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Valuing flexibility in build-operate-transfer (BOT) toll road projects : a real options approach Harley, Frank 1998

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VALUING FLEXIBILITY IN BUILD-OPERATE-TRANSFER (BOT) TOLL ROAD PROJECTS: A REAL OPTIONS APPROACH by FRANK HARLEY BComm., The University of British Columbia, 1994  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (BUSINESS ADMINISTRATION) in THE FACULTY OF GRADUATE STUDIES Department of Transportation and Logistics  We accept this thesis ^ig^nf^niiQg to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1998 © Frank Harley, 1998  In  presenting  degree  this  at the  thesis  in  University of  partial  fulfilment  British Columbia,  freely available for reference and study. copying  of  department  this or  thesis by  for scholarly  his  publication of this thesis  or  her  the  I agree  requirements  for  may  representatives.  It  be is  Department The University of British Columbia Vancouver, Canada  advanced  that the Library shall make it  I further agree that permission  purposes  an  granted  for extensive  by the head  understood  that  for financial gain shall not be allowed without  permission.  DE-6 (2/88)  of  of  my  copying  or  my written  ABSTRACT  The goal of this thesis is to show how option valuation techniques can be used to value managerial flexibility in Build-Operate-Transfer (BOT) toll road projects. It begins by discussing the dramatic shift towards public-private partnerships and BOTs in the infrastructure industry. It then looks at how traditional capital budgeting techniques often fail to capture important sources of value created by flexibility. It discusses real options and introduces Contingent Claims Analysis (CCA) as a means of valuing flexibility which takes into account the opportunity to construct replicating portfolios in the market. It applies CCA to the real options present in each phase of a BOT toll road project. During the build phase, it looks at the option to abandon, the option to change scale/technology, and time-to-build flexibility. During the operate phase, it analyzes tollsetting flexibility, development gain options, and project financing flexibility. At the transfer phase it considers arrangements with option-like features. In conclusion, this thesis emphasizes the relevance of real option valuation techniques to BOT toll road projects and points the way to further research fusing the fields of transport economics and financial economics.  T A B L E OF CONTENTS Abstract Table of Contents List of Figures Acknowledgements  ii iii vi vii  1. Introduction  1  2. The infrastructure revolution 2.1. Introduction 2.2. The changing nature of infrastructure 2.3. The spectrum of public-private partnerships 2.4. Build-Operate-Transfer Host government Sponsors Lenders Consumers 2.5. BOT toll roads  5 6 6 8 11  3. The real options revolution Introduction 3.1 Traditional capital budgeting 3.1.1. Introduction 3.1.2.Why is net present value the best investment decision criteria? The financial objective of the firm Time-preference utility curves Capital markets Productive opportunities Investing in both productive opportunities and capital markets Internal Rate of Return (IRR) Payback period Average return on book value Profitability index 3.1.3. Evaluating investment opportunities under uncertainty Risk-adjusted discount rate approach (RADR) Certainty equivalent approach Sensitivity analysis Monte Carlo simulation Decision-Tree analysis 3.1.4. Problems with traditional capital budgeting 3.2 Real Options 3.2.1. Introduction What is a real option? The real options revolution  17 18 19 19 19  15  27  40 41 41  iii  Financial options versus real options Call options Put options The analogy between financial options and real options 3.2.2. The basics of real option valuation An example of real options Contingent claims analysis (CCA) Using CCA to value real options The option to defer investment The option to default during construction The option to expand The option to contract The option to temporarily shut down The option to abandon for salvage value or switch to best alternative use The option for corporate growth 3.2.3. Summary  49  63  4. Real options in BOT toll roads Introduction 4.1. Build phase 4.1.1. Introduction 4.1.2. The option to abandon midstream 4.1.3. The option to change scale or technology 4.1.4. Time to build option 4.1.5. Summary 4.2. Operate phase: toll-setting flexibility 4.2.1. Introduction 4.2.2. Deriving the toll demand curve 4.2.3. Stochastic demand curves and the value of fixed tolls 4.2.4. The value of complete flexibility 4.2.5. The value of partial flexibility 4.2.6. Trigger adjustments 4.2.7. Profit-maximization versus economic efficiency 4.2.8. Summary 4.3. Operate/transfer phase: other real options 4.3.1. Introduction 4.3.2. Development gain options 4.3.3. Financial flexibility 4.3.4. Transfer options 4.3.5. Summary  65 66 68 68 68 75 77 78 79 79 80 93 96 103 104 106 112 113 113 113 115 118 119  5. Conclusion  121  6. Bibliography  124  7. Appendices 6.1. BOT toll road survey 6.2. Value of Travel Time Savings (VTTS) 6.3. Traffic congestion functions 6.4. Mathcad derivation of toll demand curve  128 129 132 136 138  V  LIST OF FIGURES 2.1 Structure of a BOT project 3.1 Time-preference utility curves 3.2 Market opportunity line 3.3 Productive opportunities curve 3.4 Optimal and suboptimal investments exploiting market opportunities 3.5 Diversification eliminates unique risk, but leaves market risk 3.6 Efficient Portfolio frontier with borrowing and lending 3.7 The CAPM security market line (SML) 3.8 Sensitivity analysis: A simplified toll-bridge spider graph 3.9 NPV probability distribution 3.10 Decision tree of whether or not to invest in AVI technology 3.11 Managerial flexibility introduces payoff asymmetry 3.12 Asymmetric payoffs from options 3.13 Value of a call (C) before its expiration date, . . 3.14 Value of a put (P) before its expiration date 3.15 Possible price movements over one year 3.16 A portfolio replicating the value of the investment opportunity 3.17 Possible price movements over one time period 3.18 Salvage value or best alternative use (A) 4.1 The life cycle of a BOT toll road project from the concessionaire's perspective 4.2 Stochastic movement of project value 4.3 A twist to traditional Social Cost-Benefit Analysis 4.4 Deriving the toll demand curve assuming everyone has the same VTTS 4.5 Demand for travel to D for groups with different values of travel time savings 4.6 Dynamic process to derive the toll demand curve for two groups with different VTTS 4.7 Toll demand curve derived by Mathcad for groups H and L 4.8 Dynamic process for four VTTS groups 4.9 Binomial movements of B over time 4.10 Profit-maximizing toll at time zero B =100 4.11. Present value of opportunity to receive uncertain cash flows at time one 4.12 Sensitivity of complete flexibility option value to (3 estimates (from 0 to 2) 4.13 Sensitivity of partial flexibility option value to beta estimates (from 0 to 2) 4.14. Trigger adjustment option payoff diagram 4.15. The marginal time cost toll demand curve 4.16. Average time cost toll demand curve 4.17 Movements in shopping mall value III. 1. A congestion function for a highway and a central business district street H  H  12 20 21 22 23 28 29 31 34 36 38 42 45 46 47 51 51 54 62 66 70 81 83 85 86 91 92 94 94 98 101 104 105 109 110 115 137  ACKNOWLEDGEMENTS I would like to thank Dr. Bill Waters for his guidance and encouragement throughout the preparation of this thesis. My gratitude also goes to Dr. Burton Hollifield for his valuable insights into financial economics and to Dr. Tae Oum for his teachings on transport demand analysis. Finally, I would like to thank my parents for all their support.  CHAPTER ONE  INTRODUCTION  1. INTRODUCTION The goal of this thesis is to show how real option valuation techniques can be used to quantify the value of flexibility in Build-Operate-Transfer (BOT) toll road and bridge projects. A BOT is a complex public-private partnership in which the host government grants a private-sector consortium a concession to build an infrastructure facility, operate it long enough to pay back project debt and equity, and then transfer ownership to the host government. Traditionally, infrastructure projects have been supported through public sector agencies. These projects often do not recover their costs but are justified on social grounds or because of their broader economic benefits. However changes in technology, demand, and economic philosophy have led to a surge in the popularity of BOT as an efficient approach to providing infrastructure. Within the rapidly changing infrastructure environment, policy-makers and investors are struggling to understand the key factors to a successful BOT project. However, many of their analytical tools are coarse and fail to reveal the opportunities for profit or loss created by complex contractual arrangements in BOT projects. Fortunately, innovations in financial economics have a created a new tool — real option valuation — which can provide insight into the dynamics of value in BOT toll road projects. This thesis begins by discussing the revolution which is taking place in the provision of infrastructure around the world. Chapter Two looks at how severe fiscal constraints and growing demand for infrastructure have stimulated governments in both developed and lesser developed countries to explore public-private partnerships (P s) as a 3  2  means of coping with infrastructure bottlenecks in transportation, power, and telecommunications. The chapter introduces the BOT concept and its many variations reflecting different degrees of risk sharing between the public and private sector. It discusses the key ingredients of a BOT project and the pros (additionality, acceleration, efficiency, externalities) and cons (complexity) of the BOT approach. Chapter Two concludes by looking at the role of BOTs in the area of toll roads and points out that while BOTs have the potential to improve a nation's transportation system, they can be overwhelmingly complex to value with traditional capital budgeting techniques. Chapter Three looks at how the "real options revolution" in financial economics has provided investors and policy-makers with a set of powerful new valuation tools. The chapter begins with an overview of traditional capital budgeting techniques such as riskadjusted discounting, simulation, and decision-tree-analysis and points out their inability to correctly value managerial flexibility. It then introduces option valuation as means of quantifying the value of risky cash flows which have asymmetric payoffs. It draws a comparison between financial options which involve the right, but not the obligation, to buy (call) or sell (put) financial assets and real options which involve claims on real assets (a factory, an oil field, revenues from a toll road). It shows how Contingent Claims Analysis (CCA) and the concept of arbitrage-free replicating portfolios can be used to price real option premiums. Finally, it works through a generic oil field example to illustrate the mechanics of real option pricing. Chapter Four focuses on the application of real option valuation techniques to BOT toll road projects. Real options in a BOT toll road project include the flexibility to  3  abandon or postpone the project during construction, change the scale or technology of the project, set tolls, develop ancillary facilities, alter the pattern of equity injection and debt repayment, and transfer the facility. By using real option valuation techniques to value these different types of flexibility, the parties to a BOT project can gain a better understanding of the tradeoffs they are making when they negotiate a BOT contract. This should facilitate the development of economically sound BOT projects. Chapter Five concludes by summarizing the major points of this thesis. It argues that the timely convergence of two revolutions — one in transportation economics and one in financial economics — has profound implications for both the suppliers and the consumers of infrastructure around the world. The real options approach to capital budgeting has provided policy makers and investors with valuable analytical tools. By applying these tools in an innovative and theoretically sound manner, the government and private sector should be able to improve the chances of success for a BOT project. This in turn should help societies around the world cope with the tremendous pressures mounting on their road infrastructure.  4  CHAPTER TWO  THE INFRASTRUCTURE REVOLUTION  2.1. Introduction Unprecedented economic growth in the twentieth century has created severe infrastructure bottlenecks around the globe. Rapid population growth and urbanization have placed tremendous strains on basic infrastructure such as water and sewage systems, power, telecommunications and transportation networks in both the developed and the developing world. Furthermore, mismanagement of infrastructure has damaged the environment, hurt people's quality of life, and undermined the economic competitiveness of many nations. Cashed-strapped governments realize the need for change, but are hardpressed to come up with the necessary resources. Faced with intense pressure to improve infrastructure services, the public and private sectors have begun to explore various types of partnerships. This chapter focuses on one of these partnerships: Build-Operate-Transfer (BOT). It begins by discussing how and why the infrastructure market is changing. It then looks at several types of publicprivate partnerships which have emerged as alternatives to traditional public sector infrastructure provision. Next, it examines the dynamics of BOT projects and discusses their pros and cons. The final section provides a brief introduction to the BOT toll road market and sets the stage for the application of real option valuation techniques to BOT toll roads.  2.2. The changing nature of infrastructure Traditionally, infrastructure has been thought of as a "public good" which should be provided by the government. The argument for government control is based on two  6  observations. First, infrastructure tends to be a natural monopoly because it involves large fixed costs and economies of scale. Government control is therefore necessary to prevent abuse of monopoly power. In addition, the government must keep the price low to encourage the utilization of sunk facilities, even though this may not lead to cost recovery. Second, infrastructure is the backbone of many economic activities and involves significant positive externalities and network effects. Government control is thus necessary to ensure a stable and sufficient supply of vital infrastructure services. This philosophy has been the driving force behind infrastructure policy since the turn of the century. However, since the 1980s, developments in the public and private sector have been challenging the traditional approach to infrastructure. The main forces behind this change are: •  Disenchantment with publicly provided infrastructure: In many cases, public monopolies and state agencies have been largely ineffective in meeting the growing demand for infrastructure. Voters and politicians want change.  •  Fiscal constraints: Accumulated debts, aging populations, and increasing pressures on health and education systems have left many governments with fewer resources to spend on infrastructure.  •  Construction firms looking for work : The aftershocks of the end of the Middle 1  East construction boom in the early 1980s are still being felt as construction firms scour the globe in search of new infrastructure projects to take up the slack.  Augenblick M. and Scott-Custer B. The BOT approach to infrastructure projects in developing countries. World Bank Working Paper, August 1990 1  7  •  Financial innovations and the globalization of financial markets: The rapid growth in limited recourse project financing, improvements in risk management techniques, and greater access to global capital markets have stimulated governments and financiers to look for innovative ways to finance infrastructure.  •  Technological developments: New technologies such as IPP (Independent Power Producers), cellular phones, port containerization, and AVI (Automatic Vehicle Identification) systems are radically changing the nature of infrastructure provision. These technologies have reduced natural monopoly characteristics, facilitated unbundling, encouraged private entry, and stimulated competition in many infrastructure services. These forces have converged to make "privatization" the buzzword of the 1990s.  Governments around the world are promoting "privatization" strategies as a means of tapping into private sector efficiencies while improving government finances. Unfortunately, the word privatization is often used rather crudely and fails to reflect the wide range of possible partnerships between the public and private sector.  2.3. The spectrum of public-private partnerships (P s) Public-private partnerships exist along a continuum between complete public sector ownership and operation and complete private sector ownership and operation. The goal of P s is to promote economic efficiency in infrastructure services by allocating project risks to those parties best able to mitigate the risks and by tapping into private  8  sector managerial efficiency and discipline. The following is a list of the major P types which are being used to inject life and efficiency into moribund infrastructure services . •  Service contracting: the public sector provider sets performance criteria, evaluates bids, supervises the contract, and pays lump-sum fees. A common example is road maintenance.  •  Management contracting: like service contracting, except the contractor has more autonomy in day-to-day management and shares some of the commercial risk of the enterprise.  •  Design-Build (DB or turnkey): the private sector designs and builds a facility to meet public-sector performance specifications. The private sector often bears the risk of cost overruns.  •  Design-Build-Maintain: the public sector operates the DB facility, but the private sector still has certain maintenance responsibilities.  •  Design-Build-Operate (DBO or "super turnkey"): the private sector builds and operates the facility, but ownership remains with the public sector  •  Lease-Develop-Operate (LDO): the public sector leases an existing facility to a private operator who expands, and operates the facility. The facility remains publicly owned and is transferred back to the public sector at the end of the lease.  •  Build-Lease-Operate-Transfer (BLOT): a private operator leases land from the public sector to build and operate a new facility. The facility is transferred to the public sector at the end of the lease term.  British Columbia. Task Force on Public-Private Partnerships. Building Partnerships: report of the Task Force on Public-Private Partnerships. Crown Publications, Oct. 1996 2  9  •  Build-Transfer-Operate (BTO): a private developer designs, finances and constructs a facility which is transferred to the public sector upon completion. The public sector then leases the facility back to the private developer. BTO enables the developer to get a reasonable return on investment while avoiding the liability complications of private ownership.  •  Build-Operate-Transfer (BOT): a private developer receives a concession to finance, design, build and operate an entirely new facility for a specified period. The facility is transferred to the public sector at the end of the concession period.  •  Build-Own-Operate (BOO): a private developer finances, builds, owns, and operates a facility in perpetuity subject to public sector intervention as stipulated in the concession contract.  •  Transfer to Quasi-public Authority: a quasi-public authority takes control of a public sector asset under contract that it will perform public services using private procedures and financing. This is an incomplete list of the evolving terminology in the infrastructure  market. Other points on the P spectrum include BOOT (Build-Own-Operate-Transfer), 3  BOOST (Build-Own-Operate-Subsidize-Transfer), BROT (Build-Rent-OperateTransfer), ROT (Refurbish-Operate-Transfer), RTO (Refurbish-Transfer-Operate). In Britain, P s are referred to as PFI (Private Finance Initiative). All of these terms reflect different degrees of risk and responsibility sharing between the public and private sector. Amidst this explosion of acronyms, BOT has gained the most notoriety and is often used  10  casually to refer to many of the forgoing P s. We will use BOT in a stricter sense 3  described in the following section.  2.4. Build-Operate-Transfer In a BOT project, the government grants a concession to a private consortium to build an infrastructure facility, operate it for a specified period (usually 20 to 30 years) to earn a reasonable return on investment, and then transfer the facility to the government at the end of the concession period. The roots of BOT can be traced back to the concessions of the eighteenth and nineteenth centuries. One of the earliest BOT prototypes was a concession granted to the Perrier brothers in 1782 to supply water to Paris. Egypt is home to perhaps the most famous early BOT fossil — the Suez Canal. Opened in 1869, this canal was built and financed by a British-French consortium under a concession granted by the Egyptian government. The properly-structured modern BOT has evolved considerably from its primitive ancestors. While earlier concessions gave the private sector free reign over the facility, today's BOT strives to achieve an optimal sharing of risk and responsibility between the public and private sector. As shown in figure 2.1, the key ingredients to a BOT are: Host government: Support from the host government is crucial to the success of a BOT project. There are several reasons why the host government may prefer the BOT approach:  11  Figure 2.1. Structure of a BOT project  Host government  Multilateral agencies Government agencies  Project agreement Special agreements  Lenders  Loan agreements  Insurers  Insurance policies  Construction contract  Contractors  Supply contract  Suppliers  Operation and Maintenance contract Shareholders' agreement  Sponsors  •  Technical, financial and legal advisors  Operator  Consumers of infrastructure service  Additionality: projects are realized which otherwise would not get built due to constraints on public spending. Commercial banks may be reluctant to lend to certain countries, but willing to lend to specific projects within those countries.  •  Acceleration: projects can be accelerated which otherwise would have to wait for scarce funds.  0  Efficiency: private sector capital, initiative, and know-how may reduce construction costs and time while improving operating efficiency.  •  Credibility: the scrutiny of private sector investors may weed out any "white elephants".  12  °  Positive externalities: society can benefit from technology transfer and training, the strengthening of local capital markets, and an improved investment climate.  •  Strategic control: the government maintains a certain degree of control over important infrastructure networks. The major disadvantage of BOTs is that they are extremely complex to negotiate  and manage. If the host government decides that the advantages of the BOT approach outweigh the costs of its complexity, then the host government must decide what kind of role to play in order to ensure the project's success. This role can take a variety of forms such as providing direct loans, standby loan agreements, revenue guarantees, loan guarantees, or tax breaks. The host government must decide how to regulate rate-ofreturn, tariffs, and safety. A controversial issue is whether or not the host government should take an equity position in the project. While a government equity position improves the project's transparency and could prevent the consortium from being exploited by any one sponsor (for example, the lead construction company may favour subcontracting to its own subsidiaries whereas other sponsors with smaller equity stakes may want to look for cheaper external subcontractors), such an arrangement may also result in the loss of private sector efficiencies due to the interference of inexperienced government officials. Sponsors Construction companies have typically taken the lead in forming BOT project companies. The inclusion of independent non-construction-related investors is recognized as an important means of imposing discipline on the project company. The  13  consortium should be well balanced: David and Goliath do not mix well. Project companies tend to be international consortiums. This has two main advantages. First, it improves the political acceptability of the project because the concession is not seen as benefiting any one country. Second, it spreads the risks and gives the project company access to funds from a number of different export credit agencies. A project company should also include well-connected local contractors as sponsors. Sponsors typically take equity stakes in a BOT project ranging from 10 to 30 percent. Their stake can take a variety of forms such as common shares, preference shares (guaranteed dividend), or convertible preference shares. The size of the equity stake is important because it is often interpreted by the host government as an indication of the sponsors' degree of commitment to the project. A large equity stake lessens the interest burden. However, a large equity stake also increases the pressure on the project company to impose high tariffs in order to maximize dividends. This may go against the government's desire to keep tariffs at a politically acceptable level (i.e., a level which maximizes social welfare). Lenders Project financing is the foundation of any BOT project. In project financing, lenders look to the project's assets and revenue stream for repayment. Lenders have limited or no recourse to the project sponsors' other assets. Project financing relies on financial engineering to allocate project risks optimally among lenders and equityholders. Financial engineering uses debt of varying seniority, convertible bonds, and derivatives  14  (foreign exchange and interest rate options and futures) to construct complex mezzanine financing arrangements which are tailored to the risk structure of the project. Lenders tend to be large international and domestic banks. Host governments can also play significant roles as lenders. Multilateral agencies such as the International Finance Corporation and the Asian Development Bank have begun to raise their profile as BOT lenders. Infrastructure funds such Global Power Investments ($2.5 billion joint venture between GE Capital, Soros Fund Management, and the IFC) and AIG Asian Infrastructure Fund ($1.2 billion) have also entered the BOT market. Consumers The direct consumer of the output from a BOT project can be the government (off-take agreements for electric power plants), large corporations (airlines using a BOT airport) or the general public (toll roads). The uncertainty of demand varies dramatically depending on the type of customer. This should be reflected in the financial structuring of the project. The general public may support the BOT approach because it does not involve the raising of taxes. Furthermore, the "user-pay" approach may appeal to those who rarely use the facility.  2.5. BOT toll roads Toll roads and bridges have become one of the hottest areas in the BOT market. Developed and developing countries alike are rushing to introduce BOT projects into their transportation networks. Appendix I lists some of the major BOT toll road and  15  bridge projects around the world. However, BOT toll roads should not be viewed as a panacea for a nation's transportation problems. Poorly managed BOT toll roads can exacerbate traffic congestion and damage a country's overall investment environment (witness the Bangkok Second Stage Expressway fiasco). Nevertheless, the efficiencies of a prudently planned BOT toll road project can benefit society immensely by releasing resources (government finances, travellers' time) which can be put to productive use in other sectors of the economy. The success of a BOT toll road hinges on identifying and correctly pricing important sources of project value. This enables all parties involved to gain a better understanding of what they are getting into. Chapter Three shows how traditional capital budgeting fails to recognize the value of managerial flexibility. Managerial flexibility in a BOT toll road is created both by the inherent nature of large infrastructure projects and by negotiated contractual arrangements between the public and private sector. Chapter Three introduces the option valuation technique of Contingent Claims Analysis as a means of valuing managerial flexibility. This discussion sets the stage for the main focus of this thesis: How to use real option valuation techniques to value flexibility in BOT toll road projects.  16  CHAPTER THREE  THE REAL OPTIONS REVOLUTION  INTRODUCTION To understand the far-reaching implications of the real options "revolution", it is necessary to review the basics of traditional capital budgeting. This chapter begins by reviewing the economic rationale behind net present value (NPV) analysis, first under certainty, and then under uncertainty. It looks at how traditional approaches such as riskadjusted discounting, Monte Carlo simulation, and decision-tree-analysis fail to capture the value of managerial flexibility in many situations. It then introduces the concept of real options as a means of valuing managerial flexibility and gives a brief overview of their academic evolution. Finally, it works through a generic oil field example to show how Contingent Claims Analysis (CCA) can be used to value real options.  18  3.1. TRADITIONAL CAPITAL BUDGETING 3.1.1. Introduction This section looks at traditional approaches to capital budgeting and project risk evaluation. It begins by discussing the economic theory behind the fundamental rule of investment: maximize net present value (NPV). It shows how NPV is superior to other investment criteria such as internal rate of return (IRR), payback period, average return on book value, and profitability index. It then looks at how risk is handled by the traditional tools of capital budgeting: the risk-adjusted discount rate and the Capital Asset Pricing Model (CAPM), the certainty equivalent approach, sensitivity analysis, traditional simulation (Monte Carlo), and decision-tree analysis. It concludes by pointing out the shortcomings of traditional capital budgeting tools and the need for an approach which systematically captures the value of managerial flexibility under uncertainty.  3.1.2. Why is net presient value the best investment decision criteria? The financial objective of the firm The financial goal of a firm should be to maximize the utility of its stockholders. Stockholders have different utility curves reflecting their desires:  3  •  to maximize wealth  •  to transform that wealth into preferred consumption patterns over time  •  to choose the risk characteristics of those consumption patterns  3  Brealey, R., and S.C. Myers. Principles of Corporate Finance, fourth edition. McGraw-Hill. 1991. p. 23  19  The firm does not need to know the individual utility curves of all its shareholders in order to maximize their utility. If the firm maximizes shareholders' wealth by undertaking all projects with a positive net present value, then in competitive capital markets, shareholders will be able to maximize their utility through their own actions. They can achieve the desired intertemporal pattern of consumption by borrowing and lending. They can also adjust the riskiness of their consumption pattern by investing in assets with different degrees of risk. Thus, within capitalist economies where the separation of ownership and management is a practical necessity, managers of firms should follow one simple instruction: maximize net present value. The logic behind this fundamental rule can be illustrated graphically as follows. For simplicity, we will consider a riskless, two-period world. Time-preference utility curves Individuals must choose between current and future consumption. In figure 3.1, time-preference utility curves represent combinations of current and future consumption which leave an individual equally satisfied. These curves are convex to the origin Figure 3.1 Time-preference utility curves  C  U-  c,  20  indicating that individuals prefer current consumption (C ) to future consumption (Q). 0  An individual's objective is to achieve the highest utility possible. Capital markets Capital markets facilitate the transfer of wealth across time; dollars today are traded for dollars in the future at an exchange rate known as the interest rate (r) which is represented by the slope of the market-opportunity line in figure 3.2. The relationship of a dollar today (C ) to a dollar in the future (d) is: 0  C  -  C  l  1+r Figure 3.2 Market-opportunity line Q  |  j  prefers to lend for future consumption prefers to borrow for current consumption C  0  Borrowing and lending through the capital market removes the obligation to match consumption and cash flow. Individuals will invest where their indifference curves are tangential to the market-opportunity line. Productive opportunities Individuals can also invest in real assets such as plants and machinery. The returns from investing in productive opportunities are depicted in figure 3.3. Starting from P, the initial investment lj may yield returns higher than those in the capital market.  21  However, this curve exhibits diminishing returns because "unless the individual is a bottomless pit of inspiration, the line will become progressively flatter" . The slope of 4  this curve is the marginal rate of return on productive investments which is identical to the internal rate of return (IRR). In the absence of capital markets, individuals will invest at the point of tangency between their utility curves and the productive opportunities curve. Figure 3.3 Productive-opportunities curve  Diminishing returns: investment Ij yields return of Rj Equivalent amount I only yields R 2  2  Investing in both productive opportunities and capital markets We will now see how an individual's utility can be maximized by investing in both productive opportunities and capital markets. Looking at figure 3.4, if people could only invest in capital markets, then they would have to choose a point along DH. Likewise, if they could only invest in productive opportunities, then they would have to choose a point along DL. However, by investing amount JD in productive opportunities  4  Brealey and Myers p. 17  22  and lending OJ in capital markets, they could increase their future income by GM. By borrowing against OG, they could increase their current income by JK. By investing an optimal amount JD in productive opportunities and then borrowing or lending in capital markets, individuals have increased their utility to any point along KM. Figure 3.4 Optimal and suboptimal investments exploiting market opportunities Q M  S. N. \  L H  \  With an optimal investment, utility is maximized to any point along line KM  /  \  / /  >w  G \ ^  /  i / Yv N. • \ \ >.  0  Optimal investment  ^"""O^  J  Suboptimal investment M )  /  'V/\ N D  yields only DQ Q  K  C  o  Why is JD the optimal amount to invest? This can be explained by the Net Present Value Rule. JK is the present value of OG. JD is the cost of the investment and the net present value of the investment is JK-JD-DK.  JD is the amount which  maximizes this difference DK. If we only invested ND, then the NPV would be A©NQ=DQ which is less than DK and thus suboptimal. The preceding discussion has shown how maximizing NPV will lead to an optimum level of investment from a shareholder's perspective. However, while NPV is widely considered to be the superior investment rule, there are several other investment  23  criteria available to managers: internal rate of return (IRR), payback, average return on book value, and profitability index. Internal Rate of return (IRR) The Internal Rate of Return is the discount rate that makes the NPV of a project equal zero. The IRR investment rule states simply "Accept investment opportunities offering rates of return in excess of their opportunity costs of capital." IRR, which is a 5  profitability measure determined by the amount and timing of project cash flows, should not be confused with the opportunity cost of capital which is a standard of profitability determined in the market by the expected rate of return on assets with similar risk profiles. In figure 3.4, the optimality of JD in a two-period certain world can be explained using IRR. Moving along the productive-opportunity curve from P, we invest in productive opportunities as long as the marginal return on investment is greater than the slope of the market-opportunity line. In other words, we invest as long as the internal rate of return (IRR) is greater than the opportunity cost of capital. Although NPV and IRR criteria often lead to the same results, the IRR rule contains several pitfalls: •  Lending or borrowing: In some projects, we pay out money now and receive money later which is equivalent to lending. In other projects, we receive money now and pay out money later which is equivalent to borrowing. Some projects are hybrids which exhibit features of both borrowing and lending over time. Depending on the timing of  5  Brealey and Myers p. 83  24  cash inflows and outflows, all such projects could end up having the same IRR leaving us at a loss as to which was the best investment. NPV would clearly indicate the optimal investment. •  Multiple Internal Rates of Return: In some cases, there are several discount rates which make NPV equal zero. According to Descartes "rules of signs", "there can be as many different internal rates of return for a project as there are changes in the sign of the cash flows."  6  •  Mutually exclusive projects: IRRs do not add up so that if you combine a bad project with a good project, you may end up with a higher IRR than the good project on its own. In addition, IRR can be misleading when you try to rank projects of different scale and with different patterns of cash flow over time. However, these difficulties can be overcome if you look at the IRR on incremental investment.  •  Curved interest-rate term structures: The short-term opportunity cost of capital often differs from the long term rate. The IRR rule tells us to invest if IRR is greater than the cost of capital, but how do we compare the IRR to several different rates? We are now faced with the difficult task of trying to come up with a weighted average of these rates. Despite these problems, IRR can be finessed to produce the same results as NPV  analysis. However, NPV is a simpler method less prone to pitfalls. Payback Period  6  Brealey and Myers p. 87  25  Firms arbitrarily choose a payback period — the time it takes before cumulative forecasted cash flows equal the initial investment — and then accept any projects which make this cutoff. Payback is inferior to NPV analysis because it ignores all cash flows after the cutoff point. Average return on book value Projects are judged on their book rate of return — the average forecasted profits of a project after depreciation and taxes divided by the value of the investment. This method has several serious defects. First, it doesn't consider the time value of cash flows. Second, it uses accounting definitions of capital investment and depreciation rather than real cash flows. Finally, the choice of a cutoff average book rate of return is completely arbitrary. Profitability index The profitability index, also known as the benefit-cost ratio, divides the present value of future cash flows by the initial investment. The rule is to accept all projects with an index greater than one. Profitability index closely resembles NPV, but suffers from the same pitfalls as IRR when comparing mutually exclusive projects. In conclusion, within the framework of traditional capital budgeting, the NPV investment rule is the most reliable approach to maximizing shareholder utility. However, as will be emphasized throughout this paper, straight discounted cash-flow NPV analysis does not fully capture the dynamics of project value under uncertainty. We now look at how traditional capital budgeting has dealt with the problem of evaluating investment opportunities under uncertainty.  26  3.1.3. Evaluating investment opportunities under uncertainty The preceding discussion assumed away risk. However, risk is present in any investment decision. Sources of risk include uncertainty over costs, demand, competition, and government policy. Under uncertainty, NPV input variables are characterized by a probability distribution rather than a specific value. Given an NPV function, the distribution of primary input variables can be used to generate a probability distribution of NPVs. Rational individuals are risk-adverse and will require higher returns for taking on projects which have greater NPV variability. Evaluation of risk is an integral part of capital budgeting and has traditionally been dealt with using a variety of approaches: •  risk-adjusted discount rate approach  •  certainty equivalent approach  •  sensitivity analysis  •  Monte Carlo simulation  •  decision tree analysis  Risk-adjusted discount rate approach (RADR) RADR adjusts the discount rate (r) to factor in both the time value of money and a risk premium for the project. Future expected cash flows E(c) are discounted at this adjusted rate (r) to calculate the NPV:  NPV  =± ^ - I £k (l + r) T  r  27  where / is the initial investment. According to the rules outlined above, firms should take on positive NPV projects to maximize shareholder utility. Some firms use discount rates based on a rather ad-hoc set of investment risk categories; 30% discount rate for speculative ventures, 20% for new products, 15% for the expansion of existing business (current cost of capital), and 10% for cost improvements using known technologies . However, this is a coarse approach which 7  often fails to recognize the risk characteristics of specific projects. The Capital Asset Pricing Model (CAPM) is a more systematic approach which relates the return of a project to its relevant nondiversifiable (systematic) risk. The logic behind the CAPM can be summarized as follows. As shown in figure 3.5, overall risk can be reduced by holding a portfolio of assets which are not perfectly correlated. Losses in one position tend to be offset by gains in another. The variance of returns for a well-constructed portfolio will be less than the sum of the individual variances. Unique or firm-specific risk (eg. the health of a CEO) can be diversified away leaving only nondiversifiable risk (also called market or Figure 3.5 Diversification eliminates unique risk, but leaves market risk. Portfolio o unique risk market risk  1  7  5  10  15  number of assets  Brealey and Myers p. 201  28  systematicrisk)which is driven by general economic forces (eg. inflation). By combining different assets, investors seek to create a portfolio which maximizes return for a given level of risk. In figure 3.6, the crosses represent inefficient mixtures of assets because a higher return could be attained for the same level of risk. The thick curve represents the frontier of efficient portfolios. This curve can be calculated using a variant of linear programming known as quadratic programming which requires estimates of the expected return and standard deviation of each asset and the correlations between each pair of assets as inputs. Figure 3.6 Efficient Portfolio frontier with borrowing and lending Expected return  _ borrowing  lending  /  x  x  [ X x Standard deviation  We can extend the range of investment possibilities through borrowing and lending. By investing some money in asset portfolio 5 and some in riskless treasury bills (i.e., lending), we can achieve any risk-return combination along line R S. If we can t  borrow at the risk-free rate, by investing the proceeds in S, we can achieve risk-return combinations beyond S.  29  With opportunities for diversification, an investor should only require a risk premium to compensate for a new asset's marginal contribution to the total risk of the market portfolio. The CAPM determines this risk premium using "betas" (|3) which are a function of the covariance of an asset's return to the return of the securities market as a whole: £(r)=r +6  [E(r )-r ] m  f  where E(r) is the required return from the asset, E(rJ is the expected return from the market portfolio, r is the risk-free interest rate and f  =  cov(r,r ) m  var(r ) m  is the asset's volatility relative to the market. Betas are estimated by regressing an asset's returns against the returns of the entire market (a stock index like the TSE 300): r = a + (3xr +8 m  where r is the return on the asset, r is the return on the market and a and e are m  parameters characteristic of the asset. Beta is estimated using ordinary least-squares (OLS) regression. There are a number of brokerage and advisory services which regularly publish beta estimates. The CAPM formula relating the expected return on an investment to its beta can be represented graphically with the security market line. A riskless investment has a beta of zero while the market portfolio has a beta of one. As depicted in figure 3.7, the CAPM postulates a linear relationship between these two benchmarks with the risk premium of an investment varying in direct proportion to its beta.  30  Figure 3.7 The CAPM security market line (SML)  Expected return on investment I  SML=E(r)= +(3[E(r )-r ] rf  m  f  The CAPM provides a sound theoretical basis for deriving an appropriate discount rate. It assumes there is a set of efficient asset portfolios which provide the highest expected return for a given standard deviation. The market portfolio is the point of tangency between the frontier of efficient portfolios and the borrowing-lending line extended from the risk-free rate of return. Investors are interested in the marginal impact of an asset on the risk of a portfolio. This impact depends on the stock's sensitivity to changes in portfolio value. This sensitivity is measured by betas. The appropriate discount rate (i.e., the opportunity cost of capital) for a given project is calculated using the project's beta, the risk-free rate, and the risk premium (expected market portfolio return minus risk-free rate) as inputs. This discount rate is then used for NPV analysis. Let's look at how RADR could be used to evaluate a toll road project. First the betas for similar toll road projects would be estimated by regressing the returns on toll roads to the returns on a market index. Although what constitutes a similar project is often subject to debate, we will assume that these toll roads are exposed to basically the  same economic, political, and operational risks and have similar debt-equity structures. Given an average beta estimate of 0.80 (perhaps reflecting stable demand and a monopolistic position for the toll roads), a risk premium on the market of 7.4% , and a risk free rate of 5%, we would use the CAPM as follows: E(r)=r +$x[E(r )-r ] f  m  f  Eir) =0.05 + 0.8(0.074) Eir) =0.1092 to calculate an opportunity cost of capital for this project of 10.92 %. We would then use this rate to discount the expected cashflows, assuming the beta doesn't change dramatically over the lifetime of the project (a questionable assumption). Summing up the present value of future cashflows,  NPV = ±-^L-I we should