/133 0 INDUSTRY AND ENERGY DEPARTMENT WORKING PAPER ENERGY SERIES PAPER No. 39 Decision Making Under Uncertainty- An Option Valuation Approach to Power Planning FILE COPY Report No.:11330 Type: (MIS) Title: DECISION MAKING UNDER UNCERTAII Author: Ext.: 0 Room: Dept.: August 1991 AUGUST 1991 The.World Bank Industry and EnergyDepartment.PR. . x, . . : . , ., > , . . .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~U Th Wol BankX *nur and Enrg D*atmn PRE DECISCON MAKING NDER UNCERTAINTY AN OPTION VALVATION APPROACH TO POWER PLANNING by Enrique Crousillat and Spiros Nartzoukos August 1991 Copyright (c) 1991 The World Bank 1818 H Street, N.W. Washington, D.C. 20433 U.S.A. This report is one of a series issued by the Industry and Energy Department for the information and guidance of Bank staff. The report may not be published or quoted as representing the views of the Bank Group, nor does the Bank Group accept responsibility for its accuracy or completeness. This paper addresses the problem of irreversibility of investments, a topic related to the recently evolving "real options" literature. In the presence of alternatives, the opportunity to invest in a large "irreversible" project with uncertain benefits and costs is similar to a financial "call" option. In exercising this option one forgoes the potential gains from postponing the investment decision. Conventional economic appraisal techniques ignore this opportunity cost, thus usually tending to accept larger inflexible projects. This problem is particularly important in power investment decisions, where the presence of uncertain demand, fuel prices, construction cost, etc. makes the opportunity cost too high to ignore. The present paper includes an exposition of the options approach, a discussion on the main model and its parameters, and IENED's experience in applying this innovative approach in four power planning case studies. ACKNOWLZDGKBMNT8 The authors are grateful for the contribution of Witold Teplitz- Sembitzky and James Paddock (Tufts University) who provided advice from the init4.al stages of our work and supervised several parts of this analytical effort. We also thank John Besant-Jones for his careful review of an earlier draft. The valuable contribution of Hernan Garcia and Rama Skelton, who reviewed the paper, is also acknowledged. Table of Comtents EXECUTIVE SUMMARY. . . . . . . . . . . . . . . . . . . . . .i 1. INTRODUCTION . . . . . a . . . . . . . . . . . . . . . . . 1 2. REAL OPTIONS ........... . . ...... ... . 5 The Perpetual option . . . . . . . . . . . . . . . . . . 6 3. CASE STUDIES . . . . . . . . . . . . . . . . . a . . . . 12 3.1 Costa Rica Case Study . . . . . . . . . . . . . . 13 Costs and Benefits . a . . . . . . . .*. . . ..a 14 Estimation of Option Model Parameters . . . . . . 16 Results .............. .*.*. . 18 3.2 Hungary Case Study . . . . . . . . . . . . . . . 19 Costs and Benefits ............... 20 Estimation of Model Parameters . . . . . . . . 22 R*esults .. .. .. .. .. .. .. .. .. . .. 22 3.3 Manantali Hydropower Plant . . . . . . . . . . . 23 Costs and Benefits .. .. ..... ..... 24 Estimation of Model Parameters . . . . . . . . . 25 Results . .. .. .. .. .. .. .. .. . . .. 26 3.4 Optimal Timing of Transmission Line Investments . 27 Costs and Benefits . . . . . . . . . . . . . . . 28 Model Parameters ............................... 29 4 Oesults . . . . . . . . . . . . . . . . . . . . . 29 4. CONCWaSIONS .................31 ANNEX 1 - The Perpetual Option . .* ....... . . . .. 33 ANNEX 2 - Costa Rica Case Study . a . . . . . . .* . . .. 48 ANNEX 3 - Hungary Case Study . . . . . . . . . . . . . . . 56 ANNEX 4 - Optimal Timing of Transmission Investment . . . 68 ANNEX 5 - Glossary . . . . . . . . . . . . . . * . . . . . 69 REFERENCES . . * . . . a . . . . . . a . . . . * * . . . . 71 List of Figures 1 Sensitivity of Critical Ratio on 6V. . . . . . . . . . 8 2 Sensitivity of Critical Ratio on . . . . . . . . . . 9 3 Critical Ratio.as a Function of Standard Deviation . . 10 4 Unit Lognormal Stochastic Variable . . . . . . . . . 11 1.1 U - Shape of Investments Present Value . . . . . . . 35 1.2 Unit Lognormal Stochastic Variable (a = 5%) . . . . . 40 1.3 Unit Lognormal Stochastic Variable (a = 10%) . . . . 41 1.4 Unit Lognormal Stochastic Variable (a = 15%) . . . . 42 1.5 Unit Lognormal Stochastic Variable (a = 20%) . . . . 43 1.6 Unit Lognormal Stochastic Variable (a = 25%) . . . . 44 1.7 Unit Lognormal Stochastic Variable (a = 30%) . . . . 45 2.1 Costa Rica: Monte Carlo Simulation . . . . . . . . . 55 3.1 Hungary: Nonte Carlo Simulation (Nuclear) . . . . . . 66 3.2 Hungary: Monte Carlo Simulation (Coal) . . . . . . . 67 List of Tals 1 Costa Rica: Economic and Technical Data . . . . . . . . 15 2 Costa Rica: Costs and Savings for the two Expansion Plans 16 3 Costa Rica: Critical Ratios . . . . . . . . . . . . . . 18 4 Hungary: Economic and Technical Data . . . . . . . . . 21 5 Hungary: Costs and Savings ... ........... 21 6 Hungary: Critical Ratios ............... 23 7 Manantali: Costs and Benefits . . . . . . . . . . . . . 25 8 Manantali: Critical Ratios . . . . . . . . . . . . . . 27 9 Optimal Timing of Transmission Line: Economic and Technical Data . . . . . . . . . . . . . . . . . . . . 28 10 Transmission: Net Savings and Actual Ratio . . . . . . 29 1.1 Difference between Continuous and Regular Discounting . 46 2.1 Costa Rica: Power Expansion Scenarios . . . . . . . . . 48 2.2 Reject - Acceptance Matrix: Hydro . . . . . . . . . . . 50 2.3 Reject - Acceptance Matrix: Geothermal . . . . . . . . 51 2.4 Sensitivity Analysis: Hydro . . . . . . . . . . . . . . 52 2.5 Sensitivity Analysis: Geothermal . . . . . . . . . . . 53 3.1 Hungary: Power Expansion Scenarios . . . . . . . . . . 56 3.2 Costs and Savings under different Scenarios . . . . . . 58 3.3 Sensitivity Analysis: Nuclear . . . . . . . . . . . . . 60 3.4 Sensitivity Analysis: Coal . . . . . . . . . . . . . . 61 3.5 Sensitivity Analysis: Nuclear (dual fired) * . . . . . 62 3.6 Sensitivity Analysis: Coal (dual fired) . . . . . . . . 63 Deislomnauking Under Uncertainty: An Option Valuation Approach to Power Planning Executive Summary Development agencies play a major role in financing the ension of power systems among their member countries. Since the 1960s and 1970s, decsionmakers in these agencies have relied on models such as WASP to find least-cost solutions to problems of meeting expected demand. The problem that has arsen is not in the accuracy of these models in finding least-cost solutions but in the accuracy of the ftrecasts on which the solutions are based. Once aen as a given, forecasts have become increasingly inaccurate-a tendency reinforced during the 1980s, when energy prices and growth varied abruptly and unpredictably. In fact, recent evaluations of demand and investment cost forecasting by the World Bank have demonstrated a trend among forecasts to overstmate the growth of demand for power and underesimate the costs, and yet also a significant dispersion of errors that indicates a high degree of uncerainty in making any particular investment c cision. An *optimal expansion plan" designed for a scenario that fails to materalie is no longer optimal. Overestimating demand and underestmating cost of a capital-intensive power project can lead to overdimensioned or underfimded projects-either one potentially disastrous to a developing country. Hence, the need for an improved method of appraising power investments has become apparent, and the World Bank has sought to find a methodology for taking into acoount elements of risk and uncertainty in power planning. This paper focuses on one approach to decisionmaking under uncerinty, the optan valuation appxvach. The options approach is particularly useful in dealing with the problem of irreversible' investments, in which the political and economic reecussions of abandoning a project once it is well under way-such as a typical power project, with its high proportion of sunk costs-are so high as to make abandonment impracticable. Generally, when an environment is uncertain, investors can forgo or deay investment untl more favorable conditions are at hand. In either case, the investor has kept the options open. But in exersing the opportunity to invest, the decisionmar loses this flexibility. Conventional analysis techniques, including standard least-cost power planning, ignore the cost of the lost option. Yet the value of the option to -wait to invest' could be large enough to invalidate the usua decsin rule, to invest when benefits exceed costs. In effect, the correct decision rule under such circumstances should be to invest when benefits exceed costs by an amount at least equal to the value of the lost (forgone) option. i How can the analyst or decisionmaker calculate the value of the option to invest? The financial literature provides a methodology for calculating the value of an option and the "critical ratio of benefits to costs at which it becomes optimal to invest. The model used here is the perpetual option model or the option to %Wzt to invest. Derived from the work of McDonald and Siegel (1986), the perpeual option model provides a relatively simple methodology (requiring only two formulas for application) for deriving the critical ratio. The critical ratio so derived includes a penalty equal to the opportunity cost of the lost option as the irreversible decisionr to invest is exercised. Thus, the decision sile for investment is that investment is viable when the benefit-cost ratio exceeds the critical ratio. How does the perpetual option model work in practice? To test this, the authors sought to compare relatively large and irreversible projects, such as hydro plants or nuclear plants, with a nodular-type expansion of smaller thermal plants, such as gas turbines or smaller diesel plants. Tle loss of investment flexibility related to undeatkng the large, "irreversible" investment decision would be modeled as a perpetual option and addressed in quantitative terms. Costs (e.g., of construcion) and benefits (e.g., fuel ani. operation and mainteace savings, capital costs avoided on alterative expnsion plans) would be considered uncertain, but generally to relate to fuel pricis, demand growth, and capital costs. These would be estimated through a simple z7mulation of alternative investment programs or obtained from power system planning models. The flexibility variable concerns the ability to wait instead of malkng the i-vesible investment in the hydro or nuclear plant, and the product of this analysis is the critical ratio derived from the perpetual option model. The actual ratio of savings to costs then would be compared with the critical ratio, and the comparison would indicate whether the actWal ratio of benefits to costs was equal to or greter than the critic raio. If the actual ratio was less than the critical ratio, the investor would be better off wating, since the value of the flexibility lost would exceed the project's net benefits. The authors applied the model to cases of lage power projects in four countries-Costa Rica, Hungary, Mali, and Senegal-drawing on previous data collection and analyses by Martzoukos and Paddock (1991), Martwoukos and Teplitz-Sembitzky (1991), and Schramm (1989). In all of the case studies, the option valuation approach was easy to handle and provided a simple and meaningful decision rule that should appeal to analysts and decisionmakers. The accuracy and reliability of the model still depends on the extent to which the model's assumptions hold in pracdte and on whether the simplifications undertak during the modeling are faithful to the actual circumstances. The case studies show, generally, that the penalty of forgoing flexibility can be considerable. In the Costa Rica and Hungary studies, the penalty, reflected in the ii critical ratio, varied betwee 21 percent and 26 percent of total costs. In the Manantali Dam case, a regional project on the Senegal River Basin in Mali, alterative long-term expansion programs were compared, and uncertainty was higher. The penalty thus was higher, increasing to a value of 35 percent of total costs. In the fourth study, which concerned the capital-intensive task of connecting rural areas currently serviced by decentralized plants to main power grids, uncertainty created an incer-five to delay investment in the costly connection to te main grid by almost three years beyond the "optimal" date to avoid losses from exercising the investment option. Fuller explanations of the methods and details of analyses are provided in text and Annexes. The study thus suggests that the option valuation approach addresses a significant problem with risks of a large order of magnitude; that it would pose a significant change in, and improvement on, classical benefit-st approaches; and that the critical ratio provides a relatively practical and easy-to-apply tool for analysts and decisiomakers to better anticipate the risks of forgoing flexibility in large investent decisiom The approach is applicable to investment problems outside the power subsector as w'ell. iii DECIXSON MARING UNDER NCERTaXWfN - AN OPTION VALIUaON APPROACE TO POwER PLANNING 1. INYRODuCTxON Most development agencies have an important role in financing power system expansion plans among their member countries. The methodologies for appraising these expansion plans were developed and refined largely in the sixties and seventies, and generally consist of finding the least-cost solution to the problem of meeting a forecast demand, which is usually thought of as a given. Power systems planning models - such as WASP - are widely used in almost all developing countries to formulate least-cost ex_-ansion plans. While these methodologies have become very sophisticated, there has been a growing realization that they may be inadequate or insufficient, not because they fail to achieve their limited objective, but because it is impossible to make accurately the forecasts they require. This perception has been reinforced during the last decade, after a period of abrupt changes in energy prices and disruptions to economic growth, a decade through which forecasts have become more unreliable. Studies carried out by the Bank show that the track record of forectisting in the power sector is poor. Bank ex-post evaluations of forecasts on demand growth (Sanghvi et.al., 1989), implementation time and capital costs of new plants (Merrow et.al., 1990), fuel costs and technological innovation reveal the following: - Forecasting errors have been in general high and clearly biased in one direction. There is clear evidence of an optimistic bias reflected in the overestimation of demand growth and the underestimation of costs and implementation time. - The great aispersion found in forecasting errors indicates a high degree of uncertainty in making particular power investment decisions. - The wide spread of errors found is very seldom captured by the narrow margins usually adopted in sensitivity analysis. A so-called optimal expansion plan designed for a scenario that fails to materialize is no longer optimal. Therefore it is difficult to escape the conclusion that if the methods of appraisal do not take into account elements of risk and uncertainty, they cannot be relied upon to yield truly optimal expansion plans. Furthermore, the forecasting trends - i.e. the overestimation of demand growth and underestimation of costs - suggest that these methods would tend to prescribe overdimensi^Ded and/or underfunded plans while choosing larger and less flexible plants. This problem is the more worrisome because of the capital intensiveness of power projects and the fact that governments and utilities of developing countries are planning huge power investment programs for the 1990Q, which will require around US$1 trillion of financial