Multiple discriminant analysis

What Is Multiple Discriminant Analysis?

Multiple discriminant analysis (MDA) is a statistical technique used within quantitative finance to classify observations into predefined groups based on multiple independent variables. This multivariate analytical method aims to find a linear combination of variables that best separates two or more groups. Financial professionals frequently employ multiple discriminant analysis to assess and categorize entities, such as companies or investments, into distinct classifications like financially healthy or distressed, or different bond rating categories. It is a powerful tool for data analysis when the goal is to predict group membership.

History and Origin

The foundational work in applying discriminant analysis to financial problems emerged in the mid-20th century. A significant milestone was the pioneering research by Edward I. Altman, a finance professor at New York University. In his seminal 1968 paper, "Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy," Altman introduced the Z-score model, which utilized multiple discriminant analysis to predict corporate bankruptcy. His model combined various financial ratios to classify manufacturing firms into distinct categories of financial health and financial distress. This application marked a pivotal moment in the adoption of advanced statistical methods for risk assessment in finance.⁶

Key Takeaways

• Multiple discriminant analysis (MDA) is a multivariate analysis technique for classifying observations into predefined groups.
• It seeks to derive a discriminant function that maximizes the separation between group means.
• A primary application in finance is bankruptcy prediction and credit scoring.
• MDA can help evaluate potential investments by simplifying complex data with multiple variables.
• Assumptions about data distribution and multicollinearity should be considered when applying multiple discriminant analysis.

Formula and Calculation

Multiple discriminant analysis aims to derive one or more discriminant functions. For (G) groups, a maximum of (\text{min}(G-1, p)) discriminant functions can be derived, where (p) is the number of predictor variables. Each discriminant function (D_k) (for (k=1, \dots, \text{min}(G-1, p))) is a linear combination of the independent variables:

Where:

• (D_k) = the score on the (k^{th}) discriminant function
• (w_{ki}) = the discriminant weight for the (i^{{th}) independent variable for the (k}{th}) function
• (X_i) = the (i^{th}) independent variable (e.g., financial ratios from a company's balance sheet or income statement)

The weights (w_{ki}) are chosen to maximize the ratio of the between-group variance to the within-group variance, effectively creating a new axis along which the groups are maximally separated.

Interpreting Multiple Discriminant Analysis

The interpretation of multiple discriminant analysis involves examining the discriminant functions and their associated classification rules. Once the discriminant functions are derived, a score is calculated for each observation (e.g., a company) using these functions. These scores are then compared to "cutoff" or "centroid" values for each group. A centroid is the mean discriminant score for all observations within a particular group. An observation is assigned to the group whose centroid is closest to its discriminant score.

Analysts typically evaluate the significance of the discriminant functions and the predictive accuracy of the model. Variables with larger absolute discriminant weights contribute more to the separation between groups, providing insights into which factors are most critical for distinguishing between categories like healthy and distressed firms. This process is crucial in financial modeling for effective categorization.

Hypothetical Example

Imagine a venture capital firm using multiple discriminant analysis to evaluate potential startups for investment. They want to classify startups into three groups: "High Growth Potential," "Moderate Growth Potential," and "Low Growth Potential," based on three key metrics: (1) revenue growth rate, (2) customer acquisition cost (CAC), and (3) market share expansion.

The firm collects historical data from numerous past startups, each categorized into one of these three groups. They run an MDA, which develops two discriminant functions.

• Function 1 (D1): Primarily distinguishes between "Low Growth Potential" and the other two groups, heavily weighting revenue growth rate and market share expansion.
• Function 2 (D2): Further separates "High Growth Potential" from "Moderate Growth Potential," with a strong emphasis on a low customer acquisition cost.

When a new startup is evaluated, its revenue growth, CAC, and market share expansion are plugged into these two functions to generate D1 and D2 scores. If a new startup yields a high D1 score and a low D2 score, it might fall into the "Moderate Growth Potential" category. This allows the firm to systematically apply an investment analysis framework based on quantitative factors.

