk averting

1 preference, of course, merely means that the decision maker is willing to give up some expected income for a smaller variance of income, where the terms of this trade depend on the decision maker's subjective preference. What interests us for project appraisal is the fact that the evaluation of a project within an overall plan in terms of riskiness requires knowledge not only of the variance of that project but more importantly of the sources of the variance, ~lhich pe~its those who are charged with evalu­ating the variance of the total investment plan to calculate the covariance among various project alternatives.

23. The Impact of Failures and Succ~sses of a Project not Compensated by Failures and Successes of Other Projects. Perhaps the most important reason for having to worry about uncertain outcomes from a project is the fact that the expected utility derived from a government's total investment plan is not merely a function of total income, but rather is an aggregation

- 56 -

of the expected utilities derived from each projecti 1/ Hence, the compen­sation for other than expected outcomes fro~ll a project, whether through compensating errors or the fact that a project mereiy supplements a large income (or wealth) from other sources, can but shou1d not be taken for granted. Some reasons why compensation may at least be incomplete, i.e., income may not be additive are:

(a) The value of additional income may not be the same for different incividuals or regions and projects typically affect different individuals and regions.

(b) Failure o·f individual projects may have psychological and political impacts which are not compensated by unusual successes of other projects and vice versa.

(c) Projects may have to meet specific supply targe~s which due to various capital or foreign exchange restrictions can not be reflected additively when distributions of returns from various investments are aggregated. For the same reason variability o'f returns from year to year may be a consideration in evaluating the expected utility of a partic­ular project. 1/

24. There are then at least some uncertainty environment and decision problems for which it i;; sufficient to know the expected (average) outcome of each of the alternative courses of action to choose between them. In such cases, the preference function for or against risk'of the decision maker is at best of minc)r significance. However, for the evaluation of many projects the decision might well depend on the variance of the re­tur~sc:"!41fcft:he attitude towards risk.

11 Expected utility is viewed here as the utility assigned to a certain probability distribut~on of income. (This ,concept merely implies ~hat preferences can be expressed in the sense descr~,~ed by Van Neuman and Morgenstern and does not require measurability of utility). Since only the variance of total:income approaches zero, the compensation principle required income to be additive, regardless from which project it is de­rived, or in other words, the utility of income must be the same regarp­less of where the income is earned.

'<",

/1/ Some of this kind uf variability of outcome can be adequately reflected ff in calculating expected re~urns and does not need to be analyzed in

terms of subjectiye preferences. For instance, a sugar project which is designed to supply the raw material for a factory with a fixed capacity will yield a smaller expected return if sugar yields vary greatly from year to yea; than if yields are stable, since higher than normal yields can not be processed. and, henc~, do not compensate for lower than normal ~lelds.

• Ij>'

/ / if

•

"

- 57 -

50 Selected Decision Problems

25. A more.specific use of gathering risk information (based on a , -2riori or subjec·tive' probabilities) is for evaluat:ing activities designed to eliminate or 'reduce the chances of unfavorable results. One such acti-· vity might be"the gathering of more information, another the timing of investments or scheduling of various activities, and still another is the specific consideration of sequentially related outcomes. Much re­search is currently under way on all of these problems in the operations research and decision theory fields and a comprehensive revie't~ would be beyond the scope of this report. However, by way of some simple illus­trations, we may gain some appreciation of the kind of problems and methods of solutions suggested by these theories.

Value of Information

26. Consider firs t a simple problem in which 't~e ask whether or not an investment should be held up until further information can be gathered. Basically, this is an expected benefit-cost analy~is of the information gathering activity, where the expected benefits a're detenuined on the 'basis of an ~ priori estimated probability function. What are the expected bene­fits? Assume that a project has been appraised in tenus of a probability distribution of the present value of net benefits as outlined in Chapter III, and graphically shown in Figure IV-I. Now assume that on the basis of this probability distribution, the decision would be to go ahead with the invest­ment. Assume furthermore, that a consultants' firm suggests to us that for a (probably high), cost, c, it can gather sufficient intelligence to tell us in fact precisely (highly unlikely!) what the present value from this investment will be. If the precise estimate turns out to be greater than 0, we would still go ahead with the investment and the gain from having

.employed the consultants' firm would be zero. If, however, the precise estimate of the present value turns out to be less than zero, we would not undertake the investment and we would be spared a loss due to the consult­ants' activity. On the basis of the probability distribution, feR), the expected benefit of the information gathering activity (avoidance of losses due to the WTong decision) would equal the weighted are. under the probabiiity curve to the left of R • 0, or '.

(26.1) E(Benefit) .. Lf(R) R dR'11

1/ The' approximate benefits can be evaluated, of course, quite quickly by approximating the area under the probability distribution and neg­lecting the area to the left of some R which has an extremely small chance of occurrence. Furthermore, the probability distribution need not be, of course, a no~a1 distribution.

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f(R)

(V(R»1/2

o E(R) Present.Value (R)

Figure IV-I: Probability Distribution of Present Value

If indeed we believed that the consultants can give us a precise estimate, we would employ them only if the expected benefits exceed their cost, i.e., if E(B) > C. Of course, it is quite likely that the gathering of additional information activity will take time, which iS,then ap additional cost item to be taken into consideration.

