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Timing of major transportation investments

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Title:
Timing of major transportation investments
Physical Description:
1 online resource (xiii, 75 p.) : ill. ;
Language:
English
Creator:
Chu, Xuehao
Polzin, Steven Edward
United States -- Dept. of Transportation. -- University Research Program
University of South Florida -- Center for Urban Transportation Research
National Urban Transit Institute (U.S.)
Publisher:
University of South Florida, Center for Urban Transportation Research
Available through the National Technical Information Service
Place of Publication:
Tampa, FL
Springfield, VA
Publication Date:

Subjects

Subjects / Keywords:
Transportation -- Capital investments -- Econometric models   ( lcsh )
Net present value   ( lcsh )
Genre:
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographical references (p. 71-75).
Funding:
Performed by the National Urban Transit Institute and supported by the U.S. Dept. of Transportation, University Research Institute Program under grant no.
Statement of Responsibility:
Xuehao Chu, Steve Polzin.
General Note:
"August 1997."

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University of South Florida Library
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University of South Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 029179627
oclc - 754641936
usfldc doi - C01-00160
usfldc handle - c1.160
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SFS0032268:00001


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TIMING OF MAJOR TRANSPORTATION INVESTMENTS XuehaoChu Steve Polzin August 1997 National Urban Transit lnstrt ute Center for Urban Transportation Research University of South Aorida 4202 E. Fowler Avenue CUT 100 Tampa FL 33620-5375

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TECHN ICAL REPORT STANDARD TrTLE PAGE .._ NUT I 96USF4 2. Acc:eMiorl N!>. 3. 4. TlUo 11M Subtltlo s. Aepoo1 oa' August 1997 TIMING OF MAJOR T RANSPO R TATION I NVESTME NTS e. P.eominq Code 7 A.IJhc:lt( ') 8 Ab;XItt No.. Xuehao Chu a n d S t eve Pol zin 1 Performing N $m0 &nd Adtteu 1 0 Nationa l U r ban Transit Center f o r Urban T r ansportat ion Research University o f South F l orida eoo ...... ,., DTRS 93-G-0019 4202 E. Fowler Avenue CUT 100, Tampa, Florida 33620-5375 13. Office of Research and Spec ial Programs U.S. Dep artme n t of Transpo rt ation Washington D.C 20690 14. 1 $. $>.1ps:4emfii'Wy Kolle$ Supported by a grant from the U.S Department of Transportat i on University Research Institu t e Program 1 6 AbW&(t This report offers a b r oad overview of timing research as i t applies to majo r transportation i nvestments. Spec i fic emphas i s is given to major publ i c transit investme n ts. The report is designed to provide p l anners and decision-makers with a better u nderstand i ng of ti m i n g research. The report emphasizes basic econom i c principles o f investment t i ming rather than detailed techniques. The purpose of this particula r report is to describe the k i nd of i nvestme n t timin g ru les, most useful in maki ng in vestment decisions so that transportation i nvestments are made most effic i ently. The report is divided into n in e chapters Chapter 1 is an introduction. C h apter 2 revi ews basic concepts related to econom i c analysis of investment t i ming Chapter 3 discusses the perceptions and attit u des of the planning professio n toward investment timing C h apter 4 shows w i th e x amples both quantitativ e and qua li tative sign i ficance of t i m in g. Chapter 5 describes conditions under w hich waiti n g can create a value to invest C h apter 6 discusses t i m i n g niles under different scenarios i ncluding trad i t ional rul es. rules w i t h certa i nty, a n d ru les with uncertainty. Chapter 7 presents two approaches to t ime s u bsequent analysis of a project follow i ng an ini t ial build-later decisio n Chapter 8 ident i fies what type of data economic principles requ i re what federal regu lations o n i n ves tment a n alysis requ i re and what is i nadequate in curre n t practice Chapter 9 provides a number of recommendations rega r ding what needs to be don e i n order to use these econom i c princip l es of i n vestment ti m i ng in pract ice. Re f erences are i ncluded a long with a techn ical appendix on models of investment tim ing. 17 Kt)' Wbtd5 18. Oimtlution bletntnl Investmen t Timing Discount Rate, Net Available to the public t h rough the Nationa l Technica l Informat i on Service Present Val u e CostBenefit Ana l ysis (NTIS), 5285 P ort Royal Road, Springfie l d, VA 2218 1 ph (703) 487-4650 Uncertai nty t9. 2'1. 22. PW'ic Uncl assified Unclassified 88 Fonn DOT F 1 7 00 .7 (8)

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ACKNOWLEDGMENTS This project is made possible through a grant from the U.S. Department of Transportation, University Research Institute Program. The following persons provided comments and suggestions on earlier versions of this report: Michael Baltes, Center for Urban Trans portation Research (CUTR) Patrick DeCorla-Souza, Federal Highway Administration (FHWA) Patricia Henderson, CUTR F Ron Jones CUTR Edward A. M ierzejewski, CUTR Robert Peskin, KPMG Peat Marwick Donald Pickrell, Volpe Tran sportation Systems Center Samuel Zimmerman, FHWA. v

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TABL E OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . v TABLE OF CONTENTS ... . ......................... ...... ...... vi i LIST O F TABLES . . . . . . . . . . . . . . . . . . . . . . . . . x i LIST OF FIGURES ... .... .... . ....... ... ...... .... ................ x iii Chapter 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . 1 Purp9se and Scope . . . . . . . . . . . . . . . . 1 Fundamental F la w in Ex isti ng Process . . . . . . . . . . 2 The Build Later Alternative . . . . . . . . . . . . . . . . . 3 A Debate about Build-Later . . . . . . . . . . . . . . 4 Timing Decis io ns are Common . . . . . . . . . . . . . 5 Literature . . . . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2 BAS I C CONCEPTS . . . . . . . . . . . . . . . . . 11 Cost-Benefit Analysis . . . . . . . . . . . . . . . . . 1 1 Annual Net Benefits . . . . . . . . . . . . . . 11 Discount Rate . . . . . . . . . . . . . . . . . 1 1 Net Present Value . . . . . . . . . . . . . 12 Real Values . . . . . . . . . . . . . . . . . . . 12 Study Years . . . . . . . . . . . . . . . . . . . 12 Inv estment Timing . . . . . . . . . . . . . . . . . . 13 First-Year Net Benefits .... ........................ 13 Project Age . . . . . . . . . . . . . . . . . . . . 13 Project Value . . . . . . . . . . . . . . . . . . . 1 3 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 3 WHY TIMING IS NOT CONSIDERED ...... ................ 15 Transit is a Long-Term Investment ........ ......... .. ...... 15 The Chicken or Egg Dilemma . . . . . . . . . . . 16 We'll Grow Into It ... . ....... .. ... . ..... ......... 17 We'll Lose the Opportunity . . . . . . . . . . . . . 17 vii

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Right of-W ay Avai l a b ility . . . . . . . . . . . 1 8 I nflation . . . . . . . . . . . . . . . . . 18 O pport u n it y Knoc k s .... .'. . . . . . . . . . . . . . 19 T im e to Q ui t P l ann i ng and Start Build ing . . . . . . . . . . 19 I t I s a Build Late r Decis i o n . . . . . . . . . . . . . . . 1 9 Don't Fo rget the Non-Econo m ic Benefi t s . . . . . . . . . . 20 We N eed Balanced T r ans p ortat i on ....... ................ 21 Summary .... ......... .......... . . . .......... . 2 1 C h apt e r 4 T I M I N G CAN BE S I GNIFICANT ..... .... . . . ... ......... 2 3 Wai t in g Can be V a lua b le ... . . . .... . . ............ 23 Under Certain t y ............ . .... ........... .. ... 2 3 Under Uncertainty .... . . . ......................... 24 The Value Can be Sig n ifican t . . .......... .... ........ ...... 24 Chapter 5 DETER M INI N G IF WAITIN G MAY BE WORTHWHIL E ........... 29 Time Effects ........... . .... . .... .... ...... .. .. ...... 29 Growth Conditi ons . . . ........ . .... ................ 30 Growth without Sh i ft .... .... .... ................. 31 Upward Shift .. ......... .... . . . . .......... ..... 32 Ho r izont al Shift ... . . . . ....... .... . ... . . 34 U n ce rt a i nty Cond i tions . . . . . ... ; . . . . . . . . . 35 U n certainty over Cos t s .................. . ....... . . 35 Uncertainty over Discoun t Rate ....... . ....... . ..... 36 Cha pter 6 DECID E WHEN TO STOP WAI TING . ........... .... . .... . 37 T h e B a s i c Ru l e ........ . ....... . . . ..... . ..... ....... 37 De r i ved Rule s .... ..... .... ... ... . ................. ...... 38 Tradi t iona l Rules .... ...... . ...... .... ...... ........ 38 Rules under Certa in ty . ... ... . . . . . .. .......... 40 Rules under U nce rt ainty .... . . ......... . ...... 40 Re l ationships . . . . . . . . . . . . . . . . . 40 App l icat i on . . . . . . . . . . . . . . . . . . . . 41 E x ample ..... .... ... . ..... ..... ....... .... ... 4 2 viii

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Specification ... . . ..... ................... ... .. 42 Correct Usage . . . . . . . . . . . . . . . . . . 45 Incorrect Usage .... ............. .... ....... .... . 46 Chapter 7 WHEN A PROJECT IS POSTPONED . . ....... . ........ . 49 Chapter 8 THEORY, REGULATIONS AND PRACTICE ........... ...... 51 Data Required . . . . . . . . . . . . . . . . . . . . 51 Net Present Value ............... .... . .............. 51 Annual Benefits and Costs .............. ......... . 51 D i scounting ................. ... . ................... 5 2 Start Date . . . . . . . . . . . . . . . . . . 52 Regulations .......... ................ ... ........... 53 Executive Order 11893 . . . . . . . . . . . 53 OMB Circu lar A-94 ............... ...... . ... ........ 53 Criter i a for New Starts .... . ...... .... ...... ..... 54 Current Practice ....... .......... ............... .... .... 55 Chapter 9 RECOMMENDAT I ONS ................. . .......... ... ... 57 Use Net Present Value as an Acceptance Criterion ......... ...... 57 Im prove Cost-Benefit Analys is ... .............. . . ...... ..... 58 Annua l ize Benefits and Costs ........................... 58 Discount Benefits and Costs Appropriate l y .............. 58 Use an Appropriate Ana l ysis Period .............. .. ... 59 Consider Investment Timing in Decisi o n-Making ............... 59 Consider Timing in Investment Ana l ysis .............. .... .... 59 Reflect on Timing Issues .................... . ..... . . 59 Use Economic P rin cipals of Optimal Timing ...... ... ....... 60 Include Build-Later AHernatlves ................. .... . 60 Use Proxy Variables for Tim ing . .......... ........ ... 60 Deal Uncertainty .. .. .. .. .. . . .. . . .. .. .. .. .. 61 Use Traditional Approaches ....... ................... 61 Account for the Value of Waiting ... . .................. . 61 Sponsor Nationa l Forums ... .... ...... ...... ...... ..... 62 ix

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Appendix A MODELS ......... .... . ......... . ... ......... ..... 63 Basic Model ... ....................... ............ 63 The Case of Positive NPV . . . . . . . . . . . . . . 66 The Case of Certainty and Max i mizing NPV ............... ..... 66 The Case of Uncertainty and Maximizing Expected NPV .... ...... 68 Comparisons . . . . . . . . . . . . . . . . . . . . . . 69 Growth Conditions ....... ..... ..................... 69 Appendix B REFERENCES ............ . .................. 71 X

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Table 1 Table 2 Table 3 Table 4 Tabl e 5 Table 6 Table 7 Table 8 LIST OF TABLES Quantitative Significance of Timing ................. ..... .... 25 Timing Rules ............................................... 39 Critical Va l ues .... . ............ ... ............... ..... 43 Variation of variables under Certainty . ........... ... ...... 43 Evolution of variables under Uncertainty ....... ............... 44 Summary Resul t s of Correct Usage . . . . . .. . . . .. . . . . 45 Summary Results of Incorrect Usage ................. .... 46 Errors from Incorrect Usage ..... ........................ 47 xi

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LIST OF FIGURES Fig u re 1 AHernat ive S treams of Net B enefits ........... ............ ...... 26 F i gure 2 Cumu l a t ive Distrib u tion of D isco u nt e d Ne t Be n efits ... ... .......... 26 F igur e 3 N et P r esent Value ............. . . ...... ..... ............ 27 Figure 4 Growth Wi t hout Shift ........... .... ... .. .. ......... ..... .. 31 F igure 5 Upward Shift wi t h Gro wt h ......... . ... .. ... . . ....... .... .. 32 Figure 6 Upward Shift Without Growth ... ........... . . ............. .. 33 Figure 7 Horiz o ntal Shift w"h Growth ........ ........ ........ ....... 34 xiii

