Comparisons
with Risk
In some sense, selecting among mutually
exclusive alternatives is the essence of life. Decisions made
by our parents or ourselves have a great deal to do with how
our lives turn out. We decide where to go to college or whether
to go at all. On graduation we hope to have several opportunities
for employment, from which we will choose one. Rather than live
alone, we decide to choose a mate. At points in life we choose
a car, house, and a life style. For most people, these selection
problems involve choosing from among mutually exclusive alternatives.
For almost all, there is risk.
Decisions are difficult because of the uncertainty
or variability of the outcome. Short term decisions, like selecting
from a restaurant menu, may involve uncertainty, but the value
of the decision is quickly realized. The gain from a good choice
or the loss from a poor one is immediately apparent. Long term
decisions involve much more uncertainty and often the possibility
of much more gain or loss. Decision makers want to make a good
return without too much risk.
We consider in this lesson the problem of choosing
the best from two or more mutually exclusive alternatives where
the alternatives are described by their cash flows. Our evaluation
measures are net present worth (NPW) and net annual worth (NAW).
The measure must be valid for the problem at hand. In particular,
the NPW method is only appropriate if the alternatives are compared
over a common study period. The NAW is more flexible in this
regard because it is valid even if the alternatives have different
lives.
We do not consider the internal rate of return
here because it is not valid to select among mutually exclusive
alternatives by comparing their IRR values. Rather, it is necessary
to use incremental analysis. Because incremental analysis is
a sequential decision process involving pairs of random variables,
we do not address this complex issue.
In the following we try to include the variability
of the measures as part of the decision process. We do this by
describing various measures of risk and proposing ways to include
risk in the decision process. We cannot describe a complete method
for making decisions under risk, however, because balancing risk
with return depends on the decision maker's response
to risky situations. Although utility theory is advanced
to measure that response, we do not attempt to cover that subject.
This is a very basic introduction.
Decision making under uncertainty is a topic considered by many
fields using many different names. Interested students will find
related information in the subjects of decision analysis, stochastic
programming, game theory, goal programming, and multicriteria
decision making.


Goals 



Given a list of mutually exclusive
alternatives with random features of their cash
flows find the distributions of the evaluation
measure (NPW or NAW).

Given the distribution of the evaluation measure,
compute risk measures: standard deviation, information
ratio, shortfall probability, risk percentile,
and value at risk.

Given probability distributions
for the alternatives, plot a mean/standard deviation
scatter diagram.

Use the mean value criterion
to make decisions about risky situations. Know
when the mean value criterion is appropriate
for deciding between alternatives.

Use dominance to eliminate solutions. Know when
dominance is valid and when it is an approximation.

Use the risk measures to demonstrate the tradeoffs
between risk and return.




Text 

Utility Theory is an approach to making
decisions under uncertainty that reflects the decisionmakers
response to the range of possible outcomes of the decision.
We do not include this subject in this lesson, but the referenced
section provides an introduction for interested students.
Chapter 5 is about project selection and Section
5.6 describes the ways to manage risk, nonprobabilistic approaches
and riskbenefit analysis. Section 5.7 is about decision tree
analysis, a method for dealing with multistage decisions with
uncertainty. The materials are not required for this course.


5.6
Issues Related to Risk 




Model 

The problem is to select one from a set of mutually
exclusive alternatives.
An alternative is described by a single measure (NAW or NPW).
For convenience we use the term NPW in the following discussion,
but the measure might as well be NAW.
(We note,
however, that the likeforlike assumption implicitly assumed
for the NAW comparison for alternatives with different lives
may not be appropriate when uncertainty is included in the analysis.
We neglect this problem in this lesson by assuming that likeforlike
replacements have the same statistical properties as the first.)
The
parameters of the NPW are based on estimates
of events that happen in the future. Since some of these are
uncertain, we model the NPW as a random variable.
The cumulative distribution describes the variation of the NPW
and the distribution has a mean and standard deviation. The NPW
is computed with revenues positive and costs negative,
so a larger value is better.
For simplicity, we assume that the random variables
are pairwise independent. In some of the following we
assume that the distributions are normal. For cases
when normality is not assumed, we may associate the NPW with
one of the named distributions. An alternative is to describe
the distribution by a histogram generated by Monte Carlo
simulation.
The Risk lesson in the Evaluation section
describes how to compute the mean and standard deviation when the
NPW is a linear function of the random variables. Formulas are
in the linked document.


Moments
for Linear Models





Example 

To illustrate, consider the five alternative solutions below.
We assume the NPW of each solution has a normal distribution
with mean and standard deviation values in the table.

