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Multi-zone mine valuation using modern asset pricing (real options) techniques Samis, Michael R. 2000

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MULTI-ZONE MINE VALUATION USING MODERN ASSET PRICING (Real Options) TECHNIQUES by M I C H A E L ROBERT SAMIS B.A.Sc. The University of British Columbia, 1989 M.Sc. The University of the Witwatersrand, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Mining and Mineral Processing Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A August 2000 © Michael Robert Samis, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MxW>Tj *4~ M^eAtf ff^OCfiS^i^ l^^i/l^cri/lj The University of British Columbia Vancouver, Canada Date 13 (October; <^0OC) DE-6 (2/88) Abstract The project structure models used within the modern asset pricing (MAP; also called real options) valuation framework often rely on a few fixed production plans to represent the many production strategies that management may choose from. These models may be extended to include global forms of project flexibility, such as temporary closure of the full project, or choices between competing production plans, such as those made within a project decision-tree. However, many project structures allow more production strategies than currently published MAP models would suggest. Mine valuation exercises are particularly vulnerable to restrictions on the number of production plans since they often consist of multiple zones that differ by quality (mineral concentration), size and spatial orientation. These zones increase the number of possible production strategies because they allow management to consider strategies that reflect the multi-zoned nature of mineral deposits such as the temporary closure, or the delayed development, of less attractive zones. This dissertation proposes a model of project structure, called the Flexible Discrete Mine Production (FDMP) model, for use within the MAP valuation framework. The different zones within a mineral resource project are explicitly recognized and assigned a fixed development and production profile. Management operates the project for discrete intervals by choosing, at the start of each interval, an operating mode from a set of competing operating modes. Each mode specifies the combination of zones that will be active and the amount of mineral processing capacity that is built, abandoned or temporarily closed during the next period. Constraints may be placed on the choice of operating mode because of geological structure, reserve depletion or capacity restrictions. The FDMP model is demonstrated with a MAP valuation of a stylized two-zone mine in which management must decide between competing development strategies for a satellite low-grade zone, given that current operations are focused on a developed high-grade zone. One strategy, production expansion (parallel development), combines the early development of the low-grade zone with the expansion of project mineral processing facilities. The other strategy, production replacement (sequential development), delays development of the low-grade zone until the high-grade zone reserves are exhausted. The model is run in both reverting and non-reverting mineral price environments and the results are compared to three high-grade and low-grade zone Set Production Plans (SPP; high-grade zone i i operations only, immediate low-grade zone development, and delayed low-grade zone development) valued within the DCF and the MAP pricing frameworks. A comparison between valuation method results show that, when a 10% discount rate is used, the DCF technique returns a much higher value than the FDMP and SPP MAP models. The project has less value when the MAP approach is used because, in the given price environment, it is more risky than is reflected by the 10% discount rate used in the DCF valuation method. Comparing MAP model results, the project values determined with the SPP project model and about 10% less than those determined with the FDMP model. The FDMP model also provides more detailed operating policy insights to the SPP MAP methods because the FDMP model outlines management action boundaries over extended project periods for each zone. Finally, a discrete geological (grade) uncertainty model is created for the low-grade zone to determine its influence on low-grade zone development policy. Geological uncertainty is found to have a small impact on development decisions when these decisions are associated with large expenditures but a more substantial effect when development expenditures are smaller. The low quality outcomes from the geological uncertainty model are also used to investigate selective temporary zone closure strategies. Temporary closure of the low-grade zone is shown to be preferred to abandonment in some low mineral price environments. i i i iv Table of Contents Page Abstract ii Table of Contents iv List of Tables viii List of Figures fx Acknowledgement x Dedication xi Chapter 1 Introduction 1 Important characteristics of the mineral resource environment 2 Conceptual examples of the mineral resource project environment 4 The Triple Z mine 5 Randfontein Estates 7 Cambior Incorporated - The Rouyn-Noranda Division 9 Quantitative measures of project attractiveness 10 Dissertation contribution and overview 14 Chapter 2 Literature Review 17 Conventional valuation techniques 17 Assessment of future possibilities 19 A review of finance theory 20 Equilibrium asset valuation models 22 Modern asset pricing models 26 Financial theory as a framework for valuing real assets 27 Financial market theory applied to real assets 28 The influence of uncertainty and project structure on value 28 The influence of project stakeholder interaction on value , 29 V Page The influence of management flexibility on value 31 MAP applications in the petroleum industry 34 Exploration leases 35 Project design, development and operation 36 MAP applications in the mining industry 37 Exploration and information gathering 37 Development and operating options 38 Conclusion 40 Chapter 3 The Flexible Discrete Mine Production (FDMP) Valuation Model 41 Modern asset pricing valuation methods 42 Underlying market uncertainty 42 A description of a multi-zone mine within the FDMP project structure model 45 Zone mine plan 46 Deposit state space 47 The project capacity model 48 Project state space 53 Operating mode 54 Operating policies 54 Valuation of the multi-zone mine within the FDMP project structure model 55 Grade (geological) uncertainty 63 Grade uncertainty and valuation 64 Limitations of the FDMP project structure model 66 Extensions to the FDMP project structure model 66 Conclusion 67 Chapter 4 Valuation of Production Replacement and/or Production Expansion Strategies for a Two-Zone Mine 68 Project overview 68 Economic environment 68 vi Page Deposit description 69 Current HG Zone production plan 71 LG Zone development proposals 72 Sources of management flexibility 79 Valuation results 81 Discounted cash flow valuation results 81 Modern asset pricing valuation results 82 Comparison of the standard DCF and MAP valuation results 93 Conclusion 94 Chapter 5 Two-Zone Mine Valuation Results with Price and Project Parameter Variations 96 Mineral price volatility '. 96 Price of mineral risk 100 Dual-zone production economies-of-scale 103 Comments regarding price and project parameter variations 106 Conclusion 108 Chapter 6 Two-Zone Mine Valuation Results with Geological Uncertainty and Temporary Closure Considerations 109 The influence of LG Zone geological uncertainty on project assessment 109 LG Zone geological uncertainty model 109 The influence of geological uncertainty on MAP valuation models I l l Comments regarding the influence of geological uncertainty 115 Temporary zone closure behavior across a range of LG Zone grade outcomes 115 Individual zone closure policy 116 Comments regarding individual zone closure policy 116 Conclusion 122 Chapter 7 Conclusions and Further Research 124 Recommendations for further research 126 vii Page Bibliography 128 Appendix 1 Cost Parameters for the Two-Zone Mine Valuation Example 139 Overview of cost parameters 139 Mine capital expenditures 139 Mill capital expenditures 141 Daily operating costs 141 Scenario cash flows in reverting price environments 141 Appendix 2 The Effect of Volatility on Expected and Forward Mineral Prices 148 Appendix 3 Overview of the FDMP project structure valuation computer model 151 Overview of the computer model 151 Construction of the project state tree 151 Valuation of the project state tree 155 The upwind projected SOR finite difference method 157 Vlll List of Tables Table Page 4.1 Economic environment parameters 69 4.2 Production parameters and costs for the high-grade and low-grade zones 71 4.3a Current production plan for HG Zone depletion (NREV price process) 73 4.3b Current production plan for immediate LG Zone development (NREV price process) 74 4.3c Current production plan for delayed LG Zone development (NREV price process) 75 4.4 Non-production cash flow components 77 4.5a Project state transition costs 78 4.5b Miscellaneous project state transition costs 79 4.6 Project capacity states and permissible capacity state links 79 4.7 Project values calculated by DCF and MAP methods (mineral price = $1.00/unit) 81 5.1a, b DCF and MAP project values - volatility sensitivity 97 5.2a, b DCF and MAP project values - price of mineral risk sensitivity 101 5.3a, b DCF and MAP project values - economies-of-scale sensitivity 104 6.1 Discrete grade uncertainty model for the LG Zone 110 6.2 MAP project values with LG Zone grade uncertainty (current mineral price = $1.00/unit).. 110 A 1.1 LG Zone-specific development expenditures 140 A 1.2 Multi-purpose LG Zone mine capital expenditures incurred during production expansion.. 142 A1.3 Multi-purpose mill capital expenditures incurred during production expansion 143 A 1.4 Daily operating cost calculation for single-zone operations 144 A l .5a Current production plan for HG Zone depletion (REV price process) 145 A1.5b Current production plan for immediate LG Zone development (REV price process) 146 A1.5c Current production plan for delayed LG Zone development (REV price process) 147 List of Figures Figure Page 1.1 Triple Z Mine schematic layout 6 1.2 Randfontein Estates project components 8 1.3 Operational relationships between the Rouyn-Noranda Division assets 10 4.1a, b Price process profiles (Current spot price = $1.00/mineral unit) 70 4.2a, b Low-grade zone development and project abandonment boundaries 84 4.3a, b Low-grade zone abandonment boundary 88 4.4a, b Zone closure boundaries for 17.5% economies of scale 89 4.5a, b SPP and FDMP model MAP value boundaries 92 5. la, b The influence of volatility change on the LG Zone development boundary 98 5.2a, b The influence of a price of mineral risk change on the L G Zone development boundary.... 102 5.3a, b The influence of EoS change on the LG Zone development boundary 105 6.la LG Zone development boundary with geological uncertainty - NREV model 112 6.1 b LG Zone development boundary with geological uncertainty - REV model 113 6.2a, b L G Zone abandonment boundaries with geological uncertainty 114 6.3a, b Zone closure boundaries for an expanded project (NREV model; LG=0.42%, 0.51%) 117 6.4a, b Zone closure boundaries for an expanded project (REV model; LG=0.42%, 0.51%) 118 6.5a, b Zone closure boundaries for an expanded project (REV model; LG=0.60%, 0.69%) 119 6.5c Zone closure boundaries for an expanded project (REV model; LG=0.78%) 120 A2. la, b Rates of mineral price appreciation and risk adjustment 150 A3.1 FDMP project structure valuation program components 152 A3.2 Overview of the project state tree construction algorithm 153 A3.3 Program flow to determine links between the current EPS and all adjacent EPS 154 A3.4 Program flow to calculate project value and operating policy 156 X Acknowledgements The author would like to offer his sincere gratitude and appreciation to the members of his Ph.D. committee. Dr. Scott Dunbar and Professor Allan Hall of the University of British Columbia provided many helpful comments and discussions that furthered the author's research and study. Dr. David Laughton of the University of Alberta contributed much needed guidance and counsel regarding numerical methods and the use of finance theory in both the petroleum and mining industries. The author believes his efforts exceeded those of many Ph.D. advisors from outside institutions and that this dissertation would have been a much less fulfilling personal endeavor without Dr. Laughton's steadfast support. Finally, Professor Richard Poulin of the University of Laval organized invaluable financial support and dispensed advice on the management of the research process. This dissertation would not have been possible without Professor Poulin's contribution. The author also had valuable interaction with the mining industry while conducting this research. Mr. Gardner Joe (then of Gibraltar Mines Limited) provided project data that allowed the initial model to be tested and refined. Mr. Joe Ringwald of Steffan, Robertson and Kirsten Consulting and Mr. Andy Graetz of Placer Dome Incorporated provided additional industry data to test later versions of the model. This interaction produced many useful insights and focused the author's attention on the practical implications of this research. During the course of this Ph.D., the author relocated to the east coast of the United States. Dr. Barbara Algin, Professor Nickolas Themlis and Professor Tuncel Yegulalp of the Earth Engineering Center (formerly the Henry Krumb School of Mines) at Columbia University graciously organized access to the university libraries. The completion of this dissertation would have been much more difficult without their assistance. The author is especially grateful to Dr. Jacob Sagi of the University of British Columbia for many far-ranging discussions regarding the practical applications of finance theory. These discussions were most valuable in extending the author's understanding on this subject. The author is also thankful for the comments and suggestions Dr. Sagi provided on early versions of several dissertation chapters. Above all, the author is highly indebted to his wife, Linda, for her unfailing tolerance, support, and encouragement. Dedication In memory of my father, Robert Bruce Samis For my mother, Sherrill Anne Samis And, most of all, for my wife, Linda Anne Nockler 1 Chapter 1 Introduction The use of mineral resources has been closely associated with the improvement of human welfare over the past several thousand years. The introduction of new and superior minerals into society has permitted the production of increasing quantities of better quality food, fuel, shelter and clothing for a given amount of labour (Myers and Barnett, 1985). Great disparities in human welfare still exist and the minerals industry will continue to be an important part of the process leading to the decrease in living standard differences. The minerals industry will facilitate this process by providing the large quantities of minerals necessary for improving living standards and by allowing under-developed countries to use their mineral endowment to improve their citizens' standard of living through technology transfer, the earning of foreign exchange and the development of infrastructure and human capital. The efficiency of mineral exploitation is greatly complicated by the presence of many sources of uncertainty and risk. These sources of uncertainty may be broadly classified as economic, physical, or political/social. While the individual risks may be classified, their interaction and impact are, however, not so readily defined. Individual uncertainties and risks often interact with each other such that their effects on project viability are magnified or mitigated. For example, the risk of government intervention, such as new regulations or expropriation, may increase with the resolution of the physical uncertainty surrounding a mineral deposit (e.g. when an exceptionally valuable discovery is made). Over the last ten years, a common argument has been that the private sector is more efficient and better able to absorb the risk of mineral resource exploitation than government entities. Canada's abandonment of its national energy policy and the opening of South America's mineral resources to private investment demonstrate a move away from direct government involvement in the extraction of mineral resources. Research into government involvement with the minerals industry now tends to focus more on managing the government's exposure to risk as one of several participants in the minerals industry. Given the devolution of responsibility for mineral resource development to the private sector, it has become crucial that the private sector continually improve the efficiency with which it provides the world economy with mineral resources. 2 Improvements in private sector efficiency can be made with efforts in two interrelated areas. These are: 1) the development of better technology to exploit and use the available mineral resources; and, 2) the improvement of methods used to direct investment (allocate resources) within the minerals industry. The focus of this dissertation is the refinement of valuation methods, derived from the theory of economics and finance, which improve the process of allocating resources at a company level within the minerals industry. This is a pertinent topic because financial analysis methods are an integral part of the evaluation framework used by managers to formulate, communicate and discuss proposals for action. These methods effectively form a medium or "language" for the formulation and transmission of opinions among decision-makers and, as such, will affect the presentation and communication of the results of project definition and influence the structure of discussions among decision-makers (Laughton, 1987). Because of this fundamental role, the choice of such methods is important: any systematic biases in asset evaluation can have a significant impact on the direction of the organization. 1.1.0 Important characteristics of mineral resource projects From a valuation perspective, the important characteristics of a mineral resource project are: 1) a finite economic resource stock that will ultimately be exhausted by mining activities. 2) the partition of the resource into zones or sections that are differentiated by the spatial position of each zone, the amount and quality (mineral concentration or resource grade) of ore, and the mineralogical properties. 3) distinct stages of development and operation that extend over long periods of time. 4) large and irreversible capital expenditures that are directed towards either zone-specific purposes (e.g. developing access to a particular mineral reserve) or towards providing flexible services for all the project zones (e.g. construction of a mineral concentration plant). 5) the presence of geological and technical uncertainty that can only be fully resolved by developing and exploiting the mineral resource. 6) the ability to revise operating strategy with the partial or full resolution of individual project uncertainties. 3 These six characteristics are intimately and dynamically related so that any valuation method will, by necessity, only incorporate an approximation of the actual project characteristics. Underlying any mineral resource project is a geological anomaly in which the concentration of a specific mineral is greater than the surrounding host rock. Such concentrations occur when geological processes, such as movements of the earth's crust, cause the host rock to fracture or to be exposed to weathering. Host rock fracturing allows igneous intrusive rocks or mineral solutions to enter the host rock and form a mineral deposit while the weathering process transports eroded minerals to environments whereby the mineral is preferentially situated (e.g. eroded gold-bearing material concentrated in placer deposits). The deposit forming process is finite (on a geological time-scale) in that this process often ends when the conditions encouraging mineral deposition change. This process is also erratic which leads to haphazard zones of mineralization within the deposit that are distinguished by the size, quality and mineralogical characteristics of the zone. The initial stages of a resource project are devoted to the discovery and exploration of a geological anomaly. These stages seek to determine the geological characteristics, mineral concentration and amount of the mineral resource through geophysical tools, extensive exploratory drilling, and the assaying of samples. This period of resource exploration may last several decades, as discovery and delineation are dependent on the insight and skill of exploration geologists, the refinement of technology for exploration, mining and mineral processing, and the ability to defer exploration decisions. The middle stages of a resource project are devoted to project design and development. Additional exploration is completed to improve the understanding of the mineral resource characteristics. Alternate project designs are compared to determine the exploitation strategy that maximizes value. Project development may begin with the selection of a project design. Development expenditures may be broadly categorized by their functional purpose. Development costs may be zone-specific in that the assets created by the expenditures are usable for production operations within a particular zone. The costs incurred when developing access to a particular zone is one such example. Development costs may also be project-specific in that their associated assets may facilitate the production operations of any project zone (i.e. these assets are flexible). The costs associated with building a mineral processing facility are an example of such costs since this facility may process the ore from any zone. The design and development 4 phase of the project may take fifteen years or more and is dependent upon market and political considerations together with the analysis of additional information. Project operation and closure are the final project phases. The length of the operation period is highly variable and may range from several years to decades. The duration of operations is dependent upon market and political considerations as well as the discovery of new reserves during operation. Additional development costs may be incurred during the operational phase, if new zones are developed to replace depleted mineral zones. Resource quality tends to decrease as the project matures since higher quality areas are exhausted first. Finally, large costs are often incurred with project closure due to regulations requiring the rehabilitation of the environment to its pre-mining state. Mineral companies and their investors are exposed to many types of risk that change in magnitude over the economic life of a mineral resource. In the early stages, there is great uncertainty regarding the characteristics of the mineral resource and even its existence. In later stages, market, operating and political risks often become relatively more important as geological uncertainty is resolved. The characteristics of mineral industry risk combined with the nature of capital investment and the deferral provisions in the mineral leases have a potent effect on investment behavior and project value. Management is often able to revise their operating strategy in response to new information such that implications of downside risk are minimized and upside potential of the project enhanced. For example, when a deposit is nearly exhausted, the viability of the remaining marginal reserves is sensitive to downside price movements. A 15% decrease in mineral price may cause management to revise their mine plan such that project is closed early. Conversely, a 15% increase in the price may cause management to increase the expected life of the project by including more marginal reserves in the mine plan. 1.2.0 Conceptual examples of the mineral resource project environment The types of managerial problems confronted in the mineral resource industry can be demonstrated with several examples. The characteristics common to all three examples are the zone or component nature of each project and the ability of management to adjust operating strategy in response to uncertainty resolution. Otherwise, the project environments differ substantially. The first example, named the Triple Z Mine, illustrates some of the choices facing management of a mine that is nearing the end of its economic life and where project-specific uncertainty is reduced. The Triple Z Mine also incorporates 5 choices between zone-specific and multi-purpose capital expenditures. The second example is the Randfontein Estates Gold Mine in South Africa where management must decide on a plan to develop strategic reserves while operating existing high-cost reserves. The final example is the Rouyn-Noranda division of Cambior Incorporated. This division consists of several new mines where there is a substantial amount of geological uncertainty, a large number of development possibilities, and a degree of cost interdependence. 1.2.1 The Triple Z mine1 The Triple Z Mine is an open-pit gold mine located in the western United States. Its operations are currently focused on a single open pit of high-quality reserves, called the Main Pit. The current life-of-mine (LOM) plan envisages that the reserves of the Main Pit will be exhausted within five years and the mine will be forced to close. There are two reserve areas that may allow the mine life to be extended. One of these zones is a down-dip extension of the Main Pit reserves, called the UG Zone, and it is believed to be of higher quality than the material currently being mined. The UG Zone is expected to cost $15.0 million and take 1.5 years to develop. Development of this zone can only begin once the reserves of the Main Zone are exhausted since management believes that the best method of accessing these reserves is by a ramp starting at the final Main Zone pit bottom. This zone will provide six years of reserves. The other zone is a satellite area of low-quality reserves that is located adjacent to the Main Pit called the Pushback Zone. Development of the Pushback Zone may be started at any half-year interval over the operating life of the mine and it is expected to cost $8.0 million to expose the reserves. This zone's development will require a year to complete and will add 6 years of reserves to the mine. In addition, a further $10.0 million will be required to expand mineral processing capacity at the Triple Z Mine, if either of the other zones is producing when this zone is brought into production. A conceptual schematic of the Triple Z Mine is provided in Figure 1.1. This example was investigated by the author with an early version of the model described in Chapter 3. The name of the mine is not revealed due to confidentiality agreements. 6 Pushback Zone\ * (pit expansion ^opportunity) Main Zone (Current LOM plan) J 1 I | Underground Zone | | (mine-life extension I. ' I > Ramp." Figure 1.1 Triple Z Mine schematic layout. The additional zones provide Triple Z management with two options for creating additional reserves. The first option is a capacity expansion opportunity whereby the Pushback Zone can be brought into production while either Main Zone or UG Zone operations are continuing. The second option is a reserve replacement opportunity in which the exhausted Main Zone reserves are replaced by the UG Zone reserves. The key decision for management regarding both these options is at what gold price are they to be taken advantage of. Other operating policy issues that management may face include: 1) if the Pushback Zone is developed, in what situations should it be temporarily closed or abandoned while one of the higher-quality zones remain in operation? 2) how should development of the auxiliary zones be managed? Are there any environments in which development is temporarily halted? 3) in what situations should the Triple Z Mine be abandoned early? 7 1.2.2 Randfontein Estates2 Randfontein Estates is a large gold mine located in the Randfontein, Westonaria and Roodepoort districts of the Gauteng province in South Africa (approximately 60km west of Johannesburg). The mine was first incorporated in March of 1889. At its largest, the mine employed nearly 27 000 persons and extracted ore from more than 13 gold-bearing reefs. Starting in 1969, the mine was substantially restructured and production operations were gradually transferred to the Cooke shaft complex that was brought into operation between 1973 and 1983. An additional shaft, called the Doornkop shaft, was started in 1984 to exploit the Kimberley Reef located 1100 meters below surface. The quality of this area was found to be poor and, in January 1993, the decision was made to extend the Doornkop shaft to 2000 meters below surface to access the higher-quality South Reef. During late 1997, Randfontein Estates acquired the Lindum Reefs operation, which consisted of a small open pit mine and a tailings recovery operation, and the North Shaft complex (renamed No. 4 shaft) from the neighboring Western Areas Gold Mine. At the start of 1999, Randfontein Estates was producing some 850 000 oz of gold annually and milling approximately 830 000 tonnes of ore per month. Its operating strategy was to exploit its remaining marginal reserves as a high volume operation that made flexible use of an efficient network of concentration plants. Production was being generated from the Cooke shaft complex, the No. 4 shaft and the surface ore sources (open pit and tailings sand reclamation). The lowest cost ore on the basis of cost per gold ounce was from the Cooke Shaft complex. The No. 4 shaft and surface ore sources had a significantly higher unit cost and their continued operation was dependent on the gold price remaining above $300 per ounce. The Doornkop South Reef project was in progress and production from this area was expected to commence during 2001. At the start of 1999, over R1.0 billion had been spent while developing the Doornkop South Reef and it was estimated that a further R1.0 billion would be required to complete the project. Four separate operational zones can be identified within this example. These are the Cooke shaft complex, the No. 4 shaft area, the surface operations, and the Doornkop South Reef region (see Figure 1.2). Operational efficiency is dependent upon all producing areas maintaining production such that cost efficiency would be reduced with the closure of any active zone since the remaining zones would carry 2 Additional details regarding Randfontein Estates can be found in the Mining Magazine (1999), Engineering and Mining Journal (1998), Business Day (1998), The Mining Week (1988, 1992, 1993, 1999) and at the company web site http://www.randfonteinest.co.za. 8 Lindum reefs: open pit and tailings sands. Cooke Shafts #1. #2, #3: underground operations. No. 4 Shaft: underground operations. Doornkop South Reef Project: underground operations. Mineral resource zones Figure 1.2 Randfontein Estates project components the fixed costs associated with an under-utilized mineral processing facility. The management decisions being confronted at the start of 1999 included: 1) should the development of the Doornkop South Reef region continue or should it be stopped? 2) if the South Reef development is stopped, should it be stopped temporarily by putting it into a care-and-maintenance state or should it be irrevocably abandoned3? 3) in what order should the remaining areas be shut-down in the event of significant downward movement in the gold price? 4) should sections of the mineral processing facilities be permanently closed in response to the closure of any of the producing zones or put into a care-and-maintenance state? 5) under what conditions should any of the above decisions be reversed once they are made? Surface ore processing plant Underground ore processing plants Mineral processing plants 3 The Doornkop South Shaft project was halted due to low gold prices in May 1999. It was placed on a care-and-maintenance basis because management felt that it could be re-started at low cost. 9 1.2.3 Cambior Incorporated - The Rouyn-Noranda Division4 Cambior Incorporated was one of Canada's larger gold producers in 1990 and it had exploration and mining activities in both Canada and the United States at this time. In the Rouyn-Noranda region of Quebec, its underground gold mining assets comprised the Pierre Beauchemin (PB) Mine, the Silidor Mine, and the Mouska Pre-Production Project. The PB Mine was built on the re-opened workings of a mine that had been abandoned in the early 1960s due to low gold prices. The original mine shaft complex was deepened during the mid-1980s to access a promising area, called the No. 5 Zone. Production operations started in August 1988. At the start of 1990, the PB Mine had approximately 4.5 years of reserves with the No. 5 Zone offering the possibility of finding additional reserves to the north and south of the current reserves and below the deepest working level of the mine. The Silidor Mine is jointly owned with Noranda (55%) and Nova-Cogesca (20%). It was brought into production in April of 1990 and it has an estimated 13 years of reserves. Substantial development of the ore body remains to be done including development of additional access drifts from the main shaft to the gold-bearing zone and further deepening of the shaft below its current 570-meter depth. The production decision of the Mouska Pre-Production Project was announced in June 1990. The project requires an additional $6.0 million to be spent on development during the next 12 months before production can begin. Production will be sourced from two zones located on either side of the main shaft. Ore from the mines in Cambior's Rouyn-Noranda Division are processed by the Yvan Vezina Mill. This mill originally processed ore from the Yvan Vezina Mine until the mine was closed due to depleted reserves in 1988. The mill is now used to process ore from the neighboring Cambior mines. It also accepts a gold-pyrite concentrate from the Lucien C. Beliveau Mine for further processing5. The mill is considered cost efficient due to its compact layout and the use of semi-autogenous grinding. Figure 1.3 outlines the operating relationships between the Rouyn-Noranda Division assets. The operational decisions facing management of the Rouyn-Noranda Division are complex. The cost structures of the mining operations are interdependent because a common mill processes their ores. The closure of one operation can potentially increase the operating costs of the other mines, as additional 4 Further details of the Rouyn-Noranda Division of Cambior Incorporated can be found in Gignac et al (1990a), Gignac et al (1990b), and Gignac et al (1990c). 5 In 1990, the Lucien C. Beliveau Mine was part of Cambior's Val d'Or division. It had an on-site processing plant consisting of a gravity circuit for recovering free gold, and a flotation circuit for producing a gold-pyrite concentrate. 10 Lucien C. Beliveau Mine: gold-pyrite concentrate. Pierre Beauchemin Mine: underground operations. Silidor Mine: underground operations. Mouska Mine: pre-production project. Rouyn-Noranda Division mine assets Yvan Vezina Mill : ore processing plant Rouyn-Noranda Division mill assets Figure 1.3 Operational relationships between the Rouyn-Noranda Division assets. overhead costs must be carried. Other complexities include the large reserve uncertainties that are associated with the PB mine and the staged-development requirements of the Silidor Mine. Some questions regarding operating strategy include: 1) in what environments should the individual operations be closed, given the operating cost interactions between the mines? 2) what impact do operating options and technical uncertainties at a mine level (e.g. temporary closure of one zone at the Mouska Mine) have on the division as a whole? 