Practical Applications

Multiple discriminant analysis finds various practical applications across finance and investing:

• Credit Risk Assessment: One of the most prominent uses, as exemplified by the Altman Z-score, is predicting the likelihood of corporate default or bankruptcy. Lenders and investors use MDA-based models to assess the financial health of borrowers and make informed lending decisions.
• Bond Rating: Rating agencies may use multiple discriminant analysis to assign bond ratings, classifying bonds into different risk categories based on financial characteristics of the issuing entity.
• Portfolio Management: Financial professionals leverage MDA in portfolio management to evaluate and select investments. It can help in creating Markowitz efficient sets by categorizing securities based on factors like expected return, market volatility, and other fundamental metrics, enabling investors to maximize returns for a given level of risk. For example, analysts can use multiple discriminant analysis to group stocks with similar risk-return profiles for specific investment strategies.⁵
• Mergers and Acquisitions (M&A): MDA can assist in identifying potential acquisition targets by classifying companies based on their financial strength and strategic fit.
• Fraud Detection: In finance, multiple discriminant analysis can be used to distinguish between fraudulent and non-fraudulent transactions or financial statements by identifying patterns that predict fraud.

Limitations and Criticisms

Despite its utility, multiple discriminant analysis has several limitations and criticisms:

• Assumptions of Data: MDA assumes that the independent variables are normally distributed within each group and that the covariance matrices for all groups are equal. Violations of these assumptions, particularly the homogeneity of covariance matrices, can reduce the accuracy and reliability of the discriminant functions.⁴ While some studies suggest MDA is robust to minor violations, significant deviations can lead to suboptimal classification.³
• Linearity: Multiple discriminant analysis is inherently a linear classification method. If the true relationship between the independent variables and group membership is non-linear, MDA may not effectively separate the groups.
• Sensitivity to Outliers: The technique can be sensitive to outliers, which may unduly influence the discriminant function and lead to inaccurate classifications.
• Multicollinearity: High correlation among predictor variables (multicollinearity) can complicate the interpretation of the discriminant weights and potentially reduce the predictive power of the model.²
• Sample Size: For reliable results, particularly when dealing with multiple groups and variables, MDA requires an adequate sample size. The size of the smallest group should ideally exceed the number of predictor variables to ensure valid estimations.¹

These limitations highlight the importance of careful model validation and a thorough understanding of the underlying data when applying multiple discriminant analysis.

Multiple Discriminant Analysis vs. Altman Z-score

Multiple discriminant analysis (MDA) is a broad statistical technique used for classification, while the Altman Z-score is a specific, well-known application of MDA in finance. MDA provides the methodology to derive a discriminant function that separates groups. The Altman Z-score is the actual discriminant function developed by Edward Altman using MDA, designed to predict corporate bankruptcy.

The primary difference lies in their scope: MDA is the general statistical method, applicable to various classification problems across many fields, not just finance. The Altman Z-score is a concrete financial model derived from multiple discriminant analysis, specifically tailored for bankruptcy prediction using a set of financial ratios. Therefore, while all Altman Z-scores are a result of MDA, not all applications of MDA result in an Altman Z-score.

FAQs

How does multiple discriminant analysis differ from regression analysis?

While both are statistical techniques, multiple discriminant analysis is used for classification (predicting which group an observation belongs to), whereas regression analysis is used for prediction (predicting a continuous outcome variable). MDA's dependent variable is categorical, while regression's dependent variable is continuous.

Can multiple discriminant analysis be used with qualitative data?

Multiple discriminant analysis primarily works with quantitative, interval, or ratio scale data for its independent variables. Qualitative data would typically need to be converted into numerical form (e.g., through dummy coding) to be included in an MDA model.

What is a "discriminant function"?

A discriminant function is a linear equation derived by multiple discriminant analysis that combines independent variables with specific weights. This function creates a score for each observation, and these scores are then used to classify observations into predefined groups based on how they cluster around the group centroids.

Is multiple discriminant analysis still relevant today?

Yes, multiple discriminant analysis remains relevant, especially as a foundational technique in predictive analytics. While more advanced machine learning methods exist, MDA provides a clear, interpretable model for classification problems and is widely used in areas like credit scoring and financial risk assessment due to its historical success and transparency.

What kind of "groups" can multiple discriminant analysis classify?

Multiple discriminant analysis can classify observations into any predefined, mutually exclusive groups. In finance, this commonly includes classifying companies as "financially healthy" vs. "financially distressed," or "investment-grade" vs. "non-investment-grade" bonds, or even customers into different risk profiles for lending.