Financial Risk Analysis of Infrascructure Debt

Financial Risk Analysis of

Infrastructure Debt

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Infrastructure Debt

THE CASE OF WATER AND POWER INVESTMENTS

C. Vaughan Jones Foreword by David Bendel Hertz

QUORUM BOOKS Westport, Connecticut • London

Library of Congress Cataloging-in-Publication Data

Jones, Clive Vaughan. Financial risk analysis of infrastructure debt : the case of water

and power investments / C. Vaughan Jones ; foreword by David Bendel Hertz,

p. cm. Includes bibliographical references and index. ISBN 0-89930-488-5 (alk. paper) 1. Public utilities—Finance—Case studies. 2. Risk management—

Case studies. 3. Water resources development—Finance—Case studies. 4. Power resources—Finance—Case studies. 5. Infrastructure (Economics)—Finance—Case studies. I. Title. HD2763.J59 1991 363.6'068'1-KJC20 90-45144

British Library Cataloguing in Publication Data is available.

Copyright © 1991 by C. Vaughan Jones

All rights reserved. No portion of this book may be reproduced, by any process or technique, without the express written consent of the publisher.

Library of Congress Catalog Card Number: 90-45144 ISBN: 0-89930-488-5

First published in 1991

Quorum Books, 88 Post Road West, Westport, CT 06881 An imprint of Greenwood Publishing Group, Inc.

Printed in the United States of America

The paper used in this book complies with the Permanent Paper Standard issued by the National Information Standards Organization (Z39.48-1984).

10 9 8 7 6 5 4 3

Contents

Figures and 1 ables

Foreword David Bendel Hertz

Preface

1 Introduction

2 Concepts and Procedures

3 Financial Risks in the Construction Period

4 Revenue Risk—Rate and Demand Factors

5 Revenue Risk—The Customer Base

6 Applications

7 Reflections on the Method

Appendix

Bibliography

Index

vn ix

xiii

1

17

47

69

85

97

129

143

155

165


Figures and Tables

FIGURES

1.1 Risk Profile of Rates of Return 7 2.1 Steps in Conducting a Risk Simulation 18 2.2 Triangular Probability Distribution 28 2.3 Denver, Colorado, GCD 31 2.4 Cumulative Probability Distribution 35 2.5 Event Tree for Rate Increase 36 2.6 Comparison of Risk Profiles 39 2.7a First Order Stochastic Dominance (relative

probability) 40 2.7b First Order Stochastic Dominance (cumulative

probability) 41 3.1 Overlay of Risk Profiles Developed with Uniform and

Triangular Distributions and Table 3.2 Ranges 58 3.2 Comparison of Cumulative Distributions 59 4.1 Total Revenue Curve 76 5.1 Stochastic Models of Population Growth 93 6.1 Default with and without a Sinking Fund 104 6.2 Debt Service Schedules 106 6.3 Cost Overrun Probability 108 6.4 Default Risk—Level and Tipped Debt Service 111 6.5 Present Value of Investment Costs 120

Vll l Figures and Tables

TABLES

3.1 Cost Estimates and Realized Nuclear Power Plant Costs, 1966-1972 48

3.2 Range Estimates and Expected Costs by Cost Component 55

4.1 Error in Revenue Forecast from Neglecting Consumer Price Responses 73

5.1 Typical Presentation of Population Forecasts in High, Medium, and Low Variants 91

6.1 Cash Flow Model for Wholesale Producer 99 6.2 Minimum Present Value Capacity Investment

Sequences 115 7.1 Payoff Matrix Illustrating Alternative Choice Criteria

under Uncertainty 137

Foreword DAVID BENDEL HERTZ

It is my pleasure to write a few words preceding C. Vaughan Jones's Financial Risk Analysis of Infrastructure Debt. When I began working on risk analysis and its simulations in the 1970s (eventually publishing "Risk Analysis in Capital Investment," Harvard Business Review [September-October 1979]: 169-181, and writing various books on the subject), the procedures of using computers to simulate investment outcomes were first becoming available. However, early technology was time-consuming and awkward. Today, rapid simulations can run on microcomputers, making multiple analyses easy and relatively quick. In this well-written and lucid book, Jones demonstrates some surprising results such as the fact that such simulations in the utility industry

readily show that contingency funds should be calculated against total construction costs rather than against individual cost components at given confidence levels. . . . [and] least cost capacity plans developed for average or expected population growth may be different from those suggested by a probabilistic or simulation point of view; for example, deterministic procedures may lead to higher expected discounted costs than plans developed within a framework allowing for variation in population growth and other key variables (p. 2).

Risk abounds, and in risk analysis we are always considering planning for an uncertain future, even in time intervals as different as a month, a year, or a century. Risk analysis simulations might

x Foreword

lead to effective risk "management"—that is, reduced numbers and magnitude of undesirable outcomes at reasonable (or even negative) input costs. Reductions in risk usually come at a price, usually in deferred profits of dollars or dividends. Muddled decision-making approaches make well-informed expectations unlikely. Where the investment is millions of dollars, and the results arise from projects whose foreseeable "lives" and usefulness are measured in decades and impact on a large population and its surrounding environment, decisions should be based on the strongest and most accurate methodologies available. While my original work was intended to approach the risks of investment in general, I now realize that specific scenarios such as the water and power industries require their own analyses for effective understanding and application. Cases must be evaluated on their own merits and characteristics; the estimation of probabilities for various events in these individual cases is critical to effective decision making. As Jones points out, "Risk simulation . . . makes explicit what is implicit in many qualitative appraisals, attaching numbers that, at a deep level, guide thought about the prospects of an investment situation. Inconsistencies in qualitative appraisal can be ferreted out, and the very process of inputting ranges and probabilities to variables forces a deeper integration of information and understanding about the process in question" (p. 10).

Jones creates a risk simulation model based on a good analysis of the complexity of financing utility projects. The discussion is well developed and follows step-by-step the development of agreements and contracts, the issuance of bonds, and the management of construction. On this well-researched and thought-out base, Jones then analyzes the unique characteristics of utilities, such as excess capacity, growth demands, and extension into other peripheral plants. He does this without becoming obscure or pedantic, and he backs his logic strongly with computer-based simulations and stochastic modeling of the risks involved in each crucial phase, as well as an outline of several applications based on these projections.

If life is composed of decisions and uncertainty, these decisions involve varying degrees of risk. It is not possible to make a decision, business or otherwise, without incurring a certain amount of risk. All decisions are risky because a stake is placed on the uncertain

Foreword xi

future consequences resulting from acting on these decisions. There have been all too few books on the subject of risks involving large-scale investment in a utility project, such as C. Vaughan Jones's Financial Risk Analysis of Infrastructure Debt. A modern treatment such as this one is much needed by the industry, and Jones's book fills this need admirably.


Preface

The origins of this book date to conversations with A. G. Hart at Columbia University in the late 1960s and, earlier, to a seminar in Monte Carlo analysis at the University of Colorado.

Throughout the 1970s and 1980s, I had an opportunity to see the relevance and to develop my understanding of financial risk analysis in projects carried out for the Iowa State Commerce Commission, the Colorado Public Utilities Commission, and the Denver Water Board. Since then, I have had access to the considerable resources in the area of risk analysis of the Environment and Behavior Program at the University of Colorado at Boulder, as well as the good offices of research librarians at the University of Colorado School of Business.

I also have received invaluable comments, encouragement, and support from a number of individuals, especially including John J. Boland, Department of Geography and Environmental Engineering, Johns Hopkins University; David B. Hertz, Department of Intelligent Computer Systems, University of Miami; James Man-ire, assistant vice president, Boettcher and Company; Margaret Ludlum, senior economist, Seattle Water Department; William A. Steele, financial analyst, Colorado Public Utilities Commission; Timothy Tatam, vice president, Municipal Finance Department, Standard & Poor's Corporation; Fred J. Leonard, financial analyst; and my wife, Carol Eileen Ryan.



Infrastructure Debt


1

Introduction

This book is about how to measure financial risk associated with investments in specific infrastructure—electricity generation; power transmission; and water storage, treatment, and distribution systems. Such evaluations are especially accessible with respect to these types of investments and with respect to infrastructure generally. In addition, power and water enterprises showed signs of financial strain in the 1980s and confront critical decisions in the 1990s.

As one of a small number of extended discussions of financial risk analysis, the book should be of interest to students of business, finance, and economics. Readers can fix in mind basic principles of quantitative risk analysis or risk simulation and consider the interplay between context or industry specific information and global approaches to risk evaluation.

This discussion has special and practical relevance for an audience of investment analysts, engineers, utility administrators and financial personnel, regulatory officials, bond underwriters, and public interest advocates. These groups need to develop a common language to address risks associated with bond issuance in a decade that promises significant departures from past patterns in population and economic growth, the natural environment, and regulatory politics. To this end, this book attempts an integration of material from business, engineering, economics, demography, probability theory, computer simulation, and policy studies. The

2 Analysis of Infrastructure Debt

presentation also supports hands-on analysis with spreadsheet and simulation examples.

The materials are organized around risk factors affecting costs and revenues. Separate chapters present findings relating to the variability of construction costs, customer demand, and population growth. Together with qualitative information about risks, these chapters offer suggestions about quantitative representation of relevant patterns of variability of key risk sources. The techniques are integrated in simulation models dealing with contract risk, the evaluation of sinking funds and amortization schedules, and long-run capacity planning. The concluding chapter recapitulates major findings, considers issues of reliability and validation, and discusses the manner in which the analysis developed in this book can be generalized for a variety of infrastructure investments.

Risk simulation, widely available for the first time due to new microcomputer technology and computer software, can lead to surprising results. Simulation examples readily show that contingency funds should be calculated against total construction costs rather than against individual cost components at given confidence levels. Similarly, it is easy to demonstrate that tipped amortization can run afoul of inflation, offering little, if any, risk mitigation, even when substantial chances for construction cost overruns exist. Most surprising of all, least cost capacity plans developed for average or expected population growth may be different from those suggested by a probabilistic or simulation point of view; for example, deterministic procedures may lead to higher expected discounted costs than plans developed within a framework allowing for variation in population growth and other key variables.

The point is that a great deal more is possible than running a few "what ifs" in cash flow models. Consensual estimates of the ranges of key risk sources can lead to simulations that are more informative than relying on a "standard case" or a "sensitivity analysis."

The following discussion suggests a perspective on financial risks and electric power and water investments, considers the rationale for financial risk analysis in general and risk simulation in particular, and includes further information about the organization of the book.

Introduction 3

THE PROBLEM

Slower population growth, coupled with erratic and slower economic growth, took a toll on the financial viability of the electric power industry in the 1970s and 1980s. Capacity expansion typically involves long lead times. Designs in the pipeline since the 1960s advanced to construction in the 1970s and 1980s. These projects often were configured to take advantage of scale economies, and they embodied, in the case of nuclear facilities, optimism vis-a-vis new or frontier technology. In a sense, many projects were tailored to a different economic environment than they met when they came on line. Pervasive overcapacity was one result and only recently has been overcome in some regions of the nation.1

One upshot was deterioration in electric utility bond ratings. Dividends declined along with interest coverage ratios. In the mid-1980s, investor-owned public utilities accounted for 30 to 40 percent of the dollar volume of junk bonds. Many were "fallen angels" or issues whose default probability was considered sufficient by bond rating organizations to be dropped from the list of investment grade securities.2

The most spectacular recent example of financial risk in the power industry was the default of the Washington Public Power Supply System (WPPSS) on interest payments on $2.25 billion worth of bonds in January 1984. Commentary on this event, the largest default on utility bonds since the Great Depression, focuses on construction delays, poor project planning, and the practice WPPSS adopted, for a time, of capitalizing several years of interest in bond issues. Financing costs escalated, and projected electricity demand did not materialize. Questions arose concerning the legal status of WPPSS's so-called take-or-pay contracts with buyers.3

Less well-known examples also exist. A Colorado utility serving some 200,000 customers filed for Chapter 11 protection in 1990, following years of erratic revenue and questionable investment and management practices; it was the largest bankruptcy filing by an electric utility in that year. As noted later in this discussion, many other examples can be culled from Standard & Poor's CreditWeek where particularly questionable bond issues are put on "Credit Watch."


Financial problems in the urban water industry have been less dramatic in recent years, although there are signs of financial strain as utilities confront new water quality standards and growing scarcity of new supplies. Institutional problems may be of decisive importance to this industry in the future. In many metropolitan regions, central water systems service static or declining populations, while a patchwork of districts meet the needs of outlying areas with more robust growth. Since near and cheaper reservoir sites and water sources already have been exploited, transbasin diversions must be anticipated to meet future needs. How will these massive, long distance projects be financed by consortia whose members have mixed financial profiles and differing views about their commonality of interest?

This point generalizes for a variety of infrastructure investments. With tighter federal budgets, airport, highway, bridge, and port facilities must rely more on debt issues for initial construction and rehabilitation funds.

Issues

While there are many sources of financial risk for power and water utilities, overbuilding plants and equipment relative to population growth is a significant risk, given current demographic realities. The nation's population growth has been slowing for some time. Almost all projections indicate this will continue, possibly resulting in negative growth or diminishing population by the second or third decade of the next century.4 This suggests problems for utility planning. Traditionally, facilities have been capital intensive and have required long gestation periods. In order to bring a central power station or water facility with attractive economies of scale5 on line, initial operation at less than full capacity usually is countenanced. This is a gamble if population in a community or region cannot be expected to grow at historic rates. The extent of the gamble is underlined by the literature on the accuracy of forecasting models, which stresses the unreliability of predictions of turning points.

Just as the perils of overbuilding and relying on large, central generating or water storage facilities are coming to be perceived generally, the situation shows signs of shifting. Oil prices may

Introduction 5

rebound to the 1970s levels in the coming years, rendering recent, more incremental additions to electric capacity expensive indeed. Regrets may surface about lost opportunities in developing water infrastructure if global climatic warming occurs as rapidly as some scientists anticipate. There are indications, also, that reliance on existing supplies and optimizing existing transmission, construction of smaller units such as gas turbines, and purchase of cogeneration power or wells can promulgate "false economies" and be "penny wise and pound foolish."

There is no general answer to the question, "Should large central facilities be built, or should incremental additions to capacity be sought?" Cases must be evaluated individually on their merits. It is suggested here that investment evaluation should take into account the potential variability of key factors. Rather than relying upon inflexible forecasts of conditions five, ten, and twenty years in the future, financial appraisal of infrastructure investments should consider the economic tradeoffs of projects, recognizing random influences on parameters such as construction costs, population growth, and the like. Concern with the variability of parameters affecting the success or failure of an investment is vital to steering a middle course between reckless devotion to the grandiose and overweening caution and piecemeal solutions associated with a falling standard of living.

In addition, competitive pressures from deregulation in the electric power field may impact the credit strength of municipal and cooperative systems, and revision of the Clean Air Act promises to have financial repercussions for many electric power producers.

RISK ANALYSIS AND RISK SIMULATION

Financial risk analysis shares a great deal with risk analysis as practiced in engineering or natural hazards policy. Expert opinion can be polled for anticipated ranges for risk factors and the values most likely to be realized. Using estimates of the range and distribution of a risk factor, financial risk analysis considers the performance of investments and the balance sheet of an enterprise. Financial risk analysis may contribute to (1) project selection, (2) the selection of financing options relevant to a given project, and (3) perspectives concerning company risk, based on an examina-


tion of cash flows. One basic motivation is to identify or develop risk management plans for investors or a utility organization to cope with uncertainty concerning future market conditions.

Two issues arise at the outset of discussion of financial risk analysis. These relate to: (1) the relation between financial risk analysis and financial market processes, and (2) the rationale for quantifying the analysis.

Financial Risk Analysis and Financial Market Processes

The concept of risk as defined in this book is measured by the distribution of the returns or revenues from an investment. This is called a risk profile and is a probability distribution. Thus, analysis might portray Figure 1.1 as a risk profile of rates of return from a particular investment. Probabilities are marked on the vertical axis. The horizontal axis lists rates of return that may be achieved under the varying conditions encompassed in the analysis. As is discussed more extensively in Chapter 2, one major consideration is usually the variation or dispersion of this risk profile around its mean or expected rate of return. Risk averse individuals prefer tighter distributions around the mean, even, perhaps, at the cost of a slightly lower expected value. Other decision makers may favor investments offering more dispersed returns, using the rationale that they might achieve very high rates of return. The idea is to present information as full and complete as possible on the distribution of returns from a prospective investment to inform choices about whether to go ahead with construction, whether to adopt modifications in project configuration and financing, or the like.

Modern portfolio analysis contributes an intriguing twist to financial risk analysis. The capital asset pricing theory and similar approaches focus on the covariation of an investment's returns with the payback from a portfolio of investments representative of the market as a whole.6 The interrelation of company risk, market risk, systematic risk, and unsystematic risk largely displaces individual investment risk profiles in deciding how to buy stocks. One implication is that risk management can be achieved through suitable diversification.

Introduction

Figure 1.1 Risk Profile of Rates of Return

The problem with diversification at the level of utility projects, however, is the potential size of electric power and water investments. At the level of project or financing selection, there may be no recourse except the comparison of risk profiles. The significant impact, particularly among certain classes of investors, occasioned by the default of WPPSS also illustrates the dangers of complacency

7


concerning single projects or single companies vis-a-vis the national bond market.

Another notion is that, if a project can be financed, its market valuation summarizes all relevant information about its prospects. At best, therefore, financial risk analysis "begs the market," and, at worst, its findings might deter investors who otherwise would be pleased to place their money down.

The first thing to observe about such a careless interpretation of "rational expectations" is that financial risk analysis is conducted, in any case, by agencies responsible for credit ratings. Thus, as an example, Standard & Poor's CreditWeek affirms several issues on the grounds that 'The ratings reflect the sound economic base of the service area, strong historical and projected financial performance, and the strong cash position of the electric system. The St. John's River Power Park project remains on schedule with commercial operation dates of April 1987 and October 1988 for units 1 and 2 . . . . "7 A less favorable report lowered ratings because,

As of June 30, [1986,] Gulf States had about $3.2 billion of debt and preferred securities outstanding. Continued absence of rate relief and the prospect of an additional rate cut in Louisiana due to disallowance of Southern Co. contract capacity requirements will result in further deterioration in measures of creditor protection. The highly political regulatory environments in both Louisiana and Texas and a severely depressed service territory substantially heighten bondholder risk.8

These assessments are supported by analysis of cash flow models and the exploration of what-ifs. Thus, if a utility is interested in a target bond rating, or if a regulatory body is concerned with the supply of capital to agencies under its supervision, prefiguring the implications of various local and regional developments seems advisable.

In other respects, skepticism seems warranted by recent events in financial markets. Thus, the entrapment of major investment figures in a web of insider trading and a pattern of collapse and bailout vis-a-vis a number of lending institutions do not seem to vindicate the invariable "rationality" of financial markets. Some means of checking expectations vis-a-vis various investments, especially ex ante their final selection, seem advisable.

Introduction 9

The Rationale for Quantifying Risks

This book is primarily concerned with a type of analysis called risk simulation.9 This is the most complete and comprehensive type of risk analysis and is the next logical step after identifying major risk factors and the examination of some what-ifs or a sensitivity analysis. A risk simulation seeks to weight various configurations of risk factors by their probabilities. Then, in a sense, all possible configurations and values for the risk factors are summarized in a risk profile for the investment under consideration.

Although not a new concept, risk simulation is now more accessible with advances in microcomputer technology and simulation software. In this regard, the examples in this book have been prepared with Lotus 1-2-3 and an add-in program called @RISK.10

No claim is advanced that this is the best or most up-to-date spreadsheet program. Rather, it is popular and widely used in business, and the learning curve in the discussions of actual program sequences that follow (called macros) should not be too demanding for many readers. Hardware requirements for the simulation computer programs are satisfied by an 8088 PC system, although the added speed of execution with an 80286 or 80386 system is well worth the upgrade.

The trick in a risk simulation is to estimate the probabilities for various events. The important thing is that, insofar as people are willing to make decisions, these probabilities already exist. In other words, risk simulation operates with subjective probabilities based on expert opinion supplemented by data about the objective frequencies of events, where available. This approach aims at as complete a picture as possible in order to make the most informed decision possible.

Simulation is useful when the relationship between variables is too complex to derive analytic or "closed form" solutions for the likelihood of processes and occurrences. Analytic derivations are possible when there are relatively simple formulas describing the relation of random variables to a criterion variable. Thus, suppose revenues (R) and costs (C) are known to be normally distributed around their expected values E(R) and E(C). This implies that net income (NI) in the relation,

NI = R - C (1.1)


also is normally distributed around its expected value E(R) — E(C), owing to certain well-known results of probability theory (see Appendix). Of course, costs may not be normally distributed—in fact, they may be skewed in one direction or another; for example, their probability distribution may have a long tail rightward from the expected value. Furthermore, only rudimentary information about their probability distribution may be available, perhaps only an idea of absolute upper and lower bounds or the range of costs and revenue. Simulation becomes a tool of mathematical research in such contexts.11

Risk simulation, it might be said, makes explicit what is implicit in many qualitative appraisals, attaching numbers that, at a deep level, guide thought about the prospects of an investment situation. Inconsistencies in qualitative appraisal can be ferreted out, and the very process of imputing ranges and probabilities to variables forces a deeper integration of information and understanding about the process in question.

The Staging of the Analysis

Like other evaluation techniques, financial risk analysis makes different contributions at the project level at various stages in the planning and building of a project. Early on, designs and cost estimates are largely conceptual. General discussion of risk sources is particularly useful at this point. Simpler analytical methods such as sensitivity analysis and scenario development can indicate whether or not financial problems are in any sense likely.

In a sensitivity analysis, each risk factor is allowed to vary over a range, such as plus or minus 15 percent, indicating, in rough terms, which variable contributes most importantly to the success of the project.

Scenario development involves developing a plausible story or "scripting" the future. In the best situation, reviewing several imaginative scenarios helps produce agreement about what is likely to happen and how key variables can change.

The major product at this juncture is a list of risk factors and an indication of what impact they might have, ignoring syncretisms or joint influences that surpass the sum of individual effects and

Introduction 11

ignoring the question of what probabilities can be associated with various growth rates and the like.

Later, as project alternatives are narrowed and detail comes into focus, risk analysis can shift to actual designs and the implementation process. Milestone events with regard to a project can include (1) signature of agreements and contracts committing a company or public body to this option, (2) issuing bonds to finance construction, and (3) managing construction of the project. At each such juncture, consideration of the range of future values for key parameters is critical. Early acceptance of a project configuration can elicit confidence in its financial feasibility. After preliminary cost estimates for a project are forged, risk simulation is well suited to inform choices regarding financing options, including the structure of debt service, the value of bond insurance, bank letters of credit, and sinking funds. Continuous monitoring of risk factors can provide a kind of early warning system for isolating cash flow problems that might impact future bond ratings, the stability of power or water rates, or other sensitive factors.

There is also what might be termed a "generic" planning problem particularly suited to risk simulation. Larger facilities, which may exhibit economies of scale, often imply long periods of excess capacity. If bond interest rates are high or demand growth slows, financing large central facilities may present problems, despite their low unit costs when fully utilized. Of course, insufficient revenue in some future year is not by itself an unmitigated disaster. To evaluate this problem, one has to consider the responses available to the organization building the facility. Such an analysis can consider the priority among payments and the feasibility of solving cash flow problems with higher rates or rescheduling of debt. Thus, one would want to know whether a sizeable rate increase can cover the shortfall or whether proposed rates, in fact, would be so large that both demand and revenues would be constrained.12 Another option may be to reduce principal payments with new borrowing, which adds to interest charges but extends the payback period. Other major risk factors include construction cost overruns and revenue shortfalls due to financial difficulties of wholesale customers.

This can be broadened into a framework for considering the general financial risk facing a company or public concern. A power


system may have several baseload plants, intermediate and peaking facilities, and may exist within a context of physical reliability and distribution net factors. An urban water system may have complex linkages between multiple reservoirs, treatment plants, and pumping facilities. These systems also can possess a complex structure of debt and multiple financing options. Just as many Fortune 500 firms maintain and continuously revise financial risk models pertaining to their general operations, water and power utilities may find extension of these techniques useful.

Another direction for generalizing these techniques is in capacity planning. Regulatory requirements and the need to prepare public opinion and to reserve future options lead many power and water utilities to engage in very-long-run planning, specifying the trajectory or sequence of capacity expansion over thirty years or more. Given some thought as to likely reactions to changed circumstances, lag and lead times, and the like, risk simulation can be a tool to explore the adequacy of these capacity plans.

ORGANIZATION OF THE BOOK

At the simplest level, there are two groupings of financial risk factors: those affecting costs and those affecting revenues. This classification influences the structure of this book.

Chapter 2 presents a reconnaissance of basic concepts and procedures. The exposition considers default risk, surveys methods for developing information about the probabilities of financial and other events, and considers how decision makers react to and utilize information from a risk simulation. The discussion at many points is general and can be applied usefully to the evaluation of financial risks for a variety of infrastructure and other investments.

Chapter 3 focuses on financial risks during the construction period and their evaluation. The discussion lists generic factors identified by cost engineering as major contributors to cost escalation in capital construction projects. Range estimation is introduced as a technique for developing probability imputations and is shown to apply to cost escalation, scheduling delays, and the timing of expenditures in the construction period. This chapter also considers the sizing of construction contingency allowances.

Chapter 4 considers customer demand patterns, an important

Introduction 13

element of revenue risk. The discussion underlies the effect of the price elasticity of demand on revenue projections and suggests specific tactics for incorporating consumer demand patterns in risk simulations of cash flows.

Chapter 5 deals with another major element of revenue risk— forecasting the customer base. The discussion notes the dismal track record of economic and demographic forecasts and, following a brief discussion of population forecasting methods, outlines methods for incorporating uncertainty into forecasts of population in utility service areas.

Chapter 6 pulls the various strands of stochastic modeling together with several applications. Chapter 6 provides quantitative analysis of problems related to sinking funds, tipped versus level amortization schedules, contract risk, capacity planning, and the interface between financial and benefit-cost analysis.

Chapter 7 recapitulates the findings and reflects, more generally, on the promise and limitations of the method.

The Appendix discusses rules for finding the probability distributions of sums, products, and other combinations of random variables and lists a number of commonly encountered probability distribution functions.

SUMMARY AND CONCLUSIONS

The value of financial risk analysis is that it leads decision makers away from a favored and fixed set of assumptions about an essentially uncertain future. A more comprehensive view helps identify the desirable size of contingency funds,13 leads to reconfigurations of a project, or promotes new strategies of meeting an objective or responding to difficulties.

Coming to grips with risk and uncertainty is a new and challenging aspect of business policy. As noted above, an organization can juggle investments to minimize the risk from a portfolio. Yet electric power and water projects can be large enough to present special situations deserving of specific analysis. Power and water utilities also characteristically derive the preponderance of their revenues from investments within a particular geographic area and, thus, are subject to systematic risk following from regional economic developments.


Wholesale change in investment policies may well be a response to the recognition of financial risks associated with such projects. Indeed, it can be argued that, for many organizations, new policies emphasizing smaller incremental approaches to capacity expansion, joint venturing, or reliance on conservation and wholesale purchases already reflect responses to the new financial climate in which utilities find themselves.

In addition to helping decision makers explore the performance of an investment project and their preferences for various types of returns, risk simulation integrates information that might otherwise remain the special province of certain experts and various specialists in a project team. Clearly, pulling together information about the likely range ot risk factors is superior to a decision-making process in which prejudices develop about particular numbers—for example, "the plant is going to cost exactly $756 million dollars" or "population and demand will grow at 2.4 percent for twenty years." Compiling the information to conduct such an analysis is itself valuable, especially since this process can provide focus and can integrate disparate elements in a project team.

This technique is useful in appraising the significance of contingencies from the standpoint of early decisions about a capital construction project, and it has relevance to project management. In addition to bearing on the sizing or financing of a specific proposal, risk simulation is useful in project comparison as well as the appraisal of individual companies and their financial prospects.

