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Considering build-later for major transit investments
Transpn Res.-A, Vol. 32, No. 6, pp. 393±405, 1998 # 1998 Elsevier Science Ltd. All rights reserved. Printed in Great Britain 0965-8564/98 $19.00+0.00
Pergamon PII: S0965-8564(98)00123
CONSIDERING BUILD-LATER FOR MAJOR TRANSIT INVESTMENTS XUEHAO CHU* and STEVEN E. POLZIN
Center for Urban Transportation Research, University of South Florida, 4202 E. Fowler Av., CUT 100, Tampa, FL 33620, U.S.A. (Received 6 August 1995; in revised form 17 January 1998) AbstractÐThis paper uses a theoretical model of bene®t±cost analysis to consider the timing of major transit investments. The model takes into account the net bene®ts of a project, the variation of net bene®ts with project age and investment timing, capital cost and its growth, and the discount rate. Three questions are examined: (1) when might build-later be evaluated? (2) how do changes in the discount rate and other parameters of an investment aect its optimal timing? and (3) how signi®cantly do dierences in the stream of net bene®ts from an investment aect its optimal timing? The ®rst two questions are examined analytically, and the last question is examined numerically. Implications are discussed with respect to planning for major transit investments. # 1998 Elsevier Science Ltd. All rights reserved Keywords: major investment study, build-later, net present value, investment timing, discount rate 1. INTRODUCTION
Investing in a major transit project results in costs to society to construct the project and produces a stream of net bene®ts (net of operating, maintenance, and other societal costs) over the project's lifetime. To quantify these net bene®ts, the stream of net bene®ts is ®rst discounted and summed (this sum is called the present value of the stream of net bene®ts); this sum is then compared with the present value of construction costs, and the dierence is called the net present value. Time aects the net present value of a project in at least two ways. First, as a project ages, its net bene®ts may change with changes in the economy or age-induced operation and maintenance costs. For example, growth in the economy may increase the net bene®ts 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 increase. Also, physical deterioration may require expensive maintenance and replacement to maintain a given level-of-services and, as a result, drive down net bene®ts. The second way that time aects the net present value of a project is through the timing of investment. On one hand, postponing an investment may require a dierent level of construction costs because of changes in real costs for construction. Postponing a project may also result in a dierent stream of net bene®ts because of changes in the demand for and supply of its services. On the other hand, postponing also reduces the present values of a given amount of construction costs and a given stream of net bene®ts. The net result of postponing can be signi®cant. 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 its net present value from a negative amount if built today to a positive amount if built later. This paper is motivated by a concern that the current planning process for major transit investments does not adequately consider the timing of investments (USDOT, 1984, 1993). This process is part of the Metropolitan Transportation Planning Process (USDOT, 1993), which includes three components: major investment studies (MIS), the transportation plan, and the transportation improvement program (TIP). Many major projects, especially transit guideway projects, are often conceived by sta or decision-makers and are usually ®rst explored as a scenario in the development of the transportation plan. Projects moving from the transportation plan toward TIP for implementation are those perceived to be of the highest priority. However, a systematic approach seldom exists to determine whether the `highest' priority in the transportation *Author for correspondence. Fax: 001 813 974 5168; e-mail: [email protected]
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plan is the highest priority for immediate implementation. Major investment studies typically look at performance in the design year, usually 15±25 yr from now. 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 of investments or results in erroneous decisions by favoring a build-now alternative in the absence of a build-later alternative as an option in the choice set. This paper has three objectives. One is to explore the timing issue by answering three questions. First, when might build-later be evaluated? For example, should build-later be evaluated when an area is facing a declining or stagnant economy or when an area is facing a growing economy? Second, how do changes in the parameters of an investment project aect the time at which it should be built? The Federal Transit Administration (FTA) changed the discount rate for major transit investment projects from 7% for all projects to 4.9% for projects with a lifetime of at least 30 yr and to a percentage between 4.2 and 4.9 for projects with a lifetime less than 30 yr and, recently, changed it back to 7%. One might expect that these changes in the discount rate could have a signi®cant eect on the time at which a project should be built. How these changes aect investment timing cannot be assessed without considering build-later. Third, how do dierences in the pro®le of a project's impacts aect the time at which it should be built? By answering these questions, we hope encourage further discussion about the timing issue. The second objective is to help planners and decision-makers realize the potential bene®ts of considering build-later for major transit investments. One way to help avoid the potential biases toward early implementation of major transit investments is to increase the awareness of the issue of investment timing. The issue has had little discussion in the literature on major transit investment projects. An exception is Polzin (1992), who suggested in a workshop sponsored by the Urban Mass Transit Administration (UMTA) that build-later be considered in alternatives analysis. The third objective is to assist agencies charged with conducting investment analyses by providing them with a mathematical framework to analyze investment timing in general and a buildlater alternative in particular. The ultimate objective is to have a mechanism in place that helps consider not only what project should be built and whether it should be built, but also when it should be built. Fortunately the literature on the timing of public infrastructure investment has plenty to oer on how to model investment timing (Marglin, 1963; Porter, 1984; Leconte et al., 1987; Szymanski, 1991). The value of having a technical capability to consider build-later alternatives is particularly important in light of election cycles and the strong preferences of decision-makers for a build-now alternative. It is sometimes perceived that the congruence of factors that may enable a build decision or a positive 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 the apparent economic or logical arguments regarding investment timing. The presence of a supportive city council, a strong local legislative delegation at the state level, or the presence of the right person in the right committee position at the federal level is compelling justi®cation for a build-now decision. The election cycles bring opportunities that may not be duplicated in the future when the analytically optimal time arrives. The discretionary nature of project funding perpetuates this sensitivity of decision-makers. Another factor biasing decision-making to build-now decisions is the strong emotional appeal of a `do something' mentality. Frustrations with congestion, a cynical attitude toward government, a disdain of bureaucracy and process, and a perception that we `plan projects to death' often result in a strong predisposition to an action-oriented decision. This creates a populist sense of action and serious eorts to actually solve problems. It implies a decisiveness and leadership level that can be very appealing for decision-makers. On a more pragmatic note, it also increases the chances that some of the bene®ts 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 planning and design or initial eorts to buy right-of-way as opposed to ribbon-cutting ceremonies; yet, these actions can provide a strong constituency for decision-makers. The paper is also related to the general literature on public infrastructure planning and the literature on evaluating and planning for major transit investment projects. The strong interest in
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public infrastructure planning 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 al. (1976), Johnston et al. (1988), Johnston and Deluchi (1989), Euritt et al. (1990), Hirschman (1991), UMTA (1989), and FTA (1994), have evaluated the planning process for major transit investment projects from a variety of perspectives. None of these authors, however, considered the timing issue. The issue of investment timing is conceptually not unique to transit investments. The principles of investment timing presented in this paper apply to major transportation investments across many modes. However, it may be more relevant for transit, given empirical data on the performance 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-ful®lling 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 and perhaps intelligent transportation systems (ITS), 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 justi®ed. In this paper, we ®rst present the model of bene®t±cost analysis for investment timing. Then we analyze the model for a number of special cases in order to answer the ®rst two questions mentioned earlier. We then analyze the model for a number of numerical examples to answer the third question. Finally, we discuss the implications of the results to the practice of major transit investment analyses and decision-making. 2. THE MODEL
