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lnvestment under Uncertainty

Avinash K. Dixit and Robert S. Pindyck

Princeton University Press Princeton, New Jersey

Investment under Uncertainty

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Copyright @ 1994 by Princeton University Press Publlshed by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex A1l Rights Reserved Library of Congress Cataloging-in-ablication Data Dixit, Avinash K. Investment under uncertainty / by Avinash K. Dixit and Robert S. Pindyck P. Cn1. Includes bibliographical references and index. ISBN 0-691-03410-9 making. 1. Pindyck, Robert S. 1. Capital investments-Decision II. Aritle. 11G4028.C4D58 1993

658.15'54-dc20

93-26321 CIP

This book has been composed in Times Roman Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources http://pup.princetcm.edu Printed in the United States of Merica 10 9 8 7 6 5 4 3

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Contents

Preface

Part 1. Introduction A New View

l 2 3 4 5

()f

Investment

3 4 6 8 10 23

The Orthodox Theory The Option Approach Irrcversibility and the Ability to N'Vllit An Ovcrviewof the B()()k Noneconomic Applications

2 Developing the Concepts Through Simple Examples I

2 3 4 5 6

Price Uncertainty Lasting Two Periods Extending thu Example to Three Pcriods Uncertainty over Cost Uncertainty over Interest Rates Scale versus Flexibil ity G uide to the Literature Part II. Mathematical

26 27 41 46 48 51 54

Background

3 Stochastic Processes and lto's Lemma l Stochastic Processes 2 The Wiener Process 3 Generalized Browninn Motion Ito Processes 4 Ito's Lemma

59 60 63 70 79

5 Barriers and Long-lkun Distribkltion 6 Jump Processes 7 Guide to the Literature Appendix

The Kolmogorov Equations

Dynamic

Optimization under Uncertainty Dynamic Programming l Contingent Claims Analysis 2 Relationship between the Two Approaches 3 4 Guide to the Literature Appendix A Recursive Dynamic Progralnming Appendix B-optimal Stopping Regions Appendix C Smooth Pasting

83 85 87 88 93 95 114 l20 l25 l26 128 130

Part 111.A Firm's Decisions

5

Investment

1 2 3 4

Opportunities and Investment Timing The Basic Model Solution by Dynamic Programming Solution by Contingent Claims Analysis Characteristics of the Optimal Investment Rule Alternative Stochastic Processes Guide to the Literature

l35 l36 140 147 l52

6 The Value of a Project and the Decision to Invest 1 The Simplest Case: No Operating Costs 2 Operating Costs and Temporary Suspension 3 A Project with Variable Output 4 Depreciation 5 Price and Cost Uncertaipty 6 Guide to the Literature

175 l77 l86 195 l99 207 21 l

7 Entry, Exit, Lay-up, 1 Combined Entry

213 215 229 242

and Scrapping

and Exit Strategies Lay-up, Reactivation, and Scrapping

Guidc to the, Literature

Part lV. Industry

Equilibriunl

8 Dynamic Equilibrium in 11 Competitive l The Basic Iattlition 2 Aggregate Uncertainty 3 I ndtlstry Eqtlilibrium with Exit 4 Firm-specific Uncertainty 5 A Gene ral Mode I 6 Guide to the Literature 9

Policy Intenrention

lndustry

247 Ozttl

252 'R

6l 2(:7 277 280

and Imperfect Ctpmpetition

2 82 283 ?)(b

1 Social Optimality 2 Analyses of Some Commonly Used Policit!s 3 Example of an Oligopolistic lndtlstry 4 G u id e to t he Lit e ra tu re

3t)i) 3 l4

Appendix

3 I5

Stlme Expected Present Valut!s Part V. E xtensitlns

and

Applictltilylls

10 Sequential Investment l Decisions to Start ltnd Ctlmplete :t Nltlltistagd Pnlject 2 Contilluous Investment and Timc to Btlild 3 The Lutarning Curve anl Optilnltl Prlldtlction Dccisons 4 Cost Uncertainty and Learning 5 Gu ide to the Literatu re Appendix Numerical Solution D iffe r (., n t ia l E q u a t io 1-1

()f

328 339 345 35l

Pltrtial

Incremental

1 2 3 4

3l9 32l

Investment and Capacity Choice G radual Capacity Exptnsion with Diminish ing Returns Increasing Rttturns and Lumpy Additions to Capacity Adj ustment Costs Guide to the Literature

12 Applications and Empirical Research 1 Investments in Offshore Oil Reserves 2 Electric Utilities' Compliance with the Clean 5 Tilt: Timing of Environmentai Poiicy

357 359 377 38l 39I 394 396

Air Act

4()5 4 I2

(?ontents 4 Explaining Aggregate Investment Behavior 5 Guide to the Literature

4 19 425

References

429

Symbol Glossary

445

Author Index

449

Subject Index

455

Preface

Bool provides a systematic treatment of a new theoretical approach capital investment decisions of firms. stressing the irreversibility of most to and decisions the ongoing uncertainty of the economic environinvestment which in those decisions are made. This new approach recognizes the ment value waiting of better for (but never complete) information. It option analogy with options in financial markets, which thc theory of exploits an much richer dynamic framework than was pllssible with the permits a of theory investment. traditional The new view of investment opportunities as optitlns is the product of over a decade of research by many economists, and is still an active topic in journal articles. It has led to some dramatic deptrtures from the orthodox theory. It has shown that thc traditional present value'' rule. which is taught to virtually every business school student and student of economics, can give very' wrong answers. The reason is that this rule ignores irreversibility and the option of delaying an investment. For the same reason, the new theoly also contradicts the orthodox tcxtbook view of production and supply going back to Marshall, according to which firms enter or expand when the price excceds long-run average cost, and exit or contract when price falls below average variable cost. Policy prescriptions based on the traditional theory, for example. the use of interest rate cuts to stimulate investment and antitrust policies based on price-cost margins, are also called into question. In this book we have tried to present the new theory in a clear and systematic way, and to consolidate, synthesize, and in some places extend the various strands of this growing body of research. While there is a Iarge and burgeoning Iiterature of journal articles, including a survey article by THls

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xii

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each of us, :1 book l'ormat has d istinct advan tages. l t tll/fe rs tIs the space t() develop the different themes in more detail and better order. and to place them in relation to one another. It also gives us the opportunity to introduce and explain the new techniques that underlie much of this work, to economists. We hope that the result is 11 but that are often unfamiliar better pedagogic treatment, of use to students, researcllers, and practitioners. However, perhaps even more important than pedagogy is the ability of the book tbrmat to provide a broad vision of the subject, and of the mechanisms of the dynamic, uncertain economic world. Our main aim is to clarify and explain the theory, but we think this is often best done by applying it to the real world. Therefore we often obtain numerical solutions tbr our theoretical models using data that pertain to some specific industries or products. We believe that the cumulative weight of these calculations constitutes strong prima facie evidence for the validity and the quantitative significance of the new theory of investment. However, more rigorous econometric testing, and the detailed work that is needed to devise improved decisionmaking tools for managers, await further research. We believe that this is an exciting and potentially important subject, and hope that our book will stimulate and aid such

research.

Who Should Read This Book? The first audience This book is intended for three broad audiences. consists of economists with an interest in the theory of investment, and in its policy implications. This includes graduate students engaged in the study of micro- and macroeconomic theory as well as industrial organization, and researchers at universities and other institutions with an interest in problems relating to investment. The second broad audience is students and researchers of financial economics, with an interest in corporate finance generally, and capital budgeting in particular. This would includc graduate students studying problems in capital budgeting (that is, how firms should evaluate projects and make capital investment decisions), as well as anyone doing research in finance with an interest in investment is finance decisions and investment behavior. Finally, the third audiece practitioners. This includes people working in financial institutions and concerned with the evaluation of companies and their assets, as well as corporate managers who must evaluate and declde whether to go ahead with large-scale investments for their firms. Some parts of this book are fairly technical, but that should not deter

r/-(!/c(.z the interestetl rellder. Tllc first twt) cllapters provide 11 lirly brie ( kllltl self-con t:linetl introtluct itdn to thtr tlltltlr.y of irreve rsible investnl e 1) t u ntlc r uncertainty. Tllese chapters convey many of tllc basic ideas, wllile llvtlitling technical details anl any l'nathenliltical tbrmalisln. Rttading these trhapters that is a low-risk investment we can alnlost gtlaralltee will have a high those practitioners whose knoWledge ot- econol-nics and fireturn. Even nance textbook's is vefy rtlsty should be able to follow tllese two chapters without too m uch effort t)r di fficul ty. %9eanticipate that many read ers will want to go into tlle tlltlory in mtre detail and address some of the technical issues. but Iack sonle ()f the necessary mathematical tools. With those readers in mind we hllvtl included two chapters (Chapters 3 and 4) that provide a self-contained introduction to the mathematical ctlncepts and tools that underlie this work. (These tools have applicltbility tllat goes well beyond the theo!z of investment under uncertainty, and so we anticipllte thltt some readcrs will find these chapters useful even if their applied interests lie elsewhc re, for example, macroeconomics, international trade, ()r labor economics.) .

However. wtl think that techniques are best Iearned by usi ng thcnl. Theretbre we do not attempt to bc very rigorllus t)r thorough in the mathematics as such. Nv'e rcly on intuition as far as possiblc. sktz tch stlmtl simple formal arguments in appendiccs, and reikr the reatlers whl) wish greater mathematical rigor or depth to mllre advanced treatises. Ftlr mtlst readers. we rccommend reading Chapters 3 and 4 onct:, and prllceeding t() the later chapters where thtt techniques arc used. Nve tllink tlltly will ()f thtl mathematics in this way tllan l7y emergc with a mtlch better grasp trying to master it first and in the abstract. Finally, we expect that many rcaders will wilnt to sec the ncw view t)l' investment develtlped in detail, including as many of its ram ificatitlns as possible. along with examples and applications. Chapters 5 through l2 provide just that. building up slowly from a basic and fairly simple model of irreversible investment in Chapter 5, to more complete modcls in Cilapters 6 and 7 that acctlunt lbr decisions to start or stop producing. t() the modklls in Chapters 8 and 9 that account tbr the interactions ()f firms within industries, and finally to the more ad vtnced extensions of thc theory and its applications in Chapters 10, l 1, and l 2.

Acknowledgments This book is an outgrllwth of the researcll that cach of us hlts been doing and over the past several years on the theory of investment. That research

xiv

Prelce

ormously from our interactions with hence this book-have benefitted colleagues and friends at our home institutions and elsewhere. To try to list all of the people from whom we received insights. ideas. and encouragement would greatly lengthen the book. However, there are some individuals who have been especially helpful through the comments, criticisms, and suggestions they provided aer reading our research papers and draft chapters of this book, and they deserve special mention: Giuseppe Bertola, Olivier Blanchard, Alan Blinder, Ricardo Caballero, Andrew Caplin, John Cox, Bernard Dumas, Gene Grossman, Sandy Grossman, John Leahy, Gilbert Metcalf, Marcus Miller, Julio Rotemberg, and Jiang Wang. In addition, we want to thank Lead Wey for his outstanding research assistance throughout the development of this book. Our thanks. too, to Lynn Steele for editorial help in preparing the hnal draft of the

manuscript.

The second printing of the book gave us the opportunity to correct several errors. We thank al1 the readers who brought these errors to our attention, most particularly Marco Dias of PUC, Rio de Janeiro, and David Nachman of Georgia State University. We are sure that more errors remains but we practice the preaching of capital theory: manuscripts should not be improved to the point of pertkction. but only to the point where the rate of further improvement equals the rate of interest. Finally, we want to thank Peter Dougherty, our editor at Princeton University Press, for his encouragement and advice as we prepared manuscript, and his guiding hand in seeing the book through to the

production. Both of us also received hnancial support for which we are very grateful. Avinash Dixit acknowledges the support of the National Science Foundation and the Guggenheim Foundation. Robert Pindyck acknowledges support from the National Science Foundation and from MIT's Center for Energy and Environmental Policy Research. Avinash K. Dixit Robert S. Pindyck

l part

lntroduction

Chapter

1

A New View of lnvestment

investnlent lts tllc ltct tlf illctlrring an inlnlcd itte cost in thc expectation o' ftlture rewitrtls. Fi rms thllt cllllstruct plants and instkyll equipment. mercllllnts whll Ilty in 11 stock of gllllds lr sltle. llnd persons whl) spend time ()n voclltitllll cdtlclltitln klrtr tll investors in tllis sense. Stln-lewhlt less obviotlsly. :t firm that slluts dlpwn lt ltlss-lnltking plant is :lls() the p:tyments it mtlst mllk-tl t() extract itself l'rllm cllntratrttlltl ctllmmitnlents. incltltlingseverance pllynlents t() I:tl7()r.ltre tlle initiltl expcllditu rtl. lnd tlle prllspcctive rcward is the retltlctitll, il1 l'uttlre lllsses. Viewed frllm tllis perspective. investnlent detrisilllls lre tllliqui tlltls. Ylltlr purchase ()f this tntltlk' wlls an investment. Tlle rewltrd, we I1()pe. will be 1111 ()f understanding invcstment decisitlns if ytlu are 1kn ccontlmist. and imprtlved ()1' lbility to makd sucll decisilllls in the imprllved cklu rse ytltlr l'tlture carecr an dtEtIines

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a business schotll student. investment decisillns sllare

in three imptlrtant characteristics comple the In partilll Iy degrees. Fi i is Iy te rst, nvestmen t varying or other wllrds, the initial cost ()f investment is at least partially sunk; yotl cannot recover it alI should you change ytltlr m ind. Sectlnd, t he rut is ttlt.'ertll'v over the futtlre rewards from thtt invdstment. Tlle best y()u can do is t() assess the probabilities of the alternative outcomes that can me:tn greater or smaller profit (or loss) for yllur vcnture. Third, you have somc leeway of your investment. Yotl can postpone action to get about the tnlg ()f about the certainty) coursc, complete more intbrmation (but never, Most

-rcb'o-sible.

future.

'Tllese th rct: cllari.ti-iit.: ritii itisill ic I itt;i ((? dcfcl I l 111lc i.llc t?I?(ilI lul dcttisioldsol' investors. is interacton is the focus ()f t his bf.)tlk.bl'.t dklvclllpth e theof'y ()f 'l-h

lntrodttcliolt irreversible investment applications-l

under uncertainty,

and illustrate it with some practical

The orthodox theory of investment has ntt recognized

the important

qualitative and quantitative implications of the interaction between irreversibility, uncertainty, and the choice of timing. We will argue that this neglect explains some of the failures of that theory. For example. compared to the predictions of most earlier moels, real world investment seems much less sensitive to interest rate changes and tax policy changes, and much more sensitive to volatility and uncertainty over the economic environment. We will show how the new view resolves these anomalies, and in the process offers some guidance for designing more effective public policies concerning investment. Some seemingly noneconomic

personal decisions also have the charac-

teristics of an investment. To give just one example, marriage involves an up-front cost of courtship, with uncertain tbture happiness or misery. It may be reversed by divorce, but only at a substantial cost. Many public policy decisions also have similar features. For instance. public opinion about the relative importance of civil rights of the accused and of social order lluctuates through time, and it is costly to make or change Iaws that embody a particular relative weight for the two. Of course the costs and benehts of such noneconomic decisions are difhcult or even impossible to quantify, but our general theory will offer some qualitative insights for them, too.

1 The Orthodox Theory How should a tirm, facing unccrtainty over future market conditions, decide whether to invest in a new factory'? Most economics and business school students are taught a simple rule to apply to problems of this sort. First, calculate the present value of the expected stream of prohts lhat this facto:y will generate. Second, calculate the present value of the stream of expenditures required to build the factory. Finally, determine whether the difference between the two the net present value (NPV) of the investment-is greater than zero. If it is, go ahead and invest.

' Some dccisions that are the opposite of investment gettingan immediate benefit in return for an uncertain future cost- are also irreversible. Prominent examples include the exhaustion uf Ilaiural rain forcsts. 0ur methods appjy to thcse resuurces and (he dcstruclion of lropicai decisions, too.

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Of course, there are issuds that arise in calculating this net present value. Just how should the expected stream of profits from a new factory be estimated'? How should intlation be treated'? And what discount rate (or rates) should be used in calculating the present values'? Resolving issues like these are important topics in courses in corporate finance, and especially capital the NPV of an budgeting. but the basic principle is fairly simple--calculate whether and it is positive. investment project see The net present value rule is also the basis for the neoclassical theory of investment as taught to undergraduate and graduate students of economics. Here we lind the rule expressed using the standard incremental or marginal approach of the economist: invest until the value of an incremental unit of capital is just equal to its cost. Again, issues arise in determining the vaiue of an incremental unit of capital, and in determining its cost. For example. what production structure should be posited? How should taxes and depreciation be treated? Much of the theoretical and empirical literature on the economics of investment deals with issues of this sort. We ind two essentially equivalent approaches. One, following Jorgenson ( l 963). compares the per-period value ofan incremental unit ofcapital (its marginal product) and an percost'' that can be computed from the purchase pcriod rental cost'' ()r price- thc intercst and depreciation rates, and applicable taxes. The (irm's desired stock ofcapital is tbund by equating the marginal product and the user cost. The actual stock is assumed to adjust to the ideal. either as an ad hoc Iag process.or as the optimal response to an explicit cost of adjustment.The book by Nickell ( 1978) provides a particularly good exposition of developments of 'bequivalent

'&user

this approach.

The other tbrmulation. due to Tobin ( l 969), compares the capitalized value of the marginal invcstment to its purchase cost. The value can be obscrved directly if the ownership ()f the investment can be traded in a secondary market', otherwise it is an imputed value computed as the expected present value of the stream of proGts it would yield. 71e ratio of this to the cost) of the unit. called Tobin's ty, governs the purchase price (replacement should be undertaken or expanded if q exdecision. lnvestment investment should undertaken. and existing capital should be reduced, if it 1 be not ceeds ; < 1 The optimal rate of expansion or contraction is found by equating the q adjustment which depends on the difference of benefit, marginal cost to its T:IX rules alter somewhat, and this but the basic principle 1. between ty can of investis similar. Abel (1990) offers an excellent sulwey of this fy-theory mcnt. In alI of this, the underlying principle is the basic net presen: valuc rule. .

Ilttrollltctioll

The Option Approach

2

llusiness practice lil'td tllltt stltrl'l Iltlrtlltl tes ()r Iltlrdle--- rattl. (lbstt l-vers Illes l 11 t)t Iltlr Nvortls. lil-llls tlt) typical lu i ly llree tlle ctlst of capittl rt t or are not invest until price ristls substantially above Itlng-rtln average cllst. (311tlle downside, lirlns stay in busi ness for lengthy periotls svllile absorbi ng opktryting t)f

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net present value rule, however. is based on some implicit assumptions that are often overlooked. Most impllrtants it assumes that either the investment is reversible, that is, it can somehow be undone and the expenditures recovered should market conditions turn out to be worse than anticipated.

or, if the investment is irreversible. it is :1 now or never proposition, that is, if the firm does not undertake the investment now, it will not be able to in the future. Although some investments meet these conditions, most do not. lrreversibility and the possibility of delay are very important characteristics of most investments in reality. As a rapidly growing Iiterature has shown, the ability to delay an irreversible investment expenditure can profoundly affect the decision to invest. It also undermines the simple net presentvalue rule, and hence the theoretical lbundation ofstandard neoclassical investment models. The reason is that a firm with an opportunity to invest is holding an analogous to a hnancial call option it has the right but not the obligation to buy an asset at some future time of its choosing. When 11 firm makes an irreversible investment expenditure, it exercises, or its option to invest. It gives up the possibility of waiting for new information to arrive that might affect the desirability or timing of the expenditure; it cannot disinvest should market conditions change adversely. This Iost option value is an opportunity cost that must be included as part of the cost of the investment. As a result. when the value of a unit of capital is at least as large the NPV rule and its purchase installation cost'' must be modised. The value of the unit as must exceed the purchase and installation cost, by an amount equal to the value of keeping the investment option alive. Recent studies have shown that this opportunity cost of investing can be large, and investment rules that ignore it can be grossly in error. Also, this opportunity cost is highly sensitive to uncertainty over the future value of the project, so that changing economic conditions that affect the perceived riskinessof future cash tlowscan have a large impact on investment spending, Iarger than, say, a change in interest rates. This may help to explain why neoclassical investment theory has so far failed to provide good empirical models of investment behavior, and has led to overly optimistic and tax policies in stimulating forecasts of effectiveness of interest itoption''

'kills,''

''invest

lll substantially below average vllriable cost without in(ar alsla exit. Tllis disinvestmen! ducing seenls tt) conllicl vit 11sttllltlfkrd tlltloryNvill tlnctt explained irrcversibility ltntl optioll valtle arc be see. it can but as sve

Iosses. and price can

accotlnted tbr.

Of course, one can always redehne NPV by subtracting frol'n the convtl ntional calculation the opportunity cost of exercising the option to invest- and if NPV is positivo-' holds once this ct'rrection has then say t hat t he rule made. However. to do so is to accept otlr criticisnA. To highlight the inlbeen values, in this book we prefer tt) keep them separate fron) option portance of NPV'- terthe conventional NPV I f others prefer to contintle to use u-arcful option nclude all rclevllnt they tt) are minology, that is 5ne as long as ()f who readi tllat ly PV. N Readdrs usage can values in their deti nition pre fer translate our statenlents into tllat language. ()f In this book we develop the basic theory irreversible investlnent under Ginvest

'-positive

uncertainty, emphasizing the optitln-like charllctdristics ()f investnltlnt ()pptlr(lbtained fronl nlethods tunities. Nv show how optimal investment rtlles can bt2 options financial nlark in ets. Skc also dethat have been developed for pricing ()1* optimal matllematical theor,y velop an eqtlivalent approach based on the tlncertainty illustrate dynamic programming. sequentiai decisions under the optimal investment decisions of Iirms in :1 v:triety of sittlatiolls nt!w entry. determination of the initial scale of the hrm and futtlrc talstly changes of scale, choicc between diilk re nt forms of investment that offer d iftkrent dc01* succttssive stages grees of llexibility to meet future condititlns. completion of a complcx multistage project. temptlrary shutdown and rcstart. pcrmanent exit. and so forth. Nv'ealso analyze how the actions ofsuch f'irmsare aggregated to determine the dynamic eqtlilibrium of an industry. 'bgc

Iinancial assets. thtt (lppllrtunities t() Therefore optitlns.'' called this book sometimes real assets are acquire could be titled S'The Real Options Approach to lnvestment-''

To stress the analogy with options

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-treai

'rate

investment.

The option insight also helps explain why the actual investment behavior wisdom taught ln business schools. Firms to yield a return in excess ()f a required.

ol' firms differs from the received invest in projects that are ttxpected

zsummcrs ( l987. p. 31)())fllund hurdle nlltts ranging frtlm 8 !t) 3f) pcrcent, wlh a median :tl of 15 and a mcan of l 7 pcrcent. 7-he cost of riskless capital was mucll lllwer; ltlwing ftlr the and rcal rate cltlse tlltt 4 intcrest the nominal rate was percellt, deductibility of interest cxpenscs. i'tlr investments with a!. (s!9911 p f. I Thd. bllrlltt rilttt Jlpprtlpri:lle tc zere. See alse Dertuuztts systematic risk will exceed tht: risk Iess rate. bu t not by enough t() jusliIk the n umlltrrs used by many companies. .ct

Illtrziltctizlll

3

lrreversibility and the Ability to Wait

Before proceeding, it is important to clarify the notions of irreversibility, the ability to delay an investment, and the option to invest. Most important, what makes an investment expenditure a sunk cost and thus

irreversible'?

Investment expenditures are sunk costs when they are firm or industry specific. For example, most investments in marketing and advertising are firm specihc and cannot be recovered. Hence they are clearly sunk costs. A steel plant, on the other hand. is industry specihc it can only be used to produce steel. One might think that because in principlethe plant couid be sold to another steel company, the investment expenditure is recoverable and is not a sunk cost. This is incorrect. If the industry is reasonably competitive. the value of the plant will be about the same for all lirms in the industly so there would be little to gain from selling it. For example, if the price of steel falls so that a plant turns outa expost, to have been a ttbad'' investment for the tirmthat built it, itwill also be viewed as a bad investment by othersteel companies, and the ability to sell the plant will not be worth much. As a result, an investment in a steel plant (or any other industrpspecific capital) should be viewed as largely a sunk cost. Even investments that are not hrm or industly specific are often partly irrcversible because buyers in markets for used machines, unable to evaluate the quality of an item, will offer a price that corresponds to the average quality in the market. Sellers, who know the quality of the item they are selling, will be reluctant to sell an above-average item. This will lower the market average quality, and therefore the market price. This problem (see Akerlof, 1970) plagues many such markcts. For example, office equipment, cars, trucks, and computers are not industry specific. and although they can be sold to companies in other industries, their resale value will be well below their purchase cost, even if they are aimost new. lrreversibility can also arise because Of government regulations or institutionai arrangements. For exampie. capital controls may make it impossible for foreign (or domestic) investors to sell assets and reallocate their funds, and investments in new workers may be partly irreversible because of high costs of hiring, training, and Iiring. Hence most major capital investments are in large part irreversible. Let us turn next to the possibilities for delaying investments. Of course, srms do not always have the opportunity to delay their investments. For example, there can be occasions in which strategic cons' iderations make it imperative for a 5rm to invest quickly and thereby preempt investment by '%lemons''

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existingor potential competitors.3 However. in most cases. delay is at Ieast feasible. There may be a cost to delay tlle risk of entry by other tirms, or simplyforegone cash tlows but this cost must be weigled against the benehts of waiting for new intbrmation. Those benetits are oen large. As we said earlier. an irreversible investment opportunity is much like a

financialcall option. A call option gives the holder the right, forsome specified amount of time, to pay an exercise price and in return receive an asset (e.g.. a share of stock) that has some value. Exercising the option is irreversible: although the asset can be soid to another investor, one cannot retrieve the option or the money that was paid to exercise it. A lirm with an investment opportunity Iikwise has the option to spend money (the price''). in the future. in of for value. project) return Agains asset an or some now (e.g..a the asset can be sold to another tirm, but the investment is irreversible. As with the financial call option. this option to invest is valuable in part because the tkture value of the asset obtained by investing is uncertain. lf the asset rises in value. the net payoff from investing rises. If it falls in value, the tirm need not invcst, and will tnly lose what it spent to obtain the investment opportunity. The models of irreversible investment that will be developed in texercise

Chapter 2 and in later chapters will help to clarity' the optionlike nature of an investment opportunity. Finally. one might ask how firms obtain their invcstment opportunities. that is. options to invest. in the first place. Sllmetimes investment opportunities result from patents, or ownership of land or naturai resourccs. More generally, they arise from a firm's managerial resources. technological knowledge, reputation, market posititn, and possible scale, alI of which may have been built up over time. and which enable the tirm to productively undertake investments that individuals or other lirms cannot undertake. Most important. these options to invest arc valuable. Indeed, for most firms, a substantial part of their market value is attributable to thefr options to invest and grow in the future, as opposed to the capital they already have in place.'l Most of the economic and financial theory of investment has focused on how hrms should (and do) exercise their options to invest. To better understand investment behavior it may be just as important to develop better models of how firms obtain investment opportunities, a point that we will return to in Iater

chapters.

3See Gilbert ( 1989) and Tirolc ( 1988, Chapter 8) for surveys of the literature on such strategi aspects of investlnegt. 4For discussions of growth options as sources of firm value, sec Mycrs ( 1977), Kester ( 1984). and Pindyck ( 1988b).

Illtrotlltctioll

4

An Overview of the Book

In the rest of this chapter, we outline the plan of the book and give a flavor of some of the important ideas and results that emerge from the analysis. 4.A

A Few Introductory

Examples

The general ideas about treal options'' expounded above are simple and intuitive, but they must be translated into more precise models betbre their quantitative signiEcance can be assessed and their implications for firms, industries, and public policy can be obtained. Chapter 2 starts this program in a simple and gentle way. We examine a firm with a single discrete investment of two decision periopportunity that can be implemented within a of undergoes ods. Between the two periods, the price the output a permanent shift up or down. Suppose the investment would be profitable at the average price, and therefore a tbrtiori at the higher price. but not at the Iower price. By postponing its decision to the second period, the firm can make it having observed the actual price movemcnt. It invests if the price has gone up. but not if it has gone down. Thus it avoids the loss it would hjwe made if it had invested in the hrst period and then seen the price go down. This value of waiting must be traded off against the loss of the period- l profit flow. The result-the decision to invest or to wait-depends on the parameters that specify the model. most importantly the extent of the uncertainty (whichdetermines the downside risk avoided by waiting) and the discount rate (which measures the relative importance of the future versus the present). We carry out several numerical calculations to illustrate these effects and build intuition for real options. We also explore the analogy to financial options more closely. We introduce markets that allow individuals to shift the risk of the price going up or down, namely contingent claims that have different payoffs in the two eventualities. Then we construct a portfolio of these contingent claims that can exactly replicate the risk and return characteristics of the hrm's real option to invest. The imputed value of the real option must equal that of the replicating portfolio, because othemise there would be an arbitrage opportunity an investor could make a pure profit by buying the cheaper and selling the dearer of the two identical assets. We also examine some variants of the basic example. First, we expand window of investment opportunity to three periods, where the price can the down between periods 2 and 3 just as it could between periods 1 go up or and 2. We show how this changes the value of the Option. Next, we examine uncertainty in the costs of the project and in the interest rate that is used to 'window''

discotlnt futtt re p rtlh-t Ilows. F inallv. &ve consiler clloictl bc tNveel) prtlittcts (-81* differen t scltles. 11 Iarce r oroiect llllvil1c hicller Iixed u'tlsts btlt ltlvver()7e rtttilltt costs. Even with tllese extensions

antl variations,

the analysis remaills

at the

level of an illustrative and very simple example rather than that of a theory with some claim to general ity. In Iater chapters we proctrtld t() ulevelop a broader theoretical framework. Btlt the example does yield some valuable insights that survive the generalization- and we summarize tllcm ht, re. First, the example shows that the opporttlnity cost of thk! option to invest

is a significant component of the firnl's investment decision. The option value increases with the sunk cost of the investment and with the degree of uncertainty over the future price, the downside component of the risk btling the most important aspect. These restllts are conlirmcd in more general models in Chapters 5-7. Second- we wiIl see that the optilln va Itle is not affected if t he firm is able the risk by trading in forward or futtlres marke ts. In t2flicien t markets hedge to risk is fairly priced, so any decrease in risk is offset by the decretstl in such The fonvard transaction is a financial operation that Ilas nt) eflkct ()n return. firm's real decisions. (Tl1is is another tlxample the of the Nlodigliani-M iIlt!r

theorem at

work

.

)

Third, whcn future costs arc tlncertltin- thoir dflktzt on the investment decision depends ()n thc particular llrm t)f the tlnccrtainty. l 1*tlle uncertainty pertains to tlle pricc the Iirm must plty for :111 inptlt. the eftkct is jtlst like that of output price risk. The freedom not to invest if the input price turns out t() have gone up is valuabld, st) immediat.e investment is Iess readily matle. l-lowever, instcad suppose that the project consists of several steps. the uncertainty pertains to the total cost of investment. and intbrmation about it will l7e rttvealed only as the first few steps of the project are undertaken. Then these steps have information value twer and above their contribution t() the c()nventionally calculated NPV Thus it may be desirable to start the project evcn if orthodox NPV is somewhat negative. Nvereturn to this issue in Chapter l () and model it in a more general theoretical framewllrk. Fourth, we will see that investment t)n a smaller scale. by increpsing future flexibility,may have a value that offsets to some dcgree the advantage that a larger investment may enjoy due to economies t)f scale. 4.B

Some Mathematical Tools

In reaiity, investment prqects can have ditlerent windows ol opportunity, and of the future can be uncertain in different ways. Thereforc the

various aspects

Itltrltittf:tit)tj

simple two-period examples of Chapter 2 must be generalized greatly before tlley can be applied. Chapters 3 and 4 develop the mathematical tools that are needed for such a generalization. Chapter 3 develops more general models of uncertainty.

We start by

explaining the nature and properties of stochastic processes. These processes combine dynamics with uncertainty. In a dynamic model without uncertainty, the current state of a system will determine its future state. When uncertainty is added. the current state determines only the probability distribution of future states, not the actual value. The specifcation in Chapter 2, where the current price could go either up or down by a Iixed percentage with known probabilities, is but the simplest example. We describe two other processes that prove especially useful in the theory of investment Brownian motion and Poisson processes-and examine some of their properties. Chapter 4 concerns optimal sequential decisions under uncertainty. We begin with some basic ideas of the general mathematical technique for such optimization: dynamic programming. We introduce this by recapitulating the two-period example of Chapter 2. and showing how the basic ideas extend to more general multiperiod choice problems where the uncertainty takes the form of the kinds of stochastic processes introduccd in Chapter 3. We establish the fundamental equation of dynamic programming, and indicate methods of solving it for the applications of special interest here. Then wc turn to a market setting, where the risk generated by the stochastic process an be traded by continuous trading of contingent claims. We show how the sequential decisions can be equivalently handled by constructing a dynamic hedging strategy a portfolio whose composition changes over time to replicate the return and risk characteristics of the real investment. Readers who are already familiar with these techniques can skip these chaptersaexcept perhaps for a quick glance to get used to our notation. Others not familiarwith the techniquescan use the chapters as a self-contained introduction to stochastic processes and stochastic dynamic optimization-even if their interests are in applications other than those discussed in the rest of the

book.

.,d Nev Izclv o)' /?lTr.y/nTc?l/

uncertainty. Then the investment decision is simply the decision to pay the sunk cost and in return get an asset whose value can lluctuate. This is dxactly analogous to the financial theory of call options the right but ntlt the oblivalue tbr a preset exercise price. gation to purchase an asset of ouctuating

Theretbre the problem can be solved directly using the techniques developed in Chapter 4. The result is also familiar from financial theory. The option can is the money'' . when the value of the asset rises be profitably exercised above the exercise price. However, exercise is not optimal when the option is onlyjust in the money, because by exercising it the lirm gives up the opportunity to wait and avoid the Ioss it would suffer should the value fall. Only when the value of the asset rises sufsciently above the exercise price the option is in the money''--does sufhciently its exercise become optimal. An alternative tbrmulation of this idea would help the intuition of economists who think of investment in terms of Tobin's q. the ratio of the value of a capital asset to its replacement cost. In its usual interpretation in the investment literature. the value of a capital asset is measured as the expected present value of the prolit tlow it will generate. Then the conventional criterion for the firm is to invcst when (/ equals or exceeds unity. Our option value criterion is more stringent', q must cxceed unity by a sumcient margin. It must equal or excced a critical or threshold value (?*. which itself excceds unity, beibre investment becomcs optimal. We perform several numerical simulations to calculate the value of the option and the optimal exercise rule, and examine how these vary with the amount of the uncertainty, the discount rate. and other parameters. We (ind that tbr plausible ranges of these parameters, the option value effect is quantitativelyvery important. Waiting remains optimal even though the expected rate of return on immediate investment is substantially above the intcrest rate ()k' return on capital. Return multiples of as much as rate or the two or three times the normal rate are typically needed before the 5rm will exercise its option and make the investment. 'in

Gdeep

wtnormal''

4.D The Firm's

lnvestment

Decision

These techniques are put to use in the chapters that follow. Chapters 5-7 constitute the core theory of a hrm's investment decision. We begin in Chapter 5 by supposing that investment is totally rreversible. Then the value of the project in place is simply the expcctcd prcscnt valuc of th stream of prolits (or losses) itwouid generate.This can be computed in terms of the underlying

Interest Rates and Investment

Once we understand why and how (irms should be cautious when deciding whether to exercise their investment options, we can also understand why interest rates seem to have so Iittle effect on investment. Econometric tests of the orthodox theory generally 5nd that interest rates are only a weak or insignihcant determinant of investment demand. Recent history also shows that inlerest rate cuts tetld to llavr unly a limited stimuiative effect on investment', the experience of 1991-1992 bears the latest witness to that. The

IntroflLtctiolk

'

options approach offers a simple explanation. A reduction in the interest rate inakes the ftlture generally more important relative to the present, but this increases the value of investing (the expected present valtle of the stream of proEts) and the value of waiting (the ability to reduce t)r avoid the prospect

of tkture losses) alike. The

net effect is weak and sometimes even ambiguous. real apprtach also suggests that various sources of unThe options

certainty about future prolits quctuations in product prices, input costs, exchange rates, tax and regulatory policies have much more important effects on investment than does the overall Ievel of interest rates. Uncertainty about the future path of interest rates may also affect investment more than the general level of the rates. Reduction or elimination of unnecessary uncertainty may be the best kind of public policy to stimulate investment. And the uncertainty generated by the very process of a lengthy policy debate on alternatives may be a serious deterrent to investment. Later, in Chapter 9, we construct specisc exaniples that show how policy uncertainty can have a major negative effect on investment. 4.E

Suspension and Abandonment

Chapters 6 and 7 extend the simple model. As future prices and/or costs Iluctuate, the operating profit of a project in place may turn negative. In Chapter 5 we assumed that the investment was totally irreversible, and the hrm was compelled to go on operating the project despite Iosses. This may be true of some public selwices, but most firms have some escape routes available. Chapters 6 and 7 examine some of these. In Chapter 6 we suppose that a lossmaking project may be temporarily suspended, and its operation resumed Iater if and when it becomes profitable again. Now a project in place is a sequence of operating options; its value must be found by using the methods of Chapter 4 to value all these operating options. and then discount and add them. Then the investment opportunity is itself an option to acquire this compound asset. In Chapter 7 we begin by ruling out temporary suspension, but we allow permanent abandonment. This is realistic if a live project has some tangible ()r intangible capital that disappears quickly if the project is not kept in operation-mines Ilood, machines rust, teams of skilled workers disband, and brand recognition is Iost. If the hrm decides to restart, it has to reinvest in alI these assets. Abandonment may have a direct cost; tbr example, workers may have to be given severance payments. More importantly, however, it often has aIl uppurtunity cusi-ihe loss uf the optiun to preserve the capitai so it can be used prohtably should future circumstances improve. Theretbre a firm with

Nvill

tlllt!ratc some Iosses to k'ektp th is opt ion a proiect in place sufficientlyextren'le losses will il'ldtlctl it to abandon.

al ive, and ollly

have he re an interl i1) ketl pa ir of options. svlltlll t he fi 1-11-1exe rcises its option ttl invest, it gets :1. proiet:t i11 place antl an opt il'n to abandtln. If it exercises the option to abandllll. it gets the option to invest again. The two options must htl priced simultltndously to determine tlltl opti nlal investment and abandonment policies. Tlle link age has important implitrations', for examples11 higher cost t)f abandonment makes the firm even more cautious about investing, and vice versa. 'We illtlstrate tlle theory using numbers that are typical of the copper industry. and tind a very wide range of fluctuation of the price of copper between the investment and abandonment thresholds. 'We also consider an intermediate situation. where both suspension and at di fferen t costs. A project in suspension requ ires abandonment are llvailable expenditurt't. ongoing for exannplc, keeping a ship Iaid up. but restarting some is cheap. Abandonment saves on the rnaintenltnce cost. and may even bring some immediate scrap value. but then the l'ull investment cost must be incurred again if proti t potential recovers. Now we must determine the optimal switches between tllrce altcrnlltives an idle jirm, an operating project. and project. suspended We tllis. antl dk) illustrate it lbr the case ()f crudd oil a I n tct

&ve

tankers. 4.F

Temporary

versus

Permanent

Employment

While most ofour attention in this book is k)n tirms-capital investment choices. similar considcrations apply to their hiring and Iiring decisions. Each of these choices entails sunk costs, each decision must be made in an uncertain environment, and each allows some freedom 01' timing. Thereforc the above ideas and results can be applied. For exampltt, a new worker will not be hired until the value of the marginal product of labor is sufliciently above the wage rate, and the required margin or multiple above the wage will be higher tlle grcater the sunk costs or the greater the unccrtainty. The U.S. Iabor market in mid- 1993 offered a vivid illustration of this theory in action. As the economy emerged from the rccession of the early 199()s.hrms increased production by using overtime and hiring temporary workers even for highly skilled positions. But permanent new full-time hiring was very slow to increase. The current Ievel of prolitability must have been high, since firms were willing to pay wage premiums of 5() percent for overtime alld io usc ugeflcies that suppiied temporar.ywgrkors and charged fees Of wufk 25 percent or more of the wktge. But these same firms were not willing to make

lntrlllctioll

z.l Ne' Izc'w o)- /ll I'tz.$;f/?

the commitment involved in hiring regular new workers.s Our theory offers 11 natural explanation for these observations. A high level of uncertainty about future demand and costs prevailed at that time. The robustness and durability of the recovery were unclear-, it was feared that inllation would return and lead the Federal Reserve to raise interest rates. Future tax policy was very uncertain, as was the level of future health care costs that employers would have to bear. Therefore we should have expected firms to be very cautious, and wait for greater assurance of a continued prospect of high Ievels ofprofitability before adding to their regular full-time labor force. in the meantime, they would prefer to exploit the current proit opportunities using less irreversible (even if more costly) methods of production, namely overtime and temporary work. That is exactly what we saw. 4.G

IcpTf

The path dependence can leatl to the lbilowing kind of sequence ot' events. When the tirm (irst arrives on the scene and contemplates investment. the current protit is in the intermediate range between the two thrcsholds.

Theretbre the lirm decides to wait. Then proiit rises past the upper thresllold.

so the lirm invests. Finally. profit falls back to its old intermediate Ievcl. btlt that does not take it down to the Iower threshold where abandonment would occur. Thus the underlying cause (currentproiitability) has been restored to its old level. but its effect (investment)has not. Similar eftkcts have Iong been known in physics and other sciences. Tlle most familiar example comes from electromagnetism. Take an ironsar andt' loop an insulated wire around it. Pass an electric current throtlgh the wire--k the iron will become magnetized. Now switch the current off. The magnetism is not completely lost; some residual effect remains. The cause (the current) was temporary, but it Ieaves a longer-lasting eftkct (the magnetized bar). This phenomenon is called Ilvsresis. and by analogy the failure of investment decisions to reverse themselvcs when the underlying causes are fully reversed can be called cconomic hysteresis. A striking example occurred during the l 980s. From l98() to I984. thtt dollar rose sharply against other currencies. The cost advantage of foreign lirms in US markets became very substantial. and ultimately Ied to a Iarge rise in US imports. Then the dollar fell sharply. and by l987 was back to its l t)8() Ievel. However, the imptlrt penetration was not fully reversed: in fact it hardly decreased at :111. It took :1 largcr fall in the dollar to achieve any signihcant rcduction in imports.

''

Hysteresis

When we consider investment and abandonment (orentry and exit) together. a firm's optimal decision is characterized by two thresholds. A suffciently high current level of proEt, corresponding to an above-normal rate of return on the sunk cost, justilies investment or entry, while a sufficiently large Ievel of current loss Jeads to abandonment or exit. Now suppose the current level of protit is somewhere between thcse two thresholds. Will we see an active firm? That depends on the recent history of protit iuctuations. If the protit is at its current intermediate level having most recently descended from a high level that induced entry. then there will be an active firm. However, if the intermediate Ievel was most recently preceded by a 1owIevel that induced exit, then there will not. In other words, the current state of the underlying stochastic variable is not enough to determine the outcome in the economy'. longer histor.y is needed.a-he economy isgathdevendent. ideaofpathdependence and illustrated, hasbeen recentlyexplored rominently and by Artlaur 19aa). p (l9a6) oavid(19as, vheyallow an possibility: evenverylong-run propertiesof theirsystems more extreme altered by slight differences in initial conditions. uere we have a more kind of path dependence.-rhe long-run distribution of the possible o t economy is unaltered, but ti:e short- and medium-n,n evolution still be dramatically affected by initial conditions.

4.H 1 .1

a 'rhe t mos even are moderate stateshe can

'j 1,

!

-...

,.z.''

.

Industry Equilibrium

In Chapters 8 and 9 the focus turns from a hrm's jnvcst mtl n t decisit) ttl ns the equilibrium ot a wbolu industry composcd ot many such urms. o ne.s srstreaction might be tat tlae competition among urmswill destroy anyone firm's option towait- eliminating the efocts ofirreversibilty and uncertainty that we found in 5-7. does each destroy firm's option chaptcrs competition to wait, but this does not restore the present value approach and results of the orthodox theoe. on tlae contrary, caution when making an irreversible decision remains important- but different reasons. orsomewhat Iirmcontemplating its investment-knowingthat the considerone oture of induste demand and its own costs are uncertain, and knowing that thereuqre many other firms facing similar decisions with similar uncertainty. The Iirrlais ultimatelyconcerned with tue.eonsequences of its decision for its Own profift, but it must recognizc htok't'he similar-dcisions of otllrr Iirms will affect it. In this respect, two types ofcertainty must 6'v distinguished because q j j

patla

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.

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lntrodttctil

they can have different implications for investment: aggregate uncertainty that affects all hrms in the industly and Iirm-specilic or idiosyncratic uncertainty facing each Iirm. To see this, tirst suppose that investment is totally irrcversible, and consider an industrrwide increase in demand. Any one Iirm expects this to lead to a higher price, and so improve its own prolit prospects, making investment firms are making more attractive. However. it also knows that several other effect of the dea similar calculation. Their supply response will dampen the mand shift on the industry price. Therefore the upward shift of its own protit potential will not be quite as high as in the case where it is the only rm and has a monopoly on the investment opportunity. However, with investment being irreversible, a downward shift of industry demand has just as unfavorable an eftkct in the competitive case as in the monopoly case. Even though other competitive hrms are just as badly affected, they cannot exit to cushion the fall in price. Thus the competitive response to uncertainty has an inherent asymmetry: the downside exerts a more potent effect than the upside. This asymmetry makes each tirm cautious in making the irreversible investment.

The ultimate effect is very similar, and in some models identical, to that ofthe option value for a firm possessing a monopoly on the investment opportunity and waiting to exercise it. In fact, the theory of a competitive industry can be formulated by giving each tirm the option to invest, valuing these real options the competitive as in Chapters 5-7, and fnally imposing the condition that in should value be option this zero. equilibrium If we allow some reversibility, the exit of other lirms does cushion the effect of adverse demand shocks on price. But then each firm*s exit decision recognizes this asymmetric effect of demand shocks in an initially poor situation: their upside effect is more potent than the downside. Thus competitive firms are not quick to leave when they start to make operating losses; they wait a while to see if things improve or if their rivals leave. The overall effect is just like that for a single monopoly firm's abandonment decision that we found in Chapter 7. ln fact, the competitive equilibrium model of joint entry and exit decisionswith aggregate demand shocks thatwe analyze in Chapter 8 has exactly the same critical Ievels of high and Iow prices to trigger entry and exit as the corresponding monopoly model of Chapter 7. Firm-specific uncertainty does not lead to this kind of asymmetly. lf just experiences a favorable shift of its demand, say some idiosyncratic hrm one switch of fashion, then it knows that this good fortune is not systematically shared by other Erms, and therefore does not fear tht entry ofother firmswill erode its prol'it potentiai in the same way. However, then llc vuluc uf waitillg reemerges in the older familiar form. The lucky hrm does have a monopoly

./4 Nebv P'c! $.' (

)j-

11l l

'(j'.b'tlt

ltv l

t

on the opporttlnity to enter witll its Iovvcost. Tl'lcrt.lt'oreit alstl llas 1111 optitln value Of waiting-v it can thereby avoid a Itlss if its Itnv cost sllould ttlrn otlt to be transitory. Tllus tirm-specitictlncertainty in industry equilibritlnl also Ieads to investment decisions si milar lo those fotl ntl in Cllaptc rs 5-7 l'or :111 isol a tetl srrn. 4.I

Policy Toward Investment

Some readers might interpret the rcsult that uncertainty makes li rnls Iess eager to invest as indicating a need for government policy intervention to stimulate investment. That would be a hasty reaction. A social planner also gcts information by waiting, and therefore should also recognize tlle opportunity cost of sinking resources into 11 project. A castl rorpolicy inte rvention will arise only if firms face :1 diftkrcnt value (.)f waiting than does society as :1 whole. in other words, if some market failure is associated with the decision

process.

Chapter 9 focuses on these issues. Our lirst result is :1 confirmatitln

01'

the standard theory of gene ral eq uil ibrium. lf markots for risk are complctc, and if Iirms behave as competitive price- takers (in this stochastic dynanlic context this must be interpreted to mean that ttach firm takes as given the stochastic proccss of thc pricc and has ratillnal expectations about it). then the equilibrium evolution ol' the industry is stlcially cfficient. A stlcial planner would show the same degree of llesitancy in making the investment dccision. If markets for risk are incompletc, bcnelicial plllicy intervtlntions dt) exthe correct policy needs some carcful calculation and implemttntlltion. The biunt tools that are often used for handling unccrtainty can havc adverse ()1* . effects. We illustrate this by uxamining the consequences prce ccilings and floors. For example, price supports promote investment by reducing the downside risk. However, the resulting rightward shift of the industry supply function implies lower prices in good times. Averaging over gotld times and bad, we tind that the overall rcsult is a Iower long-run averagtt price. In othllr words, the polic-yis harming the very group it sets out t() hclp. Pricc tll.ltlror ceiling policies. for example, urban rent controls and agricultural prictl stlpports. are usually criticized because they reduce overall econtlmic efficiency. Our finding is perhaps a politically more potent argument against thtm: their distributional effect can be pelwerse, too. We also study the effect of uncertainty concerning future policy itself. For example, if an investment tax credit is being discusscd, firms will recognize iiiot'e kalue 11'1wl'tl'llg, bcuausu llt;lc i: u pl ubabiliiy tilat the ctlst t)iinvestment to the firm will fall. We hnd that such policy uncertainty can have

ist, but

llttroelltctiolt

,,4New Iztl'v of

wish to a powerful deterrent eftkct on immediate investment. lf gtwernments spend a long stimulate investment, perhaps the worst thing they can do is to right way to do so. time discussing the

4.J Antitrust

Nrmscontinue their export operations at :1 loss, domestic firms allege predatory dumping and call for the standard trade policy response of countervailing import duties. However. our analysis suggests that the foreign firms may be simply and rationally keeping alive their option of future operation in our market, with no predatory intent whatever. Only a sufficiently Iong time series of data will allow us to test whether the supposed collusive or predatory actions are merely natural phases in the evolution of a competitive industry or genuine failures of competition.

and Trade Policies

Chapters 8 and 9 paint a very diftkrent picture of competitive equilibrium than the one familiar from intermediate microeconomics textbooks. There lhe industry if the price rises to equal the we are told that firms will enter long-run average cost, and they will exit if the price falls as low as the average variable cost. Our theory implies a wider range of price variation on either side. For example. in the face of aggregate uncertainty, firms' entry as soon will not constitute an industry as the price rises to the long-run average cost equilibrium. Each tirm knows that entry of other similar firms will stop the price from ever rising any higher. while future unfavorable shifts can push the price below this Ievel. Also, a future price path that sometimes touches the long-run average cost and otherwise stays below this level can never offer normal return on the firm's investment. Only if the price ceiling imposed

a by entry is strictly above the Iong-run average cost can the mix of intervltls of supernormal proit and ones of subnormal profit average out to a normal return. Similarly, tirms will exit only when the price ills sufhciently far below the average variable cost. They will tolerate some losses, knowing that thc cxit of other firms puts a Iower bound on the price. The equilibrium level of this qoor is determined by averaging out the prospects of future losses and profits to Zero.

Thus we hnd that competitive equilibrium under uncertainty is not a stationary state even in the long run, but a dynamic process wherc prices can luctuate qute wfdely. Periods of supernormal profits can alternate with perfods of losses. A similarviewofdynamic equilibrium as astochastic process has become quite common in macroeconomics, but is sumrisingly uncommon in microeconomics. particularly with regard to its implications or antitrust polic'yor international trade policy. The conceptual framework of such policies in practice are based on obseris generally static, and the recommendations particular of of industry instant. We :nd that the at a vations an rethinking of view substantial both the theory and the calls for a dynamic tsnapshots''

practice.

For example, in industrial organization theory, excess prohts suggest collusion or entry barriers, calling for antitnlst action. ln our dynamic perspective, substantial prriods of supernormal prohts w'ithout ncw cntry can occur cvcn though al1 hrms are small price takers. In international trade, when foreign

J?l1.t;'./?'?7t.vIl

4.K

. :: ..

Sequential and Incremental

Investment

In Chapters 10 and l 1 we return to a single tirm-s investment decision and examine some other aspects of it that are important in applications. Chapwhich must ter 10 deals with investments that consist of several stages, aIl of be completed in seuence before any output or prolit flow can commence. The 5rm can constantly obselwe some indicator of the future protit potential, and this fluctuates stochastically. At any stage. the firm may decide to continue immediately, or wait for conditillns to imprtwc. At an early stage ()f the investment sequence, most of the cost remains to be sunk. Therefore the will go ahead with the program only if it sees a sufficiently high threshold 517)1 level of the indicator of profitability. Gradually. as more steps are completcd and less cost rcmains to be sunk, the next step is justified by ap ever smaller threshold. In this sense, bygones affect the decisions to come. In Chapter 10 we examine another effect of current decisions on the f'uture, namely the learning curve. According to this theory, the cost ofproduction at any instant is a decreasing function of cumulated output experience. Thus the current output t1()wcontributes to a reduction in alI future production costs. This additional value must be added to the current revenues before comparing tem to the current costs of production to decide on the optimal level of production. We examine the dynamic output path under these conditions. We 5nd that greater uncertainty lowers the value of future cost reductions, and that Ieads to a reduction in the pace of investment. In Chapter 11 we turn to the study of incremental investment, where outand profit flow are available al1 the time as a I'unction ot' the installed stock put capital. The aim is to characterize the optimal policy for capacity expansion. of When production shows diminishing returns to capital, we can regard each new unit of capacity as a fresh project, which begins to contribute its marginal product from the date of its installation. Then the criterion derived in ChapiIl tel 6 for illvrstlllellt suel'l it pfojet contknues to apply. If pi'oduction shows increasing returns to capital over an interval, then all the units of capacity in

22

Iltlrodttctiolt

:tn appropriately constructed range must be regarded as a single project. and the criterion tor its installation is again a natural generalization of thpt tbr a .j, single project described in Chapter 6. When the Iirm can choose its rate of capacity expansion, we must specify'

how the costs of this expansion depend on its volume and pace. Different assumptions in this respect imply different optimal policies. We construct a general model that places the alternatives in context. and in particular shows the relationship between the adjustment costs models that have been the mainstay of theoretical and empirical work over the last decade and the irreversibility approach that has been the focus of our book. Empirical

and Applied Research

In Chapter 12 we turn to some examples that illustrate applications and extensions of the techniques developed throughout the book. We also discuss the relevance of the theory for empirical work on investment behavior. We begin Chapter 12 with a problem of great interest to oil companes how to value an undeveloped offshore oil reserve, and how to decide when to invest in development and production. Aswewill see, an undeveloped reserve is essentially an option; it gives the owner the right to invest in development of the reserve and then produce the oil. By valuing this option we can value the reserve and determine when it should be developed. Oil companies regularly spend hundreds of millions of dollars for offshore reserves, so it is clearly important to determine how to value and best exploit them. We then turn to an investment timing problem in the electric utility industry. The Clean Air Act calls for reductions in overall emissions of sulfur dioxide, but to minimize the cost of these reductions, it gives utilities a choice. They can invest in expensive to reduce emissions to mandated Ievels, or they can buy tradeable that Iet them pollute. There is considerable uncertainty over the future prices of allowances, and an investmcnt in scrubbers is irreversible. The utility must decide whether to maintain Pexibility by relying on allowances or invest in scrubbers. We show how this problem can be addressed using the options approach of this book. To show how the principles and tools developed in this book have relevance beyond hrms' investment decisions, we address a problem in public policy when should the government adopt a policy in response to a threat to the environment, given that the future costs and benehts of the policy arc uncertain? We will argue that the standard cost-benefit framework that cconomists havc traditionally uscd to uvaluate envii-onnpental policies is dellcient. The reason is that thcre are usually important irreversibilities associated with environmental policy. These irreversibilities can arise with respect to '<scrubbers''

'allowances''

environlnental dnlnage itself. btlt also with respect to tlld costs of adapting t() policies to reduce the daluage. Since tlltr atloption of ::n envinlnnlental policy is rarely a now or ntwer proposition. the sanle tdchniques used to sttldy tllt.l to the optinlal tim ing of an optimal ti m ing ot' an investment can be lpplied environmental policy. At the end of Chapter 12. we disctlss some of the enlpirical inlplicat itlns

of irreversilnility nnd unccrtainty for investl-nent behavior. Tllere is trollsitltlrable anecdotal evidence that firms make investment decisions in :1 way tllat is at least roughly consistent with the theory developed in this book. tbr tlxalmple, the use of hurdle rates that are much larger than the opportunity txlst of capital as predicted tnythe capital asset pricing model. Some ntlmdrical simulations based on plausible parameters for costs and tlncertainty are found to replicate features of reality. However, more systematic econolnetric testing of the theo!y is still at a very early stage. Nve review some work of this kind. point out its difficulties, and suggest ideas for ftlture research.

5

Noneconomic Applications

Our tbcus in the book will be ()n investment decisions implications for industry equilibrium. Tllese matlers are

()f

lirms antl their

01'

primary interest

to economists. their conditions (technologyand rcstlurcc avktilability) antl of the value t)f the firm) arc gcnerally agrced uplln, antl criteria (maximization theory leads to quantiliable predictions about them. lnvestment decisions of consumers (purchas:ofdurablcs) and workers (educationand human capital) have obvious parallels. Research on these issues already cxists. and wtl will sketch some of this literature in Chapter I 2. Many other personal and societal choices are made undcr the same basic

conditionss namely irreversibility, ongoing uncertainty, and some leeway in timing. Therefore we can think about them in terms of some tlf the models and results outlined above, and offer some qualitative speculations. We believe there is scope for more seriotls research along these Iines. This would amount to extending work such as that of Becker ( 1975. l 98()) on econtlmic approaches to social phenomena by incorporation of option values. Indecd, some recognition Qf this aspect can be read nto Becker's own remarks on the subject, for example, in Becker (1962,pp. 22-23), but a more thorough and formal analysis along these lines remains a potentially fruitful project tor the future. Although we cannot spare much splce in an already lengthy book, we believe that readers will Iind some speculations along tllese Iines both interesting and thought-provoking.

