Decision Tree Analysis (DTA) is one of the important tools available that takes flexibilities – left out from the NPV analysis – into account. It was first advocated by J. Magee in 1964 and has remained an important tool for capital investment decisions. DTA is basically a tool that can depict strategic future pathways an investor can take based on a number of different future outcomes. It shows graphically a decision road map of an investor’s and manager’s strategic initiatives and opportunities over time. DTA can be used when future outcomes are uncertain and investors have tool to react when new information is arrived in the future.
Prior to exploring DTA in detail, it is necessary to examine the concept of the expected return, which is basis of the conventional DCF analysis. According to Bodie et al (2002), the expected return of an asset is a probability-weighted average of its return in all future scenarios. Let Pr(s) be the probability of scenario s and r(s) be the return in scenario s, the expected return .E(r) can be expressed as:
∑
⋅
=
ss
r
s
r
E(
)
Pr(
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For example, consider the return for a downtown office building that is sensitive to the overall regional economy. For the sake of the analysis, we can assume the following scenario for the next year:
Bullish Economy Bearish Economy Economic Crisis
Probability .4 .4 .2
Return 30% 10% -30%
Applying these values to the formula defined above, the expected return of the office building is:
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10%The conventional NPV method blindly uses this 10% as the basis of the investment analysis. However, it is critical to note that for actively managed investments like real estate development projects, managers can take action to prevent losses when the future outcome turns adverse. The main idea of DTA is to map out potential decisions so that it would enhance future outcome of a project. If an investor acquires an option to buy the office building, instead of buying the office immediately, she would not exercise option for a loss. Therefore, the investor’s average return would be – without considering the option price and the lost time value of delaying the commitment:
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As shown, the ability to make decisions when future outcome is known creates value to a project. DTA is designed to help investor to maximize the benefit of the sequential decision making.
Decision Tree Analysis incorporates the value of flexibility by explicitly laying out the structure of a project in such a way that all uncertainties and the potential decisions to be made on the uncertainties are represented as a tree form. According to de Neufville (1990), DTA leads to the following three results:
• Structures the problem, which otherwise would be very confusing due to the complexities introduced by uncertainty.
• Defines optimal choices for any period through an expected value calculation based on the consideration of the probabilities and the outcomes of each choice.
• Identifies an optimal strategy over many periods of time.
These benefits of DTA can be used to correct shortcoming of the DCF-based analyses as previously identified. DTA illustrates how future decisions could be made as uncertainties regarding a project reveal themselves over time. Therefore, it does not assume pre-committing all the decisions at the time zero. Unlike the NPV analysis, DTA assumes that investors will learn new information about the project and they have flexibilities to change course of action as the project proceeds.
Following de Neufville’s approach, a Decision Tree is composed of three basic nodes:
• Decision nodes (square), where possible decisions are contemplated and a decision made. • Chance nodes (circle), where outcomes are determined by events or states of nature.
Chance nodes have probability of each chance happening, and the sum of the probabilities in each chance node equals one.
• Terminal nodes (triangle), where a project is completed or abandoned. They are the end points of the decision tree branches, and they are typically accompanied by terminal value of the path.
In its most basic form, a Decision Tree has series of decision nodes and chance nodes branching out to form a tree shaped structure. By assigning probabilities in chance nodes and terminal payoffs at the terminal nodes of each branch, it is possible to value the project at each decision node. Described formally, the expected value of a risky decision Di is the outcomes weighted by their estimated provability of occurrence:
ij j j i
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O
D
EV(
)=∑
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When there are a number of alternatives choices to be made, the decision rule in DTA is to choose the one that offers the best average value, defined as the expected value (EV) above. When multiples nodes and branches are involved, EV is calculated backwards from the terminal
nodes to the initial node. The following is a simple example of the decision tree regarding an investment decision:
The simple analysis in Figure 4.1 identifies T-Bill as a superior investment than risky real estate, based on the expected value calculation6. This example is deterministic model in that it assumes all the future probabilities of outcomes are already known. It is possible to estimate probabilities and payoffs based on past data and experience, and it would not possible to know for sure in most cases. The real world application of Decision Tree Analysis would have much more complex form and involve numerous variables. The strongest virtue of DTA is that it exposes all the uncertainties and the accompanying flexibilities of a project wide open, which otherwise would have been treated as a “black box” that only gives a single value estimation.
6 The example in Figure 4.1 does not account for discount rate and time, and therefore does not factor in
investor’s risk aversion. In this case, the expected value of investing in risky real estate is lower despite the potentially greater risks, and thus investing in T-Bill obviously superior to real estate. Another way to incorporate risks in this framework would be using a risk adjusted probabilities so that the probability of each outcome is adjusted for the risk. However, it would be hard to implement this approach in practice due
Figure 4.1: Simple Decision Tree Analysis of Investment Decision
4.1.1 Structuring of Decision Tree