K-Means Clustering for Arbitrage Portfolios

• Arbitrage portfolio K-means clustering is a method that segments financial assets based on cross-sectional and time-series features to identify candidate portfolios.
• It integrates unsupervised learning with statistical tests and hybrid strategies to optimize profitability and improve risk-adjusted return profiles.
• The approach offers robust asset grouping and dimensionality reduction, contributing to more stable portfolio construction and effective statistical arbitrage.

Arbitrage portfolio K-means clustering is an advanced methodology in financial modeling that leverages unsupervised learning to segment financial assets according to cross-sectional or time-series characteristics. This segmentation aids in the construction of candidate arbitrage portfolios, with optimization or filtering methods further enhancing both profitability and risk-adjusted return profiles. Across multiple studies, K-means clustering has served as both a stand-alone and hybrid technique—particularly effective when integrated with statistical tests and alternative grouping algorithms—to address challenges in statistical arbitrage and portfolio optimization (Zhang et al., 2014, Park, 21 Jan 2025).

1. Methodological Foundation: K-Means Clustering in Portfolio Construction

K-means clustering operates by partitioning N financial assets into K groups according to similarity in a defined feature space. The feature representation may consist of raw fundamental metrics (e.g., P/E ratio, price to sales, price to EBITDA), principal components from PCA, or vectors of historical log-returns. The canonical objective function seeks to minimize the within-cluster sum of squared distances:

$$\min \sum_{k=1}^K \sum_{i: x_i \in C_k} \|x_i - \mu_k\|^2$$

where $x_i$ is an asset’s feature vector, $\mu_k$ is the centroid of cluster k, and $C_k$ is the set of assets in cluster k (Zhang et al., 2014, Park, 21 Jan 2025). Selection of K is dictated by the specific application: for arbitrage portfolios, K is often chosen so that resultant clusters yield 2–4 assets per group; clusters of size one are ignored, and larger clusters are subdivided.

2. Arbitrage Portfolio Discovery via Multi-Factor Clustering

The clustering approach in arbitrage portfolio formation departs from traditional time-series correlation-based strategies by identifying candidate portfolios based on cross-sectional similarity across multiple factors. Each cluster provides a set of assets hypothesized to be conducive to future cointegration and, hence, statistical arbitrage. Specific steps include:

• Calculation of feature representations (fundamental/momentum factors, principal components, or historical log-returns).
• Execution of K-means clustering with a pre-specified K to form groups of “similar” assets.
• Ignoring singleton clusters, using clusters of 2–4 assets as candidate portfolios, and subdividing larger clusters.
• Subsequent statistical tests (Johansen cointegration) to validate whether clusters exhibit the required arbitrage properties (Zhang et al., 2014).

This approach tends to yield fewer candidate portfolios, but those selected typically possess higher average net profitability and enhanced cointegration characteristics compared to methods relying solely on price-based dependency structures.

3. Comparison with Graphical Lasso and Hybrid Strategies

Graphical lasso (Glasso) identifies candidate portfolios by sparsifying the inverse correlation matrix among assets, primarily using historical price time series (Zhang et al., 2014). K-means clustering, in contrast, operates on cross-sectional features. Comparative findings include:

Method	# Portfolios	Avg. Net Profit	Arbitrage Test Outcome
K-Means Clustering	Lower	Higher	Mixed (raw-factor alone)
Graphical Lasso	Higher	Lower	Some pass
Hybrid Approaches	Highest	Highest	All pass (p ≈ 0)

Hybrid approaches integrate both methods in two possible sequences:

• Clustering-Glasso: Cluster first (small K), then apply graphical lasso, and retain only lasso-identified relationships within-cluster.
• Glasso-Clustering: Lasso first, then cluster, filtering lasso-generated relationships by cluster membership.

These hybrids generate more candidate portfolios and achieve higher average profitability and trade win rates. The Glasso-Clustering variant may result in excess portfolio candidates, necessitating additional ranking or filtering (e.g., by the sum of the absolute values of nonzero entries) (Zhang et al., 2014).

4. Clustering for Sharpe Ratio-Based Portfolio Optimization

Beyond arbitrage detection, K-means clustering can enhance portfolio optimization by grouping assets for subsequent risk-adjusted return optimization (Park, 21 Jan 2025). The foundational steps are:

• Calculate historical log-returns: $R_{i,t} = \ln(P_{i,t} / P_{i,t-1})$.
• Apply K-means clustering to asset log-return vectors.
• For each cluster, optimize portfolio weights $\mathbf{w}$ to maximize the Sharpe ratio:

$$\max \frac{\mathbf{w}^{T}\mu - r_f}{\sqrt{\mathbf{w}^{T}\Sigma \mathbf{w}}}$$

subject to

$\sum_{i=1}^N w_i = 1, \quad w_i \geq 0$

where $\mu$ is the mean return, $\Sigma$ the covariance matrix, and $r_f$ the risk-free rate.

This approach mitigates estimation error and enables more stable allocation. Results demonstrate that cluster-optimized portfolios can significantly outperform equal-weighted benchmarks in cumulative return and Sharpe ratio, with empirical findings showing annualized returns and Sharpe ratios exceeding benchmark portfolios (Park, 21 Jan 2025).

5. Adaptive and Validation Protocols

Adaptive portfolio methods recompute asset clusters and candidate portfolios during the trading period to capture evolving asset relationships. In tests, the adaptive approach involved closing all trades at a period boundary, recalculating the clusters with updated features, and trading the newly formed portfolios in the subsequent window (Zhang et al., 2014). Empirical evidence suggests that such adaptive rebalancing can diminish average net profit per trade and per portfolio due to premature closure of profitable trades and disturbed mean reversion cycles. Statistical arbitrage significance is tested via the Johansen cointegration test and JTTW p-value analysis (with AR(1) noise term and Treasury bill risk-free rates). Hybrid models consistently pass these arbitrage tests at significance levels $p \approx 0$, supporting rejection of the null hypothesis of no statistical arbitrage.

Validation on an independent dataset, employing identical formation and trading period segmentation, demonstrates that cluster-derived and hybrid portfolios consistently outperform Glasso-derived portfolios on net profitability and arbitrage detection metrics (Zhang et al., 2014).

6. Implications and Comparative Advantages

Employing K-means clustering as a portfolio discovery mechanism yields:

• Higher average profitability despite fewer candidate portfolios relative to correlation-based methods.
• Robustness through grouping by fundamental or momentum factors, aiding generalizability across market regimes.
• Dimensionality reduction and enhanced interpretability.
• Statistical confidence validated through arbitrage testing and temporal cross-validation.

Hybrid models harness both behavioral (cross-sectional similarity) and temporal (time-series dependency) information, yielding superior results in out-of-sample validation (Zhang et al., 2014).

A plausible implication is that clustering-based segmentation paired with risk-adjusted optimization can lead to more stable, interpretable, and profitable portfolio strategies compared to naïve global dependency modeling or equal-weighted portfolio construction.

7. Considerations and Limitations

K-means clustering for arbitrage portfolio construction is sensitive to the selection of K, the choice of feature space (raw factors, principal components, historical log-returns), and the specific sequential or hybrid integration with other filtering/optimization methods. Overly frequent adaptation may disrupt long-term profit accumulation. Hybrid and segmented models require additional ranking and filtering steps when portfolio candidate proliferation occurs. Nevertheless, the systematic validation across multiple datasets confirms the robustness and generalizability of these clustering-based methodologies for statistical arbitrage and risk-adjusted portfolio optimization (Zhang et al., 2014, Park, 21 Jan 2025).