Introdttciotl

5.A Marriage and Suicide Marriage entails signilicant costs of courtship, and divorce has its own mon-

etary and emotional costs. Happiness or misery within the marriage can be only impertkctly tbrecast in advance. and continues to lluctuate stochastically even aler the event. Therefore waiting tbr a better match has an option value, and we should expect prospective partners to look for a sufhciently high initial threshold of compatibility to justifygetting married. We should expect the option value to be higher in religions or cultures where marriage is less reversible. Therefore we should expect that. other things being equal, individuals in such societies will search more carefully (andon the average, search Ionger), and will insist on a higher threshold of the quality of the match. On the contrary. when divorce is easy. we should see couples entering into marriage (or equivalent arrangements) more readily. Of course. other things are not equal. Societies that make divorce more

difficult presumably attach higher value to marriage. Therefore we should expect them to counteract the rational search and delay on the part of individuals technologies.'' Emby social pressure. and provision of bettcr of these will carefully ideas designed have be tests to pirical to distinguish the and only raises the interest and opposing often influences. but that separate research. such of challenge ''matchmaking

Perhaps the most extreme example of economics applied to sociological phenomena is the Beckerian theory of suicide developed by Hamermesh and Soss (1974). According to them. an individual will end his or her own life when expected the present value of the utility of the rest of Iife falls short ofa benchstandard cutoff Most people react by saying that the model mark or view rational of what is an inherently irrational action. gives an exccssively opposite. Whatever its merits or demerits as the Our theory suggests exactly descriptive theofy, the Hamermesh-soss model is not rational tr/ltll/q/l from the prescriptive viewpoint. because it forgets the option value of staying alive. Suicide is the ultimate irreversible act, and the future has a lot of ongoing will uncertainty. Therefore the option value of waiting to see if improve'' should be very large. The circumstances must be far more bleak than the cutoff standard of the Hamermesh-soss model to justifypulling the trigger. This is true even if the expected direction of life is still downward; all that is needed is some positive probability on the upside. Now return to the argument that most suicides are irrational, and ask exactly how they fail to be rational. There are' several possibilities, but one seems especiaiiy pertinent. Suicidesproject the bieak present into an equaiiy bleak future, ignoring uncertainty, and thereby ignoring the option value of 'tzero.''

ttsomething

,4 tge.v Piku,

tp-/' lllb

'tl-//'?lt.vl/

life. Then religious or social proscriptions against suicide selwe a useftll ftlllction as measurcs to compensate for this tiltlre of rationality. By raising the perceived cost of the act. these taboos lower the threshold of quality of litk that leads to suicide when option value is ignored. This can correct the failure of the individuals' forethought, and bring their threshold in contbrmity with the optimal rule that recognizes the option value. 5.B

Legal Reform and Constitutions

Finally, consider law reform. Different t-undamentalconstitutional or legal principles often conllict with one another in specific contexts', for examplethe civil rights of the accused often stand in the way of better enforcement of law and order. Public opinion as to the appropriate rdlative weight to be attached to these contlicting principles can shift over time. How quickly sllould the 1aw respond to the latest swing of opinion'? G iven some legislative antl administrative costs of changing laws. our theory suggests that the option to wait and see if the trend ()f opinion will reverse itself has some valutl. Reform should be delayed until the forcc of current opinion is sufticicntly overwhelming to offset this value. But there is good reason to believe that market failure.-' At the proccss is subject to a kind of myopia or all timess people and politicians tend to believe that at last they have got it right; the balance ()f opinion that has just been reached will prevail tbrcvor. Thereforc they will ignorc futurc unccrtainty and (lption values, nnd change the laws too frequently. The time to anticipate and defcat this tendency is when the cllnstitution is framed. Knowing that futurc generations will be too trigger-happy in changing laws. the founding fathers can artilicially raisc the cost ()f change, thereby making the politically flawed thresholds for Iegal changc coincide with the truly optimal thresholds. Thus various supermajority requirements for c()nstitutional change can be seen as a commitment device that corrccts for the shortsightedness of future generations. These are just a few examples of how the theory and ttlchniques ddveloped in this book are applicable to some fairly far-reaching problems. Wk have deliberately stated them in speculative and provocative ways. hoping to attract interest and research along these Iines from a broader spectrum of social scientists. In te rest of the book, we will focus largely on investmenta but ask the reader to keep in mind the much broader potential applicability of the thcory. t&ptllitical

Illttlttllillg

Chapter

2

Developing the Concepts Through Simple Examples

illvestnldllt Ilrl'jl'llt'tllls il1clllltintllptls tillle. Fillktlly. ve hvill stage for ilxlhn-lille 01* t l1eillvestlllellt vllell tlle illvestlnent tlecisitlns brielly (:lst'p:ll'!tl tlnccrttti posed to its payllff) is n- when futtl 1.t? i1-1ttt rest rates 1.1re tlllce rt :1 111.().t

when one must choose the scllle

01-

an investnlent.

1 Price Uncertainty Lasting 'lWo Periods Consider a firm thllt is trying to ddcide whether to invest in :1 widget factory. The investment is colnpletely irreversible the facto!y can only l7e tlsed to make widgets, and Should the nlarket for wdgets evaporate. t he firlu cannot teuninvest''and recover its expenditure. To keep Inattors as silnple :'s possible. we will assume that the factory can be built instantly, at a cllst / and will produce one witlget per year forever. with zero operating cost. Currently the price of a widget is $2()()-but ncxt year the price will change. Witll prtlbltbility t/it will rise to $3()().and with probability ( l t/ ). it will 1*1111 to $ 1()(). Tlpe price will then remain at this new level tbrever. (stltlFigtlre 2. l Again. to keep things simple, we will assunle that the risk ('ver tlle futtlre price of widgcts is fully divcrsitiable (that is. it is tlnrdlatcd tt) wllat llappens with the overall economy). I-lence the Ii rln slltltlltl tliscoullt futtl rc cash sows tlsing the risk-frce ratc ()f interest. wI1ich wtl will tlk'e t() be l () -

-

.)

FlltMs

MAKE,

implement, and sometimes revise their investment decisions con-

tinuously through time. Hence much of this book is devoted to the analysis problems. Idowevdr, it is best to of investment decisions as continuous-time begin with some simpl examples, involving a minimal amount of mathematics, in which investment decisions are made at two or three discrete points in time. In this way, we can convey at tle outset an intuitive understanding of tlf the basic concepts. In particular, we want to show how the irreversibility and how it requires an investment expenditure affects the decision to invest, modihcation of the standard net present value (NPV) rule that is commonly taught in business schools. We also want to show how an opportunity to invest is much Iike a hnancial option, and can be valued and analyzed according.

This chapterbegins by discussing these ideas in the context of a simple example in which an investment decision can be made in onlyone oftwo possible periods now or next year. We will use this example to see how irreversibility creates an opportunity cost of investing when the future value of the project is uncertain, and how this opportunity cost can be accounted for when making the investment decision. We will also see how the investment decision can be analyzed usng basc option pricing techniques. We will examine the characteristics of the firm's option to invest in some detail, and see how the value of that option, and the investment decision, depend on the degree of uncertainty over the future value of the project. Extending the example to three periods will provide more insight into the problem o investment timing, and set the

26

percent.

For the time being we will set / ().5. (Lalcr we will slltl 1($1)()and t/ how the investment decision depends ()11 / and t/ G ivcn tlltlstl vlllutls Iklr / and q, is this a gool.l invcstment'? Should the firm invest nllw, or wtluld it lptt better to wait a year and sce whether the price ()1' widgets gtles up ()r tlown? Suppose we invcst now. Calculating the net present value ()f th is investnlent in the standard way (and noting that the expeaetl future price of widgets is =

=

.)

l

1

=

l

P,

=

$300

P:

=

$10 O

%

----'-

q Po

=

=

2

=

30O

=

1 O0

$200 (1

-

q)

2b*i;: Iz p-tl

.33.

j/

pz lJt?bz't.

.

ryj4'

je'itlltk.t.

'--'*-

Illtrlltitt:til';

always S200), we get NPV

=

-

1600 +

x

x() V (1.1)

=

t

J=()

-

1600 + 2200

=

$600.

(l)

It appears that the NPV of this project is positive. The current value of a widget factory, which we will denote by f.$l, is equal to $2200,which exceeds the $ 1600 cost of the factory. Hence it would seem that we should go ahead with the investment. This conclusion is incorrect, however. because the calculations above ignore a cost the opportunity cost of investing now. rather than waiting and keeping open the possibility of not investing should the price fall. To see this. let us calculate the NPV of this project a second time, this time assuming that instead of investing now. we will wait one year and then invest only if the price of widgets goes up. (Invsting only if the price goes up is in fact ex post optimal.) In this case the NPV is given byl NPV

=

(0.5) -

1600

l.1

'''Z Y'N

+

''

I

=

l

300 (1.1)/

=

850 1. j

=

Developing ?/lc Cllll()epls F/l/-f?/(/l

f-rtl/?l/p/t'-

How much is it worth to llave the flexibility to makd the investment decision next year, rather than llaving to invest either now or never'? (NVc know that having this tlexibilityis of some value, because wtt would pretkr option-f is easy to wait rather than invest now-) The value of this to calculate; it is just the difference between the two NPV-S. that isp$773 $600 173. In other words, we should be willing to pay $ 173 more tbr an investment opportunity that is llexible than one that only allows us to invest biflexibility

-

=$

n0W.

Another way to Iook at the value of tlexibility is to ask the following question: How high an investment cost l would we be willing to accept to have a llexible investment opportunity rather than an inllexible or never'' one? To answer this, we tind the value of 1, which we denote by 7, that makds the NPV ()f the project whtln we wait eqtllkl to the N PV whe n I $ I (A()() and we invest now. that is. equal to $6t)().Substituting l for the $ l )()() and substituting $600 for the $773 in etluation (2): --now

=

N PV

$773.

(Note that in year (), there is no expenditurc and no revenue. In year 1. the $1600 is spent only if the price rises to $30(),which will happen with probability 0.5.) If we wait a year betbre dcciding whether to invest in the factory, the project's NPV today is $773.whereas it is only $600 if we invest in the factory now. Clearly it is better to wait than to invest right away. Note that if our only choices were to invest today or never invest. we would invest today. In that case there is no option to wait a year, and hence applies. no opportunity cost to killing such an optionv so the standard NPV rule We would likewise invest today if next yearwe could disinvest and recover our $ 1600 should the price of widgets fall. Two things are needed to introduce an irreversibility, and the ability to opportunity cost into the NPV calculation invest in the future as an alternative to investing today. There are, of course, situations in which a firm cannot wait, or cannot wait very long, to invest. (One example is the anticipated entry of a competitor into a market that is only large enough for one hrm. Another example is a patent or mineral resource lease that is about to expire.) The less time there is to delay, and the greater the cost of delaying, the less will irreversibility affect the investment decision. We will explore this point later in this book as we develop more general models of investment.

vb-llplc

(().5)

=

-7 l l

x

-/

3(j()

+

.

=

l

( l l )l

=

$()1)

.

.

-1

$ l t.)8(). In tlther Solvingthis for yields (l/lf/ ()lllv ltobv llow :tt a ctlst factory widgct the widget llctory llob' build opportunity to

vvords.

the (lpportunity to build :t v:tlue 1 the $ t)()()has sanle as an (,1, fJr llext S 1t?8(). yctlr at a ctlst with the fulr widgets. Finally, suppose therc exists 11 l'utures market tlxpected (ne equal thu price for delivery futtlrc sptlt t() tures year thlm ntlw price. that is. $2()().2N'Wluldthtt ability t() hetlge on the futurcs market changc =

tlt-

our investmcnt decision'? Specilically, wlluld it Iead us to invest now, rlther than waiting a year? The answer is n(). see this. consider investing now ttnd hedging the price risk with futures cllntracts. X) hedge all price risk. we wtluld need to sell short futures for l 1 widgets; this would exactly offset any fluctuations in the NPV of our project next year. (If the price of widgets rises to $3()(). our project would be worth T33()().but we would lose Sl l ()() on thc futurcs contract. If the price tkll to $ 1()t),our project would be worth only $ 1l ()()- but we would earn an additional $ l l()()from the futures. Eithcr way, wll end up *1'il

:ln this exampie. the futures pric? woultl equai thc expcctcd future pricc because wc assumed that the risk is fully diversiliablt!. ( If the price ()1* widgets were ptlsitivttly ctlrrclatcd wilh the markct portfolio, thc futures price wtluld be less than the expected future sptlt price.) Ntpte that if widgets were storablc and aggrcgllte storage is pllsitive. the marginal ctlnveniencc yield from holding inventory would havc to be I 1) percent. The rcason is that since thc futurus price equals the current sptlt price. thc nct htllding cllst (thc intcrest ctlst ()f I t) pcrcvnt Icss thc marginal conveniencuyield) lnust be zerl).

Llevelqalg

Illl-ltlllcitlll

3()

with 11 net project value of $220().)But this would also mean that the NPV of less the 1&)0 cst ()f the investment), exactly our project today is Slt) ($2200 withotlt hedging.s what it is Hence there is no gain from hedging. and we are still better off waiting until next year to make our investment decision. This result is a variant of the .$

Modigliani-Miller ( l 958) theorem. Operating in te futures market is only policy. and barring the possibility of bankruptcys it has a form of snancial value of the tirm's no real consequences for investment decisions or for the investment opportunities.

to Financial

Analor

1.A

Options

Our investment opportunity is analogous to a call option on a common stock. It gives us the right (whichwe need not exercise) to make an investment expenditure (theexercise price of the option) and receive a project (a share of stock) the value of which lluctuates stochastically. ln the case of our simple the lz?/n(.,y,'' meaning that if it were example, we have an option that is option is said to be payoff. (An would yield positive net a exercised today it negative exercising of tlte lnoney'' yields net payoff.) We today if it a it is money,'' better the to wait rather found that even though the option is will exercise our rises to $300. we than exercise it now. Next year if the price will worth fz'l which receive be $33()() option by paying $1600and an asset will be worth only $ l 1()(), asset falls 10(),this If the price 300/( to S 1 )f Ej'''j and so we will not exercise the option. that We found that the value of our investment opportunity (assuming the actual decision to invest can indeed be made next year) is $773. It will be helpful to recalculate this value using standard option pricing methods, because laterwe will use such methods to analyze other investment problems. To do this, let F'tl denote the value today of the investment opportunity, that is, what we should be willing to pay today to have the option to invest in the widget factory, and 1et Fl denote tbe value of this investment opportunity nextyear. Note that Fl is a random variable', it depends onwhat happens to )Jthe price of widgets. lf the price rises to $300,then Fk will equal S'') 300/( 1 1600 $1700.On the other hand, if the price falls to $100,the option to invest ''f??

Lkout

nin

=

=

.1

.

.1

-

=

aexnl.q., tc horizens of a uea.r or so. If tb.ere wf.wre f' .3Ntzst futures markets r' r . for widgets over the indeh' nite future, an equivalent hedge wou ld be to sell short one each futu re year, that is, to sell forward tht! enti re output stream. The resul t wou Id be tznlv

.

J

:k

Itl,res

4,nrd rk'et

widget

in

the same.

//1tJ

oll(.'t:ilts F/l?'t'pI/g/?

'?llp&,

E--b-tllllil..b'

willgo unexercised. and in that case /--1will equal (). Tllus wtl know al I ptlssible values for F3 The problem is to Iind 6-). the value of thd optitln today.i .

To solve this problem, wc will creattl a portlbl io that has two tron-lptlntt 1-1ts: the investment opportuni ty itsel anl :1 certain Iltllnbe r of widgets. NVe wil I pick this number of widgets so that the portfolio is risk-fret!, that is- so that its value next year is independent of wllether the price ()t' widgets gtes up t)r down. Since the porttblio will be risk-frce. we know that tlle rate of rettlrn (lntt can earn from holding it must be the risk-gree ratc of interest. (I f the rettlrll on the portfolio were Iligher than the risk-frec rate. nrbitragetlrs cotlld earn unlimited amounts of money by borrowing at the risk-frt!tl rnte and btlying the portfolio. If the portfolio's return were Iess than the risk-free rate. arbitrageurs could earn money by selling short the porttblio and investing tlle funds at the risk-free rate-) By setting the portfolio's return equal to the risk-free rate, we will be able to calculate the current value of thc investment opporttlnity. Specitically. consider a portfol io in wllich one l:()ItIs the investment (3pportunity, and sells short n widgets. ( Ifwidgets were a traded commodity. such as oil, one could obtain a short position by borrowing frllm nnllther prodtlceror by going short in the futures market. For tlle moment, howevera sve need not be concerned with the actual implementation of this pllrtfolio-) The value of this porttblio today is m4) /$) tl J.1) /$) 2()()??. The value of the ptlrtF 1 lt /$ This depends ()n l9b I1' Pl turns llut to be folio next year is * I that Fi l 7()0, then * l l 7()() 3t)()??. If /RI turns out t() be $ l ()(). $300. so 1t. that then F ()() * l (). N()w, let us chotlse 11 s() that the ptlrtfolil) I so risk-free, that that * I is independent t)f what hitppens t() tlle price. is, s() is do this, just set =

=

=

-

.

=

.

=

=

-

-

=

-

-

-l')

)() 3()()11 177( --

=

--

l ()()l l

.

whether the price ()t' 8.5. With /1 (lhosen in this way, tp I widgets rises to $3()()or falls to $ 100. Now let us calculate the return from holding this portfolio. That return is the capital gain, *1 *), minus any payments that must bc made t() hold tht.l short position. Since the expected rate of capital gain ()n a widget is zero (the expected price next year is $200, the same as this year's price). n() rational investor would hold a Iong position unless he or she could expect to earn at least 10 percent. Hence selling widgets short will require a payment of -85()-

or, n

=

=

-

4In this example. aII uncertainty is resolved next year. Therefllre an ()p(i()n to wllit has Ilt) convttntillnal NI'V critcrilln at that timc. Waiting has relevance only this year. ln Iatt!r chapters we will ctlnsider mllre gencnll sl'tuatitlns to wait rtltllins value al :111 times. where uncertainty is never fully resolved. and the tlption

value next year. and investment is made accordina t() the

/?l/p-t'pt/t..li'f..)/l 0. l J1) $20 perwidget peryear. (This is analogous to selling short a dividendpaying stock; the short position requires payment of the dividend. because no rational investor will hold the offsetting Iong position without receiving that dividend.) Our portfolio has a short position of 8.5 widgets, so it will have to pay out a total of $ 170. The return from holding this porttblio over the year is theretbre =

-850

= = 680 Because this return is risk-free.

free rate, which we have assumed is portfolio, *4) 6) 11 C3: =

6,

-

-

+ 1700

-

l70

Fi).

we know that it must equal the risk10 percent. times the initial value of the

-

680

-

6)

=

0. 1 (6)

l700).

-

=

Ektllllples

dynamic programming approach. In this case it gives exactly the same answer. because all price risk is diversitiable. Later in this book we will explore this connection between option pricing and dynamic programming in more detail. 1.B Characteristics of the Option to Invest We have seen how our investment decision is analogous to the tlecision to exercise an option. and we were able to value the option to invest in much the same way that financial call options are valued. To get more insight into the nature of the investment option. Iet tls see how its value depends on various parameters. ln particular, we will determine how the value of the option and the decision to invest depend on the direct cost of the investment. /. on the initial price of widgets, Jl,, on the magnitudes of the up and down movements in price next period. and on the probability t/ that the price will rise next period.

Changink /e Cost ofthe Investment So far

we have Iixcd the cost of the

investment. 1, at $ 16()(). I-low much would were above or below this number'? We can Nnd out by going through the same steps that we did before. Doing so, it is easy (lbtain a risk..frce porttblio depends to see that the short position nacded to

the option to invest be worth if /

on l

as follows--s

11 16.5

=

-

up??/.'?/c

(5)

We can thus determine that 6) $773. Note that this is the same value that we obtained before by calculating the NPV of the investment opportunity under the assumption that we follow the optimal stratebry of waiting a year before deciding whether to invest. We have found that the value of our investment opportunity, that is, the value of the option to invest in this project. is $773. The payoff from investing (exercising the option) today is $2200 $ 1600 $600. But once we invest. our option is gone, so the 5773 is an opportunity cost of investing. Hencc the /1// cost of investing today is $ 1600 + $773 $2373 > $2200. As a results we should wait and kecp our option alive. rather than invest today. We have thus come to the same conclusion as we did by comparing NPV's. This time, however, we calculated the value of the option to invest, and explicitly took it into account as one of the costs of investing. Our calculation of the value of the option to invest was btsed on the construction of a risk-free portfolio, which requires that one can trade (hold a long or short position in) widgets. Of course, we could just as well have constructed our portfolio using some other asset, or combination of assets, the price of which is pertkctly correlated with the price of widgets. But what if one cannot trade widgets. and there are no other assets that span'' the risk in a widget's price'? In this case one could still calculate the value of the option to invest the way we did at the outset by computing the NPV for each today versus Mzait a year and invest if the price goes investment strategy (invest up), and picking the strategy that yields the highest NPV That is essentially the =

Deb'elopil'g Ille Ctp/lctz/p/-Tllrottgll

=

The currcnt

value

of the option to invcst is thcn given by

6) Equation

0.4)05 1.

-

(7)gives the value

=

l5t)() t).455 1. -

of the investment opportunity as a tnction of $16()(),it is of / for which

the direct cost of the investment, 1. We saw earlier that if l better to wait a year rather than invest today. Are there values investing today is the pretrred strategy'? To answer

=

this. recall that we should invest today as Iong as the payoff

from investing is at least as Iarge as thel// costs that is, the direct cost, 1, plus the opportunity cost 6). Since the payoff from investing today is ), $22()0. =

5As before. the valud of the portfolio next year is * j = F, t If Pl = $3(1(),F = 336#)- /, = 33t)t) l 31)4)l1. lf Pb = 51()1).and 1 is not so low that we wlluld invcst anyway, Fb = (). l0() ?l. Setting the 4)1 for each price scenario equal gives the equation above for tt and *1 = -n

,

so tl)h

-

-

's

-

.

34

Iltlrodllction

we should invest today if 2200 we should invest as Iong as

6,. Substituting in equation

I +

>

I + 1500

0.455 /

-

(7) for FI.).

22()().


$1284, 6) 150() ().455 l > l$j /, so the option should be kept alivea that is. we should wait until next year betbre deciding whether to invest. However. if I < $ l284. 6) 1500 ().455 / < 1$) /. s() the option should be exercised now- and hence the valtle ()f the option is jtlst its net payoff, K) I In the terminology of options. when / is small. tht, nttt payllff thlm imin the mtlney.'' At mediate investment becomes large. ()r the (lption is critical threshold where point or it is stltliciently deep. the ctlst of waiting a -

=

-

-

=

-

-

-

.

(?1%

tllltyvtriglls

-

.

=

=

.

.

,$

-l

-

=

.5

-.().5

.5

=

-

's

-

K&dtlep

/1)+ 81)1)whtlther tlle price gtles Note that with 11 chosen this way. tp I up t)r down. Ntlw let us calculate the return ()n tilis ptlrtllitl- rel-nembcring that the short position will require a paymdnt ()1- (). 1 ?? J1) I /1) l ()t). That return 6:1(). Since the return is risk-free, i t mtlst cqtlal (). l *t) is 6.6() /-1) -8.25

=

2000

W l

.4,5

-

.-.1 r

tt

1600

=

ww

-

--w

12Oo 80O

y-1500

-

--.

'.0.455 / ..-

.-.

--

I I

1 l I '

400

80O

=

-

Fz l )

--

400

O O

-

.6)

l

xw

N

x. xv

N

1600 12O0N 1284 /

Pa

'-'->' Nw

N

N

N

N

2000

=

2

=

1.58

=

o

--*'

Pz 1 2

r

t

n r E) o2 0 1 f'-izure 2.3. Prdrt?f (' IVlqcL =

N.z

.

%.J I

'

.

%''

'

q p

.h#'

.

t)

36

llltrodttclioll

0. 1 /7i) 1

.Jl)+ l60. Solving

.65

-

tbr F() gives the

Fll

=

7.5 fl)

value

Table 2. 1-

of the option to invest:

727.

-

We have calculated thevalue of the investment option assuming thatwe would only want to invest if the price goes up next year. Howevera if Jl) is Iow enough we might never want to invest, and if C) is high enotlgh it might be better to invest now rather than waiting. Below what price would we never invest'? From the equation above, we see that 15) owhen7.5 C)= 727, or /$ = $97. If /l) is less than $97, 1Z1 will be Iess than the $1600cost of the investment even if the price rises by 50 percent next year. Forwhat values of /$ should we invest now rather than wait? Once again, should invest now if the current value of a widget factory, K),exceeds its we . total cost, $1600 + &). Hence the critical price, which we will denote by C*) satisfies F4) 1600 + 6), that is. 11 #t*) 1600 + 7.5 C7 727, or #(*j= $249. If f1) exceeds $249. we are better off investing today rather than waiting. The option is sufficiently deep in the money that the cost of waiting (the sacrifice of period-o profit) outweighs the benefit.: We obtained this critical price by tinding the value of the investment option, but we also could have found it by calculating (as a function of /1)) the NPV of the project assuming we invest today. and equating it to thtt NPV assuming we wait until next ycar and then decide whether to invcst based on the outcome for price. (We leave this for you as an exercise.) The value of the investment option is thus a piecewise-linear function of the current price, lj, and the optimal investment rule Iikewise depends on Jl). If C) s $97. 6) 0. and one should never invest in the factory. If $97 < C) S $249, then 6) = 7.5 J$) 727, and one should wait a year and invest if the price goes up. lf J1) > $249. then 6) = 11 /1) 1600, and one should invest immediately. This solution is summarized in Table 2.1. In Figure 2.4 we have plotted 6) as a function of Jl,. Suppose our only choice were to invest today or else never invest. Then the option to invest would be worth the maximum of zero and 11 C) 1600, that is. the value

Lxalllples

cj/?l/?/t!

Tllr()ltgll Developing tlle Ct?plcf-zp/of Izis/t/t.r

OJJf#?/l

Region

Option Value

/70S 97 97 < f1) S 249

6) 6)

C)> 249

/-1)=

= =

/t)

Iltbbesttlllt/ lllb?estlnentR!//t: Optimal Investment Rule

Never invest. Invest in period l only if price

0 7.5 /-1) 727 -

l l C)

has gone up. lnvest in period ().

1600

-

=

,

=

=

-

of a widget factory today, Pi), less the $ 1600 cost of building the factory. (In Enancialjargon.this is the ltrinsic pflftl of the option the maximum of zero and its value if it were exercised immediately-) Also, we would nvest today > > 4 as long as 11C) > $ l600, that is. as Iong as /1) $ 1 6. When Jl) $249. the lhen it is f)(), value l worth f1) I because of l intrinsic its invest optil)n to shown l 60() l wait. is today rather than 1 fl) Hence a solid invest as optimal to < $249, however- the option valucs of J1) When than /!) $249. for grcater line unexorcised- at to invest is worth lnore than I l J$) 1()()1),and should be Ie until next least year. '

-

-

-

y'

1800 1600

=

1400

-

1 1 Pz

1200

-

ur

z

1000

.

.

800

,'

7.5/% 727 -

-

'Ybu might think that immediate investment woultl be justifie only if we wouitl invest ncxt

year irrespective of whether the price went up or down. In fact. thc critical price for immediatll investment is Iower. Assuming we wait, we will invest next year if f''j 16t)0 1l 1$ 1600 > (). 0.5 Jl,, we would invcst ()nly if 5.5 /l) l6t)0 > (). t)r so if the price gocs down. that is. P, /1) > $291 which exceeds Jl; $249.Tlle problem with the faulty intuition is that it ignorcs thc protit we could elkrn thiq year by inves4 ir!g now lndt-txd f. a .!; of. say, $260. it is better tt) invust today even though if we did wait, we would ?7:J/ invest next year if it turned klut that tlle price wen down, -

=

:

=

=

for

,

.--..w.

.

6O0

.'

.

40O

0

40

80

120

160 ra

Cz

1 1 l j l l

I I l l

,'

'

O

'

,

1

.'

200

.

.

l

' ''

-

-

1600

-

200

24O

280

320

38

Developtj; J/rf.! C-llllw.'pts7W/'(Jf

Il3rotLtctl'llll

Note from Figure 2.4 that 6) is a convex function k)f /1), and that 6) is greater than or equal to the net payoff from exercising the option today. 1$)- 1, up to the optimal exercise point ($249in this examplc). As we will see, the value of an option to invest typically has these charactcristics.

Changing the Probabilitiesfor Price We can also determine how the valtle of the option to invest depends on

t/,

the probability that the price of widgets will rise next year. (So far we have assumed that this probability is 0.5.) To do this. we will let the initial price, J1,, be some arbitrary value, but we will fixthe cost of the investment, /, at $ 1600.

We can then follow the same steps as before to find the value of the option and the optimal investment rule. The reader can verify that the short position in widgets needed to construct a risk-free portfolio is n 8.5, and is independent of (The reason is that n depends on the possible values tbr the portfolio in period 1, * 1 and not on the probabilities that the portfolio will take on those values.) However, the payment required for this short position does depend on /, because the expected capital gain from holding a widget depends on t/. X) calculate thisa !et 7()4#1 ) denote the expected price of widgets next year. calculated conditional on the knowledge of the period-l) price.? Thus rt)(Pb) ((/ + 0.5) f'1). Therefore the expectcd rate of capital gain on 11 widget is g-()( P3) Jl)J/C) q 0.5. Hence the required payment per widget in the short position is 0.5)1 /l) (0.6 f/ ) J1).Setting * 1 *t) ((1.6 t/ ) l /1) t).I *()s t0. l ( and setting 11 8.5, we find that if Jl) > $97, the value ()f the option is =

.

,

=

=

-

-

-

=

-

-

-

-

=

-

=

8)

=

15 (? Jl)

l455 q

-

.

Note that 6) increases as q increases (aslong as f1) > 97). This is as we would expect, because a higher q means a higher probability that the price will go up and the option to invest will be exercised. If C) S $97,we will never invest. whether or not the price goes up, and 6) 0. How does the decision to invest depend on :/? Recall that it is better wait rather than invest today as long as /-1) > 1)) 1. In this case. )'() = to /X'4 f1)/( l (6 + 10 q) /$. Hence it is better to wait as fl) + S l q + 0 5) =

-

.1)1

.

Igh

t'trl//lp/t.,, LL-

-t'//'l/a#(.z

long as I 5 /1) l 455 q > (t) + l () t/ ) /1) - l (',(2)(),t llat is. as Iong as /1) < :1..; t/ increasds. tllat is- :t ( l ()()() 1455 (1 )/((' - 5 t/ ). >Nlso-l'*11 tlecreases #* () higher probability of a price increase intluces the lirnl t() invest nlortl reatlily. Why'?The cost of waiting is the revenue foregone by not selling a widget this with /''(). Since a Iligher ty makes a batl otltcollle next Period. which increases period less likely. it rctluces the valtle of vvaiting. Tllus :1 higher q implies tllat of svaiting exceed the value of waiting. a smaller 3-1)suffices to make tlle cost (/

-

-

Increasing the Uncertaint.y over f'Wt'c When we changed the probability t/ wllile keeping alI tlle othttr parametors flxed.we changed the expected price in period l Suppose the expectttd price in period l remains fixed at the initial price Ievel. #(), btlt we increase the size of the up and down changes so that the variance of tlle period- 1 price spreatl in tlle distribtltion increases. %vhat e ffect wotlld such 11?zI:*tl?l-J??-c'.c'?7.'/Ig PI have of value the investment option. 6-). and on the critical price. for on the svhich rather t han wait? optimal it is to invest inAmedilltely #*, above 0 will (/ is ().5. btlt now the pricc wi 11t2ithe r rise 'We assume as before that rather than l7y5() percent as before. Thus or fall in ptlriod 1 by 75 percent, expected value is st ill fl1. To lind /$ ). the variance of l>b is grcate r. but its port f(llio and we go t hrllugh the sanle steps as be tbre. creat ing a risk equating its return to the risk-free return. Again. let the ptlrttblio bc Iong the 1$) 11 /!). I11 investment option and short 11 witlgets. s() its vClltltl ntlw is thl will bc worth period l thtl price will rise t() l /!), in which cilse the prlject (h wil ?? /1). ()r l tltl ual l 9.25 /.:1)- l (k)() ) 1zr1 l 1 141 19.25 J1l so that I which will equal -().25 11 J:1y.Equating case t' l the price will fall to ().25 P(,.in these two possible values ()f tp 1 and solving for ?? gives .

.-free

=

-

.75

,

.75

=

=

-

n (Then, *

-

C) + 267 irrespective

-3.21

1

l 2.83

=

=

1()f)7/ /!)

.

t)f 141 Remembcring .)

/'$) position will require a payment of (). l n J'll l portfolio is 8.34 Jl) 8) 693. Setting tllis equal to t).l 107 and solving for 6) gives .28

=

-

-

-

that thtl shtlrt l ()7, the return ()n the /El) (). 1 1$) - l + *4) .28

=

=

/$) 8.75 /-1) 727. =

7We will use C to denote the expectation

(mean)of a random

variable,

on C will indicate that the expectation is conditional on information

and a time subscript

available

as of that time. Similarly, we will use b' to denote variances. These symbols are used consistently throughtlut thc book. Other notation is specific to each chapter ur even section. A symblll glossary at the end of the book collects the signiticant symbtlls and states what they denote.

-

If Jl) is $200, 6) is $1023, which is substantially larger than the value of 5773 that we found before when the price could only rise or l'all by 5() percent. Why does an increase in uncertainty increase the value of the option to invcst'?

Because it increases the upside potential payoff from the option. leaving the

lntroductioll

40

downside payoff unchanged at zero (sincewe will not exercise the option if the price falls). We can also calculate the critical initial price. 131. that is sufhcient to warrant investing now rather than waiting. Again, just equate the current value of the widget factoly Ftl 11 J$, to its total cost, $1600 + ). Utilizing equation (12)for F(), this gives Pl $388,which is much larger than the value of $249 that we found before. Because the value of the option is larger, the opportunity cost of investing now rather than waiting is larger, so there is a greater incentive to wait.

Developing

Ille

Cf-nlcz/pf.Tllrollgll .ki7lJ?/t3Erflp?lp/p-j'

It is easy to show (andshould be intuitively clear) that the point of indift-erence between investing now and waiting occurs in the range of Jl) where investment in period l is warranted if the price goes tlp but not if it goes down. In this case the NPV in equation ( 14) simplifies to

=

NPV

=

tbad

'ibad

Suppose that the initial price is J$, but in period l the price becomes

/1

(1 +

ul/$

with

probability q,

(1

J) J

with

probability l

= -

=

-

/ + #; + q

''= (1 + !/)/1) (1.1) f l ,

(1

-

(1

q)

=

JIIJ.

I/)

t

=

j

J) J1) (1.1) , -

On the oiher hand, if we wait the NPV is

Equation

I

=

( j( 0. 1

t'. 1 + ( I

0. l + ( l

l.1

-

t/)

(F)( l

-

-

J)

t'bad

srstspelled

out by Bernanke

j

.

(16) has one detail that is important

#(*jdoes not depend $ to note upward the size of It only depends in any way on an move. on the size of (/ and of 11 downward d, the probability downward l ( ) move. move. Also. a the critical the magnitude of the is ds larger is the prices Jl'j it is Iarger the ; wait-g news'' that drives the incentive to Possible spread in l in a We can also examine the effect of a mean-preserving J/( lt + J). Then more general way than we did betbre. Suppose we set q (/ #(2j lt and tt J S /5 ) Jl), and k'4P ) Hcnce if we increase proportionally. we keep fy and 7( P3) unchanged while increasing the variance of 1) Observe from cquation ( 16) that if J is largdr but (y is unchanged. C; incrcases. Again. we hnd that a mean-preserving spread increascs the incentive to wait. tt.

-

ttbad

=

=

=

.

.

Extending the Example to Three Periods In our example, we made the unrealistic

assumption

that there is no uncer-

tainty over the price of widgets after the first year. in most markets. future prices are always uncertain, and the amount of uncertainty increases with the time horizon. In otherwords, while the expected future price of widgets might always equal the current price. the variance of the future price will typically be greater the farther into the future we look. Later in this book we will model 9If current proht can be negative and the 5rm is contemplating

B'Fhis news principle'' was be found in Cukierman (1980).

( l5)

.

'>D

+

+ 1l ( 1 +

-

To keep things general, we will 1et the cost of the investment be 1. In this case, the NPV if we invest now is NPV

(- l

Equating the NPV of equation ( l3) tbr investing now with the NPV of equation ( l5) for waiting and solving for J1)gives Jl;

tgood

q l l .

X ''Bad News Principle'' We can take this one step t'urther by allowing both the probability q of an upward price move as well as the sizes of the upward and downward moves news'' (an upward move) to vary. ln so doing, we can determine how news'' (a downward move) separately affect the critical price, C7, and that warrants immediate investment (inthe above calculation the upward and downward moves had to increase or decrease together). We will see that /l; depends only on the size of the downward move, not the size of the upward of it is the ability to avoid the consequences move. The reason is that wait-S news'' that leads us to

=

( 1983), and

the ideas can also

a costly disinvestmcnt

or

abandonment of a project, the bad news principle turns into a good news principle: the size and probability of an upturn are the driving forces behind the incentive to wait. Rccalling our discussion of suicide in Chapter 1. we should emphasizc this point. If the potential bad outcomes become even worse. that does not increase the incentive for immediate abandonment. However, if potential good outcomes become better, that increascs the value of staying alive.

Illtr(;tltt'tl'tlll

the stochastic evolution of price in just this way. At this point, however. we dan obtain additional insight into the nature of the investment problem by extending our example so that there arc three periods in which the investment decision might be made. We will asstlme some level Jl1,and at /71 l C) or /71 again either increase .5

=

('!

h

1

l i .

=

() the price of widgets begins at as before that at ? 1 it will either increase t)r dccrease by 50 percent (to 2. it will 0.5 J1,), each with probability 0.5. Then, at t =

/

=

=

or decrease by 5() percent with equal probability. Hence values tbr Pj-.2.25 f1)-0.75 C), and 0.25 C).The price three possible there are remains this level for :111 l t 2. (See Figure 2.5.) We will again fix the at then of 1, at $ 1600. investment, the direct cost adding of price uncertainty, our investment problem period By one more becomes quite a bit more complicated. One reason is that there are now Iive different possible investment strategies that might make sense and must be considered. In particular. it might be optimal to (i) invest immediately; (ii)wait a year and then invest if the price has gone up. but never invest if the price has gone down; (iii)wait 11 year and invest if the price has gone up, but if it went down wait another year and invest if it then goes up; (iv)wait two years and only invest if the price has gone up both times; or (v)never invest. Which rule is optimal will depend on the initial price and the cost of the invdstment. and the value of the investment option must be calculated for eat:h possible rule. The second complicating factor is that while we can still compute the value of the investment option by constructing a risk-free portfolio, the makeup of that portfolio will not be constant over the two years; we will have to

1,/1 It-t.zr?fs -L'121-(.33 lrl f /1t!C't?? Si' ?lp/c E.a-tl?) llllctb flcvtrrff-lp?

change the at t

l

=

number

of

'widgets

in the short posititln after tlle price cllklnges

It)

.

We wil l again approach this problenl tlsi ng option prici ng n'let hotls. %Ve ()- /7().as :1 l'tlnction want to obtain the value of tlle option to invest at / of the initial price. /1,, as well as the optinaal investnient rtlle. Tllk) trick is to work backwards. We will solve tsvl) separate investment prllblenls looking l first for l ().5 /-1),and then for l fonvard from t l /1)-asstl lning in both cases that Nve have not yet invested. In each case we will tltlttlrlnine Fb, the value of the investment option at t = 1 by constructing a risk-free portfolo and calculating its rttttlrn. Given the tsvo possble values for (one for PL = t).5 1) and one for J''! = l /,1,(),we then back up to / () and determine 6) by again constrtlcting a risk-free portfol io and calculat ing its =

.5

=

=

.

=

,

-l

.5

=

return().5 J'l). and that /1) is such tllat we wtptlld Suppose that at / l PI invest in period 2 if the price goes up, but not if it goes down. Now construct a portfglio that incltldes the option to invest and is sllort slln-le ntlmber /1 I t'f J-.1 11 l /31 I f tlle price goes up widgets. The value ()f ths portlblio is * 1 in period 2 (to (3.75 /l), we wi lt invest. so Fz wiII ttqual (3.75 Jl)/ ( l l JJ (p: will cqual 8.25 C) lt()() ().75 11 1 1)p. lf t htlprice 1600 8.25 f-1) I()()()-and ().25 down in pcriod 2 /1) t() ), we will not invest. fC':will eqtlll ()allntl *2 will ( goes /1 ! f$,. Eqtlating the expressillns )r (pa under thc twl) sccnltrills. equal llio the wi be find hat risk t d' 21 11 free I6 )/ /$)-, t Ilen *2 i 324)1 ptprt , we will equal 8(3() 4. l 25 /l, whether tl3e prit:e gtes up t)r tltJwn. Caltrulnting tlle portfolio's return (ma t'p I (), l 11 ! 1) ) and se tt ing it eq tlal t() thc risk return (().l ml ) gives us the valtle of the investment optitln: /th 3.75 /1) 727.3. Also. note from this tllat F 1 () Nvhen J1) l 93.94. I-ldnce if the price ' has gone down in pcritld l and /$) < 193.94, we will never invest. We must now repeat this exeruise assuming that priue has gone up in period l that is, that /71 = l /1). Y()u can verify that in this casc the riskfree portfolio requires a short position ()f n 1 = 16.5 1()67/ J'1)widgets, and the value of the investment optilln is F I lI J$) 727.3. In this casc Fl = () when J'l) = 64.65. I-lence if /$) < 64.65 wtt will never invest at 1111, even if the price goes up in both periods. Als(?. suppllse we invest in peritld 1 rather than waiting until period 2. Then &ve would obtain a net value 1z'1- l 11 ( 1 #()) 1600. Setting this equal to h and solving for /1)gives /'!) l 66.23. =

=

,

=

-

.

)-N't7

-

.

=

-

-

-

-().25

.5

=

-

-

-

-fr(2tl

-

-

=

E)

=

l 0

l= 2

=

l

=

3

-

=

.5

J2 J2

Pj

=

1

Pa

=

2.25/%

-'--'>'

Pz = 2.25P

-'-'''>-

.

-

.25

=

.5/%

L 2

Pv

Pz = 0.75/%

'-''-'>'

Pz

=

Q.75Pz

'--'->'

=

..

J2

2

%

=

-

.5

-

=

0.5/%

L 2

%

=

0.25/%

-''-*'

Pz = 0.25/%

'--'*-

''l''l''hcmethod of kecping a ptlrtl'lllitl riskless by changing its ctlmpllsititln through repeatcd hedging str:ltetzy.'' llnd is of considerable irnpl3rtanctt in financial trading is called :1 economics. We will devktlopit in a gtlneral setting of continutlus tirnt in Chaptttrs 4 and 5. bsdynamic

Figure 2.5.

Prc: of Widgets

Itllrodttctioll

44

Hence if J1) > 166.23, we should invest in period 1 if the price has gone up, rather than wait another year. we have We now know Fl and the optimal investment strategy (assuming /1 of possible we can the tbr each so outcomes two invested) for not yet and calculating portfolio risk-free constructing again Fi) by a determine once its return. Since the optimal investment strategy depends on Jl,, this must be which done for diterent ranges of Jl), that is. for 64.65 < & S 166.23 (in periods), for both tbe in price only if period in 2 up invest goes case we would the price would if 1 which invest in period 193.94 < we case 166.23 C) :jE (in would never invest), and for J'1)> 193.94 (in goes up, but if it goes down we whichcase we wouid invest in period l if the price goes up, but if it goes down wait and invest if it goes up in period 2). The solution, which the reader might

E-b'tllllplf.n.

up/?/l/tr

Dcvc/trp/s/lt#?c Collt'.'pt-b-Tlll-otqqll

2800 ?400 11Pz

,

want to verify, is summarized in Table 2.2. Figure 2.6 shows 6) plotted as a function of f'1).As in the two-period model, it is a piecewise-linear function. but now therc are more pieces. each corresponding to an optimal investment strategy. Also, suppose the only choice were to invest today or else never invest. Then the option to invcst would be worth l 1 /1) l60t and we would invest today as long as 1l f1) > $1600,that is. as long as /l) > $146.When l) > 301.19, the option is worth this much, because thcn Fu 9.2 C) 1057.9 l 1 /1) l 600. and it is optimal to invest today rather than wait. Hence 11 Jl) 1600 is shown as a solid

2000 u!:'

1600

-

-1

1600

' .

.* -

1 200 ..

400

'

*

0

40

80

120

. I 160

1'

. ,

l

-1 I

j

I

I

200

I 1 1 l

.'

.*

l 1, ,1

j

1 )

e

.*

800

O

--.hw

240

280

320

360

400

-

=

=

-

-

-

Iine for J$) > 3()l 19. l7ut is shown as 11 dotted line for l 46 < /1) < 3()l l 9. Finally, notc once again that /$) is 11 convex function of Jl,, and that f;;.)ls grcater than or equal to thtl nct payoff from exercising the (lption today, l$) /, up l l 9 in this case). to the optimal exercise point ($3() As butbrc. we could examine the dependenc: of 6) and the optimal exercisc point #*() on the cost of the investment. /, or on the variance of price ' changes. (Going through these caculations is no'w a bit more laborious. howexample, we ever. because three periods are involved instead of twt).) For could show that if the variance of the price changes increases, while the expected price changes rcmain the same. the value of the option /-1)will increase, reader might want as will the critical exercise price C*)As an exercise. the ()r decrease by 75 percent in each period. check P this Ietting increase by to instead of by 50 percent as in Figure 2.5. If we wanted to, we could now extend our example to four periods. allowing the price to again increase or decrease by 50 percent at t = 3. We could then work backwards. sndingFz for each possible value of Pz, then hnding Fi for each possible value of P3 and hnally hnding Ff). In Iike manner, we could then extend the example to five periods, to six periods, and so on. As we did this, we would tind that the curve for 8) would have more and more kinks. We will see in Chapter 5 that as the number ot- periods becomes Iarge, the .

.

-

Table 2.2.

ofoption 1//1/14:

/kvt!Jl

to

Optimal Investment Rule

Option Value

Region

Never invest.

=

0

=

5.11 Jl)

166.23 < C) :; 193.94 5)

=

7.5C) 727.3

301.19 Fo

=

9.2 C)

C) S 64.65 64.65 < C) :; 166.23

8) 6)

attd lnvestment Rttle

-

330.6 Invest in period 2 only if price goes up in period 1 and in

period 2.

193.94 < C)

.s

-

-

Invest in period l if price goes up. If price goes down in period 1, then never in-

vest.

1057.9 Invest in period l if price goes up. If price goes down in period 1, then wait and invest in period 2 if price

goes up.

/% 301 19 >

.

Fi)

=

11 J$

-

1600

Invest in period 0.

.

.

,

46

llltl-oflttcioll

curve fklr 6) will approllcll a smootll culwe tllat starts at zert) and rises to nleet the curve showing tlle net payoff from irnlnedilte investnlent ( fz$) I ). In ltct th two curves meet tangentially, and the point where they meet dehnes the threshold #4,*where immediate investment is optimal. -

Adding more and more periods

will make

our example

tlnreasonably

complicated. howevcr, and in any case wtluld be less than satisfactory because ultimately we would like to allow the price to increase or decrease at every future time 1. Thus we need a better approach to solving investment problems of this sort. In Chapter 5 of this book we will extcnd otlr example by allowing the payofffrom the investment to tluctuatecoltiltllottsly over time. As we will see, this continuous-time approach is quite powerful, and ultimately quite simple. However, it will require some understanding of stochastic processes, as well as Ito's Lemma (whichis essentially a rule for differentiating and integrating functions of stochastic processes). These tools, which are becoming more and more widely used in economics and qnance. provide a convenient way of analyzinga broad range of investment timing and option valuation problems. In the next two chapters we provide an introduction to these tools for readers who are unfamiliar with them.

Uncertainty over Cost We now return to our simple two-period example and examine some alternative sources of uncertainty. In this sectiona we will consider uncertainty over the cost of the investment. Urkertainty over cost can be especialy important for Iarge projects that take time to build. Examples include nuclear power plants (where total construction costs are very hard to predict due to engineering and regulatory uncertainties). large petrochemical complexes, the development of a new line of aircra, and Iarge urban construction projects. Also, large size is not a requisite. Most R&D projects involve considerable cost uncertainty', the development of a new drug by a pharmaceutical company

is an

example.

In the context of our rwo-period example, suppose that the price of a widget is now $200, and we know that it will always remain $200. However, the direct cost of building a widget factory, 1, is uncertain. We will consider two different sources of uncertainty regarding 1. The first, which we will call input cost lfactrrftzfn/y, arises because a widget factory requires steel. copper, and labor to build, and the prices of these construction inputs fluctuate stochastically over time. ln addition. government regulations

jleveloping

F/l/'f'.?l/t/l

//?c Qhlll-)lL

vsitttill'

C-b'tlll'lillfu'vb'

t llis call cllltllgtr tl'le reqtlil'trtl tltltlltititrs change unpredictably (lver tilntl, llesv salkty regtllllt illlls of one or more constrtlctitlll inputs. ( Ft'r tlxlllllple. add to labor requirenzents. cllallging envirtlllnltlntal rcgtllltilllls I'lltty or may require more capital.) Tllus. altlltlugll I l'nigllt be kntlNvn today- its vllltltl llext year is uncertain. kllltl

As one might expect- this kind of

tlncertainty

has the sal'ne eflct

()11

the

investment decision as uncertainty over the l'tlturc value of the paytpff fronl the investment, IZ it creates an opportunity cost of investing now rather tllan waitingfor new intbrmation. As :1 result- :1 projoct could have :1 conventitlllally measured NPV that is positive, but it might still be tlneconomical to begin investing. As an example. suppose that I is $ l6()()today. but next year it will increase $2400 or decrease to $8t)().each vvith probability ().5. As before. the interest to is l0 percent. Should we invest today, or wait until next year? lf wtl invest rate NPV is again given by equation ( l ). that is- l ()() + 22()t) the $()()(). today. =

-

This NPV is positive. but once again it ignores an (lpportunity cost. X) stle tlntil &ve recalculate this time the assulming wait NPV.but let next year. this, us in which case it will be ex post optimal to invest tlnly if / falls to $81)().l n this case the NPV is given by N PV

...-.84)() (().5) + l l t

'- $ l 0.

=

.05)f

=

$42()t)

with probability 0.5.

l2ln technical languagev this result is an implication of Jensen lnequality. combined with the fact that the prcsent value of a future cash tlow is a convex function of the interest rate. 'x

Jensen's lnequality says that if x is a random variable and 7'(-r)is a convcx function of then 7(./-4x)1> 7'(-(.,r1). Thus if the cxpected value of remains the same but (he variance of increases.ft/t-rllwill increase. In the case at hand. if the expectedvalue of nextyear's interest rate remains fixed but the uncertaine around that value increases, the expected present discounted value of a payoff reccived next year will increastt. .v&

.r

.r

Illtrothloqioll vltlue

Llence the expected

(.)1'

(().5)(42()()) $2867,wllicll

1.z')is (().5)(l 533) +

=

rdtes are certain. the future interest rate (holding the increases the expected value of the constant) value ot rate interest expected uncertainty the investment decison? First. affect l4ow project. But dlles the that if there were no uncertainty over interest rates, we wotlld clearly

is higher than it is when ftlture intcrest We sec. thens that

uncertainty

(lver

notc want to invest today. The N PV of the 5:/

! 13

=

:!()()()

-

project if we invest today is 7e ky ()()

-#-

:::::1

l

whereas the N PV today if $2000 $0.

l

J ( l l)

zuz

$:!1)() .

.

we wait until next year is only

$2200/ l 1 .

-

N PV

If we invest now.

is different if interest rates are uncertain.

The situation the NPV is

'

+ 20() +

-20()()

=

C Z1)

=

-

1.1

l 800 +

2867 I.1

=

=

(0.5)

1 -2()()() + l 1 l 1 .

'M

200 .05)J

.

l

=1)

(1

=

Scale versus Flexibility

t)l' scllltt trltll As students of ectlnonlics or btlsiness IklltI-I1 etlrly 0n- tltztllloluit?s be an important sotlrce of cost sltvings. By btlilding 11 Iarge plant instelttl tlf I'nigllt btt :tl7I(l to retltlt!e its avtlrltge tnlst lllltl two or three smaller ones, a lirm increase its prolitability. This stlggests tllat lirlus sllotlld respond tt) groNvtll iI1 demand for their prodtlcts by btlnching their investments, that is. investing in neNv capacity only infrequently, btlt atltling ICrgll ltntl efticient plllnts eatlll

time. Wllllt sllotlltl firnls dlla Ilowevtlr. when t Ilcre is untrertlkin ty

$806.

( I f we invest today we can mak: and sell :1 widget now tbr $2()().and we will have a factory whose expected value next year is $2867.)This NPV is positive, but suppose we wait until next year before deciding whether to invest. lf the interest rate rises to 15 perccnt, thtl value of the factory wfll be only $ 1533which is less than the $2000cost 01- the investment. Hence we will only invest if the interest rate falls to 5 percent. Since there is a t).5 probability that this will happen, the NPV assuming wtl wait is NPV

5

$ 1()()().

Tllis NPV is higher, so it is better to wait than invest now. This is a simple analysis of interest rate uncertainty. but it has some important implications. First, mean-preserving volatility in interest rates will value ofa project. but will also create an incentive to wait ncrease the expected rather than invest now. The reason is that, as with unccrtainty over future cash flows.uncertainty over future interest rates creates a value to waiting for new information. Second. if an objective of public polic.yis to stimulate investment, the stability of interest rates may be more important than the Ievel of interest rates. Policies that Iead to lower but more volatile interest rates could end up depressing aggregate investment spendingal; Aswe will illustrate in Chapter 9, with

tlelll:tntl

'Tllis prlllllttln is an inlpllrtant tlI1e 1-4.)1' l I'le e lect ric ut ility intltlst ry. I t is cheaper per unit (lf cllp:lcity t() btlild :1 Iargc cllal-li red plhver plltnt tlllkn it is tt) add capacity in srnall anltltln ts. Istlt ltt the sllnle tin-ltl. tltilities litce (.rtlnlr tlleir electricity siderable uncertainty over the rrlte Iy vvllicll tlle denAClnd gives the utility Ilexilpil ity. lltlt will gro:v.l4 jNtlding capacity in slnall ilnltlunts is also frlortt ctlstly. I-lence it is inlptlrtant t() be able tl) vllltle this llexillil ity. 7-heoptions approach used in t Ilis bt)()k'is sve II suited t() tI() t 11is. I Ie rc Tvt.t vi II illustrate the basic idtla with a sinlple exllnlple in wllich lemantl gnlwtll is certain. but there is uncertainty (lver relative fuel prices. 15 Consider a uti lity that faces constant dc mand grllwt 11()f I()() n'lcgawrt tts (&l'W) per year. l-ldncc the utili ty mtlst add to capacity; thc qtlestilln is l)(3w. There arc tsvo altcrnatives. The utility can build a 2t)()-MSV ctllll-lirtld pllnt (enough capacity for two ycars worth ()1- additional demand) :tt a capital ctlst of $190 million (Plant A), t)r it can build a l())-MW (lil-fired plant at a capital Inuch

14The reason is nllt justthat theril is ullcertainty tlvcr llle grtlwtll

'3lngersoll and Ross ( l992) have developed a continuous-time model of invejtment interest rate uncertainty, and it leads to more or Iess the same cllnclusions.

(lver

growth (as therc tlstlally isl'? I1*tlltl Iirln irrevklrsibly invests in :t lltrge adtlit i(1l1 llnly sltlvly ()1- even sh ri 11ks. it &v Il IiI'ltl itse If to capaci ty- and tlenlltnd grow,s holding capital it dllcs ntlt neetl. I Ience svllell t l1e grllwtl'l ()f tlel-nlntl is tlncertain. tllere is a tradeofl' between scltle ectlnlll'n ies antl the llexilnility tllktt is gained by invcsting I'ntlre frcqtlcnt ly iI1 slnilll increnlents t() ctptcity lls tlley are neetled.

tlj'

ttltal eluctricity denlltlltl. ric pllwk!r

but also bccause utilitics now (lftcn Iind thcnlselvcs ctlmpet ing with (lther slltlrct:s flftllecl (such as ctlgenerlltion). ls-rhis is 11 nltldified vcrsitln t)f an example prcsented in liwhill ( It?89).

Illtrltlla

frlrl/p

Annual Fuel Costs

Plant A

t 1

t 0 19

l= 2

=

=

19

--*

--*'

19

'''-*'

the second year. To decide which choice is best. let us calculate for each the P resent value of the expected cost of generating 100 MW of power per year forever starting this year, and an additional 100 MW per year starting next year.

First, suppose we commit the full 200 MW to either coal or oil. Then, if coal. the present value of the tlow of cost is choose we

200 MVV

(21) Plant B l= 2

l= 1

l 0 =

I

30

--*-

30

--+-

Note that 180 is the capital cost for the full 200 MW and that 19 is the annual operating cost tbr each 100 MW the hrst of wbich begins now and the second next year. Next. suppose we choose oil. Since te expected operating cost for oil is $20 million per year, the present value is

2

100 MVV

20

PVn

l00 +

=

10() + 1l .

L 2

Figure 2.7.

=

lo

Cloosing zlrltlllg Eleclric

-->

t

---

PIants r()w't.r

cost of $100 million (Plant B). At current coal and oiI prices. the coal-fired plant is not only more economical in terms of its capital cost ($90million per 100 MW of capacity), but also in terms of its operating cost; operating Plant A will cost $ 19 million per year for each 100 MW of power, whereas Plant B will cost $20 million per year. We will assume that the discount rate for the utility is 10 percent per year. and that each plant lasts forever. In this case, if fuel prices remain constant, Plant A is clearly the preferred choice. Fuel prices are unlikely to remain constant, however. Since what matters is the relative price of oil compared to coal (andsince the price of coal is in fact much less volatile than the price of oil), we will assume that the price of coal will remain fixed, but the price of oil will either rise or fall next year, with equal probability, and then remain constant. If it rises, the operating cost for Plant B will rise to $30 million per year, but if it falls, the operating cost will fall to $10 million per year. (See Figure 2.7.) The choice of plant is now more complicated. Although Plant As capital cost is lower because of its scale, and its operating cost is lower at the current oil price, Plant B affords the utility more tlexibility because it only requires a commitment to one year's worth of demand growth if the price of oiI falls, the utility will not be stuck with the extra l00 MW of coal-burning capacity in

20 + )J ( 1.1 .