27. For another example, consider that our organization is quite conservative in its attitude towards risk and that on the basis of the same probability distribution presented in Figure IV-I, the decision would have been not to go ahead ,.,ith the investment. It is quite intuitively clear that the expected value of getting a precise (certain) estimate of the present value of net benefits is greater for this org~nization than in the first case. The expected benefits of the info~ation gathering activi.ty (avoidance of .nissing the opportwtity for, a profit due to the wr.ong decision) equals .now the weighted area under the probability curve to the right of R = 0, i.e.,

(27.1) E(Benefits) - [OOf(R) I. dR

Ie

From this simple illustration, it is seen that both the probability distri­bution and the organization's attitude to'~ards risk (i.e., the del::ision function) would affect the value of addi~i0t:lal i.nformation giving a precise

- 59 -

and certain forecast of R. Usually, additional information will simply lead to a more precise,-but still uncertain estimate. Hence, (26.1) or (27.1) represent rather the'maximum (or most optimistic) estimate of the benefits from the information gathering activity.

Alternat~ive Project Strategies

28. More typically, project appraisal involves weighing of alternative strategies with respect to particular project asp(~cts: how many roads are

\-,~\o be paved, how many bridges, how many farmers a're to receive fertilizer, how much water per acre, etc. Once having made explicit how various ele­ments interact to yield a measure of the e.conomic or financlal returns of the project it is usually quite easy to calculate the projec.t's performance with alternative strategies. Consider, for instance, the probability distri­butions of the present value of net benefits with project alternatives (A), (B) and (C) shown in Figure IV-2. (A) is the "all-inclusive" project pro­posal. In (B) and ce) project coverage has been reduced to exclude primarily some activities w'ith highly uncertain results. If tole were very unwilling to take a chance of getting a negative result, project alternative (B) may be more attractive than (A) in spite of a lower expected (mean) present value. However,alternative (e) may be ruled out completely ,because not only is the mean present value lower than for (A) and (B), but also the chances for

'getting a negative present value are higher than for (A) and (B).

Time Related Problems

29. Another specific application of the probabilistic approach is to time related problems, that is, where the results of a particular course of action in one period depend on uncertain events in another period or when a future course of action will be determined by the realization of an uncertain event in an earlier period. Such problems particularly arise in planning of optimtnn storage facilities. Related, however somewhat different, is the general problem of sequential scheduling of investments when future benefits are uncertain.

30. The problem can be illustrated by the following highly simpli-fied example. Consider, that the object is to build a grain storage facility and we wish to compare the expected net benefits. The benefits from having a storp,g,e facility will depend, of course, on the size of the grain harvest in a particular year (which in turn may be largely detennined by climatic conditions). Let us asstmle that the p,robability distribution of the amount of storable grain is:

Probability

1/3 1/3 1/3

Grain for Storp~

2 1 o

That is, ,in some years grain production '''ill be too low to leave any surpluses to store. Just, to simplify the problem asswne that the planning horizon is only two periods and for some reason the grain can

r : 1

1

~ - • '1'

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..:-.. ~.l_~._ :' .. 4 t .. ~ t

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• I t 7 .. ,;. .

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~_.f""",,- "' __ . _: ; __ "~*_. ; . </ '

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t r ~

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: ., i l .i ; , "I" "

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60 t .l .. I, i' .-

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~:

.: • 1

,,"'1-.... ; _~,_ ... ,.l' ... '"

: ,I; . I

~ f J t '

'O ......... _."" ..... ..,'t' -- ...... -

; ; ~ r f • t , t f . ,

, : 1 1 • I

~ 1

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t •

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... ,~J

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r ! I

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I 1 .. : , I-; ! r ! ... ~ ~ J~

:~-~- -:. ~ --.-t-~~ t 'I If

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t f

'f : I '! 1· : 1 ~'"

l' , • t ! . . I. ; ~ . j.1 - . T-t-~--t--J. -1 •• , .• ~ ..;.-:._.l....!..~.'

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- 61 -

be sold only at the end of the t"'0 periods. Firs t, we wish to determine the expected amount of grain stored for different sizes of storehouses, since the storage capacity and the supply of grain determine the expected amount of grain stored.

31. The potential amounts of grain for stor~lge can be determined by observing the "probability tree" l"1here each branch has a probability of 1/3 (in our case) in the first year and (1/3}2 - 1/9 in the second year. The numbers in circles are the potential cumulative amounts of grain for storage.

Year 1:

Year 2:

Total

\

1.'he expected p()tential storable grain is, of course:

[4 + 2(3) + 3(2} + 2(1) + 0] I 9 = 2

If the storage capacity is 2, the amount of grain stored could never exceed 2 hence, the expected amount of grain stored is:

[ 6 (2) + 2 (1) + 0] 9 - 14/9

Simila,rly ~ if the storage capacity is 3, the expected amount of grain stored~~s :

[3(3) + 3(2) + 2(1) + 0] / 9 - 17/9

32. Obviously, a storage capacity of 4 would be sufficient to accom-modate all the expected grain available for storage, i.e., 2. To determine the storage capacity which maximizes expected. net benefits, one would then merely have to know the value per unit of grain stored and subtract the c;ost (including interest charges) of each storehouse, respectively, from ~'i:s expected revenue. One further consideration could then be th~ varia-

, bility of benefits II which is not the same for the two storage operations.

