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 multi-criteria decision making.

Utility Theory is an approach to making decisions under uncertainty that reflects the decision-makers 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.

3.7 Utility Theory

Chapter 5 is about project selection and Section 5.6 describes the ways to manage risk, non-probabilistic approaches and risk-benefit analysis. Section 5.7 is about decision tree analysis, a method for dealing with multi-stage decisions with uncertainty. The materials are not required for this course.

5.6 Issues Related to Risk

5.7 Decision Trees

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 like-for-like 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 like-for-like 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

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.

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

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.

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 alpha-percentile. It indicates how much the value may fall below the mean estimate.

The links below illustrate the risk measures.

Shortfall Probability

Percentile

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.

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.

Risk Measures

For the examples on this page, the alpha-Percentile 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.

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.

Dominance

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.

Efficient Frontier

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.

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 multi-criteria 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 trade-off between return and variability, but it is ultimately the decision-maker'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.

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.

The risk measures were computed with the help of the Random Variables add-in and are presented below.

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.

Risk Measures

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.

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 five-year 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.

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.

Dominance

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.

Efficient Frontier

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 decision-maker to select a solution.