)

=(

'X'

l=

l

20 ( l 1)l'

=

$61 1.

.

(22)

Thus it would seem that Plant A is preferred. afforded by the smaller oilBut this calculation ignores the oexibility sred plant. Suppose we install 10() MW t)f oil-red capacity now. but then if the price of oiI goes up next year, we install 20() MW of coal-fired capacity. rather than another oil-hred plant. This would give us a total of 30() MW of capacity, so to make the cost comparison meaningful, we must net out the present value of the additional l ()()MW. which would be utilized starting two ' years from now: PV' 1.1=

100 +

'x'

2()

(j j 1............::1) .

+

1

''

1 +i

l ()0

1..1 180 l.l

;

I

'X'

+ I

=

l

10 (l.1)f

9() -

(1.1)2

-'*

19

+ l

=

l

( 1.l)f

=

$555.

Note that the second line in equation (23)is the present value of the capital and operating cost fqr the second 10()-MW oil plant (whichis built only if the price of oil goes down), and the third Iine is the present value of the capital and operating cost of thehrst 100 A'/1zIZ of a 200-MW coal plant. This present value turns out to be $555 million, so installing the smaljer oil-hred plant and thereby retaining flexibility is the preferred choice.

Illtl-lltt-tilltl

One way to value th is llexibility is to ask how much lower wou ltl the capital cost of Plant A have to be to n-lnke it t he prc ferred choice. Let l be the capital cost of Plant A. Then the gresent value t)f the costs of installing and operating Plant A is .1

C*

+ /-1 '

/

=

()

''M

l9 + 1 ( 1)t

19

I + 399. .1

.

/

=

l

(' I l )l

'

.

The present value of the cost of providing the 2()()MW of power by installing Plant B now and then next year installing either Plant A or B (dependingon whether the price of oiI goes up or down) is

*

10()+

(.1 .1)1

f (1 =

+

20

1

i

I

.1 ' -

1.l

+

'M

10()

1 ''i

1.1

0.5 IA + (1.1)2

(=

-.:%D

t

=

l ()

+

l

l

(1.1.),

l9 (1.1)/

To find the capital cost that makes the utility indiffcrent betwllen thdse choices. just equate these present valtles and solve for /..1:

/z1+ 399

=

5 I 0.5 + 0.248 /..1.

$148.3 million. Hence thkleconomies

of scale wlluld

have to be quite Iarge (so that a 2()()-MW coal plant was less than 75 perccnt ()f the cost of two 100-MW oiI plants) to make giving up the flexibility of the smaller plant

or,

1)

=

economical.

6

Guide to the Literature

The net present value criterion and its application to investment decisions is an important topic in corporate Nnance courses, and is the starting point for much of what we do in this book. Readers unfamiliar with NPV calculations, including the use of the capital asset pricing model to determine risk-adjusted discount rates, may want to review a standard textbook in corporate hnance. A good choice is Brealey and Myers (1992). Although we largely ignore the implications of taxes in this book, they affect the choice ot discount rate for NPV calculations. Taggart ( 1991) can

provitles a rtzvievv of tlltl varitltls apprllacllk!s t(.)c:llctllating tliscotlnt rlttes (adtaxes) Ibr tlstl in tlle st tntlct rtl NI7V Illtpde l Rtlbltck' ( l 9S()) J'ustetl for risk antl Il()l'l1inlll tnlsll I1()&vs llfttlr-tax sllolyld ltlsvilys 17t)dist-tluntctl shows that riskless I'y at the after-tax risk-free rate (l'or cxanlple. tlle Treasu bill rate tinlt!s 1 Iu illtls ttnd Nlytl rs antl iuback l 992) derive t siluple ltlltl ratd tax ( ). the corporate lloNvs in N PV c:llculations. robust rule for discounting risky cash .

betNvettn

invcstThroughout this book Nve vvill enlphasize the tronnectiolls valuatitln exercising of linancial opt itns. Al t htlugll and the and decisions ment ('ptitlns itnd option pricing tecllccrtainly not necessary, stlnle l'anlliarity vvitll niqtles will be helpful svhen reading tllis botlk. Brealtly and N'lyers ( l9t)2) provide :1 simple introduction: so do thtl expository stllweys by Rubinstein tlctIlilttd treatnlonts, sec Ctlx and Rtlbin( 1987) and Varian ( 1987). F()r more JarrtlNv ltnd Rudtl l t)83). Nlthotlgh soluewhat 989'). and 1 l I-lull ( stein ( 985). ( also nally. for hdtl ristic 97f) ftl1. Fi article is tlse Sn1it11 1 by ( datcd. the survey ()pl illns. ;ts Nlllslln and N'jttrttln Kester of 1984 ( see )discussions investmenls ltnd t)f Clpe land. K()lIe ltnd ltstln l Chapter 2 l 987), :1 r. ( ( 1985), Trigeorgis and N1u rrin ( l gg I ).

part11

Mathematical Background

3 chapter

Stochastic Processes and lto's

Lemma

THls

t()()Is stllcllastic c:llctlcl lAIvI'IIl and thc next prtwide the mathematical lussdynamic prtlgranlming. and cllntingcnt claims an:tlysis that will be ttsed throughout the rest t)f this b()()k. Witll these t()llls. wc clln study investment decisions using a continuous-time apprtlach. which is both intuitiveiy tppealing and quite ptlwcrful. I11 addition, thtl concepts and techniqtles that we introduce herc are becoming widely used in 1: 11tlmbcr ()f arcas ()' ccllnomitrs and finance. and so are worth learning even ap:trt fnlm tlleir ttpplicatilln t() investment problems. This chapter begins with a discussitln of stllchastic proeesses. We wiil begin with simplc discrctc-time processesp and thu!n ttlrn t() the Svicnt!r prllcess (or Brownian motion), an important ctlntinuous-time process that is a fundamental building block Ibr many of the models that we will develllp in this book. Nvewill cxplain the mcaning and properties of the Wiener pnlcessa and shtlw how it can be derived as the continullus Iimit of a discrete-time random walk. Wc will then see how the Wiener process can be generalized to a broad class of continuous-time stochastic processes, callo,d Ito processes. lto processes can be used to represent the dynamics of the value ()f a prtlject, output prices, input costs. and other variables that evolvc stochastically over time and that affect the decision to invest. As we will see. these processes do not have a tmc derivative in thtl conventional sense. and as a result, cannot always be manipulated using the ordinary rules of calculus. To work with these processes. we must make use

slalllelllalictll

Xtz-/fyrcv//lz/ t)f

t

of Ito's Lemma. This lemma, sometimes called the Fundamental Thcorem stochastic calculus, is an important result that will allow us tt7 diftkrentiate and integrate functions of stochastic processes. We will provide a heuristic derivation of Ito's Lemma and then, through a variety of examples. show how it can be used to perform simple operations on functions of Wiener processes. We will also show how it can be used to derive and solve stochastic differenthat make tial equations. Next, we will introduce jumpproccsses processes continuously and show infrequent but discrete jumps.rather than tluctuate the ApFinally, in how they can be analyzed using a version of lto's Lemma. equations.which describe pendix to this chapterwe introduce the Kolmogorov the dynamics of the probabilitydensity function tbr a stochastic process. and show how they can be applied.

1 Stochastic Processes A stochastic process is a variable that evolves over time in a way that is at Ieast in part random. The temperature in downtown Boston is an example: its variation through time is partly deterministic (risingduring the day and falling winter). and partly at night, and rising towards summer and falling towards examplc: it anther unpredictable.l of is stock lBM The price and

random suctuates randomly. but over the Iong haul has had a positivc expected of growth that compensated investors for risk in holding the stock.

rate

Somewhat more formally, a stochastic process is detined by a probability over time t Thus. for given times law for the evolution xt of a variable < < t, etc., we are givcn. or can calculate. the probability that the h fl etc.. lie in some specitied range, for example corresponding values .r

.

-r2.

.rl

Stochaslic f's-tpcc--c-and Ito

'.

Lemlna

The temperature in Boston and the price of IBM stock are processes differ in an important respect. The temperature in Boston is a slatiolull' that This means, roughly, that the statistical properties of this variable process. long periods of time.'s For example. although the expected twer are constant temperature tomorrow may depend in part on today's temperature. the expectation and variance of the temperature on January 1 of next year is Iargely independ'ent of today's temperature, and is equal to the expectation and variance of the temperature on January 1 two years from now. three years from llonsuliolltlty now,etc. The price of IBM stock, on the other hand, is a process. value of this price can grow without bound- and. as we will soon The expected see, the variance of price F years from now increases with F. The temperature in Boston and the price of IBM stock are both contlltotts-tinle stochastic processes, in the sense that the time index / is a continuous variable. (Even though we might only measure the temperature or stock price at particular points in time, these variables vary continuously through time-) Although we will work mostly with continuous-time processes in this book, it is easiest to begin with some examples of t/-crc/c-/?ne processes. that is, variables whose values can change only at discrete points in time. Similarly, the set of aII Iogically conceivable values for (often called the states) can be continuous or discrete. Our dehnition above is general enough to allow alI these possibilities. One ()f the simplest examples of a stochastic process is the discrete-ltle t-crc/c-'/trl/crandom walk. Herea xt is a random variable that begins at a known takes a jump of size l cither up or down. value and at times t l 2. 3. each with probability 1. Since the jumps arc independent of ttach other, we 2 can describe thc dynamics of with the following equation-. -p

-r(),

=

.

.

.

.,

.

,p

.r3.

,

arrives and we observe the actual value xl we can condition the of probability future events on this information.z

When time

fl

r/

where 6/ is a

random variable

with

xt

I

-

+

6t

.

probability distribution

.

probte/

.!

1)

=

probt6t

=

=

-

1)

=

2

(J

1 2.

=

,

.).

.

.

We call a discrete-state process because it can only take on discrete values. For example, set m) = ()- Then for odd values ()f t, possible values /), and for even values of ?, possible values of of xt are (-f l l 0, 2. t ). The probability distribution for xt is found xt are (-/, -r?

' One might argue that the randomness is a rcdcction of (he limitations uf metcortlltlgy, and that in principlc it could be eliminatcd if we could build sufhcicntly complete and accurate meteorological models. Perhaps. but from an operational point of view, ncxt wcek's tempcraturc is indeed a random variable. In this book we will not attempt any detailed or rigorous treatment of stochastic processes. offering instead the minimal explanations and intuitions that sufhce for our applications. R)r detailed and general treatments. see Cox and Miller ( 19fi5), Feller ( 197 l ). and Karlin and Taylllr ( 1975).

-

.

.

,

.

-

,

,

.

.

.

.

.

.

.

,

-2.

.

.

.

,

3'I-his ignores the very Iong-run possibilitics of giobal warming

()r

cooling.

/3t?-/kj(?-J?/7?/ ibltltllzltltlliz-tll

fronl the binornial distribu tion. Fllr / steps. t he prllbllbi Iity tllat there downward

Jklmps

ltlld

/

-

11

(1

tl()Wll

Jtlmps is

are

1/

,

.

.

-r/

.t'f

--a

-

11

will take t7n the val ue J

.I

-r,

.

(')--.

Therelbres the probability that

for xt+ l depdntls on Iy (711 antl lltlt lttltl 1t itpnklIly (31! vvh:lt llappelletl bt.?l'(l1,t.? tirrleI For exalnplt!- in t htl cllse t)f t 11esi l'nple rlt 11t1tlnl hval k give 11 l)y eq uat it)11 ( 1), if xt zzz t''. t he 11 xt .y.I ca 11 etI tli.t I 5 (lr 7- (2 ltcll v it 11p rlllplb iIity J.a Tl) t, va ltlcs of xt I -Fl1eN'llll'kt'v prtlperty is e tc.. ltrtl irrelevltllt (lnce we k Ilow becausd it cllll greatly silnplify tlle :tl'lktlysis (41*:1 sttpcllttstic prklctlss. j-mportant svill see tllis shortly lts we tu rn tk) ctntilltltltls-t inlt, prtlcesses. e

2ll 11t time

-

J

is

.

-

.

2 The Wiener Process 'Wk will use this probability distribution in thtl next section when we derive the Wiencr process as the continuous limit of the discrete-time random walk. At this points however. note that the range of possible values that can take .%-t

t)n increases

with

?,

t,11'

as does the variance

Hence

-r,

.

is a nonstational'y

-r,

PVOCCSS.

Because the probability of an upward or downward jtlmp is 1. at time 2 value of is zero for alI t. (Likewise, at time tn the = () the expected for T > / is xt One way to generalize this process is expected value of by changing the probabilities for an upward or downward jump. Let p be the probability of an upward jump and t/ ( l #) the probability ()f tt downward //n7 f), the i; 1lt time J jump. with p > t/. Now we have a randonl ww/ wilh with and is incrcasing t expected value of xt for t > () is greater than zero. Another way t() generalize this process is to let thc size of the jumpat each time t b e :1 continuous random variable. For example. we might let thc sizd ol' each jump be normally distributed with mean zero and standard deviation fr. Then we rc fer to m as a discrcte-tinle ctpn/nlttxf'-xltl/f.l stocllaslic prfpctzw. stochastic process Anothcr example of a discrete-time continuous-state is the (zt-order ta//tr/rcgrt.'.s'.s'/r ll/-fpcc.-.abbreviatcd as AR( 1). l t is given by lhe -r,

.)

-rr

=

-

=

.

equation x?

=

t5

+ pxt

-

I

(t

+

(3)

,

where J and p are constants, with l < p < l and ts is a normally distributed hs the random variable with zero mean. This process is stationary, and value. of g'I-his value p), expected its current irrespective long-run /( 1 equation value setting in expected is found by (3) xt- l x; long-run and solving for -r.JThe AR( 1) process is also referred to as a mean-reverting expected value. We process, because xt tends to revert back to this long-run version of this process Iater in this chapter. will examine a continuous-time -

,

.r/

-

=

=

.r

Both the random walk (withdiscrete or continuous states, and with drift or without) and the AR( I ) process satisty' the Nlarkovproperty, and are thereThis property is that the probability distribution fore called &Ial-k-ov,;;l')ces-.e.b..

A

svienerprocess

also called 1.: Bl-ovltitlll ??lt)/?')?? is :t colltilltltltls-tilutt stochastic process t hree important prllpe rt ies.4 Fi rst- it is :1 :Il-l()b.' //?wcess. As explained abllve. th is rneans that t hc prtlbllbil ity tl ist ribtltitln llr :111 future values tlf tlltl prtlccss depends (Jnly t'I1 its ctlrrent vltle. llntl is tlI)llIL fected by pllst valtles 01' thc prtlcess ()r l)y ltlly tltller ctlrrent il'll'tlrllllttitlll. As :1 result. the current value ()f the prtlcess is ltIIt'lntl needs t() Illtlk'tl :1 Ilest lklrttcrtst of its futu rut vllltlt). Sectlnd, tlle Wie l'le r prllcess l'lrlsd'/lt/t.z/?t.vit(?/// l('l''lll,ll'. This means tllllt tlle probllbil ity distribtltitln jklr t 11tlcllktnge il) tlle prllcess (lveI()1tinle illte 1ar111. tny ('t Iler ( ntlnllverllppillg) any time interval is independent ()!' Third. cllllllges in tlle prtlctlss (lvtlr ally fin ite intervkll t ill'le llre ll()l'l3l(lIh' distribttted,with a variance that increuses Iinettrly vvith the tinle intef-val. The Nlarkov propcrty is particularly iI'npllrtltnt. Agltin. it impIies t I11tlt'Illy current inlbrm:ttilln is tlscll I )r lrec:lst ing the l'tlttlril plttl) ()f thc pnlcdss. Stock prices arc (llktln nltldtzlltld as Nlltrktlv prllcesses. ()11 tlle gnltl nds tllttt public in fllrmatilln is quickly incorptlrated in thc ctl rrent prictt ()f t 11estllck'- s() vllltle. (Tl1is is called tl'!c vveltk' that the past pattcrn of prices has n() tklreclksting forrn of market dfliciency. I1*it did not h()Id. investlprs clltlltl in principle the market'' through technical analysis. that is- by tlsing the pltst prkttel-n ()1* prices to forecast the futurtl-) The fact that a viener pnlcess has independen t increments means that wc can think of it as a ctpntintltltls-tinle versilln ()f 11 random vvalk. a point tllat we will return t() belosv. The three conditions discussed above tht.t Mtrk'tlv pftlperty. independent increments, and changes that are normltlly distributetl may seem tltlite restrictive, and might suggest that there arc vefy ft)w real-wllrld variallles Nvith

--lpeltt

41n 1827. the blltanist Rtlbtzrt Brllwn lirst (lbscrvetl and dttscribcd thtt nplltitln tll' snpltll particles suspendud in a Iitluid. resllting I'r()n1tlle Clppllrent successive llnd r:lntltlnl inlplcls ()1' neighbllring pilrticles; hcnce thc term Brllwniltn mtplitln. In l t?()5, Alltzrt l inslcin pr( )pt lsctl lt mathematical thet'r.y ()1' Brtpwnian mtltilln. which was furthtlr develllpetl illld nl:tdtt nltlrt.l rigt )r( 3tls by Ntlrbert Wiener in l923.

f'ltzc/cgrol//lf

Matllematical

64

while

that can be realistically modelled with Wiener processes. For example, it probably seems reasonable that stock prices satisfy the Marktw property and have independent increments, it is not reasonable to assume that price stock ehanges are normally distributed; after all, we know that the price of a changes in that can never fall below zero. It s more reasonable to assume logarithm stock prices are Iognormally distributed. that is, that changes in the of the price are normally distributed.s But thisjust means modelling the logarithm of price as a Wiener process, rather than the price itself. As we will see. used through the use of suitable transformations, the Wiener process can be extremely broad range of variables that vary as a building block to model an continuously) almost and stochastically through time. continuously (or of a Wiener process somewhat more useful properties the restate lt is to z(J) then Wiener is any change in z, Az, corresponding process, formally. If a conditions: following satishes the A/, interval to a time l The .

relationship

between Lz and Lt is given by Az

=

where Et is a normally distributed and a standard deviation of 1.

random variable

with a mean of zero

1 0 for 2. The random variable Et is serially uncorrelated, that is, t # s. Thus the values of Lz for any two different intervals of time with independent are independent. (Thus z(J) follows a Markov process increments.l -(6t6.&.

=

Let us examine what these two conditions imply for the change in z over this interval up into n units of some hnite interval of time F. We can break change the = in z over this intelwal is given length Lt each, with n F/f.Then

by z(. + r)

z(.)

Ei

=

1lo 'J Lelttlna

We will make considerable use of this property later. Also. note that the Over the long run its variance will go to Wiener process is nonstationary.

insnity.

By Ietting ht become intinitesimally small, we can represent ment of a Wiener process, dz, in continuous time as Jz

dt.

6J

=

the incre-

0- and Fttcj since et has zero mean and unit standard deviation. :((z)2) dt. Note, however, that a Wiener process has no time erivatiye, i Et (11)-1/2 which becomes inh ite as in a conventional sense; hz/zt . XZCUCS 20 FO@ app -ltz)

v''-f.

=l

The 6's are independent of each other. Therefore we can apply the Central is normally LimitTheorem to theirsum, and say that the change z(J+ F) which point, Iast tt fhis and variance F. with n distributed mean zero particularly and Lt, is Lt Lz depends fact not on the that on follows from important; the variance ofthe change n a Wienerprocessgrows Iinearly with the time horizon. -z(.)

=

=

=

'$

At timeswe maywant to workwith two or more Wienerprocesses. and we be will interested in their covariances. Suppose that zl (J) and ca(;) are Wiener h .. dz?) p1: JJ, where pIc is the coejhcieltt processes. Then we can write of ctprrtl/zlfon between the two processes. Because a Wiener process has a (z-(((/z)21/JJ 1)a variance and standard deviation per unit of time equal to 1 also the covariance per unit of time tbr the two processes.b is p!c -ttzl

=

2.A

Brownian Motion with Drift

We mentioned

earlier

that the Wiener process can easily be generalized into generalization of equation (5) is the

more complex processes. The simplest Brownian motion w#7 tryi: dx

a

=

(t

+ c dz.

where Jz is the increment of a Wiener process as defined above. In equatfon (6), a is called thtt drift parameter, and f.r the variance parameter. Note ihat is normally disover any time intelwal L, the change in denoted by a2 Lt. tributed. and has expected value CL-%.b a AJ and variance )J(-r ) ..r,

-t7,

=

Figure 3.1 shows three sample paths of equation (6),with trend a = 0.2 per year, and standard deviation t'z = 1.0 per year. Although the graph is shown in annual terms (overthe time period 1950 to 2000). each sample path wasgenerated by taking a time intelwal, Lt, of one month. and then calculating a trajectory for x(/) using the equation

'' Rccall that if and F' are random l CovfzrF )/(cx (.v ). In this case t'z r Jv

variables.

zr

5we always use natural logarilhms. that is, those with base e.

=

=

=

n -

(??lt/

=

l.

Et

-//c/ltlyffc Processes

=

=

.

their coefscient

of correiatitln

is pxt.

=

6(t

l Jltlt.'q?'r/tlplt/

lltlthelltatictl

Stochasic /''/'tzt:'cxxtts.(1,1,-1

'.$'

//tJ

l-cltlllla

>'! 20

!'

66% Confidence Intewal

15

-,----

-

10

z e t -

'

,,.'

e

-

-

.

-

-

-

.

-

-

-

.

.

.

L

..

-

/

(*- r >' l C 1-xk -

jl

'J

.

k

#

-

-'

-

-

#

!l

'

->--.-ys

?..

w ..

-

.

-jj ..

.

.

''

-

--

-

--'

-

-

--

-

-

-w

4 'hr $

5

.e

.j

( ,

..-. .,.. -

''

,.

...

-

..-

-

...1

.'

/

l

'- Reallzatlon .

.

O -5

55

50

60

70

65

Time

=

80

85

90

95

1O0

Time

(). In equation (7), at eah time ? 6, is drawn fnlm a normal with with distribution zero mean atld tlnit standard deviation. (AIs() note that the t'z have been put in monthly terms. A trend of ().2 and parameters a per ytlllr of ().()!67 per mllnth. A standard deviation of l per year implies a trend (1.()833 per implies a variance of l per year- and hence 11 variancc of i1 month, so that thc standard deviation in monthly tcrms is ().()833 ().2887.) -t1()5()

75

,

t ikln prllcess grtlvvs linea rIy wi tl1 t l1e t inle Iltlriztlll. ! Ile stlplldlt rtl (41' tlle ?-r?f?/ fitltl Iltle i Iltlrizlpll.l'1 he ellctl t f')(')-I7e rck!n t ctpn t rne grows t I1esqllfllv ltllelttl lrtlcklst is F ;t given I)y t I-ls interval t'llr nn1111

tlevii.t

Wiener

;.s

.()

A' 1 ) 74

+ () () l t''l()7 F + () 2887 .

.

r

.

.()

=

=

Also shown is a trend Iine, that is. equation (7) with (qt (). Figure 3.2 shows an optimal forecast of the same stochastic process. Here, a sample path was generated from l 95() t() the end of l 974, again using b)-15 equation (7), and then forecasts of to 2()()(). ) were constructcd tbr l (For comparison, a rcalization, that is. continuation of the sample path, was also generated.) Recall that because of the Markov property. oll-%' the value of -r(J ) for December l 974 is needed to construct his forecast. The forecasted value of for a time T months beyond December 1974 is given by + 0.t)1667 F. -l974+r =

.r(/

-r

=

-rlt974

The graph also shows a 66-percent forecast conndence interval, tllat is, plus or minus one standard deviation. (A the forecasted trajectory for gs-percent conhdflnce interval would be given by the forecasted trajector.y plus or minus 1 standard deviations.) Recall that since the variance of the -r(/)

.96

One can sinlilarly clpnstrtlct 9()- t)r tls-percent ctln litltrllce illtervltls. (ll-lselwe klld alsl' l'nlnl 3.2 tllat. in tlle ltlng rtl n. tlle On(l can Figures 3. 1 ()f nltltitlnvvllereas in the shtlrt the Brllwnian dominant dcternlinilnt ' trend is Atgain. t'lf volatility the pnlcess dtlminates. t 11is is :tn inlplicat itpl) t)I' run. the ()t' (.p ) is a t and the stand:krd tleviat ikln is f'.r ? the fact that the mean hlllds. for large / that is- thc lklng run. /

60

65

.wd

..-

o

w

# =

0 and therefore exponential should be rising toward the upper barrier. and if it y > 0), the negative drift rate, the density should be falling to the right. If y. > t), has a :!. and consider a process with we can in fact Iet the lower barrier go to stationary distribution whose will have long-run 2) it a only an uppcr barrier at < (), if left the of 7. exponentially Likewise, we can Iet the y to density falls Ef upper barrier go to x. We will have occasion to make use of this distribution at severl points in Chapters 8 and 9. On a couple of occasions we will generalize the argument to death.'' More substantial extensions include allow ajump process of general Ito process, and examination of the acfollows a the case where distribution o x rather than only its Iong-run tual dynnmics of the probabilily require extensions the development of a differstationary state. Both these ential equation for the probability density the Kolmogorov equation and treat this in an Appendix to this chapter. -cx)

So far we have considered only diffusion processes. that is. stochastic processes that are everpvhere continuous. Often. however, it is more realistic to model an economic variable as a process that makes infrequent but discrete jumps. An example would be entry by a new competitor in a market with tkw value of :1 rms. so that price suddenly drops. Likewise. one might model the patent as subject to unpredictable but sizable drops in response to competitors' success in developing related patents. Or, one might view the price of oil as a mixed Brownian motion-jump process', during normal times the price tluctuates continuously, but the price can also takd Iarge jumps or falls if a war or revolution begins or ends. In this section. we discuss Poisson (>lpllp) processcs' and we introduce a version of Ito's Lemma that will help us to work with them. A Poisson process is a process subject to jumpsof (ixed or random size. which the arrival times follow a Poisson distribution. We call these jumps for lrrrfl/ rate of an event. during a time Letting denote the lneall the dt, probability that an event will occtlr inhnitesimal length of interval will and probability that not occur is given by by dt, the an event is given which l/, of size itself be is dl The event can a random variable. a jump 1 with analogy the Wiener by denote Poisson Let q a proccss; in process 'Ievents.''

4(7).

Substituting the general solution into this, we hnd B = 0. The same conclusion could have been obtaincd by considering the Iower barrier &. must be choscn to cnsure that the whole probaFinally, the constant bility mass between ,. and T must sum to one. This givcs =

'.j'

.

2 ajo'l.

=

11t1 //('? Lcntttla t'f Stochaslic farrpcctswfzt'

t Then

we write

with probability l with probability

() lf

=

tlt,

-

tlt

.

the stochastic process for the variable x as a Poisson differential corresponds to the Ito process of equation ( 1l ) as follows:

equation, which

Jx

=

J'x. t ) dt

+

,g(-r.

t ) dq

(38)

.

t ) and gt-r, f ) are known (nonrandom)functions. function of and t, and Suppose that S(-r. f ) is some (differentiable) expression expected change in H, that would for the like derive that we an to expand follows: d H as is, CdHl. To do this.

where fx,

-r

6IH

=

1)H )J

dt +

pH = )t dt

+

/?H 3 -):

pH

px

dx

(39)

g.t.r. /)

J/ + g(-t. t) dqj.

(Note that higher-order terms go to zero faster than dt because. unlike for the Ito process. dx does not depend on dt Thus changes in cause changes in S in two ways. First, Hx. t ) will change continuously and deterministically .)

-r

sltlllletnatical /.?fl()'/v-l?-t'Jd/'?ltl

86

! ;L.

.

in rcsponse to the dri in Second, thertt is 11 possibility that a Poisson event willoccur; if it does, will change by the random amoullt d/ gt-r. f ), and Hx. t ) will change accordingly. Since the probability that :1 Poisson event will occtlr over the interval dt is dt, we have .r.

-r

(40) where the expectation to the size of the jump lt. given by ()n

'tT

Hj

'H

=

)t

+

.

J (x. / )

side of the equation is wth respect Hence the expectation of the differcntial of H is

-tlz'

)

=

(J Iz-/f 11/3

=

ps J-

dt

-r

+

u (( Hx

+ g(-v'.t4 l/. /)

H (-r. t )1l (lt

-

dt

tl (.r t ) .

H (x +

pH

+ 1? /J a (x t ) .

ar ,g(.v .

/)

11 -

t)

-

;)2 H

lU

a-ra

Hfx t ) 1) tl .

P

(42)

Example: Present Value of Wages. Suppose an individual who lives forever receives a wage F41) which rises by a constant amount 6 at random points in time. If l is the mean arrival rate of raises, we can write the differential

-

t//',

.

so that

E:

+

#2

.

wage

Suppose a nlachi ne prt'd tlcos 11 kxlnstan t tltnv Exanlple: Value ef a Nlachine. :' it It reqtli res no ma i11tenance. btl t at son'le plli nt Iong operates. t of pro as as will will and have to be discartldd. I1' is the arrival rate time it break down in I'nach breakdown. and the discotl intl va Itle'? is /? of a n t ratd. what is t he Tht value of the machine follows the prtlcess 's

where an becomes

%eevcnt''

is lt

p P'

.

Note that the second-order derivative is relevant only for the variance contributed by the continuous part of the process. The jump part contributcs the last term on the right-hand side involving a difference in values of H at discretely different points'.

/) tt' tlle

Thus l'' is equivalent to 11 perpettlity that pays out forever the current 11/plus the capitalized valtle of the averagtl raise per tlnit tinle.

f

(lt

Stl I )

(1 6/p)

=

.

(4 1)

i)H + i)/' + Cu((

equation for the individual's

E:

fz-(Izl'''l

value of the change in the function Hfx, / ) is given by

' (J H 1

1/-

-

lz' J(/

.

=

l with probabil ity l Tllen the asset return equatilln

t/l

=

.

zr (/J + C(JJ,'' )

Thus, F=

7r

=

dt -

1/ J/

.

X .

p+

Hence we can treat the profit flow as a perpetuity, and valuc it by increasing This is a very goneral idea: if a profit llow can the discount rate by an amount when with arrival rate occursa then wll can calculate the a Poisson event stop value of the stream as if it never stops, but adding t() the expected present will discount rate. We come across this in many applications in later chapters. .

wage as

J )F 1 with probability l with tt earnings stream? expected =

In this case.

IZUIJ) t// +

=

=

.

tlse

then the expected

JJ

p 1ze

the right-hand

equation (4l ) in the same way that we used Ito's Lemma when with continuous processes. working Sometimes we meet a combination of an lto process and :1 jump process. The tbrmer goes on aII the time; the Iatter occurs infrequently. Then the appropriate version of Ito's Lemma also combines the two effects. Thuss if We can

We call treat 1..-as an llsset and cquate the nllrnnktl rettlrn on it at rate sum of the dividelld (currentwage) and tlle expected capital gttin:

.

=

(F (lq

,

(43)

What is the present value of the individual's

Guide to the Literature Our treatment of stochastic processes and Ito's Lemma has been at an introductory and heuristic Ievel. For a more in-depth development of stochastic

Stochastic f'Yrpce.yc.5'alld /'/(p Ltavl?//?t'l '.

fzci-qrlr/l/?lt

Matltematical

processes and

their properties. see Cox and Miller ( 1965), Feller ( 1971), and

and XtyKarlin and Taylor ( 1975. 1981). Cox and Miller ( l 965) and Karlln equations. of Kolmogorov the nice treatments lor (1981)provide particularly fora discussion ofthe use ofcontinuous Also, see Chapter3of Merton (1990) modelling. For a much more and economic and jump processes in snancial and Shreve ( 1988). Karatzas stochastic processes, see rigorous approach to of Ito's Lemma and discussions For more detailed but still introductory and Brock ( 1982), Malliaris Chow (1979), its application, see Merton (1971), of Poisson discussion nice also and Hull (1989).(Merton (1971) prtwides a Kushner rigorous treatments, see (1967), processes,with examples.) For more Also, for and Chapter 4 of Harrison (1985). Arnold ( 1974), Dothan (1990), of expected present values, see more detailed discussion of the calculation

a

(1993a).

Dixit

point

+ tll

-r

.

Hence

We recognize this as a dynamic generalization of the stationary-state computation in the text; see equation (35). Lll t Now expand $ (m),f(); x t ) in a Xaylor series around t) (-'E0 r(1 -'E J 1: -

-

.

.-

'.

.

/

(.r()/() ,

.'

-t -

lXh /'

t

-

.

t)

$ (-r()tt ;

=

,

-r

.

/)

$

2. /

-

Lt

-

2:./7

2)4 p-r

+

al

(t h )

'z

Note that third- and higher-order terms are of order (Al)3/2

hence will go to zero faster than Lt. Expand 4 (.r4).?()'. and substitutc these expressions into equation (44):

+ Lh

-r

:24

2

px2

(AJ)2, .

/

-

+

.

.

.

etc.. and

t.t ) Iikewise.

Appendix - (p

The Kolmogorov Equations

A

follows a At timeswe will want to answer questions of the following sort: If what probability the value is is x), particular stochastic process and its current that it will be within a certain range at time t later'? Or, what is the probability will have reached a point xl within a time f :G F? To answer questions that like these we will need to describe the probability distribution tbr x and its evolution over time. This can be done using the Kolmogorov eqttations. We will derive the Kolmogorov equation for the simple Brownian motion .x(l)

Finally, we use p + q

also substitute

Lll

=

=

o'

-

(1

;)4

) L lt

i)t.

+

,

2.

2

(p +

ty

) (25./?)

a

/924

-

.

2)v..'.!

1 and from equation ( 1()), p f/ ) Lt. We t(y/t7' J, divide thrllugh by Lt, and rearrange'. -

.

=

.,:(1)

representation with drift of equation (6).using the discrete-tme random that we introduced in Section 2.B of this chapter. Recall that we broke a with time interval of length t into n tjLt discrete steps, and in each step, probability p, would increase by an amount h, and with probability q 1 p, it would decrease by hh. Finally, to keep the variance of (m x()) f o. independent of the particular choice of Af. we set Lh Let /(m), /0; J) be the probability density function for -(I), given that walk

Equation (45)is called the Koltnogorovl-ward ctzllt?ltpn for the Brownian motion with drift. and it describes the evolution over time of the probability density function f) (m)- /4)', x. f ). In a similar manner. one can derive the Kolmogorov fonvar d e q uation for the gene raI Ito process of equation ( l 1): l 2

=

=

-r

-

-

=

.

.z.

at an earlier time

f() we

have x(0)

=

-v().

Thus

(46) Equations (45)and (46)are called t'forward'' equations because they have at time /#), and are solved forward as bounda:y conditions the initial value for the density function for future values of One can Iikewise describe the evolution of the density function backward in time, that is. taking at time t as the boundaq conditions and solving for the density function for previous .rt)

-r.

.r(/)

Over the interval of time t A/ to J, the process can reach the pont in one Lh, or by decreasing from the of two ways. by increasing from the point -x'

-

.r

-

I2Sce Karlin and Tayltlr ( l98 1) for dcrivations mogorov forward and backward cquatitlns.

.

and more dctailed discussion!i

()1'

thc Kt)l-

t'

'' .

i

l

!

1

:7

I 2

.4:

1

, '

. .

1i y t !

@ ) ). :j l ' kE ' jt : '! $

:k

Iue S

va

Of

PI-OCCSS

time

1t

-r()

lxlckbllll r. The K///?rltl,l.lt//-&t'

f4)
1, realizing a net payoff Fj

=

max

gP'j

-

l 0) ,

.

This could be called the expected continuation value. orjust the continuation value- with the expectation being understood. Now return to the decision at period 0. The irm has two choices. If it invests immediately, it gets the expected present value of the revenues minus the cost of invcstment. 1z$) 1. If it does not, it gets the continuation value &)() Iderived above, but that starts in period 1 and must be discounted by the factor l /( 1 + r) to express it in period-t) units-The optimal choice is obviously the one that yields the larger value. Therefore the net present value of the whole investment opportunity optimally deployed, which we denote by &, is -

1;.)

=

m ax

l.$)

-

l

l ,

l

+ r

'.)-4 ) (Fb1

.

The firm's optimal decision is the one that maximizes this net present value. This capturcs thc essential idea of dynamic programming. We split the whole sequence of decisions into two parts: the immediate choice. and the remaining decisions. aII of whose effects are summarized in the continuation value. To tind the optimal sequence of decisions we work backward. At the last relevant decision point we can make the best choice and thereby tind the continuation value (h in our example). Then at the decision point betbre that one, we know the cxpected continuation value and therefore can optimize the current choice. In our example there were just two periods and that was the end of the story. When there are more periods. the same procedure applies

repeatedly. The decision where the investment opportunity remains available at period 1is Iess constrained than the one where it must be made on a now-or-never basis in period (). Equation ( 1) shows the net payoff f24)for this latter case; since that situation terminates the decision process at time 0, Iet us call it the termination vtrl/lleut time 0. N()w we have the net worth 6) of the less constrained decision problem from equation (3).The difference (Fi) f2()) is just the value of the extra freedom, namely the option to postpone the decision. ln Chapter 2 we calculated the value of the investment opportunity, Fi), and the termination value, Q(), for some specihc cases of this general model, -

khltllenlaticalStlcqrtplfnt/

98

where the parameters C), etc., were given numerical values. Readers can now refer back to those examples and place them in the context of th general theory. Hcre we point otlt one fcature of those results. by reference to Figure 2.4, which shows these values as functions of the initial price, C).When C) exceeds the critical level of 249, the tirm finds it optimal to invest at opce. Then the option to postpone is worthless, and 6) coincides with f2f), which equals 1-r() I in this range of the price. Nvllen 1) < 249, it is optimal to wait; then the graph of 6) lies above that of ftl. A similar property holds as other parameters are varied in other figures in Chapter 2.The idea is that the critical point where immediate investment just becomes optimal is tbund where the lines representing the value of the full opportunity, ), and the termination f2(), meet. value, ,

-

To get a better idea of the factors that affect the value of the option to postpone, let us examine more closely the sources of the dit-ferences between -()and Q(). First, by postponing the decision the firm gives up the period-o revenue /-1).This difference favors immediate action. Second, postponing the decision also means postponing the cost of investment; this favors waiting since the interest rate is positive. (More generally, the cost of investment could itself be changing over time, and that would bring new considerations; for example, if the firm expects capital equipment to get cheaper over time. that is an additional reason for waiting-) Third. and most important, waiting allowsa separate optimization in each of tht: contingencies of a price rise and a price fall, whereas immediate action must be based on only the avcrage of the two. This ability to tailor action to contingency, specifically to refrain from investment if the price goes down, gives value to the extra freedom to wait-l 1.B

Many Periods

1)t)

they can be regarded as limits ()f randlln) each time period and ()f each stcp 01' the W2y.

Nvith our application

I In technical terms- the mltximum is a convex function. so by Jensen's Inequality the averof age the separate maxima in equation (2) is greatttr than the maximum o1' the corresponding

avera gCS,

to investment in mind. we will retkr to the decisitllls

of a firm, but the theory is ofcotlrse perfectly general. The firm's curront stattls as it affects its operation and expansion opportunities is described by a state variable F()rsimplicity ofexposition we taktt this to be 11scalar (realntlmberlp but the theofy extends readily to vector states of any dimension. At any date is known. btlt ftlttlrc or period t the current value of this variable random variables. We values + l are suppose that the process is .r.

.rf

.

erf

-p+a,

,

.

.

.

Markov. that is. aIl the intbrmation relevant to the determination of the probability distribution of ftlture values is summarized in the ctlrrent sta t e xt At each period t, some choices tre Ctvltilable to the firm. and we represent them by the control variablets) l/. In the Ill3llve cxample where the only clloice was whether to invest at once or wait. we could let lt btl a scalar binary variable, whose value () reprcsents waiting and l represents investing at once. In other applications, for example. if the scale of inveslment is a matter of choicc. lt can be a continuous variable. I f the firm has choices in :ltldition to those bearing on invcstment, lbr cxample. hiring labtlr at time / then lt can be a vectllr. Thtl value lIt of the control at time / must be chllsen tlsing (lnly thll inlbrmation that is available at that time, namely The state and the contrlll :lt time / aftkct the firm-s immttdiate prtllit I1()wwhich we denote by n.t (m lIt ). I'Iere thtt relevant contnll variable s/, migllt be the quantity of labor hired t)r raw matcrials purchased. The and ltI ()f period t also affect the probability distribution ()f future states. I-lere ltt can be the amount of investment or Rct D. or evcn a decision to abandon the enterprise. Let *? ttt ) denote the cumulative probability distributilln l'unction ! 1 of the state next period conditional upon the current information (state and .

,

-rf

.

.

-rf

.%-t+

We now generalize the two-period example above. 0ur applications in subsequent chapters are mostly to situations where time is continuous and thc uncertainty takes the form of Wiener processes or more general diffusion processes for the state variables. However, in this subsection we develop the theory of dynamic programming in a setting where uncertainty is modelled using discrete-time Markov processes. Some general properties are easier to demonstrate in this format. Also, the setting ofthe rest of the book is a limiting case. Diffusion processes are Markov processes, and as we saw in Chapter 3,

walks in discrete tinne :ls tlle ltlngt11ol' snlall in a stlitClble t'leta'lllle

walk both

-rf

.

control variables).

The discount factor between any two periods is I /( l + p). where p is the discount rate. The aim is to choose the sequence tli' controls jlff) ovflr time so as to maximize the expected net present value of thc payoffs. Sllmetimes we will force the decision process to end at some periotl T, with a hnal payoff that depends on the state reached; we de note th is termittation J,vl.vf.#/' function by Q r fx.r ) We are ready to apply the basic dynamic programming technique. Rcmember the idea is to split the decision sequence into two parts, the immediate period and the whole continuation beyond that. Suppose the current date is t and the state is xt Let us denote by F)(.p ) the outcome - the expected net '

.

.

s'latllel:lilticalJ'ltkckqrtpl/llt/ present value of a1lof the firm's cash flows when the Iirm makes all decisions optimally from this point onwards. When the 5rm chooses the control variables

ltt,

it gets an immediate Optimal

profit Ilow n.t xt, ur). At thtz next period (1+ l). the state will be decisions thereafterwill yield, in the notation we have establisheds

l)y,/lfl??lcOplilnizaion I/nJt!?-Ullcertall' lf the many-period problem has a tixed finite time horizon T. we similarly start at the end and work backward. At the end of the horizon firm gets a termination payoff f2rl-rrl. Then the period before,

can the

-p+1

.

(-rf+l).

';+1

This is random from the perspective of period /. so we must take its expected value ',(F;+l (m+l)).That iswhatwe called the continuation value.2 Discounting back to period t, the sum of the immediate payoff and the continuation value is (-rI

n't The frm will choose #) xt ). Thus

ut

ttt

,

)+

1

1+

-,

p

(&+l (-r?+ l )1.

to maximize this, and the result will be

just the

value

Thus we know the value function at F 1. That in turn allows us to solve the maximization problem for llr-c, leading to the value function Fr-al-rr-a), and so on. At one time this was thought to be too complex a procedure to be practicable. and aIl kinds of indirect methods were devised. However, advances in computing have made the backward calculation remarkably usable, and several of our numerical simulations in later chapters use it. Later in this chapter we will oftkr an example of it. -

lnhnite Horizon

1.C The idea behind this decomposition is formally stated in Bellman*s Principle lil/fl/fltffl/l, Of Optimality-.adl Optnalpolicy has J/l(?J7rt@t!r@l/lflJ. Wlliletrtlle Ihe

rtrrrlflrlrl#

ChOiCC.% Conlitute

(ln

Optimulpolic'y

114//1

rlMpo':t

10 J/lfJ SlbproblClll

starting at the state that results rrrl //lfJ initial fltW(?ll.. Here the optimality of the remaining choices uf+I l/+2, etc., is subsumed in the continuation value, so only the immediate control ttt rcmains to be chosen optimally. ,

The result of this decomposition, namely equation

(4), is called

the Bell-

Ltjelndamental equation sfpplfrnl/l-y. Tt) reiterate, the hrst man eqttation. or right-hand side is the immediate proht, the second term constithe term on continuation value. and the optimum action this period is the one the tutes

maximizes the sum of these two components. In the two-period example, immediate investment gave P'4) /, waiting & )/ had no period-o payout but only a discounted co'ntinuation value (1 + r), and the optimal bina:y choice between these alternatives yielded

that

-

'()(

the larger of these two. Thus our earlier equation the general Bellman equation (4).

(3) is a special case

tf

lf there is no tixed finite time horizon for the decision problem, there is no known final value function from which we can work backward. Instead, the problem gets a rccursive structure that facilitatcs theoretical analysis as well as numerical computation. The crucial simplilication that an intinite horizon brings to equation (4) is independence l'rllm timc t as such. Of course the current state xt matters. but the calcndar date / by itself has no effect. This works provided the Ilow profit function :., the transition probability distribution function *. and thu discount rate p arc themselves al1 independcnt of the actual Iabel of the date. a condition that is satisied or assumed in many economic applications. In this setting, the problem one period hence looks cxactly like the probIem now, except of course for thc new starting state. Therefore the value hmction is common to all periods, although of course it will be evaluated at Therefore we writc the function as Fxt ) without any time different points symbol. The Bellman equation for any f becomes the function label on -rf

.

Flxt ) tn71eexpectation notation is generally clear enough. However, to make it precise that the information at time t includes the state and the control at that time. we slatc it formally once for reference'.

Since

=

max ,,,

n' (.,p u: ,

)+

1 l + p

(

I

-(.rI+

l

)l

and xt-vbcould be any of the possible states: write Then, for all we get form as and -rr

-r'.

.,r

.r,

.

them in general

rlfz,?? ic.

!()2 wherc we have now denoted the expectation as conditioncd ()n tlltl kpowledge Illlinitely ot. the current period s and ll. This is the Beliman eqtlation Ior the repeating, or recursive. dynamic programming problem. ?

.

;

,

-t-

.

,

-r,

equations

,

as the

hmciollul

Despite superticial appearances, this equation is not Iinear. The optimal choice of lt depends on all the values Flx'j that appeara weighted by the appropriate probabilities. in the expectation on the right-hand side. When this optimal control is substituted back. the result can be nonlinear in the Fx') values. In general we do not know whether nonlinear functional eqtlations have solutions, let alone unique ones. Fortunatcly. the recursive Bcllman equalion has a very special structure that allows one to prove existence and uniqtleconditions typical of our economic ness of a solution function F(-r) under side technical issue for our more practical concerns. applications. This is a sketch the ideas in Appendix A to this chapter. and of We include a brief excellent readers treatments in more theoretical books, can find interested with and example, Lucas Prescott ( 1989, Chapters 4,9). But the Stokey for technical argument does have an indirect payoff: it is essentially a practical solution method. This takes the form of an iterative procedure. Start with any guess tbr the l' side of equation (5) true value function, say Ft (-r). Use it on the right-hand l which rule choice tt can now be expressed and hnd the corresponding optimal right-hand side becomes the as a function of x alone. Substituting it back()f Ft2)(-r). Now use it as the next guess t)f the true call it a new function value function, and repeat the procedure. Then the successive guesses F3 (x), F4b ( v), etc., will converge to the true function. Convergence is guaranteed with a good initial guess no matter how bad the initial guess, but of course the procedure will reach the desired accuracy of the approximation in fewer ,

.r;

steps.

lil ? li--tl

t rl/ l

:tl

Jt/t/?-

Ivllrc't','/:, ilI

b'

Tlle key lies in tlle ltcttlr l / ( l + /)) 017 tlle rigllt-hand siule. Tllis beillg less than l it scales down, or contracts, any errors i11 the guess from one stcp to the next. As Iong as the profit llows are botllldetl. any errors in tlle clloice of soltltion is Ieft. u cannot blow up. Gradually. only the correct .

Now that we have no fixed terminal date from which to work backwarda we value function F seem to have Iost an explicit or constructive way to tind the And without knowing the function F we cannot find the optimal control tt by solving the maximization problem on the right-hand side of the Bellman equation. Thus we need assurance that a solution actually exists. and a way to find it. Luckily, neither question is vey diflicult. The recursive Bellman equation (5) can be thought of ts a whole list of equations. one for each possible value of with 71 whole Iist of unknowns, namely all the values F(.r). If x took on only a tinite number of discrete values

-rf this would be a simultaneous system with exactly as many number of unknowns Fxi ). More generally. we can regard (5)as F as its unknown. equation, with the wholehtltction

t';I/J

This procedtlre is very easy to

understand-

progranx- and conlpute.

I t can

take :1 long tilne on tlle computer. especially if tlle discount rate p is very small doNvn. I-losvever. that is no Ionger so that each step does only a little scaling consideration when individtlals can Ieave their (nvn personal prohibitivd a computers running for days without tying up scarce mainframe resources.

Theret-ore this method is increasingly used in many applications- and even in econometric work 'Wewill givtt a nunnerical exanlple of this method, too. Iater in the chapter. .

1.D Optimal Stopping One partictllar class of dynamic progrilnlnling problems is very importltnt for our applications. Here the choico in any period is binary. One alternative corresponds t() stopping the prllcess to taktl the termination paytll-- and the other entails contintlation for onc periotl. when ttnother similar binaf'y choice will be available. In the investment nlodel tlf Chapter 2 and the tirst setrtitln of this chapter. stopping correspllnds t(' mllking tlle investment. ltnd continuation correspllnds to waiting. l-lere continuatitln dtles not generate any prlllit flow within the period. Howevera in other contexts thtlre may I7e sonle such flow. Ftlr example- for a firm in bad ttconom ic ctlndi titlns t hat is con te nl piatltnd ing shutdown, continuation nlay givc a prtltit flow (positive ()r negative). eqtlipment. vltlue minus ol' tllc and plant termination may yicld some scrap tlther any severancc payments thc lirm is required t() make to its workers and restorlltitln, of site breaking contracts. etc. costs Let zr (-r) denotd the Ilow prolits antl t'''z(.r ) thc tcrnlination payoff. Then the Bellman equation becomes F( ) -r

=

m ax

t' (x )

zr ( ) + .r

,

l

For some rangc ofvalues of-r. the maximum

,

1. (F ( ) ..

I+ p

()n

-r

i

-r

1

.

the right-lland

(f)) side of this

will be achieved by termination. and for other values ()f it will be achieved through continuation. In general this division could be arbitrary; intervals where termination is optimal could alternate with ones wherc contintlation is optimal. However, most econtlmic applications wili have more structure. -r

with torminatitln optimal on one side and There will be a single cutoff continuation on the other. For example, in the invttstment problem ()f Chapter 2 we had a critical Ievcl of thc initial price. /1) 249, such that in perilld (), -r*,

=

Matllcmatical Stzc/grlfnt

104

investmentwas optimal to its right butwaitingwas optimal to its Ieft. All ofour applications will have a similar property, and in each case it will be intuitively clear which action is optimal on which side of the threshold or cl:toff point. To complete the reader's understanding of this result. we should explain the general conditions that lead to it. We stat these intuitively here, and explain them somewhat more formally in Appendix B. For sake of dehniteness we concentrate on the case where continuation for x > x' and stopping is optimal for x < x*. Let us examine the forces that will make continuation more attractive relative to termination for higher values of First, the immediate protit from continuation should become larger relative to the termination payoff. Since the former is a :ow and the latter is a stock, we need to express them in comparable terms. The precise condition turns out to be that

ttlltler naltlic O/p/l'l?lzt7fibpl

f.7?,(7n-Jt.2j??(v

1()5

time is continuotls. We Lt goes to zero n.(a- lf 1) for t he rate of the profit llow, so that the actual protit over the time period of Iengtll AJ is gx. l/. /) Lt. Similarly, Iet p be the discount rate per unit time. so the total discounting over an interval of length L is by the factor 1,/(l + p lf ). and

write

The Bellman equation

(5) now

becomes

and rearrange

to write

,

,

is optimal

Multiply by ( l + p

1)

-r.

n.(x) + ( 1 +

p4

-

l

-

(7)

(Q (x lx1 t'-z(-v) ')

-

should be increasing as increases. Second. any current advantage should not be likely to be reversed in the near future. For this, we need positive serial To correlation, or persistence, in the stochastic process of evolution of be more precise, if this period's rises. the conditional distribution *(.r'1-r) of next period's values x' should give greater weight to larger values, that is. it should shift evelywhere to the right. (In technical language, this is the dominance.'') These two conditions together concept of are sufEcient to ensure the desired result. lf the expression in (7) is decreasing as x increases, then continuation will Note that the second be optimal to the left and termination to the right of require negative persistence condition stays unchanged', we do not switch it to of the stochastic process.

Divide by

l and

let it go to zero. We get

-r

p F(

-r

-r.

-r

thrst-orderstochastic

-r*.

that both of these properties will always bd satished for our applications. The srstis easily veried in each instance', the second is tnle for random walks, Brownian motion. mean-reverting autoregressive processes, and indeed in almost all economic applications we can think of. For ease of notation in conveying te concept, we did not allow the protit payoff to depend on time t as such, but that extension presents no terminal or difhculty.The threshold simply becomes a function of time, (/). This will be the case in much o the rest o the chapter and in many o Our appliations. We repeat

-r*

1.E

Continuous Time

Now return to the general control problem of Section 1.B, but suppose each time period is of length Lt. Ultimately we are interested in the limit where

.

t)

=

m ax 1/

;'r

(

-r

.

1/

.

t)+

l tlt

t.-gtF j

,

where ( 1/Jl ) JII(IFl is the limit of 7qty Fj l We must remember that the expectation is conditioned on the current and II, and we must rcmember to include the inlluence of changes in both x and / when we calculate the change in Fx, 1) ovcr the interval Jl. .

.r

This form of the Bellman equation makes explicit the idea that the en-

titlement to the Ilow of prolits is an asset, and that t) is its value. On the left-hand side we havc the normal return per unit time that a decision maker. using p as the discount rate. would require for holding this assct. On -(-r.

' thv right-hand side. the first term is the immediate payout or dividend from the asset, while the second term is its expected rate of capital gain (loss if negative). Thus the right-hand side is the expected total rcturn per unit time from holding the asset. The equality becomes a no-arbitrage or equilibrium condition, expressing the investor's willingness to hold the asset. The maximization with respect to tt means that the current operation of the asset is being managed optimaily, of course bearing in mind not only the immediate payout but also the consequences for future values. The limit on the right-hand

side depends on the expectation

correspond-

ing to the random f a time Lt later. There are two classes of stochastic processes in continuous time that allow such limits in a form conducive to further analysis and solution of the function Flx. t) in the continuation region.

Luckily they are particularly useful for many economic applications. In fact and Poisson processes we discussed in Chapter 3. We will

they are just the Ito

develop the theory of dynam ic programmi next two subsections.

ng in the ir spet:i Iic contxts

i1,1the (

valtle function F

,

we Illlve

.

The above analysis is local to the sllort time interval (/ t + dt ), and the resulting equation holds for any 1. We can complete the analysis by choosing :1 terminal payoff, gr lctting the horizon a finite time horizon T and imposing using rccursive the structurea or some other way. ln any of be infinite and of existence and uniqueness of solutions mathematical proofs these. rigorous continuous Since the details are immaterial ftlr time. become quite hard in applications, we omit them and refer the reuder to Fleming and Rishel our ( 1975) or Krylov ( 1980). Our mathematics has been simplilied in another respect. We have treated will conthe limit to continuous time in a very casual and heuristic way, and tricky reader quite that some tintle to do so. However, it is fair to warn the rigorous carcfully treatments. in more issuesare hidden, and must be handled In discrete time, we stipuiated that the action llt taken in the current period t but not on the random could depend on the knowledge of the current state the two coalesce. We have to be careful l In continuous time future state about not to allow choices to depend on intbrmation abotlt the futtlre. evcn with of would hindsight. benefit the acting be tthe next instant-'' Otherwise we requiring and could make insnite profits. Technically this can be avoiddd by strategies while the time right'' in from the the uncertainty to be in thc stochastic processu!s left-'' Then the from any jumps are change until//-/ alter Jf:' installt. occur at an n-s-fclnf,while the actions cannot 39-44)1. For a discussion and rigorous analysis, See Duffie ( 1988. pp. l -

-k

,

-p+

.

t'continuous

''continuous

t-. !

1.F

Ito Processes

stochastic process that yields a simple form for (8) The lirst continuous-time is the Ito process we discussed n Chapter 3. Equation ( 11) of that chapter dehned the formula for its increment. which we recapitulate here, but now allowing the drift and the diffusion parameters to depend on the control variable as well as the state variable:

dx

=

a (-r ,

1/

.

t ) dt + bx

.

11

,

t ) Jz-

where dz is the increment of a standard Wiener process. As before, we the proht llow as zrt-r, tt. t) and the value of the hrm (asset)as Fx. t ). ='

.x

.

tt

.

t ) b;. ,. (a. t ) .

) .

We can express the optinal If as function of Fl a (.t f )a Fx(.v ). F a. (.' / ) well as x. f and thc various paramete rs t hat govtlrn t Ilc functional lbrm ()t' Jr as s a, and b. Substituting this expression for the ('ptimal 1/ back into the right-hand side ofequation ( l ())- vve get a partial diftrcntial eqtlltion of the second tlrder. with F as the dependent variable and x tnd / as tlle independcnt variables. In general this equation is very complieated. However. in many applicatillns we can develop wasys to solve it analytically or nunaerically. . The solution methods are gencrally clllltl(.lglltls to thf lse I'(lr discrete time. lf there is a (ixed time Iimit F when a terminatilln payllff f2 fxv F ) is tlnlklrctttlthen the eqtllltion has 11 boundllry contlititln -

,

.

,

.

.

-.( r .

We can start at timc

r

.

r)

and work

f2 (

.r

()u

.

F)

r w:ly backward to find F (-r J ) for al I .

earlier times. In fact. in practice we have to chollse 11 discretc grid t)I' values of A. and t on which to calcuiate the slllu tion. We will oflkr twt) examples ()f this procedurca onc later in this chapter when we directly solve the underlying dynamic programming problem. and one in Cllapter l t) where we solvc the partial differcntial eq uation itsel f. If thc time horizon is inhnite and the l'tlnctions

zr t/, ttnd ,

/?d() not deptlnd

explicitlyt)n time the neither does the value i'unction depend equation ( 10) becomcs an ordinary differential equation with independent variable'. ,

()n .r

time, and as its only

write

+ dx the Let x be the known starting position at time f and x' ftar Lemma A/. lto's of time of small interval end the a random position at the of Applying it to 3. equation Chapter stated in (25) such a process was ,

4..2.! bl (-r

Note that we have followed standard calculus notation and used primes tt) denote total derivatives of a function ofone independcnt variable, and subscripts to denote partial derivatives of a function 01' several independent variables.

