On the Use of Principal Components Analysis in Index Construction

Research by: Daniel Broby, William Smyth

Executive Summary

This paper explores the application of principal component analysis (PCA) in constructing investment indices and introduces a method for building a factor model index (FMI). The study is theoretical and provides a structured approach to PCA-based index construction. It proposes an alternative approach to construction of traditional equity indices.

The method presented enables the selection of sub-groupings and clustering of asset proxies based on factor exposure, resulting in a more flexible weighting scheme. It leverages PCA’s dimension reduction properties, first identified by Pearson, to extract meaningful factors from historical market data. Specifically, the paper explains how the eigenvector coefficients of the first principal component, within an equity asset sub-class, can be used to determine index weights.

Traditional equity indices, such as those based on market capitalisation or equal weighting, are often criticised for biases, including over-concentration in certain sectors or stocks. Factor-based indices offer an alternative but typically rely on pre-defined factors that may not fully capture market variance. In contrast, PCA provides a data-driven method that extracts orthogonal factors directly from observed market returns, avoiding the limitations of predefined factor structures.

PCA has several mathematical properties that make it useful for index construction. The most significant is that the first principal component captures the largest portion of variance in an equity index. This aligns with the concept of systematic risk in the capital asset pricing model (CAPM). By structuring the index around the first principal component, the approach ensures consistency with the self-consistency condition. That is that a market proxy should comprise assets whose returns it seeks to represent. The proposed alternative provides a systematic way to align index construction with market dynamics and finance theory.

The steps involved in constructing a PCA based FMI index are as follows:

1. Identify the relevant equity instruments and obtain their historical return time series.
2. Standardise the return series and compute the covariance matrix to measure relationships between asset returns.
3. Compute the eigenvalues and eigenvectors from the covariance matrix. The eigenvectors indicate the principal directions of data variance, while the eigenvalues quantify the variance explained by each component.
4. Select the top 𝑘 eigenvectors based on the magnitude of their eigenvalues. These eigenvectors determine the dimensions retained in the index.
5. Assign weights to the original assets based on their loadings in the selected eigenvectors. Each asset’s weight corresponds to its eigenvector coefficient.
6. Construct the index as a weighted sum of the original assets using these assigned weights.

This methodology addresses a gap in the literature by providing a systematic approach to index construction that does not rely on traditional proxies. By grouping assets into relevant sub-sectors based on factor exposure, the FMI framework enhances performance attribution. Additionally, the method mitigates common entropy issues associated with return time series, improving the index’s ability to approximate the market portfolio.

To cite this article: Broby, D., & Smyth, W. (2025). On the use of principal components analysis in index construction. Financial Statistical Journal, 8(1). https://doi.org/10.24294/fsj10858

To access this article: https://doi.org/10.24294/fsj10858

About the Journal

Financial Statistical Journal (FSJ, eISSN: 2578-1960) is a prestigious journal dedicated to advancing the frontiers of knowledge at the intersection of finance and statistics with an open access model. FSJ is committed to publishing high-quality, original research articles and review articles that contribute to the understanding and application of statistical techniques in financial contexts.

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