- 62 -

33. In addition, however, we could also investigate a sequential decision strategy~, whereby we \Olould initially build a storage capacity not tQ exceed the first year's potential storable grain. Depending on hO~'J mhch grain \vas actually stored in the firs t year lie could subsequently add an addit.ional storage capacity. For instance, one strategy might be to build initially a storage capacity of 1 and add another 1 if this storehouse fills up in the first year~, The expected amount of grain st~ll~ed is th~n:

= 12/9

Similarly, one could build an initial capaci.ty of 2 and an.other 1 only if the storehouse fills up in the first year. The expected amount of stored grain would then be:

[3 + 3 + 2 + 2 + 1 + 2 + 1 + 0] / 9 • 16/9

These sequential decision strategies must ~hen be evaluated in terms of costs to determine the course of action which maximizes expected net benefits and the mOEit desirable distribution of benefits.

34. Actual problems i,nvolve. of course, much more complicated deci-sion environments with many more combination of outcomes and actions to be considered. Nevertheless, sometimes first crude appro:timations of optimum solutions can be obtained by ra ther simple techniques. More precise­solutions can be attained ~IY using the dynamic programming technique. 1/ Another solution technique'recently applied to such problems, after the problem has been properly conc.eptualized, is computer simulation. Y

// 11 Fot:;-dynamic programming see: Bellman, Dynamic Progranuning •••••••••••

Adaptive Contr21.Process: A Guided Tour, Princeton University Press, 1961 and o. R~ Btltt and P. R. Allison, "Farm Hanagement Decisions with Dynamic Programmitlg", Journal of Farm Economics, 1963, 45:121-136.

Y For simulation methods see: P. Zusman and A. AlDiad, "Farm Planning under Weather Uncertainty", Journal of Farm Economics, 1965, 47:574-594, and G. H. Orcutt, "Simulation of Economic Systems", American Economic Revie'tl, 1961, 50:893-907.

..

- 63 -

•

APPENDIX A

Tables for Calculating t-tean and Variance of Present Values

,(1

- 6L -

Appendix A, Table 1

~ 2t (' ' ·.)-1 L...J a ,t = 1, •.•• , n, where a = 1'" te

-------....-----------........,;---.;;.,-,~----

t),

Number of Years

(n)

1 2 3 L 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 2L 25 26 27 28 29 )0

6%

. 89CCC 1.68209 2.387c5 3.~1446 3.57285 'L .c6982 ~·51212 4.9C577 5.25611 5.,56791 5.RL5L2 6.C92Lc 6.)122] 6.5c78L 6.68195 6.83691 6.97L82 7.C9756 7.2C68c 7.30L02 7.39055 7.L6756 7.53610 7.5971C 7.65139 1.69971 7.7L211 1.78c98 7.8ISel.1 7.8L535

Rate of Interest (r)

',8% 10%

.85734 .826~5 1.59231 1.5C9L6 2.2225L 2.07393 2.76281 2.5LoLL 3.22600 2.92598 3.62311 3.2~~61 3.96357 3.5079~ L.255L6 3.72557 4.50571 3.905L3 L.12026 L.05407 L.90L20 L.17692 5.0619C 4.278L5 5.19710 L.36236 5.31301 L .L3170 5.L1239 L.L8901 5.L9759 L.53637 5.5706L L.57551 5.63326 L.60786 5.68695 L.63L59 5.73298 L.65668 5.772LL L.67L9~ 5.8c627 L.69003 5.83528 L.70250 5.A6C15 L.71281 5.881L7 L.72133 5.89975 ll.72837 5.915L2 L.73L19 5.92886 L.13900 5.9Lc38 L.1L297 5.95026 L.7L625

•

'f"

- 65 -

Appendix A, Table 2

2j (t at), t = 1, •••• , n, where a = (1 + r)-l

Rate of Interest (r) ,Number of ,

Years : .. (n) 6% 8% 10%

1 O.9L3L 0.9259 C.9C91 2 2.723L 2.6L05 2.5619 3 5.2L2? 5.('.219 L.8158 L 8.L1C6 7.9619 7.rJL78 5 12.1L71 11.36~9 10.6523 6 16.3771 15.1L61 1L.C393 7 21.(328 19.2306 17.6317 8 26.0520 23.5530 21.3637 9 31.3791 28.c5L8 25.1B06

10 3609631 32.6868 29.C356 11 L2.7579 37.t1OLl7 32.8911 12 L8.7219 L2 •. 1699 36.71L3 13 5L.8163 L6.9500 La.L8aL 1L 61.0085 51.7170 LL.1666 15 67.2680 56.LL59 L7.7576 16 73.5656 61.115L 51.2392

·'17 79.8794 65.7105 5L.6018 18 86.18L8 70.21L1 57.8LOC 19 92.L6L3 7L.616L 60~9L65 20 98.7003 78.906L 63.9185

.-,. t'

- 66 -

APPENDIX B

REVIE\~ OF SONE BAS I C CONCEPTS AND RULES

FRO}! PROBABILITY CALCULUS AND SPECIFIC APPLICATIONS

- Some Basic Probability Concepts

- Hathematical Expe!ctatio!l$

- Properties of Hathernatical Expectations

- Special Hathematical E.xpectations

- Special Mathematical Expectations Useful in Project Appraisal

- 67 -

REVIE\-l OF SOHE BASIC CONCEPTS ~~D RULES

FROH PROBABILITY CALCULUS AND SPECIFIC APPLICATIONS

1. The concepts and rules discussed briefly in this appendix should be primarily useful for.deriving the mean and variance of a variable which 1s a function of two or more other variables measured in terms of their probabili ty dietributions '.1 1/ The presentation aims a t providing a. handy reference, not at mathematical rigor.