Mathematical fkckgrtxlnf,l

108

We will generally adhere to this, but will occasionally use subscripts even for superscript for total derivatives, for example if the function symbol needs a some other reason. will ln most of our models of investment throughout Chapters 5-9, we will and develop have occasion to fofmulate and solve equations like (11), appropriate solution methods for them gradually. We turn to one special kind of control, namely optimal stopping of an Ito process. that is of particular

importance in all of our applications. 1.G

Ullcertatty

Now the Bellman equation

l ()9

for the optimal

stopping problem,

(6),

becomes Fx

.

t)

=

( 2(.r. t )

max

zr (.,r t ) +

,

-

l ( 1 + p dt )-

z.-(Ft-r

+ dx t + ,

In the continuation region, the second term on the right-hand

of the two. Expanding it by Ito's Lemma and simplity'ing as partial differential equation satished by the value function: 12 bl ( v' t ) Fra.(.v t j + 9

Optimal Stopping and Smooth Pasting

,

.

tl

tx I ) F).(.x t ) + .

.

/7)(.: t ) ,

-

JJ

) I A'l

)

.

side is the larger above, we get the

p F(x t ) + zr j..r t ) ,

,

=

0

.

(l 3)

where subscripts denote partial derivatives. > This holds for (/), and we must Iook for boundary conditions From the Bellman equation, we know that in the that hold along stopping region we have F(x, / ) Q (x. /). so by continuity we can impose the condition F(.r* (J ) J ) (-r*(/ ) t ) for alI t (l4) .r*

.r

Here we consider a binary decision problem. At every instant, the 5rm can either continue its current situation to get a proft Cow, or stop and get a termination payoff. Both the prost flow xx. t) and the termination payoff Q (x. t) can depend on a state variable x and on time f where x followsan Ito ,

process

Optilllizatiolt I/lltcr ??t'???lc

-r*(/).

.r

=

=

''z

.

=

,

.

This is oten called thc condition'' because it matches the values of the unknown function F(.r. / ) to those of the known termination pyoff function f2 (a'. t ). But the boundary itself is an unknown'. the region in (-r. tq space over which the partial differential equation ( 13) is valid is itself endogenous. The boundary of tllat region. namely the culwe -E*(/), is called a boundary,'' and the whole problem of solving the equation and determining its region of validity is called a free-boundary problem. It is clear that we need a second condition in addition to ( l4) if we are to find x*(1) jointlywith the function F(x, I). The general mathematical theory of partial differential equations is of Iittle help in this regard; the conditions applicable to free boundaries are specific to each application and must come from economic torphysical or biological. as the case may be) considerations. For us the right condition turns out to require that for each t, the values F@. t) and 62(-t, t), regarded as functions o'f x, should meet tangentially at etvalue-matching

dx

=

a (x t dt + bx ,

.

(12)

t ) #z.

The most obvious example is of a 5rm deciding whether to cease operation and sell its equipment for its scrap value. lnvestment decisions can also be put in this form: continuation means waiting, and the flow payoff is zero; stopping expected present value means investing. and the termination payoff isjust the minus the of investment. cost of future protits from the project Intuition suggests that for each t there will be a critical value x*(f ), with continuation optimal if xt lies on one side of x. (J), and stopping optimal on the other side. We saw in Section 1.D that some conditions must be imposed on the proht flow and termination payoff functions to ensure this. Continuation will be relatively more attractive for larger values of if the expression in (7) is an increasing function of x. In continuous time, if x follows the Ito process (12), then the analysis of Appendix B shows that .x'

xtxl -

p

f2(x t4 + ,

(1(.x,

t)

tt + t'-2xt.x.

lj bx. J)2 fzxxtx. tj + fzf(x, f )

should be increasing in x for each t. Similarly, continuation will be less attractive relative to stopping for larger if the expression is decreasing. In each of will hold. For sake of exposition we our applications, one of these conditions the former takc up case. (/) for various Given such conditions, we can regard the critical values ?) space into two regions, with contindivides frming the that (-r, a curve t as uation optimal above the curve and termination optimal below it. Of course = x.(t) in advance, but must hnd we do not know te equation of the curve x problem. programming solution of of the dynamic the it out as a part .x

.x*

ttfree

the boundary

-r*

(J),

or b (x*(/ ) t ) ,

=

Q.r(-x*( I ) t ) .

for all

I

.

( l5 )

This is called the contact'' or condition'' because it requires not just the values but also the derivatives or slopes of the two functions to match at the boundaly. t'high-order

t'smooth-pasting

While continuity is very intuitive, continuity of slopes or smooth pasting

is more subtle and remarkable. However, the argument for it is somewhat technical, so we relegate it to Appendix C. Here we merely illustrate how it works.

Nlathenlatical f'ltlc/v-qrot//?t/ 1.H

Optimal Abandonment

Example

1;h.p?7:l??1/c(jlilliltli--flkl'llllI/??r/cr

of a Machine

IvlllQ'Q,t.t(lillI.

x

that follow are full t)f applications of dynamic programming and contingent claims analysis. Here. however. having developed the theory at a general and rather dry Ievel. it wtptlld be useful to offer a concfete example. We do not solve it in detail, but merely state the solution backed by some intuition. We hope this illustrates the various steps in a specihc context, and prepares the reader for more detailed agplications to come. Suppose the asset is a machine used to make widgets, with a total physical life of T years. Its profitability declines over this Iife, both because it wears out gradually and thus produces less output or rcquires more maintenance, and because technical progress elsewhere in the economy makes this machine less competitivewith newer ones.rrhere are also random shocks to its productivity, because of the general business cycle, or because o idiosyncratic variations in the demand for widgets. Let the state variable x be the current operating and suppose it evolves according to profit The chapters

0 05 .

0

2

4

6

-0.05

10

8

x

-0.1 0 -0.15

now,

J-r

=

t? dt +

b t/z.

0 to rellect the gradual decline over the Iifetime. At any time during the physical life. the firm can abandon the machine. If the current prost tlowbecomes negative. this may seem an attractive alternative. Once abandoned, however. the machine will rust quickly and be very costly to restart should the protit llow recover. Therefore the abandonment decision will have to Iook ahead to such future possibilities. The tirm will condition. Of course, acccpt some losses to keep the machine in operating Iife of the machine is physical accepting losses will be less compelling if the variables, the currcnt drawing to a close. Thus we must keep track of two there will that profit, x, and the age of the machine, t. Intuitively, we know this curve the be a threshold culwe x't ) such that if the current x falls below machine will be abandoned. 10 The actual parameters used in the calculation are F = l() years. p which = standard implies a 0.2, l per year, and b percent per year, a deviation of 0.2 over one year, or a variance of 0.04 per year. To obtain a numerical solution, we use a discrete approximation to Brownian motion, 0.01, or 3.65 days. Correspondingly,each discrete step of the profit with Lt

where a


-t?''' ( r ) the limiti ng functilln F.( ) is of the Iixed pllint iteration step. l t obviotlsly satisfies the functitlnal equatitln a ()f The (5). geometric reductilln in crrors (technicallytlle property mapping tion property'-) has alltlwed us to prtlve existence and un iqueness of the solution. and the iterative procedurd ctlnsti tutes a numcrical algorithm. tettrrllrss-

-t'''

Here we sketch some tcchnical arguments that prove the existent:e and uniqueness of the solution ()f the Bcllman equation (5) for infinitc-horizon ()f reference'. dynamic programming. We restate the equation for ease

I

-h /? )

+

'

'

.r

-bcllntrac-

B

Ffl(XXrf)N?1t/

Nltlletntlilull

128

Here we consder the case of a bina:y choice between continuation and stopwhich we repeat for ease of reference'. The Bellman equation is

(6),

F(x)

=

max

t' (x) z (-v)+ .

1111(1e1.fl/'lc-tz/-&lpl/)' f.?J7???li'-:'t??'&,?

(11:The

Assulnptioll

Optimal Stopping Regions

ping.

ilyllalltic

1 l+p

'

1-(Fx )

I -K!

.

Continuation is optimal for those values of for which the maximum on the right-hand side of (6)is attained at the second argument, that is, .v

region'' Call the corresponding divisions of the range of the ttstopping of the interested respectively. in structure We region.'' are and the &continuation

.r

these regions.

For arbitrary speciscations of zr(.r), f2(.r), and *(.r'l-r), the regions could be any sequence of alternating intelwals. Thus continuation may be optimal for a lowest range of values of stopping optimal for a range above that, then of continuation again. and so on. However, one expects that in many problems and high low into the division of will range be a clean economic interest, there such that continuation is optimal for values separated by a threshold. say, and stopping optimal for x > (orperhaps the other way around). -t' < We hnd some conditions on the payoff and distribution functions that yield such a division. f) (-r) by G (x) Subtract Q (x) from both sides of (6),and denote Fx) for brevity. Then -v.

-v*,

-v'*

-x*

-

expression

n'(a-) + ( l

+ p )-

1

'

'

Q ( r ) d * (a-I ) -r

.

-

f2 (-v)

is a monotonic

function of x; tbr dehniteness. make it increasing. This is just the difference between the value of waiting for exactly one period before stopping, and that of stopping right away. The nice point is that the advantage of waiting for exactly one period translates into an advantage of waiting until an optimally chosen stopping time. If the function is increasing, it, we expect continuation to be optimal for high a' and termination forjow decreasing, the other way around. $. ,'

-r;

.

.'

Assltlnption

and immediate termination is optimal when the opposite inequality holds.

l29

There (21-.

is positive persistence of uncertainty, in the sense that shifts uniprobability distribution *(.v'l.r) of future values right when the current value x increases. If this failed. a larger current relative advantage for high x would be more likely to be reversed in the near future. This assumption will hold for alI the processes we will considcr.

the cumulative formly to the

.v'

Given these two assumptions,

the solution

function G(x) for

(3l) must

bc increasing. To see this. note that thc second argument of the max operator on the right-hand side consists of two parts. The tirstswhich is just the expression in Assumption I has been directly assumed to be increasing. The other. namely the integral, is incrcasing if G(.v) is. To see this, note that by Assumption (2J,a larger x shis the probability weights attached to the increasing set of values G(-r') to the right. and thereforc raises the expected value. Thus. startingwith an increasing function. the right-hand side yields another increasing function. Then the lixed point of the ittlration step, namely the solution of (31), is itself an increasing function.7 ,

We have proved that the second argument

in (31)is increasing. Therefore

there is a unique x' such that the second argument is positive if and only if and stopping is optimal Then continuation is optimal to the right of x> to the Ieft, as we set out to prove. -v*.

.r*,

In continuous time, we must replace p by p dt and zr(.v) by zr(-v) dt. Suppose x' follows the geometric Brownian motion dx + J.r, and =

+ (1 +

p)

-

:

-v

=

J* (x'I.r) G (-x'/)

Now make two assumptions that together sufhce to establish the desired

property.

-v

7To be technically prccisc. the operator

is closed on the convex cone of nondecreasing

functions. and therefore has a fixed point in this subspace. That is also a sxedpoint on f)f space functions. However, wc have already shown in Section A of the Appendix that xed point is uniquc.

the whole the latter

z'

l 3()

t'Yltllllolltltictll /.?tl('/v',r'pll?l(/

Then. on expanding j''z (A') using It()-s Lelnma and sinlplifying. (l I becomes silnply the requirement tlllt

dt + rr Assumption Jz

tlz.

-r

.v'

l 3.l

.n'(-r)

-

p f2 (

.

2

pz S''z(-r) + a t'r

)+

-r

-''

'

.

.

'

f2 (.r)

be monotonic. +'

C

Smooth Pasting

Here we consider the optimal stopping problem with a finite horizon and time

dependence, when the state variable follows an Ito process. We demonstrate somewhat more formally the value-matching and smooth-pasting conditions for (14)and (15)that determine the free boundary that separates the continuation and stopping rcgions. Over a short intcwal Fx

t)

.

=

max

time dt, the Bellman equation

()f

( f2 (.r)

zr (x / ) .

.

+ (l

(//

(6) becomes p dt ) Fx / ) + -(J F'J)

-

.

.

Stopping is optimal if the first term in the braces on the right-hand side is the Iarger of the two. and continuation is optimal if the second dxpression is larger. Fix a particular t. For dehniteness we consider the case where continuation is optimal tbr x > x* t ) and stopping for a- < x* (/ ) First suppose. contrary to the assertion in ( 14), that F'(x*(? ). t ) < f) (.r*(?)- J ). By continuity, we will have Fx, t ) < fz (.v t ) tbr just slightly to the right of (/ ). The second expression on the right-hand side diffdrs from Fx ? ) only by terms of order d so for sufficiently small t//, it. too, will be Iess than 2 (-r t ) over this range. Then immediate stopping wili be optimal tbr such contrary to the definition of (?) as the thrcshold. Next suppose that Ffx' t ) t ) > t'l (-r* (?). / ). By continuity, we will have F(.r / ) > (.v t ) for just slightly to the left of JJ, small continuation will yield a value that is x*(/). Then, for sufhciently /). also greater than f) (.v. so stopping cannot be optimal there, contrary to the definition of x* t ). The argument for the smooth-pasting condition also proceeds by contradiction. We develop it with the aid of Figure 4.2. Again consider just the case where continuation is optimal to the right of ) and stopping to the Ieft. If the functions F(x, /) and f2(.r. / ) do not meet tangentially at x*t ), they must meet at a kink. This cannot be an upward-pointing kink as in part (a) of the hgure; else by continuity f2(-r '/ ) would exceed F(-r / ) for slightly greater than (/), and termination rather than continuation would be optimal tor such x, contrary to the desnition of as the threshold. Next consider a downward-pointingkink as in part (b) of the figure. Here we show that (J) .

-r

.r

*

,

.

,

,

.t-.

.r

indil-l'klrtl'llccI'ltltsvtlel'ltlle twl) cllklices: contintlatilln for cannot be :1 pllint short interval of t in)e A / is tltl li ni tely t he be tter p()Iicy. 'I-he in tui tive idell is 11 thltt l7yNv:iting 11 little l7itItlnger. vtl c:ln (.'ll'lstlla!ll the next step of and cl)lltlse ()1* the twro dlles better tllan ptlsitillns t)l1 tlitht?r sitle (.)1' the k illk 'Xn llverkge tlllltlgh Intlst be discotllltetl kink itself. tllllt ltver:lge ptlint is trtlc the even zN/ tlccurs I'(1r it inle is th:lt lkter. Brllwnirtn t n'llltillll t he bccause rtlllsl')n ()f lntl ltre tlle A/ prtlpllrtitlnal ()n the elkct tlleir t() stltllrtt rollt s() is steps tl vvhilc t)f eflkct tlle A/ isxltln Wllcn ting is pnlpllrtillllal t() tl valtle. is snlltl 1. the prrner effect is rclatively nlucll lilrger. argunlellt needs t() be spelled ou t in :.1 Iitt Itl nlorut tlgebraic dc tail ('lt'

.t'

.

'l-llis

-1-11t:

-

.

''rllis

.

C.s

*

.

'z

.v

.

,

-v*(/

steps

in Chapter 3sizu A/? /)(.r

yvtl

(:,1'

JIl + tl (a-

/7

.

I

)

.

treat the pnlcess ('1* as lt rtntltlnl wltl k t hat cltn take ) A / tl p ()r dtnvll wi t 11respttot ive prtllatbi lities .r

/

zx t

/ /7(

/)

-v.

1

kI n Chapter 3 the drift they are mtlre general

1.2 l

I

q

.

tl

(

/)

-r -

Al

/ b(

-t-

.

/)

l ,

and diffusion ctpefficittnts u and t'r werc cilnstant: httre lnctitlns u (.r / ) and /?(-t. / ) l llltlwed cllnsider thc alternative Ntlw ptll icy: continuatilln j'tlr tin'le tt Clnd stopping t() take by further continuation if the next step of is upward. .

,

,

.r

the ttrmination payoff if it is dllwnward. Witll the appropriate probability weighting and d iscounting, this yields

.r

.

,

.

.x

n' (

-r

*

( ) t ) L J + ( 1+ p L / ),

1

gp F (

*

A-

(t ) +

tf:5.

/1 / + A t ) + .

.v*(/)

-r*

Expand this in a Tayltlr series anlund

condititln just above, and

remtll-nbering

(/

)(

.r

*

(/ )

-

f

/? / + ,

f.

/

)

J

(-r* (/ ) t ), using tlltt val ue-matclling that A/ is ()1* order ( A/; )2. Tl'1tlfirst ,

,

Matllematical Slci-qrt/l/?lr

132 >0

terms are F(x* t ) / ) + .

) !'Fr (-r*t )

,

t)

-

Q v (x t ) J )12./l *

,

.

If the functions meet with a downward-pointing kink as in part (b) of Figure 4.2, then lr > f2.v at -r*(J), so the second term is positive. Therefore the alternative policy does better than the common value of continuation or contradicting its dehnition as the threshold where the termination at optimal policy is a matter of indifference between the two. -r*(f),

partlll

A Firm's Decisions

5

Chapter

lnvestment Opportunities and lnvestment Timing

Wl-rl I -rI IIE nlathematical

preliminidries

bellind

tls, we can

now turn to the anal-

(.)f

I1,1this chapter and tllrotlghtlut ysis investment decisions under tlncertainty. will witll main cxpenditures tlur that have booka be investment this cllncern characte ristics. the expendittlres Ieltst important vttry First. partly are at twt) llther wllrds-.u//lktllltt l7e recovered. coxts in Second. these cllllnllt irrcversible; llle oppllrtunity tllat b(l the I'irnl has delayed. so investments tlan to wait for (ltller arrive abtltlt and nlarket conditions information prices, costs. t() new betbre it commits resourccs. As the simple ktxamples in Chapter 2 suggdsted. the ability to delay an ircxpenditure revcrsible invcstment can prolbundly affect the decision to invest. simple In particular. it invalidates the net present value rule as it is commonly studcnts in busincss schools: Invttst in a project when the prcsent taught to expected cash tlows is at least as large as its ctlst.-' This rule is value of its incorrcct because it ignllrcs thc (lpptlrtunity cost of making a comnlitment waiti ng for new in formation As and the option of thereby givi ng up now, we saw in Chapter 2, that opportunity cost must be included as part of the total cost of investing. In this chapter and those that follow. we will examine this opportunity cost and its implications for investment at a greater Ievcl of generality and in more detail. In this chapter, we will set forth and analyze in considerable detail one models of irreversible investment. In this of the most basic continuous-time model, which was originally developed by McDonald and Siegel ( 1986), a firm t-

.

adFirm l5-Decisiolts must decide when to invest in a single project.The cost of the investments /, is known and fixed, but the value of the project. 1Z, follows a geometric Brownian motion. The simple net present value rule is to invest as Iong as > /, but as '

McDonald and Siegel demonstrated, this is incorrect. Because

htturevalues

of V are unknown, there is an opportunity cost to investing today. Hence the optimal investment rule is to invest when P' is at least as large as a critical value F* that exceeds /. As we will see, for reasonable parameter values. this critical value may be lwt? or three times as Iarge as /. Hence the simple NPV rule is not just wrong; it is often very wrong. After describing the basic model in more detail, we will show how the

optimal investment rule (thatis, the criticalvalue 1Z*) can be found bydynamic programming. An issue that arises, however, is the choice of discount rate. If capital markets are (in a sense that will be made clear). the viewed problem be investment as a problem in option pricing, and solved can the techniquesof contingentclaims analysis. Wewill re-solve the optimal using of the examine the characteristics problem in this and then investment way, option Finally, invest and its dependence key frm's we will to parameters. on value model considering stochastic for the the by alternative extend processes will V. characterize optimal the project, find and the In particular, we of investment rules that apply when P' follows a mean-reverting process, and then when it follows a mixed Brownian motion/poisson jump process. Stcomplete''

1 The Basic Model Our starting point is a model srstdeveloped by McDonald and Siegcl (1986). They considered the following problem: At what point is it optimal to pay a sunk cost l in return for a project whose value is lz', given that V evolves according to the following geometric Brownian motion:

J?l !.,-t?-'???lt?/

1/ t)/'-yJt'p?-?'l/?lities

f llli

Ill !.zt!.???lt..'? l / Til'l ll

costs are positive and managers have the option to shut down the factory temporarily when tlle price of output is below variable cost, and/or the option to abandon the project completely, l'' w'ill not follow a geometric Brownian motion even if the price of widgets does. (We will develop models in which the output price follows :1 geometric Brownian motion and the project can be temporarily shut down and/or abandoned in Chapters 6 and 7.) If variable cost is positive and managers do not have the option to shut down (perhaps because of regulatory constraints). Pr can become negative, which is again in conflict with the assumption of lognormality. In addition, one might believe that a competitive product market will prevent the price from wandering too far from Iong-run industrpwide marginal cost. or that stochastic changes in price are Iikely to be infretluent but large. so that F should follow a meanreverting orjump process. For the time being we ignore these possibilities in order to provide the simplest introduction to the basic ideas and techniques.

We allow exogenously specihed mean rcversion in Section 5(a) of this chaptera and consider industry equilibrium in Chapters 8 and 9. Note that the tirm's investment opportunity is equivalent to a perpetthe right but not the obligation to buy share of stock at uaI call (lption a a prespecified price. Thercfortt the decision to invest is equivalent to deciding when to exercise such an (lption. Thus. the investmcnt decision can be viewed as a problem of optilln vtluation (as we saw in the simple examples presented in Chaptttr 2).1 Altern:tively. it can be viewed as a problem in dynamic programming. Wc will derive the (lptimal investment rule in two ways, first using dynamic programming. and then using option pricing (cntingent claims) methods. This will allow us to compare these two approaches and the assumptions that each rcquires. We will then examine the characteristics of

the solution.

In what follows. we will denote the

value

of the investment opportunity

(that is, the value of tht) option to invest) by F( P'). We want a rule that maximizes this value. Since the payoff from investing at time I is /'1 J, we want to maximize its expected present value: -

dF

=

a )' dt +

.>'

/' Jz.

where dz is the increment of a Wiener process. Equation ( 1) implies that the current value of the project is known, but future values ate lognormally distributed with avariance that grows linearlywith the time horizon; the exact formulas are in Section 3(a) of Chapter 3. Thus although information arrives over time (the hrm observes V changing), the future value of the project is always uncertain. Equation

ample, suppose

(1) is clearly

an abstraction from most real projects. For exthe project is a widget factory with some capacity. If variable

F( p' )

=

max

k6jt

Zr

-

/ )d-/'F

1

where i denotes the cxpectation, T is the (unknown) future time that the investment is made, p is a discount rate, and the maximization is subject to lThe investmcnt opptlrtunity is analogous to a perpetual call option on a dividend-paying stock. (The payout stream from the completcd project is equivalent to the dividend on the stock.) A soI utitln to this option valuatiun and cxtlrcise problem was (irst found by Samuelson ( 1965).

zl Iirln

138

'.

Inecisiolls

eqtlation (1) for P- For this problem to make sense. we mtlst also assume that a < p,. othenvise the integral in equction ( l ) could be matle indehnitely larger by choosing a larger F. Thus waiting Ionger would always be a better policy,and the optimum would not exist. $Ve will let denote the differncc > p e; thus we are assuming J 0. .

Finally- by substituting expression so Iu t ion f0 r F-( 1. ) :

ing

-

(4) illto eqtlatilln (.3).we

obt:lin the

llltlsv-

(

-

1.A The Deterministic Case

(1z

=

euT

14

-

e-PT

(3)

or fall over time- s() it is clearly > and 1, l'' if never invest otherwise. Hence optimal to invest immediately (P/. 01. Fj max What if 0 < a < p? Then F(F ) > () evcn if currcntly lz' < 1, because eventually P' will exceed 1. Also. even if )' now excceds 1, it may I7ebetter to with wait rather than invest now. To see this, maximize F( lz') in equation (3) condition is respect to F. The lirst-order

o--rhenPe(/) will remain constant

=

.

=

-

.

*

=

-

.

=

=

F(Iz') a :s

*

Figtlre 5. l shows F 1/-) as a ftlnct ion of I for / l p ().1()- antl a = (10.03, and 0.()(' ln each ease. the tangency point of F IS ) with the line F I (.r). is at the critical value Pr Note that F F ) incrcases when a p //(p increases. as does the critical valtle lz'e G rowth in lz' crcates 11value to waiting. and increases the valtle of the investnAent opportunity. =

future time T is Suppose

l''

'

Although we will be mostly concerned with the ways in which the investment decisionis affected by uncertainty, it is useful to first examine the case in which there is no uncertainty, that is, t'z in equation ( 1) is zero. As we wiil see, there can still be a value to waiting. P-(()). Thus given a ctlrrent P-, )'0 cal, where P'() 0, P' (/) lf c assuming opportunity we invest at some arbitrary value investment the of the =

for U' >

l.B

The Stochastic Case

'sve will now return tt) the general case in which cr > (). The prl'blelu is t() determine the ptlint at which it is optinlal to invcst / in rettlrn for 1111 asset worth V Since l'' evtllves stochastically. we will not be able to determine 4 time T as we did ltbllve. Instcad. our investment rule will take tlle form of .

-

2

.0

1

.8

.6

# F( )' ) JF

-(p

=

a

-

whichimpliesz

1

) p'

r + P l e

f?-t/l-tz,

-p

r

=

(;

1 .

1

.4

p/

() (p a ) )' a Note that if IZ is not too much larger than J, we will have F* > ().The reason for delaying the investment in this case is that in present value terms. the cost e-PT whereas the payoff is reduced of the investment decreases by a factor of e-sn-x'r of factor smaller the F*

=

max

-

Ioi!

,

%-'

.

-

by

Fo/ what values of )' is it optimal to invest immediately'? By setting T* 0, we see that one should invest immediately if lzr k: P' where :,

=

*

rz-

7

=

#-a

l

>

J

.

1 l

1.2 K-

(g'

1

'

I I I I i

o

0.8 0

l

I l

.6

a

O.4

=

0.06

1

l

ct 0.03 =

02 0.0

j

Ct

=

0

I

0.0

O.5

1.0

1.5

2.0

1 1 I I

2.5

3.O

Tinkillg lnveslntelttOpptrp?-/ll/ld-sand p?lvc-j-p?'lfa>n/

140 will

As we a critical value P'* such thtt it is optimal to invest tlnce F : value to that /'*, is. will result higher greater value in a of tr a see, a higher growth both genera. mind, that in however, waiting. It is important to keep in thereby waiting and value > uncertainty to (a > 0) and ((z 0) can create a affect investment timing. In the next twosections.we will solve this investment problem in twowayss following the techniques described in Chapter 4. First, we will use dynamic programming, and then we will solve the same problem over again using contingent claims methods. This will enable us to carefully compare these two approaches. F*.

the deterministic case), we assume that a < p, or J > (). With this the Bellman equation becomes the tbllowingdiftkrential equation be satished by F( r' ):

(9) In additions F F) must satisty' the tbllowingboundary conditions: F(0) F( F

*

)

Condition

ln the terminology of Chapter 4. we have an optimal stopping problem in continuous time. Because the investment opportunity, F( 1/,), yields no cash flows up to the time F that the investment is undertaken. the only return from holding it is its capital appreciation. Hence. as we saw in Chapter 4, in the continuation region (valuesof l?' for wllich it is not optimal to invest) the Bellman equation

is

=

C(dFj.

Equation (7)just says that over a timc intelwal t/. the total expected return equal to its expected rate of capital on the investment opportunity. p F f, is

appreciation.

We expand iIF using Ito's Lemma, and we use primes to denote derivatives. for example. F' dl-hlll, F'' = dlb-/dl'' etc. Then =

F'( p-) dv + 12 F''( p-) (J)')2

db-

=

gives

( 1) t'or#lz into this

*)

=

*

1

-

1

.

( 10) arises from the obselwation that if

)' goes to zero. it will stay of the stochastic is implication at zero (this an process ( 1) for P' J.Therefore when will of value option P' invest be 0. The other two conditions to the no consideration optimal from Z* is the price at which it is investment. of come in investp the Ianguage of Chapter 4, the free boundary of the optimal to or region.Then continuation ( 1l ) is the value-matching condition: itjust says that rcceives investing, the lirm a net payoff 1z'* 1. Finally, condition ( l2) is upon condition. discussed in Chapter 4 and its Appendix C. the If F( V ) were not continuous and smooth at the critical exercise point P-'a one could do better by exercising at a different point. Note that equation (9) is a second-order differential equation, but there are three boundary conditions that must be satissed. The reason is that aI()) is known. the position of though the position of the first boundary ( other second words, boundary is not. In the boundary'' /'* must be the determined as part of the solution. That needs the third condition. Equation ( 11) has another useful interpretation. Write it as P'* F( P'*) When the hrm invests, it gets the project valued F, but gives up the oppor1. fu?i/.yor option to invest, which is valued at F( F). Thus its gain, net ()f the F Z). The critical value F* is where this net gain opportunity cost. is P equals the direct or tangible cost of investment, 1. Equivalently, we could write the equation as V* = / + F( F*), setting the value of the project equal to the full cost (directcost plus opportunity cost) of making the investment. We will discuss this point in more detail later. To find F( P), we must solve equation (9)subject to the boundary conditions ( 10)-4 12). ln this case a solution is easy to find; we can guess a functional form, and determine by substitution if it works. We Iirst state the solution and derive some of its properties, and then discuss it in more detail. =

-

'

=

-

*

expression and noting that f'ttfz)

a P' F?( Z ) dt +

Hence the Bellman equation

1 (7.2

becomes

)

(7.2

P' 2 F''( Izr) dt

=

0

.

by dt):

(afterdividing through

72 F'' 1z) + a P' F'/l

-

p F

:::s

=

tefree

-

SLJF')

2

/'

( l0)

ttsmooth-pasting''

p Fdt

Substituting equation

0.

=

=

F' ( P'

Solution by Dynamic Programming

notation. that must

(8)

0.

Itwll be easier to analyze the soiuton and to compare it to that obtained using 6. To ensure p contingent claims analysis if we make the substitution with connection in explained already existence of an optimum (for reasons =

-

=

:'

il-l?l lLlc'isit'-u lzl 1-2 -.

'

'T'osatisfy tlle blltl ndary cond ition ( 1())-tlltl sol u tion

t:tktl t Ile fornl

fntlst

.I J /' l

F ( )' )

a I'Id

( 13 )

'

where z.l is a constant that is yet to be detc rl'n ined, and pl > I is lt k ntlwn constant whose val ue dklpends on the paramete rs fz /)- nnd J o f the di ffe rd ntial equation.

t). so t Ile gel'le ral

.!. a

( /)

*

*

Pr

#l

=

p

l -

l

( l4)

1.

2

g( /)

) yr)'

stll tl t ioll tk) eq tlttt itln F-(

.

The remaining boundary conditions. ( l l ) and ( l 2)- can btl tlsed to solvtz for the two remaining unk nowns the constant z1, and the critical value )' at which it is optimal to invest. By substituting ( l 3) into ( 1 l ) and ( 12) antl rearranging, we Iind that

f5

'

.!. S + a

1

'

/rT

)

-

(9) c:tl'l 14e vri ttel'l

zzl

)

t

I

l''-# ' +

z 1a

'n

-

/)/f'' 2

:ts

l-'01

where Ctnd zla are constants tt) be deterluilltttl. ln otlr pnlbldnl. tlle blltllldary (). leltving tlltl soltltion ( I 3). condition ( I()) implies that zrh Tg answer our econom ic quest ions conctt rn ing t I1eInu It iple p I / (#1 I ) we nAust therefore examine t he qtladratic equation (.l 6) in mtlre dtlta i1.Si ntre this equation- or something closely sinAilarp Nvill appear in ltlnnost every chapter. it he lps to establish 11 standa rd terfn inology and obtain sonle general rcsults at the tlutset.

,41

-

.

vvill general Iydtlnllte tlltt variable in the eq uation by p- and the wllold quadratic expression (thc Ieft-hltnd side) by t'2.Thus O-wist function 017 the vari%9Q

and

able /J, as well as the parameters /.)aand unless it is important to d() so. explicitly (''r,

Equations ( 13)-( l 5) give the value of the investment opportunity and the optimal investment rulep that is, the critical value 1.! at which it is optimal to invest. We will examine the characteristics of this solution in some detail later. For the time being, the most important point is that since pb > l we have /31/4/$1 l ) > l and r' > l Thus the simple N PV rule is incorrect; uncertainty and irreversibility drive a wedge betwecn the critical vltlue P' and 1. The size of the wedge is the I-actor pb/(Jl1- 1). and it becomes important t() examine its magnitude for realistic values of the undcrlying parameters, and its response to changes in these parameters. Tt) do that we must examine the solution (l 3) in more detail. *

,

*

-

.

*

2.A The Fundamental

qwlplp

-

5)p

p

-

=

p,

=

-

< () ren-lklnlbt)r Iy parabtlla that gtlcs to c.'o as p glles t() Als(). C'')( ( we are assuming J > ()) ltnd k?(()) -p < (). Tllerelbre the grilph crllsstls tlle horizlpntalaxis llt llne pllint t(.) thc rigllt t)l' I and ltnlpthtl!- t() t i1e lel't ()1' (). ''I-llat is. (lne rtlot, clll it p, excecds I and the (lther. pz, is Ilegative. sve lcus on tlltl ptpsitive rlult pl I'I()w dtles it changtl il' 11 pllrameter. sty changes'? Tllis is answered l)y standard ctlnlparative stilt ics. Diflrentilte c .,.l-':x).

-

,

.

.

,

f!a

(p

-

.5l/c,.

2+

-

tl/'-..,.

2

-

J. 2

2

l

0.

1 + 2p/r.r

2

>

l

.

$,

1

-p ()(/J

tlependence

.

I

The two roots are .)

sllow this

O

Since the second-order homogeneous differential equation (9) is Inear in the dependent variable F and its derivatives, its general solution can be expressed as a Iinearcombination of any two independent solutions. Ifwe try the function .,zlP'/3, we see by substitution that it satisfies the equation provided p is a root of the quadratic equation 1) + (p

svewill not

Althtlugh the r()()ts ()f a tltlltdratic are k ntlwn in explicit algebraic ftlrm. helps t() show them gellmetrically. F igtlre 5.2 shlnvs kl as lt ftlnctilln ()f p it The coetlicient of p? in k1(p ) is positive. so the graph is lln upsvard-pllinting

Quadratic

-

J.

-8

(3

zl l

-j

Dectxlolls

lll

tlplt/ Ill k'(-z'/p?l (?? It Fj/?litlg

7fJ-J??1cnt f?p/7',)?'/I t ? lities

and tll'' is given by equation long as Vt : 1, or n't : (p the firm should invest when

totally:

exprcssion

the quadratic

Irnl

( l ). The

e) 1.

-

) Q )/$1 + p Jc all derivatives are evaluated

where at pk Also

i) Q

(),

=

i)c

#1. Figure 5.2 shows that ) (? lp

at

>

pp

t'z

=

1)

-

at pj > 1. Therefore p3yl(7'< 0. In otherwords, as tz increases. #! decreases. and therefore /21/(Jl 1) increases. The greater is the amount o uncertainty wedge between 1Z* and 1, that is, the over future values of V, the larger is the

Iarger s the excess return the firm will demand before it is willing to make the irreversible investment. Readers can likewise verify two other properties of this quadratic. First, 1). increases as 6 increases, so a higher means a lower wedge pk/(/1 p, Second, p, decreases as p increases. so a higher p implies a larger wedge. We will discuss these results in greater detail and offer some numerical values in Section 4 of this chapter. Some limiting results concerning p, are also intbrmative. We merely state oa, them; they are easily verised using the algebraic formula. First, as (.. f.r is inlinite. lhe invests if is, firm F* that never and 1 x, we have pt (). We have Next consider what happens as t'z -

-..+

-.+

-.

If a

>

0. then /31

-.+

p,

If a S 0. then

-.+

p/tp (x7

t5)

-

and P'*

and 1!%

(p/)

.-->.

pI pI

Relationship to Neoclassical Investment Theoc

To push this analysis a bit further, suppose that the project itself is an inhnitely lived factory that produces a profit tlow, zt, that follows the process

(1n.

(z

n' dt +

f;r =

dz.

X =

C

Jrs. I

e

--- /,

( 16'

---

l

) jj

t

s

=

e) I

>

p

tz) /.

-

a)

-

=

! 2 p + a o'

x! ,

p-a

to

#1.

2 p + 1a t'z pt ) /

=

p 1.

>

=

JTJ

=

Jr()

e

/'?

max F

p

-

=

r

.7r1)

/

-

a :1

time T n'

can verit

t ,.-

that a

r

=

>

v

-

1 t;p- p-tz

=

)F

/ e

-

p

a

-

-

P

y. .

when

tzr

7r1)C'

=

# /

.

() is the sccond-order

condition

for this

maximization.) Therefore the firm should wait to invest even if there is no uncertainty, because waiting allows the postponement (and thus discounting) of the payment /.4 As equation ( 18) shows. with unccrtainty there is an additional 1 v2 pk term, so that the firm must wait even Ionger betbre investing. 2

This additional

term can be thought of as a correction

investment model.

Hence V is given by

)$

-

Since we have assumed zero depreciation, pl is the Jorgensonian user cost of

(The readcr

=

(p

capital'. the Jorgensonian rule is to invest when n.t p 1. Equation ( l8) says that when t-uturcprolits are uncertain. the threshold 7r* must exceed this user cost of capital. ln the absence of uncertainty. the Jorgensonian investment rule has the firm investing when (p a) 1. As we saw before. this p /. ll() when zt viewed timing rule. agains the firm must choose F be optimal Oncc as an can to maximizc

The solution is to invcst at

earlier.

(p

I

-

zr*

These results conform to those of the deterministic case that we examined

2.B

1

-

Thus the critical profit Ievel x* can be written as

/.

/.

-.>.

p

,

-

-.+

#

=

investment-3From

0

>

A: n'*

Another way to Iook at this is in terms of the Jorgensonian approach the quadratic eqtlation ( l6) satistied by p, we have

.

p Q /pf.r

xt

0

usual Marshallian rule is to invest as I-lowever. equation ( l 4 ) tells us that instead

slorgenson ( I 963) shtlwt!d that absent uncertainty. proht from an extra unit of capital cquals the user cost for suggesting this viewpllint in this cllntcxt.

to the neoclassical

thc tirm should invcst when the marginal to Giuseppe Bertola

of capital. Our thanks

Z-Ii.)our knowledge. this pllint was Iirst noted by Marglin

( 1963,

Chapter 2),

1 Frr/l

146 2.C

Relationship

''

Dccisiot

to Tobin's q tthe

value introduced 11 magnittlde q, defined lts the ratio of reprtduction ctlrrent lo of existing capital goods. or of titles them'' to cost.'' This haj become a central concept in the orthodox theory of investment. The idea is that if this ratio exceeds unity. :1 5rm can incrcase its market value by increasing its capital stock. Hence we should see a firm investing when the value of q tbr the irm is above 1 but not when it is Iess than l Furthermore, market value of f/ we can aggregate by calculating a based on the average firms in an industry (orthe whole economy) and the corresponding average replacement cost of capital. We should then observe at the aggregate level that investment spending is potitively reiated to the value of this (/. and interA number of issues arise with respect to the measurement pretation of q. An important one is that what should matter for investment is lnarginal q, that is, the (2 that applies to the (irm or industry's incremenproject, as opposed to the ab'erage c? that applies to the entire investment tal stock of the firm or industry. We will address the concept of capital and value in marginal q more detail in Chapters l l and l2. Here we want to emphasize a different problem. As stated. the numerator in t/ is the market value of existing the llc.rl assets. What is relevant for thc investment dccision is the eftkct of project on the value of the hrm. We have seen that to get this. the cost of using subtracted from the valud of a project. Thereup the option to invest must be and option value F Pr) is installed. the value value )' when project of a fore, should 1/ by F P' ). not V Correspondingly, q should increase of the firm ratio F 1. 1z') be defined as the (1z' ( 1/ Then the critical value of q that justifies will indeed be 1, as we see from equation ( 1l )swhich dehnes the investment

Tobin

( l969)

fnvc-sl/llcll f

.

-

.

ies

perspective on the decision.

tI?l(/ IlLbestllelz

effect

I

TilkI'??t)

of irreversibility

on the

hrm's

investmeut

In Chapters l 1 and l2, we will have occilsion to discuss the Iiterattlre

-btller

,

l-lpporttLllil

()11

investment that uses the t/ conccpt. Some articles have used the lirst t)f tht! above definitions of q and others the secontl. Thcretbre we will have to bkl careful to distinguish them. We will call the first net of the option valtle. antl of the tirm--concept, and the sotlontl having l as the upper Iimit the omitting the option value, and having pk/(/1 l ) as the critical Ievtll the .

tvalue

-

Gvalue

of assets in place'' concept.

Solution by Contingent Claims Analysis If the firm doing the investing is publicly held and its managers

w:lnt thcir

decisions to reoect the shareholders' interests- they should try to maximize the market value of the firm. Llow do we know that thtl invcstment rtllc derived above will do this'? One problem with this investment rule is that it is based on an arbitrary and constant discount rate. p. It is not clear where this disclltlnt rate should come from. ()r cvcn that it shlltlld be constant over time. As we saw in Chapter 4. we can use contingent claims (optionpricing) methods to derive a slightly modised invdstment rule that will indeed maximize the mllrkct value of the (irm. In this sectilln. we will go thnltlgh the reqtli red steps in detail.

-

critical F*.

However. it is difficult to allocate the portion of a firm's market value that (/ comes from the marginal unit of capital. Partly because of this, has come to namely, differently. measured, somewhat as the ratio be dehned. or at least would flow from value profits that of a completed of the expected present lo the value construction. using This corresponds investment, to the cost of its exercised and been already where has the option to invest of an existing asset. V(I. defining notation, this means q as its opportunity cost is a bygone. In our We can express the correct investment criterion in terms of this notion of q : the threshold q. that justihesinvestment is given by

(/* Jl/(Jl =

-

IZ

Reinterpreting

the Model

The use of contingent claims analysis requires (lne important assumptitln: V must be spanlled l7y existing assets in the ecllntlmy. Specihcally, capital markets must be sufliciently so that, at Ieast in principle, one could Iind an asset or construct a dynamic porttblio of assets (that is, a portfolio whose holdings are adjusted continuotlsly as asset prices change). the price of which is perfectly correlatcd with )z' This is etluivltlent to saying that markets are sutxciently complete that the firm's decisions tlo not affect the opportunit'y set available to investors.s The assumption of spanning should hold for most commodities, which typically traded on both spot and futures markets. and lbr manufactured are

stochastic changes in

etcomplete-'

.

1) > 1.

lluctuates, there will be stretches of time when the conventionally measured q exceeds 1 without attracting investment. This gives a new

As

3.A

5For a more rigorous and deta iled d iscussion oI' spanning and its ifnpl ica titlns. sce 1.Iuang and Litzenberger ( I 9t)()) and Du fhe ( I992). Du ffic and I'juang ( I 985) Iay ()u ( thk! lb 11sc t ()1* conditions needed for spanning.

ald l/l!./f;w/r?/tz/ll Invesllellt tEl/?ptr.l/-/'l/pljljf;t Til,llg

goods to the extent that prices are correlated with the values of shares or portfolios. However, there may be cases in which this assumptivn will not hold; an example might be a project to develop a new product that is unrclated to any existing ones, or an R&D venture, the results of which may be hard to P redict.

We will assume in this section that spanning holds, that is, that in princiuncertainty over future values of V can be replicated by existing assets. With this assumption, we can determine the investment rule that maximizes the hrm's market value without making any assumptions about risk preferences or discount rates. Also, the use of contingent claims analysis will make it easier to interpret certain properties of the solution. Of course, if spanning does not hold, dynamic programming can still be used to maximize the present value of the hrm's expected llow of profits, subject to an arbitrary discount rate. See the discussion in Chapter 4, Section 3 for more on the relationship between the two approaches. We follow the theoryof contingent claimsvaluation outlined in Chapter4, Section 2, but repeat some details for reinforcement and clarity. Let be the price of an asset or dynamic porttblio of assets perfectly correlated with V and denote by px,,, the correlation of with the market portfolio. Since is perfectly correlated with P' px,?, pf.,,, We will assume that this asset or portfolio pays no dividends, so its entire return is from capital gains. Then

ple the

.,t

.

.r

-

=

,

.

-r

evolves according

to

dx

/.z

=

-r

J/ +

tr

x dz.

where #t, the drift rate, is the expected rate of return from holding this asset or portfolio of assets. According to the Capital Asset Pricing Model (CAPM), risk. As explained in g should reflect the asset's systematic (nondiversihable) Chapter 4. Jz wiil be given by

Jz

=

1-

+

4 pxm t7',

where r is the risk-free interest rate, and 4 is the market price of risk.fi Thus, y. is the risk-adjusted expected rate of return that investors would require if they are to own the project. We will assume that a, the expected percentage rate of change of V, is Iess than this risk-adjusted return p.. (As we will see, the hrm would never invest if this were not the case. No matter what the current where r. is the expected rcturn on the markct. and fr,,, is the b'rhat is. 4 (r?a r/a standard deviation of that return. If we take the New York Stock Exhange lndex as the market. 0.08 and tz. ().2. so / Qz 0.4. For a more detailed discussion of the Capital Asset rm r =

-

rkC

-

,

i:k;

Pricing Model, see Brealey and Myers

( 199 I ) or

Duffie ( 1992).

level of V the firm would always be better off waiting and simply holding on to its option to invest.) We will let J dcnote the diftkrence between pt and a. > (), and this plays the same role that is. a. Thus we are assuming y corresponding assumption the the in dynamic programming tbrmulation as of Section 2. .

(5

=

-

(

The parameter ( plays an important role in this model. We discussed its role as an explicit or implicit dividend in Chapter 4'. here we elaborate on those remarks. It will be helpful to draw upon the analogy with a hnancial call option. If lz'were the price of a share of common stock, J would be the dividend rate on the stock. The total expected return on the stock would be yz = ( + a. that is. the dividend rate plus the expected rate of capital gain. If the dividend rate were zero, a call option on the stock would always be held to maturity. and never exercised prematurely. The reason is that the entire return on the stock is captured in its price movements, and hence by the call option. so there is no cost to keeping the option alive. However. if the dividend rate is positive. there is an opportunity cost to keeping the option alive rather than exercising it. That opportunity cost is the dividend stream that one forgoes by holding the option rather than the stock. Since 6 is a proportional dividend ratc. the higher is the price of the stock. the grcater is the tlow of dividends. At some high enough price, the opportunity cost of foregone dividends becomes great enough to make it worthwhile to exercise the option. For our investment problem./z is the expected rate ()f return from owning the complcted project. It is the equilibrium rate established by the capital market, and includes an appropriate risk premium. If f > (), the expected rate of capital gain on the project is less than p.. /'/zTc, J is an opportttny cto'l of delaying constntction rJ the project. tIITJinstead keep'g /c option to inves alive. If ts were zero. there would be no opportunity cost to keeping the option alive, and one would never invest. no matter how high the NPV of the project. > 0. On the other hand, if J is That is why we assume very large, the valuc of the option will be very small, becausc the opportunity cost of waiting is large. As x, the value of the option goes to zero; in effect, the only choices arc invest now or never, and the standard NPV rule again applies. to Tbe parameter t5 can be interpreted in other ways. For example, it could re:ect the process of entry and capacity expansion by competitors. (However. in Chapter 8 we will discuss more complete models that endogenize the process of rivals' entry, and hnd that the resulting equilibrium cannot be well described by simply raising the J for each hrm.) Or it can simply renect the cash flows from the project. If the project is inhnitely lived, then equation ( 1) can represent the evolution of F during the operation of the project, and P' is the rate of cash flow that the project yields. Since we are assuming that 6 is -..

onjtant. this is consistent wi th future ot the prqect's market value.'

cash llows being

11 constallt

proportion

When some other parameter of tlle model (suchas t'r) varics, wtl must ask what happens to J. Various possibilities can be imagined. We will always suppose that the riskless interest rate ?- is tixed by the larger considerations of the whole capital market. independently of what happens to any one asset (()r hrm or even industry). The aggregate market price of risk $ is likewise held fixed. Now suppose o' increases. This raises the rsk-adjusted discotlnt rate g. To preserve equilibrium in the market for either a or J must change. Two extreme cases are logically possible. First, might be a t'undamental fact about x, so that must respond to the change in Jz (forexample, the dividend Alternatively, j rate might depend on the quantity of the commodity held). change of-r price and the must process might be a basic behavioral parameter, that both a and t take up so that a does the adjusting. A third possibility is part of the adjustment. In our numerical or comparative static exercises we will often regard as a basic parameter independent of o., but will mention alternative possibilities where they make a material difference to the results. -r.

changes. /V'( l'' ) I'nay change h-()I1-l one short illttl 1w:11 ()15 tinle t(.)tlle ntlxt. s() th:tt the conlposition of the portfolio vvill l7e cllangetl. l-losvtlveraovt)r eacll sllol't Id ?, fixcd. interval t)f length t/ vvtt 114.) .

The short position in this portfol io Nvil I requ ire a payl-nellt ol' l F. ( l ) dollars per tirne period-. othenvise no rational invdstor will cnter into thd IilI1g side of the transaction. We discussed tllis in Cllitpter 2, Sectilln I (:t). allt.l recapitulate thtl calculation brietly. An investor holding lt Iong ptlsititln il'!tlltt project will demand the risk-adj usted return p.t lz' Nvh ich eq ua ls t l1e capi ta I gain a Jz-phts the dividencl stream (5 V Since the short position inclutlos li-'$l ) units of the project- it will req uire paying out t P- F' ( 17) 'Tttk'ing t his p:tylne 1)t into account. the total rettlrn from holding the porttblio over a sllort tinle interval dt isS '

'

'

.

-

.

.

To obtain an expression t

lklr t' F, tlse Ito's Lelnlma: J-- ( 1'-) (1 l -F. .!. 17 ( 1,-) (t l lz')2 '

l J'--

'

'

'

?

.

Hencc the total rettlrn on the ptlrttblio s 3.B

Obtaining a Solution

Let us now turn to the valuation of our invcstment opportunity, antl the optimal investment rule. Once again, we will lct F( V ) bc the value of the firm's option to invest. We will determine F(1') in much the same way that we did in the two-period example of Chapter 2 or tbe general theory of Chapter 4, Section 2 by constructing a risk-free portfolio, determining its expected rate ()f return to the risk-free ratc of of return, and equating that expected rate

interest. Consider the following portfolio: Hold the option to invest. which is F'( F) units of the project torequivalently, worth F(V), and go short ,1 of the asset or portfolio that is perfectly correlated with F). The value of F F'( V ) P' Note that this portfolio is dynamic; as 1/this portfolio is *

..!.J'J 2

2 ( l'' ) (t/ P' )

''

(5

.-

2 1. 2 1/-1--.( P' ) t l t 2 f'z ''

17

.12'

( I-' ) t //

.

'

-

J f' /-..( 1..,) tlt

.

Note that this return is risk-free. I-lencc t() avllid arbitragtl pllssibilities, it m ust eq uaI 1- tp t/J 1- gF. F' ( 1z') )' 1(ll : =

-

=

.x

=

-

.

7A constant payout rate, J, and required expected Letting n. denote the Ilow of profit from the project. I

=

1)

Jz,

imply an infinite prllject life.

F

F 1$

returna

'T e -

Ht

/

l/

J 6j e

=

tl

l J'

-1

V

e

-J'l

/t

.

which implies F = x. lf the project has a finite life. equation ( 1) cannot rcprescnt the evlllution of f'' during the operating period. l-lowever, it can represent its evoltltitln prior to ctlnstruction of the project, which is all lhat matters for thc investmcnt decisilln. See Majd and Pindyck ( l 987, t)f th is point. pp. l 1- l3), for a morc detailctl discussion

Dividing through by dt and rdarranging tion that F P' ) must satisfy:

1 fr2 2

1,/2 F ( P' ) + (l ''

gives the following differential equa'

' -

5) P' F ( P' )

-

r f-- =

()

.

Observe that this equation is almost identical to dquation (9) obtained only difference is that the risk-free in terest

using dynamic programming. The

,z1

152

Firnl 'J Decisils

Itlvcstlttent Clpptprfktl llkft.-s

rate r replaces the discount rate p. The same boundary conditions ( 10)-( 12) will also apply here, and for the same reasons as before. Thus the solution for F P' ) again has the tbrm -(Z)

except that now and theretbre

1-

=

zd

Vl$'

replaces p in the quadratic equation for the exponent

pb ,

(24) The critical

value

l'* and the constant

,4

are again given by equations

(14)

and ( 15).

Hence the contingent claims solution to our investment problem is equivalent to a dynamic programming solution. under the assumption of risk neutra lity (thatis the discount rate p is equal to the risk-free ratel.g Thus whether or not spanning holds, we can obtain a solution to the investment problem. but without spanning. the solution will be subject to an assumed discount rate. In either case. the solution will have the same form. and the effects ofchanges in t'z or J will Iikewise be the same. One point is worth noting. however. Withvalue for the out spanning, there is no theory for determining the assumptions about make restrictive investors' or discount rate p (unlesswe managers' utility functions). The CAPM, tbr example. would not holds and so it could not be used to calculate a risk-adjusted discount rate in the usual ,

'Ecorrect''

Way.

tz/l

Itl k't.r.sf lntlllf Tl? tilg

by equations ( 13-). ( 14). (15).and (24).Some ntlmerical solutions will llclp tt) illustrate the results and show how they depend on the values of the variotls parameters. As we will see. these results are qualitatively the same lts those that come out of standard option pricing models. Unless othenvise noted, in what follows we set th cost of the investment.

0.04. 0.04, and (.T 0.2 (at annual rates). (N()te that /, equal to 1, r we do not need to know g or a, but only the difference between them- t5.) Payout rates on projects vary enormously from one project to another. so this valtle of 4 percent for t should be viewed as reasonable, but not nec representative. As for o', the standard deviation ot the rate of return n the 1 stock market as a whole has been about 20 percent on average. Although $ this represents a diversitied portfolio of assets, it also includes the effects of Ieverage on eqtlity returns. and so might be a reasonable number for an average asset. =

=

=

assarily

'

1. Thus Given these parameter values, pj 2. l'- = 2/ 2. and the simple NPV rules which says that the lirm should invest as long 4as l is at least as large as /, is grossly in error. For this reasonable set of parameter values, V must be at least twice as large as l beforc the Iirm should invcst. The value of the firm-s investment tlpptlrtunity is F )' ) 1 1,'2 for l,' < 2. and g V F( 1/-) l for P- > 2 (sincethtl lirm exercises its option t() invest ltnd the #' > wht!n payoff net P' 1 receivcs 2 ). Figure 5.3 plots F ( P' ) as a function of l.' tbr these parameter values. ltnd 0 and c also for ().3. In each case. the tangency point ot' F ( l'' ) with the line P- l gives the critical valutl P-*. The Iigure also shows that thtt simple NPV rule must bc moditied to include the opportunity cost of investillg now rather than waiting. That oppllrtunity cost is exactly F( P' ). When 1'- < 1F( P' ) > IZ l and the re forc lz' < l + F ( l'' ) : the value of the pnlject is less than its htll cost, the direct uost l plus the opportunity cost F P' ). When fr (J. V* = /, and F( P' ) () for V :; l .J Note that F( IZ'Jincreases when o' increases. as does the critical value l-' Thus greater uncertainty increases the vitlue of a rm's investment oppllrtunities, but (tbrthat very reason) decreases the amount of actual investing that the firm will do. As a result, when a firm-s market or economic environment becomes more uncertain. the market value of the tirm can go up, even though the firm does less investing and pcrhaps produces Iess. The dependence of 1z-*on c is also shown more directly in Figure 5.4. Observe that P'* increases sharply with cz Tlltts investment highlv setlxitive &.l volatilityin project vfz/lzd', irrespective ta/l-npes/on or managers risk JJre/rf//7cto'. *

=

z..1

=

=

'

=

-

=

-

-

'z

=

=

-

4

Characteristics

of the Optimal lnvestment

*.

Rule

-

Let us assume that spanning hljlds, and examine the characteristics of the optimal investment rule and the value of the investment opportunity. as given

=

=

.

.

gThis result was (irst demonstrated by Cox and Ross ( 1976). Also. note that equatilln (23) is the Bcllman equation for the maximization (>f the net payoff to the risk-free portfolio that wc constructed. Since the portfolio is risk-frec. the Bellman equation for that problem is

?-* dt

(i)

-J 1/ F' ( F ) dt + 7(J(p ).

=

is that is, the return on the portfolio equals the per-period cash 0ow that it pays out Iwhich negative. since F F'( F) must be paid in to maintain the short positionl. plus thc expccted rate F F'( /' ) P and expanding dF as before. one can see that of capital gain. By substituting * and not r equation (23)follows from (i).Alsov note that in equation (i), Jz so one must still have an estimate of the risk-adjusted expected return that applies to F This is an example risk-neutral valuation'' procedure discussed in Chapter 4. Section 3.A of the .

=

-

(5

=

-a

-a.

.

'*equivalent

.

.

'

and irrespective of the c-v/crlf to the market. Firms can be risk

wltich

neutral.

tlle r'/crlcw

'

of P' is correlated

and stochastic

''p'1k/1

changes in P- can

zzlFintz

'.

Decisiotls

16

2.0 .8

1 1

.2

10

I

'

5- 1 o C'

12

l I l

1.4

1

14

I 1 I

.6

l

'

l I 1

0.8

(.r O.3 =

0.6 (r

O.4

=

1

(r 0 (when ?, > p)

0.2

0.0

1

0.5

0.0

6

l l

4

1

?$= 0.02 ?h= 0.04

s

=

0.08

.

2

1

1

3.0

2.5

2.0

1.5

.0

l l

I I l !

=

8

l

I

0.2

11

will still increase F

*

and hence

be completely diversiliable: an incrcase in t'.r tend to depress investment.it' (5. Obselwe that Figures 5.5 and 5.6 show how F lz') and P' depend ()n ().08 results in a decrease in A'(F ), and hence from to in J t).04 increase an (5 () ftr value V'. (In the Iimit as x, F P' ) a decrease in the critical IZ < J, and 1z'* /, as Figure 5.6 shows.) The reason is that as 6 becomes else constant exccpt for tz), thc cxpected rate of everything Iarger (holding growth of P' falls, and hence the expected appreciation in the value of the wait rather option to invest and acquire Pefalls. ln effect. it becomes costlier to than invest now. To see this, consider an investment in an apartment building, where P' is the net flow of rental income. The total return on the building. this which must equal the risk-adjusted market rate, has two components income flow plus the expected rate of capital gain. Hence the greater the income flow relative to the total return on the building the more one forgoes *

0

0.0 O.1

by holding itself.