This section presents a model of bene®t±cost analysis to examine timing for major transit investment projects. As discussed earlier, time aects the net present value of a project through both project age and the time at which the project is implemented. These two eects are incorporated in the bene®t-cost model, which consists of the following assumptions: 1. The project costs K t to construct at time t: K t is incurred entirely at time t: K t is a nondecreasing function of time t : K0 t50 2. The right-of-way costs M; M is incurred entirely at time 0 (`now') 3. The lifetime of the project is T 4. The project generates a stream of net bene®ts given by B s; t if it is built at time t and t4s4t T: s ÿ t may be referred to as project age 5. All bene®ts and costs are measured in the prices at time 0 6. The planner chooses a time, t, to maximize the net present value of the project, NPV t. That is, the planner faces the following problem: tT
Maxt NPV t
B s; teÿrs ds ÿ K teÿrt ÿ M
1
t
where the ®rst term measures the present value of the stream of net bene®ts and the second term measures the present value of construction costs. This model is based on the framework by Marglin (1963) and extends those of Porter (1984), Leconte et al. (1987), and Szymanski (1991) by allowing the stream of net bene®ts to shift with the timing of investment. In addition, it extends all these models by allowing construction costs to grow over time and a lump-sum cost to incur at time 0 regardless of the timing of investment. 3. ANALYTICAL ANALYSIS
This section considers ®ve special cases of the above model. The purpose is to answer the ®rst two questions mentioned earlier: When might build-later be evaluated? How do changes in the
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discount rate and other parameters of an investment aect optimal timing? The approach is analytical as opposed to the numerical analysis that follows. 3.1. Case 1. B s; t B s ÿ t; t4s4t T In this case, the stream of net bene®ts from investing at time t2 is the same as the stream from investing at time t1 (Fig. 1). In other words, one stream is a parallel shift of the other. Under this case, the solution to the problem is independent of time, and the optimal rule is to invest now or never. There is no advantage to build later when the stream of net bene®ts repeats itself with timing changes. NPV t for this case can be written as follows: 0T 1 2 NPV t eÿrt @ B ueÿru du ÿ K tA 0
If the integral 4K 0 M, never invest. This is because the net present value is non-positive for any t under the assumption that K t is a non-decreasing function. If the integral > K 0 M, the investment should be made now. This is because NPV 0 > 0 but NPV t declines with t. 3.2. Case 2. B s; t B s In this case, the stream of net bene®ts does not shift with the timing of investment. The stream of net bene®ts from investing at time t2 is part of the longer stream from investing at time t1 s; t2 (Fig. 2). This case would be relevant when there are external forces that aect the stream of net bene®ts in a particular way. There are three outcomes of such external eects on the stream of net bene®ts as described in the next three cases. 3.3. Case 2a. B0 0 In this case, net bene®ts stay constant with project age. The solution to the problem is independent of time, and the optimal rule is to invest now or never. The intuition is that there is no advantage for investing later when net bene®ts stay constant. The NPV t in this case can be written as follows: B o ÿ 1 ÿ eÿrT ÿ K t ÿ M 3 NPV t eÿrt r
Fig. 1. Parallel stream of net bene®ts.
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Fig. 2. Increasing stream of net bene®ts.
If the ®rst term in the parentheses 4K 0 M, never invest. If that term > K 0 M, invest now. This case would be relevant for metropolitan areas that expect a stagnant demand for services provided by the investment. 3.4. Case 2b. B0 s < 0 In this case, net bene®ts decline with project age. The solution to the problem is again independent of time, and the optimal rule is to invest now or never. The intuition is that there is even less advantage to invest later when net bene®ts decline with project age than when net bene®ts stay constant with project age. This case would be relevant for metropolitan areas that expect a declining demand for the services provided by the investment. 3.5. Case 2c. B0 s > 0 In this case, net bene®ts grow with project age. This case would be relevant for urban areas that expect a growing demand for the services provided by the investment. This case is particularly interesting because build-later can be optimal. The ®rst-order condition of the problem is: ÿ ÿ dNPV t ÿ eÿrt B t ÿ B t TeÿrT rK teÿrt ÿ K0 teÿrt 0 dt
4
That is, changes in net bene®ts and construction costs from postponing are equal at the margin. This ®rst-order condition can be written alternatively as B t K0 t B t TeÿrT rK t
5
The left-hand-side is the cost of postponing, including net bene®ts foregone, B t , and increases in construction cost, K0 t. The right-hand-side is the bene®ts of postponing, including bene®ts gained at the end of the lifetime and interest saved. Whether build-later is optimal is determined by whether (4) has an interior solution. A necessary condition for an interior solution is the following: B 0 < Bc rK B TeÿrT ÿ K0 0
6
That is, B 0 should be relatively small; otherwise it is better to invest now or never. The parameter Bc may be interpreted as the critical level of B 0 at which build-now would just barely be optimal.
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3.6. Case 3. B s; t B sea sÿt ; B s Begs ; g50 This case becomes cases 1 and 2 when B0 s 0 and a 0, respectively. In order to obtain an explicit solution for optimal timing, it is also assumed that K t Kebt ; b50 and that T 1. The solution is independent of time when g b or a5r ÿ g or r4g or r4b, i.e. when: (1) net bene®ts and construction costs grow at the same rate; (2) the discount rate is no greater than the growth rate of net bene®ts; (3) net bene®ts shift with project age at a rate no greater than the dierence between the discount rate and the growth rate of net bene®ts; or (4) the discount rate is no greater than the growth rate of construction costs. The following analysis further assumes that none of the above conditions occurs. The ®rst-order condition of the problem is r ÿ bKebt Begt
rÿg rÿgÿa
7
Construction costs saved from postponing equal net bene®ts foregone at the margin. The secondorder condition requires that g > b, that is, net bene®ts grow faster than construction costs. Solving (7) for t results in the optimal timing given by: 1 r ÿ bK rÿgÿa ln ln 8 t gÿb B rÿg where r is the discount rate, b is the growth rate of construction costs, g is the growth rate of B s, and a is the shifting rate of net bene®ts with project age. It is interesting to determine the direction of change in optimal timing from an increase in the value of a given parameter. The results are shown in Table 1, which are summarized as follows: 1. Optimal timing becomes later when the discount rate increases. An increase in the discount rate both increases the bene®ts and decreases the costs of postponing. 2. Optimal timing becomes later with an increase in the base construction costs at time 0, K. An increase in the base construction costs increases the bene®ts of postponing because of savings in construction costs. 3. Optimal timing becomes earlier with an increase in the net bene®ts at time 0, B. An increase in the base net bene®ts increases the costs of postponing because of net bene®ts foregone. 4. An increase in the growth rate of capital costs aects optimal timing indeterminately. This ambiguity results from two opposite eects of an increase in the growth rate of capital costs on the bene®ts of postponing. 5. Optimal timing becomes earlier with an increase in the growth rate of net bene®ts. This is because an increase in the growth rate of net bene®ts increases the costs of postponing due to net bene®ts foregone. 6. Optimal timing becomes earlier when the shifting rate of net bene®ts, a, increases. An increase in the shifting rate of net bene®ts increases the costs of postponing. 4. NUMERICAL ANALYSIS
This section solves the problem for a range of three examples of Case 2c, where net bene®ts grow with project age. The purpose is to answer the following question: How signi®cantly do differences in the stream of net bene®ts aect optimal timing? The approach is numerical. Table 1. Comparative static analysis Parameters r:discount rate K:base capital cost b:construction cost growth rate B:base net bene®ts g:net bene®t growth rate a:shifting rate of net bene®ts
Direction of changes in optimal timing from an increase in a parameter in column 1 Later Later Indeterminate Earlier Earlier Earlier