invest in the project if the NPV is positive. Certainty equivalent approach While RADR incorporates adjustments for risk and time into the discount rate r, the CEQ makes separate adjustments for risk and time. The certainty equivalent (CEQ) is the smallest certain payoff we would accept in exchange for a risky cash flow C . In t  other words; we have two identical expressions for present value (PV):  PV =  C, 1 +r  CEQ  l+r  f  where r is the risk-free rate. f  Brealey and Myers p. 177  32  We can use the CAPM to derive certainty equivalents. The CAPM tells us that the return on an asset with present value PV should be: C l + r = — - = l + r +6 (r py f ' f  m  -A>) J  To calculate beta, C  ^  PV  j  f  cov(r,rJ 0  Since  K  CT m  m  is known and thus does not "covary", we can rewrite the expression as, 1 cov(C rJ lt  Plugging this into the CAPM return-on-asset equation, C\  cov^.r^,)  = 1 + r, +  PV  f  The expression  J — ^ -  r -r x - oi M  7 y  is known as the market price of risk and is written as"k.It  indicates the risk premium per unit of variance. With some simple manipulation, the PV equation can be written as, C -'kco\(C ,r ) PV = l+r 1  1  m  f  The deduction we make from a risky cash flow to calculate the CEQ thus depends on the market price of risk and the co variance of the risky cashflow with the market. The CEQ and RADR approaches are equivalent under the assumption that cumulative risk increases at a constant rate over time (i.e., the risk borne per period is  33  constant). However in practice, the risk-per-period is usually not constant and the market risk premium is difficult to determine. Sensitivity analysis NPV is a function of a number of variables. Sensitivity analysis assesses how changes in each variable affect NPV. A base-case scenario is developed using the mostlikely variable forecasts. Each variable is then changed by a certain percentage above and below its base-case and the impact of this change on NPV is measured. This process forces managers to identify the crucial variables which contribute most to the riskiness of the project's NPV. In addition, it indicates where additional information would be useful and helps to expose inappropriate forecasts. Sensitivity analysis can be represented on spider graphs as in figure 3.8 which evaluates a simplified version of a toll bridge project. Assume that NPV is a function of  Figure 3.8 Sensitivity analysis: A simplified toll-bridge spider graph  construction costs \  construction time  % change in NPV toll levels traffic  operating costs % change in variable  34  the following primary variables: construction costs, construction time, operating costs, traffic, and toll levels. Looking at this graph, we see that increases in toll levels and traffic have a positive impact on NPV whereas increases in construction time, construction costs, and operating costs have a negative impact. Furthermore, we see that changes in construction costs and toll levels have a strong impact on NPV whereas operating costs have a small effect on NPV. Sensitivity analysis has its limitations. Because we are looking at the effect of variables in isolation, we may be overlooking important interactions between variables. In the toll-bridge example, there is probably a strong correlation between the level of tolls and bridge traffic. Construction time could be decreased through fast-tracking at the expense of increased construction costs. Another drawback is that we cannot easily capture the impact on NPV of simultaneous changes in combinations of variables. Monte Carlo simulation Sensitivity analysis looks at the effect of a limited number of likely variable combinations. Monte Carlo simulation is a method of evaluating all possible combinations. A probability distribution of NPV is generated through repeated random sampling from probability distributions of the crucial NPV input variables. The steps for a Monte Carlo simulation are as follows: 1. Model the project: Express NPV as a mathematical function of relevant variables. Include interdependencies between variables and across time. Models can be developed in a discrete or continuous-time framework.  35  2. Specify the probabilities of relevant variables: Use sensitivity analysis to determine which variables are important so that special care can be taken in obtaining their precise probability distributions. Probabilities can be derived from past empirical data or elicited subjectively. 3. Simulate cash flows: Use a computer random number generator to draw random samples from each of the probability distributions for the relevant variables. Input these numbers into the NPV formula to calculate NPV. Repeat the process many times to generate a probability distribution for the project's NPV as shown in figure 3.9. Figure 3.9 NPV probability distribution NPV probability  0  Expected NPV  Although Monte Carlo simulation can be useful in evaluating expected cash flows and risk, it has a number of limitations. First, it is very difficult to correctly model all the interdependencies and functional forms of the project variables. Second, it may be difficult to obtain unbiased estimates of probability distributions for the variables. Third, it is difficult to interpret the information provided by a distribution of NPVs. Cashflows for each iteration of the model are discounted at the risk-free rate rather than the cost of capital. Using a risk-adjusted rate in the simulation would amount to double counting. Risk is reflected in the dispersion of the NPV distribution. However, our traditional  36  interpretation of NPV — the price at which an asset would sell in a competitive market — is meaningless when we are faced with a number of possible NPVs. Management "has no rule for translating that profile into a clear-cut decision for action." Furthermore, simulation ignores the investor's opportunity to diversify and fails to capture the interaction between the distribution of returns on a project and the distributions of returns on other assets. Decision-Tree Analysis Decision-tree analysis (DTA) is useful in analyzing sequential investment decisions when uncertainty is resolved at discrete points in time. The mechanics of decision-tree analysis can be illustrated using a simple example. Let's look at a toll road operator considering whether or not to invest in the development of AVI (automatic vehicle identification) technology. There is a 30% chance that the $0.5 million R&D investment will lead to the successful development of a technically feasible AVI technology. The operator can then implement this new technology for $3 million. The marketing department has forecast there is a 60% chance that this new technology will be well-received by the marketplace resulting in cash flows with an NPV at year two of $20 million and a 40% chance that it will be poorly received resulting in cash flows with an NPV at year two of $6.5 million. If the firm doesn't go ahead with the R&D investment, then it will continue to receive cash flows with an NPV at year zero of $8 million. If it doesn't implement the new technology (i.e. the R&D fails or management decides the new technology is too risky), it will receive cash flows with an NPV at year one of $8.8  9  Trigeorgis, Lenos. Real Options MIT Press 1997. p.56  37  million. Assume the cost of capital is 10%. Figure 3.10 depicts this scenario using a decision-tree chart where squares represent decision nodes and circles represent outcomes determined by fate. Figure 3.10 Decision tree of whether or not to invest in AVI technology  The optimal initial decision of whether or not to invest in the new technology is determined by starting at the end of the tree and working backward to the beginning. The dynamic programming approach works as follows: First, we calculate the expected NPV assuming the R&D investment in AVI is successful and the new technology is implemented.  M(2W+04teS 1.10  38  Because the expected value of implementing the new technology ($10.27m) is greater than the value of maintaining the status quo ($8.8 m), the optimal decision at year one would be to implement the new technology. However, this decision is contingent upon the success of the initial R&D investment. Working backwards to time zero, the expected NPV of the R&D investment is: N  0.4 0.27)  p  (1  0  +  0.6(8.8)_ ^ 3 0  80  1.10  Since the expected NPV from investing in the AVI project is greater than that from maintaining the status quo ($8.0m), the NPV rule tells us to go ahead with the initial R&D investment. While decision tree analysis can be a useful tool in analyzing the value of events which are contingent upon the outcome of other events, it suffers from several drawbacks. One problem is practicality. When trying to model complex sequential investment opportunities, decision trees quickly become unwieldy decision "bushes". Another problem lies in DTA's treatment of the discount rate. In the presence of operational flexibility, the risk structure of the project changes and the appropriate opportunity cost of capital becomes unclear. In the above example, if the company could invest in the $0.5 million R&D through installments (e.g., $250,000 immediately and $250,000 six months from now), then it could abandon the R&D at an early stage if the investment didn't look promising. This option to pull out before making the full $0.5 million commitment makes the R&D project less risky than if the company had to make the full commitment upfront. We must somehow adjust the 10% discount rate to reflect  39  the change in the riskiness of the R&D cash outflows. Traditional DTA, while on the right track to valuing complex sequential decisions, unfortunately offers little insight into the discount rate problem.  3.1.4. Problems with traditional capital budgeting Managers have long realized that DCF methods such as NPV analysis are illequipped to value projects rich in flexibility and strategic importance. How can one value the role of active management? What is it worth now to be able to adapt to the arrival of new information in the future? How do we quantify the portion of a project's value created by its interdependence with other projects? Rigid NPV analysis based on the assumption of passive investment gives little insight into these questions. Sensitivity analysis, Monte Carlo simulation and decision tree analysis shed some light on the importance of flexibility, yet they also lack the ability to fully model the dynamics of project flexibility. The dissatisfaction with traditional capital budgeting techniques, combined with breakthroughs in financial economics, gave rise to the subject of the next section: "the real options revolution."  40  3.2. REAL OPTIONS 3.2.1. Introduction What is a real option? In the parlance of finance, an option is the right, but not the obligation, to buy or sell an asset at a predetermined price within a specified time period. Whereas financial options are written on financial assets (stocks, bonds, derivative instruments), real options are on "real" assets (a factory, a patent, a technology, a line of business). Real options can be broadly divided into two categories: operating options and strategic options. Operating options include the flexibility to defer, expand, contract, or abandon projects. Strategic options are created by a project's interdependence with future and follow-up investments. The real options approach to capital budgeting attempts to quantify the value of managerial flexibility in capital investment projects. Managers are able to react to changes in the marketplace and respond to the arrival of new information. This ability to adapt to future market conditions has value because it expands an investment opportunity's upside potential while limiting its downside losses. This asymmetrical payoff pattern is at the heart of an option's value. Figure 3.11 shows how this asymmetry skews the probability distribution of a project's NPV. This skewed probability distribution is also known as expanded NPV because it includes both static NPV and the value of active management. A project in which a manager can defer investment, contract or expand operations, abandon the project for a salvage value, switch inputs or  outputs, or expand into other projects has more value than a project in which the manager can do nothing but passively watch the project unfold after making the initial investment.  Figure 3.11 Managerial flexibility introduces payoff asymmetry expected static NPV expected expanded NPV  : ^  0 \  NPV  * option premium  The real options revolution The real options revolution arose as managers and academics became dissatisfied with traditional investment decision-making techniques. After the second World War, two schools of thought on corporate resource allocation emerged: capital budgeting and strategic planning. Capital budgeting focused on measurable cash flows and valued them using discounted cash flow (DCF) techniques such as Net Present Value (NPV) analysis. NPV had originally been developed to value passive investments such as bonds and was ill-suited to analyze the value of managerial flexibility present in complex capital investment projects. Strategic planning recognized the limitations of capital budgeting and used a qualitative approach focusing on intangibles such as competitive advantage, market leadership, and industry structure. The Boston Consulting Group's growth matrix (1970s) and Porter's diamond are examples of strategic planning techniques.  By the 1980s, the shortcomings of traditional passive DCF analysis had become glaringly apparent. Academic critics such as Robert Hayes and William Abernathy argued that such techniques were systematically undervaluing projects by ignoring strategic concerns. James Hodder and Henry Riggs argued that DCF was frequently 10  being misused and that the incorrect treatment of inflation and risk adjustments was leading to poor investment decisions. Myers concluded that while part of the problem 11  lay in the misapplication of the underlying theory, DCF methods were inherently limited in their ability to value investments with significant operating and strategic options. He 12  proposed option pricing as a means of valuing such investments.  The pioneering work of Black and Scholes , Merton , and Cox, Ross, and 13  14  Rubinstein has given investment analysts the basic tools to begin quantitatively valuing 15  real options. However, the application of option pricing techniques to investment analysis has encountered some resistance. First, many professional managers are intimidated by the high-powered math of option pricing and fear that a mysterious "black-box" approach to project evaluation will further complicate investment decision-making. Second, many decision scientists argue that decision tree analysis is already sufficient to capture the value of managerial flexibility.  R. Hayes and W. Abernathy, "Managing Our Way to Economic Decline," Harvard Business Review. July-August 1980 J. Hodder and H. Riggs, "Pitfalls in Evaluating Risky Projects," Harvard Business Review. JanuaryFebruary 1985. pp. 128-135 S.C. Myers, "Finance Theory and Financial Strategy," Midland Corporate Finance Journal, Spring 1987, pp. 6-13 F. Black and M. Scholes, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy May/June 1973, pp. 637-659 R.C. Merton, "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science Spring 1973, pp. 141-183 J. Cox, S. Ross, and M. Rubinstein, "Option Pricing: A Simplified Approach," Journal of Financial Economics, September 1979, pp. 229-263 1 1  1 2  1 3  1 4  1 5  43  Lenos Trigeorgis and Scott P. Mason counter these arguments in their 1987 paper "Valuing Managerial Flexibility" , which shows the workability of option pricing 16  techniques and presents option pricing as a special economically-corrected version of decision tree analysis that recognizes market opportunities to trade and borrow. Our introduction to the mechanics of discrete-time real option valuation is based largely on this article and other works of Lenos Trigeorgis. However, before we look at the dynamics of real option value, it is useful to review the basics of financial options and draw an analogy between financial options and real options. Financial options versus real options A call option is the right to buy a specified financial asset (stock, bond, future, other option) at a specified exercise or strike price on or before a specified expiration or maturity date. A put option is the right to sell a specified financial asset at a specified exercise price on or before a specified expiration date. If the option can only be exercised at maturity, it's called a. European option; if it can be exercised before maturity, it's called an American option. As emphasized earlier, the value of an option lies in its asymmetrical payoff. Figure 3.12 contrasts the symmetrical payoff position from buying the underlying asset (a long position) with the asymmetrical payoff positions from buying a call or put on the underlying asset. What is the price or premium we should pay or expect to receive in order to enter an option position? There are many mathematically sophisticated approaches to option  L. Trigeorgis and S.P. Mason, "Valuing Managerial Flexibility," Midland Corporate Finance Journal, Spring 1987, pp. 14-21  44  pricing. At this point however, we will only point out the main factors influencing the value of an option. Figure 3.12 Asymmetric payoffs from options Buy asset  Buy call on asset  Buy put on asset  + Position value  X=exercise  X  X  price  Call options The price or premium (C)of a call option depends on the following variables: •  stock price (S): an increase in S causes an increase in C. Upon exercise, the payoff max(S-X,0) becomes greater with a larger S.  •  exercise price (X): an increase in X causes a decrease in C. Upon exercise, the payoff max^-XO) becomes less with a smaller X  •  interest rate (ry): an increase in r causes an increase in C. Options enable investors to buy an asset on installment credit; they pay a premium today, but don't pay the full purchase price until a time in the future. The value of being able to delay payment increases with higher interest rates.  •  time to expiration (t): an increase in t causes an increase in C. There is more time for the stock price to move upwards. Also, the value of the installment credit feature of a call option increases with longer maturity.  45  variance of returns on stock (a ): an increase in volatility causes an increase in C. There is a higher probability of the stock price making large upward movements. As long as the stock price isn't zero, a call option will always exceed its minimum value; i.e. the payoff to immediate exercise of max(S-X,0). This provides the lower bound for the option's value. The share price provides the upper bound for the option's value. This is because the payoff from the call option of max(S-X,0) is less than S by amount X if exercised and is zero which is less than S if unexercised. As depicted in figure 3.13, the value of a call option lies along a curve between these two bounds. The shape of this curve is determined by r, t, and a . Figure 3.13 Value of a call (C) before its expiration date  X  Share price S  A n American call option on a non-dividend paying stock should never be exercised early and should have the same value as a European call option. This is because if the option is kept alive, there is always a probability that S will move up to increase the payoff from exercising the call.  Put options The price of a put option (P) depends on the following variables:  46  •  stock price (S): an increase in S causes a decrease in P. Upon exercising, the payoff ma.x(X-S,0) becomes less with a larger S.  •  exercise price (X): an increase in X causes an increase in P. Upon exercising, the payoff max(X-S,0) becomes greater with a larger X.  •  interest rate (jy): an increase in r causes a decrease in P. Upon exercising the put, the holder receives amount X A higher interest rate means a lower present value of X and hence the value of P decreases.  •  time to expiration (t): an increase in t causes an increase in C. There is more time for the stock price to move downwards.  •  variance of returns on stock (a ): an increase in volatility causes an increase in C. There is a higher probability of the stock price making large downward movements. Unlike a call option, it may be optimal to exercise a put option before its  maturity. This is because for low values of S, it is highly likely that the put will be exercised to receive X. However, waiting will bring at most X, but at a later time and with a lower present value. Thus there is a critical point (S*) where the value of waiting for a potential further drop in S is outweighed by the loss in present value of X from  Figure 3.14 Value of a put (P) before its expiration date Option value  S*  X  priceS  47  waiting. At this point, the option value curve joins the lower bound which is the payoff from immediate exercise; maxC^-S'.O). Figure 3.14 shows the shape of the put option value curve which is also determined by r, t, and a .  The analogy between financial options and real options  An opportunity to invest in a corporate project resembles a call option to invest in a corporation's stock. We can map project characteristics onto the variables which determine a call option's value as follows Project Expenditures required to acquire the asset Value of the operating assets to be acquired Length of time decision may be deferred Riskiness of underlying operating assets Time value of money  Variable X  Call option Exercise price  S  Stock price  t  Time to expiration  a  Variance of returns on stock  f  Risk-free rate of return  2  r  Investment opportunities can also exhibit features similar to a put option. For example, a government loan guarantee is essentially a put option on loan payments written by the government to the party taking out the loan. The ability to scale back operations during a market downturn can also be thought of as put with the exercise price being the part of planned expenditures which can be scaled back. Given the similarities between financial options and many investment opportunities, the pricing tools developed to value financial options can also be used to  Luehrman, Timothy A. Capital Projects as Real Options: An Introduction. Harvard Business School  48  analyze the value of complex investment opportunities. The next section will work through a simple example to illustrate the principles behind real option valuation.  3.2.2. The basics of real option valuation An example of real options 18  We will use Trigeorgis' example  of an oil extraction and refinery project to  explore the nature of real options. An oil company has a one-year lease to develop an untapped oil reserve. Operation can begin only after the development stage (exploration, construction) is completed. Once the refinery is operating, the company can adjust to market conditions by either contracting, expanding, or temporarily shutting down operations. At any time during operations, the company can switch its equipment to an alternative use or salvage it for a salvage value. The options present in this example are: •  the option to defer investment  •  the option to default during construction  •  the option to expand  •  the option to contract  •  the option to temporarily shut down  •  the option to abandon for salvage value or switch inputs and outputs  •  the option for corporate growth  Trigeorgis, Lenos. "Real options and interactions with financial flexibility." Financial Management 22 1993, no.3. 