Some Basic Probability ConFepts '/

2. The problem soli'ring procedures for measuring the mean and variance of the present value of a, stream of benefits from a project are specific applications of a fe'-l gen!:eral rules from the probability calculus. Hence, a brief review of a.f2'h general rules are given below. II

3. First, let us define a probability distribution. Let all possible outcomes of an event be denoted by xi' i = 1 .••• n, and let the probabilities

of each outcome be denoted by Pi' i.e.,

f(x i ) lII: Pi

then f(X) is a probability distribution if

and (i = 1, .... , n)

> 0

In words, the probabilities must sum to one and negative probabilities are not admissable.

4. The concept of probability distribution can be generalized to two or more variables. Let xi and Y

j be the outcomes of two events, and

let the probabili ty of each j oint outcome xi' Y j be denoted by P ij' i. e. ,

(i =- l, .... , n, j l1li 1, .... , m)

1/ As a useful reference, in terms of more mathematical rigor yet brief presentation, the reader may consult R. V. Hogg and A. T. Craig, Introduc"tio.o to Hathematical Statisitcs, Chapt~r 1 for probahility and mathematical expectations concepts, and B. l~agle, "A Statistical Analysis of Risk in Capital Investment Projects", Operations Research Quarterly, Vol. 18, No. 1 for special applications in risk appraisal. See also F. S. Hillier, "The Derivation of Probabilistic Information for the Evaluation of Risky Investments", Management Scienc~. Vol. 9, 1963.

Y For simplicity of representation, we shall discuss here only discrete probability distribution.

- 68 -

'then f(X, Y) is a probability distribution if

L Pij = 1 and (i • 1 ••••• , n, j • 1, •••• , m)

Pij :> 0

Mathematical EX2ectations

s. One of the more useful concepts in problems involving probability distributions is that of a mathematical expectation. tet X be a set of events with a properly defined probability distribution f(X) and let g(X) be any function of X, then the mathematical expectation (or the expected value) of g(X) is~

(i • l, .... , n)

For instance, if g(X) = X, then,

(i =- l, •... , n)

')

if g(X) = X~, then ') 2

E(X"):: L Pixi (1 • 1, •••• , n)

or if g(X) = (X + 4), then

(1 =- l, ••.. , n)

6. The concept of mathematical expectations generalizes to two or more variables. Let X and Y be sets of events \-lith a properly defined prob­ability distribution, an.d let g(X, Y) be any function of X and Y, then the expectation of g(X, Y) is:

E[g(X, Y)] K

(i c 1, •••. , n, j • 1, •..• , m)

For instance E(XY) is then,

E(XY) = (1 ~ 1, •••• , n, j = 1, •.•• , m)

Properties of Mathematical ExpectatioAs

7. Note the fol1o\~ing properties of mathematical e}~pectations:

(a) If k is a constant,-

E(k) = k

(since ~(k) = L Pik

= k L Pi and L Pi .. 1)

..

- 69 -

(b) If k is a constant and X is a variable,

E(kX) = k EOn

(since E(kX) = L Pil~i = k E Pixt )

(c) If kl and k2 are constants and Xl and X2 arc variables

Special Hathematical Expectations

8. Some frequently used mathematical expectations are:

2/

(A) The~: E(X)

E(X) = L Pix i

" (E) The variance: VeX) = E(X - E(X»'

VeX) = E Pi (xi - E(X»)2 !

The ',ariance can be .computed by the formula: 1/ 2 ? 2 2

E(X , - (E(X»- = E Pi xi (L Pi xi)

(e) The covariance: 2/ cov. (~y) = E(X - E(X» (Y - E(Y»

cov. (XY) = L L Pij (xi - E(X» (Yi - Eey»

Making use of the properties of expectations, it can be shown that the covariance is also

E(XY) - E(X) E(Y) ~

(D) The mean of a linear functio~:

If

Y :: aX

then

E(Y) :: aE(X)

The derivation is as follows:

E(X - E(x»2 3 E [x2 - 2E(X) X + (E(X»2]

:: E (X2) - 2(E(X»2 + (E(X»2 ? 2

• E (X-) - (E(X» Als!~JJ, cov. (XY) a r VeX) v(Y), ,·,here r is the correlation coefficient.

