Lee Merkhofer Consulting Priority Systems

Technical Terms Used in Project Portfolio Management (Continued)

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z


Term
Explanation

E

e (Euler's number)

The letter e is a symbol representing a constant number, the first few digits of which are 2.71828182, that appears frequently in mathematics, especially in formulas dealing with continuous growth. For example, if you invest $1 at a rate of interest of 100% a year with interest compounded continually, you will have the amount e = $2.71828... at the end of the year: The number e is the base of the natural logarithm. It is sometimes called Euler's number after the Swiss mathematician Leonhard Euler who is credited to be the first to use the symbol in a manuscript written in 1727.

earned value management (EVM)

An acronym-rich method for measuring progress on projects and indicating any variances in planned accomplishments, schedule, and cost expenditures. EVM, also called earned value analysis (EVA, not to be confused with economic value added) is primarily used as a way of reporting project progress to stakeholders, and government regulations often require that contractors providing services to government agencies comply with standards for using EVM. In the context of project portfolio management (PPM), EVM may be incorporated as a means for reporting progress on individual projects and for demonstrating compliance with government requirements for EVM.

The basic concept with EVM is that project work be planned, budgeted, and scheduled in time-phased, "planned value" increments. Typically, these work increments are defined in a hierarchical fashion as a work breakdown structure, but for a smaller project the work elements might simply be individual project tasks. The work elements define a schedule and cost/value baseline for the project. As project work is conducted, project value is "earned." Various indices are computed to summarize project status based on comparing earned value with planned and actual costs.

With EVM, the value that is assigned to each work element is termed its planned value (PV). The PV is meant to be a quantity or weighting factor that indicates the portion of the project value that, according to the plan, will be contributed by that work element at a specified time. Usually, the PV for a work element is set equal to its cost, however, the PV might alternatively be defined as the number of labor hours required or as a subjectively assigned number of "points."

The value of the work element is earned as the work is completed. For example, the earning rule might be that 25% of the value is earned when the task is started, and the remaining 75% is earned upon completion.

Progress against the plan is reported on a regular basis (e.g., weekly or monthly) by accumulating earned value (EV) based on the earning rules. By subtracting the value of the work planned (PV) from the value of the work performed (EV), a schedule variance (SV) may be computed at any point in time during the project:

SV = EV - PV.

(Some EVM documents alternatively define SV = PV - EV. With the definition given above, negative numbers are "unfavorable," and positive numbers are "favorable.")

Similarly, a schedule performance index (SPI) may be computed by dividing the EV by the PV:

SPI = EV/PV.

If the SV is greater than zero (SPI is greater than 1), the work is ahead of schedule. If the SV is less than zero (SPI is less than 1), the work is behind schedule. Schedule variances can be rolled up to any level in the work breakdown schedule to provide higher-level indicators of schedule compliance.

Since a work element's PV is traditionally chosen to be the scheduled cost of the work, the traditional term for a work element's planned value is the budgeted cost for work scheduled (BCWS). The traditional term for earned value is the budgeted cost for work performed (BCWP). The actual cost of conducting each work element is termed the actual cost of work performed (ACWP). In this context, where value and cost are both measured in dollars, a cost variance (CV) can be computed by subtracting the actual cost of work performed (ACWP) from the budgeted cost of work performed (BCWP):

CV = BCWP - ACWP = EV - AC.

EVM defines many additional indicators of technical, schedule, and cost performance that can also be calculated, and guidance is available for interpreting and addressing the various discrepancies that the indicators may reveal. As you can no doubt appreciate, EVM can be confusing because of the many acronyms that are used.

Although EVM is a well-established and effective means for managing the completion of complex projects, it's major limitation from the standpoint of PPM is that it does not provide indicators for tracking the anticipated ability of the project to deliver benefits to the organization. Because EVM is unconcerned with project changes that might impact the ultimate value derived from a project, it provides no signals that might suggest that the project plan should be reconsidered. EVM might, for example, indicate that a project is under budget, ahead of schedule, and within scope, but that project could nevertheless be in trouble with regard to achieving the benefits that motivated the decision to fund it.

economic value added (EVA®)

A financial project valuation metric and related management framework developed by consulting company Stern Steward founders Joel Stern and G. Bennett Steward III (EVA® is a registered trademark of Stern Steward). The EVA® of a project is calculated by taking net operating profit and subtracting a charge for the capital or assets deployed. The deducted amount is the "cost of capital"—what shareholders and lenders could obtain by investing in securities of comparable risk.

