# Expected value analysis

Expected value analysis is a special way of determining severity in risks. To do this, we must measure the probability of the risk in numbers between 0.0 and 1.0. Of course the numbers 0.0 and 1.0 themselves are not used since these would mean that the risk was either an impossibility or a certainty. If the risk is a certainty, it should be put into the project plan as a required task; if it is an impossibility, it should be ignored.

The values for the impact of the risks are estimated in dollars or some other monetary value. By evaluating the impact and probability this way, we can multiply the two values together and come up with what is called the expected value of the risk. This value for severity has quantitative meaning. The resulting value is the average value of the risk. In other words, if we were to do this project many times, the risk would happen some of the time and not happen some of the time. The full cost of the risk each time it happens is the impact of the risk. Of course, since the probability is less than 1.0, the risk does not occur each time. Adding up the cost of the risk each time it occurred and dividing by the number of times the project was done would give an average value. This is the expected value.

The expected value is extremely useful because it gives us a value that could be spent on the risk to avoid it. If the cost of avoiding a risk is less than its expected value, we should probably spend the money to avoid it. If the cost of the corrective action to avoid a risk is greater than the expected value, the action should not be taken.

The same is true with the other risk strategies. If the difference between the expected value of the unmitigated risk and the mitigated risk is less than the cost of the mitigation, then the mitigation should not be done. If the difference between the expected value of the nontransferred risk and the transferred risk is less than the cost of the transfer or insurance premium, then the transfer should not be done.

The expected value of several risks can be summarized by their expected values into best-case, worst-case, and expected-value scenarios as well. The best-case scenario is the summation of all the good things, but none of the bad things, that can happen in the project or subproject. It assumes that all of the opportunities will occur but that none of the risks will materialize. The worst-case scenario is the situation that assumes that none of the good things will happen but that all of the risks will happen.

The following example illustrates the use of expected value and a best-case, worst-case scenario:

Figure 8-5: Example expected value

Suppose a project has a 65 percent chance of being completed successfully and earning \$2,000,000. It also has a 15 percent chance of earning an additional \$3,000,000 in revenue, and it has a 20 percent chance of an additional cost of \$700,000. It can be seen in Figures 8-5, 8-6, and 8-7 that for this project the total expected value is \$1,610,000. The best of all situations that can occur is that the project earns \$5,000,000. The worst possible situation is that the project loses \$700,000.

# Decision tree analysis

Another technique that allows us to make risk management decisions based on evaluating expected values for different possible outcomes of

Figure 8-6: Example best case

Figure 8-7: Example worst case

the risk event is called the decision tree. This technique is a way of looking at interdependent multiple risks. It also allows us to evaluate risks with multiple outcomes. For a project environment, this technique becomes extremely useful because one chosen unplanned event can often result in multiple outcomes of various levels of severity depending on the situation and on decisions made by people who are responsible for risk management.

The decision tree can also be useful for us in our further work of developing workarounds in case of active acceptance of risk event (see risk response, later in this chapter).

As shown in Figure 8-8, decision tree diagrams are composed of boxes, which identify decision choices that must be made, and circles, which represent places where probabilistic multiple outcomes are possible. From the boxes, lines are drawn showing each possible decision. The lines lead to other decisions or probabilistic multiple outcomes. On the probabilistic circles, notice

Figure 8-8: Decision trees

that the sum of the probabilities of all the possible outcomes of this point equals 1.0. This is because all of the possible outcomes are included.

Here is an example of decision tree analysis:

Suppose a farmer must decide what to do with his land for the next growing season. He can choose to plant corn or soybeans or to not plant anything at all. If he plants nothing at all, the government farm subsidy will pay him \$30 per acre.

If the farmer decides to plant corn or soybeans on his land, there is some risk involved. The yield per acre depends on the amount of rainfall. Too much rain or too little rain will give poorer results than the right amount of rainfall. There is a 40 percent probability that the rainfall will be low; there is a 40 percent probability that the rainfall will be medium; and there is a 20 percent chance that the rainfall will be high.

If the farmer decides to plant corn, the yield per acre will be \$0, \$90, and \$50, respectively, if the rainfall is low, medium, or high. If the farmer decides to plant soybeans, the yield per acre will be \$40, \$70, and \$20, respectively, for low, medium, and high amounts of rainfall.

As shown in Figure 8-9, the decision to be made is whether the farmer should plant corn, soybeans, or nothing at all. There are three lines coming out of the decision box to indicate the three choices. Each choice leads to a probabilistic occurrence—how much rainfall will occur.

Figure 8-9: Example: Decision tree

Each probabilistic occurrence has three possible outcomes—low, medium, or high amounts of rainfall. For each of these events there is an associated payoff. The payoff amount multiplied by the probability of that event occurring is the expected value of each occurrence.

In order to evaluate the decisions, we must add the expected value of each event associated with each decision to get the expected value for each decision. For corn, low rainfall means that no money will be made from the crop. For medium rainfall there is a 40 percent chance and a \$90 yield, giving an expected value of \$36. For high rainfall there is not as much yield per acre at \$50 and there is a 20 percent probability of that occurring. The expected value for high rainfall is thus \$10 per acre. Adding the expected values for the events gives us the expected value for the decision. This is \$46 per acre.

Using the same calculation for the soybeans and for not planting at all, we see that of the three decisions, planting soybeans has the greatest yield.