Clustering based on Stochastic Dominance with application for risk averters and risk seekers

Abstract: Stochastic Dominance (SD) theory provides a rigorous framework for selecting superior assets tailored to the asset allocation needs of investors with varying risk preferences (i.e., risk-averse, risk-seeking, and risk-neutral). However, traditional stock clustering methods typically rely on geometric metrics such as Euclidean distance, which often fail to effectively capture the intrinsic risk dominance relationships among assets. To address this limitation, this paper proposes an innovative clustering analysis framework based on SD test statistics. Methodologically, this study deeply integrates SD theory with machine learning algorithms. Transcending the limitations of traditional reliance on geometric distance, we innovatively utilize test statistics from first-, second-, and third-order SD to construct a "Stochastic Dominance Coefficient Matrix." Building upon this matrix, we modify the classic K-means and Hierarchical Clustering algorithms. Specifically, we derive 12 distinct algorithm variants tailored to different orders of SD relationships. Simultaneously, we construct the SD-SC coefficient and the SD-DBI index as specialized validity indices to evaluate the clustering performance. Empirically, we analyze constituent stock data from a representative developed market (the US NASDAQ Index) and an emerging market (China's CSI 100 Index). The results verify the effectiveness and robustness of the proposed method. Furthermore, we apply the clustering results to the modification of the Single Index Model and the construction of Global Minimum Variance Portfolios (GMVP). The findings demonstrate that the proposed method effectively facilitates customized asset allocation for investors, holding significant theoretical value and practical implications.

• The paper develops twelve clustering variants using bootstrap-based stochastic-dominance coefficients, enabling stock groups tailored to risk averters through ascending dominance and risk seekers through descending dominance.
• The method produces positive, preference-aware silhouette scores across NASDAQ 100 and CSI 100 data, often outperforming classical clustering, although its SD-DBI results are weaker for CSI 100.
• The paper shows that dominance-based clusters support economically meaningful portfolio selection, including lower-risk CSI 100 portfolios for risk averters and higher-risk, higher-return groups for risk seekers.

Overview

The paper addresses a structural gap in stock clustering methodology: conventional approaches rely on geometric dissimilarity measures (Euclidean or Mahalanobis distance) computed over price or return features, and therefore encode no information about the risk preferences of the investors who will ultimately consume the clusters. The authors—Li, Jia, Kang, and Wong—propose a clustering framework in which the pairwise dissimilarity between stocks is defined by test statistics from first-, second-, and third-order Stochastic Dominance (SD) tests, separately constructed for ascending SD (ASD), which ranks prospects for risk averters, and descending SD (DSD), which ranks prospects for risk seekers. Because ranking assets via ASD or DSD is equivalent to expected-utility maximization for the corresponding investor class, the resulting clusters are directly interpretable as preference-consistent asset pools.

The methodological contribution consists of three components: (i) a "Stochastic Dominance Coefficient" derived from Bootstrap empirical $p$-values of SD tests, used as the distance metric; (ii) twelve algorithm variants—SD-K-means and SD-Hierarchical (Agglomerative, Average Linkage) at each of three SD orders for each investor type; and (iii) two preference-aware validity indices, the SD-SC coefficient and the SD-DBI index, obtained by substituting the SD distance into the classical Silhouette Coefficient and Davies-Bouldin Index. The framework is applied to NASDAQ 100 and CSI 100 constituents and used to modify the Single Index Model analysis and construct Global Minimum Variance Portfolios (GMVP).

Statistical foundations

The framework builds on the ASD/DSD integral hierarchy, where the $j$-th order ascending integral is obtained by repeated left-integration of the CDF and the descending integral by repeated right-integration. Dominance of $X$ over $Y$ at order $j$ requires the corresponding integral of $F$ to lie pointwise below (ASD) or above (DSD) that of $G$ over the support. The authors adopt the Davidson–Duclos style test statistics $T_j^A(x)$ and $T_j^D(x)$, which normalize the difference of estimated SD integrals by its standard error at each grid point, and rely on the limiting Gaussian process result of Bai et al. under the null $F \equiv G$.

