In the previous article in this series, probability distributions and the expected value concepts were introduced. A probability distribution can completely represent someone's judgment about the likelihood of an uncertain event. If one or more inputs to an analysis are probability distributions, then project value and other outputs also will be probability distributions.

Expected value (EV) was shown to be the only unbiased predictor and, thus, the best single-value estimate for use in forecasting. EV is simply the probability-weighted average of all possible outcomes represented in a probability distribution. In this article, we will discuss building a logical, consistent decision policy around the EV concept. An example uses two approaches to calculating EV: a payoff table and a decision tree.

This series is oriented toward project management decisions. The decision analysis techniques are fully general and can be applied to three types of problems:

- Choosing between alternatives
- Buy, make or build, or lease
- Size or number of units of equipment to purchase
- Best use or disposition of asset

- Appraising value
- Project or venture value, or elements of projects or ventures
- Estimating transaction or project proceeds

- Determining the optimal value for a decision variable
- How much to bid to maximize the value of the bid opportunity
- Optimal capacity or configuration of a facility or equipment

Most decisions fall into one of these types. Decision analysis techniques are universal. They apply to all professional disciplines and to problems in your personal life.

**ATTITUDE TOWARD TIME VALUE OF MONEY**

In business, and for money-dominated decisions elsewhere, it is the impact on the organization's after-tax net cash flow that is important to decision making.

It is well established that there is value to receiving money sooner rather than later. Conversely, we would like to defer any payment obligations.

Present value (PV) discounting is the generally accepted method to recognize time preference. The general equation, and one that always works, is:

PV = CF/(1 + i)^{t}

where

PV | = | present value at the time of decision |

CF | = | a cash flow amount realized at a future time |

t | = | the time between the effective, or as-of date, and when the cash flow is realized |

i | = | discount rate |

The discount rate, i, represents the decision maker's attitude toward time value of money. I recommend choosing i so that Expect Monetary Value (EMV) measures increase to company net worth by doing the project. Often, doing the project is compared against a “do-nothing” or “current plan” alternative for which we assume a zero value. The PV discount rate is usually fixed in decision policy, but there is no reason that i cannot change over time (e.g., as a function of capital markets and inflation).

It has been my experience that most people involved with project evaluation sometimes make serious errors in present value discounting. Just because “PV” functions are ubiquitous on hand calculators and in computer programs does not mean the calculation is correct. Here are some guidelines for proper discounted cash flow analysis:

**The cash flow projection is the net of investment and net of all taxes.** Ideally, we want the incremental cash flow effect on the company from doing the project (or whatever alternative is being considered). Be careful that overhead burdens are legitimately incremental. In most cases, exclude financing costs from the analysis.

**The objective present value discount rate is the company's marginal, after-tax cost of capital.** If the stockholders would agree that your cash flow projection is objective, one can reason that the discount rate should be the after-tax rate these investors would demand for a risk-free investment of similar duration.

Your chief financial officer should be able to provide this rate. Do not adjust the discount rate for project risk! That is the investment bankers’ approach and a poor way to deal with uncertainty. In decision analysis, we handle risks explicitly, and separately, with probabilities.

**Be careful about timing.** Cash flows are discounted to a single date, usually the next decision point. Be sure that your calculation tool properly recognizes the timing of when your cash flows are realized.

**Ignore sunk costs.** Such previous investment costs, are not relevant to decisions about the future. Only future cash flows should be considered in today's decision.

Expected value present value (EV PV) is most often the recommended value measure for organizations seeking to maximize monetary value. This measure is so important that it has its own name: Expected Monetary Value (EMV).

**ATTITUDE TOWARD RISK**

Risk attitude, or preference, is a fascinating area of decision analysis. This is because most people, unknowingly, make choices that are inconsistent with their objective. You will do better in project evaluations, decisions, and negotiations by understanding the concept of risk preference.

What we see most often among the uninitiated is hyper-conservatism. For example, consider the manager of a large project who would give up a chance to “invest” $1 million to gain a 50 percent chance at $10 million in present value (PV) savings. That is, there are equal chances between a $1 million loss and a $9 million PV net gain. This behavior may seem rational if the project had a modest budget. However, a better perspective is to consider project outcomes to the size of the organization's capital budget. Or, better, compare this investment to the collective net worth of the company's stockholders. The expected monetary value for this project decision is:

A decision maker who is risk neutral would be indifferent between (1) having $4 million cash in hand and (2) having an opportunity to do this project.

Once people realize the implications of their conservativ_{e} selections, most will make their future decisions more objectively. An investment like this example will, on average, reduce the present value of the project's cost by $4 million. This is clearly a good decision in a portfolio of many similarly-sized decisions. If the project's value is large in comparison to the outcomes of a particular decision, generally the project manager will want to he objective toward risk.

If you want to have a conservative risk attitude as a part of your decision policy, there is a straightforward way to do this. A utility function is specified to represent the company's risk policy. This defines how the company makes trade-offs between dollar value and risk. This topic will he further explained in a future installment in this series. For most decisions, however, risk neutrality is a reasonable assumption.

**DECISION POLICY**

Logical, consistent decisions require using a clear decision policy. Setting up decision policy is straightforward. Yet, most organizations have failed, in varying degrees, to do this.

In your company, discounted cash flow analysis probably is used for project evaluations. Dollars measure value. Present value discounting accounts for time value of money.

