Conceptual Issues in Financial Risk Analysis: A Review for Practitioners ppt

It begins by examining three of the main approaches to assessing risk: the use of the probability distributions ofproject outcomes, such as the NPV, the use of a single risk-adjusted dis

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 andGovernment, John F Kennedy School of Government, Harvard University, whereshe is involved in developing and managing training programs in variousdeveloping countries, including Vietnam and China From 1998-2000, she wasthe Academic Director of the Fulbright Economics Teaching Program (FETP) inVietnam, a teaching center funded by the U.S State Department and managed byHarvard University

Joseph Tham is a Project Associate at the Center for Business andGovernment, John F Kennedy School of Government, Harvard University.Currently, he is teaching at the Fulbright Economics Teaching Program in Ho ChiMinh City, Vietnam Before moving to Vietnam, he taught in the Program onInvestment Appraisal and Management at the Harvard Institute for InternationalDevelopment for many years He has also served as a consultant on variousdevelopment projects, including working with the government of Indonesia oneducational reform in 1995-96

The authors would like to thank the following individuals for their helpfulfeedback: Tran Duyen Dinh, Le Thi Thanh Loan, Brian Quinn, Nguyen BangTam, Cao Hao Thi, Bui Van, Nguyen Ngoc Ho, Graham Glenday and Baher El-Hifnawi Responsibility for all remaining errors lies with the authors Criticalcomments and constructive feedback may be addressed to the authors by email atlora_sabin@ksg.harvard.edu and ThamJx@yahoo.com

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 ofproject outcomes, such as the NPV, the use of a single risk-adjusted discount ratefor the life of the project, and the use of certainty equivalents The first twoapproaches are very common, while the third is used less often Next, it proposes

an approach based on annual “certainty equivalents” that is conceptually similar tousing multiple risk-adjusted discount rates and which involves specifying the riskprofile of a project over its lifetime Finally, this approach is illustrated with asimple numerical example

The certainty equivalent approach is compelling because it clearlyseparates the time value of money from the issue of risk valuation While theauthors point out the analytical challenges of the certainty equivalent approach,they note that its informational requirements are no greater than those posed bythe 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

Key words or phrases

Risk Analysis, Monte Carlo Simulation, Cash Flow Valuation, ProjectAppraisal

Available for free download from the Social Science Research Network on

the internet at: papers.SSRN.com

It would not be exaggerating to argue that financial risk analysis is one ofthe most important and most difficult components of project appraisal Suchanalysis is especially important because the financial viability of a project may becritical for its long-term sustainability and survivability Its particular difficulty isdue to the inherent challenge of pricing risk with market indicators, an exercisewhich even in developed countries, where capital markets are mature and functionwell, is far from simple In such countries, capital markets can play an invaluablerole in providing general market-based assessments of risk and bounds to the price

of risk for given projects In developing countries, where inadequate andimmature capital markets predominate, lack of reliable market-based informationabout 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 hasmade it increasingly easy and fashionable to conduct certain types of analysis,such as Monte Carlo simulation, in the financial risk analysis of projectevaluations.2 The popularization of (Monte Carlo) simulation analysis, however,should be viewed as a mixed blessing On the one hand, the ability to performsophisticated 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.

information about the character of a project and in understanding the effect ofcertain variables - contractual arrangements, for example - on important projectoutcomes.3 On the other hand, analysts have increasingly relied upon the use ofcomputer simulations to carry out financial risk evaluations without acorresponding appreciation of the serious limitations of such analysis.4 Inpractical project appraisal, there is a tough balancing act to maintain betweenrigorous techniques and user-friendly applied techniques.

Computer simulation analysis is fundamentally limited by the nature of itsfinal output – typically a probability distribution of the project outcome inquestion, such as the financial Net Present Value or Internal Rate of Return - andthe difficulty of its interpretation Although the probability distribution suggests

to the analyst the likelihood that the project will have an undesirable outcome, thetrue relationship between this probability and the inherent risk of the project is farmore complicated The current danger of the popularity of Monte Carlosimulation analysis is precisely this temptation to confuse the rather simple use of

an output produced by a powerful and sophisticated computer technique with ameaningful understanding of project risk.5 Ironically, the limitations of computersimulation analysis, and the problems in interpreting the probability distributionsthat it yields, are well understood in the theoretical literature Many practitioners

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.

of project evaluation, however, have failed to recognize the inadequacies of thistype of analysis when it comes to assessing and modeling the level of financialrisk associated with a given project.6

It is also common practice to make use of a single, risk-adjusted discountrate when analyzing the long-term financial risk of a project In this case, therewould 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 moneyand the valuation of risk – in a single parameter Again, this is an issue that hasbeen 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 theconceptual issues involved in accounting for financial risk analysis in projectappraisal Part One discusses three of the main approaches to accounting for thepotential 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.”