resources (Moore and Smith, 1990). Therefore, errors in the timing and scale of power investment decisions or in the choice of technologies have a potential of impacting negatively the overall economy of developing countries. Generally speaking, the World Bank's approach to the economic appraisal of individual projects is theoretically sound. Hovever, the Bank's practice departs considerably from its guidelines, apparently due to institutional reasons and the difficulties that some analytical tools place for their application (Little and Mirrlees, 1990). This situation appears to be particularly true ir. the treatment of risk and uncertainty. The theory urierlying the Bank's approach to project appraisal under uncertainty is quite conse-vative. It proposes an allowance for uncertainty only for projects where profits were expected to be correlated with the general state of the economy (Little and Mirrlees, 1974). It is sometimes argued that this allowance should be made also for specially large projects, relative to the country economy. Clearly, the approach is more concerned about the effects of risk at a macro level and thus understates the problem of relative flexibility and risks associated to competing projects. However, Bank's guidelines go a step further (World Bank, 1980, 1980a and 1984). They recognize that some elements in the cost and benefit streams of projects are uncertain, and therefore recommend the use of sensitivity analy?is and, if necessary, formal risk analysis for project appraisal . Unfortunately, sensitivity and risk analyses face two main practical problems that have hampered their effective use: (i) sensitivity analysis is usually undertaken at a late stage of the project cycle only to confirm the robustness of the evaluation conclusion to arbitrary, and unduly narrow, variations in a few planning parameters, and (ii) the guidelines for risk analysis are insufficient since they just recommend the use of a probabilistic risk method, a costly simulation approach with strong elements of black box, without providing much guidance on how to do it. I It should be noted also that current Bank Guidelines for power, water supply and telecommunications -(World Bank, 1978) consider a least cost planning approach as an alternative to project benefit/cost analysis when economic benefits are difficult to quantify. The case studies presented in this paper are undertaken along those lines, since in all cases avoided costs are estimated as a proxy for benefits. This approach, though conceptually correct, does not include all benefits thus excluding the consumers surplus. 2 To address this problem, the World Bank is undertaking a program designed to incorporate risk and uncertainty into power planning. A first stage of this effort consisted in a survey of the literature and current day practices, and the identification of suitable approaches (Crousillat, 1989). It was found then that power planners in industrialized countries have largely abandoned formal least cost planning as an explicit goal and are giving increasing attention to flexible or robust plans able to adapt to a changing world. The literature review found three promising approaches to deal with the problem, namely: (1) stochastic optimization, (2) strategic models, particularly risk-tradeoff analysis, and (3) the option valuation approach. A second stage of the Bank's program on risk in power planning consisted in testing the three identified approaches in several case studies. This stage concerns the undertaking of country level case studies on dissimilar power systems with the objective of ultimately answering the following questions: (1) is it advisable to abandon the methods currently in use in favor of an approach that addresses the question of risk and uncertainty?, (2) wbat are the policy implications of incorporating risk and uncertainty in power planning?, and (3) are any of the identified methods suitable to produce a practical tool which Bank staff and/or country power planners might use in appraising power expansion plans? The results of the case studies, plus a final comparative evaluation, are expected to answer the above questions. This paper reports on World Bank's experience in the study and application of one of the approaches, namely the option valuation approach to power planning2. Chapter 2 addresses the basic theory of options, describes the standard model developed for real options, and discusses its parameters and assumptions. Finally, it examines the critical assumptions underlying the estimation of the main decision indicator: the critical ratio that should be compared to the conventional investment benefit-cost ratio. To facilitate its reading, some sections are presented in separate annexes which include the mathematical formulation of the optiors model, its underlying assumptions and a glossary. Chapter 3 describes four case studies. The first two undertake comparisons of generation projects on a project-to-project basis, without the use of a systems approach. The first case is a typical hydro vs. thermal project comparison in the Costa Rica power system, and the second a comparison of thermal generating technologies in Hungary. The third case study is a specially interesting exercise of combining the use of a power system 2 Initial investigative work was done by Teplitz-Sembitzky (1989). Case studies are drawn from Martzoukos and Paddock (1991), Nartzoukos and Teplitz-Sembitzky (1991) and on further in-house analytical work. 3 planning model with the options model. Here, the comparison is done on two alternative expansion plans for the Mali and Senegal power systems. The last case refers to the problem of determining the optimal timing of transmission line investments for their extension to isolated systems. Finally, Chapter 4 draws conclusions from this experience and provides some practical recommendations. 4 2. 2 AML OPTIONS The term "real option" refers to a concept analogous to that of financial investment options. Real options are faced in many activities. They can be related to a choice of technology, a production decision, decisions on investment timing, the option to temporarily or permanently shut down a plant, as well as to many other decisions dealing with real (physical) assets. The options approach addresses the problem of "irreversibility" of investments. Very often investment decisions are irreversible, since they imply embarking on rigid programs with a subsequent large proportion of sunk costs. This is particularly true in the capital intensive power sector where once we are committed to a particular investment, the political and economic costs of abandonment can be extremely high thus making almost impossible to reverse a decision. Generally speaking, prior to making a particular investment decision, we can always "wait to invest" and learn more about the parameters that affect the economic viability of this investment. Once an irreversible decision is made, the flexibility gained with time from the potential reduction of uncertainty is lost. As a rule, the higher the uncertainty related to critical variables like energy demand, fuel prices and construction costs, the higher will be the value of the flexibility loss. In an uncertain environment, an investment could be delayed until more favorable conditions occur. In the case of worsening conditions though, there would be no obligation to invest and we could keep our options open - an opportunity - for undertaking alternative decisions that, in that case, would result in a more favorable outcome. This opportunity is similar to a "stock option". In exercising the decision to invest we forego the potential gains of postponing the decision, i.e. we incur in an opportunity cost - a lost option - as the irreversible investment reduces our choices, thus reducing our flexibility. Conventional economic analysis techniques, including standard least cost power planning, ignore the cost of the lost option and tend to accept less flexible projects too easily. The value of this foregone option3 could be large enough to invalidate the usual decision rule: to invest when benefits exceed costs. In contrast, the correct decision rule should be: to invest when benefits exceed costs by an amount at least equal to the value of the foregone option. The financial literature provides option models to calculate the 3 The value of the option "to wait to invest". 5 value of an option and the critical ratio of benefits to costs at which it becomes optimal to invest. The model used in this paper is known as the perpetual option' or the option to wait to invest. This model was developed by McDonald and Siegel (1986) as an extension of a previous work done by Samuelson (1965). A complete mathematical formulation of the perpetual option model, including a discussion of its parameters and assumptions, is presented in Annex 1. Here, we include the main expressions of the model, which are in fact the only two formulas needed for the application of the perpetual option model. The Perpetual Option The critical ratio provided by the perpetual option is given by: C = t/(1t-3) where the parameter r is given by: I = (0.5 - (6f-6 )/c2) + [((6f_6v)/a2 _ 0.5)2 + 2Sf/02] and "a'", the total effective variance, is an overall measure of uncertainty and equals a v + af - 2r f0vaf. The critical ratio includes a penalty equal to the opportunity cost of the lost option as we exercise an irreversible investment decision. The value of the investment opportunity is given by: w = (C-1-)F(V/WCY where: V 5 present value of benefits ("stock price equivalent" in the finance terminology). F = present value of costs ("exercise price"). °v = standard deviation of benefits, on the expected rate of change. af = standard deviation of costs (on expected rate of change). r = correlation between benefits and costs. Sv , 8fZ effective discount rates representing the opportunity cost on the benefits and costs of deferring investment, respectively, equal to the difference between the corresponding discount rate and the growth rate of benefits and costs. Values C and W constitute the main contribution of the model. C 4 It was originally called "perpetual option" because it has no finite expiration (maturity) date. When the option can be exercised at any time, it is called an American option. A European option can be exercised only at the expiration date. 6 is the critical ratio that should be exceeded by the actual Benefit-cost ratio in order to make the investment viable under uncertainty. Therefore, the decision rule will be: Benefit/Cost Ratio > Critical Ratio W is a quantitative measure that indicates in monetary terms the value of the investment5 plus an option penalty due to the lost flexibility. The relationships between the critical ratio and the main parameters as well as the model's assumptions are discussed in Annex 1. In summary, these relationships are as follows: a) The critical ratio is a decreasing function of 6v and an increasing function of Sf. If the effective discount rate on the benefits (6,,) is higher, the opportunity cost of the foregone savings would be lower, therefore the required flexibility premium would be also lower. Figure 1 displays this relationship. On the other hand, if the effective discount rate on the costs (68) is higher, the required flexibility premium is also higher because the opportunity costs of deferring construction are lower. This relationship is presented in Figure 2. b) The critical ratio is an increasing function of the overall uncertainty (standard deviation a). If uncertainty is higher, the value of waiting to receive more information is higher and the required flexibility ,remium should be higher too. Figure 3 presents this relationship, keeping other model parameters constant (i.e. discount rate: 10%, and savings appreciation and cost appreciation: 1.3%). It can be seen that the critical ratio and thus the investment decision are very sensitive to the estimate or perception of the underlying uncertainty. It is important to understand the implications of the random walk hypothesis used in the model. For illustrative purposes we provide in Figure 4 a distribution of future values for a variable that has no growth (drift) but fluctuates with a standard deviation of 15% per year. Assuming an initial value of 1 and that the variable follows a random walk, i.e. that after fluctuating in one period, the variable has no tendency to revert to a previous "normal" value, the extreme values of this variable after 5 years would vary between 0.52 and 1.96 for a confidence level of 95% (and between 0.72 and 1.4 for a confidence level of 65%). For a standard deviation of 10% the corresponding range for the 95% confidence interval would be between 0. 67 and 1. 54. A set of graphs for these and other s In fact, the net present value of the investment with optimal deterministic timing. 7 Figure 1 SENSITIVITY OF CRITICAL RATIO ON 6v 6v - effective discount fote on soving: 110 100 _ 90 _ 80- 0 70 - 600 750 ._*0 40 - 30 0 20 - 60 *~~ _0 40 0.1 0.087 0.07 0.05 0.04 0.03 0.02 0.01 0.005 0.0025 0.001 dv = Discount rate - growth rate uncertainty levels is included in Annex 1. These figures reveal the wide range of values that can be expected in the future even for relatively low standard deviations, and are thus an indication of the need to be specially cautious when quantifying uncertainty in any practical appiication of the model. c) The critical ratio is also a decreasing function of the correlation between benefits and costs (rv). Regardless of the underlying uncertainty, if benefits and costs fluctuate together and in the extreme, if they are perfectly correlated, the critical ratio will be considerably reduced. If benefits and costs are poorly correlated, the effects of uncertainty would increase since there will be always the possibility of having simultaneously higher than expected costs and lower than expected benefits. 8 Figure 2 SEN'.JTIVITY Or CRITICAL RATiO ON 6 f 6f - effective discount rate on costs 2.6 2.5- 2.4- 2.3 - 2.2 - Cc 2.1 2 co 1.7 _-). 16 1. 1.3 1.6 (-I~~~~~~~~~~~~~~~~~ 0.2 0.15 0.12 0.1 0.087 0.07 0.05 0.04 0.03 0.02 0.01 4.f - Discount rote - growth rote The main assumptions of the perpetual option model are: a) Variables are log-normally distributed. The practical implication of this first assumption is that negative values should be excluded, therefore all benefits and costs should have a positive sign. If for example an investment in a hydro scheme results in avoiding construction costs in an alternative development, these costs should not be subtracted from the construction costs but should be added to the benefits. This has, of course, a direct implication for the estimation of the relevant uncertainty of costs and benefits. 9 Figure 3 CRITICAL DATIO AS A FUNCTION OF "a" AMERICAN TYPE, iNFINITE MATURITY OPTION 2.9 2.8 2.7 2.6 2.5- 2.4- 2.3 - 2.2- O 21 -2 2 1.9 c 1.8 1.7 1.6 1.5- 1.4- 1.3 1.2 1.1 0.030 0.060 0.090 0.120 0.150 0.195 0.255 0.345 0.450 TOTAL EFFECTIVE N"o b) The model is defined in continuous time and assumes constant rates of change, which implies a perpetual exponential growth of benefits and costs. For practical purposes we apply a "rotations" approach, i.e. the repetition of a finite time profile of benefits and costs. It should be noted, however, that under the normal practice of heavy discounting, the length of the rotations is not crucial if the life of the irreversible alternative if sufficiently long. 10 C) The model assumes that the dynamics of the uncertain variables can be described in terms of a Brownian motion in continuous time. This implies continuous discounting for the estimation of present values. Compared to regular discrete time discounting, continuous discountinc -.nderestimates" present values. The difference between -.. th approaches is an increasing function of the discount rate and the period of analysis. However, the impact of these differences in power planning problems tends to be minor since it will affect similarly the present value of benefits and costs. Figure 4 UNIT LOGNORMAL STOCHASTIC VARIABLE 95X ANO 65% CONFICENCE NtERVALS 3.4 3.2 :3 2 8 2.6 W 2.4 >41. 