1.3.0 Quantitative measures of project attractiveness Mining companies may use an evaluation process that incorporates several measures of project attractiveness to determine the economic viability of a mineral deposit or to choose between competing managerial actions. Research by Goucher (1992), Laas (1992) and Bhappu and Guzman (1995) found that mining companies often use a valuation framework that incorporates a combination of payback period, internal rate of return (IRR), and net present value (NPV) valuation measures. However, the use of either payback period or IRR within a valuation framework is undesirable because these measures 11 produce investment signals that are either equivalent to the results of the NPV method, ambiguous, or incorrect, depending on the project cash flow characteristics (LeRoy, 1989; Brealey and Myers, 1991). The preferred quantitative measure of project attractiveness is NPV because of the deficiencies of the other two measures. Much of applied project valuation research is currently focused on developing methods of calculating NPV that incorporate desirable valuation characteristics. Laughton (1987) outlines the characteristics for a financial valuation method that are desirable. These are: 1) the financial valuation framework must be, insofar as possible, both consistent with the principles of economic theory and compatible with the characteristics of a company's valuation process. This stipulation may require a trade-off between economic rigor and the requirements of the organization; 2) the valuation framework must be understood by all parties in the process such that the framework can be used to communicate ideas about proposed projects; 3) the valuation framework should be standardized within the organization so that comparisons can be made both between mutually-exclusive projects and with projects that have been previously analyzed and understood; 4) the valuation system should (at least implicitly) generate information that is relevant to the decision-making of all participants in the valuation process; 5) the valuation framework should be able to minimize the development and utilization of misleading information about project proposals. A common method of calculating NPV in the mining industry is the discounted cash flow (DCF) method. At its most basic, this method calculates annual net project cash flow from forecast (expected) project parameters. Revenues are determined as the product of annual mineral production and the expected mineral price. Expected operating costs, taxes and capital expenditures are subtracted from revenue to obtain net project cash flow. NPV may be then calculated using an aggregate discount factor to account for the value influence of risk and time on the individual project cash flows. The DCF exhibits several positive characteristics. These include recognizing the influence of risk and time on value, broad-based understanding within the organization due to its long history, and the ability to be standardized such that the same procedure can be used to apply it throughout an organization. Negative characteristics associated with the DCF method are the inclusion of systematic biases that may unduly promote or 12 denigrate a project and the inability to generate decision information in addition to project value, such as project operating policy insights based on action signals. Efforts to correct the disadvantages of the DCF method, such as Monte Carlo simulation and decision-tree methods, have not been fully successful (see Chapter 2 for further discussion). An alternative approach to project valuation has been proposed that arises from advances in finance theory. This approach is called Modern Asset Pricing (MAP) 6 and it is noted for its ability to combine project structure with uncertainty models, generated from the financial markets where appropriate, to produce relatively unbiased calculations of project value. It also has the advantage of being able to provide operating strategy insights in the form of parameter signals (e.g. close the project if the mineral price drops below a specific level). MAP's main disadvantage is a consequence of it being a relatively new valuation approach. The first MAP academic applications appeared in the late 1970s (Tourinho, 1979; Lessard, 1979) and few corporations within the mining industry have used it due to perceived unfamiliarity with its underlying concepts. MAP research can be divided into several subject areas (Laughton, 1998a, 1998c). The first research area is focused on developing more sophisticated uncertainty models for project parameters. Most research in this area has concentrated on improving the understanding of the relationship between spot mineral prices and forward prices (i.e. the term structure of commodity prices). The earliest price models used in MAP applications were single-factor models such as models of geometric Brownian motion with constant drift (e.g. Brennan and Schwartz, 1985) or reverting price processes that featured a constant long-term risk adjusted futures price (e.g. Laughton and Jacoby, 1993). More advanced multi-factor price models have been introduced that include interest rate effects (Schwartz, 1997), stochastic long-term futures prices (Schwartz and Smith, forthcoming) and jump diffusions (Hilliard and Reis, 1998). Less work has been done on other types of project uncertainty such as project costs and technical uncertainty. The second area of research explores the numerical methods used in the MAP valuation approach. The two numerical approaches that are commonly used in MAP valuations are discrete recombining trees and finite difference partial differential equation methods. Discrete recombining trees originate from the 6 This approach is more commonly known as Real Options due to its origins in the pricing of financial options and the focus of academic literature on valuing managerial flexibility (options). The term Modern Asset Pricing is used here because this approach is still relevant when there is no management flexibility. Other names that have been used in the literature include contingent claims analysis (CCA) and derivative asset valuation (DAV). 13 binomial option pricing method of Cox, Ingersoll and Ross (1979). This approach has been extended to specific multi-factor processes (Boyle, Evnine and Gibbs, 1989; Hull and White, 1994b) and one-factor risk-adjusted processes that exhibit reversion (Hull and White, 1994a)7. The finite difference approach has been used extensively in the finance industry to price complex financial derivative products8. This method is limited because of practical computational issues associated with the size of the project state space. The price process state space is limited to two or at most three dimensions while the auxiliary state space may be limited to several (tens of) thousand. Newer approaches are being introduced that combine Monte Carlo simulation with creative search techniques to overcome the limitations of discrete trees and the finite difference approach. Boyle, Broadie and Glasserman (1997) provide a review of early research into this approach. Castillo-Ramirez (2000) provides a demonstration of the Monte Carlo approach to solve the Brennan and Schwartz (1985) model of temporary mine closure. The third area of MAP research focuses effort on developing increasingly realistic models of project structure. Most models in the literature do not account for the heterogeneous nature of a mineral deposit. These models, called Set Production Plan (SPP) project structure models, may be misleading because they rely on the ability of the analyst to represent the project's most important operating scenarios with exogenously set production plans. Early mineral resource models, such as Brennan and Schwartz (1985), allowed an open project to follow one production policy. Later models (e.g. Pickles and Smith, 1993; Samis and Poulin, 1996; Laine, 1997) rely on decision trees, where each branch represents a unique production policy, to describe disparate project characteristics such as multiple deposit zones and staged exploration and development. Exceptions to the SPP model approach are Cortazar and Casassus (2000) and the cut-off grade models of Mardones (1991, 1993) and Sagi (forthcoming) in which management may alter the proportion of mined material that is processed as ore in response to price fluctuations. This dissertation concentrates solely on the third area of MAP research. Laughton et al (2000a) and Laughton et al (2000b) note that discrete trees do not necessarily converge to the multi-factor continuous processes that they are meant to approximate. They note that Duffie and Huang (1985) show that the number of replicating assets needed for a multi-factor process is one more than the number of factors while for a discrete process this number is equal to the number of branches originating from each state. In some situations, these numbers will not be equal and the discrete tree will not necessarily converge to multi-factor process even though the state pricing calculations do. See Wilmott, Howison and Dewynne (1993, 1995) and Wilmott (1998) for an overview regarding the application of finite difference techniques to the valuation of derivative products. 14 1.4.0 Dissertation contribution and overview The primary objective of this dissertation is to extend the project structure model describing a mineral project's physical environment within the MAP framework. The project structure model proposed for this task is called the Flexible Discrete Mine Production (FDMP) model. It is differentiated from other MAP mining applications by two characteristics: first, the mining project is structured as a portfolio of production assets in which each asset represents a deposit zone and, second, by the explicit recognition of capacity as an auxiliary project state variable9. This approach allows the FDMP model approach to extend the project structure state space such that it includes, for each zone, a variable to identify the stage of zone development and depletion and three capacity variables (open, closed and under-construction). Previous project structure state spaces used within the MAP valuation framework are more limited because their state spaces are delineated by auxiliary state variables that summarize either global deposit reserves or position within a decision tree. The FDMP model makes the assumption that the project will be operated in discrete management intervals of fixed duration. • At the start of each interval, management chooses from a set of competing operating modes the method of project operation for the next interval. Each operating mode specifies which zones are active and any changes that are to be made to project capacity. The model allows the number of zones that may be active simultaneously to be restricted due to geological considerations (e.g. one zone may not be active when another is producing) and capacity limitations. The types of capacity changes that are recognized by the model include temporary closure of a portion of project capacity, the abandonment of project capacity, and the addition of new capacity as outlined by a construction program. Project cash flows are generated over discrete management intervals (i.e. the period between operating decision points) and they are determined by the project state at the beginning of the interval and the operating mode chosen for the interval duration. These cash flows comprise many components such as zone-specific capital expenditures, multi-purpose capital expenditures, economies-of-scale benefits, and capacity under-utilization charges. 9 Mardones (1991, 1993) and Sagi (1998, forthcoming) also include project capacity as an auxiliary state variable. However, their models maintain capacity at a constant level and maximize project value by allowing allow cut-off grade (the quality of mineral resource being processed) to fluctuate. This dissertation allows management to maximize value by capacity levels to fluctuate through new construction, temporary closure or abandonment of capacity while maintaining constant zone production profiles. 15 The secondary objective of this dissertation is to investigate and characterize the operating decisions that may be reached within the FDMP framework. Mineral resource projects potentially embody many types of flexibility that management may use to respond to the resolution of uncertainty. However, not all forms and combinations of flexibility are of equal importance. For example, Pindyck (1988) showed that the ability to temporarily close a unit of capacity has less impact on project value than the ability to choose the timing of the installation of that unit of capacity. This dissertation explores operating policies when mine management has the ability to defer the development of individual zones over an extended period, suspend capacity construction programs versus abandoning them, temporarily close individual zones, and abandon the project irrevocably. The dissertation is organized as follows. Chapter 2 reviews the relevant literature for the FDMP model. This chapter outlines the concepts of finance theory that are used to construct the valuation arguments and discusses the factors influencing the value of real assets. MAP applications from the petroleum and mining industry are reviewed. Chapter 3 develops the FDMP model. It outlines how development and production plans for each zone may be combined with possible levels of project capacity to create a project state space and decision tree. A valuation partial differential equation is presented to calculate the evolution of project value over a management interval. Boundary and initial conditions are provided for valuation equation that allows the project to be continuously abandoned during the interval and for cash flows to be generated either discretely or continuously over the interval. The model is extended to include zone geological uncertainty in which zone grades are independently and discretely distributed. Chapter 4 demonstrates the FDMP model with a stylized two-zone mine example. In this example, a high-grade zone is being mined and management must decide whether to develop a satellite low-grade zone as part of production expansion or production replacement strategy (parallel versus sequential development strategies). The project is valued within a MAP framework using both FDMP and SPP project structure models and within a discounted cash flow (DCF) value framework using the SPP models and the results are compared. Project operating policy is determined with the FDMP method and then discussed in the context of the investment signals provided by the SPP structure models valued with DCF and MAP valuation approaches. Chapter 5 investigates the value sensitivity of the three valuation methods to changes in project and price parameters. The effect of parameter fluctuations on the signals for low-grade zone development is also 16 examined. Chapter 6 introduces low-grade zone geological uncertainty to the two-zone mine example presented in Chapter 4. The effect of zone geological uncertainty on project value and operating policy is explored. Temporary zone closure is studied as a result of the different low-grade quality (grade) outcomes generated by the geological uncertainty model. The dissertation concludes in Chapter 7 with recommendations for further research. 17 Chapter 2 Literature Review A firm's success is critically dependent upon making decisions that add value. Corporate decision-makers use an evaluation process to assess such decisions. When applied to projects, this process is called capital budgeting and it evaluates the project on the basis of the project's technical structure and external economic environment. Much of the capital budgeting research in the past 40 years has been directed at finding better methods of calculating project value with due consideration to the influences of value. This research has frequently used concepts from finance theory to devise better methods of calculating value. Developments in the equilibrium asset pricing models (e.g. Cox, Ingersoll and Ross, 1985a) show that research in both areas is converging with the integration of production opportunities into asset pricing models. However, Laughton (1987) noted that general equilibrium models cannot be used to value projects without making simplifying assumptions that are unrealistic. At present, the mathematical tools do not exist to implement a full general equilibrium valuation of a project. 2.1 Conventional valuation techniques Conventional valuation approaches have followed one of two related calculation philosophies. These are the certainty-equivalent (CEQ) approach and the risk-adjusted discount rate (RADR) method. Early capital budgeting research embellished these philosophies with techniques designed to assist managers to understand and incorporate project risk into the value calculation. These techniques include sensitivity and scenario analysis, Monte Carlo simulation and decision-tree analysis (DTA). Both the CEQ and RADR value calculation philosophies define project value, or net present value, as the difference between the project's cash flow present values and its capital expenditure present values. However, these philosophies use different methods of incorporating the effects of time and risk into the value calculation. The CEQ valuation approach was first proposed by Robichek and Myers (1966). This approach separates the effects of time and risk on cash flow present value. A CEQ is defined as the magnitude of a time t certain cash flow, c,, that has the same present value as an expected (i.e. risky) time t cash flow, E[c t ]. Stated as an equation in which r is the riskless interest rate and k a risk-adjusted discount rate: 18 Cash flow t = ct.e-"=E[ct].e .-kt (2.1) By rearranging equation 2.1, the certainty equivalent cash flow is calculated as: c , = E [ c t ] . ^ (2.2) The risk premium associated with the uncertain cash flow is equal to the expected value of the uncertain outcome minus its certainty-equivalent amount. Trigeorgis (1996) notes that one disadvantage of the CEQ approach is that it is difficult to determine certainty equivalents by traditional methods when risk fluctuates over the life of the project. The more commonly used value calculation technique is the (RADR) method. The present value of a cash flow is determined by multiplying the cash flow's expected value by a discount factor. This factor is derived from a formula that incorporates the RADR. In an equation, the present value of a time t cash flow is: The RADR includes the effects of both time and project risk. It is calculated as the sum of the riskless interest rate and a risk premium. The risk premium is determined from the risk characteristics of the project and not from the risk and financing characteristics of the firm considering the project. A key consideration for both value calculation techniques is the determination of an appropriate project risk premium. It is tempting to argue that a project's risk premium is influenced by the risk of the firm as a whole. However, it is often pointed out in corporate finance texts (e.g. Brealey and Myers, 1991; Trigeorgis, 1996) that investors can diversify more efficiently through the stock market and will not reward the firm for diversifying for them. Hence, the risk interaction between the different projects within a firm should not affect the risk premium used in the valuation of a particular project. This implies that the risk premium should be determined by the project's exposure to systematic market risk. The appropriate risk premium is thus equal to the return that would be demanded from a comparable investment by investors in the financial market. P V Cash flow t -RADR.-1 •e 1 (2.3) 19 2.1.1 Assessment of future possibilities Four techniques have been proposed to expand project valuation beyond the calculation of value. These are sensitivity analysis, scenario analysis, Monte Carlo simulation and decision tree analysis (DTA). Their purpose is to provide project analysts with a better understanding of the risks and uncertainties that the project is exposed to. Sensitivity analysis is utilized as a method of identifying the key parameters of the project and then determining the "value" consequences of these parameters being incorrectly specified. The main benefits of this procedure are that it forces analysts to identify the most important underlying variables, it indicates where additional information is required and it helps to expose confused or inappropriate forecasts (Brealey and Myers, 1991). This procedure is also limited by its inability to consider more than one variable at a time and to account for the interdependencies and serial dependencies of project parameters (Trigeorgis, 1996) and by the assumption that risk is constant over a parameter's range of variation, which is usually not the case. Scenario analysis is an attempt to overcome the deficiencies of sensitivity analysis and is used when project parameters are interdependent. Alternative project scenarios are used to determine a range of possible project values where the scenarios are constructed such that the combinations of variables are consistent in terms of their interaction and variation over time. The primary disadvantage of this technique is that project analysts must rely on subjective interpretations of the valuation results (Jacoby and Laughton, 1992 ) since this approach includes no clear method of interpreting the results. Monte Carlo simulation (Hertz, 1964) is considered an improvement on scenario analysis because it considers the impact of all combinations of variables. The primary project parameters are each described by a probability distribution. These distributions are randomly sampled repeatedly to produce a probability distribution of the individual cash flows or project NPV for a given management strategy. Trigeorgis (1996) noted that this technique can handle complex decision problems under uncertainty with a large number of interacting input parameters. However, this technique also suffers from interpretation problems. It provides no clear method of interpreting the final project value distribution and no method of accounting for the dynamic nature of risk. In addition, the final value distribution may be suspect because it can be extremely difficult to correctly capture parameter interdependence. 20 DTA provides management with a tool to consider both the underlying project uncertainties and the possible operating strategies that may be implemented in response to uncertainty resolution within the same valuation framework. The tool is based on a tree-like structure that is constructed of nodes. These nodes indicate either an information point, where project information is revealed according to a probabilistic measure, or a decision point, where management may choose from one of several responses to new information. Within the tree, values are calculated based on expected outcomes of information events. At decision nodes, management is expected choose the operating policy that maximizes project value. The advantage of DTA is its application requires management to consider their (implicit) operating strategy and to recognize explicitly underlying project uncertainties and the interdependencies that exist between current decisions and subsequent decisions. The disadvantage of DTA is its tendency to become overly complex in realistic situations (e.g. projects that permit abandonment at any point in time). It also does not offer a method of calculating an appropriate discount rate as the project risk evolves within the decision tree. 2.2.0 A review of finance theory A financial asset is a legal contract that allows the transfer of economic resources at different times. Their contractual provisions may also provide the owner of the asset with discretionary power over the asset's payoff as well as possibly dictating their owner's actions (Ross, 1989). Many financial assets have value because they are direct claims to a portion of the economic benefits generated by one or more real assets. This link suggests that the concepts used to price financial assets can also be used to price real assets. Laughton (1987) notes that these concepts offer the only relatively complete framework for asset pricing within an organization that strives to act in the best interest of its stakeholders. This observation is accompanied by a proviso that financial economic theory provides only a limited description of asset evaluation because it simplifies the behavior of organizations, their internal structure and their links to external markets. The markets that trade financial assets generate enormous amounts of data and observations regarding the pricing of these assets. Financial theory is the result of efforts to develop a framework in which to assimilate this data such that the trading of assets may be understood. Ross (1989) outlined four concepts upon which financial theory is built. These are: 1) markets make efficient use of available information; 2) there is a relationship between risk and return; 21 3) no arbitrage can exist when the markets are in equilibrium; and 4) corporate finance is a mechanism to distribute value among stakeholders and does not itself create value. The first concept is known as the efficient market hypothesis (EMH) (Fama, 1970) and it refers to the process of information distribution within the market place. This hypothesis assumes that, at any point in time, an asset's current price incorporates (or "fully reflects") a specific information set1. The intuition behind the EMH is that individual traders process the information available to them and then take a position in the asset based on their understanding of the information and their personal risk preferences. The market is the mechanism that aggregates this information such that the net effect is an equilibrium asset price that will not change until new information enters the market. Equilibrium economic models incorporate the EMH by stating conditions of market equilibrium in terms of expected return that is conditional on some relevant information set. Capital budgeting texts (e.g. Giammarino et al, 1996) often note that most people are risk-adverse and require the provision of an additional return before assuming additional risk. Finance theory has focused on developing this observation into an evolving quantitative framework, called capital market theory, to explain the risk premium (the difference between expected return and the riskless interest rate) in the capital markets (Ross, 1989). There are two types of risk that an asset may be exposed to. There is unsystematic risk that refers to risks that are not correlated with the underlying state variables of the economy. It may be mitigated by diversification and hence does not affect an asset's risk premium. The other type of risk is called systematic risk and this includes risks that are correlated to the underlying economic state variables and may not be avoided by diversification. A central feature of capital market theory is a quantitative relationship that explains the risk premium associated with an asset in terms of the systematic risk to which an asset is exposed. An arbitrage opportunity is an investment strategy that requires no net investment and creates an opportunity for a positive payoff with no possibility of a negative payoff. The absence of arbitrage implies that assets with equivalent future payoffs (i.e. are substitutes) will have the same price. This condition is necessary for the establishment of equilibrium asset prices when investors have increasing An important implication of market efficiency is that it is impossible to make economic profits by trading on the basis of the reflected information set. 22 preferences. Arbitrage research showed that the no arbitrage rule (Ross, 1976; Ross, 1978; Harrison and Kreps, 1979) implies the existence of a linear pricing rule, which says that in many spaces, including finite state spaces, there exist positive state prices that correctly price all assets. This was an important development because it showed that the linear pricing rule could be extended beyond market assets to include all assets. The fourth concept is due to Modigliani and Miller (1958) who used an arbitrage argument to show that the value of the firm or project is unaffected by its financing decision. Their work implied that there is no optimal financing structure for a company since value is independent of financing. The focus of this paper is to extend the model of project structure used to determine the project value that underlies financing decisions. Hence, the role of corporate finance will not be considered further. The four underlying concepts of finance theory may be used to develop asset valuation models. These models may be broadly categorized based on their input parameters. One type of model is the general equilibrium model and models of this category require the identification of an underlying core of economic state variables before implementing the valuation argument. The other type of pricing model was originally developed to price short-term derivative securities such as call and put options on stocks. They are called modern asset pricing (MAP) models and they use existing financial assets as parameters for valuation. 2.2.1 Equilibrium asset valuation models Equilibrium asset pricing models were initially developed as a framework in which to price assets trading in uncertain financial markets and were later extended to permit the inclusion of real assets. Equilibrium asset pricing models have experienced three stages of development. These stages include the initial one-period mean-variance models, an intermediate stage continuous-time financial market models and the current stage of continuous-time asset pricing models. One period asset pricing models The origins of equilibrium asset pricing models can be found within the portfolio selection model of Markowitz (1959) and the liquidity preference model of Tobin (1958). These models used the assumptions of quadratic investor utility and normal asset prices to show that the portfolio choice problem of an individual investor could be reduced to the analysis of the properties of the mean-variance efficient 23 set. Essentially, an investor would choose a portfolio with the highest mean return for a given level of variance. Tobin (1958) also provided an early version of the two-fund separation theorem. This theorem states that an individual can attain any mean-variance efficient portfolio by holding a linear combination of any other two mean-variance efficient portfolios. When a riskless asset is available, portfolio separation allows an investor to obtain greatest utility by investing only in combinations of the riskless asset and the market portfolio. This was an important result because it allowed asset demand in equilibrium pricing models to be characterized in terms of only price distribution parameters; other parameters, such as individual preferences, wealth distribution or age distribution, could be ignored. Sharpe (1964), Lintner (1965) and Mossin (1966) aggregated the intuitions and results of prior portfolio choice models to develop a model of capital market equilibrium. This model became known as the Capital Asset Pricing Model (CAPM) and it is comprised of two principal components. The first component is a model of investor portfolio choice, called the capital market line (CML). The CML provides a graphical depiction of the relationship between risk and return for efficient portfolios in expected return - return standard deviation space. It can be used to analyze efficient portfolio choice in the capital markets. The second component is called the security market line (SML) and it is a linear restriction on expected asset returns in equilibrium. The SML is used to calculate equilibrium asset returns in financial markets and stipulates that the expected asset return required by investors is equal to the riskfree return plus an adjustment for risk. The risk adjustment is defined as the market risk premium weighted by a factor (called the asset's beta) that measures of the asset's contribution to the overall risk of the market portfolio. An alternative to the CAPM is the Arrow (1964) market model that considered the determination of asset prices in a discrete time pure exchange economy. This model structured the economy such that there are a finite possible number of future states of nature with an associated unit security that pays one monetary unit if the state occurs and nothing otherwise. The model allowed individuals to have different subjective probabilities regarding the occurrence of each possible state and it provided a market mechanism in which investors could buy and sell unit securities to maximize their consumption utilities across time. This An efficient portfolio is defined as the portfolio that has minimum variance for a given expected return or, equivalently, the portfolio that has the maximum return for a given variance. Investors elect to hold efficient portfolios because these portfolios maximize investor utility when investors are risk averse and have increasing preferences. 24 model is considered a one period model because, when the market is complete, all investors achieve a Pareto optimal consumption pattern with one trading date (i.e. trading at time zero). Intermediate continuous-time asset pricing models The continuous-time framework is characterized by continuous trading in the financial markets and the use of diffusion processes to describe the dynamics of the assets traded in the markets. It is a powerful framework that enables intertemporal equilibrium asset pricing models to combine the portfolio-separation characteristics and the SML of one-period mean-variance models with the optimization of investor consumption-investment programs. The foundation of continuous-time models is the optimization of the consumption-investment program of a single investor (Merton, 1969; Merton, 1971) •a that is formulated using a stochastic version of the Bellman dynamic programming equation . The first continuous-time equilibrium asset pricing model was the intertemporal capital asset pricing model (ICAPM) (Merton, 1973b). This model was considered a better reflection of the capital markets than the one-period CAPM because it allowed the investor opportunity set to vary over time in addition to having the previously stated benefits of the continuous-time framework.4 The ICAPM model also exhibits portfolio separation but requires three assets (funds) to achieve this. These assets are a riskless asset, the market portfolio and a third fund consisting of a single risky asset that allows investors to hedge against unfavorable intertemporal shifts in the investment opportunity set. The ICAPM also included a version of the one-period CAPM SML that demonstrates that an asset's return must compensate investors for bearing market risk and unfavorable shifts in the investment opportunity set. It related the equilibrium excess returns of all traded assets to the expected excess returns of the market portfolio and the hedging portfolio. The use of dynamic programming requires that the state variable dynamics be expressed as a Markov stochastic process that is defined over any arbitrarily small time interval (Merton, 1971). A Markov stochastic process has the property that the conditional probability distribution for future values of the process, conditional on the time state t, depends only on the current value of the process. This conditional probability is not altered by the inclusion of further information as of time t (Merton, 1990). These processes are not differentiable in the usual sense and require their dynamics to be described by a more general type of differential equation. Ito's lemma provides this type of equation if the Markov stochastic process is an Ito process. Ito's lemma shows that any twice-differentiable function of an Ito process is also an Ito process. This result simplifies the dynamic programming problem by reducing it to a second order partial differential equation (PDE) that only involves the first two moments of the price process. 4 The original ICAPM used one state variable to model shifts in the investor opportunity set. Merton (1990) extends the ICAPM with an economy described by m generic state variables. This extension permits investors to hedge against other economic events in addition to changes in the investment opportunity set. 25 Equilibrium models have also been developed with a priori theoretical reasoning. Breeden (1979) used this approach to develop the Consumption-Based Capital Asset Pricing Model (CCAPM). In this model, exogenous changes in the economy are described by a single state variable of aggregate consumption. Excess asset returns are determined in the CCAPM by an SML that is based on the consumption-betas5 of the asset of interest and another arbitrary portfolio. Breeden (1979) noted that the CCAPM may be a better equilibrium pricing model than the ICAPM because aggregate consumption measures cover a greater proportion of the true consumption than the portion of the true market portfolio that is described by market portfolio measures. However, Merton (1990) has noted that empirical results of CCAPM tests are mixed and that the CCAPM's lack of success may be due to discrepancies between model preference structures and actual preference structures or the nature of real world consumption. Current equilibrium market models Cox, Ingersoll and Ross (1985a) developed an equilibrium asset pricing model that may be considered representative of the best current asset pricing models. This model assumes that the dynamics of the economy are the result of n underlying stochastic state variables. These state variables are used to describe uncertain production and random technological changes within the economy and the current level of these variables determine all future production opportunities. The relationship between the state variables and production opportunities allows the model to integrate the financial and real asset markets. The Cox, Ingersoll and Ross (CIR) model showed that the excess expected return on an asset traded in the economy is equal to the negative of the covariance of its rate of return with the rate of change in the marginal utility of wealth. This result implied that individuals are willing to accept a lower rate of return on an asset that tends to have a higher pay off when the overall economy is unfavorable and an individual's marginal utility is higher. The CIR model also provided a PDE with which to price all assets in terms of the economy's underlying real variables. This PDE had two unique solutions. One solution involved discounting an asset's expected payoffs with respect to a randomly varying rate of return. The second solution obtained results by discounting the asset's expected value at the riskless interest rate. The asset's expected value was determined with respect to a risk-adjusted process for wealth and the state variables in which the risk adjustment was achieved by reducing the expected return or drift of each underlying state variable by its factor risk premium. Cox, Ingersoll and Ross (1985a) noted that the first 5 A consumption-beta describes the relationship between changes in an asset's return and changes in overall consumption. 26 solution is of only intuitive value since a practical solution is only possible if the equilibrium expected rate of return is known in advance. The second solution was considered to be more practical because a solution can be found with only the interest rate and the factor risk premiums; these parameters are common to all securities. 2.2.2 Modern asset pricing models Merton (1989) noted that MAP models have had a tremendous impact on asset valuation research for three reasons: 1) only relatively weak assumptions are required to develop these models; 2) the model's input variables and parameters are either directly observable or relatively easy to estimate; and 3) MAP models may be easily extended to pricing other types of securities and assets displaying option-like characteristics. MAP models assume that markets are competitive and free of transaction barriers. In such a market, an equilibrium condition of no arbitrage ensures that different assets with the same cash flow results have the same price. This result allows a complex asset to be priced by a trading strategy involving a portfolio of simple assets, if the trading strategy replicates the cash flow results of the complex asset (Jacoby and Laughton, 1992). The prime attribute of these models is the lack of information required to value an asset. Many of the concerns associated with equilibrium asset pricing, such as the joint distribution characteristics of all available securities, may be ignored because they are reflected in the stochastic price processes of the underlying simple assets.6 For example, the only information required for simple option pricing models pertain to the underlying stock and the default-free bond. Development stages of modern asset pricing The valuation model underlying the MAP model was refined in three stages of development. The first stage of MAP model development was the Black and Scholes (1973) option pricing model which valued European put and call options. Their model was based on the insight that a dynamic trading strategy in the underlying asset and the option could be used to produce a riskless hedge. By observing that a riskless hedge must earn a riskless return, a PDE was produced that the option value must satisfy in 6 This does not imply that equilibrium asset pricing issues are avoided when developing a price process for the underlying simple assets. 