202-224  49  Before exploring each of these options in more detail, it is necessary to provide a brief introduction to an option-pricing technique know as Contingent Claims Analysis (CCA). Contingent claims analysis (CCA) Contingent Claims Analysis is used when the value of a claim on an asset is contingent upon the outcome of a probabilistic event. CCA is operationally similar to decision tree analysis. However, CCA usesrisk-neutralprobabilities rather than actual probabilities to calculate expected values and it discounts these expected values at the risk-free rate rather than a risk-adjusted rate. The basic idea behind CCA is as follows. First, through market transactions we create a portfolio which exactly replicates the payoffs from the investment opportunity under all possible future states. From this portfolio, we calculate risk-neutral probabilities and use these probabilities to calculate certainty-equivalent (CEQ) cash flows. We discount the CEQ cash flows using the risk-free rate to determine the current value of the investment opportunity. CCA assumes there is a "twin security" which has the same risk characteristics as the project and is traded in financial markets. In the oil field example, a "twin security" could be the stock price of a similar operating unlevered oil company. The value of the project (i.e. V; the present value of expected cash flows) and the value of the twin security (S) are expected to move up with probabilityp and down with probability (1-p). The value of the opportunity to invest in the oil field {E) should not be confused with the gross value of the fully developed oil field (V). Movements in the value of the  50  opportunity to invest in the oil field are perfectly correlated with movements in V and S. Figure 3.15 shows the possible movements in value over one year. Figure 3.15 Possible price movements oyer one year Value of completed project  Value of investment opportunity  Value of twin security  P  /  Through open market transactions, we follow a standard option pricing hedging strategy to create a replicating portfolio. By carefully combining n units of S and B units of a riskless investment, we can create a portfolio which exactly replicates the payoff from investment opportunity E. The mechanics of the replicating portfolio involve borrowing amount B and using the proceeds to finance a portion of the purchase price of n units of S. The difference between the price of n units of stock and the amount borrowed is the cost of the replicating portfolio. Figure 3.16 shows this graphically. Figure 3.16 A portfolio replicating the value of the investment opportunity  E =nS*-(l+r )B +  f  E=nS-B Borrow B and put the proceeds towards buying amount n of S.  E=nS'-(l+r )B f  51  We now invoke the principle of "no arbitrage" which states that investments with equivalent payoffs and risk characteristics must have the same present value. If the future payoffs from our replicating portfolio and our investment opportunity are the same under all possible future states, then their current prices must also be the same. If the prices diverged, we could make infinite riskless profits through arbitrage by continuously buying low and selling high. The market doesn't allow such "free lunches" and astute market players are quick to force prices to a no-arbitrage equilibrium. Under the assumption of no arbitrage, we can use a replicating portfolio of traded assets to price a non-traded asset such as an investment opportunity. From figure 3.16, treating the conditions of equal payoffs as two equations, we can solve for the unknowns n and B: (E -E-) n— (S -S~) +  +  (E S~  -E~S )  +  B=  +  (S -S-)(l  + r)  +  f  Plugging n and B into the equation E=nS-B, we can calculate the current value of the investment opportunity (E -E-) (S -S~)  (E S~-ES ) (S -S-)(l + r )  +  E=  +  s  +  +  +  f  Rearranging, (S(l + r )-S-) f  (S -S~) +  E=  E +\l+  (S(l + r )-S-) f  ( s  +  - s - )  E J  1 + r, /  52  and making p' —  ^p'E  +  +  E  (S(X +  r )-S~) f  (S -S~) +  , we can express the value of the investment as,  (l-p')E-  1 + r, where p' is called the risk neutral probability. Another way of looking at p' is to ask what probabilities would make the expected return on asset 5 equal to the risk-free rate. Expressed mathematically,  s  f  Solve for p',  p' F  (l +  r )S-S~ f  s  +  - s ~  Note the difference between the risk-neutral probability p' and the actual probability p. Instead of using actual probabilities to calculate expected future values and then discounting these values at a risk-adjusted rate, we use risk-neutral probabilities to calculate CEQ values which allows us to discount at the riskless rate. So far, our approach to CCA risk-neutral option valuation has been based on the ability to use a traded underlying security and riskless borrowing to create a portfolio which replicates the payoff of the option. For real options however, the underlying asset is often not traded. Despite this obstacle, contingent claims on an asset can be priced in a world with systematic risk by replacing the expected cash flow with a CEQ calculated using the market price of risk A, (recall CEQ approach in capital budgeting). Using C C A to value real options  53  We will now work through a simple one-period numerical example of the oil field investment opportunity to compare traditional discounted cash flow decision tree analysis (DCF/DTA) with contingent claims analysis. Assume that the expected price movements of the project value and the twin security value are as shown in figure 3.17 with the actual probability of an up movementp-0.5, the risk-free interest rate ry=8%, the riskadjusted interest rate r=20%, and the required investment outlay at time zero 7 =$104. 0  Figure 3.17 Possible price movements over one time period Value of completed project 0.5  .^  V +  =  1 8 0  0.5  'y ,  S =36  \  S"=12  +  S=20 < ^  V=?? <Q 0.5  Value of twin security  \  V"=60  0.5  First, we find V, the value of this investment opportunity, assuming there are no options. Traditional DCF/DTA analysis using expected values calculated with actual probabilities yields a result of, pV + (l-p)V+  r  =  1+r 05(180) +0.5(60) 1.2 7 = 100 CCA uses risk-neutral probabilities calculated from a perfectly correlated twin security S (in our case, the traded oil company),  54  ,  =  {S(l + r)-S~) (S -S~)  P  +  ,  20(1.08)-12  p = 36-12 p' = 0A The certainty equivalent values are then discounted at the risk free rate, p'V+ + (l-p)yl+r f  0.4(180)+ 0.6(60) 1.08 V = 100 In the absence of options, traditional DCF/DTA and CCA yield the same result, 7=100. Given the initial required investment of $104, the NPV rule tells us to reject this project because it has a negative NPV (100-104=-4). In the presence of options however, DCF/DTA and CCA yield different results. Managerial flexibility creates asymmetrical payoffs which DCF/DTA is ill-equipped to value. The following examples explore the options present in the oil field case and show how CCA succeeds in valuing these options. The option to defer investment Assume the company has a one-year lease on the oil field. Although undertaking the project immediately has a negative NPV, the company can wait one year to see what happens to oil prices. If prices go up, the project may become profitable. If prices go down, the company has no obligation to invest and will end up losing only the cost of the lease. This position is analogous to a call option on project value V with an exercise price equal to the required outlay ^=112.32 (104x1.08).  55  E = max(F -/ ,0) = max(180-112.32,0) = 67.68 +  +  1  E~ =max(V~ -I 0) lt  = max(60-112.32,0) = 0  Using the risk-neutral probabilities derived earlier to calculate expected value and then discounting back at the risk-free rate, p'V  +  v  =  +  (l-p')V-  1 + r, 0.4(67.68)+ 0.6(0) 1.08 V = 25.07 we find that the value of this opportunity to invest E is $25.07. An NPV which includes both the value of passive investment and the value of active management is often called expanded NPV. The expanded NPV of this investment opportunity is derived from two sources: traditional static NPV (-4) and the option to defer (29.07). Expanded NPV = option to defer + static NPV  The value of the option to defer can thus be expressed as, Option to defer = expanded NPV - static NPV  It is interesting to compare CCA valuation of this option with DCF/DTA valuation. Using actual probabilities and a risk-adjusted discount rate, DCF/DTA may value the option to defer differently to CCA. In this example, E  =  E+ + (X-p)E-  P  l +r  f  „ 05(67.68) + 0.5(0) E = 1.20 E = 28.20 Why is the CCA value of $25.07 correct and the DCF/DTA value of $28.20 incorrect? Recalling the principle of no arbitrage, clearly nobody would pay $28.20 for a  56  future payoff which can be replicated in the market for $25.07. In this case, the replicating portfolio would be, (E -E~)  67.68-0  +  n=  =  (S -S~) _ (E S~-E~S ) +  +  +  " (S -S-)(l+r ) +  f  = 2.82  36-12 (67.68x12)-(Ox36) =  ~  =  3  J  3  3  (36-12)(1.08)  By buying 2.82 shares at $20 each for a total of $56.40 and borrowing $31.33 at the riskless rate, next year our portfolio would be worth E =nS -(1 + ^)5 = 2.82x36-1.08x31.33 = 67.68 +  +  E~ =nS~ -(1 + ^)5 = 2.82x12-1.08x31.33=0 which is exactly the same as the payoffs from our investment in the oil field. The current cash outflow required to create this portfolio is the price of the stock purchase ($56.40) less the amount borrowed ($31.33) which is $25.07, the value of the investment opportunity calculated using CCA. This example shows how DCF/DTA fails to take into account market opportunities to buy or sell and borrow or lend. CCA corrects for this by creating a replicating portfolio and using it to determine the expanded NPV of the investment opportunity. The option to default during construction If the investment is made in stages, management can abandon the project during construction if further outlays are expected to exceed the value of continuing the project. Suppose the $104 investment consists of two stages: construction of basic infrastructure costing $44 and then construction of the oil refinery costing $60. First, 7 =44 would be 0  paid upfront and the remaining $60 would be placed in an escrow account earning the  57  risk-free rate (8%) to be paid as 7 =64.8 (1.08x60) at year one. At the end of the first Z  year, management has the option to decide whether or not to proceed with construction of the plant. If the value of the plant goes up, management should continue with the project for a payoff of, E = max(7 -7 0) = max(180- 64.8,0) = 115.2 +  +  lf  If the value of the plant decreases, management should default on the payment and abandon the project for a payoff of, E~ = max(V~ -1, ,0) = max(60 - 64.8,0) = 0 The expanded NPV of this investment opportunity with the option to default on future outlays is,  rr  h  °  y  „ 0.4x115.2 + 0.6x0 E = 44 1.08 A A  E = -1.33 The value of the option to abandon by defaulting is, Option to abandon by defaulting = expanded NPV - static NPV = -1.33-(-4)=2.67  While the option to default during construction has value, the expanded NPV is still negative so we shouldn't undertake this project.  The option to expand If market conditions turn out to be favourable after one year, management may want the flexibility to expand operations. Suppose that after one year, a follow-on investment of I[ =$80 would double the scale and value of the plant. At year one,  58  management has the option to either expand operations and receive a payoff of 27  +  -1\  or maintain the current scale and receive V. This investment opportunity can be thought of as a call option on a future opportunity to expand. The payoffs from this investment opportunity are, E  =  +  max(V ,2V -/*) = (180,360-80) = 280 +  +  E~ = max(7-,2F- - J, ) = (60,120-80) = 60 e  The expanded NPV of this investment opportunity with the flexibility to expand operations is, p'E  +  +  (l-p)E-  r~ <> l +r „ 0.4x280 + 0.6x60 . . . E = 104 1.08 E = 33.04 h  -  7  f  The value of the option to expand is, Option to expand = expanded NPV - static NPV =33.04-(-4)=37.04  The option to contract The option to contract is similar to the option of abandoning by default during construction. However, instead of completely abandoning the project, management merely scales down the operations. Suppose that an initial investment of IQ=50 would enable the refinery to run at half the planned scale, 0.5V. An investment of I± =$58.32 (the future value of $54) at year one would bring the plant up to full capacity. If market conditions next year turn out to be unfavourable, management may want to exercise its option to halve the scale of operations by not investing I . This option to contract is X  analogous to a put option on the part of the project which can be contracted with an  59  exercise price equal to the planned expenditures which can be canceled. The payoffs from this investment opportunity are, E = max(F -1, ,0.5V ) = (180 - 58.32,90) = 121.68 +  +  +  E~ = max(V~ -1, ,0.57') = (60 - 58.32,30) = 30 The expanded NPV of this investment opportunity with the option to contract is, „ 0.4x121.68+0.6x30 E= 1.08  c  n  50  £ = 11.73 The value of the option to halve operations is, Option to contract = expanded NPV - static NPV=11.73-(-4)=15.73  The option to temporarily shut down If cash revenues are insufficient to cover variable costs in a given year, management may have the flexibility to temporarily shut down operations until they become profitable again. Each year can be viewed as a call option on that year's cash revenue with the exercise price equal to the variable cost of operating. We can breakdown the components of project value into certainty equivalent amounts (CEQ) of revenue (R), variable costs (VO and fixed costs (FC). Assuming a lifespan of T years, the value at year n=l of the project is given by: „  ".-E  ^CEQ(R-VC-FC)  q  + r /  r  For simplicity we will treat R, VC, and FC as riskless after year one and will ignore the value of any options after year one. If the value of the project goes up to 180 in year one, assume that the cash flows for year one are ^=100, 76^=35 and FCi=5. Under good market conditions, year one thus contributes 60 to the project's total value of  60  180. The remaining cash flows from year two to year T contribute 120 to the project's total value at year one. If the value of the project goes down to 60, assume that the cash flows for year one are ^=20, FC =35, and FCj=5. Under poor market conditions, 1  operating in year one results in a loss of 20 (20-35-5=-20) implying that the remaining cash flows after year one contribute 80 to the project's total year-one value of 60 (8020=60). What is the value of being able to avoid this 20 loss by temporarily shutting down? If we operate, we pay VC and FC to get R X  pay FC  X  v  If we temporarily shut down, we  avoid paying VC and forego receiving R . The payoff from this option to  lt  X  x  temporarily shut down is, E = max(/? - VC - FC, ,-FC^) + V 1  X  r  where V is the value of the cash flows after year one. r  £  +  = max(100 - 35 - 5,-5) +120 = max(60,-5) +120 = 180  E~ = max(20 - 35 - 5,-5) + 80 = max (-20,-5) + 80 = 75 The expanded NPV of an investment opportunity which allows management to avoid any losses in year one due to poor market conditions, but still continue the project by paying fixed costs is _ 0.4x180 + 0.6 x 75 1.08 E = 4.33  The value of this option to temporarily shutdown is Option to temporarily shut down = expanded NPV - static NPV=4.33-(-4)=8.33  The option to abandon for salvage value or switch to best alternative use  61  If market conditions become unfavourable, management may have the option to abandon the project and receive its salvage value or switch the refinery's facilities to their best alternative use (i.e. convert the crude oil into a variety of products). Assume that the project's salvage value or its best alternative use value (A) moves over time as in figure 3.18. Figure 3.18 Salvage value or best alternative use (A)  A =144 +  Currently, .4=90 is less than the 7=100 so management would not want to switch or abandon. However, if prices go down in year one, management would want to switch or abandon for a value of A' =72 which is greater than V =60. The payoffs from this investment opportunity with the option to abandon or switch are: E = max {V ,A ) = max (180,144) = 180 +  +  +  E~ =max(7",iO = max(60,72) = 72 The expanded NPV of this investment opportunity is, „ 0.4x180 + 0.6x72 E = 1.08 E = 2.67  104  The value of this option to abandon for salvage value or switch to the best alternative use is, Value of option to abandon or switch = expanded NPV-static NPV=2.67-(-4)=6.67  62  The option for corporate growth Corporate growth options are similar to the option to expand operations, but have much deeper strategic implications. Some projects are unattractive in isolation, but play a crucial role as a link in a chain of interrelated projects. The value of such projects "may derive not so much from their expected directly measurable cash flows, but rather from unlocking future growth opportunities." For example, suppose the proposed oil refinery 19  is a pilot project to develop a new technology. The infrastructure and experience gained from undertaking this project could act as springboard to further process improvements and to applications in new areas. Real option techniques attempt to quantify the value of such strategic growth options.  3.2.3. Summary  The preceding discussion has provided an introduction to the basics of real option valuation. In reality, Contingent Claims Analysis is complicated by the effects of multiple interacting options. For example, the option to change the scale of the project during construction affects the value of the option to temporarily shut down. If a project is a collection of calls and puts which interact, their combined value may differ from the sum of the separate option values. However, with some adjustments, the risk-neutral backward valuation procedure outlined above can be extended into a multiperiod interacting options setting. As the number of periods increases, the discrete solution approaches the continuous time Black-Scholes solution if it exists. However, for  19  Trigeorgis, Lenos. "Real Options and Interactions With Financial Flexibility" Financial Management Autumn 1993 p. 202-224 63  simplicity, this thesis will work within a discrete time framework. In the next chapter, we look at how real option valuation techniques can be used to understand the dynamics of value in BOT toll road projects.  64  CHAPTER FOUR  R E A L O P T I O N S IN B O T T O L L  ROADS  INTRODUCTION  The value of a BOT toll road project is affected by a variety of real options. Some of these real options arise from the inherent nature of large-scale infrastructure projects involving sequential investments. Others are created by contractual arrangements negotiated by the parties to a BOT. This chapter looks at how contingent claims analysis can be used to value BOT real options. Figure 4.1 depicts the typical life cycle of a BOT toll road project from the concessionaire's perspective.  20  Figure 4.1 The life cycle of a BOT toll road project from the concessionaire's perspective money BUILD  OPERATE  TRANSFER -><>  : loan drawdown '• pre- design design Tendering/design field  Revenue  .-7  time  operation and maintenance debt servicing  commisioning mgmt of design/construction  m g m n t o f operation and maintenance  During the build phase, important real options include the option to abandon midstream, the option to change scale or technology, and "time-to-build" options. During  Wahdan, Mohamed Y., Russell, A.D., and Ferguson, D. "Public Private Partnerships and Transportation Infrastructure" paper presented at 1995 Annual Conference of the Transportation Association of Canada, Victoria, British Columbia.  66  the operating phase, significant real options include the flexibility to set tolls, the option for development gain, and financial flexibility created by project financing. At the transfer stage, there are number of possible arrangements between the public and private sector which exhibit option-like features. Analyzing the value of these options is important to understanding the tradeoffs involved in negotiating a BOT toll road. This understanding is crucial to the successful development and management of a BOT toll road.  67  4.1. BUILD PHASE 4.1.1. Introduction Important real options during the build phase of a BOT project include the option to abandon midstream, the option to change scale or technology, and "time-to-build" options. Because BOT projects are negotiated between the public and private sector, there are likely to be many contractual arrangements which affect the value of flexibility during the build phase. For example, a BOT contract may stipulate certain penalties for abandoning a project during the build phase. This would affect the value of the option to abandon midstream. This section outlines how contingent claims analysis can be used to value various types of flexibility during the build phase.  4.1.2. The option to abandon midstream: If the value from continuing the project is less than cost of the investment required to continue, then management may wish to abandon the project. To analyze this option within the context of a BOT toll road, we develop a two-factor model which captures the stochastic nature of project value and construction costs at each stage of the project. Kensinger's analysis of the option to exchange one asset for another is a useful 21  starting point. We adapt his model which looks at the option to transform an input commodity into an output commodity (for example, soybeans into soybean meal or crude oil into gasoline). In our case, we move up the value chain and consider construction costs (labour, materials) as the input and a completed construction stage as the output.  Kensinger, John W. "Adding the Value of Active Management into the Capital Budgeting Equation." Midland Corporate Finance Journal. Spring 1987 p. 31-42 2 1  68  Project value is the present value of future operating profits plus any operating options upon completion of the project. We assume there is no correlation between project value and construction costs and that there is no correlation between the construction costs of different stages. This is to simplify the explanation, although such correlations could be factored into the model. Consider a BOT toll road build phase consisting of three one-year stages. The first stage involves detailed design work and the costs are certain at $3 million. The second stage involves preliminary ground preparation and the costs are uncertain with an expected value of $11.25 million and a standard deviation of $5.44 million. The third stage involves major construction and the costs are also uncertain with an expected value of $105 million and a standard deviation of $15 million. These expected values and standard deviations are derived from the following distributions:  Stage one Stage two  Stage three  Construction costs (C) $ 3 20 10 5 130 100 90  Probability of C 1.00 0.25 0.5 0.25 0.25 0.5 0.25  These distributions are based on past data and expert opinion from engineers. We assume the variance of construction costs is largely driven by the probability of technical problems rather than the price volatility of inputs (concrete, labour). In other words, construction costs are not correlated with the market.  69  To find the value of the option to abandon midstream, we begin at the final stage of construction and work backwards. We are interested in the spread between construction costs and expected value upon completion. We model the project's value which is derived from the stream of expected toll revenues over the concession period and includes the value of any options during the operating period. We treat movements in project value as binomial and assume these movements are highly correlated to the movements of a traded asset (perhaps an index of toll road companies). In our example, we assume that both our project's value and the value of the twin security have an actual probability of 50% of moving up by a factor 1.2 and a 50% probability of moving down by a factor of 1/1.2. These movements are shown in figure 4.2. Figure 4.2 Stochastic movement of project value u = 1.2  r = 5% f  d=l/u  Value of project  Value of twin security: toll index 144  P*  288  120  p* 100  100  200  1-p** 83  240' 200  ^  1-p*  167 139  69  We use this twin security to calculate the risk-neutral probability of an up movement: (l + r)-d  (1.05)-0.833  u-d  1.2 - 0.833  = 0.59  70  If we complete stage three, we have the following risk-neutral probabilities for the values (V): P*(V)  V  144 0.349174 100 0.483471 69.44444 0.167355  where p*(144)=(p*) , p(100)=2xp*x(l-p*), and p*(69.44)=(l-p*) . 