2.2 i:,J 2 .D- i.8 - _ f16 OD1.4 (A 9 1.2 0.8 0.6. .. ...... 0.4 0.2 i I ' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time- stondard deviclion * 15X 0 Inner (outer) 2 curves - 65X (952) confidence intervol. 6 See Annex I page 33, for the model formulation or Karlin and Taylor (1975) for mathematics relevant to stochastic processes. 11 3. ORs S¶IUb This chapter presents four case studies on the application of the perpetual option model. The first two cases undertake comparisons of generation projects pertaining different technologies (on a project-to-project basis, i.e. they do not follow a systems approach). The first case is a classic hydro vs. thermal comparison, in the Costa Rica power system. The second case compares different thermal technologies: i.e. combined cycle, coal and nuclear, in the Hungarian power system. The third case study is an effort to combine the use of a traditional power system model with the options model. The analysis is done on several alternative expansion plans for the Mali and Senegal power systems, including as a main generating alternative the Manantali hydropower plant. The last case addresses the problem of optimal timing for the extension of a transmission line to an isolated area, a frequent decision problem in power systems. The Costa Rica and Hungary cases are drawn from Martzoukos and Paddock (1991), an IENED internal report that discusses extensively theoretical and practical issues of the options approach. The Nanantali hydropower case study is an extension to the economic analysis undertaken by a consultant, Association Nomentanee, and uses the results of their power systems analysis (draft report of Mar. 1991). The transmission line case study is drawn from work by Martzoukos and Teplitz-Sembitzky (1991), which is an extension to theoretical work by Schramm (1989). All these applications compare a relatively large and irreversible project, usually fuel-efficient but capital intensive (such as a hydro plant, nuclear plant or the interconnection with a major system), versus a modular type expansion based on smaller thermal plants (a set of gas turbines, combined cycle modules or smaller diesel plants). The loss of flexibility related to the large investment decision is modelled as a perpetual option, and is assessed in quantitative terms. In the general case, both costs (construction costs of the irreversible project or program) and benefits (mostly fuel and O&M savings, plus some capital costs avoided on alternative expansion) are considered uncertain. The sources of uncertainty are case specific, however they usually refer to fuel prices, demand growth and capital costs. The type of flexibility addressed is related to the ability to wait instead of investing immediately in the hydro or other irreversible project, and thus have the option of investing later when it may be more valuable to do so. The method analyzes the convenience of delaying the investment decision on a large and/or irreversible project taking into account that an alternative expansion plan could become more advantageous. In the analogy between the real option and the financial option, the benefits or savings correspond to the equivalent "stock" and the project costs correspond to the "exercise price". 12 In order to proceed with the application we need two types of data: a) The actual costs and savings associated to the large investment. This data was either estimated through a simple simulation of alternative investment programs or obtained from power system planning models. b) Variables relevant to the options model, such as quantitative measures for the relevant uncertainty (on costs and benefits), specifically the standard deviation of uncertain variables and the correlation between these variables, as well as relevant discount rates and the appreciation rates of benefits and costs. The actual ratio of savings to costsT from part (a) is compared to the critical =atio given by the options model from part (b). This comparison deteraines whether the investment should be exercised, i.e. if the actual ratio is equal or greater than the critical ratio. Otherwise, we would be better off waiting since the value of the flexibility lost would exceed the projectes net benefits. The first application presented below refers to the Costa Rica case. In the effort to present a complete illustrative example, this case includes an extensive description of the computational process and its assumptions. To avoid repetition, the remaining case studies are presented briefly, supported by separate annexes when necessary. 3.1 Costa Riaa Case Study The Costarrican power system had in 1989 an instalLed capacity of 772 NW, about 90% of this capacity consisted of hydropower plants and the rest by small thermal plants of different types (Instituto Costarricense de Electricidad - ICE, 1989). During the 1980s demand for electricity grew at an average rate of 7%. However, this rate of growth is expected to decline to annual rates between 3% to 5% during the 1990s. The investment lecision to be analyzed is that of a large 177MW hydropower plant and a 104MW geothermal program, both irreversible and fuel-efficient investments, compared to a sequence of diesel- fired gas turbines (36MW units). In both cases the question to be answered is: should we invest now in the large project, or should we wait, keep our options open and meanwhile, if necessary, install a small gas turbine? Applying the options model we will assess the value of the flexibility loss, i.e. the benefits foregone as we commit to a specific project and can no longer pursue a more t That is, the standard benefit-cost ratio obtained through regular deterministic present value techniques. 13 flexible plan which, under certain future conditions, could turn out to be more advantageous. The main variables considered uncertain are fuel prices, demand growth and construction costs. The analysis consists of comparing the benefit-cost ratios obtained for the two major investments using a standard deterministic approach, to the critical ratios -.alculated by the options model. Costs and Benefits To determine the costs and benefits (savings) for each alternative, we need to simulate the investment expansion. This was done by considering the following three alternative plans: a) The hydro plant complemented by gas turbines. b) The geothermal program (2x52MW) complemented by gas turbines. c) A program of 36MW gas turbines, to be compared with the hydro and geothermal programs. The comparison of programs (a) and (b) with (c) permits the estimation of the required costs and benefits. These are defined as follows: - costs: construction costs for the hydro and geothermal programs. - Benefits: fuel and O&M savings plus avoided construction costs of gas turbines. Fuel and O&M savings are computed as the difference in these two variables when comparing (a) and (b) to (c). Avolded construction costs are obtained directly from program (c) . The estimation of costs and benefits was done following standard practices. This includes: - A reserve available capacity margin of 10% was considered above the maximum load. This value was obtained from ICE's use of the WASP model taking into account the effective capacity of hydro plants during the dry period. 8 It should be noted that as it is assumed that variables are log-normally distributed, benefits and costs should have a positive sign (see Chapter 2, p.8, 1st assumptio i). In this particular case we are calculating the differences (savings) in fuel and O&M costs, instead of adding them separately on each side, because these differences are positive. However, this is not always the case. As it should be expected, these choices on the costs and benefits componente will affect directly our analysis on the relevant uncertain-ties. 14 Since alternative generating technologies usually have different economic lives, the replacement of short-life plants was taken into account. A salvage value was included at the end of the period of analysis using straight-line depreciation. Energy dispatch rules followed an economic criterion. In cases of excess capacity, least fuel efficient plants were affected. Accordingly, it was assumed that the hyydro plant would be able to dispatch all its energy after four years of operation, while the smaller geothermal plants would achieve the same state after three years. A diesel oil price of $ 25 per barrel was considered for the base year. A real cost increase of 1.3% was assumed for fuel, construction and maintenance costs. The uncertainty was modelled around these values. A discount rate of 10% was used. The basic data used for the comparison of expansion programs is presented in the following table. Table . Costa Rica: Economic and Technical Data. Gas Turbine Hydro Geothermal Installed capacity 36 NW 177 MW 104 KW (2x52) Economic life 15 yrs 40 yrs 25 yrs Max. capacity factor .72 .64 .81 Fixed investment cost 17.08 M$ 289.26 M$ 226.98 M$ Variable fuel cost 0.049$/kwh ----- ---- 0 + M 2.87$/kw-mo 0.7$/kw-mo 1.21$/kw-mo Construction period 1 yr. 5 yrs 3 yrs The three expansion plans are presented in Annex 2. It was found that the hydro plant would reduce future investment by 5 gas 9 This case study, as well as the Hungary case, was undertaken without the recourse of an operation simulation model. It was considered that, due to the project-to-project nature of the comparative approach, simple dispatch rules were sufficient. 15 5 gas turbines, while the geothermal program would reduce it by 3 gas turbines. Present values of costs and savings for the hydro and geothermal plans are presented in the following table. Table 2 Costa Rica: Costs and Savings for the two Expansion Plans. (present value in million US S) Hydro Geothermal Costs 289.3 227.0 Benefits - Fuel savings 303.7 251.9 - 0+M savings 28.8 14.5 - Avoided cost 64.8 4o.5 Total Benefits 397.3 312.9 Actual Ratio 1.374 1.379 Rate of Return 12.2% 13.05% Batimation of the Option Model Parameters The model parameters to be estimated are the standard deviation and correlation of the underlying uncertain variables (a., af and rvf), and the two effective discount rates 8v and 6f (see formulation of the model in Chapter 2, p.6). The estimation of the uncertainty related to benefits is rather delicate since these benefits include fuel and O&M savings, and avoided construction costs. In addition, fuel savings are a function of fuel prices and demand uncertainty. The standard deviations used for each of these variables are: a) Fuel price uncertainty; afue- fce= 10%. A conservative value considering that direct measu6es give a log-relative value of arour.d 20% for crude oil prices. The rationale for using a lower value is based on the fact that the options model assumes that fuel prices follow a random walk behavior, which tends to extend excessively the range of possible future values (see Figure 4). b) Demand uncertainty was estimated to be around 20%. However, it should be noted that the effects of this uncertainty on the 16 fuel savings attributed to an individual plant are considerably reduced. In fact, this uncertainty would affect fuel savings only during the relatively short period when plant capacity is not fully utilized. Accordingly, in order to represent the periods when demand uncertainty does count we weighted this value (of demand uncertainty) by the ratio between this initial period to the plants total life. Resulting ratios were: for hydro 4 out of 40 years = 0.10; for geothermal, 3 out of 25 years - 0.12. Although this may be an arbitrary way of estimating uncertainty, it offers the advantage of being intuitively rational and consistent in its application. c) The aggregate fuel savings uncertainty is computed as follows: For the hydro project; a,,,,- [(.1)2+(.10*.20)2] = 0.102 For geothermal; fu = -/(.l)2+(.12*.20)2] = 0.103 This calculation assumes that the two uncertainties (fuel prices and demand) have a multiplicative effect on the total fuel savings uncertainty; they are also assumed to be non- correlated. d) Total savings uncertainty would be equal to the weighted average of fuel savings and avoided costs uncertainties. Considering a cost uncertainty of 5.6% (see next paragraph), final benefits uncertainties will be: Hydro project; v = t((1-.1631).*.102)2+(.1631*.056) 2] rvy 0.086 Geothermal; - ¶/((1-.1486)*.103) 2+ (.1486*.056) 2] UV 0.088 The estimation of costs uncertainty is straight forward since they only include construction costs. A standard deviation of u= 5.C6% was chosen for hydro projects, based on the annual (at overruns inferred from Merrow's paper (1990). This value was also used for geothermal costs, assuming that the uncertainty surrounding geothermal costs is similar to that of hydro developments. The correlation between construction costs and savings is assumed to be 15% for the base case. This value was chosen on the basis that fuel costs account for approximately 10% - 15% of construction costs and about 90% of total savings. The effective discount rates, that represent the opportunity costs on the benefits and costs of deferring the investment, are equal to the difference between the discount rate and the rate of growth of benefits and costs (escalation). As explained before, the discount rate selected is 10% while a real cost increase of 1.3% 17 was assumed for all costs10. Therefore effective rates for both the hydro and geothermal analyses will be equal to 8.7%. Results The critical ratios obtained for the hydro and geothermal expansion plans (always lower than the actual ratios) reveal that in the face of uncertainty, the decisions to proceed remain optimal when compared to the option to wait to invest. However, results indicate that in both cases the value of the flexibility loss is quite considerable since it reaches 26% of total costs. Hence, these two projects, which have actual (benefit/cost) ratios of 1.37 and 1.38 under a deterministic approach, are no longer so attractive. Table 3 Costa Rica: Critical Ratios Actual Ratio Critical Ratio Optimal Decision ------------------------------------------------------------__-- Hydro 1.374 1.256 Build Geothermal 1.379 1.261 Build It is interesting to examine the sensitivity of results, particularly the optimal decision, to changes in our perceptions .or estimates of the underlying uncertainty. This sensitivity is summarized as follows: a) For the hydro project; - the decision is reversed, i.e. it is no longer optimal to invest now, if the uncertainty about savings (standard deviation, oav) exceeds 13%, and - the decision is quite robust to the uncertainty about costs and the correlation between costs and savings. It is not affected for values within the analyzed range, i.e. an uncertainty on costs up to 8.6% and a correlation coefficient (which is inversely related to the critical ratio) down to 5%. b) For the geothermal program, the results are quite sj.milar; - the decision would be reversed if the uncertainty about savings is slightly higher than 13%, and 10 It should be note4 that this cost escalation could be country specific and that benefits and costs are likely to have different escalations. In that case 6. and 6f would not be equal. 