NOTES

1. See, for example, the North American Electric Reliability Council, 1987 Reliability Assessment: The Future of Bulk Electric System Reliability in North America 1987-1996 (Princeton, N.J.: North American Electric Reliability Council, September 1987).

2. See Edward I. Altmann and Scott A. Nammacher, Investing in Junk Bonds: Inside the High Yield Debt Market (New York: John Wiley & Sons, 1987); and Scott Fenn, America s Electric Utilities (New York: Praeger, 1984).

3. See James Leigland, "WPPSS; Some Basic Lessons for Public Enterprise Managers," California Management Review (Winter 1987): 78-88; Gene Laber and Elisabeth R. Hill, "Market Reaction to Bond Rating Changes: The Case of WHOOPS Bonds," Mid-Atlantic Journal of Busi-

Introduction 15

ness (Winter 1985/1986): 53-65; Darryl Olsen and Robert J. Defillippi, "The Washington Public Power Supply System—A Question of Managerial Control and Accountability in the Public Sector," Journal of Management Case Studies (Winter 1985): 323-343; and David Myrha, Whoops/ WPPSS: Washington Public Power System (Jefferson, N.C.: Mcfarland, 1984).

4. U.S. Bureau of the Census, Projections of the Population of the United States by Age, Sex, and Race: 1983 to 2080, Current Population Reports, Population Estimates and Projections, Series P-25, No. 952, U.S. Department of Commerce (Washington, D.C.: Government Printing Office, 1984).

5. A facility exhibits economies of scale if, for example, it has twice the service capacity at less than twice the capital and operating costs of another plant. Scale economies can have a physical basis. Thus, F. M. Scherer writes,

The output of a processing unit tends within certain physical limits to be roughly proportional to the volume of the unit, other things being equal, while the amount of materials and fabrication effort (and hence investment cost) required to construct the unit is more apt to be proportional to the surface area of the unit's reaction chambers, storage tanks, connecting pipes, and the like. Since the area of a sphere or cylinder of constant proportions varies as the two-thirds power of the volume, the cost of constructing process industry plant units can be expected to rise as the two-thirds power of their output capacity, at least up to the point where they become so large that extra structural reinforcement and special fabrication techniques are required.

Industrial Market Structure and Economic Performance, 2d ed. (Boston: Houghton Mifflin, 1980).

6. John C. Hull notes the difference between the variance of returns view and portfolio risk in '4Risk in Capital Investment Proposals: Three Viewpoints," Managerial Finance (1986): 12-15.

7. Standard & Poor's CreditWeek, October 27, 1986, p. 9. 8. Ibid., p. 52. 9. See, for example, John C. Hull, The Evaluation of Risk in Business

Investment (London: Pergamon Press, 1980). 10. @RISK is a computer program produced by the Palisade Corpo

ration of Newfield, New York, telephone (607) 564-9993. 11. The literature on analytic probabilistic features of complex financial

computations is developing apace and should not be ignored in this connection. Interesting work has been done on the expected present value of serially correlated revenue streams—a realistic characterization, if one considers that good and bad business years tend to occur in runs. Unfortunately, the derivations underlying the analysis of time-interrelated cash


flows are presently unable to accommodate fluctuations like the business cycle and other varying influences on revenues over the period of analysis. See Carmelo Giacotto, kkA Simplified Approach to Risk Analysis in Capital Budgeting with Serially Correlated Cash Flows," Engineering Economist (Summer 1984): 273-286; and Chan S. Park, kkThe Mellin Transform in Probabilistic Cash Flow Modeling," Engineering Economist (Winter 1987): 115-133.

12. This is a situation of an elastic demand response. For moderate price increases, the econometric evidence regarding, for example, residential water or electricity suggests that the consumer response is price inelastic. Thus, higher prices cause the consumer to purchase less—as suggested by the law of demand—but the consumer's total expenditure on this lesser quantity actually increases. Larger price increases, however, can push demand responses into the price elastic region of the demand curve. This point is discussed further in Chapter 4.

13. Sang-Hoon Kim and Hussein H. Elsaid, "Safety Margin Allocation and Risk Assessment Under the NPV Method," Journal of Business Finance and Accounting (Spring 1985): 133-144.

2

Concepts and Procedures

Risk simulation has been widely applied to investment evaluation, including corporate planning models with Sears, Roebuck and Company data;1 World Bank loans;2 computer leasing;3 petroleum investment decisions;4 plant expansion proposals;5 hotel construction;6 and the analysis of insurance companies. In the late 1970s and early 1980s, the Electric Power Research Institute (EPRI) sponsored studies to assess demand uncertainty and expansion plans for electric power systems.7 Risk simulations of Sizewell, a British pressurized water nuclear reactor, focus on its projected financial and economic advisability and timing in the capacity plan.8

Parallel techniques can be identified in engineering reliability studies9 and natural hazard assessment.10 The mathematical basis of risk simulation dates to World War II and a Manhattan Project analysis of the diffusion of neutrons in fissionable material, developed by simulation methods and code named Monte Carlo."

The method of risk simulation has four basic steps, indicated schematically in the flowchart of Figure 2.1. These include:

1. the identification of risk factors

2. the appraisal of the likely range and probability distribution of risk factors

3. the simulation of investment performance with parameters sampled from the probability distributions developed for the various risk factors


Figure 2.1 Steps in Conducting a Risk Simulation

4. the summary of the results of the analysis in a risk profile for the investment performance measure or criterion.

This process supports the formulation of risk management policies and tactics. The analysis as a whole is rendered more effective by attention to the facts and idiosyncracies of risk communication.

This chapter discusses this procedure in some detail, defining and motivating key concepts that are relevant and useful in this type of analysis. These concepts include default risk; random variable; frequency and subjectivist interpretations of probability; techniques of probability elicitation, including juries of executive opinion, the Delphi method, and interview techniques supporting probability encoding', the bootstrap; time series analysis; structural models; simulation sampling strategies; random numbers and pseudo random numbers', and more about risk profiles and risk preferences. The purpose of this chapter is to lay a foundation for subsequent discussion and the series of examples presented in later chapters. The reader may want to check his or her comprehension, accordingly, returning to this chapter to pin down a basic point discovered in a later chapter.

Concepts and Procedures 19

IDENTIFYING RISK FACTORS

Generally, the objective of utility planning is to meet demand in a defined geographic service area and, possibly, to supply other utility systems through bulk or wholesale contracts. This often implies that new capacity is planned for completion at the time that system demand is projected to exceed available capacity. Utilities rely on capital markets for financing that cannot be met by internally generated funds. For publicly owned companies, options include debt, preferred stock, and common stock. New equity is generally more costly than debt, due to capital gains and dividend taxation, transactions costs, and tax deductibility of interest on debt.

One specific reading of what financial risk means for power and water utilities, therefore, is default risk on debt. Larger, riskier capital construction projects almost inevitably are financed by issuance of bonds. New debt service increases total obligations and usually is associated with pledges in the bond ordinance drawn up prior to the sale.

While it may seem extreme to focus on breaking points, there are several good reasons—historical and conceptual—for doing so. It is helpful to recall that widespread misallocation of utility financial resources in the 1930s prompted public regulation in the first place.12 Thus the current push for deregulation may resurrect some of the old problems. There was also the pervasive downgrading of investment ratings of investor-owned utility bonds in the 1980s. Uneven patterns of regional growth resulted in a number of problem power and water systems, particularly in the oil patch and in the southwestern part of the nation, and the newfound freedom to diversify investments led to some of the same excesses witnessed in the thrift industry. In general, times seem to be growing more complex with uneven impacts of business restructuring and foreign competition and changing social patterns. The conceptual argument, on the other hand, identifies default risk as the bottom line against which bond ratings and other assessments of financial condition occur.13 Movements from slight to slightly more risk of default on debt obligations can prove significant to a company's access to capital markets and its ability to provide high quality service. Clearly, the appraisal of default risk should be the


first study in financial risk analysis, possibly to be supplemented by research vis-a-vis other measures of financial performance.

Relevant Accounting Categories

Several accounting categories, which are outlined in Text Box 2.1, are relevant to appraising default risk. Thus, net income is commonly defined as total revenues net of purchases from other systems, operating and maintenance (O&M), and other expenses. Typically, a utility pledges to sustain a debt coverage ratio, defined as net income divided by the debt service. The debt coverage ratio is commonly specified at about 1.25 or 1.3. When debt coverage falls below the preassigned figure, the utility is in technical default and must attempt to increase revenues through rate increases.

The word "attempt" stresses the fact that there can be constraints on a utility's ability to increase revenues through higher rates. Thus, many power and water producers are subject to regulation regarding their rate of return or earnings, where adverse developments in rate hearings have earned the appellation "political risk." Rate increases in these proceedings tend to be evaluated against a revenue requirement and a rate of return judged to be adequate (or "fair") to attract investment capital to the firm.I4

In simple terms, the relationship

RR = C + d + T + (V-D)r

where RR = the revenue requirement, C = operating expenses or costs, d = annual depreciation, T = taxes, V = gross valuation of utility property, D = accrued depreciation, and r = rate of return, governs identification of the allowable average price P*,

P* = (C + d + T + (V-D)r)/Q

where Q is the quantity sold in a representative and recent year. This creates a difference between actual and allowable expense in the rate base. Hence, the valuation of utility property, depreciation procedures, and what constitutes a fair rate of return become hotly contested issues.

It is less often recognized, however, that enhanced revenues


Text Box 2.1

Revenues

Sales Retail Customers Wholesale Customers

Invested Funds

Expenses

Purchases from Other Systems Operation & Maintenance (O&M) Administration Taxes Debt Service on Bonds

Amortization Schedule

Debt Service Interest Principal

Demand/Supply Balance

Demand Additions to Demand Total Capacity Excess Demand

may not result from price increases for a product, even if higher rates are allowed by a regulatory body. This is especially likely if large increases are contemplated, since consumer demand responses may be price elastic, as discussed in detail in Chapter 4,


"Revenue Risk—Rate and Demand Factors." Actual default occurs when a utility fails to pay the debt service

in a given period, that is, it fails to set aside funds for payment of bond interest and principal payments. In many cases, the situation may be highly dynamic, in the sense that there can be considerable uncertainty in the short term about whether revenues will be sufficient to meet obligations or whether short-term cash can be raised to save the day. Given the capital intensity of water and power systems and the potential for holdings of land as well as plant and equipment, it may be possible to arrange a sale of assets to generate short-term cash to meet debt service. Another option is the refinancing and rescheduling of existing debt, although, clearly, caution may be necessary on the part of the utility in broadcasting the extent of their duress. The argument here always must be one of temporary exigency and the promise of future growth in revenues.

A deeper level of distress occurs when a utility cannot meet operating expenses and interest payments. This is generally viewed as irremediable without recourse to reorganization or liquidation, but it is interesting that there may be room for maneuver even in this circumstance. Thus, under normal conditions, interest payments are sometimes capitalized in loans or the issuance of bonds, that is, there is an initial grace period in which there are no interest payments. So, again, if the promise of future revenues is extraordinary, interim expediencies may be negotiated.

Financial exigencies are caused by higher costs or lower revenues than anticipated when debt and other obligations were assumed.

On the cost side, construction cost growth and delay in construction schedules are a major source of financial risk. If interest is capitalized during the construction period, project delays exact high penalties. As contingency funds are exhausted, new debt or other, usually more costly, funds must be obtained, adding to the level of subsequent debt payments.

Revenue shortfalls due to actions of bulk customers also may be a consideration. Utilities may be lulled into a false sense of security by so-called take-or-pay contracts without verifying recognition of such contracts in local jurisdictions. Continuation of specific bulk contracts may be risky in regions of abundant electric supply in which transmission is being erected to support more power pooling.


A pervasive financial risk is related to long periods of excess capacity that can accompany construction of large facilities designed to reap economies of scale. If bond interest rates are high or if demand growth slows, the financing of these projects may become difficult, despite prospects of their low unit costs when fully utilized.

Other Measures of Investment Performance

Default risk is conditional on servicing demand in a defined geographic service area (since the safest option may be to liquidate productive assets and buy Treasury bonds), and its appraisal may be conditioned on other investment performance criteria also. Achieving a high or target rate of return on capital, for example, is a standard yardstick of business success.15 In general, the goals and motives of a large electric power producer or water purveyor can be complex.

From a planning and regulatory standpoint, emphasis in recent years has been placed on the present value of capacity investments projected by a utility. This is a broad performance measure linking financial risks and categories of benefit cost analysis and economic optimization in utility systems. With suitable accumulation of reserves, the investment sequence with the minimum present value of costs often is the expansion path with the minimum financial risks.16 When utility rates are flexible, the minimum present value of costs is associated with a capacity expansion trajectory in which marginal cost pricing is applied and maximum social benefits are attained, insofar as these may be measured. Considerable literature exists concerning the application of mathematical algorithms to optimize capacity investment plans, where the attainment of these optima are gauged by the present value of the costs criterion.

APPRAISING RANGES AND PROBABILITIES

The problem addressed by risk simulation is simply that it is desirable to have some way to compare cash flow projections developed upon optimistic or pessimistic assumptions or with various intermediate values for key risk parameters. This involves associating probabilities with various values of these risk factors.


What are probabilities? One answer is associated with random variables. A random variable is the outcome of a random experiment, that is, a procedure that is repeatable under similar conditions in which the outcome characteristically varies, such as flipping coins or tossing dice. Thus, a random variable might be the number coming face up on a toss of a die. Given certain, intuitive physical conditions, the probability that, say, one toss shows a six face up is 1/6. Another way of looking at this probability of one in six is to consider it to be approximately the proportion of sixes that come face up as a die is tossed a very large number of times (see Appendix). This reasoning is the basis of the frequency interpretation of probability, upon which, to a large extent, the mathematics of probability have been developed. To an extent, historical data support frequency assessments. Thus, if one is interested in the probability of cost overruns on construction projects of a certain type, it makes sense to consider the past record of such projects.

There are several alternative views of the probability concept. Some argue, for instance, that repeated trials under similar conditions are an almost pointless abstraction in many contexts. Nevertheless, they grant that people persist in making and acting on assessments like, "Her chances of promotion are about 50:50," or 'Td give him a 60 percent chance of getting the contract." What do these statements mean? The subjectivist interpretation suggests that such attributions of probability signify a degree or intensity of belief that conditions our responses, if any, to the events in question. Quite remarkably, subjective probabilities conform to the same mathematical laws governing probabilities viewed as frequencies. Subjective probabilities, in addition, can be viewed as being responsive to new, objective information about an event or process.

These interpretations reinforce each other. Rarely do we have the opportunity to run controlled experiments in social or financial situations. Nevertheless, there are several techniques of quantitative analysis, such as statistical regression procedures, that enable us to consider processes and events under comparable situations. Thus, we may abstract the effects of estimated total costs from estimated completion time on construction cost overruns on power or water projects if we can assemble data on dozens of projects in


which these variables have different relative values (see Chapter 3). We may even have success in linking financial events to probability density functions for failure rates of equipment (exponential distribution), the likelihood of rare events (Poisson distribution), the chance that errors of measurement will cumulate into significance (normal distribution), or that binary or binomial processes or other probability densities will predominate. More generally, immersion in the facts produces judgments about the relevant bounds of events and expected values relevant to decision making. Thus, frequency analysis can contribute to subjective probability evaluations. The use of objective or historical data also limits the influence of typical biases in processing risk information, such as overoptimism, letting recent events dominate responses, and the like.

As a general rule, objective and subjective probability assessments can be put to best use when a complex event or process is broken into relatively independent constituent occurrences and processes. Thus, until the final hour, the "probability of default" may not be a very available notion, even to direct participants in the investment process. The probabilities of construction cost overruns, losing an important buyer, and lower population growth, however, are concepts that are more intuitively accessible. Decomposition of events into simpler occurrences imparts focus to probability appraisals and elicitation.

Gathering Probability Information

The collection of probability information is a key step in risk simulation. The availability and effectiveness of methods for eliciting subjective probabilities and analyzing objective data are perhaps less understood than they should be. This section considers several such methods, their key features, and provides some sources for further reading.

Subjective Assessment Methods

The general tenor of subjective probability elicitation is conveyed in David Hertz's classic discussion of financial risk in manufacturing, where analysts are advised to


probe and question each of the experts involved—to find out, for example, whether the estimated cost of production can be said to exactly equal a certain value or whether, as is more likely, it should be estimated to lie within a certain range of values. . . . The ranges are directly related to the degree of confidence that the estimator has in the estimate. Thus certain estimates may be known to be quite accurate. They would be represented by the probability distributions stating, for instance, that there is only 1 chance in 10 that the actual value will be different from the best estimate by more than 10%. Others may have as much as 100% ranges above and below the best estimate. Thus, we treat the factor of selling price for the finished product by asking executives who are responsible for the original estimates these questions. . . . Given that $510 is the expected sales price, what is the probability that the price will exceed $510? . . . Is there any chance that the price will exceed $650? .. . How likely is it that the price will drop below $475? Management must ask similar questions for all of the other factors until they can construct a curve for each.. .. Often information on the degree of variation in factors is easy to obtain. For instance, historical information on variations in the price of a commodity is readily available. Similarly, management can estimate the variability of sales from industry sales records.17

Subjective assessment methods attach quantitative tags to people's perception of various risks and probabilistic relationships, where "subjective" need not have the connotation of "capricious" but can refer to opinions formed against the backdrop of special expertise and experience. Some standard methods distinguished in the literature include juries of executive opinion, Delphi methods, and probability encoding techniques based on interviews.

Jury of executive opinion. This simple, widely used method involves group decisions about the best estimate for a risky or uncertain item. It is essentially a committee approach where participants meet "to resolve this thing once and for all." One drawback is that face-to-face interaction can result in weights or probabilities for events that depend on a person's role in an organization rather than, strictly speaking, their knowledge about the process being considered.

Delphi method. This technique aims to elicit a consensus from a group of experts about an uncertain event or development while minimizing undesirable elements of group interaction. The basic method involves circulating a questionnaire, summarizing expert


evaluations in an anonymous format, and repeating this process.18

Initial responses are summarized without identifying their source and are circulated in a second round. The same group is asked whether they are inclined to change their estimate. Estimates are supposed to converge in a few rounds.

Encoding probabilities. Techniques for eliciting appraisals or evaluation of the probabilities of events—called encoding probabilities—have been widely applied in risk analysis.19 The interview is a one-on-one situation between someone with privileged knowledge about an element of risk and an individual skilled in eliciting information without imparting his or her biases in the process. The literature contains guidelines for introducing the basic topic and tactics to avoid or minimize bias. The ultimate objective is to elicit the subject's assessment of the cumulative probability distribution of a risk factor.

The direct approach, of course, is to ask the subject to draw the probability density function or to inquire about the chance that the variable in question will be greater than or less than various values. A simple approach, for example, is to focus on three items of information: the lower bound for the risk factor deemed at all within the realm of possibility, the upper bound, and the most likely value. Then the probability distribution associated with this risk factor can be approximated by a triangular probability density function, such as that presented in Figure 2.2. Note the distinction here between "most likely" and "expected." The most likely value of a random variable is its mode or the most frequently occurring value, while the mathematical expectation is its mean or expected value. Only when the probability distribution is symmetric do the mode and mean or expected value coincide.

Psychological research suggests people can have problems with direct approaches to probability elicitation.20 Alternative procedures involve questions to which the subject can respond by choosing between simple alternatives or bets. Options can be adjusted until the subject is indifferent between them. This indifference can then be translated into a probability or value statement. A probability wheel, for example, facilitates these comparisons. This is a disk with adjustable sectors of two different colors and a fixed pointer in the center. When spun, the disk stops with the pointer in one or the other color sector (usually blue or orange). The


Figure 2.2 Triangular Probability Distribution

relative size of the two sectors can be altered by the interviewer, changing the probability that the pointer will be in one or the other sector when the disk stops spinning. The subject is asked which of two events he or she considers more likely—the event relating to the uncertain quantity, for example, construction costs will be 50 percent greater than the base case estimate, or the event that the pointer ends up, for example, in the orange sector. The amount of orange is then varied until the subject judges the two events equally likely. The probability of a cost overrun then can be read


off from calibrations on the back of the disk. While seemingly artificial, this approach may be useful in prompting thought in the early stages of an elicitation process.

Reference can also be made to fixed probability events such as drawing a royal flush in poker or tossing ten heads of a coin in a row.

Verbal encoding is another method of probability elicitation. It ascribes verbal descriptors such as high, medium, and low to events but has been shown to be relatively variable across individuals.

Developing Probability Assessments from Objective Data

Complementary methods of characterizing the probabilities of various risks rely on the analysis of data from comparable projects or situations and historic data on the variation of key factors. These techniques, which include the bootstrap, time series analysis, and structural regression analysis, are technically challenging but, again, are more accessible with new software and new computer capabilities.

The bootstrap. A relatively new method for developing confidence intervals for estimates of distributions or parameters, the bootstrap makes few prior assumptions about the probability distribution generating observed data and figures in important recent applications.21 Bootstrap methods, for example, are used by statistical consultants to the Rogers Commission in analyzing data from the space shuttle Challenger disaster and the question of whether a relation between low temperature and O-ring failure should have been seen as a risk before the launch.22 These methods also have been applied to the characterization of probability distributions of peak electricity demand (see Chapter 4).

The idea is to sample repeatedly from the existing empirical distribution of a random variable and to use these synthetic samples in establishing confidence intervals for the underlying distribution that generates the complete sample data in the first place. This approach is particularly useful when an apparent functional relationship between variables exists, but it is not expected that the variability of the dependent variable around the explanatory variables is normally distributed. An example is discussed in the following chapter vis-a-vis the simulation of total construction costs,


when interdependencies between cost categories or between costs and time must be acknowledged.

Time series analysis. Early discussions of financial risk simulation were concerned primarily with variables representing discrete or, at least, independent events. However, financial risks can be related to ongoing processes, such as population growth and inflation, and the existence of time interdependencies can affect outcomes, for example, when one is drawing down limited reserves. Thus, instead of simply considering the probability distribution for a variable, it is desirable to be able to characterize the band of variation of a process over time and time interdependencies between stochastic components of this process. Time series analysis provides important techniques for accomplishing this type of characterization.

A time series is simply a set of observations on a variable taken at various times. Examples include daily closing prices of a common stock over a period of time, hourly blood pressures of an individual, or, as illustrated in Figure 2.3, average annual water consumption in gallons per capita per day (GCD) over seven decades.

Historically, researchers have been intrigued with the possibility of identifying periodicities in such series. Early efforts applied methods such as the periodiogram or regression fitting of sinusoidal functions to this type of data and led to development of a branch of mathematics called spectral analysis. Often, however, the variation at hand was really almost periodic or pseudoperiodic and was not adequately described as the result of a fixed oscillation and period with superimposition of random effects.

In 1976, George E. P. Box and G. M. Jenkins wrote a groundbreaking book titled Time Series Analysis: Forecasting and Control.23 This showed how, given certain stability conditions relating to the mean and variances of the variable being analyzed: (1) deterministic and stochastic components of a time series could be identified, and (2) stochastic or random components could be further decomposed into autoregressive and moving average processes, or some combination of these two processes.

An example shows the superiority of this method. Consider, for instance, the water consumption data in gallons per capita per day in Figure 2.3, which are derived from a western metropolitan water system over the period 1918-1989. Note the wavelike pattern and

Figure 2.3 Denver, Colorado, GCD


its irregularity. This wavelike pattern suggests autocorrelation or serial correlation and can be confirmed by various diagnostic tests. If GCD is above trend in year j , it is likely that the year j + 1 GCD also is above trend, and vice versa for below-trend observations.

Construction of diagnostic graphs called the autocorrelation and partial autocorrelation functions suggests a simple model:

where et is the GCD expressed as a deviation from the trend line. This model appears to reduce the detrended series to white noise residuals, a concept explained below.

Accordingly, there are two components: the GCD trend, and what standard regression analysis would call the error term or residuals et. These residuals et can be analyzed further into a first order autoregressive process, described by equation 2.1 above, and white noise or Gaussian residuals.

Note this is a peculiar type of explanation in the sense that it tells how to reproduce the data series and, by implication, how to generate representative future terms. With respect to causes, however, one must appeal to technological and attitudinal change leading to more water-using appliances in households, particularly since World War II, and other systematic factors, where this type of explanation motivates the general trend in the data.

Many processes lend themselves to this type of analysis. Time series may be decomposed into deterministic and stochastic components with weighted averages of white noise components and/ or correlations between the current value of the variable and its previous values in the series (autocorrelation). Diagnostic tests are available to determine, at least at a gross level of detail, which characterizations of stochastic structure are most plausible.24 When combined with a priori information about contributing processes and factors, time series models can help arrive at the relevant band and likely pattern of variability in a risk source.25 Standard time series modeling approaches emphasize stationarity conditions, or constant mean and variance, which often can be enforced by first differencing data or through logarithmic transformations of the time series variable.

White noise refers to a purely random time series exhibiting no


systematic time interdependencies or lag effects. Note that white noise processes have a special significance in financial studies since researchers believe this type of randomness characterizes the movement of stock prices over time.26 A white noise process is a random variable determined by a normal or Gaussian distribution having a zero mean value. In time period 1, one value of this random variable is sampled. In time period 2, another value of the random variable is sampled on a completely independent basis, that is, the value of the random variable in period 1 does not affect the value sampled for period 2. While recent studies reveal some pattern in the movement of stock prices,27 for anyone but the largest investors, transaction fees make the slight correlations identified over time difficult to exploit.

Structural models. The deterministic components in time series may be described by equations relating a dependent variable to independent or explanatory variables. These are called structural equations in econometrics, a leading example being consumer demand models. Thus, the amount of a commodity a consumer is willing to buy at various prices typically is presented as a linear system in which price (P), income (Y), and other factors (Z) influence the quantity (Q) a consumer would like to purchase:

Q = b„ + b,P + b2Y + b,Z

Here b0, b,, b2, and b3 are coefficients usually estimated by multiple regression procedures, which assume a particularly simple characterization of the affiliated random component of the observations on the consumer's purchases. Often such models are developed on cross-sectional data, where observations on households for the same time period are available.

CONDUCTING THE SIMULATION

Ultimately, all risk factors are represented, either directly or in more complex formulation, as random variables characterized by probability distributions. Given this information, risk simulation involves sampling the possible values of these variables based on the representation of their cumulative distributions.


Sampling Strategies

To illustrate the sampling procedure, assume we consult a random number table or rely on a computer program to produce a random number between 0 and 1 called n*. This random number n* will guide our selection of the value of some risk factor for one run of the cash flow model and contribute to development of the performance index for this investment. Assume that Figure 2.4 represents the cumulative distribution of the risk factor, that is, the probability that the risk factor is at most the value of the x axis. Then, we associate a value C* with n* by reading from the probability axis to the cumulative distribution F(x) and down to the x axis. Repeating this sampling process with many random numbers assures that the repetitions of the cash flow model are computed with a random sample of this risk factor, determined by this particular cumulative distribution.

The question of how many samplings ought to be made depends on the mathematical complexity of what is being sampled, the sampling procedure, and the penalty function established to assess the consequences of error in the risk profile. A crude test for the adequacy of the sample size is to vary the number of samplings by one and two orders of magnitude to determine whether there are noticeable differences in the resulting shape of the risk profile.28

The risk profiles in the examples in this book seemed to achieve stability in the 10,000 iteration range.

Computer programs, it should be noted, produce pseudo-random numbers, usually a repeating series with some relatively long recurrence number. The Lotus 1-2-3 (a RAND function has a recurrence cycle of over 1 million, which makes it a relatively good random number generator for a microcomputer.29 For advanced applications, short computer programs exist to purge undesirable features of stock pseudo-random number generators.30

Management Responses

Some of the most difficult questions in simulation involve policy or management responses over longer time periods. Thus, random factors leading to actual or prospective deficits in income can trigger a search process on the part of management. The analog of


Figure 2.4 Cumulative Probability Distribution

this in the computer simulation is a goal-seeking routine programmed to go into motion when predefined thresholds are crossed in a cash flow model. Thus, if the debt coverage ratio falls below a key level, a rate increase can be triggered whose time lag, extent, and effect would be determined by assumptions about regulatory and demand constraints. Often, multiple possibilities can be represented by an event tree,31 such as that presented in Figure 2.5.