A key methodological decision is the move from a binary reject/accept decision to a continuous measure of evidence. The Bootstrap procedure pools the two samples, resamples with replacement, recomputes the supremum of the test statistic $j$0 times, and yields an empirical $j$1-value. The Stochastic Dominance Coefficient is then defined as $j$2, so that a large coefficient indicates a statistically significant dominance relationship (dissimilar distributions) and a small coefficient indicates distributional similarity. This construction is what allows SD test output to serve as a distance-like quantity in clustering, though the authors do not formally verify metric properties (symmetry is immediate, but the triangle inequality is not established)—a point that bears on the theoretical standing of the "distance" but not on the empirical performance reported.

Algorithms and validity indices

The SD-K-means variant follows the standard Lloyd iteration but assigns stocks to clusters by minimizing the SD coefficient to the cluster center, where the center is updated as the cross-sectional mean return series—an equally weighted portfolio of cluster constituents. The SD-Hierarchical variant uses Agglomerative clustering with Average Linkage, defining inter-cluster distance as the arithmetic mean of all pairwise SD coefficients across two clusters, a choice motivated by robustness to outliers relative to Single or Complete Linkage.

The SD-SC coefficient replaces Euclidean distances in the Silhouette formula with SD distances, computing intra-cluster cohesion $j$3 and nearest-neighbor-cluster separation $j$4, with higher values indicating better clustering. The SD-DBI index analogously substitutes SD distances into the Davies-Bouldin construction, with lower values preferred. The optimal number of clusters $j$5 is selected by maximizing the SD-SC over a search range, jointly with the chosen algorithm.

Empirical design

The dataset comprises weekly log-returns of NASDAQ 100 constituents (March 2014–March 2024, 83 stocks retained after filtering for complete trading records) and CSI 100 constituents (March 2013–March 2024, 76 stocks retained), sourced from the Choice Financial Terminal. Weekly aggregation is used to suppress microstructure noise while preserving distributional detail. Although the theoretical framework extends to third-order SD, the empirical analysis restricts attention to first- and second-order dominance, on the grounds that the first two orders capture non-satiation and risk attitude, while higher orders add interpretive complexity with marginal empirical gain.

Clustering results

The procedure is sequential: first-order clustering is applied first, and second-order clustering refines the pools where first-order relationships are ambiguous or absent. Several findings stand out.

NASDAQ, risk averters. First-order ASD-K-means yields three clusters, with a 68-stock cluster dominating the single-stock cluster (PepsiCo) at first order but no clear relation to a 14-stock cluster. Granular comparison identifies three non-dominated stocks (Costco Wholesale, Paychex, Xcel Energy) that are merged forward. Second-order SASD-K-means on the refined 71-stock pool yields two effective classes, with the 13-stock cluster strictly dominating the other in second order; this cluster contains defensively oriented names (Microsoft, Costco, ADP, Texas Instruments, T-Mobile US). For CSI 100, no significant first-order dominance exists among the 76 stocks; second-order clustering isolates a two-stock cluster—China Yangtze Power and ICBC—whose center strictly dominates the other cluster's center, implying substantially higher expected utility for risk-averse investors from this pair.

NASDAQ, risk seekers. The authors report that FASD and FDSD clustering produce identical results, so second-order SDSD-K-means (with $j$6 for NASDAQ, $j$7 for CSI 100) is applied directly. The first cluster (42 NASDAQ stocks; 72 CSI 100 stocks) dominates all others in second order and is selected for risk-seeking portfolios.