O.2

0.3

O.4

O.5

0.6

O.7

an option to invest in the buildi ng rather than

0.8

tlwni

0.9

1.0

ng the builtling

We have treated t'z and (5 as independent parameters. It' insteatl wtt allow adjust t5to as c changes, then each unit increase in f-z requres an incre:se in of 4 p..,n units. because

-.>.

-+

-.>

0. we have P' = ! if J : 0.()4s but P' discussion of the Jorgenstlnian criterion.

dtlhlotethat for our

earlicr

q

*

=

*

>

l if J


/

In pfJ.j'/??1(.!??/ ()pptlt'l d/l/i'(!-

wo have (.3()').

-

p')

=

rt

-

q

( 17 vl.

(27)

-

1tl P'/4.'r-a.Nvtz call tral-s'll-nl eqtlatitlll (32) By making the substitution ?y/cr2 (2 ) a/ (.r ) llnd ) so t h a t 11( 1.--) into 11 standard form. Let h ( P' ) = h'' )' ) (2n/c.r2):g''(-r ). Then (32) beconles -r

(27)substi-

The dift-erential eqtlation (23)will again applya but with eqtlation tuted for Hence F( 1z-)must satist .

=

r g'' (-r) + b

where

le +

b=

=

F l')

z'1P6?11 ( lz'),

=

(29)

(28) and

rcarranging

x ) g' (.t-)

2 (r

t? gt-t- )

-

/.z +

-

=

)/rr2

??

().

.

,

,

15

rep resentation:

3

.r2

H ( ; 0 /7)

where and 0 are constants that will soon be chosen in such a way as to make h( V ) satisfy' a diftrential equation with a known solution. Substituting this equation

=

,

Equation (33) is known as Kummer's Equation. l ts solutitln is the conlltlent hypergeometric function 11(.,r: 0 /?(p) ) wh ich has the following se ries

zd

expression for F( k' ) into equation :

-

'

.g(.t-

=

(28) Also: F F) must satisfy the same boundalz conditions ( 10)-(12) as before, and for the same reasons. gNote that 1,' 0 is an absorbing barrier for the process (26),so F40) 0.1 Finding a solution to equation (28)is a Iittle more complicated than it was for equation (23).We define a new function 11( P' ) by

=

'

-r

.

gives the tbllowing

=

l+

0

+

-r

J

0 (0 + 1)

bb + l ) 2!

Nve have verified that the slllution solut itln is

+

p (f? + l )(f? + 2 )

/?(/?+ l $fl, + ?)

to equatilln

.r

+

.3!

(28)is indeed

.

.

. .

of the forn-l

of equation (29). The

1-2 (Z )

(3())

=

zl )'

'/

2p

lI

J

.'J

l

. '

f?

'

%

*

1)

,

where is a constant that is yet t() be determined. We can find zt :ts wcll as the critical value V at which it is optimal to invest, l'rom the remaining / and F;. ( P' ) l Because two boundary conditions. that is, F ( )' ) P' the conll' uent hypergeometric function is an infinittl seritts. zl and V must be found numerically. ,4

*

of P' so the bracketed terms in Equation (30)must hold for equation the and second lines of must equal zero. First we choose both the Iirst first line of the equation equal to zero: 0 to set the bracketed terms in the any value

1 fz: 0(0 a

-

1) + (r

-

/1 + r? P ) (

,

-

r

=

0.

*

#

*

=

-

=

.

#

We can gain some insight into the effects of mean reversion by looking numerical solutions. Unless otherwise stated. wtl will set / 1 at will vary ?? and P Ntlte, however, that We and 0.2. 0.08. 0.04. c Jz r the dependence of ( l /tl ) C d )' )/ Z on t) depends on thtr scaling 01* P Wc will work with values of P in the range of 0.5 to l so a value of r? of ().5 or ablwe implies a ver.y high rate of mcan reversion. Figure 5.1 1 skows the value of the investment opportunity F tfZ ) and the 0.05 (whichimplies a relatively Iow rate of mean critical value P'* for several

This quadratic equation has two solutions for 0, one of which is positive and 0, we use the other negative. To satisly' the boundar.y condition that F(0) =

the positive solution: (31)

=

=

=

=

.

.5,

n

'4We have not developed a model that explains why l'' is mean reverting, and unless there is

a payout strcam that is mean reverting, P' will have a rate of return that is below the equilibrium rate Jz, See McDonald and Sittgel ( 1984) For a discussion of this ptlint.

,

.

=

l5See Abramtlwitz and Stcgun ( l9f)4), Sectitln 7.9 ()f Pearst'n ( l(J9()), ()r Slater ( 1$9f)1)) for discussions ()f the cllnflucnt hypergeofnetric ftlnction and its prtlperties.

..glFrp'?l Decisiotls 's

0.8

0.8

0.7

0.7

l/- /

0.5

0

1 I 1 I

0.3

;

=

1.5 1

0.2

.

;=

0.1

j

j

.0

O.5

I

I

1

1 1

I I

l I l I

1

0.0

0.2

F igure 5. l l

0.4

0.8

O.6

1

.0

1

.2

1

=

1

t

1

1

.6

2.0

.8

0

I I

I

C

l l

1.0 I

t

V=0.5

0.1

I

1 l p 1

.5

0.2

1

.4

g =

I

I

.5

0.3

1

I I I

0.0

1 1

l j

I I l I l

l

'

I l

.0

0.0 0.2

0.6

0.4

0.8

1

.0

1

.2

1

1

.4

.6

1

2.0

.8

.

reversion) and P 0.5. 1 and l For comparison. note that if ?) were zero, the model would correspond to the basic one in the previous section, with In Figure 0.08,. in that case, l'* would equal l Jz a 0 and hence close each value P' is fairly enough of P, to 1 so that for 5.1 1, r? is small .5.

.0.

=

(5

.39.

=

=

.39.

*

As we would expect, the larger is P. the larger is F )') and the higher is Z*. Other things equal, a larger P implies a higher expected rate o growth ()f so that an option to buy P- will be worth more. Figures 5.12 and 5.13 also show the value of the investment opportunity F( /') and the critical value P'* for P = 0.5, 1.0. and 1 but in Figure 5. l2, ',

.5,

P is 1 n 0.1, and in Figure 5. 13. ?? 0.5. As tese figures show, when P is 0.5 (smaller (larger than /), a larger value of n increases F( Izr). but than /), a larger value of ?? reduces F( 1z').(If l7' < I and n is Iarge. it is unlikely that V will exceed / for very much time, and the optfon to invest will not be worth much. On the other hand, if > I and ?? is large, even if )' is initially small, it is Iikely to quickly rise above / and remain above / most of the time, so that F Z) will be large.) =

l

-

I 1 I I 1

R C- 0.4

l

O.4

=

1

0.6

0.6

C

l'

I'kvesl'leltt Oppotlltltitiesflrlt/ Illvestlnetlt T/'rlll!,'

.5

when

=

'

Figures 5.12 and 5.13 also show that if J? and n are Iarge. F( P' ) will no longer be uniformly convex; it will be concave for small valtles of P' This .

is a result of the particular stochastic proccss (26)that we used to describe the evolution of lz' That process has an absorbing barrier at 1/ () (sothat -(()) 01. but the absolute rate of mean reversion rises rapidly tbr small but positive values of IZ so that F( V ) likewise rises rapidly. This is most evident (). but the expected rate of in Figure 5. l3 for the case of V- 1 F(0) ()f P' becomes large growth once P' is even slightly greater than zero, so that =

.

=

,

.5-

=

Ff P' ) rises rapidly. Figure 5.1 zlshows

3

=

F* as a function of the mean-reversion Obselwethat 1z'*increases with ?? when parameter ?? for ' is large. but decreases with r; when I is small. This is an implication of what when P s large, a larger ?? increases F F) (and hence F*), we saw before'. I but when is smail, a larger ?? rcduces F( I'' ). Figure 5. 14 suggests that thc critical value 0.5, 1 and l .0,

=

.5.

'-

=

with

is the dividing line, but in on the risk-adjusted expected

also depends

?? V. rises or falls rate of return, 1J.. In Figure 5.14, /.t

fact, whether

=

0.08.

0.04. Other things Figure 5.15 also shows 1z'*as a function of ;, but for p, equal, a lower value of Jz implies a lower value for the expected rate of capital 6 ( l /J/ ) E IJZ )/ 1z',and hence a larger value of FV ). yz gain This increasc in F P' ) will be most pronounced when n is small. (When z? is =

'shortfall,''

=

-

166 0

Illt

l.)ili.x,ulltllilit'vb'p/lt/ lll I 'tM'p??t'??/ lltllg

-t?.j'????t,???

.8

2.2 I 1

0.7

2.0

l l

0.6

7

=

1

V= 1

j

.5

I

1.8

I 1

0.5

.5

ls 1 6

K- o 4 u.

O .3

O.2

v-

.

1

o

O1

0.0

j

j

I l

l

I

1

0.4

0.6

0.8

1.2

1.4

1.6

1.8

.)

'*

.()

=

and Figure 5.17 shows the same. but for declines with y; again. a higher /2 implies

n

,

l Note that in cases. lz' higher capital gain J( F), and hence a Iower F( 1z') and lower )'*. However, the rate of decline depends on P and n-When r7 is small, Z* begins at a higher value (again.if tt 0 were zero, the model would reduce to that of the previous section. with a x), and declines more rapidly (becausethe and 6 Jz so that Iims-t) rate of reversion to P and hence the expected rate of capital gain for V is small). Also, as we would expect, the larger is P the largcr is )' (andF )' )1, whatever the values of n and /z. Our choice of the mcan-reverting process (26) for l'' was convenient in that it led to a quasianalytical solution for the value of the investment opportunity and the optimal investment rule. This should not be viewed as particularly restrictive. We could just as well have specified some alternative mcan-reverting process for V (for example, 0ne in which the absolutes rather .5.

1111

=

=

=

''e

=

*

,

.0

0.1 O.2

0

.3

0

.4

0

.5

0

.6

0.7

0

.8

09

1.0

.

thltn pttrcentage. rate of melln reversitln is Iinear in Iz'). Depentling t)n the I'1tlt hltve prllcess. the restll ting tl il)k rential equa t itln tbr Ii-(lz') nligllt t)r lm igl)t :1 k nknvn series solutikln; in any case it clltlld be slllved numerically, ustlally wi th Iittle di fficulty.

*

t-shortfall''

:1

0

2.0

;z' is in any case expected to revert quickty to Henee if pz is small. V* will decrease with l unless 17 is substantially larger than /. Finally, Figures 5. 16 and 5. l 7 illustrate the dependence tlf )' on /t. Fig= ().()5. (). 1 and ().5, I and ure 5.16 shows )'* as a function of p. for

large.

.0

V=O.5

1.O

1.0

1

.2

7::: 0.5 0.0 0.2

)=

1.4

I l

S.B

Cdlmbined Brownian Motion and Jump Process

basic model in which P' follows a getlmetric Brtlwnian wtl will allow for the pllssia Poisson jtlmp downward. This version (:,1' the model could describe a situatilln in which a conlptny has a patent that gives it the option to invest in a project whtlse value is P' l7ut s'uccessful will allow them ot hcr compan ies are a lso doing research which if to invest in a similar project. If and when one of those other companies is successful. the resulting competition will reduce profits. and hence V. To modity- our basic model, we will assume that V folltlws the mixed Brown ian motion/jump process-. Let us now return to

(lur

motion, and extend it in a diffcront way. This time bilitythat at some random poi nt in time, P' will take

,

.

,

(11,'

=

a Izrt/J +

f'r

Z

(1.z

-

IZ tt/

.

168

Z

j*V tl '

J) PI- 151011.% '

'

nl

S

'

*

2.2

2

1 1

.6

V=1.5

1

l 69

.8

2.4 2.2

*

N

Clllpt?rfltllif ies tz?lt.d Illvestn-tetltTbnillg

2.6

2.0 .8

l/lI,zt.!sfl?lcllf

2

=

.0

1

V=1.O

.4

'r) 0.05

.8

1

.6

1.2

1

V=0.5

1.O

0.0

0.1 O.2

0

.3

0

.4

0.5

0.6

0

.7

0.8

0.9

1.O

T1

=

() j

Tl

=

0

.

.4

1

.2

1

.0

.5

0.02 0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

m

and where dq is the increment of a Poisson process with mean arrival rate ()j. We will assume that if an dq and Jz are independent (sothat t'event'' occurs, q falls by some xed percentage / (with0 4 .'.'.S1) with will equation 1. that P' Thus Nuctuate as a geomctric probability (36)says ,

for Brownian

motion. let us

tbr simplicity set e

=

() and write

-ldzdqq

=

Brownian motion. but over each time interval tlt there is a small probability dt that it will drop to (1 /) times its original value. and it will then continue fluctuating until another event occurs. (Poisson jump processes of this kind are described in Chapter 3.) It is important to be ciear about the meaning of equation (36).First, note that the expected percentage rate of change in P' is not a. but instead is V (1/J/) a 4, because over each interval of time dt there is a that will Pfall by 100 4 percent. Thus incrcases in reduce probability J/ the expected rate of capital gain on /' by increasing the chance of a sudden drop in V. Second. because a Poisson event occurs only infrequently, most of the time the variance of dVj I'' over a short interval of time dt is just that of the Brownian motion part. c2 dt. However. if the event occurs. it contributes a very Iarge deviation, so its contribution to the variance calculated given the information at t cannot be neglected. Using the random walk approximation

JJ

c.rV

.'.'.:.!

with

l probability 1( 2

-

tlt

)

.

ilt

)

,

.!.

tF

=

J/ with probability

P'

-fr

with

-() V

1

2( probability ). dt

-

.

-

Then

b-fdv1

=

'((JP' )2j

=

t//

-

(1

-

/

Iz

,

dt ) v2 F2 dt +

.

t//

42 /2

.ldvjj

=

-

FIJP'J =

=J

-((Jlz')2j

2:2

t-gtlzrjjz 42 ),2 dt

dt +

in (JJ)2 etc. Note that this variance (7.272 Jf is the itlstantaneous

ignoringterms

-

,

has two components.

(or

4&local'')

variance

The first component. of dv which comes from ,

k',t:z.J'/??t,l?/

lll

2 2

.4

'llill'g

't,./??ltvl/

%Vewill now proceed t() solve for the

2.8 .6

t??lt/ Ill ! tll'.lltll-tlttll'ties

tlptilmal

investment rtlle

namic programming. :Ve will ltsstllntl tllat the firm is risk neutr:ll. discount rattt is /? ' 1. Then the BeIIn'llln etltlat ion for 1--( )' ). the investment opporttlnity. is .

t

2.2

We now expand

F tlt

t- (( l f--)

=

s() tll:tt its val tle of the

.

F using thtl vcrsion of Ito's Lemmil tbr combined

and Poisson processes (seeSection 5 of Chaptcr 3):

2.0 1

f/

'

dy-

tlsing

Brownian

.8

m 0.5 =

1

.6

1 1 1

'l

.4

=

Replacing e with

0. 1

,

0.05

0.06

0.08

2

?

'

I,z y ( ;p') .j. l - (

.0

0.02 0.04

this can be rewritten as

-

1. (T 2

.2

=

1-

0.1 O 0.1 2 0.14

0.1 6

0. 1 8 0.20

j)

-

) F (P ) + -

+

'

(t.

r,' y' ( p-) F g( 1

-

4 ) P' 1

=

()

(3t)) .

The same boundary conditions ( 1())-( l 2) apply as betbre. The soI u tion to (39) is again oI-the form F 1z-) zl )' #' bu t now thc positive solution to a slightly more complicated nonlinear equatikln: =

,

pt is (4())

Thc value ()f p that satisfies (4t1)l'nd also satisties the condition /J'(()) t) can be lbund numerically. Then. given /.11P' and zl can agtin be lklund frllm equations ( I4) and ( 15), which in turn follow from boundary conditions ( 1I ) and ( 12).1f' Figure 5. 18 shows the critical value P- as a function ()f rr for / = (), and l (In each case. = 0.4, (). k r = 5 = ().(14. and l = 1 Note that the larger is 4. the smaller is The reason is that a larger value of 4 implies a smaller valuc of thc investment opportunity (whenan event occurs, ' will fall by a Iarger fraction), which means a smaller opportunity cost of investing now rather than waiting. Xlble 5. 1 shows p! P' and /1 for various values of for the case in which / l (so that P' falls to zero when an event occurs). In this table, ( =

the Brownian motion part of the process, and is conditional t)n n() jump occurring. The second component. 42 )'2 dt accounts for the pllssibiiity ()f a jump. Shortly we will want to use Ito's Lemma to 5nd the diftkrential of a function of V. As we saw in Chapter 3, when applying Ito-s Lemma to :1 comtt' this bined Brownian mrion/jump prtlccss, it is t'lnly llle Iirst ctlmptanent variance that contributes to the new term involving second-order derivatives. ,

jump part contributes a different term involving a diftkrence in values at discretely different points. we will want to know Finally, in order to gauge the effects of changing IZ tluctuates continuously the expected value of F, the amount of time that before dropping. To determine &-T), we use the fact that the probability that e-k 7-. Therefore the probability that no event occurs in the interval (0. T) is T /F. Therefore the hrst event occurs in the short interval ( F. F +JF) is e-h until Ptakes is Poisson expected time jump the a

The

,

*

.

*

.)

.

.

'

*

.

*,

,

,

=

l If 4 I (s() that thc event is that b' falls to zero. where it remains forever), equatitln (41)) simpliiies to a quadratic cquatitln. which is just like our ttarlier equatilln except that the Poisson parameter l gcts addcd to thc intercst rate in the constant tcrm. The gositive solution is =

pt

=

!

2

-

(r

-

:

5)/fz +

gfr

-

J )//z2

-

!

2

17+ 2(r' +

)//.z'2 .

z1'Firnz l Dccisiolu.

lll rtM'???lt.'??/ (lpilol-lltllitics tl/lt/ Ill ptr.s'/pzlt;'/lf Tiltlillg olwpj, P-*, alttl Table 5.1. Dtrpflvlt/fzzllcfz' oll (NoIe.. I 0.04, tllld (.7. 0.2 1, / 1, 1- = t ..,1

14

=

12 0 f).()5 0. l 0.2 ().3 0.5 1

10

Y 0.0 =

8 6

th 0.4 =

.0

=

=

.)

=

p1

P' *

2.00 2.70 3. 19 4.00 4.65 5.72 7.73

2.00 1 1 1.33 1.27 1 1

.,z1

0.250 0. 169 0. 138 0. 105 0.009 0.007 0.005

.59

.46

.2

1.15

4 1.0

=

2

an increasc in would be cquivalent to an increase in the 7 would Iead to an lcrease in F( V ) and )'

risk-free

rate r, and

*.'

0

0.6

0.5

0.4

0.3

0.2

0.0 0.1

0.8

0.7

0.9

1.0

G

Figure 5.18. = ().l

Crifitfl/ Vltle V

*

Flfrlclrlll

flJ tl

(

?T

I'oi.Y.LmlnroWnQn /WfJllWll.Willt JWIXIW

jOr

.

affects the

value

of

of 1.) A positive 0.2. and I 0.04, tz r reduces the expected rate of First, it the investment opportunity in two ways. which F(l'). reduces Second, it increases capital gain on (froma to a ), intervals P' of time, and this changes hnite in over the variance of percentage is effect to reduce F( P'). tends to increase F(V). As Table 5.1 shows, the net value /'*. Furthermore. this net effect is and therefore to reduce the critical quite strong; small increases in lead to a substantial drop in P'*. For example, 0.2, !he expected time for F to remain using equation (37)we know that if positive, '(F), is 5 years, but compared to when is zero, falls by more than half, and V* drops from 2 to 1.33. =

value

=

=

=

'

-

=

,4

As we said earlicr, it is important to be careful when interpreting comparative statics results. In this case we have increased while holding a iixed. expected rate of return on V One could argue that the market-determined Should r) remain constant, so that an (which in this case is the risk-free rate,

The particularjump process given by equation

(36)Ieads

to a differential

equation for F'( 1Z) that is easy to solve. Onc could. of course. specify alternative proccsses for J''. For example. a firm holding a patent might face many potential competitors, each of which is trying to develop its own patent. The success of a competitor might cause V to fall by some randoln, rather than fixed.amount. Ovtlr time. additional competitors may succeed in entering the market. so that P' continues to fall. The calculation of the optimal investment rule for a model of this sort will be more difhcult, however, and would likely require a numerical solution method.

6

Guide to the Literature

Antdcedents of the McDonald-siegel

(1986)model

include Myers

(1977),

who showed that hrms' investment options are a component of their market value, Xlurinho ( 1979), who showed that natural resource reserves can be viewed as options to produce the resource and valued accordingly, and work by

.

increase in should be accompanied by a commensurate increase in a (oth1. Then erwise no investor would choose to hold this project). Suppose / remains constant, we would if a increases exactly as much as so that ce t5) in equation (40)with r + ). In this case have to replace the terms (r =

-

.

-

-

''Mertlln ( 1976) derivcd a formula for the value of a call option on a non-dividend-paying sttlck whose price folltlws a mixcd Bruwnian motion/puissonjump process, and showed that if the ' C, and can be valued accordingly. The other extreme possibility, which we will consider in Chapter 7, proibits such temporary suspension by supposing that the full investment cost / must be incurred ovcr again if operations are ever rcsumed. Then some Iosses will be sustained to keep the option of future operations alive. but if the losses grow sufhciently large, the project will be abandoned. Of course reality lies somewhere between these two extremes. Ongoing projects generally build up specific assets workers' if opskills, customers' loyalty, etc. that will gradually disappear, or resumption involves a cost, but less than the cost eration is suspended. of starting anew, and the difference depends on the nature of the product and the duration of the suspension. Our analysis of the extreme cases yields results that can be suitably combined to tit particular applications that Iie in between. -

ttrust,''

'fhc Uildc of

(1

Project

fJ/1(1

tlte Dflcl'sftp/l/t) Ilb-est

transient price tluctuations. Now the prolit tlowbecomes a nonlinear of the price. which alters the effect of uncertainty on investment.

t'unction

All ofthe analysis up to this point assumes that the projectp ont:e installed, goes on producing the output llow forevcr. This unrealistic assumption is made only to convey the basic ideas of option values in a simpler Inanner. In Section 4 we relax this assumption by introducing depreciation. We show that the effect on option values depends not k)n the mortality of one projecta but on how we specit'y the opportunities available to the firm after its initial project has reached the end of its Iife. We also show that option values remain of considerable signiticance even with fairly rapid depreciation. In the concluding section of this chapter we consider a situation where variables that affect the firm's investment decision the output price and two . the investment cost .. are both random. Here the value of the option to invest is a function of both of these independent variables, and theretbre it satisties a partial differential equation. In general such equations are difhcult and must be solved numerically. A special feature of homogeneity helps us reducc the problem to an ordinary diftkrential equation and solve it analytically. Now investment is optimal only when the ratio of output price to investment cost exceeds a threshold inlluenced by the option value ()f waiting. Throughout this chaptcr, the insights about option values which we gained from thc analysis of Chaptcr 5 will remain valid and valuable as we introduce new reaturesinto the model. The techniques developed there will also continue to be useful. In future chapters we will continue the program of generalizing the mlldels and posing new issues. In Chaptcr 7 we will consider the possibility of temporarily mothballing or permanently abandoning a project if its cash flow turns negative. Then. in Chapters 8 and 9 we will move to the level of industry equilibrium, where each lirm has the opportunity to invest in a single project. In Chaptcr l () we will return to the perspective of a single lirm. but generalize the nature of the project, letting it consist of a number of investment steps, all of which must be completed before the proqt tlowsbegin. Finally, in Chapter l 1 we consider incremental investment. where each unit of addition to capacity begins to yield its marginal revenue product as soon as it is installed.

'rhus

and

In Section 3 we allow some instantaneous variation of inputs like labor to vary the output flow from the project in response to

raw materials,

1 The Simplest Case: No Operating Costs In this section the hrm's investment project, once completed, will produce a fixed flow of output forever. For convenience, we will choose the units so that the quantity of output from the project is equal to one unit per year.

Suppose the inverse 19 F' D( Q),where ()f costs production )' D 1). Hence. P =

=

the stochastic

dtlmanll function giving price P in terms (.)1' qtlantity Q is )' is 11stocllastic shift vnriablc. ln this section the variltble are assumed t() be zero, so the Iirm-s prolit llow is just without further loss of generality. we can take P itself as

variable.

For mtst of the time in this chapter we will assume the simpldst stochastic process for #, and one closest to the framework- of Chapter 5. namely, tlle geometric Brownian motion:

(1P

P (lt +

a

=

r.r

P dz.

The profit Ilow is # in perpettlity, and its expected value grows at the trend rate a. If future revenues are discounted at the rate lx. then the expected present value V of the project when the currcnt price is # is just given by F #/t/z a). In this case I'r, being a constant multiple of P, also follows a geometric Brownian motion with the same parameters x and t'z Hence the investment problem rcduces to the model we studied in Chapter 5, but we will rework it directly in terms of P to set the stage for the generalizations to =

-

.

COm0.

1.A The Risk-Adjusted Rate of Return --spanned''

=

r +

/

ty

psp,

,

(2)

where r is the discount rate appropriate to riskless cash flows. p/,,,, is the coeffcient of correlation between the asset that tracks P and the whole market portfolio, and 4 is the market price or risk. For the value of the project, P', As in Chapter 5, we will denote the to be bounded, we must have p. > difference Jz a by .

-

.

Investors will hold the output or asset perfectly correlated with P only if they get a total expected rate of return, /z, from it. Of this, a comes in the form of expected capital gain. The rest, must accrue as some kind ()f dividend. If the output of the project is a storable commodity (forexample, oil or copper), ,

lce rese 11t t l'ltlllet p?lfl?-tt:/lfl/ :(211 f5 11re 1.-., t 11:t t is. t 11k,l of benetits ( Iess storagc costs) tllat tlle nltrgillal stored tlnit providcs. l NV:will generally Iet J be 111-1exogentltlsly specilidtl pllranldter. I-lowevcr. in prnctict!(lver tinle and/or in response to the convtrniellce yield can valy (stochllstieally markdt-widevariables stltrh as total sttlrage). ttnd otlr nlodels can be adaptdd to account for this--' .tlll'tlxqc.

Wlxen slll-ntl untlerlying paraneter changes, the equilibritlm relatiilnsllip J nltlst cont intle to hold. btlt which of the tl1ree nlagn ittldes adjust tt) Jz equilibriunl depends on the underlying technology and behavior. We restore

assune that the riskless rate 1- and the market price of risk /, being properties of the whold Iuarke t. are totally exogenous t() our analysis. Nosv. when t 11et-z of the P asset increases, y nlust incrcase. lf is a fundanlental market constante then a must change one for (lne with /.t I-lowever. if a is 11 fundamen tal I'narket constants then must adjtlst'. )r examplea tlle total amtltl nt t)l' storagc n-light change. Wllerl wll sttldy the effects of chltnges in f'r i)n the hrm-s investment decision. our ansqvttr will depend on which of these viewpoints we adopt. (5

.

Generally

another this out. 1.B

The capital asset pricing model allows us to determine the risk-ltdjustcd discount rate y.. For this we need the stochastic tluctuations in P to be by tinancial markets, that is, there is a traded asset, or one could construct a dynamic portfolio of assetss that s pertctly correlated with #. For simplicity of exposition we suppose that the project output is dircctly tradeable. In this case, the discount rate Jz will be the market risk-adjusted expected rate of return on P. As in Chapter 5 we have the CAPM tbrmula p.

l''tz/lt-v !'.t!/f/7h't'???? !1()N'v Nvi

we will tak'kl J to be the blksic paranleter ltnt.l Iet t; adjust. Wllen specification Icads to an impllrtant different rdstllt- we will point

Valuing thtl Prlyjttt!t

Our prlljetrt is 11 cllntingent

()r

derivative

ltsset.

whose pltytlffs depend

t)n

tlle

value ('f thtt nlllre bksic ltsset P. Tllen we can de rivc the vllltle ()1* thc pnect ()1' as a functitln 1/( J'') thc price ()f the basic ltsset. We l'ollllw tlle prllcedure ()f contingent claims valuatitln that was disctlsscd in tliflkrent ctlnttlxts in Cllapters 4 and 5. N'Vconstrtlct l riskless ptlrt f(llio by taki ng sui table cllmbinat ions of the asset t() be valtled (the project) and btsic asstlt ( 134. Since th is pllrtt'tllio is riskless, it must earn the riskless rate ()1* rcturn. Tllat condition yields a diffe ren tial equation f()r thut tl 11known value of (lur project. The equation can then be solved given appropriate boundary conditions. ' Thesc benests can includc ak1 increklsetl :lbility to smllllth prtlductiiln- avllid stllcktyuts- tnd facilitate the scheduling (3f prlptluctitln rknd s:lles. Ct3nvtlnitlnce yield is tlle reastln t hat Iirnls l1()ld inventories in the lirst plactta even when the expected capitlll gain )n tlltlstt invcntlpries is l'lelf tw lr I'ntpst the risk-adjusted ratc. ()r even negative. As (,11e wlluld tlxpcctp clplnnltltlities- mklrginal conveniencc yield varies inverscly with thc ttltal amount (1f sttpragtr. For tlmpirical studics t't' convcnience yield and its rtlle in cllmmodity price ftlrmation, see Pindyck ( !993 c.d ). see. for example, G ibson and Schwartz ( l9t?().l 99 I ). whf.) shllw l1()wtlil-pnlduci ng prtljccts lows 11 geonletric Brownian mflt ilhn, and the ratc ()1' ctlnvecan be valtled when the price t)f ()iI 1*()1 nience yield also llltlws :1 sttlchastic prtlcess. AIs(). Brennan ( l 99 l ) estimlltes and tests altcrnlltivc i'unctionsand stochastic prtlcesses j'llr convenience yieltl and its tlepentlence t)n price tlnd tinlc.

zl Firm

l 8()

''

Decisiolls

Suppose we construct a portfolio at time t that contains one unit of the project, and a short position of n units of output. where we will choose n to small make the portfolio riskless. We consider hclding this portfolio over the intervaj of time (I. f + dt. The holder of the project will get the revenue or profit llow # dt over the time interval of length fX.A1so, a holder of each unit ofthe short position equai to must pay to the holder of the corresponding long position an amount namely, earned. the dividend or convenience yeld that the Iatter would have (5 Pdt. Thus holding our portfolio yields a net dividend CP n P4 l. It also capital gain, which is equal to yields a (stochastic)

Tlle I-'il/ff e tp- tl Proju-t

tl/lt

J/ltz

i/ecisioll

/f.) lllj.'e.b1

importantly for otlr immediate purpose. given our economic and t > (), the two rtlots satisfy pk > l and p? < ().

conditions

1. >

0

Then the general solution of the homogeneous part of the equation is a '/1 and Bz P#:. To it linear combination of the two independent solutions B3 we add any particular solution of the full equation: the easiest to spot is #/J. Theretbre

-

dv

-

n f,I#

=

( et#l P (lr,(,) + #

(1.''(/:) -

-

,,)

nl c(P)

#2 v''P,

apsz

+

) t/

1.C

tz.

(Note that we have used Ito's Lemma to express dv in terms of the price F'(#) so that the terms in dz disappear and the process.) Now choose n risklessv3 The total return to the portfolio is then portfolio becomes =

(P

-

t5

,2 P P'/4 P) + 1

z

Equating this to the riskless return have the differential equation

t-

gI''( #)

192 Ize/'(#) n

-

PIJ/

and collecting terms, we

=

!. 0.2 2

pp

-

1) + (?. J) -

p

-

Fundamentals and Speculative Bubbles

The term P/45in the solution has an immediate interpretation-. it is just the expected present value ofthe revenuc stream Pt when the initial level is J'--f-his tt.-( l is because Pt 1 # tz'' and discounting at the appropriate risk-adjusted rate =

.

# g ives;

j d(.

Simplesubstitution shows that the homogeneous part of the equation has AP'. provided p is a root of the fundamental solutions of the form l'( #) equation quadratic

(2=

where the constants S1 and B1 remain to be determined.

r

=

0.

(4)

This might be callud the fundamental comptlnent of the value of the project, in the sense that it is justihedby the prospective profit tlows.Thtt other two terms must be spcculative components of value. We can eliminate them by invokingeconomic considerations to rule out speculation. First. it makes sense to require that IZ (()) = 0. If the price is ever zero in the geometric Brownian motion ( l ). it will lbrcvcr remain zcro'. in technical terms. zero is an absorbing barrier for the proccss. With no prospect of a profit now, the asset should have zert) value. I-lowever, sincd pz < (), that power of P goes to intinity as P goes to zero. To prevent the value from diverging, we must set the corresponding coeflicient Bz = (). The othcr term, B3 #/J. is not so easy to get rid of. It represents a compox. People might value nent of P' attributable to speculative bubbles as P the asset above its fundamentals if they expected to be able to resell it later at a sufhcient capital gain. That is exactly what this Iirst term ensures. #/J' To see this, we show that an asset that is always valued at yields its risk-adjusted capital alone. expected from its gain By lto's return appropriate ,

This is very familiar from Chapter 5', there we gave a detailed account of f5, and c. Most its roots and their dependence on the three parameters r, composition of the portfolio is held constant over the short intewal (/. t + dt ); thus lt remains equal to V'( #ff )) cven though P' P) itself changes over this interval as P changes. Over hedging strategy'' will adjust the portfolio at successive a Ionger period of time. the t/I. I ldt ) we will sct n at F'( P(t + Jf ) ), and so on, rather like Thus intelvals. (1 + + over short of such strateges in the limiting case of continuous a chained price index. Rigorotls formulation () needs great care; sce I'larrison and Kzeps ( 1979) and Dufhe ( 1988, pp. 138-147). time as J/ A'l''he

'(

'Adynamic

.-..+.

..-.

zl l lrp??

elstons

.j

/'t..) l'zl/t.z

(.)/- (1

Ile

/5't4.jtz(.? tll

?t //1c

I.l,ci-b.illll Ilt !

'(,.b.(

optitln to invest and J.t sllort pllsition ('f 11 Folloyving tl'ltt sanle steps as be fore, t 1ltl reatle r diffe re n t ial tlqtla t io 11

of one

Lemma we have

.!.(.r 2 l.': 2 t'r dzpvm (T l dt + /711 = ( 1- + / /.11

where the third line follows from the fact tllat /$1satisties the quadratic equastandard tion Q (). and the Iast Iine uses the CAPM tbrmula (2).Thtls the of covariance times that of P. The deviation of the return on P#' is exactly /:11 market with the #/' with the market porttblio also becomes pt times that of P portfolio. With the covariance and the variance both multiplied by the same PI$b and the market ptlrtfolio is the factor, the correlation coefficient between namely, ps,, Theretbre same as that between # and the market porttblio, the risk-adjusted rate of return for Pth is (r + 4 p/w, Jl fr ), which is exactly the expected rate of return in the Iast line above-s In the remainder of this chapterwe will rule out such speculative bubbles. (.)f value fotlnd by direct we are Ie with the fundamental component integration above, namely, Pl. P' ( P$

Then

=

'

) 13 ( 1') -

-

'(

l

'

F-( P )

() .

This is jtlst like eqtlat ion (3) for the value of the project, btlt of cllurse the option has no divide nd or prolit llow. This is a homogeneous Iinear eqtlation of second order. so its solution is :1 Iincar combination of any two linearly independent soltltions- say,

=

.

' '

1-- ( /7 ) + (1.

J-/ I') units klt' the otltptlt. trill'1 check t hat xve gttt t he

J--( 13)

zl

Pl'' +

I

:-12

/.:/$2

where z11 and are constants to be determined. This solution is valid over of prices tbr wllich it is optimal t() hold the tlption. Since hightlr the range make investnlent more attractive. the range in question extends from prices threshold investmcnt P* Of course P* is itself an unknown to btt zero to an tlf solution. the :1 Thus we have three unknowns, zll determined' as part conditions and need three to complete the solution. and J''*, zzh

.

,4-a,

,

The Iimiting behavior of F ( l>j near zero gives tls (ll1e condititln. Wllen ()f it rising t() the exercise thresholtl P. is qtlite prllspect option should be almost worthless at this tlxtren-lc. X) remote. Tlltlrctbre the lls 19goes t() zttro. we should stlt the cllefhcient enstlre that F ( P) goes to zero ()f P equal negative of the to zero'- thus zlc (). power ()f cllntlititlns, F( /:') ;lt P* At other F()r thc we consider the behltvior two the option, and thereby Ctcqtl ire optimal exercise thrcshold it becomes this to exercise the of value V ( #) by incurring price (stlnkcost ) an asset (the project) of invcstment /. As in Chapter 5, twl) conditions govttrn this. First. the value of the option must equal tht! nct value obtained by exercising it; this is the condition: valtte-lnatcllilbq P is very small. th

=

1.D

Valuing the Option to Invest

.

Once we know the value IZ of an installed project as a function of thtl current Ito's price P, we can obtain the diftksion process of )' from that of P by using allow would find 5 Chapter us methods to of Lcmma. Then, in principle, the ()f P' I-lowcver, function the in project as a the value F of the option to invest the drift and diffusion parameters of the process of P' are generally quite complicated expressions. making it hard to solve the differential equation linking F and V. An alternative and generally simpler approach is to tindthe value of the option to invest as a function of the price, F( P), using the above solution for ;'' ( P) as the boundary condition that holds at the optimal exercise .

F (P )

*

Seconds the graphs of F P) and is the slnooth-paslb? condi tion:

threshold.

We will now employ this method for our simple project. Once again we followthe steps of contingent claims valuation. Now the porttblio will consist 5The fact that the risk-adjused discount rate for P equals thtt cxpectcd ratc ot' growth of Pt6when p is a root ofthe fundamental quadratic was also shown using the equivalent risk-neutral valuation procedure in Chapter 4, Section 3.A.

IZ

V ( 19 )

*

=

( P)

F' P* )

-

I

.

/ should meet tangentially

-

at #* this ',

Z'( #*).

=

Using the specihc functional forms of F( P) and Z ( #), we ctn write the

value-matchingand

smooth-pasting

conditions

z1l( P. )/', pl

-d1

(

.'*)/$1

-

i

=

#*/J

=

1//5.

as -

/

./z-lI'-uil-lllllecisiolb' '-

l84

1.E

These yield

For reference

The 'Llltteo)' (1 Pl-tljct

we also state the solution for

./4

(#:

=

-

zzll

1)/1:-1 /-6/.-11

; it is

/ (Jpl )/3,

(10)

.

Using the relation (5),we can express the price threshold equivalently in terms of a value threshold,

//?(.zilecisioll t() /,1!

t//lf/

185

-t,.j'?

Dynalnic Programming

If the risk in P cannot be spanned by existing assets. then we eannot construct a riskless portfolio and use it to obtain a differential equation tbr Pr( #). As explained in Chapters 4 and 5. we can instead use dynamic programming with an exogenotlsly specitied discount rate p. although we will not be able to relate this discount rate to the risk-less rate and the market price of risk using CAPM. Here is a quick summary of the steps. The value of the project at time t can be expressed as the sum of the operating protit over the interval (/ l A-dt ) and the continuation value beyond t + dt Thus -

.

1,( #) This is exactly the equation ( l4) of Chapter 5. Thus our approach expressing ()f the option in ternls of the both the value of the project and the value result could have obtained by the as we same underlying price has produced instance, )' is the In directly. value project with present of the the starting approaches is equivalence of the of and multiple #. the two just a constant will We result general. pert-ectly is the easy to demonstrate directly. However. generally find it somewhat more convenient to work in terms of #, as it is economicallythe more basic variable. The important point stressed in Chaptur 5 was that P' > 1', the option value ofwaiting to invest implies an action threshold where the xpected value from investing exceeds the cost. Here the corresponding idea is that P*/ > l /5 t5 the llow-equivalent (per unit of time) cost of or P' > 1. We could call l investment'.that is the Ievel the initial prot flow must have if its subsequent expected value is to cover the cost of investing.f' excceds /, ln Chapter 5 we discussed at length the factor by which magnitude calculated its 1 multiple'' value namely the pk/(/51 ). We /5. We do not need to for a range of variation of the parameters l'. c, and calculations for corresponding will perform repeat those points here, but we chapter. this in models to be developed later the new and more general the Likewise, in Chapter 5 we defined Tebin's q in sense of the ratio of namely V! 1. This allows their replacement cost. the value of assets in place to investment occurs us to interpret the effect ofwaiting as the possibility that no exceeds 1 as long as it remains below p /4#, 1). We can now though q even similarly detine q P/, /') and obtain the same interpretation.

Expanding the right-hand

=

,-l

P J/ +

lz'( # +

tl

#)

JJ f-z-7'

1

side using Ito's Lemma. we have

where odt ) collects terms that go to zero faster than f/. Simplifving, dividing (). we get the differential equation by JJ, and proceeding to the Iimit as t/J -.+.

*

'*

toption

-

This is exactly Iiktt eqtlation (3) that we had earlier. except that 1- is rcplaced by the (arbitrary)discount rate p and ( 1- /5 ) by (x. The equation can be solved by c). similar methods, and ruling out bubbie solutions, we get ' ( P) #/tp For this t() make economic sense we netld p > (x. Then the option to invest can be analyzcd similarly. Start with a P in the range ((). P* ), where the option continues t() bc held. Split the future into the immediate intelwal (l. t + tJ ) and thd continuation beyond that. Expanding and simplil ing as above yiclds the differential equation -

=

12 c2

Pl F'' Pj +

/.' F' /a)

(x

P)

-(

p

-

=

-

().

Now consider the quadratic equation

-

,

Q- )

=

Since p cost

45 6lf we Icave out the uncertainty and the trend in price. we get = r and thc flow-equivalent :he the of opportunity cost amount investeda 1. 1. btcomes just the interest cost or

/,2

pp

-

1) +a J$ p -

=

0.

> a, the Iarger root p, of this exceeds 1 Since p > 0. the other negative. is Then the solution for the option value takes the form root pz F( #) ##' where the constant z1l remains to be determined. =

.

,41

,

!' ./'??

z'l

186

'J

17irl'l Igeciiolls

Finally we use value matching and smooth pasting between F ( #) and l'' ( /2) at #* to complete the soluton. The result is

J'5-(?jf..('/ 'I-lle P/l(.,

2.A

(q1-

t'l??(/ 111e

fl

The Value

()1*

f..ltlta-hf?/lI()

'est

the Project

flnce again sve consitler the porttblio that consists ()1' a unit (.)f tlle prtlject :kl1tI lt )-/, ( f-') units of :1 short position in the assct tllat spans P. Nvhen lleltl for the short tinle interval t t + dt ), the owner of this portfolio can exercise the current operation option. That is prolitable if P > C; the resl.l Iting prof i t llosv rate is just zrt P4 max ( 17 C, ()). The other aspects of the portfolio (capital gains. dividend payment for the short position. etc.) are as before. Tlle re fore the diftkre n tial equation for t he va ltle of the project is .

which is the natural analog of (9)above. For most of the rest of this chapter we will assume that spanning holds and use contingent claims methods, Ieaving to the rcader the obvious modihcations that apply when dynamic programming is used instead. Occasionally. for variety and simplicity of exposition, we will do the opposite.

2

Operating Costs and Temporary Suspension

-

1.

(,,2

P? I.z!'(1o4+ (r

J) Pv' /'.')

-

-

?-

)' ( P4 + zr ( #)

=

().

This is solved by familiar methods. The homogeneous part has two intlependent solutions Pl$' and J4/2 exactly as above. The only new feattlre is tllat tlle nonhomogeneous part. or forcing ftlnction ( P). is defined diflkrently when # < C and when P > C. Tllerelre we solve the equation separ:ttely lbr P < C and P > C. and then stitch together the two solutions at the pllint P C. In the region 17 < C. wc have n. ( #) () and only the homogenellus part of the equation rcmains. Therefore tlle general stlltltion is just a linear combinatilln of the two ptlwer solutions corrcspllnding t() the two .n'

Suppose once again that the output price follows the geometric Brownian (q motion of equation ( 1). Then a. t'r, y.. and > Jt a are all constants. If the option of investing in the project is evergoing to be exercised. we need g, > a, will also assume or > 0, and we will assume that this is indeed the case. We operation can entails that C. but the project the tlow operation of cost a that and costlessly C, when below and suspended 19falls costlessly temporarily be the llow from proht rises Therefore. instant at above C. any resumed later if P -

this project is given by =

max

(#

-

C.

( l 1)

t)J.

McDonald and Siegel (1985)pointed out another useful way to Iook at such a project. It gives the owner an infinite sct of options. The option at time 1, if exercised, means paying C to reccive the P that prevails at that instant. Since each option can only be exercised at its specified instant, these European call options.; They also showed that the project can be valued

are by valuing each of these options (usingthe standard Black-scholes formula), and then summing these values by integrating over t. We will find it easier to value the project as a simple contingent claim that depends on P. We will go through the steps of obtaining P' ( P) in the following subsection, and then afterwards we will turn to the problem of valuing the option to invest.

can

=

roots: )' ( 13)

zr(#)

7A European option can be exercised tlnly at the time of cxpiration. at any time up to and including the time of expiration.

be exercised

=

An American

option

=

K3 P/'i + K? P/:

where the constan ts S! and K? rutm ai n to be determ ined. I11 thtl region P > C, we take another linear combination ()f thtl pllwer stllutillns of the homtlgtlneous parts and add on any particular slllution of the full equation. A sinApltt substitution shows that ( P!( - C-jl-l satisties the equation. Tllerefore the general solution for P > C is ( P) f/z-

where the

=

Bt

Pl$I

+ Bz /.'/$:+ P/ds

-

C/ r.

constants B, and Sa arc to be determined. These solutions have straighttbrward economic interpretatitlns. In the region P < C, operation is suspended and the project yields no current proht tlow. However, there is positive probability that the price process will at some futurc time move into the region P > C, when operation will resume and /J < Q.is jtlst the expectcd presen t prots will accrue. The val ue 1' ( #) wht!n value of such future ows.

I't'-l? l l-?-?';v krllh (-'(-.Lf'-b '--. '-b.

a'il

11k l 119

.j7

Next consider the region P > C. Suppose tbr a moment that tlle hrm is forced to continue operation of the project tbreverseven during those times worth of such a project'? when the risky revenues fall below C. What is the net the rate value of at a, and is discounted the revenues grows The expected expected present value is the risk-adjusted appropriate rate so y' back at the is at the riskless discounted = C #/.The sure constant cost stream Pl;k constitutes worth #/ C/r) value The ( net C/r. rate r, yielding a present impose solution did not that Since the solution above. the last two terms in (he other terms losses, despite two any requirement to continue operating operations the future in the suspend value option to of must be the additional should # fall below C. The constants in the solutions are determined using considerations that apply at the boundaries of the regions. Begin with P < C. As P becomes very small, the event of its rising above C becomes unlikely except perhaps in the vaiue t)f future profits should then very remote future. The expected present value of the project. However, with pz negative, go to zero, and so should the Pth goes to cx:)as P goes to (). Therefore the constant multiplying this term, namely, K:, should be zero. Now turn to # > C. When P becomes vel'y Iarge. in the very the suspensioll option is unlikely to be invoked except perhaps rule should out the remote future. so its value should be zero. Rlr this we ().8 This Ieaves of P, by making Bb = positive

The Wl/llt;'ol' f? p/-p./tzc/t???(/

//lc

llccisioll t() /?ll.t,-/

These are two Iincar cquations the solution

in the unknowns

cl-pl

Sl

pg

=

Jl

pz

-

yy

-

and Bz,'they readily yield

SI

j

-

,

J

?-

-a)

c1-#:

S2

-

r To verif

C

tives of the two component KL

c/f, =

-1

/31K I cll'

=

=

at C, we have

solutions

Bz Cz + C/J

-

>

/31(l-

C/r.

p? B1 c/l:-l+ 1/J.

Q(?-/(r Therefore

le

))

-

tlut

1 c2

=

speculative bubblcs jtlst as wc tlid l7el'tlrc.

j

-

.

t

pz (r

?- >

c

,.t5/4,-

J)

-

.

Qps at p j)2

-

>

=

r/(r

-

t5).

We

().

(5 t-jqr ) must Iie either to the right of the larger root pl or to the of the smaller root pz. First suppose 1- > J, so 1-/(,J) > (). Then -

-

r/(r ar

done. Next, suppose

)

(

-

r

/1 > p?,

>

J, so r/ (r




-

-

Since the term in A'1 captures the expected profit from the option to resume operations in the futurev and that in Bz the value of future suspension options, both the constants should be positive. For that. we need

power

if P

yj

=

=

-

=

l 9()

400

4OO

350

35O

300

300

(r 0.4 =

250

250

& 200

& 200

r

r

=

$ 0.02 =

Le

0.2

150

150

15 0.04 =

1O0

100 50 0

G

-

0

2

8

6

4

10

14

12

=

16

8

50

0 18

20

0

0

2

6

4

8

10

0.08

=

12

14

18

16

20

P

2.B

smooth-pasting conditions thttn give

The Value of the Option to Invest

Now that we know the value of the project, P' ( P), we can tind the value of tle option to invest in the project. F ( #). as well as the optimal investment rule. Since the price # follows the geometric Brownian motion ( 1), we can go through the same stcps as in the prtwious sdction to establish that the value of the option to invest takes thc form F( #)

=

zd1

#/1 +

.,42

P# =

=

-

.

IZ-

/J 1

l

pb- 1

(p ) .

./1

=

=

/.ya ( p+ )p? + /a*/ t 1 /12/y2 ( # )p? - + j j j

-j

?. -

/

.

.

.

Recal I that l$zis k nown frtlrn ( I4), s() tl'lc pai r 01* equat ions ( I 5 ) :.1ntl ( l ()) be we are Ieft with an equation for can solved for z1j and P* Eliminating the investment threshold: ad1

.

(p I

.

0 is an absorbing barrier. so that F'40) (),we know that zh = ().At smooth-pasting the optimal exercise point #* we have the value-matching and conditions linking F #) with the appropriate P' ( #) from equation ( l 2). Of exercised when # < C; there is no reason to course the option will not be to keep the project idle for some time. This only l investment cost incur the zdl Pts' verified cannot satist value matching and smooth formally'. can be KL J'/$' Theretbre with in equation ( 12) we use the solution fbr I pasting and region. operating that is, for P > C. The value-matching the ( P) in

Since P

X1 ( P +)ts,

-

p?) B? ( P* )X

.

+

t.I

-

I ) P* /t

-

Isb( Q-/?'

+ l)

=

().

Equation ( l 7l.which is easily solvcd numercally. gives the (lptimal investment rule. The reader can check first, that ( I 7) has a unique positivc slllutilln for #* that is Iarger than (C + 1- I ), which is the Marshallian full cost (operating cost plus interest on the capital cost of investment ), and second, that l,' ( P' ) > 1, so that the project must have a N PV that exceeds zerl) beftlrc it is optimltl t()

invest.

Returning to our n ume rical example, th is s()ltl titln lr F ( P ) :, nd 1'. is t5 ().3 f'or ,1).2- and I ().()4 rz I(/).

shown graph ically in Figure

=

=

,

=

=

z1Firm 'J Decisions 400 350 300 l l

7 250

C 2o0 L,..c1s0 %.
0. The reason is that tbr any P, the value of the investment option (andthus the opportunity cost of investing), F( P), increases even more than V ( P4. Hence as with the simpler model developed in Chapter 5, increased uncertainty reduces investment. This is illustrated in Figure 6.4. which shows F #) and 1z'( #) I 0. the critical price is 14, which just makes 0, 0.2, and 0.4. When (z for (r equal to its cost of l 00. As fz is increased, both P' (#) the value of the project and FP) increase; P* is 23.8 for t:z 0.2, and 34.9 for t'r = 0.4.

50O 45O

-

-

-

=

=

=

I

-I

l l I l l l 1 1 l 1 I

tr 0.2 =

400

C 350 L' 300 250

(F

C' 20O

=

CT

0.4

I

I 1

j

X 1 P*

-50

-100

t 0

4

8

12

I l

1

100

50 0

0

1

h

150

=

16

=

14.0

((r

20

=

0) IXP* I

24

I

p-

=

:.p,9 ((r 0.4) I =

N

=

23.8

28

(r

=

0.2)

32

I 1

36

),

1000

500

900 400

--I

800 :

3OO

600

I

C

KVQ

700

=0.04

Y tt-

l

R

i

200

% N .(;.()a

gj

100

'

0 -100

I l

/*=23.8 0

4

8

12

16

20

24

28

4OO

300

I

2O0

1

100

/a*=29.21

0

30

=0.04

500

1 j

l

@$

1$=0.08

0.05 0.10 0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

(F

P

50

40

3

=0.08

$ 30

8=0.04 20

10

0

0.05 0.10 0.15 Figttre 6.6.

0.20

0.25

0.30

0.35

0.40

0.45

0.50

A Project with Variable Output

The basic model ()f this chapter can be cxtended and generalized i11 several ways. One is to allow a more general price prtlcess. for example, a meanreverting process such as the one we discussed in Chapter 5, ()r an even mtlre general Ito process. The (lnly diftkrdncc this makes is that in the diffdrential equations for the value of the project and that ()f thc option- the coefficients become morc complicated ftlnctions of P. In almost 1,11 such cases we mtlst rely on numerical solutions. Since there are n() new general econtlmic insights to be had from such calculations, wc will not develop this Iinc of models here. A different direction of extension is worth some attention. Suppose the single discrete project, oncc installed, allows some I'lexibilityin its operation at any instant, by varying some inputs, such as labor or raw materials, that do not require any irreversible commitments that extend ovdr time. Thttn tht! optimal amounts ()f tllesklltt any instant will dttpend ()n thtt output price at that

./g1F lrm '

l96

'

'

'

'

D CCl%%IOn establisllel, conditions we have p1 lz > ()- so the multiple again exceeds tlnity. Now we can examinc the effect of uncertainty on investnlent. ln earlier work. &ve regarded 6 :s a pllran-leter indtlptzndent 01' ry; I(2t tls continue to do that for a while. We already know one effect of an increase in t'r : it lowe rs pj and theretbre increases the n'ltlltiple. pb/(/.11 y ). Tllis contribtltes to incrcasing #*. Greater volatility raises the value of the option to invest. and therefore requires a higher ctlrrent thrcshold of protability to bring forth investment. However, now we have an added eftkct. As increases. holding ( fixed- we see from equation (2l ) that decreases. Then equation (23)shows that this added e ffect leads to a lower P* and tllere fore it con tribtltes :1 grea ter induceme nt to invest. Th is new aspect arises fron'l the ctlnvexity of protit in price. As we look farther ahead into the future, the variance ()f the distribtltion of price increases. By Jensen's Inequality. the expectcd value of :1 convex function increases. Ito-s Lemma makes this precise: there is an added tern; (7.2 y ( y l ) in the expected groqvth ratc t)f tlle prkltit ltpw.Tllereforc :t Iarger c mcans 1 Iarger expetzttldprdsent vltltle ()1' thc prlltit strtlan-l. ltnd tlltls 1t grcater incelltivc tt) Inequttlity effkct'- ()11 nvestment invest. We will come acrtlss tilis again in Cllapter l l

J) y

-

-

1 /.2 p,( p,

a

-

I)

.

With this, our particular solution isjust K #F/J', which is the expected present risk-adjusted using tlle appropriately

value of the protit stream calculated discount rate.

and set the value of the project

Once again we rule out bubble solutions equal to the fundamental'.

t

-

Kilensen's

)' ( P4

K

=

PY

/5

' .

.

For all this to make cconomic sense we need J' > t). That can be lpotcd at from another perspective. We can recognize the expression for 6' asjust the negative of our t'undamentalquadratic evaluated at y. Theretbre by requiring that 6' > (), we are requiring Q(y) < 0. Then y must Iie between the two roots of the quadratic, specihcally y < pl In turn, this amounts to a restriction 0 < pb l )/pL on the permissible power in the Cobb-Douglas production .

-

function.

The solution for the value of the option takes the familiar form F P)

=

yzl1

##'

.

and smooth pasting,

Finally. the usual conditions of value matching

4

Depreciation

We have assumed thus I'ar that thtt investmcnt, once madtt, Iasts forever. I n reality. physical decay or tcchnological obsolescence Iimit the Iife of a project. In other words, capital deprecitttes th rough agc. ()r ustl. or advance of competing technologies. It is not difficult in principle to modify our analysis to takc this aspect into account, although that does complicate the algebra to some extent. In this section we will indicate how stlch modification can be made for relatively simplc patterns of decay that are commtlnly used in economic

theory. options'' apDepreciation also has conceptual relevance for our proach to investment. One would cxpect that an opportunity to invest in a depreciating project would be less valuable, and therefore that allowance for depreciation would reduce the importance of the issues we have stressed. Our analysis will show that this intuititln must be interpreted With carc. The value of an option to tako an action depends on the dcgree of irreversibility of the itreal

F P*j

can be solved to investment:

=

p' ( #*)

characterize

-

F'( #*)

/.

=

1z''4#*),

the threshold of price P* that triggers the

K ( P*4Y J'

= p,

#! -

/ )z

.

z'1

200

Firm

'.

Decisions

action.This depends not only on the life expectancyof one project, but also on the opportunitics that may or may not remain available after the Iirst projcct comes to the end of its life. 4.A

Exponential Decay

We begin with a form of depreciation that is most Often used in economic theory Iargely for its analytical convenience. Here the lifetime is random and follows a Poisson process. At any time F, if the project has lasted that long, there is probability JT that it will die during the next short interval of time dT. Now, from the initial perspective, the probability that the project dies before r, or the cumulative probability distribution function of the random T'. lifetime F, is 1 e-k The corresponding probability density function of F -

is

e-k

T

Suppose that during its lifetime the project will produce a unit flow of output with no variable costs. The initial price is P. and its subsequent path fl follows a geometric Brownian motion with growth rate a. The risk-adjusted discount rate appropriate to P is p.. If the project lasts exactly F years, the expected present value of its proht Eows is

'

r

e-#t /1 dt

()

r

# e'l

=

f,-''

'

= #(1 =

-t/1

-

-

-t'

e

1F

1/(p.