27 equilibrium. The solution of the PDE was found, with the appropriate specification of boundary conditions, that states the value of an option as a function of the underlying asset prices and time. The second stage of MAP valuation models was the self-financing portfolio valuation argument (Merton, 1973a). The self-financing portfolio argument used Ito stochastic integrals to extend the Black-Scholes model to include stochastic interest rates and stock dividend payments. Merton (1973a) also derived a smooth-pasting condition that controls the early exercise of American put and call options. This condition requires that the derivative of the option value function with respect to price be continuous at the exercise boundary. The replication argument (Merton, 1977) was the third refinement of the MAP valuation argument. This argument demonstrated that the derivation of a MAP equation does not depend upon the existence of an option on the underlying asset. It used a trading strategy involving the underlying asset and riskless bonds to derive a pricing function for an option on the underlying asset. This trading strategy and the pricing function provided the means and the cost of creating a synthetic option when such an option is not available as a traded security. The MAP model has been modified to relax the particular assumptions of the Black-Scholes-Merton model. These modifications included allowances for the effect of differential tax rates on income and capital gains (Ingersoll, 1976), the effect of transaction costs in frictionless markets (Leland, 1985; Merton, 1990) and the restriction of trading to discrete time intervals (Merton and Samuelson, 1974). 2.2.3 Financial theory as a framework for valuing real assets Jacoby and Laughton (1992) noted that the key result from the advances of finance theory is that real asset valuation may be carried out, to a good approximation, as if financial markets were competitive and free of transaction barriers. The implications of such a market are two-fold. First, all asset prices are determined by the risk preferences of investors, as reflected in the markets. Thus the basic assets that have some direct interaction with future economic variables can provide information about risk discounting. In the mining industry, these assets are mineral forward contracts, which are related to the future spot prices of minerals and correlated with the state of the economy. The second implication is that different assets or portfolios with the same cash flow consequence must have the same price. Hence, the cash flow consequences of complex assets, such as a mine, can be valued as if it were a portfolio of 28 simpler assets, such as bonds and mineral forward contracts, with an arbitrage valuation argument under certain circumstances. These two results allow MAP project valuations to incorporate a market view of risk and uncertainty that is not always incorporated into conventional valuation approaches. 2.3 Financial market theory applied to real assets Project value and the operating policy derived from the value calculation are dependent upon the interaction of such factors as the sources of project uncertainty, investor risk preferences, project structure (e.g. timing and magnitude of output production and input consumption), the presence of non-equity stakeholders, and the ability to alter operating policy in response to the resolution of uncertainty. A valuation model can be built that recognizes these factors based on information from financial markets and valuation concepts provided by advanced finance. Financial markets provide the information necessary to identify the important characteristics of output and input price uncertainty7, such as long-term reversion and the magnitude of price fluctuations. Finance theory provides both a framework within which to (imperfectly) translate this information into a valuation methodology that can handle the variations of uncertainty and risk created by project structure and management flexibility. 2.3.1 The influence of uncertainty and project structure on value A primary cause of project uncertainty and risk variation during the life of the project is its structure. Project structure incorporates the project's production profile, degree of operating leverage8, and development expenditures. Uncertainty propagates through this structure such that its impact on value is amplified in some periods and reduced in others. During development phases, project risk fluctuates such that a larger risk discount is used for periods in which larger amounts of capital are expended than those periods where a lesser amount of capital is expended. During operational phases, project risk varies with the degree of operating leverage such that larger risk premiums are associated with periods of higher operating leverage. Jacoby and Laughton (1992) discussed the influence of project structure within a valuation model of an offshore petroleum project. Their model considered a field in which there was no management flexibility 7 Project-specific uncertainty must still be modeled based on input from the persons knowledgeable with the project environment. Finance theory may provide insight into how investors price their exposure to this uncertainty. 8 Giammarino et al (1996) defined the degree of operating leverage as the ratio of the percentage change in profits to the percentage change in sales. This dissertation adapts this definition for a project valuation environment in which mineral price is the only source of uncertainty. The degree of operating leverage is redefined as the ratio of the percentage change in project value to the percentage change in mineral price. 29 and the only source of uncertainty was the oil price that followed a non-reverting log-normal price process. They demonstrated that project risk fluctuates over the life of the project with changes in operating leverage and the proportion of total capital expenditure sunk. Project risk was shown to decrease if the field's recoverable reserve were larger. The valuation results for various field sizes were compared to a standard DCF valuation and it was shown that the DCF method undervalued the larger fields (excessive risk adjustment) and overvalued the smaller fields. Salahor (1998) extended this discussion by examining the trade-offs between incurring higher development costs in exchange for lower operating costs. He showed that the degree of operating leverage is important when choosing between project designs since a bias may be introduced against designs built around this trade-off when operating leverage is incorrectly accounted for. Uncertainty and risk profiles may also vary between projects because of differences in the characteristics of uncertainty. Some cash flow components will display uncertainty that grows at a constant rate over time while the uncertainty of other components may grow at a decreasing rate over time. Consequently, components with a declining rate of uncertainty growth tend to be much less uncertain in the future than those with a constant uncertainty growth rate. Salahor (1998) examined the influence of price uncertainty characteristics on design choice when management could choose a design that shortened the production profile through higher initial development costs. He showed that in non-reverting price environments the most valuable design is the one that shortens the production profile because highly uncertain long-term cash flows are avoided. In reverting price environments, the most valuable design was the one with a longer production profile because initial development costs are smaller and long-term cash flows are less uncertain (thus requiring a smaller risk adjustment per unit time). 2.3.2 The influence of project stakeholder interaction on value Project financing and taxation distribute project risk and value between the stakeholders. The value of a stakeholder claim to a portion of the project cash flows is determined by the claim's inherent underlying characteristics, the specific terms of the claim, and its interaction with the claims of other stakeholders. For example, valuation of a project creditor's claim requires assessment of future interest and principal payments with consideration of possible equity stakeholders actions such as project abandonment in low output price environments. 30 Mason and Merton (1985) discussed the MAP approach to capital structure analysis and provided an example of how such a valuation exercise would be constructed for a large-scale energy project. Within this example, the valuation equations of equity, senior debt, guaranteed junior debt and a government price subsidy were developed and discussed9. Mason and Baldwin (1988) discussed the contingent nature of government subsidies to energy projects and the challenges that this produces for valuation. They explained how the MAP framework is especially suited for this type of analysis. Samis (1994, 1995) applied the MAP approach to the capital structure analysis of a marginal South African gold mine that was financed by several issues of debt, government subsidies and a loan guarantee. He compared the results to government analysis that used conventional accounting procedures and showed that the government analysis underestimates the default probability associated with the mine's capital structure. The influence of fiscal regimes on valuation has attracted more research. Cavoulacos (1987) considered the impact of political risk on project value where political risk originates from the ability of government to appropriate a greater share of excess profits than that specified by initial contract terms. He showed that contract stability is less in fiscal regimes using ad valorem royalties or production sharing agreements than those using service contracts or a resource rent tax because there is more incentive to expropriate windfall discoveries. Mackie-Mason (1990) examined the influence of the percentage depletion allowance (PDA) on investment decisions involving development leases. The PDA increases the complexity of investment decisions because it introduces a non-linearity into the tax cash flow, even if there is significant income to absorb tax losses, because it is subjected to an income cap. Mackie-Mason showed that an increase in the tax rate and a decrease in the depletion allowance rate can induce immediate investment when there is an option to defer development. This behavior was observed because the changes in tax rate and PDA reduce the value of delaying investment more than the value of immediate development. Lund (1992) investigated how government taxation creates a disincentive to invest in petroleum through the asymmetric treatment of losses. He compared the Norwegian taxation policy during 1980 to 1986 to the tax policy introduced in 1987 and showed that the tax policy changes introduced in 1987 substantially reduce distortions. Jacoby and Laughton (1992) demonstrated that the structure of government taxation has a significant effect on the distribution of project risk. Resource royalties were shown to increase the 9 The equations were not solved numerically within an example. 31 leverage of operating cash flows and lead to an increase in operating cash flow risk. They also noted fiscal conditions when the after-tax cash flows accruing to equity were significantly less risky than the pre-tax cash flows. This transfer of risk occurs because the special taxation features such as tax loss carryforwards, oil allowance and uplift provisions ensure that tax revenues are highly sensitive to unfavorable price paths. This effect was noted to decrease with the life of the project. Bradley (1998) studied the effect of five different royalty systems on the risk distribution of six "now-or-never" projects that vary in field size. These systems included a pure rent royalty, an ad valorem royalty on gross revenue, an accounting profits royalty and two types of rent resource royalties. The results of the MAP valuation were compared to a DCF valuation. The DCF valuation method was shown to overestimate the project value accruing to the project owner under the ad valorem royalty and to underestimate the project value distributed to the developer under the other taxation systems. He also showed that the value of the tax claim increases with the level of underlying uncertainty, if the pretax value is kept constant, since the upside potential of the tax stream improves with increasing uncertainty while the tax stream downside is limited. 2.3.3 The influence of management flexibility on value Most projects offer management the opportunity to revise their operating policy in response to the resolution of uncertainty. This ability adds value because it allows management to limit downside losses and to participate in upside gains as uncertainty is resolved. The value influence of management flexibility is complex for several reasons. First, the ability to defer investment decisions10 and their largely irreversible costs create an additional opportunity cost that is not considered in standard incremental cash flow calculations (Pindyck, 1988, 1991; Dixit and Pindyck, 1994; Trigeorgis, 1996). This opportunity cost is the result of forgoing a portfolio of future management options that are associated with maintaining the current project state when the option to change the project's current state (e.g. open to closed; undeveloped to developed) is exercised. Its effect can be seen when the investment decisions, such as temporary project closure, fail to reverse themselves when the underlying causes reverse themselves or when an otherwise attractive projects are not developed. The investment deferral decision is often associated with delaying the initial development of a project. There are other less obvious forms of this decision that include delaying abandonment or temporary project closure. 32 Valuation complexity is also increased by the presence of multiple types of flexibility because management options interact in a non-additive manner. Trigeorgis (1987, 1993) demonstrated that the value of a project with multiple options is not equal to the sum of the underlying project value and the individual incremental option values. Laughton, Frimpong and Whiting (1993) and Kulatilaka (1995) showed that the incremental value of individual management options can be reduced by other options that are a partial substitute for the first option (e.g. reduction in project scale versus temporary closure) or enhanced by other options that are a complement to it (e.g. growth options with downside losses limited by shutdown options). Finally, valuation complexity is promoted by management's ability to affect the resolution of uncertainty. Laughton (1998c) showed that the opportunity cost associated with implementing investment decisions is reduced when such a decision assists the resolution process of a particular uncertainty. In particular, he showed that managers initiated development of a petroleum concession earlier when geological uncertainty is resolved by such development than for an identical concession that has no geological uncertainty associated with it. The economic literature discusses the influences of many examples of management flexibility in generic settings. Many of these models assume that the primary source of project uncertainty is the present value of the cash flows generated by the operating project. These models posit that the cash flow present value follows a lognormal stochastic process and that the project owners receive project cash flows through a dividend process. Such models are interesting because they are able to illustrate important valuation concepts without resorting to sophisticated numerical methods. However, they may be of limited use for actual applications because the assumption of lognormality implies that the present value of project cash flows is never negative. The project will then never produce a negative operating cash flow if a positive dividend process is defined for the project. This type of project characteristic is unrealistic when the project cash flow exhibits a high degree of operating leverage since there is an opportunity to make operating losses. The ability to defer investment was discussed by McDonald and Siegel (1986) when the present value of the underlying cash flows generated by a completed project and the cost of development are both stochastic. They showed that standard investment rule of developing a project as soon as the present value of project cash flows is greater than the development costs can cause a significant reduction in 33 project value. Majd and Pindyck (1987) developed a variation of this model in which project construction takes time and such construction can be temporarily halted at no cost at any time. They also showed that traditional DCF investment signals may lead to sub-optimal investment decisions. Trigeorgis (1986, 1990) developed a model of project expansion in which management has the ability to expand the scale of a natural resource project by paying a fixed expansion cost. Pindyck (1988) described a model of capacity expansion where the price received for project output is dependent upon a stochastic demand function. This model showed that the traditional rule of adding a unit of capacity when its marginal benefit is equal to or greater than its marginal cost must be modified to reflect the management options associated with the new unit of capacity. In this model, the marginal benefit of an additional unit of capacity is equal to the sum of the expected flow of profits from the unit and the option to shutdown the unit should output prices fall. The marginal cost of this unit is its installation cost and the portfolio of options allowing management to install the unit in the future. He and Pindyck (1992) developed two models of capacity expansion of a project producing multiple outputs. In one model, capacity was product-flexible such that each unit of capacity may produce any of the outputs. In the other, capacity was product-specific in that each unit of installed capacity can only be used to produce a particular type of output. Dixit (1995) introduced a model of capacity expansion for which scale economies are gained from expanding over a limited lower range of capacity levels and decreasing returns-to-scale are received when adding capacity at levels above this range. This model showed that, when expansion is triggered, capacity is added in a discrete amount when the initial capacity level is part of the capacity range that exhibits scale economies. Otherwise, capacity was added incrementally according to the capacity expansion model of Pindyck (1988). Temporary closure and project abandonment are two possible strategies for managing downside losses. McDonald and Siegel (1985) produced a model where a project can be temporarily closed at no cost when output price falls below unit operating cost. They showed that the value of each project cash flow is equal to that of a European call option whose expiry date corresponds to the timing of the cash flow. Trigeorgis (1986) described a model where the project can be abandoned for a non-stochastic salvage value. Myers and Majd (1990) discussed an extension to this model where salvage value follows a stochastic process. 34 Real assets may allow management to change the composition of project output as market conditions evolve. Triantis (1988) developed a model of a fixed-capacity manufacturing system that produces multiple products where management may change the output mix at known cost. The general model assumed that the unit operating profit (price less production cost) follows a stochastic process and it is noted that such a model requires sophisticated numerical methods to determine a solution. A closed-form solution to the model is determined by making simplifying assumptions such as unit operating costs are known, the operating life of the manufacturing system is infinite, costless switching is permitted and that each output price follows a lognormal stochastic process with constant volatility and expected return. Triantis and Hodder (1990) demonstrated this model with a two-product manufacturing project of finite life and fixed capacity. Companies may also be confronted with investment decisions involving portfolios of interrelated projects. Childs (1995) developed a valuation model that demonstrates the differences between sequential development and parallel development of multiple projects when uncertainty about the value of an individual project follows a lognormal stochastic process. Project value is shown to increase when investment policy is periodically reviewed and management has the option to alter the composition of active projects. Childs, Ott and Triantis (1998) used this valuation model when the underlying project values are normally distributed. They showed that sequential development is favored when project uncertainties are highly correlated, uncertainties have low variances, development costs are high and expected project values differ significantly. Interestingly, they found that a project with lower expected value may be developed first if it has high variance and its development provides significant learning about both projects. 2.4 MAP applications in the petroleum industry The structure of petroleum projects and the existence of developed financial markets for the project outputs have made these projects the focus of many MAP studies. Project structures often produce non-linear payoffs, due to multiple stages and operating alternatives, that are difficult to assess with standard DCF techniques but may be more easily incorporated within a MAP framework. The financial markets provide information on price movements that can be used to construct reasonable price uncertainty models. The petroleum MAP applications are broadly grouped as exploration lease valuations and the analysis of project design / development / operational phases. 35 2.4.1 Exploration leases Exploration leases are structured such that they allow the lease-holder exclusive rights to explore for petroleum within the lease area. These rights usually exist only for a fixed time period. They may also have additional conditions attached to them such as the specification of a minimum amount of annual exploration activity or terms of renewal. A lease often has an embedded development option that allows the lease-holder to defer the development decision until either exploration activities reveal an economically attractive oil field or the lease expires. Siegel, Smith and Paddock (1987) (a shorter version can be found in Paddock, Siegel and Smith, 1988) provided the first example of a petroleum lease valuation using MAP principles. Uncertainty entered their valuation model through the market value of a developed oil field and the quantity of reserves to be discovered within the lease. The development period was assumed to be instantaneous in order to simplify the valuation calculations. This approach permitted the development period to be collapsed into the exploration phase of the project. They used their model to calculate the value of petroleum leases in the Gulf of Mexico. The MAP lease values were shown to be more compatible with the actual industry bids for the lease than the lease values calculated by the US Geological Survey using DCF methods. Kemna (1993) developed a MAP model to analyze the decision to extend a petroleum lease that is expiring. The option to extend the lease had value in this model because there of the possibility of future oil price increases. This value was compared to the cost of extending the lease (e.g. additional exploration and lease fees, forgone operational revenues) to determine whether the lease should be abandoned, maintained or developed. Other MAP models have combined the exploration phase with the development and operational phases of an oil field. Pickles and Smith (1993) provided a valuation model that includes wet and dry hole uncertainty. They used their model to estimate the maximum amount that should be spent on exploration. Smit (1997) discussed a decision tree approach to pre-production project phases that include exploration and appraisal drilling. Chorn and Carr (1997) created a model that analyses the influence of price and reserve size uncertainty on capacity choice. Reserve size uncertainty was shown to have the greatest impact on the initial decision to defer the capacity choice with their model. Laughton (1998b) demonstrated that, when there is an option to defer exploration, geological uncertainty causes management to initiate exploration activities earlier than when there is no geological uncertainty. Lund (1999) provided an additional study of the influence of geological uncertainty on initial management 36 decisions. His model showed that operator flexibility, such as choice of project capacity, is valuable when there are high levels of reserve uncertainty. Amram and Kulatilaka (1999) discussed the setup of a "green field" valuation model that allows management to choose between alternative exploration methods and incorporates both oil price and geological uncertainty. They presented a strategy space that delineates possible exploration and operational decisions for different combinations of oil prices and estimates of reserve size. 2.4.2 Project design, development and operation The development and operational phases of an oil field incorporate many forms of management flexibility. These include early abandonment, investment deferral, temporary closure, and expansion possibilities. The ability to defer development of an oil field was initially studied by Bjerksund and Ekern (1990). They developed valuation equations, based on the Black and Scholes option pricing model, that could be used to analyze a variety of development deferral situations. These equations could be solved without numerical procedures and they provided break-even oil prices that signaled when to invest. Bjerksund (1991) analyzed the economic cost of a government promise to develop a petroleum reserve by a fixed future date. The economic cost was shown to be the difference between the value of a perpetual development deferral option on the field and the value of the field if it must be developed by the fixed future date. Operational flexibility can have a significant impact on project value. Ekern (1988) developed a two-period model that considers investment in additional capacity for an existing field when there is an opportunity to develop a marginal satellite oil field. Laine (1997) considered a similar project but develops a more detailed model that incorporates abandonment, expansion and investment deferral over a longer time horizon. The expansion option was shown to be most important if the adjacent field was profitable. When the adjacent field was truly marginal, the deferral and abandonment options were most important. Oguztoreli (1996) used a detailed simulation model of primary and secondary pumping operations to study optimal field depletion strategies when management has the option to temporarily closed, abandoned early and to select the time to irrevocably switch from primary to secondary recovery operations. Smith and McCardle (1998) developed a model in which management has the option to drill additional wells in the presence of both price and well production rate uncertainty. They noted that their model can easily extended to include additional types of uncertainty, such as cost and exploration, and temporary closure. Lund (1999) discussed a related valuation model of an offshore oil field. The 37 valuation model includes exploration, well rate and price uncertainty. Management flexibility included early abandonment, investment deferral, and capacity choice (platform and additional wells). 2.5 MAP applications in the mining industry The MAP approach to project valuation was applied first in the mining industry with the Brennan and Schwartz (1985) model of temporary mine closure. Mining projects are similar to investments in the petroleum industry in that primary extraction projects of both industries have distinct stages of development and operation that provide management with alternative responses to the resolution of uncertainty, they are exposed to significant project-specific uncertainty, and they produce output for which there are developed financial markets. Unlike the petroleum industry, mining operations tend to be unique in that there is no underlying production model, such as the "tank" model used for oil reservoir depletion, to describe orebody exploitation over the life of the project. 2.5.1 Exploration and information-gathering Frimpong (1992) used the MAP framework to ascertain the effect of a multiple-stage feasibility study, feasibility study duration, geological (grade) uncertainty and commodity price on project value. An important feature within the model was a link between each stage of the feasibility study and ore reserve uncertainty. By completing the next stage of the study, management could reduce the overall level of geological uncertainty. The model provided management with the ability to perform, defer or forgo any portion of a multiple stage feasibility study, the ability to defer the project investment decision and the ability to abandon the project at any time. The results of the analysis showed that the ability to implement multiple staged feasibility studies adds value to a project in situations of high geological uncertainty and low to medium-high commodity prices. This model allowed management to investigate and design an appropriate feasibility study for the project environment. Laughton, Frimpong and Whiting (1993) investigated how the value of an option to undertake project-level research (e.g. exploration of a mineral deposit) is affected by the research objectives. In their model, the sources of project uncertainty are mineral price and the total mineral content of the deposit. Managers may select research programs that are differentiated in terms of cost, duration and objectives such improved knowledge of effective ore (output) and effective average cost or the effective average cost. Their results showed that project value is created if the objective of the research program was to: 38 1) reduce uncertainty surrounding the effective average cost; 2) increase the effective output; and 3) reduce the effective average cost. Their model suggested that it is possible for resource project managers to design information-gathering programs (e.g. exploration drilling, seismic surveys) such that their costs are justified by the value that they create. 2.5.2 Development and operating options Brennan and Schwartz (1985) developed a mine valuation model in which management can, by incurring an action-specific transition cost, temporarily close an operating mine, re-open a closed mine or irrevocably abandon the whole project. The valuation model consisted of two valuation PDEs; one PDE determined the value of the mine when it was open and the other valued the mine when it was closed. These equations were derived by forming a non-stochastic portfolio that comprised a long position in the mine and a short position in copper future contracts. Their model was significant for two reasons. First, their model provided a value for a mine that was temporarily closed. Second, their model produced an optimal operating policy for the mine that was stated in terms of the current operating status of the mine, the current copper spot price and the remaining mineral reserves. This policy showed at what price the mine should temporarily close if it is currently open and at what price the mine should re-open if it is currently closed. The policy also indicated at what price the mine should be abandoned. Brennan and Schwartz also developed valuation PDEs for mines that required development and mines that were producing under a long-term supply contract. Palm, Pearson, and Read (1986) used the Brennan and Schwartz (1985) model to analyze the effect of cost structure (the ratio of fixed to variable operating costs) on the value of a mine that may be temporarily closed. They found that the value of the temporary closure option was intimately linked to the proportion of operating losses avoided since there may be little advantage in temporarily closing a project with a large component of fixed costs. Palm et al suggested that reducing the ratio of fixed costs to total costs can increase project value. However, their results showed that this strategy has little value influence if fixed costs comprise more than 25% of total operating costs. 39 Cortazar and Casassus (2000) developed a valuation model of a two-stage mining project11 in which one stage represents mining operations and the other represents mineral processing. The unit operating costs were different for each stage (mineral processing costs are higher than mining costs) and either stage could operate without the other being active. This allowed the mining stage to continue operating and feed an intermediate stockpile if the mineral processing section is closed. Conversely, the mineral processing stage may be operated without the mining stage operating as long as the intermediate stockpile contains material. Their model showed that, for a project of this structure, it may be optimal initially to build up intermediate inventories by operating only the mining stage and then, once the mining reserves are low, open the mineral processing stage to run-down inventories and the final mining reserves. Management can vary the quality of material that is sent to the mineral processing plant in response to mineral price fluctuations. The level of ore quality that divides processed material from discarded material is called the cut-off grade (COG). The choice of COG influences the value of the mine since it influences the rate at which ore reserves are depleted. A high COG allows the mine to generate high revenues and deplete the ore reserve quickly but it creates an opportunity cost since much of the mined material will not be processed. A low COG depletes the ore reserves more slowly but generates low revenues since capacity constraints are fully utilized by the bulk of the reserves extracted. Mardones (1991, 1993) provided a model that combines a Monte Carlo simulation using a risk-adjusted copper price distribution with a forward-looking COG model based on a DCF valuation of remaining reserves. His results showed that COG flexibility increases in importance as reserve quality decreases and that its value impact is small. Sagi (forthcoming) provides a MAP model based on a non-linear PDE that identifies the underlying structure of the depletion opportunity cost. His model shows that this opportunity cost resembles a call option, written on the underlying mineral, that expires when the reserves are depleted. Mining projects are heterogeneous assets in the sense that the underlying mineral deposit often comprises a set of distinct ore zones. These zones differ in quality, production rate and the effort required to develop them. The sequence in which combinations of these zones are extracted is determined by the mine design and geological structure. Most mining MAP models have treated a mining project as a homogenous entity in that management is limited to a single operating strategy when producing mineral. This strategy is often the current mine plan that describes the sequence in which the remaining zones are to be mined. Samis and Poulin (1996, 1998) have used decision trees to incorporate the heterogeneity of 1 1 They show how to extend their model to a general n-stage investment model. 40 mineral deposits into the valuation model. Their model considered a two-zone project where management may develop the high-grade portion of the project and then elect to extend production, when the high-grade reserves are depleted, by developing a satellite low-grade zone. Their model showed that the ability to replace exhausted zones with auxiliary reserves can add great value in non-reverting price environments and modest value when mineral price reverts to a long-term median. 2.6 Conclusion The MAP applications that have been described in the literature for economics, the petroleum industry and the mining industry provide a foundation for valuing mining projects with MAP valuation methods. However, each application discussed only incorporates part of the diverse characteristics that mining projects display. For example, He and Pindyck (1992) discussed product-specific capital and multi-product capital within a MAP framework. Their model could be transformed such that product-specific capital represents expenditures required to develop specific areas of a mineral deposit and multi-product capital corresponds to expenditures required for developing facilities shared by all zones (e.g. mineral processing facilities). However, each unit of capital is generic within their model and this contrasts with the program-specific nature of mining capital expenditures. The portfolio of projects approach developed by Childs (1995) and Childs, Ott and Triantis (1998) could also provide a method of dealing with mining complexity. Within their model, each section of the mine could be considered as a separate asset that is part of a project portfolio. Their model ultimately proves to be unsuitable because it details only the development period of each asset and not the asset's operational phase. Finally, the decision-tree methods presented by Samis and Poulin (1996, 1998), Smit (1997), Chorn and Carr (1997), and Laine (1997) could also provide a method of representing mining complexity. Unfortunately, these methods quickly become cumbersome because they require the mine planner to specify all possible operating scenarios. A valuation model is required that ties together these models while remaining practical. Such a model must recognize the multi-zonal nature of a mining project, allow capital to be directed within expenditure programs to either zone-specific or project-wide applications, and recognize both the development and operational phases of zone exploitation. The model presented in Chapter 3 meets these requirements. 