EVA®, also sometimes termed earned value added, provides a useful input for prioritizing projects because it quantifies the direct financial component of project value. However, other techniques are needed to account for the indirect or non-financial components of project value. Also, depending on the characteristics of projects, it may be more convenient to account for the cost of capital using the more traditional calculation of net present value (NPV).

While there are other financial metrics that likewise account for the cost of capital, the appeal of EVA® is that it does so in a conceptually simple and intuitive way that is easy for non-financial managers to understand. Since EVA® starts with familiar operating profits and then deducts a charge for the capital employed, it can be interpreted simply as "net profit minus the rent."

EVA® has become popular because it highlights the importance of the cost of capital when financially evaluating projects. EVA® may show, for example, that despite increasing earnings, a project is destroying shareholder value because the cost of capital associated with the required investment is too high. By assessing a charge for using capital, EVA® forces managers to think about managing assets as well as income.

As indicated above, a major weakness of EVA® is that it fails to account for non-financial project impacts (such as improved employee knowledge) that are difficult to express in terms of incremental cash flows. Also, accounting for opportunity costs by subtracting a capital charge is conceptually simple only if project start times, durations, and spending rates aren't very important (if they are, then the NPV approach of discounting cash flows using hurdle rates is computationally and conceptually simpler). Like classic NPV, EVA® does not explicitly address cash flow uncertainties, and it can be very difficult to determine the appropriate charge for the capital used by a project.

efficient frontier

In the context of modern portfolio theory, the efficient frontier is the bounding curve obtained when portfolios of possible investments are plotted based on risk and expected return. The efficient frontier shows the investment combinations that produce the highest return for the lowest possible risk. A portfolio that is not on the efficient frontier is said to be "inefficient" because another portfolio exists that has lower risk for the same return.

MPT efficient frontier

Efficient frontier as defined by modern portfolio theory


In the context of project portfolio management, the efficient frontier typically refers to the bounding curve that is obtained when portfolios of projects (or sometimes individual projects) are plotted based on cost and some quantity that is intended to represent portfolio (or project) attractiveness (ideally, the y-axis should represent the value or worth of the portfolio to the organization). In this context, a portfolio that is not on the efficient frontier is inefficient because another portfolio exists with greater value for the same cost. For more explanation see the paper chapter on the finding the efficient frontier.

PPM efficient frontier

Efficient frontier as defined by project portfolio management


eigenvalue, eigenvector

Concepts from linear algebra relevant to prioritization because of their use in AHP, a multicriteria decision making approach. A scalar (number) λ is called an eigenvalue of the n n matrix A if there is a nonzero vector x such that Ax = λx. In this case, x is called an eigenvector corresponding to the eigenvalue λ.

PPM efficient frontier

An eigenvector for a matrix is scaled when multiplied by the matrix


In the language of matrix algebra, a vector can be multiplied by a matrix to produce another vector. An eigenvector of a matrix is a vector that when multiplied by that matrix simply scales the entries in the vector; in other words, the vector doesn't change direction, it merely changes size. The eigenvalue associated with the eigenvector is the amount by which the vector entries are scaled.

ELECTRE

A decision aid that involves comparing pairs of potential actions based on multiple criteria. ELECTRE, like PROMETHEE, is a so-called outranking method, representative of what has been referred to as the "European school" of multi-criteria methods. Unlike multi-criteria analysis methods such as multi-attribute utility analysis (MUA) and the analytic hierarchy process (AHP) (the so-called "American school"), outranking methods do not involve developing or assessing from decision makers a utility function (see decision theory) for quantifying decision-maker preferences. Instead, with ELECTRE and other outranking methods, preferences are determined indirectly by having decision makers express relative preferences between pairs of options.

The results of the comparisons are organized into a matrix of values that show the "concordance" and/or the "discordance" between the candidate actions. The matrix is analyzed to produce various results and to choose, rank, or sort the alternatives.

The ELECTRE method was developed in France in the late 1960s and the term is an acronym for ELEmination et Choix Traduisent la Realite—elimination and choice reflecting reality. Like PROMETHEE, ELECTRE comes in various "versions" that indicate refinements and whether the version is meant for selecting or classifying options.

A strength of ELECTRE is its ability to account for uncertainty and vagueness. One reported weakness is that, due to the way preferences are incorporated, the lowest performances under certain criteria are not displayed. The outranking method results do not show the strengths and weaknesses of the alternatives, nor results and impacts to be verified. ELECTRE has been used in energy, economics, environmental, water management, and transportation problems.

