Conceptual Issues in Financial Risk Analysis: A Review for Practitioners Joseph Tham and Lora Sabin February, 2001 Lora Sabin is Senior Program Officer at the Center for Business and Government, John F. Kennedy School of Government, Harvard University, where she is involved in developing and managing training programs in various developing countries, including Vietnam and China. From 1998-2000, she was the Academic Director of the Fulbright Economics Teaching Program (FETP) in Vietnam, a teaching center funded by the U.S. State Department and managed by Harvard University. Joseph Tham is a Project Associate at the Center for Business and Government, John F. Kennedy School of Government, Harvard University. Currently, he is teaching at the Fulbright Economics Teaching Program in Ho Chi Minh City, Vietnam. Before moving to Vietnam, he taught in the Program on Investment Appraisal and Management at the Harvard Institute for International Development for many years. He has also served as a consultant on various development projects, including working with the government of Indonesia on educational reform in 1995-96. The authors would like to thank the following individuals for their helpful feedback: Tran Duyen Dinh, Le Thi Thanh Loan, Brian Quinn, Nguyen Bang Tam, Cao Hao Thi, Bui Van, Nguyen Ngoc Ho, Graham Glenday and Baher El- Hifnawi. Responsibility for all remaining errors lies with the authors. Critical comments and constructive feedback may be addressed to the authors by email at lora_sabin@ksg.harvard.edu and ThamJx@yahoo.com. 2 Conceptual Issues in Financial Risk Analysis: A Review for Practitioners Abstract: This paper presents a critical review of the conceptual issues involved in accounting for financial risk in project appraisal. It begins by examining three of the main approaches to assessing risk: the use of the probability distributions of project outcomes, such as the NPV, the use of a single risk-adjusted discount rate for the life of the project, and the use of certainty equivalents. The first two approaches are very common, while the third is used less often. Next, it proposes an approach based on annual “certainty equivalents” that is conceptually similar to using multiple risk-adjusted discount rates and which involves specifying the risk profile of a project over its lifetime. Finally, this approach is illustrated with a simple numerical example. The certainty equivalent approach is compelling because it clearly separates the time value of money from the issue of risk valuation. While the authors point out the analytical challenges of the certainty equivalent approach, they note that its informational requirements are no greater than those posed by the older, more traditional approaches, while avoiding the numerous inadequacies of the latter. JEL codes D61: Cost-Benefit Analysis D81: Criteria for Decision-Making Under Risk and Uncertainty G31: Capital Budgeting H43: Project evaluation Key words or phrases Risk Analysis, Monte Carlo Simulation, Cash Flow Valuation, Project Appraisal. Available for free download from the Social Science Research Network on the internet at: papers.SSRN.com 3 INTRODUCTION It would not be exaggerating to argue that financial risk analysis is one of the most important and most difficult components of project appraisal. Such analysis is especially important because the financial viability of a project may be critical for its long-term sustainability and survivability. Its particular difficulty is due to the inherent challenge of pricing risk with market indicators, an exercise which even in developed countries, where capital markets are mature and function well, is far from simple. In such countries, capital markets can play an invaluable role in providing general market-based assessments of risk and bounds to the price of risk for given projects. In developing countries, where inadequate and immature capital markets predominate, lack of reliable market-based information about the price of risk makes financial risk analysis a truly daunting undertaking. 1 At the same time, the rapid decline in the cost of computing power has made it increasingly easy and fashionable to conduct certain types of analysis, such as Monte Carlo simulation, in the financial risk analysis of project evaluations. 