18 - the decision is not affected by the uncertainty about costs nor the correlation between costs and savings, for the same range of values applied to the hydro project analysis. More complete results on the sensitivity of the decision to the main uncertain variable, savings, is presented in Annex 2, which includes accept-reject matrices for savings uncertainty and discount rate, together with tables that display the sensitivity analysis undertaken for all relevant variables. In addition, Annex 2 includes the results of a Monte Carlo sensitivity analysis aimed at studying the overall impact of variations in the model parameters to the critical ratio. This analysis reveals that there is considerable uncertainty regarding the estimates of the critical ratio. However, they also indicate that the decision recommended by the model is quite robust since it concludes that the probabilities of having a critical ratio greater than the actual ratio (i.e. of reversing the decision) are very low, being 1.4% and 1% for the hydro and geothermal projects, respectively. 3.2 Hungary Case Study The Hungarian power system has an installed capacity of about 700OMW. This capacity is composed of a great variety of thermal generating technologies, from oil and gas fired plants, to coal, lignite and nuclear. In addition, Hungary has been importing 1,850MW through a long term agreement with the Soviet Union, in order to satisfy their power needs. Peak demand reaches 6,500MW and is expected to continue growing at a modest rate of about 1.3% during the current decade. However, there is uncertainty about this growth. Other and apparently more important uncertain variables are construction costs and fuel prices. The investment decision to be analyzed in this exercise is that of a large 480MW nuclear plant and a 270MW coal-fired plant, both considered irreversible and relatively efficient in terms of fuel, compared to a program of combined-cycle 135MW units. As in the Costarrican example, the question to be answered is whether we should invest now in the nuclear or coal plant, or would we be better off waiting. The application of the options model will assess the value of the flexibility loss on exeruising either of the two investment decision so as to appLy a simple decision rule under the face of uncertainty. 19 Costs ind Benefits The analysis consists in comparing two expansion plans that include alternatively a nuclear and a coal-fired plant, to a third pi," based on a sequence of distillate-fuelled combined cycle plants'. These comparisons produce the required costs and benefits which ire defined as follows: - Costs: Construction costs for the nuclear and coal expansion programs. - Benefits: fuel and O&M savings plus avoided construction costs. Fuel and O&M savings are computed as the difference between the nuclear and coal program compared to the combined- cycle program, i.e. the scenario without the project being analyzed. Avoided construction costs are those corresponding to the combined-cycle program. The estimation of costs and benef its was undertaken taking into account the following considerations: - A reserve capacity margin of 25% was considered above the maximum load. This coincides with the reliability standards used by the Hungarian Electricity Board (Magyar Villamos Muvek Troszt - MNVMT) in its power planning practices. - End-of-period adjustment similar to those used in the Costa Rica study were applied (see p.15). - Fuel :rice: Coal $ 40/ton Distillate S 25/barrel - Discount rate: 10% - Exchange rate: 63 Forints per US.$ Technical and economic data on the alternative generating technologies were drawn from MVMT's data base and a recent World Bank report (Moore and Crousillat, 1991). These data are presented in Table 4. The three expansion plans are presented in Annex 3. This annex includes, in addition, the results of a comparison based on dual- fired combined cycle units, considering both distillate and natural gas in equal shares. 11 A second set of analyses considered a combined-cycle program using two fuels, distillate and imported natural gas, in equal shares. This particular case is presented in Annex 3. 20 Table 4 Eungary: zaonomic and Teohnical Data Combined-Cycle Nuclear Coal Capacity 135 MW 480 MW 270 NW Economic life 20 yrs 25 yrs 25 yrs Max. capacity factor 65 % 65 % 65 % Fixed investment cost 35 106.2 83.2 (in 1000 Forints/KW) Variable fuel cost .032 0.008 0.014 (in $/KWH) Operations & maintenance 4 % 2 % 2.5 % (% of inv. cost per year) Construction time 3 yrs 7 yrs 5 yrs Present values of costs and savings for the nuclear and coal plans are presented in Table 5. Table 5 Nungary: Costs and Savings (Present value in million Forint) NUCLEAR COAL Gross fixed cost 50976.0 22464.0 decommissioning 242.9 ------- Total Fixed Cost 51218.9 22464.0 Fuel savings 22541.6 10446.6 O+M savings -1952.9 -1191.8 Avoided Costs 9402.6 6978.5 Total savings 29991.3 16233.2 Actual Ratio .586 .723 Rate of Return 6.10% 6.95* 21 Estimation Of Model Parameters The model parameters to be estimated are the standard deviation of benefits and costs (7v and of, both measures of uncertainty), the correlation between these two variables (rv) and the two effective discount rates Sv and 6f. The estimation of costs uncertainty is straight forward since they only include construction costs. We have chosen 5.6%, the same value used for the Costa Rica case. However, it should be noted that this uncertainty should be a function of the technology analyzed. Benefits are considered to be a function of fuel savings and avoided costs. In this case fuel savings will not be affected by demand variations because both the nuclear and coal plants are expected to have lower variable costs compared to existing and other future thermal plants. Therefore, their energy dispatch, and thus the fuel savings attributed to these plants, will not be affected by a lower demand. Accordingly, fuel savings uncertainty will be equal to fuel price uncertainty: 10%. Total savings uncertainty is computed as a weighted average of fuel savings and avoided costs. Nuclear Project; a, = ,[((l-.6865)*.10)2+'.6865*0.056)2] = 0.071 Coal project; ClV = 4[((l-.570)*.10)2+( .570*0.056)2] = 0.062 The correlation between costs and benefits is assumed to be 7% (lower than in Costa Rica) considering that nuclear and coal generating plants are less dependent on fuel during their construction. The effective discount rates used are the same of those chosen for the Costa Rica case (see pp.17-18), i.e. S.7% for both benefits and costs. Results The actual ratios for the two expansion plans are lower than one, which indicates that these projects are not viable even under a deterministic scenario. Furthermore, when analyzing these projects in the face of uncertainty, they become even less attractive since the critical ratios reveal a penalty of around 22%. These results are shown in Table 6. Knowing the deterministic result, the option analysis may appear redundant. However, the application of the options approach provides additional insight. It reveals option prices (denoted by X in Tables 3.3 to 3.6, Annex 3) that, for the base case, are equal to 229.7 and 249.7 million Forint for the nuclear and coal plant, 22 respectively. These figures are useful to value the rights for developing the large plants, nuclear and coal-fired, if they are to be sold to a third party. Therefore, the investment opportunity still has some value, and it may be reasonable to "keep alive" although it is currently not profitable. Table 6 Hungary: critical Ratios -------------------------------------------------------------__- Actual Ratio Critical Ratio Optimal Decision ------------------------------------------------------------__-- Nuclear 0.586 1.232 Don't build Coal 0.723 1.213 Don't build ---------------------------------------------------------------- Sensitivity analyses on all variables are presented in Tables 3.3 to 3.6, including those for a dual fuel combined cycle program. A Monte Carlo simulation analysis on the critical ratio was also performed. All these analyses confirmed the study's initial findings, i.e. that the two large projects are not viable and that under the face of uncertainty they would become more unattractive. 3.3 Nanantali Hydropower Plant The construction of the Manantali dam, completed in 1988, was undertaken with the purpose of developing a capacity to regulate river flows for irrigation, power supply and navigation in the Senegal River. It has been proposed to build a hydropower plant at the dam site, taking advantage of the head (fall) and water regulation provided by the existing reservoir. The plant is designed for an installed capacity of 200MW in five units of 40MW; however, its implementation program is a function of the scope of the development to be chosen. Two development programs have been proposed. A relatively small development constrained to Mali's power market, and a large development that considers the construction of a 900 Km long, 225- kV transmission line to the west coast, reaching Dakar, Senegal, in order to supply power also to Senegal and Mauritania. While tpe first alternative, named "Hydro-Mali", implies a phased program , the second alternative ("Large-Hydro") considers the installation of all five units by 1996. 12 The installation of two generating units (2x40MW) in 1996, one in 2005 and a fourth in 2008. This case does not consider the installation of the fifth unit during the 25-year period of analysis. 23 The present application of the options model is based entirely on the data set and simulation results obtained from the consultant's draft report (Association Momentanee, 1991). The analysis is done for two investment decisions, (i) the decision to invest in the smaller Hydro--Li alternative, and (ii) the decision to extend the smaller development to a larger one undertaking a major trans- mission investment oriented towards the power markets ot the three countries, i.e. Senegal, Mali and Mauritania. The small Hydro-Mali project is compared to a hydro-thermal development for the corres- ponding electrical system (Bamako-Segou, in Mali), while the Large- Hydro is compared to the Hydro-Mali development complemented by thermal plants. These two comparisons are aimed at assessing the economic viability of each alternative. In the first case, the costs and benefits of the hydropower development in Manantali are confined to the regional market. In the second case, we measure the economic merits of building a major transmission line to Dakar, thus allowing a full utilization of Manantali's hydro energy during the initial years and a better dispatch of thermal generation. costs and Benefits In both cases, costs and benefits are computed as follows: - Costs: Investment costs plus fixed operating costs. - Benefits: Avoided investment costs (in altepative developments) plus fuel and non-served energy savings . All values were obtained from the consultant's simulations of the system(s), which rest on the following assumptions: - Energy demand growth rates of 6.5%, 6.6% and 3.5% for Mali, Mauritania and Senegal, respectively. - Discount rate: 10% - Expansion period: 25 years. - Economic analysis period: 50 years. - Fuel prices computed for each location, based on an international crude oil price of US$ 20/barrel. - Energy dispatch followed strict economic efficiency rules, i.e. plants with lowest variable costs had priority in dispatching their energy. 13 Non-served energy savings are measured as the difference in this variable when comparing the situation with and without the project. 24 Present values of costs and benefits for the Hydro-Mali and Large- Hydro programs are presented in Table 7. Table 7 ianantali Projeots Costs and Benefits (Present value in million US $) …--------------------------------------------------------------- Hydro-Mali Large-Hydro Costs 68.1 194.1 Benefits - Savings1' 31.9 44.2 - Avoided investment 83.5 148.8 Total benefits 115.3 193.0 Actual Ratio 1.695 0.994 Rate of Return15 30.0% 9.9% ---------------------------------------------------------------- Estimation of model parameters The model parameters to be estimated are the standard deviation and correlation of benefits and costs (ov,, of and rv), and the two effective discount rate Sv and &. Costs are a function of construction costs and demand growth. Demand is specially relevant since this case refers to the costs of 25 year expansion plans, and not to a single project. Assuming that the uncertainty related to construction costs and demand is reflected by standard deviations of 5.6% and 20% (see Costa Rica case), costs uncertainty for both cases should be computed as follows: af = 9(0.0562 + 0.202) = 0.2077 Benefits are assumed to be a function of fuel savings (including savings in non-served energy) and avoided costs. In addition, fuel savings are a function of fuel prices and demand uncertainty. The standard deviations for case 1, i.e. Hydro-Mali alternative, were 14 Fuel and non served energy savings. 15 The consultant's estimate for rates of return were 35% (they actually reported "above 30%") and 10.6%. The difference resides in the fact that the option model discounts in a continuous manner, thus increasing slightly the effects of discounting. 25 estimated as follows: a) Using the same approach as in previous cases, fuel savings uncertainty is computed as follows: OfueI-savfng = 4E((9/50)*.20)2 + (.10)23 0.1063 considering demand and fuel price uncertainties of 20% and 10% respectively, and a period of 9 years during which the hydroplant's energy dispatch will be affected by demand variations. b) Total savings uncertainty are computed as a weighted average of fuel savings and avoided costs. aV = vL((l-.7237)*.1063)2 + (.7237*.2077)23 0.1525 The correlation between costs and benefits in case 1 is assumed to be equal to the ratio of avoided costs to total savings, i.e. 73%. Benefits uncertainty in case 2 (Large-Hydro) were estimated as follows: a) 0fu.-.lavings 4E(.10*.20)2 + .102 ] = 0.1020 considering the same uncertainties as before and that demand variations will affect fuel savings during the initial 5 years of the hydroplant's operation. b) Total savings uncertainty: ov = 4[((l-.7710)*.1020)2 + (.7710*.2077)2] 3 0.1618 The correlation between costs and benefits in case 2 is assumed to be equal to the ratio of avoided costs to total savings, i.e. 77%. Dividend terms are equal to the difference between the discount rate and the rate of growth (escalation). As no escalation on any costs is considered, dividend terms will be equal to the discount rate, i.e. 10%. Results The critical ratios estimated for the two cases indicate that, in the face of uncertainty, the Hydro-Mali program is a robust investment decision, while the Large-Hydro alternative - an 26 economically marginal project under a deterministic point of view16 - is highly risky. These results are shown in Table 8. The critical ratios obtained indicate that penalties of 38% and 34% should be added to the program's costs to incorporate the additional costs of the lost flexibility. Table 8 Manantali Project: Critical Ratios …-----…------------------------------------------------------___ Actual Ratio Critical Ratio optimal Decision …-----------------------------------------------------------__-- Hydro-Mali 1.695 1.376 Build Large-Hydro 0.994 1.343 Don't build ---------------------------------------------------------------- The estimated critical ratios correspond to overall uncertainties (a) of 14.3% and 13.2%. As stated before, these deviations capture the effects of uncertain demand growth, construction costs and fuel prices. The sensitivity of the critical ratio upon variations in the overall standard deviation was analyzed for both cases. In the Hydro-Mali case a range between 11% to 17% was studied, concluding that in all cases the investment was appropriate under the face of uncertainty (the critical ratio ranged from 1.28 to 1.46, always below the actual ratio: 1.69). For the Large-Hydro case, the critical ratio varied from 1.25 to 1.42 for standard deviations of 10% and 16%, concluding that in all cases the investment should be rejected. 