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4.1. Model speci®cation The model is speci®ed as follows. The lifetime T is 40 yr. The discount rate r is 7%. The project costs K 1; 000; 000 if built now. Construction costs grow at a constant rate of b 0:01 or K t Kebt . The cost to acquire the right-of-way M is $100,000. These values are shown at the bottom of Table 2. In addition, Table 2 shows the example streams of net bene®ts. They are convex (growing at an increasing rate), linear (growing at a constant increment), and concave (growing at a decreasing rate). The horizontal stream is included for comparison. The constants in these functions, bi i 1; 2; 3; 4, are determined as follows: the constant for the linear one is arbitrarily chosen to be $10,000; and the others are chosen such that these streams result in the same net present value if the project starts now. The resulting values are shown in Table 2 under `Net bene®t at time 0.' Also shown in Table 2 for each steam of net bene®ts are the values of Bc , the critical level of B 0 at which build-now would just barely be optimal. The four streams of net bene®ts are plotted in Fig. 3. Alternatively, these streams may be compared with respect to the cumulative distributions of their discounted values shown in Fig. 4. The farther away a distribution is from the bottom right-hand corner, the larger is the proportion of the bene®ts that materialize in the early years. These cumulative distributions capture the dierences among these streams of net bene®ts better than a simple plotting of them as shown in Fig. 3. 4.2. Results For the speci®cations of the model shown in the note to Table 2, the problem is numerically solved for each of the four streams of net bene®ts. The results are shown in Table 2. The optimal timing is 13, 9, 3, and 0 yr from now for the convex, linear, concave, horizontal streams of net bene®ts, respectively. If the project is to be built now, these alternative streams would generate the same net present value of $603,400 because of a constraint imposed on the constants of the net bene®t functions shown as in Table 2. If the project is to be built at its optimal time, however, the net present value would increase by 92, 44, 8, and 0% for the four examples, respectively. These dierences in optimal timing and improvements in net present values are better re¯ected in Fig. 5, which shows how the net present value varies with the time of investment for each of the four streams of net bene®ts. First, these curves have the same value at time 0 because of the constraint mentioned above. Second, the net present value for the horizontal stream decreases over time, implying that build-now is better than build-later. Third, the other curves reach their maximum after time 0 (at years 13, 9, and 3, respectively), implying that build-later is better than build-now. Fourth, when build-later is optimal, build-later would be better than build-now during some period around the optimal timing. The later the optimal time is, the wider this period is. Table 2. Numerical comparisonsa Streams of net benefits Convex Net bene®t function:B t Net bene®ts at time 0:B 0 Critical B 0 : Bc Optimal timing (yr):t Improvement in NPV (%) a
Linear 1:5
b1 t 1 2,321 97,053 13 92%
b2 t 1 10,000 84,932 9 44%
Concave
Horizontal 0:5
b3 t 1 38,520 74,999 3 8%
b4 127,000 67,732 0 0%
NPV of investing at time t is computed as follows: NPV t
tT t
B seÿrs ds ÿ K teÿrt ÿ M
K t Kebt The parameters are set as follows: Construction cost growth rate
Discount rate
Lifetime (yr)
Right-of-way cost ($)
Initial capital cost ($)
b 0:01
r 0:07
T 40
M 100; 000
K 1; 000; 000
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Fig. 3. Example streams of net bene®ts.
Fig. 4. Cumulative distribution of discounted net bene®ts.
5. REGULATIONS AND PRACTICE
5.1. Regulations Federal regulations on economic analyses of transportation investments require some of the elements necessary for an economic analysis of investment timing, but fail to require analysis of investment timing. Three such regulations are discussed below.
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Fig. 5. Net present value.
5.1.1. Executive Order 12893 (Federal Register, 1994). This document sets forth principles for Federal Infrastructure Investments. The order requires all Federal agencies with infrastructure responsibilities to conduct systematic analysis of expected bene®ts and costs for all infrastructure investments, including both quantitative and qualitative measures, in accordance with the following guidelines: (a) Bene®ts and costs should be quanti®ed and monetized to the maximum extent practicable. All types of bene®ts and costs, both market and non-market, should be considered. To the extent that environmental and other non-market bene®ts and costs can be quanti®ed, they shall be given the same weight as quanti®able market bene®ts and costs. (b) Bene®ts and costs should be measured and appropriately discounted over the full life cycle of each project. Such analysis will enable informed tradeos among capital outlays, operating and maintenance costs, and nonmonetary costs borne by the public. 5.1.2. OMB Circular A-94 (OMB, 1992). This circular gives Guidelines and Discount Rates for Bene®t±Cost Analysis of Federal Programs. The circular recommends bene®t-cost analysis as the technique to use in a formal economic analysis of government projects. The circular recognizes net present value as the standard criterion for making decisions on government projects on economic principles. The circular requires the use of a real discount rate of 7% in discounting future bene®ts and costs measured in constant dollars. Restated below are two related sections: 5a. The standard criterion for deciding whether a government program can be justi®ed on economic principles is net present valueÐthe discounted monetized value of expected net bene®ts (i.e. bene®ts minus costs). Net present value is computed by assigning monetary values to bene®ts and costs, discounting future bene®ts and costs using an appropriate discount rate, and subtracting the sum total of discounted costs from the sum total of discounted bene®ts. Discounting bene®ts and costs transforms gains and losses occurring in dierent 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.
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8b1. Constant±dollar bene®t-cost analyses of proposed investments and regulations should report net present value and other outcomes determined using a real discount rate of 7%. This rate approximates the marginal pretax rate of return on an average investment in the private sector in recent years. 5.1.3. Criteria for New Starts. The Federal criteria for new starts relied on cost-eectiveness measures with little attention devoted to the criterion of net present value from 1976 to 1984 (Johnston and Deluchi, 1989). This is re¯ected in UMTA's policy statements (USDOT, 1976, 1984). Despite the OMB Circular A-94 and Executive Order 12893, Federal Transit Administration continues to rely on cost-eectiveness measures (FTA, 1994; USDOT, 1996, 1997). FTA considers bene®t±cost analysis as the desirable basis for project evaluation. The agency, however, rejects the use of bene®t±cost analysis in actual evaluations because it believes that the problems of quanti®cation are too great. Johnston and Deluchi (1989) believe that FTA underestimates the practicality of conducting bene®t-cost analysis for major transit investments. 5.2. Current practice Federal regulations thus recommend that net present value be used as the evaluation criterion; that annual bene®ts and costs be measured in constant dollars for the full life-cycle of a project; and that annual bene®ts and costs be discounted at a real discount rate of 7% to calculate net present value. However, current practice of investment analysis for major transportation investments, including major transit investment, poorly meets these regulations. According to Lewis (1992), a 1990 survey of 35 transportation projects conducted for the National Corporative Highway Research Program indicates that: . About one third of the projects use net present value as a basis for evaluation. . Most projects fail to express costs and bene®ts on an annual basis over the life-cycle of the project. . A large number of studies failed to use an appropriate analysis period. . Only about 5% use adequate discounting techniques and properly justi®ed discount rates. The sample of 35 projects includes 10 airport and air trac control-related projects, 10 highway projects, 2 high speed rail systems, 5 ports, 2 inland waterway projects, and 6 public transit proposals. The sample was drawn from a larger universe with a four-factor strati®cation: location and scale, mode, type, and point of approval. Location and scale include national, regional and local projects and project size. Mode includes highway, public transit, rail, ports, airports and inland waterways. Type includes construction, reconstruction, and repair. Point of approval covers appraisals in progress, projects rejected, and projects approved (including projects completed, projects in-progress, and those not started). In addition, analysis is 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 recognize the importance of investment timing, though some federally-sponsored conferences and research projects do (Lewis, 1992; FHWA, 1996). 6. TOWARD AN OBJECTIVE APPROACH
The current practice of analysis and decision-making for major transit investments fails to systematically consider the timing of such investments. Many factors contribute to this failure. Election cycles, discretionary project funding, and politicians' desire for action now all have the tendency to create biases toward early implementation of major transit projects. Federal regulations do not help either by failing to recognize the importance of investment timing and even bene®t±cost analysis in the case of transit new starts. In an era of scarce resources, it is increasingly important to improve the economic worth of major transit investments. One eective way to achieve such improvements is to develop an objective approach for better analysis and decision-making regarding the timing of these investments. The following discusses some of the elements that might be part of the approach.