2 The popularization of (Monte Carlo) simulation analysis, however, should be viewed as a mixed blessing. On the one hand, the ability to perform sophisticated computer simulations is clearly helpful in providing valuable 1 Some project analysts may not appreciate that fact that in many developing countries, especially transitional economies, the application of risk-pricing models, such as the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) is particularly difficult due to unreliable or nonexistent data. 2 For a recent example, see Dailami et al. (1999). Also see, Jayawardena, et al. (1999) and Jenkins and Lim (1998). For an early discussion of Monte Carlo simulation in the context of project appraisal, see Savvides (1988). For references to earlier literature, see the citations in Dailami et al. (1999). At a practical level, in many developing countries, more advanced techniques, such as contingent claims analysis, would simply be out of the question. In addition, one cannot simply specify the cash flows as Brownian motion with certain values for the key parameters and solve the stochastic differential equation with Ito calculus or numerical methods. 4 information about the character of a project and in understanding the effect of certain variables - contractual arrangements, for example - on important project outcomes. 3 On the other hand, analysts have increasingly relied upon the use of computer simulations to carry out financial risk evaluations without a corresponding appreciation of the serious limitations of such analysis. 4 In practical project appraisal, there is a tough balancing act to maintain between rigorous techniques and user-friendly applied techniques. Computer simulation analysis is fundamentally limited by the nature of its final output – typically a probability distribution of the project outcome in question, such as the financial Net Present Value or Internal Rate of Return - and the difficulty of its interpretation. Although the probability distribution suggests to the analyst the likelihood that the project will have an undesirable outcome, the true relationship between this probability and the inherent risk of the project is far more complicated. The current danger of the popularity of Monte Carlo simulation analysis is precisely this temptation to confuse the rather simple use of an output produced by a powerful and sophisticated computer technique with a meaningful understanding of project risk. 5 Ironically, the limitations of computer simulation analysis, and the problems in interpreting the probability distributions that it yields, are well understood in the theoretical literature. Many practitioners 3 See Glenday (1996). 4 For instance, at the click of a mouse, an analyst selects the probability distributions for the relevant risk variables. The computer will then conduct a comprehensive Monte Carlo simulation and produce a mountain of outputs, most typically probability distributions of desired outcomes. 5 For example, as Savvides writes, “Project risk is thus portrayed in the position and shape of the cumulative probability distribution.” See Savvides (1988), pp. 12-13. For more recent practical applications in risk analysis, see Dailami, et al., (1999), p. 5; Jayawardena, et al., (1999), p. 46; and Jenkins and Lim (1998), p. 57. 5 of project evaluation, however, have failed to recognize the inadequacies of this type of analysis when it comes to assessing and modeling the level of financial risk associated with a given project. 6 It is also common practice to make use of a single, risk-adjusted discount rate when analyzing the long-term financial risk of a project. In this case, there would appear to be a misunderstanding of one of the key issues in risk analysis, the specification of risk over a broad time horizon, and its year-by-year resolution, or the “intertemporal resolution of uncertainty.” 7 Here, the main problem is that it is impossible to capture two independent dimensions – the time value of money and the valuation of risk – in a single parameter. Again, this is an issue that has been raised by theoreticians, but apparently without leading to significant progress in assessing and modeling long-term risk in practical project appraisal. 8 In view of these trends, this paper seeks to present a critical review of the conceptual issues involved in accounting for financial risk analysis in project appraisal. Part One discusses three of the main approaches to accounting for the potential financial risk of an investment project: the use of probability 6 For a critical assessment of economic risk analysis, as contrasted with financial risk analysis, see Anderson (1989) and Dixit and Williamson (1989). In this paper, we do not address the equally important and relevant issue of economic risk analysis and the determination of the economic opportunity cost of capital. For general textbook discussions of risk analysis, see Brealey and Myers (1996, Chapter 9), Haley & Schall (1980, Chapter 9), Levy and Sarnat (1994, Chapter 10), Zerbe & Dively (1994, Chapter 16), Eeckhoudt & Gollier (1995), Benninga and Sarig (1997, p. 11), and Vose (1996). 