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Chapter 1 INTRODUCTION Purpose and Scope This report discusses perceptions and attitudes toward investment timing, presents economic principles of optimal timing and offers practical consideration of the timing of major transportation investments The report addresses a number of issues, includ ing: 1) The inadequa te consideration of timing in the current planning process; 2) The significance of timing, both qualitatively and quantitatively; 3) General conditions under which it may be worthwhile to postpone a n investment ; 4 ) Rules f or detenmining the optimal timing of investment projects under both certainty and uncertainty; 5) Criteria for subsequent steps following a postponed invest ment under uncertainty; 6) Data requirements for investment t i ming analysis; 7) Procedures recommended by federal regulations on investment analysis; and 8) The state of current practice of investment analysis. The report provides a number of recommen da t i ons for better analysis and dec i sion-making regarding the timing of major transportation investments. This report addresses one of three basic questions invo lved in making major transportation inves tments : 1) Should any project be built? 2) What particular p rojec t should be built? And 3) When should the project be built? Properly answering these questions is im portant partially because these investments are often the single largest public works projects in a given area. T his report is in three ways. First, the t iming of investments may be addressed under different perspectives ranging from economic to environmental to political. This report focuses on timing of investments only in terms of their economic worth, which presumes that projects be built when their net present value is positive and maximal. In the decision to invest depends not only on economic worth but also on social and environmental considerations that are beyond the scope of this report. Second, timing rules can differ, depend ing on whe the r individua l pro ject s are being considered in isolation or whether there are budgetary constraints. Procedures for timing projects vary according to the presence and nature of budget limitatio ns and 1

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the mutual exclusivity of proJe. cts or alternatives. In all cases projects can be timed through a linear or dynamic program that takes into account such budget constraints and project interdependence (Marglin, 1963). This report deals with the simplest case where one single project is being evaluated against the "do-nothing" alternative Third, the timing rules under uncertainty are derived under a particu l ar form of uncertainty Specifically, today's annual benefits are known with certainty, and future annual benefits are uncertain and lognonnally distributed with a constant variance This fonn of uncertainty is used only for analytical convenience. Regardless of the form of uncertainty, however, the basic resuH holds: there is a value of waiting to invest under uncertainty. The issue of investment timing is conceptually not unique to transit investments. The principles of investment timing presented in the report apply to major transportation investments across many modes. However, it may be more relevant for transit given empirical data on the perfonnance of transit investments relative to their roadway counterparts. If one believes the numerous needs estimates for roadway investment and looks at project histories it is apparent that In the vast majority of cases we are building roadway capacity to meet historic or existing demands. Indeed, the highway engineer is often accused of building roadways that are immediately or very soon self fulfilling prophesies That is, they are utilized at or near capacity soon after they are completed Thus, the issue of investment timing may be less critical for roadway investments where the project is far less likely to depend on growth in the demand Other modes, particularly transit, pemaps ITS investments, air, and water port investments, might be strong candidates for investment timing analysis. These modes are more frequently dependent on future demand to be economically just i fied. Furthennore, the degree of uncertainty about the Mure demand for these modes Is likely to be higher than that for highways. Fundamental Flaw in Existing Process The interest in investment timing is motivated by a concem that the current planning process for major transit investments does not adequately consider investment 2

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timing. A brief revie w of the current process reveals this problem (UMTA, 1984; USDOT 1993 ). The initial consideration of most major capital investments occurs as part of the long-range p lan development. Many major capital especially transit guideway projects, are often conceived by staff or decision-makers and are usually first explored as a scenario in the development of the long-range plan. In pract ice projects moving from the IQng-range plan toward implementat ion are those perceived to be of the highest priority. However, there is seldom a systematic or analytical method of determining whether the "highest" priority in the long-range plan is the highest priority for immediate implementation. Often, major projects pass into the phase of major investment studies (MIS) based on their ranking in the long-range planning process. The Build-Later Alternative While a project may be very promising for meeting longrange needs, it may not be best that the project should be implemented immediately. The MIS stage of plann ing typ ically looks at performance in the context of a 15to 25-year time frame. It is implicitly assumed, by virtue of the fact that evaluation focuses on design year performance measures, that if the project performs well in the design year, then implementation now is an appropriate action. This creates strong process biases toward early implementation and can result i n erroneous decisions by favoring a build now alternative in the absence of build-later alternatives as an option in the choice set. Consideration of build-later alternatives is particularly important in light of the strong decision-making p refer ence fo r a build-now alternative. Even if there are no obvious transportation needs, seldom will a decision-ma ker favor a do-nothing alternative. L ow-cost opt i ons can be part of major investment studies; however, these options often under perform build options and, evaluation in the context of design-year performance does not fully reflect the prospect that a low-cost option could be coupled with build-later options. This composite scenario may offer a superior overall alternative; however, it is usually not in the choice set in a major investment study. One way to address this potentially significant option is to include the issue of investment timing in major investment studies. 3

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The ultimate objective is to encourage more explicit consideration of investment timing. The analytical approaches presented in this report provide possible methods of addressing the optimality of investment timing. While the concept of investment timing is not new, only recently has an analytlcal framework for evaluating this concept for public transit investments been explored. This report summarizes efforts to develop analytically the concept of explicitly evaluating investment timing, i.e., considering build later alternatives. However, simply recognizing the issue of investment timing and reflecting on it as one carries out major investment studies is a very important first step. A Debate about Build-Later A great deal of concern exists r egarding the effectiveness and efficiency of major investments especially light rail systems, in some of the rapidly growing urban areas in the US. The significance of these projects is heightened because they represent an alternative to the historical pattern of addressing transportation capacity problems by building additional roadways. Thus. transit guideway options represent not only major investments but provide an important test of fundamentally different transportation investment strategies In some cases the investment also provides a test of a significantly different urban vision and urban lifestyle. Thus, there is strong interest in evaluating the performance of these investments New guideway investments have frequently been characterized by serious local debate regard in g project merit and the Investment worthiness of the projects. These debates can become polarized discussions with participants being quickly labeled as pro or anti transit, or at least pro or anti rail This pol arization of discussion suggests that some aspects of project worthiness or some considerations i n the evaluation of projects are not being adequately captured in traditional evaluations. Even the nomenclature of traditional alternatives in the Alternatives Analysis process, now the MIS process, highlights the polarization as we have the "do-nothing alternative and a variety of build' alternatives. In at least some instances, the critics of a project were not necessarily against transit or even rail transit but rather concerned that the necessary market to support the 4

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investment was not adequate. A potential strategy that can capture the fact that the market may not yet be sufficient for a particular project to operate cost-effectively i s the introduction of t he "do something later" alternative, or investment timing analysis. Timing Decisions are Common Investment timing decisions are common in households and businesses. Newlyweds, for example may decide to buy a large h ouse with the attitude that they will grow into it. Those having lived through the housing price inflation of the seventies and early eighties may look back with pride at how shrewd they were to invest in the larg e home. On the other hand, a couple may not choose to afford the cost (taxes, mortgage, insurance maintenance, etc.) of the large house or their situation might change so many times that the buy-now decision might result in very negative co nse quences They might be relocated, might not lik e the neighborhood as it changes, never have children and the need for the space, wish they ha d bought a modest home and reserved resources for a second vacation home, or any number of other that might make them favor an incremental approach to housing investment. Business analogies abound as well and range from the shrewd decision to make an investme nt that stretches resource no w with the opportunity of a big payoff to the overextended firm going under in a mild downturn because everything did not go just right and it was too leveraged to survive in any but the most optimistic conditions. Literature This report builds on four streams of The first is the limited transportation literature on investment timing. Georgi (1973) argued for the of dynamic investment planning and showed a simple timing rule due to Marglin (1963): the annual benefrts of an investment should exceed the interest costs for the first year for the project to be worthwhi le. Szymanski (1991) investigated how differences in public and private sector incentives lead to differences in the optimal timing of infrastructure investment. Polzin (1992) suggested that build-later be considered in 5

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alternatives analysis in a workshop on aHematives analysis sponsored by the Urban Mass Transit Administration. Lewis (1992) and FHWA (1996) suggested that major transportation investments should be subject to the simple timing rule derived by Georgi and Marglin. Chu and Polzin (1996) extended the model by Szymanski to address three ques tions: Under what conditions build-later might be optimal? How do changes in the parameters of an investment affect its optimal timing? How significantly do differences in the stream of annual benefits affect optimal timing? These authors do not provide a systematic treatment to timing rules, nor do they address the issue of uncertainty. The second stream of literature is a growing one in the field of economics about the timing of irreversible investments under uncertainty (McDonald and Siegel 1986 ; Crousillat and Martzoukos, 1991; Martzoukos and Teplitz-Sembitzky, 1992 ; and Dixit and Pindyck, 1994) The central argument is that there is a value of waiting to invest when the project is irreversible and its profile of Impacts is uncertain. This value of waiting ex i sts because waiting maintains the option to invest and makes it possib le to adopt a better decision when new informati on arrives. The third stream is the transportation literature on uncertainty. The general ro le of uncertainty in the planning and decision-making for major transportation investments has been widely recognized in the transportation literature (Pearman 1977; Ashley, 1980; Pell and Meyburg, 1985; Gifford et al., 1993; Khisty, 1993; Lewis, 1995; FHWA, 1996; and Mierzejewski, 1996). Traditional approaches to addressing uncertainty include sensitivity analysis, scenario analysis, and risk analysis Sensitivity analysis evaluates how sensitive numerically the initial investment timing and the corresponding net present value are to changes in one of the many assumptions i n an analysis. Scenario analysis on the other hand evaluates this sensitivity with respect to a set of assumptions that represent likely future scenarios. Unlike sensitivity analysis or scenario analysis risk analysis assigns a distribution on each assumption and produces distributions for investment timing and net present value, respectively (Pouliquen, 1970 ; Lewis, 1995) These traditional approaches do not lead to timing ru le s nor do they account for the value of wait ing to invest under conditions of uncertainty. The report is also related to the general lit erature on pub li c infrastructure planning and the literature on evaluating the planning process for major transit 6

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investment projects. The strong interest in public in frastructure p la nning is evidenced in the two recent special issues in the Annals of Regional Science (Snickars, 1989; Rietveld, 1995). A number of authors, including Deen et at. (1976), Stowers ( 1983), Johnston et al. (1988), Johnston and Deluchi (1989), UMTA (1989), Euritt et al. (1990), Hirschman et at. (1991), and FTA (1994), have evaluated the planning p roces s i n the United States for major transi t investment projects from a variety of perspectives. None of these authors, however, considered the timing issue. Summary This chapter covers the purpose and scope of the report, discusses lim itations of current practice describes the concept of build-later alternat i ves, and reviews the related literature. Chapter 2 reviews a number of basic concepts of planning for major transportation investme nts These concepts are separa ted into three groups : those r elated to general cost-benefit analysis those related to investment timing and those re lated to uncertainty. Chapter 3 discusses the perceptions and attitudes of the p l anning profession, particularly transit planning, toward investment timing. It focuses on those factors that may be responsible for failing to consider investment timing in current pract ice For example, electlon cycles, discretionary project funding, and politicians' desire for action now tend to create a bias toward early implementation of major transportation projects. Chapter 4 illustrates the im portance of investme nt timing both quantitativelY and qualitatively through three examples. In one example, where the annual net benefits from a $1,200 investment are assumed to increase from $100 to $114 by waiting for one year, the net present value would increase by almost 40 percent by waiting. In another example, aMemative growth patterns in annual net benefits result i n dramatically different optimal timing and net present values. The second example also indicates that there can be a wide window of opportunity for late r impleme nta tion that would resuM in higher net present values than immediate impleme ntation. 7

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Chapter 5 presents condnions for waning to invest. Genera lly, either growth in benefits or uncertainty can create a value to waiting Waning saves interest costs but at the same time may preclude realizing some benefits. When benefrts are relatively small today and grow over time, the savings in I nterest costs will more than offset the losses in benefits As a resull, waiting creates a value. Four forms of growth in benefits are illustrated Under uncertainty, on the other hand, there is an opportunny cost of making an irreversible investment now by giving up the option of waiting for new information. It is true that waiting in general does not resolve uncertainty However, waiting could increase the value of an investment. Two examples are used to illustrate the value of waiting under uncertainty. Chapter 6 presents three types of tim ing rules that are applicable under different conditions, depending on whether the objective is to maximize the net present value of an investment and whether annual benefits of the in vestment are uncertain. Traditional rules apply if the objective is simply to get a positive net present value. Certainty rules apply if net benefrts are known with certainty and the objective is to maximize net present value Uncertainty rules apply if future net benefits are uncertain and the objective is 'to maximize expected net present value. Each type of timing rule is stated in three forms. The first is as a ratio of project value and capital costs of an investment. The project value of an inves tment measures the total value of its stream of annual benefrts discounted to various years of imp lementation. This form is an extension of the traditional benefit-cost ratio. The second form is in terms of annual benefrts, which is net of annual variable costs of an i nvestment, includ ing operating, maintenance and other societal costs. The third form is in terms of the year of implemerrtation. The timing rules are compared analytically and illustrated with an example. The analytical results indicate that maximizing net present value would delay investments beyond what achieving a positive net present value would suggest; and that uncertainty in annual benefits would delay In vestments longer than what certainty would suggest. This is true regardless the direction of uncertainty i n annual net benefrts. The numerical results show that the different sets of conditions are quantitatively significant. 8

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These timing rules may serve three purposes: 1) to determine whether an investment being proposed for implementation in a particular year is premature or overdue; 2) to determine the optimal timing for implementation under conditions of certainty; and 3) to determine an appropriate t ime for reevaluation of a p ro ject after it is postponed. Chapter 7 offers two approaches to determining subsequent steps that might be followed under uncertainty when a project is postponed The discussion focuses on a choice between time planning, in which subsequent steps are taken on a fixed schedule, and event planning, in which subsequent steps may be trigge red by particular events Chapter 8 discusses data requirements for invest ment t im ing analysis, procedures recommended by federal regulations on investment analysis, and the state of current practices of investment analysis as revealed in a survey of 35 transportation projects throughout the country It appears that the procedures in current regulations are poorly followed in practice. Chapter 9 makes recommendations for incorporating tim ing into the current planning and decision-making processes for major transportation investments. The recommendations are In three groups: th ose on improv ing general cost-benefrt analysis; those on considering t im ing in investment analysis and decision-making; and those on dealing with uncertainty. In an e r a of in creas in gly scarce resources, it is important to improve the economic worth of our investments through better timing. Appendices A and B contain models of investment timing and references, respectively. The models are used to derive the condit ion s for waiting in Chapter 5 and the timing rules in Chapter 6. 9