Figure 2.5 Event Tree for Rate Increase

Here, possibilities associated with application for a rate increase by a regulated utility are portrayed and indicate branches of the event tree leading to or supporting financial problems that require further action to resolve.

It would be helpful if unambiguous bounds to risk could be established by specifying response modes in certain ways. Thus, good management and honest employees are usually assumed in risk simulation. Matters become more complex, however, when it becomes necessary to specify how quickly responses will occur. Usually, there is debate within an organization about whether a change is transitory or whether it signals the beginning of a new trend. Another question is whether the basis decision makers use to establish policies will change in the future. The technical basis of specifying management responses in the simulation is a series of if-then or conditional statements. Examples are presented in Chapter 6.

THE RISK PROFILE

The risk profile measures investment performance by incorporating probabilistic information and appraisals. Indeed, this curve, which follows from assumptions and imputations in the risk simulation, is a probability distribution. Hence, discrete and continuous versions exist, and it is convenient sometimes to present the risk profile as a cumulative distribution rather than as a probability density function.

Figure 1.1 shows a continuous probability density function. The horizontal axis charts the value of the random variable selected as


the performance measure. For continuous distributions, the height of the risk profile at a particular rate of return measures the relative frequency of a rate of return in a small interval around a given point rather than the absolute numeric probability of the random variable attaining precisely this particular value. In other words, the probability distribution for a continuous random variable indicates the relative likelihood of attaining a particular value or something close to this value. The important thing, however, is the area under the curve or the probability density. Thus, about 75 percent of the area under the curve occurs to the right of the 10 percent point on the horizontal axis. Based on this relationship, therefore, decision makers might be advised there is a three in four chance, or a probability of about 0.75, that the internal rate of return will exceed 10 percent.

The default risk in any particular year is determined by the chance that some variable, such as the ratio of net income to debt service, exceeds a preassigned number. Note that net income in any year is the sum of a number of separate components, many of which can be relatively independent from each other. Accordingly, we might appeal to the Central Limit Theorem of statistics and approximate the probability distribution of net income by a normal curve (see Appendix). Debt service, once a bond is issued, is predetermined or nonstochastic, except when it is tied to a variable interest rate. Accordingly, the probability distribution of the debt coverage ratio of net income to debt service also usually may be approximated by a normal distribution. Thus, the probability of default can be visualized as an area in the tail of a bell-shaped distribution to the right of the minimum debt coverage promised in the bond covenant.

Risk Preferences and Risk Profiles

Risk profiles support project comparisons, subject to the risk preferences of decision makers. Leaving aside love of gambling per se—relish in the thrill of staking it all and having one's fate determined by the wheel of chance—we can rank people's willingness to risk losing various amounts in return for the chance to gain other amounts.

This connotation is illustrated with a simple example. Suppose


a credible party offers you a chance to win $1,000 if a coin, predetermined to be fair, comes up heads on a toss. If the flip comes up tails, you pay a given amount. One possibility is that you flatly refuse to participate. Or you might be willing to risk something to have the chance of winning $1,000. Individuals, accordingly, can be ranked as risk averse or risk seeking and along a gradient between these opposites. Similarly, decision makers, responsible for taking one or another course of action, can be risk averse or willing to accept higher risk in order to have potential access to higher earnings.

The classic comparison is with respect to the variances of continuous risk profiles having the same expected values. Suppose Figure 2.6 shows the probabilities of rates of return on projects A and B. Both greater gains and losses are possible with project B, graphed with the crosses in Figure 2.6, than with project A, the risk profile indicated by the boxes in the figure. Risk-averse individuals would select A. On the other hand, some investors would be willing to accept a higher probability of low returns for an approximately equal chance to earn higher returns.

There also are situations in which one risk profile seems superior more or less independently of risk preferences. Figures 2.7a and 2.7b illustrate a relationship between investment projects known as first order stochastic dominance with risk profiles and their associated cumulative distributions. If project A is delineated by the boxes and project B is indicated by the crosses, it is clear that project A dominates project B in an important sense. This relationship is perhaps more compelling when we examine the cumulative distributions of the rate of return for these two projects, which, in this case, show the chance of attaining a given rate of return or less. The cumulative probability distribution associated with project B, indicated by the crosses, is always above or to the left of the cumulative distribution of project A for any given rate of return. Pick any probability on the vertical axis in Figure 2.7b, say, 0.10. Then, following this probability level over to the two cumulative distributions, one can see that the chances of attaining a higher return are always greater with A than with B. Clearly, project A is superior, under a range of risk preferences. The analogue for default risk distributions, of course, is simply that one course of action produces a lower default probability than another.

Concepts and Procedures

Figure 2.6 Comparison of Risk Profiles

SUMMARY AND CONCLUSIONS

Risk simulation recognizes that precise forecasts of variables are impossible but holds that actual values may conform to specifiable stochastic processes. A utility with cash flow problems may raise rates and charges, borrow or reschedule debt, declare a moratorium on bond principal repayment, default on interest or premium


Figure 2.7a First Order Stochastic Dominance (relative probability)

payments, seek the protection of Chapter 11, or liquidate assets. Evaluation of these options is a matter of informed judgment and analysis, which can be supported by computer simulations made more accessible by recent advances in microcomputer technology and computer software.

Ultimately, all risk factors are represented, either directly or in more complex formulation, as random variables characterized by probability distributions. Given this probability information, risk simulation involves sampling the possible values of random variables based on the representation of their cumulative distributions. The product of a risk simulation is a picture of the likely variation of some target variable, termed here the investment performance measure. Comparison of investment options is most straightforward when one risk profile exhibits stochastic dominance over another. Otherwise the choice of investments cannot be decoupled from the risk preferences of decision makers.

In general, there are two sources of information about probable variation in risk factors: expert opinion or evaluation, and statis-


Figure 2.7b First Order Stochastic Dominance (cumulative probability)

tical and historical, comparative analysis. Juries of executive opinion, Delphi methods, and formal probability elicitation techniques can be helpful in polling expert opinion.

One useful distinction is between essentially "one-shot" variables, like construction costs and multiple period variables such as labor costs, interest rates, or population growth. Single period variables may be characterizable by a simple probability distribution. Frequently, discussions of risk simulation stop with identification of this simple type of stochastic process. More complex, time-interrelated processes, however, can characterize the time path of multiple period variables. General features of these stochastic processes can be analyzed by time series methods.

There are styles of thought regarding risk, in addition to risk preferences, that make risk communication particularly important. Executives or top administrators may favor decisive, seemingly deterministic modes of thought. Numerous studies, furthermore, show the prevalence of bias in reasoning about risky situations, even with highly trained subjects.12 Simplicity is probably the key


to effective communication. The results of an analysis must be stated in a sentence or two—something like, 'The chance of a 50 percent cost overrun with the nuclear baseload plant is about four times greater than with a conventional coal-burning facility," or, "The rate of return with option A is about twice as likely to be above 15 percent than with option B." Simple graphics also can be useful. If this information touches a nerve, assumptions and the structure of the risk analysis model can be discussed.

In many respects, the process is the product. A risk simulation, in other words, draws together diverse strands of information about costs, physical contingencies, organizational and financial responses, and socioeconomic forecasts. In doing this, management, the investment community, and regulators increase their overall awareness of the context of a project or a utility's financial position.

NOTES

1. Pamela K. Coats and Delton L. Chesser, "Coping with Business Risk through Probabilistic Financial Statements," Simulation 38 (April 1982): 111-121.

2. L. Y. Pouliquen, Risk Analysis in Project Appraisal (Baltimore: Johns Hopkins Press for the International Bank for Reconstruction and Development, 1970).

3. A. M. Economos, "A Financial Simulation for Risk Analysis of a Proposed Subsidiary," Management Science 15 (1968): 75-82.

4. P. D. Newendorp and P. J. Root, "Risk Analysis in Drilling Investment Decisions," Journal of Petroleum Technology (June 1968): 579-585.

5. L. Kryzanowski, P. Lustig, and B. Schwab, "Monte Carlo Simulation and Capital Expenditure Decisions—A Case Study," Engineering Economist 18 (1972): 31-47.

6. D. A. Cameron, "Risk Analysis and Investment Appraisal in Marketing," Long Range Planning (December 1972): 43-47.

7. Charles E. Clark, Thomas W. Keelin, and Robert D. Shur, User's Guide to the Over/Under Capacity Planning Model, EA-1117, Final Report (Palo Alto, Calif.: prepared for the Electric Power Research Institute; October 1979); Martin L. Baughman and D. P. Kamat, Assessment of the Effect of Uncertainty on the Adequacy of the Electric Utility Industry's Expansion Plans, 1983-1990, EA-1446, Interim Report (Palo Alto, Calif.: prepared for the Electric Power Research Institute, July 1980). See also Ian S. Jones, "The Application of Risk Analysis to the Appraisal of


Optional Investment in the Electricity Supply Industry," Applied Economics 3 (May 1986): 509-528.

8. Nigel Evans, "The Sizewell Decision: A Sensitivity Analysis," Energy Economics 6 (January 1984): 15-20; and Jones, "The Application of Risk Analysis to the Appraisal of Optional Investment in the Electricity Supply Industry."

9. E. J. Henley and H. Kumamoto, Reliability Engineering and Risk Assessment (Englewood Cliffs, N.J.: Prentice-Hall, 1981), discuss these earlier engineering applications.

10. For an interesting selection of articles on this topic see Paul R. Kleindorfer and Howard C. Kunreuther (ed.), Insuring and Managing Hazardous Risks: From Severso to Bhopal and Beyond (New York: Springer-Verlag, 1987).

11. Hence, the other name for risk simulation—Monte Carlo simulation or analysis.

12. See Charles F. Phillips, Jr., The Regulation of Public Utilities: Theory and Practice (Arlington, VA: Public Utilities Report, Inc., 1984).

13. See Standard & Poor's Corporation, Municipal Finance Criteria (New York: Standard & Poor's, 1989). R. Charles Moyer and Shomir Sil list factors affecting bond ratings as follows: "The level of long-term debt relative to the firm's equi ty. . . the firm's liquidity, including an analysis of accounts receivable, inventory, and short-term liabilities. . . the size and economic significance of the company and the industry in which it operates . . . [and] the priority of the specific debt issue with respect to bankruptcy or liquidation proceedings and the overall protective provisions of the issue." "Is There an Optimal Utility Bond Rating?" Public Utilities Fortnightly, May 12, 1989, pp. 9-15.

14. Frederic H. Murphy and Allen L. Soyster, Economic Behavior of Electric Utilities (Englewood Cliffs, N.J.: Prentice-Hall, 1983), provide a comprehensive survey of public utility commission rate standards as of the early 1980s in Tables 2 and 3.

15. The internal rate of return r satisfies the equation,

where the an i = 1 , . . . , n, are expected annual payback amounts for an investment C. The rate of return r is equivalent to an interest rate at which C dollars could be invested to produce a stream of discounted earnings equivalent to that actually anticipated for the investment in question.

16. Here, as elsewhere in the discussion of this book, the phrase "min-


imum financial risks" will be subject to the condition that supply and demand are in balance. Thus, obviously, absolutely minimum financial risks might be attained by liquidating utility investments and buying U.S. Treasury securities.

17. David B. Hertz, "Risk Analysis in Capital Investment," Harvard Business Review (September-October 1979): 174-175.

18. See H. A. Linstone and M. Turoff, The Delphi Method: Techniques and Applications (Reading, MA: Addison-Wesley, 1975).

19. See M. W. Merkhofer, "Quantifying Judgmental Uncertainty: Methodology, Experiences, and Insights," IEEE (Institute of Electrical and Electronic Engineers) Transactions on Systems, Man, and Cybernetics SMC-17, no. 5 (September/October 1987): 741-752.

20. See Detlof von Winterfeldt and Ward Edwards, "Cognitive Illusions," Decision Analysis and Behavioral Research (New York: Cambridge University Press, 1986).

21. An early discussion of the approach is found in B. Efron, "Bootstrapping Methods: Another Look at the Jackknife," Annals of Statistics 1 (January 1979): 1-26. See also B. Efron, "Nonparametric Standard Errors and Confidence Intervals," Canadian Journal of Statistics 9 (1981): 139-172; and B. Efron and G. Gong, "A Leisurely Look at the Bootstrap, the Jackknife, and Cross-Validation," The American Statistician 37 (February 1983): 36-48.

22. Siddartha R. Dalai, Edward B. Fowlkes, and Bruce Hoadley, "Risk Analysis of the Space Shuttle: Pre-Challenger Prediction of Failure," Journal of the American Statistical Association 84 (December 1989): 945-957.

23. George E. P. Box and G. M. Jenkins, Time Series Analysis: Forecasting and Control (San Francisco: Holden-Day, 1976).

24. John M. Gottman, Time Series Analysis: A Comprehensive Introduction for Social Scientists (New York: Cambridge University Press, 1981), presents a nice discussion of these graphic patterns.

25. See, for example, C.W.J. Granger, Forecasting in Business and Economics (New York: Academic Press, 1980); Richard McCleary and Richard A. Hay, Jr., Applied Time Series Analysis for the Social Sciences (Beverly Hills, CA: Sage Publications, 1980); and William W. S. Wei, Time Series Analysis: Univariate and Multivariate Methods (Redwood City, CA: Addison-Wesley, 1990).

26. Burton G. Malkiel, A Random Walk Down Wall Street, 4th ed. (New York: W. W. Norton, 1985), is still an entertaining and informative introduction to the subject.

27. See Stephen Taylor, Modeling Financial Time Series (Chichester, England: John Wiley & Sons, 1986).


28. This test is applied, for example, by Peter Pflaumer, "Confidence Intervals for Population Projections Based on Monte Carlo Methods," International Journal of Forecasting 4 (1988): 135-142.

29. See D.T.R. Modianos, C. Scott, and L. W. Cornwall, "Testing Intrinsic Random-Number Generators," Byte (January 1987): 175-178.

30. See William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling, Numerical Recipes: The Art of Scientific Computing (New York: Cambridge University Press, 1986), p. 197.

31. Event trees are a standby of risk assessment. See Risk Assessment: Report of a Royal Society Study Group (London: The Royal Society, January 1983).

32. See, for example, P. Slovic, "Perception of Risk," Science 236 (1987): 280-285.


3

Financial Risks in the Construction Period

Financial risks during construction have proved significant for nuclear power plants (see Table 3.1). Cost overruns, scheduling delays, and unanticipated changes in the disbursement of construction payments have triggered multiple and large rate increases, utility reorganization, and abandonment of nuclear projects in some instances. Cost overruns for central, coal-fired power stations or large, capital-intensive waterworks seem less of a problem. Their technology is "tried and true," and construction techniques are well understood. Capital costs associated with all such facilities, however, are large enough to recommend attention to risk management tactics, such as the sizing of construction contingency allowances.

Simulation methods to assess cost overrun potential are a natural extension of existing practices in cost estimation and project scheduling. Cost estimates get fine-tuned as design details come into focus, and the anticipated range or interval of costs tends to narrow through this process. Such range estimates of construction costs can be deployed to indicate the risk of cost overruns and to support the sizing of contingency funds with given confidence levels. The scheduling of large projects has long relied on tools such as Gantt diagrams, critical path modeling (CPMs), and program evaluation and review technique (PERT) analysis. As a review of project management computer programs illustrates,1 simulation with these


Table 3.1 Cost Estimates and Realized Nuclear Power Plant Costs, 1966-1972 (nominal dollars per kWh of generation capacity)

Year Construction Started

1966 1967 1968 1969 1970 1971 1972

Average Estimated

Cost

147 150 155 179 228 258 418

Average Final Cost

299 352 722 890

1,331 1,313 2,258

Average Percent Overrun

103 135 365 397 484 409 440

network models is increasingly important in appraising the likely variability of completion times and expenditure patterns.

This chapter: (1) surveys general and qualitative factors contributing to cost overruns in capital construction projects; (2) considers the literature on cost overruns, specifically with regard to water and power projects; and (3) discusses risk simulation in assessing the likelihood of cost overruns, scheduling delays, and the characterization of the pattern of construction expenditures.

Seven general factors contributing to cost escalation on capital construction projects are identified in the next section. A review of construction cost literature pertaining to water and power projects indicates the importance of technological factors, as well as competent management in accounting for construction cost performance.

Following that, the discussion of quantitative technique considers risk simulation of construction cost. A simple example is developed to illustrate the tendency of the total cost distribution to be approximated by a normal probability distribution, even though component cost distributions are not characterized by normal dis-

Risks in Construction 49

tributions. This result, which depends in part on the way construction costs are classified, is useful in sizing contingency allowances. An interesting finding supported by these exercises is that contingency funds designed to cover each construction cost component at a given confidence level would sum to a total contingency fund that would be larger than needed to cover total construction costs at that same confidence level. Additional topics considered in the discussion of quantitative techniques in this chapter include simulation of the construction schedule and expenditure curves and the problem of stochastically interdependent costs and construction schedules.

A primary objective of this chapter is to show that generating construction cost risk estimates is straightforward, at least as a first approximation. Methods in this chapter are applied, with other techniques identified in the following two chapters, to several integrated simulation applications in Chapter 6.

FACTORS ASSOCIATED WITH CONSTRUCTION COST OVERRUNS

In broad terms, several factors are linked with cost overruns in capital construction projects.2 These include:

1. the stage of the product cycle at which the cost estimate is developed 2. the type of technology 3. the size and complexity of a project 4. the competence of product management 5. regulatory or political considerations 6. contracting arrangements 7. the volatility of prices and other economic variables

The Stage of the Product Cycle at Which the Cost Estimate Is Developed

There is a classification of construction cost estimates based on the stage of the product cycle, which suggests an avoidable risk— making irreversible decisions based on initial or early cost estimates. Initially, conceptual design cost estimates can be based on


analogous or comparative information about similar units (e.g., power plants, water facilities). Later, preliminary or engineering cost estimates are prepared as major design decisions are resolved and preliminary site plans and system flow diagrams become available. This preliminary estimate need not imply a complete scope of work, although it is often useful to prepare estimates of contingency allowances at this point. Finally, there is the definitive, baseline, or official cost estimate, generally developed to coincide with the start of construction activity on site. This official estimate ''becomes the basis for evaluating subsequent cost performance by management and regulatory agencies," even though engineering design work may not be fully complete at that point.3

The Type of Technology

Novel technology may be associated with flagrant overruns in capital construction projects. Thus, attempts to fast-track new technologies on a large and previously untested scale have led to memorable cost escalation in the San Francisco subway system (BART), the Eurotunnel under the English Channel, the trans-Alaskan pipeline, and nuclear power plants. By the same token, routine engineering in fossil fuel power plant construction or in water facilities may be undertaken with a higher expectation that completion can be achieved at or under estimated cost and scheduled time.

The Size and Complexity of a Project

Along a somewhat different line of comparison, there is a relationship between cost overruns and the size and complexity of projects. Thus, a classic parametric analysis developed by James F. Tucker for 107 civil works projects, 39 water resource projects, 39 highway projects, and 29 building projects states,

R = .0233L - .0092(T - 1940) + .0019C - .0066t

where R = cost growth, L = project length in years, T = calendar year of estimate, C = estimated project cost, t = fraction of project length completed at the time of the cost estimate.4 This


indicates that projects with higher initial cost estimates and lengthy construction periods have more cost growth and that cost estimates made later in the project cycle are more accurate. Similar results are produced by other, more recent studies.3

The Competence of Project Management

Management and organization problems are widely cited contributors to cost overruns. An investigatory report about the trans-Alaska pipeline, for example, a project whose costs escalated from $900 million to final costs in 1977 of $7.7 billion, states that,

the project was virtually run by committees; it was structured with vertical and horizontal duplication of supervision and decision making, cumbersome decision chains, unclear lines of authority, and fragmentation of responsibility. Compounding this were significant communication, coordination, and liaison problems between project groups. The result of this duplicative management structure was paralysis of the project management decision making process.6

Regulatory or Political Considerations

Regulatory or political considerations became more important with enactment of the National Environmental Protection Act (NEPA) of 1971 and related legislation. Local interests object to the plume of smoke from power plants or potential radioactivity. New water storage sites are increasingly scarce as real estate development locks up the countryside. Regulatory conflicts and bad initial planning often lead to numerous change orders that cause "ripple effects" throughout the timing and cost of activities in a power or water project.

Contracting Arrangements

Inefficiencies can be associated with contracting arrangements. Lowest-bidder rules applying to contracts let by public agencies have been known to let inexperienced firms in the door, based on "lowballing" the bid. Such companies later may have to be replaced in costly, time-consuming recontracting.


The Volatility of Prices and Other Economic Variables

Finally, the volatility of economic variables, such as prices and interest rates, can be a significant factor in large capital construction projects with a construction period of several years.7

THE RECORD OF WATER AND POWER INVESTMENTS

Studies of cost performance on water and power projects generally are consistent with the preceding discussion. Water and power investments using conventional technology are likely to be associated with small or negative cost overruns, while fast-tracking technical innovation in the nuclear field has definite hazards.

Studies of water resource development show improvements in construction cost control in federal agencies after World War II. Studies by Edward G. Altourney,8 Maynard M. Hufschmidt and Jacques Gerin,9 and Robert H. Haveman10 indicate progress by the U.S. Army Corps of Engineers and the U.S. Bureau of Reclamation. The Corps of Engineers' actual to estimated (A/E) cost ratio improved from 2.24, as of a 1951 study of 182 projects, to about 1.0 in a 1964 study considering 68 subsequent projects. The Tennessee Valley Authority (TVA) maintained good cost control from the beginning. Over the period 1933 to 1967, the TVA A/E cost ratio was .947 with approximately one-third of its projects showing overruns. In addition to good management practices, this is probably attributable to the relatively familiar technology embodied in these projects.

Glenn J. Davidson, Thomas F. Sheehan, and Richard G. Patrick note that fossil fuel power plants usually finished on time and within budget over the period from the late 1940s to the early 1960s. These plants had unit sizes ranging from 75 megawatts (MW) to 400 MW and, generally, were designed by either the owner or an architect/engineer and were built under contract to the owner. In most cases, it has been noted, "the design was almost complete before construction started."11

Relying on conventional technology can overcome some adverse factors, such as lengthening regulatory delay. An EPRI-sponsored


study of construction lead times notes that coal plant construction has been able to adjust to

a fairly continuous stream of regulations aimed primarily at decreasing the environmental impact of plant emissions. . . . They lengthen the licensing period significantly by imposing stricter requirements for evaluating the environmental impacts of the plant before it begins construction. Second, the environmental regulations have required an increased level of air pollution control. . . [such as] electrostatic precipitators . . . and flue gas desulfization devices (scrubbers).12

The importance of technological factors in nuclear power is underlined by cases in which cost overruns appear to be associated with (1) larger, more complex plants; (2) new or frontier technology; and (3) an evolving regulatory climate. Table 3.1 summarizes an appalling cost performance early in the period during which a move to larger plants was made. The average estimated and final costs in the table are in nominal dollars and so partly reflect the increasing pace of inflation in the early 1970s. Nevertheless, an important influence is the fact that larger capacity plants in this period required designs, containment vessels, and foundation systems that challenged the limits of structural engineering knowledge. In addition, wholly new reactor designs were, in some cases, fast-tracked (e.g., WPPSS), and regulatory attitudes favored more vigorous intervention.

In this regard, there has been a study of the pattern of regulatory delay on construction schedules of fossil fuel and nuclear power plants. Construction lead time research sponsored by EPRI distinguishes out-of-scope work—involuntary delays caused by the actions of an agency other than the constructing utility—from deliberate delays caused by the voluntary actions of the utility. Based on a survey of twenty-six nuclear units, out-of-scope delay was identified in 78 percent of the cases. Redesign and rework problems (53 percent) dominated in the sample. The situation was reversed for fossil fuel plants, where deliberate delay caused 68 percent of the total delays in a sample of twenty-eight units.13

This study reached other relevant conclusions. For example,

A large part of the financial risk to a utility constructing a large generating facility is the direct result of long and uncertain leadtimes.. .. Two decades


ago, the capital cost of most plants was less than $400 per kilowatt for a nuclear plant. Even taking inflation into account, the real cost per kilowatt has tripled. . . . Today, the combination of large capital expenditures over a long period and high interest rates cause time dependent charges to make up a substantial portion of the total capital cost of the plant.14

In addition there are investment risks because "the cost of large power plants, especially nuclear plants, is now so high that they make up a large percentage of the total assets of many utilities."15

RANGE ESTIMATION AND SIMULATIONS TO DETERMINE CONTINGENCY FUNDS

How can the likelihood of construction cost overruns be evaluated? Range estimation is a major approach to this question, as noted in the cost engineering literature.16 This method is supported by historical cost data or reliance on judgmental factors when a database of comparable facilities does not exist. Range estimation operates with cost summaries of major items in a construction project, estimates of their range of variability, and other information, where available, about the likely distribution of component costs. The term "range estimation" derives from first approximations that operate with lower and upper bound estimates of component costs. Additional information usually shrinks the estimated variance of total costs, reducing estimates of contingency funds needed to cover changes in the total cost at a given confidence level or a given percentage of the time. In this sense, range estimation provides a yardstick for measuring the adequacy of contingency funds and the value of information about the variability of component costs.

Let us illustrate the power and generality of this method. Table 3.2 lists the major construction cost categories on some project. The first column simply names these cost categories generically, as c,, c2, c3, and so on. Columns 2 and 3 tabulate the anticipated lower and upper bounds for these cost categories, defining a cost range. These ranges may be absolute, encapsulating all possible values of the cost components, or can incorporate a given percentage of likely variation of these cost categories, for example, five and ninety-five percentile costs. (The five and ninety-five per-


Table 3.2 Range Estimates and Expected Costs by Cost Component (millions of dollars)

Cost Component

(D

d c2 C3 c4 C5 c6 c7 c8 c9 C10

TOTALS

Lower Bound Costs

(2)

100 60 30 20 10 10 10 10 10 5

265

Upper Bound Costs

(3)

140 90 50 40 40 40 40 40 40 30

550

Expected Costs

(4)

121.7 75

41.7 28.3

25 25 25 25 25 15

406.7

centile points on a probability distribution mark off events likely to happen only one time in twenty. Thus, the chance of a value below the five percentile point or above the ninety-five percentile mark of a probability distribution is one in twenty.) The assumption, of course, is that these range estimates are produced by polling expert opinion and that such expertise is specialized by cost component.

The fourth column lists expected or average costs by component and the expected total cost. These expected costs, mathematically the average costs that would be achieved in multiple realizations of the same situation, are assumed to be the cost estimates developed by engineers for this project.

Note that there are only a few major costs and more numerous smaller components in Table 3.2—an illustration of Pareto's Law


of the significant few and insignificant many. For purposes of discussion, the estimates in the tables are assumed to be in millions of dollars.

Stochastically Independent Costs

The standard assumption in this type of analysis is that the costs are stochastically independent. In other words, there can be no correlation between the cost overruns experienced or realized by the various cost components. In Table 3.2, c2 is 20 percent higher than its expected value in column 4, there is no added chance that c3 or any of the other cost components will come in higher than expected.

Although this is a strong assumption, cost engineering suggests that this condition can be approximated by suitable aggregation.17

Thus, substitutability between construction tasks is generally clustered in significant groups. Overall, subcontracting of different phases of the project to different firms (e.g., site preparation, foundation work, erection) limits the degree to which cost overruns or slowdowns in one phase can be made up by economies or speed-ups in another phase. Even within groupings of tasks, statutes pertaining to overtime pay, surcharges for immediate delivery of materials, and the like limit speedup opportunities. Once estimated costs are exceeded, therefore, it is seldom possible to cheapen other aspects of a project. For purposes of discussion, then, let us assume the cost categories of Table 3.2 are grouped so as to be stochastically independent. Later, we will comment on more sophisticated tactics for dealing with correlated costs.