-

a

and 6 is the return shortfall or where as beforewe have abbreviated p. the risk in P. Now we replicates traded convenience asset that yield on a the can use the probability density of the lifetime for a Poisson process to obtain the expected value of the project: -a

=

l P

1

J

y

= #/(

=

(I?If/ //lc oftl F'a/-f.?.jccf Dccisioll to /'?1I't?.J

20 l

This formula has alternative interpretations. First. the project might have infinite lifes btlt it might function less and Iess well as it ages. producing an an output tlowe-kt at time t after installation.-fhat will yield the same discounted present value as the expectcd value calculated above. Second. the machine might need an increasing stream of maintenance expenditures as it ages. We can then regard P e-kl as a prox'y or reduced-form expression for the projit tlow it can produce at time 1. Next we value an option to invest in such a project. We must distinguish possibilities. The option may give the lirm the right to invest in this project two the project dies, the hrm has no further rights. Or the tirm just once; aer right to invest in perpetuity', after one project dies it gets the may have the original opportunity to invest back again. We begin with the hrst case. Let F( P) denote the value of the option. As before. we construct a portfolio consisting of one unit of the option, and F' /7) units of a short position in the asset that spans the risk in #. Note that such a traded asset is quitc exogenous to the project or the firm; it does not die with the project. Therefore its convenience yield is t5 /.1 (z. Then the calculation goes through as betbre, and yields the functional form for the option value =

-

F( #)

=

z-1l

P/1

.

and

/1 is the positive

F P)

(.

1 -

where l is the

=

IZ

( P)

-

1.

F' P)

=

P''( P).

sunk cost of investment. A simple calculation gives

This is almost identical to our earlier formula (9)for the investment threshold for an infinite-lived project-The only difference is that the 6 on the riyht-hand side is replaced by + This can be understood by considering the special t5 r, and r l isjust the case of risk neutrality and zero trend in the price. Then annualized or flow equivalent of the cost of the insnite-lived investment. Then at the threshold the proft flow should be a multiple of this cost sow,to rellect the value of the option that is sacrihced when the investment is made. When we introduce deprcciation, the flow equivalent cost is increased by the Poisson death parameter because the sunk cost of investment must be recouped over a .

+

=

+ J).

Formally. we can regard the project as infinite-lived, but augment

the rate at

which future prohts are discounted by adding the Poisson death parameter, so that the discount rate increases from Jz to p. + This conforms to our general analysis of Poisson processes in Chapters 3 and 4. .

root of the

The investment threshold P* and the constant z1l are jointly determined by solving the value-matching and smooth-pasting conditions

)

T')/.

e-6

I-za/fJ

where z1l is a constant to be determined, familiar qtladratic equation (4).

dt

()

P g1

Tlle

zl 17//'??l

:!():?

'.

flc't.'/'.tpn.

shorter expected Iifetime. However. the same option value multiple applies to this new llow cost: the equation tbr pl is unaffected by depreciation. Tllerefore the intuition that depreciation would reduce the importance of option values is not valid in this context. to invest is available only onccp its exercise isjust as

Since the opportunity

irreversible even though the project has a finite Iife. If we take the alternative perspective where the physical life is infinite but the otltput flow decreases exponentially, then the depreciation has no bearing on the irreversibility of the action. Matters are different if the option to invest is available in pemetuity, so that the hrm regains the right to start another identical project aer the first one expires. Of course the randomly evolving price might be too low to justify investment at the instant the lirst project dies. but the firm once again has the option, and can start a second project when the price rises again to the threshold. We now turn to this case. In this analysis we must consider revenues and values that are various nonlinear functions of Ps and the risk-adjusted discount rate appropriate to each is different. We take the dynamic programming perspective with an exogenously specilied discount rate p. However, a similar analysis can be conducted using the contingent claims approach provided the Poisson risk of the project's death is fully diversifiable; this point was discussed in the context of Poisson processes in Chapter 4. Let #* denote the investment threshold. and Iet F( Pj denote the value the option to invest. While unexercised. that is, in the range () < P < P*, of option merely has an expected capital gain this F( fJ)J CLd =

h

P F'( P4 + 1.c2 /72 F/'4 p) j (J/.

2

Setting this equal to the normal return p F #) (lt, we get a familiar differential equation for F Pj, and an equally familiar solution F #)

is where quadratic ..dl

=

P#'

adl

-

current project F J7). 'Thus

??lt/

tlle /)(.,(--j-&?,l :t.) /?l!-e/

will die and the Iirnx svilI go l7:lck to Iltlltling tlle opt itlll svllI't 11

Expanding the right-lland side using Ito-s Lemma

and simplii'

ing, we havtl

This has the solution J( #)

where #1 is a quadratic

B6 P p'' + #/(p

=

(z)

-

to be determined, and

constant

2!

+

4,.2

p(p

1) +

-

(z

p

l

P p,

.

J'1 is the positive

(/) +

-

.4

+

)

=

root of the

().

(26)

Over the range P

project dies

> P', a similar analysis applies, btlt if the current a new one will be immediately started. Therefore

This becomes

The solution

is J( P)

where Bz is a constant quadratic (25).

=

Bj

JJ/h

+ P( p

-

a

to be determined. and

)

-

). //p

p? is the

.

root of the

negative

#) must meet tangentially at the common Now the two branches of point P. of their ranges of validity. Also, since P* is the investment threshold, F( P) must satisfy value matching and smooth pasting with J( P) / at #*. Thus we have four equations to determine the constants Cl Bb Bn and the threshold #*. This completes the solution. The scemingly complicated procedure yields 11very simple answer. Since the two branches of the function J( #) to the Ieft and right of P* meet and tangentially at P*, we can use either branch for the value-matching smooth-pasting conditions that link F(#) and J P) / at #*. The left-hand branch gives the solution more easily. The value-matching condition is .74

-

a constant to be determined.

12 (7.2 pfp

,

Tlle billtle t7./-l Ih-oj'ct

1) +

a

p is the positive root of the

and

p

-

p

=

0.

(25)

0 to rule out the term with the As usual, we have considered the Iimit as # negative root. Let JP4 denote the value of an installed project along with that of all -+

future replacement options. First consider the range P < P*. Over the next short interval of time Jl, the proht flow is PJl. Then with probability dt, the

.

-

,

204

..dFirln Decisiolts '

Table 6.1.

lz/?eItl

Option

Mltltiples

Depreciatioll

ksf

&

4.B

tl?7t/ Ihe

'lcc-tpll

o lll !

205

't?.y?'

Sudden Death

Next to the exponential or Poisson decay. the tbrm of depreciation most popular in economic analysis is a Iixed hnite life during which (he project continues in perfect health, followed by a sudden instantaneous death.''

4

).

().l

().:!

t).:5

().ul

0.00 0.01 0.05 0.10 0.15

1.4215 1.3706 1.2651 1.2077

2.0000 1.8632 1.5954 1.4561 1.3813

2.7631 2.5000 2.0000 1.7500 1.6193

3.7321 3.2966 2.4868 2.0938 1.8927

1.1759

Tlle Idl/c oj- fl Project

Let T denote the fixed finite life. The prtect

continues to produce a

unit of output, and tllerefore the protit tlow lC), for F years, at which point it suddenly stops functioning. Let the initial price be P. The value of the project at installation is the discounted present value of expected profits over it s life t irne : r

l''( #)

=

= '/'',

B1

The smooth-pasting condition

' Bl pb

+ #/(p +

a)

-

1.

=

a)

().

=

/ J/

X (l

-

-

(?

1F

t 3t -t

e-'s

7-

1/(M

-

(')

jj.

where we have used the risk-adjusted discount rate p, and recognized that the expected value of the price, and thereforc the protit tlow,grows exponentially at rate a.

-1

-

'

:-7. ()

= # (l

is

J'/$', + l/(p +

t-

In equation (5) wc found the value P-( #) P/t5 for an infinite-lived project. The above formula is now seen as a very natural generalization; the inlinite-life case is obtained by taking the limit ps F goes to x. The value ()f an optilln to invest can now be found lbllowing the same steps as in the previous scction. and distinguishing thc cascs ofa one-time and a perpetual option. We Ieave this as an excrcisc. =

Solving these and writing p

-

x

=

J, we find

(27) Contrast this with the formula (24)for the case where the right the to invest in the project just once. The identical, but the option value multiple is different. The root #( an equation (26)different from equation (25)for pj The latter

option gavc cost part is now-equivalent comes from has the depreciation parameter added to the discount rate p. Therefore p'l > pb and 1) < J1/(J1 1), so that in this case depreciation does lower the J'ylp'j option value multiple. If the option to invest is perpetually available, its exercise on any one occasion is a less irreversible act when the project depreciates .

-

4.C

The General Case

We can subsume the above common

forms of depreciation into a very genzr( P. /). The Poisson a profit t1()w

eral analysis. Suppose the project produces

-

faster.

We illustrate in Table 6.1 the numerical signiscance of this effect. As in the central case of ourearliercalculations in thischapter,we take p r 0.04 =

=

0. We then show the option value multiple J'j/(#'j 1) for various and a values of tr and Depreciation has a signihcant effect on the option value multiples. But the multiples rcmafn substantially above 1, particularly when (r is not too Iow. =

-

.

''In economic theory such dcpreciation

by sudden death is slymctimcs labelled the Wllnderful One-hoss '-one-hoss

shay.'' The term comes frllm thc pocm '*The Deacon's Nlasterpiecc: or Shay'' by 01 iver Wcndcll Ilol mes, Sr.: Havc you hcard of thu wonderful onc-hoss shay Which was built in such a logical way That it ran a hundred years to a day And then, of a suddcn, it PiCCCS all at Once ... WC rlt to ...

We give this quotation in an attcmpt to dcmonstrate

culturc.

that economists are not entirely dcvoid of

/ I-'il-t';

l.lpl..t'b.illts

'.

shlly-*case decay case has x declining exponentially. while in the gtlnerlllly. and then :1 More to zr migllt while drops for constant zero. n' stays physical suffcrs gradtltll capital either because time. gradually decline over technology its of conembodying date at the project. the because decline, or the the In latter with and projects. later case, newer struction, must compete when J drops P. endogenotls date will zr ) to ( the suspend operation at project This obsolescence-'' termed zero; this phenomenon is sometimes advance and mlAnotonic was modelled in the framework of a deterministic and Bliss Yaari and l967) Weizsacker, ( in technology by Solow, Tobin. von research uncertainty of extension exerto the case as a ( 1968),. we suggest its cise for interested readers. -tone-hoss

-eeconomic

The value of such a project can be expressed as a present value integral. but for general forms of zr that is hard to evaluate. We describe an alternative approach. lt has the added advantage of yielding a general formula tbr the value of a project at any moment during its lifetime, and not just at the initiai

date of construction. Let Pr( P, t) denote the value of a project as a function of the current price P and the current time f We consider the usual portfolio that holds the project and goes short /1 units of an asset perfectly correlated with P, for dividends a short interval of time dt. Over this interval, the portfolio earns shortfall'' or zCP. t ) dt pz a is thd n J Pdt, where as usual J convenience yield on the P asset. and the second term in the expression for the dividend represents the payment the holder of the short position must make to the holder of the corresponding long position. The porttblio has a capital gain given by .

'return

=

-

-

where we are using subscripts to denote partial derivativcs sincc P' now has 11 1z',4#. /) two independent variables; thus Vp 1)V1L)P, etc. By choosing expected return we can make the portfolio riskless. Then we can set its total riskless the return'. to equal =

=

7r( P t ) ,

=

-

(5

# Ypl P, t ) +

r (I''' ( P, t)

-

c2

/22 Vpp P, t) + P',( P, ; )

P Fpt #, t )1,

Or

This is a partial differential eqtlation that can be solved numerically. If there is a known muimum possible lifetime T, then the solution can be started

at ? r wi t 11t I1e ct'lntlitildn l'' ( I'. T ) () lklr tlIl 1*. y'v'.twill illtlstr:.lttl stlcll stlltllillns il'l:lI)()t Iltlr ctllltext il1Clllptttr lllltl

trtpn-lplctctl

bllckwltrd.

I().

poillt kltlt tllc klctlllklnlikz siglliticltllcc ()f tlle nevv ternl l''f( 13. / ) t Ilklt vltlue (.)t' tllt'tprljetrt i2:lI1 l'1tlw cllllllge l'or tNvt) I'eastllls: ente rs the picttl re. -'F11t., llitittl t)f tlle sttlcllklst ic price, alltl ftkre i Itle v:y nt pure pllssage o1*t in)e be:.1di of cllllnges tlle ftlture proti Ie proli t tltlws.Tlle latter t? l'ftltrt is dxactly caus't tllat econonzists l j ( 13.t ) gives tls a qulklltitanletll by deprecikttioll. 'Tlltls what dtltlnt'jnlilz o1' depreciation. Tllis concept was well eltlcidated tive Ineasure by Sanluelson ( l 964) in the context of trt't rtktillty and pe rftlct l'oresigllt, and tllat illtrorptlrates tlnce rtain ty ktnd rational Ilere &ve Ilave a nattlral klxtonsitlll Ilere

&ve

expectations.

Price and Cost Uncertainty Thus ttr we have allowetieonly one randol'n variable. nllmely. the tlutptlt price (or a demand shi variabltl). ktlcping :111 (lther paralmeters bearing on the investment decision kntlwn lnd constant. Sv tlid this to develop tlle analytical methods in a relatively simple setting. The same methods can be employt!d in more general situations where twl) or more randon) variables affect the firm's decision. F()r exanlplc, if b()tl1the investment cost I lknd the outptlt price 13 (,1' tlle pnjcct lknd the villtle are tlncttrtain. thcn wc have tt) exprtlss the valtlc ()f lxltI) ()1* tllcsc variltblcs, lz'( P, / ) and of thtl klption to invest as functillns whtlle rtlgitln ()f vltlues of ( P. I ) where intlle Nve l find P. Then llavtl t() )-( will (lccur. the whole region wllertl it will nllt llccur. and the critical vestnpent ()r separating threshold the twll regitlns. Needless t() say, this boundary curve the vltlue mathematically Nvith twt) independent variables. is morc diffictllt. eqtllltions. their satisfy and slllution partial diflkrontial functions can require (Jxalmples complexity-lt' with speI-ltlwever. methods of some numerical somc of solved homogencity can be by cial tkatures in particular. some form reducing the problem t() one state variable. We now illustrate tllis. Consider a unit-sized project whostl investment cllst / and the revenue flow P are both uncertain. We can even nllow the uncertainty in these two variables t() be correlated dtle t() sllme commlln macroeconomic shocks. Thus Inotions: we assume that P and I follow the geometric Brown ian

d #/ 1.'

=

ap

dt +

fT/,

dzl,

(1 .

11 I

=

al

J/ +

t'r/

(lzI

,

,-1F?-v? $- dqlt?t7in-itl,7 r;

208 where

CWz2s J =

:J

tti ,

gt/Fy j

dt

=

Sflzp Jz/

.

)

p Jf

=

-

tl//t/ 0J,tl r/-tp-Jct'/

rule. By now. the steps should be very familiar. Let F P. /) be the value of

Note that the (IP and dl on the right-hand side are stochastic. However. we F'p and zl Fl to get rid t7f these terms and make the can choose ??l portfolio riskless. Then the holder of the portfolio over the intelwal (z f + dt will have the sure cagital gain =

for brevity. Considcr a portfolio consisting of one unit m units short in the output. and ?, units short in capital. By Ito's

=

.

He must also make a payment corresponding to the convenience yields on output and capital. (?,1 l, P + 11 t5/ /) t/l, to hold the short position. Equating the sum of these two components to the risk-less return r ( F m P 11 14 JJ and collecting terms. we get the basic equation -

the option. We find a differential equation for it. We assume that both the risks in output price and investment cost are spanned by existing assets, and work with assets whose prices are P and 1. respectively.l l Call these assets and

+ (1-

J l ) I f--I

-

l

-

-

F

=

0

-

.

tcapital''

As there are two independent variables ( P. /). this is a partial differential equation. It applies over the region of the ( P. 1) space where it is optimal to hold the option unexercised. Over the region where the option is immediately

exercised. we have

P

/:7 p=

Invest

Xx

l

Free Boundary

At the boundary between the two regionsp this becomes a value-matching condition. The two functions must also meet tangentially at the boundary, yielding two smooth-pasting conditions Fr ( P. l )

wait Figure 6.8.

lnvrt.p./l?lt'n?with Pnke and Cb'; Unccrtainly

=

a dynamic programming

approach

Ieads to

:1

very similar

differential equation.

lZ

'

( Pj

=

l/

/,

.

F; ( P, l )

=

-

1

.

The differential equationa together with these boundary conditions, should tix the position of the boundary itselt and also yield a solution for the function F in the waiting regign. The fact that the boundary itself is an unknown makes problems of this kind quite difcult. In fact the theory of partial differential equations has little boundary'' problems in general. Analytical soluto say about this class of tions are rarely available. and numerical solution methods are mostly ad hoc, each tailored to iit a particular situation. This is in principle no different from t'free

l1Otherwise

209

.t?-$?

=

=

of the option,

l() /?I )

Lernfna. we have

t'rate-of-return

Etoutput''

tlle lt.zc-.tp/l

.

Once an investment is made, further uncertainty in the evolution of the investment cost is irrelevant. The value of a live project when the current price ap4 P/tp, where p. J> = r + / p/w, a'l, is P/p.p is P is simply P' ( P) risk-adjusted appropriate discount rate to P, and l, the Jz /, - ep s the shortfall'' in #. tconvenience yield'' or The value of the option to invest, however, depends on both P and /. Intuitively, we expect that the option will be held when P is low or l is high, and exercised when P becomes sufsciently high for given 1, or / becomes sufhciently low tbr given #. Figure 6.8 shows the suggested regions in (/. #) space corresponding to waiting and investing, and the boundaryseparating the two. Our aim is to make this intuition more precise. and develop an analytical method to find the boundary and thereby determine the optimal investment =

Tllt: I-'J/l/t.'

g' ?'r?'l .,.,.j ym. .

2 l0

. .j

.

.

.

)L t)IJ 16) j ,

.

.

.

js

tlle problem even whcn only the price is uncertain: the investment threshold P* is itself unknown and becomes the free-boundary point that separates the one-dimensional range of values of P where investment occurs from the much easier to find the range where it does not occur. In one dimension it is analytically in the present case the whether Luckily, solutions or numerically. tIs reduce it to one dimension. allows of to the problem natural homogeneity

If the current values of both P and I are doubled, that will merely double the value of the project and also the cost of investing. The optimal decision should therefore depend only on thc ratio /7 - P/ 1, and theretbre the boundvalue ary in Figtlre 6.8 should be a ray through the origin. Correspondingly, the enabling I us to of the option should be homogeneous of degree l in ( P. ),

write is now the function to be determined. Successive diftrentiation gives

where

Le t pl de n o te t he l:trge r root of t 1)is; i1'J t assume), then JI > l Tllen we tind

:1 1)

d

f,

itrllNve a re b(')t 11pos it ive (svl'l

.

This ray throuyh the origin separates thtt rtlgitlns of w:liting and investl'ption value nltlltiple ment in the ( P, J) space. I ts slope has the stalltlllrd 12 interpretation. Thus if either t'z'/, or (.'l increase. pb will dekrrease. and the multiple pj/(/31 l ) will increase. However. the multiple will decrease if p increases-,holding their variances fixeds :1 greater covariance betwecn cllanges in P and 1 implies less uncertainty twer their ratio. ltnd henctt :1 reduced incentive to wait. -

.

Fl

and

P, /)

Fra/'t P. /)

./',(

p)

=

Fl ( P- l )

,

/'''( p)/

=

1.

-11 ( P

/)

.

=

Guide to the Literature /'(

=

.1''

/J)

F'/ ( #. /) = ,2 /'''( p)/ 1.

-

a

2,p

cp o'l + c/ a )

a

/7

.

p .1''' p)/ 1.

-

.

(28)and grouping terms,

Substituting this in the partial diffcrential equation we find J. (c/a c

?)

p

-

.,,

J (P) + (t/

.1

(/,)

-

.,

p

( JJ)

.

./ (p)

,

-

=

() .

(c;)

This is an ordinary differential equation for the unknown function fp) of the scalar independent variable p. Moreover, it has exactly the same form uncertain. Its boundary as the familiar (6) for the case when only the price is value-matching condition becomes also The similar. conditions are

./*(/7) JJ/J/, =

conditions

The two smooth-pasting

j'p)

=

l/Jp

1 .

become ./*(JJ)

.

-

-

p ./''(P)

=

-

1.

Of these three conditions, any one can be derived from the other two. We can select the value-matching condition and the first smooth-pasting condition uncertainty case, and as the two that are exactly parallel to the pure price complete the solution as before. The fundamental quadratic is Q

=

1

2

(c/1 2p fzp -

o'l +

fT/)

pp

-

1) + 6;

-

p)

p

-

j/

=

0.

This chapter prescnted more complete models t)f investment, and also extended the important idea that the value of a lirm is Iargely the value k)f a set of options. In Chapter 5 we saw that a tirm has vltluable options t() invest. Some of these will be exercised and stlme not, so thltt the value of the tirm will equal the value of its existing projects (that is, its capital in plactl) plus the valuc ()f its options to invcst in new prlljects in thll future. In tllis chapter we to saw that an existing project can also be viewetl as a set of options--options produce and earn profits should pricklbe stlfhciently lligh rulative to operating cost and can be valued accordingly. This idea appears in Marcus and Mtldest ( 1984) in the context of agricultural productitln decisions. btlt was Iirst spellud out in detail by McDonald and Sillgel ( I985). Tlley shllwed that if prit:c folIows a geometric Brownian motion. a unit-output prlject witl) jixe llperating cost can be valued as the sum of an infinite set of European cal I options. The approach used in this chapter is morc general and mllrtt tractable we simply valued the project as a single contingent claim, and thcrtlby derived a ditkrential equation for 1/'( #) in the same wlty that we derivod an equation I'()r the value of the investment option, F( #). in Chapter 5. We then used tlle solution for )' ( P) in the boundary conditions to Iind the solutitln lr F ( /3), ajong with the optimal investment rule. This approach was dutvelllped in Pindyck ( l 988b) in the context of incremental investment. 12It may seem puzzling that thc risklcss interest r:tte Ills disappeared frllm the qtladratic equation and hcncc frtlm thc slllutitln. In fact. ?' rtlmains, cllllcuttllk!din /z /,. j/,, lnd 6: .

A F'r//l

'J

Deci-iolts

We have seen that an important starting point for the valuation of projects and investment opportunities is the underlying stochastic process for the outmotion. put price, #(f ).We assumed tlat ifprice follows a geometric Brownian expected risk-adjusted return. the y.. For than less its expected rate of growth is storable commodity, this diftkrence would have to retlect the convenience a yield accruing to holders of inventory. This point was raised by McDonald and Siegel ( 1984), and the stochastic structure of convenience yield itself is examined in Gibson and Schwartz ( 1990) and Brennan ( l99 l). This chapter introduced a number of free boundae problems, some of which could be solved analytically. and some of which require numerical solution methods. For readers seeking more background on the mathematics of these problems and their solution, we suggest Guenther and Lee ( 1988) and Fasano and Primicerio ( 1983). Bertola ( 1988, Chapter I ) has a very gendral model wfth simultaneous uncertainty in the output price. the price of capital goods. and prices of variable

Chapter

7

Entry, Exit, Lay-up, and Scrapping

inputs.

variables. and Most of these methods rely on discretization of the state solutions of multivariable numerical we have adopted the samc approach for problems in Chapters 4 and 1(). Somc recent work t)n numerical methods is whole based on an alternativep called the finitc element method. Here the small funcsplit cellsthe variables is into region of space of the independcnt in polynllmial approximated low-order by a tion we are trying to obtain is cdgtls the togcthur at each cell. and the diffcrent approximations are pastcd of the cells. This method looks promising for many economic applications. Interested readers should consult books by Johnson ( 199()) and Judd ( 1992).

previous chapter we showed how one can hrst value a project, and then value the option to invest in the project and determine the optimal investment rulc. Our starting point was a stochastic process for the evolution of the price of the project's output. and hence unccrtainty over the future 2ow of operating prolits. This f1()w of profits could sometimes becomc negative, and we assumed that at such times the firm could suspend operation. and resumc it later if the profit flow turned positive. without paying any lump-sum IN THE

restarting costs. For many projects, this assumption of costless suspension and restarting suspend and later restart s unrealistic. In some cases it is almost impossible to the operation of a project. An example is a research Iaboratory engaged in the development of a new pharmaceutical product', suspending the laboratory's operation may mean losing its team of research scientists and hence the ability to resume development ofthe product in the future. In other cases it is possible to suspend and later restart the operation, but only at a substantial cost. For example. if the operation of an underground mine is suspended. a sunk cost and an ongoing fsxed cost must be incurred to prevent the mine frorn flooding with water so that it can be later reopened, and an additional sunk cost must be incurred to actually reopen it. This chapter begins with a model that is the opposite extreme of the will assume that if the operation is one developed in the last chapter. We again jirm whole the investment cost over to must incur ever suspended, the

stopping or

,..1Firltt

'x

Decisiolls

and crtlmbles once it is unused.) Instead restart it. (This is as ifcapital suspension, active of tirm must contemplate outright abandonment. now an i.Xnitlle lirn; dxercise the option to invest. This gets it the tlow01' operating proht. plus can option to abandon. sv'ennust lind tlle rtlles for optilual exercise of tllese an options simultaneously in terms of the tlnderlying ralldtn-l variable (price). Sinxilarly. when sve consider tlle third alternative of or up,-' there are three tliscrete states. with optilnal ssvitches llnlong then; tt) be --nlothballing-'

-wl:ly-

calculated.

reentry.)

In most cases, reality lies between the extremes of costless temporary suspension and immediate total rusting. The capital sunk in most projects rusts when not used, but does so gradually. Machinery or ships rust Iiterally', other intangibles like customer mines are subject to cave-ins and nooding; loyalty and brand-name recognition fade.l Then restarting is costly. but not quite as costly as new investment. In some cases, the cost of restarting rises with the duration of the suspension.'lb model this.we must consider an optlon to restart (as distinct from the option to start ab initio), and introducc thc elapsed time since the last suspension as an explicit state variable affecting the value of this new option. The resulting partial differential equation can be solved numerically, but we will not treat this case here. Later in this chapter we will consider another intermediate possibility. 'tlkusting'' of idle capital can be prevented by undertaking literal in the case of machinery, ships, and mines, and tigurativein the case of intangibles like customer loyalty. Instead of abandoning, a firm may choose to keep its project alive by maintaining capital but not actively producing output. For example, ships are or ''Iaid up.'' This incurs an ongoing but the maintenance cost, prospect of future reinvestment cost. The saves alternatives depends on the relative magnitudes of betwen these tradeoff and likelihood of the a quick return of favorable operating the two costs, on '.maintenance''

'tmothballed''

conditions. Mathematically, in addition to the state variable (forexample. price) that evolves stochastically and affects the prostability of operation, the possibility of abandonment introduces a second discrete state variable, which takes on 1A scientist undertaking a new research project must invest capital acquiring familiarity with the literature. learning new mthematical tcchniques or Iaboratory skills. etc. Our own experience is that such capital rusts very quickly if we set the project aside for as little as a few weeks.

1 Combined Entry and Exit Strategies We conhne the disctlssion to the case of demand uncertaintya asstlming :1 geometric Brownian motion price process. lnterested readers can develop extensions to other proccsses along the Iines of Section 5 of Cllapter 5. and t)f to uncertainty in other variables along the Iines of Section 5 Chltpter 6. Nv'e the and ntltation wherever possible. The investlnent of Chapter 6 setup retain and abandonment decisions are made l7y a firm that takes price as given. and we again assume that thtl pricc lbllows a geometric Browni:ln motitln,

t/ 1.*

a /3 t/ +

=

f'z

P Jz'.

(thatis, enters the market ), it tlbtains

11prleu't tllltt produces pcr pcriod. and Iasts forcvcr or tlntil abandoned. Variable C are knllwn and constant. Tlle risklcss rate of interest is exogenouslyIixed at r. We will assume that stochastic lluctuations in price are spanned by other assets in the economy (althtlugh.as wll havtl seena if this were not the case, a solution could be obtained by dynamic programming). The appropriately risk-adjusted discount rate for the firm's revenues is

If thc Iirm invests

one unit of output (lperation costs of

pz

=

r +

4 /) tn'f

rr,

where / is the market price of risk, and pJ,,$f is the coeflicient ()f correlatign lx between the price P and the entire market portfolio. As usual, we Iet > (). and that rate-of-return price, shortfall we the assume denote on (

=

-a

(

The firm must incur a Iump-sum cost I to invest in the project. and a lump-sum cost E to abandon it. This latter cost might include legally rcqured termination payments to workers, or costs of rttstoring the site of a mine to its natural condition. It might be the casc that part of the investment cost l is not sunk. so that E is ncgative, reflecting !he portftln of lhc fnvestment that

z4Ffrzn:. Dections can be recouped upon exit. Of course we need l + E 0 to machine'' of rapid cycles of investment and abandonment. >

rule out a

ttmoney

Ently

'-rf,

Lflv-Uz

(l/lt

Scrapplg

of equations and boundary complete the solution.

conditions

contains

just enough information

to

In Chapter 6 we began by hnding the value IZ of a live project, and then went on to the value F of the option to invest. Now that sequence becomes which is an a full circle. The live project is really a composite asset, part of exercised, goesback option 5rm abandon. that the is to the inactive If option to acquires another option words, namely, the asset, it to invest. state. In other project. exercised Thus the live back in turn, it leads When this option is to a determined be interlinked, and must values of a Iive firm and an idle firm are

%Mebegin with the idle firm. To obtain a differential equation for l''(j( P). construct a portfolio with one unit of the optiop to invest, and a short position of Zt1'1#) units of output. The steps that follow are exactly the same as those in Chapters 5 and 6, so we omit them and leave them as an exercise for the reader. The resulting equation is

simultaneously.

This has the general solution

Intuition suggests that an idle hrm will invest when demand conditions become sufsciently favorable. and an active firm will abandon when they become sutxciently adverse. Indeed, we will see that the optimal strategy for investment and abandonment, or for holding or exercising the two options, will take the form of two threshold prices, say, #s and Pg, with #s > Pg. An idle 5rm will hnd it optimal to reman idle as long as P remains below Pu, and will invest as soon as P reaches the threshold Pu. An active lirm will remain PL,. ln active as long as P remains above PL, but it will abandon if P falls to PL and #s, the optimal policy is (he range of prices between the thresholds whether with operation or waiting. We it be active the continue staus quo, to verify this intuition. Of course we must find the values of these proceed to now thresholds in terms of the exogenous data.

1.A Valuing the

K)(#) where z11 and of the quadratic

and

=

pz

=

!

z1l ##' + zh ##2

are constants to be determined. and pt and equation familiar from Chapters 5 and 6:

a4:

p,

=

-

a

(p

-ta (p -

-

-

l/J

2

l/z

2

+

-

((p

t5)/c

g(p

.)//r2

-

Ztlt #)

.

-

-

-

2 1. cl +

!2)

2

the roots

zp/fr

2

>

1.

+ 2p/cz z




t

() >

,

#a.

at /5/ in =

(15)to write

0,

which yields -

(pbf'sp'

ppz

-1

g

Since G P) is concave at #//, Gpp(H)

dl > 0. The investment threshold rises expect. Similarly, Pg falls as E rises.

-

pz s/h-l s

) # PL

=

-(Jl

-

pp, t

j d; j

ks

.

is negative and then dPu > 0 when with the investment cost, as we should

Similarly, the lower smooth-pasting condition G p#(

+#a-I

#2) PL/f!

again should the price process ttlrn stllliciently I'avllrable in tlle ftlturc. Tllerefore, the larger is the investlnent cost. tbe larger is this option valuc antl thtt greater is the reluctance to abandon. The nlirror inlltge restllt. namely. that the investment threshold Ptl rises as tlle abantlonnlent cost E increases. is perhaps even clearer. The tirl'nis nlore reluctant to undertake the prlject if it might have to incur a Iarger cost to shut it down in the ftlturt!.

If we evaluate these conlp:trative static derivatives as / antl lL2 both go to zero, %ve find P:l P:- both go to C. btlt tl P// /t/ / cx'aand / 1':./f/ / and lik-ewise with respect to E'- Thus the entry and exit thresholds begin to spread apart very rapidly even for small costs of entq llnd exit. Exp:tnding thtl four equations of the set ( i4) and ( 15) in Tayior series and carrying throtlgh some tedious algebra, Dixit' ( 199 l a) finds that --v

-cx3.

.

lOg(

' /3 l:'/// PL ) = 16* (/ + E)

.

where k is a constant. ln other wtlrds, entr.y and exit costs tllat nre smal I and ()1* third order (proportionalto 63 where 6 is small) produce :1 gap bdtween the entry and exit thresholds that is of lirst order (proportillnalto 6 ). Thtls vttry small sunk costs huve a disproportionately large ekct on the hrm-s dccisions. ()f

smal I costs on the thresholds is entircly sym-

metric: the entry threshold is affectetl by the exit cost just as strongly as l7y the entry cost. For Iarger magnitudcs. each type (.)f cost will affect its threshold more strongly. The reason is intuitively clear. Cilnsidcr :1 firm contemplating entry. Tht: entry cost must be paid immedilktely, while the cxit cost affects the firm's entry decision only through the prospect that it will havd to pay that some time in the future. Because ()f discounting. the immediate effect is the stronger ()ne. Howcver. if the ctlsts are vttr.y small, the thresholds are very close togethcr, and the Brownian motion is almost surc to reucll the other threshold very quick-ly.Therefore the diflkrence madc by discounting is small, and vanishes in the limit. S*()wn-'

G pP S) d f5/ + G ,zl (S) d,'l 1 + G #s( S) dBz

=

.$ft.'rtl////l?

Morcover, the cffect

.

Now differentiate the smooth-pasting condition

G />1, (S) J &/

l

..--...

-

,

atlt

.

Note that the terms Gp(H) J#s and GpL) J#/. have vanished because of the smooth-pasting conditions (lsl.Therefore the general comparative staticsystem in the four endogenous changes dA 1, dBz, J#&, and #J5f in fact separates into a simpler system. First we solve the above two equations for the changes in the option value coefficients 6lA 1, dBz. Then we can totally differentiate the smooth-pasting conditions to gei te changes in the thresholds dpu, t//N. /'' e tc the solution is Noting that Grt ( S) 1% =

E-llttl'. h/, Ltlv- L/JA

gives cjj

ja

.

Since Gppl-, > 0, we have dPL < 0 when dl > 0: the abandonment threshold falls when the investment cost rises. This important interaction between the costs and thresholds should also be intuitive upon renection. The 5rm abandons an ongoingprojectwith some reluctance because of its optionvalue. By keeping the project alive, it avoids having to incur the investment cost once

We leave it as an exercise for the readcr to vcrify thrtt botll IL; and P;. rise as C rises: as one would expect, a prllject with higher operating cost is undertaken more reluctantly and abandoned sooner. 1.D

An Example: Entry and Exit in the Copper Industry

We have seen that uncertainty

over future demand conditions

increases the

firm's zone of inaction; thlt is, t causes the optimal investment and abandonment thresholds to be spread farther apart than the lraditillnal Marshallian ones. How much larger does this zone of inaction become in practice? Is it really necessary to account lbr irreversibility and unccrtainty as we have, t)r

zl f'*il-t't might the simple Marshallian rules provide a good enough most investment and abandonment decisions'? To answer is useful to look at a specihc example. We

will

examine

'-

llecisiolls

approximation

for these questions. it

EllIl'.

Fzit. Ltlv-Up. tl?7f/

tscrappbq

2 25

variable cost at C $0.80per pound, abtlut the average for U.S. producers in 1992.but we will also vary this cost to determine its impact on the ent:y and exit thresholds. (Ft)r comparison, the average price ofcopper was about $1 in 1992 but twer the 1985-1992 period it fell to as low as $0.60per pound a-nd rose to over $ 1.50 per pound-) =

.00

the decision to invest in a new copper

production

facility a combined mine. smeiter, and rehnery and the decision to permanently abandon a facility that is currently operating. The price of copper has historically been quite volatile. (Tl1e standard deviation of annual percentage changes in the price of copper has been 20 to 50 percent over the past two decades.) In addition, opening or closing a mine or smelter involves large sunk costs, so that copper producers need to make these entry and exit decisions very carefully, taking uncertainty into account.

In reality. a producer with an operating copper mine has an alternative option to permanent abandonment or continued operation. A copper mine and Iater reactivated should the price rise. can be temporarily Mothballing and reactivation involve sunk costs (constructionis needed to prevent the mine from llooding or caving in while it is inactive. and an additional expenditure is needed to reactivate the mine). as well as an ongoing fixed cost (to pump out water, prevent unauthorizcd entry, etc-). However, if reactivation in the not-too-distant future is Iikely, this can still be cheaper than abandoning and later building a new mine from scratch. In the next section. we will expand our basic model of entry and exit to include the possibilities of mothballing and reactivation. For the time being. however, we will ignore Likethis additional option and consider only investment and abandonment.s refinery, mine could shut but for down a but not a wise, a producer open or t)f refined will the intcgratcd production simplicity we copper as one treat ttmothballed,g'

operation. We will consider a facility that produces l0 million pllunds of relined copper per year. To keep the analysis simple, we will ignore the fact that the mine's reserves are limited and will eventually run out; we will assumd instead that the mine can operate forever. (This is not too extreme an assumption, since most copper mines can operate for at Ieast 2() or 30 years.) A reasonable number for the cost of building such a mine, smclter. and refinery is I $20 million, and for the cost of abandonment (largelyfor cleanup and environmental restoration) is E $2 million. (These and all other numbers are in 1992constant dollars.) Average variable cost of production varies across srms in the United States, and even more so across different countries. We will set =

=

,

for the real risk-adjusted annual rate of return for a copper mine or refinery is /.1 = 0.06, for the average rate of convenience yield (or return shortfall) is J Jz a 0.04, and for the real risk-free interest rate is r 0.04. Finally, we will take 0.2 as a base value for the volatility parameter, fy,but we will also consider values of that parameter in the range of 0.1 to 0.4, consistent with estimates that differ in different periods of time-t' Given these parametervalues, equations (9)to ( 12) can be solved numericallyfor the constants z1l and Bz and the entry and exit thresholds PH and P:. Figure 7.2 shows tke critical entry and exit thresholds. P$t and #/.. as functions 0.2, these thresholds are of the volatility parameter o'. Observe that for fr about $1.35 and $0.55,respectively. For comparison, if there were no uncertainty over tkture prices (c ()). the thresholds would be $0.88 and $0.79.7 Hence a very moderate amount of unccrtainty causes the zone of inaction ().55 from 0.88 0.79 $0.()9to l $().8(). to increase dramatically Observe that this zllne increases as o' increases; if o. is 0.4. the width of this zone is about $ l Figures 7.3 and 7.4 show the dependence of the entry and exit thresholds the operating cost, C. and on the sunk cost of exiting. E Observe that as on operating cost increases, both Pu and PL increase. A highcr operating the reduces expected tlow of profit from. and hence the value ot the the cost that project, so a higher pricc is required before the firm is willing to invest. In addition. thc firm will abandon at lt higher threshold price. because it will lose more money when C is higher. A reasonable

value

=

=

-

=

.

=

=

.35

=

-

=

-

.3().

.

.

f'See.for examplc. Bodie and Rosanski ( 198(3)and Brcnnan ( 199I ) rcgarding rz The use t)f ts (1.(34for thc rate of ctlnvenienctt yield is close t() its avcrage value twer the past two decades, but tlne shtluld keep in mind that this paramctcr l3as fluctuated widcly ovcr timu. There have been sustained periods when it has been close to zero, and shorter periods (when the total stock ()f inventories has been l()w) during which it has been as high as 3() f)r 4() percent per year. We havc assumcd a constant to simplify the analysis. For discussions of conveniencc yield and its behavillr, in general and for copper. see Brennan ( 199 l ) and Pindyck ( l993c.d). 7sincc the prtlject produces l () million pounds of copper per year. the NPV of investing 8/1).4)4.tThe rcvenue llow is discounted + 1() #/().04 (expressed in millions t)f dollars) is 0.t)2. and the operating cost is discounted at at rate Jz (9.1)6 but is expected t() grow at rate (), the tirm should invest if this NPV > 0, that is, if P > $0.88. the riskless rate 1, ().04.) If fr .

=

f

-2()

-

=

=

=

5Brennan and Schwartz ( 1985) use cuntingent claims methods focus on the options to mothball and later

reactivitte

the mink!.

t() value

a copper minev and

=

Likewise. once the firm is in the market. the NPV of exiting is -2 positive whcn /4 < $().79.

-

f-'/0.()4 + 8/0.()4, which is

t!

.,4F irm Decisiolls '.$'

1.8

1.6

1.6

1.4

Ps

1.4 W

1

.2

u 1 0.

.0

c

r c (Q.

Elltl'. E'a/, Ltlb'-U/7.(111t1 A-crfl//J?/l

1.2

Ps

c = 1.0 o o-

c. 0.8

C

J

o 0.6

(D 0.6 Q

Q

pL

0.4

0.2

0.2 0.O

0.10

0.15

0.20

Figure 7.2.

0.25

0.30

0.35

Cfppptzn'E rl/ry unil Exl Thrcxllolilx (1.$. F Ilzl(.'i#vvl.s'

())'

=

=

=

0.2

O.3

0.4 O.5 0.6 O.7 C (Dollars per pound)

0.8

0.9

1

.0

Fiqure 7.3.

/'z

=

=

=

=

0.0 0.1

1.6 1.4 Ps

.2

K

1

o 1

.0

c.

c. 0.8 o 0.6 Q

Pz

0.4 0.2

=

=

rz'1 K)+

0.0

0.40

Obselwe that when the abandonment cost E increases. the entry threshalso increases. The reason is that tbr any price P, a higher E reduces Pu old value of the option to abandon an active jroject, and hence reduces the the of value the project, which in turn implies that the price must be higher before the 5rm is willing to invest in the hrst place. Likewise, an increase in E reduces the abandonment threshold P:.,.the (irm must pay more to exercise its abandonment option, so the price must fall more before it is willing to abandon. Observe, however, that while #/f rises and Pg falls when E increases. they do not rise and fall by very much. The reason is that the value of the option to abandon is largely determined by tz and by the much larger entry cost 1, and does not change very much as E is varied. Figure 7.5 shows the value of an idle Iirm, F(,4#), and the value of an active firm, F1 (#), both as functions of the price #. (We have used the base case parametervalues'. l $20 million, E $2 million, C $0.80 per pound, 0.04, and o' 0.2.) Also shown are the thresholds Pu and PL.. Note r Pg, fG(#) exceeds 1zrl( P) by the abandonment 2, since cost E that at # option. giving up at that price it is optimal to exercise the abandonment E + Pr1 and receiving 1r). Likewise, at P Pti it is optimal to invest, so =

Pt

0.4

1.

O.O

-5

0

5

s (hzlillitlrl (1()1111rs )

1O

15

228

z4Firm

250

I l 1 1

2O0

K'

l

t

I I

I

l

I

l

1 l

t/1 (8)

I I

I

1

1 I l l l

I I

l

t/o(8)

50

I 1

O

0.0

0.2

0.4 Figure 7.5.

'

1

I I

I

I 1

Pv

0.6 0.8 1 P (Dollars per pound) .0

P'ljf P # und

'l

( P 3 u.

F'l1?!ctt)?I.s

1

.2

JL 1

.4

1.6 Fkure 7.6.

flJP

Finally, Figure 7.6 shows the function G( Pj 1z'1 F()( Pj. Note the ( #) S shape of this curve in the region of inaction between Pt. and Plt, and the tangency with the horizontal line at / at # Pu, and with the horizontal line at E at P JN This example can help us understand the behaviorof copper producers in the United States and elsewherc during the past two decades. During periods of very Iow prices (forexample. in the mid-1980s, when copper prices had fallen to their lowest levels in real terms since the Great Depression), lirms often continued to operate unprohtable mines and smelters that had been opened when prices were high. At other times when prices were high, hrms failed to invest in new mines or reopen seemingly profitable ones that had been closed when prices were low. This response of producers to uncertainty had a feedback effect on the price Ievel itself. The reluctance of lirms to close down mines during the mid-1980s when demand was weak allowed the price of copper to fall even more than it would have otherwise. We have assumed here that the price process of copper is exogenous to the srm.In Chapter 8 we will see how the price of a competitively produced commodity can be made endogenous in an equilibrium model of industry =

=

-

=

s'crtk/p/aaq

I

!

10O

W/?) k/. Lav- U/?,(111t1

Decixiolu-

I l

j

1so &-

''

-

G( P )

=

1.-1( P )

1. ( P ) '4,

-

behavior. At that point we will ruturn to this example of entry and exit in the copper industry. but in thu context of :& competitive equilibrium where price is endogenous.

.

2

Lay-up, Reactivation,

As mentioned above. a copper

and Scrapping producer has other options besides perma-

nently abandoning an operating mine when the price of copper falls. Instead, the mine could be put into a state ()f temporary suspension, allowing it to be reactivated in the future at a sunk cost much less than the cost of building a new mine from scratch. Plants that are or ships that are up'' are examples of tis state of temporary suspension. '4mothballed''

''laid

Mothballing, Iike permanent abandonment, requires a sunk cost, which will by maintaining the denote in addition, E.u. is plant mothballed, once a we capital requires a cost flow /W.The operation can be reactivated in the future

at a further sunk cost R. Mothballing only makes sense if the maintenance cost ij'I is less than te cost C of actual operation, and if the reactivation

4 F.r/l? llccisiolts -.

cost R is Iess than the cost of fresh investment l ; we will assume that these conditions are indeed met. Our objective is t() determine how the value of an operating project, the value of the opportunity to invest in such a prtlject. and the decision rles for investment. mothballing. reactivation, and scrapping are affected by the various costs 1. Est, /W,and S, as well as the volatility of the output price. As before, we will assume that the price follows the geometric Brownmotion ian ( l). The hrm must decide whether and when a plant should be mothballed, taking into account this uncertainty over future prices. Intuition

suggests the fbllowing general scheme. Starting from a state in which it does not have any kind of capital installeda the tirm will make the investment if the price rises to a threshold Pu. The firm will mothball an operating project if the price falls to another threshold Pjt. Given a project in mothballs. the lirm will reactivate it if the price rises to yet a third threshold PR. Since the cost of reactivation is Iess than that of investing from scratcha we expect Pl < f5/ .

If instead the price falls. making reactivation a sufficiently unlikely or remote event, there is a fourth threshold 1%at which the mothballed project will be scrapped altogether to save on maintenance cost. Then the firm will revert to

the original idle state.

Of course all these thresholds P/?, C/, PR, and P.vare endogenous. and must be determined in terms of the basic paralndters. Even more fundamentally, we must ask if the lirm will find it optimal to use the mothballing option at all. If the maintenance cost M is sufciently high. or the reactivation cost R not sufsciently Iess than the full investment cost /, then the tirm might tind it better to scrap an operating project directly if the price hits a Iower threshold PL.; in that case we are back to the model of the previous section. We must determine endogenously whether mothballing figures in thc firm's optimal

Similar consitlerations apply to tlle investment cost l In prillciplc- one imagine installing a project in the n'ltltllballtld state at 11.cost s:ty. and can activating this it later at a cost /?. I-lovvever.we see Iittle reastln then cheaper ltn tllan silnply operating be investing in indirect route shotlld ever project. I-lence the firnl :vi 11never Iind it optinl:ll to take t llis rtltlte-. it &vi11 never invest into a mothballed projcct. By postponing investnlent tlntil tlle instant of operation. it can d elay spending the first tranche J of capital costand save on the llow 54 of maintenance cost.s Th is Ieaves us onc less switch to consider anlong the possible six ssvitches five switches that are conceivable acrtlss the three states. Of tlle I'enlaining idle to live. live to motllballctl. nlllthballed to scrltp. lnothb:ll Ied to Iivcpand live to scrap the first lur are usetl if nlothbltlling s acttlally a part 01' the optilual strategy. Otherwise only the Iirst and thtl last artl tlsed. We proceed for a while on the assumption that when price falls t() :1 ctlrtain point. mothballing is tlsed. and then determine its Iimits 01* vltl idity dtlring thc ctpurstt (1f the analysisWe will contintlc t() denote the idle and tlperating states l)y the Iabels Iabel ??l for the mllth0 and I rcspectively- and we introdtlce the :lddititlnal balled state. We lind the vllue ()1* the lirm in each state as the apprtlpriatc combinations ()f the expected profit ()r ctlst strcams ltnd thtl (lptions tt) switch. The method is exactly the same as that tlsed earler in this chapter, and in Chaptcrs 5 and ('). s() we wi lI sketch out t htt analysis antl om it m any of the .

-l-

'wlly

,

details. The firm can btl in thtl idle state over the intervlll ((). again givcn by equation (4) alxlv(.!:

strategy. 2.A

when done via the stttge ol' mtltllball illg. Cllsts of preparing a sllip and luovillg it to and frol'n the Iay-tlp Iocation n'llty llavtl to btl inctlrred tvvice-but there may be solne saving in Iktbor Iiring ctlsts if tlle Inllre gr:ltlultl route al losvs tlle Iabor tbrce reduction to btl ach ievetl lnyreti rdnaent or qtlits. NVeleave it to the reader to exanline the issues raised I)y stlch nonatltlitivity of cllsts.

its value is once

ILl) t)f prices. Tllen

Rules for Optimal Switches

In the previous section we denoted the cost of abandoning a Iive project by E Now we are letting Eu be the cost of mothballing an operating project, and Es be the cost of scrapping a project already in mothballs. (ln the case tf a mine, the former may be the cost of hring the miners and the latter the cost of site restoration. With a ship, the latter may be negative, representing the scrap valtle.) To keep the exposition simple. we will assume that E,u + Es E, so the cost of directly abandoning an active project is just the sum of the costs of mothballing it rst and then abandoning the mothballed project. In practice, going from an operational project to total scrapping may be more or less costly .

is a constant to bc determined. This is just thc value of the option whcre We invest. have as tlsual eliminated thd term in the negativc power pz by to the fact that Jr)( P) must go t() zero as 13goes t() zero. using z41

=

'iI n an ()Iigtlplll istic ind ust r.y t he re m:ly bt2 strategic rtrasl )ns l'tr ht )ld ing a pr( jcct in n1()t 11 but we d() n()t ctlnsidcr (hat markct structu rc hcrc. balls. -

zz1F'1,lll s Decisiolts '

232

-

Similarly, the operating state can prevail over the interval ( P./. (6) above: the

(x)ls

with

value of the firm again given by equation

1,'1 ( P)

B1 fM2 + P/5

=

where the constant Bz remains to be determined. In the model developed in the previous section, Bz J':2 represented the value of the option to abandon. Now this term is the value of the option to mothball. As before. the other two terms give the expected present value of continuing operations forever. Of course' the mothballing option derives its value from further possibilities of reactivation or scrapping. The mothballed state can continue over some range of prices ( #., PR). Since neitherzero nor insnity is included in this range.we cannot eliminate either the positive or the negative power in the option value part ofthe solution. Therefore the value of the mothballed project is given by

b' ( #) ?ll

=

J-'/31 + Dz P*

DL

M/r.

-

where the constants D! and Da remain to be determined. The first term in equation (17)is the value of the option to reactivate the mothballcd project. The second term is the value of the option to scrap the project. Finaily, the last term is simply the capitalized maintenance cost. assuming the project remains in the mothballed state forever. value-matching and At each switching point, we have appropriate smooth-pasting conditions. For the original investment. these conditions

are

Z()(PH )

Zl ( PlI

=

the conditions

For mothballing,

Z1( P5f)

I'

=

( P'u )

)

-

Z(;( PI1 )

1.

ZI'

=

( Pli )

E AJ

VI( P.u)

.

=

( P.bf) Z,J?

.

For reactivation, Vm( PR

)

=

( P8 )

1Z1

-

1z',;, ( PR )

R-

=

sense. and thereby determine the Iimits on the range mothballing option is actually used.

Z1'(PR )

-

D l J#'R +

(Bz

Dal Pgtk +

-

-#1 D1 PRyl

l

-

+

/32( s 2

PR/6

-

E 5.

(C ) U)J,

.

-#1 Dl Pu/3.-1+ pz ( s 2

lz't( lb' )

.

pp?-

l

+ j/j

R

S,

=

.() .

o2 )

-

=

,

-

-

investment-abandonment system, each expressed as a fgnction of thc lumpsum and llow costs. Remember our comparative static results: both functions H and 1 are increasing in the flow cost argument

PR

=

H R

.

EM

,

C

A/v

'/) -

.

C. the function H is in-

L (R E

=

,

,v

C

,

-

M4

.

Turn now to the remaining four equations in the eight-equation system above.Thevalue-matching and smooth-pastingconditions fornew investment are familiar:

- A 1 l3'+ -p3 Wl Pupb

# c ,/12 + # // /J s

/$ s 2

+

-

pp-t + u

c/r jyj

1,

=

.().

(22) (23)

.

This system of eight equations determines the four thresholds Pu, J''./, Pe, Ps and the four option value coefhcients zdl B?, D1 Da. Wc can solvc the equations formally, but then we must ask whetherxthe solution makes economic ,

/W)/r

-

ptk-3 + j/j (). (c1) st regard of We can this as system four equations in the four unknowns D1 (S2 Dz), PR, and Pu, and solve it on its own. Furthermore. the system has exactly the same form as the one we worked with in the previous section for the case of investment and abandonment without the mothballing option: compare equations ( 18) to (21) with equations (9) to ( 12) above. We need only reinterpret R as the cost of investment, Est as the cost of abandonment. and (C Ms as the cost of operation-Then PR is like the investment tizrcshotd from the previous section. and #,v is like the abandonment threshold. Tocontinuewith the analogy to the earlier model. let usdeline H( 1. E. Cj as the upper threshold and Ll. E. C) as the lower threshold that solve the

')

=

(C

-

oz)

-

-1

li ('Ps)

the

(20)

J's

=

of parameters where

and reactivation. Using the functional tbrmsabove, two thresholds become

Finally. for scrapping, Z,,,( C )

233

creasing in the lump-sum cost arguments I and E, while the function 1 is decreasing in these two. Now the reactivation and mothballing thresholds of our present model can be written in the form

.

are -

EW/, Ltly- Up. tll?# Scrapping

The most promising starting point is the interaction between mothballing the four equations at these

C/?',

-

fl/r-M

Those conditions

at the scrapping threshold become

( D?

zzj

-

,

p l ( Dl

-

2

) Pvpt +

A I ) Psp

o2

pp?

+

s,

.-/vyr

.-

s

-1

pz o2

.

p/32-1 s

=

() .

(24) (25)

234 tlle tbresllolds //, /3$.and tllll ctleffiThese equations hrtve six unknowns z.1l z41 and the soltltion to the Iirst group of D?. Howevera Bj, D ). ( cients 1w() the cotlflicients; w know relations equations abtlve among us lbur gave vzll complete the sol ution. and )a fore Therc Bz can we + Dl D! ( ) ( ) This system is too complicated to grasp analytically. so wtl will present the nature of the some numerical simtllations that provide more insight into of the solution. properties general inttlition suggcsts However, some solution. mothballing tantamount is to costless and then both R First, if M zero. are 1 model ()f 985) McDonald-siegel basic back the and to we are ( suspension, Chapter 6. Then the mothballing and reactivation triggers both converge to C, and the scrapping trigger collapses to zero. Now consider raising R and 5'I gradually, one at a time. When R is raised holding NI constant. the reactivation threshold Pll will rise and the mothballing threshold P5f will fall. just as the investment and abandonment thresholds did earlier in this chapter when thc investment cost increased. The threshold Pll for new investment will rise: when reactivatfon is more costly. the option of mothballing is less useful. and therefore the lirm ttlso rise: is more reltlctant to invest. Finally, the scrappng threshold Ih. will Iaid-up project will h()Id not on to a when reactivation is more costly, the firm as willingly when the price falls. J./ falls and the As we keep on increasing R, the mothballing threshold (.)1' lhe (lptimal scrapping threshold P.s rises. Fllr muthballing to be a part where these twl) strategy, we must have P5t > Jt$..Thercfore the value of R t)f the parameter space where moththresholds meet delines the boundary balling ceases to be relevant. Write F(- for the common value of the two at this boundary. Adding the value-matching conditions (2())and (24)satisfied by the common Pc. and likewise for the smooth-pasting conditions (2l ) and -

,

./11

=

-

-

.

(25), we find

',

There wi 11also be :1 tratle-off between tlle crit itlaI values of 1?:tl1(.l ) l t Illtt I'i de ne the botlndaries of t he tlse 01' nlothbal !ing. Nvlltrn S is large r, tlle cri t ictl vaiue of NI will be smaller, and vice ve rsa. 2.B

Numerical

Results

Now we turn to numerical solutions to verify these intuitions. Tlle paranActers cost 51 and the rcactivation cost R. X) focus on them, we will asstlmtl for the rest of tllis exposition tllat the lay-up and scrapping costs E and f/ are l7oth ztl ro-, t hen so is thei r stlnlthe cost ()f direct abandonment. E. Shortly wtl wil l present another ntlnlerical example that illustratcs the effkcts 017 nlltk' ing tllese parktmeters nonzero. We normalize to C l We asstlme a risk-netltral (irm. with 1().(5. (),2. Then pz = ;The price process has a ().t)5. and (5 ()ltnd f'z pt 0.()5.The Itlmp-sum cost ()t' investing is l 2. antl there is no Itlnlp-sum cost of disinvesting. so E (). Witll these nunlbersa lnd ignoring the ptlssibility of mothball ing. the investment and abltndtlnment threshtllds turn out to be ().7 I 35. f5/ l a nd J.

of greatest intercst are the 9tlw maintcnance ./

=

..

=

.

=

=

=

=

-(x

=

=

=

.5977

=

=

Now allow mllthballing. and consider twt) cases, t%1 ().()I and 54 ().45. Ftr each. we consider a range ()f val ucs ()1' S. Tlle rcsu It ing val ues ()f the four thresholds arc shown in Table 7. 1 The eftkcts ()f vktrying R for fixetl &'fcttn I7e seen separately in each casc detailed therc. In Case I thc maintenance cost is =

=

.