- 70 -

(Note, therefore, that the mean of a linear function is a function of the me~n. But the mean of a non­linear function is ~ a function of the mean, i.e., if 222 Y = aX then E(Y) = aE(X ) and E(Y) ~ a(E(X»

(E) The varianc~ of a linear function: Similarly,

v (Y) 2 = a VeX)

(F) The mean of a SUQ: E(Y)

(G)

Given that Xl and X2 are random variables andal and a2 are constants, such that Y - a1 Xl + a2 X2, the mean of the sum is

E(Y) =

The var~ance of a sum:

(H) The mean of a product if the covariance is zero:

Note from (e) that if cov. (XY) -= 0,

E(XY) = E(X) E(Y)

(1) The variance of a product if the covariance is zero: 11 V(XY) = (E(X»2 V(Y) + (E(y»2 VeX) + VeX) Y(Y)

Special Mathematical Expectations Useful in Project Appraisal

9. Assuming that the means of annual benefits are equal, let t -t a = (1 + r) ; t. = 1, 2, •••• , n, E{R) be the mean present value

of a stream of benefits, and E(X) be the mean benefit in each year, then,

11 The variance of a sum can be V(~~ = E[a1Xl + a2X2 -

\\~~:/,~[a1 (Xl - E(X1» 2 . . 2

• a1 E(XI - ~(Xl» +

derived as follows: 2

a1 E(Xl ) a 2 E(X2~1

+ a2

)X2

- E(X2»)

a; E(XI - E(X2»2 + 2a1a2 E[(X1-E(X»(X2-E(X2»)

2/ See L. A. Good'man, "On the Exact Variance of Products", Journal of American Statistical Association, 1960.

Ii

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E(R) • 2 n '

(a + a + •••• + a ) E(X)

10. Assuming that the variance of annual benefits are equal, let V(R) be the variance of the present value of a stream of benefits, VeX) be the variance of the annual benefits, then if successive benefits are independent, i.e., all covariances are zero,

VCR) -2 4 2n

(a + a + .... + a ) VeX)

if all successive benefits are perfectly correlated, i.e., cov. (Xij

) = VeX)

for i, j • 1, 2, •••• , n, then,

VCR) - (a + a 2 + ..•. + an)2 veX) I

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APPENDIX C

EXCERPTS FROM REVIEWERSiCOMMENTS

1. The author had the benefit of receiving cOialents froa .. ny colleagues in the World Bank. Some of the cODlllents and criticis1I8 have been used in re­shaping the final draft of this study. However, .ore than this, the ca.aent. reflect somewhat differing points of view and ways of stating and dealing with the problems of uncertainty in project appraisal and as such, deserve to be brought to the attention of the readers of this study.

2. The excerpts from these co_ents given below have been selected be-cause they present, in the author's judgement, insights into the problem of dealing with uncertainty in project appraisal, co.pleaentary to the exposition in the final draft of the study. Since the excerpts are not given in the con­text of the original comraents, it would be unfair to cite the author of any particular comment by nsae.

Excerpts from ReviewerarCOIIJBents

•••• Some of the advantages clai.ed for the (sugg.sted) approach suCh' as aking explicit whllt is now implicit and providing decision aaker8 with more knowledge and in~ormation could be achieved st.ply by adding explanatory' notes to the rate of return calculations without additional analysis •••• the technique could probably be used without adding materially to the work of the appraisal mission 'since much of the data is already being collected and'judge­ments are being made implicitly if not explicitly on the probability di8tri­butions of the key variables. It would be interesting to try out the technique on several appraisals toldetermine more precisely the amount of work involved and the usefulness of the approach ••••

••••. In the case of water supply projects the probabilistic approach seellS to have only limited application as the historic data and experience to fonl judge.ents of the pl:obability distribution of the par_etars uaed in the analysis are generally n9t available. Quantification of all the benefits that accrue is not also possible in the coat benefit analysis of a water supply project. Perhaps the technique might be adopted in parts of a project suCh .. in the study of population growth, and demand elasticity. Also, probability distribution could be assigned to population, water consumption, collection of revenue, etc., for different asa_ed water rates and the probability distri­bution'of the fin.ncial tate' of return could perhaps be calculated. This additional in,formation about the rate of return would no doubt be of val",e in deciding on the water rate to be Charged. A better appreciation of the uaefulness of the probab~lity techniques in project appraisal .ight be possible if an actual project is analyzed ••••

•••• Incorpbration of the method of analy.is suggested could con­tribute materially to the value of an appraisal 1Iainly by providing additional information on which to base decisions and by providing a logically rigid framework within which _1ssions would conduct their analyses. Adapta~lon of this method need not entail additional work for appraisal _is.ions. On the

•

- 73 -

contrary, once it has been tested and undergone minor adjus~ents to adapt it to the conditions met with in the field it could eventually streamline appraisal work and increase mission efficiency. There 8hould be no difficulty in pre8enting and explaining thi8 technique and its results with relative simplicity in an appraisal report ••••

; •••• The incorporation of uncertainty in return calculations would improve appraisal work: aore uniformity in approach, greater quantity of information included in the end-result, and a more logical handling of the basic: assumptions. In addition, it would stimulatl! the appraiser's awa'ceness of the probleas affecting the project, by increasing the focus on the afore­said as SUDlp tions • The opportunity could be seized of having ,a record of such assumptions kept and made available for consultation by other' staff members who would thus benefit fro. the accuaulated experience of the Bank. The in­corporation of uncertainty is theoretically feasible. The practical feasi­bility, though, remains largely to be ascertained ••••

•••• The study is a reminder that most of our appraisal work is done in a climate of uncertainty. In the gr,eat majority of cases each one of the variables in a cost-benefit and rate of return calculation is "auesstimated" by a number of people in consultants' offices, foreign ministries and Bank ai8sion8. Our own second-gues8ing variable inputs transmitted to us by others without comment as to their expected ranges of probabilities coapounds possible (and likely) errors ••••• It i8 by no means more certain that a project with a calculated rate of return of 25% will in reality yield a -- maybe lower but .till -- acceptable return than another one for which a mere 15% had been calculated to begin with ••••

•••• If, as appears from the study, the analytical methoda of ap­prai8ing and decision-making have been refined to the stage where it is po.­sible logically to allow for the input in aost appraisals being a collection of probability distributions rather than of single/-valued functions, then 1 believe that it behooves us to take the concepts and methods set out in the study seriously. I believe a180 that the "message" is there and is clear -­that if out methods of appraisal are amenable to such refinement, and we do not refine our methodology accordingly, we will be open to the criticism that our appraisals are at best imprecise, at worst misleading or even wrong!