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Chapter 2 BASIC CONCEPTS This chapter describes a number of basic concepts related to inves tment analysis. These concepts are organized into three groups: those related to general cost-benefit analysis, those related to investment timing and those related to uncertainty. Most of the definitions related to general cost-benefrt analysis are adopted from "A Manual on User Benefit Analysis of Highway and Bus-Transit Improvements" (AASHTO, 19n) and Circular NO A-94, "Guidelines and Discount Rates for Cost Benefit Ana ysls of Federal Programs" (OMS, 1992). Most of the definitions related to uncertainty are adopted from "Guidelines for Risk and Uncertainty Analysis in Water Resources Planning" (USACE 1992). Cost-Benefit Analysis Annual Net Benefits Annual net benefits are the difference between annual benefrts and annual costs (including mainly operating maintenance costs, and other societal costs but excluding initial costs) of an investment. Annual net benefits may be affected by project age as well as investment timing A project's stream of net benefits is the series of annual net benefits over its lifetime. The three terms annual net benefits, ne t benefits, and annual benefits may be used inte rchangeably throughout this report. Discount Rate The discount rate represents the rate of interest which money can be assumed to earn over the period of time under analysis. Benefits and costs are worth more if they are experienced sooner. The higher the discount rate the lower the present value of future cash flows. For typical investments with construction costs concentrated in early periods and net benefrts following in later periods, raising the discount ra te tends to reduce the net present value. The proper discount rate depends on whether the construction costs and annual n et benefits are measured in real or nominal terms. A 1 1

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real discount ra t e has been adjusted to eliminate the effect of expected inflation and should be used to discount constant-dollar or real benefrts and costs. A nom i nal d i scount rate reflects expected inflation and should be used to discount nom i nal benefrts and costs A real d i scount rate can be approximated by subtracting expected inflation from a nominal discount rate. A real discount rate of 7 percent is required for federal p r ojects (OMB, 1992). Net Present Value Net present value is a common criterion for deciding whether a program can be jus t ified on economic princ i ples Net present value is computed by assigning monetary va l ues t o benefits and costs d i scounting future i nvestment costs and net benefits using an app r opriate discou n t rate, and subtracting the discounted investment costs from the sum total of discounted net benefits. An investment with a positive net present va lue is likely to be worthwh il e in that it is likely to contribu t e to productivity and economic g r owth in an economy Net present values of diffe r ent projects can be used to reflect their relat i ve contr i butions Real Values Economic analysis is often most readily accom p lishe d using real or constant dollar values, i e by measuring benefits and costs in units o f stable pur chasing power Nominal benefits and costs are measured i n terms of the future pu r chasing p ower of the d o llar Analysis should be done in constant dollars. Study Years Study years are selected from the ana l ysis period at which benefits and costs are estimated. Benefrts or costs are estimated preferably for each year of the analysis period Since calcu l ations of year-by-year values is laborious, many anal ysts choose only one, two, or three years o f the project life f o r d eta i led study and extrapolate or interpolate for the other years The suggested practice In selecting study years is t o choose the minimum number of years that allow reasona b ly accurate interpolation or extrapolation of benefrts or costs in other years. 12

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lnvesbnent Timing FirstYear Net Benefits First-year net benefits are the annual net benefits in the first year after an investment is made. Annual net benefrts are annual benefits net of annual costs inclu ding operating maintenance and other societal costs; initial capital costs are excluded First-year net benefrts for a given project can vary with inve stment timing First-year net benefits may be used in relat ion to initial capital costs to decide whether a proposed investment is premature, overdue, or optimal in timing for realizing maximum net present value. Project Age The project age is the number of years after the construction of a p r oject It has a range of one through the li fetime of the project. Annual net benefits of a project can vary with project age. Net benefits may change w ith changes in the economy or age induced operation and maintenance costs. For example growth in the economy may increase the net benefrts of a project for a given level-of-service. A rail project may carry more passengers as the population and employment in the service area increases. Also, physical deterioration may require expensive maintenance and replacement to maintain a given level-o f -service and, as a result, re duce annu al net benefrts. Project Value Project value is the total value of a project's stream of annual net benefit s discounted to a particular year of implementation For a given project there is a project value for every potential implementation year. Project va lue differs from the p res ent value of a project's stream of ann ual net benefits in the base of d is counting. Annual net benefits are discounted to the current year in calculating present value while they are discounted to potential implementation years in computing project values Project value may be used in timing rules to help decide whether a proposed investment is premature overdue, or optimal in timing for realizing maximum net present value 13

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Uncertainty Uncertainty is broad l y defined here to include both risk and uncertainty as conventionally defined (USACE, 1992). Under of risk, we know the outcomes and we can estimate the probabil i ties of their occurrence As a result, we can do risk ana l ysis ( L ewis, 1992) and compute expected values. Under conditions of uncerta inty, on the other hand, we may not be able to identify outcomes and cannot estimate the probabilities of 1heir occurrence Sources of uncertainty can be many in making decisions for major transportation i nvestments. Investment costs can be uncertain because of delays i n construction, increases in general construction costs and i n right-of-way costs, and technological cha n ges Operating and maintenance costs can be uncertain because of increases i n energy costs and l abor costs. Potential benefrts can be uncertain because of changes in demand and a lack of knowledge as to whether aHemative tra n sportat i on projects will be built. 14

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Chapter 3 WHY TIMING IS NOT CONSIDERED This chapter discusses the issue of investment timing in the context of the decision-making environment for major transportation in vestments. It al1empts to capture the perceptions and al1itude s at work in these decision-making environments and reflect on the prospects for a more comprehensive, explicit consideration of investment timing in decision-making for major transportation projects. Specifically, this chapter addresses several of the considerations that appear to have a significant impact on the decision-making process for major investments. The arguments and factors that have resulted in resistance to considering build-later options or other treatments of the issue of investment timing are many. These arguments are legi timate and in some cases powerful motivators and are no doubt part of the reason that investment timing has received little al1ention in the mainstream of MIS policy development and process specification. Each of these factors is d i scussed bri efly in the narrative below. Transit is a Long-Term Investment Transit investments are perceived differently f rom roadway investments in many situations. Often transit inv estments a r e made with the in tention of meeting future demand and in fact creating future demand. Throughout the seventies and eighties the issue of major transit investments not realizing the benefits and serv ing the levels of demand forecast was a major point of contention within the transportation plann ing community. These disputes boiled over into the ma in stream press as new systems opened and various parties reflected on whethe r or not they were meeting their objectives. Where a project did not meet expectations, discussions often erupted regarding the true project goals and expectations. Advocates reflected on the fact that guideway investments are 50 to 100 year investments and one should not be al1empting to evaluate the contribution of recently implemented projects. Another argument has been to discuss benefits of the investment that go beyond the impacts that one might try to measure by reflecting on the near-term success in attracting riders. Comments like 15

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"This is a long term investment, we did not expect to realize the benefrts for years. This is an investment in our children's future ... are among the types of dialogue that followed If there are no near term expectations then is difficulty to argue nonoptimal investment timing. The Chicken or Egg Dilemma Another perception is that a major t r ansn investment is not an independent event in the development of urban areas and the shaping of travel behavior. Rather, a major transn investment is very much a factor in the subsequent development of an area and in the subsequent travel behaviors, specifically mode choice that will result. This percept ion has influenced the attitudes toward quantitative analysis of the costs and benefits of transit investments. Transit investments are often perceived to be the catalyst for significant changes in urban land use. These changes will uHimately create the market and demand levels that will enable transtt to deliver the transportation benefits that n was originally intended to deliver. A typical argument is that we need to make the major transit investment now in order to begin influenci n g the land use pattems to uHimately make t ransit work. The log i c continues, ... If we do not make this early investment the densities needed to make guideway transn effective will never materialize Hence, if we delay investment we w ill never grow the market that we seek." This logic is generally considered sound and, with the exception of those markets that matured at high densities because they were built before the dominance of the auto, n is often believed that an exclusive auto/bus based market will never increase densijy to the point at which guideway will operate effectively absent of some exclusive guideway transit investments. The dilemma of this assumption is the fact that there is little assurance that the market will mature or become denser even If we build the tra nsit investm ent. And, even if the market does materialize over time, is it a sound investment? If the payoff is so far i n the future do we ever capture enough benefns to compensate for the early investment and carrying costs of the investment and service provided in the early years when the system operates below economically effective condrtions? With several new rail 16

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systems implemented in the past three decades in this country, the verdict is still out reg a rding how effective rail investments can be in building transit marl
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Right-of-way Availability One of the most frequent fears is the prospect of diminishing right-of-way availability More highly urbanized areas typically have much lower right-of -wa y availability and proponents of early investment argue that if we do not build now the opportunity to build will pass as critical right-of-way is developed into other uses. At a minimum, delays in commitment might result in far higher costs for right-of-way as land prices are bid up, the need to buy and demolish existing development increases the cost of utility and maintenance of traffic increases, and the prospects of needing to build elevated or subway systems increases. While t hese sound logica l even the oldest urban areas have found ways to implement systems and it presumes that it is not possible or economical to preserve the right-of -way now fo r futu r e system development when the market is more mature. Unfortunately, the logic of these arguments is very hard to evaluate in a given context as conjectures about future costs and ava i labi lity are highly uncertain. Inflation Another f ear, most probably born i n the inflation heyday of the 70' s and 80's, is that r is ing costs will preclude the investment at a later date. Implicit in this argument is the assumption that costs w ill i n flate faster than will the revenues f ro m the funding sources. This was in fact the case in the era of high inflation in construction costs and may still be a valid concern i n situations where right-of-way or other cost components are rapid l y increasing. The source o f funding may also play a r ole in this fear as those funding sources that are not indexed to economic growth and/or inflation may not keep up with inflation costs. However, in many instances the growth in revenues due to economic growth and inflat ion exceed the pace of inflation in construct ion costs. Some legitimate concerns r egarding the rates of cost i ncreases for land, maintenance of traffic, or the prospect that new requirements such as broader citizen part ici pation, increased expectations for impact mitigation or other system elements will increase faster than general inflation, merit consideration. In most instances, these arguments sti ll deserve consideration but may not be as valid in today's planning environment where inflat ion costs are l ower. 18

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Opportunity Knocks It is sometimes perceived that the congruence of factors that may enable a build decision or a pos"ive decision on funding may not be assured in the future Often decision-makers change regularly and one cannot be assured that a favorable response to a proposal will be received in the future regardless of economic or logical arguments regarding investment timing. The presence of a supportive city council, a strong local legislative delegati o n at the state level, or the presence of the right person in the right committee position at the federal level is seen as a compelling justification for a build now decision. The election cycles bring opportunities that may not be duplicated in the future when the analytically optimal t ime arrives. The discretionary nature of proj ect funding perpetuates this sens i tivity to decision-makers. Time to Quit Planning and Start Building Another factor biasing decision-making to build-now decisions is the strong emotiona l appeal of a do-something mentality. Frustrations with congestion, a cynica l attitude toward government a disda in of bureaucracy and process, and perception that we plan projects to death often resuHs in a strong predisposition to an action-oriented decision This creates a populist sense of action, and evidences serious efforts to actually so l ve problems. It implies decisiveness and leadership that can be very appealing for decision-makers. On a more pragmatic note, it also increases the chances that some of the benefits of the project will be reaped within a political time frame that is relevant to the decision-makers. That might only mean consultant contracts for p lann ing and design or initial efforts to buy right-of-way as opposed to ribb o n-cutt ing ceremonies, yet these actions can provide a strong constituency for decision-makers. It is a Build-Later Decision Others would argue that the time frames tha t are currently required to plan, design, and impleme nt major projects are such that a decision today is in actuality a jg

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decision to build later given the reality of how long it takes to implement major urban infrastructure projects today. Not only has the time for decision-making expanded the broadening of the players and funding agencies in complex major projects, but the legal hurdles, cash flow constraints, and other factors resun in most projects taking more than a decade to adVance from the concept to concrete stage. Light rail new start projects are typically taking more than ten years to implement the first approximate 20 mile stage of a system. The first stage often costs between one-half and o ne billion dollars and carries fifteen to twenty-frve thousand passengers on an average in the early years of operation. Often these projects are part of larger system p lans and the total system implementatio n time may be measured i n decades. Thus, there is a strong desire to get started, knowing that the start of service may be many years away. Don't Forget the Non Economic Benefits Over the past several years the goal set for public t ra nsportation investments has gradually shifted from a simple focus on cost effectiveness, capacity, and safety to a much broader set of goals as far rang i ng as contributing to economic development, to aiding in the reduction of the balance of trade by reducing the need for petroleum imports to con tributing to the sense of community and social understanding facilitated by the interpersonal opportunities afforded by the "mass in mass transit and the urban environment that guideway t ransit facilitates. This diverse set of goals, specifically the contribution that guideway transit is expected to make on influencing land-use patterns has resulted in transit advocates often arguing that traditional cost/benefit or other economic impact assessments of guideway investments do not fully capture the range of impacts of transit investments and, hence, understate the benefits. Thus, the logic goes, the assessment of timing is not analytically sound since we are unable to fully capture the positive impacts and quantify t hem in a techn ical analysis. This large and sometimes abstract set of objectives does not render the consideration of investment timing irrelevant nor assure that the broadly defined cost benefit assessment always produces a positive number. It does make the analytical assessments alluded to in th i s report more difficult or may result in them being only a 20

