## 3.4 Expected value

Expected value (EV) is a way to measure the relative merits of decision alternatives. The expected value term is a mathematical combination of payoffs and probabilities. You calculate the expected values after all probabilities and payoff values are identified.The goal of the calculations is to find the EV for each decision alternative emerging from the root node. For the purposes of this primer, the decision alternative with the highest EV is the best choice. See figure 3.4.

Although you can apply the formal definition of expected value, in practice you can calculate EV calculations by applying the following rules. To calculate EV, start from the endpoints and work back towards the root. An easy way to find expected values is to calculate an EV for each terminated branch, then each chance node and each decision node.

- For a terminated decision branch, EV is equal to the
payoff.

EV = Payoff - For a terminated chance branch, EV is the product of
its payoff and probability.

EV = Payoff x Probability - For a chance node, EV is the sum of each chance
branch payoff multiplied by the probability for that
payoff.

EV = [EV_{branch1}+ EV_{branch2}+ ... + EV_{branchN}] - For a decision node, EV is the greatest EV value of
all the connected decision branches. Mark the lower value
EV branches with double-hatch marks to disregard
these branch paths. Since the root node is also the
first decision node, the decision alternative with the
largest EV is the overall best decision.

EV = the greatest EV value among all the connected decision branches. - The EV at any node becomes the payoff “input” at the node closer to the root.

Figure 3.4

**Example**

In figure 3.4, calculate the EV for the decision alternative to develop the project by following the given EV rules:

- Decision node (“international and domestic marketing” vs. “domestic marketing only”). The EV is the greatest value given by all the decision branches, $3,000,000. This value then becomes the payoff “input” for the next node to the left.
- Chance node (“all criteria” vs. “domestic criteria
only” vs. “not enough criteria”).

[$3,000,000 x 0.2] + [$500,000 x 0.5] + [(-$1,000,000) x 0.3] = EV = $550,000.

This value becomes the payoff “input” for this alternative when considering the root node. - By a similar calculation, the EV for the alternative to wait for the report before deciding whether or not to develop the project is EV = $750,000. This value becomes the payoff “input” for this alternative when considering the root node.
- The EV for the alternative to invest the capital in a fixed-yield investment is just the payoff value, EV = $50,000.
- The EV for doing nothing is EV = $0.

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