Risk Simulation

The information in Table 3.2 leads to a risk profile for total costs when we develop a characterization of the probability distributions of components' costs. Thus, given that the numbers in columns 2 and 3 represent absolute lower and upper bounds for component costs, the availability of the expected values in column 4 suggests the use of the triangular probability distribution. The bounds for the costs delineate the base of this distribution, and the height or


vertex of the triangle for each component cost distribution can be obtained by solving for the mode (M) in the formula

Expected value of a triangular distribution = (L + M + U)/3

where L is the lower bound and U is the upper bound. It is interesting to compare this triangular approximation with

a simpler probability density function assuming that any value of a component cost between its lower and upper bound is equally likely. This assumption produces the uniform probability distribution, often deployed as an uniformed prior distribution in decision analysis—that is, in the absence of better information, each contingency might be assumed to be equally probable.

Given either of these constituent probability distributions, simulations lead to a risk profile resembling a bell-shaped curve or normal distribution. This remarkable result is not accidental but follows from the Central Limit Theorem of statistics (see Appendix). The basic notion is that the sum of sufficiently numerous independent random variables has a limiting distribution that is a normal or bell-shaped curve.

Figure 3.1 shows an overlay of the resulting risk profiles. The risk profile associated with the imputed triangular probability distributions is graphed in black. The more dispersed risk profile indicated by the white bars is associated with the imputation of uniform probability distribution functions to the component costs C! through c10.

Contingency Funds

The risk profiles in Figure 3.1 help form perceptions of the size of construction contingency funds needed for this project. Here, one must employ a broad concept of contingency allowances, meaning funds to cover potential deviations of total cost above its estimated value a given percent of the time.18

Suppose we intend for the contingency fund to cover cost variation 95 percent of the time and want to include a fund of this size in the bond issue as an addition to funds determined by the construction costs estimate. We can compute the required contingency allowance with the cumulative probability distribution of


Figure 3.1 Overlay of Risk Profiles Developed with Uniform and Triangular Distributions and Table 3.2 Ranges

total costs. Figure 3.2 presents the cumulative distributions associated with the risk profiles of Figure 3.1. Here, the usual presentation is rotated so that the probability of the project totaling a certain cost or less is shown on the horizontal axis. Following the line up from the 95 percent probability level on the horizontal axis to the cumulative distributions produced by triangular or uniform distributions of component costs, one can estimate a total cost figure that is exceeded only one time in twenty. If contingency allowances are desired to cover cost overruns 95 percent of the time, and the cost estimate is the expected total cost ($406.7 million), the analysis suggests that a contingency fund of about $40 million will cover most exigencies. Here the higher variance of the risk profile of the simulation performed with component uniform distributions carries through to a higher estimate of the contingency fund to cover cost overruns 95 percent of the time.19 Thus, in determining how much extra debt one wishes to add for contin-

Risks in Construction

Figure 3.2 Comparison of Cumulative Distributions

gencies, one can see tangible evidence of the value of precision in probability estimation.

This simple exercise underlines an important and practical point. Covering total costs at a given confidence level does not require coverage of component costs at that same confidence level. Thus, a 95 percent confidence level for a uniform distribution would span 95 percent of the area between the lower and upper bound, beginning at the lower bound. Given that the range of the high variance risk profile in Figure 3.1 is just short of $200 million (between about $312.5 million and $500 million), a contingency fund of, in round numbers, $450 million is far less that $500 million minus 5 percent of $200 million, or about $490 million. The same point also follows by a simple argument if we assume that the


component cost distributions are normal (bell-shaped) as well as stochastically independent,20 and this point is true for many characterizations of component costs and their variability. Another way of understanding this is in terms of risk pooling. The chance that a series of independent random variables will exhibit large deviations in the same direction from their mean values is, intuitively, a low probability.

Correlations in Component Cost Distributions

Suppose there are correlations between cost categories that are not compelling enough to justify aggregation of the categories into a single classification but are strong enough to have implications for the resulting risk profile of total costs. In this case, one must ask how such correlations are established.

If such correlations follow from databases of costs on comparable projects, an analysis could run along the following lines. Compute an ordinary least squares (OLS) regression of a cost category on its related counterpart(s). Then, bootstrap an estimate of the probability distribution of the coefficient of this regression. If two costs are involved, this procedure runs as follows. Select a series of random subsamples from the observed residuals of the regression linking the dependent and explanatory cost categories. Combine this subsample of the residuals with the associated values of the dependent and independent variables and reestimate the regression coefficient linking these costs. Do this repeatedly and the result is that the various coefficients estimated will trace out a distribution from which confidence intervals for the coefficient can be established.21

Otherwise, one may suspect correlations, perhaps motivated by some type of engineering or economics argument, but be without hard data to back up this suspicion. In this case, one tactic is to look at the sensitivity of the risk profile to such interrelationships. If the risk profile is highly sensitive to the assumptions made about the correlation of some set of random variables, one needs to dig for more information or attempt to set these correlations at some intermediate value.


Simulation Analysis of the Length of the Construction Period

A similar analysis can be applied to scheduling problems. Construction activities are characterized by precedence relations— some activities must occur before others. Construction tasks necessarily occur in parallel or in sequence. Methods like CPM (critical path modeling) and PERT (program evaluation and review technique)22 accommodate such precedence requirements in generating estimates of completion times and information about the construction schedule. A key concept is the critical path, which is the longest chain of activities linked to each other by precedence relationships (i.e., task al must precede activity bl , bl must precede cl, and so on). In the PERT system, for example, three time estimates are obtained for each activity: an optimistic time, a most likely time, and a pessimistic time. A risk profile for the critical path is developed with these range estimates and can be linked with the time path of construction expenditures.23

Integrating Time and Costs

The issue of mutually correlated random variables reappears in connection with the question of interdependencies between cost overruns and scheduling delays.24 In some instances, stochastic independence can be preserved; that is, time and cost issues can be decoupled. Thus, interest during construction is related to the length of the construction period and can be independent of whether activities are brought in under or over cost. Nonetheless, interdependencies can exist. There are correlations between cost overruns and the construction period, for instance, since the site must be kept open and administration expenses generally continue. Delays in task completion also can be associated with overtime pay.

A mixed strategy usually accommodates the relation between time and cost. Direct linkages can be established, as between interest costs and the total construction period. Otherwise, correlation between major cost components and delays in the completion schedule can be explored by regression analysis, if data from comparable projects exist. Finally, interrelationships not sup-


ported by cost and scheduling data may be dealt with along the lines of a sensitivity analysis. If such correlations have significant impacts on the risk profile of total costs or completion time, care must be exercised in setting their precise value in the final simulation.25

The Disbursement Pattern

Cumulative expenditures over the construction period usually will be some type of S-shaped or sigmoid curve. Expenditures in the design period ordinarily will be less than initial startup costs, and payout will mount as the project goes on, probably peaking about midway through construction and then trailing off as detail tasks and finishing become the main preoccupation. Given the lag between project completion, billing, and payment, expenditure curves can be developed with the same information used for costs and completion times.

Mention also can be made of an explicit stochastic model of the payment stream, unrelated to range estimation. This derives interesting results from the assumption that the probability of the completion of any work element in any small interval within the construction period is a small number—an observation that may elicit empathy from construction project managers. Based on analogies with engineering reliability theory, payment is represented as a Poisson process, and the payment completion rate is an exponential function. The resulting probability distribution of payments is a mixture of uniform and Weibull distributions, which describes a kind of S curve.26

CONCLUSION

Text Box 3.1 lists several qualitative factors identified in the engineering and economics literature as being linked with cost escalation on capital construction projects. Evidence suggests these same factors are relevant to water and power capital construction projects. This specialized literature emphasizes, on the one hand, the dependability of conventional technology and, on the other, the hazards of fast-tracking technical innovation in the nuclear field. Accordingly, the factors in Text Box 3.1 might be taken as


Text Box 3.1

stage of the product cycle at which the cost estimate is developed

technology

project size

project complexity

competence of project management

regulatory or political considerations

contracting arrangements

volatility of economic variables such as prices and interest rates

the basis for a parametric ranking system that would appraise the relative likelihood of cost overruns on a series of projects.

Quantitative assessment of the likelihood of cost overruns and scheduling delay on power and water projects can be carried out with range estimation. Range estimation involves associating probabilities of excess with various cost or completion time estimates and identifying the expected or most likely values for component activities or tasks. With respect to costs, implementing this technique involves the following:

1. appropriate classification of costs (preserving stochastic independence) 2. identifying five and ninety-five percentile costs or absolute lower and

upper cost bounds


3. imputing probability distributions to component costs

4. simulating the risk profile of total costs

Prior to bond issuance, this procedure might be implemented with conceptual construction cost estimates; it can size contingency allowances; and, as suggested in Chapter 6, it might inform determination of optimal amortization schedule or sinking fund arrangements. Cost estimation software employing hundreds or thousands of cost categories exists, of course, and supports refinements of such early estimates of risks from construction cost overruns.

The simulation example presented in this chapter illustrates several important technical points, including (1) the operation of the Central Limit Theorem for sums of stochastically independent variables, and (2) economies from risk pooling in setting contingency allowances against total costs rather than component cost categories.

This simulation example may leave a question in readers' minds as to how skewed total cost distributions—implied, for example, by Table 3.1—are possible if sums of random variables tend to add up to normally distributed totals. While a definitive answer requires detailed examination of cases, a process that might be called amplification may be at issue. Costs, in other words, can be amplified through repeated redesign, due to regulatory or sponsor initiative, after construction begins. This essentially introduces multiplicative factors to the realization of total costs, insofar as existing work has to be removed and the same tasks done more than once. Together with positive correlations in costs that might accompany this frustrating and inefficient process, simulations show that cost amplification can produce markedly skewed total cost distributions.

Given the computer software now available, analysis of the risk profile of construction costs is increasingly accessible. Preliminary analysis, working with a few major cost categories, may provide substantial insight into the potential variability and financial risks of a project. Cost estimation software suitable for bottoms-up estimation of large power and water facilities can expand the number of cost categories manyfold. The same principles apply, however, and the added detail may support only incremental refinements to


a carefully conducted analysis with major cost categories and full design and site information.

NOTES

1. James A. Bent, Project Management for Engineering and Construction (Lilburn, GA: Fairmont Press, distributed by Prentice-Hall, 1989).

2. This expands on John W. Hackney, Control and Management of Capital Projects (New York: John Wiley & Sons, 1965), p. 17. Hackney groups the causes of cost growth under four headings pertaining to (1) changes in project scope or design, (2) problems in management and organization performance, (3) changes in economic and legal parameters, and (4) limitations and imperfections in estimating method. He draws on research and reviews in Edward W. Merrow, Stephen W. Chapel, and Christopher Worthing, A Review of Cost Estimation in New Technologies: Implications for Energy Process Plants, prepared for the U.S. Department of Energy by the Rand Corporation, R-2481-DOE, Washington, D.C., July 1979.

3. Earl J. Miller, "Project Information Systems and Controls," in Jack H. Willenbrock and H. Randolf Thomas (eds.), Planning, Engineering, and Construction of Electric Power Generation Facilities (New York: John Wiley & Sons, 1980), p. 308. Miller suggests that in typical electric power construction the official estimate is based on engineering that is 30 percent complete and preferably 40 percent to 60 percent complete.

4. James F. Tucker, Cost Estimation in Public Works, MBA thesis, University of California at Berkeley, September 1970.

5. The Japanese may be setting the standard today in their aggressive utilization of modern computing capabilities and extensive cost databases, as the following description in an article by Kurazo Yokoyama from ABI-Inform, a computer database, suggests,

Actual cost data on about 10.000 buildings constructed by 40-50 firms in Japan over the period 1975-1984 were analyzed. . . . The construction cost of a standard building design with the usual functions is calculated, which is called the median cost. Then, special feature factors are selected from the tables of various building conditions. The proposed method allows an easy calculation of the cost... necessary to simply imagine the building. . . . The workability and accuracy of the technique have been approved by applying it to over 270 cases in the past three years.

Annual Transactions of the American Association of Cost Engineers (AACE) (Morgantown, W.V.: AACE, 1988).

6. Terry F. Lenzer, The Management, Planning, and Construction of


the Trans-Alaska Pipeline System (Anchorage, AK: Alaska Pipeline Commission, August 1, 1977), Chapter II.

7. See Derek T. Beeston, Statistical Methods for Building Price Data (London: E. & F. N. Spon, 1983).

8. Edward G. Altourney, The Role of Uncertainties in the Economic Evaluation of Water Resources Projects (Stanford, CA: Institute of Engineering-Economic Systems, Stanford University, 1963).

9. Maynard M. Hufschmidt and Jacques Gerin, "Systematic Errors in Cost Estimates for Public Investment Projects," in Julius Margolis (ed.), The Analysis of Public Outputs (New York: Columbia University Press, 1970), pp. 267-315.

10. Robert H. Haveman, The Economic Performance of Public Investments: An Ex Post Evaluation of Water Resources Investments (Baltimore: Johns Hopkins Press, 1972).

11. Glenn J. Davidson, Thomas F. Sheehan, and Richard G. Patrick, "Construction Phase Responsibilities," in Willenbrock and Thomas (eds.), Planning, Engineering, and Construction of Electric Power Generation Facilities, p. 160.

12. D. S. Bauman, P. A. Morris, and T. R. Rice, An Analysis of Power Plant Construction Lead Times, Volume I: Analysis and Results, EA-2880, Final Report (Palo Alto, CA: EPRI, February 1983), p. 2-2.

13. Ibid. 14. Ibid., section 1, p. 4. 15. Ibid. 16. See, for example, Michael W. Curran, "Range Estimating: Rea

soning with Risk," Annual Transactions of the AACE (Morgantown, W.V.: AACE, 1988), n.3.1-n.3.9; R. W. Hayes, J. G. Perry, P. A. Thompson, and G. Willmer, Risk Management in Engineering Construction (Morgantown, W.V.; Thomas Telford Ltd., 1986); Karlos A. Artto, "Approaches in Construction Project Cost Risk," Annual Transactions of the AACE (Morgantown, W.V.: AACE, 1988), B-4, B.5.1-B.5.4; and Krishan S. Mathur, "Risk Analysis in Capital Cost Estimating," Cost Engineering 31 (August 1989): 9-16.

17. See Derek Beeston, "Combining Risks in Estimating," Construction Management and Economics 4 (1985): 75-79.

18. James E. Diekmann, Edward E. Sewestern, and Khalid Taher, Risk Management in Capital Projects, a report to the Construction Industry Institute (Austin: University of Texas at Austin, October 1988), pp. 63-81. The authors of this book note that some exceptions in the sources of construction cost variation covered by contingency funds often are allowed (e.g., out-of-scope variation in costs).

19. Here, the expected value of total costs is similar in both simulations, the primary difference being the variance of the implied risk profiles.


20. When the component cost distributions are normal or Gaussian as well as stochastically independent,

and

where E(.) (the period indicates the argument of the function E) is the expectation operator indicating the mean or average of a random variable and Var(.) indicates its variance. In ordinary language, these equations mean that the expected total cost equals the sum of the expected values of component costs and, similarly, that the expected variance of total costs equals the sum of the variances of the individual component cost probability distributions. Confidence levels of a normal distribution, however, are determined by the standard deviation, which is the square root of the variance. Thus, the standard deviation of total costs must be less than the sum of the standard deviations of the component cost distributions. Accordingly, a 95 percent confidence interval for total costs is less than the sum of the 95 percent confidence level values for component costs.

21. See Michael R. Veall, "Bootstrapping the Probability Distribution of Peak Electricity Demand," International Economic Review 28 (February 1987): 203-212, and the cited references for a particularly clear discussion of this method.

22. PERT (program evaluation and review technique) had a number of precursors. Its development was associated with the development of the Polaris missile system in 1958 and the earlier Gantt (bar) charts and milestone reporting systems. See Joseph J. Moder, Cecil R. Phillips, and Edward W. Davis, Project Management with CPM, PERT, and Precedence Diagramming, 3rd ed. (New York: Van Nostrand Reinhold Company, 1983).

23. One problem is that the critical path identified by a deterministic analysis may be supplanted by other, more lengthy chains of activities when task completion times are considered to be random variables. Until the advent of modern microcomputers, this was a real barrier because of the cost of core computer time to generate all possible combinations of completion times and their resulting critical paths. Currently, simulation is the favored approach to this problem, and algorithms for the efficient or near-optimal solution of this problem are available. See Thomas Byers and Paul Teicholz, "Risk Analysis of Resource Levelled Networks," in


Proceedings of the Conference on Current Practice in Cost Estimating and Cost Control, sponsored by the Construction Division of the American Society of Civil Engineers in Cooperation with the University of Texas at Austin (New York: American Society of Civil Engineers, 1983), pp. 178-186. Bruce E. Woodsworth notes some problems, in this regard, with standard PC software, which purports to take account of resource constraints. "Is Resource-Constrained Project Management Software Reliable?" Cost Engineering 31 (July 1989): 7-11.

24. Chandra S. Murthy notes that cost and schedule functions are rarely integrated in common construction projects. "Cost and Schedule Integration in Construction," in Proceedings of the Conference on Current Practice in Cost Estimating and Cost Control, pp. 119-129.

25. Note that the most widely studied joint probability distribution allowing mutual correlation of variables, in this regard, is the bivariate normal distribution. With some care, however, uniform and triangular distributions can be adapted to a two or many variable context in which there are mutual relationships between the constituent random variables. Thus, the Morgenstern distribution allows uniform marginal distributions, as might be implied by range estimates without information on the mode or central tendency, and correlation coefficients ranging between - 1/3 to -I-1/3. A good reference here is Mark E. Johnson, Multivariate Statistical Simulation (New York: John Wiley & Sons, 1987).

26. See S. N. Tucker, "Formulating Construction Cash Flow Curves Using a Reliability Theory Analogy," Construction Management and Economics 4 (1986): 179-188.

4

Revenue Risk—Rate and Demand Factors

Attitudes toward rate responsiveness in the power and water industries have shifted since the mid-1970s. Initially, it was not uncommon to meet utility professionals who doubted whether consumers paid attention to the price of electricity or water. One might hear (and there is modest published literature to the effect) that rate increases initially impact demand but that, after a year or so, consumers forget about changes in rates and return to their old patterns of usage. Of course, those advancing this thesis seldom distinguish between real and nominal rates. They may have observed inflation reducing the real or inflation-adjusted rate after a time to its prerate-change level, or, alternatively, they may have witnessed differences between short- and long-run price responses. The statistical evidence for price effects on water or electricity demand, however, is overwhelming.1 By the late 1970s, there was greater awareness among utility staff about the implications of price elasticity on per capita usage levels. The message may not have reached echelons at which key decisions were made until later, however. Thus, as noted in Chapter 1, demand projections and capacity decisions in the electric power industry were out of synch with the realities of lower usage rates and demand growth in the face of higher, real rates until recently. Even today, despite widespread discussion of system optimization, conservation, and econometric modeling of demand, bond covenants often require utilities to pledge to increase rates in the event the debt coverage


ratio falls below a certain level. This, of course, assumes higher rates lead to higher revenues, which, demonstrably, is not always the case.

Financial risk, therefore, can be associated with rate responses in a fairly catastrophic way. If the inherent variability of demand due to weather or other stochastic factors is ignored, or if analysts fail to capture long-term trends related to income growth or change in housing types, problems can develop. A real disaster potential, however, is that adverse cost conditions will push a utility to raise rates high enough to enter the price elastic region of demand. Thus, typically, residential demand and system demand for water and electric power are price inelastic in the year or so after a price or rate increase (see the following discussion for an explication of the concept of price elasticity). This means rate increases produce an increase in expenditures on the part of consumers, although, as the Law of Demand suggests, the consumption level or total quantity consumed will decrease somewhat. Generally speaking, demand becomes price elastic at high enough rate or price levels, and the opposite effect comes into play—price increases reduce the consumer's total expenditure as well as the quantity consumed. Some claim this perverse relationship between rate levels and consumer expenditure—sometimes called rate shock—can be seen at work in certain power systems attempting to finance nuclear plant cost overruns.

These issues are explored with simple examples in the following discussion. Suggestions are offered about how to capture deterministic rate effects, as well as the overall band of variation, characterizing power or water demands of utility customers. Conditioned on physical facts about households and commercial establishments, electricity and water demand patterns change relatively slowly, although use levels respond immediately to price or rate changes. Structural and time series models based on historical data, therefore, provide a guide to prospective patterns of demand variability for purposes of analyzing revenue risk, if allowance is made for price effects.

Remarks on the general concept of demand, as used in economic and other discussions, follow. Then, two rules are presented illustrating the price elasticity of demand and its impact on revenues and the limits of revenue enhancement. After provisos concerning

Rate and Demand Factors 71

the propositions, the discussion turns to several ways in which demand uncertainty can be represented in simulation models.

WHAT IS DEMAND?

Once, in the early days of the television program "Saturday Night Live," the character Father Guido Sarducci exhorted viewers to sign up for instruction at the Five Minute University. This required a minimum time commitment. The economics course consisted of two words. Anytime one was asked something about economics, one would be taught to say "supply and demand."

Of course, "demand" has a range of meanings. Demand may be presented as an achieved fact, as in the statement, "after crude oil prices quadrupled in 1974, growth in gasoline demand flattened out." More abstractly, demand is a relationship describing the amount people are willing to purchase at various prices, given other factors like income and tastes. This produces the demand curve, a downward-sloping function in price-commodity space. Its negative slope illustrates what has been called the Law of Demand. Apart from exceptional cases, that is, people tend to buy less of a good when its price increases, other things being equal.2

Readers accustomed to a probabilistic framework may question how real these textbook descriptions of consumer responses actually are. The answer is that the operation of the Law of Demand in electric power and urban water markets has been confirmed in numerous econometric studies. At the same time, there are conceptual and statistical issues pertaining to measures of consumer price or rate responses. Thus, strictly speaking, "rates" are not "prices," and there can be questions about exactly what aspects of the rate schedule influence consumer responses. In addition, individual demand patterns are subject to considerable statistical noise. Many persons are surprised to discover that, generally speaking, structural demand models or equations including price, income, family size, and other explanatory variables rarely explain more than 20 to 30 percent of the variation in individual usage.

Structural variables influencing demand are numerous and differ for households and businesses. For residential customers, family size, family income, the number of electrical or water-using appliances, and their efficiency ratings are relevant. For commerce


and industry, market factors and the current state of technology are significant. Furthermore, conservation plays a growing role in water and electricity demand studies, as utilities disseminate information about resource-efficient appliances; adopt rate forms, such as increasing block rate schedules, which encourage care in usage; and as various ordinances governing plumbing fixtures or appliance efficiency are implemented. Finally, there is the matter of the temporal distribution of demand through daily, weekly, seasonal, and other cycles.

Having said this, it must be acknowledged that consumer responses to power and water rates are important because rate setting is one of the major ways utilities implement policy objectives, such as increasing revenues. The following discussion shows, in this regard, that even small rate effects can imply significant revenue impacts.

PARABLE #1—REVENUE EFFECTS OF CONSUMER PRICE RESPONSES

Let us propound and illustrate the following rule:

Ignoring the price responsiveness of consumer demand contributes to error in the revenue forecast, which is at least equal to the percentage change in prices or rates multiplied by the price elasticity.

The price elasticity is defined as the percentage change in quantity demanded divided by the percentage change in price, where these percentages usually are taken around the average values of the variables in the sample.3 Such elasticities are estimated from empirical data, usually in connection with a statistical demand analysis. This could involve identifying explanatory variables influencing consumption (Q), collecting cross-sectional data, and estimating coefficients of a structural model, such as

Q = a„ + a,P + a2X, + .. . + anXn (4.1)

where P is the real or inflation-adjusted price of the good and the Xj are other explanatory variables. The price elasticity of demand in the simple linear model of equation 4.1 is a,(P/Q).


Table 4.1 Error in Revenue Forecast from Neglecting Consumer Price Responses (33 percent price increase)

Current Demand

Projection at Current

Price

Zero Price Elasticity Revenue Forecast,

Higher Price

Revenue Forecast,

Higher Price [Assuming

1 e - 0.3)

Unit Price

($)

1.50

2.00

2.00

Quantity Demanded

(million)

18.5

18.5

16.67

FORECAST ERROR

Revenue Forecast

($millions)

27.75

37

33.3

10%

e * price elasticity of demand, or the percentage change in quantity demanded as a result of a given percent change in the price of a good

Short-run residential-commercial price elasticities for electricity or urban water typically are on the order of .3 to .75 in absolute magnitude. This is not especially price responsive, but large rate changes, contemplated because of cost overruns or reductions in the growth of the service area, can lead to significant revenue effects. Thus, if a rate increase necessary to balance costs and revenues is approximately 50 percent, reductions in systemwide electricity or water demand of 15 percent can result in the short run. This translates directly to the bottom line in the manner shown in Table 4.1. Here, the initial price is $1.50 per unit, where we assume for simplicity that this commodity is sold at a uniform price,


rather than a schedule of rates. At that price, consumption of 18.5 million units produces revenues of $27.75 million. A revenue forecast is prepared for a price increase of $2.00 per unit (33 percent increase) on the assumption that demand is unaffected by the price change (zero price elasticity, perfectly inelastic). If the short-run price elasticity is - 0 . 3 , consumption will decrease to 16.67 million units after the price change (18.5[1 - (.33)(.3)]). The error in the revenue forecast, therefore, is $3.66 million or equals the price elasticity multiplied into the percentage change in price (i.e., 9.9 percent).

Thus, even with moderate price elasticities and price increases, ignoring consumer price responses can produce 10 percent errors in the revenue forecast. These errors cumulate over time and may increase for another reason. Producers and consumers will have added incentive to make investments in energy or water efficient equipment, appliances, or appurtenances. Thus, rate effects estimated with a short-term price elasticity provide a lower bound to the actual effects on consumption and revenues that are likely to be encountered. Rate effects, therefore, ought to be taken into account in financial risk analysis of water and power systems.

PARABLE #2—LIMITS TO REVENUE ENHANCEMENT THROUGH RATE INCREASES

A second rule or proposition can be advanced here also:

There are limits to revenue enhancement through rate increases.

Assertions about price elasticities are similar to statements found in applied physics or other sciences, where differences exist between general statements and those with working usefulness. Thus, people often are taught to associate the size of a price elasticity with whether a good is a luxury or a necessity. Luxuries tend to be price elastic (e > 1), and, interestingly, expenditures on them will fall as the price goes up. Necessities, like food, on the other hand, tend to be price inelastic (e < 1). When their price increases, the quantity demanded will drop, but expenditures will continue to increase.4 This seems logical, but there is a catch. Price elastic-


ities vary along the demand curve.5 At low prices, demand is typically price inelastic, whereas at high prices, it becomes elastic.

This is relevant to real-world planning because a utility often pledges in a rate covenant to increase rates to cover a shortfall in revenues—an oversight, in all likelihood, linked with traditionally low prices for electricity and water. Without information about price responsiveness, rate analysts tend to look at the ratio of needed revenues to existing revenues. Thus, if commodity charges produce revenues of $52,500,000, and there is a revenue shortfall of $20,000,000, a 40 percent rate increase might be judged adequate to boost revenues back up to the breakeven point.

Suppose, however, the relevant demand relationship is linear in the price of the commodity in question, according to the formula

Q = 25,000,000 - 2,500,000 P (4.2)

so that at consumption of 17,500 units, the price elasticity is between 0.2 and 0.3. Here, total demand (Q) is measured in 1,000 units, and the initial price is assumed to be $3.00 per 1,000 units. Other influences are assumed to be constant for the duration of the analysis and are reflected in the constant term.

The fact is that there is no price increase capable of generating revenues of $75 million, given these parameters. Multiplying both sides of equation 4.2 by P, the following results:

PQ = 25,000,000P - 2,500,000P2 (4.3)

or

75,000,000 = 25,000,000P* - 2,500,000P*2 (4.4)

since we are interested in attaining a target revenue of $75 million. Applying the quadratic formula leads to solutions that include the square root of - 1 or to imaginary prices—a sign that something is definitely amiss.