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single piece of information in the decision-making i nfonnation base However it should not invalidate the merits of reflecting on or analyzing the issue of investment timing. We Need Balanced Transportation Occas ionally the logic for build-now decisions reverts to emotional appeals rather than attempts to analytically or theoretically rationalize a position. This can result in turning to arguments favoring balanced transportation or critiquing the social cost or hidden subsidies of auto reliance. After all, who would want an unbalanced transportation system? We are supposed to have a balanced diet, a balance between work and play and a well-b a lanced disposition. Budgets should be bal anced and, of course we need a balanced transportation system. Balanced transportat ion seems to mean spending a lot more money on public transportation and at least some more money on pedestrian and bicycle facilities. The transit industry would like 20 percent of any new revenues in the transportation trust fund dedicated to public transit. The appea l to a balanced system or an intennodal system is emotionally compelling regardless of whether or not its merits can be substantiated with empirical data or other facts. Nonetheless, these types of positions are common in discussions about transportation investment. Summary This chapter attempted to capture the perceptions and attitu des that surround the existing major inves tment study process and identified those factors that may have resu lted in a re luctance to consider a l ternatives that might delay implem entation of major investments. These arguments are no t without merit and clearly have a basis in logic as well as strong appeal to advocacy-oriented entities that might be in)lolved in the MIS process. Among the factors discussed, only two are beyond being incorporated into economic analysis of investment timing. These two are election cycles and po l iticians desire for action now. These two factors are related because short election cycles can create an Incentive to politicians for action now. 21

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The fundamental shortcoming of the current practices and the opportunijies for rev i siting current practice merit serious consideration. At a minimum, p l anners should reflect on the issue of optimal investment timing. Preferably there would be efforts to analytica ll y evaluate the time stream of costs and benefits for build-later a l ternatives or to utilizing the information in the remainder of this report to eva l uate the consequence of del ayed imp l ementation. In an era of scarce resources this is an opportunijy to imp rove the economic worth of investments. This report presents a rationale for doing this and analytic tools to help in carrying it out. 22

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Chapter4 TIMING CAN BE SIGNIFICANT Waiting can add a value to an investment by improving the net present value of a project. Two examples are used to illustrate qualitatively the imp ortance of timing. In addition, timing can be quantitatively significant. Optimal t iming can mean substantial postponement of a project and at the same t ime dramatic improvement in its net present value. One example from Chu a nd Polzin (1996) is used to illustrate the quantitative significance of timing. Waiting Can be Valuable Waiting can increase the economic worth of an investme n t project. The objective of economic analysis of transportation investments is to help select and time investment s so that thei r net present values are maximized. The net present value of an investment project can be sensitive to its start-date. This is especially true for investments that draw progressively greater benefrts as traffic grows. A project with an estimated economic l oss from an immediate implementation can be timed to yield a positive economic worth A project with an estimated economic worth in the positive range now can often be scheduled to yield an even greater worth through adjustments to the tim i ng of the project. This is also true for inve stme nts i n which futu re construction costs and annual net benefrts are uncertain. Two examples are used to illustrate the value that waiting to invest can create. Under Certainty Consider a community that is trying to decide whether to make a major t ransportation investment. To keep the example as simple as possible, we will assume that the project can be built instantly at a cost of $1,200 Also, a discount rate of 7 percent is assumed. If built now, annual net benefits are constant at $1 00; ann ual benefits w ill change to $114 if next year Annual net benefits will then remain at this new leve l forever. Should this community invest now or would it be better to wait a 23

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year? If it decides to Invest now, the project value would be $1,529 and the net present value would be $329 The net present value would increase to $450 however, if the investment is made next yea r The value ofwaijing i n thi s case is $121. This i s an increase of almost 40 percent in net present value by waiting. Under Uncertainty Now suppose that annual net benefits are uncerta i n for next year. If buil t now, annual net benefits are $100; annua l benefrts will change if built next year With probabi l ity 0 .5, net benefrts will rise to $150, and with probabi lity 0.5, an n ual benefils will fall to $50 Annual net benefrts will then remain at this new level forever. Should this community invest now or would ij be better to wait a year and see whether net benefits go up or down? Again, if it decides to i nvest now, the project value would be $1, 529 and the net present value would be $329. It seems that the community should go ahead with the i nvestment because net present value is positive now. The conclusion is incorrect, however, because it ignores the opportunity cost of i nvesting now, rather than waiting and keeping open the possibility of not investing should annual net benefits fall. In fact the net present value wou l d be $511 i f the commun ity waits a year and decides to i nvest only if annual net benefits go up. An increase in the net present value by over 50 percent would be realized by waiting for an increase i n annual net benefrts. The Value Can Be Significant The quantitative s i gnificance of investment timing is illustrated with an example adopted from Chu and Polzin (1996). The purpose i s to see how much optima l t i ming and net present value of a project are affected by variations in its stream of net benefrts : The lifetime of the project is 40 years and the project costs $1,000,000 if built now. Construction costs grow at an annual exponential rate of one percent. The cost to acquire the right-of-way is $100,000. These values are shown at the bottom of Table 1. In addition, Table 1 shows the example streams of net benefits. They are convex 24

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Tab l e 1 Qua n titative Sig nifi cance o f Ti m in g Streams of Net Benefrts C o nvex Linear Concave Horizonta l Net benefit function: 8(1) b,(t+ 1)' b,(t+1) b,(1+1)0 b. Net benefits at time 0: 8(0) 2,321 10, 000 38,520 127 ,000 Optimal t im ing (years) : f 13 9 3 0 Improvement in NPV 92% 44% 8% 0% NPV of Investing at time t is computed as follows : fB(s)e'"ds K(QenM = Ke01 where the parameters a r e set as follows: Construction cost Discount ra t e Lifet ime (years) Rig h t -ofway cost Ini tial capi t al cost g r owth rate ($) ($) b = 0.01 r = 0 .07 T=4 0 M = 100,000 K = 1 ,000,000 (growing at an increasing r ate), linear (growing at a con stant incre m ent) and con cave (growi n g a t a decreasing ra1e). The horizontal stream is included for comparison. The constants in these functions, b, (I = 1, 2, 3 4), a r e determined as follows: the constant f o r the li near example is arbitrarily c hosen to b e $10 000; and the o thers are chosen such that these streams resu l t in the same net p re s ent value if the project starts now. The resu l t i ng va l ues a r e shown in Tabl e 1 under "Net benefit at time 0." The th ree streams of net benefits are plotted in F i gure 1 along with the horizontal A l terna t ive ly, these streams may be compared in t erms of the cumula tive distr i butions of their d is counted v alues (Figu r e 2). The farther away a distribution is from the bottom righthand co m er, the la rger i s the proportion of benefits mater i alizing in the early years These cumulative distributions capture the differences among these streams of net benefits better than a simple p l otting of them as shown in Figure 1 25

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620,000,---------------------------:. Convex Linea:. ....... ... . ......... ...... . ........ .......... ........ Concave -..... ..... ------............. -------------.... __ __ .... ------Horizontal .. -----.. -.. --... ... -.. -.. -.. -.. -.. -.. -.. ----............. --; ...... ..... .. ......... ........ ....... 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Project age Figure 1. Alternative Streams o f Net Benefits .. ---:::::.::.;;::::-::.......... -..... ........ ........ .... ............ ... .. .... _. ... .... ............ .... ..... .... ..,.... .. . .,... .,.-r.-' / ..... .,.,..... .,..,-' .... . _,. .. .// .... ..... / / ..... ,..,. / .......... ,.. _,... ..... ./ ,,.., / / ./ .> / .... _.../ _,/ ...... ./ ./ ...-,/ ... . /' / .. / /' .. ....... / ... / /' ...... .. ... .. / _.,/ // .... l /' ....... Convex Linear Concave Horizontal // ........ .. .. 0 2 4 6 8 10 12. 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Project age Figur e 2 Cumulative Distribution of Discounted Net Benefits 26

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The results are shown in Table 1. The optimal timing is 13, 9, 3, and 0 years from now for the convex, linear, concave, horizontal streams of net benefits. respective ly. If the project is built now, these aernative streams would generate the same net present value of $603,400 because of a constraint imposed on the constants of the net benefit functions shown in Table 1. If the project is to be built at rts optimal timing, however, the net present value wou ld increase by 92 percent 44 percent 8 percent, and 0 percent for the four examples, respectively These differences in optimal timing and net present values are better reflected in Figure 3, which shows how net present value varies with investment timing for each of the example streams. First, these curves have the same value at t ime 0 because of the constraint mentioned above. Second the net present value for the horizontal stream decreases over t ime implying that build-now is better than build-later. Third, the other curves reach their maximum after time 0 (at years 13, 9, and 3, r espectively), implying that build later is better than build-now. Fourth, the curves for the non-horizontal cases are higher around optimal timing than they are at year zero imp ly ing a w indow for later implementat i on that would result in higher net present values. 1,200,0001 ------=::::===r===::=-------------, 900,000t 6oo.oool?-r--.___ I I 300,000t o + I Convex Horizontal -3oo.ooo l _....L __ __; _ _L ________ _____ _j 0 2 4 6 8 10 12 14 16 1 8 20 Z2 24 26 28 30 32 34 36 38 40 Investment timing Figure 3 Net Present Value 27

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Chapter 5 DETERMINING IF WAITING MAY BE WORTHWHILE This chapter discusses some general conditions under which waiting to invest may be worthwhile. Both growth in project value and uncertainty can create a value to waiting. These two broad are discussed separately below. Growth in project value is discussed under of certainty, while uncertainty is discussed with examples that show how different types of uncertainty can create a value to waiti n g. Before proceeding, we discuss possible effects that time has on the net present value of investment projects. Time Effects Tim e affects a pro ject 's net present value at least in three ways. F irst as a project ages, its net benefits may change changes in the economy or age-induced operation and maintenance costs. For example, growth in the economy may increase the net benefits of a project for a given level-of -service A rail project may carry more passengers as the population and employment in the service area increases. Also, physical deterioration may require expensive maintenance and replacement to maintain a given level-of-service and, as a result, drive down net benefits. The second way that time affects net present value is through the timing of inv estment. Postponing an inves tment may require a different leve l of construction costs because of changes in real costs for construction. Postponing a project also may result in a different stream of net benefits because of changes in the demand for and supply of its services Postponing also reduces the present values of a given amount of construction costs and a given stream of net benefits. The net result of postponing can be significant. It is possible to increase the net present value of a project by postponing it. It is even possible that postponing a project will change net present value from a negative amount If constructed today to a positive amount if constructed later. The third way that time can affect the economic value of a projeC\ is through uncertainty in its capital costs and annual benefits. There is a value in to invest 29

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when the project can be delayed and its implementation is irreversible. This value of waiting exists because waiting maintains the option to invest and makes it possible to adopt a better decision when new information arrives. Growth Conditions Under certainty, waiting saves interest costs and at the same t ime preclude realizing some net benefrts. When project value is relatively small today but grows over time the savings i n interest costs will more than offset the losses in net benefits. As a result, creates a value A number of cond i tions in terms of net benefits can result i n growth in project value. Some of these conditions are based on the pape r by the authors of thi s report, "Considering Build-Later as an Alternative in Major Investment Analyses" (Chu and Polzin 1996). As discussed earl ier, net benefits of a project may be affected by both project age as well as investment t i ming. The following conditions are some special cases of a general relationship between annual net benefits and pro j ect age and investment timing (see Append i x A). 30

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Growth without Shiff Project value does not grow if the stream of annual net benefits does not shift with i nvestment timing and annual net benefits either remain constant or decline over time When the stream of n et benefrts is independent of investment timing, the stream from inves ting at time 1:z is part of the longer stream from investing at t (Figure 4). However project value would grow if annual net benefits remains independent of investment t i ming, but grows over time This is l ikely to be t he case for many applications Stream 2 Stream 1 Figure 4 Growth Without Shift. 31

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' Upward Shiff Project value stays constant If the stream of annual net benefits repeats Itself from inv esting at different times. In other words, one stream is a parallel shift of every other. In this case, the investment rule is to invest now or never. The intuition is that there is no advantage to invest later when the stream of net benefits repeats itself as investment timing changes. However project value grows if the stream of annual net benefrts shifts with an upward lift when investment timing changes. This result holds true regardless of whether annual net benefrts grow or stay constant with project age (Figures 5-6) "' z Stream 1 Figure 5. Upward Shift with Growth. 32 Stream 2

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Q) c Q) "" a; z Stream 1 Figure 6. Upward Shift Without Growth 12 T i me 33 Stream 2

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Horizontal Shift Project value stays constant if the stream of annual net benefits shifts parallel to itself when investment timing changes. That is, as investment timing changes, the firstyear net benefits remain the same and, annual net benefrts remain the same pattern over its lifetime. However, project value grows if th e first-year net benefits remain the same, but annual net benefits grow faster over the lifet ime when investmen t t iming changes. Figure 7 illustrates this condition. Stream 2 Stream 1 Figure 7. Horizontal Shift with Growth 34