•••• With our present technique we just do not know how far wrong we may be in(the asseSS1Rent of the outcome of a project if all the input data turn out, in the event, "worse" than anticipated; nobody else knows either at this moment (fortunately), but this happy state of affairs may not last for long~ As 1 read the report, if we recognize the existence of an element of uncertainty (ignorance) in the input data of our appraisal arit~tic, and apply the probability calculus, then pre_uaably we will be aware of the pro­bability distribution of the "solution" (the internal rate of return, or any other criterion) and if we then express this "solution" in terms of a single value, at least we will have an idea of how far in error this single value aiaht be.t This is III advance on our present method, but i. no, more, really, than. variation of the old adage "It's not what you don't knOw that is danaerou8, but what you don't know you don't know!!" ••••

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•••• The author believes that the proposed probability approach i. well .uited to the assessment 9f uncertainty and decision problelllS ariaing from large scale investment projects involving the intft,raction effects of many non-controllable, independent and exogenous variables. He, however, recognizes the unavoidability of evaluations of these variables being a matter for subjective judgment of the appraiser and rejects the "objectivity criterion" which aims at distinguishing bC!tween "right" and "wrong" formu­lation of the nature of uncertainty. According to the author, the "correct", as contrasted with the "right", formulation of probability is merely the "sum of quantified judgment" supported by relevant data and information, which cannot be refuted by actual outcomes ••••

•••• I have the feeling that the author had something impossible in mind with ,tbe report: namely, to satisfy both the scholar in this field, and the practitioner who has never heard anything before about this topic. I am afraid, neither will be very happy. The scholar may prefer a more austere presentation, describing briefly the state of the art, and highlighting the shortcomings and indicating future solutions

•••• The probability approach will lead to reliable resulte only if it is based on a solid quantitative analysis of the factors influencing pro­duction and demand and of the interrelationships between the different vari­ables affecting the rate of return. The use of a computer is not a remedy to these deficiencies. For agricultural projects, the limiting factor often is our quantitative knowledge of the factors influencing production. It would therefore be useful to il108 trate the approach by &ctual cases, where this knowledge is satisfactory • I would not subscribe to the view that the leat is known about the facts, the more necessary it is to embark upon a sophis­ticatical probability approach. I would rather say that the •• jor contribution of a probabili ty appr~ach sh9uld be to force us to a aore sys te1l8tic analysis of the factors influencing the rate of return and to a better understanding of the forces at work •

• ••• The "naive" method consists in cOlibining all the .ost likely esttaates together. The sensitivity analysis consists in measuring the differ­ence in the rates of return between the optimistic and the pessimistic esti­mates for each factor taken independently. ~is simple analysis permits to identify the error:s of estimation which have the greatest bearing on the rate of return and to (!oncentrate re~earch on the most important factors so as to reduce the overal~ ma~gin of unqertainty.

;~ •••• If the rafe of retu~:were se~~five to one factor only, a formal probability appro,ch would not~0~e required: the uncertainty of the final result would, for practical purpo~~s, reflect the uncertainty of this .pecific factor. But if the rate of ret~~ is sensitive to several factors, the above analysis does not inform us of the uncertainty in the final result because we do not know how the various/estimates of these several factors should be ca.bined. Actually, if the pessimistic, most likely the optimistic estimates of a given 'factor were each combined with the three estimates of the other sensitive factors, one would obtain 3n (243 in the case of 5 f~,ctor.) cOlibination. Obviously, the presentation of hundreds of alternatives with

1

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v~ry different p,robability levels would only confuse the decision maker who ne,eds a digestible summary. The probability curve of the rate of return may be the type of summary needed .....

•••• To convince myself that we operate under conditions 6f uncer­tainty as the author terms it, I listed the main ingredients of the calcu­lation for the rate of return in a highway project appraisal. These are:

i) present traffic ii) rate of induced growth of traffic

iii) rate of natural growth of traffic . iv) the traffic "mix" (the ratio of light vehicles to

heavy vehicles v) the project cost

vi) maintenance costs vii) vehicle operating costs (for different types of

vehicles and different types of road)

All the above are put into the rate of return calculation as single figures but all without exception are estimated, each value representing merely one ordinate in a probability distribution ••••

•••• The report intends to show how the new techniques could be used to improve the basis: (i) for evaluating a single project and (ii) for ranking different project or alternatives of a single project. I feel that these two issues ought to be dealt with separately in the report in order to avoid dis­tracting the reader continuously from one issue to another. Thus, the report could focus attention first on the appraisal of a single project and, there­after, on the problems and issues of ranking and of choice alternatives ••••