2  2  To receive these values, we must pay construction costs at stage three. For simplicity, assume that we receive the project value instantaneously upon payment of the construction costs. If construction costs were found to be correlated with the market, we could work with risk-neutral probabilities. However, because most construction projects are highly idiosyncratic and cost overruns are not likely to be related to general market movements, we choose to use true probabilities. These true probabilities may be elicited subjectively or based on past data from similar projects. As stated above, for our project, engineers consider the following costs and probabilities as appropriate for stage three: c  P*(C)  130 100 90  0.25 0.5 0.25  We are interested in the expected spread between project value and costs. With the option, the project will be abandoned if the spread is negative. The spread V - C , the associated joint probability for each spread (using risk-neutral probabilities for V and true probabilities for C), and the expected value at stage three both with the option to abandon (i.e., max(V-C,0))and without (V-C) are as follows:  71  V  C  144 144 144 100 100 100 69.44444 69.44444 69.44444  130 100 90 130 100 90 130 100 90  prob(V,C) no opt V-C option V-C E(opt V-C) 14 1.2221074 14 0.087293 44 7.6818182 44 0.174587 54 4.713843 54 0.087293 0 0 0.120868 -30 0 0 0.241736 0 10 1.2086777 0.120868 10 0 0 0.041839 -60.55556 0 0 0.083678 -30.55556 0 0 0.041839 -20.55556 14.826446  E(no opt) 1.222107 7.681818 4.713843 -3.62603 0 1.208678 -2.53357 -2.55682 -0.86002 5.25  At stage two, the value of the investment opportunity with the option is the expected spread with the option discounted back at the risk-free rate:  2  1.05 To receive this value, we must pay construction costs to complete stage two.  Given the following distribution of stage two construction costs: C  P(C) 20 10 5  0.25 0.5 0.25  The spread between the value of 14.12 and C, the associated joint probabilities, and the expected stage two value with and without the option to abandon are: V C 14.12043 14.12043 14.12043  20 10 5  no opt V-C option V-C E(opt V-C) P(V,C) 0 0.25 -5.879575 0 0.5 4.120425 4.120425 2.0602125 9.120425 9.120425 2.2801063 0.25 4.3403188  E(no opt) -1.46989 2.060213 2.280106 2.870425  The value of the investment opportunity with the option to abandon at stage one is: 4 34 V, =—— = 4.133 1.05 1  72  To receive this value, we must pay known costs of $3 million at stage one. The value of the investment opportunity with the option to abandon at any stage during the project is thus worth, 4.133-3 = 1.33 How does this compare with the value of an investment opportunity which has no option to abandon? The stage three expected value without the option to abandon is $5.25 million. Discounting this amount back to stage two at the riskless rate yields $5 million. At stage two, the expected value is: V  C 5 5 5  20 10 5  p(V,C) no opt V-C option V-C E(opt V-C) E(no opt) -3.75 0.25 -15 0 0 -2.5 0.5 -5 0 0 0.25 0 0 0 0 -6.25 0  Assuming that we cannot abandon at stage two, we discount the no option expected value of -$6.25 million back to stage one,  0  1.05  and subtract the stage one construction costs. This yields a negative value of: -5.95-3 =-8.95 The value of the option to abandon at any stage is the difference between the expanded NPV of the investment and the static NPV: 1.33-(-8.95) = 10.28 In other words, if the BOT agreement was structured so that the project company could walk away at any time when the costs of proceeding exceeded the value of  proceeding, this would add $10.28 million to the value of the BOT investment opportunity. Of course, due to the highly political nature of BOT projects, it may be very difficult for a concessionaire to disentangle itself from a project which becomes unprofitable during the build phase. The government may try to discourage the concessionaire from exercising its option to abandon by explicitly stipulating penalties for abandonment in the BOT contract. For example, suppose the BOT contract stated that if the concessionaire abandons the project at stage three, it must pay the government $65 million. Under this condition, the concessionaire would never abandon the project because the expected payoff from continuing is higher under all possible states at stage three; i.e., the greatest expected loss of $60.56 million (for V=69.44, C=130) is less than $65 million. However, if the penalty for abandoning at stage three was only $25 million, then the option to abandon would still have some value because $25 million is less than the V-C spreads of -60.56 (for V=69.44, C=130), -30.56 (for V=69.44, C=100), and -30 (forV=100, C=130). The government could also use the value of the option to abandon as a bargaining chip. For example, the government could consider asking for an upfront payment of $10.28 in exchange for granting the concessionaire the right to abandon the project. In essence, the government has sold the concessionaire a call option on the value of the project at stage three and a call option on the stage three call option. The premiums for these calls could be paid by the concessionaire in the form of higher taxes or lower tolls - whichever the government feels is socially optimal.  74  The option to abandon may interact with the option for strategic growth. Abandoning a project midstream would undoubtedly have negative a impact on the concessionaire's relationship with the government and the public. This in turn could hurt the concessionaire's chances of procuring future BOT projects and expanding further into a nation's infrastructure market. 4.1.3. The option to change scale or technology The option to expand or contract the scale of the project in response to the arrival of new market information has value. For example, the concessionaire may wish to expand the scale of the project if expected demand increases during the build phase. The ability to adapt technologies to changing demand conditions also has value. For example, a toll road company may have the option to install a more expensive but more efficient high tech AVI (automatic vehicle identification) toll collection system if demand conditions justified the increased cost. The option to change scale or technology can be valued using a method similar to the technique outlined in the previous section. In the following analysis, we will consider a situation where the concessionaire has both the option to abandon and the option to change scale or technology. Assume C, is the base-case construction cost which has a distinct distribution with an expected value of E(Cj). By paying C — where C has a k  k  distinct distribution and E(C )>E(Cj) — to increase the scale or install a more efficient k  technology, management could increase the value of the project by 50%. Likewise, by paying Q where Q has a distinct distribution and E(Ci)<E(Cj), management could decrease the scale of the operations or use less efficient technology, but avoid paying C,-  75  Ci. At year three, the payoff from this investment opportunity {W) with the option to abandon and the option to change scale or technology is,  £ £ V'  E(W ) = max 3  = 1  m  a  x  ^  -C ,0)xp*{V )xp{C ) J  i  £ £ max(1.5V, - C ,0) x p *  ) x p{C )  k  m  k  q  £ £ max(.75F;. - C, ,0) x \i=i  J  J=  l  * iy.) x /»(C,)  1=1  where m is the number of possible values, p* is the risk neutral probability associated with each of these values, n is the number of possible base case construction costs with probabilityp(C), x is the number of possible increased-scale/improved technology construction costs with probabilitypfC/J, andg is the number of possible decreasedscale/cheaper technology construction costs with probability p(C). The present value of this investment opportunity is found through the reiterative process:  ^max  E{W ) = max 2  ^  E(W )  max 1.5  t  Jt  1 + r,  k=l  /=i  •C 0 xp{C )  3  E(W ) 3  1 + r, f  C ,0 xp{C ) k  max . 7 5 ^ > - C „ 0 1 + r, /  k  xp(C,)  yielding an expanded net present value of, E (W) Expanded_ NPV = ^ - C 2  x  76  This approach assumes that we can expand/contract or change the technology of the project by the same amount each year. This is perhaps unrealistic because there may only be certain times during construction when the scale or technology of the project can be changed. There may also be upper and lower limits to the size of these changes. The model can be tailored to factor in these characteristics. It should be noted that the option to change scale or technology involves multiple interacting options. Changes in scale may affect the value of options present during the operating period. For example, toll-setting options depend on the number of possible users which in turn depends on the scale of the operation. The option to change technology could also be an important strategic option. For example, by developing hightech toll collection technology, a company could establish itself as a leader in the growing toll technology industry.  4.1.4. Time-to-build option This option is a variation of the option to abandon midstream, but rather than completely abandoning the project, management can alter the timing of the project's completion. If the market moves favourably, management can bring the project into operation earlier by fast-tracking (overlapping different phases of the project), accelerating (shortening the length of a phase), or some combination of fast-tracking and accelerating. However, fast-tracking/accelerating comes with a higher cost than a regularly paced project. If the market moves unfavourably, management may wish to wait and see if the situation improves before making further investments.  77  Saman Majd and Robert Pindyck have written an instructive article on this topic 22  called "Time to build, option value, and investment decisions" . In their article, they show how "optimal investment rules can be determined for projects with sequential investment outlays and maximum construction rates". They model the value of a completed factory using a Weiner process and then use contingent claims analysis to analyze the "market value of the entire investment program" given that sequential investment outlays can be adjusted between zero and some maximum level depending on the arrival of new information. Their analysis shows "how simple NPV can lead to gross errors." 4.1.5. Summary Real options during the build phase can contribute significantly to the value of a BOT project. When negotiating a BOT contract, it is important for the public and private sector to understand how different contractual arrangements affect the value of BOT build real options. Due to the public nature of BOT projects, they are likely to have less flexibility than similar scale projects developed purely in the private sector. However, straight NPV analysis is likely to overlook important sources of value arising from the option to abandon, the option to change scale or technology, and time-to-build flexibility. Contingent Claims Analysis can successfully value these real options.  Majd, Saman and Pindyck, Robert S., "Time to build, option value, and investment decisions." Journal of Financial Economics 18 (March) 1987: 7-27  78  4.2. OPERATE PHASE: TOLL-SETTING FLEXIBILITY 4.2.1. Introduction The setting of tolls is a highly contentious issue in BOT toll road projects. The concessionaire wants the freedom to set profit-maximizing tolls. The public wants an equitable and predictable toll structure. The government wants to ensure toll rates are economically efficient and satisfy both the concessionaire and the public. Within this framework of competing demands, the government and the concessionaire can choose from a variety of toll-setting arrangements; •  Complete flexibility: the concessionaire is free to react to changes in market conditions and set profit-maximizing tolls.  •  Partial flexibility: the concessionaire is free to set prices within specified bounds.  •  Trigger adjustments: adjustments are allowed to ensure a specified rate of return  •  Fixed: the toll structure is fixed over the life of the project Each of these options entails varying degrees of flexibility. This flexibility can be  valued using real option valuation techniques. We will develop a generic toll road BOT project to show how these techniques can be applied. First, we model a demand curve which relates toll prices to traffic levels. The derivation of this demand curve is an iterative process which reflects how groups of consumers with different willingness-topay to get to a certain destination and with different values of travel time savings (VTTS) respond to varying levels of congestion. Next, we determine the stochastic nature of the socioeconomic variables which shape the demand curve. Given the stochastic characteristics of the demand curve, we then look at profit-maximizing price-setting  79  strategies under the constraints of each of the price-setting arrangements. Finally, we use the real option technique of contingent claims analysis to value projects with varying degrees of price-setting flexibility in an environment of uncertain future demand conditions. 4.2.2. Deriving the toll demand curve We begin by deriving a toll demand function for travel to destination D through the BOT toll road . This toll demand curve relates toll levels (p) to traffic quantity (q) where quantity measures the number of vehicles which pass through the system per unit of time (hour, day, year). Our demand curve analysis in figure 4.3 puts a slight twist on 23  the traditional transport economic analysis of pricing congested roads . Unlike traditional congestion pricing which factors in congestion externalities to determine the socially optimal price and quantity, we are interested in the toll road operator's profitmaximizing toll. In figure 4.3, the vertical axis measures both the driver's willingness-to-pay in terms of money and the financial value of lost time and the supplier's costs to provide the service. The demand curve reflects consumers' willingness to pay (WTP) out-of-pocket expenses (tolls) and willingness to sacrifice time to get to destination D. The average total social cost curve (ATSC) is the vertical sum of fixed and variable infrastructure operating costs (we treat road operator average variable costs, AVC, as equal to road operator marginal costs, MC; i.e., the costs of serving one additional vehicle does not change significantly with quantity) divided by q and the user's own time costs and  Mohring, Herbert. Transportation Economics. Ballinger Publishing Company 1976, Ch. 3  80  vehicle operating costs . The user marginal social cost (MSC) curve reflects congestion 24  0  externalities and reveals how much an additional driver increases the travel time of all drivers.  Figure 4.3 A twist to traditional Social Cost-Benefit Analysis  User AC(average costs): average time and operating costs of each driver User MSC(marginal social costs): the time and operating costs an additional driver imposes on all other drivers Road operator MC(marginal Costs): the costs of servicing one additional vehicle. Equivalent to average variable costs in our case. ATSC (Average total social costs): the vertical sum of user AC, operator MC, and operator fixed costs divided by q Demand: consumer's willingness-to-pay in time and money to get to destination D  Consumers are willing to use the facility and suppliers are willing to supply the service up to point q where average total social costs equal WTP. However, this point is not optimal from society's perspective because drivers are not considering the congestion externalities they impose on others. The socially optimal toll would restrict traffic to quantity q* at a generalized price of p*. This is where user MSC intersect the demand  Throughout our discussion, we will only model the effects of congestion on driver time and not consider its effect on vehicle operating costs. This is because the full costs of congestion perceived by road users is dominated by their value of time spent rather than their vehicle operating costs. In other words, vehicle operating costs are assumed constant regardless of congestion.  81  curve. In other words, the cost of an additional driver equals the willingness to pay of the marginal driver. However, in BOT projects, we must also consider the toll-setting decision from the operator's perspective. The operator wants to maximize profits by restricting quantity to the point where the operator's marginal costs equal marginal revenue. This is shown in figure 4.3 at pointp , q . The toll operator can exploit its monopoly power to maximize t  {  profits by charging the operator optimal toll. To analyze a toll operator's profit maximizing behaviour under stochastic conditions, we require a demand curve solely in terms of WTP cash; i.e., demand net of the value of time contributed by motorists using the toll road. We achieve this by deriving a toll demand curve which relates traffic quantity exclusively to toll levels. This is in contrast to traditional congestion pricing which relates general willingness to pay in both time and money to traffic quantity. If we assume that consumers all have the same VTTS, deriving this demand curve is a straight-forward procedure in which we simply subtract average time costs at quantity q from the general WTP at q to determine the toll which generates traffic q. Taking the derivative of this toll demand function yields the toll marginal revenue curve. The profit-maximizing toll and quantity is where the toll marginal revenue curve intersects the toll road operator's marginal cost curve. For simplicity, we assume the marginal costs for providing toll road services are fixed, although in reality maintenance expenses may increase for every additional car allowed on the system if the traffic level far exceeds the facility's design capacity. The derivation of this demand curve is shown in figure 4.4.  82  Fig. 4.4 Deriving the toll demand curve assuming everyone has the same VTTS  General demand curve P  WTP  average user time costs  Toll demand curve  Profit-maximizingj9* and q*  MR  L  firm's MC  However, the assumption that all consumers have the same VTTS is questionable. There is evidence that people with higher incomes tend to have higher VTTS. See appendix II for a more thorough discussion of VTTS. Our model allows for groups of consumers to have different VTTS. It generates a toll demand curve through an iterative process in which given a toll price, consumers adjust to different levels of congestion according to their VTTS. These adjustments oscillate towards an equilibrium quantity for each toll level. If we assume that consumers all have the same VTTS, then our model will lead to the same results as the previously mentioned process of simply subtracting average consumer borne (time and operating) costs from consumer WTP. However, the above-mentioned approach breaks down when we introduce different VTTS because it fails to reveal which points on each group's time cost curves should be subtracted from which points on each group's WTP curves. Our model is capable of determining an equilibrium toll demand curve which captures congestion effects in an environment where consumers value time differently. The mechanics of this model are as follows.  83  We divide potential consumers into groups based on their value of travel time savings (VTTS). For simplicity, consider two groups of potential road users who want to get to destination D: •  Hgroup: high-income potential road users who are willing to pay $5 to reduce their travel time by one hour  •  L group: low-income potential road users who are willing to pay $1 to reduce their travel time by one hour  Once we have segmented potential consumers based on their VTTS, we derive a demand curve for each group. The demand curve reveals how much consumers in each group are willing to pay to get to destination D. We are making an important distinction between consumers' willingness to pay out-of-pocket expenses (i.e., a toll) to get to D and their willingness to sacrifice travel time. For arguments sake, assume there is no congestion and a person from group L can always get to D at the free flow travel rate (the optimal car speed and hence travel time which is determined by the design capacity of the system). Assume that free flow travel time converts to a monetary value of $1 for people from L. Under this condition, there may be one person from the L group willing to pay a toll of up to $9 on top of the $1 travel cost for a total price of $10 to use the road. Thus a $9 toll would induce one person from L to use the road. At a toll of $4 (for a total price of $5), there may be fifty people from L who would use the road. When deriving the WTP curves for H and L, it is important to bear in mind that these curves are made up of two components: the WTP a  84  toll and the WTP time costs. For simplicity, we assume these demand curves are linear as shown in figure 4.5. Fig. 4.5 Demand for travel to D for groups with different values of travel time savings Generalized price (toll and time)  Now we introduce the effects of congestion on the quantity demanded for a given toll price. As shown in figure 4.6, the derivation of the demand curve with congestion costs is an iterative process which reveals how groups with different willingness-to-pay for travel to destination D and different VTTS respond to different levels of congestion over time. At time zero, a toll pricep is set. We assume that at time zero, users expect to pay time costs based on the free flow rate of traffic in addition to the toll. At time zero, the total number of users is q^o + q . H>0  However, at total quantity q  L0  + q , there is likely to be some unexpected H0  congestion. The level of congestion is determined by a travel time-traffic flow function which is based on empirical results from traffic engineering studies. Our model uses the Kimber and Hollis function: 25  Bureau of transport and communications economics. Traffic congestion and road user charges in Australian capital cities. Australian Government Publishing Service, 1996  85  Figure 4.6 Dynamic process to derive the toll demand curve for two groups with different VTTS WTP toll and time to get to D Time Zero E(TCn  )-TC  0  E(TC )-TC U0  Hi0  E(TC )  U0  Hi0  E(TC ) Li0  <1L  Time cost function L  \  )-TC  HJ  _  Lt0  E(TCE(TC )-TC yw free flow U  )-TC  E(TC  H0  E(TC )-TC Li0  E (TC  HJ  E(TC )= free flow H>0  >, -I  UU  H0  + qui Time one  —  E(TC  )-TC  H0  H0  E(TC )-TC U0  U0  qui + qm  Time two  _ E(TC  LJ  E(TC )-TC HJ  HJ  )-TC  LJ  qui + qn2  K = K[l + ((* - 1) + a V ( * - D + f o t ) ] 2  where t is average travel time per kilometre, t is free speed travel time per kilometre, x a  0  is the volume-capacity ratio, and a and b are parameters which change depending on the  86  type of road. See appendix III for more details on traffic flow functions. We multiply t  a  by each group's VTTS to find their time costs, TC. After actually using the road at time zero, drivers from L find they are paying more than they had expected to pay because TC >E{TC ), h0  where £ ( • ) is an  u0  expectations operator. At time zero, EiTC^o) is equal to the free flow rate. Those people who had paid E{TC )-TC Li0  Li0  more than they would have been willing to pay if they had  known the costs of congestion beforehand will not use the road again at time one. The same reasoning applies to drivers from H. Thus, the number of drivers from L and H at time one will be less than at time zero. However, because the total number of drivers on the road at time one is less than at time zero, travelers from L and H who actually use the road at time one will find the congestion costs lower than they had expected. At time one, the expected time costs for drivers from L would be, EiTC^)  which is equal to the actual time costs from the  previous period TC . The actual amount drivers end up paying, TC L0  price drivers expected to pay, TC  U1  L0  is lower than the  and thus more drivers would have used the road if  they had known this beforehand. The same reasoning applies to drivers from H. Hence the number of drivers from H and L at time two will be greater than at time one, but less than at time zero. This process will converge to equilibrium points q and q which are unique to L  H  the pricep, as long as each group's WTP curve is greater than its average time cost (ATC) curve. We repeat this process for all relevantp's to derive the equilibrium demand curve. We impose the conditions that q cannot exceed the quantity at which ATC equals L  L  87  WTP and q cannot exceed the quantity at which ATC equals WTP . Otherwise the L  H  H  H  process will become unstable. This dynamic process can be modelled using a computer program such as Mathcad (see appendix IV). We will work through a simple example using Mathcad to show the mechanics of this dynamic process. First we define the variables for the two groups mentioned above; H and L. Variable A L  A  H  BL,  B  H  P q ,t L  iHt  TC TC , VTTS VTTS D k a b Ut  H t  L  H  f  Description generalized price which would force everyone from L off the road under free flow conditions generalized price which would force everyone from H off the road under free flow conditions number of drivers from L who would use the road if it were free number of drivers from H who would use the road if it were free toll price number of drivers per time unit from L who would use the road at time t expecting to pay total costs experienced at t-1. number of drivers per time unit from H who would use the road at time t expecting to pay total costs experienced at t-1 time costs for L at a given level of congestion at time t time costs for H at a given level of congestion at time t value of travel time savings per hour for L value of travel time savings per hour for H length of the road capacity of the road beyond which significant congestion arises estimated congestion parameter depending on the type of road estimated congestion parameter depending on the type of road free speed kilometer per hour  Value 5 10 120 100 varies varies varies varies varies 1 5 20 400 25 0.00089 60  We assume that the relevant time unit is one day. This means that a driver decides whether or not to use the road on a particular day based on the traffic conditions of the previous day. This assumes that information regarding traffic conditions disseminates to all potential users. The model works as follows: 1. We start from the highest generalized price anyone is willing to pay in terms of both tolls and free flow time costs to get to B. If we charged a toll of $10, nobody from H  88  OTL would use the road. Assume that we can only charge tolls in multiples of $0.25. We start therefore with a toll of $9.75. 