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A for to be worthwhile under certainty is for today's project value to be small relative to the costs. This condition can also be stated in terms of annual net benefits for some special cases. Under the case of "Growth without Shift," for example, this condition is equivalent to the f o llowin g : the first-year net benefrts should be smaller than annual in terest on the costs. That is, if the first-year net benefits are la rge because, for example, demand is already high today, the investment should be made immediately rather than delayed. Uncertainty Conditions Under conditions of uncertainty, there is an opportunity cost of making an irreve rsible investment now, and thereby giving up the option of waiting for new info rmation We already saw in Chapter 4 that uncertainty in annual net benefits can create a value to waiting In what follows we examine two alternative sources of uncertainty. These examples are based on those by Dixit and Pindyck (1994). Under a narrow definition these examples are really examples under conditions of risk. Uncertainty over Cost Continue the example in Chapter 4, where a community is trying to decide whether to i nvest in a major t ran sportat ion investment that is irreversible. Now suppose that investment costs are uncertain and the project costs, in real values, $1, 200 today, but next year the cost will increase to $1,800 or decrease to $600, each with probability 0.5. As before, the discount rate is 7 percen t and the project will generate annual net benerns of $100. Should this invest today or should it wait to make a decision until next year? If it decides to invest today, the project value would be $1,529 and the project's net present value would be $329 This is positive but once again n ignores an cost. To see this, let us recalculate the net present value, assuming the until next year, in which case it will Invest only if the investment cost falls to $600. In fact, the net present value in this case is $439 Thus, if the community waHs a year before dec iding whether to invest, the project's net present value can increase by $105 which is the value to waiting in this example. 35

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Uncertainty over Discount Rate Government agencies have guidelines for discount rates for tran sportat ion projects, but the discount rate can change. For example, a couple of years ago the Federal TransH Administration changed the discount rate for new starts from 7 percent for all projects to 4.9 percent for projects with a lifetime of at least 30 years and to a percentage between 4.2 to 4.9 percent for projects with a lifetime of less than 30 years and, recently, changed the discount rate back to 7 percent. Suppose this time that the only uncertainty is over the discount rate. Today the discount rate is seven percent, but next year it will change. There is a 0.5 probability that it will increase to 10 percent, and a 0.5 probability that H will decrease to 5.4 percent. II will then remain at this new leve l. As before. the project cost is fiXed at $1,200 and annual net benefrts are $100. If the community invests today, the project value is again $1,529 and the net present value is $329. Suppose the community waits until next year before it decides whether to invest. If the discount rate rises to 10 percent, the project value will only be $1,100, which is less than the project cost of $1,200. Hence it will only invest if the discount rate falls to 5.4 percent. The net present value assuming that the community waits is $354. The value of waiting in this example is $25. 36

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Chapter 6 DECIDE WHEN TO STOP WAITING The conditions in the previous chapter allow determining whether waiting may be a better choice than investing today. This chapter presents rules of optimal timing that allow d e cid ing when to stop Specifically, it presents three sets of timing rules, discusses their relationships, and illustrates their use and misuses with simple examples. Before proceeding these timing rules, however, the basic rule is stated. The Basic Rule The basic rule for investment timing is that investments should be made when net present value is maximized with respect to different imp lemen tation years. This rule is based on a simple model of cost-benefit analysis: Net Present Value = Present Value of Net Benefits minus Present Value of Capital Costs By invest ing in a major transportation project, society incurs costs to construct the project and enjoys a stream of net benefrts over the project's lifetime. Net benefits here are net of operat ing, maintenance, and other societal costs. To calculate net present value, the stream of net benefrts is first discounted and summed using an appropriate discount rate; this sum is then compared with discounted construction costs and the difference gives net present value. Net present value can vary with investment timing because of changes in net benefits over time. Optimal timing occurs when investment objectives are achieved. A common objective is simply to achieve a p ositive net present value. Another objective is to maximize the net present value of an investment under certainty or to maximize the expected net present value of an investment under uncertainty. 37

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Derived Rules The basic rule of maximizing net present value can be used to derive timing rules. This section presents three types of t i ming rules that are applicable under different conditions, depending on whether the objective is to maximize the net present value of an investment and whether annual benefits of the i nvestment are uncertain. Traditional ru les apply i f the objective is simp l y to get a positive net present value. Certa i nty ru les apply if future val ues of the i nvestment are known wit h certainty and the objective is to maxi mize net present value. Uncertainty rules apply i f future values of the investment are uncerta i n and the objective is to maximize expected net present value Each type of t im ing rule is stated in three forms: the ratio of project value and capital costs of an investment {V/K) ; annual net benefrts (B); and the year of action (t). These timing rules are shown in Table 2. As stated in the introduct ion, these timing rules are derived with certa i n assumptions so that they are In algebraic forms. For examp le, we only consider the situation where a s in gle transportation project is being evaluated aga inst a "do-nothing" a l ternative. It is assumed that the cost of the investment K, i s known and fixed in constant dollars. Also annual net benefits will change with investment timing at an annual rate of a Furthermore, uncertainty takes a particular form: annua l net benefrts are lognormally distributed with constant variance Traditional Rules Under rul e (1 ), invest when project value exceeds capital costs or, when the ratio of project va lu e and capital costs exceeds one Rule (1) is w idely used as the t r aditional benefrt-cost ratio. Typically this rule is used to decide whether a particu l ar investment should be made now or never. This traditional use has been ex tended here to allow a project with a low benefit cost ratio now to be worth investing In later. Unde r rule (2), invest when annua l benefits exceed a,.= (p-a)K or, when annual benefits as a proportion of capital costs exceed the difference between the d i scount rate and growth rate of annual benefits Under rule (3) invest when t ime reaches a critical year given by T = log[KN(O))Ia. 38

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Tabl e 2. Timing Rules Form of Rule Type of Ru l e V/K B t Traditional VI K C7 (1) B B, (2) r, (3) Certainty VIK > Cc (4) 8 Be (5) Tc (6) Uncertainty VIK C (7) a (}_ T (9) Notes to Tab le 2 : 1 K is t he cons t ruction cos t s in oons t a n t dollars. 2. V i s the project value a-ssocialed with a partieu l ar yea r of imp l ementation. 3. B is t he ann u al benefits assoc i ated with the year of implementation 4. t ts investment 1iming. 5 0: is the expected annual rate of growth in annua l n e t benefits 6 p i s the reaJ discount rate for both construction oosts and annual benefits 7. o i s a measure of the u ncertainty in annual benefits. 8. 13 i s a parameter determ i ned as follows : ll = 0.5 ala' J(ato' 0 .5) 2p/o' 9 Cr. Cc. and C u are critical values for the ratio. VI K. T hey a re: C7=1. Cc=PI(p-a). Cv= Mll 1 ) 10. Br. Be. and B u are critical values for annua l benefits They are : 87 = (p-a)K, Be pK, Bv = Cv(P-a)K 11. T,, Te. and Tuare optimall iming. They are g iven by: T, = l og[C,(KIV0)jta, Tc = !og [ C0(KIV0)jla, where V0 is today s project va l ue 39

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' Rules under Certainty Under rule (4), invest when the ratio of project value and costs exceeds a critical value, Ce = pl(p-cx). Under rule (5), inves t when annual benefits exceed annual interest costs given by Be = pK. Rule (5) has been advocated by a number of authors (Marglin, 1963; Georgi, 1973; Lewis, 1992). A participant of an FHWA sponsored conference on benefit-cos t applications indicate d that the World Bank uses this method (FHWA, 1996, p. 17). Rule (5) is appealing because it is simple and easy to interpret. Under rule (6), inve st when time reaches a crit i ca l year given byTe= log [Ce(KN(O))Ycx. Rules under Uncertainly Rules (7)(9) apply when future annual net benefits are uncertain in a particular way (Appendix A). Under rule (7), invest when the ratio of project value and capital costs exceed a critical value given by Cu Under rule (8), invest when the first-year net benefrts as a proportion of capital costs exceed the critical value multiplied by the difference between the discount rate and the growth rate of annual net benefits. Rule (7) has been advocated by a number of authors (Dixit and Pindyck ; 1994; Martzoukos and Teplitz-Sembitzky, 1992; McDonald and Siegel, 1986) Rule (8) is cumbersome and does not appear often. Rule (9) gives the expected value of optimal timing. Relationships As shown in Appendix A, the critical values in the timing rules have the following relationships: C,. < Ce < Cv; Br
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Application Several steps may be involved in applying these timing rules. The first step of applying these timing rules is to determine which type of rule is the most appropriate for a given problem. Rules (1)-(3) are applicable If the objective is to achieve some positive net present value. Rules (4)-(6) are applicable to situations where there is no uncertainty and the objective is to maximize net present values. Rules (7)-(9) are applicable to situations where there is uncertainty in annual net benefits and the objective is to maximize expected net present values. The second step is to determine the critical values for the rules. These critical values depend on the following parameters: growth rate of annual benefits, standard deviation of annual net benefits, capital costs, and today's annual net benefits. The third step is to determine the type of applications. There are three general applications of these rules as follows: 1. The first type of application is for projects being proposed for implementation in a part icular year under conditions of both certainty and uncertainty. The rules may be used in this application to check if a particular investment is premature, optimal or overdue in timing. Consider rules (4)-(6) for example. An investment is optimal if rule (4) or (5) is satisfied with an equality in a particular year; premature if neither (4) or (5) is satisfied; and overdue if rule {4) or (5) is satisfied with an inequality. Rules (1)-(2) and (7)-(8) can be similarly used. 2. The second type of application is for projects being evaluated for timing re evaluation under conditions of certainty. This application may be done in two ways. In one way, a particular project may be analyzed for a number of possible implementation years over an extended period. One then checks whether any of the rules stated in annual net benefits or the ratio of project value and capital costs are satisfied in each of these years. The optimal year of imp lementation is the earliest year in which one of the rules is satisfied. Another way is to calculate the critical year, which gives the optimal timing for implement at ion. 41

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3. The third type of application is for projects being evaluated for timing re eval u ation under of uncertainty. Rule (9) is based on the expected optimal timing, which is given by Tu = l o g[C,. (KN(O)))/a. Thi s expected value may be used as an appro x imation for t i ming r e-evaluat i on of an investment after it is postponed See more on the t iming of re-evaluati o n i n Chapter 7 Exam p l e Thi s section specifies an e x ample to i ll ustrate the timing rules. This example extends those used in the earlier chapters where a community i s trying to determine the optima l timing of a $1,200 p r oject. Results are shown separately for bo t h c orrect u ses and incorrect uses of the r u les. Specificat i on The example is specified so that all three t ypes of rul es can be ill u strated Suppose that a community is try ing to decide in the base yea r 1997 when to make a major transportat ion i nvestment. To keep the examp l e as s i mple as poss i b le, we assume that the project can be built instantly at a fixed cost of K=$1,200, w ith a d i s c ount r ate of p = 7 p ercent. decides t o i nvest in the base year the project value would be V(O)-K=$1,529 and a correspond ing annual benefits of 8(0) = $61 In addition project value will be assumed to grow at an annual rate of 3 percent or a = 0 03. For o f uncerta i nty, the standard deviation of annua l benefits is assumed at 0.2 o ro= 0 2 These assumptions a ll ow one to determine the crit i cal values for the timing ru l es which are shown in Table 3 To i ll ustrate the rules future annua l benefits and project values are esti mated Wrth the assumptions described earlier they can be determined depend i ng on whether future annual benefits are uncertain They are shown for the per iod 1997-20 1 7 in Table 4 for the c ase of certain annua l benefits and in Table 5 for t h e case of uncerta i n annual benefrts Also shown in these tables are the ratio of projec t value to cap ital costs an d 42

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Tabl e 3 C ritical Values. Type of Critica l Va l ue Type of Ru l e P r oject Value over Annual Benefits Timing Cap i tal Costs Trad i tional c,. = 1 B r =48 Tr0 Certa i nty Cc = 1 75 Be = 84 T0 = 11 Uncerta i nty Cu = 2 .57 Bu = 123 Tu =23 T able 4 Va l ues o f Vari a bl es unde r C e rta in ty Year P r oject Value P r o ject V a l ue I Annual Benefits Net Present ($) Capital Cos t s ($) Value($) 1997 1,529 1 .27 61 329 1 998 1,575 1 .31 63 350 1 999 1,623 1.35 65 368 2000 1,673 1 3 9 67 383 2001 1,723 1 .44 69 396 20 0 2 1,776 1 .48 71 406 2003 1 830 1 53 73 414 2004 1,886 1 57 75 420 2005 1,943 1 62 78 425 2006 2,002 1.67 80 427 ... .,. ... ,,. .. '' "'"' ; 2007 . 2 063 ''<'11' '1..72 . 83'; .,;;A29, ,. }f. ., .. ': ... c . / ifi; .. !J>: >' >: :. . .... 2009 2 ,191 1.83 88 428 2010 2,258 1 88 90 426 2011 2 326 1 .94 93 423 2 012 2 39 7 2 00 96 4 1 9 2013 2 4 7 0 2 06 99 4 1 4 20 1 4 2,546 2.12 102 409 2015 2 623 2 .19 1 05 404 2016 2,703 2 .25 108 397 20 1 7 2 785 2 .32 1 1 1 391 43