Outranking methods, like ELECTRE and PROMETHEE, which were prevalent early on in the development of MCA methods, have been largely overtaken by the use of value measurement approaches such as MUA and AHP. Still, there are a few Europe-based project portfolio management tools that advertise their use of ELECTRE.

elimination by aspects

A multicriteria method wherein alternatives are evaluating with respect each criterion, one criterion at a time, and all those alternatives that fail to reach a minimum level of performance with respect to the criterion are eliminated. The approach is non-compensatory in that it does not require specifying weights or willingness to make tradeoffs. It typically does not result in either a "best" alternative or a prioritization of alternatives. However, the approach can be an efficient method for screening projects provided that minimum levels of performance on each criterion can be specified. After screening, a compensatory approach is required to prioritize the remaining projects.

enterprise management

A collection of management principles and techniques focused on helping the organization achieve its highest-level objectives, such as increasing shareholder value. Enterprise management typically includes strategic planning, long-term investment strategy, organizing and resourcing, performance assessment, and leading and directing the organization.

enterprise project management (EPM)

A broad term that refers to processes for improving the conduct and coordination of projects across an enterprise. The term predates project portfolio management (PPM), and, like project management and program management, EPM is mainly focused on "doing projects right," not on doing the "right projects." Thus, though tools to support EPM often include dashboards that can show work progress at various level of detail, project prioritization and portfolio optimization capabilities are generally not included.

enterprise project portfolio management (EPPM)

Project portfolio management applied at the enterprise level; that is, to all projects and programs conducted by the organization. EPPM may be implemented by establishing a hierarchy of project portfolios that are managed both individually and collectively to maximize the value derived by the enterprise.

enterprise resource planning (ERP)

Refers to a process and/or comprehensive software system aimed at centrally managing and coordinating the broad set of activities needed to successfully run a business enterprise, including product planning, material purchasing, inventory control, distribution, accounting, marketing, finance, and HR. Most ERP software products are composed of modules, with each module focused on one business process — some ERP systems include a module for project portfolio management. Customers can purchase as many modules as they require. A single, central repository contains the information needed to support planning and decision making across the various modules and associated business functions. ERP systems typically use or are integrated with a relational database.

An oft cited example illustrating the ERP concept is a system wherein when the sales department records a new customer order information is routed to the inventory and warehousing department to retrieve and package the order, to the finance department to prepare an invoice for billing, and to the manufacturing department for purchased product replacement.

event tree

A graphical representation, similar to a decision tree, showing the sequence of events that might occur following some initial event. The difference between a decision tree and an event tree is that an event tree does not contain nodes representing decisions. Like decision trees, event trees are typically used to understand risk and to identify actions for improving performance.

Historically, event trees have been used most often for accident analysis. In such applications, each event is represented by two branches corresponding to the possibility that an event either does or does not occur (e.g., a safety system works or fails). The figure below provides an example.

Example event tree

Event tree for assessing the consequences of loss of coolant to a nuclear reactor


If they are quantified, event trees, like decision trees, can be used for quantitative risk analysis. Suppose, for example, that a model has been constructed to estimate some quantity or quantities of interest. An event tree can then be constructed with each event in the tree representing one of the inputs to the model that is uncertain. If an uncertain input could take on any value within some range, it is discretized, for example, by specifying high, medium and low possibilities. Some decision tree tools can create an event tree automatically from an influence diagram and link it to a model constructed with Excel.

Each path through the event tree defines a set of inputs for the model, so model outputs can be computed for each path. To quantify uncertainty over the model outputs, probabilities must be assigned to each branch in the tree. By convention, as in the case of decision trees, probabilities of the event outcomes are placed under the corresponding branches and the model outcomes are placed at the ends of the tree (illustrated here). The probability of each path can be calculated by multiplying the branch probabilities along the path. The tree thus provides a probability distribution describing the uncertain outcomes at the ends of the tree. Some project portfolio management tools, including some for prioritizing R&D projects, use event trees in this way to quantify project risks.

evidential reasoning (ER)

A multi- criteria analysis (MCA) decision-making approach based on a theory for reasoning with evidence. ER incorporates concepts from decision theory, artificial intelligence, statistics, expert systems and fuzzy logic. The approach differs from conventional MCA both in the way that uncertainty is represented and in the method used to combine assessments of alternatives against individual criteria in order to draw overall conclusions regarding the relative desirability of those alternatives.