41 Chapter 3 The Flexible Discrete Mine Production (FDMP) Valuation Model Multi-zone mine valuation is a complex exercise that must incorporate many possible development and production strategies. Each strategy description includes a (usually unique) specification of both global project characteristics, such as capacity, and individual zone characteristics, such as production rate and grade. Further complexity is added by the contingent interaction of possible management strategies. Mine planners often use scenario-based techniques to simplify the valuation exercise. These techniques require the mine planner to reduce the dynamic nature of a multi-zone project to a single development and production strategy that completely specifies the timing of the costs and benefits associated with zone development, production, and project capacity changes. Essentially, scenario-based techniques achieve simplicity by concealing the characteristics of a multi-zone project within a single development and production strategy. The cost of this approach is reduced project insight: the benefits of flexibility, the critical mineral prices signaling strategy changes, and the probability of specific project events cannot be determined. A simple example of a two zone mining project can illustrate some of these valuation complexities. Consider a project that is comprised of two zones: one producing high-grade area and an undeveloped low-grade area. The mine planner must choose a management strategy from many possible polices that will maximize project value. Each policy describes the development and production cycle of each zone (e.g. production rates and grade; operating and development costs) and the development and operation of overall project infrastructure, such as the mineral processing plant. The policy must also specify the development and operation of the overall project infrastructure such as the mineral processing plant. One decision the mine planner must make is whether to develop the low-grade zone along with the high-grade zone or delay such development until the high-grade zone is exhausted. Early development of the low-grade zone requires additional expenditure to increase project capacity to handle dual zone production. This additional expenditure is offset by the benefits of receiving the low-grade revenues earlier and possible cost savings due to economies-of-scale. Developing the low-grade zone later avoids the up-front expenditure for increased capacity but forgoes the benefits of earlier low-grade zone production. This decision is further complicated by the ability of the mine planner to wait for partial resolution of uncertainty before developing the low-grade zone. 42 3.0.1 Modern asset pricing valuation methods The modern asset pricing (MAP) method is being introduced to the mining industry as an alternative valuation method that is able to handle the dynamic nature of project valuation. However, the structure of published mine models transform a multiple zone mining project into a single zone project. This transformation occurs because the decisions to develop or abandon an individual zone are made exogenously as part of the life-of-mine plan. Only decisions to temporarily close / re-open the project as a whole or to abandon the project irrevocably are made endogenously. These types of MAP models are essentially scenario models because the project's operating parameters are set outside the model in a manner similar to scenario discounted cash flow methods (DCF). Decision-tree methods have been used (Samis and Poulin, 1996; Samis and Poulin, 1998) where the mine planner describes multi-zone mine development and operation with a set of possible project scenarios. Each scenario includes a description of an operating strategy that reflects the mine planner's development strategy, choice of producing zones, cut-off grade, and capacity choice. The dynamic nature of multi-zone project operation is reflected in the decision tree, which specifies the scenario interactions. Unfortunately, the decision tree grows quickly with the number of operational possibilities such that this approach becomes unmanageable. The Flexible Discrete Mine Production (FDMP) project structure model described in this chapter allows the extraction policy of a multi-zone mine to be set endogenously. However, it still requires the mine planner to set an exogenous extraction policy for each zone that specifies the zone's production policy when it is operated. The decisions to develop, temporarily close, re-open a particular zone are made endogenously. In addition, the model allows the planner to specify endogenously the global project decisions of temporarily closing / reopening, irrevocable abandonment, and capacity change. The solution of this model will provide an indication of the importance of allowing the valuation model to endogenously determine the project's overall operating policy. 3.0.2 Underlying market uncertainty For the purposes of this dissertation, macro-economic uncertainty affects project value only through the price received for a unit of mineral production. It is assumed that project costs are known and the riskless interest rate is taken to be known and constant. Price uncertainty is modeled by a single factor log-normal 43 price process (Laughton, 1987; Jacoby and Laughton, 1992; Laughton and Jacoby, 1993) in order to focus on the influence of project structure on value1. Mineral price is assumed to follow the diffusion: dS = OJ + Y < X 2 - / I n S* Sdt + crSdz where: S = current mineral spot price. S* = current long-term price median. a = current short-term growth rate of the price medians. a = short-term price volatility. 7 = rate of mean reversion. dz = a standard Gauss-Weiner process increment. This process is derived from a price model in which changes to each future price expectation are proportional to the current price expectation and a random standard normal variable that reflects the arrival of new information. The model allows new information to have a declining influence over the term structure of price expectations such that the mineral price tends to revert to a long-term equilibrium path. The importance of price reversion is measured by the half-life of the decay process and it is calculated from the reversion rate as: ln(2) Half-life, H=—^ (3.2) The process half-life represents the time at which the impact of a current price shock on future price medians has been reduced by one-half. If the reversion half-life is 3 years, then a current upward price shock of 20% will cause the price median 3 years in the future to increase by only 10%. An asset's market risk-adjusted expected rate of return is equal to the capital (price) appreciation of the asset and the benefit flow (dividend) produced by the asset. In commodity markets, the benefit flow is called a convenience yield and it represents the flow of services that accrue to the holder of a physical commodity but not to the holder of an obligation to buy the commodity in the future (a forward contract). 1 More sophisticated models of price uncertainty can be used with this model such as the multi-factor models described by Schwartz (1997) where the commodity convenience yield and the riskless interest rates are stochastic. 44 Brennan and Schwartz (1985), Brennan (1991) and Lund (1991) suggest that convenience yields exist because owners of the physical commodity may profit from temporary local shortages or from the ability to maintain a production process, in the presence of shortages, with a physical inventory of the commodity. The convenience yield, c, of this process is: 1 7 c = r + P R i s k M i n e r a l o - - a - - o - + y l n S* (3.3) where: r = the riskless interest rate. PRiskM i n e r a l = the price of market risk x correlation between market and mineral price uncertainty The terms r and PRiskM i n e r a l • a of equation 3.3 sum to the total expected return required to induce investors to hold physical quantities of the mineral asset. This return may be estimated within a one-period CAPM or a general equilibrium asset pricing model2. The remaining terms on the right-hand side of equation 3.3 represent the expected price appreciation of the mineral and they correspond to the price appreciation term included in equation 3.1. The current associated variance, median and expected price at time T for equation 3.1 is: Var 0[s(T)] = fi(l-exp[-2rT]) (3.4) Med 0[S(T)] = S* -7- • exp iexp(-yT) (3.5) E 0 [S (T)] = Med 0 [S (T)] • exp[o.5 • Var0 [S (T)]] (3.6) The risk discount factor used to determine the forward price from the expected price is: Risk discount factor = exp PRiskM i n e r a, • o - . ( 1 _ e x p ( _ y T ) ) (3.7) 2 Salahor (1998) details how the standard CAPM expected return model can be transformed into the total return format presented in this dissertation. 45 Derivation of these formulas can be found in Laughton and Jacoby (1993), Salahor (1998) and Chapter 3 of Dixit and Pindyck (1994). 3.1.0 A description of a multi-zone mine within the FDMP project structure model The mining project consists of a heterogeneous mineral deposit that incorporates multiple ore reserve zones indexed by a set of integers n = 1,..., N . Each zone is differentiated by the quantity of reserves it contains, the quality (i.e. grade or mineral concentration) of the zone reserves, an exogenous exploitation plan specified by the mine planner, and the capacity required to process the zone's ore and waste production. The exploitation plan is set with two assumptions. First, the variable production cost of each zone is assumed to be concave or constant. This assumption ensures that the zone is either operating at full capacity or it is not producing3. Second, the cut-off grade is set exogenously for each zone. The project operates by committing to the development and production requirements of each active zone and the temporary closure costs associated with inactive zones over a fixed management interval of magnitude, T M , (e.g. half-year)4. At the beginning of each interval, project management is confronted with a set of operating modes that define the project's level of production and the overall capacity during the next management interval. Production level change occurs through zone closure, re-opening of closed zones or developing and operating a new zone. Such changes are achieved by modifying the combination of active zones. Capacity change occurs by building additional capacity, abandoning a portion of existing capacity, or adjusting the level of closed capacity by either closing a portion of currently open capacity or re-opening a portion of currently closed capacity. The set of possible operating modes may be additionally constrained by the project environment. For example, specific active zone combinations may be precluded by geological structure, mining method, or project infrastructure (e.g. shaft capacity). These constraints are implemented based on the judgment of the mine planner. If variable production costs are strongly convex, then there may be a benefit to varying the production mix from the various project zones in response to changes in the mineral price. An active zone is defined as one in which development or production operations occur during the next management interval. An inactive zone is defined as one in which no such activities occur. Inactive zones may incur temporary closure costs associated with maintaining the zone in a state such that production activities may resume. 46 3.1.1 Zone mine plan. The mine planner specifies an exogenous operating policy for each zone that defines the development and extraction operations for each zone. This policy is called a zone mine plan. A zone mine plan, Z n , is a sequence of R+5 vectors: Z n ={zn,e: e = 0 ,l , . . . ,E„; Z n e = ( q n , e . g n , e . V n , e . d n , e . U n , e ) ; q n , e ' g n . e . V n , e ' d n , e ' U n , e G R+] (3-8) where the index e indicates the zone's stage of combined development and production. Each sequence member, z n e , is an information set detailing the parameters of zone operations. Mineral extraction operations are described by a rate of mining, q n e , the grade (mineral concentration) of the current reserves, gn,e, and a variable operating cost, v n e . Zone development is parameterized by the capital expenditure incurred, d„,e, to prepare the zone for mining operations. An example of such costs would those associated with pre-stripping a new pit but not those required to expand project infrastructure to handle the new pit's production5. The term, u n i e, represents the cost having the zone n inactive at extraction index e. The zone mine plan can often be divided into two segments. The first segment is considered to be developmental and represents the period when the zone is being prepared for production. During this period, qn,e, gn,e, and v n e are likely to be zero and the zone is assumed to have no impact on the project's overall capacity infrastructure.6 The second segment represents a period when production activities occur (qn,e, gn,e, v n e > 0). The boundary between these two regions defines the point in the zone mine plan when the zone production activities require a portion of the project's capacity infrastructure to continue. This boundary is defined as eCapBdy, n and the amount of capacity infrastructure that must be available to handle the zone's production is w 0 C a p n > m i I i e and w O C a p n j i n i l l . The costs are incurred while expanding the project's infrastructure is specified within a capacity change model developed by the mine planner. Activities associated with capacity change include purchasing additional rolling stock to handle increased ore and waste production or expanding mineral processing facilities. Obviously, this can be changed if development activities do have an impact on project capacity infrastructure such as the situation when a new zone is being developed underground and all ore and waste must be removed through a single shaft system. It is assumed in this dissertation that the impact of development activities is subsumed into the cost of zone development. An example of this would be the development of a new open pit zone by outside contractors. If the zone's development period has an impact on project capacity, then there will be a series of different e C apBdy, n that identify regions of different capacity requirements. 47 During a management interval, active zone operations advance the zone from stage e to stage e+1. Development capital expenditures generated by these operations are incurred at the beginning of the interval while mineral extraction cash flows derived from production are received at the end of the management interval8. A variable, Ae, can be defined that is derived from the difference between the zone index at the start of the management interval, e, and the zone index at the end of the interval, e'. This variable is defined: Ae n =e' n -e n (3.9) where: , fen if management does not operate zone n. " 1 en +1 if management operates zone n. e'„ < E„ This variable is an indicator variable in that it assumes a value of one if the zone is active and zero if the zone is inactive or temporarily closed. 3.1.2 Deposit state space. An ordered collection, m = (e,,...,eN), is defined where en corresponds to the extraction index of zone n. The collection, m, is called a deposit zone state because it defines the development and extraction stage of each zone. The set of all possible feasible deposit zone states, M , creates a deposit space. It is defined as: M = {m:m = (e 1,...,eN);e ne{0,...,ED}} (3.10) The total number of possible states within the deposit zone space M is: This convention reflects a common situation where development capital is required before development can begin and mineral extraction cash flows are only received once the production operations have been completed. Its net effect is that operating revenues and costs are subject to additional discounting. It serves to emphasize the structure of the production process. 48 where E n is the final schedule index of zone n. The number of nodes contained within the deposit space lattice can quickly become immense even for a deposit with a modest number of long-lived ore zones. The mine planner can reduce the size of the lattice by identifying regions that are infeasible from a mine design perspective and then imposing the appropriate constraints. An example of such a constraint is being unable to develop one or more zones until another zone has reached a particular stage of development and extraction. Such constraints produce a restricted set of feasible deposit space states M ' c M . 3.1.3 The project capacity model. The capacity state model consists of three types of capacity that are designated open, closed and unfinished. Open capacity refers to the project's current ability to process ore and waste production. Closed capacity indicates the level of capacity that is temporarily closed but which may be re-opened quickly. Unfinished capacity defines the current state of "capacity-under-construction" which may be converted to open capacity when the expansion program is completed9. The project's capacity state may be changed by advancing a capacity construction program to the next construction stage, temporarily closing open capacity, reopening closed capacity or abandoning capacity. Open and closed capacity states Open project capacity is categorized as either open mine capacity, w^e, which reflects the ability to remove in-situ ore and waste, or open mill capacity, W ^ H , which reflects the ability to process ore from mining operations. The project's current open capacity is described by the state, c 0 C a p = ( w ^ , w^,,). The set of possible open capacity states, C O C a p , is assumed to be discrete and its actual members, c 0 C a p , are predetermined by cost efficiency and operational constraints. This set is defined: C0Cap ={COCap :COCap =(Wmine. Wmill); Wmine> Wmill ^  R+] (3.11) Membership is limited with the restriction that each open capacity state must match the open capacity requirements of one of the active zone combinations. Open capacity states that do not adhere to this condition are assumed to be cost inefficient because the benefits of excess capacity are assumed not to 9 Note that implicit in the designation of the state of unfinished capacity is the minimum time at which such capacity is available for productive use. 49 warrant the additional costs incurred by maintaining excess open capacity . A capacity underutilization charge is levied in situations where the capacity requirements of the current active zone combination are less than available open capacity". Closed capacity specifies the amount of capacity that has been temporarily closed so that the costs related to capacity under-utilization are not incurred. The amount of closed capacity is described by the state, c cca P = (ymine> ymiii)' a °d the discrete set of possible closed capacity states as: CcCap = {CCCap : CCCap = (Vrnine> Ymill ) ' Yinine' Vmill e ^ } (3.12) The membership of this set is restricted to those closed capacity states whose levels are equal to the difference between the open capacities requirements of active zone combination pairings. Other amounts of closed capacity are cost inefficient because they create excess open capacity. Positive levels of closed capacity require care-and-maintenance activities and their associated costs in order to maintain closed capacity in a condition such that it may be re-opened. Capacity construction programs and unfinished capacity states. Capacity construction programs allow management to increase ore and waste production in response to price increases. Prior to valuation, the mine planner exogenously specifies one or more programs that are indexed by a set of integers, u = 1,.. .,U . A capacity expansion program is a sequence of /?+ vectors12: X u = {xUj B I : BI = 0,1,..., BIU, F i n a l ; xu, B I = (MineCu, MillC u)} (3.13) where: BI = a integer build index indicating the stage of program, X u . BIU F i n a l = the final integer build index of program, X u . MineCu = the new mine capacity created by completing the program, X u ; MineCu e R+. MillC u = the new mill capacity created by completing the program, X u ; Mil lC u e R+. 1 0 Capacity under-utilization costs are generated because unused open capacity must be maintained in such a state that it can be used at any point during a management interval. An example of such a cost would be the expense of maintaining excess rolling stock in a condition that allows it to be swapped for stock currently in use. 1 1 This charge is dropped when a zone is exhausted and the current open capacity cannot be fully utilized by production from the remaining zones. Excess capacity is assumed to be in place but not maintained in this situation. 1 2 Each element in the sequence can be interpreted as a stage in the capacity expansion program, X u . 50 The expansion programs are differentiated on the basis of the level of project capacity required before starting the program, the program's duration, the cost of advancing to the next stage of construction and the amount of mining (milling) capacity created by the completed program. The progress of a capacity expansion program is described by the state, c U C a p = (BI,,. . . ,BI U), and the set of possible unfinished capacity states is: CuCap = {CUCap = ^UCap = (BI,,..., BI„ ); BIU e 0U, 1,..., BIU, F i n a l } (3.14) The unfinished capacity state set, Cucap. defines the structure of the project's possible expansion program in which each state, Cucap, identifies the capacity expansion program that is active (if any) and the progress that has been reached. Membership to this set is limited by the assumption that only one capacity expansion program can be in progress at any given time. This assumption limits the state set to unfinished capacity states of the form, c U C a p = (BI,,..., BI U , . . . ,BI U ) , where: (0 < BIU < BIU F j n a l if expansion program, X u , is in progress. BI =i (3.15) 0orBI U i F i n a l otherwise. A care-and-maintenance charge is levied when a capacity expansion program is temporarily suspended over the next management interval. This reflects the costs associated with maintaining a partially completed capacity expansion program in a state such that it may be restarted. The project capacity state space. A project capacity state, c = ( w ^ , w^,,, y ^ , y^,,, BI,,..., BLj) , is used to describe the current ability of the project to process production, the amount of capacity that is temporarily closed, and the progress of any expansion program. The set of possible capacity states form the state space: C = {c : c = ( W m i n e , W m i l l , y m i n e , y^n, BI,,..., Blu)} (3.16) The project capacity space is not equal to C 0 C a p x C C C a p x C U C a p because many combinations are infeasible. For example, the capacity expansion programs are dependent on the levels of open and closed 51 capacity when the program is started. It is also assumed, to improve model tractability, that unfinished capacity is converted to open (at no cost) or closed capacity (at some cost)13 immediately upon completion of the expansion program. This assumption allows capacity states in which no expansion programs are in progress: C = (Wmine' Wmill' ymine' ymill'^l'-'-' ^ u ) to be considered equivalent to the capacity state associated with completing the expansion program, X u , but without converting its unfinished capacity to open or closed capacity: CEndX„ = ( W mine, End X„ ' W mill, End X„ > ymine, End X„ ' ymill, End X„ ' ^, BIU, Final'•• •' U ) when the following conditions both hold: w'mine + y'mine = Wmine, End Xu + y mine, End X„ + MineCu w'mill +y'mill = Wmill. End X„ + ymill, End Xu + M i l l C u There may be additional limitations placed on the capacity state space that the mine planner may consider based on pragmatic considerations. Capacity state change Project capacity state change is effected by completing another stage of a capacity expansion program, re-opening closed capacity, closing open capacity or abandoning a portion of the current capacity levels. Capacity state change, Ac, between a starting capacity state, c, and an ending capacity state, c', is defined: Ac = c ' - c = ( A w m i n e , A w ^ , , , A y ^ , A y ^ , , , A B I 0 , . . . , A B I U , . . . , ABlu) (3.18) The capacity produced from a completed expansion program may be temporarily closed instead of being brought into production. The cost associated with this action is assumed to be equal to the cost of closing an equivalent amount of open capacity (see equation 3.30). Temporarily closing recently completed capacity allows management to avoid the costs associated with hiring new employees to operate the new capacity infrastructure. 52 where: w • mine — w = mine the change in open mine capacity. A wmill = W'miu - Wmill = the change in open mill capacity. ^Yinine ymine — ymine — the change in closed mine capacity. Aymill = ymill — ymill = the change in closed mill capacity. A B I U - B I ; ~ B I U = the B I stage change for expansion program, X u Upon completion of a capacity expansion program, the change in open mine (mill) capacity is equal in magnitude to the amount of mine (mill) capacity created by the completion of an expansion program u, less the sum of the change in closed mine (mill) capacity and abandoned closed mine (mill) capacity and less the amount of open mine (mill) capacity abandoned. If a capacity expansion program has not been completed, the change in open mine (mill) capacity is equal to the negative sum of the change in closed mine (mill) capacity and the abandoned closed mine (mill) capacity less the abandoned open mine (mill) capacity. Stated as an equation: Awmjne - - (Ay^g + yABD,mine + WABD,mine) + Aw^n = -(Ay^n + yABD,mill + WABD,mill) + -MineCu i fBI u = BI U i F i n a l . otherwise. (3.19) -MillC u i fBI u = B I u F i n a l . ) otherwise. where: yABD, mine' yABD, mill = m e amount of closed mine and mill capacity abandoned. W A B D mine' w A B D mill= t n e amount of open mine and mill capacity abandoned. Capacity state changes may occur instantaneously or they may require a management interval to effect. Instantaneous capacity changes include the conversion of unfinished capacity to open capacity, the closure of open capacity, the re-opening of closed capacity14 and the abandonment of any type of capacity. The conversion of unfinished capacity to open capacity is considered costless. The other instantaneous capacity changes require payment of a transition cost, that is a function of the type of capacity change and the starting capacity state, to complete. Re-opened capacity or converted capacity could both be modeled as being available at the end of the management period. This would increase the numerical computations required. 53 Capacity abandonment decisions are also subject to an additional restriction that limits the manner in which capacity is abandoned. In this dissertation, capacity abandonment decisions are only valid if they produce an ending capacity state of the form: c H w ' ^ . w V , , , 0, 0, 0,0) (3.20) where: w' • < w v" mine — mine W'mill ^Wmill This restriction states that, when any type of capacity is abandoned, all outstanding balances of unfinished and closed capacity are also abandoned. This reflects the assumption that, in severe operating environments, management is unlikely to incur the cost of maintaining closed or unfinished capacity when a portion of project capacity is abandoned. The only capacity change that requires a management interval to effect is the completion of the next stage in a capacity expansion program. The cost of adding building capacity and the amount of unfinished capacity that can be built over the next management interval is determined by the current stage of the capacity expansion program. Given the time required to build unfinished capacity, the conversion of unfinished capacity to open (or closed) capacity occurs at the end of the management interval in management initiates the final stage of the a expansion program. 3.1.4 Project state space. The project's current zone and capacity state can be described by the arbitrary ordered collection: p = (m, c: m = (e,,..., e N); c = ( w ^ , w^,,, y m i n e , y^,,, BI,,..., BLj)) (3.21) The set of all possible feasible project states, P r , is defined as: P r={p:p = (m, c ) ; m e M r , c e C } (3.22) 54 3.1.5 Operating mode. Over a management interval, the project moves from a starting project state, p = (m, c), to an ending project state, p' = (m', c'). The transition between the two project states is achieved by means of an operating mode, vp. An operating mode is defined as a vector: u„ = p'-p , N (3.23) = (Ae,,..., Ae N , A w ^ , Aw^,,, A y ^ , Ay^,,, ABI,, A B ^ ) where the first N elements of the operating mode vector delineate zone development and production activity and the remaining elements describe the changes in project capacity. The feasible operating modes at project state p form a set, Y p , that is defined: Y p ={u a ;a = l , . . . ,A} (3.24) The composition of this set will vary among project states because there may be restrictions preventing an operating mode from being implemented. In particular, each operating mode requires a minimum level of open capacity to process mineral production from active zones. This requirement is calculated as: _ N | w 0 C a p n m i n e whene„ >e C a p B d y n W« mine — 2-i) r\ i_ n=, 10 whene n<e c B d (3.25) _ N J w 0 C a p n when en > eC a p B d y i n Wu„,mill - ^ 1 n , n = 1 l 0 when en < eC a p B d y n If open capacity, w U p i i n j n e and w U p i i n i l l , is not available at the initial project state p then the operating mode, i) p , cannot be selected for the next management interval. There may be other restrictions, such as exhausted reserves or geological structure, that also limit management's choice of operating mode. 3.1.6 Operating policies. There are many different operating policies that management can adopt to exploit the mineral deposit. Each operating policy, (j>, is a sequence of project states defined as: 55 0 = {p o , - ,p„ . . . ;p , cP' ,t=0,1,. . .} (3.26) These policies are contingent on the mineral price at time t. The duration of the period t between p t and p t + 1 is T m . Note that p, and p t + 1 may be the same project state if the decision at time t was to temporarily close the project. All operating policies are subject to two constraints. First, they all have the same initial project state, p 0 = (0,,. . . ,0 N , w ^ , Wroi,,, y ^ , y^,,, BI,, . . . , B L j ) 1 5 . Second, given a project state, p t , the next mine state in the sequence, p t + 1 , can only be the result of adopting an operating mode, v e T P i . 3.2.0 Valuation of the multi-zone mine within the F D M P project structure model A project conforming to the description of the previous section may be considered a directed graph16 and as such may be valued using a graphing algorithm17 and dynamic programming methods. Given an initial project state and set of possible operating modes, the graphing algorithm generates a decision tree that delineates all possible project operation strategies. Project value is calculated, once the decision tree has been generated, by searching the tree for paths to terminal or fully valued project states. When such a path is found, the project value at the path's end is used to start a dynamic program to calculate project value through the decision tree until another path to either a terminal or a fully valued project state is found. The current project value is determined once all project states in the decision tree have been valued. The project value, H, is a function of the current mineral spot price, S, given a project state p t and an operating mode for the previous management interval, . This relationship is defined: 1 5 Note that a deposit that has not been developed will have no capacity in reality. Development of base project capacity can be incorporated into the development expenditures of an initially active zone or through creation of a "NULL" zone that has no ore reserves. 1 6 A graph is a pair of sets in which the first set of the pair consists of a finite number of elements and the other comprises binary relationships linking the elements of the first set. The multi-zone mining project may be represented as a graph because the elements of the project state space, Pr, form the first set and the set of operating modes allow the binary relationships between the project states to be generated. The project is a directed graph because the project states are ordered since zone reserves once mined cannot be replaced. See Chapter 5 - Section 4 Cormen, Leiserson and Rivest (1997) for a detailed discussion of directed graphs. 1 7 Graphing algorithms are discussed in Chapter 23 of Cormen, Leiserson and Rivest (1997). H s s H t ( S t ; p t , vt_,) (3.27) 56 This relationship requires that the operating state of the previous management interval be specified because there may be path-dependent costs, such as those associated with capacity expansion or hiring more employees, associated with moving from one project state to another. These costs also induce economic hysteresis into the project valuation model such that operating decisions may not be reversed when the underlying causes have been fully reversed. For example, a multi-zone mine may close its low-grade zones in response to low-mineral prices but may not re-open these zones when prices recover to their previous levels if there are substantial re-opening costs. Instead, management may elect to exhaust the high-grade zones before deciding whether to mine the low-grade zones. At each node within the decision tree, management is confronted with a set of alternative operating modes, one of which must be chosen for the next management interval. The project valuation function, H, (S,; p t , t),_,), can be calculated with the dynamic program: H , (S t; p t , u t_,) = max (it (p,, u t_,, u,) + V (S,, T = T m ; p,, u a)) (3.28) where: *(Pt . V i - «.) = - C A P E X m i U a -CapChge C | i U , - P r o d C h g e ( p „ i>,_„ u a ) - T . C ( v , , «.) Capital expenditure, C A P E X P i U a , consists of costs incurred from active zone development at project state, p t , while in operating mode, u a . Active zone development costs are the sum of the development costs specified in the active zone mine plan. This is calculated as:. N f d n . when Ae = 1. C A P E X . v =2\ (3-29) p " U a i S l O whenAe = 0. Capacity change costs, CapChgep U a , are specified within the capacity change model. There are five types of capacity change in this dissertation for which each has its own cost function18. These cost functions are defined: Mining literature provides guidelines for cost functions related to building new capacity such as the 6/10ths rule where the cost of adding unit capacity decreases with amount of capacity being built. This rule and other capacity cost curves are discussed in O'Hara (1980) and Mular and Poulin (1998). In this dissertation, the cost of building additional capacity follows this convention when there is a pre-existing base of installed capacity. There appear to be no such comparable guidelines in the literature for the other types of capacity change other than anecdotal data. 57 CapChgePi „a /BUILD ( C ) /CLOSE (C, A W CLU b V~' mine' ^Wraill ) REOPEN (C- Ay mine, Ay,,*,, ) /BUILD (') + /CLOSE (") /ABD ( c-Aw mine, Aw mi,,, Ay: (3.30) mine' Ay mill) The cost function, / B U I L D (•), specifies the cost building unfinished capacity and it is dependent upon the current stage of the expansion program in progress. The cost function, / C L 0 S E (')> determines the cost of temporarily closing currently open capacity and it is a function of the amount of open capacity being temporarily closed. The cost of reopening an amount of closed capacity is determined by the function, /REOPEN ('), a °d it is determined by the magnitude of capacity being reopened. The cost function, /BUILD (') + /CLOSE 0 ' determines the cost completing a capacity expansion program and then temporarily closing a portion of the open capacity base. This action may occur if management elects to avoid the costs associated with opening the unfinished capacity such as hiring additional employees. Finally, the cost function, / A B D ( ' ) > calculates the cost of abandoning a project capacity and it is a function of the amount of each type of capacity that is abandoned. Production change costs, ProdChge(pt, ut_,, t) a), are incurred when mine and mill production fluctuates between the previous period and the current period. This cost is a function of the production change magnitude and it is intended to reflect retrenchment or hiring costs associated with large changes in production levels. Implicit in this term is the assumption that the mine does not retain any unnecessary production personnel. The final term, T.C.(ut_,, i>a), is a transition cost that represents the cost of changing the combination of active zones. Examples of such costs are those incurred in zones that have been exhausted or are being temporarily closed when equipment (e.g. winches or pumps) is removed and barriers built to prevent unauthorized entry. These cost are assumed to dependent only on the change in operating mode. They can be made dependent on the deposit state space if required19. For example, it could be less costly to change into an operating mode when some of the active zones are still being developed than when all the active zones are producing mineral. 