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Tab le 5. Evolution of Vari a b les under Uncertai n ty Year Pro ject Valu e Project Value I Annu a l Benefits Net Present ( $ ) Costs ($) Value($) 1997 1,529 1.27 61 3 2 9 1998 1,588 1.32 64 362 1999 1 ,809 1.51 72 529 2000 1 815 1 .51 7 3 499 2001 1,77 7 1.48 71 436 2 002 1. 9 7 1 1 64 79 544 2003 1,537 1 .28 61 221 2004 1,608 1.34 64 250 20 0 5 1 ,784 1.49 7 1 334 20 0 6 2,475 2.06 99 679 2007 2,554 2 .13 102 672 20 0 8 2,288 1.91 92 504 20 0 9 1 9 2 9 1.61 77 315 2010 1 ,948 1 62 78 302 2011 1,725 1 .44 69 19 7 2012 2,075 1.73 83 306 2 013 2,450 2 .04 98 408 2014 2,765 2 .30 111 476 ',., .... .... .,..$<)''<)'' <.(. V 'Y<>" ,-i.'<>t Hr ''* "i.3Ji.:4.T ;a 1Z: .,, .,, j 50 " 722 $_. __ <';. '* $ -_ j< t" <;' i''' > .,-!(.'; ?.-. .. A <>'t 1;>-M -><' _s,.,: ,." '" x '< ''"" _,. .. .,., ""' 2016 3 608 3.0 1 144 637 2017 3,828 3 .19 153 64 8 corre s ponding net pre sent values. For example, annual b e nefrts grow t o $ 83 i n y ea r 2 0 0 7 under certainty a nd t h e corresponding v alues for project values, the r a ti o of p r oject va l ue to c a pit a l cost s a nd n et present va lues are $2 ,063, 1.72 and $429, r e s p ecti vely. Sim i larly, an n ual benefits g r o w to $102 in year 2007 unde r uncerta i nty and t he corres ponding val u e s fo r p roject val ues the rat i o o f p r oject va l ue to capi t al costs and net prese n t values are $2 5 54 2 13, a nd $672, respect ive ly The two tables s h o w the same values for t h e b ase year because t o day's annua l benefit s are kno w n w ith certa in ty. 44

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Coffee/ Usage The resuHs in terms of timing and net present values from correctly applying the timing rules to this example are summarized in Table 6 below. Correct usage means that a particular type of rule is applied to the set of conditions that underlie this type of rule. For example, applying the certainty rules to Table 4 is correct. ResuHs for different types of rule are discussed separately below. Table 6. Summary Results of Correct Usage. Timing NPV ($) 1997 329 Timing Rules Certainty Uncertainty 2008 2015 429 722 Iradijional Ryles: Both Tables 4 and 5 can be used to illustrate the use of rules (1)-(3). In both cases the optimal ti ming i s 1997 and the corresponding net present value is $329. The resuHs happen to be the same in this example because project value in the base year exceeds capnal costs. In general, however, the results can be different if project value in the base year is less than capital costs. For the traditiona l rules, the critical value is 1 for the ratio of project value and capital costs. The investment would be made in the base year under rule (1) because the actual ratio of project value and capital costs is 1 .27 Rules (2)-(3) would result i n the same conclusions because today's annual benefrts are $61, which exceed t he value of $48 and the critical timing is 0 years. If the critical ratio is below one, however rule (1) may be used to find a better time w hen project value exceeds capital costs. This can be done by searching the earliest year when the actual rat i o exceeds the crit ica l value. Certainty Rules: Table 4 ca n also be used to illustrate the use of rules (4)-(6) Annual in te rest costs are $84 as shown i n Table 3 as Be. Annual benefits do not exceed interes t costs until 2008 when it becomes optimal to make the investment under rule (5) The corresponding net present value is $429. Using rules (4) or (6) would result in the 45

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same conclusions. For e x amp l e the ratio of project value and capita l costs does not exceed the critical value Cc = 1 75 until the year 2008 Also, the critical time given by Tc is 11 years. Ungertajntv Ry l es: Table 5 can als o be used to illustrate the use of rules (7)-(8) for determining whether a proposed investment i n a part i cular year i s premature overdue o r optimal i n t i ming. Usi n g rule (7). the actual rat i o of project value and c a pijal costs does n o t exceed the critical ratio Cu = 2 .57 until 2015 The corresponding ne t present va lue is $722. It w ould be prema t ure to make the investment before 2015, whil e it would b e ove rdue after 2016. Us i ng rule (8) would result in the same conclusion However, using rule (g) can resuH in a different result. In this case Tu = 23. That is. the expected optimal timing is 23 years from the base year. Incorrect Usage The timing rules are incorrect l y used if they a r e applied when diffe r from those that what under l ie the rules. For example, the certainty ru l es wou l d be incorrectly used if they are applied to Table 5. Similarly, the uncertainty rules are incorrectly used if they are applied to Tabl e 4 Also the traditiona l rules are in c orrect l y used if the investment objective is to maximize net present value The resultant t i ming and net present values from i ncorrect l y using the timing rules are summarized in Table 7, along with the results of correct usage f o r comparison. For example applying certainty rules to Table 5 wou l d resuH in a wrong timing of 2006 and a wrong net p resent val ue of $679 S imil arly, applying uncertainty rules to Tabl e 4 would result in a wrong timing beyond 2017, the last year included in the tab le. Table 7 Summary Results of Incorrect U sage Condit i ons Certainty (data in Table 4) Uncertainty (data in Tabl e 5) Rules Traditional Certainty Uncertainty Traditional Certainty Uncertainty Timing 1997 2008 2017 1997 2006 2015 NPV ($) 329 429 391 329 679 722 46

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The errors from i ncorrect u s age a r e shown in Table 8. It first shows the effects of incorrect l y usi ng the traditional ru les. They are incorrect l y used when t h e objecti v e i s to maxim ize net present value under certainty or expected net present v a lue u nder uncertainty Incorrectly applying the traditiona l ru l es t o certa i nty condit i ons results in a bias towa r d earty action for 11 years (from 2008 to 1997) and a reduction in net p r esent values of ove r 23 percent (from $429 to $329) while incorrectly applying t hem to uncertainty condit i ons results in a bias t o ward early action for 18 years (from 2015 to 1997) and a reduction in net pre s ent value s of over 1 19 percent (from $722 to $329). Tabl e 8. Errors fr o m l ncorrec1 Usage. Incorre ctl y App lying Applyin g Certainty or Applying Certa inty and Tradi tional Ru l es to Uncertainty Rules t o Uncertainty Rules to a T w o Sets of Conditions G i ven Set of Condi tions Certainty Uncertainty Certa i nty Unc e rt a inty Certa i nty Uncertainty Conditions Conditions Rules Ru l e s Conditions Conditions Timi ng -11 years -18 years 2 years 2 years 9 years -9 years NPV -23% 119% 58% -46% -9% -6% There are two ways to look at the effects of incorrect l y using certainty and uncerta i nty rules O n e way is to compare the r e s u lt s from applying the same type of rules to two sets of condit i ons Let us look at certainty rules first. When they are applied correctly to certainty conditions (Table 4), the tim i ng is 2008 and net present value i s $429. When they are ap p li e d incorrect l y t o uncert a inty cond i tions (Table 5), the tim i n g is 2006 and net present va l u e is $679 In this case incorrect usage results in a timing two years earlier and an increase in net present value by 58 percent. Let us l ook at the uncertainty rules next. When they are applied correctly to uncertainty conditions (Tab l e 5), the timing is 2015 and net present va lue is $722 When they a r e appli ed incorrectly to certainty conditions (Table 4), however the timing is 2017 and net present value is $391 I n thi s case, i ncorr ect usage results in a del ay of two years and a reduction of net present value by 46 percent. 47

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The other way is to compare the results from applying two types of rules to t he same set of Let us first look at both certainty and uncertainty rules being applied to certainty conditions (Table 4). While the certainty rules resutt in a timing of 2008 and net present value of $429, the uncertainty rules result in an incorrect timing of 2017 and an incorrect net present value of $391. In this case, incorrect usage results in a delay of nine years and reduces net present value b y 9 percent. Now look at these two types of rules being applied to uncerta inty conditions (Table 5) While the uncertainty rules resutt in a timing of 2015 and net present value of $722, the certainty rules result in an incorrect timing of 2006 and net present value of $679. In this case, incorrect usage resu lts in a timing 9 years earlier and reduces net present value by 6 percent. Two pattens emerge from the resutts in Table 8 One pattern relates to comparing the two ways to look at the effects of incorrectly using the certainty and uncertainty rules. If incorrect usage is examined from applying a given type of rules to two sets of errors in timing seem to be relatively small while errors in net present value seem to be relative large On the other hand, if incorrect usage is examined from applying two types of rules to a given set of conditions (certainty or uncertainty) errors in timing seem to be relatively large, while errors in net present value seem to be relative l y small. The other pattern from the resutts i n Table 8 relates to the errors from incorrectly using traditional rules. Errors in both timing and net present values can be significant when traditional rules are applied to cases where the objective is to maximize net present value. 48

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Chapter 7 WHEN A PROJECT IS POSTPONED The conditions and timing rules given in the previous two chapters help determine whether a project should be delayed under conditions of uncertainty. The next issue to address under conditions of uncertainty is to determine what follows when the current decision is build-later. That is, how should the timing of subsequent steps be determined when uncertainty exists? This chapter briefly discusses two approaches for addressing this issue. The discussion i s adopted from lntriligator and Sheshinski (1986). The basic choice is between the time approach, in which subsequent steps are taken on a fixed schedule, and the event approach, in which the timing of subsequent steps are triggered by particular events The time approach is the tra d itiona l method by which subsequent steps are being taken after a fixed time interval has elapsed. The event approach is an aHernative method by which subsequent steps are being taken after a certain event or set of events occurs. There is also the hybrid approach to following an initial decision of build-later. It combines the time and event approaches. In t h is approach, e ither time or some event or set of events can trigger subsequent steps. A subsequent step is taken if either a particular event occurs or a certain time interval has passed since the last decision of build-later. This approach has the desirable properties of both types of approach. It recognizes the existence of uncertainty by allowing events to trigger action. At the same time it recognizes that a particular event cannot embody all relevant Informat ion concerning the transportation system. What is the preferred approach to use? A simple theory of planning by l ntriligator and Sheshinski (1986) seems to ind ica te that reanalysis on the basis of events is preferable to reanalysis only on the basis of t ime Thus, if the impacts of the project are uncertain, then events should influence the timing of subsequent analysis. A major challenge of the event or hybrid approach, however, is to identify the particular event or set of events that would trigge r subsequent analysis 4g

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In the case of transr,>ortation projects, any number of logical events might be tri ggers. For example, if benefits were related to demand and demand grew over time with population and employment, one m i ght be able to se target levels for demand or development as triggers for implementation or re-analysis. In the transit industry historically, some rules of thumb evolved that indicated an adequate markel for consideration of guideway investments, such as central business district employment hitting certain level s, corridor t rave l volumes reaching certain volumes or existing bus ridership levels reaching certain levels might be the trigger for re-analysis.

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ChapterS THEORY, REGULATIONS, AND PRACTICE This chapter discusses three issues related to timing considerations in the practice of investment analysis One issue is what type of data the economic principles in Chapters 5 and 6 require for investment timing analysis. The second issue is what procedures federal regulations on the economic analysis of federal projects recommen d. The third issue is the current practice of investment analysis for major transportation investments It appears that the federal procedures are poorly followed in practice. Data Required Certain data and information are required to use the economic principles in Chapters 5-6 for analysis and decis ion -making on investment timing. Such data may not be readily available in current practice. Net Present Value Net present value is a common criterion for transportation investments Net present value is the sum of net benefits discounted to the present day at a correct discount rate, minus the investment costs also discounted to their p resent value. Any project with a positive net present value may be regarded as acceptable in that it can be expected to yield productivity and growth-related benefits in excess of the i nvestment costs. As an acceptance criterion, net present value reje cts projects in which the value of any contribution to productivity and growth is less than the economic costs to be incurred in achieving that contribution. Annual Benefits and Costs One essential element of an economic analysis of investment timing is to identify and measure annual benefrts and costs in constant dollars. Analysis should include comprehensive estimates of the expected benefits and costs to society. Social benefrts and costs should be the basis for evaluating transportation inves tments. In order to 51

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calculate project value and net present value, one is required to estimate annual benefits and costs for each year of the life of a prcject. Annual net benefits are defined as annual gross benefrts net of annual costs. Discounting In order to compute net present value from investment costs and annual net benefits, it is necessary to discount them This discounting reflects the time value of money: benefits and costs are worth more if they are experienced sooner. This discounting also allows comparing benefits and costs occurring at different points of time in comparable terms A failure to apply discount ing techniques means that decision-makers cannot determine whether the capital resources would add greater economic welfare to the economy if directed to other uses. More generally, the absence of discounting will result in the improper allocation of investment resources for the objective of maximizing the economic contribution of public infrastructure . It is important not only to discount benefits and costs but also to use the appropriate discount rate in discounting. If the rate is too hi gh, we will wrongfully reject projects whose benefits are concentrated in the later years of its life-cycle. If the rate is too low, we will accept projects whose benefits are too far in the future to justify investment today. Start-date The economic worth of an in vestment can be sensitive to the start-date. Particularly, this sensitivity can resutt because of the t iming of traffic growth, especially for investments that draw progressively greater benefits as traffic grows. This sensitivity can also result because downstream benefits are worth less than early benefits. Maximizing net present value with respect to start-date requires that many of the variables be dependent on start-date. Specifically, the streams of net benefits over the life-cycle of a project be sho uld calculated for every year over an extended period. Directly us ing the criterion of net present value requires calculating net present values for consecutive years of start-date over the extended period. Directly using t h e tim ing rules requires calculating the project value for every year over the extended period. 52