3.4 Optimal Timing of Transmission Line Investment A common problem with rural electrification programs is that remote areas characterized by dispersed demand and low loads are costly to connect and supply from a main electricity grid. Decentralized solutions on the basis of diesel generating units may prove more attractive in an early stage. However, as demand grows, the conditions tend to change in favor of an interconnection with the main grid. The present case study addresses the problem of optimal timing for this interconnection in an uncertain environment. 16 At this stage of the project, it was argued correctly that the Large-Hydro project design could be improved in its transmission component, thus increasing its economic merits. However, it is not likely that this improvement would affect the results of the present study. 27 This case study uces the perpetual option model as applied to the problem of optimal timing of investment (Martzoukos and Teplitz- Sembitzky, 1991). The model developed to estimate the expected period of time after which it will be worth investing in the transmission line is presented in Annex 4. The technical and economic assumptions of this hypothetical case are explained below. Costo and Benefits It is assumed that there is an initial peak load of 4MW, with a load factor of 0.35, which is typical for isolated rural areas. The installed capacity amounts to 6MW provided by 'existing' diesel plants. Demand is expected to grow annually at 7%. Incremental demand can be met with new diesel units of 2MW, assuming that economies of scale are negligible. In this case, after a period of 11 years the required capacity would be O1MW; after 17 years it would be 14MW. It is also assumed that diesel plants need to be replaced every 15 years. Alternatively, there is the option to invest in a 230kV transmission line that would be sufficient to supply up to 200MW from the main grid. The length of this line is assumed to be 120Km. Table 9 presents additional economic and technical data. The benefits provided by the transmission option are defined in terms of the parameters that are sensitive to changes in demand. These include fuel savings, the avoided investment costs in diesel Table 9 optimal !iing of Transmission Line Uconomic and Teoonical Data Transmission Line Diesel Program Capacity 230 kV 2 MW/unit Reserve Capacity -- 35% of peak Economic Life 35 years 15 years Construction Time 3 years 1 year Capital Costs 12 million $ 1,000 $/kW Fuel/Energy Costs $ 0.026/kWh $ 0.065/kWh Fixed O&M 2% cap. costs 2% cap. costs Variable O&M -- $ 0.26/kWh 28 plants and the savings in variable OCa costs. On the other hand, costs include the initial investment costs for the transmission line, the corresponding O&N costs minus the fixed (demand independent) costs of operating and maintaining the diesel plants. Model Parameters Assumptions have to be made on the overall uncertainty (a), and the effective discount rates on savings (6.) and costs (6f). The discount rate is assumed to be 10%. Since demand is expected to grow at a 7% rate, the effective rate on savings is 68 3%. Accordingly, assuming that the expected increase in transmission line costs is 1.3% per year, the corresponding effective rate on costs will be &f = 8.7%. The composite standard deviation is assumed to be a = 15%. Results Tab] a 10 presents the resulting values for the discounted net savings (V-F, discounted to year t), the ratio of discounted savings to costs (actual ratio V/F), and the present value of net savings for the initial year t=O, which is denoted as PVO(V-F). All these values correspond to a deterministic situation and refer to various commissioning years t. Table 10 Net Savings and Actual Ratio (savings in million US $) year (t) V - F V/F PV0(V-F) 0 15.084 2.257 15.084 1 16.891 2.389 15.283 2 18.838 2.530 15.423 3 20.936 2.678 15.510 4 23.195 2.835 15.548 5 25.628 3.001 15.544 6 28.247 3.177 15.502 7 31.066 3.364 15.427 8 34.100 3.561 15.322 9 37.364 3.770 15.191 10 40.875 3.991 15.037 11 44.650 4.225 14.863 12 48.711 4.473 14.671 ------------…------------…-------------------------------------- From the table we can observe that in the face of certain savings, the optimal timing to invest would be after 4.4 years, i.e. when the present value of net benefits is maximized. 29 The critical ratio obtained from the options model (on the basis of the assumed parameters) is 3.429. Thus with uncertain savings it can be expected that after 7.34 years it will become economically viable to exercise the transmission option. Although even in the case of no uncertainty it would pay to wait to invest, the deterministic decision rule would lead to a premature investment. The analysis therefore shows that the incorporation of uncertainty creates an incentive to delay the transmission line investment. 30 4. oONCLUSXONS We have seen in four case studies that the option valuation approach is quite easy to handle, it does not require much information, and it provides a simple and very meaningful decision rule that should appeal to analysts. However, its accuracy and reliability depend on the extent to which the model's assumptions hold in practice and whether the simplification undertaken during the modeling process does not distort the problem in question. In applying the options model to the Manantali hydropower project, the simplifying element was reduced since all costs and benefits streams were obtained directly from the results of a power system planning model. In the other case studies simplification could be an issue since the project-to-project comparison does not address the complexities of a multi-decision problem that is usually thought to require a systems approach. The options approach emphasizes that an uncertain environment imposes a penalty to reflect the flexibility foregone as irreversible investment decisions are taken. The case studies presented in this paper show that this penalty can be considerable and should not be ignored. This penalty, which is reflected in the critical ratio, is a case-specific function of the general uncertainty conditions found in the country and, most important, of the particular conditions of the problem in question. in the Costa Rica and Hungary studies, this value varies between 21% to 26% of total costs. In the Manantali case study, where we compare long term expansion programs instead of an specific generating project, uncertainty is higher and, therefore, the penalty increases to a value accounting for 35% of total costs. In the case of the transmission line, uncertainty creates an incentive to delay the investment by almost 3 years beyorid the date that would be optimal if we do not consider any loss of flexibility. Although the application of the options approach may appear to be easy due to its reduced computational effort, it is in reality a rather delicate analytical process. Some important aspects that should be addressed with special care in any application of the optiono model are: - The extent to which the model results are sensitive to variations of key parameters and the problem formulation, such as: (i) the estimation of uncertainty; a very delicate issue that should require inputs from experienced planners, and (ii) the choice of what should be included in costs and in benefits/savings, and the potential impact of these choices on the decision rule. On the basis of the experience gained in the above case studies it is recommended to analyze the sensitivity of results, and specifically of the recommended decision, with respect to variations in the key parameters. 31 - To what extent do basic assumptions of the model hold. Here we are referring essentially to the assumptions related to the behavior of key variables, i.e. the random walk hypothesis. Since in many cases it will be almost impossible to prove statistically whether the model variables follow a random walk, it is recommended to make final assumptions on uncertainty (i.e. on standard deviations) taking into account, in an iterative fashion, the impact of our assumptions on the future dispersion of values (see for example Figure 4) and compare them with our perceptions of the underlying uncertainty. - The correlation between benefits and costs. Although the case studies undertaken do not reveal a significant impact of this parameter on results, it should be recognized that the estimation of this correlation is rarely a straight forward statistical exercise. The estimation of the correlation coefficient is a case-specific problem requiring careful judgement. In general, the application of the options model should be complementary to conventional least-cost planning tools. It should be seen as an additional decision tool aimed at checking and, when necessary modifying, the results of deterministic planning models. A complete evaluation of the options approach and other risk methodologies, including a comparative analysis, will be presented in a final report on IENED's risk and uncertainty program. 32 AmX I The lernetual Option t4h valu of vaiting to invest). model Wo@aulation The solution to the perpetual option is in fact a result of stochastic optimization. The benefit-cost rotio is the stochastic variable *that follows geometric Brownian' motion; a critical barrier C is defined, at which the (positive) investment decision is optimal. If the actual ratio exceeds C we invest, otherwise investment is deferred. The result is a generalization of the deterministic optimization (say optimal timing) of the investment problem. The deterministic problem will be exposed first, since it leads to a better understanding of the model parameters. In the deterministic case, it can be shownta that even an investment with a positive Net Present Value could optimally be deferred. Consider the investment with NPV today equal to "V - F", where "V" represents the benefits and "PO" the costs; also consider that if the investment is deferred, benefits will exhibit a growth of G and costs a growth of Gf. Assuming exponential growth (anZ continuous discounting) the value after time T will equal Ve - WT and the benefit-cost ratio will equal (V/F)e,(v-G)T The sign of Q - Gf determines whether this function is increasing in T. PZositive sign gives an increasing function, negative gives a decreasing function, and 0 gives a constant function. If we use a discount rate of R, optimal timing is when the NPV is maximized1: maxT 1Ve (R 6V)T_Fe-(R-]f )T . 17 Brownian motion is a continuous time Markov stochastic process; as such, its future evolution does not depend on the complete history (past realizations) of the process, but only on the last one. The implication of this (so-called Narkovian) property is that the process has no tendency to revert to a normal mean. It was first used by the physicist Brown to model particle movements. For the relevant mathematics, see Karlin and Taylor (1975). When the natural logarithm of a variable follows Brownian motion, the variable is said to follow a geometric Brownian motion. 18For an example of a Transmission line case, see Schramm (1989). "Positive NPV is equivalent to a Benefit-Cost ratio greater than one, and for monotonically changing present value functions (with respect to time) it can be shown that maximizing NPV is equivalent to maximizing the benefit-cost ratio. ','he rate of change of the benefit-cost ratio equals Gv-Gf. 33 Let's name the differences R-GV and R-Gf as 6 and fs, Greek ntheltas" denoting the effective discount rate on Ienef its 'V" and costs "F"; they represent the opportunity cost on the benefits and costs, if investment is deferred. It is easy to see the optimum when F/V becomes equal to 6f/6v; thus We invest if V6f x F6 . The above ratio of 6S/6v defines a deterministic critical ratio that is compared with the actual. The S. should be above 0; if it equals zero the investment is never implemented since the critfcal ratio is plus infinity. Both 6 and 6f should be above zero for the problem to be well define&. The NPV function in general will plot as an inverted U-shape; it is increasing for 6f>6ve The optimum is at the maximum of the U- shape, as it appears in Figure 1.1 that plots the U-shape for an investment that at time zero exhibits (in arbitrary monetary units) benefits = .95, costs - 1, G - .03, Gf - .013, and the discount rate equals 10%. Thus the benefit-cost ratio today is .95, I equals .07, 6 .087, the NPV equals -.05, and optimal deterministic timing is after 15.81 years (given later by formula 1), when the actual ratio is projected to reach a critical ratio of 1.243 (given later by formula 4a). If the actual benefit-cost ratio is below this critical ratio, it can be shown that optimal timing occurs for: (1) T = Ln((6IF)/(6VV)]/(GV-Gf) if GV-Gf > 0, T = +co if Gv-Gf - 0, and T is undefined if GV-Gf < 0. Under uncertainty, McDonald and Siegel have shown that the above critical ratio is given by: (2) Cw = ij(r-1), where the parameter r is given by: (2a) r - (0.5 - (6f-6V2) + .25)2 + 26f/e] and "a2" equals e,, + ef - 2rvovaf where the rv, denotes correlation; thus the variance of the percentage change of the benefit-cost ratio is a function of the individual variances and the correlation among the benefits and costs' percentage change. 2This is because we assume perpetual constant growth. An asset can not grow at a rate equal to or above the discount rate for ever. For a discussion of such issues in a continuous time stochastic framework, see Brock, Rothschild, and Stiglitz (1988). 34 Figure 1.1 U-SHAPE UF- INvESTMENT'S PRESENT VALUE A FUNCTION OF INVESTMENT TIMiNG 0.07 0.06 0.05 _ 0.04- 2 0.03 _ ° 0.01 _ f 0 O / - -0.02 - -0.03 _ -0.04 1 -0.06 - -0.02 O 1216 1' ' -0.0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 1 3 5 7 9 11 13 15 17 19 Investment Timing the parameter Or" must not be below 1; McDonald and Siegel have defined it as the weighing factor that determines the required return 'll on the investment opportunity2, given the required return on assets with systematic risk similar to that of benefits and costs. The perpetual option model gives the value "WN of this investment opportunity (for actual ratio less than the critical) as: (3) W = (Cd-l)F(V/Fc)? and when the actual ratio becomes equal to the critical, W equals the difference between benefits and costs; under the deterministic 2180 gA - v*Rv + (l-v)*Rf, where RP and Rf are the required return on assets with (approximately) the same systematic risk with benefits and costs respectively; it is equivalent to going long on the benefits and short on the costs. 35 approach (02 0) this reducesU to: (4) W - (C -1)F(V/FC')t/(ftV) and (4a) C / In this case 6f should be greater than or equal to 6v, otherwise we do not wait but we invest now (if actual ratio is at least unity) or never. It has been shown2 that the optimal timing of the investment under 22It can be shown that Ve eqKation that gives r can be rewritten as r = 2f/ ( /[ (6f_6V_o) +26fa ] + (6f-6V-/2) ) and for zero variance r -> 6fl(4-6 Under no uncertainty the investment opportunity W of formula (3) also equals the Net Prpfsent Value under optimal deterministic timing -> W = Ve " - Fe''f where T is given by formula (1). 