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

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
and
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
11
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 an
appropriate 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 more
meaningful single market-based price. If a project has many so-called “present
values,” then it lacks the single, risk-adjusted value associated with a “NPV.”
In practice, the question of what discount rate to use to generate the
probability distribution of the NPV tends to be downplayed and different analysts
discount 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 discount
rate to use when discounting a future cash flow is one that embodies the
opportunity cost of capital relative to that project. There is no economic
justification, for instance, for using the risk-free discount rate to discount any
expected cash flow except for one that is absolutely certain. But the use of a
properly risk-adjusted discount rate is rarely observed in the Monte Carlo
simulations for project evaluations. Why? Because if one knew the true
opportunity 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 itself
embody the risk of the project. Thus the probability distribution approach
essentially 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.
12
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 a
particular 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 the
distribution 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 into
account the “intertemporal resolution of uncertainty” because it does not specify
the actual nature of the risk over the life of the project. That is, once we have
identified the relevant risk variables for the project, how are they expected to
change over the life of the project? Since the risk profile will change over each
year of a project, a meaningful analysis should at least anticipate the changing
nature of the risk profile over the project’s life by specifying the assumptions
regarding 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 discount
rate, 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.
13
project.
26
The equity investor may compare the expected risk-profile of the
current 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 to
compensate the equity investor for the risk in the project.
For example, let us assume that the risk-free rate, r, is 10% and the risk-
premium is 5%. If Project Risquey’s initial investment is $1,000 and annual
returns from year 1 through year 3 are $402.11 (as above), then the project would
not be acceptable at a risk-adjusted discount rate of 15% because the NPV would
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’s
NPV may be expressed as:
NPV
PR
@ρ
= -K + B
1
+ B
2
+ B
3

(1 + ρ) (1 + ρ)
2
(1 + ρ)

3
= -K + B
1
*π + B
2
*π
2
+ B
3
*π
3
= -K + Σ
t=1
3
B
t
*π
t
(3)
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 of
15% (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.”
14
Table 3. Risk-Adjusted Discount Rates
Year: 0 1 2 3
1 Expected Annual Benefits 437.98 437.98 437.98
2 Discounted @ 10.0 % 398.16 361.96 329.06
3 Difference = line 1 – line 2 39.82 76.01 108.92
4 Line 3 as % of line 1 9.09% 17.36% 24.87%
5 Discounted @ 15.0 % 380.85 331.17 287.98
6 Difference = line 1 – line 4 57.13 106.80 150.00
7 Line 6 as % of line 1 13.0% 24.4% 34.2%
PV of Benefits, Discounted @ 15% 1,000.00
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 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 deduction for risk in the later years because the
cumulative risk is increasing at a constant rate over time.
This aspect of a constant risk-adjusted discount rate is often neglected in
practice and is obviously critical to the assumption of the risk profile of the project
over its life. It is in fact only appropriate to use such a single risk-adjusted
discount rate if the project has the same market risk at each point in its life relative

15
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 discount
rate 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 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 to
the alternative method of using a risk-adjusted discount rate. This is most clearly
illustrated in the case of a single period.
28
For example, let Z = α*B where α is
the certainty equivalent adjustment factor, a value between 0 and 1; B is the
original expected value; and Z is the certainty equivalent. Z is the amount (less
than the original B) that the equity investor would be willing to accept rather than
face any degree of project risk. Let ρ again represent the required return on
equity, 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."
16
the risk-adjusted discount rate will be equal to the certainty equivalent amount Z
discounted at the risk-free rate. That is,
B = Z (4a)
(1 + ρ) (1 + r)
Z/α = Z (4b)
(1 + ρ) (1 + r)
Solving for α, we find that
α = (1 + r) (5)
(1 + ρ)
Alternatively, if we know the value of α, we can solve for the value of ρ:
α + α*ρ = 1 + r (6a)
ρ = 1 + r - α (6b)
α
Suppose now that we have determined that the risk of Project Risquey
remains constant over its project life and that the appropriate allowance for risk is
10% of the annual benefits. That is, the certainty equivalent adjustment factor is
90% of the expected annual benefits, so that α = α
1
= α
2
= α
3
= 90%. The risk-
free discount rate, r, is 10%. We can now calculate Project Risquey’s NPV using
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 risk-
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).
17
Let the certainty equivalent for year t be Z
t
. Then Z
t
= α
t
*B
t
and the NPV
can be calculated by discounting the annual certainty equivalents by r. If we
define λ as 1/(1+ r), then:
NPV
PR
@r
= -K + Z
1
+ Z
2
+ Z
3