7 Of course, if there is a known and constant beta for an all-equity claim on cash flow (together with a known, constant market risk premium and a known, constant Treasure bill rate), then it is appropriate to use a constant risk-adjusted discount rate. See Myers and Ruback (1987) or Zerbe and Dively (1994) for a fuller explanation of these conditions. 8 See, for example, Myers and Turnbill (1977) and Bhattacharya (1978). As Dailami, et al. (1999), p. 5, point out, “Specification of uncertainty through time may affect a project’s cash flow and is also an important issue in project valuation.” 6 distributions of project outcomes, the use of a single, risk-adjusted discount rate, and the use of certainty equivalents. As an alternative to the first two approaches, we argue that the most conceptually appropriate technique begins by specifying the risk profile of a project over its lifetime. 9 This necessarily involves making use of multiple, risk-adjusted discount rates, or, correspondingly, the annual “certainty equivalents” with which they are mathematically linked. In Part Two, we illustrate our preferred approach in dealing with the “intertemporal resolution of uncertainty” in financial risk analysis in a way that may be understood and adopted in carrying out project appraisals. 10 PART ONE In carrying out risk analysis, the question naturally arises, how do we take into account the annual risk of the project over its entire lifetime? The main approaches to date of dealing with this issue have made use of the following three analytical tools: 1) probability distributions of the NPV and/or IRR of a project; 2) a single risk-adjusted discount rate; and 3) annual certainty equivalents. 11 Each of these approaches will be examined in more detail below. To help focus the discussion, let us first specify a simple investment project, the three-period “Project Risquey” shown in Table 1. At the end of year 0, Project Risquey has a required investment K and will enjoy expected benefits at 9 The idea of certainty equivalents is not new. However, it is not widely used in practice. 10 To a large extent, our analysis is inspired by Myers and Robichek (1966), Chapter 5. 11 We do not explicitly discuss the Capital Asset Pricing Model (CAPM), although this model is closely related to the main ideas presented in this paper. For example, if the data were available, the required returns could be estimated with the CAPM. 7 the end of years 1, 2, and 3 of B 1 , B 2 , and B 3 , respectively (there is no salvage value at the end of year 3). 12 Table 1: Cash Flow Statement of Project Risquey. (an all-equity project) Year: 0 1 2 3 Expected Benefits B 1 B 2 B 3 Investment -K Net Cash Flow -K B 1 B 2 B 3 Discounted NCF -K B 1 *γ B 2 *γ 2 B 3 *γ 3 Annual CF less Discounted CF B 1 *(1 - γ) B 2 *(1 - γ 2 ) B 3 *(1 - γ 3 ) For now we do not consider the impact of financing and assume that Project Risquey is fully financed with equity (i.e., there is no debt financing). 13 We assume that ρ, the required risk-adjusted return on an investment with all- equity financing, is 10%. 14 In addition, we assume that there are no taxes and no foreign exchange risks associated with the project. Letting γ = 1/(1+ρ), the Net Present Value (NPV) for the equity investor of Project Risquey is as follows: NPV PR @ρ = -K + B 1 + B 2 + B 3 (1 + ρ) (1 + ρ) 2 (1 + ρ) 3 = -K + B 1 *γ + B 2 *γ 2 + B 3 *γ 3 12 For simplicity, we assume that the expected values of the annual benefits are constant over the life of the project, even though in reality, the profile of a project’s annual benefits may be nonlinear. 13 With debt financing, the cash flows to the recipients, namely the debt- and equity-holders, would be censored and this would complicate the analysis. Furthermore, with debt financing, we would have to specify the impact of leverage on the value of the levered cash flows. 13 If there was no risk, and the benefits were to occur with certainty, then the required return would be the risk-free rate. 8 = -K + Σ t=1 3 B t *γ t (1) If, for example, B t = $402.11 for 1 ≤ t ≤ 3, and K = $1,000, the project’s NPV would be equal to zero. Equivalently, its Internal Rate of Return (IRR) would be 10%. 15 How should we think of the factor γ? Basically, it is the adjustment factor for the time value of money and the cost of risk. 16 Since we have assumed that the required return on equity is constant for the life of the project, the discounted benefits decrease year-by-year at the rate of (1-γ). In this case, γ = 90.91%, so discounted benefits decrease by 9.1% from year to year. Another way to think about this is to calculate the ratio of the discounted annual benefits in year t+1 with the discounted annual benefits in year t, which yields a result equal to γ. Thus, in a simple deterministic analysis, the above project would be acceptable because the NPV at the required risk-adjusted return to unlevered equity is zero (or equivalently the IRR is equal to the required rate of return). 