2. At time zero, we use the demand curves to determine the number of drivers from H and L who would use the road at a toll of $9.75. Assuming that drivers make their decision expecting to complete the journey at the free-flow time, their expected time costs are: E(TC )= •  — xDxVTTS, f  rn  = — x 20x1 = 0.333 60  E (TC „) = — x D x VTTS „ = — x 20 x 5 = 1.666 H,0> j "60 H  At a toll of $9.75 under free flow conditions, the number of drivers from L and H would be: -P-E(7U ))x^-  q =U L  = (5-9.75-0.333) x — = -122 5  £>0  L  A  L  <1H = ( H ~P~ ( HO» A  E TC  x  —  A  H  = ( 1 0  9  -  7  5  -  L 6 6 6  ) x—  = -14  10  We cannot have a negative number of drivers so throughout the model we impose the condition q=max(q,0). At $9.75, no drivers would use the road. 3. We repeat the process with descending tolls until we have a positive quantity. At a toll of 8.25, we have,  q =(A L  L  -p-E(TC )) L0  B 120 x—^- = (5-8.25-0.333) x = -86 A 5 L  q =(A -p-E(TC ))x^=(10-8.25 A H  H  H0  H  -1.666) x ^ = 1 10  However, there is virtually no congestion at this level of traffic so we will move  89  down to a lower price to illustrate how q  L  , q  H  ,qi+qH>  VTTS , and VTTS interact to L  H  converge towards an equilibrium point. 4. Consider the price $2. Atp=$2, we would have the following number of users given that users were expecting to pay the free flow time costs. q  = (A -p-E(TC ))  L  L  L0  x ^ = (5-2-0.333) X — = 64 A 5 L  q  = (A -p-E(TC ))x^=(10-2-1.666)x^ A  H  H  = 63  H0  10  H  However, at a total quantity of 127, there is some congestion and we use Kimber and Hollis' function to determine the congestion costs forZ, and//:  TCLO=  '  TC  L0  "  TC  —  f  TC  H0  9L +QH  +b  +.  xDxVTTS,  J  J  = 0.339  ~f  fi  l+a  l + a <1L +4H  + .  + b<1L +1H xDx VTTS H  =1.698  In this first iteration, drivers from L were expecting to pay 0.333 in time costs, but ended up paying 0.339. Likewise, drivers from H were expecting to pay 1.666, but ended up paying 1.698. This will affect their decision to use the road again at time one. 5. In the second iteration (i.e., at time one), the number of L and H drivers will decrease by the amount:  90  Aq , = ^ L  A  x (E(TC ,)- TC L  Ll  Aq , =^-x(E  {TC ) - TC  H  HA  (0.333- 0.339) = -0.144  5  L  )= ^  HX  A  x (1.666 -1.669) = -0.3  1U  H  The number of L and H drivers for the second iteration will be = <IL,« +  = 64  -  0.144  «  64  1H,I =<1H,O+ ^H.I = 3 - 0.3 - 63 6  In this case, the congestion effects are extremely small and we don't need many iterations to converge to an equilibrium point. However, when there are significant congestion effects, it may take several iterations to reach equilibrium. The number of users will contract and expand with each change being smaller than the previous change. In figure 4.7, Mathcad has calculated a set of equilibrium total quantities for each price given the group demand and congestion conditions stated above. Note that the zero toll quantity is less than the horizontal sum of B +B due to congestion and L  H  time cost effects.  Figure 4.7 Toll demand curve derived by Mathcad for j'roups H and L 10  1  1  1  9 -  g 7  P  -\  -  6 5  \  4 3 2 1 0  1 50  1  100  1 150  200  q (drivers/day)  91  This method of generating demand curves can be extended to more than two VTTS groups. Figure 4.8 illustrates how the dynamic process would look for four groups with different VTTS: commercial truck drivers, high-income wage earners, low-income wage earners, and pensioners. The different VTTS are reflected in the size of each group's response to levels of congestion for each iteration. For simplicity, this diagram assumes that a stable equilibrium of qcr+lH  is reached at toll p (q and q are zero) after L  P  three iterations/ Figure 4.8 Dynamic process for four VTTS groups  ' toll commercial truckers (CT)  / ,  ICT  *  /  /  high income (H)  '  7  pensioners (P)  low income (L)  For toll p, the total quantity will be q after three iterations  CT  + q  H  The derivation of toll demand curves can be summarized as follows. First determine the generalized price-traffic quantity relationship for all significant VTTS groups assuming that travellers expect to make the journey at the free flow rate. Use the iterative process outlined above to determine the equilibrium quantity at each toll price given the congestion function and the different VTTS and demand curves for each group.  We are now ready to model the stochastic nature of the toll demand curve. This will enable us to determine the profit maximizing toll within a stochastic environment for each of the toll-setting-arrangements. With this information, we will be able to value varying degrees of toll-setting flexibility using Contingent Claims Analysis.  4.2.3. Stochastic demand curves and the value of fixed tolls In our simple model, the shape of each VTTS group's demand curve is determined by two parameters, y-intercept (generalized price) A and x-intercept (quantity) B. By ascribing stochastic properties to these parameters, we can model the random nature of the demand curve. An increase in the WTP to get to destination D among current users would result in an increase in 4. This may be due to a change in the industrial structure of D. A general increase in the population would lead to an increase in B. Predicting the nature of these parameter changes over time is the work of demographers, urban planners, and transport economists. By modelling the volatility of these parameters, we can determine the probability distribution of future demand curve shapes. Continuing with our two VTTS group example, imagine that next time period (in 26  this case next year ), B will either move up by a factor of 2 to 200 or down by a factor H  of 1/2 to 50. The probabilities of an up move and a down move are both equal to 0.5. Figure 4.9 shows these scenarios. Note we are dealing with different time periods here than in the iterative process to equilibrium. Here, we are modelling probabalistic demographic shifts over a long time period (one or more years). The dynamic process works through iterations over short time periods (one day) and assumes fixed parameters. 2 6  93  Figure 4.9 Binomial movements of B over time H  Time zero  Time one  10 Pu = 0.5  A = \Q  B = 200  u  H  \  B =100 H  H  Pd = 0.5  ^v. d 10  B = 50 H  What toll will maximize the net present value of profits from time zero and time one under this scenario given the condition that the toll set at time zero can not be changed at time one? Assume tolls can be set at increments of $0.10. In figure 4.10, Mathcad is used to calculate the profits (jr.) for each toll level (p). The nomenclature for profits is J t p ^ wherep is the toll, t is the time period, ands is the state at time period t  Figure 4.10 Profit-maximizing toll at time zero B =100  2 0 0  1  H  p*= 3.10 %.i,o - 133.62  Profit 0  - /  1  200 0  p*  5  10  P  94  which will either be up (w) or down (d). The profit formula is: Xpt =P  ~ (FC + VCxq)  Xa  where fixed costs (FC) are 100 and variable costs (VC) are $0.5 per car. Figure 4.10 shows the profit curve at time zero with £#=100. The tolls and associated profits for time zero and the up and down states at time one are as follows: Toll 3.6 3.5 3.4 3.3 3.2 3.1 3.0 2.9 2.8  Time zero % B =100 125.904 128.806 131.029 132.572 133.436 133.62 133.125 131.95 130.096 H  Time one ot £ # = 2 0 0 272.215 273.377 273.66 273.064 271.589 269.235 266.003 261.892 256.903 1|U  Time one jt B =50 52.648 56.418 59.609 62.22 64.252 65.704 66.576 66.868 66.581 lid  H  At a toll of $3.10, the present value of the fixed toll investment opportunity (i.e. the net present value of profits from time zero and time one) would be: A7 7(jt )=at  ^  3  31  31|0  +-  A W ( j t ) =133.62 +  3.1,1 ^  1+ r 05x269.235 + 05x65.704  31  1+r  167.47 NPV {% ,)) =133.62 + 1+ r 3  The question of what risk-adjusted discount rate (r) to use will be addressed later. At this point, it is sufficient to note that/?=3.10 yields the greatest net present value as long as r is less than 144% Althoughp= 3.2 has a greater expected time one value of 167.9205 (0.5x271.589+0.5x64.252), p=3.2 has a smaller time zero value of 133.436. It  95  would only make sense to choosep=3.2 if the present value of the difference between E(x ) 311  and E(x i) was greater than the difference between o t 32>  3 1 0  and  Jt .2,o3  This  condition is only met when r is greater than 144% which is highly unlikely to occur. Therefore, given the stochastic properties of B outlined earlier, if we were forced to set H  an unadjustable toll at time zero,/?=3.10 would be the toll which maximizes profits from time zero and the expected profits from time one and hence maximizes the value of the investment opportunity. To price other toll-setting arrangements with varying degrees of flexibility, we act as though there is a market for the time-one profits from the fixed-toll position. In other words, we treat the time-one cash flows as though they were a tradable commodity. We can price this theoretical commodity based on its risk profile and the market's attitude toward risk. The price we would pay for this theoretical commodity if it were traded can be calculated using an equilibrium model such as the CAPM. We will discuss the application of the CAPM in more detail in the next section. For now, it is important to accept that even if an asset (in this case claims to future annual toll profits) is not actually traded, it can still be priced in a world with systematic risk.  4.2.4. The value of complete flexibility Imagine that the concessionaire has complete freedom to set profit-maximizing tolls under any state at time one. At time zero, the operator would set the toll at $3.10 for a profit of $133.62. At time one, if B increased by a factor of 2 the operator would set H  the toll at $3.40 for a profit of $273.66 (versus $269,235 at a toll of $3.10). This reflects  the increase in the number of people with a high WTP to use the road. If B decreased by H  a factor of 1/2 the operator would set the toll at $2.90 for a profit of $66,868 versus $65,704 for a toll of $3.10).  27  This flexibility to adapt toll levels to changes in the demand curve can be valued using Contingent Claims Analysis. By combining the present value of the expected fixedtoll profits at time one with a riskless asset, we can create a portfolio which exactly replicates the payoffs of the flexible toll investment opportunity under all possible states at time one. The cost of this portfolio at time zero must equal the value of the flexible toll investment opportunity. The replicating portfolio is created as follows. First, we recognize that the net present value of the fixed toll investment opportunity has two components: a certain time zero cash flow and an uncertain time one cash flow. We are only interested in the present value of the expected time-one cash flows: PV(E(z )) 3X1  =  Pu  X  3  t  3.1,1,* +Pd  x  j  167.47  t  1+r  1+r  At time zero, we are willing to pay PV(E(it )) 3u  to receive an uncertain profit at time  one. We create a portfolio by combining PV(E(% )) with a riskless investment to 31>1  replicate the payoffs to a flexible toll-setting investment opportunity at time one. Figure 4.11 shows the payoffs for the two investment opportunities. Given these payoffs, we solve the following system of equations to determine the replicating portfolio:  It is interesting to note that if B increased by a factor of 2 rather than B „ the optimal pricing strategy would be to decrease the toll to $2.90 for a profit of $233,456. With a decrease in B by a factor of 1/2, the optimal toll would be $3.4 for a profit of $87,019. L  H  L  97  Figure 4.11. Present value of opportunity to receive uncertain cash flows at time one Complete flexibility (cflex)  Fixed toll (/)  /  ^3.1,l,u~  ^3.4,1^-  269.235  273.66 cflex.O' ,=  1+r ^.l.l.d  -  65.704  n xot  -B = %  nx%  -B = %2.9,14  ^2.9,14""  66.868  where n is the size of the position in the asset which pays off at time one and B is the amount of borrowing or lending. Solving the equations for n and B, we get: « x 269.235-5 = 273.66 nx 65.704-5 = 66.868 n = 1.016 5 = -0.111 By buying 1.016 of the time-one fixed-toll payoff position and buying 0.111 bonds (i.e. lending money) at the risk free rate, we have a portfolio with exactly the same time-one payoffs as the complete flexibility (cflex) position. Conceptually, we run into a problem here because the replicating portfolio involves buying a stake in the fixed-toll year-one profits which is greater than the whole. While it is impossible to own more than 100% of something, it is important to remember that with the CCA approach, we are treating the fixed-toll year-one profit position as though it were a traded asset in unlimited supply.  98  We assume such an asset is priced in the market based on its systematic risk and the market's price of risk. While mentally difficult to digest, this idea is crucial to the use of real options pricing techniques. The cost of this replicating portfolio (RP) to us today is: n  167.47  n  RP =lM6x-  0.111 +  a  . 1+ r  0  l +r  f  Under the principle of no-arbitrage, RP must equal the time-zero value of the time-one 0  flexible payoff position; RP = V 0  . The exact value of RP at time zero depends on  c(lexfi  Q  the risk-adjusted discount rate r and the riskless lending rate r . For arguments sake, f  assume that r =5% and r =15%. f  o=RP = 1.016 x 0 0  c f l e x 0  1.15  + M l = 148.06 1.05  This is greater than the time-zero value of the time-one fixed-toll payoff position:  V, o = '° f  E  167.47 = =^^- = 145.62 1+r 1.15 3 1 4  The difference, J'cfiex.o  - F , =148.06-145.62 = 2.44 f  0  is the value of having complete flexibility to set toll levels. The value of flexibility makes up almost 1% of the total value of the complete flexibility alternative. This difference will always be positive as long as r is less than 25xr (this is found by solving the f  inequality V , <V ). The exact value of complete flexibility is determined by the riskcnex 0  f0  adjusted discount rate (r). If the CAPM holds, then r is estimated using the familiar formula:  99  r = r, + 6 (r -r ) f  where (3 =  '- jn  a  m  Beta could be estimated by regressing percentage changes in fixed-toll profits to market returns. Whereas betas are usually estimated on assets such as company stocks, here we are treating the toll profits from each time period as an asset. For example, consider the Eastern Harbour Tunnel crossing BOT project in Hong Kong. Figures from 1992-1996 on profits, percentage changes in profits, and annual Hong Kong stock market (Hang Seng index) returns are: Year 1992 1993 1994 1995 1996  Profit^ 77,744,879 131,083,266 143,734,462 119,500,526 131,997,006 8  % change in profit  Hang Seng return^  0.686069 0.096513 -0.1686 0.104573  0.11567 -0.311 -0.22981 0.33534  The statistics from regressing percentage change in profit on the annual Hang Seng return are: (3 0.5  Op 0.76  a 0.19  a 0.20 a  R 0.181 2  Residual a # of obs. 0.399 4  The beta of 0.5 indicates that toll profits are less sensitive to general economic forces than the market. The alpha is the average percentage change in toll profits when the market return is zero. R reveals that 18% of the changes in toll profits can be 2  explained by market movements. In other words, 18% of the risk is market risk and 82%  2 8  29  from company Eastern Harbour Tunnel Crossing website: www.easternharbourtunnel.com courtesy of Rod Morreau at CIBC Wood Gundy 100  of the risk is unique risk. Residual a measures the amount of unique risk as a standard deviation. The high standard deviations for beta and alpha mean these estimates have a high probability of being wrong. There is usually a large margin of error when estimating betas for individual stocks. The above estimates could probably be improved if we regressed the profits from several similar toll road companies (i.e., came up with a Hong Kong toll road industry profit index) on the Hang Seng returns. Furthermore, the above example has only four observations; exceedingly few for a reliable regression. Despite the shortcomings of the crude regression above, the technique could be fine-tuned to produce workable risk-adjusted discount rate estimates. These estimates could then be used to value the flexibility in different toll-setting arrangements. Figure 4.12 shows how sensitive the value of complete flexibility in our example is to different estimates of beta. The value of complete flexibility appears quite robust to errors in the estimate of beta.  Figure 4.12 Sensitivity of complete flexibility option value to |3 estimates (from 0 to 2) j  1 v  1  Our estimate |3=0.5  2.5 Option(r , r , p , E V , n , B ) f  m  f  2  1.5 0  :  i 1  i 2  3  P  At this point, you may ask why not just discount the time-one flexible profits at the project's risk-adjusted discount rate:  101  Pu  V  cflex cflex  X j t  3.4,i,«  Pd  +  2.9,14  xx  = 1+r  0.5x273.66 + 0.5x66.868 = = 148.055 1.15 1 / ) O A C C  This method will yield a different answer to CCA, although in this case the difference is slight (148.06 for CCA versus 148.055 for straight CAPM r discounting). CCA is the correct approach because it recognizes that flexible price-setting arrangements change the riskiness of future cash flows. The same risk-adjusted discount rate cannot be applied to cashflows with different risk profiles. Recall the arbitrage argument from chapter three. Astute investors would not allow someone to pay $148,055 to enter a position which the market valued at $148.06. The price would be bid up to the CCA value. Using risk-neutral probabilities The preceding discussion has shown how a replicating portfolio can be used to price the value of toll-setting flexibility. The same results can be achieved using riskneutral probabilities. Recall the formula for risk neutral probabilities from chapter three: (\ + r )S-S~ f  p' —  s  +  - s ~  In this case, the twin security S is the fixed toll investment opportunity where, =  167.47 _  6  2  5  6  5  1.15 S = 269.235 +  5" =65.704 The risk neutral probability of an up movement is, , (1.05) x 145.62565-65.704 „ p' == 0.42845 269.235-65.704 n  O  A  r  102  We use the risk neutral probabilities to calculate the certainty-equivalent expected cash flows at time one for the completely flexible toll-setting investment opportunity. We then discount this value back to time zero using the risk-free rate: P'*X34iM cftex  + G-/>')xjt  0.42845x273.66+(1-0.42845) x66.868  29W  l +r  1  A  O  n  1.05  f  The risk-neutral probability approach yields the same answer as the replicating portfolio approach. This is because they are different ways of saying the same thing. Both methods depend on the market price of risk and the correlation between toll revenues and the market. These relationships are expressed through beta calculated using the CAPM. While the use of risk-neutral probabilities is perhaps more elegant, working through the mechanics of replication instills a deeper understanding of the underlying logic behind real options. 4.2.5. The value of partial flexibility Suppose the government allowed the concessionaire to increase tolls by no more than $0.10 a year. At year one, if B shifted up, the concessionaire would want to H  increase the toll from $3.10 to $3.40 for a profit of $273.66, but would be limited by the ceiling and would be forced to settle for an increase to $3.20 for a profit of $271,589. If B shifted down, the concessionaire would decrease the toll to $2.90 for a profit of H  66.728,. Using the same contingent claims approach as above, we could replicate the year-one payoffs from the partial flexibility position above by buying a stake n in the fixed-toll year-one profit payoffs and lending money at the risk-free rate.  103  rax 269.235-5 = 271589 rax 65.704-5 = 66.868 ra = 1.00585 5 = -0.7798 Working with the figures from the complete flexibility example (r =15% and r=5%), we calculate the present value of the replicating portfolio and hence the partial f  flexibility (F  ^ f i e x o pflex  '°  =  R P  pflex  o0 =  L 0  ) as: °585 x  1.15  +  —  1.05  = 147.22  The value of partial flexibility is $1.59 (147.22-145.62). The partial flexibility position (147.22) is more valuable than the fixed position (145.62), but less valuable than the complete flexibility position (148.06). Figure 4.13 shows the sensitivity of partial flexibility option value to estimates of beta. Again, it is quite robust.  