(1 + r) (1 + r)
2
(1 + r)

3
= -K + α
1
*B
1
+ α
2
*B
2
+ α
3
*B
3

(1 + r) (1 + r)
2
(1 + r)
3
= -K + α
1
*B
1
*λ + α
2
*B
2
*λ
2
+ α
3

*B
3
*λ
3
(7)
Recall that the expression for the NPV with the risk-adjusted NPV is:
NPV
PR
@ρ
= -K + B
1
*π + B
2
*π
2
+ B
3
*π
3
(8)
where π = 1/(1+ρ). For the above two NPVs to be equal, the discounting
coefficients for each year would have to be equal as follows:
Year 1: 1 = α (9)
(1 + ρ) (1 + r)
Year 2: 1 = α (10)
(1 + ρ)
2
(1 + r)
2
Year 3: 1 = α (11)

(1 + ρ)
3
(1 + r)
3
From line 9, it is clear that
α = (1 + r) (12)
(1 + ρ)
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
18
basic assumptions, we face a major contradiction. Clearly, a constant adjustment
for risk, reflected in constant adjustment factors, does not correspond to a single
risk-adjusted discount rate. Instead, we can estimate a different risk-adjusted
discount rate for each year of the project as shown below.
Year 1: 1 = α (13a)
(1 + ρ
1
) (1 + r)
Year 2: 1 = α (13b)
(1 + ρ
1
)*(1 + ρ
2
) (1 + r)
2
Year 3: 1 = α (13c)
(1 + ρ
1
)*(1 + ρ
2

)*(1 + ρ
3
) (1 + r)
3
From line 13a, we see that:
ρ
1
= 1 + r - α = 22.22% (14a)
α
Whereas combining lines 13a and 13b, and then 13b and 13c indicates that:
ρ
2
= r = 10.00% (14b)
and: ρ
3
= r = 10.00% (14c)
Using certainty equivalents with a constant adjustment factor of 90%, the
annual benefits would have to be $446.79 or higher for the project to be
acceptable at the risk-free rate of 10%. The IRR of the resulting adjusted cash
flow stream or the certainty equivalent (the original expected benefits less the
annual risk adjustment of 45) is 10%, equal to the risk-free rate, as illustrated
below in Table 4.
19
Table 4: Certainty Equivalents using Constant Adjustment Factors.
Year:0123
Original Benefits 446.79 446.79 446.79
Certainty Equivalent -1,000.00 402.11 402.11 402.11
Adjustment Factor (%) 90.00 90.00 90.00
Discounted NCF @ 10% -1,000.00 365.56 332.32 302.11
NPV @ 10% 0.00

IRR of Discounted
NCF 10.00%
Alternatively, we may use the risk-adjusted discount rates calculated in
lines 14a to 14c and obtain the equivalent NPV. The relevant discount rates,
cumulative discount factors, and NPV of this approach are shown in Table 5. As
expected, both techniques yield the same discounted Net Cash Flow as well as the
same NPV (in this case, zero). Again, it is important to emphasize that once the
risk-adjustment in the discount rate has been made in year 1, there is no need for
further risk-adjustment in subsequent years, when the appropriate discount rates
are the risk-free rates.
Table 5: Calculation of NPV using Annual Risk-Adjusted Discount Rates.
Year: 0 1 2 3
Discount Rates (%) 22.22 10.00 10.00
Discount Factors (DFs) 1.00 1.22 1.34 1.47
Cash Flow Statement
Year: 0 1 2 3
Original Benefits -1,000.00 446.79 446.79 446.79
Discounted NCF @ DFs -1,000.00 365.56 332.33 302.11
NPV 0.00
IRR 16.21%
20
The above example illustrates the poorly understood gulf between constant
risk and a constant risk-adjusted discount rate. However, it is possible to find the
IRR for the expected benefits over the life of the project. The IRR of the cash
flow in Table 5 is 16.21%, for example. But it is difficult to give a coherent
meaning to this IRR. That is, does this IRR really take into account the risk
profile over the life of the project?
Let us consider the converse situation. Given the IRR of 16.21%, we can
assume a constant risk-adjusted discount rate of 16.21% and then find the
corresponding certainty equivalent for each year, using the definitions provided in