58 The term, V(S t , T = T m ; p t, u a ) , is the project value in state, p,, and operating mode, u a , immediately after incurring the costs of zone development, capacity change, production change and operating mode transition. Investors are assumed to be able to hedge the uncertainty associated with the mineral price in the financial markets. This assumption allows a partial differential equation (PDE) to be derived by arbitrage valuation arguments that values the project given the associated mineral price uncertainty (for derivation of equation (3.31) see Brennan and Schwartz (1985), Dixit and Pindyck (1994), or Wilmot, Dewynne and Howison (1993, 1995)). The actual form of the PDE depends on whether the project cash flow is generated discretely or continuously. Note that T is a continuous-time variable over the interval 0 < T < T • i < T 2 S 2 V s s + ( r -c)SV s - V T - rV =0 (discrete cash flow) ^ • o - 2 S 2 V s s + ( r - c ) S V s - V T - r V + C.F. p i U a =0 (continuous cash flow) where: S = the current mineral spot price. a = the short term standard deviation of the mineral spot price. C.F.p „ = the instantaneous cash flow generated at project state p, in operating state, v,. c = a convenience yield as defined in equation 3.2. r = the riskless interest rate. (3.31) The valuation partial differential equation is subject to several boundary conditions. The terminal boundary condition is dependent on the nature of cash flow generation and assumes either of the following forms: V ( S , T = 0;p„ u a) = H t + 1 (S;p, + 1 , u,=u,) + C.F.(S) „ (discrete cash flow) p " a (3.32) V(S, r = 0;p t, u a ) = H t + 1 (S; p t + i , u, =u a ) (continuous cash flow) Equation (3.32) states that the project's value at the end of the current management interval is the value of the project at project state p t + 1 given the current operating mode u , . Discrete cash flows are incorporated into the terminal boundary condition to conform to the observation that productive activity 59 must occur before a (possibly unprofitable) cash flow can be generated. Note that the project's final value when all the zone schedules are complete is the cost of abandonment. B A B D p Identifying lower boundary conditions for the P.D.E. in equation (3.31) requires that the cost of project abandonment, B A B D , be calculated. This cost is defined: B A B D , P T = SiteClosure + CapABDC t + ProdChge(pt, u a , u A B D ) + T.C.(u a, u A B D ) (3.33) where: SiteClosure = the cleanup cost incurred on closure. CapABDC i = the cost of abandoning processing capacity at capacity state c t. ProdChge(pt, ua> u A B D ) = the production change costs incurred during project abandonment. T.C. (u a, t)A B D) = the cost of reducing the number of active zones to zero. The lower boundary condition is dependent upon whether the project may be abandoned continuously or only discretely at the end of a management interval. Discrete abandonment requires a two-part lower boundary condition that specifies project value during the management interval (T > 0), when the project cannot be abandoned, and at the end of the management interval (T = 0), when the project can be abandoned. During the management interval, the lower price boundary for project operations is zero since the project must be operated until the end of the period. This is an absorbing barrier for the PDE because the mineral price can never rise, once the mineral price ever falls to zero20. Given this behavior, the lower boundary condition for the valuation PDE is: I - ( B a b d +C.F.(S = 0, T = 0) )<TrT (discrete cash flow) V(S = 0 , T > 0 ; p t , u a)= V " P , ' " a / (3.34) - ( B A B D , P l ) e _ r T - j 0 r C - F - ( S = 0 ' t )p „ e " r ' a t (continuous cash flow) where: TT = T , the time remaining in the current management interval; 0 < TT < Tm. When mineral price is zero, the price increment dS of equation 3.1 is also equal to zero. This implies that the mineral price never changes and hence remains at zero. 60 This boundary condition says that the project value, when the mineral price is zero, is equal to the present value sum of the remaining interval operating cash flows and the abandonment costs. These costs are discounted at the riskless rate because they are considered known. At the end of the management interval, the lower boundary condition becomes due to discrete abandonment: V ( S < S A B D , T = 0;p„ u a ) : "BABD.P, - C R ( S ) P L , U A 'ABD, p, (discrete cash flow) (continuous cash flow) (3.35) The mineral price, S A B D , is defined as the price at which the project value is equal to the abandonment costs. As an equation: H, + , (S = S A B D ; p t + 1 , u, = u a) = - B A B D (3.36) The fixed-time interval restriction for operating decisions can be relaxed such that the decision to abandon the project irrevocably is continuous. This exception reflects the intuition that, in poor economic environments, management would consider the option to abandon at very frequent intervals. Continuous abandonment creates a free price boundary that is demarcated by a critical abandonment price, SABD- The solution of the valuation PDE must meet two conditions at this boundary. First, the operating project's value must be equal to the abandoned project's value. When cash flow's are continuous, the project's value at this boundary is the abandonment cost since the project has no value once it is abandoned. When cash flows are discrete, the abandonment value is supplemented with the value of production generated between the start of the management interval and the abandonment time21. The project value at the free boundary for both types of cash flow is: This abandonment boundary formulation assumes that production is generated at a constant rate over the management interval. All mineral produced in the current management interval prior to abandonment is sold at the prevailing market price and all costs incurred producing this mineral are due. Note that management would always abandon the project during a management interval when the cash flow term is not part of this boundary condition since negative cash flow outcomes for the current management interval could be avoided. 61 V ( S A B D , T ; p„ u a ) = - B A B D i P t +G^l.C.F. p i,„ a V ( S A B D , T ; p t , u a ) = - B A B D , p t 0 < T < T M (discrete cash flow) (3.37) 0 < T < T „ (continuous cash flow) The second condition requires the derivative, with respect to price, of the project value function to be continuous at the abandonment boundary22. The price derivative of the project value function is: (T - T ) 3 ( C . F . d „ (S)) V s ( S A B D , T ; p t , u,) = l ^ _ i . _ J S h ^ J l 0<T<T m (discrete cash flow) T m dS (3.38) v s ( S A B D . T;p„ u a) = 0 0 < T < T „ (continuous cash flow) Note that this condition is not required for transitions between non-abandonment operating modes because these transitions are only considered on a discrete basis. The project cash flow is calculated using the parameters for project state, pt, and operating mode, v,, over the next management interval. The cash flow is calculated from the equation: C . F . ^ =EOSBftm i U a -MCapC C i , U a -UUCapC p p U a + £ (gn,m, ST - v n m_ ) q n m i if Aen U( = 1. -u n,mt i f A e ^ O . (3.39) where: EOSBft m„u, MCapCCi)„a UUCapCP t,U a Sn, m. = economy-of-scale benefit associated with deposit state m t and operating mode u a . = cost of maintaining the same level of capacity at capacity state c t and operating mode t)a. = cost of having under-utilized open capacity at project state p t and operating mode u a . fl if zone n is active in operating mode u a . [0 if zone n is not active in operating mode u a . = the grade of zone n at deposit state m t . 2 2 This condition requires the value function derivative with respect to price, when the project is abandoned, to equal the value function's price derivative when the project is operating. This condition may be justified using arbitrage arguments (Sick, 1989; Dixit, 1993). 62 v n = the variable cost of zone n at deposit state m t . q n m = the production rate of zone n at deposit state m,. un_ = the cost of maintaining zone n in a temporarily closed state at deposit state m t . The capacity costs, fixed costs and production parameters (except grade) are instantaneous rates when cash flows are generated continuously and they are cumulative amounts when the cash flows are discrete. The cash flow component, EOSBftm U a , is used to introduce economies of scale to the cash flow equation. The mine planner can vary this term such that a operating mode with twice the production of an alternative operating mode may have an instantaneous fixed cost that is less than twice that of the alternative operating mode. This term is dependent on the deposit state, mt, because it is assumed that operating economies of scale are only gained from the combined production of producing zones. If any zones of operating mode, u a , are being developed at deposit state, m,, the economy of scale benefit is either reduced to reflect lower production or forgone altogether. The term, MCapCCt „ a , represents the costs associated with the project's capacity stock. These costs may be generated by having a positive level of closed capacity or by having positive unfinished capacity levels when a capacity construction program has been temporarily halted. Positive levels of closed capacity generate costs from maintaining unused equipment and production facilities, such as a pit crusher and conveyor belt system, in a viable state. The costs associated with temporarily stopping a construction program result from efforts to retain the building site, such as a shaft-deepening project or a mill extension, in a reasonable state. These costs provide management with an incentive, in periods of low mineral prices, to either continue capacity construction once such a program has begun or to abandon it. This cost is assumed to have a magnitude of zero whenever there is no closed capacity and either the unfinished capacity balance is zero or a capacity construction program is in progress (i.e. Ax^e , AX^,, >0). The term, UUCapCp , defines the cost associated with under-utilizing current open capacity. An example of such a cost would be maintaining a fully staffed shaft complex when mine production only justifies a portion of the personnel. This cost is dependent upon both the actual amount of open capacity and the active zones of operating mode v3 that are producing mineral. As mentioned in section 3.1.1, 63 active zones that are being developed do not contribute to production that must be handled by open capacity. 3.3.0 Grade (geological) uncertainty Zone grade is another potential source of project uncertainty. Exploration prior to project development provides some knowledge of each zone's actual grade but does not fully resolve this uncertainty. Additional information regarding the grade of a particular zone may be obtained by extraction operations in other zones. However, zone grade uncertainty is usually not fully resolved until extraction operations begin in the zone2 3. Grade uncertainty is introduced into the valuation model by defining a set of possible grade multiplier outcomes for each zone: K={^n,i- i = l , 2 , . . . , I n ; KnAeR+] where: E [ K n ] = 1 (3.40) Grade multipliers, Ka ,, are values that represent a revision in the zone grade due to the resolution of geological uncertainty. For example, a grade multiplier of 0.9 indicates that the grades specified in the zone mine plan are 10% lower than stated. Zone grade uncertainty is resolved all at once when a particular zone extraction index, en, G R , called a grade resolution boundary, has been reached24. A discrete marginal probability distribution, P K n ( K n = Kn ^, is specified for each zone exhibiting grade uncertainty by the mine planner. The expected value of this distribution must equal one because the grade values presented in the zone mine plan are considered expected grades in the presence of geological uncertainty. In addition, geological uncertainty associated with a particular zone is considered independent of the geological uncertainty displayed by the other zones. This allows the grade uncertainty displayed by the full project to be characterized by the discrete joint probability distribution: P ( K 1 = J C 1 > i , . . . , K N > i = v N i i ) = n P ( K n = K n i i ) (3.41) n=l 2 3 Grade uncertainty is often not fully resolved even at this point. This assumption is made in order to simplify the valuation model. 2 4 Note that the grade resolution boundary is not restricted to the commencement of mineral extraction activities. This boundary could be before this point if the development capital expenditure includes an exploration component that resolves grade uncertainty early. 64 A model of the overall project grade uncertainty can be developed such that the resolution of one zone's grade uncertainty is correlated with the uncertainty associated with another zone. This would require additional effort (possibly substantial) by the mine planner to specify a unique joint probability distribution. Specification of such a joint probability distribution is an area of future research. Grade uncertainty states. The current estimation or realization of the various zone grades for each project state, p, is summarized by a grade state. A grade state is an arbitrary ordered collection of zone grade multipliers, f = (/,,..., y N ) , where yn is either the expected grade multiplier or a realization of the grade multiplier for zone n. The set of all possible grade multipliers is defined as: F P = i f p : f p = ( ^ ' - - - ' ^ N ) ; 7 „ = 1 if e < e n, GR | K„,i i f e n > e n i G R | (3.42) The number of possible realized grade states associated with a particular project state, p, is: N j l n i f e n >e n G R . ! J i l i fe n <e n i G R . 3.3.1 Grade uncertainty and valuation. Geological uncertainty alters the formulation of the zone mine plan and the valuation equations. Zone grade multipliers are incorporated into the zone mine plan such that: Z„ ={zn,e:e = 0, 1,..., E n ; v + i ( 3 - 4 3 ) ^n, e — (fn, e'/n, e * Sn, e> V n, e'd n, e'Un, e J' fn, e> Ya. e'Sn, e> V n, e >°-n, e' Un, e ^ ^ J where: Yn.e=\ 1 i f e „ < e n G R K„,i i f e „ > e n , G R 65 With geological uncertainty, the project value function, H, requires an additional argument, fp, that represents the project grade state. The project value function is redefined: H s H ^ S , ; ? , , ^ , ^ . , ) (3.44) The dynamic program that is solved at each project state is reformulated as: H t(s t;p„f P i, u t_,)=max(;r(p t, u a , ut_,) + E[v(S t, T = T m ; p t, fpi, ua)]) (3.45) Within this dynamic program, geological uncertainty is treated as a diversifiable (unsystematic) risk since it is uncorrelated with market uncertainty25. Its impact on value is determined by calculating the value expectation with respect to the joint grade probability distribution specified in. equation (3.41). This calculation incorporates the impact of zone grade uncertainty that has been previously resolved, that is being resolved over the next management interval, and that will be resolved in the future. Note that the components of n(pt, vt, t>t_,) are independent of the grade state and do not require an expectation calculation. The structure and boundary conditions of v(st, T ; p t, fp , u a) are the same as those in equations (3.31) to (3.38) except that there is an additional grade state argument. The cash flow term defined in equation (3.39) is redefined: C.F . P i , f p , U a =EOSBft m i U a -MCapC c_ -UUCapC P t , U a f(/„, fP(, m,gn, m, ST - v n , m i )q n m i if Aen_ v< = 1. (3.46) N + 1 n=l if Ae„ .. =0. where: y n f P i = the zone grade multiplier for zone n at project state p t and grade state fp 2 5 This assumes that the resolution of zone grade uncertainty does not affect mineral prices. A risk-adjusted grade distribution would be used if the grade distribution includes outcomes that would allow the project managers to unduly influence the mineral price. 66 3.4.0 Limitations of the F D M P project structure model The FDMP project structure model presented here has two important limitations. First, the model requires a project description that exogenously sets the development and production profile of each zone. This requirement improves on current real options and DCF valuation models because current models set such zone profiles at a mine level. However, mining projects are managed such that a zone's development and production profile may change while the zone is depleted. The approach used here is a valid first attempt at developing a more dynamic project model since there are no set rules or production models that can describe the depletion of a mineral deposit26. Removing this limitation will require a significant amount of research. The FDMP model will also encounter important computational and data handling problems when more than several zones are mined simultaneously. Geological structure may reduce this problem in some instances because of zone restrictions. The computational burden may also be alleviated by the development of efficient code (e.g. problem specific algorithms) and the use of sophisticated computer facilities (e.g. multi-processors). However, even when the computational burden is manageable, the FDMP model has the potential to generate such a large amount of information that an efficient method is required to search for and compile the relevant data needed to make an investment decision. An effort is required to find value approximations within the model because of the data handling and computational issues associated with multi-block problems. 3.5.0 Extensions to the F D M P project structure model There are several extensions to the FDMP model that would improve its reflection of industry operating environments. First, a more detailed capacity change model could be developed. The capacity model presented here is used because it represents a reasonable approximation of a realistic capacity expansion program while maintaining numerical tractability. This model could be extended to include competing capacity expansion programs that require the specification of unfinished capacity in stages. For example, management may have to make a mutually exclusive choice between a capacity construction program that takes two years to complete and another that takes three years. An oil field is one example of a natural resource project where there are rough rules or production models describing the resource depletion. Tank models and other descriptions exist that model the reduction in well-head pressure as the oil field is depleted. Comparable models for the mining industry do not exist. 67 Another extension to the FDMP model would allow a decision tree of management operating strategy sets to be superimposed on the current model's structure. A decision tree representing choices between different sets of operating strategies could provide a method of introducing dynamic zone operating schedules or allowing the set of project zones to change. For example, such a decision tree could be used to allow management to abandon a zone irrevocably and avoid the costs associated with maintaining it in a temporarily closed state. A more detailed geological uncertainty model is almost certainly required. Grade information gained regarding one zone may often help to resolve grade uncertainty in a neighboring zone. This is especially important when neighboring zones may be considered geologically as one. In this case, mining operations in one zone may partially resolve grade uncertainty in the inactive zone through grade estimation techniques such as geostatistics. Project-specific information gathering activities have been shown to have a large impact on value in other valuation problems and it is expected that the resolution of geological uncertainty is no different. This model could be extended to allow exploration activities to be incorporated. 3.6.0. Conclusion The multi-zone mine model presented here provides a method of incorporating the heterogeneous nature of mineral deposits and its influence on mine design. The models structure allows global project operating policy, cash flow components and transition costs to be determined endogenously on a project state basis as opposed to being set exogenously for specific scenarios as is common in current real options valuations. Its limitations include pre-set development and production profiles for individual zones and potentially large computational and data handling issues when more than several zones are considered. These limitations can be somewhat alleviated by more sophisticated computer facilities and algorithms and model extensions such as decision trees outlining alternate sets of zone profiles. The model can also be extended to reflect more complex forms of geological uncertainty and exploration activities. 68 Chapter 4 Valuation of Production Replacement and/or Production Expansion Strategies for a Two-Zone Mine A generic mining project is described and then assessed using the proposed FDMP project structure model. The results of the exercise are compared to results obtained by analysing SPP project structure models with the DCF and MAP valuation methods. The comparison illustrates the differences between the various methods and suggests situations when a SPP project structure model can be used in place of the more complicated FDMP model. 4.1.0 Project overview An open pit mine has its operations currently focused on a single high-grade mineral deposit called the HG Zone. Recent exploration has revealed the existence of a large satellite zone (called the LG Zone) whose grade is significantly lower than the reserves in the HG zone. This discovery is considered potentially valuable but it is unclear how development of the satellite deposit should proceed. Management recognizes that there are competing development strategies and that each strategy has its own distinguishing characteristics in the presence of uncertainty. They also realize that their policy to meet every half-year (i.e. T m = 0.5 years) to consider major operating choices allows them to revise previously made decisions in response to new information. 4.1.1 Economic environment The real long-term riskfree interest rate is 3.0% and its term structure is assumed to be flat. There is no inflation. The mineral price follows a log-normal diffusion as specified in Equation 3.1. The project is assessed using both a non-reverting (NREV) and a reverting (REV) diffusion process. The parameters for each process are outlined in Table 4.1. The price medians growth rate has been set to zero for both processes to insure that the price medians of both price processes are time-independent and equal to zero throughout the price term structure1. 1 Note that the price medians of both processes are equal $1.00 per mineral unit over the price term structure. However, expected prices can vary significantly because of the relationship between median and expected prices expressed in equation 3.6. 69 1 General economic Risk-free interest rate (%) 3.0 Interest rate term structure Flat Price process Nori-reverting (NREV) Reverting (REV) Current long-term price median ($/unit mineral) 1.000 1.000 Price median growth rate (%) 0.000 0.000 Short-term price volatility (%) 25.000 25.000 Mean reversion factor N/A 0.231 Price of market risk 0.500 0.500 Mineral and market uncertainty correlation 0.500 0.500 Price of mineral risk 0.250 0.250 Table 4.1 Economic environment parameters. The primary difference between the two processes is the manner in which the magnitude of price uncertainty (i.e. the associated price variance as determined by equation 3.4) increases over time. Uncertainty grows at a constant rate over time when price follows a NREV process and it grows at a decreasing rate when it follows a REV process. Using the parameters of Table 4.1, the impact of this difference can be seen in Figures 4.1a and 4.1b2. When price follows a non-reverting process, the constant increase in uncertainty produces an expected mineral price profile, forward (risk-adjusted) price profile and 90% confidence boundaries3 that change through time at a constant rate. These profiles and boundaries increase at a decreasing rate through time when price follows a reverting process. 4.1.2 Deposit description The project is exploiting a disseminated porphyry mineral deposit where both ore zones lie close to the surface. The stripping ratio for both zones is 2:1. The HG Zone contains 14.175 million tonnes of ore with an average grade of 0.90% mineral. The low-grade zone contains 12.600 million tonnes of ore with an average grade of 0.60% mineral. 2 The price profiles were produced using equations 3.3 to 3.7. 3 The 90% confidence boundaries indicate the price range, at a particular point in the term structure, through which 90% of the possible price paths pass. 70 0.0 1.0 2.0 3.0 4.0 5.0 Year 6.0 7.0 8.0 9.0 10.0 • Expected mineral price Forward price Upper / lower 90% confidence boundary Figure 4.1a N R E V price process profiles (Current spot price = $1.00 / mineral unit). 2.00 1.80 1.60 1.40 H 1.20 .00 0.80 0.60 0.40 0.20 •c D. 0.00 0.0 1.0 2.0 3.0 4.0 5.0 Year 6.0 7.0 8.0 9.0 10.0 •Expected mineral price Forward price Upper / lower 90% confidence boundary Figure 4.1b R E V price process profiles (Current spot price = $ 1.00 / mineral unit). 71 Zone index, e H G o r L G 0 1 2 3-18 Zone time (year) 0 0.5 1.0 1.5-9.0 HG Zone parameters Mill feed ( q H G e , million tonne/period) 0.788 0.788 0.788 Grade (g H G e , % mineral) 0.900 0.900 0.900 Mineral production (million units) 15.611 15.611 15.611 Operating cost ($ million) 9.353 9.353 9.353 (vHG,e - $/tonne) 11.869 11.869 11.869 ($/mineral unit) 0.599 0.599 0.599 HG zone development d H G e $ million) Low-grade zone parameters Mill feed ( q ^ , million tonne/period) 0.788 Grade (g L G e , % mineral) 0.600 Mineral production (million units) 10.407 Operating cost ($ million) 9.353 ( v LG, e > $/tonne) 11.869 ($/mineral unit) 0.899 j LG zone development (d L G e $ million) 7.500 7.500 7.776 Table 4.2 Production parameters and costs for the high-grade and low-grade zones. 4.1.3 Current HG Zone production plan Current mining capacity is 3.15 million tonnes (9000 tonnes per day for 350 days per year) of ore and waste per year. The mill complex can process 1.575 million tonnes (4500 tonnes per day) of ore annually. Annual operating costs are $18,706 million per year of which 57.1% are attributed to labor charges4. Project closure costs are expected to include a charge of $40.0 million to cleanup the mine site, a charge of $1.5 million for abandoning project capacity infrastructure, and a charge of $3,204 million (30% of annual personnel costs) for retrenching (laying-off) mine personnel. Table 4.2 summarizes the current production parameters for the HG Zone. Table 4.3a5 details the cash flow generated by the high-4 Operating costs were calculated using the open pit cost curves outlined in O'Hara (1980) and reproduced in Mular and Poulin (1998). See Appendix 1 for details of the development and operating cost calculation. 5 Note that the rows labeled expected operating revenue (EOR) and expected operating cash flow (EOCF) pertain only to the discounted cash flow value calculation. In this calculation, EOR is the product of expected price and 72 grade zone production plan when price follows a NREV price process. The high-grade zone production plan cash flow, when the price follows a REV process, is detailed in Table A1.5a. 4.1.4 LG Zone development proposals There are two competing development strategies for the LG Zone. The first strategy initiates low-grade reserve development such that ore is produced simultaneously from both the high-grade and low-grade zones for at least part of the project time horizon. This strategy considers the low-grade reserves as a source of expansion production for the project. The second strategy considers the low-grade zone as replacement reserves for the high-grade zone6. Essentially, low-grade development in the second strategy is structured such that production from the low-grade zone is available immediately after the exhaustion of the high-grade reserves. The low-grade zone production plan and its development requirements (e.g. pre-stripping, access roads) are common to both strategies. Table 4.2 outlines the timing and magnitude of the L G Zone production parameters and development costs that are common to both development strategies. Strategy 1: Production expansion development of the LG Zone The low-grade zone can be developed at any time prior to the near exhaustion of the high-grade reserves. Capital expenditure is incurred to both develop the low-grade zone and to expand the project's capacity infrastructure to handle the additional ore and waste production from the low-grade zone. For this project, 0.788 million tonnes of mine capacity and 0.394 million tonnes of mill capacity can be added during a production period at a cost of $12,793 million. Two periods (a total time of 1 year) are required the quantity of mineral production. EOCF is the difference between EOR and project costs (operating, capital and closure) that has been adjusted for economies-of-scale benefits. The discounted cash flow NPV is the sum of EOCFs that have been discounted for time and risk at 10%. The rows labeled RA (i.e. risk adjusted) operating revenue (RAOR), risk discounted operating profit (RDOP), and risk discounted net cash flow (RDNCF) apply only to the MAP value calculation. RAOR is the product of quantity of mineral produced and the mineral forward price. RDOP is the difference between RDOR and operating costs with an adjustment for economies-of-scale benefits. RDNCF is equal to the RDOR less any capital and closure costs. 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In addition, a transition cost of $0.2 million is incurred when switching to the combined high-grade and low-grade zones operating mode and an additional hiring charge of $1,282 million is incurred for expanding the workforce when production starts in the low-grade zone. These additional expansion costs are offset by the combined benefit of receiving low-grade production earlier and the economies-of-scale benefit generated by doubling project production. In this example, the economies-of-scale benefits are 17.5% so that the operating costs associated with combined high-grade and low-grade zone production is $30,865 million per annum. The site cleanup cost remains $40.0 million. However, total employee retrenchment costs and total capacity shut-down costs are now $5,127 million and $2.5 million respectively since additional mineral processing infrastructure and personnel were needed to handle the increased production. Note that the employee retrenchment costs may be incurred at different project times if the two ore zones are exhausted on different dates. Table 4.3b and Table A1.5b detail the current expected cash flows (no flexibility) associated with the expansion development strategy. The cash flow components and transition costs associated with production expansion are detailed in Tables 4.4, 4.5a, 4.5b and 4.6. Strategy 2: Production replacement development of the LG Zone Exhaustion of the high-grade reserves and the initiation of low-grade zone production are perfectly synchronized when development of the LG Zone starts with one year of high-grade reserves remaining. If low-grade development is started at this time, the project labor force and capacity infrastructure (i.e. mine rolling stock, mill grinding and floatation capacity) can be transferred to the task of exploiting the low-grade reserves by paying only minimal transition costs totaling $0.4 million (see Table 4.3c for the timing of these costs)8. Capital expenditure is incurred to develop the low-grade zone for mining operations (as outlined by Table 4.2c) but no outlays are required to expand the project's capacity infrastructure. In addition, the project closure costs incurred for site cleanup ($40.0 million), capacity The capacity expansion program incorporates two stages. Advancing the expansion program from stage 0 to stage 1 involves costs of $12,793 million and completing the program by moving from stage 1 to stage 2 (completion) requires an additional $12,793 million. If development of the LG zone starts after this time, the project incurs additional costs for retrenching and then re-hiring employees. This is a due to a "no-production" period created by exhausting the HG zone reserves before the LG zone is ready for production. This is considered an acceptable approximation of management behavior because the cyclical nature of mining operations has made flexible personnel policies acceptable to both management and labor. 77 Cash flow component description Mill capacity (million tonnes / period) CF cost or benefit9 ($ million) Open Closed Unfinished Underutilized capacity costs: UUCapC(c = (1.575, 0, 0); u a = u L G ; e L G = 0,l) with unused open capacity of 1.575 million tonne. 1.575 0 0 1.500 UUCapC(c = (1.575, 0,0); u a = u H G , u L G ) with unused open capacity of 0.788 million tonne. 1.575 0 0 1.200 UUCapC(c = (0.788, 0, BI = 0,l); u a = u L G ; e L G =0,l) with unused open capacity of 0.788 million tonne. 0.788 0 0 1.200 Capacity maintenance costs: MCap(c = (0.788, 0, l ) ; u a = u H G , u L G ) MCap(c = (0.788, 0.788, 0); u a = u H G , u L G ) 0.788 0.788 0 0.788 1 0 1.000 0.200 Economies of scale (no capacity under-utilization): EOSBft(e H G =0,...,18;e L G =2,...,18; u a = H G + L G ) EOSBft(e H G =0,...,18;e L G =0,...,18; u a = HG, L G ) 1.575 0.788 0 0 0 0 3.274 0.000 Temporarily closed zone: u(e H G =0,...,18; e L G = 2,...,18; u a = u H G , u L G ) A l l A l l A l l 0.200 Table 4.4 Non-production cash flow components. Cash flow costs or benefits that accrue over a management period. Economies-of-scale are considered cash flow benefits and have a positive effect on cash flow by reducing overall operating costs. The other cost items are cash flow charges and have a negative effect on cash flow. 78 Open mill capacity (million tonnes) Transition cost Project state transition type Starting Ending ($ million) Capacity construction /Build (unfinished) (CStart = (0.788, 0, 0); U A = U H G ' U L G ' U H G + L G ) 0.788 0.788 12.793 /Build (unfinished) (CStart = (0.788, 0, l ) ; U , = U H G ' U L G ' U H G + L G ) 0.788 1.575 12.793 Aeopen (closed) (Cstart =(0-788, 0.788, 0); U A = U H G - U L G , U H G + L G ) 0.788 1.575 3.00 Open capacity abandonment: /ABD. open (cs,ar, = (1-575, 0, 0); A W m i l l = 0.788) 1.575 0.788 1.500 /ABD, open (c S t t r t = (0.788, 0, 0); A W r n i l l = 0.788) 0.788 0 1.500 /ABD, open (cS t a Jt = (1-575, 0, 0); A W m i l l = 1.575) 1.575 0 2.500 Unfinished capacity abandonment: /ABD, u n M * e d ( c S M = (0.788, 0, l)) 0.788 0.788 0.500 / A B D , Unfinished (CStart = (0.788, 0, l)) 0.788 0 0.500 Closed capacity abandonment: /ABD, closed (c S t a r t = (0-788, 0.788, 0)) 0.788 0.788 1.500 /ABD. cosed ( c s m = (0-788, 0.788, 0)) 0.788 0 1.500 Personnel hiring charges: ProdChge(eH G =0,...,18;eL G =0, 1, 2 ;u L G ->-u H G ,u L G) 0 1.575 2.136 ProdChge(eH G =0,...,18;eL G = 2, . . . ,18;u H G ,u L G -*vH0+LG) 1.575 3.150 1.282 Personnel retrenchment charges: ProdChge(eH G =0,...,18;eL G =2, . . . ,18;u H G + L G - » u A B D ) 3.150 0 5.127 ProdChge(eH G =0,...,18;eL G =2, . . . ,18;u H G + L G - > U H G > U L G ) 3.150 1.575 1.923 ProdChge(eH G = 0,...,18;eLG = 2, . . . ,18;u H G ,u L G ->u A B D ) 1.575 0 3.204 Table 4.5a Project state transition costs. 79 Other distinguishing Transition cost Project state transition type information ($ million) Operating mode transition: T.C.(uH G -> u L G ) or T.C.(vw ->uH G) HG only to LG only/vice versa 0.400 T .C . ( u H G ->u H G + L G ) orT.C.(u H G + L G ->uH G) HG only to both zones/vice versa 0.200 T - C . ( U H G + L G - ^ ^ L G ) orT.C.(u L G - > U H G + L G ) Both zones to LG only/vice versa 0.200 Table 4.5b Miscellaneous project state transition costs. Permissible capacity state links Capacity Mill capacity (million tonnes / A = capacity abandonment; B = build new capacity; state period) and build index C = close capacity; N = no link; R = re-o )en capacity index Open Closed Build index 1 2 3 4 1 0.788 0 0 B N N 2 0.788 0 1 A B B+C 3 1.575 0 0 A N C 4 0.788 0.788 0 A N R Table 4.6 Project capacity states and permissible capacity state links. abandonment ($1.5 million) and employee retrenchment ($3,204 million) are delayed until the low-grade zone is exhausted. Table 4.3c and Table A1.5c detail the current expected cash flows (no flexibility) associated with the replacement development strategy. 4.1.5 Sources of management flexibility The project provides management with various combinations of flexibility during its lifetime. Initially, management conducts mining operations in the HG Zone and may opt to expand the project at discrete intervals when there are between 9.0 and 1.5 years of high-grade reserves remaining. The expansion option requires the immediate payment of a $0.2 million transition cost. Low-grade zone development expenditures, mine and mill capacity expansion charges and employee hiring costs are incurred at 80 designated times as development and mining of the low-grade zone progresses. If the project is not expanded, management may elect to replace exhausted high-grade reserves by initiating development of the low-grade reserves when there are between 1.0 and 0 years of high-grade reserves remaining. Total transition costs of $0.4 million are incurred when the LG Zone is developed as part of a production replacement strategy (see Table 4.3c for details). There may also be employee retrenchment and re-hiring charges if there is a production gap between exhaustion of the high-grade reserves and initial low-grade zone ore production. The expanded project also provides management with the option of temporarily closing one of the ore zones in response to low mineral prices. This option allows management to reduce or avoid operating losses by incurring the costs associated with reducing the scale of the operation. It may also allow management to save higher quality reserves if the risk-adjusted mineral price is expected to increase significantly in the future. In this example, the cost of temporarily closing one zone (when two are producing) includes a $0.2 million transition cost and a retrenchment charge of $1,923 million. Management may elect to maintain expanded project capacity levels while only operating a single zone by incurring an under-utilized capacity charge of $1,200 million per production period. Alternatively, management may avoid the under-utilization charge by either irrevocably abandoning excess capacity after paying a $1.5 million capacity abandonment charge or by temporarily closing excess capacity and incurring a charge of $0.3 million. If management elects to temporarily close capacity, they retain the option to re-open this capacity in the future by paying a closed capacity maintenance fee of $0.2 million each period and a re-habilitation charge of $3.0 million when the capacity is re-opened. Finally, management has the ability to irrevocably abandon the project at any time. The total abandonment charge includes a fixed site cleanup cost of $40.0 million, employee retrenchment costs that are dependent on the number of employees on site and a capacity abandonment cost that is a function of the project's capacity state. The retrenchment cost is $3,204 million, if only one zone is producing at the time of abandonment, and $5,127 million, if both zones are producing at the time of abandonment. Note that abandonment is the only form of flexibility associated with the LG Zone once the high-grade reserves are exhausted. See Tables 4.5a and 4.5b for a summary of the costs associated with exercising management flexibility. Table 4.6 details the project capacity state and the permissible links between these states. 