Hierarchical variants. FASD-Hierarchical on NASDAQ produces 20 fine-grained clusters with more distinct block structure in the coefficient heatmaps than K-means. SASD-Hierarchical ($j$8) followed by merging yields two classes of 31 and 26 stocks, with the 31-stock class dominating. SDSD-Hierarchical ($j$9 and $X$0) similarly identifies dominant classes of 49 NASDAQ and 16 CSI 100 stocks. The CSI 100 silhouette coefficient for SASD-Hierarchical peaks near 0.72 at five clusters.

Validity indices. The quantitative comparison strongly favors the SD models on SD-SC. Representative values include:

Model	NASDAQ SD-SC (SD / classical)	CSI 100 SD-SC (SD / classical)
SASD-K-means	0.3838 / −0.3760	0.4827 / 0.0894
SDSD-K-means	0.1424 / −0.1541	0.1872 / −0.1645
SASD-Hierarchical	0.5460 / −0.5140	0.7167 / −0.0431
SDSD-Hierarchical	0.3667 / −0.3195	0.3263 / −0.3097

The classical models frequently produce negative silhouette values, indicating misassignment, while the SD models are uniformly positive. The SD-DBI comparison, however, is not uniformly favorable: SD models achieve lower (better) DBI than classical models on NASDAQ across all specifications, but on CSI 100 the SD-DBI values are consistently higher than classical DBI—for instance, SDSD-Hierarchical reaches 8.4478 versus 0.998 for classical clustering. The authors attribute this to weaker applicability of the SD-DBI index on the CSI 100 dataset but do not provide a formal diagnosis; this is an acknowledged inconsistency rather than a resolved issue.

Portfolio applications

Two application exercises evaluate whether the clusters are economically meaningful. First, $X$1-$X$2 plots relative to index-average returns show that SASD-selected stocks exhibit lower market-relative volatility ($X$3) while SDSD-selected stocks occupy the upper range, consistent with the intended preference mapping, though excess returns ($X$4) do not differ significantly between selected and unselected pools. Second, minimum-variance portfolios constructed by repeated random draws from optimal versus non-optimal classes show the expected preference-consistent structure. For CSI 100 risk-averse two-stock portfolios (100 draws), the optimal class averages a return of 0.0023 with risk 0.0004, versus 0.0026 and 0.0032 for the non-optimal class—lower return but roughly eightfold lower risk. For risk seekers, the two-stock optimal class averages return 0.0026 with risk 0.00109, versus 0.0017 and 0.00047 for the non-optimal class. Portfolio risk-return plots show clearer inter-class separation for NASDAQ than CSI 100, which the authors attribute to differences in market structure, and hierarchical clustering separates classes more distinctly than K-means.

Limitations and open questions

The authors are explicit about three constraints. First, the empirical scope is limited to index constituents in two markets; extension to full markets, where the computational advantage over exhaustive pairwise SD testing would be most consequential, is not demonstrated. Second, the evaluation system rests on only two validity indices, and the SD-DBI's poor behavior on CSI 100 is unexplained—whether this reflects a defect of the index or of the clustering on that dataset remains open. Third, the Stochastic Dominance Coefficient's formal properties as a distance (notably the triangle inequality) are not established, and the Bootstrap procedure assumes IID returns, which is restrictive for financial return series exhibiting volatility clustering. The restriction to first- and second-order dominance in practice, despite the third-order machinery, also leaves the empirical value of the higher-order variants unverified.

Conclusion

The paper contributes a coherent integration of SD hypothesis testing with classical clustering, replacing geometric distance with a preference-conditioned, inference-based dissimilarity measure, and supplies matching validity indices and twelve concrete algorithm variants. The empirical evidence—substantially higher SD-SC values than classical baselines on both markets, preference-consistent risk-return separation in GMVP construction, and identification of economically interpretable dominant classes (e.g., the China Yangtze Power–ICBC pair for risk-averse CSI 100 investors)—supports the claim that the method delivers customized asset preselection for risk averters and risk seekers. The unresolved SD-DBI behavior on CSI 100, the unverified metric properties of the coefficient, and the IID assumption underlying the Bootstrap are the principal points on which further work is needed.