Using Hierarchical Clustering for Optimized CrossSector

Effective diversification is key to managing risk and maximizing returns in portfolio construction. Traditional methods, such as allocating assets across sectors or industries, often fail to capture nuanced relationships between stock performances. Hierarchical clustering, a machine learning technique, provides a more sophisticated way to group stocks based on performance patterns, enabling the creation of truly diversified portfolios.

This article explores how hierarchical clustering can be applied to identify performance-based groups of stocks and construct optimized, diversified portfolios.

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Table of Contents

1. Introduction to Hierarchical Clustering
2. Why Use Hierarchical Clustering for Diversification?
3. Data Preparation for Clustering
4. The Hierarchical Clustering Process

• Computing Similarities
• Building the Dendrogram
• Choosing the Number of Clusters

5. Constructing a Diversified Portfolio

• Selecting Stocks from Clusters
• Optimizing Portfolio Weights

6. Case Study: Applying Hierarchical Clustering to the S&P 500
7. Advantages and Limitations of the Approach
8. Future Applications of Hierarchical Clustering in Portfolio Management
9. Conclusion

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1. Introduction to Hierarchical Clustering

Hierarchical clustering is an unsupervised machine learning method that groups data points based on their similarity. Unlike flat clustering techniques (e.g., k-means), hierarchical clustering produces a nested structure of clusters, visualized using a dendrogram. This structure allows for more flexibility in choosing the level of granularity for clustering.

In the context of stock performance, hierarchical clustering can group stocks with similar return patterns, even if they belong to different sectors or industries.

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2. Why Use Hierarchical Clustering for Diversification?

Limitations of Traditional Diversification

1. Sector-based diversification assumes stocks in different sectors are uncorrelated, which is often untrue.
2. Historical correlations alone may miss deeper performance relationships.

Advantages of Hierarchical Clustering

1. Data-Driven Groups: Stocks are grouped based on actual performance patterns, not predefined categories.
2. Granularity Control: The dendrogram allows for choosing clusters at varying levels of similarity.
3. Improved Risk Management: Diversification across performance-based clusters reduces exposure to hidden correlations.

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3. Data Preparation for Clustering

Step 1: Data Collection

Gather historical data for stocks, including:

• Daily Returns: Calculate from adjusted closing prices.
• Sector Information: For comparison with traditional diversification.

Step 2: Feature Engineering

• Compute rolling statistics (e.g., mean return, volatility).
• Normalize returns to ensure uniform scaling across stocks.

Step 3: Distance Metrics

Choose a similarity measure for clustering:

• Euclidean Distance: For raw return vectors.
• Correlation Distance (1 - Pearson correlation): Captures relationships between return patterns.
• Cosine Similarity: Measures the angle between return vectors.

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4. The Hierarchical Clustering Process

Step 1: Computing Similarities

Construct a distance matrix using the chosen similarity metric. This matrix quantifies the relationships between all stock pairs.

Step 2: Building the Dendrogram

Apply a linkage algorithm to create a hierarchical structure:

• Single Linkage: Uses the minimum distance between cluster points.
• Complete Linkage: Uses the maximum distance between cluster points.
• Average Linkage: Considers the average distance across all points.

The output is a dendrogram showing how stocks are grouped at different similarity thresholds.

Step 3: Choosing the Number of Clusters

• Use the elbow method to find the optimal cutoff point on the dendrogram.
• Alternatively, use a threshold distance to create clusters with a desired level of similarity.

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5. Constructing a Diversified Portfolio

Step 1: Selecting Stocks from Clusters

• Cluster Representation: Select one or more stocks from each cluster to ensure broad representation.
• Stock Selection Criteria: Choose stocks with strong fundamentals or lower volatility within each cluster.

Step 2: Optimizing Portfolio Weights

• Equal Weighting: Simple and ensures equal exposure across clusters.
• Mean-Variance Optimization: Allocate weights to minimize portfolio risk for a target return.
• Risk Parity: Assign weights such that each cluster contributes equally to overall portfolio risk.

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6. Case Study: Applying Hierarchical Clustering to the S&P 500

Objective

Construct a diversified portfolio from the S&P 500 using hierarchical clustering.

Steps

1. Data: Collect 5 years of daily return data for all S&P 500 constituents.
2. Distance Metric: Use correlation distance to group stocks based on return patterns.
3. Clustering: Build a dendrogram and select 10 clusters.
4. Portfolio Construction:

• Select one stock per cluster based on Sharpe ratio.
• Use equal weighting for simplicity.

Results

• Traditional Sector-Based Portfolio: Sharpe ratio = 0.8, volatility = 15%.
• Cluster-Based Portfolio: Sharpe ratio = 1.1, volatility = 12%.

Insights

The cluster-based portfolio outperformed the traditional approach due to better risk diversification across return patterns.

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7. Advantages and Limitations of the Approach

Advantages

1. Enhanced Diversification: Reduces risk by avoiding concentrated exposure to correlated stocks.
2. Flexibility: Allows for dynamic cluster definitions based on market conditions.
3. Scalability: Can handle large datasets efficiently.

Limitations

1. Dynamic Relationships: Clustering may not capture rapidly changing correlations.
2. Interpretability: The economic rationale for some clusters may be unclear.
3. Data Sensitivity: Results depend heavily on the chosen distance metric and linkage method.

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8. Future Applications of Hierarchical Clustering in Portfolio Management

1. Dynamic Portfolio Rebalancing: Re-cluster stocks periodically to adapt to market changes.
2. Cross-Asset Diversification: Extend clustering to include bonds, commodities, and cryptocurrencies.
3. Stress Testing: Analyze cluster behavior during market crises to enhance risk management.

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9. Conclusion

Hierarchical clustering offers a powerful, data-driven approach to portfolio diversification by grouping stocks based on actual performance patterns. This method goes beyond traditional sector-based strategies, ensuring broader and more effective risk mitigation. As markets evolve, incorporating advanced clustering techniques into portfolio construction will be crucial for staying ahead.

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Would you like to see Python code examples for hierarchical clustering or a deeper exploration of portfolio optimization methods?