,

low; Nl = ().() l Now mothballing is used tlvcr some range of prices until thll critical Iimit of R ::::: 1 once R excceds this Rs mothballing is never used. rises R As over the range 1) to R, ( l ) hf rises to its ievei in the absence ()f mothbalIing, (2) l>.bttkllsand P. rises unti 1,whe n /? R, the twl) mee t at PL (3) PR rises, only to bttcome irrclevant oncc R reaches R. In Case 2. the maintenance cost is higher'. NI ().05- Now mothballing relevant is over a shorter range; R is only a little largor than 1 The general pattorn is easier to see from Figure 7.7. Fixing 54 at a relatively low value. the various thresholds are shown as functillns of R by thc thicker culwes. Mothballing is part of the optimal strategy when S < R. Nvhen R A: R, the two thresholds P.f and Ps merge into Pl. Svhen a higer vaIue of IV is considered. the curves shift to the positions shown by the thinner lines. The Pu curve sh i s down, the Ih. cu rve sh ifts u p, and thc two meet at .

,76,.

zll Pct' + Bz Jtt: + Pc/

-p3

live prljtlct Iess roatlilys antl reactivate :1 nlot Ilballed projet)t I'ntlrd reatlily. &vi l-losvttve r, 1..'11a nd Ih. Il rise t he li rfn Nvill l'tl 1114.)re rklltl tlt:.k 1)t to iIlvcst lt t lt I1. wi Inot hbrllled prlject nlore reatl ily. Once ngain :1 1':11 and Il scrap a Iing I''bt:1 !ld rising l)v wi lI nleet svhen 54 rises to a crit icaI levc1.'tbr al'ly Iliglle vltltltls :1 r of 51- mgthball ing vil l not be uscd.

-d1

Pc(,

-1

+

pz s 2 ppz-' c

-

C/

,'

+ jyj

=

=

-

( E st + Esj

=

-

E

.

=

().

These are exactly the abandonment equations ( l 1) and ( l 2) that. together with the corresponding pair for investment (9) and ( 10), were satisfied by the thresholds Pu and Pi. earlier in this chapter when mothballing was not available at all. Thus the whole story fits together as it shotlld. For high enough values of R, the firm ignores the possibility of mothballing and switches optimally between idle and active states as before. Next hold R hxed and raise M. This reduces (C M), the flow cost saving from mothballing. Therefore both Pa and Pu fall; the firm will mothball :1 -

.

=

.

.

,4

236 Table 7.1.

(Parameters:

r

=

/kJfJf/ldm#?1g

0.05,

=

and

0.05.

o'

ucmppng

=

0.2, C

=

Firm 'J Decisions

Tlreshol 1, I 2, E =

El1tl)L

kf, Lay- U/J.

kt-.?w/p/?l(

P

0)

=

alld

Pu

6 l

Casc 1: Lower cost M R

&

P,

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.7634

1.557 1.568 1.576 1.582 1.587 1.591 1.594 1.597 1.598

1.202 1.272 l

=

f'I,

0.8322 0.7987 0.7770 0.7608 0.7478 0.7369 0.7276 0.7195 0.7135

.325

1.372 1.413 1.451 1.487 1.521 1.548

Case 2: Higher cost M

=

l

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0307

1.577

1.108 1 l57

0.8246

0.5240

0.7968 0.7783 0.7644 0.7530 0.7434 0.735 l 0.7278 0.7212 0.7135

0.5430 0.5612 0.5793 0.5978 0.6 l 70 0.6371 0.6584 0.6811 0.7 135

.590

1 1

1.194 1 1.253 .225

.594

1.278

.596

1.301 1.323 1.343 1.370

1.597 1.597 .598

1

l

I

l l

J;L

Ps I

1 l

I I

;. ;!. btltll /rilil/z-tr

Ps

&f

.587

-

I I l 1 I I I J;v I I

R

R

&

.

l I

C M

Ps I

(3.05

Ptt

.583

l I 1

0.2937 0.3171 0.3424 0.3713 0.4061 0.4498 0.5085 0.5955 0.7135

R

1 l 1 1.592

I I

l l

(1.01

a lower value W.The Pu curve shifts up to reach its hnal constant Ievel at this new lower W.The PR curve shifts down, only to end when mothballing ceases to be used. Note also that as R increases starting at zero, the restarting and mothballing thresholds spread apart very rapidly. Since we have set the 0, the sum of the costs of the pair of switches, namely, mothballing cost Eu =

R + Eu, is small, andwe have an instance of the cube root formula (16)above. One further numerical experimeht is of interest, namely, making both R and M small to drive the mothballing model to the limit of otlr model in Chapter 6. This does happen, but the approach to the limit is very slow, in

/7zJ

//?l$$;f tll

.6;t?-tl/7/7f?l.t;

kecping with the general insight that even small sunk costs matter a great deal when thcre is ongoing uncertainty. Thus. keeping C l etc.. as above. even when we reduce the costs associatcd with mothballing t() /W ().()()l and P,u ().9 l9, each about l() perccnt 1 and R ().02, we find PR value. namely l The scrapping threshold from Iimiting their common away becomes C 0.0963, again significantly above its limit of zero. =

,

=

.4)89

=

=

=

.

=

2.C

Example: Building, Mothballing,

and Scrapping OiI Tankers

The numerical solutions presented above help to illustrate the qualitative dependence of the optimal thresholds on the various cost parameters. However, it is useful to also examine optimal investment. mothballing, reactivation, and scrapping decisions for a real-world example. As with our copper industry example, this can give us a better appreciation for the importance of sunk costs and uncertainty, and aso show how the mode! can be applied in practice. We will apply our model to the oi1 tanker industry. Oi1 tankers provide particularly good example since the potential or actual owners of tankers a face considerable profit uncertainty, as well as substantial sunk costs. The uncertainty arises because the market for oiI tankers is very competitive, and tanker rates (therevenue per day for the use of a tanker. that is, the price #

Decisions

L-lltt''

in our model) lluctuatc considerably as oil prices tluctuate, as the geographical distribution of oiI production and consumption change. and as the supply of tankers changes. Also, sunk costs are important because of the considrable expense of building a new tanker and maintaining or reactivating one that is mothballed.

11

.,z1Finn

tankers.with capaities Tllere are fourgeneral sizes of oil tankers-small capacities around tankers,with medium (dWt), 3s.ooodeadweight tons around and, since the with around dwt, large capacities 140,000 dwt, tankers, 85,000 with of about capacities Crude (VLCC's). Large Carriers'' mid-1970s, and other and dwt. operating, Revenue construction, rates costs do 270,000 economics of with investment, increase linearly tanker capacity the not so mothballing, etc., will vary across these different categories of tankers. We will focus on one particular category-medium tankers with an 85,000-dwt Ktvely

capacity.g

The average cost of building a new tanker with an 85,00()-dwt capacity is about I $40 million. (All costs and revenues are expressed in 1992 dollars.) The one-time cost of mothballing a tanker is Eu $200,000, the cost of million mothballed tanker is E.v scrapping a (thatis, the tanker has a reactivating mothballed value), tanker of this and the cost of positive scrap a mothballed maintaining annual tanker is R cost of $790. 000. The size is a and operating M annual 1992 labor costs. the $515. 000. Finally, given fuel cost for this tanker is C = $4.4 million. ln 1992, this tanker would earn a gross revenue of about P $7.3 million per year. We have assumed that P followsa geometric Brownian motion. and the drift (z and volatility c for that process can be estimated from the sample mean and sample variance of an actual time series for gross revenue. Using quarterly data for 1980 through 0.15. Finally, we use a value of 0.05 mid-1992,we found that t:r 0 and c risk-free and the risk-adjusted rate #t (sothat the real interest rate r, for both 10 = 0.05.) Figure 7.8 shows the critical thresholds #s, PR, 8w, and Ps as functions of the one-time reactivation cost R. Note that for our base value of $790,000,

'aic

I .tIv-

t.//

tl//t

tb-clzqlpitlg

l

PH

10

l l

9

j

I I 1

W 8

=(3 c

7

I

I I

= 6

u: =

>

5

Q.

4

l

Pa

I I I 1 I

P.

3

I I

Ps

2

l

=

=

1 O.0

0.5

1

.0

1

2.0

.5

2.5

3.0

-$3.4

=

3.5

R (Million Dollars)

=

Izigure 7.8.

C*r#CfJ/

TllrL.'.b'ltt

?/f

l

tlN

/;il?tt'/f 111.%( )/' Ictlcli

'tl/tl/l

C).%I

/f

=

=

=

=

gFor a general introduction to the oiI tanker industry, see Rawlinson and Porter (1986). Goncalves (1992)has also used contingent claims rnethods to examine optimal investment decisions for tankers, as well as thc relation between spot and long-term contract prices. 'BWeobtained timeseriesdata for revenue,costs, and other industryvariables from Marsoft. Inc., of Boston. Our thanks to Dr. Arlie G. Stcrling, the Prtsident of Marsoft. for making this data availableand giving us advice on a variety of economic issues related to this industry. Thanks are also due to Victor Norman and Siri Pettersen Strandenes of the Nomegian Schot)l of Economics and Business, Bergen, who also provided data and advice.

the 1992 average gross revenue of $7.3 million pcr year wlluld have been sufhciently high to reactivate a mothballed tanker, but well below the threshoId revenue of about $9.5 million per year needed to invest in a ncw tanker.

Consiptent with this result, thcre

was

indeed little

()r

no invcstment in new

tankers during 1992. Observe that PR and Ps rise as R is increased. and P5f falls, but not very fast. Mothballing remains a viable option as Iong as this ctlst of reactivation is below about $2.7 million. The investment threshold P// also rises as R rises (althoughso siowly that it is hard to discern from the graphl; a higher R reduces the value of a tanker. and therefore raises the revenue that I a 5rm must expect to receive before it is willing to invest-l

Figure 7.9 shows the optimal thresholds as functions of the one-time of cost mothballing an operating tanker, Eu. Observe that the qualitative

l1 Unlike Figurc 7.7, the curves for Pg and Pbt do nflt mct!t ()n thc vertical axis in the cubc form. becausc in drawing the earlier sgurewc assumed the cost ()1' mllthballing tt) btt zero, root which is not the case in this numerical example.

J*/777-.Silecisions .,z1

240 11

Fs

1: 18 7 c c uq

>

Pa

5

PM 3

1

0.0 -iqItnl-t;F. J'.

0.5 L2*l?/ic'ttl;r-7lz-t'./lzJ/z/.

2.0

1 1 Eu (Million Dollars) .0

6Ij(

9

.5

tp/c (n6,- ir7zzzty Lntl.t /7l/zlt7rzlzll/

rJ/!'

llltll

*

:F 7

O c =

>

*

I l I

Pa

5

Q.

Pu

I 1 I

3

1 I I 1 1

PS 2.5

1 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

.0

llitl-i;d7'1/ /pzz

dependence of the thresholds on Esf is the same as it is on R. Values of E.u above$1 million are unrealistic (recallthatour base value is $200,000), butwe have included this largerrange for illustrative pumoses. Note that mothballing remains an option as long as Eu is below $2.1 million.

Figure 7.10 shows the optimal thresholds as functions of annual cost of a mothballed tanker, M. As the Iigure makes clear, this is a critical parameter in determining whether mothballing is a viable option for the rm; note that mothballing remains an option only up to an M of about $720.000.and our base value of $515,000is not far from this value. Remember that higher values of i%1 reduce the value of a tanker and thereby raise the threshold for investment. A higher value of 2%/ also makes mothballing less desirable and therefore reduces 'r and Pm,and raises #x. Figure 7.1 1 shows the four critical thresholds as functions of the annual cost of operation, C. As with a higher value of iV, a higher operating cost reduces the value of the tanker. and thereby raises the threshold #s required for investment. As one would expect, however, changes in the operating cost have a much greater effect on the value of a tanker, and hence on J5./, than maintaining

l 1 I l I I I l I

G'

l I 1 I I

PS

I 1 I I

Ps

l I l I I I 1 1 1 I 1 I 1 I 1 1 I

A

we

11

i 1 I l

9

=

k/'r); Erc Lay- Up. tl/lt/ Swapplg

do changes in the maintenance cost M. Also. because a higher operating cost reduces the value of an operating tanker, it raises the threshold PR at which a mothballed tanker is reactivated, and raises thc thresholds P5t and PR, so that the revenue # need not fall as far before the irm is willing to mothball or scrap its tanker. Finally. Figure 7.12 shows the critical thresholds

as functions of c, the

standard deviation of annual percentage changes in the revenue #. Observe that the thresholds. and especially PtI and &, are quite sensitive to rz. As we saw earlier with our copper example. for large values of o', the zone of inaction widens considerably. Also, note that if fy is Iess than about (). 1 #.J and Ps coincide, and mothballing is not an option used by the firm. The reason is that mothballing is useful only if there is a reasonable probability of a substantial increase in revenue in the near future (so that the tanker will be reactivated). With o' < 0. l the probability of a sufhciently Iarge increase in revenue is too small to make mothballing economical, given the costs of mothballing, reactivation, and maintenance. ,

,

12

10 @8

Ps

o c

E 6

X

Jp&

I l

4

1

3.0

3.5

4.0

4.5

5.0

5.5

6.0

10

I

: a

1

, I l l

6

0 0.00

Ps

t

Pp

l

PM l

2

l

2.5

I

4

Ps

I

2.0

j

Y,

I

0

'@' 12

;

Pv

I

2

14

&

l

j

I I

c; O

1 1 I

l l

16

1

=Y

7.

18

I I I I I I

l I

0.05

0.10

pS 0.15

0.20

0.25

Guide to the Literature The pioneering article on the jointtlecisions to invest and abandon is Brennan and Schwartz ( l985). They construct a vcry gdneral model of the dccision to open, c Iose, an d mo thball a mine producing a natural resllurcc whose price fluctuates over time. The finiteness t)f the total stock in the mine introduces addditional complexities beyond a11 the issues considered in this chapter. In fact the vely complexity of the modcl conceals some important concepts. Having obtained the system of equations characterizing the price thresholds Brennan and Schwartz immediately for investment and abandonment, numerical solutions. Thtty to obtain the ratio ()f the entry and exit resort threshold prices, and show that it exceeds 1 for rcasonable parameter '

values.

Dixit (1989a) isolates the entry and exit decision from isstles of Iay-up stocks. This allows some analytical results and insights. ln particular, finite or the ent:y and exit threshold prices can separately be compared to the myopic or Marshallian criteria of full and variable costs. respectivcly. This clarises the role 01- the time value of the stlparate options to invest and to abandon,

or morc generally. the tlption follows Dixit's treatnlent. Some of the earliest

valtle

work

()11

()1'

0.30

0.35

0.40

tlle stktttls qull. Our cxptlsititln

clllsely

C (Millioc Dollars)

optinAal

nllltllballing

decisions

was

by

klossin ( l 968). who developed lt IT1lltltll in whicll (lperating )IIllws reventle a trendlcss random witlk with upper and Itlwt,r retlecting bitrriers. Itnd in which there is no ptlssibility tt' scrapping. I-Ie calculated the optimai rtlvttntle levcls at which it is optimal t() nltlthball and reactivate tlle prtect. The nlore general model of Brennan and Schwartz ( I985) incltldes the possibility ()f and abandllnmtlnt. mothballing as wcll as active tlpttration Howcver. thcy the transition the confuse twt) states frlln :tn ttltive state, t3yusing thc sCtme to Iowcr thrcshold symblll lbr mothballing as

lr

abllndllnment-

Nvhcn they

fnln'll.r

to numerical solutions they specializc l() a model where the mClintttnance cllst is zcro's then scrapping is never used and they consitler switches betwden j ust two states operation and mothballing. Once again (lur approacll follllws the subsequent but somewhat clearer analysis 01* Dixit ( l t?88).

Thc models we studied in this chapter artl instlnces ()f the gcncrnl problem of optimal switching among a number of alternatives in rttspllnse t() changing economic conditions. Each swtch is an exercise of an ()pti()n- ftntl

zdFirm Decisiolls 's

244

each switch yields an asset that combines a payoff flow with the option of switchingagain. Thus we have a stlt of Iinked or compound options. and must price them simultaneously. There is a large body of literature analyzing such compound options, either from a general theoretical perspective or aiming at specic applications. Geske (1979) is an early example of this kind in financial economics;subsequent articles include Geske and Johnson ( l 984) and Carr ( 1988).

Turning to real investment decisions. Kulatilaka and Marcus ( 1988) develop a model of switches between two modes with three time periods, and indicate how it can be extended to many motles and switches. They also survey the early Iiterature on real options by placing the different models within their

framework.

Fine and Freund ( 1990) examine a general two-period model where the hrm must choose its capacity belbre uncertainty is resolved, and it can choose either specific capital (suitablefor a particular kind of output) or capital (which can produce all kinds of output, albeit at greater cost). Triantis and Hodder 1990) havc a similar model in continuous time. He and Pindyck

oexible

partW

lndustry Equilibrium

(

( 1992) go further by allowing subsequent expansion of capacity. ln all these models, option value is twen more important. because by waiting the Iirm preserves the opportunity of making a better investment Iater, and not just that of not investing at all. as was the tlase in the models of Chaptcrs 5 llnd 6. Bentolila and Bertola

( 199()) examine

employment

decisions of firms

when there arc hirfng and firing costs. Dixft ( l 989b,c) considers production and import-export choices when exchange rates tluctuate. This builds on earlier work by Baldwin and Krugman ( 1989).,see also Irugman ( 1989). Dumas ( 1992) and Krugman ( 1988) develop such models at a general equilibrium level to endogenize the exchange rate process. Kogut and Kulatilaka ( 1993) consider multinational hrms' decisions to shift production from one country to another in response to exchange rate movements. Van Wijnbergen ( 1985) constructed a two-period model of capital outflows from less-developed countries in a context of policy uncertainty.

i

l?qC1lzNr-rlus 5 through 7. we examincd a vklriety of investnlent and tlisinvestment decisions )r :1 singlc (irm. Throughout, we asstlnled tllitt the hrm has a montlpoly right to invest in a givcn project. and wc ignored the possibility of other firms entering in competititln. l Tllc profit tlows of an operatitlnal project werc subject to ongoing shocks. Since we were assuming that the project would yield a fixed output Ilow, we could model these shocks as an exogenous price process. We tbund that th0 textbook Marsllall ian present value criteria or comparisons of price nnd cost were very far fnlm the true optimal choices ()f the tirm. With moderate amounts of uncertainty, investment was justihed only when the current price excceded the long-run average cost, or the current rate of return on the sunk cost of investment exceeded tht! cost of capital, usually by a large margin. Similarly, abandonment was triggered not when the pricc fell below average variable cost. or when the currcnt operating proht became negative. but only when the loss became sufficiently Iarge. The reason was the option value of waiting bcfore making irrevcrsible decisions in conditions of evolving uneertainty.

l In Chapttt r 5 we mentioned

t hat the rate tlf return short l'alI ()n the vaI uc ()f a project, wherc l.t is the competitivc expected return ()n an asset with cquivalcnt risk. clluld rcllttct the possibility ()1' entry by other lirms. I-lowtlvcr. we did ntlt explicitly mtldel entry, ntlr did wc offer any justificationftlr why it should Iead to a ctlnstant rate of return shtlrtfall.

J

ing stochastic processes and the decision rules of other tirms. Each lirrri has the capacity to produce the llow of one unit of output, which it can activate by incurring a sunk cost. There are no variable costs of production. and the elasticityof demand is large enough to ensure that each tirm that has paid its sunk cost will in fact want to produce at its capacity level.3 The prices of different hrms' outputs can change unpredictably because of aggregate industry demand shocks, or because of hrm-specific demand shocks that reoect changing relative tastes for the tirms' products. Hence one hrm's output can sell at a price that includes an exogenous premium or zExtension to a CAPM setting is straightlbrward and mercly introduces stlrnc extra nofalign

in this contextv so we leave it to intcrcstud rcaders. 3With

Q

irmss this requires a pricc tlasticity is very weak.

by this assumption

in excess of l / Q, so thtl restriction

entailed

=

pz

-

fz,

downside risk by chllosing not to invcst if thtt price tlls. while preselwing t he upside ptltcntial to invcst if the price rises. Thus the consequence of waiting is that the payoff to the firm is a convex function of zY so the expccted value of waiting ristts with uncertainty in X. This contrast between the effects of tlncfartainty on the expectcd valucs ol- the payllff from investi ng immediately verstls value'' premium waiting explains why greater unccrtainty raistls the waiting. on -soption

Now consider intlustry-widd uncertlinty. The firm k ntlws that if F rises. en try just as attractive for othcr firms as for itsttlf. I-lowever, when

this makes

4Otlr t reatnaent t) f thtt resu lting ctln t in tl u m lf ra ndtlm va riables 11nd l heir lktws (41* largt.r numbers is dttcidedly hetlristic. btlt thc restllts are intuitively clellr. itntl il jilrmltl rigllrtlus trtrltnlent would be far t()() lcngthy lnd di flicult I'(lr thc presellt purpllse. 5tte Judtl ( l 985 ) lr the blsis (,1' the rigorous tlleofy. (

249 In practice, most (irms d() not enjoy mollopoly rights to invest, btlt insteatl must consider the possible entry of new competitors. ()r expltnsion ol' existing ones. This raises 11 fundamental doubt concerning our earlier conclusions. The opportunity to wait. and its value, depend on what the firm-s competitors do. 'With free entry. should this value not be reduced to zdro'? Wtlultl that not restore the Marshallian criteria comparing price to the Iong-rkln average cost in the case of investment- and to the average variable cost for disinvest ment? Thus the reader might suspect that the thcory of an individtlal hrm would not survive an extension of the scope of thtl analysis to thtp level of industry

llttlllstlLy )-t////'/#;v2./?)2

other firms enter, the industry supply curve sllifts to the right, and the price rises less than proportionately with F. Therefore price is a tloncave function of F, and then so is the profit tlow. Greater uncertainty in 3- now reduces the expected value of investing relative to that of not investing. That is why the firm requires a higher current prolitability (in excess of the Marshallian normal return) before it will invest.

We should stress the similarity as well as the diftrence between the two scenarios. In each. the underlying symmetric demand shoek translates into an asym'metric protit llow shock; but this happens in very different ways in the two cases. In the case of firm-specific uncertainty, the downside ofthe profit shock is cushioned by the p'ossibility of waiting. Thus greater uncertainty makes waiting more valuable relative to investing at once. In the case of industrywide uncertainty. any one firm in the mass of competitve potential firms has the a zero value of waiting. However, the upside protit potential is cut off by uncertainty reduces the of Therefore firms. other entry of greater prospect value of investing relative to that of not investing :tt all. In reality there are several other factors that can aftkct the convexity concavity of profit tlowsas a t'unctionof the underlying shock variable. l 1* or adjust some variable inputs instantaneousiy, then its protit flow tirm the can of the price, as we saw in Chapter 6. Setltion 3. In function becomes a convex will sde that when the tirm can add to its capital 1l in Chapter we addition, and is llow given by a producton functilln- thertt can be its output stock, which the marginal protitability of an incremental iilvestment by other ways I-lowever. price. the above intuition still operatt!s similarly. in becomes convex For example, with tirm-spccificuncertainty. the possibility of wlliting cuts off the downside risk and maktls thd profit flow an evml l/?tprf.r convux ftlnction of the underlying shock variable. In most of Chaptcrs 8 and 9, hllwcvcr. we will

and therefore charaeterized by ctll'lsiderablu inertia. should the government attempt to encotlrage investment'? I-low will various policy instruments aftkct investment? In particular, what will be the dffect of governnxent policies to reduce uncertainty tforexample. through the use tlt' price controlsl? 'What will be the effect of uneertainty concernilg the governfnentfs own actions (for example. uncertainty over future tax rates and regulatory changesl'? Such questitlns must be examined at the industry level if they are to be useful guides to policy in many practical situations-, therefore these chapters are the right place tbr ther study.

/Jp??f???1(7 /. tlltilillrilllt

?l tl

Illtltt-stll' trtp/zl/pt?f/rc

:!5 5

in an essential way on the joint presence of the two kinds of uncertainty examined in the context of a simple but general model in Section 5.

are

When all uncertainty is industq-wide, the multiplicativc factor X in tlle general demand curve ( l ) is constant, so we can just set it equal to I Then the industry's inverse demand curve becomes .

P

=

i'

DQt.

The aggregate shock J' will follow the geometric Brownian motion process dY

=

a )' fJ +

ty

1' dz.

On the production side. we assume that there is a Iarge number

of

risk-neutral competitive lirms. Each hrm can undertak'e a single irreversible investment. requiring an initial sunk cost 1. Once this investment is made. it yields a flow of one unit of output forever with no variable cost of production.

We embed such tirms in an industry by supposing that each unit of output is very small relative to the total industry output Q, so that each firm is an innitesimal pricc taker. When Q firms are active. the short-run eqtlilibrium price can be determined from cquation (2) above. As we discussed before, this is the simplest continuation of the model of Chapters 5-7 that serves our present purpose. Latcr and in Chapter 9. we introduce various generaiizations. whcre each hrm has some variable cost. short-run output variabilitya exit possibilities. etc.. where the shocks affect demand in more general ways, and wherc the industry has some imperfect

competition.

To set the stage for the competitive industry equilibrium. think of the static model. The industry price a single number is parafirm. The sum of the individual firms' optimum quantity re-

usual textbook metric to each

I'ltlll-bll7'I-t/l//k'/'lrl/l/? I11 an interval of t inle vvhen 11(.) nesv en try taktls plltce. ? is lixed. s() P is proportional to 1' and equation (3) gives

dP

a

#

tlt

+

f'r

13t/--.

A potential entrant observes this process. and interprets a lligh pricc as a signal of a high level of demand. Intuition suggests that tllere shotlld be an

upper threshold level P which, if reached. will trigger new entry. As soon as any one new firm enters. Q increasesp and price decreases aIong the demand culwe that applies for that instant. Thus. if the price ever climbs to P, it is immediately brought back to a slightly lower level. I n technical terms, the thrcshold # becomes an uppc r rejlectillg /?flv(!r on the pricc process.

Of course firms rationally anticipate aIl this. Thus the price process from the perspective of any one firm is 11 moditication 01' the geometric Brownian motion process of equation (4).That process is valid as Iong as P < /''. I-lowever, the price cannot go any higher; an upper rcflecting barrier is imposed a t #. In Chapters 5-7 we studied a monopoly firm's dntry tlecisilln when it faccd a geometric Brownian pricc proccss. Now we set, that a competitive lirm's entl'y decision is a similar problem- dxcept that thtl price prtlcess has a ceiling at P imposed by this firm's rational expectlttitln ()f (lther firms' entry decisions. So we mtlst rcexamine a firm-s optimal entry decisitlna taking intt) account this new pricc process. The solution is again charactcrizcd l)y a thrcsllllld /7*, stlch that the firm which we focus will choose to enter if the price rises t() 13*.Since a11 firms ()n arc identical. industry equilibrium rcquires this Iirm-s entry threshold t() equal its rational expectation of aIl (lther firms- threshold. So we start with a trial P, and solve one firm-s dccision problem to (ind its P%.Tlpe (ixed point of the process, namely, #* P, determines the industry equilibrium. Note how thtl intuition or guess that the equilibrium price process is a geometric Brownian motion with a reflecting ceiling has simplified the calculation. We need to look for the Iixed point of a chain of reasoniny that takes tls from one number to another, instead of a mapping on a whole function space of general stochastic =

processes.

not restricted upNvard by the rttllect ing hnltl-l-icl-. $ve svtlultl llavc l,( l'4 /. J f.z tll /''/(. is 11esl1() bctyvee rt re t tl t l'ltldiscotl nt rltttt t he risk- free ( intertlst rrtte in this case ) lnt.l tl'ltlexptlctetl lnlttt (,1* trtlvvt 11()1- 17 v'llell it is belknv tlle barrier-s The rellecting llllrrit)r at l'' cuts tlff stll'ntl ()1* t I1e tlpsitle potklntiitl for priot)s antl prolits- so the firnl's val tll.) nl ust 111 ftct l)e less t l1l 1) l'b/fq svil l Iind a forlutl Ia for tlle correctikln. Start a t 111-.inilqial price /2 < 1'. I11 11 stl l'liciently snlll Il i1) lA.'al t)l' inle (Il te t the price process is al rnost stl re to stay bel(.nvt he iliIlg. ve can carry out the usulll dynarnic prograrn rning or trolltillgent clllill'ls Ctnalysis to tll'ltllill the fa l'n iliar diftk re n tia l eq tlat io 11 Nvtlre

yvlltl

's've

.

.

''rllen

.e

where B is a constant fundamelltal quadratic t,-.1 ln

to be 1.f'.r : a

The Value of an Active Firm

The first step in the equilibrium calculatior is to hnd the value ofan established hrm facing a price process of the stipulated type. Since the future price path depends on the current price level #, the expected present value of the firm's future prolits is a function of P, which we denote by u( J.'). I1*the price process

p (p

-

antl

l ) + (1-

-

t

pb is t l1tt ptlsitive )p

-

1.

rtltlt

tll'

tlle

t)

.

p! > 1 tlntler the tlsual clllltlitillns ?' > t) itntl J p. (). This conforn'ls tt1 the intuition stkttetl abt3ve'. /'/J Nvtlultl btl tllc value ot' thc firnl in the absencc ()1 . the prictl ceil illg. s() 1) l ,/ rntlst jAe t jye kytly-rvqytjtju due to the ceiling. Thereforc B slltltlltl be negkttivc. 7-I1eillterpretatitln :l1s() explainswhy wc left out the ternl invlllving the negative rtdtlt ()1' the qulldratic. If the currcnt price 1.' is vcry snall. thtl barrier 19is unlikely tt) bc reached except in the far futtlre. Then the ctlrrcctitln arising frllnl tlle barrier shlluld /J become small. I-lowklver, a llegltivtl pllwttr ()1' gtlcs t() inlinity lls /' gtles t() therelre, shlluld that term not appeltr in the s()ltltion. zertl; To deterrnine the value of the conslan t B we lollk 11t the upper tlnd poi nt. From a starting point vcry close t(.)tl'ttl rellecting barrier P. the price is ktlnAtJst surc to fall during the next small time intcrval. If thtt value functitln lllts Ct negative(rcspectively. positive ) slope at tllis pllint. therc &vi 11be surc arbitrage profits (respectiveiy. losses) to be made. rulc th is ptlssibility (lut. we lmust tct

-rt)

have

, p(I ) .)

2.A

tlutttlrnAilltld.

p1 B P

#1- l

+ 1/t

().

(7)

'

/?ltII.J?y

256

Eqltilibriltnl

This looks like a smooth-pasting condition. but it is not a consequence of any optimization. Such a condition holds at any retlecting barrier for :1 diffusion PI-OCeSS.

b

Now we can solve for B from equation

(7)to get

ilyllalnic

f7t'/l///.')/-lf/?l111tl C-ollpcitivc 111(111.b.11y

the solution more fully. we must gt) into the details ()f tlle hxed-point process for constructing the equilibrium. Consider a firm contemplating entry. Write /'( P) for the value of its option to enter. As in Chapter 6. this takes the form i' X) Z PV' To understand

=

0', as explained earlier, the barrier cuts off some upside price potential, so the correction to the value is a reduction. Substituting for B into (6), We have Note that B


a; note that the Poisson death probability acts Iike a discount rate in achieving convergence, as we saw in the discussion of Poisson processes in Chapter 3,

Section 5 and Chapter 4. Section l.I. Asstllning the convklrgencc condition to we have p l > 1 as in the famil iar nlodels t)f Cha pters 5 tnd (). Our intuitive discussion in the introduction tt) th is chapter tells tls tlle reason why equation (27)is exactly analogous to the corresponding tbrmultts lor the monopolist's entfy decision in Chapters 5 and 6. When uncertainty is firm-specihc, a hrm that gets a favorable X does have an edge over its competitors. The favorable ZYis specific to this Iirm; if it does not invest at its zr and jump in. Theretbrc u positive value of once, a rival cannot waiting does survive, and the firm's optimal decision shows familiar inertia. Of coucse this is an incomplete account of the industry equilibrium until show that Q can be determined in a way that is consistent with the above we needs more steps in the development of the model. That story.

be met.

Ssteal''

4.B

I $-1

The Entry Decision

With A'*determincd as above. we can find the value of a potential firm that observes its current zr in an industry with Q active tirms:

I 1'-

'

'ir'

.'

2::.

ae

5k'''* .

This is also familiar from Chapters 5-7. The uppcr line is the optitln value for a waiting tirm,and the lower line is the cxpected present value (.)1-profits that the potential firm will get by immediate activation, net of tlle costs of the activation.Ofcourse the latter formula applies in the region where immediate activation is optimal, and the former in the rcgion wllere waiting is optimal. ln tkct .Y* is determined by starting with these two expressions and value malching and smooth pasting. That also yields

A plltential entrant can then calctllate its cxpected payllITC r! P' ( A'. ()) 1. Free entry conditioned using the knllwn distributitln 0f the initial draw 01zr.

ensures

C.v(V ( X. ? ) l

zr,

the zero expected net value condition ---p3

value function

P' (A3. Figure 8.4 shows a typical value function and its shift as changes. The intuition is that if there are more active firms in the industry, Q

any one new entrant perceives :1 smaller prospective protit flow and therefore requires a higher firm-specilic shock to prolit before committing itself to activation.

(3())

-

where R is the initial entry cost. We just saw that the left-hand side is monotonic in )-, therefore this equation determines the equilibrium Q. As an example, supposc the distribution of the initial draw is uniform zr over the interval ((). ). If the activation thrcshold X* turns out to be larger than

A highcr Q Iowers D( Q).Theretbre from (29) it Iowers z1( (2),and from (27) it raises A'*. Finally, from (28)we see that a higher Q Iowers the whole

R

=

A ( 0-.) X

(30) is

/ (1 + #I )

=

R

.

(but

In simple cases, for instance, if D Q) is isoelastic, this admits an explicit < X, we get solution for Q. 11. 'e
.dtj

#(X)d/1

.

Iilillt-itll

?1

il !

tl

(75,tll, cz//ib,e111(l/-/? )'

For the second condition. observe that the firms that hit becomc ltctive this and are lost to distribution. Consider the segnlent of the space just to # the Ieft of x centered at dh For t his segment- we must modi tk equation the balance (32)above. It does not get any incomers from the right. .r*

qNj

dhldh

Dv?lf? ? ?: ic.df7I

-

tx+

J1) dhjdh

.

.j-

*

.r

Y

o

%; s

c,A-

+

x

=

-6)

,jOo

Entry to x

=

*

-

x

dh

+

x

i

* -

dh

x

Exit from x O

o >

qN(1

-

A.dl)

j

c

N. dt

@

4

o R =

.

o

'>

Since these firms activate in the time interval t// the rate of activation is 2 N - 2 t.y 4/ ( v. ) sve illustrate this calculation for a case ttmenable to an analyticul solution. Suppose the distribution ot zr is uniform over thc intelwal ((). Then .r log has an exponential distribution over (-x. where :Iog 7t Thus ,

.!.

*

-

.

eu ,r).

.

=

PN(1

.

bxldh

-

.-*.

'Vo

%o %

cs

*

O

%z &

%

+

=

-'*

x

cyN

Similar simplification now gives 4 (-r) ().Letting (//1 go to zero, we have the condition 4 (-r ) 0. 'We can also calculate the rate at which waiting Iirms hit and so become active. These are just the fraction p ( l l dt ) of the N 4 (.r J/?) J/l tirms located in the segment just to the le of Using Taylor's theorem again, the leading term in this expression is

+

i

dh

.

o oo&

4oo +

-

-

therefore

VJ

R

-r

=

.

.

X d1)

.

fldry'. Slltjt. and Er/

(?f

G (-r)

I'p/ikg Firnts

where the last term is a particular solution of the full equation. and the hrst two terms arise from the general solution of the homogeneous part. The constants Cl and Ca remain to be determined, while pl and p'a are roots of the quadratic

2-

1 r.r2

y

2

2

-

p

y

-

=

0

.,

thus one is positive

and the other is negative. condition for determination of the constants Cl and (:2 is found The from the consideration that the total mass of waiting hrms, that is,

srst

*

.

.-)r

=

,

(')(:?l

txldh

Figltre 8.5.

-

yr

17

$ (A') dx,

-X

must be hnite. That helps rule out the negative root in the solution of the homogeneouspart, as a negative exponential would go to inhnity as-r -..+

-co.

=

g(-r )

exp ( x

=

Ji-)

-

.

.

We will simply assume that the rest of the parametcrs of the problem arc such that the activation threshold x* turns out to be Iess than leavjng the other case,which is simpler because none of the entering hrms activate immediately, to the reader. A particular integral of (33)is easily verihed to be ,

4()(.r)

=

ex-k

/

(

+ w

-

2

v2

) .

T'he denominator of this must be positive for it to make economic sense. This just says that the quadratic expression (2 evaluated at y 1 is negative. Then the positive root yq must in fact exceed unity. We assume this, and for typographic convcnience write that root as y. Then the general solution of (33)is =

$ (A')

=

C

CY''

+

/()(-r)

.

Illdusly

/(a-*)

The constant C is to be determined from the condition

=

ltl/7 I tr f 1. In other words, at the threshold that justihesestablishing the marginal 5rm (IN when there are N actual hrms, the marginal expected value of so doing exceeds the cost of the action. The reason is also familiar: the excess is just the opportunity cost to society of exercising the option. Thus the general model in many respects replicates the analysis and the results of the earlier simple case. However, one result does not sulwive; only in very special cases will the threshold level found from (1 1) imply a constant price D( Q, F'). Generally, the threshold price will be a function of the current number of lirms, N. Once again, since there are no distortions or market failures, the social planner's problem yields the same result as would a direct solution for the competitive equilibrium, but now we see te merit of our indirect approach. u/v

If the entry thresllold price is a ftlnction W(N), then tlltt endogenous pricul process of the equilibrium is much mtlre complicated. Its ceiling (reflecting barrier) shifts as new Iirms enter. To tind the equilibrium. we mtlst solve a fixed-point problem in a complicated ftlnction space. The social optimality problem remains :1 simple dynamic optimization calculation. There is more. The tbrmula ( l l ) conceals an important and simple principle that governs the entry decisions of competitive tirms.Suppose N firms are already in the industry, and the next marginal firm is contemplating entry. Suppose it has rational expectations about the stochastic evolution of F, but it assumes itself to be the Iast entrant, ignoring the fact that other tirms will enter after it if and when F rises to suitable Ievels. It will calculate the expected present value of its prolits as X

p( N. F')

zr ( N. F, ) c

C

=

()

enter, and tind

(N. 5 ) e-r dt

29 l

-''

?

Jf

=

Fv( N. F )

.

It will carry out the usual calculation for the value /'( N. 1') of its option to

X =

/??.1/?tp-/t',t'? C-t.ll'tll'tititlll

t?,!t/

'elltio'l

.

Differentiating under the integral sign, we get Fx ( N. l')

fp/cv /rl?t.v7

.l'N

1')

=

bNj F#'

where b N) is determined jointlywith the entry value-matching and smooth-pasting conditions /'( N. F )

=

t? (n

.

1/-)

-

1.

.

( N. F )

threshold

=

t.' y ( N,

F ( N) from the

1/)

.

It is easy to check that. on eliminating /J( N). these reduce to equation ( l l )p exactlythe equation for the threshold in the social optimum cum competitive equilibrium. where each frm recognizes the possibility of futtlre ttntry and of the effect of such entry on the price process and on its own prolit tlow. In other words. each firm can make its entry decision by finding the expected present value of its prohts as if it were the Iast h'nn #ItIJ would enter this industry, and then making the standard option valtle calculation. While the 5rm should entertain rational expectations about the stochastic process other lirl'n.' enlr.v t7trc'.s'l'tlrly. Not of Y, it can be totallv myopic in the matter W* would reach the if it correctly anticipated their only does it same decision as it with a far simpler calculation. entry decisions, but it gets to this answer will When the Nth firm pretends that it be the last one to enter this industry,it is ignoring two things. First, it is thinking its proht Ilowwill be given by the stochastic evolution of n'N. F) as )' changes, holding N fixed. Thus it is ignoring the reduction in the upside of its proht caused by the subsequcnt entry that would occur in response to F rising to new heights. Other things equal, this would make investment seem more attractive to this firm than it '

Ill6ittnntt?lkTylf/llrl/lzy

would if it took entry into account. However, it is also ignoring the fact that the prospect of future ent:y reduces its value of waiting. lt is pretending that it has the ltlxury of postponing its decision, and acting as though there is a positive value to its option to wait. Other things equal. this makes investment seem less attractive than it would othelwise. These two effects exactly offset each other, so that in this case, two wrongs make a right in the irm's optimal choice. We noted a special case of this in Chapter 8, Section 2(b), where the exact level of a price ceiling was immaterial to the calculation of a firm's entry threshold price, as long as the ceiling was at Ieast as high as this threshold. Now we see the general version of the effect. This remarkable was discovered by Leahy (1992).

property of the

competitive equilibrium 1.C

Implications

for Antitrust

and Trade Policy

We have seen that the standard correspondence between social optimality and competitive equilibrium holds in the kind of model we have constructed so far. As long as there are no externalities, and the relevant risks can be traded in efscient markets, dynamics and uncertainty are not by themselves sufhcient reasons for policy intervention. True, the market outcome shows a great deal of inertia ranges of shocks where no investment takes place-and in popular perception the hesitancy of firms to invest might seem reason enough for the government to intervene and speed up the pace of investment. However. the social planner would not want to invest any faster. inertia is optimal-a There are other features of our stochastic dynamic market equilibrium that are often thought to be inetsciencies requiring corrective policies. That is because the conventional textbook view of equilibrium is static, based on the Marshallian long-run picture. In that view, srmsenter an industry when the price rises above the long-run average cost. and exit when price falls below the average variable cost. A price in excess of long-nn average cost is then regarded as evidence of entry barriers. calling for antitrust measures. Similarly. a price below average variable cost is often regarded as a sign of predatory dumping, usually by foreign tirms,and thus ajustitication for trade

sanctions. We want to emphasize that such conclusions are liable to be fundamentally mistaken, because in reality economic conditions are never tranquil. It is essential to think of the industry equilibrium in such a situation as itself changing in response to evolving uncertainty. that is, a stochastic process. The natural competitive dynamics of an industry in the face of ongoing uncertainty of the industry has features that the static will have phases when a theory would interpret as deviations from competitive behavior. This idea of ''snapshot''

/2:7lic'b'/?l tewell

Jp? l t? l l (1

J/??/7t! r1Lzc/

>t)l

??/?

t'/

il

t)? t

regarding competitive equilibrium as a stochastic process has become quite common in macroeconomics', it is time for industrial organization theory and antitrust policy to recognize the same reality. Suppose such an industry comes to the attention of policy authorities at when the price is between the Marshallian long-run Ievel Jl) and instant an the equilibrium threshold W.They see established tirms making supernormal profits. but no new entry taking place. Using conventional microeconomics or industrial organization theory, they would suspect the presence of monopoly power or entry barriers, and might take antitrust action. That would be wrong; the process viewed as a whole is fully competitive, long-run expected returns are normal, and the equilibrium is socially optimal. Likewise, if the price is below thc minimum average variable cost. that need not signal predatory dumping by the hrms that are incurring the losses. Remember that if market conditions are sufsciently volatile, the Iower threshold price at which firms should optimally exit will be well below the minimum average variable cost. Thus in such a situation we may simply be observing hrms rationally riding out a bad period to keep their sunk capital alive.s The numerical example of the copper industry in Chapter 8 showed us that the market price is Iikely to be outside the Marshallian range tbr most of the time. Thus not just snapshots but time series of a tkw years' duration may also be insufticient. Only by observing the evolution of the industry for a very long time can we hope to spot genuine departurus from the competitive norm. Basing policies on snapshots can result in scrious mistakes despite the policymaker-sbest intentions. In the picture presented above. prices above long-run averagc costs and the rcsulting temporafy large profits are merely due to the swings of demand in a competitive industry that permits only a normal protit as a long-run average. However. governments oen t:y to control the supposedly excessive profits of lirms, and protect consumers from these supposedly excessive prices. Urban residential rent controls are a common instance of such policies. In Section 2 we will develop a model that describes the effect of such policies on investment in a true dynamic context. We will find that price controls can depress investment and thereby reduce industry supply to such an extent that the averaj;e price in the long run actually goes IIp. Thus a policy of price

5Of course, cven if there wcrc no uncertainty whlttsocver

over future market conditions,

pricc could be bcltlw the minimum average variable cost if lirms are moving down the steep part of a Iearning curve. Finally, price could bc below the minimum average variable cost because of (nf volatile economic ctlnditions and a learning curve for example. in the case of a combination semiconductors.

Illtlltstl E-t/l//ib/.?l-ii/pl?

294

'

controls can have perverse effects oven from the perspective of the group it is designed to help. Conversely, if the government introdtlces price floors to support (irms in bad periods, Iirms will react accordingly. They will ttnter the industry in greater numbers, and that can then makc the bad times even worse. Ieading to a large drain of government revenues. Agricultural price supports, a Iong-standing policy fixture in the United States and in E urope (throughthe Common Agricultural Policy), often have such eflkcts.

1.D Market Failures and Policy Responses We found above that uncertainty and irreversibility do not by themselves constitute market failures that warrant government intelwention. It is important to emphasize this point, because public policy debates often err on this matter. The existence of adjustment costs is itself often thought to be an economic problem requiring policy action when industries and workers suffer adverse shocks,especially those arising from international competition. Calls for government action in such cases should be based on some other genuine market

t'ailure. This does not mean that markcts always work pcrtctly

and government

intervention is never called tbr. Some forms ()f market failure do arise naturally in a dynamic environment with uncertainty. In particular, markets tbr risk are often incomplete. The rcasons for this hav: to do with asymmetric information or the sheer complexity ofcomplete contracts. Labllr income risk is particularly difficult to insure against. When such separate causes of market failure coexist with our basic issue of irreversible choices under ongoing uncertainty, the two intcract to produce some new and interesting kinds of suboptimal outcomes. We do not have the space here to provide a detailed description of models

that demonstrate such market failures. Therefore we will merely outline the economic intuition behind them, and refer the interested reader to the work on which our discussion is based. Dixit and Rob (1993a,b). The failure of risk markets is most frequent and natural

in the context

of labor income. Now labor supply decisions. for example. educational and occupational choices, also involve substantial sunk costs or irreversibilities, and must be made in an environment with ongoing uncertainty. Thus they are in essence investment decisions, and our general framework is naturally suited to their analysis. That is the setting we use. Consider an economy that offers two alternative occupations, which we call sectors and which may be dlfferent industries or cities. The relative will attractivenessof the two fluctuates over time, for example, because of random

technologic:tlslltlcks. A shil't l'n'nl ontl sector to t he t)t Iler reqtlires solntt stlnk tlirtrtll-nstances) retraining. travei. costs that can inclutle (tlependingo11thtl ptlrchase antl sale of :1 hotlse, moving hotlselloltl effectsa til-ne and eflrt involved in mak' ing nesv friendss and many othtlr tangible and intangible costs. The refore an individual will not make the ssvitcll un less the re lative attractiveness of the other sector is sufticiently higll to offse t not jtlst the normal return on these costs. but also the option value of the stattls qtlo. The relative price betwecn the outptlts of the tsvo sectors is determined in equilibriul'n by the standard equal ity of demand and supply. Call the scctors I'nobi lity. the inconles of worktl rs in 1 and 2. l 11 the absence of any Iabor each sector move in proportion to the tochnological productivi ty shock lbr that sector. This comprises the income risk that people face. Now suppose sector 2 receives a favorable shock. Wllell :11,1individual nloves fron; sector 1 to sector 2 in responsea this raises the output of 2 and so lowtlrs its equilibriunl price relative to 1 That dampens the initial increase (')f incomes in sector 2 relative to I In other words, when one person moves to the favored sectorthat reduces the extent of income risk tced by :111 the othtlrs in both sectors. If the risk were efhciently allocatetl using compltlte markets. this price change would be ()nIy a pecuniary externality. I lowevcr. because risk markets arc incomplete. the redtlction in risk has rea 1(2 ffects: it is lt bencficial cxternality conveyed by the mover ()n aIl (lthers in society. Since the mover does not take this stlcial beneiit intt) acctltlnt in his private calctllatitln. the resulting labtlr mobility is subtlptimally l()w. A governnlent that can ellcourage some extra mobility (perhaps by subsidizing jol7changes and any retraining that might be involved) can achieve a better outcome for sllciety as a wll()le-t' This mcchanism operates via the endtlgenous relative pricc. Thdrefore it is less significant the morc elastic is demand. In the limiting castl t)f a small economythat is open to world trade. the relative price is determined (perhaps as a stochastic process) by world market conditions- and the shi of Iabor across sectors in the economy has no impact on this price. Then the degrce of mobility in equilibrium is the second-best optimal given the Iimitations on the risk markets. Unless the government can devise new ways to share risk, it cannot improve upon the outcome of uncoordinated private choices. Various kinds of taxation do provide indirect risk sharing. In an open economy, trade taxes are such an instrument. Suppose the government levies .

.

'' In thc light of this analysis, it is irollic that govcrnments tlften pursue policies that actually reduce labor mobi Iity. l'tlr exam plc, public housing p()Iicics (hat favllr lllng-t imc residents of a city. I t is then nttccssary tl) u ntle rtake a variety ()1* socilll insu rll ncc meltsu res that amelioratc t hc grca tc r incomc risk t hat t hc Illcked-i n individulkls I'ak7tl,

296

11J( ltt-bntjlf/l

tilill rI d?z:

a tax (or offers a subsidy) so that the domestic relative price is not the same as the world relative price. Now the incomes in the two sectors are tied to the domestic output prices, and are furter affected by the distribution of tax revenues (or the contributions to the subsidy proceeds). This changes the allocation of risk. Then it is not hard in principle to find a policy that will improve risk bearing and thus generate a social improvement. However, the precise nature of the policy is highly situation specific; a simple recipe such as an import tariff cannot be guaranteed to be benelicial across ny broad range of circunstances.

2 Analyses of Some Commonly Used Policies Governments do employ some policies that alter the uncertainty facing firms and consumers in their economic decisions. The policies are usually motivated by some immediate political or economic reason a belief that rms are charging excessive prices, or a perceived need to stimulate investment. Now the simplest economic analysis teaches us that policies implemented with one aim often have other side effects', these are oen surprisingly detrimental. This problem is all the more severe in the stochastic dynamic environment that is the theme of our book. In this section we illustrate this by analyzing two such policies. 2.A

Price Controls

Governments often attempt to reduce price volatility by imposing controls. The immediate aim is usually to protect consumers from excessively high prices, as in the case of urban rent ceilings or the U.S. natural gas and oil price controls of the 1970s, or to protect the incomes of produccrs, as with the European Common Agricultural Policy and the plethora of agricultural price supports in the United States. Economists are generally critical of such measures. They emphasize the harmful side effects, for example, the scarcity and poor quality of rental housing in cities with rent controls, and the wheat mountains and wine lakes in Europe. The economic analysis underlying these arguments is the standard textbook supply-and-demand framework. If the price is kept below the market-clearing Ievel, there will be too little supply, and if it is kept above, too much. However. this is a very static picture, while the effects in reality are mostly dynamic, operating through the investment decisions of landlords or farmers. Our approach to investment under uncertainty permits a richer analysis of the effects of price controls, and yields a deeper understanding of their side effects. '

Policb.f/lfclaf'c/llt?ll

tl/lt/

Ilnperjltck Ct-aipllpfizf itl'll I

We continue to use our basic model. To recapitulate, each firm has the cnpacity to produce one unit output when active. a sunk cost of investment 1, and a variable cost of production C. For simplicity of notation. we assume risk neutrality and the riskless discount rate l'. The industry demand curve is

p where the aggregate shock

=

y DqQj,

F follows the geometric Brownian motion

JF

=

a J' t// +

t:r

F' /z.

The simplest and unihed treatment of price ceiling and lloor policies can be

given if the industry' equilibrium involves both entry and exit. Therefore we adopt the model of Chapter 7. where 11 firm that is not producing must lose its sunk investment. Then we know that the industry equilibrium in the absence of any controls is characterized by an entry threshold and an exit threshold. For reasons that will be evident soon. we write the tbrmer as S and the latter as .

-f5

Now suppose the government imposes kl ceiling and/or a tloorE on the price. We take this to mean that if at the cxisting level of Q and F in the industry the price given by ( l2) that would clear the market exceeds W.then the Iirm can collect only Wand the excess dcmand is rationed. If the price that would clear the market is below -P. thdn the hrms reccive -C,and government becomes the buyer of last resort to absorb the excess supply. We assume in the background that such government purchases are either destroyed, or exported or given to other countriesa with no future feedback on demand or supply in our economy. (This is broadly true for European agricultural policies.) Nor shall we be concerned here with the eftkcts on the government budget. ln Chapter 8 we studied an aspect of a firm's entry and exit choices when such policies are in effect. In fact we found that as long as exceeds the natural entry threshold S, the actual Ievel of > has no effect on S at all. A lower ceiling reduces the value of investing immediately and thc value of waiting by equal amounts, leaving unaffectcd the trade-off that governs the investment threshold. A similar situation holds when the lloor L>is actually lower than the natural exit threshold S. Now we must allow the ceilings and floors to bind, and see how that will affect hrms' choices and the industry's equilibrium. Forsimplicity of exposition we do this separately for ceilings and Iloors; then the two treatments are in principle easy to combine into a joint analysis. First suppose the price lloor is too Iow to affect the picture, say, ..P 0. but ceiling the sometimes binds. That is, sometimes the price that would that -#

=

-15

298 clear tlte nlarket exceetls --#.slv will call this llyptltlletical markct-clearing price the slladow prkt?. sv'easstlnAe that the (irms. lMlt 11actual and potential, can obselwethe degree ofscarcity (landlordssee how many prtpspective tenants are going around looking tbr apartmcnts, or read about it in the newspapers), so that they can calctllate the shadow price. This will inlluence their decisions. Even though :1 unit of output currently gets only the controlldd price P, if the shadow price is mtlch higher, they know that the control is Iikely to stay binding, and therefore the price is unI ikely to fall below M1 for 11 Iong time. Therefore a higher shadow price will make investment more attractive even though it does not alter the actual current protit Ilow. Then there will be a threshold S that will trigger new investment. As the eeiling -1 is gradually -%5 will rise. When the ceiling approaches lowered, the shadow price thresllold the Marshallian Iong-run average cost ( C + 1- /). the shadow price threshold will approach inhnity if the controls are going to allow only a normal return. hrms will invest only when they are assured that this state of affairs will last .

forever.

are the eftkcts ol' reaching the linAits of tlltl rnnge- and the const:lnts rernain Bb 1: to be deterlu ined. To help tlo thtt- %9< httve the value-matching slnootll-pasting conditions at the tlntl'y tllreshold: and

two tcrnls ,

Final Iy, the value function must be con t intlously di ftkrc n tiable lt t t I1e P where the tsvo regi mes nleet. Tllus the expressitlns l'or p ( frol'n ( l 3 ) alld ( l 4) evaluated at P must be equal, and the same must hold for the corresponding expressions for v' ( 5') ln all we have six equations to determine the two thresholds Sns tnd the four constants z1l z1:, B3 Bz. These need numerical solution. which we soon

J ) / g1og S

-

log

1.

..

Now we turn to the numerical simulation. Since price ceilings are more likely to be applied to an industry in which demand and the free market price 0.05. which are 0.02. We take t.r 0.2 and 1have been rising, we let a used earlier values Finally. of the have in chapters. 1 we set C typical we variable cost), and units that normalizes 2, / the average (this is a choice of which makes the Marshallian long-run average cost 1.1. Table9.1 shows the results. In the absence of anycontrols (orequivalently, when the ceiling is at an irrelevantly high level. in effect at inhnit) the entry threshold is 1.5324 and the exit threshold is 0.6790. As the ceiling is lowered, the most important effect is on the entry thrcshold of the shadow price. lt increases gradually at first, but then more and more rapidly. By the time the ceiling is halfway between the natural thrcshold and the Iong-run average cost, the effect gathers momentum. The long-run avcrage t'requencywith which the ceiling is binding also increases. The exit threshuld also rises, but this is quantitatively much less significant. This analysis gives us a clearer idea of the mechanism by which price controls discourage investment. Under price controls. firms wait to observe a =

=

=

=

=

Table 9.1.

(Parameters: a

=

0.02.

Perhaps the most interesting and surprising result is the eftkct of controls the long-run average price. As the ceiling is lowered. this price tcreases. In on other words. the policy fails to achieve its intended goal of reducing the prices to consumers: its etct in the long-run average sense isjust the opposite! The reason is that while the incidence of prices above the ceiling is cut off. the proportion of times the price stays at the new ceiling is increased because of the reduction in investment and therefore in long-run suppiy. In Figure 9. l we see this shift of the distribution of prices: as > is lowered, the density to its right disappears, but there is a larger point mass of probability at P, and some of the left tail also disappears. The effect on the average price is ambiguous from the picture. but from the numerical simulations we find that the effect j

ti

s perverse.

F'

The analysis of a price lloor follows similar steps. As we raise L from 0 to that the exit threshold S would have in the absence of any policy, the level the Iloor remains nonbinding. Beyllnd this point, it starts to bind for the relatively Iow states of demand. That in turn changes the entry and exit thresholds. As the price floor rises. the downside risk of investment is reduced. Tllerefore firms are more willing to enter and less eager to Ieave, so btlth thresholds fall. When the price lloor rcaches C. the exit threshold s drops to zero-, incumbent firms that have the guarantee of their variable costs being covered will never leave. When the price tloor is between the variable cost C and the Marshallian

Ejfects clfh/ic: Ceiling t'r

0.2.

=

1-

=

0.05, I

=

2. C

=

l

.)

Point masses of probability

Density W

(>3 1.50 1.40 1.30 1.20 1.15 1.12 1.l 1 1.10

'J

a

0.6790 0.6790 0.6795 0.6823 0.6930 0.7066 0.7212 0.7280 0.7364

.5324

1 1.5338 1.5608 .6573

1

1.9948

2.6123 4.0995

6.0406 (x;

Probability

'l

/21

.0484

0.()000 0.0273 0. 1307 0.2736 0.4802 0.6275 0.7467 0.8001

1.0658 l

1.0000

1.1000

l 1

.0485

.0494

1

1.0517 .0564

1 l

.0607

.0692

f Figure 9.1.

Effect of Ceiling on Price Distribution

Illtllt-tly'1Jtyl///??-l//?y

Policv /?I/f?n'tv?/Ib?l

tlllt

lltlperfec Ctp/en/?t.prf?'f?pl Table 9.2.