17 There would be no need to take into account the variances of the annual benefits. However, suppose we consider instead a stochastic analysis and specify constant expected values and variances for the annual benefits. 18 That is, 15 The IRR is obtained by finding the discount rate at which the NPV is zero. 16 Later in the paper, we will show how we can use certain equivalent to separate the compensation for the time value of money and the risk. 17 We recognize that in the presence of options, the simple NPV rule for capital budgeting and project selection may be inadequate. See Dixit & Pindyck 91994) and McDonald (1998). In this paper we will assume that the simple NPV rule is appropriate and will not address these additional complications. 9 µ B1 = µ B2 = µ B3 = µ B (2a) (σ B1 2 ) = (σ B2 2 ) = (σ B3 2 ) = (σ B 2 ) (2b) We also assume zero serial correlation between the annual benefits, meaning that benefits in any year s are independent of benefits in year t, or Cov(B s , B t ) = 0 for all s and t, s ≠ t. There is no uncertainty about the cost of the initial investment; the only uncertainty concerns the annual benefits as indicated by the annual variances (see Table 2). Table 2: Expected Values and Variances for the Annual Benefits. Year: 0 1 2 3 Variance (σ B1 2 ) (σ B2 2 )(σ B3 2 ) Expected Benefits E(B 1 )E(B 2 )E(B 3 ) For simplicity, we assume that the annual variances are constant over the life of the project, although, in reality, it is more likely that they would vary. 19 We may also specify that the probability distributions for the annual benefits are normal (Gaussian), though this assumption is not a necessary one. In a more complex cash flow statement with many different line items, it may be a practical impossibility to specify the functional form of the annual cash flow since it will be dependent on many line items in the cash flow statement. 18 We assume that these expected values were obtained from a Monte Carlo simulation conducted with sufficient runs to obtain estimates within the desired level of accuracy. 19 In other words, we are assuming that the stochastic process for the benefits is stationary in the mean, variance, and covariance. 10 (1) Probability distribution of Project NPVs and IRRs This approach involves producing and analyzing probability distributions of a desired project outcome, typically the NPV or the IRR. These distributions are obtained by running computer simulations of possible future cash flows, based on specifications of the probability distributions of the risk variables previously identified as having an impact on the project’s cash flow, and then discounting the resulting cash flows by the risk-free discount rate. 20 The final step requires the analyst to examine the probability distributions and, most typically, to determine the likelihood that a particular project outcome will be a certain value, for example, the probability that the NPV will be negative. 21 The use of these probability distributions in risk analysis is appealing because they appear to be easy to explain and interpret, while containing a lot of information. After all, what could be more useful than a range of possible project outcomes? On closer examination, however, this apparent attractiveness is misleading. First and foremost is the difficulty of interpreting the “probability distribution” of the NPV. In short, what does such a probability distribution really mean? 22 With capital markets, we would expect a single risk-adjusted price for 20 A particularly difficult issue that we will ignore is the determination of the appropriate intermporal probability distributions for the key risk parameters that have been identified through sensitivity or scenario analyses, especially in the presence of sparse or no historical data. 21 In the case of our simple Project Risquey, we would not need to carry out simulation analysis to determine the expected value and the variance of the NPV. In this case, the Expected NPV Proj or µ NPV Proj = - E(K) + Σ t=1 3 E(B t )*γ t and the Variance NPV @ρ Proj = [σ NPV Proj ] 2 = Σ t=1 3 Var(B t )*[γ t ] 2 = {Σ t=1 3 [γ t ] 2 }*(σ B 2 ). If we were to specify particular probability distributions for the annual benefits, however, then simulation analysis would help us determine the probability that the project’s NPV would be negative, just as with more complex projects. 