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 specifyingthe risk profile of a project over its lifetime.9 This necessarily involves makinguse 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 andadopted in carrying out project appraisals.10

PART ONE

In carrying out risk analysis, the question naturally arises, how do we takeinto account the annual risk of the project over its entire lifetime? The mainapproaches to date of dealing with this issue have made use of the following threeanalytical 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 ofthese approaches will be examined in more detail below

To help focus the discussion, let us first specify a simple investmentproject, 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.

the end of years 1, 2, and 3 of B1, B2, and B3, respectively (there is no salvagevalue at the end of year 3).12

Table 1: Cash Flow Statement of Project Risquey.

(an all-equity project)

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.

How should we think of the factor γ? Basically, it is the adjustment factorfor the time value of money and the cost of risk.16 Since we have assumed that therequired return on equity is constant for the life of the project, the discountedbenefits decrease year-by-year at the rate of (1-γ) In this case, γ = 90.91%, sodiscounted benefits decrease by 9.1% from year to year Another way to thinkabout this is to calculate the ratio of the discounted annual benefits in year t+1with 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 acceptablebecause 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 specifyconstant expected values and variances for the annual benefits.18 That is,

Table 2: Expected Values and Variances for the Annual Benefits.

For simplicity, we assume that the annual variances are constant over thelife of the project, although, in reality, it is more likely that they would vary.19 Wemay also specify that the probability distributions for the annual benefits arenormal (Gaussian), though this assumption is not a necessary one In a morecomplex cash flow statement with many different line items, it may be a practicalimpossibility to specify the functional form of the annual cash flow since it will bedependent on many line items in the cash flow statement

(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 distributionsare obtained by running computer simulations of possible future cash flows, based

on specifications of the probability distributions of the risk variables previouslyidentified as having an impact on the project’s cash flow, and then discounting theresulting cash flows by the risk-free discount rate.20 The final step requires theanalyst to examine the probability distributions and, most typically, to determinethe likelihood that a particular project outcome will be a certain value, forexample, the probability that the NPV will be negative.21

The use of these probability distributions in risk analysis is appealingbecause they appear to be easy to explain and interpret, while containing a lot ofinformation After all, what could be more useful than a range of possible projectoutcomes? On closer examination, however, this apparent attractiveness ismisleading First and foremost is the difficulty of interpreting the “probabilitydistribution” of the NPV In short, what does such a probability distribution reallymean?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 NPVProj or

µ NPV Proj = - E(K) + Σ t=13 E(Bt)* γ t and the Variance NPV@ρProj = [ σ NPV Proj]2 = Σ t=13 Var(Bt)*[ γ t ]2 = { Σ t=13 [ γ t ]2}*( σ B2) 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

any asset This means that the market has taken into account the opportunity cost

of capital relative to that asset by discounting its future cash flows by anappropriate discount rate But a probability distribution of a project’s NPV

presents a multiplicity of possible prices rather than a single, risk-adjusted price.

Moreover, it is not possible to reduce the probability distribution to a moremeaningful single market-based price If a project has many so-called “presentvalues,” then it lacks the single, risk-adjusted value associated with a “NPV.”

In practice, the question of what discount rate to use to generate theprobability distribution of the NPV tends to be downplayed and different analystsdiscount the cash flows by a variety of different discount rates, including the risk-free rate.23 The problem with this practice is that the only appropriate discountrate to use when discounting a future cash flow is one that embodies theopportunity cost of capital relative to that project There is no economicjustification, for instance, for using the risk-free discount rate to discount anyexpected cash flow except for one that is absolutely certain But the use of aproperly risk-adjusted discount rate is rarely observed in the Monte Carlosimulations for project evaluations Why? Because if one knew the trueopportunity cost of capital, it would not be necessary to run simulations designed

to reveal the project’s risk, since that opportunity cost of capital would itselfembody the risk of the project Thus the probability distribution approachessentially dodges the lack of information about a given project’s risk by glossing

project’s opportunity cost of capital would fall to the risk-free rate The distribution of NPVs represents the distribution of possible project values on that second day of the project’s life.”

23 In many developing countries, poor capital markets preclude the modeling and estimation of the required return to equity In addition, one must be explicit about the extent to which the project analyst believes that the M & M world is an appropriate approximation of reality.

over the need to discount at the appropriate (risk-adjusted) discount rate Instead,

it places the “analysis of risk” in a final determination of the likelihood of aparticular project outcome, such as the probability of a negative NPV As Stewart

C Myers has written, “ it is very difficult to interpret a distribution of NPVs Because the whole edifice is arbitrary, managers can only be told to stare at thedistribution until inspiration dawns No one can tell them how to decide or what

to do if inspiration never dawns.”24

In addition, the probability distribution approach does not take intoaccount the “intertemporal resolution of uncertainty” because it does not specifythe actual nature of the risk over the life of the project That is, once we haveidentified the relevant risk variables for the project, how are they expected tochange over the life of the project? Since the risk profile will change over eachyear of a project, a meaningful analysis should at least anticipate the changingnature of the risk profile over the project’s life by specifying the assumptionsregarding the process of change in the risk profile.25

(2) Risk-adjusted discount rates

The second approach, namely the use of a single risk-adjusted discountrate, is extremely popular and widespread Again, the idea is deceptively simple

To obtain the required return on equity, the analyst adjusts the return on equity by

a certain (subjectively determined) percentage to account for the risk of the

24 See Myers (1996), p 255.

25 We recognize that this is much easier said than done In practice, the best that can be done would be some form of stochastic scenario analysis based on the results of deterministic scenario analyses or sensitivity analyses.

project.26 The equity investor may compare the expected risk-profile of thecurrent project with the risk-profiles of other projects and come to the conclusion,for instance, that a risk-premium of five percentage points would be sufficient tocompensate the equity investor for the risk in the project.