Iong-run average cost ( C + r I4, the exit threshold s stays at zero, while the entry t hre shold S remains above (C +r l4. This is because (irms contemplating new investment still need some periods of stlpernormal prolit to balapce the periods of below-normal protits. Finally, as the price lloor rises to (C + r/). the entry threshold falls to that same Ievel. Tt characterize the equilibrium in more detail, we proceed as before. First suppose the price floor -C.is high enough to be sometimes binding, but below C. When the shadow price S Iies in the interval (.t, -P). the Iirm's profit C) and its value is flow is

.2j! 0 0.75 0.80 0.85 0.90 0.95 0.97 0.99

.l..t

-

''!).

ln the interval (Z. the floor does not bind, the actual price equals the shadow price, the protit flow is (S C), and the value is -

f./'ects

t#'

:

j

(1.7462

1

0.7462 0.7398 0.7 l84 0.6722 0.5735 0.4982 0.3543

1.6668

.6667

l

.6665

l

.6645

.6584

1

1.6433 1.6323 I .6090

303

Ihice J--lo0l.

I'rtltnkltxilily

()

0.0 126 (). l 804 0.35 l 1 0.529 I 0.7238 0.8 l 85 0-916()

r7l1' I .()3()9

l 1.()3t)9 .4)34)3

1

1

.0287

l

.0256

1.02()l 1.0l 64 l

.()(')9t)

Then the vaiue-matching and smooth-pasting conditions at and S, and the continuous differentiability at Z, complete the solution. We can also compute the probability that the lloorwill bind in the long run. and the Iong-run average price, as before. Next suppose the price ooorZ Iies in the range (C. C + r/). Here ... (). For S in the range (0. f.), we get =

rlxl

=

jl

-

C)/r

+

S#'

zz!

.

This is just like the above solution except that only the positive root is retained by consideration of the Iimit as S goes to (). For S in the range (Z. X). the solution is exactly as above. Now the value-matching and smooth-pasting

conditions at S and the requirement of continuous four equations to determine the three constants threshold S.

differentiability ./dl

,

at Z give #1 and B? and the ,

Calculation of the Iong-run average price is different in this case. Since

firms never leave once they have entered. the long-run number of lirms goes to inhnity and the distribution of the shadow price has the whole probability piled up at zero. Then in the Iimit the ceiling is always binding, and the average price equals the ceiling price.

We show a numerical simulation in Table 9.2. The parameters are as in the ceijing case, except that since price qoors are more likely to be prevalent in industries with declining demand, we set a Once again the most remarkable feature is the decline in the long-run average price as the lloor is raised from its Iowest nonbinding Ievel up to the variable cost C. The reason -0.02.

=

is the same: the price is higher in the Iowest states of demand. but this is more than offset by the effect of the induced extra entry leading to a lower price in the higher states of demand. The effect is quantitatively somewhat more modest, but it is unmistakable. Thus the Iong-run effect of the price Iloor policy may be harmful even to those it is intended to help. The observation that farmers in many countries are so dependent on farm price supports and yet so dissatished with their outcomes may have this explanation. Of course once the price Iloor rises above the variable cost Ievel, it is always binding.

Then the average price equals the ooorand rises one-for-one with it, but this is a rather pathological state of affairs. 2.B

Policy Uncertainty

Governments can not only deploy measures

to reduce the uncertainty

facing

potential investors. they can also create uncertainty through the prospect of policy change. This feature of the polic'y process is particularly relevant in the United States, where changes in t:tx policy are constantly suggested and

/pltl/-f?)/iltlililzriltll

304

even after a specihc new tax legislation is introduccd in the Congress it goes through months and even years of debate and modilication. It is commonly believed that expectations of shis of policy can have powerful effects on decisions to invest. Our theory of investment under uncertainty can

altd fpn//c#zc'/ Ctlptqitil Policy fnftrnzfswff-vl

Over the

discussed, and

examine the validity of this belief. However, policy uncertainty is not likely to be well captured by a Brownian motion process; it is more likely to be a Poisson jump. We now develop an example, based on Metcalf and Hassett ( 1993), to show how such policy uncertainty affects investment. We adapt our basic model of Chapter 6. Begin with a firm contemplating discrete investment with sunk cost /. The project will produce one unit of a flow forever after with no variable costs of production. The output output price t'ollowsthe geometric Brownian motion

(1P

u

=

P dt +

tr

P Jz.

1a. Then the The firm is risk neutral and ?- is the discount rate. Let J when the initial completed project. value from of revenues a expected present is #, given P/t5. is by price The policy instrument we consider is nn investment tax credit at a given rate 9. When this policy is in effect, the sunk cost of investment to the 5rm is lowered to ( l 0) 1. However, the government can switch between two policy regimes, one wherc the credit is not available, and the other where it is. We will identify the state without the tax credit by the subscript 0, and that with the tax credit by the subscript l The switches between the two policy regimes are Poisson processes. Starting with a state when the credit is not in effect, the probability that it will be implemented in the next short interval of time dt is l dt and when the credit is initially in etect, the corresponding probability that it will be withdrawn is () dt. Intuition suggests the following form for the tirm's investment policy. Over an interval of low values of #, say, ((). /71), the firm will not invest irrespective of whether the credit is in etect. Over an interval ( J$ f1)). the hrm will invest if the credit is in effect. but if not, the hrm will hnd it preferable to wait in the hope that such a policy will be implemented. Beyond Jl). the prospect of immediate revenues will be so large that the tirm will invest irrespective of the current policy. To determine the thresholds PI and Jl), we procded as usual. Consider the net payoff to the investment opportunity as a function of the price, 1$)(P) ( P) when the credit is in effect. We obin the absence of the credit and 1zr1 tain expressions and equations for these, and conditions they satisfy at the =

-

-

range ( Jl3.x),

3()j

the Iirm always invests at once, so we have

/'()(P)

Pj

=

/.

-

and F1 ( P)

Pj

=

(1

-

0j 1.

-

(18)

Over the range (#l C)),the firm invests at once if the credit is in effect, 1z-1 so ( #) is given by ( l8) as above. However, P'4)(P) is more complicated. Over the next short interval of time dt, with probability l l the credit will be implemented, the 5rm will invest. and its value will become lzrl( P + J P). Othemise its value will be )'(,4 # + t P4. Thus ,

r)( #)

e-rd'

=

(

l

dt

'tp'l

(P +

tI#)J+ ( l

l

-

tlt,

z7(K)(P

Expanding the right-hand side using lto's Lemma. retaining in dt. and simplifying as usual, we have

+ dP4j

) .

leading terms

This is the usual dift-erential equation for a hrm waiting to invest, except for the additional term involving l which captures the expected capital gain from institution of the credit in the immediate future. Note that in the range of prices ( Pl Jl)). we know the expression for 1z'1( P4. Then we can obtain the general solution to the differential equation: ,

,

.

.

.

thresholds.

) ( #) lzh

=

B3 P#t ' '' + Bz 'Vt

where Sl and Sa are positive and negative

1':

+

/5(

lP +

l -

l)

(1

-

()) l

r + 2.1

constants to be determined, and roots of the familiar quadratic

v

J( 1)1 and J( l )a are the

The seemingly complicated notation for the quadratic and the roots serves to distinguish them from some other related expressions and roots that will

appear soon.

Finally we address the region (0, 771 ). Here the firm waits in both policy regimes. and each regime can switch to the ther. Following the same steps as above that 1ed to the lifferential equation (19)for the waiting Iirm, we have a pair of differential equations:

t

-

t;r2

P1 pe'?(#) + (r ()

/7.2

p2

v''p) l

+

(/.

-

-

j) # pr't'j () J) #

-

''p)

l

-

r

K)(#) +

r gj

(#)

+

l

(#) gprj

o gIr)( /a)

-

-

p'()(#)j

=

0,

prj( p)j

=

().

bldltxtryf'tr/l/l.?nb//7

3()6

that can be solved easily. Detine two new

These yield two Iinear combinations

functions, say,

:

l (P )

1z',( P4

=

-

lz))( P4

.

'C'f (L>()l ,1/? t'I ili()ll lic-' /zl/ c,?7 l Jilnltt/ 11(1 11?lJ?(,?7' fa/? .el

smooth-pasting condit ions. F()r 1,$)( P) t 11is is not 11 klecision t llrtlshtlltl. but l Ilc function has to be ctlntintlotlsly different itl7le across it- s() (2l ) and (.2()) slltltlltl have equal values and derivatives tllttre. Finallya for tlle optinltllu clltlice ()t' the threshold /$, the expressions for lz$)( l3j to the Ie (2())and the rigllt ( l 7) should satist the value-matching antl srflooth-pasting contlitions. In all Nvt) have six equations to deternAine the thresllolds /31 C) and the l'otlr collstants ,

1:! 1

:!:

(7.2

#2

v'' :b.

(,.2

P1

P) + (r

v'' tl

-

P) + (?. j) P Iz-?(P) -

?.

-

il

K,( #)

=

0,

St B?, Ca , and Ds We illustrate tlis calculation with a numerical sklltltion. Xlke a (). = tz 0. l and r = 0.05.va1ues that were fairly typical in our earlier calculations. Then J r a = 0.05. Let / = 20: this is just a clloice of tlnits that gives thc Marshallian investment threshold l = l and allows easier interprctation ot' the numbers to come. 'With these numbers. and no tax credit policy at aII, tlle optimal investment threshold would be #* 1 Thus the normal tlption value premium over the Marshallian thresllold is i).37. We consider a l o-percent (). I If this credit were alwfys tax credit. s() 0 in effect, the threshold would be lowered by l() percent to l l I-lowevttrwhen the two regimes can switch back and forth in a Ptlisson process. botll thresholds are affected. 'We examine the effects on the twl) thresholds, /ly when the credit is not currently in effect. and /3l wllen it is currently in effect. as the probability rates of enactment ( ) ) and removal (&)) val'y tlver a range from 0 to ().5 in each case. The results are shown in Tables 9.3 ktnd 9.4 First consider thc situation when the crcdit is not currently in effect. Table 9.3 shows that as the probability of enactmcnt I witllin the next year increases, the threshold /$)increases. This is intuitively klbvious; the prospect of a reduced cost of investment increases the value ofwaiting. The impressive aspect is the magnitude of this effcct. The usual option value premium over the normal return ()f l if the credit had never been mentioned. was ().37., a 3o-percent probability of enactment of the 1(l-percent tax credit more than doubles this premium to 0.8 l Remember that the tax credit had thc aim of lowering the premium to 0.23. However, while the policy is being discussed and enactment is uncertain. the effect is to deprcss investment very .

.

=

)'J( #)

j) #

(?. +

-

.b

()

+

!

) 1Z.$.(P)

=

0.

=

Each of these equations yields a solution with powers of P that are the roots of a familiar quadratic equation. ln each case we have an interval of # that extends to 0, so we include only the positive riot. Thus U ( #)

Ca P /l(t'3!

=

-

.,.

.

( #)

I). #/lt2)l

=

,

.&

-

.

.3702.

=

=

where C:, and D.s.are

constants

Q(0) - 1a

v2

p(0)I is the positive

to be determineda

pp

-

1) + fr

-

j)

p

-

r

=

root of

0,

and /(2) ! is the positive root of

.

.233

.

.

J4())comes from the quadratic 24())where the constant term includes neither () norl p 1) comes from the quadratic k?(l ) whose constant term has l alone, and J42) comes from the quadratic Q(2)whose constant term has both () and l As beforea the subscript l refers to the positive root and 2 to the negative root. Then the relevant roots satisfy the following chain of inequalities: Mnemonically,

,

.

,

.

/3(2)1> /(1)1 > #(.) l

>

1> 0

>

p( l )a.

With this notation, we can write down the solutions for 1.$)(#) and 1zh( #) in the range (0, Pl ) as

(2 1) (22) Now we can relate the expressions for the value in the different regimes. /71 the hrm invests if the credit is in effect, so the expressions P'1 for ( P4 to the left (22)and right ( 18) should satisfy the value-matching and

At the threshold

,

strongly.

Even when the credit is not in place. thtl threshold

J1)is affected by the

probability () of its removal. This is because the lirm calculates that at the random future time when the t;kxcredit is enacted, the economic cllnditions might and the credit might be removed before they irnprove be very unfavorable, it for invest. sufhciently This reduces the value of waiting now. However, to effect negligible. is quantitatively the Next consider thc situation when the credit is currently in effect- Now the relevant threshold is f-'1in X'tble 9.4 We see that it decreases as () increases: .

dTyz//f/l/-fl//hl

/A?tlttstl?l

308

lJllit'

)'

fnvtr-frrlc?,fThreshold w/1(?/lTax Credit is (Parameters: a = 0, tz = 0.1, r = 0.05, I

Table 9.3.

in Effect

,1:)/ =

(JlU

20.)

1 4)

0.0 0. 1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

1.371 1.371 1.371 1.371 1.371 1.371

1.498 1 1.492 1.491 1.491 1.490

1.642 1.641

.494

1.640

1.639 1.638 l .638

0.3

0.4

0.5

1.8 13

2.003

2.201

13 1.813 1.812 1.812 1.812

2.003 2.20 1

l

.8

2.003 2.003 2.003 2.002

2.201 2.201 2.201 2.200

readily now.This

the prospect of Iosing the credit induces hrms to invest more effect is quantitatively not as strong as the delaying effect of

on Jl) above. premium the before 0.5 all the from to 0 way () in need Here we an increase 17. is halved from 0.233 to 0.1 with a higher Moreover, an increase in l increases /:1 The point is that removed, it is effect is in credit if the now tax enactment probability, even immediately likely to be restored fairly quickly. so the imperative to invest = quite becomes but 0, if 4) irrelevant is is less strong. Of course this issue investment-promoting = the then 0.5, If values of (). l signihcant for larger 1()) is very small. credit (higher effect of the prospect of removal of the current l

.

( /$ )

-ccl

lh Investment Threshold wcn Tax Credit is (Parameters: a = 0, o' = 0. 1, r = 0.05, 1 = 20.)

Table 9.4.

I

)

0.0 0.l 0.2 0.3 0.4 0.5

0.0

1.233 1.177 1. l52 1. 135 1.125 1.117

0. l

1.233 1.196 1.176 l l 153 1.145 .162

.

0.2

1.233 1.209 1.193 1.l82 1.174 1.167

1.233 1.216 1.204 1.195 1.188 1.183

0.5

0.4

0.3

1.233 l 1.212 1.204 .221

l

.198

1.194

.233

1

1.224 1.216 1.210 1 1.201 .205

/a J??1k'tr,?Jt?z7

tll 7:/

,?7/7 t'l il 11z7/7t>r/,t-/t7&

These results suggest that uncertainty

fa/

l

about the enactment

309 of stimulus

policies is likely to have a very detrimental effect on investment. In fact, if a government wishes to accelerate investment, the best thing it can do is to enact a tax credit right away, threaten to remove it soon, and swear never to restore it (high () and l()w l ). The credibility of such a policy is. of course, open to doubt. While our analysis s conducted at the tirm level, we can hnd its industry implications using the methods of Chapter 8. If we consider a competitive industryconsisting of numerous such hrms, the price threshold for each firm's investment decision will simply become the ceiling for the indujry's prie process ip the stochastic dynamic equilibrium. Now the increase in J$ will $ be interpreted as the effect of the reduction in industry supply due to the individualfirms' reluctance to invest. While the debate on whether to institute media, firms a tax credit proceeds in the administration. the Congress,and the waiting mplies and to their consumers in the a cost wait upon the outcome tbrm of higher prices. Metcalf and Hassett ( 1993) have a more general model where the scale of investment is itself a matter for choice. They hnd that policy uncertainty not only rases the threshold at which the firm invests, but also lowers the scale ot' its investment. When we develop the theory of investments of varying scale in Chapter l 1, the reader will be able to handle this extension as an exercise.

3

Example of an Oligopolistic Industry

We have thus farconsidered two extreme market structures, namely monopofy in Chapters 5-7 and perfect competition in Chapter 8 and Chapter 9 to this

point. The reason is practical rather than fundamental. Oligopolistic industries in our stochastic dynamic setting present formidable difficulties. The development of stochastic game theory for such applications is quite recent, and tractable models using that theory are rarer still. We will develop a particularlysimple example, based on Smets (1991),that brings out some of the issues. More general and more richly detailed treatments must await further research.

The general point is not difhcult to state. On the one hand, uncertainty imply an option value ofwaiting, and therefore greater hesirreversibility and each investment decisions. The fear of preemption by a rival, hrm's in itancy hand, other the suggests the need to act quickly. Which of these considon is the more important depends on the parameters of the problem, erations

3l 0

/-Llttl'lillrittllj //1(/!/:1r)7

and on the ctlrrent state illustrate this tension.

of the underlying

shock. Our simple model selwes to ' -

We consider two firms, each with the potential t() produce

:1

Iicy /?7t c/3 /7t?

?c,77

?ill ?

l 11??/7c,,7'

fl/lt

','t

C/J?? ?/?('J/

i()1l

%

unit outptlt

llow, which it can activate by incurring the sunk cost /. There are no variable costs of production, and we suppose that the industry demand is sufficiently elastic to ensure capacity production. Thtls industry output is 0, l or 2 depending on the number of active firms. The price is given by the demand function ( 1), which we restate here: #

>'

I

y 1)( Q),

=

Z

and the multiplicative shock

)' follows the geometric Brownian motion (2). For simplicity of notation we assume that the firms are risk neutral, or that the risk in )' has zero correlation with the overall market risk. Thus the discount rate for all future costs and revenues, certain or uncertain, is the riskless

rate

xe...

/2 D (2 ) where

#l

=

pl

-

I

l

and J have the usual meanings. If F zr Fa, the tbllowerwill invest at once, and get the value i' D(2)/J /. If F < i'a, the follower will wait until the threshold is hrst hit, and at that point /. Therefore its expected present value is get /2 D(2)/J -

-

'

J( l'a D(2)/J

-

I

z

F1

z

z'

z

z

z

z

F(

z'

z' zz

z

Z

z'

f

z I I I

FJ

'z

Fa

Fa

F

- /

(23)

.

p,

E gc-r

l

..e-

r.

Dynamic games are usually solved backwards, and this one is no exception. We begin by supposing that one of the firms has already invested. and find the optimal decision of the other, which we now call the tbllower. Then we look at the situation where neither firm has invested. and consider the decision of either as it contemplates whether to go first, knowing that the other will react in the way just calculated. The tbllower'sprot tlowwill be F D(2). Following familiar steps, we can hnd the threshold level Fa that will trigger its investment. lt satisfies

I I I .,,e ..e' l l I I l/'a( ?') l I I l l I I l I I

%( ?)

,

1 ,

hrst time when the stochastic process of the demand where F s the (random) shock reaches L starting at i'. We calculate this expectation in the Appendix. Using it, the follower's value can be stated as

if

F

i'c

The form of this function can be seen in Figure t).2. Nllte that the twl) branches meet tangentially at F:-,this is a smooth-pasting-like property of present values of Brownian motion at points of :1 switch of regimes.? Now suppose neither (irm has invested. and one of them is contemglating becoming the Ieader. gOf course this need not actually arise for some rangets) of values of F, but we must consider thtt hypotheticaf sccnaril) precisely in grder to determine when immediate investment for at Ieast one firm is optimal and so hnd just these ranges of i'.) In making this calculation, this tirm wilt take into ttccotlnt the actitln of the other tirmafter it Sees that its rival has invested. That is just tlle follower's

,

(24)

?In fact this is just an alternative way t)f dcriving rule and the val ut formula in Chapter 5, cquatiun (()).

and expressing

lltlr

very Iirst invcstrllcnt

ltlttililll-itlt't J?lt/!/-f?)7

51;!

decision we studied abtwe. So if )' 1 L, the follower will invest at once, the leader's protit llow will also be F D(2), and its valtle will be the same as the tbllower's. If F' < F2, then the follower will wait until l'a is hit. In the meantime. the leader will have the larger profit tlow 3eD l ). and its expected value will be

skj' ()

where, as before, T is the srsttime the stochastic process ofthe demand shock reaches :2 starting at F'. Again we compute the expectations in the Appendix and state the result here. The leader's vlue is P'a( J'')

1zr1 ( F)

=

i' D (2 )/ (

=

( 1/) F D( 1) + ( Y/

(1

l'a)/1

-

I

-

( Y!

l'a)/J1

r? D(2)/J

'

-

j

(25)

I

-

,

=

B1f pk is large enoughv negative slope. but this does

-

l'l

/''

() () r My a

+

--v

r

2

ya 0(2) -

j

f5

Thus

r

D 1)

>

/+

p'

y' X

D 1)

-

0(2)

wj

/2

-

J 1'' the hrst investment does not occur until the current

profit flow of the first investor provides a supernormal return on the sunk cost. The reason is similar to that in the case of perfect competition. Even though the value of waiting is zero, the firm contemplating being the srstto invest recognizes that future entry by the other hrm will reduce the upper end of the distribution of protit Qows.Therefore it requires enough of a current premium in compensation. Unlike the pertctly competitive case, though, the expected present value of the firm at this point is positive. With just two irms and no free entry, this makes intuitive sense. Extending this analysis to N tirms is simple in principle but messy in practice. However, the results should be evident without doing the formal work. At the smallest )' that triggers investment by the first firm, say, ih we )'.v ( Fl). The common value will will have )'l D( 1) > I and l'1 ( Fl ) N infinity. go to zero as goes to .

.

l

r I 1.,)(1)

111(..311 C(?l?IJ?t!f

(5

This has a more complicated shape, whic is shown in Figure 9.2. It is con8 cave over the range ?' < /, and its slope is discontinuous at Fi. The Iatter that the follower's decision changes is of course a consequence of the fact discontinuously at J'2. For a range of )' to the le of F2, the Ieader's value cxcceds that of the follower because it enjoys a higher prot tlow before the follower invcsts. However, tbr a range of very low values of F, the leader's value is less than the follower's because the leader incurs the investment cost up front but has a low profit flow initially. The two curves cross at the point Fl We have not designated one particular firm as the leader. and each tirm's own proht considerations should govern whether it wants to Iead or tbllow. The outcomes are different depending on the initial circumstances. lf the initial i' is below l'l neitherhrm will invest. When l'1 is reached, one will invest at once and the other will wait until Fa, but the two are indifferent between these two roles. Note that at i'1, we have P'l ( J'l ) P'2(FI ) > 0. Then equation (25)gives

1,' 0(1) j

Policy fllfc/-t'cllfo/l (ll1t Itnpellct

l

>

0

.

1z'1( l') can actually peak to the left of :2 and then approach not affect the qualitative results that follow.

F with

Next suppose

=

)' starts somewhere

.

.

.

=

in the range ( FI F'2). Then each hrm afford the option of .

stands to gain by seizing the Ieader's role. Neither can waitingabecause the other will invest if it does not.

Note that if both hrms invest at once. each will have a value D(2)/J - 1. This is even Iower than the follower's lz'a(F) while )' is in the range ( ih l'a), from their joint perspective. Of so simultaneous investment is a such mistakes equilibria. If the game were played can occur in game course the equilibrium discrete would mixed time, in be in strategies, where each independently chooses a probability of investing at once. Given that the other firm is choosing the equilibrium mix, cach is indifferent between investing and not inveting. However, with positive probability (the product of the two individualinvestment probabilities) the two invest together and get the lower value. In continuous time we would Iike the probability of such coincidences to go to zero. Doing that correctly requires some delicate limitingconsiderations. We omit these details. and refer the interested readers to Fudenberg and Tirole ( 1985). We merely state the outcome in the resulting continuous-time equilibrium. One firm, chosen at random, gets to invest at once, and therefore gets the leader's (larger)value. The other then waits until l'a is hit and so gets the follower's (lesser)value. Tllere are really two equilibria of this kind, where the two roles are interchanged between the frms. Since the firms are economicallyidentical, the two equilibria are indistinguishable for our purpose. '

,

Gmistake''

l//dlrl/'z/ Ill6ltt-tl?'/J* el

ln practicep some minor which one invests first.

tlnspecihed

Qatters are ditferent it the .

!

diftkrence between the lirms can govern '

'

roles of leader and..

.

tollower are

txogenously

preassigned to the firms. Now the follower cannot invest until the Ieader has (aption value done so, and the leader has the ability tt wait and recognizes an V/$. usual for values forms. have the a range of F tf doing so. The option neither zero Y# 11 F/31 includes that for range + B that includes zero, and Bt of the qsual negative and root pz the nor infinity, where pkis the positive root values these option as dashed quadratic in the parameters. Figure 9.2 shows value tlnce. They of investing at meet curves smoothly pasted to the leader's the latter at the points F1:, Fa', and Fn as Fhown. Then the designated Ieader will invest at once if F is in the range ( F,/. FJ)(whenthe follower will wait until /2 is reached), and when F' exceeds F3 (whenthe other will tbllow at once). z41

of J' either below F1/or in the range ( FJ F3), the Ieader will prefer entirely above to wait. For some parameter values. (he option valtle cklrve lies waits for the whole of (0. /3 ) 1/1(F); then the Ieader For

4

values

.

Guide to the Literature

The general idea of the optimality of competitive equilibrium in a dynamic and uncertain environment as long as markets are complete goes back to Arrow ) present the earliest spectic modei of and Debreu. Lucas and Prescott (1971 explicitly. The fact that irreversible costs this that demonstrates investment themselves constitute a market failure shifting by do not capital Iabor of or of economic adjustment to much-needed in emphasis the given context a was conditions Mussa by international trade changing ( 1982), but I1econsidered unexpected shock. terministic single to d a response on1y a e Lucas and Prescott ( 1974) studied a stochastic dynamic equilibrium with costly intersectoral movements of labor. They assumed risk neutrality, so the equilibrium was again socially optimal. Dixit and Rob ( 1993a,b) introduced risk aversion and incomplete risk markets. and examined some policies that can improve upon the suboptimal equilibrium in these circumstances. The fact that competitive hrms would make the optimal entry decision even if they acted myopically and ignored aI1 future entry was discovercd by Leahy (1992). Our treatment of the dynamic effects of price controls follows Dixit 199 lb). See also Newbery and Stiglitz (1981)for a thorough analysis of price ( stabilization policies in agriculture.

al3(l /,??/7t.,/./- C-tJ,??/7t,????,l Policb'J?'l/tnn'f?pl/i'tppl 't'l

Our treatment 01, policy unccrtainty was based on Mettutlf Clnd Hassett 1993). Additional work on tax policy and investment from a real tlpt ions per( spective includes Majd and Myers ( l986), and Maclie-Mason ( I t)9()). Using different technical approaches. Rodrik ( l99 l ) examined the eftkcts of uncertainty over policy reforms designed to stimulate investment (for example. a tax incentive), and showed that if each year there is some probability thrtt the policy will be reversed. the resulting uncartainty can eliminate any stimulative effect that the policy would otherwise have on investment. Aizenman and Marion ( 1991) developed a similar model in which the tax rate can rise or fall, and showed that this uncertainty can reduce irrevcrsible investment in physicaland human capital, and thereby suppress growth. The trade-off between the strategic incentive to invest early in an oligopoly and the value of tlexibility in the face ()f uncertainty has been studied by several writers using a two-period mtxlet'. examples are Appttlbaum and Lim ( l 985), Spenccr and Brander ( l 9t?2), and Kulltil aka and Perotti ( l992). The Iast of these pairs of authors make an interesting new point. In a quantity-setting duopoly the lirst mover gets lk lltrger market sllare. Therefore the leader's profit is a more convex function ()f a demitnd shock variable than is the foliower's. As a resulta an increase in tlncertainty increases the relative value of early investment. Models of duopoly in :1 continuous-time framewllrk' are rarer. because the underlying theory of stochastic games in contintlous time is itsclf an ongoing research topic. A recutnt paper by Dutta llnd Rustichiui ( 199 l ) tlflrs a promising framework, and Smets ( l 99 l ) builds 11 dullpllly model using it. Our treatment follows his.

Appendix

A

Some Expected Present Values

Here we establish the formuas stated in section 3 for F

-r 7gf:r

1

and

(f7

.r

-''

t!

7-

)z'dl

()

when )' follows the geometric Browian motion (2), and T is the random first time the process reaches a fixed level Fa starting from the general initial position F'.For a more general approach tt) the calculation ofsuch exprcssions, see, for example, Harrison ( 1985, p. 42) or Karlin and Taylor ( 1975. p. 362).

tlltilillrittlll Illtltt-tlq' lJ*

Write /'( F) for the first of these expectations. As long as F < L, we l'c in the next short can choose Jf sufficiently small that hitting the level time interval dt is an unlikely event. Then the problem restarts from a new recursion level (l' + dY4. Theretbre we have the dynamic programming-like

expression j '( F')

e-rdl

=

'j

y' + Jy)

)'

).

Expanding the right-hand side, recalling that F follows the process using Ito's Lemma. this becomes

Simplifying and letting dt 2 1. 2 c

/2

(2),and

partV

0, we get the differential equation

-.+

''

+ a y'

f ( y)

.'(

F)

r

-

.J( F)

Extensions and Applications

().

=

This has the general solution

f't

=

z11

F#' +

FX

-d2

.

root and pzthe negative root of the standard quadratic. and are determined by a pairof boundary conditions. T close F':, F is likely to be small and e-'' to l therelbrc As i' approaches 7* close and Iarge e-'' small, F likely is be F' to (); F2) is very When to 1. /( F2/'' l and z1: these, () so 0. Using we get therefore /(0)

where Jl is the positive The constants

.?d!

..42

.'

=

,41

=

=

f This is used in equations Similarly, dehne

F)

=

( F/

=

F2)/J1 .

(24)and (25)in the

text.

This satishes the differential equation !

2

2

r.r

y2

,'

I (F) +

,

y g ( y)

r g(

-

y) + y

.() .

This has the general solution g ( y)

and the boundary

=

conditions

Sa

=

1.11F/$1+ Bz /11 + F/(r are g(F2)

Sj

0

which is used in (25)in the

text.

0, g(0)

=

=

-

=

1-/31

y'c

-

a),

0. Therefore

/(r

-

a).

.

10 chapter

Sequential lnvestment

IN

''rldls

and the following chapters

we return

to the investment decisions of

:1

single firm. In Chapters 5, 6, and 7, we developed a series of models in which the :rm must decidc when (:tndwhether) to invest in a single grojcct. I11 Chaptcr 5 we assumed that the value of that project

dvlllvcd

as an exllgenotls

stochastic process, and we derived thtt optimal investment rule. In Chapter 6 we allowed the price of the project's output to evolve as an cxogenotls stocllastic process. and then, given a variable cost of protltlction. wc derivcd l7tltllthe value of the project and the investmcnt rule. Finally, in Chnptcr 7 we exttlnded the model to allow tbr the mothballing and later reactivation of the prtlject, as well as scrapping. In all three of those chapters we considered a single, discrcte project, 11 single initial investment decision. In hence and many situations, however, investment decisions are made setptenliallv, and in a particular order. R)r example, investing in new oi1 production capacity is a two-stage process. Firsta reserves of oiI must Le obtained, either through exploratitln or otltright purchase. Second, development wells and pipelines must be built s() tllat the oil can be produced from these reservcs. An oiI company might invest in thc Iirst stage (forexample, by buying proved reserves of oil). but decide to wait rather than immediately invest in the second stage. Investing in a new Iine t)f aircraft is also a multistage process that begins with engineering, and continues with prototype production, testing, and the final tooling stages. An investment in a new dnlg by a pharmaceutical company begins with research that (withsome probability) Ieads to a new compound, and continues with cxtcnsivc testing until FDA approval is obtained, and concludes with thc construction ()f a pr()duction facility and the marketing of the product. Both the aircraft company

320

Eztensions

and the pharmaceutical companycould decide to go ahead of these investments, and then wait on the later stages.

tlnt

Applicatils

with the first stages

Even investments that appear to involve only a single decision can turn especially large out to be sequential. The reason is that many projects (and midway and halted ones) take considerable time to complete. and can be think of such temporarily or permanently abandoned. As a result, we can option projects as having many stages; each dollar spent gives the tirm an which it may or may not exercise to go ahead and spend the next dollar. The key characteristic of sequential investments is the ability to temporarily or permanently stop investing if the value of the completed project falls, or if the expected cost of completing the investment rises. (lf one had no started, investing would choice but to complete the project once it had been This possibilty of stopping midonce again involve oniy a single decision.) compoltnd options; each stage stream makes these investments analogous to the firm invested) gives an option to complete the next completed (or dollar problem boils down to hnding investment dollarl.The stage (orinvestthe next sequential these (andirreversible) expenditures. a contingent plan for making kinds of sequential investdifferent examine several In this chapter we analogies, will draw to as we have done try ment problems. In each case we options. of valuation tinancial exercising and with the throughout this book, sequential investments. three-stage and simple twoWe begin with some fairly and use them to illustrate a basic approach to valuing the Iirm's option to inrule. Next, we turn to problems of vest and hnding the optimal investment spends money a dollarat a time, it takes time continuous investment-the iirm investing in response to to complete the projcct, and the lirm can always stop problems, we will investment these conditions. In studying changing market the completed project value of uncertainty the over draw distinctions between and output), the project's the of price uncertainty over (due, tbr example, to project. completing actually the uncertainty over the cost of In each of these investment problems, the Iirm does not earn any cash only the flows from the project until the project is complete. (Investing in sufticient aircraft is not engineering and prototype production stages of a new will also examine for the hrm to actually sell airplanes and earn money.) We the related problem of a tirm moving down a learning curve. Here current llow, and also production serves two functions: it yields an immediate profit Chapter 11 we discuss lowers future costs. The latter is like an investment. In each unit of where usual sense, modelling of incremental investment in the capital contributes to the proht flow as soon as it is instailed. The mathematical models of this chapter and the next involve two state variables. One is the number of stages completed or the amount of capital

Sequetltial

/'?lI.'c-/?7Ic?I/

installed; the other is the price or some other indicator of protitability of investment. Then the diftkrential equations that emerge from dynamic programmingor contingent claims analysis are partial differential equations, with the value of the project or the option as the dependent variable. and the two state variables as the independent ones. Solution of such equations typicaliy requires numerical methods. For the class of models we consider, such solution is relatively easy. In an appendix to this chapter we brietly discuss the numerical procedure, and in the text we apply it to our specihc model.

1 Decisions to Start and Complete a Multistage Project In Chapter 6 we worked baclwards to hnd the value o a project together with the optimal investment rule. Given a stochastic process for the project's output price, P, we derived the value of the project. P' ( P). Then, given )' ( P), in project, F P4, we were able to find the value of the option to invest the that Recall which invest. optimal it is we needed to the critical price P* at and P' ( P4 to 5nd F( #) dnd #* because l'' ( P*) appeared in two of the boundary conditions that accompany the differential equation for F( #). We can use this working baclwards to solve sequential investment problems same approach of which the investment occurs in two or more discrete stages. in this how To see can be done, considr a two-stage investment in new oil capacity. First, reserves of oil must be obtained. through exploproduction at purchase. some cost I Second, development wells (andpossibly ration or built, be at an additional cost /z. Suppose that the price of oil, pipelines) must specihed exogenous stochastic process. Then the tirm begins #, follows some worth option. with an i ( P), to invest in reselves. Making this investment worth Fat #), to invest in development wells. another option, the tirm buys second yields production capacity, worth F ( #). investment Making this work backwards to find the optimal investment rules. First, as we We can P' have shown in Chapter 6, ( P) is the value of the srm'soperating options, and can be calculated accordingly. Nexta >t P4 can be found in exactly the value of a single option to invest in Chapter same way that we obtained the 6,' it wll satisfy a differential equation subject to boundary conditions for an matching'' and example, Fa(0) = 0!. and for endpoint should which make lhis investment the 5rm pasting'' at the critical price 'a* at in development wells. Finally, Fl ( P) can be tbund. It will satisfy a similar differential equatien and boundary conditions, but now there will be another critical price. J$*. at which the hrst-stage investment should be made. Note that because the payoff from the tirst-stageinvestment is t #), that is, the .

tfor

K C. so we use the solution for F(#) in the operating region, that is, tbr # > C, in the boundary conditions (8) and (9).We then obtain the solution Fz ( #)

=

D? #/1

From boundary conditions

(8)and (9)we

and that #z*is the

to the equation

solution

( l0)

.

determine that

Recall from Chapter 6 that equation ( l2) must be solved numerically for PJ. The solution given by equation (10) applies for # < #a'. When P t Pa' Iirm exercises its option to invest. and Fa(P) = )' ( #)- /2. This is important the what follows, so we restate it t'ormally: for Fz( #)

Dz ##,

for P

=

Pe( #)

-

Iz

x.

8

6 4 2 0 0.

:

0.50

1

=

0.12

1

.00

2.

.50

2.50

3.00

Maximum Investment Rate, k (millionsof dollals per year) llltte.

l7ij;ttre 1(3.33. (7rJcJ/ (?-az

The Value of Construction Time Flexibility

How valuable

'

=

> z:

tetlttelltill /?lv,c-/??,c?l/

r'

*.

().()0-. rr

ils az

/7!l?1cJ/a

())' #/(a??7Ill?T

(1.20va n tl JJ

=

ltlte

is tl4e ability to speed up construction of a project'? Many projects can be built with alternative construction technologies that differ in terms of nexibility over the rate of construction. Technologies offering greater tlexibilitytend to be more costly, so the increased cost must be balanced against the value of the greater construction time tlexibility.We can use this model to determine the value of that greater ilexibility. Since greater construction time tlexibilitycorresponds to a higher k, we simply calculate the value of the investment opportunity, F r''.K4, for different values of k. The change in F corresponding to a change in k then measures the incremental value of extra llexibility. Of course this value of extra tlexibilitywill depend 4 on P' and K. as well as the other parameters. Figure 10.3 shows this calculation for the base case parameter values r 0.06. It plots F 1z'. K.' as a function of holding P0.02. t'r 0.20. and constant (lirstat $ 10 million and then at $ 15 million) and holding K constant (at $6 million). For each value of F, the incrementai value of construction that is. by the slope of the curve. As the time flexibility is given by A sgure shows, the incremental value of llexibility is always positive, but falls as k increases. Alsoa thc horizontal lines in the Iigure show the value of the investment opportunity when there is maximum tlexibility,that is, J' x,. when IZ = 10, this value is 4.(), and whcn P' l5. it is 9.0.5 Consider two different construction technologies with the same 1(), the same total cost K 6. but different maximum rates of investment'. k 0.5 tbr the first. and 1 for the second. As can be seen from the figures the incremental value of the more gexible technology is 2 F/s 0.98/0.5 ;:k: 2. Hence one should be willing to pay up to $2 million to have access to the more oexible technology. The incremental value is higher if V is higher; if 1z' 15, the incremental value is about 5.5. Of course, n general greater flexibility might be accompanied by a different total required investment, K. In that case, we can rank the technologies by comparing F fz'. K., k') for each. =

-)

=

f?/'

Itlb'e.b'tttlz,ttt, k

1ti millio r!)

=

'

-//,

=

Then, lower values of k (thatis. longer minimum construction times) imply a lower present value of the payoff from completing the project. and hence a higher cttrrent critical value F*. As above, this second effect could have been muted by considering F**; we leave this for the readers. We have assumed that k is constant. which may be unrealistic for some projects. where the maximum rate of investment can depend on the stage of the construction. However. this model can easily be modifed to allow the l'unction of the total remaining maximum rate of investment to be a (known) substituting investment, K. Thisjust means a function of K rather than a conequation numerical procedure can be used to stant for 1 in (24).The same maintain the assumption that obtain a solution. Likewise. it is not necessary to 1 1: 0. There might be some positive lower bound on the rate of investment (for example, to maintain a construction site), so that the constraint becomes I S l S 1, with I or k possibly dependent on K. This constraint can be interpreted in two ways: either that the 5rm is forced to proceed at this minimal rate once it embarks on the project, or that any smaller expenditure is tanif not properly tamount to abandonment because existing stages will maintained. The optimal decision rules will be different in the two cases: the initial entry threshold will be stricter if there is no way out Iater. ' C( Q).

Then the optimal production rule becomes l if # A #*( Qj and x () if P < P* Q). Also, since cost falls as Q increases, we will have dP*/dQ < 0. We now proceed to derive this solution. The solution to (37)must satisfy the following boundary conditions: -r

,

if Q >- Q

Q).

The reader should be able to easily verify that )' ( #. followingpartial dit-ferential equation:

tically evolving output price are thus closely related to the kind of continuous investment decisions we just examined. At each instant the firm must observe price and decide whether to produce (andthereby invest in future cost reductions). knowing that it can later cease producing should the price fall sufGciently, and then resume producing should the price rise. To show how an optimal production rule can be lbund and to illustrate the close parallel with the investment problem, we make use of a recent model by Majd and Pindyck ( 1989). In this model the tirm sells its output at a price P that follows the geometric Brownian motion ofequation (1).Marginal production cost is constant with respect to the rate of output. up to a capacity constraint arbitrarily sct unity. However, there is a learning curve; marginal cost declines with cuat mulative output, Q, until it reaches a minimum Ievel, Letting c denote the initial marginal cost, and Q,nthe level of cumulative output at which learning ceases, we can write the marginal cost function as

34l

Z (0, Q4

=

Iim Vp P.

Qj

P-w(r

P*(Q4 C(Q4+

=

0. =

(39) 1/.

P:( P.

-

F(#.

=

(.?) 0. =

Qm) F( #), =

(41) (42)

condition that P' ( #, along with the value-matching Qj is continuous at P P* Conditions (39)and (40)are analogous to boundary conditions (26)and (27) in the mbdel of investment with time to build in the preceding section; 0, it remains 0, so that )' (39) just says that if P 0, and (40) says that will P becomes high, the firm almost surely always produce, and then if very the incremental value of a $1 increase in price is just the present value of $1 =

=

=

342

Exellsiolls

J/lt.f

v'.6illllictliolts

Seqlttnltial

Condition (41) follows frol tlpe per period i'orever, discounted at ;L a maximizatfonof value with respect to production it is just the intuitive detl inition of #*( Q) above, and replaccs thc smootll-pasting condition. Finally. condition (42)is analogous to boundary condition (25)in the model ()f investand production cost ment with time to build. It says that once Q reaches becomes constant, cumulative production can no longer affect the value of the firm, so that V is a function only of P. Nvhat is F in this case? We already derived it in Section 2 of Chapter 6 or eqtlation (3) above: =

-

.

343

'eslllellt

Pil/i/t.z(2j' Fp-,,.1(lllfl t'J/)/??7t',/I.'l.otlltntillll ///t?

Table 10.3.

.r;

.?,,,

J?I !

tNote:

Entries shoqv vklltle ()1- the tirn1 l,' ( /?. Q)$Starretl ent ries ind ica te t)ptimal prodtlction rule; tlle price correspllnding t() tlacll sta rretl ent ry is tlle 0.05. c critical price P* ( l-'). Paranleter valtles Ctre 1. J 4(). l 0a (...?,a 2() :111d t'z t) 2() j .

=

,

.

.

Curntllative Prodtlct itln in U llits trtlst in dk')lltrs)

(current marginal

Price

if P

(43) >

where Jl and pz are, respectively, the positive and negative roots of damental quadratic, and /1l and Sa are constants given in equations (5) above. When P


t', a > 0. =

=

=

().()()

8.0()

l2.t)()

1f).()l)

(4(3.0())

(31).32)

(22.98)

( I7.4 l )

( l3.2())

( l ().()t) )

27.66 26.58 25.53 24.53 23.57 22.()5 2l 2().9l 2().()9 l 9.34) 18.54 17.8 1 l 7. 12 l 6.44 15.80 l5. 18 l4.5t) 14.01 13.46 12.94 12.43 11.94 11 11.02 1().59 10.18 9.78 9.39

2 12.39 l 94.75 l 78.53* l63.76 15().22

270.08 25().l () 23 l l 5 2 l3.24 196.35

3 16.33 295.7() 276.()() 257.2 l 239.29 222.24

348.03 327.24 3()7.36 288.33 270. 15 252.78

366.()4 345.23 325.3 l 3()6.25 288.t)3 27().6l

37 l 35().95 33 I 3l I 293.75 276.32

2()6.t)3

236.2()

253-96

259.67

190.65 I 76.4)9 l62.35

220.39 2(5.32 194).97 l 77.33 l 64.38 152. I l 14().5i) 129.55 l l 9-26

238.()6 222.89 2()8.42 l 94.62 l8 l

243.77 228.6)() 2 l4. l 2 2()().32 l 87. l 8 174.67 l 62.69 15 l 14().79

.76

When P > P' and x l (37)does not have an analytical solution and must be solved numerically. Once again.a tinitedifference method is used, whereby the partial differential equation is transformed into a diffcrence equation that can be soived algebraically. This method is essentially the same as that used in the preceding section, and described in the Appendix. =

in Iltll lars

.47

4.(.2)

.

137.79

18().5()

l 26.40 I 15-94

165.7() 15 1

1()6.35 97.56 89.49 82.09 75.3() 69.07 63.36 58. I 2 53.31 48.9() 44.86 4l 37.74 34.62 31 29.13 26.72 24.51 22.48 20.62

139.32* 127.8() l l 7.23 l07.53 98.64 9t).48 83.()() 76. 13 69.83 64.06 58.76 53.9() 49.44

.15

-76

.97

l 49.42 l 37.3 I I 2fi.()l

45.35 41.60

115.54 I (5.92* 97. l 6 89. 12 81.75 74.99 68.79 63. l() 57.88 53.()9

38.1 6 35.00 32. l 1 29.45 27.()2

44.67 40.98 37.59 34.48

48.7()

1(39.6l

1()().62 92.29 84.6 l * 77.62 7I .2()

.49

l 69.0() 157. 13

145.86 l 35. 19 l25.09 l 15.55 106.57

98. l3 t)().22

82.84

65.31

76.00

59.9 1 54-95 50.41 46.24

69.68* 63.92 58.63 53.78

42.4 1

49.33

21).(1()

.76

.()3

.t?8

.50

13(1.65

l2 l

.06

1 12.00

:03.47 95.45 87.93 80.89 74.33 68.24 62.6 1* 57.43 52.68

344

Eteltsiolu

flnl

Applications

cktylfcvl/ffz/ /?lI.'c.;??lc??f

22 20

beeause the priee may rise in the future. As Q increases, the value of the tirm costs have been reduced), and #* falls. The critical price falls to rises (because long-run the cost of $10 as Q reaches 20..at that point the tirmhas reached the of the learning curve and the shadow value of cumulative production bottom is zero. Of course, even if there is no uncertainty, when there is a learning curve a should produce at a price that can be substantiallybelow current marginal lirm because of the shadow value of cumulative production.8 To see how uncost affects the hrm's production decision. we can examine this shadow certainty V, and its dependence on P and (r. Figure 10.4 shows, for Q 0, I''c value, 0, 0.05, 0.1, 0.2, 0.3, and 0.5. Also shown is the as a function of P for a equation line c #; note from (41)that #* satisies P CQ) V, and so, when Q = 0, is given by the intersection of this line with the Vc culwe. When 0, so Vc is zero up to the critical price of $19.00 (if P is J r t'r 0, a below this price the tirm will never produce), and then is constant at $21.00.9 Note from this sgurethat the larger is c. the larger is #*. For example, when r.ris 0.5, #* is about $31. The effect of uncertainty on Vc dcpends on the current price. The possibility of future increases in price raises r%.and the possibility of decreases reduces it. At low priccs. the possibiiity of increases in price dominates, so uncertainty increases Vc. To see this. note that if o. = 0, the price can never increase (because r), so future cost savings have no value when price is low. However, if o' > (), the price may later rise enough to justify production, so reductions in future cost have some value. The greater is tr, the greater is the probability that the firm will begin to produce over some finite horizon. and thus the greater is the present value of reductions in future cost. At high prices the effect is just the opposite. Suppose P is high and the firm is producing. If c > 0. the pric may fall to the point where the firm shuts down, and the higher is o', the sooner this is likely to occur. Thus for high prices, a higher tr implies a lower I/. It is this higher price region that is relevant for the production decision. Thus. as Figure 10.4 shows, an increase in c raises P-. This result can also be understood by remembering that with a learning curve. part of the srm's production cost is actually an irreversibke investment in reduced future costs.

C

'% N

=

=

-

-

BAs Spence

( 1981) has

shown, if the discount rate is zcro

(and there

is no uncertainty). a

competitive firm should produce when price exceeds the long-run marginal cost that will prevail when the lirm rcaches the bottom of its Iearning curve. gWhen fz 0, Pb and P* can be found analytically by integrating the tlow of discounted profit. =

.

16 14

O

(J!

o5

l I I I 1

12

$ l

6

l

4 2

l I 1

I

0

u/l/

l7igttre l (l..6.

(for r

=

J

=

fltllv

0.1)5,

When future price is uncertain.so

fz

bIlllIe =

.

(r l I 1 i

:)?C

o.2

=

G

0.3

=

'

l

I

I l $ 1 I l I I I I l I l I I I I 1 l

$ 1 I I I l l I

20 Price

10

() j

ZQ

I

10

O

=

=

()-

=

=

8

=

=

(y

18

=

-

345

ty l I 1 l I

.

C

-

P

I

1 1

30

littttlitib'z.

0.5

=

40

/3r/a:/l/c/itpll

0x().()5,().l 0.2. (3.3,and 0.5) .

is the payoff from this investment. As usual,

this implies an opportunity cost to investing now rather than waiting to see how price will evolve. The ntlt benefit of this investment which is measured by Vo thus falls, pushing the critical price up closer to where it would be if there were no Icarning curve.

4

Cost Uncertainty and Learning

ln most of the investment problems that we have examined so far. it is the future payoffs from the investment that are uncertain. However. sometimes the cost of an investment is more uncertain than the payoffs. Nuclear power plantssforwhich construction costs are hard to predict due to engineering and regulatory uncertainties. are an example. Although the future value of a completed nuclear plant is also uncertain (becauseelectricity demand and costs of alternative fuels are uncertain), construction cost uncertainty is greater than revenue uncertainty. and has deterred utilities from building new plants. Other examples include the development ofa new line of aircraft, urban con-

struction projects, and many R&D projects, such as the development of a new drug by a pharmaceutical company.

346

-lellsiolls

-,',-

d'//lt

zjpplicaliollx

When projects take time to complettl. the Iirm facuts11 seqtlential investbtlth ment problem that can involve two differcnt kinds of cost tlncertainty, of which were discussed briefly in Chapter 2. Tlle first is lechllical uncertainty, and relates to the physical difficulty of completing the project: Assuming prices of construction inputs are known, how mucll timea effort, and materials will ultimately be required'? Technical uncertainty can only be resolved by tlndertaking the project; actual costs and construction time unfold as the project proceeds. These costs may be greater or less than anticipated if impediments arise or if the work progresses more quickly than planned. but the total cost of the investment is only known for certain when the project is complete. The second kind of uncertainty, which we will call inpllt ccu'l unccrtainty, external is to what the Iirm does. It arises when the prices of Iabor, land, and needed materials to build a project fluctuate unpredictably, or when unprechanges in government regulations change the cost of construction. dictable and regulations change regardless of whether or not the tirm is investPrices and uncertain the farther into the future one Iooks. Hence inptlt ing, are more cost uncertainty may be particularly important for projects that take time to complete or are subject to voluntary or involuntary delays. Both technical and input cost uncertainty increase the value of an investment opportunity for the same reason that uncertainty over future cash tlows increases it the net payoff from the investment is a convex function (Jt' the cost of the investment. I-lowever, as %9e saw in Chapter 2, these two types of uncertainty affect the investment decision differently. Technical unccrtainty makes investing more attractive; a project can have an expectt!d cost that makes its conventional NPV negative. but it can still be economical to begin investing. The reason is that investing reveals intbrmation about cost, and therefore has a shadow value beyond its direct contribution to the completion of the project; this shadow value Iowers the full expected cost of the investment. (In Chapter 2, we illustrated this in the context of a simple two-period

example.)

On the other hands input cost uncertainty

makes

it Iess attractive

to

invest now. A project with a conventional NPV that is positive might still be uneconomical, because costs of construction inputs change whether or not investment is taking place, so there is a value of waiting for new information before committing resources. Also, this effect is magnilied when fluctuations in construction costs are correlated with the economy, or, in the context of of cost is high. The reason the Capital Asset Pricing Model, when the is that a higher beta raises the discount rate applied to expected future costs, which raises the value of the investmect opportunity as well as the benefit from waiting rather than investing now. ttbeta''

Since tcchnical alld inptlt cost tlncertainty have diflkrent eftkcts ()n investment, it can be important to incorporate botll in tbe analysis. l-lere Nve sulnmarize a nlodel developed l7yPindyck ( l 9t)3b) tlltlt yiclds decisiol) rules for irreversible investments subject to both types t)f cost uncertainty. Tllt! basic idea is that thll project in qtlestion has ttn actual cost ()f conapletitln that - only the expected is a random variable, &', cost & t ( Kj is known. As in the mtldel of Section 2, tlle project takcs time t() complete the maximum rate at which the lirm kzan (productively)invest is I Upou completion. the tirm receives an asset (tbr example, a factory or new drug) whose value, l'' is k'nown wi t h ce rtain ty. To allow for both teclnical and factor cost uncertainty. we wil! assume the expected cost K evolvcs according (() that -

-

.

-

=

.

,

lK

=

-

/ dt +

1,

(I K)

l ''

(/::

+ y KJ

t.l

-

where dz and tlw are the increments of uncorrelated 'Wiener processes.ltl I t is unlikelythat the technical difficulty of completing :1 project will have much to do with the state of the overall economy, but tllis may not be the case lbr the evolutien of constrtlction costs. Henk:e we will assume that atl risk associated with Jz is divcrsifiable, (thatis, (/z is uncorrelated with the economy and the stock market ), but (l may be ctlrrelated with the market.

Note that the second term on the right-hantl side of tquation (45) tlescribes tcchnical uncertainty. If y (). K can change only if the hrm is investing, and the instantaneous variance ()1' t/ &/ &-increases Iincarly with // K. When the firm is investing. the expected change in K twer an intervtll Lt is l l/. but the realized change can be grcatt!r or less than this, and K can even increase. As the project proceeds, progress will at times be slower an d 2 t times faster than expected. The variancc of &V falls as K galls but the ? . fl. is only known when the actual total cost of the prolect, / prqect is comJ() pleted. The last term ()n the right-hand side t)f equation (45)describes input cost uncertainty. If w 0, the instantaneous variunce of ts/ K is constant =

-

,

.

=

''C-rhis

is a special casc

()f

the folltlwing ctlntrtlllttd diffusion prtpcess: JK

=

-

I d t + g( f K ) tI .

-.

wherc g; > (). glt :S((), and gs. : 0. R)r this equation to makc cctlntlmic sttnsta ccrtain conditions should hold: (i) F ( K 1-h is homogeneous of degree ! in K 8- ztnd L' ( ii ) F. < (), that iss an s. increase in the expected cost of an investmcnt ilways rcduccs its value: (iii) the instantantlous variance of d K is bounded for all finite K and approachcs zttrl) as & (); and ( iv) if thc (irm ''

'.

',

.

.

,

-.+.

' . at the maxim um ratc k unt il the prqcct is cllm plcte. li f) jb 1' d the expectcd txlst (() ctlmplttitln. Equation (45) mccts thcsc ctlndititlns.

invests

'x

.

=

&- s() that K is indced ,

Etensions TJZIJApplications

Seqttelltial

/'?l:'tM-/?'nt??l/

349

and independent of 1. Now K will lluctuate even when there is no investment; ongoing changes in the cosis of labor and materialswill change K irrespective of what the firm does. The problem is to hnd an investment policy that maximizes the value of investment opportunity, FK4 FCK.. V, /c): the =

F(S)

f

&) Ve ->'

max

=

l (1)

-

/(;)(,-ltl dt

.

()

Iim

A'

along with the value-fnatching condition that FK4 is continuous at K*. Condition (50)says that at completion. the payoff is Condition (51) says that when K is very large. the probability is very small that over some finite time it will drop enough to begin the project. Condition (52)lbllows from (49),and is equivalent to the smooth-pasting condition that F.'( Kj is continuous at K*. ''.

When l

=

Problem

We will assume that spanning holds, so that Jz can be eliminated from the or dynamic porttblio of assets perfectly

problem. Let x be the price of an asset correlated with lz?. so that dx follows J-t

=

axr A' Jf

+ cx

Jlz?

-r

.

Then, the reader should be able to conlirm that F A-) must satisly' the following differential equation: J. 2

:!

p

l KF

''

K4 +

- 4.vy K

.l.

:!

I

-

0s equation

=

'

r

=

( K)

p

1 F K4

-

-(K4 where p? is the negative root mental quadratic equation

F A).

(48)

-

4.B

=

.

Because equation (48)is linear in 1, the rate of investment that maximizes F( K4 is always either zero or the maximum rate k.'

1

2

() otherwise.

.

p

y

'-

-

0.

=

Characteristics of the Solution

When only technical

-

F'K)

-

1 t ().

uncertainty

: P 1. 2

-r-asset

for 1p2 K F''K4

4:

-

(51)) of the funda-

condition

The remaining boundary conditions are used to determine A along with K. and the solution for F S) for K < &*. Tllis is done numerically. which is not difficultonce equation (48)has been appropriatcly transformed-l 1 A family of solutions for K < S* can bc found, each of which satislies condition (50)vbut a unique solution, together with the value of z1, is determined from condition (52) and the co n ti n u ity ()f F ( K ) at K

=

k

l)

-

solution

X SX

=

(fromboundary

?-2 pp .!. a

analytical

.

where 4.r > (r.r rl/qr, and rx is the risk-adjusted expected rate of return on r + 4 pxa, (h, where / is the market price of risk and pxm is ..:. namely, rx and the market portlblio. Thus the correlation coefficient between the yvr / pxm. Since / is an economy-wide parameter, the only project-specific parameter needed to determine y.r is pxm

=

(48) has the simple

*

2

K F ay

F' &)

0,

=

(52)

0. Here Jz is an appropriate subject to equation (45),0 S l (/) k, and K?4 risk-adjusted discount rate, and the time of completion, ?, is stochastic.

Solving the Investment

F( S)

cxl

-+

is present, equation

J K F'' ( K )

'

I F (K)

-

I

-

=

(48) reduccs to Ff K)

1,

.

Then K can change only when investment is occurring, so if K > K* and the firm is not investing, K will never change. and FK4 0. When r 0, equation (54)has an analytical solution'. =

=

(49)

Equation (48)therefore has a free boundary at a point K*, such that lts k 0 othenvise. The value of K. is found along with when K S K. and It) F(S) by solving (48)subject to the following boundary conditions:

l' When /

=

=

(50)

make

=

k. equation (48) has a first-degree singularity at K = 0. To eliminate = F J.( y'). where T' = 1og K Then equation (48)becomes

the subslitutitln

-)

.

''(

./ )') and boundary conditillns

-

/? (