ER is based on the concept of belief structures as defined by Dempster-Shafer (D-S) theory. A belief function is a generalization of a probability function that allows distinguishing different types of uncertainty. For example, uncertainty associated with the random behavior of a well-understood system may be viewed as being different than uncertainty due to insufficient knowledge. Such distinctions are retained through the analysis, and the different types of information are interpreted as bits of evidence relevant to revising beliefs. The use of belief functions gives ER the ability to quantify ignorance and the impact of incomplete, potentially conflicting, and unreliable information in a way that is more complete and explicit than what can be accomplished with classical probability theory.

One way to contrast ER with more conventional MCA methods has to do with the way the multi-criteria analysis is commonly represented. With conventional MCA, the analysis can be viewed as a decision matrix where numbers are assigned to each alternative to measure the alternative's performance with regard to each of some number of criteria. If, for example, there are M alternatives and N criteria, the decision matrix will have M rows and N columns and a total of M x N cells. Each cell contains a number indicating the evaluation of the corresponding alternative relative to the indicated criterion. Depending on the MCA method, the numbers in the cells might be scores, values, or utilities. The individual evaluations are combined via an aggregation equation, for example, by weighting and adding the individual assessments. The numbers produced by a conventional MCA approach can be regarded as summarizing an average performance metric that, according to the supporters of ER, fails to communicate the diversity of performance and quality of information on which conclusions are based.

With ER, the decision problem is represented by a belief decision matrix. A belief decision matrix is similar to a conventional decision matrix except for the fact that the assessment of each alternative against each criterion is represented by a two-dimensional variable defined as a belief structure. The first element indicates the value or "grade" assigned by the assessment. For example, the grade assigned to an alternative's performance with respect to a criterion might be "excellent," "good," "average," or "poor." The second element indicates the degree of belief underlying the grade assessment. The degree of belief is a number, which, like a probability, lies between zero and one. Thus, for example, an alternative might be assigned a grade of "good" with a degree of belief of 0.4. With ER, each criterion can have its own set of evaluation grades and the criteria can be arranged into a hierarchy. To integrate the belief structures established for the different criteria, ER uses rules for evidence pooling and belief updating. Instead of simply aggregating average scores, the ER approach employs an evidential reasoning algorithm to aggregate belief degrees according to the evidence combination rules of D-S theory.

The evidence combination rules of D-S theory employ "fusion operators" that apply specific rules for integrating different sorts of evidence. For example, one kind of evidence would imply a constraint for the set of possibilities containing the belief. If there are multiple such constraints, the fusion operator would be the intersection of the constrained sets dictated by independent belief sources. The application of the fusion operators can be interpreted as being analogous to, but more general than, Bayes theorem for updating probability distributions based on new data.

When the degree of belief is such that it is not possible to characterize uncertainty with a precise measure such as a probability number assigned to a point in the space of possibilities, ER views probability as being assigned or spread across an interval or sub-set of the set of all possibilities. As with fuzzy logic, the theory of belief functions is addressed through a set-membership approach. The idea is that partial knowledge about some variable X is described by a set of possible values E. The set defines a constraint over the variable. As a result, utility is spread across some interval or set. The ER approach is, therefore, characterized as a distributed modeling framework capable of representing both precise data and ignorance, with the use of interval utility for characterizing incomplete assessments and incomplete knowledge. The ER approach is in this way a hierarchical evaluation model with rules for synthesizing evidence specified by the D-S theory of evidence.

The major advantage of ER is its ability to handle uncertainties associated with quantitative and qualitative data. It provides a straightforward way of quantifying ignorance and is therefore a useful framework for handling incomplete, uncertain information with varying levels of support or credibility. Computer programs have been developed to facilitate the application of ER to real-world problems. Applications include cargo ship design selection, marine system safety analysis, organizational self-assessment, supplier pre-qualification assessment, and environmental assessment. The applications demonstrate that ER is able to handle both deterministic and random systems with incomplete, missing information, or vague (fuzzy) information, large numbers (hundreds) of criteria arranged in a hierarchy, and many alternatives.