81 Project value ($ million) Project design NREV price model REV price model SPP project structure (DCF valuation method; no abandonment available) HG only production. 79.522 63.498 HG + Early development of LG. 88.793 62.164 HG + Late development of LG. 106.589 71.368 SPP project structure (MAP valuation method; no abandonment available) HG only production. 32.163 39.509 HG + Early LG development. 19.304 31.541 HG + Late LG development. -4.616 20.967 FDMP and SPP project structure (MAP valuation method; abandonment available) HG only production. 41.019 40.185 HG + Early LG development. 35.036 32.827 HG + Late L G development. 52.945 38.246 FDMP model. 56.616 42.302 Table 4.7 Project values calculated by DCF and MAP methods (current mineral price = $1.00/unit). 4.2.0 Valuation results Project net present value (NPV) was calculated using FDMP and SPP project structure models within the standard DCF and MAP valuation frameworks. Possible project operating policies were delineated using the FDMP model. 4.2.1 Discounted cashflow valuation results. A DCF NPV was calculated using both price processes and a 10% risk-adjusted discount rate (RADR) for each of the following low-grade zone development SPPs: 1) the current HG Zone production plan (no low-grade zone development), 2) immediate development of the LG Zone to expand project production, and 3) delay LG Zone development until there is one year of high-grade zone reserves remaining. 82 The DCF NPV was determined by calculating expected revenue as the product of mineral production and the expected mineral price (given a current price of $1.00 per mineral unit) for each half-year period. Operating costs, transition costs, and capital expenditures were subtracted from expected revenue to generate a net cash flow. A constant 10% RADR 1 0 was used to calculate each cash flow's present value. Tables 4.3a, 4.3b and 4.3c detail the DCF value calculation (including expected price trends) for the current HG Zone production plan and each low-grade zone development strategy when mineral price follows a non-reverting process. Tables A1.5a, A1.5b, and A1.5c in Appendix 1 detail the expected cash flows and the present value calculation for the production plans when mineral price follows a reverting process. The DCF project values are presented in Table 4.7. These results indicate, when either price model is used, that the best investment policy is to delay low-grade zone development until just prior to high-grade exhaustion. Early development of the LG Zone is not indicated by the DCF method unless the current price is above $1.36 ($1.26) when the price follows a NREV (REV) process. 4.2.2 Modern asset pricing valuation results. The MAP valuation determines both project value and possible operating policies given a pre-defined project structure model and exogenous uncertainty. Two sets of MAP project values are calculated for the low-grade zone development strategies outlined in the previous section. The project structure underlying the first value set does not allow management to abandon the project in response to low mineral prices. The second value set is for a project structure that does allow management to abandon the project. Project value is then calculated using the FDMP model within the MAP framework in which management has a full range of operating alternatives that are described in section 4.1.511. The various value results are compared so that an indication of the relative importance of each type of flexibility can be determined. Project operating policies are also described in terms of critical price signals. These The RADR is not varied with changes in price uncertainty and project structure because several studies of valuation practices within the mining industry have revealed that a constant RADR is often used to value many of the projects being evaluated by a company. A 10% RADR was considered representative for this example. Note that each low-grade zone development strategy, as outlined in section 4.2.1, is assumed to be a complete description of project operation. Each of these strategies is just one of the many possible project operating policies considered within the FDMP project structure model. 83 descriptions provide insight into future management actions and provide a qualitative indication of which decisions are most significant12. Project values The MAP project values are presented in Table 4.7. When there is no abandonment and the low-grade development strategy must be immediately decided, the MAP values for the SPP models indicate that the most prudent operating policy is to mine the H G Zone until it is exhausted and then close the mine. Late development of the L G Zone is the least favorable strategy because the cash flows generated by the stand-alone low-grade zone operation are highly leveraged. When the low-grade zone is developed early, the expense of adding capacity and the increase in operating leverage produced by the low-grade zone cash flows is only partially mitigated by the economies-of-scale benefit derived from operating both zones together. Each L G Zone development SPP increases in value (substantially when the price follows a non-reverting price process) when management is able to limit downside losses by abandoning the project. The SPP models determine that late development of the low-grade reserves is most valuable when price follows a non-reverting process. Closing the mine upon exhaustion of the high-grade reserves is still most valuable strategy when there is an abandonment option and price follows a reverting process. The FDMP model assigns the most value to the project. When price follows a non-reverting model, project value increases by $3,671 million (an 6.9% increase) when compared to the most valuable low-grade development SPP and abandonment configuration. Project value increases by $2,117 million (a 5.3% increase) when compared to the most valuable low-grade development SPP within a reverting price environment. Operating alternatives could also be quantitatively ranked in terms of event probabilities. An example of such an approach is determining the probability that the low-grade zone would be developed as part of a production expansion strategy versus the probability of it being developed as production replacement strategy. 84 3.00 2.75 2.50 2.25 2.00 1.75 1.50 X 1.25 1.00 0.75 0.50 0.25 0.00 X .—x- -x-9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 2.0 1.0 0.0 -)K - - • LG development - full flexibility • Abandonment - full flexibility Figure 4.2a Low-grade zone development and project abandonment boundaries - N R E V model. 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 2.0 1.0 0.0 •>K • • • LG development - full flexibility • Abandonment - full flexibility Figure 4.2b Low-grade zone development and project abandonment boundaries - R E V model. 85 Description of operating policies There are three critical price boundaries that delineate possible project operating policies. These are: 1) the initial development boundary for the low-grade zone, 2) the project abandonment boundary, and 3) a temporary zone closure boundary when both zones are operating. Figures 4.2a and 4.2b outline the initial low-grade zone development boundary for each type of price model. A dashed line is used to demarcate this boundary because the option to develop the LG Zone may only be exercised at discrete times (every half-year in this example). These times are indicated on the development boundary by data markers. If the mineral price is above the boundary at this time, then it is optimal for management to initiate development of the low-grade zone. Low-grade development is deferred if the mineral price is below this boundary but above the abandonment price (indicated by the lower solid line). A "no capacity expansion boundary" is included in Figures 4.2a and 4.2b to indicate when the option to expand production becomes irrelevant. Capacity expansion becomes a non-issue at this boundary because the high-grade reserves will be exhausted before the low-grade zone is ready for production. Note that a period of no mineral production will be created if low-grade development is initiated with less than a year of high-grade reserves remaining because the low-grade zone will require further development after the high-grade zone has been totally exhausted. The low-grade development price is determined by comparing the present value of revenues directly attributed to low-grade zone operation11 to the costs associated with developing this zone. The cost of developing the low-grade zone has three components. The first component is the present value of non-revenue cash flows (i.e. cash flows whose magnitude is not dependent on the prevailing mineral price) produced by developing and operating the low-grade zone12. These cash flows include the obvious capital expenditures such as those associated with developing the low-grade zone and expanding project capacity. They also include cash flows such as low-grade zone operating costs, economies-of-scale benefits, and miscellaneous transition costs. The second component is the difference between the present value of the high-grade zone cash flow when the low-grade is not developed and the present value of the 1 1 This does not include the revenues generated from the operation of the high-grade zone. 1 2 This present value calculation includes an adjustment for the timing of the cash flow and, if there is operating flexibility such as abandonment or temporary zone closure, the risk-adjusted probability that the cash flow occurs. 86 high-grade zone cash flow when the low-grade zone is developed . The third component is the opportunity cost associated with forgoing a portfolio of European barrier call options that give management the opportunity to develop the low-grade zone at specific future times conditional on the low-grade zone having not been developed previously14. Prior to the "no capacity expansion boundary", the low-grade development boundary rises as the high-grade reserves are depleted because the costs of development are rising in relation to the present value of low-grade revenues. The relative total cost of low-grade development rises due to a decrease in the absolute value of the economy-of-scale benefit and an increase in the relative value of the European barrier option portfolio that gives management the opportunity to develop the low-grade zone in the future. The low-grade development boundary begins to rise steeply within three years of the "no capacity expansion boundary" because there is an increased probability that the "no capacity expansion" period will be reached without the low-grade zone being developed15. Development options that expire during the "no capacity expansion" period have greater intrinsic value than earlier development options because they do not require capacity expansion expenditures when exercised. An increase in the probability that this period will be reached causes the value of the future low-grade development options to increase relative to the benefits of early low-grade zone development. When there is a year or less of high-grade reserves remaining, the early development boundary decreases because no capacity expansion is required. A personnel retrenchment and re-hiring charge16 is incurred if This term will be zero when the only available form of management flexibility is the low-grade zone development decision. The value difference will also be small when the other forms of management flexibility, such as project abandonment, become important only after a drastic change in mineral price. 1 4 A European barrier call option is similar to a call option except that its value immediately becomes zero when an upper price boundary is reached. The underlying asset associated with each barrier call option is the present value of low-grade revenues generated by initiating low-grade development upon the option's expiry. The exercise price of this barrier call option is the present value of non-revenue cash flows generated from low-grade zone operations, the forgone portfolio of future low-grade development barrier call options and the incremental present value of the high-grade zone cash flow. Note that this exercise price is the "zero-value" barrier price boundary for the remaining LG Zone development options. 1 5 The LG Zone development boundary, when there is 1.5 years of high-grade reserves remaining, is presumed not to be infinite. The upper price boundary used in the valuation's finite difference grid was $6.00 per mineral unit and the value difference between developing and not developing the LG Zone at this point was small and decreasing. Previously run models with much smaller expansion costs have shown the development boundary to behave in a similar manner except that the boundary peaked before the upper price boundary was reached. 1 6 There will be a period of no production if low-grade development is started when there is less than a year of high-grade reserves remaining. The model requires that production employees be retrenched when there is no mineral production and re-hired when production re-starts. These personnel costs decrease the value of starting low-grade development when there is 0.5 years of high-grade reserves remaining. 87 low-grade development is started when there is only 0.5 years of high-grade reserves remaining. This additional charge sharply increases the critical price at which the low-grade is developed. The low-grade development boundary then drops because the final option to develop the low-grade zone is expiring. The project is abandoned when the mineral price falls at any time below the solid line17 in Figures 4.2a and 4.2b. This boundary rises slowly because the probability that the mineral price will return to levels such that the project becomes profitable is decreasing as the high-grade zone is exhausted. The abandonment boundary18 rises sharply immediately before the high-grade zone is exhausted because management is being confronted with the prospect of developing and operating a reserve in which operating leverage is much greater. Figures 4.3a and 4.3b display the project abandonment boundaries when the high-grade zone has been exhausted. The solid line indicates the critical price at which the project is abandoned if operations during the previous period were conducted only in the low-grade zone. This boundary initially declines as zone development expenditures are sunk and then increases as the low-grade zone reserves are depleted. The boundary demarcated by the dashed line indicates project abandonment when the operations during the previous period were being conducted in both zones (i.e. both zones were previously operating but the high-grade zone has now been exhausted and only the low-grade zone is producing). This boundary is slightly higher than the low-grade only operations abandonment boundary because management must incur a small transition cost to re-focus joint operations solely on the low-grade zone19. When mineral production begins in the low-grade zone (i.e. when there is 8.0 years remaining of the low-grade zone schedule), the abandonment boundary for joint operations is below the boundary for low-grade-operations-only because it does not incur a re-hiring charge. At this point in the low-grade zone schedule, when previously there was joint zone activity, personnel working in the now-exhausted high-grade zone can be shifted to low-grade production without the project incurring any A solid line is used to indicate the abandonment boundary because this option is available continuously. The abandonment boundary is shown in both figures to rise sharply when there is only 0.5 years of high-grade remaining. This is inaccurate in that the abandonment boundary is continuous and should actually continue to rise slowly until the high-grade zone is completely exhausted. At this point, the abandonment boundary becomes vertical when management must decide whether to develop the low-grade zone. The abandonment boundaries is presented as shown because the valuation program calculates, but does not record, the abandonment boundary when there is only a small amount of high-grade reserves remaining. This transition cost can be waived by the project analyst if it is deemed inapplicable in situations where the project operating mode is changed due to the exhaustion of zone reserves. 88 1 30 1 20 1 10 1 00 '3 3 0 90 " « <U c 0 80 g 0 70 u 0 60 o o . 0 50 15 c 0 40 s 0 30 0 20 0 10 0 00 Low-grade abandonment boundary when the high-grade reserves are exhausted and the low-grade zone is currently 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining low grade reserves (years) 2.0 1.0 0.0 •Low-grade abandonment boundary X - HG_LG abandonment when HG exhausted Figure 4.3a Low-grade zone abandonment boundary - NREV model. ed g 0.40 -0.30 -0.20 -o.io - • -0.00 -) ' 1 1 1 ' 1 1 1 1 1 ' 1 ' 1 1 1 ' 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Remaining low grade reserves (years) Low-grade abandonment boundary - - -)K - - HG_LG abandonment when HG exhausted Figure 4.3b Low-grade zone abandonment boundary - REV model. 89 Management action horizon legend - 8.5 years of H G reserves .First zone closure opportunity when L G development starts with 9 years Q H G production only; capacity expansion suspended. | H G production only; capacity expansion abandoned. j Full project abandonment. -of H G reserves. Management action horizon legend - 8.0 years of H G reserves |Erj H G production only; temporarily close L G zone and excess capacity. | Full project abandonment. I - I I- H h H h - 1 h -t-6.5, 6.5 5.5, 5.5 4.5, 4.5 3.5, 3.5 2.5, 2.5 1.5, 1.5 0.5, 0.5 7 ^ 75 Remaining zone reserves box. Left-hand number indicates the I amount of H G reserves and the right-hand number indicates the remaining of L G reserves. 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Remaining high-grade reserves (years) • Abandonment Figure 4.4a Zone closure boundaries for 17.5% economies of scale - N R E V model. - First zone closure opportunity when LG development starts with 9 years of HG reserves. AbffliJflaBgflent action horizon legend - 8.5 years of HG reserves [73] HG production only; capacity expansion suspended. {ji l HG production only; capacity expansion abandoned. H Full project abandonment. Management action horizon legend - 8.0 to 5.0 years of HG reserves HI LG production only; temporarily close HG zone and excess capacity. (Hi HG production only; temporarily close LG zone and excess capacity. EH Full project abandonment. 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 2.0 1.0 0.0 H— Abandonment Figure 4.4b Zone closure boundaries for 17.5% economies of scale - R E V model. 90 personnel charges. If there was only low-grade zone activity during the previous period, the project incurs an employee re-hiring charge since the low-grade zone was previously under development and the project had retrenched any on-site production personnel. Figure 4.4a outlines the temporary zone closure and project abandonment boundaries when L G Zone development is started with 9.0 years of high-grade reserves and the mineral price follows a N R E V diffusion. This figure shows that management only considers non-abandonment alternatives to dual-zone production during the period when the low-grade zone is being developed. Vertical bars are used to demarcate alternative operating strategies to dual zone production and project abandonment over a range of prices. When there is 8.5 years of H G zone reserves remaining, management elects to switch to only H G Zone production with suspension of the capacity expansion program when price is between $0.43 and $0.47 per mineral unit and when price is between is between $0.81 and $1.05 per mineral unit20. Management switches to only H G Zone production with abandonment of the capacity expansion program when price is between $0.47 and $0.81 per mineral unit. When there is 8.0 years of H G Zone reserves remaining, there is only one non-abandonment operating alternative in the price region of $0.41 to $0.85 per mineral unit in which management shifts to H G Zone production only with abandonment of the capacity expansion program. Figure 4.4b outlines the temporary zone closure and project abandonment boundaries when the mineral price follows a R E V diffusion and L G Zone development is started with 9.0 years of high-grade reserves. Unlike the N R E V diffusion, there is an extended horizon over which management considers non-abandonment alternatives to dual-zone production. When there is 8.5 years of H G Zone reserves, vertical bars are used to indicate price ranges over which two non-dual zone production strategies are optimal. These bars show that it is optimal to shift to only H G Zone production with abandonment of the capacity expansion program when price is between $0.29 and $0.61 per mineral unit and to shift to only high-grade zone production with capacity expansion suspension when price is between $0.61 to $0.75 per mineral unit. When there is less than 8.0 years of H G Zone reserves remaining, there is a significant region where it is optimal to temporarily close the L G Zone and excess capacity in order to limit operating losses. There is also a price range, when there is 7.5 years of H G Zone reserves, in which it is The lower action region is due to the characteristics of the project cost model. In the lower price range, it is less costly to suspend the capacity expansion program than to abandon it since at such low price levels the project will likely be abandoned in the near future. 91 optimal to close the HG Zone and excess capacity. This occurs because the expectation of future price increases provides an incentive to preserve the high-grade reserves despite incurring higher operating losses from only producing in the L G Zone. MAP valuation results discussion The low-grade zone is a valuable region of the deposit depending on the underlying characteristics of the price process. It adds $15,597 million (38.0% increase) in value when price follows the NREV diffusion and $2,117 million (5.3% increase) in value when price follows the REV diffusion. This reflects the observation that management options are generally more valuable in non-reverting price environments where there is more long-term uncertainty in the price. Figures 4.5a and 4.5b compare the various value payoff profiles for the SPP and the FDMP project structure models over a range of mineral prices. These two figures show that, in high and low price environments, one of the SPP payoff profiles closely tracks the payoff profile of the FDMP approach. This result demonstrates that, when only a project value is needed, an experienced analyst can reduce the complexity of the FDMP in some situations with several well-chosen set production plans that include project abandonment. The two project structure models can also be compared on the price signals they provide for the early development of the low-grade zone. The FDMP model indicates that immediate L G Zone development should occur if current mineral price is greater than $1.49 ($1.31) per mineral unit when the price follows the NREV (REV) diffusion whereas the SPP model signals immediate development when the mineral price is above $1.25 ($1.17) for a NREV (REV) price process. The SPP model underestimates the low-grade zone development price significantly because it does not incorporate the opportunity cost associated with the option to develop the low-grade zone at discrete intervals in the future. If low-grade zone development does not occur immediately, then it is deferred until the mineral price rises above the critical price boundary delineated in Figures 4.2a and 4.2b. The FDMP model indicates that the production expansion development strategy is more likely to be implemented when the amount of remaining high-grade reserves and its attendant benefit of economies-of-scale from dual zone production are large. When amount of remaining high-grade reserves is small, the incentive for developing the low-grade zone as part of a production expansion strategy is small and low-grade development is better deferred until a production replacement strategy can be implemented. Surprisingly, when either price 92 -60.0 Mineral price ($ / unit mineral) HG only scenario — - - — Early LG development scenario — - — - Late LG development scenario 1 •Full flexibility Figure 4.5a SPP and FDMP model MAP value boundaries - NREV model. Note: Scenario DCF payoffs are NOT shown. Payoffs are for MAP valuations only. The critical price for LG early development is $1.26 per mineral unit using the SPP model and the DCF method. DCF calculates a project value of $93,294 million. ^ ^ A ^ The critical price for LG early ' development is $1.17 per ' | mineral unit using the SPP j I model and the MAP ' I framework. The critical price for LG " early development is $1.31 per mineral unit using theFDMPmodel and MAP framework. Do 0.20 0.' 0 . 60^ ' 0.80 1.00 1.20 .1,40 .1.60 1.80 Mineral price ($ / unit mineral) HG only scenario — - - — Early LG development scenario — - — - Late LG development scenario •Full flexibility Figure 4.5b SPP and FDMP model MAP value boundaries - REV model. 93 model is used, management's optimal production replacement strategy is more likely to involve deferring the low-grade development decision until the high-grade is completely exhausted even though significant additional costs from production gaps are incurred21. The SPP model does not provide these insights explicitly unless additional scenarios are generated at lower high-grade reserve levels. The LG Zone development and operation policy as part of a production replacement strategy is outlined in Figures 4.3a and 4.3b. These figures illustrate that the low-grade zone is less likely to be developed as part of a production replacement strategy when the mineral price follows a reverting diffusion. However, once this zone is developed, it is optimal to maintain operations over a wide range of prices due to the possibility of future price increases and the delay of large closure costs until the low-grade zone is exhausted. The SPP model, for the production replacement strategy, provides an identical abandonment profile once the low-grade zone development expenditures have been incurred. Figures 4.4a and 4.4b demonstrate that the ability to temporarily close either of the zones, when both zones are producing mineral, appears to be unimportant in most situations due to the presence of economies-of-scale. The ability to defer or avoid the costs associated with production expansion programs during the initial development period of the low-grade zone is important when either price diffusion is used. The SPP model approach is unable to generate these results. 4.2.3 Comparison of the standard DCF and MAP valuation results. The standard DCF and MAP methods show a wide divergence in the values calculated for the project due to differences in assessing the influence of operating leverage on project risk. The influence of operating leverage on project risk enters the DCF valuation process through a RADR set by the project analyst. Both expected revenues and known operating costs are discounted at the same RADR. The MAP method, however, applies a risk adjustment only to the uncertain cash flow component (in this case, mineral price and, by extension, revenue) such that an explicit risk-adjusted operating profit can be calculated. A time discount is then applied to determine a MAP NPV of the cash flow. In this example, the DCF method This result can be ignored by assuming that the option to develop the low-grade zone expires when there is 1 year of high-grade reserves remaining. Such an assumption is also not realistic since it requires management to ignore the existence of the low-grade zone, if it hasn't already been developed, in the final year of high-grade production. The project-state MAP model requires additional work (possibly producing a small marginal benefit) to overcome deficiencies in this situation. 94 over-estimates the value of the project because the ad-hoc RADR provides an insufficient adjustment for operating leverage. Both valuation methods indicate that the low-grade development decision should be deferred when the current mineral price is $1.00 per unit. However, this result is deceptive since the MAP investment recommendation is sensitive to changes in correlation between mineral price uncertainty and general macro-economic uncertainty. This sensitivity may cause the MAP method to produce a different investment signal than the DCF method. In this example, if mineral price correlation is decreased, the MAP method may change its investment recommendation to developing the LG Zone early. The DCF method would fail to register this change in project risk unless management makes a conscious choice to reduce the RADR 2 2 . In addition, the MAP method is able to provide greater insight into project operating policy because it shows, with critical price boundaries, under what conditions it is optimal to develop the low-grade zone as part of a production expansion or production replacement strategy. 4.3.0 Conclusion. A stylized two-zone mining example was used to demonstrate the valuation model introduced in Chapter 3. The FDMP model found additional project value, through more refined managerial flexibility specification, that SPP option pricing techniques missed. The FDMP model also permitted cash flow and transition cost components to be specified on a project state basis as opposed to exogenous production plan specification that usually occurs when conventional MAP and DCF methods are used. Comparing the MAP results to those obtained by standard DCF methods show that a DCF approach, when using a 10% RADR, provides a smaller risk-adjustment than either MAP approach. The FDMP model was also able to provide insights into project operating policy that a SPP options approach would miss. In particular, the FDMP model outlined operating policies for temporarily zone closure and multiple discrete capacity expansion opportunities that would be difficult, at best, to incorporate into a SPP project structure model. In addition, the operating policies produced by the FDMP model are broadly consistent with practical intuition in that early low-grade development boundary increases as the remaining high-grade reserves decrease. A change in correlation between mineral price uncertainty and general macro-economic uncertainty does not affect the trajectory of the expected mineral price! It only affects the magnitude of the MAP risk adjustment to the expected mineral price. 95 For this valuation exercise, the benefits of using the FDMP model appear to be small when compared to the simplicity of the SPP approach. The differences in value generated by the two project structure models were small at a price of $ 1.00 per mineral unit. However, valuation problems can be conceived where the value difference between the two methods is greater than 10%. The choice between the two project structure models is best made on the basis of cost. The FDMP approach will be preferred because it generates more project information when the costs incurred from using each method are the same. When the costs are different, then the choice becomes more difficult because benefits gained (e.g. more information) from using a more sophisticated method must be offset against its costs (e.g. increased training and reliance on specialists, more expensive computer facilities). 96 Chapter 5 Two-Zone Mine Valuation Results with Price and Project Parameter Variations The two-zone mine valuation problem presented in Chapter 4 is re-evaluated given variations in the price process parameters of volatility and the price of mineral risk and in the project structure parameter of economies-of-scale. The results are used to illustrate the importance of using a valuation method that is sensitive to the differences between project environments. 5.1.0 M i n e r a l price volatili ty Volatility sensitivity analysis was completed with mineral price volatilities set at 10%, 25% and 40%'. This parameter is important because it describes the uncertainty associated with future mineral prices (i.e. the spread of possible mineral prices). It affects the value of future cash flows through three mechanisms. First, as volatility increases, the expected mineral price also increases because the underlying price processes are log-normal2. Second, if the underlying mineral price has positive non-zero correlation with macro-economic uncertainty, the risk adjustment necessary to compensate investors for holding the mineral increases with growing uncertainty. At low volatilities, the risk-adjustment component is more sensitive to changes in uncertainty and any increases in expected price due to volatility increases are overwhelmed by larger risk-adjustments. As uncertainty increases to higher levels, the expected mineral price becomes more sensitive to volatility changes and future cash flows values are influenced more heavily by increases in the expected mineral price. Appendix 2 describes this relationship between volatility and future mineral prices. Third, increasing volatility causes the migration of probability towards outer price outcomes. This increases the value influence of project values in the tail of the value distribution. Tables 5.1a and 5.1b present the influence of volatility change on value. Given the same DCF RADR across volatility scenarios, increasing volatility has a positive effect on project value when the DCF The volatility sensitivity analysis conducted here does not maintain the same mineral forward price curve for each level of volatility. This would require the use of time-dependent price processes. These types of processes have been specifically avoided to simplify the valuation model. 2 Expected prices of log-normal variables are equal to the product of the underlying distribution's median and the exponential of one-half the associated price variance. 97 Project value ($ million) Project design o=10% o=25% o=40% SPP project structure (DCF valuation method; no abandonment available) H G only production. 58.003 79.522 129.447 H G + Early development of L G . 53.311 88.793 171.268 H G + Late development of L G . 65.696 106.589 216.594 SPP project structure (MAP valuation method; no abandonment available) H G only production. 42.901 32.163 42.901 H G + Early L G development. 37.032 19.304 37.032 H G + Late L G development. 17.843 -4.616 17.843 FDMP and SPP project structure (MAP valuation method; abandonment available) H G only production. 43.290 41.019 61.446 H G + Early L G development. 37.786 35.036 69.520 H G + Late L G development. 42.998 52.945 94.647 F D M P model. 47.113 56.616 99.850 Table 5.1a DCF and M A P project values - volatility sensitivity (NREV price model) Project value ($ million) Project design o=10% o=25% o=40% SPP project structure (DCF valuation method; no abandonment available H G only production. 55.703 63.498 78.973 H G + Early development of L G . 49.489 62.164 87.351 H G + Late development of L G . 60.399 71.368 92.226 SPP project structure (MAP valuation method; no abandonment available) H G only production. 50.837 39.509 36.751 H G + Early L G development. 50.238 31.541 26.850 H G + Late L G development. 41.728 20.967 13.841 FDMP and SPP project structure (MAP valuation method; abandonment available) H G only production. 50.875 40.185 40.399 H G + Early L G development. 50.302 32.827 33.467 H G + Late L G development. 45.527 38.246 42.247 F D M P model. 51.380 42.302 46.869 Table 5.1b DCF and M A P project values - volatility sensitivity (REV price model) 98 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 is A A - - - A A - - ' -A* ~ A — ~ Ar . - - -A--x- X 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 2.0 1.0 0.0 • O — LG development (10% Vol.) -Abandonment (10% Vol.) —X - - LG development (25% Vol.) Abandonment (25% Vol.) -A- - L G development (40% Vol.) Abandonment (40% Vol.) Figure 5.1a The influence of volatility change on the L G Zone development boundary - N R E V model. c .3 —. u o *C o. "ei u c 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 2.0 1.0 0.0 O LG development (10% Vol.) Abandonment (10% Vol.) X- - - LG development (25% Vol.) Abandonment (25% Vol.) -A- - L G development (40% Vol.) Abandonment (40% Vol.) Figure 5.1b The influence of volatility change on the L G Zone development boundary - REV model. 