Using Decision Trees in Finance

Decision trees are major components of finance, philosophy, and decision analysis in university classes. Yet, many students and graduates fail to understand their purpose, even though these statistical representations play an integral role in corporate finance and economic forecasting.

Decision Tree Basics

Decision trees are organized as follows: An individual makes a big decision, such as undertaking a capital project or choosing between two competing ventures. These decisions, which are often depicted with decision nodes, are based on the expected outcomes of undertaking particular courses of action. An example of such an outcome would be something like, “earnings are expected to increase by $5 million.” However, since the events indicated by end nodes are speculative in nature, chance nodes also specify the probability of a specific projection coming to fruition.

As the list of potential outcomes, which are contingent upon prior events, becomes more dynamic with complex decisions, Bayesian probability models must be implemented to determine priori probabilities.

Binomial Option Pricing in Decision Tree Analysis

Decision tree analysis is often applied to option pricing. For example, the binomial option pricing model uses discrete probabilities to determine the value of an option at expiration. The most basic binomial models assume that the value of the underlying asset will rise or fall based on calculated probabilities at the maturity date of the European option. 

However, the situation becomes more complex with American options, wherein the option can be exercised at any point until maturity. The binomial tree would factor in multiple paths that the underlying asset’s price can take over time. As the number of nodes in the binomial decision tree increases, the model eventually converges onto the Black-Scholes formula.

Although the Black-Scholes formula provides an easier alternative to option pricing over decision trees, computer software can create binomial option pricing models with “infinite” nodes. This type of calculation often provides more accurate pricing information, especially for Bermuda Options and dividend-paying stocks.

Using Decision Trees for Real Option Analysis

Valuing real options, such as expansion options and abandonment options, must be done with the use of decision trees, as their value cannot be determined via the Black-Scholes formula. Real options represent actual decisions a company may make, such as whether to expand or contract operations. For example, an oil and gas company can purchase a piece of land today, and if drilling operations are successful, it can cheaply buy additional lots of land. If drilling is unsuccessful, the company will not exercise the option and it will expire worthless. Since real options provide significant value to corporate projects, they are an integral part of capital budgeting decisions.

Individuals must decide whether or not to purchase the option prior to the project’s initiation. Fortunately, once the probabilities of successes and failures are determined, decision trees help clarify the expected value of potential capital budgeting decisions. Companies often accept what initially seems like negative net present value (NPV) projects, but once the real option value is considered, the NPV actually becomes positive. 

Decision Tree Applications for Competing Projects

Similarly, decision trees are also applicable to business operations. Companies are constantly making decisions regarding issues like product development, staffing, operations, and mergers and acquisitions. Organizing all considered alternatives with a decision tree allows for simultaneous systematic evaluation of these ideas.

This is not to suggest that decision trees should be used to contemplate every micro decision. But decision trees do provide general frameworks for determining solutions to problems, and for managing the realized consequences of major decisions. For example, a decision tree can help managers determine the expected financial impact of hiring an employee who fails to meet expectations and must be fired.

Pricing of Interest Rate Instruments With Binomial Trees

Although not strictly a decision tree, a binomial tree is constructed in a similar fashion and is used for the similar purpose of determining the impact of a fluctuating/uncertain variable. The upward and downward movement of interest rates has a significant impact on the price of fixed income securities and interest rate derivatives. Binomial trees let investors accurately evaluate bonds with embedded call and put provisions using uncertainty regarding future interest rates.

Because the Black-Scholes model is not applicable to valuing bonds and interest rate-based options, the binomial model is the ideal alternative. Corporate projects are often valued with decision trees that factor various possible alternative states of the economy. Likewise, the value of bonds, interest rate floors and caps, interest rate swaps, and other types of investment tools can be determined by analyzing the effects of different interest rate environments.

Decision Trees and Corporate Analysis

Decision trees let individuals explore the ranging elements that could materially impact their decisions. Prior to airing a multimillion-dollar Super Bowl commercial, a firm aims to determine the different possible outcomes of their marketing campaign. Various issues can influence the final success or failure of the expenditure, such as the appeal of the commercial, the economic outlook, the quality of the product, and competitors’ advertisements. Once the impact of these variables has been determined and the corresponding probabilities assigned, the company can formally decide whether or not to run the ad.

Limitations of Decision Trees

One major drawback is their tendency to become overly complex, especially when dealing with a large number of variables and possible outcomes. This is clearly the case with the ‘Corporate Analysis’ example above. As the number of branches increases, this can lead to what is known as the “curse of dimensionality”, meaning too much information may make a decision overly complex.

Another limitation is the reliance on subjective probability estimates. Many financial scenarios, such as predicting market movements or estimating default risk, involve assigning probabilities based on historical data or expert judgment. This may not only materialize or be a reliable function; for example, the causes of past recessions may not necessarily indicate the probability of a future recession.

Decision trees also struggle with handling correlated variables, which are common in finance. For example, things like interest rates, inflation, and stock prices often influence each other. Standard decision tree models do not inherently account for these interdependencies, potentially leading to oversimplified conclusions.

Finally, decision trees may not effectively capture continuous financial data or non-linear relationships between variables. Financial markets often exhibit complex patterns that are better modeled using advanced statistical methods or machine learning approaches like neural networks. For instance, when trying to assign credit risk, there may be too many interconnected variables all reliant on each other to be useful in a tree-analysis structure.

Pruning Decision Trees For Financial Analysis

Pruning in a decision tree is a primary way to refine the model by removing branches that do not significantly contribute to decision-making. By pruning unnecessary branches, the decision tree becomes more streamlined, improving its ability to make accurate predictions in real-world scenarios.

There are two primary types of pruning: pre-pruning and post-pruning. Pre-pruning, also known as early stopping, involves setting constraints during the tree-building process. For instance, you could limit the maximum depth of the tree or require a minimum number of data points per split. This prevents the model from growing excessively large in the first place.

Post-pruning involves growing the tree fully and then systematically removing branches that do not add substantial predictive value. This is typically done using validation data to determine which branches can be eliminated while maintaining accuracy. Not that this post-pruning stage is where you’re at the most risk to overfit your data.

Software Tools Commonly Used for Decision Tree Analysis

Financial analysts can use different software tools to construct and analyze decision trees. Microsoft Excel is one of the most widely used tools, particularly for basic decision tree modeling. Analysts can manually build decision trees using Excel’s built-in functions or use add-ins like XLMiner for more advanced analysis.

For more sophisticated decision tree analysis, R is a popular choice. The rpart package in R allows you to build, prune, and optimize decision trees using machine learning techniques. R generally has a better ability to handle large datasets and perform more complex calculations. Python, particularly with the Scikit-learn library, is another option. Scikit-learn allows you to implement decision tree classifiers and regressors. It can also usually be integrated with other financial modeling tools.

In addition to general-purpose programming languages, specialized financial modeling software such as Palantir, SAS, and MATLAB also provide decision tree analysis capabilities. These types of tools might be better suited for institutions needing to not only do more robust analysis but having the analysis at the core of what the company does (i.e. corporate analysis or underwriting).

The Bottom Line

These examples provide an overview of a typical assessment, which can benefit from utilizing a decision tree. Once all of the important variables are determined, these decision trees become very complex. However, these instruments are often an essential tool in the investment analysis or management decision-making process.