Figure 4.13 Sensitivity of partial flexibility option value to beta estimates (from 0 to 2) 1.8 I  Option(r ,r ,P,EV ,n,B) f  m  f  L  T  T  6  4.2.6. Trigger adjustments The objective of trigger adjustments is to ensure that the concessionaire receives a rate of return acceptable to both the public and private sectors. Downside risks are limited so as to encourage private sector participation in the project. If there is a return  104  shortfall below a specified threshold (X) the government allows the concessionaire to increase tolls. The government has written the concessionaire a put. Upside potential is bounded to prevent excessive exploitation of monopoly power. If the return exceeds a specified value (Y), the concessionaire is compelled to reduce tolls. The concessionaire has written a call to toll-road users. The upside and downside potential is thus limited between the bounds shown in Figure 4.14. Figure 4.14. Trigger adjustment option payoff diagram.  The value of this position can treated as the sum of a put with strike price X and a call with strike price Y. As with financial options, the values of the put and call depend on their strike price, the current value of the toll profits (the underlying asset), the volatility of the toll profits, and the maturity of the guarantees. However, unlike financial options which are valued under the assumption that traded assets can be used to create a replicating portfolio, these toll-road real options are valued using theoretical assets. The  105  value of these theoretical assets is determined through equilibrium models such as the CAPM. Trigger adjustment toll-setting arrangements may seem similar to partial flexibility arrangements which cap the size of toll increases. However, the two are fundamentally different. Partial flexibility sets no limit on the size of the return. Under partial flexibility, although the concessionaire may not be able to set the profitmaximizing price, by lowering costs it may still end up achieving high rates of return — perhaps too high for the government's and public's liking. If the government is concerned with regulating the concessionaire's rate of return, then trigger adjustment arrangements are clearly the more effective alternative. However, by regulating rates of return, the government is damaging the concessionaire's motivation to innovate and improve operations. If the government is worried that the public will perceive the concessionaire as a monopolistic price-gouger, yet does not want to dampen the concessionaire's incentives to strive for lower costs and better service, then the partial flexibility alternative has some advantages. It prevents the concessionaire from jacking up tolls dramatically, yet also encourages the concessionaire to push for efficiency so as to maximize its rate of return.  4.2.7. Profit-maximization versus economic efficiency The discussion so far has focused on the toll operator's profit-maximizing behaviour. A toll demand curve was generated under the assumption that consumers decided whether or not to use the road based only on their own time costs (i.e., average  106  time costs) and the level of tolls. The operator then maximized profits by setting the toll at the point of intersection between the marginal operating cost curve and the marginal revenue curve derived from the toll demand curve. How does the profit maximizing toll compare to the socially optimal toll? The socially optimal toll occurs where price equals social marginal costs. The social marginal cost curve consists of both consumer-borne marginal time costs created by congestion and the supplier's marginal operating costs. Our earlier toll demand curve model assumed that marginal time costs did not factor into a driver's decision. Drivers were only concerned with their own time costs and did not think about how much they would increase the travel time of other drivers if they joined the traffic stream. To find the socially optimal toll, we must force drivers to internalize such congestion externalities. This requires adjustments to our multi-VTTS model. First, we force drivers to bear the marginal congestion costs they impose on others. To do this, we calculate the derivative of total travel time . Total travel time (7) 30  is T = q xt T  a  where t is the Kimber and Hollis congestion function and q a  T  is the total  number of users. The derivative of t with respect to q is: a  T  The marginal travel time function has the form:  Bureau of transport and communications economics. Traffic congestion and road user charges in Australian capital cities. Australian Government Publishing Service, 1996.p. 81  107  dt.  For q , we enter the quantity q + q to determine the marginal increase in total traffic T  L  H  time for one additional driver at the total traffic level. To determine the monetary marginal time costs (MTC) for each group, we multiply the components of t  m  M  T  C  l  A  ?  X«' *  r  ^  dq  q  T  M T C h  =  £V  +  L  x  t  r  dq  x''  X  W  T  x  L  V  T  T  S  "  L  H  x  m  5  »  +  t  .  XVTTS„  +q  H  We multiply —— by the weighted average — dq T  xVTTS  +q  ^*'  O  q  T  ? «  as follows:  q  —— q  — because this term  +  L  H  shows how much an additional driver increases the time costs to everyone in the traffic stream. Therefore, we should use a weighted average V T T S to reflect the fact that different components of the traffic stream have different V T T S and that the breakdown of these components changes with the overall level of traffic. The problem of how a driver could know the weighted average V T T S at a specific traffic level is irrelevant to our analysis. We want to derive a toll demand curve under the assumption that drivers behave in a socially optimal manner considering both their own average time costs and marginal time costs. By equating the marginal operating cost to this marginal cost toll demand curve, we can find the socially optimal price and quantity. In the adjusted multi-VTTS model, for each iteration, drivers decide whether or not to use the road in the next time period based on the difference between the sum of expected average and marginal costs (i.e., the actual average and marginal costs in the  108  previous time period) and the sum of actual average and marginal costs in the current time period. As in the calculation of the toll demand curve, these differences oscillate towards an equilibrium total traffic quantity for each toll level. In figure 4.15, Mathcad has calculated the marginal time cost toll demand curve using the numbers from our earlier example. Figure 4.15. The marginal time cost toll demand curve  10  1  1  1  9  g 7 6 P_  5 4  —  -  3 2  Socially optimal  v  quantity =108  Operator marginal costs = 0.5  —\^  30  60  \ \ ^ ^ \  1 0 0  90  q(Al,Bl,A2,B2,a,b,k,TVl,TV2,f,D,n)  „,  q*  120  The socially optimal quantity occurs at point q*= 108 where the operator's marginal cost ($0.50) intersects the marginal time cost toll demand curve. At this point, consumers have reached an equilibrium through a decision process in which they consider both their own time costs and the time costs they impose on others. In reality however, drivers do not consider the costs they impose on others and are only concerned with their own time costs. This is the toll demand curve shown in figure 4.16 which was derived earlier based only on average time costs. Assuming the operator behaves in a socially optimal manner and charges a toll equal to its marginal  109  <  operating costs, drivers will use the road up to point q. However, this is an excessive amount from society's point of view. To a reach a socially optimal quantity, we must force drivers to internalize time cost congestion externalities by charging a toll 7 *. 1  Figure 4.16. Average time cost toll demand curve 10  200 q ( A l , B l , A 2 , B 2 , a , b , k , T V l ,TV2,f,D,n)  In our example, the operator's marginal cost curve is constant at $0.5 and intersects the average time cost toll demand curve at q - 176. From society's perspective, this is an excessive amount with the optimal amount at g*=108. Assuming that people behave based only on their own average time costs, the government should force the operator to charge a toll of />*=2.55 to reach the socially optimal output. However, a toll of $2.55 will result in profits of only $122,489. This is significantly less than the profits of $133.62 at the profit maximizing toll of $3.10. The government and concessionaire should negotiate a BOT toll-setting arrangement between these two bounds. To tap into private sector efficiencies, the government should maintain attractive profit incentives for the private sector. At the same time, the government wants to prevent the concessionaire from abusing its monopoly power. For example, the  110  government may grant the concessionaire a certain degree of pricing flexibility in exchange for the concessionaire setting the price closer to the socially optimal level. This could keep the public happy while still motivating the private sector with the possibility for future profits. Real options can be used to value varying degrees of flexibility. This will enable BOT participants to better understand the nature of the tradeoffs they are making. For example, consider a concessionaire operating under a fixed toll regime at the profitmaximizing price p=$3.10. The government makes the following proposal: "Drop your toll this year to $2.90 and we will give you complete toll-setting flexibility next year." The concessionaire would lose $1.67 in current profits (^ -3t . =133.62-131.95 =1.67), 31  2 9  but would gain $2.44 in option value. The concessionaire should accept the government's offer. However, if the government asked the concessionaire to drop its toll to $2.90 in exchange for partial flexibility (i.e., increases limited to $0.10), the concessionaire should decline because it would lose $1.67 in current profits, but only gain $1.59 in option value. In our scenario, the toll road happens to be profitable at the socially optimal output. However, if the toll road had large,fixed costs relative to demand, it may lose money at the socially optimal output. In such cases, the government traditionally tries to make up for this shortfall by subsidizing the loss or by regulating the firm to charge a toll covering average total operating costs. It may be difficult to attract the private sector to manage an unprofitable infrastructure facility on a BOT basis. However, the possibility of future toll-setting flexibility may entice the private sector to take on a facility which is  ill  currently unprofitable. The government may force the concessionaire to operate close to the money-losing socially optimal output level for a certain period of time in the hope that during this period, the efficiencies of the private sector concessionaire will transform the toll road from a money-losing natural monopoly to a profitable economic activity. The concessionaire may be willing to operate at an unprofitable output level for a period of time in the hopes of being able to charge profit-maximizing tolls in the future By dangling the "carrot" of toll-setting flexibility in front of the concessionaire, the government may be able to inject BOT efficiencies and discipline into certain types of infrastructure services while at the same time maintaining output close to the socially optimal level. Real options can be used to value such "carrots".  4.2.8. Summary The setting of tolls is often a source of friction in BOT projects. While governments and concessionaires realize intuitively that flexibility in toll-setting has value, there are few tools available to systematically quantify this value. In combination with sophisticated transport demand modelling techniques, the real options method of contingent claims analysis is well-equipped to tackle this problem. By identifying and attempting to quantify elusive sources of value such as toll-setting flexibility, real options valuation techniques provide important insight into the complex dynamics of value in public-private partnerships.  112  4.3. OPERATE/TRANSFER PHASE: OTHER BOT REAL OPTIONS 4.3.1. Introduction In addition to toll-setting options, there are other real options during the operate phase of a BOT toll road such as development gain options and options arising from limited recourse project financing. Project financing also creates real options during the build phase. At the transfer stage of a BOT project, real options can be used to analyze the value of different transfer arrangements between the concessionaire, the government, and third parties. This section presents a brief look at these types of BOT real options.  4.3.2. Development gain options An important source of value in BOT projects is often the potential for "development gain". Development gain refers to an arrangement in which a concessionaire is given the exclusive right to develop ancillary facilities in return for providing much needed, but perhaps unprofitable, infrastructure services. The giant construction and real estate development company Hopewell Holdings of Hong Kong is famous for taking this approach. The company has constructed toll roads in China on condition that it have the right to develop real estate running alongside the roads. The company has also worked on a $3.2 billion mass transit project in Bangkok (Bangkok 31  Elevated Rail and Transport System — BERTS)  which involved the development of  real estate nearby stations. Another famous example involving development gain is the Channel Tunnel. The tunnel operator Eurotunnel was granted first option on a concession Cancelled by the Thai cabinet in September 1997 due to the Asian currency crisis. "Thai Cabinet Derails Hopewell transit project" Globe and Mail. October 1, 1997 3 1  113  to operate a parallel road link and was also given the right to propose expansions to the existing rail routes.  32  BOT projects with the potential for development gain have option-like features. If the government grants a concessionaire the exclusive right, but not the obligation, to develop an ancillary facility, then the government has written a call option to the concessionaire on the value of the ancillary facility. The flexibility inherent in such arrangements can profoundly affect the value of BOT projects. For example, imagine that the government makes the following offer: "If you build and operate toll road X, we will give you the exclusive right to develop a shopping mall alongside toll road X next year." For simplicity assume the toll road can be built and brought into operation in one year. After calculating all the build and operate options, the expanded NPV of the toll road project turns out to be a paltry $5 million. On an investment of $100 million, this return of 5% is not very attractive. However, we must also include the value of the option to develop a shopping mall. Suppose the value of the shopping mall next year moves as shown in figure 4.16. The risk-free interest rate is 5%, w=1.5 and d=l/l.5. The riskneutral probabilitiesp* are also shown in figure 4.17. The cost of developing the shopping mall next is 100. For simplicity, assume we pay 100 and receive the value of 33  the project instantly . The expanded NPV of this investment opportunity is:  "Second Channel tunnel proposed: Road link would parallel rail route." Globe and Mail. December 30, 1997 In reality, there is of course a delay between when we exercise this real option and when we receive the value from exercising the option. 3 3  114  Figure 4.17 Movements in shopping mall value  p*= 0.46  150  max(150-100,0)=50  100 p*= 0.54  shopping_ exp_ NPV —  67  max(67-100,0)=0  (0.46) x 50+(1-0.46) xO 1.05  = 21.9  Adding our shopping mall expanded NPV of 21.9 on to our toll road without development gain expanded NPV of 5 substantially increases the expanded NPV and the expected net return for our overall involvement in the project. Note that if we were obligated to develop the shopping mall under money losing conditions (67-100=-33), our static shopping mall investment opportunity would only be, shopping _static _NPV =  (0.46) x 50+(1-0.46) x (-33) 1.05  = 4.933  The option not to develop the shopping mall is worth 16.97 (21.9-4.933) and contributes significantly to the overall value of our toll road project with development gain.  4.3.3. Financial flexibility  115  Project finance is the backbone of BOT projects. Limited recourse project financing was originally developed in the oil industry, but is now common in many other industries. Project financing can be defined as: "A financing of a particular economic unit in which a lender is satisfied to look initially to the cashflowsand earnings of that particular economic unit as the source of funds from which a loan will be repaid and to the assets of the economic unit as collateral for the loan"  34  BOT projects are typically financed by mezzanine capital which is a mixture of equity and limited recourse debt of varying seniority. Projects such as toll roads which are exposed to market risks tend not to exceed 60-65% debt leverage . The introduction 35  of debt financing gives the concessionaire the option to default on debt payments. This option has value as illustrated by the following example. Consider a BOT toll road which involves a cash outlay of 100 million at time zero to receive cash flows at year one. The project which currently has a value of 100 million will have a value of either 170 million or 59 million (w=1.7,/>*=0.415) at year one. Under complete equity financing, the NPV of this project is zero. However, if 60 million of the time-zero 100 million outlay is project financed with debt at an interest rate of 12%, then the equity holders must put up 40 million of their own at time zero and pay the debtholders 67.2 (60x1.12) million at year one. The expanded NPV of the project with the option to default becomes:  NevittK.P. Project Financing. Euromoney Publications, London, 1989 (reprint), p. 13 United Nations Industrial Development Organization. UNIDO BOT Guidelines 1996. p. 185  116  ENPV_def  =  (0.415) x max(170 - 67.2,0) + (1 - 0.415) x max(59 - 67.2,0) 1.05  _ 40 = 0.63  The right under limited liability to default and surrender the projects assets to debtholders has given this project a positive NPV. Financial flexibility may interact with operating flexibility. Take the previous example and imagine that construction expenditures are made in two installments. We pay $60 million in year zero and then $40 million in year one to receive the year one value of the firm (either 170 or 59). Equityholders now have the financing option to default on the loan payment and the operating option not to make the required $40 million year-one expenditure to complete the project. The expanded NPV with these two options is, ENPV_op_def  =  (0.415) x max(170 - 67.2 - 40,0) + (0.585) x max(59 - 67.2 - 40,0) 1.05  = 24.8  In this case, the value of the investment opportunity with both operating and financial flexibility is substantially greater than the investment opportunity with only financial flexibility. BOT financing arrangements are often extremely complex as governments, export credit agencies, international agencies such as the World Bank and the IMF, construction companies, contractors, banks, and investment funds can all end up having both debt and equity stakes in a project. Ideally, BOT financing arrangements should be able to change during the life cycle of the project so as to reflect its changing risk profile.  117  Creating financial flexibility through project finance plays a crucial role in achieving this goal. In the words of Lenos Trigeorgis: "...the flexibility to actively revalue the terms of a financing deal to better match the evolution of operating project risks, whether increasing or decreasing, as the project moves into its various stages creates value..."  36  4.3.4.  Transfer options The transfer phase of BOT projects is still largely uncharted territory. This is  because most BOT projects are relatively young and no one is sure what will happen at the end of a twenty or thirty year concession period. There are several possible transfer arrangements: 1. concessionaire turns over the facility to the government for free and the government manages the facility 2. concessionaire must bid on equal footing against other potential operators if it wants to continue operating the facility 3. concessionaire turns over the facility to the government for a predetermined price 4. concessionaire turns over the facility to the government for its market value at the time of transfer 5. concessionaire has the option to pay a specified value to continue operating the facility Most BOT transfer stages are type 1 or 2 arrangements which don't have any option-like features. In type 3 arrangements the government has essentially sold the  Trigeorgis, Lenos. "Real options and interactions with financial flexibility." Financial Management 22 1993p.219  3 6  118  concessionaire a bullet bond (i.e., one payment at maturity). The price of this "bond" is the present value of the termination payment discounted at the government borrowing rate. The concessionaire could pay for this "bond" through higher taxes or lower tolls, whichever the government feels is socially optimal. Type 4 transfer arrangements have no option-like features and the concessionaire should be willing to pay the expected residual value discounted at the appropriate riskadjusted rate. Again, there are various ways the concessionaire could pay for this arrangement, some of which may improve social welfare more than others. A type 5 transfer arrangement involves the government writing a call to the concessionaire on the residual value of the facility with the predetermined price as the exercise price. At the time of transfer, the government owns the facility, but must sell the facility back to the concessionaire if the concessionaire wishes to exercise its option (i.e., if the residual value exceeds the strike price). The premium of this call could be determined using option pricing techniques, although long maturities and difficulties in modelling the stochastic nature of the underlying asset (toll profits) may pose some problems. It is interesting to consider the implications of allowing trading in this option before its maturity date. Could an option market emerge for residual values of BOT toll roads?  4.3.5. Summary  While this section has provided a cursory treatment of development gain options, financial flexibility, and transfer options, the brevity of the discussion by no means  119  indicates that these options are unimportant. A large part of a BOT project's value may be derived from these options, especially development gain options and financial flexibility. By valuing these different types of flexibility, governments, financiers, and concessionaires will be better equipped to create BOT projects which effectively meet the needs of the public and private sector.  120  CHAPTER F I V E  CONCLUSION  CONCLUSION The goal of this thesis was to show how real option valuation techniques can provide insight into the complex dynamics of value in BOT toll road projects. BOT toll roads are a promising part of the infrastructure revolution, yet they are fraught with new uncertainties and risks. As policy-makers and investors struggle to understand the nature of BOT risk-reward tradeoffs, they are finding that traditional methods of capital budgeting are ill-equipped to analyze many important sources of value in a BOT project such as the option to alter construction plans, the option to change tolls, the option to develop ancillary facilities, the flexibility created by project financing, and the options involved in transferring the facility. The real options approach promises to give policymakers and investors a deeper understanding of the nature of value in BOT toll road projects. This thesis has a presented a simplified discussion of the mechanics of real option valuation in a discrete time framework. In the field of financial economics, there are many sophisticated continuous-time mathematical models which can be used to price real options. Such models were beyond the scope of this thesis, yet an understanding of them is crucial to the actual implementation of real option pricing. It is hoped that financial economists will turn their attention to the emerging BOT market as an area in which they can test and fine-tune their valuation techniques. Furthermore, it is hoped that they will continue to proselytize the real options revolution in a manner which is accessible to politicians, bureaucrats, investment bankers, and engineers alike.  