99 method is used because the expected mineral price rises with an increase in volatility when the price medians are held constant. When the M A P approach is used, the volatility effect is more complex because project value is dependent upon the term structure of mineral forward prices. These prices are a function of both expected mineral prices and the risk adjustment associated with the mineral. At low volatilities, the risk adjustment has a greater influence on project value and project value falls with increasing in volatility. At higher levels of uncertainty, the influence of the expected mineral price predominates and project value rises with volatility increases. This effect can most clearly be seen on M A P values in Table 5.1a when there is no flexibility and price follows a NREV model. Project value decreases when volatility increases from 10% to 25% and then it increases when volatility is rises to 40%. The influence of volatility change on the low-grade zone development boundary is presented in Figures 5.1a and 5.1b. An increase in volatility causes the opportunity cost associated with early L G Zone development to grow because the value of the option to defer L G Zone development increases with greater mineral price uncertainty. This increase in value is driven by increased future mineral price expectations and the increased possibility of higher future prices. The influence of the positive drivers of deferred option value are offset somewhat by an increase in discounting that the higher volatilities attract. The net effect is for the low-grade early development boundary to rise with increasing volatility except in situations where the high-grade zone will be exhausted before L G mineral production starts. When the high-grade zone is exhausted, the ranking of low-grade zone development prices depends on the price process used. When price follows a NREV process and there is either one year or a half-year of high-grade reserves, the option value associated with delaying low-grade zone development maintains the previous ordering of low-grade development boundaries. When the high-grade zone is completely exhausted, the ordering of low-grade zone development prices reverses such that development prices decrease with increasing uncertainty4. Developing the low-grade zone becomes more attractive, in NREV price environments, when volatility is high because the possibility of consistently higher prices5 in the future is greater while the impact of lower prices is mitigated by the option to abandon. When price follows a REV process, it is more difficult to determine the ordering of development prices from price 3 The impact of greater uncertainty enters the DCF value calculation through the RADR selected to discount the net project cash flow. This figure will not reflect changes in uncertainty if the valuation policy is to use a single RADR for all projects, as is assumed here. 4 Note that this ordering may change if management has the option to temporarily close the whole project. 5 When price follows a NREV process, there is no reverting effect pulling the current price towards the long-term price median. 100 volatility because the convexity of the mineral price term structure due to volatility dominates the value calculation. 5.2.0 Price of mineral risk The price of mineral risk (PMR) was set at 0.10, 0.25, and 0.40. This parameter indicates the extra return an investor requires for each percentage of uncertainty in mineral price return he or she is exposed to. A low PMR (e.g. PMR = 0.1) is associated with an asset whose price does not closely follow price movements in the general economy. Investors require less of a premium to be exposed to this type of risk because diversification strategies are more effective at mitigating the effects of negative price outcomes when there is little correlation between asset price moves and prices in the general economy. Conversely, a high PMR (e.g. PMR = 0.4) is associated with an asset whose price closely follows price movements in the general economy since it is more difficult for investors to mitigate the effect of negative asset price moves. The PMR affects valuation through the calculation of an appropriate risk adjustment. The larger the PMR, the greater the magnitude of risk adjustment applied to a project cash flow with a given level of uncertainty. The influence of PMR variation on project value for both valuation approaches is presented in Tables 5.2a and 5.2b. Variation in mineral price risk is shown not to affect project values when the DCF method is used because it is assumed that a single corporate RADR is used in all valuation situations. When the MAP valuation framework is used, PMR affects the valuation through the risk-adjustment applied to the expected mineral price. As the PMR increases, there is a decrease in the value of future revenues at a given time because the magnitude of the risk-adjustment increases. This effect is displayed by project value when either price process is used. The presence of management flexibility dampens the impact of an increase in PMR only to a small degree. Figures 5.2a and 5.2b display the influence of the PMR on the LG Zone development boundary. The response of the low-grade development boundary to PMR changes depends on which price process is used. When price follows a NREV process, the early low-grade zone development boundary initially falls as the PMR is increased from 0.1 to 0.25. The increase in risk adjustment rate has a disproportionate effect on long-term cash flows. Hence, the value of early LG Zone production is less affected by the PMR increase than the opportunity cost created by forgoing the options to develop the 101 Project value ($ million) Project design PMR = 0.1 PMR = 0.25 PMR = 0.4 SPP project structure (DCF valuation method; no abandonment available H G only production. 79.522 79.522 79.522 H G + Early development of L G . 88.793 88.793 88.793 H G + Late development of L G . 106.589 106.589 106.589 SPP project structure (MAP valuation method; no abandonment available) H G only production. 70.936 32.163 1.052 H G + Early L G development. 83.359 19.304 -32.001 H G + Late L G development. 81.199 -4.616 -64.253 FDMP and SPP project structure (MAP valuation method; abandonment available) H G only production. 75.482 41.019 16.134 H G + Early L G development. 91.456 35.086 -4.992 H G + Late L G development. 114.880 52.945 18.517 F D M P model. 116.742 56.616 21.129 Table 5.2a DCF and M A P project values - price of mineral risk sensitivity (NREV price model) Project value ($ million) Project design PMR = 0.1 P M R = 0.25 PMR = 0.4 SPP project structure (DCF valuation method; no abandonment available H G only production. 63.498 63.498 63.498 H G + Early development of L G . 62.164 62.164 62.164 HG + Late development of L G . 71.368 71.368 71.368 SPP project structure (MAP valuation method; no abandonment available) H G only production. 61.856 39.509 20.071 H G + Early L G development. 67.155 31.541 -0.356 H G + Late L G development. 58.036 20.967 -11.674 FDMP and SPP project structure (MAP valuation method; abandonment available) H G only production. 61.401 40.185 21.753 H G + Early L G development. 67.578 32.827 2.807 H G + Late L G development. 65.368 38.246 18.722 F D M P model. 71.039 42.302 21.972 Table 5.2b DCF and M A P project values - price of mineral risk sensitivity (REV price model) 102 3.00 2.75 2.50 2.25 2.00 1.75 i .50 jjfr- r & r --yjft- >r 1.25 1.00 0.75 0.50 0.25 0.00 0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 2.0 1.0 0.0 LG development (PCorr=0.2) - Abandonment (PCorr=0.2) - -)K - - LG development (PCorr=0.5) — -A- - LG development (PCorr=0.8) Abandonment (PCorr=0.5) Abandonment (PCorr=0.8) Figure 5.2a The influence of a price of mineral risk change on the L G Zone development'boundary - N R E V model. 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Remaining high-grade reserves (years) — - O — LG development (PCorr=0.2) Abandonment (PCorr=0.2) -X£ - - LG development (PCorr=0.5) Abandonment (PCorr=0.5) •6r - LG development (PCorr=0.8) — Abandonment (PCorr=0.8) Figure 5.2b The influence of a price of mineral risk change on the L G Zone development boundary - R E V model. 103 low-grade zone in the future. This effect is more balanced when the PMR is increased from 0.25 to 0.4. At this stage, the loss in value of earlier low-grade production due to increases in the PMR is offset by the reduction in opportunity cost associated with future L G Zone development. Note that development boundaries for 0.25 and 0.4 PMR cross when there is 5 years of high-grade reserves remaining because the operating leverage associated with early L G Zone development much more important when the PMR is 0.4. When the high-grade zone reserves are totally exhausted, the critical price signaling low-grade zone development decreases with reductions in the PMR reflecting the reduced risk associated with future low-grade cash flows. PMR variations have an opposite effect on the early L G Zone development boundary when price follows a R E V process. Figure 5.2b shows that, as the PMR decreases from 0.4 to 0.25, the L G Zone early development boundary falls because the value of early development increases at a faster rate than the value of deferring low-grade zone development until the H G Zone is exhausted. This difference in sensitivity to PMR change is due to uncertainty-leveraging differences between the L G Zone early development SPP and the H G Zone mining SPP. The L G Zone early development SPP initially leverages the underlying price uncertainty to a greater extent because of capacity expansion expenditures and, thus, the value of this development strategy increases at a greater rate when the PMR decreases. The option to develop the L G Zone in the future has little effect on this dynamic because option value is generally reduced in a R E V price environment. This ordering of L G Zone development boundaries is maintained when the H G zone is exhausted. 5.3.0 Dual-zone production economies-of-scale Economies-of-scale (EoS) sensitivity analysis was completed with dual-zone production EoS set at 0%, 17.5%, and 35% of total zone production costs. This parameter specifies the benefit, through reducing redundant costs and greater purchasing power, of producing from both zones simultaneously. An EoS of 0% would indicate that there are few benefits to dual production. This may occur for a project in which the two ore zones are located such that production resources cannot be easily shared (e.g. the two pits are located several miles apart) and their mineralogy requires two separate processing circuits to produce mineral concentrate. An EoS of 35% would indicate significant benefits to dual production derived from being able to easily transfer production resources between zones and the use of identical mineral processing techniques to concentrate the mineral. 104 Project value ($ million) Project design EoS =0% EoS=17.5% EoS=35% SPP project structure (DCF valuation method; no abandonment available H G only production. 79.522 79.522 79.522 H G + Early development of L G . 56.976 88.793 120.601 H G + Late development of L G . 106.589 106.589 106.589 SPP project structure (MAP valuation method; no abandonment available) H G only production. 32.163 32.163 32.163 H G + Early L G development. -25.554 19.304 64.148 H G + Late L G development. -4.616 -4.616 -4.616 FDMP and SPP project structure (MAP H G only production. 41.019 41.019 41.019 H G + Early L G development. 4.769 35.036 70.797 H G + Late L G development. 52.945 52.945 52.945 FDMP model. 54.543 56.616 70.829 Table 5.3a DCF and M A P project values - economies-of-scale sensitivity (NREV price model) Project value ($ million) Project design EoS =0% EoS=17.5% EoS=35% 1 SPP project structure (DCF valuation method; no abandonment available H G only production. 63.498 63.498 63.498 H G + Early development of L G . 30.346 62.164 93.972 H G + Late development of L G . 71.368 71.368 71.368 SPP project structure (MAP valuation method; no abandonment available) H G only production. 39.509 39.509 39.509 H G + Early L G development. -13.316 31.541 76.385 H G + Late L G development. 20.967 20.967 20.967 FDMP and SPP project structure (MAP valuation method; abandonment available) H G only production. 40.185 40.185 40.185 H G + Early L G development. -7.532 32.827 76.625 H G + Late L G development. 38.246 38.246 38.246 F D M P model. 41.258 42.302 76.625 Table 5.3b DCF and M A P project values - economies-of-scale sensitivity (REV price model) 105 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 6 - - -.. * - O - - - ~ 0 ~ - - = - - - 0 - * - " * - X - - - X * -x; .:x' j. _ A A A-A — A - — Ar'^or--- - -. A ' 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Remaining high-grade reserves (years) O — LG development (0% EoS) -A- - LG development (35% EoS) - - -X - - LG development (17.5% EoS) Abandonment (All EoS) Figure 5.3a The influence of EoS change on the L G Zone development boundary - N R E V model. 0.00 I 1 1 1 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 Remaining high-grade reserves (years) - ~0 ~ LG development (0% EoS) - - -X - - LG development (17.5% EoS) — -A- - LG development (35% EoS) 1— LG dev./HG close (35% EoS) Abandonment (All EoS) Figure 5.3b The influence of EoS change on the L G Zone development boundary - R E V model. 106 The valuation results are presented in Tables 5.3a and 5.3b. A l l valuation approaches demonstrate that project value improves with increasing EoS. The results also show that the added detail of the FDMP project structure model is more important at intermediate levels of EoS. When EoS benefits are low, the H G Zone only or the late L G Zone development SPPs provide a lower bound for project value and the option to develop the L G Zone early is worth little. When EoS benefits are high, the L G Zone early development SPP provides a lower bound for value and the options to defer the L G Zone development or temporarily close one of the zones later is worth little. It is only at intermediate EoS values that the options to develop the L G Zone early or temporarily close one of the zones becomes significant. The influence of EoS change on the early low-grade development boundary is displayed in Figures 5.3a and 5.3b. These figures indicate that L G Zone early development price is high when there is little benefit associated with dual zone production and that it decreases as EoS benefits increase. EoS variations affect the L G Zone development price through increasing the value of immediately developing the low-grade zone as well as increasing the cost of forgoing the opportunity to develop the low-grade zone in the future. The critical price boundary decreases with improved EoS because the value of immediately developing the low-grade zone is more sensitive to changes in this parameter. The value of deferring low-grade development is less sensitive to EoS variations because the decision to defer development decreases the horizon over which EoS benefits can be generated. In R E V price environments with 35% EoS, a price horizon is created where it is optimal to cease mineral production by temporarily closing the high-grade zone and developing the low-grade zone (this region's lower bound is the dashed line with cross data markers and its upper bound is the dashed line with triangle data markers). This strategy is used to preserve high-grade reserves and the opportunity to gain the benefits of future EoS on the expectation that the mineral price will increase in the future. 5.4.0 Comments regarding price and project parameter variations Parameter variations have a significant impact on the valuation results when either the DCF or M A P valuation approaches are used. The results show that the policy of using a single RADR, when valuing several projects with the DCF method, may be undesirable when the characteristics of mineral price uncertainty are different for each project. In particular, the single-RADR policy does not reflect the changes in expected price and risk that are caused by increasing or decreasing volatility. Nor does this policy reflect changes in risk caused by variations in mineral price correlation. Finance theory and the 107 MAP results suggest that corporate policy should require that the RADR vary in response to changes in the parameters of underlying uncertainty. The MAP valuation methods demonstrated the ability to reflect changes in price and project parameters. When no flexibility was available, the convex relationship between volatility and mineral prices was reflected in the valuation results. The impact of variations in the PMR and EoS was also clearly shown. When flexibility was considered, the degree of under-valuation of the SPP model in comparison to the FDMP model depended upon the underlying price process and the degree of price parameter variation. When a NREV process was used, under-valuation (on a percentage basis) by the SPP model tended to increase as volatility decreased and the PMR increased. An opposite trend appeared for the REV process. When EoS was varied, a concave pattern of the SPP model under-valuation was discerned. At 0% EoS, the ability to make flexible LG Zone development decisions provided little additional value benefits. The value of full flexibility increased when there were moderate EoS benefits to dual-zone production but decreased at large EoS due to the attractiveness of immediate LG Zone development. The LG Zone development boundary generated by the FDMP model method was also greatly influenced by project and price parameter variations. This boundary's behavior is complex because it is determined by the relative impact of parameter variation on the value of immediately developing the LG Zone versus the value of deferring this zone's development. The low-grade early development boundary reacted to variations in price volatility and EoS variations in an obvious manner. Increases in volatility increase the opportunity costs associated with early LG Zone development and thus cause the early development boundary to rise. Increases in EoS levels increase the incentive for early development and cause the early development boundary to fall. However, the behavior of this boundary appeared dependent on the type of price process when the PMR was varied. The early low-grade development boundary fell with PMR increases when price followed a NREV process and rose with PMR increases when price followed a REV process. This is due to the relative importance of the value of the option to defer LG Zone development in each pricing environment. In NREV price environments, this value forms a significant component of the low-grade development cost. Increases in PMR have a strong negative effect on the value of this option and as a result the development boundary falls as PMR increases. In REV price environments, the option value of deferring LG Zone development is a smaller proportion of the opportunity cost associated with early LG Zone 108 development. As a result, the influence of the PMR on the value of the developing the LG Zone now and the value of continuing to mine only the HG Zone was more important in placing the low-grade early development boundary. The SPP of developing the LG Zone immediately is more risky than continuing to mine only the HG Zone so decreases in the PMR cause larger value increases in this scenario. This differential increase in SPP value causes the low-grade development boundary to rise as the PMR increases. 5.5.0 Conclusion This chapter assessed the impact of project and price parameter variations on project operating policy and value. A policy of using a single RADR for DCF valuation performed poorly when price uncertainty parameters varied since such a policy does not reflect changes in project risk. When the MAP valuation framework was used, both project structure models responded to changes in project and price parameters but the SPP model was found to undervalue the project by up to 10% in some situations. In other project and price environments, the SPP model calculated MAP project values that were similar to those calculated by the FDMP model. Project operating policy was often greatly influenced by parameter variations. Volatility and EoS level changes caused similar reactions in NREV and REV price environments. The effect of varying the PMR was found to be dependent upon the mineral price process due to the importance of option value in each environment. 109 Chapter 6 Two-Zone Mine Valuation Results with Geological Uncertainty and Temporary Closure Considerations The two-zone mine valuation problem presented in Chapter 4 is re-evaluated with a discrete model of LG Zone geological (grade) uncertainty and the results used to describe the influence of geological uncertainty on management behavior. The low-grade zone quality outcomes from this exercise are also used to examine the decision to temporarily close one of the zones to determine the environments in which this decision is important. 6.1.0 The influence of L G Zone geological uncertainty on project assessment Mining operations in one area of a mineral deposit often do not fully resolve the geological uncertainty associated with other mineralized zones. This uncertainty may only be fully dissipated through developing and mining the unexploited zone. Zone grade uncertainty affects project value and operating policy through its impact on operating leverage. Grade outcomes that are higher (lower) than expected will decrease (increase) project operating leverage. The project structure model that permits management greater latitude when reacting to changes in operating leverage should assign higher values to projects than a project structure model that does not. 6.1.1 LG Zone geological uncertainty model A simple geological uncertainty model is used to investigate the impact of low-grade zone geological uncertainty on project value and operating policy. The geological uncertainty associated with the low-grade zone is assumed to be discrete such that there are five possible grade outcomes (0.42%, 0.51%, 0.60%, 0.69%, 0.78% mineral) with equal probabilities of occurrence. The resolution of actual low-grade zone quality is considered at three stages of LG Zone development. These resolution boundaries include geological uncertainty resolution immediately upon the start of LG Zone development (LG index = 9.0), resolution at the mid-point of low-grade zone development (LG index = 8.5) and resolution upon the start of low-grade zone development (LG index = 8.0). Table 6.1 displays the parameters of the low-grade zone uncertainty model and the variation of operating cost with grade outcomes on a per unit mineral basis. 110 Parameter Possible grade outcome Outcome probability (%) 0.20 0.20 0.20 0.20 0.20 Grade multiplier 0.70 0.85 1.00 1.15 1.30 L G Zone grade outcome (% mineral) 0.42 0.51 0.60 0.69 0.78 Production cost ($ / lb. mineral) L G only operation LG+HG operation (economies of scale) 1.284 0.674 1.057 0.631 0.899 0.593 0.781 0.560 0.691 0.530 Table 6.1 Discrete grade uncertainty model for the L G Zone. Project value ($ million) NoLG L G Zone geological uncertainty (grade geological uncertainty resolved at L G reserve index) Project design uncertainty LG=8.0 LG=8.5 LG=9.0 FDMP and SPP project structure (MAP valuation method; abandonment available) - NREV price model H G only production 41.019 41.019 41.019 41.109 H G + Early L G development 35.404 35.404 35.430 35.485 H G + Late L G development 52.945 53.397 53.539 53.794 Full flexibility 56.616 56.874 56.997 57.218 FDMP and SPP project structure (MAP valuation method; abandonment available) - REV price model H G only production 40.185 40.185 40.185 40.185 H G + Early L G development 32.827 33.078 33.079 33.080 H G + Late L G development 38.426 39.817 40.410 41.517 Full flexibility 42.302 42.843 43.183 44.285 Table 6.2 M A P project values with L G Zone grade uncertainty (current mineral price = $1.00/unit). Ill 6.1.2 The influence of geological uncertainty on MAP valuation models The presence of low-grade geological uncertainty increases project value when both the SPP and FDMP project structure models were used within the MAP valuation framework. The results are presented in Table 6.2. These results show that this type of geological uncertainty model only has little influence on project value when either project structure model is used. Under-valuation by the SPP model appears to decrease when geological uncertainty is present and price follows a NREV process. It appears to increase when price follows a REV process. Low-grade zone geological uncertainty has some influence on the LG Zone early development boundary. This influence is illustrated in Figures 6.1a and 6.1b. These figures show that early development is dependent upon the underlying mineral price process, the amount of capital expenditure required to bring the low-grade zone into production (i.e. whether capacity expansion is required), and the time at which geological uncertainty is resolved. Geological uncertainty appears to have the most influence when price follows a reverting process, development capital expenditure is minimized and the uncertainty is resolved sooner rather than later. Both figures also display, when geological uncertainty is resolved in the period after development starts and when there is more than 1.5 years of remaining HG Zone reserves, a significant region where the LG Zone is developed without the construction of additional capacity. This region is created because the low-grade development program essentially becomes a short-term information-gathering (i.e. exploration) exercise. Once the HG Zone is completely exhausted, development of the low-grade region becomes more likely the earlier LG Zone geological uncertainty is resolved. The presence of geological uncertainty is more important for the decision to start and continue LG Zone development once the high-grade zone has been exhausted. This result is illustrated between the LG Zone schedule indices of 9.0 and 8.0 in Figures 6.2a and 6.2b in which the solid gray line indicates the abandonment boundary when the LG Zone quality is assumed certain and the lower solid black line indicates the abandonment boundary when there is geological uncertainty. These figures show that the presence of geological uncertainty lowers the critical price at which development of the LG Zone is abandoned. The impact of geological uncertainty becomes more pronounced when price follows a REV process and the closer the LG Zone development program is to completion. Once geological uncertainty is resolved, the position of the abandonment boundary depends on the actual quality of the LG Zone. The abandonment boundary of the developed LG Zone rises as the quality of this zone decreases. XjBpunoq uoisuedxa /CuoEdsooN .5 O O C o 3 "S c x : •B * •> "O 53 o xi S ert & ui C 13 ° H u 03 c .2 .2 -B U Qi 5 2? £ '§•8 a. u C3 c U o N ci x W U o o in cs -k :f¥. V. 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" - N y -^\ . —— " ""I--Resolution boundary for low-grade zone geological uncertainty. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Remaining low-grade reserves (years) 2.0 1.0 0.0 • LG abandonment (LG grade=0.42%) •LG abandonment (LG grade=0.51%) •LG abandonment (LG grade=0.60%) LG abandonment (LG grade=0.69%) LG abandonment (LG grade=0.78%) ————— LG abandonment (No geological uncertainty) Figure 6.2b L G Zone abandonment boundaries with geological uncertainty - R E V model. 115 6.1.3 Comments regarding the influence of geological uncertainty The presence of geological uncertainty appears to have little effect on project value and low-grade zone development decisions, especially when there are substantial development expenditures to be incurred before such uncertainty is resolved. The results also suggest that information-gathering activities, such as exploration, are potentially an important part of mine valuation models. This conclusion can be inferred from the behavior of the LG Zone development boundary when geological uncertainty is resolved during the period of initial development. The boundary splits such that, in some price scenarios, it is optimal to begin LG Zone development without initiating a capacity expansion program. It is not often observed in the mining industry that additional low-grade reserves are developed when substantial higher quality reserves exist without ensuring that there is sufficient capacity to process these new reserves. It is more likely that management would start an extensive exploration program to reduce the uncertainty surrounding the LG Zone quality before committing to the actual development of the low-grade zone. A more informed development decision could then be made once new information had been obtained from the exploration program. The decision dynamics associated with an exploration program are complex. The FDMP model could be modified to incorporate such a program by expanding the project state space to reflect zone quality information levels. Managers could move between information levels by incurring exploration costs. The results presented here only suggest that gathering project-specific information is important in some situations (e.g. REV price environments). More sophisticated modeling of geological uncertainty and information gathering, such as lagged information-gathering and partial uncertainty resolution, is required before drawing more definitive conclusions. 6.2.0 Temporary zone closure behavior across a range of L G Zone grade outcomes Individual zone closure is a possible low-mineral price operating strategy that appears to be implemented more often in the mining industry than temporary closure of the full project. The previous chapter presented zone closure policy for the expanded project when the low-grade zone quality was 0.60%. The results were inconclusive in that temporary zone closure was not implemented except in select circumstances. The inclusion of LG Zone geological uncertainty allows the temporary zone closure policy of the expanded project to be studied across a broader range of LG Zone grade outcomes and some broad conclusions be made about selective zone closure. Note that the results presented here represent 116 the expanded project's zone closure policy when the LG Zone is brought into production at the earliest possible time (i.e. LG Zone development is started immediately and initial production is not delayed). 6.2.1 Individual zone closure policy Figures 6.3 and 6.4 display zone closure policies for lower LG Zone grade outcomes in NREV and REV price environments respectively. In both sets of figures LG Zone closure is significant as an operating alternative to project abandonment. In both sets of figures, the optimal policy is to close the LG Zone when the mineral price is below the data marker on the dashed line at management decision points. Figures 6.3a and 6.4a show that the preferred policy is low-grade zone closure with capacity abandonment when the L G Zone grade is 0.42%. This policy maintains its importance over the dual-zone production horizon until the reserves of each zone are almost exhausted. Figures 6.3b1 and 6.4b show that LG Zone closure policy is associated with temporary capacity closure when the remaining reserves in each zone are large and the LG Zone grade is 0.51%. This association changes to capacity abandonment as dual-zone mining continues. At highly reduced levels of zone reserves, low-grade zone closure ceases to be an attractive alternative to project abandonment. Figures 6.5a, 6.5b and 6.5c present the zone closure policy in a REV price environment when the LG Zone grade is 0.60%, 0.69% and 0.78%. These figures show that the preferred operating policy at large reserve levels is to close the high-grade zone and continue LG Zone production when the mineral price is below the data markers on the dashed line. Low-grade zone closure combined with temporary capacity shut-down is the preferred policy at these grades in only very select environments2. When price follows a NREV process, this type of behavior was not observed and full project abandonment is the preferred strategy at low mineral prices and higher LG Zone grades. 5.2.2 Comments regarding individual zone closure policy The importance of the ability to shut down individual zones is dependent the difference between the unit mineral operating costs3 of dual-zone production and those of each single-zone production strategy. This difference can be negative (dual-zone unit costs are less than the unit costs of both single-zone strategies) 1 The management action bar shown in Figure 6.3b, when there is 5.5 years of HG Zone reserves remaining, is due to the characteristics of the project cost model. In the price range of $0.45 to $0.47 per mineral unit, it is better to temporarily close excess capacity since at such low prices the project will likely be abandoned in the near future. 2 This behavior is consistent with other low-price situations where capacity is temporarily closed instead of abandoned. 3 Unit mineral operating costs are calculated as a total operating costs divided by the quantity of mineral produced. 117 1.00 0.90 0.80 § 0.70 c E .ts 0.60 c 3 ^ 0.50 u f 0.40 •a g 0.30 0.20 0.10 0.00 . — X - — - X - - — - X - - -TX- - - X - - - X - - - X - - - X - - - X - — X , 7.5, 7.5 6.5, 6.5 5.5, 5.5 4.5, 4.5 3.5, 3.5 2.5, 2.5 7.5,7.5 Remaining reserves box: The left-hand number indicates the amount of remaining HG Zone reserves and the right-hand number indicates the remaining LG Zone reserves at the critical price boundary. 8.0 7.0 6.0 5.0 4.0 3.0 2.0 Remaining high-grade and low-grade reserves (years) 1.0 0.0 1— - HG only/close capacity — -X — HG only/abandon capacity H Abandonment Figure 6.3a Zone closure boundaries for an expanded project (LG = 0.42%) - N R E V model. 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Management action horizon legend - 5.5 years of HG reserves | [' HG production only; temporary capacity closure: - - - -7 5 7 5 Remaining reserves box: The left-hand number indicates the _ amount of remaining HG Zone.reserves and the right-hand number indicates the remaining LG Zone reserves at the critical price boundary. 0.5, 0.5 7.5,7.5 6.5, 6.5 5.5, 5.5 4.5, 4.5 3.5,3.5 2.5, 2.5 1.5, 1.5 8.0 7.0 6.0 5.0 4.0 3.0 2.0 Remaining high-grade and low-grade reserves (years) 1.0 0.0 HG only/close capacity — -X — HG only/abandon capacity H Abandonment Figure 6.3b Zone closure boundaries for an expanded project (LG = 0.51%) - N R E V model. 118 7.5, 7.5 Remaining reserves box: The left-hand number indicates the amount-of remaining-HG Zone reserves.and.fhe rightrhand number . . indicates the remaining LG Zone reserves at the critical price boundary. - -x- - - * —x— -x 5.5, 5.5 4.5,4.5 3.5, 3.5 7.5, 7.5 6.5, 6.5 8.0 7.0 6.0 5.0 4.0 3.0 2.0 Remaining high-grade reserves (years) 0.0 h- - HG only/close capacity •X — HG only/abandon capacity H Abandonment Figure 6.4a Zone closure boundaries for an expanded project (LG = 0.42%) - R E V model. 7.5,7.5 Remaining reserves box: The left-hand number indicates the amount of remaining HG Zone reserves and the right-hand number indicates the remaining LG Zone reserves at the critical price boundary. - - - w — X - . X-^ - - X - - -X - - *X-5.5,5.5 4.5, 4.5 3.5, 3.5 7.5,7.5 6.5,6.5 7.0 6.0 5.0 4.0 3.0 2.0 Remaining high-grade and low-grade reserves (years) 1.0 0.0 HG only/close capacity — - X — HG only/abandon capacity H Abandonment Figure 6.4b Zone closure boundaries for an expanded project (LG = 0.51 %) - R E V model. 119 7.5, 7.5 Remaining reserves box: The left-hand number indicates the • • amount of remaining HG Zone reserves and the right-hand number indicates the remaining LG Zone reserves at the critical price boundary. 0.5, 0.5 4.5, 4.5 3.5, 3.5 2.5,2.5 7.5, 7.5 6.5, 6.5 5.5, 5.5 — i — 2.0 7.0 6.0 5.0 4.0 3.0 Remaining high-grade reserves (years) 1.0 0.0 HG only/close capacity - - -X - - LG only/close capacity H Abandonment Figure 6.5a Zone closure boundaries for an expanded project (LG = 0.60%) - R E V model. 7 5 7 5 Remaining reserves box: The left-hand number indicates the . . . . . .amojint of remaining HG Zone reserves and the right-band.number. . indicates the remaining LG Zone reserves at the critical price boundary. 7.5, 7.5 6.5, 6.5 5.5, 5.5 4.5, 4.5 0.5, 0.5 3.5,3.5 2.5, 2.5 1.5, 1.5 7.0 6.0 5.0 4.0 3.0 2.0 Remaining high-grade and low-grade reserves (years) 1.0 0.0 — - f — - HG only/close capacity — -O — LG only/close capacity H Abandonment Figure 6.5b Zone closure boundaries for an expanded project (LG = 0.69%) - R E V model. 120 7.5,7.5 Remaining reserves box: The left-hand number indicates the amount of remaining HGLZone reseryes and the .right-hand number . . indicates the remaining LG Zone reserves at the critical price boundary. 