Decision trees are useful for projects that proceed in stages where investment decisions may change over time. In this application brief, we will use decision tree analysis to evaluate a research and development project where we are uncertain if a commercial product can be produced as a result of the research portion of the project.

Our project will initially start with a research project to determine commercial viability. If the research shows promise we have the option to proceed with development or to kill the project. Likewise, if the research shows little commercial viability, we could kill the project or continue with more research.

A decision tree with our possible outcomes is shown below.

Starting from the left hand decision node, we can either start the research or not do the project altogether. We need to include not doing the project as an option. If expected value of the project is negative, we would not want to do the project since we are not making the cost of capital.

The initial research project is expected to take one year. At one year we assess the results and make a decision on the next step. If we decide to develop the product, it is assumed development will take an additional year, so revenue starts in year 3. If we decide to do additional research, it's assumed that the research will take another year. If after doing the additional research we decide to develop, we will not see revenue until year 4 (two years of research and one year of development).

Note in the spreadsheet above, we have several cash flow projections based on market demands of high, medium, and low. The expenses of high demand are lower as a percentage of revenue than low demand because we are assuming economies of scale. These cash flow projections are simplistic, and don't take into account things like depreciation, but you get the idea based on this simple example.

All cash flows for this decision tree analysis are present values using a discount rate of 10%.

## Risk Neutral Decision Tree Analysis

The decision tree in Figure 1 was rolled back using expected values. This assumes that we are risk neutral. This might be an appropriate approach if the project was part of a very large company where project failure would have little danger of putting the company in peril. If this project were part of a small company, then project failure may put the company in peril so we would likely be risk averse. We will talk about risk averse analysis later.

The expected value of the project is $1.55 million. The optimal paths through the tree are marked TRUE. If the initial research shows promise, we should go ahead with development. If the research lacks promise, we should kill the project.

### Sensitivity Analysis

Doing a sensitivity analysis of our decision tree allows us to see what factors are most important in the determination of expected value. DTace has a sensitivity analysis tool where node values and chance node probabilities can be changed one at a time. Each time a change is made, the tree is rolled back to calculate the new expected value. For our tree we will vary the node values by +/- 15% and vary chance node probabilities by +/- .2. The sensitivity report is shown below.

From the tornado chart we see that node 3.1 and 4.1 probabilities have the greatest impact on expected value. This means our estimates of research success probability is the most important assumption in our analysis. The estimates of low and high demand payoffs for going to market in nodes 9.1 and 11.1 are also important to expected value.

The sensitivity analysis tells us that we should focus our efforts on making the best estimates possible for research success and our market projections.

## Risk Averse Decision Tree Analysis

If we are risk averse, we can use a utility function instead of expected value to account for our risk tolerance. In risk averse behavior, utility is increasingly downward sloping as payoff values decrease. In other words, the shape of the utility curve is concave and low value payoffs are penalized heavily depending on the risk tolerance. In this example we will use exponential utility. The risk tolerance constant, R, is a measure of our risk tolerance. The curves shown below are for an exponential utility function with R values of 5 and 10. Note that R = 5 is more concave than R = 10. An R value of 5 means we are less risk tolerant than when R equals 10. Low payoff values result in a lower utility.

DTace can easily change calculations from expected value to exponential utility. To do this we click the Tree Settings button on the ribbon. When the tree settings form appears, click on the Roll Back menu item.

We will use a risk tolerance constant, R, of 5 million. Once the changes are applied, we roll back the decision tree using utility instead of payoffs.

Using the utility function now results in a negative utility if we do the project. Since not doing the project has zero utility, we should not do the project. Remember earlier that using expected value the decision tree analysis shows that we should do the project, and now using utility, we should not do the project. This is based on our risk tolerance of 5 that we entered earlier.

Let's say we have a higher risk tolerance of 10. If we change the risk tolerance constant, and roll back the tree we get the following values.

With our higher risk tolerance, the decision tree analysis shows that we should go ahead with the research.

## Conclusion

In this application brief we performed a decision tree analysis of an R&D project using both a risk neutral and risk averse position. In a risk neutral situation, we should proceed with the project. In a risk averse situation, we may or may not proceed depending on our risk tolerance.