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6.1. 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 living. 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 transit investments. 6.2. Improve bene®t±cost analysis The validity of the net present value criterion, however, hinges on an adequate bene®t±cost analysis. As the survey for NCHRP 2-17(1) indicates, current practice of bene®t±cost analysis needs improvements, particularly in the following three areas. 6.2.1. Annualize bene®ts and costs. A key requirement of any investment analysis under the net present value criterion is an accounting for annual bene®ts and costs realized over the life-cycle of a project. 6.2.2. Discount bene®ts and costs appropriately. Because a dollar tomorrow is worth less than a dollar today, future costs and bene®ts must be discounted to comparable worth today. The accepted approach is to calculate the present value of bene®ts and costs using a discount rate. The same rate should be used for both bene®ts and costs. The choice of the correct discount rate is also important. A rate of 7% is recommended for all federal projects when bene®ts and costs are in constant dollars. If the rate is too high, we will wrongfully reject projects whose bene®ts are concentrated in the long run. If the rate is too low, we will accept projects whose bene®ts are too far in the future to justify investment today. 6.2.3. Use an appropriate analysis period. A properly done bene®t-cost analysis requires using the life-cycle of a project as the analysis period. If an analysis period is too short, the project under consideration would in fact generate bene®ts well beyond the analysis period. As a result, these bene®ts would be excluded. 6.3. Consider investment timing in decision-making Many factors may have contributed to a reluctance to consider timing in the current process of decision-making for major transit investments. Most factors can be incorporated into a formal analysis of investment timing, except two factors. It is often perceived that the election cycles bring opportunities that may not be duplicated in the future. Also, decision-makers tend to have a strong emotional appeal that they need to do something and do it now. Both factors create a bias toward early implementation of investments. One eective approach to overcome these may be to require through regulations that all major transit projects pass a test on the net present value criterion. 6.4. Consider timing in investment analysis For investment timing to enter the decision-making process for major transit investments, a critical factor is to consider timing issues in investment analysis. The fundamental shortcoming of the current process and the opportunities for revisiting current practice merit serious consideration. 6.4.1. Re¯ect on timing issues. At a minimum, planners should seriously re¯ect on the issues of investment timing. These include the importance of investment timing in improving the economic worth of investments, barriers that prevent timing from being considered in the planning process, and how investment timing may be incorporated into the current planning process for major transit investments. Planners should be prepared to educate decision-makers on these issues. 6.4.2. Use economic principles for optimal timing. Preferably, there would be eorts by planners to use economic principles of investment timing to ®nd the optimal timing. This would include
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applying the framework developed in this paper to determine whether a proposed project should be delayed and how much it should be delayed. 6.4.3. Include built-later alternatives. To ®nd the optimal timing for an investment, one would need to estimate a series of net present values across an extended period of possible investment timing. The timing is optimal when net present value reaches its maximum. Doing this could mean enormous eort because, in order to estimate this series of net present values, one ®rst needs to estimate series of annual bene®ts and costs over the project's life-cycle for each net present value estimated. One way to reduce these eorts is to only estimate net present values for a few years over an extended period. For example, net present value may be estimated for every 5 yr over a period of 30 yr. One can then choose the year that gives the largest net present value. This less extensive approach may not result in the optimal timing but will result in an improvement over what can be achieved if investment timing is ignored completely. These build-later alternatives can be analyzed along with those currently required for major transportation investments. 6.4.4. 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. In the initial analysis, planners may evaluate the sensitivity of the project's net present value to variables that are closely related to market conditions and are readily measured. These could include population or employment in a given market area, roadway congestion, parking price, bus transit ridership levels or market share, and population density. The purpose is to determine the level of a particular proxy variable at which the project reaches its maximum net present value. Thus, rather than directly using net present value or time as the ¯ag for implementation, one could establish performance 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 ¯ags for decision makers and the public.
7. FUTURE RESEARCH
This paper is limited in many ways. First, uncertainty is not addressed. The general role of uncertainty in the planning of major transportation investments has been widely recognized in the literature (Pearman, 1977; Ashley, 1980; Pell and Meyburg, 1985; Giord et al., 1993; Khisty, 1993; CUTR, 1995; and Lewis, 1995). The role of uncertainty in the timing of transportation investments has not, however. An extension to incorporate uncertainty could bene®t from a growing literature in economics on the timing of irreversible investments under uncertainty. Dixit and Pindyck (1994) review much of this literature. The central argument is that there is a value of waiting to invest when the project is irreversible and its pro®le of impacts is uncertain. 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. Second, the timing of investment projects may be addressed under dierent perspectives from economic to environmental to political. This report focuses on timing of transit investment projects only in terms of their economic worth, which presumes that projects be built when their net present value is positive and maximal. In reality, the decision to invest depends not only on economic worth but also on social and environmental considerations. Third, timing rules can dier, depending on whether individual projects are being considered in isolation or whether there are budgetary constraints. Procedures for timing projects vary according to the presence and nature of budget limitations and the independence or mutual exclusivity of projects 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 paper deals with the simplest situation where one single project is being evaluated against the `donothing' alternative. Finally, the paper provides a framework for addressing investment timing, but does not provide timing rules and practical guidelines for planners and decision-makers involved in the analysis and
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decision-making of major transit investments. Another extension of the analysis would be to develop such practical timing rules and guidelines. AcknowledgementsÐWe thank Ken Small and three anonymous referees for comments on earlier versions and Michael R. Baltes and Patricia Henderson for editorial suggestions.