A1 
A2 
A3 
A4 
A5 
A6 
Mean 
60 
150 
130 
25 
160 
5 
Standard Deviation 
80 
126 
83 
55 
220 
69 
The problem is to select one and only one of these investment
alternatives. They all have positive NPW values so all satisfy
the MARR requirement. The link shows a scatter chart
of the parameters of the alternatives.


Mean/Standard
Deviation Scatter





Mean Value Criterion 

One criterion for selecting the best alternative
is to choose the one with the greatest mean value. Although this
approach obviously neglects uncertainty,
it is a respected and not unreasonable approach in many contexts.
When an organization has many relatively small project decisions,
where no single project is a great threat to the organization's
financial viability, variations between projects, especially
when they are relatively independent, tend to balance out. Making
decisions that maximize the mean, or expected value, of each
decision, maximizes the expected value of all decisions.
Although not considered in this course, utility functions,
measure a decision maker's tolerance to risk. Maximization of
expected utility is the best course when the utility function
is properly defined. If one assumes a linear utility function,
the utility of money is proportional to the amount of money.
This is often a reasonable assumption and one that leads to the
conclusion that the best solution is the one with the greatest
mean, regardless of the variance. This criterion would select
alternative A5 for the example.
Many individuals and organizations do consider risk in the context
of selecting between alternatives, and this lesson provides an
introduction to the risk measures and how to use them to help
make decisions. 

Risk Measures 

If the decision maker is to consider uncertainty, there must
be quantitative measures to describe the risk. We list several
in the table below. They are adapted from similar measures used
to quantify portfolio selection. We should emphasize however,
that selecting a portfolio is quite different from the problem
considered here. The portfolio selection problem is to find the
optimum mix of several investments. The problem of this lesson
is to select a single investment from a set of available alternatives.
Measure 
Definition 

Standard Deviation/
Variance 

The standard deviation and variance measure the variability
of the estimate. With the same mean, the alternative with the
greater standard deviation would be the more risky one. 
Information Ratio 

The information ratio includes both the mean and standard
deviation measures. It seems preferable to have a high information
ratio that a low one. 
Shortfall Probability 

The shortfall level is a specified value of the NPW
that indicates an undesirable solution. Zero is
an important level. If the NPW or NAW falls below 0, the solution
does not provide the minimum acceptable rate of return. The
risk measure is the probability that the evaluation falls below
the shortfall level. A lower value is better than a higher
one. 
Risk Percentile 

Alpha is a small probability (0.01, 0.05 ...
). The percentile is
the value such that the probability that the NPW
falls below this value is alpha. Thus it is a conservative
estimate of the NPW for the project. 
Value at Risk 

The value at risk, VaR, is the difference between the mean
and the alphapercentile. It indicates
how much the value may fall below the mean estimate. 
The links below illustrate the risk measures.


Normal Distribution 

When we assume that the NPW has a normal distribution, the
measures can be computed entirely from the mean and standard
deviation parameters. The table shows the general expressions
for the normal distribution as well as values of the inverse
standard normal for some common values of the risk level.
Shortfall Probability 
Value at Risk 
Inverse Standard Normal 



When the shortfall level is chosen as 0, the shortfall probability
is the probability that the net present worth is negative, or,
equivalently, the probability that the solution does not meet
the minimum acceptable rate of return. It is not hard to show
that the shortfall probability decreases with an increase in
the information ratio. It is also easy to see that the value
at risk is proportional to the standard deviation.
The table shows the measures computed for the solutions of the
example.

A1 
A2 
A3 
A4 
A5 
A6 
Mean 
60 
150 
130 
25 
160 
5 
Standard
Deviation 
75 
110 
80 
55 
100 
69 
IR 
0.80 
1.36 
1.63 
0.45 
1.60 
0.07 
PS(0) 
0.212 
0.086 
0.052 
0.325 
0.055 
0.471 
10%Percentile 
36.12 
9.03 
27.48 
45.49 
31.84 
83.43 
90% VaR 
96.12 
140.97 
102.52 
70.49 
128.16 
88.43 
Some of the measures can be illustrated on the
scatter chart. A line passing from the
origin to the point representing the solution describes the information
ratio. The line is drawn for A3. The slope
of the line is the information ratio. Clearly A3 has the greatest
information ratio because the lines for all other solutions would
fall below the one for A3. This also assures that A3 has the
smallest shortfall probability (0.009). The red dots show the
10% percentiles for the six alternatives, so the lengths of the
vertical lines show the 90% VaR values.