lines 13a – 13c. As shown 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:0123
Expected Benefits 446.79 446.79 446.79
Certainty Equivalent -1,000.00 422.91 400.31 378.92
Adjustment Factor (%) 94.66 89.53 84.81
Discounted NCF @ 10% -1,000.00 384.47 330.84 284.69
NPV 0.00
IRR 10.00%
This simple numerical example has shown that the use of a single risk-
adjusted discount rate can be problematic because it does not necessarily take into
account the complete risk profile of the project. A more coherent way to take
account of the time value of money and project risk is to determine the certainty
21
equivalent of annual expected benefits and then discount the cash flow at the risk-
free rate.
One of the major difficulties with the certainty equivalent approach is with
the determination of the appropriate adjustment factors for each year of the
project. However, the specification of the adjustment factors for the annual
certainty equivalents is identical to the specification of different risk premia over
the life of a project. With different annual risk-adjusted discount rates we can
determine the corresponding annual certainty equivalents, and conversely, given
certainty equivalents we can determine the associated risk adjusted discount rates.
In general, the following relationship between annual risk-adjusted and
certainty equivalents can be shown. Let α
t
be the certainty equivalent for year t

and ρ
t
the risk-adjusted discount rate for year t. Then:
1 = α
t
(15a)
(1 + ρ
t
) (1 + r)
1 = α
t+1
(15b)
(1 + ρ
t
)*(1 + ρ
t+1
) (1 + r)
Substituting line 15b into 15a, we obtain that:
1 = α
t +1
_ (16a)
(1 + ρ
t+1
) α
j
and: ρ
t+1
= α
t
- 1 (16b)

α
t+1
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.
22
PART TWO
In view of the discussion in Part One, the question naturally arises: how
should we handle the valuation of risk in practical project appraisal? One
approach is to directly specify the “intertemporal resolution of uncertainty”. In
this section, we will illustrate this approach with some simple numerical examples
drawn from the fictional Project Risquey introduced in Part One. However, we
now need to add some new notation to denote the expected value, in year 0, of
future benefit streams that begin at different points in time.
As shown in Table 7, let B
t
signify the actual value of Project Risquey’s
benefits at the end of year t, where this value is not revealed until the end of the
year t. Before year t, there are only expected values of the annual benefits for year
t and beyond. These are expressed as E(B
t
). Thus, from the perspective of year 0,
the investor will expect a stream of annual benefits beyond the current year, but
only after the passage of time to year 1 will the actual value of project benefits for
year 1 be revealed. This new information may or may not cause the investor to
revise her views about the expected values of the annual benefits beyond year 1.
Table 7: Cash Flow Statement of Project Risquey.
Year: 0 1 2 3
Investment -K
Actual Benefits B
1

B
2
B
3
Expected Benefits E(B
1
)E(B
2
)E(B
3
)
Certainty Equivalent α
1
*E(B
1
) α
2
*E(B
2
) α
3
*E(B
3
)
23
To denote this temporal nature of expectations, we add a “time of
assessment” dimension to the annual benefits. That is, E(B
ts
) is defined as the
value of the expected benefits in year t as of year s. To find the expected value of

the future benefits with respect to year 0, all the expected benefits from years 1 to
3 in Table 7 are assessed as of year 0 and expressed as E(B
10
), E(B
20
), and E(B
30
).
Similarly, the certainty equivalent adjustment factors are given with respect to
time, so that α
10
refers to the adjustment factor in year 1, from the perspective of
year 0. The expression for the present value of future benefits, as of the end of
year 0, is therefore:
V
0
= α
10
*E(B
10
)*λ + α
20
*E(B
20
)*λ
2
+ α
30
*E(B
30