The basic reason can be seen in the revenue curve implied by this demand function, presented in Figure 4.1. This is a parabola with the maximum revenue occurring at a price of $5.00 per unit, at which rate level revenues total a little more than $60 million.


Figure 4.1 Total Revenue Curve

Normally, a utility operates within the inelastic region of the electricity or water demand curve in the short run of one to three years. Price responsiveness is low in percentage terms and consumer expenditures increase with higher rates. Sufficiently large rate increases, however, can move consumption into the price elastic region of demand. Then, consumers buy less, and their total outlay is reduced. At the same time, the size rate increase necessary to bring this situation to pass can cause public resentment.


PROVISOS

These relationships must be qualified with respect to several factors. The following relationships bear mention because they have been intensively researched in recent years.

Rate Versus Price

The primary problem of rate versus price is that it is unclear exactly what people respond to when faced with a schedule of rates as opposed to a uniform price for a commodity. A typical rate schedule has a fixed fee or charge for connection to the system plus rates applying to various quantities of consumption. Electric power rates also might include a demand charge for consumption during some defined peak demand period. Water rates usually are described by a schedule applying to various quantity intervals— that is, purchases up to and including 10,000 gallons in a billing period are charged at one rate, while consumption in excess of 10,000 gallons in a billing period is charged at a second rate, and so on.

What's the price? Economists tend to identify the marginal price as the critical decision criterion—the commodity charge for the final unit consumed in a billing period. In applied studies and general discussion, the average price of water or power is often cited as important. Yet surveys show only a small percent of water or power customers know the rate schedule,6 and their imputations of average price may be wide of the mark, at least as this quantity is computed from the actual bill. Indeed, the only certain thing is that consumers, at some point, look at their utility bill. It seems reasonable, therefore, that they may react when bills rise above a certain threshold, making discretionary adjustments in usage in the short run and, if high bills continue, contemplating purchase of efficient appliances, new landscaping, and so on in the longer run. This type of behavior may look like a price response, but actually it does not have to be mediated by knowledge of rates.7

This issue is not trivial in an era of rate reform. Many power and water companies are switching from declining rate block schedules to ascending or increasing rate block schedules. This conversion is considered to offer incentives to conserve, since if use


increases, the bill goes up more rapidly. Until the problem of deciding which aspect of the rate schedule elicits action is solved, it does not seem possible to fine tune predictions of how rate restructuring will affect usage. Our fallback position in conceptual discussions is the standard uniform price model; empirical work, on the other hand, usually employs some type of average price or, if most household purchases are contained within a broad rate interval in a season, the marginal rate applying to most households. To the extent that these categories or price variables move together, as in the usual rate increase that multiplies rates and charges by the same factor (e.g., a 10 percent rate increase), there is no problem. As soon as there is some claim to be able to discriminate the effects of rate restructuring, however, these conceptual and statistical matters become controlling.

Statistical Estimation Issues

Along with conceptual problems, there are issues concerning how regression or curve-fitting can get around the facts that the applicable rates are determined by the quantity used and that the quantity used is influenced by the rates. This is called simultaneity in the technical language of econometrics. Rate effects, of course, are part of the deterministic portion of time series on electric power and water purchases over time. To assess them, comparisons between similar customers facing different rate levels must be made, either across geographically distinct customer service areas or through some period of time in which rate levels for a particular service area have varied appreciably. The topic has been extensively treated in the literature, and a consensus appears to be evolving concerning an approximation technique.8

Conservation Effects

Over long periods of time, price or rate increases provide incentives to substitute more resource-efficient equipment for the current stock. Thus, long-term price elasticities are higher than short-term price elasticities. Indeed, studies dating from the 1970s suggest that long-run electricity demand for residential and com-


mercial users can be price elastic, a finding that is controversial in the urban water demand field.9

What does seem clear is that merely having the technical opportunity to conserve is a less potent influence on demand without price or rate incentives. Often, public discussion of more efficient air conditioners or washing machines is in terms of end use savings predicted on the basis of engineering criteria. A household runs an air conditioner an average number of hours in a summer day. Hence, the logic goes, the electricity saved should be predictable from technical characteristics. Econometric studies have detected a flaw in this argument, however. If the air conditioner is more energy efficient, the marginal cost of using it during peak demand periods may be lower. Some people, recognizing this, will increase its hours of operation. This is sometimes called the rebound effect, and has been documented also with respect to the installation of insulation and thermostat settings.10 Thus, to trigger the full savings potential of resource-efficient appliances, price or rate increases may be necessary.11

ACCOUNTING FOR UNCERTAINTY

Planning studies often try to cope with uncertainty about future electric power or water demand by presenting high, medium, and low forecasts. These ranges might be taken as a starting point for developing a stochastic model of demand suitable for risk simulation, except that blind application of such ranges ignores responses of consumers to rate effects that may become necessary to support coverage ratios.

There are basically two approaches to this problem. One approach is to use historic data and develop multivariate time series models of demand, thereby capturing deterministic components such as price effects and stochastic elements such as autocorrelation between usage levels over time. Many electric power utilities and some water utilities have developed econometric demand models, often disaggregated by customer class and primarily for forecasting applications. Some recallibration of these models may be required, where attention should be given to the "error term," so that the structure of random components in demand, which define the band of variation of demand over time, is adequately represented.


The other approach relies more on judgment and, possibly, demand studies produced for comparable communities. A simple, often defensible form of a synthetic demand relationship is

qt - ERt + pqt_, + et = Q, - C(t) (4.5)

where qt is the deviation in per capita demand (Qt) from some trend C(t) in period t, E is the price coefficient operating against the rate level Rt, E < 0, p is the autocorrelation coefficient against the previous period's per capita deviation in usage from trend qt_,, and et is a white noise process. Then, total utility system demand equals population in year t multiplied by Qt. The price coefficient E in this equation can be based on empirical studies from the utility service area or from documentary sources. Statistical research also should establish a reasonable value for the autocorrelation coefficient (which, conceivably, may be zero) and the variance of the random error term et (which may be a more complex moving average). It can be noted, in general, that first order autocorrelation is a component of many behavioral time series, perhaps the major component.12 The trend C(t) can incorporate judgmental factors, such as how conservation is anticipated to affect the trend in per capita usage. Since community income changes gradually, its influence on demand also can be summarized in the trending specification. Weather, on the other hand, exhibits considerable short- and long-run variability, within the context of general seasonal patterns, contributing to the random shock component e. The interrelation between the level of per capita demand over adjacent and nearby periods represented in the autocorrelation coefficient p can be related to the business or weather cycles.

The simplest approach of all, of course, is to effect a determinate, nonstochastic adjustment in customer usage levels according to a price elasticity judged to be representative of the service area. This crude method is perhaps superior to assuming that demand patterns will be completely inflexible to changes in rate levels.

CONCLUSION

Demand patterns of individual customers contribute to revenue risks in at least two respects. First, underestimates of the price


responsiveness of demand translate directly into similarly sized errors in revenues. A point estimate of such errors in the revenue estimate is produced by the product of the percentage change in prices or rates and the price elasticity. Second, there are limits to revenue enhancement through rate or price increases because extreme rate increases may drive demand into the price elastic region.

These processes are subject to provisos relating to how people actually respond to the schedule of rates typically governing sales of water and power, how these rate effects may be captured in statistical terms, and how long-term price impacts can be sorted out. An analysis is on safest grounds when all the possibly relevant rate factors—the marginal price, the bill, and the average price— move in concert. This occurs with rate increases in which all rates and charges increase proportionally.

Although after any particular price increase, stochastic factors can wash out the effect of consumer price responses, over the long run and statistically, price effects have an impact on revenue and should be included in the analysis of financial risks. Furthermore, diminishing supplies of energy in the form of coal or natural gas, ever more scarce landior reservoirs and catchment areas, and new water requirements may lead to higher relative prices for electricity and water. Hence, risks of perverse total expenditure or revenue effects from rate increases are likely to increase over coming decades.

NOTES

1. Earlier statistical studies of water price effects are reviewed in John J. Boland, Benedykt Dziegielewski, Duane D. Baumann, and Eva M. Optiz, Influence of Price and Rate Structures on Municipal and Industrial Water Use, Contract Report 84-C-2, Engineer Institute for Water Resources, U.S. Army Corps of Engineers (Washington, D.C.: U.S. Government Printing Office, June 1984). J. Kindler and C. S. Russell, have useful discussions of industrial and agricultural water demands, as well as modeling methodology. See their Modeling Water Demands (Orlando, FL: Academic Press, 1984). An early but still pertinent review in the electric power demand field is Lestor Taylor, v'The Demand for Electricity: A Survey," Bell Journal of Economics 6, no. 1 (1975): 74-110. For a more recent synopsis, see the University of California at Berkeley, Price Elasticity Variation: An Engineering Economic Approach, EM-5038, Final


Report (Berkeley, CA: Electric Power Research Institute, February 1987).

2. Matters are complicated by the practice of selling water and electric power according to a schedule of rates, but a similar generalization applies. See C. Vaughan Jones, "Nonlinear Pricing and the Law of Demand," Economics Letters 23 (1987): 125-128.

3. Another definition of the price elasticity of demand uses the differential calculus:

This is simply the limit, in the mathematical sense, of the arc elasticity defined in the text as the percentage change in the price becomes smaller and smaller.

4. This follows in a straightforward way from the definition of price elasticity. Thus, if demand over some interval (ql, q2) is price inelastic, this means that

{(ql -q2)/(ql +q2)}/{(pl -p2)/(pl +p2)} > - 1

and

{(ql -q2)/(ql + q2)}/{(pl -p2)/(pl + p2)} < 0

It readily follows from the first expression that

(ql-q2)(pl+p2) < (pi-p2)(ql+q2)

or that

qlp2 < q2pl

However, since, by assumption, p2 > pi ,

qlpl < qlp2 < q2pl < q2p2

where the first and last of these expressions are the initial and final expenditures, respectively.

5. The exception is a logarithmic demand curve. 6. Thus, despite a recall initiative triggered by rate restructuring in

the 1970s, only 21 percent of several hundred persons responding to a question about the type of rate schedule applied for water sales in Tucson knew that this schedule was increasing block rates. See Donald E. Agthe, R. Bruce Billings, and Judith M. Dworkin, "Effects of Rate Structure Knowledge on Household Water Use," Water Resources Bulletin 24 (June 1988): 627-630.


7. Along these lines, it is interesting that research seems to indicate the viability of a hybrid price hypothesis, that is, responses that fall between marginal and average price responses. See, for example, David L. Chicione and Ganapathi Ramamurthy, "Evidence on the Specification of Price in the Study of Domestic Water Demand," Land Economics 62 (February 1986): 26-32; and Michael L. Nieswiadomy and David J. Molina, The Perception of Price in Residential Water Demand Models Under Decreasing and Increasing Block Rates, paper for the 64th Annual Western Economics Association International Conference, June 21, 1989.

8. This is the technique of instrumental estimation. A good exposition is presented in Steven E. Henson, "Electricity Demand Estimates Under Increasing Block Rates," Southern Economic Journal 51 (July 1984): 147-156. See also C. Vaughan Jones and John R. Morris, "Instrumental Price Estimates and Residential Water Demand," Water Resources Research 20 (February 1984): 197-202.

9. Kent Anderson computes the long-run residential electricity price elasticity to be - 1 . 2 . Residential Demand for Electricity: Econometric Estimates for California and the United States, R-905 NSF (Santa Monica, CA: Rand Corporation, January 1972). D. Chapman, T. Tyrell, and T. Mount report a commercial long-run electricity demand price elasticity of - 1 . 5 . "Electricity Demand Growth and the Energy Crisis," Science 178 (November 1972): 703-708. Philip H. Carver and John J. Boland find long-run price elasticities to be higher than short-run price elasticities but still in the inelastic range. "Short and Long-Run Effects of Price on Municipal Water Use," Water Resources Research 16 (August 1980): 609-616.

10. See, for example, Jeffrey A. Dubin, Allen K. Miedema, and Ram V. Chandran, "Price Effects of Energy-Efficient Technologies: A Study of Residential Demand for Heating and Cooling," Rand Journal of Economics 17 (August 1986): 310-325.

11. Raymond S. Hartman and Michael J. Doane, "The Estimation of the Effects of Utility-Sponsored Conservation Programmes," Applied Economics 18 (1986): 1-25; and Raymond S. Hartman, "Self Selection Bias in the Evaluation of Voluntary Energy Conservation Programmes," Review of Economics and Statistics 70 (August 1988): 448-458. These two articles argue that, in many instances, the effects of nonprice conservation measures have been overestimated.

12. See John M. Gottman, Time Series Analysis: A Comprehensive Introduction for Social Scientists (New York: Cambridge University Press, 1981).


5

Revenue Risk— The Customer Base

Financial risks are associated with population in a utility service area. If population growth is slower than anticipated, the costs of capacity investments will be paid by fewer customers. This can require rate adjustments and can be politically volatile. In some instance, efforts to compensate for revenue shortfalls resulting from slow growth may push rates into the price elastic range or trigger customer protest and resistance.

Attention to this type of risk is recommended by changes in fertility and death rates, family formation, and the age structure of the U.S. population. After World War II, there was an increase in family formation, and the U.S. birth rate surged. Five decades later, the demographic picture looks very different. Many central locations in U.S. metropolitan areas are losing population. Sudden reversals have affected whole regions, such as portions of the west around the Rocky Mountains. At least one instance of an electric utility being pushed into receivership by loss of energy investment projects and subsequent depopulation can be cited. Depopulation also has been discussed as a problem with respect to financing multibillion-dollar rehabilitation of older utility systems in core urban areas.

A risk that is possibly more potent, because it is more subtle, is gradually slowing population growth. Studies of the accuracy or errors of forecasting models suggest that turning points—the timing of a switch from positive to negative growth of a variable—are the


hardest item to predict. Thus, how slower growth will play itself out presents a formidable problem, creating substantial risks for central facilities promising economies of scale at the cost of excess capacity in the near term. Yet gradually slowing population is the basic forecast for the forseeable future and is related to a number of factors. The current low U.S. birth rate developed in the 1970s. At the same time, advances in modern medicine and lifestyle changes have led to longer lives for nonminority populations. One consequence is that the U.S. population is aging. An older population, in turn, is more likely to be settled, to migrate less. Thus, not only is population growth for the nation anticipated to slow in the next century, possibly diminishing in absolute numbers, but sizeable migration, which has buoyed population growth in many areas over past decades, cannot be counted on in the 1990s or the first decade of the twenty-first century.

This chapter confronts these problems with information that may help in the evaluation of demographic projections and proposes ways to represent uncertainty and inevitable variability in population time series. The following section considers evidence relating to the accuracy of economic and personnel projections that, by general consensus, drive population growth in a community or region. There is agreement that population forecasts are not very accurate, especially when longer time periods and smaller geographic or population units must be considered. Then, a synopsis of standard population projection techniques commonly applied by agencies supplying utilities with forecasts is presented. The discussion thereafter advances suggestions regarding stochastic modeling of population change for purposes of risk simulation. These methods run the gambit from extremely sophisticated multivariate time series models to simple stochastic models that capture subjective estimates of high, medium, and low growth.

THE ACCURACY OF ECONOMIC AND POPULATION FORECASTS

The single most important factor in population growth probably is the economic situation. Employment opportunities operate as a magnet for new migration, supporting the rate of family formation and the natural increase of the population. Lacking jobs, commu-

The Customer Base 87

nities and whole regions (such as Wyoming, Idaho, Montana, and the Dakotas) experience significant depopulation. Unfortunately, there is no golden road to forecasting the size of the workforce, despite the availability of mathematically complex econometric forecasting models.

Most planners and engineers are aware of a range of private and public sector forecasting models offering some guide as to what course the economy, regions of the nation, and various industrial sectors will take over the short, medium, and long term. Data Resources, a Cambridge, Massachusetts, firm founded by the late Otto Eckstein, is preeminent among the private services that typically allow on-line access to forecasts and associated databases on an annual subscription basis. The Bureau of Economic Research OBERS model is one of the main public long-range economic forecasting models. It utilizes a "step-down" method in which national economic growth is forecasted based on manpower projections from the Bureau of Labor Statistics and other data. National projections are then allocated, on some basis or other, to individual states and to industrial groupings.

Since 1968, records have been kept by the American Statistical Association and the National Bureau of Economic Research on the accuracy of about fifty forecasting operations. In reviewing these, William Ascher notes that macroeconomic models produced errors "about a quarter of the magnitude of change for nominal quarterly forecasts, and about an eighth of the change for nominal annual forecasts," while for real GNP (gross national product) changes "the annual errors are about a third as great as the annual changes."1 Ascher, furthermore, presents evidence that forecasting accuracy has not improved appreciably since development of the first multiequation econometric models in the 1950s.

Persistent and large errors in economic forecasting models were recognized during the adjustment to the energy crisis of the early 1970s. One reaction was examination and comparison of alternative forecasting methods. Among the most comprehensive of these studies was led by Spyros Makridakis, who concludes that no matter what forecasting technique is applied c*nd regardless of its pe-riodization (monthly, quarterly, or annual), forecast errors as measured by the mean absolute percentage rise to 20 to 30 percent after about five forecast periods.2


Since jobs and population are linked, population forecasts evince similar size errors. At the level of U.S. Census projections for the national economy, errors in the range of 20 to 30 percent in forecasted to actual growth rates can be documented for periods f\\c to thirty years after a projection is published.3 Errors increase for smaller population units and subregions of the nation.4

Population growth is not just a matter of job availability but has an inner dynamic depending on adjustments in fertility, mortality, and the age structure. Thus, the expanding economy and high rate of family formation led to high birth rates and an age cohort—the "Baby Boom"—that swelled college enrollment and real estate sales in the 1950s, 1960s, and early 1970s. Baby Boomers have controlled various markets and investments in the economy and society as they have matured. First, there was a surge in the demand for public schools, then colleges experienced extraordinary expansion in the 1960s. As the Baby Boomers hit the housing market in the late 1960s and 1970s, home and land costs escalated. Curiously, this generation was itself reluctant to raise children. For one thing, the recent entry of large numbers of women into the workforce has been associated with postponed childrearing and smaller families. Lower fertility rates, combined with a bulge in the middle-age population segment, and extended life expectancy for segments of the population with access to jobs and medical care result in a gradual drift upward of the median or average age of the population. The other major demographic force—migration—also is affected by a generally aging population. Migrants from one region to another in search of employment typically are younger, usually in the twenty-four to thirty-five age group, although there is also some movement of the sixty-and-over crowd to retirement residences, communities, and, eventually, facilities and places where they may receive full care.

The new population trends add up to a startling fact. In many areas of the nation and in many communities, the future will not be like the past. The U.S. Census Department, by the mid-1980s, began to forecast a leveling off of national population in the first decades of the next century followed, perhaps, by a period in which the total population will be absolutely reduced. In cities of the eastern seaboard and midwest, older, core urban areas already are experiencing population loss.


POPULATION FORECASTS—BASIC METHODS

Electric power and water utilities often rely on external organizations such as cities, counties, regional, and state authorities for population forecasts. These forecasts usually are based on techniques such as: (1) extrapolation, curve-fitting, and regression-based techniques; (2) ratio-based techniques; (3) land-use techniques; (4) economic-based techniques; and (5) cohort-component techniques. Each has strengths and weaknesses, discussion of which is helpful in establishing an idea of the confidence nonexperts should place in these forecasts.

Extrapolation techniques often are used for the short term and generally involve projection of total population. A common assumption is that past rates of change can be linearly extrapolated into the future. Typically, high, medium, and low forecasts are generated by varying an underlying annual growth rate. In many communities, however, simple extrapolation of patterns from the 1960s and 1970s will not produce a plausible account of population change in the 1990s or thereafter.

Ratio-based and land-use-based techniques are widely employed in projecting populations of component areas within a larger region for which total population projections have been obtained. Using alternative ratios, trends in ratios, density limits, and other similar factors, these methods can be applied in small jurisdictions in cities or counties. Their accuracy depends, in large part, on the accuracy of the projections for the larger area.

Economic-based procedures use projections of trends in economic factors, usually employment, to project population change over the projection period. There is a range of such models from those using simple population to employment ratios, such as the U.S. Bureau of Economic Analysis model, to those that balance employment supply and demand, based on separate economic and demographic models, and forecast migration. Some account of future economic conditions seems essential in developing any picture of future population growth, although in the final analysis these methods are no more accurate than the economic forecasts on which they are based.

Cohort-component procedures utilize the fact that populations change as a result of fertility, mortality, and migration. Because


rates for these components vary by age and sex, separate assumptions are developed about each age group. The most difficult component to project in subnational projections is migration,5 and most cohort-component procedures utilize either past migration rates or absolute numbers of migrants. The advantage of cohort-component procedures is their conceptual completeness relative to the demographic processes and population structure. Their disadvantages derive from the fact that only population factors are considered in the projection of future events. Most texts on demography acknowledge that cohort-component or cohort-survival methods are not more accurate than simpler extrapolations or judgmental estimates, although they have the potential to provide a better picture of how trends may play themselves out.

ACCOUNTING FOR UNCERTAINTY

There are several methods to account for uncertainty in population projections. These include use of sensitivity analysis and the construction of confidence intervals, either with empirical data derived from errors of past population forecasts or from stochastic models in which age-specific fertility and mortality are random variables.6 The choice of methods depends, in large part, on the assessment of how stable population change patterns are within a particular utility service area. Thus, time series models are appropriate when past variations of population parameters are anticipated to be similar to those in the future. When there are substantial shifts in age structure, fertility, and mortality, stochastic models incorporating judgmental factors may be the best way to assess the likely range of variation of future population. Stochastic models also are recommended for smaller population areas, such as communities, as opposed to states or the nation as a whole. Hence, a risk simulation method that accommodates current expectations of future population change without holding the future to past patterns may be advisable for utility service area population.

When high, medium, and low population forecasts are provided, a kind of range estimation is available to simulate the possible future population growth in an area. The range of population growth rates can be identified, and population trajectories can be generated depending on a sampling procedure from this estab-


Table 5.1 Typical Presentation of Population Forecasts in High, Medium, and Low Variants

Year

1995

2000

2005

2010

High

200,000

231,855

266,184

301,163

Medium

200,000

226,281

249,833

272,471

Low

200,000

210,202

220,924

232,193

lished range. As an illustration, Table 5.1 shows high, medium, and low population projections for a fifteen-year period for a hypothetical community whose size in 1995 is 200,000 persons. These numbers can be taken to imply different underlying growth rates in five-year periods. Thus, the increase in population for 1995 to 2000 in the high forecast presented in the first column of Table 5.1 can be produced with an annual growth rate of 3 percent. Let us assume that the high and low projections, therefore, define the bounds on expected population growth, much the way previous range estimates established bounds for other risk factors. Then, given these bounds, how do we impute a probability distribution to the implied range of growth rates for purposes of a risk simulation?

Since there seldom is any reason to believe one number rather than another, when these figures pertain to abstract events (e.g., growth) several years hence, the uniform probability distribution seems a natural choice. As noted in the previous chapter, this is a probability density function that assumes that all values between a given lower and upper bound are equally likely to occur.


Now a trick can be employed to create interdependency between population growth rates in adjacent time periods, as is characteristic of empirical growth rate series. Instead of developing a time series analysis to determine the coefficient of autocorrelation, say, this alternative approach starts with preassigned and absolute limits for the annual growth of population over a period. A second condition requires that population growth in a subsequent period be sampled from an interval around population growth in the previous period. Thus, referring to Table 5.1, this would mean that in year 1 of the simulation, a growth rate is randomly sampled from the bounds 1 percent to 3 percent per year. Depending on what value is selected for this first growth rate gl, the second growth rate g2 will be sampled from a permissible growth rate range subject to the condition that Igl - g2l < k, where, for example 0 < k < 1 percent. Similarly, each subsequent growth rate will be sampled according to a uniform distribution over the range indicated by the population forecasts in the first and third columns of Table 5.1, subject to the condition that the absolute value of its difference with the preceding growth rate is less than some number k. The effect of k on the simulation can be seen in Figure 5.1. The ''tightly-wound" series, indicated by the boxes in the diagram, has a smaller value of k (.25 percent) than the more variable series indicated by cross marks (k = 1 percent). Here, selection of the time interdependency parameter is partly a matter of judgment but can be informed by reference to past series of growth rates.7

Application of this method implicitly assumes there is some hom-eostatic process coming into play when growth nears its upper or lower bound. This is reasonable in respect of the fact that communities growing very rapidly experience various types of diseconomies that tend to discourage further increases in the growth rate. On the other hand, citizens usually mobilize to halt dramatic drops in population growth rate in their area.

Another note may be added with regard to disaggregation of the population growth model. Disaggregation beyond the level of total population may be advisable if migration plays a major role in population change in the utility service area. Population change in a geographic area is a result of natural increases through births, deaths, and the balance of immigration and emigration. It is widely recognized that some of these components tend to be more volatile

Figure 5.1 Stochastic Models of Population Growth


than others. Of these components, the death rate appears to be the most stable, reflecting long-term trends toward increasing life expectancy due to medical advance, except in certain inner-city and disadvantaged populations. Fertility rates fluctuate more—the Baby Boom of the immediate post-World War II years and the baby bust of the 1970s being notable examples.8 In relative and sometimes in absolute terms, however, migration rates tend to be the most volatile element of regional or local population change and are hardest to predict. Thus, the range of variation of migration can be large compared to the total population change contributed by natural increase—the net of births over deaths.9 In this case, separate stochastic models for natural increase in the population and migration may be desirable.

CONCLUSION

Water and power utilities face a formidable planning problem, insofar as they attempt to have new capacity ready in a timely fashion for population growth in a service area.

Risk simulation has special advantages in this context since, as noted earlier, the accuracy of population forecasts is low and decreases rapidly with the length of the forecast period. Given future prospects in many U.S. communities, there is need to consider financial models in which the customer base may peak and then shrink, models in which large components of pure uncertainty, stochastic variation, and jumps or shifts in behavior can be incorporated. Typically, these will allow for random variation and time interdependency in population growth rates or in both migration and natural increase of the population.

The modeling choices again revolve around the application of judgment or formal statistical analysis of historic data. The stochastic model suggested here is a variant of range estimation. Maximum ranges reasonable for population growth in a series of years are established first. Then, the analyst focuses on the time interdependency with which he or she feels comfortable for these stochastically generated series of population growth rates.

While this method leaves considerable discretion in the model setup, it is recommended in risk simulation because, as this chapter


has stated, the future is likely to be different than the past when it comes to American population patterns.

NOTES

1. William Ascher, Forecasting: An Appraisal for Policy-Makers and Planners (Baltimore: Johns Hopkins Press, 1978), p. 74.

2. Spyros Makridakis, "The Art and Science of Forecasting: An Assessment and Future Directions," International Journal of Forecasting (1986): 15-39.

3. Michael A. Stoto, "The Accuracy of Population Projections," Journal of the American Statistical Association 78 (March 1983): 13-20.

4. See Donald B. Pittenger, Projecting State and Local Populations (Cambridge, MA: Ballinger, 1976).

5. Errors in national economic forecasts are most closely related to variability in fertility, in part because of immigration barriers to migration. At the regional level, however, migration becomes the dominating influence in many cases.

6. Peter Pflaumer, "Confidence Intervals for Population Projections Based on Monte Carlo Methods," International Journal of Forecasting 4 (1988): 135-142.

7. See Juha M. Alho and Bruce D. Spencer, "Uncertain Population Forecasting," Journal of the American Statistical Association 80 (June 1985): 306-314.

8. Some researchers have proposed a generational argument based on the relative standard of living to account for the decline in fertility in recent decades. Fertility is said to be related to the difference between currently attainable standards of living and the standard of living of the family of origin. Given declining real incomes per worker since about 1970, younger families are by this account less prone to raise children. This explanation has some credibility in light of widely reported difficulties of young couples, brought up in the suburbs, in buying their first home. But does the theory explain the surge in births in inner city areas with ethnic populations?