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Regulations Federal regulations on economic analyses of transportation investments require some of the elements necessary for an economic analysis of investment timing Three such regulations are discussed below. Executive Order No. 12893 (1994) Thi s document sets forth principles for Federal Infrastructure Investments. The order requires all Federal agencies with infrastructure responsibilities to conduct systematic analysis of expected benefits and costs for all Infrastructure investments, inc l uding both quantitative and qualitative measures, in accordance with the following guidelines: (1) Benefits and costs should be quantified and monetized to the maximum extent practicable. All types of benefits and costs, both market and non-market, should be considered. To the extent that environmental and other non-market benefrts and costs can be quant i fied, they shall be given the same weight as quantifiable market benefits and costs. (2) Benefits and costs should be measured and appropriately discounted over the full life cycle of each project. Such analysis will enable informed tradeoffs among capital outlays, operating and maint enance costs, and nonmonetary costs borne by the public (3) When the amount and timing for important benefits and costs are uncertain, analyses shall recognize the uncertainty and address it through appropriate quan!Hative and qualitative assessments. OMB Circular A-94 (OMB, 1992) This circular gives guide l ines for cost-benefit analysis of Federa l programs. The Circular 1) recommends cost benefrt al\alysis as the techn i que to use in a formal economic analys i s of government projects; 2) recognizes net present value as the standard c r iterion for making decisions on government projects on economic principles ; 3) requires the use of a real discount rate of 7 percent i n discount ing future benefits and costs measured in constant dollars; and 4) requires that the effects of uncertainty be analyzed and reported. Restated below are three related sect i ons from the Circular: 53

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Sa. The standard criterion for deciding whether a government program can be justified on economic principles is net present value -the discounted monetized value of expected net benefits (i.e., benefrts minus costs) Net present value is computed by assigning monetary values to benefits and costs, discounting future benefrts and costs using an appropriate discount rate, and subtracting the sum total of discounted costs from the sum total of discounted benefrts. Discounting benefrts and costs transforms gains and l osses occurring in different time periods to a common unit of measurement. Programs with positive net present value increase social resources and are generally preferred Programs with negative net present value should generally be avoided. 8b1. Constant-dollar benefrt-cost analyses of proposed i nvestments and regulations should report net present value and other outcomes determined using a real discount rate of 7 percent. This rate approximates the marginal pretax rate of retum on an average investment in the private sector in recent years. 9 Estimates of benefrts and costs are typically uncerta i n because of imprecision in both underlying data and modeling assumptions. Because such uncertainty is basic to many analyses, its effects should be analyzed and reported. Usefu l information in such a report wou l d include the key sources of uncertainty ; expected value estimates of outcomes; the sensitivity of resutts to i mportance sources of uncertainty; and where poss i ble, the probability distributions of benefits, costs and net benefits Criteria for New Starts The Federal criteria for new starts during the period 1976-1984 re l y on cost effect i veness measures with little attention devoted to the criterion of net present value (Johnston and Deluchi, 1989). This is reflected in UMTA's policy statements (UMTA, 1976; 1984). Despite OMB Circular A-94 and Execut i ve Order 12893 Federal Transit Administration continues to rely on cost-effectiveness measu r es (FTA, 1994). The FTA policy paper on selection criteria now describes cost-benefit analysis as the desirable basis for project evaluation The agency, however rejects the use of cost-benefrt ana l ysis i n the actual evaluation because it believes that the prob lems of quantification are too great. Johnston and Oeluchi (1989) believe that FTA overest i mates the p r oblem of quantifying benefits and costs in conducting cost-benefit analysis for major transit investments. 54

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Current Practice Current practices of investment analysis for major transportation investments do poorly in meeting Federal regulations. These regulations recommend that: net present value be used as the eva lu ation criterion ; annual benefits and costs be measured in constant dollars for the full life of a p r oject; annua l benefits and costs be discounted at a real d i scount rate of seven percent to cal cu late net present value ; and uncertainty be analyzed. However, a 1990 survey of 35 trans p ortation projects conducted for NCHRP Project 2 17(1) (Lewis, 1992) indicates that: only about a t h i rd o f the projects examined use net present value as a basis for evaluation; most p r ojects fail to express costs and benefits on an annual basis over the lif e-cycle of the project; a large number of studies failed to use an appropriate analysis period; only about five percent use adequate discounting techniques and property justified discount rates; and issues related to uncertainty were largely ignored. 55

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The samp l e of 35 projects includes 1 0 a i rport and a i r traffic control re l ated projects 10 highway projects 6 public transit proposals 2 h i gh speed rail systems five ports, and 2 inland waterway projects The sample was drawn from a larger universe with a f our-factor stratification: l ocation and scale, mode type, and point of approva l. Location and scale covers national, regional and local projects, and project size; mode covers highway, public transit, rail, ports, airports, and i nland waterways Type covers construction reconstruction, and repair Point of approval covers appraisa l s in progress, projects rejected and projects approved ( i ncluding projects completed pro j ects in-progress and those not started). I n adaition, ana l ysis i s typically not undertaken in current practice to determine the most appropriate timing or start year of projects. This is not surprising Federal regulations on investment analysis fail to r ecognize the importance of investment timing, though some federally-sponsored conferences and research projects do (Lewis, 1992; FHWA, 1996). 56

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Chapter 9 RECOMMENDATIONS This report has shown the importance of investment timing and presented theoretical principles for considering timing of major transportation investments. However, the report has also revealed a number of issues tha t need to be resolved in order to incorporate investment timing in evaluating and making decisions for major transportation investments. These i ssues include: Election cycles, discretionary project funding, and polijicians' desire for action now create biases toward early implementation of major transportation projects Federal regulations on Investment analysis for major transportation investments fail to recognize the importance of investment timing and even cost benefit analysis in the case of transit new starts Current practices of investment analysis appear insufficient to meet Federa l requirements for cost-benefrt analysis, deal with uncertainty, a nd con sider inves tment timing. Traditional rules reinforce the bias from election cycles, discretionary p r oject funding, and polijicians' desire for action now toward early implementation. In an era of scarce resources, it is important to improve the economic worth of investments in transportation infrastructure. One approach is through better analysis and decis ion-making regarding the t iming of these investments. To address these issues, the following are r ecomme nded: 57

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Use Net Present Value as an Acceptance Criterion Any project with a positive net present value may be regarded as acceptable under the objective of improving economic welfare and standard of liv in g. A positive net present value means that a project contributes positively to both productivity and growth. As an acceptance criterion, net present value rejects projects In which the value of any contribution to productivity and growth is less than the economic costs to be incurred in achieving that contribution. The criterion of net present value may be supplemented by other criterion. However, it should be used for all major transportation investments. Improve Cost-Benefit Analysis The validity of the net present value criterion however, hinges on an adequate cost-benefit analysis. As the survey for NCHRP 2-17(1) indicates, current practice of cost benefit analysis needs improvements, particularly in the following three areas. Annualize Benefits and Costs. A key requirement of any investment analysis under the net present value criterion is an accounting for annual benefits and costs realized over the life-cycle of a project. Discount Benefits and Costs Appropriately. Because a dollar tomorrow is worth less than a dollar in hand today future costs and benefrts must be discounted to comparable worth today. The accepted approach is to calculate the present value of benefrts and costs using a discount rate The same rate should be used for both benefits and costs. The choice of the correct discount rate is also important. A rate of seven percent is recommended for all Federal projects when benefits and costs are in constant dollars If the rate is too high, we will wrongfully reject projects whose benefits are concentrated in the long run. If the rate is too low we will accept projects whose benefrts are too far in the future to justify investment today. 58

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Use An Appropriate Analysis Period. A properly done cost-benefit analysis requires using the life-cycle of a p r oject as the analysis period If an analysis period is too short the project under consideration would in fact generate benefits well beyond the analysis period As a result these benefits wou l d be excluded Consider Investment Timing in Decision-Making Many factors may have contributed to a re l uctance to consider timing in the current process of decision-making for major transportation i nvestments. All but a few factors can be incorporated into a formal analysis of i nvestmen t timing. It i s often perce i ved that the election cycles bring opportunities that may not be duplicated in the future. Also, decision-makers tend to have a strong desire to do something and do it now. Both factors create a bias toward early implementation of investments. One effective approach to overcome these may be to require that all major transportation projects pass a test on the net present value criterion. Consider Timing in Investment Analysis Fo r investment timing to enter the decision mak i ng process for major transportation investments a critical factor is to consider tim i ng issues in investment analysis. The fundamental shortcoming of the current process and the for revisiting current practice merit serious consideration. Reflect on Timing Issues At a minimum planners should seriously reflect on the issues of investment timing These include the i mportance of investment timing in improving the economic worth of investments, barriers that prevent timing bei ng considered i n analysis and decision-making, and how investment tim i ng may be i ncorporated i nto the current process of investment analysis for major transportation projects. Planners should be prepared to educate decision-makers on the issue of investment t i ming. 59

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Use Economic Principles for Optimal Timing Preferably, there would be efforts by planners to use economic principles of investment timing to find the optimal timing. This would include applying these principles to determine whether a proposed project shou l d be delayed and how much it should be delayed Chapters 5, 6, and Appendix A offer ma n y of these princip l es under conditions of certa i nty. Include Built-Later Alternatives To find the opt i mal timing for an investment, one would need to estimate a series of net present values across an extended period of possib l e i nvestment timing. The timing is optimal when net present value reaches its maximum. Doing this could mean an enormous efforts because, in order to est i mate this series of net present values, one first needs to estimate a series of annual benefits and costs over the project's life-cycle for each net p r esent value estimated . One way to reduce these efforts is to only estimate net present values for a few years over an extended period. For example, net present va lue may be estimated for every five years over a period of 30 years. One can then choose the year that gives the largest net present value. This less extensive approach may not resuH in the optimal timing but will result in an improvement over what can be achieved if investment timing is ignored completely. These build-later aHematives can be analyzed along with those currently required for major transportation investments. Use Proxy Variables to Time Implementation As an approximation proxy variables for net present value may be used to time Investments after an initial decision of build-later I n the i nitial analysis, planners may evaluate the of the project's net present value to variables that are c l osely related to mar1
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implementation, one could establish perfonnance or condition targets as triggers for implementation. This type of indicator might reinforce the logic of the delay, provide an incentive for policies designed to help build transit market and provide clear flags for decision makers and the public. Deal with Uncertainty Uncertainty prevails in project appraisal. The importance of uncertainty in transportation planning is increasingly being recognized (Mierzejewski, 199 6) Use Tradftional Approaches Traditional approaches to addressing uncerta inty include sensitivity analysis, scenario analysis, and risk analysis. Sensitivity ana ly sis eva luates how sensitive the initial investment timing and the corresponding net present value are to changes in one of the many assumptions in an analysis. Scenario analysis, on the other hand, evaluates this sensitivity with respect to a set of assumptions that represent likely future scenarios. Unlike sensitivity a n a lys is or scenario analys is, risk analysis assigns a distribution on each assumption and produces distributions for investment timing and net present value, respectively (Pouliquen, 1970; Lewis, 19g5). Account for the Value of Waiting None of the traditional approaches to dealing with uncertainty, however, account for the value to wait i ng. When a project can be delayed and is irreversible once built, t his .value can be large. There is an opportunity cost of making an investment today by giving up the option of waiting for new information. T here is the possibility that new information is so unfavorable that the investment should never be built. One way to capture the value of waiting is the option valuation approach (Dixit and Pindyck, 1g94; Martzoukos and Teplitz-Sembitzky, 1992) The Wortd Bank has studied it for power plant planning (Crousillat and Martzoukos, 19g1). The results of a simple model under this approach are discussed in Chapter 6 and Appendix A. 61

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The approach also offers a timing rule for determining whether a today's investm ent is premature, overdue, or optimal in timing. If the resulting decision is waiting, the timing rule is used again in the same way in a subsequent analysis of the investment. There are two basic approaches to determine when a subsequent analysis should be done: the time approach, in which subsequent analysis is done on a fixed schedule, and the event approach, in which the timing of subsequent analysis is triggered by particular events (lntriligator and Sheshinski, 1986). The re is also the hybrid approach, in which either time or some event or set of events can trigger a subsequent analysis. Generally it is preferable to have events influence the timing of subsequent analysis. Sponsor National Forums The Transportation Research Board should sponsor workshops symposiums, or sessions in annual transportation meetings on issues relat ed to investment timing. These could include theories applications, deCision-making, case studies problems and guidance. 62

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Appendix A MODELS This appendix presents a basic model, derives the timing rules as shown in Chapter 6, compares these timing rules, and derives the growth conditions discussed in Chapter 5. Basic Model From in vesting in a major transportation project, society incurs capital costs to construct the project and enjoys a stream of annual benefds (net of operating, maintenance, and other societal costs) over the project's lifetime. T o quantify these benefrts, the stream of annual benefits is first discounted to the year of implementation and summed (the sum is called the value of the project, or simply project value). This sum is then compared with capital costs and, the difference is called net project value Net project value becomes net present value ifthe discounting is to the current year. Timing decisions are based on either net project value or net presen t value depending on investment objectives. Time affects net project value or net present value i n at least three ways. First, as a project ages, its annual benefrts may change with changes in the economy or ageinduced operation and maintenance costs. For example, growth in the economy may increase the annual benefrts of a project for a given level-of-service. A rail project may carry more passengers as the population and employment in the service area increa ses. Also, physical deterioration may require expensive maintenance and replacement to maintain a given level-of-services and, as a result drive down annual benefits. The second way that time affects the economic value of a project is through the timing of investment. On one hand postponing an investment may require a different le ve l of const ruction cost because of changes in real costs for construction. Postponing a project also may result in a different stream of annual benefits because of changes i n the demand for and supply of its services. To simplify matters, this paper focuses on annual benefits as the dominant source of change. Chu and Polzin (1996) consider 63