Delay | Medium Crane | Large Crane | ||

Probability | Delay Days | Probability | Delay Days | |

Long | .2 | 10 | .2 | 3 |

Medium | .5 | 5 | .5 | 2 |

You may have never seen or heard of a utility function. Fortunately, that is not serious for most decisions. If the outcomes are small in comparison to project value, then risk aversion should have little affect on your decisions. For most purposes, expected value of present value dollars, i.e., expected monetary value, is a good measure of value.

This leads us to the EMV decision rule which is a complete decision policy statement suitable for most projects:

*Choose the decision alternative having the greatest expected monetary value (EMV).*

If there is a need for a conservative risk policy, then an expected utility decision rule is used instead:

*Choose the alternative having the greatest expected utility.*

Delay Outcome | Medium Crane | Large Crane | ||||

probability | Delay cost | Pr x cost | Probability | Delay Cost | Pr x Cost | |

No Delay | .70 | 0 | 0 | .80 | 0 | 0 |

Long Delay | .06 | $50,000 | $3,000 | .04 | $15,000 | $600 |

Medium Delay | .15 | $25,000 | $3750 | .10 | $10,000 | $1,000 |

Short Delay | .09 | $10,000 | $900 | .06 | $5,000 | $300 |

Expected Value Delay Cost | $7,650 | $1,900 | ||||

Crane cost | $10,000 | $15,000 |

**CRANE SIZE DECISION**

Here is an example of using the EMV decision rule. The approach in this example is fairly generic. Other potentially constraining resources could be analyzed similarly, such as computer, copy machine, or fork lift capacity.

Assume you are managing a project which requires use of a crane for activity A15. “Medium” and “Large” cranes are available via lease. The medium crane is suited to A15 and would do the job. However, a large crane would accomplish the activity sooner. You want to decide which size crane is the best alternative

A15 is not on the project critical path using either crane size (affecting A15‘s duration). Thus, a conventional project analysis sees no benefit of greater crane capacity on the project completion time. However, a probabilistic project model, a topic of future articles in this series, shows a potential for A15 to lie on the critical path. Suppose that the probability that A15 will become critical is .30 (30 percent) with a medium crane. There is thus a .70 probability that it will not become critical. With the greater capacity of a large crane, A15 has only a.2 probability of becoming critical and a probability of .80 of no delay. Table 1 represents best judgments about delay times versus crane size in the event the path containing A15 becomes critical.

This situation can be analyzed using either a payoff table or a decision tree.

*PMs*

**Payoff Table**

Each day's delay in project completion costs $5,000. A medium crane can be leased for $10,000, and a large crane can be leased for $15,000. For simplicity, we'll ignore the crew cost savings with the more efficient Large crane.

Which crane size should the project manager choose?

The Payoff Table approach is shown in Table 2. It is a convenient way to portray simple problems such as this. In this problem, only costs apply, so the expected monetary value (EMV) decision rule reduces to:

*Choose the alternative having the lowest expected value cost.*

If the decision is made to use the medium crane, the probability of no delay is .70. The probability that there is a long delay is .06( .30x .20), etc.

The cost that is incurred if there is a long delay with the medium crane is $50,000 (10 days @ $5,000 per day).

EV for this decision is calculated by multiplying delay cost outcomes times the respective probabilities of occurrence, then summing. EV delay costs are $7,650 for the medium crane and $1,900 for the large crane. These are added to the crane lease costs to arrive at total EV crane costs of $17,650 and $16,900. The large crane is the preferred solution. Its EV cost is $750 less than the medium crane.

**Decision Tree**

A standard calculation tool in decision analysis is the decision tree diagram, which is suited to more complex problems. The diagram is a graphical representation for calculating expected values. Decision points are drawn as squares, and chance events are drawn as circles. Figure 1 shows a decision tree analysis for the crane size problem. Decision trees are more flexible in expressing the logic of a complex decision. Decision tree analysis will be featured in a later article in this series.

For most of us, combining values and probabilities is difficult to do with unaided intuition. This example shows the power of simple decision analysis where intuition and conventional analysis fails.

### SUMMARY

Two decision models have been presented, payoff table and decision tree. The payoff table is especially useful in less complex situations involving discrete outcomes, probabilities for each outcome, payoffs (costs) associated with each outcome, and the application of expected value. Alternatively, the decision tree can be used for any problem for which the payoff table is applicable. The more complex the decision, however, the more clearly can be expressed the decisions versus the probabilistic events and the relationships between them. Indeed, for multi-phase decision analysis, the decision tree is required.

There are many other tools for decision making which provide increased flexibility, continuous probability distributions, and more sophisticated logic. These will be introduced in future articles.

In the next installment, I will discuss project modeling. This is the most credible way to generate cash flow projections used to evaluate decision alternatives. Scenario and sensitivity analysis can be used to identify the important chance and decision variables.

**John R. Schuyler,** *PE, CMA, is a planning and evaluation consultant in Aurora, Colorado. He teaches decision analysis courses worldwide in association with Oil and Gas Consultants International. His consulting focuses on modeling capital investments, acquisitions, and other corporate planning decisions. He received BS and MS degrees in engineering from Colorado School of Mines and an MBA from the University of Colorado. His prior experience includes vice president and evaluation engineer with the nation's fifth largest bank, planning and evaluation analyst for a major oil company, and senior management consultant with a national CPA firm.*