9. This is particularly true in relatively sparsely populated areas of the west and the southwest. In Colorado, for example, annual net migration since 1950 has fluctuated between a loss of about 24,000 persons and a net gain of 90,000 persons while natural increase has had bounds varying between 18,000 and 35,000 people. See Colorado Division of Local Government, Colorado Population Growth (Denver: Colorado Division of Local Government, 1989), Table 1.


6

Applications

This chapter presents five applications of risk simulation to utility investment evaluation. The topics include

1. contract risk and the potential loss of a bulk customer,

2. the impact of sinking funds on financial risk,

3. risk comparisons of alternative debt service schedules,

4. risk profiles for the present value of capacity expansion plans, and

5. benefit-cost analysis and financial risks.

The discussion focuses on how simulation of multiple risk sources can be orchestrated, integrating material from the preceding chapters. While the examples can be refined with respect to encoding probabilities or representing components like population growth or consumer demand, the detail developed illustrates the gains in insight supported by this type of analysis.

First, we consider a simple binary risk relating to renewal of a contract for bulk electric power. This allows us to present a cash flow model and outline steps in developing a relatively simple probability estimate of financial risk. The second example focuses on the advantages of sinking funds and requires a more dynamic or process-oriented analysis of default risk. A third and related application looks more intensively at level and tipped amortization arrangements, where construction cost overruns, rate shock, in-


flation of O&M costs, and variability in the growth of revenues are incorporated into the analysis. This is the most ambitious simulation model developed in these pages. Then, we turn to the topic of risk simulation in capacity planning, where the criterion is the minimization of the present value of costs. We consider the tradeoff between lower unit costs from a large facility having economies of scale and flexibility of response allowed by smaller, high unit cost capacity investments. Brief mention is made, in this regard, of stochastic aspects of benefit cost analysis. The chapter finishes with a summary of conclusions indicated by these applications.

Several applications incorporate techniques for simulating the overall variability and trend in the general price level applying to expenses. These inflation models appeal to time series analysis and the observation that periods of high inflation or deflation tend to occur in runs of several years duration.

POTENTIAL LOSS OF A BULK CUSTOMER

The following example, which is based on data from a recent bond prospectus, concerns a joint-ventured wholesale power company that supplies its product to several municipalities and another power company according to two different types of contracts. Currently, the cities who created this power company cannot absorb the output of its single, relatively new baseload plant. Nearly half the productive capacity of the wholesale producer is currently sold to another power company under a contract scheduled to expire in 1995. Table 6.1 displays the standard case cash flow projection associated with this situation, allowing for gradual increases in expenses and revenues. The critical year in which the bulk contract with the other power company must be extended, renegotiated, or lost is shaded by a vertical gray band.

If the contract for bulk power is not renewed after 1995, revenues will fall 45 percent. The rate covenant requires increases in the rates quoted to the municipalities to maintain the debt service coverage ratio. Analysis suggests that rates in mills per kilowatt hour (kWh) for municipal customers would have to increase by at least 80 percent to make up the shortfall in 1995. The question is whether this would push consumer responses into the price elastic

Table 6.1 Cash Flow Model for Wholesale Producer

($1000)

1990 1991 1992

OPERATING REVENUES Primary Service Area (retail cuttomert) 60.000 61.200 62.424 Bulk Contract 45.000 46.800 48.672

OPERATING EXPENSES Operation & Maintenance 35.000 37.100 39.328 Adminittration 4.500 4.770 5.056

INCOME Net Operating Income 65.500 66.130 66.714 Other Income (Inverted Fundt) 12.000 12.000 12.000

DEBT Debt Service on Bondt 62.000 62.000 62.000 Debt Service Coverage Ratio 1.25 1.26 1.27

DEMAND AND RATES Annual KWh (million kWh) 1.500 0 1.530.0 1.560.6 Millt per kWh (primary tervice area) 40 40 40 Bulk Contract Rate (Mlllt per KWh) 50.000 52.000 54.080

1993 1994 1995 1996 1997 1998 1999 2000

63.672 64.946 66.245 67.570 68.921 70.300 71.706 73.140 50.619 52.644 54.749 56.939 59.217 61.586 64.049 68.611

41.686 44.187 46,838 49.648 52.627 55.785 59.132 62.680 5.360 5.681 6.022 6.383 6.766 7.172 7.603 6.059

67.246 67.722 68,134 68.478 68.745 68.928 69.020 69.012 12.000 12.000 12.000 12.000 12.000 12.000 12.000 12.000

62.000 62.000 62.000 62.000 62.000 62,000 62.000 62.000 1.28 1.29 1.29 1.30 1.30 1.31 1.31 1.31

1.501.8 1.623.6 1.656.1 1.689.2 1.723.0 1.757.5 1.792.6 1.828.6

56.243 68.493 60.833 63.266 65.797 68.428 71.166 74.012 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0


demand region and, short of that, whether such a rate increase would trigger consumer protest.

Fortunately, the initial charges to the municipal buyers in mills per kWh are relatively low in this case. Because of this and the fact that wholesale power costs constitute only about 20 percent of total distribution costs to final users in these cities, chances are good that an 80 percent rate increase to these municipal distributors can be absorbed without dramatic effects (this estimate ignores, for the moment, the price elasticity of consumer responses). On these assumptions, something like a 20 percent rate increase to final customers could generate the needed revenues. Given current levels, this does not seem likely to elicit price elastic responses, and the total effect on per customer usage might be on the order of a few percent reduction. If rates increase, in short, revenues are likely to increase also.

Provisos, therefore, might relate to competing power sources available to the municipalities.1 There is the possibility that with an 80 percent jump in wholesale costs, some municipalities might consider alternative wholesale sources despite their historic relationship with this supplying organization. This possibility is compelling in this case, perhaps, because of substantial excess producing capacity in continguous electric power systems.

Imputing probabilities to these contingencies is straightforward. It involves encoding subjective assessments of key players and observers. The contract renewal occurs in a single time frame. Its successful negotiation is associated with a simple probability (e.g., 0.50), which may vary from period to period before the event, depending on negotiations, power demands, and so on. An event tree with suitable probabilities along its branches can neatly summarize the situation as viewed from a contemporaneous perspective.

EVALUATION OF DEBT SCHEDULES

If the preceding analysis pointed to rate shock as a real possibility, creation of some type of reserve, funded, possibly, by immediate rate increases, would be a logical step. Such reserve or sinking funds are a classic risk management tactic, and their rationale requires adoption of a perspective that acknowledges ex-

Applications 101

pectations of variability in revenues. Sinking funds generally operate by capturing extraordinary but uncertain revenues to make debt service payments at future dates. Their funding is more flexible than ordinary repayment of debt, although their creation may be mandatory.

Sinking funds can reduce default risk in various ways, depending on how they are structured. One tactic is to reduce near-term financial risk by lowering year-by-year debt service. This creates, of course, tradeoffs related to the chance that fund accumulations will be insufficient to meet a compensating end-of-period lump sum payment. Another tactic might be to skim accumulations in high income years and use the sinking fund to make up shortfalls in low income years. Interest earnings on sinking fund accumulations can be exploited by tax-free municipal issues in some instances. For these reasons, sinking funds may enhance the chance that debt can be paid off without default, where the relevant probabilities depend on parameters of (1) capital construction costs for a project, (2) low and high forecasts for the growth of revenues and expenses, (3) some grounds for estimating revenue and inflation variability, and (4) relevant capital market information. Text Box 6.1 summarizes key parameters.

Our problem is to conduct a "with and without" analysis of a sinking fund arrangement, estimating the relative impacts on financial risk associated with repaying a $100 million bond.

Without a sinking fund, suppose the bond is amortized over fifteen years with a level debt service. Assume that subjective anticipations are that revenue growth will vary between - .5 and + 3 percent per year. In addition to these overall bounds for revenue growth, it is assumed—perhaps on the basis of inspection of past records—that year-to-year variation in the revenue growth rate does not exceed.0025 or one quarter of one percent. Given this interdependency, a uniform probability distribution is allowed to govern selection of the annual revenue growth rates in the simulation.

Inflation is characterized by a simple time series model. This first determines the overall trend or expected average increase in the level of prices—here assumed to be 5 percent per year. Then, variation around the trend is introduced with a serially correlated

Construction Cost Estimate $100,000,000

Payback Period 15 years

Interest Rate 10 percent

Debt Coverage Ratio 1.3

Initial Revenue $18,500,000

Revenue Growth low: -1.0 percent per annum high: 3.0 percent per annum

Inflation low: 3.0 percent per annum high: 8.0 percent per annum

Initial Operating Expense $1,000,000

Interdependency Factor for Revenues: established by assuming variation is within a 0.25 band around the previous year's growth rate

T e x r b o x 6 . 1

Applications 103

residual. This leads to a rate of inflation falling between about 3 and 8 percent per year, applying to production, distribution, and administration expenses. The autocorrelation coefficient of - 0 . 5 corresponds to the perception that inflation and deflation tend to occur in runs of several years duration.

Net income is calculated as revenues minus expenses, and the debt coverage ratio is net income divided by the debt service.

The object is to compare the default risk in the case with no sinking fund with risks likely to be realized when a sinking fund is in place. This, of course, requires two simulations. Assume, then, a second situation in which $90 million of the $100 million bond debt is amortized according to a level debt service. A sinking fund is set up to pay off the remaining obligation, which, after 15 years at 10 percent interest, amounts to an additional $41,772,482. The sinking fund is allowed to skim off up to $2 million a year, whenever the debt coverage ratio exceeds 1.4.

There is a remarkable difference in the default risk in these two cases. The results are graphed in Figure 6.1, which displays the risk profiles, or, more precisely, fifteen risk profiles from the no sinking fund case—the bars with sloping slashes—and fifteen risk profiles from the sinking fund case—the solid bars in the diagram. Each bar in this graph indicates the probability of default in a particular year.

Always positive inflation of expenses as well as the potential of zero or negative revenue growth appears to create a risk equilibrium after the first four or five years of the payback period in the no sinking fund case when there is relatively constant default risk. With a sinking fund in force, the annual debt service is lower, but there is a kind of "balloon" payment that must be made out of the sinking fund. Accordingly, the risk profiles for this case indicate lower default risk in the early years of the payback period and markedly higher risk of default in the final year of the payback period.

What is best? Overall, the probability that this project is in default one or more years in the payback period cannot be read from the chart because the debt coverage may be inadequate in multiple years in a realization of the variables in the cash flow model. A second criterion variable must be defined to answer this question, a variable that is 0 if the coverage ratio is met every year


Figure 6.1 Default with and without a Sinking Fund

of the payback period and 1 otherwise. This variable indicates that the overall probability of default is higher without the sinking fund, where the financial risk in both situations is exaggerated for purposes of illustration.

We do not include rate adjustments to improve debt coverage levels in this example. One way this might be justified is by assuming that the analysis is carried out in terms of maximum revenues. In other words, year-by-year demand levels could be set at price levels determined to fall at the dividing line between inelastic

Applications 105

and elastic price responses. By definition, then, raising rates would prove self-defeating, and the simulation would indicate an irreducible default risk of some type. The other option is to introduce an additional component or module to mimic probable rate responses of the utility to information about narrowing coverage ratios—the approach adopted next.

In any case, the point is to develop an initial estimate of financial risk. Such an analysis can be refined in various ways. With the exception of the representation of the time interdependency of revenues, many of these refinements add little to the point of this analysis, which is that sinking funds can reduce default risk. The time interdependency of revenues, however, is critical because runs of above or below trend earnings have disproportionate effects on outcomes because of interest on fund accumulations. Accordingly, further attention to the characterization of time interdependency may be warranted, where, for example, time series models of historic series on utility revenues may be considered.

Note also that there appears to be room to fine tune the debt service level and end of period payout, in a sense, to optimize or minimize overall default risk. Thus, along with refinement of the risk simulation model, attempts to optimize the sinking fund arrangement become increasingly plausible.

Deferred Versus Level Debt Service

A publication of the First Boston Corporation concerning hydroelectric facilities, notes that,

There is a significant degree of flexibility available to a hydroelectric project developer in designing a financing plan and debt structure to best fit the economies associated with a given project. It is generally believed in the power development field that hydroelectric projects have the most tentative economic feasibility in the first four or five years of operation. . . . The project may not be deemed economically feasible when perfectly reasonable and accepted capital financing techniques... are left unexplored. . . . It is possible to produce "accelerated," "level," or "deferred" debt service payments.2

Let us consider, then, the effect of a deferred schedule of debt service on project feasibility, where a comparison is drawn with a


Figure 6.2 Debt Service Schedules

conventional level amortization. The debt payment schedules to be compared are shown in Figure 6.2. Schedule B (indicated by the boxes in the figure) defers debt payment, beginning at a lower level than the level payments in schedule A (indicated by tick marks), but rises above the schedule A level at around the middle of a typical bonding period of fifteen years. Both schedules amortize a $100 million bond over fifteen years. Text Box 6.2 summarizes the assumed parameters, and the following discussion develops the details of the simulation.

In addition to revenue and price level uncertainty, the application allows for uncertainty in construction costs and rate adjustments and stochastic variation in population growth. Variability in the demand and inflation series is introduced through

Construction Cost Estimate $100,000,000

Payback Period 15 years

Interest Rate 10 percent

Debt Coverage Ratio 1.3

Initial Revenue $36,000,000

Initial Per Capita Usage 90 units per year

Price Elasticity approximately -0.1 in a range of $1-$2 price per unit

Initial Population 200,000

Population Growth Trend 1.02 per annum

Interdependency in Demand and Inflation determined by a serially correlated residual

Interdependency in Population determined by k = 1 percent change per year in a band from 0 to 4 percent

Texr box 6.2


Figure 6.3 Cost Overrun Probability

time series models. Time series models determine a fuzzy, rather than absolute, band of variation outside of which it is unlikely a variable will be found in a realization. Variability in population growth is developed with a stochastic model such as that described in the previous chapter. The population growth rate is assumed to occupy a band between 0 and 4 percent per year, where a time interdependency factor k = 1 percent is allowed.

Construction cost overruns. We assume that prior simulations provide a risk profile for construction cost growth, as indicated in Figure 6.3. This is generated by a normal distribution, truncated at the $50 million level. Note there is a high probability of costs running above the assumed construction cost estimate of $100 million.

Construction costs in excess of $100 million are presumed to be

Applications 109

funded by additional debt amortized according to a level debt service for fifteen years. Additional realism could be incorporated at this point by assuming that some of the cost escalation must be funded by shorter term debt, possibly at higher interest rates. All debt in this application is assumed to be issued at 10 percent interest, and the analysis disregards sinking funds or construction contingency allowances.

Revenue risk. Revenues are broken down into population, per capita demand, and rate level components. Bands of random variation are introduced around anticipated trends in demand and inflation with simple time series models, while a more judgmentally developed stochastic model produces the population series. Rates are determined to cover average costs, where instantaneous adjustment is assumed for year-to-year changes in revenue requirements. Consumer price responses are incorporated into the demand model, where a threshold below which demand is completely inelastic or totally unresponsive to rate change is acknowledged also.

The baseline per capita demand (PCD) per year is

a function which represents a minimum demand of fifty units per year and embodies a price elasticity of around - 0 . 1 when price varies between $1.00 and $2.00 per unit. Variability around this level is introduced by a norrrrally distributed random component with a zero mean and standard deviation of ten units. The first order serial correlation coefficient is assumed to be 0.3.

A stochastic model is used to generate the population series, where the annual population growth rate is assumed to fall between 0 and 4 percent, and year-to-year changes in growth are limited to 1 percent—that is, if year t growth is 1.5 percent, year t + 1 growth is constrained to be greater than zero percent and less than 2.5 percent. Sampling from these ranges is determined by uniform probability distributions, as described in Chapter 5.

Rates that are set on an average cost basis are assumed to adjust instantaneously to revenue and cost conditions. The rate module


totals expenses and debt service from a year, applies a factor of 1.3, and divides by the product of total population and per capita demand for this same year. The resulting rate is adjusted upwards by a tolerance factor of 15 percent, and a fixed component of $0.50 per unit is added, which roughly corresponds to the fixed charge portion of the rate schedule. A ceiling on rates of $5.00 per unit, adjusted by inflation from the first year in the payback period, also is assumed. This derivation simulates some features of rate setting, but in allowing for instant annual adjustments, it represents the best responses possible, given the demand response patterns in the customer service area. Accordingly, default risks are lower by this procedure than they might be with response lags and errors in forecasting the revenue needs of coming years.

Expenses. Initial expenses of $1 million are adjusted each year by an inflation rate that averages three percent annually. Variation around this trend in the overall level of prices is determined by a normal random error term and a first order serial correlation coefficient of 0.5.

Risk Profile. The criterion is whether or not the debt coverage ratio falls below 1.3 in a year. The results are presented in Figure 6.4, which summarizes fifteen discrete, binary variables representing the risk of technical default in particular years of the payback period. The bars with sloping marks in Figure 6.4 represent year-by-year default risks associated with the level amortization schedule. The solid bars indicate risks associated with the tipped debt service alternative.

The early problems with construction costs, it would seem, are not decisive in this formulation. This is surprising in view of the immense variance of construction costs and the allowance for relatively rapid population growth. Again we cannot simply add these annual default risks together to arrive at a total default risk. Hence, we must add a binary variable to the simulation (0 if no default in period, 1 otherwise). If we do this, the tipped debt service schedule embodies the more significant risks. Thus, the overall risk of default is lower with the level debt service. The growth in revenues is simply not adequate to cover both inflation and the jump in the debt schedule after year 7. The deterioration in performance of the tipped amortization schedule at the end of the payback period is related to the chance that the end of period growth rates are

Applications 111

Figure 6.4 Default Risk—Level and Tipped Debt Service

consistently below trend, while end-of-period inflation is above trend.

The assumption is presenting this simulation example is that the ranges and values for key parameters are those best supported by judgment and available data. In a real world application, this may or may not be so. Thus, it is interesting that the conclusions indicated by this analysis are robust to changes in the basic modeling


strategy. Instead of judgmental and absolute ranges for population growth, population may be represented by a time series model. Providing that this time series model incorporates a comparable variability and time interdependence, roughly the same results are obtained. Alternatively, the time series models in the above application can be replaced by wholly judgmental models imposing absolute bounds on demand growth and so on without fundamentally changing the conclusion—again provided that similar variability and time interdependence are built in.

Related Applications

A comparable simulation framework can apply to questions concerning tradeoffs between new capacity and power or water purchases from other systems. Problems, for example, have been noted in the northeast region of the nation, where

Most [electric power] purchase agreements are signed using a least cost alternative formula that provides only minimal savings over the building of base load units. However, the life of the contracts is anywhere from one-half to one-fifth the life of a typical base load unit (such as a coal-fired plant), and even shorter if life-extension programs are taken into account.3

This issue can be explored by a risk simulation in which future power demands and the costs of new generating units are allowed to be partly stochastic, given the exact contractual arrangements in the purchase agreements. If power purchase contracts allow considerable flexibility in the amounts taken each year, and substantially uncertain and volatile demand is anticipated, this simulation could confirm what might appear as a suboptimal choice on the part of the northeastern U.S. electric power utilities.

CAPACITY PLANNING

The preceding analysis is concerned with fairly work-a-day problems in utility finance and business strategy. Let us now move to what appears to be a higher level of abstraction. The topic is capacity planning and the utilization of risk simulation in deter-

Applications 113

mining least cost capacity plans, where costs usually are measured in terms of present values. The present value (PV) of a series of investments (KM K2,. . . , Kn) is defined as

where r is generally taken to equal current or projected interest rates on debt issues for the projects under consideration.4

Least-cost planning procedures have received attention in regulatory hearings and other contexts in recent years. Although there are varying interpretations of what constitutes a least-cost plan,5

the primary emphasis is on the minimum present value of costs criterion. One of the reasons this criterion has received so much attention is that it lends itself to mathematical optimization or programming procedures. Because electricity cannot be stored, optimization of power systems focuses on the mix of generating resources to serve peak and off-peak demands and the merit order of operating units.6 Linear programming7 and Monte Carlo simulations8 contribute to the identification of optimal power loads and reliability factors. Water resource planning, on the other hand, focuses more on the sequence of supply projects, their safe annual yields, and anticipated annual demand, appealing to dynamic programming or similar formulations.9

Another factor recommending the minimum present value criterion is that this criterion identifies the expansion plan with the lowest financial risks. There are provisos to this observation, and at least one electric power planning model includes subroutines to check the financial feasibility of investment series, given financing from internally generated funds, debt, preferred stock, common stock sold under nondilution constraints, and common stock sold below book value.10 Nevertheless, with ready access to debt and equity markets, the minimum PV path has the lowest default risk, providing that special risks of construction cost overruns do not exist and feasibility constraints, such as debt to equity ratios, can be met. Thus, the minimum PV expansion path produces the maximum present value of net revenues and should lead to favorable access to financial markets for capital.

Some simple examples of how capacity investments fit into de-


mand projections are presented in Table 6.2, and the following discussion makes one or two further analytic points regarding the minimum present value criterion. Then we turn to applications of risk simulation in this type of planning exercise and develop an example showing tradeoffs between economies of scale and flexibility of response.

The Minimum Present Value Criterion and Changes in Interest Rates and Demand Growth

A salient feature of the minimum present value criterion is that reversals in project evaluation can accompany changes in the discount rate or changes in the rate of growth of demand. Thus, if the real significance of long-run planning is to determine which project to build next, immediate construction of a large facility exhibiting economies of scale might be favored by one regime of interest and demand growth rates but be inferior for different values for these variables.

This point can be made straightforwardly, if somewhat abstractly, with simple algebra. Thus, a discussion in the engineering literature postulates a linear growth of demand and an infinite planning horizon, avoiding valuation problems associated with excess capacity at the end of a period of analysis.11 It employs the cost function K(x) = axb, where K is capital cost and the present value of all capacity-related O&M costs of providing capacity x and economies of scale are indicated when the elasticity of cost parameter b > 1. If we assume operating and maintenance costs are negligible, it is possible—given the discount or interest rate r—to identify the optimal size facility and intervals t* at which it should be constructed. For this design period t*, we have the relation

b = (rt*)/(ert* - 1) (6.1)

where e is the base of the natural logarithm system (2.718. . . ) . Solving this equation for t* (by numerical approximation) implies the corresponding optimal size for a facility since, ex hypothesis, we know the demand growth. Interestingly, implicit differentiation of equation 6.1 indicates that dt57d < 0. In other words, the optimal construction interval t* decreases as the discount rate increases.

Table 6.2 Minimum Present Value Capacity Investment Sequences

Period Years

CASE1

CASE 2

CASE 3

CASE 4

Demand Capacity Investment ($million)




1 0

100 58

100 25

100 25

100 58

2 5

130 25,25

110 25

110 58

120

3 10

150

120

150

135

Present Value (10 percent discount)

89.05

40.52

61.01

58.00


Hence, smaller projects with higher unit costs produce the least present value expansion path when the discount rate increases.

We can study how acceleration and deceleration in the growth of demand affect the optimal investment sequence with Table 6.2. This table presents the optimal capacity investments for several scenarios of nonlinear demand growth. Investment sequences are built with two types of capacity projects: a $58 million design with a capacity of 30 demand units, and $25 million projects with capacities of 10 units apiece. Table 6.2 pairs demand paths and the costs of the optimal sequence of capacity investments, listing the present value of these sequences in the last row, assuming a discount rate of 10 percent. There is an initial demand of 100 capacity units and a planning period of 10 years. Two point estimates of projected demand are shown in each case, for the end of a 5-year period and at the end of the planning period.

If demand growth is rapid, as in case 1 in the table, initial construction of a large project is economic and has the lowest present value of costs—$89.05 million.

If demand growth is low, as in case 2 of Table 6.2, staging two smaller capacity additions has the lowest present value of costs, even when their unit cost, as measured by their total capital cost divided by their capacity, is fully one and one half times greater than the thirty-unit project.

Case 3 illustrates the resorting of investments in an optimal sequence as the growth of demand changes. There is slow initial growth in demand, followed by a rapid surge in demand. A smaller, higher unit cost project is best suited to periods of slow demand growth, and the larger project with economies of scale is best for more rapid growth of demand.

Finally, case 4 shows a demand projection that traces a path precisely halfway between the upper and lower bounds established by the other cases in Table 6.2. This could be average or expected demand growth and is conceptually often taken to be the standard case in planning exercises. Building the large project initially produces the lowest present value of costs for this demand trajectory.

Note this "capacity plan" does not satisfy demand at the end of the ten-year period. There are five additional demand units that must be met somehow in the average demand scenario of case 4.

Applications 117

In any case, the initial thirty units of demand are satisfied in the cheapest way by the $58 million project.

Risk Simulations of the Present Value of Least-Cost Capacity Investments

Risk simulation is relevant to capacity expansion planning because of uncertainties in the growth of demand and other factors. Analysis of demand uncertainty can be traced to work by H. Bal-eriaux and E. Jamoulle in the 1960s12 and studies prepared for the California Energy Resources Conservation and Development Commission and the U.S. Federal Energy Administration in the 1970s.13 Several citations reference similar methods, including simulations of uncertainty in the availability of hydroelectricity14 and an ambitious study of long-term capacity needs for the Electric Power Research Institute.15 In the United Kingdom, a related approach was developed by D. V. Papaconstantinou.16 Studies associated with public debate over the Sizewell project systematically apply many of these techniques to the evaluation of the timing and advisability of construction of Britain's first pressurized water reactor.17

As an illustration of these methods, it is interesting to develop a risk simulation to consider data, such as in Table 6.2. The results produce some surprises and show the dangers inherent in not thinking in terms of stochastic process.

To consider these investment options in a stochastic framework, suppose the high and low demand projections in cases 1 and 2 of Table 6.2 establish the lower and upper bounds pertinent to each of the five-year time periods considered. Thus, we presume that after the first five years of the planning period, total demand is between 110 and 130 units. Let us also assume that: (a) chance variation in demand follows a uniform distribution between the bounds implied in any five-year period by these projections, and (b) variation between adjacent five-year periods is limited to 25 percent of the allowable range.

The End of Period Capacity Valuation Problem

To set up this risk simulation, we must make decisions about the end-of-period valuation of excess capacity. Note that the first


three case examples in Table 6.2 avoid this problem by considering end-of-period projections that exactly dovetail with capacity additions. We confronted but did not elaborate upon this difficulty in case 4, which presents the expected or average demand profile over this ten-year planning period.

Note what would happen in case 4 if we prorated the cost of five units of needed and unmet capacity at the end of the planning period. Thus, we would allocate one-sixth of the costs of five units of capacity from the thirty-unit project and half of the costs of the ten-unit project to these five demand units. This would indicate that the cheapest alternative is to build the large project because its prorated costs would be $9.66 million, as compared with $12.5 million from the smaller project. Therefore, such a prorating of capacity costs to excess demand in a period means that a large project generating substantial excess capacity is usually selected over a smaller, higher unit cost project under this type of end-of-period convention, particularly if there is high excess capacity in this project. This is unsatisfactory, especially if further study indicates that system demand will peak in the planning period, declining thereafter, or if there is considerable uncertainty as to whether growth will occur following the period under analysis.

Another option is to simply allocate all investment costs as of the date of a bond issue to meet excess capacity, no matter how small this excess capacity may be. Thus, we may include a large investment that is almost completely unutilized but is in the capacity plan as one alternative way to meet demand until the end of the planning period. This could make such a plan seem unduly expensive when compared with another plan with no excess capacity at the end of the planning period. This is unsatisfactory if our expectation that the planning period currently being considered will be followed by a period of more rapid growth or if, somehow, not utilizing this large project in the current period will result in this investment option being lost altogether.

It is clear that a part of the problem is solved by selection of the planning period. Thus, a prior analysis should look at alternative plans and the expectation for demand growth over future periods. The length of an appropriate planning period will be influenced by the typical gestation period for the investments, ex-

Applications 119

pectations for growth, and the point at which uncertainty becomes almost absolute, that is, more than a decade or two hence.