ER's main disadvantage is its complexity and the corresponding large computational requirements needed to apply the combination rules of D-S theory. Because the belief structure requires two-dimensional values, calculations required for the aggregation processes are naturally more involved than with traditional methods using an additive utility function. ER has also be criticized for a tendency to produce counterintuitive results in cases where there is conflicting evidence, and researchers continue to develop modifications to the method to address such problems. Also, even though ER has been applied to a wide variety of problems, the specific types of problems that warrant ER's more intense level of computation remains unclear.

expected commercial value (ECV)

A method often used to assign a value to a project that is intended to create a new product. Also called, estimated commercial value, ECV represents an application of expected net present value (ENPV). Scenarios are defined to represent possible project outcomes. Each scenario is assigned a probability to indicate its likelihood, and a project value is estimated for each scenario. The expected commercial value is obtained by multiplying each scenario's value by the scenario probability and adding the results. ECV the prioritization metric most often used by project portfolio management tools aimed at new product development projects.

Typically, the scenarios defined for computing ECV are highly simplified. In particular, it is common to represent the product development project as having two stages, which may be represented in a simple decision tree.

Decision tree for computing ECV

Decision tree for computing ECV


The first stage is the product development stage. Recognizing uncertainty, the probability of the project being technically successful is Pts. The second stage is the product launch, the success of which is likewise uncertain. The probability of commercial success (assuming the project is technically successful) is Pcs. If D is the development cost, C is the cost of commercially launching the project, and PV is the present value of future earnings for a commercially successful project, then ECV may be computed using the formula:

ECV = [(PV*Pcs-C)*Pts]-D

In reality, of course, technical and commercial success are not yes/no outcomes. There may be varying degrees of technical success and, assuming the product is launched, commercial sales could be anywhere within a range of possibilities. Thus, the simplifications typically used for the calculation of ECV may lead to inaccurate project valuations. Also, because ECV is a simplified version of ENPV, it has the limitations of the more general approach (including the potential for omitting non-financial sources of project value and inadequate accounting of risk and organizational risk tolerance). On the other hand, depending on the application, the simple ECV formula may provide a reasonably adequate method for ranking product development projects.

expected internal rate of return (EIRR)

A modification of the internal rate of return (IRR) sometimes used to prioritize projects (such as new product development projects) whose costs and future cash flows are highly uncertain. In the formula for computing IRR, project costs are replaced by the expected value of initial-year project costs, and project cash flows are replaced by the year-by-year expected value of project cash flows. Thus, The EIRR is the solution to the equation:

Formula for computing EIRR

In other words, to use the EIRR, alternative project cost and future cash-flow scenarios are defined. For example, the various stages and associated cash flows for the project (such as development, testing, and commercialization) may be represented in a decision tree. Probabilities are assigned to each scenario. The expected value of project costs and expected value of each year's net cash flow are computed by multiplying probabilities by cash flows and adding. The EIRR is then computed as the discount rate that equates the discounted value of expected future cash flows with the expected project cost.

When applied to multi-stage, high-risk projects, the EIRR behaves in an intuitive way. For early stage projects with a low probabilities of ultimate success, expected cash flows tend to be low so the EIRR tends to be low. However, if such a project is funded, its EIRR tends to grow (assuming initial project outcomes are successful) as project costs are sunk and early-stage failure scenarios are avoided. A late stage project (one that has successfully avoided early and middle stage risks) tends to have a very high EIRR. Because of the strong influence of project stage on the EIRR, typical advice is that project-by-project comparisons using EIRR be conducted only for projects at the same stage of development and that separate budgets be established for funding projects within the different stages.

As a project prioritization metric, the EIRR has the advantages and disadvantages described for the IRR, plus the advantages and difficulties associated with assigning probabilities to alternative scenarios.

expected net present value (ENPV)

An enhancement of the net present value approach that explicitly addresses uncertainty. Depending on how it is applied, ENPV can produce estimates of uncertainty in the value of the overall project portfolio and adjust project value to account for risk. It can also be coupled with methods for quantifying the non-financial or indirect components of project value. It is, therefore, a useful tool for computing project and project portfolio value. However, the computations necessary to compute ENPV can be difficult, and the method is often best reserved for very large and risky projects.

With ENPV, rather than calculate a single time-stream of project cash flows and other project impacts, alternative scenarios are defined representing the range of possibilities. Simulation techniques are often used to generate the alternative scenarios, which may be represented in as a decision tree (a graphic structure wherein alternative sequences of choices and outcomes are displayed as branches in the tree and the various paths through the tree represent the alternative scenarios) or event tree (similar to a decision tree, but without nodes and branches representing alternative choices). Probabilities are associated to each scenario in the tree. A project NPV is computed for each scenario, and the ENPV is the probability-weighted sum of the values.