For example, let us assume that the free rate, r, is 10% and the premium is 5% If Project Risquey’s initial investment is $1,000 and annualreturns from year 1 through year 3 are $402.11 (as above), then the project wouldnot be acceptable at a risk-adjusted discount rate of 15% because the NPV would

risk-be negative In this case, if we let ρ, the required return to the equity investor,represent the new risk-adjusted rate, where π = 1/(1+ρ), then Project Risquey’sNPV may be expressed as:

We can then easily verify that the value of the three annual benefits would need to

be at least $437.98 to make the project acceptable at the new risk-adjusted rate of15% (see Table 3)

26 Assuming that there are no relevant financial data, the determination of the risk premium is fraught with difficulties In principle, with well-developed markets for investors, the nonsystematic risk is diversifiable and the risk premium only applies to the systematic risk But in developing countries, is it reasonable to assume that the investors hold well-diversified portfolios? Many analysts use various informal rules of thumb for the risk premium Some might advocate, for instance, using a risk premium of 6% wherever in doubt As quoted in Benninga & Sarig (1997, p 90), “Whenever in doubt as to what is the right P/E to use, use 10 If you don't know the RADR, use 10 percent The answer to almost any troublesome finance question should include the world "risk." When in doubt, blame the accountants.”

Table 3 Risk-Adjusted Discount Rates

in year 2, the percentage rises to 17.4%; and in year 3, it is 24.9%

In contrast, as shown in Line 5 of Table 3, the risk-adjusted discountedbenefits are declining at an annual rate of 13.0% In absolute terms, theadjustment 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 theconstant risk-adjusted discount rate does not imply a constant deduction for risk.Rather, it implies a larger deduction for risk in the later years because thecumulative risk is increasing at a constant rate over time

This aspect of a constant risk-adjusted discount rate is often neglected inpractice and is obviously critical to the assumption of the risk profile of the projectover its life It is in fact only appropriate to use such a single risk-adjusteddiscount rate if the project has the same market risk at each point in its life relative

to the previous period.27 If the cumulative risk is not increasing at a constant rate,

then it is better to break the project into segments within which the same discountrate can be reasonably used or to use certainty equivalents

(3) Certainty equivalent approach

Whereas the risk-adjusted discount rate discussed above adjusts future

cash flows for both time and risk, the 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 riskycash 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 tothe alternative method of using a risk-adjusted discount rate This is most clearlyillustrated in the case of a single period.28 For example, let Z = α*B where α isthe certainty equivalent adjustment factor, a value between 0 and 1; B is theoriginal expected value; and Z is the certainty equivalent Z is the amount (lessthan the original B) that the equity investor would be willing to accept rather thanface any degree of project risk Let ρ again represent the required return onequity, adjusted for risk, and r the risk-free rate Then the original B discounted at

27

As noted by Sugden and Williams (1978, p 62), the use of a single risk-adjusted discount rate assumes that “ the divergence between the expected value of a return and its certainty equivalent increases systematically the further into the future it occurs On the face of it, this is a somewhat arbitrary procedure.” See also Myers & Ruback (1987, p 17) and Fama (1977, p 23) Fama writes: " it might be reasonable to assume that the risks in the reassessments of the expected value

of a cash flow are constant through time… If the market parameters are likewise constant through time, ….a single risk-adjusted discount rate or cost of capital can be applied to all the cash flows of

a project or firm."

the risk-adjusted discount rate will be equal to the certainty equivalent amount Zdiscounted at the risk-free rate That is,

the certainty equivalent values and find the single risk-adjusted discount rate that

yields the same NPV

28 See Levy and Sarnat (1994, p 273) For a discussion of the relationship between the adjusted and the certainty equivalent in the context of the Capital Asset Pricing Model, see Brealey and Myers (1996, p 229) and Zerbe (1994, p 332).

risk-Let the certainty equivalent for year t be Zt Then Zt = αt*Bt and the NPVcan be calculated by discounting the annual certainty equivalents by r If wedefine λ as 1/(1+ r), then:

Substituting the above expression into line 10, the risk-adjusted discount

rate is equal to the unadjusted discount rate Since this is impossible given our

basic assumptions, we face a major contradiction Clearly, a constant adjustmentfor risk, reflected in constant adjustment factors, does not correspond to a singlerisk-adjusted discount rate Instead, we can estimate a different risk-adjusteddiscount rate for each year of the project as shown below.

Table 4: Certainty Equivalents using Constant Adjustment Factors.

Table 5: Calculation of NPV using Annual Risk-Adjusted Discount Rates.