3See Martzoukos and Teplitz (1991). The result seems intuitive, but the proof requires use of the martingale stopping theorem; for such applications in a Brownian motion context, see Karlin and Taylor (1975) pp. 357-365. In addition, the introduction of an arbitrary lower "absorbing barrier" where the investment option becomes insignificant (i.e., actual ratio of 0.001) gives the following modified formulae for the optimal timing: E(T) - (Ln]eiP - Ln[V/F] + *(l-P])/(G4-Gf) if Gv-Gf > O, E(T) = 2 ( cj]p + In2[V/F] - 2LOEC"]Ln1V/F]P + *(l-P)(#-2Ln[V/F]))/ if Gy-Gf = 0, and B(T) is undefined for G,-Gf < O where P is the probability that the actual ratio becomes equal to the critical before it drops to the lower value *, and is given by: p _ {1 _ [(F/V)0l-n]-2 Gv-Qf)/b/}il - [lO0^/C ]'2(VGf)/b) if Gv - Gf > 0, P = (Ln[V/F]/n + Lnl10])/{Ln(C']/n + Ln[10]) if G. - GO = ° b equals the variance "a2" and * equals Ln[l0"]; for example for a lower benefit-cost ratio of 0.001 the parameter n equals 3 and 0 equals -6.9078. Such a lower barrier has the significance that low value options can be abandoned due, for example, to the high administrative costs of keeping them alive although theoretically options always will have a positive value. Another significance of these formulae is that, even if Gv - Gf = 0, uncertainty is expected to be resolved in finite time; here optimal timing is redefined to mean the time that the actual ratio either reaches the critical, 36 uncertainty is given by the following, where "E" denotes the expectation operator: (5) E(T) = Ln(CeF/V)/(Gv-Gf) if Gv-Gf > O, E(T) = +0 if Gv-Gf = 0, a;.d E(T) is undefined if GV-Gf < 0 where C* denotes the critical benefit-cost ratio under uncertainty. The decision rule whether to invest in both cases is given by the comparison of the critical ratio and the actual ratio; we invest if actual ratio is greater than or equal to the critical ratio. The impact of uncertainty is that the critical ratio is higher than in the no uncertainty case; uncertainty thus defers investment assigning a premium to waiting so as to receive more information about the future evolution of critical uncertain variables. The Model Parameters and Assumptions (perpetual option to vait to invest). The decision whether to invest or not is a result of the comparison between the critical and the actual rati-. The main model parameters for the critical ratio are the dividend yields "6 " and "6,", and the variance "a'" of the rate of change of the benefit-cost ratio. The dividend terms (effective discount rates or opportunity costs) equal the difference between the corresponding discount rate and the rate of growth as: 6v - R - Gv and 6f = R - G,. The a2 equals a2v + a2f - 2rfavaf; it is a function of the variance of the benefits and costs' rate of change and their correlation. The critical ratio is an increasing function of 6 (positive first derivative) and decreasing function of Sv (negative first deri- vative). If the effective discount rate on the costs is higher the required flexibility premium is also higher since the opportunity costs (of deferring construction) are lower; if the effective discount rate on the benefits is higher the required flexibility premium is lower, since the opportunity cost on the foregone savings is higher. or falls to the lower barrier and in any case uncertainty about the investment is resolved. 37 The two dividend terms should always be positive24. By construction the perpetual option model considers the underlying savings and costs as the present value of perpetual cash flows. If the effective discount rate were zero, the value of the perpetual stream of cash flows (ratio of an annual cash flow over the effective discount rate) would be infinite. Also if the effective discount rate were negative, the value of the perpetuity would be negative too; thus both cases are not feasible. In the deterministic case if condition 6, > 6 is satisfied, thX investment should be deferred; 6S / 6v equals the critical ratio "Cd" If this critical ratio is not above unity, the investment is either profitable to exercise now (NPV > 0) or is never.exercised. In the presence of uncertainty the critical ratio "C " is even higher ("C* " is greater than or equal to "Cn) If the two dividends are equal, then the parameters "T" and C are simplified: (6) t .5 + 4(.25 + 26f/02) (6a) Ce -.5 + 4(.25 + 26,/a2)] / [/(.25 + 26,1a2) - .5] Figure 1 (in the main text) F;hows the critical ratio as a function of 6v, keeping the other mod l parameters constant (i.e., discount rate 10%, cost apprecia ion 1.3%, total effective standard deviation 15%). Figure 2 (also in the main text) shows the critical ratio as a function of Sf,, keeping the other model parameters constant (i.e., discount rate 10%, savings' appreciation 1.3%, total effective standard deviation 15%). The critical ratio is also an increasing function of the total effective variance *la2"; if uncertainty is higher, the value of waiting to receive more information is higher and the required flexibility premium is higher too. Note that this is not always true for the individual standard deviations, since the total effective standard deviation is a (quadratic) function of the two standard deviations and the correlation. Figure 3, in the main text, shows the critical ratio as a function of total effective standard deviation, keeping the other model 24In the case of stock options and in the absence of dividends we would have 6 - 0 and it can be shown that the perpetual option value equals the stock value; the critical ratio becomes infinite, and the option is never exercised. The same result stands for finite maturity stock options, thus the European and the American options are equivalent; the option is exercised at maturity only. 2'Lets denote the critical rat.io in the deterministic approach and under uncertainty as ed and Cu respectively. 38 parameters constant (i.e., discount rate 10%, savings' appreciation 1.3*, cost appreciation 1.3%). As was discussed before, in the absence of uncertainty (a2 = 0) the formulas for the critical ratio reduce to the ones under the deterministic approach; (2) as a function of (2a) reduces to (4a) as regards the critical ratio; (3) to (4) as regards the investment opportunity; and (5) to (1) as regards the optimal timing of the investment. As we see from Figure 3, the critical ratio and thus the investment decision is very sensitive to our estimate of the underlying uncer- tainty. Our unwillingness to invest unless a flexibility premium is received is equivalent to hedging against this uncertainty. It is important to understand what this uncertainty implies. For exposition we provide a set of graphs (Figures 1.2 to 1.7) for a lognormally distributed variable (with value of 1 at time zero and standard deviation varying from 5% to 30%) and the 65% and 95% confidence intervals for the possible values this variable could take in the future. The graphs assume zero drift for the underlying Brownian motion process which is equivalent with accepting that benefits and costs exhibit the same growth rate, since the growth rate of the benefit-cost ratio equals the difference of the two groath rates. The extension to the case with arbitrary drift is straightforward. Critical Assumptions Affecting the actual Ratio a) The lognormalitv for the distributional characteristics of the underlying assets. b) The existence of perpetually constant rates of change (that imply constant effective discount rates). c) Continuous time assumption as a result of modeling the underlying assets with Brownian motion. The first assumption is relevant to the way the benefit-cost ratio is estimated. Both benefits and costs individually are (approxi- mately) lognormally distributed; thus negative values should be excluded. To avoid violating the above assumption, all sub-parts of the benefits and costs should have a positive sign (i.e.,, they should be additive). If for example an investment in a Hydro scheme results in avoiding construction of thermal generation facilities, the avoided construction costs should not be subtracted from the costs but should be added to the savings. Note the implication on estimating the relevant uncertainty of the costs and savings. The second assumption is relevant to the way savings and costs are estimated. It can be shown that a constant rate of change implies a perpetual stream of cash flows; the benefits and costs due to the investment should either be expressed in the equivalent perpetuity 39 UNIT LOGNORMAL STOCHASTIC VARIABLE 95% AND 65% CONFIDENCE INTERVALS 1.6 1.5 1.4 1.3 14 ; 1.2 1.1 t i 0.7 _ _ __;, 0.6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time; standord deviation = 5% Inner (outer) 2 curves = 65% (95%) confidence intervol. UNIT LOGNORMAL STOCHASTIC VARIABLE 95% AND 65% CONFIDENCE INTERVALS 2.3 2.2 2.1 2 1.8 1.7 1.6 g: 1.5 --- 1.4 tL 1.3 -,_ W 1.2 X I-- to 1. 0 1 C; 0.9 _ 0.8 - ^ 0.7 _ i- -------- 0.6__:v ; _. 0.5 . l _ _ _l : I - L_ 0.4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Times; stondord deviation = 10% Inner (outer) 2 curves = 65% (95%) confidence intervol. UNIT LOGNORMAL STOCHASTIC VARIABLE 95% AND 65% CONFIDENCE INTERVALS 3.4. 3.2 3 2.8 2.6 2.4 2.2 2 I' 1.8 1.4 1.4 o 1.2 0.8 0.6 0.4 4 0.2 I I I I - I I . I L I . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time; stondord deviation 1 157 Inner (outer) 2 curves = 65% (95%) confidence interval. UNIT LOGNORMAL STOCHASTIC VARIABLE 95% AND 65% CONFIDENCE INTERVALS 5 4.5 4. 3.5 .1 2 iL 0.5 ------a Li '-a to~~~ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time: standard deviation = 20% Inner (outer) 2 curves = 65% (957e) confidence intervol. UNIT LOGNORMAL STOCHASTIC VARIABLE 95% AND 65% CONFIDENCE INTERVALS 8 7 6 v 4S 4 _ 2 0t / a 1 2 3 4 5 6 7 8 9 10 11 ;2 13 14 15 rime; stondord devatto-P 25X Inner (outer) 2 curves = 657* (95%) confidence interval. UNIT LOGNORMAL STOCHASTIC VARIABLE 95% AND 65% CONFIDENCE INTERVALS 10 A 9 a 7 6 4 _ _ _ _ ____ 0(_ w - -J 4 O I o 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time; standord deviation = 30% Inner (outer) 2 curves = 65% (957.) confidence inteval. form or we should consider several "rotations"26 of similar investments and increase the benefits and costs accordingly. We adapt the second method because it is intuitively easier to quantify the underlying uncertainty. For practical purposes, this adaptation is not crucial if the life of the irreversible alternative is sufficiently large. The third assumption is important27 for the way present values of the cash flows are estimated; the need for continuous discounting is obvious where the effective discount rate is the difference "S" between the discount rate and the growth rate. The following table shows the difference between continuous and regular discounting at three levels of discount rate, notably 8%, 10%, and 12%. The regular discounting would overestimate cash flows relative to the continuous discounting. For the relevant formulae for lump-sums, annuities and growth annuities under continuous discounting see Copeland and Weston (1983). Table 1.1 % DIFFEP ENCE BETWEEN CONTINUOUS AND REGULAR DISCOUNTING. annuities Discount rate: 8% 10% 12% 5yrs 0.0087 0.0132 0.0186 lOyrs 0.0149 0.0223 0.0309 2Oyrs 0.0246 0.0355 0.0473 3Oyrs 0.0312 0.0434 0.0559 4Oyrs 0.0354 0.0477 0.0598 5Oyrs 0.0379 0.0498 0.0614 lumpsums Discount rate: 8% 10% 12% 5yrs 0.0153 0.0237 0.0339 lOyrs 0.0309 0.0480 0.0690 20yrs 0.0627 0.0983 0.1427 30yrs 0.0955 0.1511 0.2216 40yrs 0.1293 0.2063 0.3058 50yrs 0.1641 0.2643 0.3959 26This Is referred to ae the rotations problem, and results in a different Net Present Value and Benefit-Cost ratio; if the effective discount rates "w6" and "Ef are the same, the benefit- cost ratio remains unchanged. This adaptation is used very often when comparing projects with unequal lives. "Note that the same assumption also implies continuous trading. We can reasonably assume this to hold without practical implications. 46 Note that the above table exhibits the percent of overestimation of the present value of cash flows when the regular discounting method is used. 47 A11B3 2 costa Rica Case StudY After considering the investment expansion of the local utility, the following alternative scenarios were considered: a) all Gas Turbines (denoted by GT), b) the Hydro plant plus Gas Turbines (denoted by HYD), c) the Geothermal plant plus Gas Turbines (denoted by GEO). The following table shows the forecasted energy demand and total capacity expansion under each scenario for the years 1988 to 1998, with #T, #H, and #G denoting the number of Gas Turbines, Hydro, and Geothermal plants constructed under each scenario. We assume that construction is to begin in 1991, with decision taken by the end Of 1990. Table 2.1 Power Uxpansion scenarios. Max load forecasts (+ margin 10%) and investment plans (T, H, G) Demand GT #T HYD #T #H GEO #T #G 1988 729.3 751.0 751.0 751.0 1989 751.9 751.0 751.0 751.0 1990 774.5 787.0 1 787.0 1 -- 787.0 1 -- 1991 798.5 823.0 1 823.0 1 -- 891.0 1 -- 1992 824.0 823.0 -- 823.0 -- -- 891.0 -- -- 1993 859.5 859.0 1 1000.0 1 -- 891.0 -- 1 1994 901.6 895.0 1 1000.0 1 -- 927.0 -- -- 1995 951.2 967.0 2 1000.0 -- 1 963.0 1 -- 1996 992.1 1003.0 1 1000.0 -- -- 999.0 1 -- 1997 1040.7 1039.0 1 1036.0 -- -0 1998 1092.7 1111.0 2 1108.0 1 -- ..... . Notes: The increase in maximum forecasted load was .031, .03, .031, .032, .043, .049, .055, .043, .049, and .05 from year 1989 to year 1998. 2The installed capacity was 751NW at the end of 1988, and maximum load demand vas 663MW for the same year. These figures were the starting point for our demand and capacity projections. We assume that during 1990 a diesel plant will be added to the network to increase capacity. 48 The capacity expansion considers the addition of 177 NW for the Hydro plant, 104 KW for the Geothermal plant, and 36 MW for each Gas Turbine. We can see that the Hydro plant will reduce the future investment by 5 Gas Ttrbines and the Geothermal plant will reduce future investment by 3 Gas Turbines. Sensitivity analysis The Acceptance - Rejection matrix that follows in Tables 2.2 and 2.3 for the Hydro and the Geothermal, respectively, shows the actual and critical values for different combinations of discount rates and uncertainty, defining two regions in the plane: one where the investment is accepted and one where it is rejected. Tables 2.4 and 2.5 present complete results of the sensitivity analysis. 49 Table 2. 2 REWJCTION - ACCEP!ANCZ MATRIX IT SHOWS THE PAIRS OF CRITICAL BENEFIT CST RATIO (UPPER) AND ACTUAL BENEFIT COST RATIO (LOWER) COSTA RICA, HYDRO - PROJECT (I.R.R.=12.2%) MAIN UNCERTAINTY: FUEL SAVINGS (WHILE COSTS HAVE CONSTANT STD 5.6% AND 15% CORRELATION WITH FUEL SAVINGS) STANDARD DEVIATION OF PERCENTAGE CHANGE OF SAVINGS DISCOUNT II .05 .10 .15 .20 .25 RATES % .075 1.994 2.074 1.944 .08 1.901 1.682 .085 1.747 1.654 .09 ACCEPT 1.609 1.452 .095 1.485 1.292 1.437 .10 1.374 1.374 1.283 .105 1.273 1.170 .11 1.182 1.166 .115 1.100 .12 REJECT .13 50 Table 2.3 R8EJCTION - ACCEPTANCE UlTRIX IT SHOWS THE PAIRS OF CRITICAL BENEFIT COST RATIO (UPPER) AND ACTUAL BENEFIT COST RATIO (LOWER) COSTA RICA, GEOTHERMAL - PROJECT (I.R.R.