5.5, 5.5 4.5,4.5 3.5, 3.5 7.5, 7.5 6.5, 6.5 1.5, 1.5 8.0 7.0 6.0 5.0 4.0 3.0 2.0 Remaining high-grade and low-grade reserves (years) 0.0 HG only/close capacity •G — LG only/close capacity •4 Abandonment Figure 6.5c Zone closure boundaries for an expanded project (LG = 0.78%) - R E V model. or positive (dual-zone unit costs are greater than unit costs of the only H G Zone production strategy) depending upon the quality difference between the H G and L G Zones. Large quality differences between the zones create a positive difference between unit operating costs of dual-zone production and the H G Zone only production strategy. In this example, when the L G Zone grade is 0.51% (0.42%), the unit operating cost of dual-zone production is $0.63 ($0.67) per mineral unit and the unit cost of only high-grade zone production is $0.60 per mineral unit. Smaller quality differences between the zones create a negative unit operating cost difference since the EoS associated with dual-zone production reduces this strategy's unit operating cost below that of the H G Zone only production strategy. When the L G Zone grade is 0.69% (0.78%), the unit operating cost of dual-zone production is $0.56 ($0.53) per mineral unit and the only H G Zone production unit cost remains $0.60. Positive unit operating cost differences between dual-zone and single-zone production create a management incentive to close the poorer qualify zone. This incentive is the result of the operating cost savings generated by this action. For example, when L G Zone grade is 0.51% and there are 7.5 years of H G Zone reserves (see Figure 6.3b), management closes the L G Zone and excess capacity when the 121 mineral price drops below $0.57. The project's total operating losses change (net of transition costs) from $1,492 million when both zones are producing to $0,453 million when only the HG Zone is operating. Figures 6.3 and 6.4 delineate the zone closure boundaries when there is a positive unit operating cost difference in NREV and REV price environments. These figures show that the decision to selectively close a low-quality zone is dependent upon the magnitude of the unit operating cost difference such that the decision to close the LG Zone is made at higher mineral prices as the unit cost difference increases. Figures 6.3 and 6.4 show that the zone closure boundary is also influenced to a lesser extent by the size of zone reserves. Increasing reserve size increases the incentive to close the lower quality zone because larger reserves provide a longer production time horizon over which the costs savings necessary to offset the costs of zone closure can be generated. In these figures, the zone closure option is important over the full dual-zone production horizon when LG Zone grade is 0.42% while it is only important over part of this horizon when the LG Zone grade increases to 0.51%. The choice of between alternate methods of managing excess capacity is another aspect of zone closure that is dependent upon differences in unit mineral operating costs. Large differences in unit mineral operating costs reduce the incentive to maintain closed capacity because the possibility that mineral prices will increase to a level necessary to support dual-zone production are less likely. As differences in unit operating costs decrease, it becomes preferable to maintain excess capacity in a closed state since the probability that mineral prices will increase enough to support dual-zone production is greater. This result can be observed in Figures 6.3 and 6.4. When low-grade zone quality is 0.42%, excess capacity is abandoned upon LG Zone closure whereas, when LG Zone grade is 0.51%, excess capacity is maintained in a closed state upon LG Zone closure at larger reserves levels. Figures 6.5b, 6.5c and 6.5d show selective zone closure boundaries, in a REV price environment, when there is negative unit cost difference between dual-zone and single zone production. The zone closure boundaries presented in these figures differ from those previously because the most prominent zone closure boundary requires closure of the HG Zone instead of the LG Zone. This behavior indicates that, at large reserve levels and higher LG Zone grades, the optimal policy is to preserve the high-grade reserves by closing the HG Zone. In this situation, the operating losses created by mining the LG Zone are offset by possible higher profits gained by re-opening the high-grade zone in higher mineral price environments. Sagi (1998, forthcoming) has previously shown that, in REV price environments, mine 122 management may find it optimal to decrease the project's cutoff grade in response to low mineral price. This action was justified by the observation that the expectation of future price increases, derived from the mineral price reverting to the long-term price median, causes management to preserve higher quality reserves until price levels improve. Temporary closure of all ore zones is also a possible strategy for low mineral price situations. The addition of a full project closure strategy would affect management policy at low mineral prices in that it would become the preferred strategy at some price outcomes where project abandonment or selective zone closure is now preferred. In particular, it may have its greatest impact on the strategy of closing the HG Zone in low REV price environments because full project closure may dominate high-grade zone closure in all low price outcomes (i.e. it is potentially better to preserve the reserves of both the HG and LG Zones until prices improve). However, it is not considered in this example because the dynamics and costs of either full or temporary project closure in the mining industry have not been studied in detail1. Cost curves exist to estimate the costs associated with building and operating mining projects but there do not appear to be similar such tools for estimating the costs associated with either maintaining or abandoning excess capacity. Given the lack of closure cost data, it was considered better to observe selective zone closure in isolation with "guess-timate" cost figures than to risk underplaying the role of selective zone closure with an ill-conceived project closure model that incorporated both partial and full closure. 5.3.0 Conclusion This chapter investigated the influence of discrete geological uncertainty and the factors favoring selective zone closure in low mineral price environments. Discrete zone geological uncertainty, that is fully resolved at a particular zone state, was determined to have little value effect, given the project and price parameters used. It had a more ambiguous influence on operating policy in that the critical price at which the LG Zone is developed was not affected significantly until the HG Zone reserves were exhausted. Extension of the FDMP model to include exploration programs may be warranted since such It has been this author's experience that temporary closure costs and the strategy of temporary mine closure are not considered an important aspect of mine valuation until low mineral prices force the issue. A few years ago, the author initiated a general discussion regarding real options at a large mining company only to have the discussion dismissed by senior management as irrelevant since, in managements opinion, temporary closure was not a serious consideration for mines that were not exploiting small placer deposits. Moel and Tufano (2000) have completed econometric research that demonstrate that temporary mine closure is a viable strategy in the mining industry but they do not provide an indication of the costs associated with this action. 123 programs may increase project value if they allow management to avoid wasting development capital on low LG Zone grade outcomes. Selective closure of the L G Zone was demonstrated to be a viable strategy in low mineral price environments when there is a positive unit operating cost difference between dual-zone and single-zone production. This strategy became more important as zone reserve levels increased or the magnitude of positive differences in unit costs between dual-zone and single-zone production increased. When there was a negative unit operating cost difference between strategies, management occasionally elected to close the HG Zone as a method of preserving high-grade reserves until mineral prices recovered in REV price environments. 124 Chapter 7 Conclusion This dissertation has presented a new project structure model, called the Flexible Discrete Mine Production (FDMP) model, which explicitly recognizes the influence of a mineral deposit structure on the economic management of the deposit. This influence is recognized through the division of the mineral deposit into zones that are differentiated by quality, size and spatial orientation. The mine planner then specifies an exogenous discrete development and production plan for each zone that defines zone operations when the zone is active. Within the FDMP model, project operation is broken into discrete intervals that correspond to the interval length of the zone plans. Management operates the project by choosing an operating mode, at the start of each project interval, from a set of possible operating strategies whose membership may be limited by geological structure and by capacity restrictions. Each operating mode indicates the zones that will be active and the changes that will be made to the project's capacity stock during the next project period. The cash flow generated over a project period within the FDMP model reflects individual mineral production, zone-specific capital expenditures, multi-purpose capital expenditure for mineral processing facilities, economies-of-scale benefits and charges such as the costs associated with capacity under-utilization or temporary zone inactivity. The model may also be extended to include independent discrete zone geological uncertainty where management only resolves zone quality at a particular stage in the zone plan. A solution is obtained for the valuation model by first creating a project state tree using a graphing algorithm and then determining the project's value and operating policy with a dynamic programming approach that uses the upwind projected SOR finite difference technique to solve a valuation partial differential equation. The model was demonstrated in Chapter 4 with a stylized two-zone mining example. In this valuation exercise, management must determine the development timing of a satellite low-grade zone given that there are continuing production operations in nearby high-grade zone. The FDMP value results are compared to three low-grade zone development SPPs (no low-grade zone development, immediate low-grade zone development, and delayed low-grade zone development) that are valued within the DCF and MAP valuation frameworks. The comparison showed that the DCF method, using a 10% discount rate, 125 placed a higher value on the project than values calculated with the MAP framework because the MAP method applied a greater risk adjustment to the project cash flows due to the operating leverage. Both project structure models assigned comparable MAP values to the project but the extra flexibility incorporated into the FDMP model added approximately 6% to the SPP project value, depending on the mineral price environment. The FDMP model results also provided greater operating policy detail in that it delineated a low-grade development boundary that spanned the full operational horizon of the high-grade zone while the SPP model was only able to give a signal that indicated immediate or delayed development. The choice of project structure model also had a significant effect on the immediate low-grade zone development decision. The SPP model signaled immediate low-grade zone development at a price that was 11 % to 16% less than the development price determined within the FDMP model. This result was due to the SPP model assessing a smaller opportunity cost to the immediate development of the low-grade zone. This finding could become important as other risk management techniques, such as value-at-risk, are adapted to the management of real assets. Chapter 5 investigated the sensitivity of the valuation results to fluctuations in price process and project parameters. Increases in price volatility caused the low-grade development boundary to rise because the opportunity cost associated with early low-grade zone development increases with volatility. The effect of changes in the price of mineral risk on the low-grade development boundary depended on the underlying price process. In non-reverting (NREV) price environments, the development boundary fell as the price of mineral risk initially increased due to the risk-adjustment changes having a disproportionate effect on the long-term cash flows that are part of the early low-grade zone development opportunity cost. In reverting (REV) price environments, the development boundary decreased with the price of mineral risk due to the value of early development being more sensitive to changes in mineral price risk than the value of delayed development. Changes in economies-of-scale benefit, which are generated when both zones are operating together, cause the early development boundary to drop substantially as the benefit increased. Chapter 6 introduced discrete low-grade zone geological uncertainty to the FDMP model. Discrete zone geological uncertainty was shown to have little effect on overall project value but an increasing influence on the low-grade zone development decision as the cost of development decreased. Geological 126 uncertainty appears to have the most effect when price follows a reverting process, development capital expenditure is minimized, and the uncertainty is resolved sooner rather than later. This relationship was observed most readily when the high-grade zone reserves had been exhausted and low-grade zone development was in progress. For example, in a reverting price environment, the presence of geological uncertainty reduced the development price by 8% at the start of low-grade zone development while this reduction increased to almost 20% immediately before the last development expenditure. Selective zone closure in low-mineral price environments with dual-zone operations in effect was also investigated with the various low-grade zone quality outcomes that were generated by the geological uncertainty model. The temporary closure of the low-grade zone was shown to be a viable strategy in this environment when there was a positive mineral unit cost difference between dual-zone and single-zone production. This strategy increased in importance with a increasing positive unit cost difference between dual and single zone operations and expanding reserve levels. In addition, the excess capacity created by closing one of the zones was more likely to be abandoned when the unit cost difference was large than when it was small. Excess capacity also tended to be closed temporarily when there was a small positive unit cost difference between dual and single zone production. In a reverting price environment, the high-grade zone was sometimes closed in preference to the low-grade zone when there was a negative unit cost difference between the two operating modes. This action was associated with the desire to conserve the high-grade reserves until the mineral price improves. 7.1.0 Future research The description of the FDMP model and the valuation results generated in the two-zone mine example suggests several areas of future research that include extensions to FDMP model framework and more detailed analysis of mine management in low mineral price environments. The FDMP model framework, as presented, does not allow management to endogenously alter the parameters of the zone plan. In some situations, zone operations are inter-dependent in that the closure of one zone may allow production from another zone to be increased. One solution may be to develop a decision tree structure that overlies the project state space described in the FDMP model. In effect, the project state space becomes just one of several project state spaces that are represented by branches within the decision tree. In such a model, movement between the branches of the decision tree would (partially) describe the production relationship between ore zones. 127 The FDMP model also has the potential to become exceptionally complex as the number of zones increases. Further research is required to reduce this complexity through the better management of information and the creation of suitable approximations for project structure. Efforts to improve information management techniques would include developing advanced search algorithms that assist with data mining and creating techniques for presenting decision information in a format that can be understood by managers (e.g. operating policy is difficult to assess when it is derived from a three-factor price process). Better approximations of project structure may include a coarse project state space that is paired with value interpolation or extrapolation for situations that are not aligned with state space grid. Selective zone closure was shown to be a possible strategy in low mineral price environments. However, there appears to be little public information regarding transition costs (e.g. capacity abandonment) and cash flow charges (e.g. the maintenance of closed zones) that are incurred when reducing the scale of mining operations. 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Howison (1993). Option Pricing: Mathematical Models and Computations. Oxford, UK, Oxford Financial Press: 457p. Wilmott, P., J. Dewynne and S. Howison (1995). The Mathematics of Financial Derivatives - A Student Introduction. Cambridge, UK, Cambridge University Press: 457p. Wilmott, P. (1998). Derivatives - The Theory and Practice of Financial Engineering. New York, NY, John Wiley and Sons: 739p. 139 Appendix 1 Cost parameters for the two-zone mine valuation example The details of the calculation used to determine the costs in Chapter 4's valuation example are presented. The cash flows in a reverting price environment for the current mine plan (HG Zone depletion only) scenario, the production expansion scenario (immediate LG Zone development) and the production replacement scenario (delayed LG Zone development) are also presented in section 2. A l . l Overview of cost parameters Project cost parameters were calculated using the open-pit cost curves presented in Mular and Poulin (1998). Their cost curves have the same structure as the cost curves detailed in O'Hara (1980) but their cost data has been updated to reflect inflation. O'Hara's cost data used 1978 as a base year with a cost index of 565. The costs in Mular and Poulin are based on a cost index of 1400 and reflect a cost multiplier of 2.48 (i.e. the cost figures are 2.48 the magnitude of the O'Hara cost figures). This example uses cost parameters for a base year of 1998 and a cost index of 1100. They are 0.786 (1100/1400) the magnitude of the Mular and Poulin cost figures. The project is operated 350 days per year. The project produces 9000 tonnes per day (3.15 million tonnes annually) of ore and waste when either zone is operating alone and 18000 tonnes per day when both zones are operating. The project has a stripping ratio of 2.00 (i.e. half of the mined material is waste) and the mill processes 4500 tonnes of ore per day (1.575 million tonnes annually) if only one zone is operating and 9000 tonnes per day of ore if both zones are operating. ALIA Mine capital expenditures Capital costs for the LG Zone are categorized into zone-specific and multi-purpose expenditures. The LG Zone-specific expenditures are common to both the production expansion and replacement scenarios. These expenditures are incurred specifically to prepare the LG Zone for production operations and they are represented by the cost categories of site preparation, pre-production stripping, feasibility study (site preparation), construction supervision and support staff. The calculation of LG Zone-specific capital costs is detailed in Table Al . l . n U 'J « 9 , H S u •tt o I a 140 v© VO oo O II "3 o v^ CJ 5 3 O ry cd 2- -° U CL) cj S X X . -ap a '"3 e t .3 u cd <= 00 g. erf s rt* a a M o> 1) rt 60 C X cj cj X, o CJ > 5 60 § .2- e 3 . § CA K a) O c 2 ej X > o •— cj ~ m > ° B O W o cj "3 > o £ CJ ej T3 l-i 3 X CJ > O cj O Crf CJ c o N cd D. X cj ^ CA <D .3 CA 3 cd 5 X 3 3 1) - E IS '3 o a -cj c -3 cj cd Li .2 . 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'3 cr u oo S c .2 u cd § ^ CJ cj C CM 01 U B cd s .5 w B O 'in 'E a. s IA B O •r; u 2 o U o cd cd o 3 o 60 C cd erf vo (--CN o | to 2 tt o, Q. a s 3 o H •a c CD a. x CD rt a a, o > CD •a o CD a, o N a < 141 Multi-purpose mine capital expenditures are only incurred if the LG Zone is developed while HG Zone mining operations are in progress. They reflect the expenditures necessary to increase the mining capacity of the project and they may include such items as additional trucks and drill rigs. Table A1.2 outlines the capital expenditure calculation for immediate LG Zone development. The multi-purpose capacity expenditures are found in the open-pit equipment and feasibility (mine equipment) cost categories. The total multi-purpose capacity expenditure incurred is $3,606 million and this number is calculated by determining the difference between the total LG Zone expenditures in Table A1.2 and the total LG Zone-specific expenditures outlined in Table Al . l . This figure is one component of the total mine/mill expansion expenditures found in the expansion scenario cash flows of Table 4.3b and A1.5b. A 1.1.2 Mill capital expenditures Mill capacity expenditures represent multi-purpose capital expenditures in this example because the resulting facilities can be used to process ore from any part of the mine. Table A1.3 outlines the total mill expenditures incurred if the LG Zone is developed as part of a production expansion strategy. Note that this figure is calculated from the cost curve structure as an incremental amount necessary to raise the mill capacity from 4500 tonnes per day to 9000 tonnes per day. The total multi-purpose capital expenditures for the example are $25,586 million and this figure represents the sum of the multi-purpose mine ($3,606 million) and mill ($21,979 million) capital costs. Al.l.3 Daily operating costs Daily project operating costs when there is only a single zone active are calculated in Table A1.4. If both zones are operating, economies-of-scale benefits may be produced such that doubling production does not require twice the operating expense. Labor costs in this example are $30 519 per day ($10,682 million annually) when there is only a single zone operating. This represents 57.1% of the total operating costs. A1.2 Scenario cash flows in reverting price environments Project scenario cash flows are presented in Tables A1.5a, A1.5b and A1.5c. 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In the MAP framework, the forward price may be interpreted as a risk-adjusted expected price. For non-reverting (NREV) price environments, this interpretation takes the form: FP0 (T) = E 0 [P(T)] • exp (-PRiskm<rT) (A2.1) where: FP0 (T) = the current mineral forward price at time T. E 0 £ P ( T ) ] = the expected mineral price at time T. PRiskm = the price of mineral risk, cr = mineral price volatility. T = time in years. The risk adjustment in equation A2.1 is produced by taking the limit of equation 3.7 as the mean reversion factor approaches zero. The expected mineral price can be calculated from the long-term price median, such that in NREV price environments: E 0 [ p ( T ) ] = Med 0 [p(T)]-exp ' 1 2^ - C T T 2 (A2.2) where: Med 0 [ P ( T ) ] = the long-term median mineral price at time T . Equation A2.2 is derived by taking the limit of equation 3.4 as the mean-reversion factor approaches zero and substituting the resulting associated variance into equation 3.6. 149 Combining equations A2.1 and A2.2 produces the following relationship between the long-term median price and the forward price in NREV price environments: The term 1/2<T2 - PRiskmo" drives the convexity relationship. Initially, the effect of risk adjustment dominates the forward price calculation. This dominance changes once o~ equals PRiskm in magnitude. At this point, the expected price appreciation factor re-asserts itself causing the forward price to increase towards the long-term price median. Figure A2.1 illustrates the convexity relationship between the rates of expected price appreciation and mineral risk adjustment using the NREV price data of equation from Table 4.1 and a time horizon of 2.5 years. (A2.3) A similar relationship holds in reverting price environments. This is shown in Figure A2.2. 150 20% 15% 4 c 10% 4 5% c c < 0% Volatility at which the long-term price median and the forward price are equal. Maximum difference between the rates of price appreciation and risk adjustment. ote 10% 20% 30% 40%. 50% 6d% -5% Price volatility — - — -Expected price appreciation Risk adjustment • Rate differential Figure A2.1a Rates of mineral price appreciation and risk adjustment (NREV price process). 20% 15% -5% Volatility at which the long-term price median and the forward price are equal occurs at a volatility of 64%.. Maximum difference between the rates of pricg^ appreciation and risk adjustment is_ at a volatility of 32% Price volatility Expected price appreciation Risk adjustment •Rate differential Figure A2.1b Rates of mineral price appreciation and risk adjustment (REV price process). 151 Appendix 3 Overview of the FDMP project structure valuation computer model The organization of the F D M P project structure computer model and the details regarding finite difference method used to solve equation 3.29 are provided in this appendix. A3.1 Overview of the computer model The computer model consists of three components that are presented in Figure A3.1. The first component allows the input of project information such as zone mine plans, possible operating modes and their interaction, the capacity stock levels and the links between these levels, and parameters for the numerical method such as grid boundary prices. The second component builds the project state tree using the data input provided by the first component. The third component determines project value and operating policy once the project state tree has been delineated. A3.1.1 Construction of the project state tree The project state tree is generated using the breadth-first graphing search algorithm detailed in Chapter 23 of Cormen, Leiserson, and Rivest (1997). A graph, G, is defined as the union between a set of objects, V , and a set of binary relationships, E, that specify the interaction between the members of V . Given a starting point in G, a graphing search algorithm systematically analyses the set of binary relationships to determine the graph's structure. In the F D M P model, the project state space, P, forms the object set, V , and the possible active zone combinations and project capacity stock changes that are available at each project state form the binary relationship set, E. For this discussion, a project state node (PSN) is defined as a point in the project state space, P r , and it defines the development and depletion stage of each zone and the level of open, unfinished, and closed project capacity stock. It exists within the program as an object that holds information on zones plan stages, capacity levels, feasible operating modes and the links to adjacent PSN that management may reach during the next period. An extended project state (EPS) is defined as a point in the project state space that has the project's current operating mode defined. An EPS exists within the program as a data structure that is only present while the project state tree is being built (i.e. only while the second program component is executing). 152 Start ( l ^ R e a d proiect i n fo rma t ion^ ) c Bui ld project state c Value proiect I End Figure A3.1 F D M P project structure valuation program components. Figure A3.2 provides an overview of the procedure used to build the project state tree. A linked list1 is created to hold all the PSN that are generated when the project state tree is built and then a queue2 is created to organize the EPS while the project tree structure is explored. An initial PSN is created and used to seed the PSN linked list and the EPS queue. A loop that comprises three actions is executed that searches the project state space for the structure of the project tree. The first action retrieves an EPS from the head of the queue. This EPS is then analyzed to determine the links between it and adjacent PSN. These links may be created over the next management interval by retaining the current EPS operating mode, by switching the project to an alternative active zone combination or by changing the level of project capacity stock. Once all the current EPS links have been discovered, the current EPS is removed from the queue and the loop is restarted. This loop is executed until the EPS queue is empty, which 1 A linked list is a data structure in which objects are arranged in a linear order. The order of the list is determined by a pointer, which links neighboring objects. The FDMP project structure valuation program uses a double-linked list that allows the PSN list to be traversed from in either direction. 2 A queue is similar to a list in that it is a linear data structure that provides links between data objects. However, objects are added to and removed from queues in a first in, first out policy. This policy requires new data members to be added to the tail of the queue while members that are to be removed are taken from the head of the queue. 153 Start: Build project state tree (see Figure A3.1) Create list for PSN c Create queue for EPS C r e a t e initial PSN and seed EPS queue and P S N l l s t ^ End M -EPS queue empty EPS queuq not empty C^Retrieve first EPS in queue^> eProcess EPS (see Figure A 3 ^ ) C^_Remove EPS from queuej^>~ Figure A3.2 Overview of the project state tree construction algorithm. indicates that the project structure has been fully defined. At this point, the second program component stops executing. Figure A3.3 outlines the procedure used to determine the links between an EPS and its adjacent EPS. The first step of the procedure is the creation of a list that contains all adjacent EPS indices. These indices define the project state and the operating mode of the adjacent PSN. A loop is then executed that retrieves each member of the adjacent EPS index list in turn. The PSN list is searched to determine if a PSN exists with a matching project state index. If such a PSN does not exist, then a PSN object is created 154 Start: Process EPS (see Figure A3.2) I •^Create list for adjacent EPS indices) Adjacent EPS l i s l ^ v ^ fiffi is empty. < S > * Adjacem EPS list is not ;mpty. (^Retrieve next adjacent EPS index^) <C^ L i n k current PSN to adjacent I>S^> ^Search PSN list for matching PS inde IF> Matching PSN •(^Create new PSN and add to PSNlisT> ^ does not exist. Matching PSN does < ;xist. 4> Adjacent EPS operating mode has not been activated at the Adjacent EPS adjacent PSN. bperating mode has been ac ivated at the adjace: it PSN. C^Activate operating mode at adjacent PS^-} d^Create adjacent EPS and add to tail of EPS queued Figure A3.3 Program flow to determine links between the current EPS and all adjacent EPS. 155 and added to the PSN list. If such a PSN already exists, the matching PSN is checked to determine if the adjacent EPS operating mode has been activated at this PSN (or in graphing terminology, the adjacent EPS has been previously discovered)3. If the adjacent EPS operating mode is not active at the adjacent PSN (this includes an adjacent PSN that was created in the current loop), then it is activated and a new EPS is created with the adjacent PSN and the operating mode of the adjacent EPS index. The new EPS is then added to the tail of the EPS queue and the program flow recombines with the path followed by an adjacent PSN with a previously activated operating mode. The loop finishes by constructing a link between the current PSN and the adjacent PSN using the adjacent EPS operating mode. A3.1.2 Valuation of the project state tree Figure A3.4 details the procedure used to value the project state tree. The first step is the creation of a stack data structure4 to hold the PSN that are currently part of the value path. The value path outlines an operating policy through the project state space. It is anchored by the initial project state and it ends with either a terminal PSN or a PSN for which all adjacent PSN have all been valued. The value path is seeded with the initial project state. Project valuation begins with the start of an exterior valuation loop that continues until there are no unvalued PSN remaining in the value path. The first step in this loop is a search algorithm that runs until either a terminal PSN or a PSN with all adjacent PSN fully valued is found. Any unvalued PSN that the algorithm finds before the stop condition is reached is added to the value path. When the search algorithm has finished, the first of two interior loops is started to value the last PSN in the value path. This loop determines the present value (PV) of each operating mode that is valid at the current PSN. The second loop is then started to determine first the alternative payoffs that are available to each operating mode and then this loop calculates the final payoff of each operating mode. The current PSN is removed from the value path and the exterior loop condition retested. The valuation procedure stops when the value path is empty because the project has been fully valued. 3 Note that the activation test ensures that an EPS can only be activated (or discovered) once while the project state tree is built. 4 A stack data structure is similar to a queue data structure except that it is managed with first-in, last-out policy. This implies that PSN are added and removed only from the tail of the stack. 156 Start: Value project (see Figure A3.1) 1 Cjcreate a stack to hold value-path PSN§^ Value path P S N / v E n d « « all valued. Value phth PSN not all valued. (^Remove PSN from value-path stack^> Find PSN in project state tree to valued PV calculated for ^ some operating modes^ PV cd all operating modes. Calculate PV of retaining current operating mode lculated for Calculate alternative payoffs for current operating mode , Final payoff . — <fe>udtul-Uul fui • r ^ ° m P * r e operating mode PV to alternate J— . ^payof fs to calculate final payoff ^ operating modes. ±-J-—*• Final for al some payoff calculated operating modes. Figure A3.4 Program flow to calculate project value and operating policy. 157 A3.2 Upwind projected successive-over-relaxation finite difference method The partial differential equation (PDE) used to value the project was approximated with the upwind projected successive-over-relaxation (PSOR) finite difference method (Huang and Pang, 1998)5. This method was chosen because it does not become unstable when there is an adverse sign change in the coefficient of the first partial derivative with respect to price (called the convection term) in the valuation PDE (equation 3.30). Adverse sign changes in the convection term may be a problem for mine valuation exercises when the price process of output mineral is mean reverting. The finite difference method approximates Equation 3.30 over an area delineated in (T, S) space. A grid may be superimposed over this area by dividing the T -dimension into "m" equally spaced nodes at a distance of A T apart and by dividing the price dimension into "n" equally spaced nodes at a distance of AS apart. The lower boundaries for this grid are T = 0 in the time dimension and S = S L B > 2AS in the price dimension. The grid's upper boundaries are T = T M in the time dimension and S = S L B + NAS in the price dimension. Points within this grid have an index of the form (S L B +nAS, mAr) where m = (0, 1,...,M) and n = (0, 1,...,N). The following convention is used to simplify expressions involving project value at (S L B + nAS, mAr) : V n m = V(S L B +nAS, mAr) The implicit upwind PSOR finite difference method6 uses the following difference equations to estimate the terms of equation 3.30: 9V(S, r ) _ V n m - V n m A T (A3.1) av(s, T ) as 4V n m + 1 -3V n " n+2 2AS 4 V m , - 3 V m - V m , ^ v n - l - ' v n vn-2 2AS if ( r - c )S>0 if ( r - c )S<0 (A3.2) 5 Wilmott et al (1993), Wilmott et al (1995) and Wilmott (1998) provide detailed discussions regarding the use of the PSOR method to solve the American option valuation problems. 6 See Huang and Pang (1998) for the upwind difference equations used within the explicit and the Crank-Nicolson finite difference methods. 158 a 2 v(s ,T) = v ; + 1 - 2 v ; + v n n as2 ( A S ) 2 (A3.3) By substituting equations A l . l , A1.2 and A1.3 into equation 3.30, the following finite difference approximation is produced for the grid point, (S L B +nAS, mAr) : e„V n m + 2 +a nV n m + 1 +b n V n m + c n V n m 1 +d nV n m_ 2 = (A3.4) where: Q - 2 S 2 A T 2 ( r - c ) + A t a " ~ 2(AS) 2 AS ( r - c ) + = max(r -c , 0), the non-negative part of the convection term. . , CT 2 S 2 At 3 ( r - c ) S b „ =1 + + ——,—T—+ rAt (AS) 2 2(AS) o- 2S 2At 2 ( r - c ) "SAt C n ~ 2(AS) 2 AS ( r - c ) = m a x ( - ( r - c ) , 0), the non-negative part of the convection term. ( r - c ) " S A r d„ s 2 (AS) _ ( r - c ) + S A r 2(AS) Note that in this finite difference equation A3.4 the terms d n and en are never both non-zero at the same price. 159 Using equation A3.4, project value can be determined at each grid point using the matrix system: jyj Upwind (A3.5) where: jy|Upwind _ N c N d N 0 • 0 0 0 0 0 0 N- l b N - l C N - 1 d N - i • • 0 0 0 0 0 0 N-2 a N - 2 b N - 2 C N - 2 • 0 0 0 0 0 0 0 e N - 3 a N - 3 b N - 3 • . 0 0 0 0 0 0 0 0 0 0 • e3 a 3 b 3 c 3 d 3 0 0 0 0 0 • 0 e 2 a 2 b 2 - 2 0 0 0 0 • 0 0 ei a, b, c, 0 0 0 0 • 0 0 0 e 0 a 0 b 0 vm = V N V, V, V, N-l N-2 N-3 v3n v," V m - n V m - P V m V N a N VN+1 C N V N -V m - p vm V N - 1 C N VN+1 V N - 2 V N - 3 v3m v2m V 0 m -CoV™ - d 0 V _ m 2 The project values V N + 2 , V N + 1 , V_,, andV_ 2 are not found on the finite difference grid and must be determined. These values are calculated as: V 0? = E[v(S, T = ( m - l ) A T ) | s ? =S L B +n'As] e'^ (A3.6) 160 where: n' = N + 2 , N + 1, - 1 , - 2 E [ ] = the expectation operator for the risk-adjusted mineral price distribution, r = riskless interest rate. The finite difference equation system A3.5 can be solved using the iterative PSOR method7. If there is an abandonment option available, this equation system is supplemented with a vector constraint that corresponds to the project's abandonment value as calculated in equation 3.34. This vector comprises N elements for which the value of each element is: g(S L B +nAS, mAT)-g™ = < The PSOR method begins with an initial guess, V A r ' 0 , of the solution at time slice x = A T and then successively improves on this solution until the error between two successive iterations is within a specified bound. The PSOR method then advances to the next time slice x = 2Ax to begin the iterative solution process again. Project value is estimated at each successive time slice using as input the project values from the previous time, V m _ 1 , until the time slice x = M A T is reached. For a given time step mAT, the initial guess for project value at mineral price, S L B + nAS , is: max (V n m _ 1, g"1) continuous abandonment available V ' (A3.8) V n m ~ ' no abandonment available At time step mAT, each value iteration begins at the lower price boundary and works its way towards the upper price boundary, S L B + NAS . The k , h iteration of the PSOR algorithm at mineral price S L B + nAS first calculates an intermediate project value U™'k by rearranging equation A3.4 such that: , ( T M - I T I A T ) . . | ~ B A B D + — - C F . (S L B + nAS) discrete cash flow T M (A3.7) I - B A B D continuous cash flow Iterative methods work by starting with an initial guess for the solution and then successively improving on the estimated solution until it converges to within a reasonable limit of the exact solution. Wilmott et al (1995) discuss the relationship between the Jacobi and Gauss-Seidel iterative methods and the PSOR method. 161 U m, k n n m-1 -e„V: m, k-l - c n VJ d nv;_ 2 k] (A3.9) Note that the project values for prices higher than S L B + nAS use the iteration index k - l because they have not yet been updated by the k t h iteration. An over-correction calculation is then used to increase the rate at which V™ converges. This calculation increases the difference between the previous project value, V n n k ~ 1 , and the intermediate value, U™k, by an over-correction factor, co , such that8: At this point, the iteration calculations begin for price S L B+(n + l)AS if there is no continuous abandonment option. When abandonment is available, the iteration completes the following calculation before moving to the next price point: (A3.10) where: 0<a><2. v: n m, k _ = max (V*" ,g?) (A3.ll) The optimal value of CO for rapid convergence is between 1 and 2 since a factor of less than 1 would cause an under-correction. Details of determining the optimal over-correction factor can be found in Morton and Mayers (1994). 

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