)*λ
3
(17a)
where λ = 1/(1 + r), and r refers to the risk-free discount rate, as before. To
summarize, we are discounting the annual certainty equivalent at the risk-free rate
from the point of view of year 0.
In similar fashion, we can write the expressions for the present value of the
stream of future benefits from the perspectives of year 1 and year 2. Thus, E(B
21
)
and E(B
31
) represent the expected values of benefits in year 2 and year 3,
respectively, from the perspective of year 1. It may well be the case that E(B
21
) is
different from E(B
20
), and E(B
31
) is different from E(B
30
). The revision of future
expectations will also occur in year 2 with the availability of the actual value of
the benefits at the end of year 2. As a result, E(B
32
) represents the expected value
of the future benefit in year 3, from the perspective of year 2, and it may be
different from E(B
31

). The expressions for the present values of future benefits
from the perspectives of years 1 and 2 are given below (see also Appendix A).
24
V
1
= α
21
*E(B
21
)*λ + α
31
*E(B
31
)*λ
2
(17b)
V
2
= α
32
*E(B
32
)*λ (17c)
Assume now that Project Risquey’s benefits in each year, from the
perspective of year 0, are expected to be 446.79, while the certainty equivalent
adjustment factor in each of the three years is 90%. Thus, E(B
t0
) = 446.79 and α
t0
= 90%. For now, we also assume that the expected value of future benefits and

the certainty equivalent adjustment factors remain constant as time passes.
With these two assumptions, it follows that:
E(B
20
) = E(B
21
) and E(B
30
) = E(B
31
) = E(B
32
) (18)
α
20
= α
21
and α
30
= α
31
= α
32
(19)
Z
20
= Z
21
and Z
30

= Z
31
= Z
32
= 402.11 (20)
We can now verify the following equations:
V
0
= Z
t0
*[λ + λ
2
+ λ
3
] = 402.11*(0.91 + 0.83 + 0.75) = 1000.00 (21a)
V
1
= Z
t1
*[λ + λ
2
] = 402.11*(0.91 + 0.83) = 697.89 (21b)
V
2
= Z
t2
*[λ] = 402.11*(0.91) = 365.56 (21c)
(1) The meaning of changing valuation over time
We clearly observe a decrease in the valuation of Project Risquey over
time. The yearly decrease, in turn, represents the amount the investor gains each

year as repayment on the equity implicitly invested at the beginning of each
period. More specifically:
25
(V
1
- V
0
) = 697.86 - 1000.00 = -302.14 (22a)
(V
2
- V
1
) = 365.56 - 697.86 = -332.30 (22b)
Since future benefits are directly related to the equity holdings in the project, the
decrease by 302 from year 0 to year 1 in expected future benefits represents an
equivalent decline in the value of the equity investment as the cash flow generated
by the project is paid to the equity holder. A similar explanation accounts for the
reduction in expected future benefits from year 1 to year 2.
This change in the project’s value to the investor, or the repayment of
equity, may be viewed as one component of the actual payment received by the
investor at the end of each year. The other component is the return on the equity
investment. In general, we can thus express the actual annual benefits as follows:
B
t
= (V
t-1
- V
t
) + Y
t

(23)
where (V
t-1
- V
t
) represents the decline in the value of future expected benefits and
Y
t
is the equity return.
29
If we now assume that each annual payment for this
project is equal to expected benefits (B
t
= E(B
t
) = 446.79), we can derive the value
of the equity return, Y, in year 1 as follows:
Y
1
= B
1
+ (V
1
– V
0
) (24a)
= 446.79 + (697.89 – 1,000)
= 446.79 + -302.11 = 144.68

29

For comparison, we can consider the case with a loan repayment. With the latter, part of the
repayment is for the interest accrued and the rest is for repayment of principal.