REFERENCES Ashley, D. J. (1980) Uncertainty in the context of highway appraisal. Transportation 9, 249±267. Center for Urban Transportation Research (CUTR) (1995) A New Strategic Urban Transportation Planning Process. University of South Florida, Tampa. Deen, T. B., Kulash, W. M. and Baker, S. E. (1976) Critical decisions in the rapid transit planning process. Transportation Research Board 559, 33±43. Dixit, A. K. and Pindyck, R. S. (1994) Investment under Uncertainty. Princeton University Press, Princeton, NJ. Euritt, M. A., Homan, M. A. and Walton, C. M. (1990) Conceptual model of the ®xed-guideway decision process. Transportation Research Board 1266, 152±162. Federal Highway Administration (FHWA) (1996) Exploring the Application of Bene®t/Cost Methodologies to Transportation Infrastructure Decision Making. U.S. Department of Transportation, Washington, DC. Federal Register (1994) Principles for federal infrastructure investments. Executive Order 12893 of January 26, 1994. Codes of Federal Regulations, Vol. 3. National Archives and Records Administration, Washington, DC. Federal Transit Administration (FTA) (1994) Revised Measures for Assessing Major Investments: A Discussion Draft: An FTA Policy Paper. U.S. Department of Transportation, Washington, DC. Giord, J. L., Uzarski, D. R. and McNeil, S. (eds) (1993) Infrastructure: Planning and Management, Part II, Guessing the Future: Coping with Uncertainty in Infrastructure Planning. American Society of Civil Engineers, New York. Hirschman, I., Hoover, J. and Benz, G. (1991) Social equity implications of UMTA's major capital investment policy. Transportation Quarterly 45, 43±53. Johnston, R. A. and Deluchi, M. A. (1989) Evaluation methods for rail transit projects. Transportation Research-A 23, 317±325. Johnston, R. A., Sperling, D., Deluchi, M. A. and Tracy, S. (1988) Politics and technical uncertainty in transportation investment analysis. Transportation Research-A 21, 459±475. Khisty, C. J. (1993) Matching transportation planning methods to cope with uncertainty. Selected Proceedings of the Sixth World Conference on Transport Research, Lyon, France, pp. 2141±2147. Leconte, R., Hughes, C. and Narayanan, R. (1987) Economic eciency and investment timing for dual water systems. Water Resources Research 23, 1807±1815. Lewis, D. (1992) Primer on Transportation, Productivity and Economic Development. National Cooperative Highway Research Program, Report 342. Transportation Research Board, Washington, DC. Lewis, D. (1995) The future of forecasting: risk analysis as a philosophy of transportation planning. TR News 177, 3±9. Marglin, S. A. (1963) Approaches to Dynamic Investment Planning. North-Holland Publishing Company, Amsterdam. Oce of Management and Budget (OMB) (1992) Guidelines and discount rates for bene®t-cost analysis of federal programs. Circular A-94. Pearman, A. D. (1977) Uncertainty and the transport investment decision. Transport Decisions in a Age of Uncertainty: Proceedings of the Third World Conference on Transportation Research, Rotterdam, Netherlands, pp. 431±437. Kluwer Boston, Hingham, MA. Pell, C. M. and Meyburg, A. H. (1985) Sources of error and their implications for uncertainty in urban transportation planning forecasts. Paper prepared for the 64th Annual Meeting of the Transportation Research Board. Polzin, S. (1992) Alternative evaluation. Proceedings of the UMTA/APTA Workshop on Fixed Guideway Planning, Philadelphia, Pennsylvania, pp. 35±39. U.S. Department of Transportation, Washington. Porter, R. C. (1984) The optimal timing of an exhaustible, reversible wilderness development project. Land Economics 60, 247±254. Rietveld, P. (1995) Introduction to the special issue: infrastructure and spatial economic development. Annals of Regional Science 29, 117±119. Snickars, F. (1989) Infrastructure: a treat for regional science approach. Annals of Regional Science 23, 251±253. Szymanski, S. (1991) The optimal timing of infrastructure investment. Journal of Transport Economics and Policy 25, 247± 258. U.S. Department of Transportation (USDOT) (1976) Major urban mass transportation investments. Federal Register 18, 41512±41514. USDOT (1984) Urban mass transportation major capital investment policy. Federal Register 49, 21284±21291. USDOT (1993) Metropolitan transportation planning and programming. Federal Register 58, 58070±58079. USDOT (1996) FTA new starts criteria. Federal Register 61, 67093±67106. USDOT (1997) FTA new starts criteria. Federal Register 62, 60755±60758. UMTA (1989) Urban Rail Transit Projects: Forecast versus Actual Ridership and Costs. U.S. Department of Transportation, Washington, DC.
Pergamon PII: S0965-8564(98)00123
CONSIDERING BUILD-LATER FOR MAJOR TRANSIT INVESTMENTS XUEHAO CHU* and STEVEN E. POLZIN
Center for Urban Transportation Research, University of South Florida, 4202 E. Fowler Av., CUT 100, Tampa, FL 33620, U.S.A. (Received 6 August 1995; in revised form 17 January 1998) AbstractÐThis paper uses a theoretical model of bene®t±cost analysis to consider the timing of major transit investments. The model takes into account the net bene®ts of a project, the variation of net bene®ts with project age and investment timing, capital cost and its growth, and the discount rate. Three questions are examined: (1) when might build-later be evaluated? (2) how do changes in the discount rate and other parameters of an investment aect its optimal timing? and (3) how signi®cantly do dierences in the stream of net bene®ts from an investment aect its optimal timing? The ®rst two questions are examined analytically, and the last question is examined numerically. Implications are discussed with respect to planning for major transit investments. # 1998 Elsevier Science Ltd. All rights reserved Keywords: major investment study, build-later, net present value, investment timing, discount rate 1. INTRODUCTION
Investing in a major transit project results in costs to society to construct the project and produces a stream of net bene®ts (net of operating, maintenance, and other societal costs) over the project's lifetime. To quantify these net bene®ts, the stream of net bene®ts is ®rst discounted and summed (this sum is called the present value of the stream of net bene®ts); this sum is then compared with the present value of construction costs, and the dierence is called the net present value. Time aects the net present value of a project in at least two ways. First, as a project ages, its net bene®ts may change with changes in the economy or age-induced operation and maintenance costs. For example, growth in the economy may increase the net bene®ts 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 increase. Also, physical deterioration may require expensive maintenance and replacement to maintain a given level-of-services and, as a result, drive down net bene®ts. The second way that time aects the net present value of a project is through the timing of investment. On one hand, postponing an investment may require a dierent level of construction costs because of changes in real costs for construction. Postponing a project may also result in a dierent stream of net bene®ts because of changes in the demand for and supply of its services. On the other hand, postponing also reduces the present values of a given amount of construction costs and a given stream of net bene®ts. The net result of postponing can be signi®cant. 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 its net present value from a negative amount if built today to a positive amount if built later. This paper is motivated by a concern that the current planning process for major transit investments does not adequately consider the timing of investments (USDOT, 1984, 1993). This process is part of the Metropolitan Transportation Planning Process (USDOT, 1993), which includes three components: major investment studies (MIS), the transportation plan, and the transportation improvement program (TIP). Many major projects, especially transit guideway projects, are often conceived by sta or decision-makers and are usually ®rst explored as a scenario in the development of the transportation plan. Projects moving from the transportation plan toward TIP for implementation are those perceived to be of the highest priority. However, a systematic approach seldom exists to determine whether the `highest' priority in the transportation *Author for correspondence. Fax: 001 813 974 5168; e-mail: [email protected]
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plan is the highest priority for immediate implementation. Major investment studies typically look at performance in the design year, usually 15±25 yr from now. 