# How to Calculate Expected Value in Decision Trees

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A decision tree is a visual way of thinking through the business decisions you make every day. Suppose you're debating whether it's worth investing in more efficient equipment or if it's better to pay off some debt. You list the possible outcomes of your decision, evaluate which looks best and pick that one. A decision-tree solver gets the same results as working through it in your head, but the approach is usually more analytical and thorough.

## Branches, Nodes and Leaves

The decision tree gets its name because of the way it branches out from the root node, which is the initial question. Decision-tree examples could include:

• Is it better to issue dividends or reinvest this year's profits?

• Is it better to buy new equipment or squeeze a couple of extra years out of the old machinery?

• Will buying your own building work out better over time than renting?

• Is expansion an option in the current economy, or is it better to run lean for a while?

Whatever the question, the process of drawing the decision-tree solver is the same.

• From the root node, draw branches for the different options. There may be two or several. List all the ones that come to mind.

• Use chance nodes when the outcome from an option could go more than one way. For example, suppose you're thinking of firing your sales manager. The possible outcomes include not finding a replacement, hiring a great replacement, hiring someone incompetent and so on.

• Use decision nodes when an option requires a further decision — for example, whether to promote a new manager from the sales team or hire an outsider.

• Follow each branch until you reach the final possible outcomes. These are known as leaves or terminal nodes.

• Use the expected value formula to calculate the potential gain or loss at each possible terminal node. Coupled with the probability for each outcome, it can show you the right path.

## Expected Value for a Decision Tree

Calculating expected value for a decision tree requires data. It may also require good business judgment. If you want to compare the cost of buying diesel vehicles vs. the fuel savings, that's a dollars-and-cents question. Whether to promote a team member calls on your judgment of his abilities.

The first step in applying the expected value formula is to figure out the potential costs and benefits of each terminal node. For example, if you're considering whether to launch a promising new product, the terminal nodes might include:

• You test-market the product and it tanks. You lose \$15,000 making and testing the product and get nothing back.

• You test-market the product and it's modestly successful. You begin a small-scale manufacturing line that costs \$75,000 for the first year plus the \$15,000 test-marketing. The revenue for the first year is \$225,000.

• The test marketing is a huge hit, and you invest \$150,000 in manufacturing, bringing in \$350,000 in revenue.

• If you have other terminal nodes, repeat the calculation. For example, the possibility of competing products or a recession killing consumer spending might lead to more nodes.

## Decision-Tree Percentages

The next step is to assign probabilities to the various outcomes, either as percentages or fractions. Take each set of leaves branching from a common node and assign them decision-tree percentages based on the probability of that outcome being the real-world result if you take that branch.

For example, suppose on one decision-tree branch, you buy your only regional competitor to boost your own revenue and eliminate competition. The terminal nodes you foresee are that you boost your revenue significantly, that you fail to manage the larger company and that new competitors enter the market and undercut your prices.

Based on your analysis and number crunching and your judgment of your abilities, you set the probabilities: 15% overreach, 30% new competitors and 55% that you maintain an area monopoly and become more profitable. Perform the same analysis for each group of terminal nodes. Make sure that for each full set of possible results, the percentages add up to 100%.

## The Expected Value Formula

Once you have the probabilities for the leaves in your decision tree, you can apply the expected value formula to figure out which path promises the biggest payoff. Start with the terminal nodes and move back up the tree.

If you have any chance nodes, assign them probabilities too. Keep going until you reach a decision node and then apply the formula. For example, if you're looking at the outcomes of "what if I buy the competition?" you'd multiply the outcome of each leaf node by the decision-tree percentages:

• You become a successful monopoly: The returns are \$2.4 million after the cost of the purchase. Multiplied by 55% chance of realization, that's \$1.32 million.

• New competitors enter the market: Return is \$1.1 million. Multiplied by 30% chance of realization, this gives you an outcome of \$330,000.

• The purchase is a failure: You lose \$1.7 million. The loss is \$255,000.

Consider the three together, and the total for this decision node is \$1.395 million. Comparing that to the alternative decision nodes shows which branch offers the best chance of a good payoff.

## Performing a Sensitivity Analysis

Sensitivity analysis looks at how elements in a what-if scenario respond if you tinker with the variables. For example, suppose you're thinking about opening a new store in a new shopping center. You've calculated the costs and the returns, but you're not sure of some of the projections, such as the number of visitors to the shopping center.

With a sensitivity analysis, you adjust one of the factors and reevaluate your terminal nodes. Say you set an 80% probability of the shopping center succeeding, but you're not sure of that figure. If you lower the estimate to 70% or 60%, would that substantially change the cost/benefit analysis of opening the store?

Excel and similar spreadsheet software can help you with a lot of the number crunching in a sensitivity analysis and other parts of the decision tree.

## Decision-Tree Drawbacks

The great advantage of a decision tree is that when you're considering possible outcomes in your head or taking notes on paper, it's easy to overlook something. The decision tree's systematic approach makes it easier to visualize every possible outcome, even ones you wouldn't normally have imagined. Decision trees are only an approximation of reality, however, so they don't always give you good answers.

• You have to know the problem well before you can use a decision tree. If your knowledge is superficial, mapping out options on the decision tree may still miss a lot, or your estimates of outcome gains and losses may be way off.

• To make an effective decision-tree solver, you have to isolate the key elements of the decision. If you focus on the wrong issue, a decision tree for how to solve it won't help with the real problem.

• Simple mistakes can mess up the result. The percentage probabilities for each clump of terminal nodes should add up to 100%. If you get that wrong, the expected value formula will be wrong too.

• The longer the time frame, the greater the number of chance nodes that will probably come into play and the more numerous the outcomes. That can make decision trees cumbersome.

Decision trees can greatly improve your judgment, but they can't substitute for it.