As described under net present value, selecting discount rates is often problematic. If risk is important, risk-adjusted discount rates are often used, with different risk-adjusted rates being appropriate for different scenarios. Alternatively, techniques based on risk tolerance can be used to account for risk (these techniques generally involve using a risk-free discount rate for computing EPNV).

In addition to the difficulties mentioned above related to selecting the discount rate, another limitation of ENPV is that historical data is generally unavailable for estimating probabilities. Thus, probabilities must typically be assigned subjectively.

expected value

Term used to represent the result of a mathematical computation performed using probabilities. Suppose there is an uncertain (random) variable X that may produce various "payoffs" (values). Suppose the possible payoffs are denoted x1, x2,..., xN, and suppose that these alternative payoffs occur with probabilities p1, p2,...pN, respectively. The expected value of the variable is sum of each possible payoff multiplied by its probability:

Expected value is the sum of possible payoffs weighted by their probabilities

If instead of there being a finite number of payoffs, the uncertain variable can take on a continuum of possible values (e.g., any value between 0 and 1), then its expected value is computed by weighting the possible values using the variable's probability density function and using integral calculus.

The expected value may be interpreted as the average return one would expect over many "trials" or opportunities for the uncertainty to occur. See expected commercial value and expected net present value for examples of measures based on expected value.

expert

A person recognized by others to have superior knowledge or skill in a specific area. As used on this website, the term expert is used to refer to a person who, due to great experience and understanding of an organization's projects, or a subset of its projects, is tasked with providing estimates needed as inputs by a project selection decision model. Because the process of providing project-specific inputs to a project selection model is frequently called scoring, an expert who provides scores may be referred to as a scorer.

expert system

A computer system programmed to behave like a human with expertise in a particular field or problem area—it uses human knowledge and reasoning techniques to provide advice for solving problems. Expert systems represent an application or subfield of artificial intelligence. Although various methods can be used to simulate the performance of an expert, most expert systems consist of two components: (1) a knowledge base that contains subject matter expertise and (2) an inference engine that applies heuristics or reasoning rules similar to those used by experts in the given field. Expert systems are typically used as an aid to human workers or to supplement some information system. Some project portfolio management tools are advertised as including components that operate as expert systems.

exponential function

A mathematical function of the form: f(x) = ax (a raised to the power x). where x is a variable, and a is a constant. The most commonly encountered version of the exponential function is where the constant a is Euler's number denoted e, which is equal to approximately 2.78128. In that case the function is often expressed as f(x) = exp(x).

exponential utility function

Also called the relative risk aversion function and the negative exponential utility function, an often used and very practial (ordinal utility function) for valuing uncertain projects. A decision maker's utility function will be exponential if the delta property holds and may be written:


U(V) = 1 - e-V/R


where U is utility, V is value, and e denotes the Euler's number (~2.78128). The parameter R, called "rho", is the decision maker's risk tolerance..

With the above form of the exponential utility function, the utility numbers are all negative, but this should be of no concern since the relative utilities are what matters. If you would like to scale the utility nunbers to go from zero to one is, use this version:


Exponential utility function scaled

In this form of the equation, Low is the lowest value of V that can be obtained and High is the highest value. As with the previous equation, this equation applies assuming that the the utility of V is monotonically increasing (more value is better) and R is positive (the decision maker is risk averse) and not equal to infinity.

The figure below shows a plot of the exponential utility function with utility and value both scaled to go from zero to one. As shown, the lower R (the more risk averse the decision maker is) the more the utility function curves:


Exponential utility scaled so that value and utility go from zero to one

As suggeted by the figure, situations where the uncertainty in possible project values approaches the decision maker's risk tolerance are the situations where accounting for risk tolerance matters.

The reason that the exponential utility function is so often used is that it leads to a simple equation for computing the risk adjusted value of projects. If the decision maker has an exponential utility function, the certain equivalent of a project having an uncertain value V is given by the equation:


CE = -R × ln{1-E[U(V)]}


where E{U(V)] is the expected utility of the project, R is the decision maker's risk tolerance, and ln is the natural logarithm. Thus, the exponential utility function provides an easy way to compute the risk adjusted value of a project. Simply compute the probability distribution for the uncertain project values (e.g., by using a decision tree, with project values being the possible discounted net present values of the project). Convert the possible values to utilities using an exponential utility function, and then use the above formula to compute the project's risk adjusted value.

externality

An undesired (or desired) impact associated with the production or consumption of of a good that affects the welfare of a third party without any compensating payment (or compensation) being made. In project analysis, an externality is an effect of a project not reflected in its financial accounts and consequently not included in the valuation of the project.