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 of investments or results in erroneous decisions by favoring a build-now alternative in the absence of a build-later alternative as an option in the choice set. This paper has three objectives. One is to explore the timing issue by answering three questions. First, when might build-later be evaluated? For example, should build-later be evaluated when an area is facing a declining or stagnant economy or when an area is facing a growing economy? Second, how do changes in the parameters of an investment project aect the time at which it should be built? The Federal Transit Administration (FTA) changed the discount rate for major transit investment projects from 7% for all projects to 4.9% for projects with a lifetime of at least 30 yr and to a percentage between 4.2 and 4.9 for projects with a lifetime less than 30 yr and, recently, changed it back to 7%. One might expect that these changes in the discount rate could have a signi®cant eect on the time at which a project should be built. How these changes aect investment timing cannot be assessed without considering build-later. Third, how do dierences in the pro®le of a project's impacts aect the time at which it should be built? By answering these questions, we hope encourage further discussion about the timing issue. The second objective is to help planners and decision-makers realize the potential bene®ts of considering build-later for major transit investments. One way to help avoid the potential biases toward early implementation of major transit investments is to increase the awareness of the issue of investment timing. The issue has had little discussion in the literature on major transit investment projects. An exception is Polzin (1992), who suggested in a workshop sponsored by the Urban Mass Transit Administration (UMTA) that build-later be considered in alternatives analysis. The third objective is to assist agencies charged with conducting investment analyses by providing them with a mathematical framework to analyze investment timing in general and a buildlater alternative in particular. The ultimate objective is to have a mechanism in place that helps consider not only what project should be built and whether it should be built, but also when it should be built. Fortunately the literature on the timing of public infrastructure investment has plenty to oer on how to model investment timing (Marglin, 1963; Porter, 1984; Leconte et al., 1987; Szymanski, 1991). The value of having a technical capability to consider build-later alternatives is particularly important in light of election cycles and the strong preferences of decision-makers for a build-now alternative. It is sometimes perceived that the congruence of factors that may enable a build decision or a positive 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 the apparent economic or logical arguments regarding investment timing. The presence of a supportive city council, a strong local legislative delegation at the state level, or the presence of the right person in the right committee position at the federal level is compelling justi®cation for a build-now decision. The election cycles bring opportunities that may not be duplicated in the future when the analytically optimal time arrives. The discretionary nature of project funding perpetuates this sensitivity of decision-makers. Another factor biasing decision-making to build-now decisions is the strong emotional appeal of a `do something' mentality. Frustrations with congestion, a cynical attitude toward government, a disdain of bureaucracy and process, and a perception that we `plan projects to death' often result in a strong predisposition to an action-oriented decision. This creates a populist sense of action and serious eorts to actually solve problems. It implies a decisiveness and leadership level that can be very appealing for decision-makers. On a more pragmatic note, it also increases the chances that some of the bene®ts 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 planning and design or initial eorts to buy right-of-way as opposed to ribbon-cutting ceremonies; yet, these actions can provide a strong constituency for decision-makers. The paper is also related to the general literature on public infrastructure planning and the literature on evaluating and planning for major transit investment projects. The strong interest in
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public infrastructure planning 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 al. (1976), Johnston et al. (1988), Johnston and Deluchi (1989), Euritt et al. (1990), Hirschman (1991), UMTA (1989), and FTA (1994), have evaluated the planning process for major transit investment projects from a variety of perspectives. None of these authors, however, considered the timing issue. The issue of investment timing is conceptually not unique to transit investments. The principles of investment timing presented in this paper apply to major transportation investments across many modes. However, it may be more relevant for transit, given empirical data on the performance 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-ful®lling 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 and perhaps intelligent transportation systems (ITS), 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 justi®ed. In this paper, we ®rst present the model of bene®t±cost analysis for investment timing. Then we analyze the model for a number of special cases in order to answer the ®rst two questions mentioned earlier. We then analyze the model for a number of numerical examples to answer the third question. Finally, we discuss the implications of the results to the practice of major transit investment analyses and decision-making. 2. THE MODEL
This section presents a model of bene®t±cost analysis to examine timing for major transit investment projects. As discussed earlier, time aects the net present value of a project through both project age and the time at which the project is implemented. These two eects are incorporated in the bene®t-cost model, which consists of the following assumptions: 1. The project costs K t to construct at time t: K t is incurred entirely at time t: K t is a nondecreasing function of time t : K0 t50 2. The right-of-way costs M; M is incurred entirely at time 0 (`now') 3. The lifetime of the project is T 4. The project generates a stream of net bene®ts given by B s; t if it is built at time t and t4s4t T: s ÿ t may be referred to as project age 5. All bene®ts and costs are measured in the prices at time 0 6. The planner chooses a time, t, to maximize the net present value of the project, NPV t. That is, the planner faces the following problem: tT
Maxt NPV t
B s; teÿrs ds ÿ K teÿrt ÿ M
1
t
where the ®rst term measures the present value of the stream of net bene®ts and the second term measures the present value of construction costs. This model is based on the framework by Marglin (1963) and extends those of Porter (1984), Leconte et al. (1987), and Szymanski (1991) by allowing the stream of net bene®ts to shift with the timing of investment. In addition, it extends all these models by allowing construction costs to grow over time and a lump-sum cost to incur at time 0 regardless of the timing of investment. 3. ANALYTICAL ANALYSIS
This section considers ®ve special cases of the above model. The purpose is to answer the ®rst two questions mentioned earlier: When might build-later be evaluated? How do changes in the
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discount rate and other parameters of an investment aect optimal timing? The approach is analytical as opposed to the numerical analysis that follows. 3.1. Case 1. B s; t B s ÿ t; t4s4t T In this case, the stream of net bene®ts from investing at time t2 is the same as the stream from investing at time t1 (Fig. 1). In other words, one stream is a parallel shift of the other. Under this case, the solution to the problem is independent of time, and the optimal rule is to invest now or never. There is no advantage to build later when the stream of net bene®ts repeats itself with timing changes. NPV t for this case can be written as follows: 0T 1 2 NPV t eÿrt @ B ueÿru du ÿ K tA 0
If the integral 4K 0 M, never invest. This is because the net present value is non-positive for any t under the assumption that K t is a non-decreasing function. If the integral > K 0 M, the investment should be made now. This is because NPV 0 > 0 but NPV t declines with t. 3.2. Case 2. B s; t B s In this case, the stream of net bene®ts does not shift with the timing of investment. The stream of net bene®ts from investing at time t2 is part of the longer stream from investing at time t1 s; t2 (Fig. 2). This case would be relevant when there are external forces that aect the stream of net bene®ts in a particular way. There are three outcomes of such external eects on the stream of net bene®ts as described in the next three cases. 3.3. Case 2a. B0 0 In this case, net bene®ts stay constant with project age. The solution to the problem is independent of time, and the optimal rule is to invest now or never. The intuition is that there is no advantage for investing later when net bene®ts stay constant. The NPV t in this case can be written as follows: B o ÿ 1 ÿ eÿrT ÿ K t ÿ M 3 NPV t eÿrt r
Fig. 1. Parallel stream of net bene®ts.
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Fig. 2. Increasing stream of net bene®ts.
If the ®rst term in the parentheses 4K 0 M, never invest. If that term > K 0 M, invest now. This case would be relevant for metropolitan areas that expect a stagnant demand for services provided by the investment. 3.4. Case 2b. B0 s < 0 In this case, net bene®ts decline with project age. The solution to the problem is again independent of time, and the optimal rule is to invest now or never. The intuition is that there is even less advantage to invest later when net bene®ts decline with project age than when net bene®ts stay constant with project age. This case would be relevant for metropolitan areas that expect a declining demand for the services provided by the investment. 3.5. Case 2c. B0 s > 0 In this case, net bene®ts grow with project age. This case would be relevant for urban areas that expect a growing demand for the services provided by the investment. This case is particularly interesting because build-later can be optimal. The ®rst-order condition of the problem is: ÿ ÿ dNPV t ÿ eÿrt B t ÿ B t TeÿrT rK teÿrt ÿ K0 teÿrt 0 dt
4
That is, changes in net bene®ts and construction costs from postponing are equal at the margin. This ®rst-order condition can be written alternatively as B t K0 t B t TeÿrT rK t
5
The left-hand-side is the cost of postponing, including net bene®ts foregone, B t , and increases in construction cost, K0 t. The right-hand-side is the bene®ts of postponing, including bene®ts gained at the end of the lifetime and interest saved. Whether build-later is optimal is determined by whether (4) has an interior solution. A necessary condition for an interior solution is the following: B 0 < Bc rK B TeÿrT ÿ K0 0
6
That is, B 0 should be relatively small; otherwise it is better to invest now or never. The parameter Bc may be interpreted as the critical level of B 0 at which build-now would just barely be optimal.