=13.05%) MAIN UNCERTAINTY: FUEL SAVINGS (WHILE COSTS HAVE CONSTANT STD 5.6% AND 15% CORRELATION WITH FUEL SAVINGS) STANDARD DEVIATION OF PERCENTAGE CHANGE OF SAVINGS DISCOUNT | .05 .10 .15 .20 .25 RATES% ii 2.199 .06 2.252 2.119 .065 2.108 1.713 .08 1.744 1.682 .085 1.642 .09 ACCEPT 1.452 .095 1.459 1.437 .10 1.379 1.283 .105 1.304 1.274 .11 1.234 1.160 .115 1.208 1.156 .12 1.134 REJECT .13 51 Table 2.4 SENSITIVITY ANALYSIS, COSTA RICA BYDRO VIS THERMAL UNITS. FLEXIBILITY OPTION TO WAIT TO INVEST V F DV DF SV SF RVF C X 397.31 289.26 0.042 0.087 0.086 0.056 0.15 2.26512 149.433 397.31 289.26 0.057 0.087 0.086 0.056 0.15 1.71718 121.541 397.31 289.26 0.072 0.087 0.086 0.056 0.15 1.42125 108.591 397.31 289.26 0.087 0.087 0.086 0.056 0.15 1.25613 114.833 397.31 289.26 0.102 0.087 0.086 0.056 0.15 1.16594 151.802 397.31 289.26 0.117 0.087 0.086 0.056 0.15 1.11631 246.174 397.31 289.26 0.132 0.087 0.086 0.056 0.15 1.08739 462.545 397.31 289.26 0.087 0.042 0.086 0.056 0.15 1.09351 389.148 397.31 289.26 0.087 0.057 0.086 0.056 0.15 1.1250r 217.841 397.31 289.26 0.087 0.072 0.086 0.056 0.15 1.17620 143.529 397.31 289.26 0.087 0.087 0.086 0.056 0.15 1.25613 114.833 397.31 289.26 0.087 0.086 0.086 0.056 0.15 1.24980 115.882 397.31 289.26 0.087 0.117 0.086 0.056 0.15 1.50124 111.096 397.31 289.26 0.087 0.132 0.086 0.056 0.15 1.64983 118.032 397.31 289.26 0.087 0.087 0.056 0.056 0.15 1.19103 134.412 397.31 289.26 0.087 0.087 0.066 0.056 0.15 1.21075 125.835 397.31 289.26 0.087 0.087 0.076 0.056 0.15 1.23256 119.425 397.31 289.26 0.087 0.087 0.086 0.056 0.15 1.25613 114.833 397.31 289.26 0.087 0.087 0.096 0.056 0.15 1.28120 111.689 397.31 289.26 0.087 0.087 0.106 0.056 0.15 1.30761 109.673 397.31 289.26 0.087 0.087 0.116 0.056 0.15 1.33521 108.534 397.31 289.26 0.087 0.087 0.130 0.056 0.15 1.37569 108.051 397.31 289.26 0.087 0.087 0.150 0.056 0.15 1.43692 108.958 397.31 289.26 0.087 0.087 0.086 0.026 0.15 1.22860 120.406 397.31 289.26 0.087 0.087 0.086 0.036 0.15 1.23471 118.922 397.31 289.26 0.087 0.087 0.086 0.046 0.15 1.24396 116.960 397.31 289.26 0.087 0.087 0.086 0.056 0.15 1.25613 114.833 397.31 289.26 0.087 0.087 0.086 0.066 0.15 1.27093 112.804 397.31 289.26 0.087 0.087 0.086 0.076 0.15 1.28809 111.055 397.31 289.26 0.087 0.087 0.086 0.086 0.15 1.30736 109.687 397.31 289.26 0.087 0.087 0.086 0.056 0.05 1.27094 112.802 397.31 289.26 0.087 0.087 0.086 0.056 0.08 1.26656 113.347 397.31 289.26 0.087 0.087 0.086 0.056 0.12 1.26063 114.156 397.31 289.26 0.087 0.087 0.086 0.056 0.15 1.25613 114.833 397.31 289.26 0.087 0.087 0.086 0.056 0.17 1.25309 115.321 397.31 289.26 0.087 0.087 0.086 0.056 0.20 1.24849 116.114 397.31 289.26 0.087 0.087 0.086 0.056 0.25 1.24068 117.619 Actual ratio: 1.374 Note: if the actual ratio is greater than the critical ratio, the option price "X" is not valid, and should be equal to the difference between savings 'V" and costs "F" 52 Table 2.5 SZNSI!XTVZY ANaXIYSZ, COomTAa RICA UOH aL VIa !REoKAL UNIT8. FLUXIBZLITY OPTZON TO WAVT TO INVUBS V F DV DF SV SF RVF C X 312.91 226.98 0.042 0.087 0.088 0.056 0.15 2.27139 118.260 312.91 226.98 0.057 0.087 0.088 0.056 0.15 1.72287 96.447 312.91 226.98 0.072 0.087 0.088 0.056 0.15 1.42665 86.354 312.91 226.98 0.087 0.087 0.088 0.056 0.15 1.26103 91.131 312.91 226.98 0.102 0.087 0.088 0.056 0.15 1.17005 119.305 312.91 226.98 0.117 0.087 0.088 0.056 0.15 1.11963 190.332 312.91 226.98 0.132 0.087 0.088 0.056 0.15 1.09007 350.518 312.91 226.98 0.087 0.042 0.088 0.056 0.15 1.09653 295.054 312.91 226.98 0.087 0.057 0.088 0.056 0.15 1.12878 168.600 312.91 226.98 0.087 0.072 0.088 0.056 0.15 1.18068 112.898 312.91 226.98 0.087 0.087 0.088 0.056 0.15 1.26103 91.131 312.91 226.98 0.087 0.086 0.088 0.056 0.15 1.25469 91.932 312.91 226.98 0.087 0.117 0.088 0.056 0.15 1.50571 88.272 312.91 226.98 0.087 0.132 0.088 0.056 0.15 1.65390 93.646 312.91 226.98 0.087 0.087 0.058 0.056 0.15 1.19479 106.343 312.91 226.98 0.087 0.087 0.068 0.056 0.15 1.21496 99.649 312.91 226.98 0.087 0.087 0.078 0.056 0.15 1.23714 94.679 312.91 226.98 0.087 0.087 0.088 0.056 0.15 1.26103 91.131 312.91 226.98 0.087 0.087 0.098 0.056 0.15 1.28638 88.709 312.91 226.98 0.087 0.087 0.108 0.056 0.15 1.31304 87.160 312.91 226.98 0.087 0.087 0.118 0.056 0.15 1.34087 86.290 312.91 226.98 0.087 0.087 0.130 0.056 0.15 1.37569 85.932 312.91 226.98 0.087 0.087 0.150 0.056 0.15 1.43692 86.535 312.91 226.98 0.087 0.087 0.088 0.026 0.150 1.23421 95.2270 312.91 226.98 0.087 0.087 0.088 0.036 0.150 1.24012 94.1512 312.91 226.98 0.087 0.087 0.088 0.046 0.150 1.24913 92.7113 312.91 226.98 0.087 0.087 0.088 0.056 0.150 1.26103 91.1315 312.91 226.98 0.087 0.087 0.088 0.066 0.150 1.27555 89.6080 312.91 226.98 0.087 0.087 0.088 0.076 0.150 1.29244 88.2806 312.91 226.98 0.087 0.087 0.088 0.086 0.150 1.31145 87.2303 312.91 226.98 0.087 0.087 0.088 0.056 0.050 1.27599 89.5677 312.91 226.98 0.087 0.087 0.088 0.056 0.080 1.27156 89.9873 312.91 226.98 0.087 0.087 0.088 0.056 0.120 1.26558 90.6105 312.91 226.98 0.087 0.087 0.088 0.056 0.150 1.26103 91.1315 312.91 226.98 0.087 0.087 0.088 0.056 0.170 1.25796 91.5070 312.91 226.98 0.087 0.087 0.088 0.056 0.200 1.25332 92.1169 312.91 226.98 0.087 0.087 0.088 0.056 0.250 1.24544 93.2736 Actual ratio: 1.379 Note: if actual ratio is greater than the critical ratio, the option price "X" is not valid, and should be equal to the difference between savings "V" and costs "F" 53 Monte Carlo simulation. The following figure shows the results of the Monte Carlo simulation. The computer work was done with SAS (on the World Bank's mainframe, CMS operating system). The parameters were as follows (for the perpetual warrant "option to wait to invest"): V = the "stock price equivalent" (total savings), F = the "exercise price" (construction costs), DV = the "dividend equivalent" of total savings, DF = the "dividend equivalent" of construction costs, SV = the uncertainty of the total savings, SF = the uncertainty of the construction costs, RVF = the correlation between total savings and costs, C the critical ratio, and x = the option price (to wait to invest) The Monte Carlo simulation was based on the following assumptions for the underlying distributions the characterize our uncertainty on the model parameters: Option to wait to invest, Hydro and Geothermal vis thermal alternatives. DV -> triangular distribution between .0787 and .0987 DF -> triangular distribution between .0787 and .0987 SV -> triangular distribution between .067 and .107 SF -> triangular distribution between .046 and .066 RVF -> triangular distribution between .13 and .17 The Monte Carlo simulation results show that there is considerable uncertainty regarding the estimates of the critical benefit-cost ratio, but in both cases the conclusions stay the same. The simulation results are synopsized as follows: option to wait to invest for hydro or geothermal vis thermal #replications mean std minimum maximum critical ratio: 30000 1.26048 .04626 1.12803 1.47356 Percentage of replications where the critical ratio exceeded the actual benefit-cost ratio: - hydro: 1.4% - geothermal: 1.0% 54 Figure 2.1 MONTE CARM ON CRITICAL C,N30000, COSTA RICA, NYDRO - GEOTNERMAL N Mean Std Dev Minimum Maximum _____________-____-_--------------- _____________-- - --------- 30000 1.2604795 0.0462615 1.1280266 1.4735634 ---------------------------------------------------------__-- Percentage ] ~~~** 6 + 5 + 4 + 3 3 + 1 2 + ********* 1 + ************ ] ************************* actual benefit-cost 3 **************************s ratio ] ******************************** * --------------------------------------------- 11111111111111111111111111111111111111111111 11111111122222222222223333333333334444444444 23456678900123445678890122345667890012344567 86420864208642086420864208642086420864208642 C Midpoint 55 A1533 3 Iunaar! Case Study The 3 scenarios under consideration are: CC (only combined cycle units), NUCL (combined cycle units with the addition of a nuclear plant), and COAL (combined cycle units with the addition of a coal plant). Table 3.1 Power Nzpansion scenarios. YR Demand Capac. CC #CC NUCL #N #CC COAL #C #CC 1991 8000 7643 8048 3 8048 - 3 8048 - 3 1992 8104 7643 8048 - 8048 - - 8048 - - 1993 8209 7643 8183 1 8183 - 1 8183 - 1 1994 8316 7643 8318 1 8318 - 1 8318 - 1 1995 8424 7643 8453 1 8453 - 1 8588 1 - 1996 8534 7445 8525 2 8525 - 2 8525 - 1 1997 8645 7369 8584 1 8929 1 - 8584 - 1 1998 8757 7343 8693 1 8903 - - 8693 - 1 1999 8871 7316 8801 1 8876 - - 8801 - 1 2000 8986 7289 8909 1 8984 - 1 8909 - 1 2001 9103 7120 9145 3 9085 - 2 9145 - 3 2002 9221 7093 9253 1 9133 - 1 9253 - 1 _e_r_.r,- -- - --- -- -- - Notes: The capacity at year 1991 equals 7643 NW after reducing the existing 8393 NW by 750 NW. Hungary was importing 1850 MW power from the Soviet Union, but from 1991 only the 1100 NW are ensured by contracts. Demand refers to (projected) maximum load demand pluS 25% margin during any year. Capac. refers to capacity available during the year, taking into account the retirement of existing plants, before any additional capacity. CC stands for capacity after expansion only with combined cycle units. NUCL stands for capacity after expansion with a nuclear plant and combined cycle units. COAL stands for capacity after expansion with a coal plant and combined cycle units. #CC, #N, and #C stand for number of combined cycle units, nuclear plants, and coal plants respectively. 56 The #CC (new investment in combined cycle) in the nuclear and the coal scenario respectively subtracted from the first #CC (new combined cycle units in all combined cycle scenario) denote the construction savings in number of combined cycle units. As the final capacity under the nuclear scenario is 9193 NW (instead of 9253 under the other coal scenario) we consider the additional savings of one unit of combined cycle with half the capacity assumed for combined cycle units that could be available at the start of year 2001, so as to give the same reliability to the power system. This way we assume away economies of scale for this part of the capacity expansion, which is acceptable due to the heavy discounting. Practically at the end of year 2001 we consider the installation of 2.5 CC units. In either icenario the investments are fully utilized (to the limit of their capacity factor) from the first year of their operation; this implies the existence of older plants with higher variable costs that would be displaced in the energy dispatch priorities so demand uncertainty is not important for these "irreversible" investments. 57 Table 3.2 Costs and Bavings under Different 8cenarios. CC ALTERNATIVE DISTILLATE ONLY 50% DISTILIATE-50% GAS VIS NUCLEAR COAL NUCLEAR COAL in mill Forints - Gross fixed cost 50976.0 22464.0 50976.0 22464.0 decommissioning 242.9 ----- 242.9 ----- Total Fixed Cost 51218.9 22464.0 51218.9 22464.0 Fuel savings 22541.6 10446.6 16436.6 6674.2 O+K savings -1952.9 -1191.8 -1952.9 -1191.8 Total fuel SavingsD 20588.7 9254.7 14483.7 5482.4 Cost savings 9402.6 6978.5 9402.6 6978.5 Total Savings 29991.3 16233.2 23886.3 12460.9 Actual Ratio .586 .723 .466 .555 IRR 6.10% 6.95% 4.3% 4.3% Notes: The actual ratio is that between the sum of the fuel and O+M savings plus the avoided diesel CC-construction costs over the total fixed cost of the investment. The above appraisal is based on discount rates of 10%. The cost of the combined cycle units is assumed to appreciate by 1.3% (real) per year. The fuel prices (and O+M costs) are assumed to appreciate by 1.3% (real) per year. 2The total fuel savings are calculated after taking into account: a) the fuel (and O0+) savings will materialize after the fuel- efficient plant will be in operation, for the life time of this plant, discounted to the present. b) a constant appreciation rate for the savings of 1.3% per year. c) the use of negative O+M savings violates the rule stemming from the assumption that savings are log-normal; however, the effect of this violation is negligible (the effect on the actual ratio for the distillate fuel case is very small: nuclear .601 instead of .586, coal .738 instead of .723). 58 The decommissioning cost for the nuclear plant is assumed30 to be 130 $/KW (or 8190 For/KVs), appreciating by 1.3% (real) per year. The nuclear and coal plants are fully utilized from the first year of their operation, being used for both base and peak load; they replace other thermal type units. The current fixed cost for the different units is 50976 mill-For for the nuclear plant, 22464 mill-For for the coal plant, and 4725 mill-For for each combined-cycle unit. The remaining model parameters are the variance terms and the drift terms (dividend equivalent) used to calculate option values which are discussed in the main text. 3See Traiforos et.al., "The Status of Nuclear Plant Techlology - An Update", IEN Energy Series Paper No.27 59 Table 3.3 SENSITIVITY ANRLYSIS8 NW3RY NUCLA VIS CC UOSING DISX!LL&TB FUL. FLEXIBILITY OPTION TO WAIT TO INVEST v F DV DF SV SF RVF C X 29.99 51.22 0.042 0.087 0.071 0.056 0.070 2.23557 5.6045 29.99 51.22 0.057 0.087 0.071 0.056 0.070 1.69003 2.6351 29.99 51.22 0.072 0.087 0.071 0.056 0.070 1.39517 0.9438 29.99 51.22 0.087 0.087 0.071 0.056 0.070 1.23231 0.2297 29.99 51.22 0.102 0.087 0.071 0.056 0.070 1.14605 0.0385 29.99 51.22 0.117 0.087 0.071 0.056 0.070 1.10043 0.0051 29.99 51.22 0.132 0.087 0.071 0.056 0.070 1.07465 0.0006 29.99 51.22 0.087 0.042 0.071 0.056 0.070 1.07924 0.0010 29.99 51.22 0.087 0.057 0.071 0.056 0.070 1.10726 0.0076 29.99 51.22 0.087 0.072 0.071 0.056 0.070 1.15462 0.0497 29.99 51.22 0.087 0.087 0.071 0.056 0.070 1.23231 0.2297 29.99 51.22 0.087 0.086 0.071 0.056 0.070 1.22605 0.2102 29.99 51.22 0.087 0.117 0.071 0.056 0.070 1.47988 1.4085 29.99 51.22 0.087 0.132 0.071 0.056 0.070 1.63050 2.2850 29.99 51.22 0.087 0.087 0.031 0.056 0.070 1.16031 0.0581 29.99 51s22 0.087 0.087 0.041 0.056 0.070 1.17417 0.0819 29.99 51.22 0.087 0.087 0.061 0.056 0.070 1.21067 0.1660 29.99 51.22 0.087 0.087 0.071 0.056 0.070 1.23231 0.2297 29.99 51.22 0.087 0.087 0.081 0.056 0.070 1.25573 0.3091 29.99 51.22 0.087 0.087 0.091 0.056 0.070 1.28068 0.4043 29.99 51.22 0.087 0.087 0.101 0.056 0.070 1.30697 0.5150 29.99 51.22 0.087 0.087 0.071 0.026 0.070 1.19350 0.1226 29.99 51.22 0.087 0.087 0.071 0.036 0.070 1.20335 0.1467 29.99 51.22 0.087 0.087 0.071 0.046 0.070 1.21642 0.1819 29.99 51.22 0.087 0.087 0.071 0.056 0.070 1.23231 0.2297 29.99 51.22 0.087 0.087 0.071 0.066 0.070 1.25065 0.2910 29.99 51.22 0.087 0.087 0.071 0.076 0.070 1.27111 0.3666 29.99 51.22 0.087 0.087 0.071 0.086 0.070 1.29342 0.4567 29.99 51.22 0.087 0.087 0.071 0.056 0.020 1.23894 0.2511 29.99 51.22 0.087 0.087 0.071 0.056 0.035 1.23696 0.2446 29.99 51.22 0.087 0.087 0.071 0.056 0.055 1.23431 0.2361 29.99 51.22 0.087 0.087 0.071 0.056 0.070 1.23231 0.2297 29.99 51.22 0.087 0.087 0.071 0.056 0.085 1.23030 0.2234 29.99 51.22 0.087 0.087 0.071 0.056 0.115 1.22623 0.2108 29.99 51.22 0.087 0.087 0.071 0.056 0.150 1.22142 0.1964 Actual ratio: .586 Note: if actual ratio is greater than the critical ratio, the option price *X" is not valid, and should be equal to the difference between savings *V" and costs "F" 60 Table 3.4 UNSITIVITY AMLYSIS, NUNGARY COAL VIS CC USING DISTILLaTE FVL. FLEXIBILITY OPTION TO WAIT TO INVUST V F DV DF SV SF RVF C X 16.L3 22.46 0.042 0.087 0.062 0.056 0.070 2.21250 3.5344 16.23 22.46 0.057 0.087 0.062 0.056 0.070 1.66849 1.8598 16.23 22.46 0.072 0.087 0.062 0.056 0.070 1.37401 0.7925 16.23 22.46 0.087 0.087 0.062 0.056 0.070 1.21275 0.2497 16.23 22.46 0.102 0.087 0.062 0.056 0.070 1.12988 0.0597 16.23 22.46 0.117 0.087 0.062 0.056 0.070 1.08772 0.0124 16.23 22.46 0.132 0.087 0.062 0.056 0.070 1.06458 0.0024 16.23 22.46 0.087 0.042 0.062 0.056 0.070 1.06810 0.00333 16.23 22.46 0.087 0.057 0.062 0.056 0.070 1.09315 0.01625 16.23 22.46 0.087 0.072 0.062 0.056 0.070 1.13711 0.07171 16.23 22.46 0.087 0.087 0.062 0.056 0.070 1.21275 0.24973 16.23 22.46 0.087 0.086 0.062 0.056 0.070 1.20653 0.23218 16.23 22.46 0.087 0.117 0.062 0.056 0.070 1.46279 1.11873 16.23 22.46 0.087 0.132 0.062 0.056 0.070 1.61523 1.67233 16.23 22.46 0.087 0.087 0.032 0.056 0.070 1.16154 0.11956 16.23 22.46 0.087 0.087 0.042 0.056 0.070 1.17574 0.15204 16.23 22.46 0.087 0.087 0.052 0.056 0.070 1.19299 0.19545 16.23 22.46 0.087 0.087 0.062 0.056 0.070 1.21275 0.24973 16.23 22.46 0.087 0.087 0.072 0.056 0.070 1.23458 0.31440 16.23 22.46 0.087 0.087 0.082 0.056 0.070 1.25816 0.38870 16.23 22.46 0.087 0.087 0.092 0.056 0.070 1.28325 0.47171 16.23 22.46 0.087 0.087 0.062 0.026 0.070 1.16992 0.13835 16.23 22.46 0.087 0.087 0.062 0.036 0.070 1.18102 0.16488 16.23 22.46 0.087 0.087 0.062 0.046 0.070 1.19546 0.20199 16.23 22.46 0.087 0.087 0.062 0.056 0.070 1.21275 0.24973 16.23 22.46 0.087 0.087 0.062 0.066 0.070 1.23243 0.30783 16.23 22.46 0.087 0.087 0.062 0.076 0.070 1.25413 0.37572 16.23 22.46 0.087 0.087 0.062 0.086 0.070 1.27757 0.45263 16.23 22.46 0.087 0.087 0.062 0.056 0.020 1.21892 0.26754 16.23 22.46 0.087 0.087 0.062 0.056 0.035 1.21708 0.26220 16.23 22.46 0.087 0.087 0.062 0.056 0.055 1.21461 0.25507 16.23 22.46 0.087 0.087 0.062 0.056 0.070 1.21275 0.24973 16.23 22.46 0.087 0.087 0.062 0.056 0.085 