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both factors. On the other hand, postponing reduces the present values of a given amount of construction costs and a given stream of annual benefits. The net result of postponing can be significant. It is possible to increase the net present value of a project by postponing tt. It is even possible that postponing a project will change its net present value from a negative amount if constructed today to a posit ive amount if constructed later. The third way tha t time can affect the economic value of a project i s through uncertainty in its capttal costs and annual benefrts. There is a value of waiting to invest when the project can be delayed and is irreversible. This value of waiting exists because waiting maintains the option to invest and makes it possible to adopt a better decision when new information arrives. To simplify matters, we focus on annual benefrts as the do minant source of uncertainty. Crousillat and Martzoukos (1991) consider uncertainty for both costs and benefits. The following model, adopted from Dixit and Plndyck incorporates these effects of time on the economic value of a major t ransportation investment. The following are assumed: 1. Suppose that a community must decide when to invest in a single project, which has two important characteristics First, the costs are at least partly irreversible; in other words . sunk costs that cannot be recovered. Second, the project can be delayed so that the community has the opportunity to wait for new information to arrive about mar1 0. Both a and o are fixed. Mathematically, annual benefrts foll ow a geometric Brownian motion. Brownian motion is a continuous time Mar1
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state of the variable detennines what may happen to the variab l e in the Mure. When the natural logarithm of a variable follows a Brownian motion, the variable is said to follow a geometric Brownian motion. One advantage of this particu lar fonn of uncertainty is that the problem of maximizing expected net present value has a closed so l ution 3. The cost of the investment in today's dollars, K is known and fiXed. As ment i oned earlier, we focus on annual benefrts as the dominant source of changes and uncertainty 4. The commu nity detennines a pcint at which it is optimal to invest. How the community detennines this depends on its objective and whether uncertainty exi sts. Its objective may be simply to achieve a positive net present value, to maximize the net present value of the project under conditions of certainty, or to maximize the expected net present value under conditions of uncertainty. The net present value of the project is given by NPV(t) = ( V(t) -K) e (1) where p is a discount rate and V(t) is the va l ue of the project i f the investment is made at time t. It can be shown that V(t) relates to B(t) in the following way (Dixit and Pindyck, 1994, p. 144): V(t) = E J B(s)e
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The Case of Positive NPV If the community's objective is to achieve a net present va lue, it will invest when NPV(t) > 0 or the following is true: V(t) > K (3) rega rd les s of whether uncertainty exists. If Ieday's project value V(O) > K the community would invest now, even though project value will grow later. If V(O) < K, it would wait until project value exceeds capital costs. There is a value to waiting when V(O) < K because eventually V(t) will exceed K The net present value of the project Wr. at the time of inves tment Tis: w, = (V(7) -K ) e -or (4) T im in g rule (1) is from equation (3), which extends the basic investme nt rule that invest if the net present value of a project i s at least as large as its capHal costs; never invest otherwise. Rule (2) can be derived from rule (1) using the re latio nship between project value and annual beneffls in equation (2). Rules (1) and (2) apply under conditions of both certainty and unce rtainty Ru l e (3) can be derived from rule (1) and equation (6) below. Specifically, one can solve fort in rule (1) by first substituting V(t) i n equation (6) Rule (3) is only applicable under certainty. Under uncerta in ty, one can only determine the expected le ngth of the period at wh ic h the net present value is posHive This expected length can differ from Tr s h own in Table 1. Rules (1) and (2) apply under both certa i nty and u ncertainty The Case of Certainty and Maximizing NPV When annual benefits are certain, the standard deviation, o, becomes zero. I t can be shown then that Mure values of annual benefits become ( Dixit and Pindyck 1994): 66

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B(t) = 8(0) e t (5) where 8(0) is today's annual benefits. That is, annual benefrts grow at an annual constant rate of a. This is a common assumption in the literature of investment timing under certainty. Combining equations (2) and (5) gives the following equation for future project values: V(t) = V(O)e ot (6) The community can determine a future time to invest f rom maximizing the net present value of the project given by equation (1 ) The net present value of the p roject will become positive at some point even if today's project value V(O) < K, because eventually V(t) will exceed K. One difference between this case and the case of positive NPV is that even if V(O) now exceeds K, it may be still better for the community to wait rather than i nvest now. The maximum net p resent value of the project, We, is V(O) -K if V(O) > pKI(p-a); it is the following otherwise: W = aK [(p-a)V(O)]PI"' c p-a pK (7) Timing rules (4) -(6) can be derived from maximizing net p resent value in equation (1) with future project values given in equation (6). Specifically, the first-order condition is -[(p-a)V(t)-pK]e-P' = 0 (8) As long as today's project value V(O) is not too much larger than costs K, it is optimal to waH. Altematively, as l ong as today's annual 8(0) are not too much larger than annual interest costs, it is optimal to wail. Rules (4) and (6) result from solving equation (8) for the ratio of V(t) and K and for timing t, respectively. Rule (5) can 67

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be obtained from rule (4) and the relat ionship between project value and annual benefits in equation (2). The Case of Uncertainty and Maximizing Expected NPV Under uncertainty, B(t) evolves stochastically. One will not be able to determine a future time for investment as it could from maximizing expected net present value. Rather, one can derive a value of project value, at which it is optimal for the community to invest. Using methods of dynamic prog ramm ing or contingent claims analysis, Dixit and Pindyck (1994) show that is optimal to invest when the value of the project exceeds a critica l value given by: where v = LK ll-1 ll = 0.5 :. + ( ; 0.5) + 2.. a (9) (10) Thus, rule (7) holds: invest when V(t)/K Rule (8) results from rule (7) using the relationship between annual benefits and project value shown in equation (2). Unlike under certainty, where the c ritical value for timing i s optimal, the critical value for timing in rule (9) is the expected value of optimal timing. As shown by Martzoukos and Templitz-Sembitzky (1992), the expected optimal timing is given by: Tu = llog[L --.!S...] a ll-1 V(o) (11) The maximum expected net present value of the project is given by the following: Wu = (V' (12) 68

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Comparisons One way to compare these timing rules is through comparing the critical values. In fact, the following relationships are true : Cr < Cc < C0 ; Br a> 0. To see Ce < C0 and Be< B0 we use a relationship from Dixil and Pindyck (1994, p 145): _L ll-1 = _e_ + p-a ..!. o'll 2 p-a (13) As a result we have the following relationships between see Ce and Cu and Be and B0 : (14) and (15) Thus, maximization of net present value under certainty defers i nvestment so as to take advantage the possibility that annual benefrts and hence net present value of the project will grow later. Furthermore, unc ertainty defers investment so as to receive more information about the future evolution of uncertain annual benefits and project value Growth Conditions To show the three growth conditions unde r certainty in Chapter 5, lets represent project age (or more precisely, s-t present project age) and B(s,t} be annual net benefits, depending on both investment t iming and project age. The three specific growth conditions, growth without shift, upward shift, and horizontal shift, correspond to 69

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the following three special cases of annual net benefits: B(s,t) B(s) (16) B(s t) B(s-t)e' (17) B(s,t) = B(s-t)e (18) where both a and b are positive parameters In each of these cases, project value grows with investment timing. Annual net benefits and project value relate as follows : V(l) JB(s,t)e-<-Ods assuming a very long life cycle for the project. 70 (19)

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Appendix B REFERENCES American Association of State Highway and Transportation Officials (AASHTO) (1977) A Manual on User Benefit Analysis of Highway and BusTransit Improvements. Washington D.C. : AASHTO. Ashley DJ (1980) Uncertainty in the context of highway appraisal. Transportation 9: 249-67. Chu X and Polzin S (1996) Considering Build-Later as an Alternative in Major Investment Analysis. Transportation Research Board Preprint No. 9600669. Crousillat E and Martzoukos S (1991} Decision Making under Uncertainty: An Option Valuation Approach to Power Planning. Washington, D.C : Industry and Energy Department, World Bank. Deen TB, Kulash WM, and Baker SE ( 19 76) Critical decisions in the rapid transit p lanning p r o cess. Transportation Research Board 559: 33-43. Dixit AK and Pindyck RS (1994} Investment under Uncertainly. Princeton, New Jersey: Princeton University Press. Executive Order No.12893 ( 1994) Principles for federal infras t ructure investments Federal Register 1994 Compilation and Parts 100-102: 854-857. Euritt MA, Hoffman MA, and Walton CM (1990) Conceptual model of the fixed-guideway decision process. Transportation Research Board 1266: 152-62 .. Federal Highway Administration (FHWA) (1996) Exploring the Application of 71

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Benefii/Cost Methodologies to Transportation Infrastructure Decision Making. No. 16, Searching for Solutions: A Policy Discussion Series. Washington, D.C.: U.S. Department of Transportation. Federal Transit Administration (FTA) (1994) Revised Measures for Assessing Major Investments: A Discussion Draft. An FTA Policy Paper Washington, D.C.: U.S Department of Transportation Georgi H (1973) Cost-Benefit Analysis and Public Investment in Transport: A Survey london: Butterworths, pp. 175-81. Gifford JL, Uza r ski DR, and McNeil S ed. (1993) Infrastructure: Planning and Management Part II, Guessing the Future: Coping with Uncertainty in Infrastructure Planning New York: American Society of Civil Engineers Hirschman I Hoover J and Benz G (1991) Social equity implicat ions of UMTA's major investment policy. Transportation Quarterly 45: 43-53. lntriligator MD and Sheshinski E (1986) Toward a theory of planning. In Heller, Walter P., Starr, Ross M., and Starrett, David A., ed. Essays in Honor of Kenneth J. Arrow. New York: Cambridge University Press, pp. 135-58. Johnston RA, Sperling D, Deluchi MA, and TracyS (1988) Politics and technical uncertainty in transportation investment analysis. Transportation Research 2 1A: 459-75 Johnston RA and Deluchi MA ( 1989) Evaluation methods for rail transij projects Transportation Research-A 23A: 317-25. Khisty CJ (1993) Matching transportation planning methods to cope with uncertainty In Selected Proceedings of the Sixth World Conference on Transport Research: Lyon '92, 2141-47.' Lyon, France 72

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Lewis D (1992) Primer on Transportation, Productivity and Economic Development. National Cooperative Highway Research P r ogram, Report 342. Washington, D.C.: Transportation Research Board Lewis D (1995) The future of forecasting: risk analysis as a philosophy of transpo rt ation planning TR News 177: 3-9. Marglin SA (1963) Approaches to Dynamic Investment Analysis Amsterdam: North Holland Publishing Company. Martz o uk o s SHand Tepl itz-Sembitzky W (1992) Optimal timing o f transmission line investments in the face of uncertain demand: An o p ti o n val uati o n a ppro ach Energy Economics 14: 3-10. McDonald Rand Siege l D (1986) The value of waiting to invest. Quarterly Journal of Economics 101: 707-27 Mierzejewski EA (1996) An Assessment of Uncertainty and Bias: Recomme nded Modifications to the Urban Transportation Planning Process. Unpublished Dissertat i on Tampa, FL : Department of Civil and Environmental Engineering, University of South Florida. Office of Management and Budget (OMB) (1992) Guidelines and discount rates for benefit cost analys i s of federal programs. Circular A 94. Pearman AD (1977) Uncertainty and the transport i nvestment decision In Transport Decision in an Age of Uncertainty. Proceedings of the Third World Conference on Transport Research, Rotterdam, 26-28 April, 1977 Boston: The Hague pp. 431-37. Pell CM and Meyburg AH (1985) Sources of error and their imp li cations for uncertainty in urban transportation p l anning forecasts Paper prepared for the 64th Annual Meeting of 73

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the Transportation Research Board. Polz in S (1992) Alternative evaluation, in Proceedings of the UMTAIAPTA Workshop on Fixed Guideway Planning, Philadelphia, Pennsylvania, June 12-14, 1991. Washington D.C.: U .S. Department of Transportation, pp. 35-39 Pouliquen L Y (1970) Risk Analysis in Project Appraisal. World Bank Staff Occasional Papers, No. 11. Distributed by The John Hopkins Press, Baltimore. Rietve ld P (1995) Introduction to the special issue : infrastructure and spatia l economic development. Annals of Regional Science 29: 117-19. Snickers F ( 1 989) Infrastructure: a treat for regional science approach. Annals of Regional Science 23: 251-53. Stowers JR Reno AT, and Boyar VW (19 83) Improving Decision-Making for Major Urban Transit Investments. National Cooperative Transit Research and Development Program, Report 4 Washington. D.C.: Transportation Research Board Szymanski S (1991) The optimal liming of infrastructure Investment. Journal of Transport Ec onomics and Policy 25: 247-58. Urban Mass Transportation Administration (UMTA) (1976) Major urban mass transportation investments. Federal Register41: 41.512-41.514. UMTA (1984) Urban mass transportation major capital investment policy Federal Register49: 21284-21291. UMTA (1989) Procedures and Technical Methods for Transit Project Planning. Washington D.C.: U.S. Department of Transportation. 74

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. U.S. Department of Transportation (1993) Metropolitan transportat ion p l anning and programm i ng Federal Register 58: 58070-58076 US Army Corps of Engineers (USAGE} (1992} Guidelines for Risk and Uncertainty Analysis In Water Resources Plann i ng, Volume I-Princip les with Technical Append i ces USAGE, I n stitute for Water Res o urces, Ft. Belvoir. 75