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3.6. Case 3. B s; t B sea sÿt ; B s Begs ; g50 This case becomes cases 1 and 2 when B0 s 0 and a 0, respectively. In order to obtain an explicit solution for optimal timing, it is also assumed that K t Kebt ; b50 and that T 1. The solution is independent of time when g b or a5r ÿ g or r4g or r4b, i.e. when: (1) net bene®ts and construction costs grow at the same rate; (2) the discount rate is no greater than the growth rate of net bene®ts; (3) net bene®ts shift with project age at a rate no greater than the dierence between the discount rate and the growth rate of net bene®ts; or (4) the discount rate is no greater than the growth rate of construction costs. The following analysis further assumes that none of the above conditions occurs. The ®rst-order condition of the problem is r ÿ bKebt Begt
rÿg rÿgÿa
7
Construction costs saved from postponing equal net bene®ts foregone at the margin. The secondorder condition requires that g > b, that is, net bene®ts grow faster than construction costs. Solving (7) for t results in the optimal timing given by: 1 r ÿ bK rÿgÿa ln ln 8 t gÿb B rÿg where r is the discount rate, b is the growth rate of construction costs, g is the growth rate of B s, and a is the shifting rate of net bene®ts with project age. It is interesting to determine the direction of change in optimal timing from an increase in the value of a given parameter. The results are shown in Table 1, which are summarized as follows: 1. Optimal timing becomes later when the discount rate increases. An increase in the discount rate both increases the bene®ts and decreases the costs of postponing. 2. Optimal timing becomes later with an increase in the base construction costs at time 0, K. An increase in the base construction costs increases the bene®ts of postponing because of savings in construction costs. 3. Optimal timing becomes earlier with an increase in the net bene®ts at time 0, B. An increase in the base net bene®ts increases the costs of postponing because of net bene®ts foregone. 4. An increase in the growth rate of capital costs aects optimal timing indeterminately. This ambiguity results from two opposite eects of an increase in the growth rate of capital costs on the bene®ts of postponing. 5. Optimal timing becomes earlier with an increase in the growth rate of net bene®ts. This is because an increase in the growth rate of net bene®ts increases the costs of postponing due to net bene®ts foregone. 6. Optimal timing becomes earlier when the shifting rate of net bene®ts, a, increases. An increase in the shifting rate of net bene®ts increases the costs of postponing. 4. NUMERICAL ANALYSIS
This section solves the problem for a range of three examples of Case 2c, where net bene®ts grow with project age. The purpose is to answer the following question: How signi®cantly do differences in the stream of net bene®ts aect optimal timing? The approach is numerical. Table 1. Comparative static analysis Parameters r:discount rate K:base capital cost b:construction cost growth rate B:base net bene®ts g:net bene®t growth rate a:shifting rate of net bene®ts
Direction of changes in optimal timing from an increase in a parameter in column 1 Later Later Indeterminate Earlier Earlier Earlier
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4.1. Model speci®cation The model is speci®ed as follows. The lifetime T is 40 yr. The discount rate r is 7%. The project costs K 1; 000; 000 if built now. Construction costs grow at a constant rate of b 0:01 or K t Kebt . The cost to acquire the right-of-way M is $100,000. These values are shown at the bottom of Table 2. In addition, Table 2 shows the example streams of net bene®ts. They are convex (growing at an increasing rate), linear (growing at a constant increment), and concave (growing at a decreasing rate). The horizontal stream is included for comparison. The constants in these functions, bi i 1; 2; 3; 4, are determined as follows: the constant for the linear one is arbitrarily chosen to be $10,000; and the others are chosen such that these streams result in the same net present value if the project starts now. The resulting values are shown in Table 2 under `Net bene®t at time 0.' Also shown in Table 2 for each steam of net bene®ts are the values of Bc , the critical level of B 0 at which build-now would just barely be optimal. The four streams of net bene®ts are plotted in Fig. 3. Alternatively, these streams may be compared with respect to the cumulative distributions of their discounted values shown in Fig. 4. The farther away a distribution is from the bottom right-hand corner, the larger is the proportion of the bene®ts that materialize in the early years. These cumulative distributions capture the dierences among these streams of net bene®ts better than a simple plotting of them as shown in Fig. 3. 4.2. Results For the speci®cations of the model shown in the note to Table 2, the problem is numerically solved for each of the four streams of net bene®ts. The results are shown in Table 2. The optimal timing is 13, 9, 3, and 0 yr from now for the convex, linear, concave, horizontal streams of net bene®ts, respectively. If the project is to be built now, these alternative streams would generate the same net present value of $603,400 because of a constraint imposed on the constants of the net bene®t functions shown as in Table 2. If the project is to be built at its optimal time, however, the net present value would increase by 92, 44, 8, and 0% for the four examples, respectively. These dierences in optimal timing and improvements in net present values are better re¯ected in Fig. 5, which shows how the net present value varies with the time of investment for each of the four streams of net bene®ts. First, these curves have the same value at time 0 because of the constraint mentioned above. Second, the net present value for the horizontal stream decreases over time, implying that build-now is better than build-later. Third, the other curves reach their maximum after time 0 (at years 13, 9, and 3, respectively), implying that build-later is better than build-now. Fourth, when build-later is optimal, build-later would be better than build-now during some period around the optimal timing. The later the optimal time is, the wider this period is. Table 2. Numerical comparisonsa Streams of net benefits Convex Net bene®t function:B t Net bene®ts at time 0:B 0 Critical B 0 : Bc Optimal timing (yr):t Improvement in NPV (%) a
Linear 1:5
b1 t 1 2,321 97,053 13 92%
b2 t 1 10,000 84,932 9 44%
Concave
Horizontal 0:5
b3 t 1 38,520 74,999 3 8%
b4 127,000 67,732 0 0%
NPV of investing at time t is computed as follows: NPV t
tT t
B seÿrs ds ÿ K teÿrt ÿ M
K t Kebt The parameters are set as follows: Construction cost growth rate
Discount rate
Lifetime (yr)
Right-of-way cost ($)
Initial capital cost ($)
b 0:01
r 0:07
T 40
M 100; 000
K 1; 000; 000
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Fig. 3. Example streams of net bene®ts.
Fig. 4. Cumulative distribution of discounted net bene®ts.
5. REGULATIONS AND PRACTICE
5.1. Regulations Federal regulations on economic analyses of transportation investments require some of the elements necessary for an economic analysis of investment timing, but fail to require analysis of investment timing. Three such regulations are discussed below.
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Fig. 5. Net present value.
5.1.1. Executive Order 12893 (Federal Register, 1994). This document sets forth principles for Federal Infrastructure Investments. The order requires all Federal agencies with infrastructure responsibilities to conduct systematic analysis of expected bene®ts and costs for all infrastructure investments, including both quantitative and qualitative measures, in accordance with the following guidelines: (a) Bene®ts and costs should be quanti®ed and monetized to the maximum extent practicable. All types of bene®ts and costs, both market and non-market, should be considered. To the extent that environmental and other non-market bene®ts and costs can be quanti®ed, they shall be given the same weight as quanti®able market bene®ts and costs. (b) Bene®ts and costs should be measured and appropriately discounted over the full life cycle of each project. Such analysis will enable informed tradeos among capital outlays, operating and maintenance costs, and nonmonetary costs borne by the public. 5.1.2. OMB Circular A-94 (OMB, 1992). This circular gives Guidelines and Discount Rates for Bene®t±Cost Analysis of Federal Programs. The circular recommends bene®t-cost analysis as the technique to use in a formal economic analysis of government projects. The circular recognizes net present value as the standard criterion for making decisions on government projects on economic principles. The circular requires the use of a real discount rate of 7% in discounting future bene®ts and costs measured in constant dollars. Restated below are two related sections: 5a. The standard criterion for deciding whether a government program can be justi®ed on economic principles is net present valueÐthe discounted monetized value of expected net bene®ts (i.e. bene®ts minus costs). Net present value is computed by assigning monetary values to bene®ts and costs, discounting future bene®ts and costs using an appropriate discount rate, and subtracting the sum total of discounted costs from the sum total of discounted bene®ts. Discounting bene®ts and costs transforms gains and losses occurring in dierent 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.