Our approach here is to assume that a prior analysis has indicated ten years as the appropriate planning period. We, furthermore, assume that, because of uncertainty or an expectation of zero growth after this period, we have no grounds to prorate projects. Thus, to avoid a bias toward the smaller project sequences, we will simply select the investment sequence that meets the maximum length of time in the ten-year planning period so as to leave no excess capacity at the end of the planning period. In some runs, this means the schedule of investments does not satisfy demand for a full ten years. If there are no limits to the number of small, $30 million projects that can be built, this presents no problem to meeting the physical needs over the foreseeable future. Furthermore, the result indicates that the average period of time covered by investment sequences determined in this fashion, where we separate those beginning with the large project from those beginning with the small project, is roughly comparable.

Determining the investment series with the lowest present value is simplified, also, by the fact that the $58 million project is a multiple of the capacity of the $30 million project. A simple way to create an optimal sequence, therefore, is to first locate the smaller, higher unit cost projects into a time sequence that meets the demand over the planning period. Then, this series of investments can be inspected to determine whether it contains any segments or subsequences that are "tightly packed." For example, building $25 million investments each year for two years when the discount rate is 10 percent produces a present value of approximately $62.74 million as of the time of the first of these three investments. This is, therefore, more costly than satisfying the same thirty-unit increment in capacity with a single $58 million project.

A Risk Simulation

The practical problem usually is to identify the next project to be placed on the construction schedule and brought into operation. Let us focus, therefore, on comparing capacity plans that result after an initial selection of either the large or small project is


Figure 6.5 Present Value of Investment Costs

made.18 Then, we obtain the interesting information depicted in Figure 6.5. The two risk profiles there refer to the minimum expected present value of investment costs, given an initial commitment to the thirty-unit or ten-unit projects as the initial project to be built.19 The solid bars are associated with sequences in which the large project is built first, whereas the bars with the slashes mark out the risks of building the small project first.

Building the small project first has the lowest expected present value of costs ($59.17 million versus $61.31 million), even though the average or midpoint demand path would indicate that the cheapest tactic would be to build the large project first. At the same time, this lower expected present value of costs is associated

Applications 121

with a somewhat shorter expected period covered by investment sequences beginning with the small project first (8.54 years versus 9.02 years). In fact, the concentration of present value when the large project is built initially at $58 million is due to the fact that in most of the simulation runs, the large project completely satisfied demand during the planning period.

This example underlines the potential complexity of decisions about what project in a chain of projects ought to be next in the construction schedule. Thus, there are tradeoffs not only in the cost dimension but also in the time dimension. There are also issues presumed to be addressed before the analysis can properly take place at all, such as that of the appropriate planning period.

Further attention might be given to refining the probability distribution imputed to the demand forecasts. For example, is lower rather than higher demand considered more likely in the forecast range? If demand growth is negatively skewed, there is an even greater chance costs can be finessed with a smaller investment. Of course, this simple analysis abstracts from project lead times, response lags, and the impact of shortages on customers.

BENEFIT COST ANALYSIS AND FINANCIAL RISKS

The challenge in financial risk modeling is to develop appropriate detail and complexity. In some cases, this requires choosing between refining existing components of a model and adding new linkages and elements. Thus, in the previous application, in addition to fine tuning probabilities, it is possible to develop more realistic assumptions concerning management responses and the valuation of supply options. Among the more important of these new assumptions are lags in management response to changing demand and the opportunity cost of electric power or water shortages.

The preceding simulation assumes foresight on the part of utility managers, which may not exist. Sizeable demand fluctuations may not elicit immediate management responses. Existing plans may be adhered to until decision makers are persuaded that recent shifts in demand growth are more than transitory. Accordingly, two effects commonly accompany shifts in the growth rate of demand. When there are sharp upward revisions in growth, shortages may


be recurrent throughout the utility system. Long periods of excess capacity, on the other hand, may accompanying a downward shift in demand growth and impose higher rates and charges on a smaller than anticipated volume of production or fewer customers.

Here, then, there is a problem of what metric to select to express a diverse set of possibilities. Originally, we took the view that financial risk is evaluated, subject to the condition that new facilities are brought on line in a timely manner to meet growing demand in a utility system. In a long-term planning context, however, the situation is complicated by the possibility of response lags or selection of projects that turn out to be inappropriate to the conditions that develop.

The solution suggested by economics is to rely on a demand-based metric called the consumer willingness to pay or consumer surplus. In dollar terms, U.S. average outage costs for electricity have been estimated from $1.00 to $3.80 per kWh in 1978 prices.20

The effects of water shortages have been subject to empirical study21 and have been variously estimated with econometric models distinguishing summer or outdoor from winter or indoor water use of residential users. The imposition represented by excessively high utility rates that can result when there is excess capacity for prolonged periods also is amenable to this metric.

Models can be developed, for example, that balance off the possibility and imputed costs of utility rates that are excessively high with the chance that noisome shortages will impose disamen-ities on customers. Initially, what usually must be a search algorithm identifies the investment sequence which is optimal from a benefit/cost standpoint, given project lead times, likely management response lags, and imputed costs for shortages. This investment sequence represents a minimum feasible present value of costs, given the range of contingencies that may arise during the planning period. Then, financial risk analysis can consider the optimal investment path, as costs are disaggregated from the present value framework into a cash flow framework, to identify the best way of implementing it.

Integrating financial and benefit/cost analysis is a fertile research area. Thus, in the minds of many utility managers, there appears to be a presumption that, over the longer period, per customer costs are likely to be less for persisting surpluses than for shortages.

Applications 123

This could be because projects will be dropped from the construction schedule if high excess capacity persists. On the other hand, the acceleration of construction plans may favor quick fixes, increasing the costs of meeting demand over the longer period. There is a sense here of an ensemble of probability-governed relationships, although the question is largely unexplored.22 Of course, financial debacles surrounding many ill-fated nuclear power installations suggest that a large project bias could contribute to financial risk.

CONCLUSION

The examples in this chapter underline one point above others— the probabilistic point of view leads to insights into financial and planning problems, insights that may not be mere restatements of the results of simpler "what-if" or sensitivity analysis.

The first application illustrates what commonly is regarded as the subject matter of risk analysis—the appraisal of probabilities of an event essentially bounded in time.

The other applications exhibit more dynamic and time-dependent meanings of financial risk, summarized by a series of default probabilities over the payback period or by a distribution of minimum present values.

Interestingly, with regard to sinking funds, there is some presumption that people always think in probabilistic terms. Thus, it is almost impossible to justify and appraise a sinking fund without regard to variability factors and the advantage of capturing income during exceptionally good years. Nevertheless, risk simulation suggests a curious equilibrium of default risk after the first few years of the payback period in the no sinking fund case, an equilibrium or stabilization apparently due to an interaction between inflation and the potential for zero or negative revenue growth. It is difficult to see how anything other than a stochastic framework could indicate this result, which is of some interest because it is a circumstance likely to be encountered in some utility systems in coming years.

The simulations concerned with the relative risks of level and tipped amortization schedules also produce an unconventional result. Despite extraordinary chances for construction cost overruns,


the total default risk associated with a tipped amortization schedule exceeds that associated with a level debt service schedule. The culprit here is inflation or too ambitious a reduction of initial debt service. The clear challenge is to devise an optimum debt schedule for the perceived circumstances, and the apparatus indicated in this chapter can inform this pursuit.

Similarly, the risk simulations of the capacity planning problem, when the initial capacity investment is "locked in," indicate a result that could not have been guessed at by a "what-if" analysis. Here, the average or expected demand profile is best met by building the larger project first, the project embodying economies of scale. Nevertheless, the risk simulation indicates that an investment plan in which the smaller project is selected for initial construction is superior, namely with respect to the present value of costs. This approach is usually cheaper and attains a roughly comparable, although slightly lower, time coverage in the planning period. This conclusion, it is noted, is more emphatic if the probability distribution for demand in a particular year is positively skewed.

Generous use of the uniform probability distribution and a type of range estimation pertaining to growth rates is apparent in the applications discussed in this chapter. This corresponds to what Bayesian statistical theory calls a "uniform prior" distribution; that is, in the absence of any other information, assume that all acknowledged options are equally probable. If the true probability distribution has a distinct and single peak, the uniform distribution overstates the variance of the process under study. This may provide incentives to further explore the anticipated variation of growth rates. Alternatively, one may "back in" to the probability distribution and time interdependence characteristics of growth rate series that produce an acceptable level of risk. Then, the question can be whether this distribution is realistic or plausible or whether, for example, its variance is too small to be believed.

In recent years, utilities have brought a number of newer instruments and tactics into play, which include, as Charles F. Phillips, Jr., notes,

convertible securities, issues with warrants, preference stock, short-term notes (five years) and intermediate-term bonds (seven to ten years), pollution control facilities financed with industrial development bonds or

Applications 125

pollution abatement revenue bonds, and employee stock option and dividend reinvestment plans. Leasing and project financing became viable options. With high interest rates, refunding and debt-equity swaps were carried out. A few utilities have offered common stock to their customers through a monthly installment plan; a few electric utilities have established an energy trust to finance nuclear power plants or the purchase of nuclear fuel.23

Restrictions on arbitrage of tax-exempt debt issues in the Tax Reform Act of 1986 and innovations such as interest rate futures or swaps add to the complexity of financial analysis for power and water utilities. In principle, each of these financial instruments, tactics, or options can be evaluated within risk simulation contexts such as these developed in this chapter.

The first task of risk simulation models is to get an answer. Then, refinements and extensions can be explored to determine their impact on the risk profile.

NOTES

1. About one half of all states protect municipal and cooperative service areas by exclusive franchise rights or state law. Elsewhere, no specific provisions shield service areas. See Malachy Fallon, "Municipal Electric Credit Review," Standard & Poor's CreditWeek, June 5, 1989, p. 8.

2. First Boston Corporation, Financing Hydroelectric Facilities (Boston: First Boston Corporation, April 1981), pp. 10-11.

3. Robert Woodard, "Power Shortage Threatens Northeast," Standard & Poor's CreditWeek, June 5, 1989, p. 9.

4. Operating costs here are assumed to be capitalized in the values of K,

5. See Daniel J. Duann, "Alternative Searching and Maximum Benefit in Electric Least-Cost Planning," Public Utilities Fortnightly, Decemb21, 1989, pp. 19-22.

6. This means scheduling the lowest cost unit to provide baseload capacity and intermediate and peaking units to come on line in order of increasing cost.

7. A pioneering paper applying this method is P. Masse and R. Gibrat, "Application of Linear Programming to Investments in the Electric Power Industry," Management Science 3 (January 1957): 149-166.

8. See Ralph Turvey and Dennis Anderson, Electricity EconomicsEssays and Case Studies (Baltimore: Johns Hopkins University Press, published for the World Bank, 1977), p. 259.

126 Analvsis of Infrastructure Debt

9. See, for example, Yacov Y. Haimes, Hierarchical Analysis of Water Resources Systems: Modeling and Optimization of Large-Scale Systems (New York: McGraw-Hill, 1977).

10. Martin L. Baughman, Paul L. Joskow, and Dilip P. Kamat, Electric Power in the United States: Models and Policy Analysis (Cambridge, MA: MIT Press, 1979).

11. This exposition relies on the discussion and derivations in Daniel P. Loucks, Jery R. Stedinger, and Douglas A. Haith, Water Resource System Planning and Analysis (Englewood Cliffs, N.J.: Prentice-Hall, 1981), pp. 121 passim. See also A. S. Manne, "Capacity Expansion and Probabilistic Growth," Econometrica 29 (1961): 632-641.

12. H. Baleriaux and E. Jamoulle, "Simulation de l'exploitation d'un pare des machines thermiques de production d'electricite couple a des stations de pompage," Revue Electricitie, edition SRBE, 5 (1967).

13. Stanford Research Institute, Decision Analysis of California Electrical Capacity Expansion, report submitted to California Energy Resources Conservation and Development Commission, Menlo Park, February 1977; Gordian Associates, Optimal Capacity Planning Under Uncertainty in Demand, report submitted to the U.S. Federal Energy Administration (Washington, D.C.: U.S. Government Printing Office, November 1976).

14. Arun P. Sanghvi and Dilip R. Limaye, "Planning Future Electrical Generation Capacity," Energy Policy 1 (June 1979): 102-116.

15. Martin L. Baughman and D. P. Kamat, Assessement of the Effect of Uncertainty on the Adequacy of the Electric Utility Industry's Expansion Plans, 1983-1990, EA-1446, Interim Report (Palo Alto, Calif.: prepared for the Electric Power Research Institute, July 1980).

16. D. V. Papaconstantinou, Power System Planning Under Uncertainty Using Probabilistic Simulation, Internal Report EPU/DP/1 (London: Energy Policy Unit, Department of Mechanical Engineering, Imperial College of Science and Technology, May 1980). See also Nigel Lucas and Dimitrios Papaconstantinou, "Electricity Planning Under Uncertainty," Energy Policy 10 (June 1982): 143-152.

17. Nigel Evans, "The Sizewell Decision: A Sensitivity Analysis," Energy Economics 6 (January 1984): 15-20; and Ian S. Jones, "The Application of Risk Analysis to the Appraisal of Optional Investment in the Electricity Supply Industry," Applied Economics 3 (May 1986): 509-528.

18. Note that this will not violate our convention of never selecting a project that will leave excess capacity at the end of the demand period since the minimum growth of demand exceeds thirty units over this twenty-year period.

19. Because one project is a multiple of another and the minimum

Applications 127

demand increment exceeds the capacity of the largest project, the periods to which these expected present values refer are identical in expected value and distribution.

20. A. Kaufmann, Reliability Criteria—A Cost Benefit Analysis, OR Report 75-79 (Albany: New York State Department of Public Service, June 1975); "Report on the Reliability Survey of Industrial Plants," IEEE Transactions on Industry Applications 1A-10, no. 2 (March 1974): 231-233. See Roland Andersson and Lewis Taylor, "The Social Cost of Un-supplied Electricity," Energy Economics (July 1986): 139-146.

21. See Mark Hoffman, Robert Glickstein, and Stuart Liroff, "Urban Drought in the San Francisco Bay Area: A Study of Institutional and Social Resiliency," in American Water Works Association Resource Management, Water Conservation Strategies (Denver: American Water Works Association, 1980), pp. 78-85.

22. C. Vaughan Jones, "Analyzing Risk in Capacity Planning from Varying Population Growth," Proceedings of the American Water Works Association (Denver: June 22-26, 1986), pp. 1715-1720. Jones explores the issue in outline.

23. Charles F. Phillips, Jr., The Regulation of Public Utilities Theory and Practice (Arlington, VA: Public Utilities Reports, Inc., 1984), p. 221.


7

Reflections on the Method

Monopolistic markets and capital intensity make financial risk analysis of power and water investments more available than, say, analysis of investment risks for a new assembly line for computer components or a retail outlet. Debt financing is the favored vehicle in financing major capacity expansion, and debt service is a major cost component. Construction cost overruns, therefore, are a primary risk factor on the cost side, along with the potential for interest rate changes prior to bond issuance. Population growth and the level of customer demand introduce risks on the revenue side. Customers are "captive," and their numbers and market responses can be analyzed separately, although deregulation is introducing new competitive possibilities.

This discussion demonstrates the potential of this approach for generating insights into financial risk, defined chiefly as default risk on debt. The vantages gained go beyond those derived from the examination of various "what-ifs" or a sensitivity analysis in various ways and are, in some instances, almost surprising.

Thus, risk simulation underlines how contingency allowances ought to be figured against total construction costs rather than, as sometimes suggested in engineering discussions, as a fixed percentage allowance against each major construction cost category. Once one conceptualizes the simulations, the logic becomes apparent—it is the logic of risk pooling, when cost overruns in component cost categories are stochastically independent.


The discussion of sinking funds in Chapter 6 led to an expected result—that such arrangements can reduce default risk. By the same token, a relatively ambitious simulation model in the same chapter indicated the superiority of level debt service over one tipped debt service alternative, even though initial construction cost overrun probabilities were large.

In capacity planning, the preceding chapter developed a risk simulation to consider whether initial construction of a large facility exhibiting economies of scale or a smaller, high unit cost facility was superior, given an anticipated range of variation for demand over ten years. This analysis generated distributions for the present value of costs, when subsequent projects were optimized to the random fluctuations of demand that occurred, and coverage time for the projects analyzed. The findings proved intriguing because of nonlinearities in the cost equations in this model. The least-cost sequence fitted to the average or expected demand profile suggested that the large project ought to be the first facility to be built, when minimizing the present value of costs was the investment performance measure. On the other hand, risk simulation indicated the superiority of building the small project first because of the extremely high and nonlinear variation in present values when demand grew slowly over this planning period.

These are stimulating findings, but questions arise about how defensible and robust they are. It is time to take stock, therefore, and reflect on the vantage gained and on the strengths and limitations of the technique.

ADVANTAGES AND DISADVANTAGES OF RISK SIMULATION

The prominent strength of risk simulation is its integrative function. It forces coordination of diverse information and coerces one to be comprehensive in accounting for risk factors and the specification of interlinked processes. A second strength is that the method directly acknowledges risk preferences, providing decision makers with maximum information about the distribution of likely gains and losses.

Attention to probability has unanticipated payoffs, and one might reasonably speak of a stochastic or probabilistic outlook.

Reflections on the Method 131

On the other hand, the complexity of risk analysis is a limitation. This complexity, of course, motivates books such as this, which attempt a review of critical issues and methods. In general, it is not clear why anyone would expect investing large sums of money to be easy.

Perhaps what needs to be emphasized is that risk simulation is a responsible way of proceeding when projects running into the tens of millions or billions of dollars are contemplated. With the advent of faster microprocessors and the growing number of computer programs to handle simulation problems, the cost of conducting a risk simulation is dropping. Whereas earlier a full-scale analysis for a several-hundred-million-dollar project could cost a few hundred thousand dollars, today a competent analysis can be carried out for much less. The techniques described in this book also support simple approaches—back of the envelope simulations, if you will.

The following discussion amplifies these remarks. It considers the significance of procedure in validating a risk simulation, the treatment of pure uncertainty, and issues having to do with the robustness of estimation. The discussion closes with a recapitulation of basic points and findings and remarks on the generalization of the methods presented here to other types of infrastructure investment.

COMPREHENSIVENESS AND THE IMPORTANCE OF PROCEDURE

Reviewing the steps taken in developing a risk analysis is the primary means of validation, short of waiting for the attendant processes to manifest themselves.

This underlines the importance of preliminary analysis of risk sources. At an initial stage of study, a list of risk factors should be developed. This should include a sensitivity analysis in the standard sense. Variants of the base case cash flow model should be explored to determine, broadly speaking, which subset of these factors exerts the most significant influence on the risk of default or other appropriate performance measures. Attention should be focused on accurately characterizing the anticipated range of risk factors.


An EPRI-sponsored study of electric power capacity needs for the 1980s illustrates what can happen when elements of this basic procedure are ignored. The primary problem with this study, which concludes that the 1980s was an era of capacity shortages, is that range for risk factors were developed from partial evidence.

At the end of the 1970s, this EPRI-sponsored study concluded that

Based on 1978 reported electricity demand forecasts, 1978 capacity expansion plans, and an assumed economic cost of $1 per kWh for energy undelivered due to shortages, the results of this study indicated that for most regions of the country, even with the narrowest range of demand uncertainty, a more rapid expansion of generating capacity would result in lower total social costs in 1990.'

This conclusion was obsolete almost the moment it was announced. What was the problem? The error appears to stem from basing range estimates for the growth of electricity demand on forecasts supplied by the regional reliability councils and excluding other information. Regional reliability councils collate member electric company forecasts. Until recently, many utilities extrapolated patterns from the 1960s and earlier to the future, producing growth rates that proved far too high in the 1970s and 1980s. Thus, the raw average growth forecast for 1978-1990 in the EPRI study from the nine reliability regions was 5.62 percent,2 while the actual summer peak electricity demand in the North American Electricity Reliability Council grew an average of 2.2 percent per year from 1981 to 1986 and at a slightly higher rate thereafter.3 Note, however, that alternative growth rate forecasts had been produced for several years and had received confirmation in growth trends of the 1970s. Authorities note, for example, that since 1974, forecasts of load growth "ranged from 2 to 7% and have resulted in a continuing controversy between utilities and various intervenors at rate-making proceedings."4

This example is revealing in another respect. It illustrates that it is possible for events to chagrin analysis in fairly short order and raises questions about a position staked out on the verification issue by John C. Hull:


Even if we know in a certain situation that the output of a risk evaluation study influenced a manager to take decision A rather than decision B, we will not know for several years how decision A turns out and even then we may not be sure about what would have happened if decision B had been taken. Furthermore, suppose that it is definitely established that decision A turned out worse than decision B would have done, an advocate of risk evaluation will, undaunted, argue along the following lines: "It was a good decision—just an unlucky outcome. We were in the extreme left-hand tail of the distribution of NPV's."5

In principle, the point in this quote must be conceded. At the same time, the focus of the quote is perhaps more philosophical than warranted by practical situations in which risk simulation is the technique selected to inform an investment decision. Errors can reveal themselves quickly, and plausible linkages between errors of omission and commission in procedure and the findings of a risk analysis can exist. Indeed, if a risk simulation is deemed necessary in the first place, there may be reason to expect that bad decisions may soon lead to adverse situations. Cost overruns in power plant construction and inadequate contingency plans become apparent before a project comes on line. Problems of population growth may be imminent, if they appear salient at all.

One safeguard, therefore, is a thorough review of existing range estimates of risk factors. Decisions to exclude some lower or upper bounds should be defended so that outlying estimates are at least acknowledged.'Qualitative analysis provides similar safeguards. Understanding of market and demographic relationships as well as familiarity with standard rate making and financial procedures are important in characterizing responses in a risk simulation.

ROBUSTNESS

Robustness in risk simulation pertains to whether imputations of probabilities and ranges of risk factors can be approximate and still indicate fundamentally the same conclusions.

Symmetric, unimodal distributions are the easiest to approximate. Since the mean, mode, and the median of such distributions are identical, judgmental approaches may be more likely to correctly estimate the central tendency.

Similarly, the number of sufficient statistics required to char-


acterize a probability distribution can be small. If the distribution is known a priori, based on an analytic argument, it may be possible to develop relatively precise characterizations with a few items of information. Thus, if the underlying probabilities are normally distributed, the risk factor is summarized by stating its mean and variance.

The principle of compensating errors also imparts a degree of robustness to risk simulations. Small errors in identifying the central tendency of one variable, for example, can be offset by errors in the other direction with respect to other, stochastically independent risk factors, such as cost or revenue components.

An important tactic can be mentioned here also—that of paying the most attention to the largest risk elements. Thus, the influences on revenues or costs generally manifest Pareto's Law—the law of the significant few and insignificant many.

An interesting series of suggestions for imputing probability distributions to risk factors is developed in an Environmental Protection Agency handbook, and bears mentioning in this context:

A uniform distribution would be used to represent a factor when nothing is known about the factor except its finite range . . .

If the range of the factor and its mode are known, then a triangular distribution would be used.

If the factor has a finite range of possible values and a smooth probability function is desired, a Beta distribution (scaled to the desired range) may be most appropriate. The Beta distribution can be fit from the mode and the range that defines a middle specific percent (e.g. 95 percent) of the distribution.

If the factor only assumes positive values, then a Gamma, Lognormal, or Weibull distribution may be an appropriate choice. The Gamma distribution is probably the most flexible; its probability function can assume a variety of shapes by varying its parameters and it is mathematically tractable. These distributions also can be fit from the knowledge of the mode and the percentiles that capture the middle 95 percent of the distribution.

If the factor assumes positive and negative values and is symmetrically distributed around its mode, then a normal distribution may be an appropriate distribution.

Unless specific information on the relationships between . . . parameters


is available, assume values for the required input parameters are independent.6

These guidelines aim at robustness in estimation. Thus, the variance of the uniform distribution is larger than a broad class of unimodal distributions. If risk is proxied by the variability of a variable that is an additive or multiplicative composite of various component risks, use of such maximum variance distributions to characterize these component risks leads to something like an upper bound estimate of a risk. Similarly, use of uniform distributions for population variability exaggerates risks of low and high population growth. If population growth is lower than anticipated, financial burdens may be placed on customers. If population growth is higher than anticipated, shortages—brownouts or restrictions in use—may occur, imposing other types of cost. If these costs can somehow be made commensurate, as discussed in Chapter 6, then maximum combined costs are associated with maximum variance distributions for population processes.

The use of the triangular probability distribution was illustrated in Chapter 3, which showed how estimates of the mode and average value of a random variable with finite range are linked.

Use of the Beta, Gamma, Lognormal, and Weibull distributions has not been discussed in this book, although these forms are mentioned in the Appendix.

Of course it should be clear from this discussion that simulations with substantially skewed risk factors are less reliable. A recent text on engineering economics considers this under the rubric of the "problem of outliers,"7 significant in accidents or other low probability, high damage events. Thus, the distribution of cost overruns for large nuclear facilities embodying new design features appears sharply skewed to the right (negatively skewed). More attention must be devoted to plotting points on cumulative or probability distributions for such risk variables.

The modeling tactic suggested here is to employ simple modeling representations and to examine the effect of: (a) successive refinement of assumptions, and (b) extensions of the model on the risk profile. This is a higher order sensitivity analysis that allows assumptions and the characterization of process to vary in addition to considering the impact of changes in the magnitude of risk

136 Analvsis of Infrastructure Debt

factors. If the acknowledged range of risk factors is faithfully recorded and realism is sought in the simulations, the risk profile should represent a best effort at prefiguring the total consequences of various investment decisions.

PURE UNCERTAINTY

The Chicago economist Frank H. Knight suggested a distinction between risk and uncertainty.8 A risk situation, according to Knight's terminology, is one where probabilities are known. When we have no knowledge of the range or distribution of a variable, on the other hand, we are faced with uncertainty. This is the difference between, say, drawing a ball of a certain color from an urn when we know beforehand there are ten red balls and ten black balls and drawing a ball when we lack essential information, such as the number of balls, their color, and so on. In the first case, the probability of drawing a red ball (with replacement) is 0.5, while in the other situation we may have no way of asserting anything about probabilities of drawing, for instance, green balls. The first case is a risk situation in Knight's terms, while the second is a situation in which there is uncertainty.

There are various ways uncertainty manifests itself, where pure uncertainty refers to the fact that not only do we not know probabilities of an event, but we also have no information about the nature of the event in the first place. Thus, history shows a sustained capacity for producing surprises, that is, occurrences for which we really have no way of imputing probabilities because we cannot even conceptualize their existence. These historical innovations can develop on various levels. Most recently, there are the profound changes in Eastern Europe and the Soviet Union. On the economic front, few anticipated the regime of high interest rates initiated by United States Federal Reserve Bank policies in late 1979. One can also refer back to Pearl Harbor and so on.

In econometric language, the issue of surprises becomes the "structural shift" problem. Specific tests are recommended to identify time periods in which the coefficients of a regression are distinct.9 Many large-scale econometric models of the U.S. economy had to be fundamentally revised after 1974, for example, as the


Table 7.1 Payoff Matrix Illustrating Alternative Choice Criteria under Uncertainty

actions 1 2 3 4

1

20 10 50

100

"state of nature"

2

40 20 40 15

3

5 30 75 40

4

100 40 8

30

regime of higher energy prices triggered cascading changes in other economic relationships.

The economist Kenneth Boulding comments that

the general conclusion... is that under uncertainty alternatives have a high value which involve liquidity, flexibility, capacity for reversal, and capacity for continual learning from mistakes and revision of images of the future; under conditions of certainty, we simply select what appears to be the best alternative and zero in on it. In uncertain situations, this often involves zeroing in on disaster.10

This conclusion is embodied in decision criteria suggested for choices under uncertainty, including the maximin rule and the Hurwicz rule.11 The payoff matrix in Table 7.1 illustrates the general operation of such rules. This lists the payoff for "actions" (a(, where i = 1, 2, 3), listed along the right side of the table, given the "state of nature" (Sj, where i = 1,.. ., 4), which prevails. Note probabilities are not attached to the payoffs. Accordingly, there is Knightian uncertainty about whether one as opposed to another state of nature will come to pass; although, in this case, we are fortunate in being able to anticipate the prospective states of nature.