•••• A computerized model of the investment problem is desirable indeed, but not only for Simulating hypothetical outcomes of the venture in order to obtain a probability distribution. Another advantage would be to run a "sensitivity analysis" of the problem, (a foraally similar but concept­ually different "si1lulation" approach) as a tool for planning the optimal information effort •••• this approach can be described as follows: (1,) The 1Iodel is fo~ulated algebraically, i.e., the basic equation for calculating the capital value of the project has to be spelled out as far as possible, showing explicitly the effect of all the (estimated) independent variables and paraaeters; (2) the model is programmed for the coaputers; (3) without any particular effort, a set of very crude first "guesses" of the values of all the independent input data is prepared; (4) the model is run with this first set of expectation values; (5) one after the other these first crude input data are modified to the "negative" side by changing them successively by certain controlled percentages or amounts. The resulting changes in over­all outcome tell a striking lesson: they show clearly which of the input data are the "critical" ones and which are of minor importance and require no more than just this guess. This could serve as the key to allocate the limited time or .oney for the appraisal of a particular proje'ct in such a way that the data exploration is concentrated on the critical ones, in order to keep the overall uncertainty at a minimum. Any improvements of info~ation during the whole process can be fed into the model immediately for continuous adj us tment and con tro! ••••

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•••• By direct statement or by implication one is presented with the juxtaposi tion of two approaches: (a) in one, the individuals charged with the appraisal of the project introduce judgments, or bias, in their evaluation of all phases of the project as they go aiong and present the "decision maker" with a one-way judgment of uncertain validity. The decision­maker is then in's position of not being able to review the judgments intro­duced in such a step-by-step fashion and his choice i.merely one of accepting or rejecting the conclusion. (b) In other, "experts: look not at the entire project but merely at its phases (e.g. the water storage capabi­lity of a reservoir, the growth potentials for a number of particular crops, etc.). These experts, without making judgments about the project as a whole, would prepare input information giving the range of possible variation of the parameter on which they have been working. This input infor.ation would then be. taken over by someone "who know best" (in the context of the paper it is reasonable to assume that this would be a mathematician-economist) who would elaborate mathematically all the information 80 obtained and pre­sent the decision-maker with an appraisal stating the probability of a num­ber of different outcomes of the project spanning a fairly wide range • •••• The subjective choices may introduce order of magnitude differences in the percentages obtained to represent probabilities. At best, therefore, what is offered is a trade-off in the manne.r and timing of introduction of a bias. On one side, this would be done by pragmatic sector experts who look at the project as a whole and make judgment based on their profe8siol~al experience of similar situations through which they have previously lived: On the other, it would be done in a fragmentary and disembodied sort of way by persons who are apt to view the problems and their parameters as abstract stQtistical information rather than problems of people in specific milieus •

•••• I am not sure that it is correct to vtsualize the decision 1I8ker as much·'better able to fu1fill his task for being told that a certain decision has an X% probability of success and the resulting ability to gauge the aao'~t of risk which should be taken. In fact, l think the decision­maker may well be blinder in the probabilistic approach becauae it could be more difficult for him to realize, let alone review, the judgment of someone down the '.line who, by picking the va:lue of a parameter, has put before him an appealingly explicit X% which could well be t:wice as much or half as much ••••

•••• The preseptation of the advantages of introducing uncertainty, as well as the criticisms about. the current method could have been made more striking in the report. The report is also rather weak in shOWing how mea­sures of risk can be used ill- making de,::isions. Chapter IV does not really say much about what one does once the complicated process of aggregation has been successfully carried out. As to the form in which results should be presented to the manage_nt, it seems that ~., relatively condensed set of indicators would provide answers for most ot'the important questions. This could include (1) the expected value (in the senae used in probability theory), (2) the.standar~ deviation, (3) the probability that the return is less than a given mark, (4) the list of the major factors contributing to the variance of the return. A more complete description would include the distributiQD of probabilities in the form of a table, or a curve

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•••• No'm~tter how the mathematicJ (of aggregation) are carried ou~, it will still be essential to define, in quantitative term~:

i)

ii)

,

the input parameters (e.g., probabiiity distribution of present traffic, or traffic growth, 'of project cost, etc.) ;

the output (i.e., the criterion which we are using to "appraise" the project - internal rate of return, benefit-cost, present value of total benefits, etc.);

iii) the mathematical relationship between each of i) and ii);

iv) any inter-relationships between the input parameters (e.g., the effect on the normal growth of traffic if the induced growth is greater than anticipated).

To defineauch parameters and to quantify such relationships would be a useful discipline in itself. The results of such an exercise might well affect the type of information we require and gather on appraisal missions ••••

( I

); /.

0 If,

1.

2.

4.

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REFERENCES

Adelman, R. M., "Criteria for Capital Investment", QE.~­ti.onal Research Quarterly, March 1965.

Arrotv, Harris and }1aishak, "Optimal Inventory Policy", Econometrica, Vol. 20, .July 1951.

Arrmv, K. J., "Alternative Approaches to the Theory of Choice in Risk-Taking Situations", Eco~ometrica, Vol. 19, 1951.

Bellman, R., Adaptive Control Processes: A Guided T,?!!!', Prince ton Uni vel's:!. ty Press, 1961.