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8b1. Constant±dollar bene®t-cost analyses of proposed investments and regulations should report net present value and other outcomes determined using a real discount rate of 7%. This rate approximates the marginal pretax rate of return on an average investment in the private sector in recent years. 5.1.3. Criteria for New Starts. The Federal criteria for new starts relied on cost-eectiveness measures with little attention devoted to the criterion of net present value from 1976 to 1984 (Johnston and Deluchi, 1989). This is re¯ected in UMTA's policy statements (USDOT, 1976, 1984). Despite the OMB Circular A-94 and Executive Order 12893, Federal Transit Administration continues to rely on cost-eectiveness measures (FTA, 1994; USDOT, 1996, 1997). FTA considers bene®t±cost analysis as the desirable basis for project evaluation. The agency, however, rejects the use of bene®t±cost analysis in actual evaluations because it believes that the problems of quanti®cation are too great. Johnston and Deluchi (1989) believe that FTA underestimates the practicality of conducting bene®t-cost analysis for major transit investments. 5.2. Current practice Federal regulations thus recommend that net present value be used as the evaluation criterion; that annual bene®ts and costs be measured in constant dollars for the full life-cycle of a project; and that annual bene®ts and costs be discounted at a real discount rate of 7% to calculate net present value. However, current practice of investment analysis for major transportation investments, including major transit investment, poorly meets these regulations. According to Lewis (1992), a 1990 survey of 35 transportation projects conducted for the National Corporative Highway Research Program indicates that: . About one third of the projects use net present value as a basis for evaluation. . Most projects fail to express costs and bene®ts on an annual basis over the life-cycle of the project. . A large number of studies failed to use an appropriate analysis period. . Only about 5% use adequate discounting techniques and properly justi®ed discount rates. The sample of 35 projects includes 10 airport and air trac control-related projects, 10 highway projects, 2 high speed rail systems, 5 ports, 2 inland waterway projects, and 6 public transit proposals. The sample was drawn from a larger universe with a four-factor strati®cation: location and scale, mode, type, and point of approval. Location and scale include national, regional and local projects and project size. Mode includes highway, public transit, rail, ports, airports and inland waterways. Type includes construction, reconstruction, and repair. Point of approval covers appraisals in progress, projects rejected, and projects approved (including projects completed, projects in-progress, and those not started). In addition, analysis is 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 recognize the importance of investment timing, though some federally-sponsored conferences and research projects do (Lewis, 1992; FHWA, 1996). 6. TOWARD AN OBJECTIVE APPROACH
The current practice of analysis and decision-making for major transit investments fails to systematically consider the timing of such investments. Many factors contribute to this failure. Election cycles, discretionary project funding, and politicians' desire for action now all have the tendency to create biases toward early implementation of major transit projects. Federal regulations do not help either by failing to recognize the importance of investment timing and even bene®t±cost analysis in the case of transit new starts. In an era of scarce resources, it is increasingly important to improve the economic worth of major transit investments. One eective way to achieve such improvements is to develop an objective approach for better analysis and decision-making regarding the timing of these investments. The following discusses some of the elements that might be part of the approach.
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6.1. 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 living. 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 transit investments. 6.2. Improve bene®t±cost analysis The validity of the net present value criterion, however, hinges on an adequate bene®t±cost analysis. As the survey for NCHRP 2-17(1) indicates, current practice of bene®t±cost analysis needs improvements, particularly in the following three areas. 6.2.1. Annualize bene®ts and costs. A key requirement of any investment analysis under the net present value criterion is an accounting for annual bene®ts and costs realized over the life-cycle of a project. 6.2.2. Discount bene®ts and costs appropriately. Because a dollar tomorrow is worth less than a dollar today, future costs and bene®ts must be discounted to comparable worth today. The accepted approach is to calculate the present value of bene®ts and costs using a discount rate. The same rate should be used for both bene®ts and costs. The choice of the correct discount rate is also important. A rate of 7% is recommended for all federal projects when bene®ts and costs are in constant dollars. If the rate is too high, we will wrongfully reject projects whose bene®ts are concentrated in the long run. If the rate is too low, we will accept projects whose bene®ts are too far in the future to justify investment today. 6.2.3. Use an appropriate analysis period. A properly done bene®t-cost analysis requires using the life-cycle of a project as the analysis period. If an analysis period is too short, the project under consideration would in fact generate bene®ts well beyond the analysis period. As a result, these bene®ts would be excluded. 6.3. Consider investment timing in decision-making Many factors may have contributed to a reluctance to consider timing in the current process of decision-making for major transit investments. Most factors can be incorporated into a formal analysis of investment timing, except two factors. It is often perceived that the election cycles bring opportunities that may not be duplicated in the future. Also, decision-makers tend to have a strong emotional appeal that they need to do something and do it now. Both factors create a bias toward early implementation of investments. One eective approach to overcome these may be to require through regulations that all major transit projects pass a test on the net present value criterion. 6.4. Consider timing in investment analysis For investment timing to enter the decision-making process for major transit investments, a critical factor is to consider timing issues in investment analysis. The fundamental shortcoming of the current process and the opportunities for revisiting current practice merit serious consideration. 6.4.1. Re¯ect on timing issues. At a minimum, planners should seriously re¯ect on the issues of investment timing. These include the importance of investment timing in improving the economic worth of investments, barriers that prevent timing from being considered in the planning process, and how investment timing may be incorporated into the current planning process for major transit investments. Planners should be prepared to educate decision-makers on these issues. 6.4.2. Use economic principles for optimal timing. Preferably, there would be eorts by planners to use economic principles of investment timing to ®nd the optimal timing. This would include
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applying the framework developed in this paper to determine whether a proposed project should be delayed and how much it should be delayed. 6.4.3. Include built-later alternatives. To ®nd the optimal timing for an investment, one would need to estimate a series of net present values across an extended period of possible investment timing. The timing is optimal when net present value reaches its maximum. Doing this could mean enormous eort because, in order to estimate this series of net present values, one ®rst needs to estimate series of annual bene®ts and costs over the project's life-cycle for each net present value estimated. One way to reduce these eorts is to only estimate net present values for a few years over an extended period. For example, net present value may be estimated for every 5 yr over a period of 30 yr. One can then choose the year that gives the largest net present value. This less extensive approach may not result in the optimal timing but will result in an improvement over what can be achieved if investment timing is ignored completely. These build-later alternatives can be analyzed along with those currently required for major transportation investments. 6.4.4. 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. In the initial analysis, planners may evaluate the sensitivity of the project's net present value to variables that are closely related to market conditions and are readily measured. These could include population or employment in a given market area, roadway congestion, parking price, bus transit ridership levels or market share, and population density. The purpose is to determine the level of a particular proxy variable at which the project reaches its maximum net present value. Thus, rather than directly using net present value or time as the ¯ag for implementation, one could establish performance 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 ¯ags for decision makers and the public.
7. FUTURE RESEARCH
This paper is limited in many ways. First, uncertainty is not addressed. The general role of uncertainty in the planning of major transportation investments has been widely recognized in the literature (Pearman, 1977; Ashley, 1980; Pell and Meyburg, 1985; Giord et al., 1993; Khisty, 1993; CUTR, 1995; and Lewis, 1995). The role of uncertainty in the timing of transportation investments has not, however. An extension to incorporate uncertainty could bene®t from a growing literature in economics on the timing of irreversible investments under uncertainty. Dixit and Pindyck (1994) review much of this literature. The central argument is that there is a value of waiting to invest when the project is irreversible and its pro®le of impacts is uncertain. 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. Second, the timing of investment projects may be addressed under dierent perspectives from economic to environmental to political. This report focuses on timing of transit investment projects only in terms of their economic worth, which presumes that projects be built when their net present value is positive and maximal. In reality, the decision to invest depends not only on economic worth but also on social and environmental considerations. Third, timing rules can dier, depending on whether individual projects are being considered in isolation or whether there are budgetary constraints. Procedures for timing projects vary according to the presence and nature of budget limitations and the independence or mutual exclusivity of projects 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 paper deals with the simplest situation where one single project is being evaluated against the `donothing' alternative. Finally, the paper provides a framework for addressing investment timing, but does not provide timing rules and practical guidelines for planners and decision-makers involved in the analysis and
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decision-making of major transit investments. Another extension of the analysis would be to develop such practical timing rules and guidelines. AcknowledgementsÐWe thank Ken Small and three anonymous referees for comments on earlier versions and Michael R. Baltes and Patricia Henderson for editorial suggestions.
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