On the transport side, it is hoped that transport economists will begin to analyze infrastructure projects within a real options framework. Such an approach will enhance their understanding of the economic tradeoffs created by public-private partnerships. While real options may seem rather esoteric to many currently involved in BOTs, they have the potential to make significant contributions to the understanding of BOTs and P s. It is hoped that this thesis points the way to further research fusing the fields of 3  transport economics and financial economics. The synergies from the convergence of these two academic disciplines will undoubtedly pave the way to exciting new developments in the world of transportation infrastructure.  123  6. BIBLIOGRAPHY Public-Private Partnerships and Build-Operate-Transfer •  "A pioneer on hold." Asiamoney. Dec. 1994/Jan. 1995 p.58  •  "A long and winding road." Asiamoney. Sep. 1993 p. 65  •  "Are Asian Projects all that enticing?" Euromoney. Jan. 1994. p. 92-92  •  Augenblick, M. and Scott-Custer, B. The BOT approach to infrastructure projects in developing countries. World Bank Working Paper, August 1990  •  Bond, Gary and Carter, Laurence. Financing Private Infrastructure ProjectsEmerging Trends for IFC's Experience. IFC Discussion Paper 23, The World Bank. 1994  •  British Columbia. Task Force on Public-Private Partnerships. Building Partnerships: report of the Task Force on Public-Private Partnerships. Crown Publications, Oct. 1996  •  "For whom the road tolls." Asiamoney. Mar 1994. p. 53  •  Fox, William F. Strategic options for urban infrastructure management. Urban management programme UNDP/UNCHS/World Bank. Washington D.C. 1994  •  HM Treasury. Private opportunity, public benefit: progressing the private finance initiative. 1995  •  Israel, Arturo. Issues for infrastructure management in the 1990s. World Bank discussion papers; 171. Washington D . C 1992  •  Kessides, Christine. Institutional options for the provision of infrastructure. World Bank discussion papers; 212. Washington D.C. 1993  •  Lall, Ashish and Tay, Richard. "Private Provision and Financing of Infrastructure in ASEAN." Logistics and Transportation Review. March 1996  •  Levy, Sidney M. Build, operate, transfer: paving the way for tomorrow's infrastructure. J. Wiley & Sons. 1996  •  Nevitt, K.P. Project Financing. Euromoney Publications, London, 1989 (reprint)  •  OECD. Infrastructure Policies for the 1990s. Paris. 1993  124  •  Scurfield, R.G. The private provision of transport infrastructure. World Bank, 1992  •  "Second Channel tunnel proposed." Globe and Mail. Dec. 30, 1997  •  "Thai cabinet derails Hopewell transit project." Globe and Mail. Oct. 1, 1997  •  Tiong, R. "Impact of Financial Package Versus Technical Solution in a BOT Tender." Journal of Construction engineering and management. Sept. 1995, p.304310  •  Tiong, R. "Comparative Study of BOT Projects." Journal of Management in Engineering. Vol. 6 No.l Jan. 1990, p.107  •  Tiong, R. "Critical Success Factors in Winning BOT Contracts." Journal of Construction Engineering and Management. Vol. 118, No. 2, June 1992, p. 217-228  •  Tiong R. The structuring of BOT construction projects. (Construction Management?) (technological monograph, Nanyang Technological University, Singapore, 1992)  •  Tiong, R. "BOT projects: risks and securities." Construction management and economics. E. & FN. Spoon. 1990. 8. 315-328.  •  United Nations Industrial Development Organization. UNIDO BOT Guidelines  •  Walker, C. and Smith, A.J. editors. Privatized infrastructure: the BOT approach. Thomas Telford Publication. 1995  •  Wahdan, Mohamed Y., Russell, A.D., and Ferguson, D. "Public Private Partnerships and Transportation Infrastructure" paper presented at 1995 Annual Conference of the Transportation Association of Canada, Victoria, British Columbia.  •  World Bank, Infrastructure for Development. Oxford University Press, New York, World Development Report 1994  Real Options •  Berk, Jonathan, Green, Richard C , and Naik Vasant. "Valuation and Return Dynamics of R&D Ventures", working paper 1997  •  Black, F. and Scholes, M., "The Pricing of Options and Corporate Liabilities," Journal of Political Economy May/June 1973, pp. 637-659  125  •  Brealey, R., and Myers S.C. Principles of Corporate Finance, fourth edition. McGraw-Hill. 1991  •  Cox J., Ross S., and Rubinstein M., "Option Pricing: A Simplified Approach," Journal of Financial Economics, September 1979, pp. 229-263  •  Dixit, Avinash K. and Pindyck, Robert S., Investment Under Uncertainty. Princeton University Press 1994  •  Grenadier, Steven R. and Weiss, Allen M., "Investment in technological innovations: An option pricing approach." Journal of Financial Economics 1997  •  Hayes and Abernathy,W. "Managing Our Way to Economic Decline," Harvard Business Review. July-August 1980  •  Hodder and Riggs, H. "Pitfalls in Evaluating Risky Projects," Harvard Business Review. January-February 1985. pp. 128-135  •  Kensinger, John W. "Adding the value of active management into the capital budgeting equation." . Midland Corporate Finance Journal 5, 1987 Spring 31-42  •  Kolb, Robert W., Futures, Options and Swaps. Blackwell Publishers 1997  •  Luehrman, Timothy A. "Capital Projects as Real Options: An Introduction" Harvard Business School class note. 1994  •  Majd, Saman and Pindyck, Robert S., "Time to build, option value, and investment decisions." Journal of Financial Economics 18 (March) 1987: 7-27  •  Merton, "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science Spring 1973, pp. 141-183  •  Morck, Randall, Schwartz E., and Stangeland D. "The Valuation of Forestry Resources under Stochastic Prices and Inventories. " Journal of Financial and Quantitative Analysis 24 (December) 1989: 473-487  •  Myers, "Finance Theory and Financial Strategy," Midland Corporate Finance Journal, Spring 1987, pp. 6-13  •  Neftci, Salih N., An Introduction to the Mathematics of Financial Derivatives. Academic Press 1996  •  Siegel, Daniel R., Smith, James L. and Paddock, James L. "Valuing offshore oil properties with option pricing models." Midland Corporate Finance Journal 5, 1987  126  Spring 22-30 •  Trigeorgis, Lenos. Real Options MIT Press 1997  •  Trigeorgis, Lenos. "Real options and interactions with financial flexibility." Financial Management 22 1993, no.3. 202-224  •  Trigeorgis, Lenos and Mason, S.P. "Valuing managerial flexibility." Midland Corporate Finance Journal 5, 1987 Spring. 14-21  •  Wilmott, Paul. et. al. The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press 1997  Transportation Economics •  Bureau of transport and communications economics. Traffic congestion and road user charges in Australian capital cities. Australian Government Publishing Service, 1996.  •  Gujarati, D.N. Basic Econometrics. Second Edition. McGraw-Hill Publishing Company. 1988  •  Mohring, Herbert. Transportation Economics. Ballinger Publishing Company 1976  •  Oum, Tae Hoon, et. al. Transport Economics: Selected Readings. Korea Research Foundation For the 21st Century.  •  Waters, W.G., "The Value of Time Savings for the Economic Evaluation of Highway Investments in British Columbia" Centre for Transportation Studies, University of British Columbia. 1992  •  Varian, Hal R. Intermediate Microeconomics. Second Edition. W.W. Norton & Company. 1990.  127  APPENDICES  128  APPENDIX I P T O L L ROADS AND BRIDGES AROUND T H E WORLD 3  3'  The following is a sampling of the P and BOT projects being developed around the world: CANADA: •  Highway 407: A 35-year concession granted by the Ontario government to Canadian Highways International Corporation (a consortium led by Monenco AGRA Inc) in 1994 to design, build, operate and maintain a new C$1.7 billion, 69 km toll highway project north of Toronto. Financing largely arranged by the Ontario Financing Authority.  •  Highway 104: C$120 million toll highway in Nova Scotia. Concession period tied to the length of time needed to repay money borrowed. Funding from federal-provincial highway program and Newcourt Credit Group.  •  Northumberland Strait Bridge: An C$840 million, 35-year BOT fixed-link to replace ferry service between Prince Edward Island and New Brunswick.  •  Lion's Gate Bridge: negotiations for a new P tunnel or bridge to replace the aging 3  Lion's Gate Bridge are underway. UNITED KINGDOM •  Birmingham North Relief Road: England's first toll road built by a joint-venture between Trafalgar House and Iretecria.  •  Queen Elizabeth II Bridge (also known as the Dartford Crossing): a toll bridge spanning the Thames river east of London. Built and financed by Dartford River Crossing Ltd. (DRC) a joint venture between Trafalgar House and Bank of America.  129  •  Skye Bridge: a toll fixed link to the Isle of Skye in Scotland.  •  Second Severn Crossing: A U.S. $549 million toll bridge linking England and Wales.  •  Channel Tunnel Crossing: U.S. $12 billion dollar tunnel between England and France. Concession granted to Euromnnel (10 U.K. and French contractors and banks) for 55 years.  ASIA: •  North-South Highway (Malaysia) : U.S. $1.8 billion, 30-year concession led by United Engineers. Noted for government guarantees against revenue short falls due to adverse exchange rate and interest rate movements.  •  Sydney Harbour Tunnel (Australia) : U.S. $540 million, 35-year BOT toll tunnel concession granted to joint venture between Transf ield Pty Ltd. and Kumagai Gumi.  •  Second Stage Expressway (Thailand) : U.S. $880 million, 30-year toll road concession in Bangkok initially granted to Bangkok Expressway Company Limited (BECL) consortium led by Kumagai Gumi. Notorious for controversy over how to share tolls between BECL and the government. Ambiguities over whether the project was BOT or BTO also caused many headaches.  •  Eastern Harbour Crossing (Hong Kong) : U.S. $442 million BOT toll tunnel by Japanese construction giant Kumagai Gumi.  •  Western Harbour Tunnel (Hong Kong) : U.S. $790 million toll tunnel built by Kumagai Gumi-Nishimatsu Construction Co. Ltd. joint venture.  130  •  Guangzhou Superhighway (Hong Kong-China) : U.S. $1.5 billion 304 km toll road connecting Hong Kong and Guangzhou built by Hong Kong's construction giant Hopewell Holdings.  •  Indonesian toll roads: many BOT toll roads controlled by Suharto's family.  LATIN AMERICA •  Autopista del Sol (Mexico): U.S. $1.3 billion BOT Cuernavaca-Acapulco toll road joint venture between Mexican construction firms Grupo Mexicano de Dessarrollo (GMD), ICA, and Tribasa. Dubbed "headache highway" due to construction cost overruns and lower than expected demand. Initial concession period of 14 years extended to 30 years.  •  Mexico City-Toluca toll road (Mexico): U.S. $313 million, 10-year concession to Tribasa. Financing arranged by Lehman Brothers.  •  Argentina, Brazil, Chile, and Columbia: actively promoting P3s and BOTs in their transport sector.  UNITED STATES •  Dulles Greenway: part of a road privatization program in Virginia  •  Caltrans (California Department of Transportation): experimenting with the BTO approach for toll roads.  I  131  APPENDIX II. V A L U E OF T R A V E L TIME SAVINGS (VTTS) The importance of VTTS to transport economics Travellers must sacrifice their time to consume a transportation service. The monetary value of a traveller's time is the subject of much research and debate in the field of transport economics. Unlike other economic activities, transportation requires that consumers also act as suppliers because they must provide a substantial amount of their own time to consume a transportation service. In transport economics, the characteristics of consumers are reflected in both the demand curve and the supply curve. The demand curve shows how much a traveller is willing to pay in out-of-pocket' expenses and value of time to get to a certain destination. The supply curve shows the operating costs of the supplier plus the time costs of the users. An understanding of how people value their travel time is thus crucial to any transport demand-supply analysis. Evaluating VTTS There are theoretical reasons to expect the value of travel time to be related to wage rates. For people with high wages, the opportunity costs of being stuck in traffic are higher and they may be willing to pay more to travel faster. There are a number of empirical approaches which can be used to estimate people's value of travel time savings. These approaches can be broadly categorized as "revealed preference" and "stated preference". Revealed preference (RP) Revealed preference looks at how people make time-money tradeoffs. There are three major markets in which time-money tradeoffs can be indirectly observed:  j  132  •  Choice of speed market: By driving faster, travellers can reduce their travel time. However, driving faster also raises operating costs and increases the chances of receiving a speeding ticket or getting into an accident. By looking at how drivers choose their speed given this speed-cost tradeoff, we can impute their implied value of travel time. Of course, this approach is undermined by the questionable assumption that drivers fully understand the nature of the speed-cost tradeoff.  •  Choice of location market: Homeowners face a tradeoff between property values and accessibility to the central business district (CBD). By examining the relationship between property values and distance to the CBD, we can get an idea of how people value travel time. However, there are many other factors which influence the location decision and these tend to cloud the analysis.  •  Choice of mode and route choice: A majority of RP studies are of this type. We impute people's value of travel time by looking at how they choose between modes with different travel times and different costs (i.e., a fifteen minutes bus trip which costs $1.50 versus a five minute car trip which costs $3.00 in gas and parking). However, it is often difficult to separate travel time from other factors which influence, choice such as comfort and convenience.  Stated Preference (SP) Stated preference asks people directly how they value travel time. Carefully designed surveys with built-in cross-checks for accuracy are used to elicit people's value of travel time. While there is a danger of bias in questionnaires, they have an advantage in that they can distinguish the effects of time from other factors such as comfort and  133  convenience. Improvements in the design of questionnaires have led to a greater reliance on SP methods in recent times. Empirical results Looking at the results from these empirical methods, there appears to be several factors which influence how people value travel time: •  Trip purpose: Work trips are generally valued at an employee's full wage rates. This ^ makes sense because an hour less travel on the job translates into an hour more productive work. Non-work trips such as commuter and leisure trips tend to be valued below the wage rate. There are substantial differences among the estimates of appropriate wage rate percentages, although most estimates range between 25-40% . 37  •  Personal characteristics of the traveller: Personal attributes such as age, sex, family structure, cultural background, and income can affect VTTS. Of these, the effects of income level appear to be the most compelling. There is some evidence which backs the following relationship between income (Y) and VTTS : 38  =l^xVTTS  VTTS  r  where Y  avg  avg  is the mean income level and VTTS  avg  is the VTTS for the average  income. In other words, VTTS rises with income, though not proportionally. •  The size of time savings: People are unable to do much with a few seconds saved. Therefore, people should value small blocks of time saving proportionally less than  37  I  Waters, W.G., "The Value of Time Savings for the Economic Evaluation of Highway Investments in British Columbia" Centre for Transportation Studies, University of British Columbia. 1992. p. xii 38 Ibid. p. xiii  134  large blocks. In other words, one second saved by one thousand people should be valued differently to one hundred seconds saved by ten people. The evidence on this issue is conflicting and it is still the subject of much debate. While the above methods are far from an exact science, they nevertheless play an important role in the evaluation of transport projects. For simplicity, many transport projects are evaluated using a homogeneous VTTS or a weighted average VTTS . 39  However, this thesis proposes that such a simplification glosses over important characteristics of the process which determines transport demand. Our multi-VTTS model attempts to rectify this problem.  Mohring, Herbert. Transportation Economics. Ballinger Publishing Company 1976. Ch. 4  135  APPENDIX III TRAFFIC CONGESTION FUNCTIONS Congestion functions attempt to describe the relationship between traffic volume and speed. Roads are designed with a capacity to handle a certain level of traffic at a free flow rate. As traffic volume approaches road capacity, drivers must slow down to avoid accidents. However, the manner in which drivers decrease their speed as traffic volume increases is by no means clear and depends on the type of road, the road conditions, and the nature of the drivers themselves. Engineers have long struggled to establish empirical relationships between traffic flow and traffic speed. However, the results have been weak.  . For our study, we follow the approach of the Australian Bureau of Transport and  Communications Economics (BCTE) and adopt the Kimber and Hollis (1979) model. This function handles "random fluctuations in traffic flow below capacity, the effect of overcapacity traffic flows of finite duration, and provides a smooth transition between the two regimes." The function has the form: 40  K =t„x l + a((jc-l)+V(x-l)  2  +&cJ)  where, t = average travel time per kilometre a  t = free speed travel time per kilometre 0  x = volume-capacity ratio a and b are parameters which describe the type of road  40  Bureau of transport and communications economics. Traffic congestion and road.user charges in Australian capital cities. Australian Government Publishing Service, 1996 p.79  136  The parameters a and b are estimated empirically and can be adjusted to reflect different road types. In figure III.l, we use BCTE estimates to show how different types of roads have different congestion characteristics. The jc-axis shows the volume-capacity ratio while the_y-axis shows how many minutes it takes to travel one kilometre at each level of traffic. The congestion curve for a highway with a free speed of 100 km/hr uses a=31.26 and Z?=0.0004. The congestion curve for a central business district street with a free speed of 30 km/hr uses a=9.26 and &=0.0350. The CBD congestion function is smoother indicating that drivers make less dramatic adjustments to increasing traffic density. This is because they are already driving at a low speed and less likely to panic at a sudden increase in volume.  Figure III.l. A congestion function for a highway and a central business district street  27 24  -  I  I  1 //  21  if  H i g h w a y _ t ( t , a , b , x ) 18 — a  o h  h  15 C B D  - a( oc. c- c ) t  t  a  b  x  1  2  9  -  y  _  // J  ~~  0  i  r-  I  ' /  6 3  t  ll  h  j  J —  '  /  ——4-^—W  1  0  1.5  0.5  1  2  X  While a number of traffic flow functions (Davidson 1978, Akcelik 1978, Taylor 1984, Tisato 1991) have been developed, we choose Kimber and Hollis's model for its robustness and proven track record.  137  APPENDIX IV  MATHCAD PROGRAM TO CALCULATE MULTI-VTTS TOLL DEMAND CURVE  138  VARIABLES Congestion function a =25  b=.00089  1 -ha-  k  /  A  k =400  D =20 f =60  y =50  1 ^•D-TV  \k  Group 1: High income Al=10  Bl=100  TV1 =5  freeflowTC1 Tfl(TVl) = y - D T V l  Group 2: Low income A2=5  B2=120  freeflow TC2 Tf2(TV2) = y-D-TV2  TV2=1  Solving for maximum values of q1 and q2. The quantity from any group cannot exceed the point where the group's demand curve equals its average time costs. Beyond this point (WTP<ATC), the model becomes unstable. High income quantity limit  H(Al,Bl,a,b,k,TVl,f,D,y)  1 -ha-  solHCAl.Bl.a.b.k.TVl.f.D.y)  :=root(H(Al,Bl,a,b,k,TVl,f,D,y),y)  :-l +  1  +b-  j-D-TVl  Al-^-y  solH(Al,Bl,a,b,k,TVl,f,D,y) =83.287 Hlimit ^ s o l H C A l . B l . a . b . k . T V l . f . D . y ) Low income quantity limit  L(A2,B2,a,b,k,TV2,f,D,y)  solL(A2,B2,a,b,k,TV2,f,D,y)  1 + a-  1 +  1  +b-  •D-TV2  A 2  -B2-y  :=root(L(A2,B2,a,b,k,TV2,f,D,y),y)  solL(A2 B2 a,b,k TV2,f,D,y) =111.969 )  )  J  Llimit =solL(A2,B2,a, b,k,TV2,f,D,y)  139  THE MULTI-VTTS DYNAMIC PROCESS Determine equilibrium total traffic quantities for the following toll prices p  10,9.9..0  Number of iterations  n=10  qT= i^O qTi -if^ (Al - p - Tfl(TVD) ;  (A2 - p - Tf2(TV2))  qT2.<-if  Bl Al B2 A2  >0, (Al - p - Tfl(TVD) >0, (A2 - p - Tf2(TV2))  Bl Al B2" A2  q T l . ^ if(qTl. <HlimitqTl.,Hlimit) qT2.<- if(qT2. <LlimitqT2., Llimit) qT.^qTl qT2 0 +  0  i^i + 1 ETC 1. _ <- Tf 1 (TV 1) x  ETC2_ *-Tf2(TV2) 1  TCI  i -  TC2.  i -  l + a- ^____  1  [ l + a-  1  ^  i  -  i  diffl.  ,^ETC1.  -TCI. ,  diff2  ,^ETC2  - TC2. ,  l - l  i-l  i - l  i - l  1 +  D-TV1  1 +  D-TV2  k  i - l  i - l  , + diffl. • — ] >0, qTl. ' - l l  qTl.^if (qTl.  1  .-1  -  1  1  +  diffl. 1  A  -  1  •—1,0 Al'  qTl.^if^qTl.<HlimitqTl.,Hlimitj, . qT2^iff ( TZ_ q  t +  diff2_ J | ) 1  >0, (qT2 _  qT2.<- if^qT2. <Llimit qT2., Llimit) qT.^-qTl. + qT2. for ie 2..n ETCl._j.-TCl._2 ETC2 i-l  ,^TC2.  , 1-2  ;  x +  difK_  ,0  (q i-i T  TCI  TCZ.  i- 1  i- 1  1 + a-  \  1-j-a-  1  k  -t-  •D-TV1  b  If,-,  •D-TV2  diff 1. ,<- E T C ! , - TCI. , l—l i—i i - i diff2 _ <- ETC2 _ j - TC2. _ l  qTl  r  x  if] (qTL_  t +  R1\  diffl,  / >0, qTl._  1 +  R1 d i f f L , , — J.0  qTl.^Hlimit if qTl >Hlimit R2\ / • — >0,• qT2. , + diff 2 i-i '-i 2 qT2.^- Llimit if qT2.>Llimit qT2.«-ifj  qT2.  1+  diff2  A  qT.^-qTl.-t-qT2.  R?  ,  ),0 A2>  Toll demand curve and profits for B1 =100  50  100  150  200  q(Al,Bl,A2,B2,a,b,k,TVl,TV2,f,D,n)  P  Profits(Al,Bl,A2,B2,a,b,k,TVl,TV2;f,D,n FC,VO  4 J3 3JS 3J_ 3^ 35 JA 33 32  107.5 113.12  H T J3 2J8  )  118.061 122.322 125.904 128.80(4 131.029" 132.57i 133.436 133.62 133.125 131.95 130.096 127.563 124.35 120.45.  200  1  Profits(Al,Bl,A2,B2,a,b,k,TVl ,TV2,f,D,n,FC,VC)  oh  1  -200'  1  27 2L6 25  142  Toil demand curve and profits for B1 =200  10  p  sh  50  100  150  200  250  300  q ( A l , B l , A 2 , B 2 , a , b , k , T V l ,TV2,f,D,n)  Profits( A l , B l , A2, B2, a, b, k, TV1, TV2, f, D, n, FC, V O  P 4  258.779*  3^9 3^8 3.7 3^6 35 3A 33 32 3A 3_ 2~9 2]8 27 2^6 25  263.457 267.253  500  270.174 272.21$ 273.377 273.66 273.064 271.589* 269.23$ 266.00J 261.892* 256.903^ 251.035 244.289^ 236.66$  'Profits(Al,Bl,A2,B2,a,b,k,TVl ,TV2,f,D,n,FC,VC)  0 h  -500'  143  Toll demand curve and profits for B1 =50  50  '  ,100  150  200  q ( A l , B l , A 2 , B 2 , a , b , k , T V l ,TV2,f,D,n)  Profits( A l , B l , A2, B2, a, b, k, TV1, TV 2, f, D, n, FC, V O  P 4  J3 3^ 37 3^6 3^5  M  33 32  31.76? 37.857 4336? 48.29? 52.648 1  ioo  1  56.418 59.609  Profits(Al,Bl,A2,B2,a,b,k,TVl ,TV2,f,D,n,FC,VC)  62.22 64.252 65.704  1  T  66.576  2$ 2& 2.7 2~6 2.5  66.868  1  -200'  1  66.581 65.714* 64.267 62.241 1  1  144  

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