For the examples on this page, the alphaPercentile
seems more important the value at risk (VaR). The VaR shows how
much the evaluation might change from the expected value. This
could be important when considering the portfolio of a number
of investments where all might be affected by the same economic
factors. If the factors are unfavorable, every investment in
the portfolio might fall below the mean and the organization
might find itself with serious financial problems. Since we are
comparing solutions to for a single project, this portfolio affect
is not apparent, but we include the VaR because it is important
in some contexts. 

Making Decisions 

The risk measures provide additional information that the
decision maker might use for the selection of the best alternative.
With the normality assumption we may be able to eliminate an
alternative because of dominance.
Dominance is illustrated on the graph. Each solution
defines a dominated region that is the rectangle below and to
the right of the solution. Any solution that is in this rectangle
is dominated. For the example, A2 is dominated by A5, and A6
is dominated by A4. It follows that for any undominated solution
the rectangle above and to the left must be empty of solutions.

If
one alternative dominates another, the dominated alternative
may be eliminated. 

The collection of undominated solutions (A4, A1,
A3, and A5) consists of a series of points with increasing values
of mean and standard deviation. When graphed the points move
upward and to the right. We have connected the points with
lines in the figure. This display is called the efficient
frontier. The points on the frontier are undominated and any
one may be selected as the solution to our problem.
How do we select a single solution? We present
again the table showing the measures for the four undominated
solutions. The last two columns show the best and worst solutions
for each measure. The solution with the greatest mean NPW is
also the solution with the greatest variability. The solution
with the smallest variability has the smallest mean. This will
always be true for the solutions on the efficient frontier.

A1 
A3 
A4 
A5 
Best 
Worst 
Mean 
60 
130 
25 
160 
A5 
A4 
Standard
Deviation 
75 
80 
55 
100 
A4 
A5 
IR 
0.80 
1.63 
0.45 
1.60 
A3 
A4 
PS(0) 
0.212 
0.052 
0.325 
0.055 
A3 
A4 
10%Percentile 
36.12 
27.48 
45.49 
31.84 
A5 
A4 
90%
VaR 
96.12 
102.52 
70.49 
128.16 
A4 
A5 
Since the probability distributions are normal,
we can make other observations. When the shortfall level is 0,
the shortfall probabilities decrease as the information ratio
increases. The best solution for PS is the best solution for
IR. For a given risk level, the VaR is proportional to the standard
deviation, so the VaR value increases with the standard deviation.
The best solution for the standard deviation is the also be best
solution for the VaR. The worst solution for the VaR is the best
solution for the mean.
With our six measures the selection problem is
called a multicriteria decision problem. There are approaches
to this problem, but we cannot consider them here. Solving the
problem is not as simple as selecting the best according to a
single measure as our deterministic models suggest. The efficient
frontier shows the tradeoff between return and variability,
but it is ultimately the decisionmaker's responsibility to choose
between the alternatives. The efficient frontier must include
the solution with the greatest mean and also the solution with
the smallest standard deviation. If these two solutions are identical,
there is only one solution on the frontier and it must be the
chosen one.
One might suggest that a reasonable selection rule
is to choose the solution with the greatest information ratio.
This selection also minimizes the probability that the solution
will fail to satisfy the MARR. For the example, this rule would
select A3. This rule may not always be appropriate. The largest
information ratio might be associated with a solution with a
very small NPW and also a very small standard deviation. It
would seem wrong to select this solution over another with a
very high NPW and some variability. Some examples may include
the null alternative, that is the decision to do nothing. Because
that solution has zero return and zero standard deviation, the
information ratio is undefined. Unless you accept the rational
for using the mean value criterion, there is no easy answer for
the selection problem with risk. 

Other Distributions 

Our discussion has been restricted to solutions
with normal distributions for the economic measure. It would
be more likely that the distributions have some other form. To
illustrate, the second example has six alternatives that have
triangular distributions for the NPW. The example has the mode
values the same as the mean values of the first example, but
the distributions are skewed to the left or right. Alternatives
A1, A3, and A5 are all skewed to the left, so smaller values
of the NPW are more likely than larger. Alternatives A2, A4,
and A6 are skewed to the right. Although the alternatives have
similar variances to the first example, skewing to the left makes
the downside risks greater for the odd numbered solutions.
Triangular Distributions 
A1 
A2 
A3 
A4 
A5 
A6 
lower limit (a) 
165 
40 
110 
30 
140 
64 
mode (m) 
60 
150 
130 
25 
160 
5 
upper limit (b) 
135 
480 
210 
190 
260 
212 
The risk measures were computed with the help of the Random
Variables addin and are presented below.