The maximin criterion is an abbreviation for selection of the maximum of the minima. It is implemented by identifying the minimum payoff associated with each action, that is, the minimum payoff in each row. Then, the action to be taken is determined by finding the maximum of these minimum payoffs. This is ax in Table 7.1.

The Hurwicz rule amalgamates this cautious criterion with an optimistic rule—the maximax criterion which states "take the action that leads to the maximum payoff in the set of maximum payoffs in each row." The Hurwicz rule involves maximizing a weighted sum of the payoffs associated with the maximin and maximax rules, where the weight is referred to as the pessimism-optimism index.

These rules seem excessively technical and reflect a general lack of consensus about what the optimal course of action is for choices under uncertainty.

Perhaps the best reply to various types of uncertainty in financial choice situations is that we must try our best under the circumstances. Risk simulation, subject, again, to procedural checks, provides decision support at least as good and probably in some instances superior to other approaches. In other respects, the ability of history to produce genuine innovations means that risk analysis, like other planning and evaluation tools, requires continuous updating.

CONCLUSION

Shifts in industrial patterns and age structure present challenging problems for utility planning. Economies of scale of large, central power and water facilities must be balanced against flexibility of response. The uncertainty and volatility of business and demand conditions have led to greater emphasis on optimizing the existing utility system, conservation, construction of smaller units, purchases, and joint ventures of production facilities. Yet some tactics can lead to regrets, such as reliance on gas turbines if oil prices rebound to 1970s levels and foregoing new structural water projects if forecasts of global warming are borne out.

Risk simulation makes explicit what is implicit in many qualitative appraisals, attaching numbers that, at a deep level, guide


thought about the prospects of an investment situation. The method promotes consistency and a deeper integration of information and understanding about the processes in question. It helps identify desirable risk management approaches, such as contingency or sinking funds, bond insurance, bank letters of credit, or, on the physical side, reconfiguring a project.

At the simplest level, factors contributing to financial risk can be classified as to whether they affect costs or revenues. On the cost side, there are construction costs and O&M costs. Revenues, it is suggested here, can be considered from the standpoint of per capita usage—where allowance is made for price effects and commonly observed chance variation—and population process. This abstracts from other segmentations of demand by customer class but can be a useful simplification for a first-cut analysis. This discussion develops and justifies methods of representing per capita usage and population growth so as to imprint their stochastic features onto the cash flows. Two questions are especially relevant in this regard: (1) Will the future be like the past, and (2) if not, what assumptions seem salient? If strong elements of continuity appear to exist in the situation, time series or structural models of risk relationships should be a mainstay of the analysis. Otherwise, judgmental methods and stochastic modeling become more important. This emphasizes, in particular, how qualitative information and modeling techniques from the behavioral sciences can contribute to risk simulation. One key to the applicability of this method is the largely monopolistic aspect of markets for water and power. Although cogeneration and power pooling introduce a degree of competitiveness into electric power markets, many utility customer areas remain "captive markets." Urban water systems appear to have relatively unchallenged customer bases and fixed supply sources also.

Ultimately, all risk factors are represented, either directly or in more complex formulation, as random variables characterized by probability distributions. Given this information, risk simulation involves sampling the possible values of these variables based on the representation of their cumulative distributions.

Range estimation is a major method of appraising risk factors. Its use is suggested in connection with cost overruns and delays in the construction schedule and in the stochastic modeling of pop-


ulation growth. Range estimation involves identifying lower and upper bounds associated with risk factors, as well as probabilities of exceedance of various levels of such risk factors. One major application is the identification of construction contingency allowances.

In addition to range estimation, time series models can characterize the variation of time-dependent variables. Thus, there are essentially "one-shot" factors, like construction costs, and multiple period variables, such as the quantity demanded and population growth. Single period risk factors may be described by a simple probability distribution. More complex, time-interrelated processes usually must be projected with time series models. The simplest of such models employ first order autocorrelation and a white noise or random shock term.

These methods generalize to a number of contexts. Market share almost always becomes an issue when we leave the "captive customers" of a power or water company, adding a dimension to the risk simulation. If a toll road or highway is contemplated, chances for diversion of traffic along alternative routes must be considered. The elasticity of this diversion to changes in the toll tends to make for more elastic price responses. An airport cannot set landing fees without considering whether major carriers will pull up stakes and centralize their operations somewhere else. The question of market share, however, is amenable to analytical approaches as mentioned here in connection with population growth or demand variation— time series and structural modeling when data permit and judgmental evaluation. The methods presented here, therefore, are broadly useful, although, as the exposition underlines, specific industry knowledge is essential for realistic appraisal of financial risk.

NOTES

1. Martin L. Baughman and D. P. Kamat, Assessment of the Effect of Uncertainty on the Adequacy of the Electric Utility Industry's ExpanPlans, 1983-1990, EA-1446, Interim Report (Palo Alto, Calif.: preparefor the Electric Power Research Institute, July 1980), p. vi.

2. Ibid., Table 4-2, p. 4-3. 3. See North American Electric Reliability Council, 1987 Reliabilit

Assessment: The Future of Bulk Electric System Reliability in North A


ica 1987-1996 (Princeton, N.J.: North American Electric Reliability Council, September 1987), p. 8.

4. Frederic H. Murphy and Allen L. Soyster, Economic Behavior of Electric Utilities (Englewood Cliffs, N.J.: Prentice-Hall, 1983), p. 75.

5. John C. Hull, The Evaluation of Risk in Business Investment (London: Pergamon Press, 1980), p. 135.

6. Exposure Assessment Group, Office of Health and Environmental Assessment, Exposure Factors Handbook, EPA/600/8-89/043 (Washington, DC: U.S. Environmental Protection Agency, March 1989).

7. John A. White, Marvin H. Agee, and Kenneth E. Case, Principles of Engineering Economic Analysis, 3rd ed. (New York: John Wiley & Sons, 1989), p. 392.

8. Frank H. Knight, Risk, Uncertainty, and Profit (New York: Hough-ton Mifflin, 1921).

9. See A. C. Harvey, The Econometric Analysis of Time Series (Cambridge, MA: MIT Press, 1989), and the discussion on model selection. The classic test in this regard was developed by Gregory Chow, "Tests for Equality Between Sets of Coefficients in Two Linear Regressions," Econometrica 28 (1960): 591-605.

10. Kenneth Boulding, "Social Risk, Political Uncertainty, and the Legitimacy of Private Profit," in R. Hayden Howard, Risk and Regulated Firms (East Lansing: Michigan State University Press, 1973), pp. 82-93. For an early but well-reasoned discussion, see Albert G. Hart, Anticipations, Uncertainty and Dynamic Planning (Clifton, N.J.: Augustus M. Kelly, 1940).

11. See G. J. Thuesen and W. J. Fabrycky, Engineering Economy (Englewood Cliffs, N.J.: Prentice-Hall, 1984).


Appendix

This Appendix focuses on a topic closely linked to the evaluation of financial risk: the logic of probability transformations, or rules for finding the probability or probability distribution of sums, products, and other combinations of random variables, each characterized by its own probability or probability distribution. This is the basis of analytic studies of financial risk and informs risk simulation in a specific sense. This discussion is undertaken without stopping in every instance to explain terms. Additional explication is presented following the main exposition of points below, where concepts are defined, including random experiment, probability distribution, random variable, law of large numbers, probabilitdiscrete and continuous distributions, and cumulative distribution. Maexpositions deal with these points, such as Norman S. Matloff's Probability Modeling and Computer Simulation (Boston: PWS-Kent, 1988). There also are works that treat the foundations of the subject, such as V. Bar-nett's Comparative Statistical Inference, 2d ed. (New York: John Wile& Sons, 1982).

The Appendix concludes by reviewing some main probability distributions, including the normal, Gamma, Beta, and exponential distributions.

PROBABILITY TRANSFORMATIONS

The basic issue discussed here is how the analytic approach to risk analysis, mentioned in Chapter 1, works. There is probably nowhere better

144 Appendix

to begin than with a discussion of the major accomplishment of the analytic approach to probability transformation—the Central Limit Theorem. The Central Limit Theorem can be stated as follows:

THEOREM 1: Suppose xlt x2, JC3, . . . are independent, identically distributed random variables with E(x) = m and Var(x) = v2. Let

Wn = (Tn - nm)l(where Tn = JC, 4- x2 + . . . + xn. Then for any real number u,

Urn P(Wn < u) = P (z < u) where x has a normal distribution with a zero mean and a variance

ofl.

We have n independent random variables, each characterized by the same probability distribution. The major finding of the Central Limit Theorem is that the sum of these n variables can be approximated by a normal distribution. This approximation becomes more and more accurate as the number of terms in the sum becomes larger. Thus, the normal distribution can be said to be a limiting distribution to the probability distribution applying to such sums of random variables. Extensions and generalizations of this have been a major preoccupation of mathematical statistics. Thus, there can be cases in which the random variables to be summed are characterized by different probability distributions or exhibit cross-correlations; that is, they are not stochastically independent and yet their sum converges in the limit to a normal distribution. Indeed, exploratory simulation suggests that it is difficult to produce anything but a bell-shaped curve when summing almost any random variables, provided sufficiently numerous terms are summed.

Part of the reason why the Central Limit Theorem applies quite broadly can be seen if one understands what is involved in summing random variables and determining the probability distribution of their sum, given their individual probability distributions. Two important concepts here are a convolution of probability distributions and the moment-generating function.

Suppose we have two stochastically independent random variables x, and x2. Random variable x, is characterized by the probability density function f(x,), and g(x2) describes the probability density of x2. Stochastic independence means that f(.) is in no way dependent on the value attained by x2, or vice versa. Given this, what can be said about the probability distribution of X = x, + x2?

One approach is to look to the cumulative distribution of the sum X, which we will denote by Fx(.). Thus, Fx(t) = P(X < t) = P(x, + x2 < t). For continuous f(.) and g(.) and nonnegative random variables, this is

Appendix 145

known as the convolution of f(.) and g(.). Note, we use the fact here that the joint distribution of two independent random variables is the product of their marginal distributions—a general form of one of the so-called laws of chance mentioned in the following section. Now, if f(.) and g(.) are normal distributions, Fx(.) will be the cumulative distribution of a normal distribution having a mean and variance that is the sum of the means and variances of f(.) and g(.), respectively. This follows from the additive property of exponents.

Another way of demonstrating this important fact about normal variables summing to normal variables is to consider the moment-generating function. The moment-generating function is defined as

mx = E [e'x]

for any random variable X. This function has a number of interesting properties. Its name obtains from the fact that the kth moment of any random variable X equals the kth derivative of its moment-generating function, if the moment-generating function exists. Otherwise the most important fact is that the moment-generating function is unique and completely determines the distribution of a random variable. Thus, if two random variables have the same moment-generating function, they have the same probability distribution. Again, application of the moment-generating function to the sum of normally distributed independent random variables indicates that this sum is itself normally distributed and has a variance equal to the sum of the component variances and a mean equal to the sum of the component variable means.

This points to extensions of the Central Limit Theorem. Thus, suppose we have a set of random variables (x, , . . . , xn) characterized either by a probability distribution f(.) or g(.) and that as n gets larger the number of random variables characterized by both these distributions increases. Then, Theorem 1 applies to sums of the xt characterized by one or another of these distributions. That subset of terms characterized by f(.) will converge to a normal distribution. Its complement among the n terms characterized by g(.) also will converge to a normal distribution. Then, on the basis of the result obtained from moment-generating functions, that is, that the sum of two variables characterized by normal distributions is itself characterized by a normal distribution, the sum of these n variables can be seen to be approximated by a normal distribution.

This additivity—where random variables characterized by one type of

146 Appendix

probability distribution add up to a sum also characterized by the same type of probability distribution—is true of some but, it should be stressed, not all probability distributions. Thus, it is also true of the Poisson distribution, but it is not true of the uniform distribution.

The point of developing this perspective on probability transformations can be seen in the following quote:

It is well-known that if the future cash flows are normal variates, the NPV [net present value] distribution would also be normal regardless of whether the returns are independent or not. Moreover, if the future cash flows can be assumed to be identically and independently distributed, the NPV distribution would also be approximately normal provided there are a relatively large number of future cash flows. Also, assuming a large number of cash flows, the NPV distribution converges to normality if the cash flows are dependent stationary Markovian random variables. In general, where inter-period dependency exists, other than the aforementioned normal and stationary Markovian cases, the NPV distribution cannot be specified.'

Then, since the net present value is a sum of the discounted cash flows in each year of some period of analysis, it is possible to apply variants of the Central Limit Theorem or the aforementioned logic of probability transformation to reason from independent, identically distributed cash flows and normally distributed cash flows to the normality of the net present value. The authors of the above quote also extend this type of result to correlated cash flows under certain circumstances. In general circumstances, however, only simulation techniques are capable of characterizing the NPV probability distribution.

Analytic methods and simulation experience suggest that it is likely that net income for an enterprise such as a utility will be approximated by a normal distribution. A large enterprise will have numerous cost and revenue accounts that are not highly interrelated. As noted in Chapter 2, this may imply that the probability distribution of the debt coverage ratio (net income divided by debt service) also will be a normal distribution, provided debt service is fixed or nonstochastic. Thus, it should be possible to compute the probability of default, given the variances of the components of net income or its overall variance.

A circumstance can be noted when debt service is determined by a variable interest rate. If, as a consequence of the variable interest rate, total debt service in a particular year is normally distributed around some mean debt service, the coverage ratio will be characterized by a Cauchy probability distribution. This is an analytic result from considering the ratio of two normally distributed random variables. For the Cauchy distribution first and second moments, the mean, and the variance do not exist as mathematical objects, although location parameters analogous to the mean and variance can be developed.

Appendix 147

Clearly, this type of thinking, although technically demanding, has promise. It is interesting that scholars also have considered the mean and variance of a lump-sum cash flow of uncertain timing,2 which might be interpreted as the arrival of a subsidy to a project, promised by some supporting, but not directly responsible, level of government, such as the federal level. The analytic approach can give us some preview of what to expect under given circumstances, expectations that may be extended by the application of simulation to related cases.

BASIC CONCEPTS

Finally, some introduction to basic probability and statistics concepts seems appropriate. Thus, to give readers a flavor of the foundations of probability theory, we consider a conceptual framework below that is suggested by the frequency interpretation of probability. The function of this framework is to motivate definitions of probability that suggest mathematical interpretation. Then, we briefly review the laws of chance, or the rules for figuring the probabilities of joint and mutually exclusive events. In addition, we provide examples of important probability distributions in Text Box Al .

Random Experiment

The first problem is to define random variable. This is usually solved by appeal to other primitive concepts that, ultimately, must be left largely undefined. Thus, in essence, a random variable is the outcome of a random experiment.

Suppose we have one red die and one blue die. If we toss them repeatedly under similar conditions and add the face numbers, we perform a random experiment that generates information about a random variable defined as, say, the sum of the two numbers coming face up. These numbers belong to the set of 36 ordered pairs (1, 1 ) , . . . , (1, 6), (2, 1), . . . , (2, 6 ) , . . . , (6, 6) delineating the sample space of this experiment. Under the frequency interpretation of probability, we accept a law of large numbers, which suggests that as the number of tosses of these two dice increases, the relative frequency of the occurrence of these pairs stabilizes at or converges to a set of ratios or fractions that are identified as the probability of occurrence of the respective pairs of outcomes. Thus, under the frequency interpretation, the probability that the random variable assumes a value of 12 converges to 1 in 36.

148 Appendix

Discrete and Continuous Random Variables, Probability Distributions, and Cumulative Distributions

There are two major types of probability distribution: discrete and continuous.

A discrete random variable is associated with a discrete probability distribution, which canonically lists the relative frequencies of occurrence of the values assumed by this random variable. In the preceding example, this is a discrete triangular distribution. Such distributions have two key properties, the discussion of which is facilitated by symbolism. Let us designate the random variable by z and agree that P(z = 12) means "the probability that the random variable z sums to a total of 12," where, when we are sure which random variable is denoted, we simply write p(12), which can be read as indicating an event has occurred in which the random variable assumes the value 12. Then, we have two properties:

P(z = i) > 0, i =1,2,... , 12

P(z = 1) + P(z = 2) + ... + P(z = 12)

Equation A.l states that probabilities are nonnegative. Equation A.2 asserts that the probabilities of any one of the mutually exclusive events in the sample space occurring is 1.

Continuous distributions are a second important type of probability distribution.3 An example of a continuous probability distribution is produced by the spin of a pointer on a circular disk with numbers along its perimeter. If we suppose this disk has a radius of 0.5 feet, the number the pointer indicates can be defined as a random variable. It will range from 0 at some arbitrary starting point to approximately 2.1412 feet.

Note there exists a curious fact involving continuous probability distributions, namely that P(z = v) = 0, where v is any number between 0 and 2.1412 in the above example. This is because under the frequency interpretation of probability, the ratio of the number of times that the pointer stops at exactly v to the total number of spins of the pointer becomes smaller and smaller, converging to zero, as the spins of the pointer increase. On the other hand, denoting P(v = z) by p(z), there exists the analogue of equation A.2 for continuous distributions, namely

Appendix 149

where instead of a summation we apply the operation of integration from calculus. The relevant question, of course, is the probability that the random variable assumes a value within a finite interval,

For any random variable z, the cumulative distribution function F is defined by

F(t) = P(v < t) (A.6)

For discrete distributions, this is the summation of all the values of the random variable v that are less than or equal to t. For a continuous distribution, this is defined as an integral.

The Laws of Chance

The rules for combining probabilities depend, to a large extent, on whether or not events are stochastically independent and whether events are mutually exclusive or can be realized together. Mutually exclusive events are like distinct values of a random variable in a random experiment; that is, it is impossible for a coin to come up both heads and tails on a single toss. Stochastically independent events, on the other hand, can occur together, but the probability of one event occurring is not related to whether another event occurs or not. Because they are so fundamental, these rules can be called the laws of chance.

Perhaps the simplest rule is that if A and B are mutually exclusive events, P(A and B) = 0, where we read P(x) as "the probability that event x occurs." Like some of the elemental propositions of Euclidean geometry, this rule follows from verbal definition and is intuitive.

More interesting is the assertion that if Al , A2, and A3 are mutually exclusive events, then P(A1 or A2 or A3) = P(A1) + P(A2) + P(A3). Note here that if the Ai are mutually exclusive events that include all possible outcomes, the sum of the probabilities of each mutually exclusive event will equal 1. A general form of this rule may be derived from the proposition that P(A or B occurs) = P(A) + P(B) - P(A and B).

Similarly, we may begin from an obvious rule for the case of stochastic independence. Thus, if A and B are stochastically independent, P(A,B) = P(A)P(B). The general form of this rule is also relatively obvious, and states that P(A and B) = P(A1B)P(B), where the P(x!y) is read as "the conditional probability that event x occurs, given that event y has occurred."

Further extensions of these rules yield interesting results such as Bayes

150 Appendix

Theorem, which determines the probability that one event occurs, given information about the prior occurrence of conditionally related events.

Probability Distributions

Text Box A.l lists the mathematical form of several probability distribution functions important in financial risk analysis. The following text discusses each of these functions in turn, noting important features and potential contexts of application.

Among the discrete distributions, the binomial distribution is basic and important. The binomial distribution describes a random experiment with two mutually exclusive outcomes in a series of repetitions or trials. Suppose P(A) = p, so that the probability that event A does not occur, usually symbolized as P(A), is, by definition, equal to 1 - p = q. Thus, if we are interested in the likelihood of three heads occurring in ten flips of a coin, the answer is given by the expression:

where ('?) indicates the combination of ten things taken three at a time and is equal to 10!/((10-3)!3!).

An interesting aspect of the binomial distribution is that it converges to the normal distribution or another discrete distribution called the Pois-son, as the number of trials increases without limit. If probabilities p and q are roughly the same size, the binomial converges to a normal distribution. On the other hand, if the probability of event A occurring is not near .5, but, rather, much nearer 0, the binomial converges to the Poisson distribution.

Note that the first two moments, the mean and variance, completely characterize the normal distribution.4 Thus, a random variable with a normal distribution may be standardized by the transformation z = (x - u,)/ a where u, is the mean and a is the standard deviation of the random variable X. A standardized variable such as z obeys the three sigma rule—the probability that the absolute difference between a normally distributed variable and its mean is greater than 3a is less than .003. Similarly, a deviation of more than one sigma from the mean is to be expected about once every three trials. Since the normal distribution is characterized by its first two moments, probability tables are easy to prepare and consult.

Another mainstay of the normal distribution is its role as a sampling distribution. Suppose, for example, we have a population listing of a human characteristic such as height and weight. Then, sample from this

Text Box A.l

Binomial Distribution

Normal Distribution i

Poisson Distribution

Uniform Distribution i

b = maximum, a= minimum

Triangular Distribution

a = minimum, b = most l ike ly , c = maximum

Gamma Distribution

Exponential Distribution

152 Appendix

population to estimate its mean with the information we obtain from the mean of the sample. Then, in repeated samplings with the same size lots, the sampling distribution will be normally distributed around the mean of the population and will have a variance determined by the sample size and the variance of the population. For these reasons, modern statistics— associated with names such as Karl Pearson or Irving Fisher—has been erected on the basis of the normal distribution.

The Poisson distribution also has application to real world processes. It is a discrete probability density function with the unusual property that its mean and variance are equal. This distribution is perhaps most important in queuing problems. Given purely random arrival times in a line, timing of telephone calls, and so on, the number of arrivals in a line or calls in an interval of time is described by a Poisson distribution.

We have discussed the uniform distribution throughout the text of the book. It is a continuous distribution possessing finite range and the property that intervals of equal size in its range always have the same probability.

The triangular distribution also is discussed in the text at some length. The triangular distribution is a rough approximating function for any unimodal probability distribution.

One way to consider probability distribution is in terms of parametric families. In this light, the binomial, normal, and Poisson distributions are linked by convergence processes. The normal distribution, furthermore, is linked by probability transformation to the Cauchy distribution (as a ratio of two normal variates) and the Chi-square distribution, as the sum of squared normal variates.

The Gamma distribution, on the other hand, is a form that is entitled to a family of its own due to the ease in which it is transformed algebraically to other related distributions. Here, the peculiar symbol V in Text Box Al in the denominator of the Gamma distribution is the Gamma function, a sort of generalization of the factorial. The Gamma function has a non-negative range and is determined by two parameters (a, (3) whose product is the mean of the distribution. Gamma functions can assume a variety of shapes, depending on the values selected for these parameters. For specific values of these parameters, the Gamma function becomes a Chi-square, exponential, Erlang, or Beta distribution.5

Finally, the exponential distribution has an interesting relationship to the Poisson in waiting time problems. Thus, in a Poisson arrival process, the distribution of waiting times is exponential. The exponential distribution often is held to be a good representation of failure processes, such as the time it takes for a part to wear out.

Appendix 153

NOTES

1. L. C. Leung, V. V. Hui, and G. A. Fleischer, "On the Present-Worth Moments of Serially Correlated Cash Flows," Engineering Costs and Production Economics 16 (1989): 281-289.

2. John H. Estes, "Stochastic Cash Flow Evaluation Under Conditions of Uncertain Timing," Engineering Costs and Production Economics 18 (1989): 65-70.

3. Mixed cases also can exist. 4. The moments of a distribution provide important summary data

concerning a distribution's central tendency, dispersion, and shape. The first moment is the mean, average, or expected value—a measure of central location. The second moment is the variance—a measure of the dispersion around the mean. The third and fourth moments are less familiar but indicate broadly whether and how the distribution is nonsy-metric (right or left skewed) and whether it is flat or peaked (kurtosis).

5. See Stephen Kokoska and Christopher Nevison, Statistical Tables and Formulae (New York: Springer-Verlag, 1989).


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Index

Amortization schedules, 105-112 Autocorrelation, 80

Benefit cost analysis, 121-123 Beta distribution. See Probability

distribution Bootstrap methods, 29, 60 Bulk power distributor, 99-100 Bureau of Economic Research

OBERS model, 87

Capacity planning, 112-121 Central Limit Theorem, 57 Conservation, 78-79 Consumer willingness to pay, 122 Contingency funds, 57-60 Critical path analysis, 61 Critical path modeling (CPM), 47

Debt schedules, 100-112 Default risk, 20-22 Demand: definition of, 71; inelas

tic and elastic, 74-77; Law of, 70; price elasticity of, 72

Disbursement pattern for construction expenditures, 62

Economic forecasts, 86-88 Economies of scale: engineering

basis for, 15; role in utility planning, 11, 114, 117-121

Electric Power Research Institute (EPRI), 17, 132

Event tree, 35-36 Exponential distribution. See

Probability distribution

Financial risks, 47-68, 121-123

Gamma distribution. See Probability distribution

Hurwicz rule, 138

Inelastic and elastic demand. See Demand, inelastic and elastic

Internal rate of return on capital, 43

Junk bonds, 3

Law of demand. See Demand, Law of

166 Index

Least cost planning procedures, 113, 117-121

Microcomputer simulation programs: @RISK, 9

Minimum present value criterion, 23, 114

Monte Carlo simulation, 17

National Environmental Protection Act, 51

Normal distribution. See Probability distribution

Nuclear power plant delays and cost overruns, 53

Pareto's Law, 56 PERT analysis, 47, 61 Poisson distribution. See Probabil

ity distribution Political risk, 20 Population forecasts: accuracy of,

87; baby boom and, 88; fertility rates and, 89; methods, 89-90; migration and, 90; mortality rates and, 89; time interdependency parameter for, 92

Price elasticity of demand. See Demand, price elasticity of

Probability: frequency interpretation, 24; subjectivist interpretation, 24

Probability distribution: Beta, 134; exponential, 25; Gamma, 134; normal, 25; Poisson, 25; triangular, 56; uniform, 57; Weibull, 134

Probability elicitation: Delphi method, 26-27; jury of executive opinion, 26; probability encoding, 27

Probability transformations, 143-147

Pure uncertainty. See Uncertainty

Random numbers, 34; Lotus 1-2-3 @RAND function, 34

Range estimation, 54; of population growth, 91-92

Rate effects: rate shock, 70; simultaneity in estimates of, 78

Revenue requirement, 20 Risk analysis, 5-8 Risk factors, indentification of,

19-23 Risk pooling, 60 Risk preferences, 36-39; first or

der stochastic dominance, 38-39 Risk profiles, 6-7, 36-39 Risk simulation, 9, 17-18, 56-57,

117-121, 130-131; robustness in, 133-136

Robustness. See Risk simulation, robustness in

Sampling strategies, 34 Scenario development, 10 Sensitivity analysis, 10 Sinking funds, 101-105 Standard & Poor's, 8 Stochastic dominance, 38, 40, 41 Stochastic independence, 56, 60 Structural models, 33

Tennessee Valley Authority, 52 Time series analysis, 30-33, 79,

106-108 Trans-Alaska pipeline cost over

runs, 51 Triangular distribution. See Prob

ability distribution

Uncertainty, 90-94; maximin criterion for decisions, 138; pure, 136-138

Index 167

Uniform probability distribution. See Probability distribution

U.S. Army Corps of Engineers, 52

Utility bond ratings, 3, 43

Verification of risk analysis, 132— 133

Washington Public Power Supply System (WPPSS), 3

Weibull distribution. See Probability distribution

White noise residuals, 32-33

About the Author

C. VAUGHAN JONES is a principal in Economic Data Resources of Boulder, Colorado, a consulting firm specializing in public utility and infrastructure issues. His numerous research studies have appeared in the Journal of Economic Issues, Economic Letters, American Economist, Atlantic Economic Journal, and Water Resources Research.

Documents

Financial Risk Analysis of Infrascructure Debt