22 Using rather harsh language, Brealey & Myers (1996), p. 255, suggest: “The only interpretation we can put on these bastard NPVs is the following. Suppose all uncertainty about the project’s ultimate cash flows were resolved the day after the project was undertaken. On that day the [...]... the case that α32 must be equal to α31 The adjustment factor α31 is based on the information available at the end of year 1, while the adjustment factor α32 takes into account any relevant new information available at the end of year 2 Similarly, α21 and α31 may not be equal to α20 and α30, respectively, after the passage of the first year of the project Thus in each year, any available information about... Table 3 shows the present values of the annual cash flows discounted at both the unadjusted rate of 10% and the risk- adjusted rate of 15% As can be seen in Line 2, the former values are declining at a constant annual rate of 9.1% In year 1, the adjustment as a percentage of the annual benefits is 9.09% (see line 4); in year 2, the percentage rises to 17.4%; and in year 3, it is 24.9% In contrast, as... below in Table 6, however, the annual adjustment factors will now vary, indicating an increasing adjustment for risk over time This once again illustrates the difference between a constant adjustment for risk and a constant risk- adjusted discount rate Table 6: Certainty Equivalents using Varying Adjustment Factors Year: Expected Benefits Certainty Equivalent Adjustment Factor Discounted NCF NPV IRR 0 -1,000.00... certainty equivalent method makes separate adjustments for risk and time The relevant question one asks using this technique is, “what is the smallest payoff for which the investor would exchange the risky cash flow?” Since that amount is the value equivalent of a safe cash flow, it may be safely discounted at the risk- free rate This approach is clearly closely linked conceptually and mathematically... By making use of certainty equivalent adjustment factors, or their corresponding multiple risk- adjusted discount rates, we are nudged away from the earlier easier techniques that rely on a single summary measure, such as the probability distribution of the NPV or a single risk- adjusted discount rate Rather than staring at the probability distributions of NPVs for a project, and waiting in vain for inspiration... shown in Line 5 of Table 3, the risk- adjusted discounted benefits are declining at an annual rate of 13.0% In absolute terms, the adjustment in year 1 as a percentage of the annual benefits is 13.0% (see line 7); in year 2, the percentage increases to 24.4%; and in year 3, it is 34.2% Thus the constant risk- adjusted discount rate does not imply a constant deduction for risk Rather, it implies a larger... annual compensation for risk is constant in each year, due to our initial assumption that the adjustment factors for calculating the certainty equivalents remain unchanged over the life of the project Thus in each year, the compensation for risk is equal to the difference between the unadjusted and adjusted annual benefit, or (1- α)*Bt = (10%)*446.79 = 44.68 (3) Revisions in the annual valuations For. .. each of the first two years of the project, there is a gradual resolution in the uncertainty of the values of future benefits and a belief that future risk has declined Accordingly, the adjustment factors are increased from 90% to 93% at the end of year 1 and again to 95% at the end of year 2.30 The results of this analysis are summarized below in Table 10 Appendix B contains more detailed calculations... the case of a single risk- adjusted discount rate, the adjustment for risk is conveniently and implicitly embodied in a single measure that also includes compensation for the time value of money In the absence of common marketbased data and measures of the price of risk, there are no truly reliable assessments for risk and any analysis will be largely based on the best “educated guesses” of experts in. .. Substituting line 15b into 1 5a, we obtain that: 1 = (1 + ρt+1) and: ρt+1 = αt αt+1 αt +1_ αj -1 (1 6a) (16b) In words, the risk- adjusted discount rate in year t+1 is equal to the ratio of the certainty equivalent in year t to the certainty equivalent in year t+1, minus one 21 PART TWO In view of the discussion in Part One, the question naturally arises: how should we handle the valuation of risk in practical . ThamJx@yahoo.com. 2 Conceptual Issues in Financial Risk Analysis: A Review for Practitioners Abstract: This paper presents a critical review of the conceptual issues involved in accounting. Conceptual Issues in Financial Risk Analysis: A Review for Practitioners Joseph Tham and Lora Sabin February, 2001 Lora Sabin is Senior Program Officer