A1 
A2 
A3 
A4 
A5 
A6 
Mean 
10.00 
223.33 
76.67 
61.67 
93.33 
51.00 
Standard
Deviation 
63.74 
93.48 
67.99 
46.74 
84.98 
58.64 
IR 
0.16 
2.39 
1.13 
1.32 
1.10 
0.87 
PS(0) 
0.403 
0.000 
0.158 
0.074 
0.163 
0.215 
10%Percentile 
82.84 
109.57 
22.36 
4.79 
30.46 
20.36 
90%
VaR 
92.84 
113.76 
99.03 
56.88 
123.79 
71.36 
The graph shows some of the risk measures. Without normality,
the decision problem is more difficult. The values of the mean
and standard deviation alone can no longer determine the values
of the shortfall probability or the percentile levels, so none
are dominated strictly in terms of the mean and standard deviation.
It happens in this case the A1 and A6 are dominated by A4 for
all the measures, so A1 and A6 can be eliminated from further
consideration.
Eliminating A1 and A6, we evaluate the best and worst for each
measure. It looks like A2 is a good choice, but A4 has a smaller
variability and a lower VaR. It might be the best choice for
the conservative decision maker.

A2 
A3 
A4 
A5 
Best 
Worst 
Mean 
223.33 
76.67 
61.67 
93.33 
A2 
A4 
Standard
Deviation 
93.48 
67.99 
46.74 
84.98 
A4 
A2 
IR 
2.39 
1.13 
1.32 
1.10 
A2 
A5 
PS(0) 
0.000 
0.158 
0.074 
0.163 
A2 
A5 
10%Percentile 
109.57 
22.36 
4.79 
30.46 
A2 
A5 
90%
VaR 
113.76 
99.03 
56.88 
123.79 
A4 
A5 


Cost Only Alternatives 

Many comparisons involve
primarily cost. In this example, we consider four alternatives
to perform some function. Each is described by an initial investment
and an annual operating cost. All have fiveyear lives. We
see that an alternative with higher initial investment has lower
operating cost. For the purpose of this lesson, we add uncertainty
to our estimate of the operating cost. Generally as the annual
cost level increases, the standard deviation also increases.
For convenience we assume that the operating costs are the same
in each year, but the value of the cost is normally distributed
with the mean in the Annual
Cost column.
The standard deviation is in the Standard Deviation
Annual Cost column. The investment values are not uncertain.
Alternative 
Investment 
Ann. Cost 
Std. Dev.
Ann.
Cost 
Mean NAW 
Std. Dev. NAW

A1 
5000 
240 
50 
1732 
50 
A2 
3000 
875 
200 
1770 
200 
A3 
4000 
500 
100 
1693 
100 
A4 
2500 
1000 
300 
1746 
300 
We compare the alternatives with their net annual worth values
using 15% for the MARR. The annual operating cost is a linear function
of the annual cost with the coefficient 1. The NAW is computed
as:
NAW = Investment(A/P, 0.15, 5)  Annual Cost
Since the annual cost is a random variable, the NAW
is also a random variable with the same standard deviation. The
moments for the NAW distributions
are in the last two columns of the table.
Considering only the mean value of the NAW we should
choose A3. With the distribution information we can also consider
risk. The graph showing the distribution parameters shows mean
values in red and 10% percentile values in green. The shortfall
probability for the value 0 has no meaning when the cash flow has
no income. The shortfall probability would have relevance for some
negative value of the shortfall level.


Mean/Standard
Deviation Scatter



Because of the assumption of normality, we can apply
dominance to eliminate solutions. The colored region is dominated
by A3. A2 and A4 have smaller mean values and greater standard
deviations than A3 and can be eliminated because of dominance.
They are also dominated by A1.
A1 is not dominated. Although it has a lower mean
NAW (so is more costly) than A3, its variability is less. The efficient
frontier for the example has only two solutions.
The table shows the tradeoff between the mean value and the risk
measures. We have not specified a shortfall level, and the information
ratio has no relevance when the comparison measure is cost. A3
has the lowest expected cost, but A1 wins for the other measures.
It is up to the decisionmaker to select a solution.

A1 
A3 
Best 
Worst 
Mean 
1732 
1693 
A3 
A1 
Standard
Deviation 
50 
100 
A1 
A3 
10%Percentile 
1795 
1821 
A1 
A3 
90%
VaR 
64 
128 
A1 
A3 


Summary 



Problems 

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