1.21087 0.24439 16.23 22.46 0.087 0.087 0.062 0.056 0.115 1.20708 0.23371 16.23 22.46 0.087 0.087 0.062 0.056 0.150 1.20259 0.22127 Actual ratio. .723 Note: if actual ratio is greater than the critical ratio, the option price OX" is not valid, and should be equal to the difference between savings 'V" and costs "F" 61 Table 3.5 SUNBITIVTTY AMhLYSIS, RIGARY NUCLXAR VIS CC USING FUEL NIX. FLEXIBIXITY OPTION TO WAIT TO INVEST V F DV DF SV SF RVF C X 23.89 51.22 0.042 0.087 0.051 0.056 0.070 2.18843 3.53303 23.89 51.22 0.057 0.087 0.051 0.056 0.070 1.64565 1.32982 23.89 51.22 0.072 0.087 0.051 0.056 0.070 1.35099 0.29988 23.89 51.22 0.087 0.087 0.051 0.056 0.070 1.19114 0.02840 23.89 51.22 0.102 0.087 0.051 0.056 0.070 1.11223 0.00105 23.89 51.22 0.117 0.087 0.051 0.056 0.070 1.07412 0.00002 23.89 51.22 0.132 0.087 0.051 0.056 0.070 1.05396 0.00000 23.89 51.22 0.087 0.042 0.051 0.056 0.070 1.05648 0.00000 23.89 51.22 0.087 0.057 0.051 0.056 0.070 1.07818 0.00004 23.89 51.22 0.087 0.072 0.051 0.056 0.070 1.11806 0.00153 23.89 51.22 0.087 0.087 0.051 0.056 0.070 1.19114 0.02840 23.89 51.22 0.087 0.086 0.051 0.056 0.070 1.18499 0.02414 23.89 51.22 0.087 0.117 0.051 0.056 0.070 1.44451 0.57797 23.89 51.22 0.087 0.132 0.051 0.056 0.070 1.59911 1.14469 23.89 51.22 0.087 0.087 0.021 0.056 0.070 1.15019 0.00766 23.89 51.22 0.087 0.087 0.031 0.056 0.070 1.16031 0.01121 23.89 51.22 0.087 0.087 0.041 0.056 0.070 1.17417 0.01768 23.89 51.22 0.087 0.087 0.051 0.056 0.070 1.19114 0.02840 23.89 51.22 0.087 0.087 0.061 0.056 0.070 1.21067 0.04492 23.89 51.22 0.087 0.087 0.071 0.056 0.070 1.23231 0.06875 23.89 51.22 0.087 0.087 0.081 0.056 0.070 1.25573 0.10120 23.89 51.22 0.087 0.087 0.051 0.026 0.070 1.14247 0.00554 23.89 51.22 0.087 0.087 0.051 0.036 0.070 1.15547 0.00940 23.89 51.22 0.087 0.087 0.051 0.046 0.070 1.17192 0.01649 23.89 51.22 0.087 0.087 0.051 0.056 0.070 1.19114 0.02840 23.89 51.22 0.087 0.087 0.051 0.066 0.070 1.21260 0.04681 23.89 51.22 0.087 0.087 0.051 0.076 0.070 1.23589 0.07327 23.89 51.22 0.087 0.087 0.051 0.086 0.070 1.26074 0.10903 23.89 51.22 0.087 0.087 0.051 0.056 0.020 1.19664 0.03259 23.89 51.22 0.087 0.087 0.051 0.056 0.035 1.19501 0.03131 23.89 51.22 0.087 0.087 0.051 0.056 0.055 1.19281 0.02963 23.89 51.22 0.087 0.087 0.051 0.056 0.070 1.19114 0.02840 23.89 51.22 0.087 0.087 0.051 0.056 0.085 1.18947 0.02720 23.89 51.22 0.087 0.087 0.051 0.056 0.115 1.18609 0.02487 23.89 51.22 0.087 0.087 0.051 0.056 0.150 1.18208 0.02227 Actual ratio: .466 Note: if actual ratio is greater than the critical ratio, the option price "X" is not valid, and should be equal to the difference between savings "V" and costs "r 62 Table 3.6 SBNSITIVITY ANILYIZS, HUNGARY COAL VX8 CC USING FMlL MIX. FLXMIBILITY OPTION TO WAIT TO INVEST V F DV DF SV SF RVF C X 12.46 22.46 0.042 0.087 0.046 0.056 0.070 2.17905 2.11277 12.46 22.46 0.057 0.087 0.046 0.056 0.070 1.63662 0.88599 12.46 22.46 0.L72 0.087 0.046 0.056 0.070 1.34171 0.23937 12.46 22.46 0.087 0.087 0.046 0.056 0.070 1.18231 0.03027 12.46 22.46 0.102 0.087 0.046 0.056 0.070 1.10509 0.00169 12.46 22.46 0.117 0.087 0.046 0.056 0.070 1.06871 0.00006 12.46 22.46 0.132 0.087 0.046 0.056 0.070 1.04978 0.00000 12.46 22.46 0.087 0.042 0.046 0.056 0.070 1.05195 0.00000 12.46 22.46 0.087 0.057 0.046 0.056 0.070 1.07227 0.00009 12.46 22.46 0.087 0.072 0.046 0.056 0.070 1.11038 0.00230 12.46 22.46 0.087 0.087 0.046 0.056 0.070 1.18231 0.03027 12.46 22.46 0.087 0.086 0.046 0.056 0.070 1.17618 0.02621 12.46 22.46 0.087 0.117 0.046 0.056 0.070 1.43723 0.42970 12.46 22.46 0.087 0.132 0.046 0.056 0.070 1.59278 0.78260 12.46 22.46 0.087 0.087 0.016 0.056 0.070 1.14672 0.01131 12.46 22.46 0.087 0.087 0.026 0.056 0.070 1.15474 0.01463 12.46 22.46 0.087 0.087 0.036 0.056 0.070 1.16682 0.02066 12.46 22.46 0.087 0.087 0.046 0.056 0.070 1.18231 0.03027 12.46 22.46 0.087 0.087 0.056 0.356 0.070 1.20062 0.04438 12.46 22.46 0.087 0.087 0.066 0.056 0.070 1.22125 0.06378 12.46 22.46 0.087 0.087 0.076 0.056 0.070 1.24382 0.08906 12.46 22.46 0.087 0.087 0.046 0.026 0.070 1.13059 0.00617 12.46 22.46 0.087 0.087 0.046 0.036 0.070 1.14464 0.01053 12.46 22.46 0.087 0.087 0.046 0.046 0.070 1.16213 0.01817 12.46 22.46 0.087 0.087 0.046 0.056 0.070 1.18231 0.03027 12.46 22.46 0.087 0.087 0.046 0.066 0.070 1.20461 0.04784 12.46 22.46 0.087 0.087 0.046 0.076 0.070 1.22863 0.07159 12.46 22.46 0.087 0.087 0.046 0.086 0.070 1.25410 0.10189 12.46 22.46 0.087 0.087 0.046 0.056 0.020 1.18745 0.03393 12.46 22.46 0.087 0.087 0.046 0.056 0.035 1.18592 0.03282 12.46 22.46 0.087 0.087 0.046 0.056 0.055 1.18386 0.03135 12.46 22.46 0.087 0.087 0.046 0.056 0.070 1.18231 0.03027 12.46 22.46 0.087 0.087 0.046 0.056 0.085 1.18074 0.02920 12.46 22.46 0.087 0.087 0.046 0.056 0.115 1.17758 0.02711 12.46 22.46 0.087 0.087 0.046 0.056 0.150 1.17383 0.02475 Actual ratio: .555 Note: if actual ratio is greater than the critical ratio, the option price "X" is not valid, and should be equal to the difference between savings "V" and costs "F" 63 Monte Carlo situlatioa. The following tables show the sensitivity analysis and the Monte Carlo simulation. The computer work was done with SAS (on the World Dank's mainframe, CMS operating system). The parameters were as follows (for the perpetual warrant "option to wait to invest"): V = the "stock price equivalent" (total savlngs), F - the "exercise price" (construction costs), DV = the "dividend equivalent" of total savings, DF - the "dividend equivalent" of construction costs, SV - the uncertainty of the total savings, SF = the uncertainty of the construction costs, RVF - the correlation between total savings and costs, C the critical ratio, and x - the option price (to wait to invest) The Monte Carlo simulation was based on the following assumptions for the underlying distributions the characterize our uncertainty on the model parameters: option to wait to invest, Nuclear (or Coal) vis CC with distillate fuel. DV -> triangular distribution between .0787 and .0987 DF -> triangular distribution between .0787 and .0987 SV -> triangular distribution between .051 and .091 (or .042 to .082 for Coal) SF -> triangular distribution between .046 and .066 RVF -> triangular distribution between .05 and .09 Option to wait to invest, Nuclear (or Coal) vis CC with distillate and gas fuel mix. DV -> triangular distribution between .0787 and .0987 DF -> triangular distribution between .0787 and .0987 SV -> triangular distribution between .031 and .071 (or .026 to .046 for Coal) SF -> triangular distribution between .046 and .066 RVA -> triangular distribution between .05 and .09 The Monte Carlo simulation results show that there is considerable uncertainty regarding the estimates of option values, but in both cases the conclusions stay the same. The simulation results are synopsized as follows: 64 Option to wait to invest for nuclear or coal via CC alternative MONTE CARLO SIMULATION ==--- #replications mean std minimum maximum Nuclear via CC with distillate fuel: critical ratio: 30000 1.2345 0.0451 1.1119 1.4328 Nuclear via CC with distillate and gas fuel mix: critical ratio: 30000 1.1940 0.0428 1.0783 1.3806 Coal via CC with distillate fuel: critical ratio: 30000 1.2154 0.0441 1.0947 1.4072 Coal vis CC with distillate and gas fuel mix: critical ratio: 30000 1.1858 0.0424 1.0692 1.3712 65 Figure 3.1 NWONT ChMeO ON CRITICAL C,N=30000,N 4A2Y* NUCLDAR VXS CC JIZNG DWSThL&TN 1UBL. N Mean Std Dev Minimum Maximum ----------------------------------------__------------------ 30000 1.2344809 0.0450760 1.1119530 1.4328322 a--------------------------------------- -…----------------- Percentage 7 +** ]+ *** 6 + 3 5 + 4 + ***** 3 **** J ********** 11111222233334 1257********** 2 +******** 1 ********** I ****************~******** 3 ****************************** *........ .... *...@..*.....e.....*...................*......... . *0* ** 11111111111222222222222233333333333344444 12234566789001234456788901223456678900123 20864208642086420864208642086420864208642 C Midpoint 66 Figure 3.2 NOMTZ CARL ON CROTXICL C,N30000,NUNGARY, COAL VIS CC UOING DZSTXLLaZ PQULs N Mean Std Dev Minimum Maximum ------------------------------------------------------------- 30000 1.2153529 0.0441309 1.0946875 1.4071990 ---------------------------------------------------------__-- Percentage ] ** 5 + 4 + 3 + 2+ 1 + ] *************************** --11--1111111-111-111111-1-11-1-1--1111-11-1-1--11111- 01111111111111111122222222222222222333333333333333344 9001123344566778990012233455667889!: 311223445567788900 51739517395173951739517395173951739517395173951739517 C Midpoint 67 am2 X 4 ODtial Timinag o Transmission Investment The expected period of time after which it will be worthwhile exercising the option to invest is expressed as: ln(C6)P - ln(V/7) + i11-P] B(T) = _ ; for G.>Gf 3(T) = [lnfC4) ln(V/r)] Cln{V/F) - 0 / o2 for G(=Gf Also, as 0 -^ -0 Z(T) - lnC4F/V)/(Gv-Gf) for GV>Gf1 and 3(T) 4- O for Gv-Gf where, 6= critical ratio provided by the option model V = present value of benefits ("stock equivalent price") F = present value of costs ("exercise price") Gy = growth rate of benefits ,. = growth rate of costs P - probability of reaching the critical ratio = lower barrier for critical ratio a - total effective standard deviation, equal to: a = , aV2 + af2 _ .rvflavaf1 (see page 6 in main text) 68 UNEsX 5 GLOSSARY American Option: an option that can be exercised any time up to the expiration (maturity) date. At-The-No9ey: an option is said to be at the money when the price of the underlying instrument is almost equal to the exercise price. Actuaj Rio: standard benefit-cost ratio obtained through regular deterministic present value techniques. Black-Scholes Model: a model developed by Black and Scholes; it is European type with finite maturity and fixed (non-stochastic) exercise price. Brownian Motion: a continuous time Markov process, as such its future evolution does not depend on the complete history of the process, but only on the last period. call ption: an option to "purchase" the underlying asset at a price that is predetermined or uncertain; see also option. Conti_ngcn Claim: a claim the value of which is dependent upon some other asset(s) or security(ies); see also option. Critical Ratio: ratio provided by the perpetual option approach that should be exceeded by the standard benefit-cost ratio in order to make the investment viable under uncertainty. Buropean Option: an option that can be exercised only at the expiration (maturity) date. Exercise Price: the price at which the option holder can purchase the underlying asset; this price can be fixed (like in Black- Scholes model) or uncertain, i.e., changing with time. EDiration: the maturity (date) of the option; if not exercised by that date, it becomes worthless. Flexibility O=tion: a real option is called flexibility option when the asset and the exercise price represent the savings and costs respectively that are incurred due to an irreversible investment. See also real option. Geometric Brownian Motion: when a variable .ollows a Brownian motion, its natural logarithm is said to follow a geometric Brownian motion. IZnhThena=: an option is said to be in the money if exercising it is profitable; the price of the underlying asset is higher than 69 the exercise price in the case of a call option, and lower in the case of a put option. Maturity: see expiration. Cption: the right (but not the obligation) to buy (call option) or to sell (put option) an asset by paying the exercise price; this price can be predetermined or uncertain. Out-Of-The-Money: an option is said to be out of the money if it is not profitable exercising it; the price of the underlying asset is lower than the exercise price in the case of a call option, and higher in the case of a put option. Put Otion: an option to "sell" the underlying asset at a predetermined or uncertain exercise price; see also option. z2epeanl Warrant: an option with no finite expiration (maturity) date; it can be exercised any time and thus is of the American type. Real OQtion: an option is called real when the underlying asset is not observable, i.e.,, is not a financial instrument tradeable in the market. Stochastic Variable: a random variable is considered stochastic when its probability distribution changes as a function of time (or in a more general sense as a function of any index; usually the index is time). Writer: the party that originally creates the option by selling it to another party; he carries the full obligation implied by the option to the present owner. Nnrrant: see option. 70 RnRC's Association Nomentanee: Tractebel, Hydro-Quebec/Dessau, Electricite de France International (Mar. 1991) "OMVS Project Energie - Etude Economique Complementaire du Reseau 225 kV de la Centrale de Manantali, Rapport de Phase III. Draft report prepared for the World Bank. Brock, W. A., M. Rothschild, and J. E. Stiglitz (1988) "Stochastic Capital Theory", Ch. 20 in Essays in Honour of Joan Robinson, Feiwal ed., Cambridge Univ. Press. Copeland, Thomas and Fred Weston (1983) Financial Theory and Corporate Policy, 2nd edition, Addison Wesley, pp. 701-706. Crousillat, Enrique (Jun. 1989) "Incorporating Risk and Uncertainty in Power System Planning", IEN Energy Series Paper No. 17. The World Bank. Instituto Costarricense de Electricidad (1989) "Programa DQsarrollo Blectrico II, Periodo 1990-1994 - Programa de Expansion de la Generacion", San Jose, Costa Rica. Karlin, Samuel and Howard M. Taylor (1975) A First Course in Stochastic Processes, 2nd ed., Academic Press. Little, I.M. D., and James A. Mirrless (1974) Project Appraisal and Planning, London: Heinemann. Little, I.M.D., and James A. Mirrless (1990) "Project Appraisal and Twenty Years On", in Proceedings of the World Bank Annual Conference on Development Economics, The World Bank. Martzoukos, Spiros H. and James Paddock (Summer 1990, revised Winter 1991) "Option Models, Flexibility, and the Power Sector" unpublished document, prepared for the IENED (PPR), the World Bank. Martzoukos, S. H. and W. Teplitz-Sembitzky (April 1991) "Optimal Timing of Transmission Line Investments in the Face of Uncertain Demand - An Option Valuation Approach", Working Paper. McDonald, Robert and Daniel Siegel (November 1986) "The Value of Waiting to Invest", Ouarterly Journal of Economics, V.101, N.4. Merrow, Edward W. and Ralph F. Shangraw Jr. (Jul. 1990) "Understanding the Costs and Schedules of World Bank Supported Hydroelectric Projects", IEN Energy Series Paper No.31. The World Bank. Moore, Edwin A. and Enrique Crousillat (May 1991) "Prospects for Gas-Fueled Combined-Cycle Power Generation in the Developing Countries", ZEN Energy Series Paper No.35. The World Bank. 71 Moore, Edwin A. and George Smith (Feb. 1990) "Capital Expenditures for Electric Power in the Developing Countries in the 1990s", IEN Energy Series Paper No.21. The World Bank. Samuelson, Paul A. (1965) "Rational Theory of Warrant Pricing", Iustrial ManaaeXent Review, Vol. 6, No. 2, Spring, pp. 13-31, reprinted in Samuelson 1972 (Merton editor). Sanghvi, Arun, Robert Vernstrom and John Besant-Jones (Jun. 1989) "Review and Evaluation of historic Electricity Forecasting Experience (1960-1985)", IEN Energy Series Paper No.18. The World Bank. Schram,, G. (July 1989) "Optimal Timing of Transmission Line Investment", Energy Economics, V. 11, N.3. Teplitz-Sembitzky, W. (1989) "Option Pricing in Power Planning", working paper (mimeo), IENED. The World Bank. Traiforos, Spiros, Achilles Adamantiades and Edwin Moore (Apr. 1990) "The Status of Nuclear Power Technology - An Update", IEN Energy Series Paper No.27. The World Bank. World Bank (1878) "Operational Maznual Statement 3.72 - Energy, Water Supply and Sanitation and Telecommunications (EWT)". World Bank (1980) "Operational Manual Statement 2.21 - Economic Analysis of Projects". 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