Real Estate Example

You have the opportunity to purchase a piece of property for $200,000. If it can be rezoned for business use, you will be able to resell the property for $250,000. You estimate that there is a 60% chance that your rezone request will succeed. If it fails, you can dump the property for $170,000 or you can choose to appeal the zoning board’s decision, which has only about a 10% chance of success. Applying for the zoning change and appealing an unfavorable ruling will each cost $5,000. Should you purchase the property?

Build the Tree

To solve this problem with DecisionPro, you first point to New in the File menu. This will give you a list of templates you can use to begin your model. The templates are essentially blank models that have all style and calculation options set properly.

There are three decision tree templates: Basic, Long Names, and Wizard. The Basic and Long Names templates are the same except that the Long Names version uses comment fields in each node to display the node name. Since there are restrictions on what characters can be used in a name, using the comment field to hold names is more flexible. For example, Buy & Rezone is not a legal node name because the ampersand is an operator. Using the Basic template, you will need to name this node Buy and Rezone. Using the Long Names version, you can use any characters you like, as well as duplicate node names. The Basic template requires each node name to be unique. Duplicating node names often simplifies building your tree, but is generally bad practice because it can lead to communication errors when discussing your tree with others. The primary disadvantage to the Long Names template is that the analysis tools report results using node names, not node comments. Unless you have a real need to use special characters in your node names, you are better off using the Basic template.

The Wizard template is designed to be used with the Decision Tree Wizard tool, which guides you through construction of a tree. This is a great tool for building small trees when you are not familiar with DecisionPro. However, building trees directly using the Basic template is still the preferred method.

Back to the example, after pointing to New in the File menu, click Decision Tree (basic). This will open a model that looks like this:

The problem states that you have two options from which to choose: buy the property for $200,000 or do nothing. To reflect these options, double-click on the triangle following Root. This will bring up the Insert Branch dialog box:

Next, click on the Decision Node button. This causes DecisionPro to add two additional branches following Root.

DecisionPro automatically assigns the names Option1 and Option2 to the new branches. You should use the more descriptive names Buy Property and Do Nothing. To make this change, click on the text Option1 that appears in the tree and then type Buy Property followed by the ENTER key. Repeat this process to rename Option2 to Do Nothing.

Note that you do not need to edit formulas in the Formula Bar. When building decision trees you create new branches by double-clicking and you enter all input information by editing directly in the tree. DecisionPro builds the required formulas for you automatically.

When you click on a node field, such as the node name, a dotted line appears around the field. If you simply begin typing, the entire field is replaced with the new text you type. However, if you click again on the field, it will turn into an edit box where you can change the existing field contents. Clicking outside the field completes the edit.

Note: You do not double-click on a field to begin editing; instead, you click twice. Double-clicking will cause the tree to page into the branch you selected. This will all make sense with a little practice.

In this example, you have only two options from which to choose. However, if there were a third option, you could add a third branch by double-clicking on the square following Root. Each time you double-click on any branch symbol (circle, square, or triangle), a new branch is added. You can go ahead and try this if you like--just click Undo to remove the branch when you are done.

If you buy the property, you will spend $200,000. To reflect this in the tree, click on the zero in the Buy Property branch and enter -200000. Note that the incremental value you enter here is negative because it is a cash outflow.

The Do Nothing branch is complete.

If you buy the property, you will next need to apply for a zoning change. To reflect this in the model, double-click on the triangle following Buy Property and click Pass Through in the Insert Branch dialog box.

As you probably expected, this inserted a new branch but its name and incremental value need to be corrected as follows:

The problem states that there is a 60% chance that the rezone request will be approved. Therefore, there is also a 40% chance that it will be denied. You reflect this situation in your tree by inserting an event node. Again, double-click on the triangle following Request Rezone and then click Event Node in the Insert Branch dialog box.

Just as before, edit the tree to include meaningful names. You also need to change the probabilities on the Approved and Denied branches from 50%/50% to 60%/40%.

If the rezone request is approved, you will sell the property for $250,000. You can show this in the tree by inserting a pass through node after Approved and enter an incremental value of 250000.

If the rezone request is denied you have a decision to make--appeal the zoning board’s decision (which costs $5,000) or just dump the property for $170,000. Show this in the tree by inserting a decision node following Denied.

The problem states that the appeal has only a 10% chance of success. You model this chance by inserting an event node after appeal.

If the appeal succeeds, you will sell the property for $250,000; and, if it fails, you will dump the property for $170,000. Note that you cannot use the node names Sell Property and Dump Property because they have already been used. A common way to get around name conflicts is to append a number to each name. For example, Sell Property 1, Sell Property 2, etc. This tree uses the names Sell and Dump.

Calculate a Solution

Finally, to calculate a solution press the F9 key or click on the Recalculate toolbar button .

Once you have calculated a solution, you might want to set number formats by pointing to Node in the Format menu and clicking Properties .

The values at the end nodes are the actual profit or loss that you will experience if you wind up at each node. For example, if you choose to buy the property, request a rezone, the request is approved, and you sell the property, you will earn $45000 (-200000 + -5000 + 0 + 250000). DecisionPro calculates the end node values by accumulating all incremental values between Root and each end node. After calculating the end node values, DecisionPro folds-back the tree to calculate EMVs for each node.

Interpret Your Results

The final step is to interpret the results of the decision tree analysis. Buying the property has an expected value of $14,200 vs. $0 for doing nothing. Therefore, if you are not adverse to risk, you should purchase the property. Furthermore, if your rezone request is denied, you should appeal.

Keep in mind that you can never earn $14,200 on this investment; you will either earn $45,000, $40,000, or lose $40,000. The expected value calculation implicitly assumes that you are not adverse to risk. That is, individual gains or losses on investments like this are of no consequence to you. This is true if you have a large amount of money and many investment opportunities. In other cases, though, you should consider the range of outcomes as well as the expected values when interpreting your results. Techniques for applying risk aversion are presented later in this chapter.

Choosing Event Probabilities

One important difference between the real estate and the coin toss examples is in where the event probabilities came from. When you flip a coin, you know precisely what the possible outcome probabilities are--50% heads and 50% tails. In the real estate example, however, the probabilities are only estimates. We have assumed that there is a 60% chance that the rezone request will be approved. Another person constructing the same model might come up with a different estimate.

Flipping a coin is truly a random event. That is, each time you repeat the event, you might get a different outcome. However, no matter how many times you issue the rezone request, you should get the same result (in theory). The probabilities you assign to this event represent your confidence in predicting an outcome rather than the probability of the outcome occurring. This distinction is not something you should be concerned about. Decision trees help you choose the best option given whatever information you have. It is perfectly valid to use estimated outcome probabilities as was done in this example.