5. Burt, o. R. and D. R. Allison, "Farm Hanagement Decisions with Dynamic Progranlmin~ It, Journal of Farm Economics, Vol. 45, 1963.

6 e Dorfman, R., "B.3.sic Economic and Technologic Concepts" J:

Chapter in 1\. Haas, et a1 Design. of Wa.ter Resource .fu'stems, Harvard University Press, 1962.

7. Farrar, D. E., The I~:lvestment Decision Under .'Uncertainty, Prentice-lIall, 196iz.

8. Fellner, Hil1iam, Probability and Profit, Richard D. Irwin~ Inc., 1965.

9. Friedman, t1. and L. T. Savage, "The Utility Analysis of Choice Involving Risk", Journ31 of Political Economy, Vol. 56, August 1948.

10. Goodman, L. A., HOn the E~act Variance of Products",

11.

12.

13.

14. (:j

Journal of American Statistical Association, Vo:t. ,55, 1960.

Hertz, o. B., "Risk Analysis in Capital Investment", illiirvard ~usiness Revie,,,, Jan.-Feb. 1964. U .if

Hillier, F. S., "The Derivation of Probabilistic Information for the Evaluation of Risky Investments", Management Science_ Vol. 9, 1963A

Hi.rshleifer, J .. , "Investment Decision Under Uncertainty: Applications of the State-Preference Approach", The Quarterly Journal of Economics, May 1966.

lIirsh1eifer, J.~ "Investment Decisi~n Under Uncertainty: ':Choice-Theorctic Approaches", The,:~Quarterly Journal of Economics, Vol. LXXIX, November 1965.

•

•

- 79 -

15. HO'(vard, R. A., ttDynamic Inferenc,e!", Operations Research Vol. 13, September 1965.

16.

17.

Lamberton, D.M., The Theory of Pr9fit, Basil Blackwell & Mott, Ltd. t 1965.

Maas, A., Design of Water Resource Systems, Harvard University Press, 1962.

18. Markowitz, H., Portfolio Selection: Efficient Di versi-fication of Investments, Wiley & Sons, New York, 1954.

19. Marshak, T., "Alternative Approaches to the Theory of Choice in Risk-Taking Situations", Econometrica, Vol. 19, October 1951.

20. Marshak, 'f., "Noneyand the Theory of Assets", Econometrica, Vol. 6, 1938.

i 21. MaSSE!, Pierre, Optimal Investment Decisions, 1962.

22. Modigliani, F., and K. J. Cohen, "The Significance and Uses of Ex Ante Data", ':hapter IV in Expectations, Uncertainty and Business Behaviour, edited by M. J. Bowman e

23. Orcutt, G. H., "Simulation of Economic Systems", American Economic Review, Vol. 50, 1961.

24. Ozga, Expectations in Economic Theory, Aldine Publishing Company, Chicago, 1965.

25. Savage, L. J., The Foundations of Statistics, J. Wiley and Sons, New York, 1954.

26. Swalm, Ralph 0., "Utility Theory - Insights i.nto Risk Taking", Harvard Business Review, Nov.-Dec. 1966.

27. '~ Re ... aDa, J. and O. Morgen~te'tn, Theory of Games and Economic Behaviour, P'~inceton University' Press, 1947.

28. Wagle, A~, "A Statistical Analysis of Risk in Capital In,vestment Projects", Operational Research Quarterly, Vol. 18, No.1.

29. Zusman~ P. and A. Amaid, "Fann Planning under 'Weather Uncertainty", Journal of Farm Economics, Vol. 14, August 1965.

[)

E~onomics Department Studies on Problems of Sector and Pro1ect Analysis

• EC-117, July 15, 1963, Herman G. van der Tak, The Economic Comparison of Hydr~electric Projects with Alternative Developments of Thermal Electric Power

EC-128, May 7, 1964, Herman G. van der Tak, The Evaluation_of Ag~icultural Projects.: A Study of Some Economic and Financial Aspects

EC-130, October 26, 1964, Lee Charles Nehrt '<I A Pre-Investment Study of the Flat Glass Industry

EC-132, January 22, 1965, Herman G. van der Tak and Jochen K. Schmedtje, Economic Aspects of Water Utilization in Irrigation Projects

EC-138, Oct.ober 21, 196.5, .Jochen K. Schmedtje, On Estimating the Economic Cost of Capltal

* EC-140a, December 16, 1965,. Jan de Weille, Quantification of Road User SavinA~

EC-147, Sel't~mber 26, 1966, Herman G. van der Tak and Jan de Weille, An Econo..1fc Reappraisal of a Road Project: The First Iranian Road Loan of 1959 (I~~-227)

EC-157, December 20, 1967, Mark Blaug, A Cost-Benefit Approach to Educational rj}anning in Developing Countries

EC-158. January 11, 1968, Alan A. Walters, The Economics of Road User Charges_

EC-160. Harch 18. 1968. Herman G. van der Tak and Anandarup Ray, The Economic Benefit!i.::of Road Transport Projects

EC-161, May 29, 1968, AyHian C!lingiroglu, Manufacture of Heavy Electrical E9uipm~.'!.~ 'in' Developing Countries

~:C-162, ~1ay 31, 1968, Jack Baranson, Automotive Industries in Developing Countries

• Reissued 'ih World Batik Staff Occasional Papers, distributed by J~hn8 Hopkins Press.