Group 9 Clustering With Correlation Coefficient Presentation

Group 9 Clustering With Correlation Coefficientthis Presentation P

group 9 : Clustering with correlation coefficient This presentation / project is about correlation coefficients used as the clustering measure. File needed to see the correlation coefficients and bi-monthly returns: returnAstcBiMonthlyEG1.mat Address in the presentation the following: 1) Given correlation coefficient from A) What is the meaning of correlation coefficient between the two set of stocks prices? B) Show why the correlation coefficient between the set of prices A and set of prices B is the same as the correlation coefficient between the set of prices B and set of prices A. 2) After loading returnAstcBiMonthlyEG1.mat in Octave, you'll have stocks with bi-monthly returns of above 0.009 : CC -> correlation coefficients between the stock returns, RTRS -> bi-monthly returns (rows) for 412 stocks (columns) A) Comment the correlation coefficients based on bi-monthly returns among 412 stocks shown in the figure P5_CC_eg1.pdf B) Assuming that clusters are created by putting all the stocks whose correlation coefficients among all the pairs are greater than 0.5, comment on the following cluster distribution : 302 clusters with only 1 stock, 28 with 2 stock, 6 with 3, 4 with 4, 1 with 5, 1 with 6 , and 1 cluster with 9 stocks in it. C) Explain why it would be good to pick just one stock from the cluster for the diversified portfolio.

Paper For Above instruction

The analysis of stock market data through correlation-based clustering offers valuable insights into the relationships among stocks, allowing investors and financial analysts to optimize diversification strategies. This paper explores the meaning of correlation coefficients between stock prices, their mathematical properties, and practical implications within the context of bi-monthly stock returns, using data loaded from the MATLAB file "returnAstcBiMonthlyEG1.mat." Additionally, it examines the clustering results derived from these correlation coefficients and discusses the rationale for selecting a single stock from each cluster to build diversified portfolios.

Understanding Correlation Coefficients in Stock Returns

Correlation coefficients measure the strength and direction of the linear relationship between two variables—in this case, the prices or returns of stocks. A correlation coefficient ranges from -1 to +1, where +1 indicates perfect positive linear relationship, -1 indicates perfect negative linear relationship, and 0 signifies no linear relationship. When applied to stock prices, a high positive correlation indicates that the two stocks tend to move together—when one increases in value, the other tends to do so as well—while negative correlations imply inverse movements. The correlation coefficient provides a quantitative measure of how similar or dissimilar stock movements are, which is crucial for portfolio diversification since holding highly correlated stocks does not significantly reduce risk.

Symmetry of Correlation Coefficients

Mathematically, the correlation coefficient between two sets of stock prices, A and B, is symmetric because the calculation involves the covariance of the two variables divided by the product of their standard deviations: rAB = rBA. This symmetry arises because covariance is symmetric: Cov(A, B) = Cov(B, A). Consequently, the correlation coefficient between stocks A and B remains constant regardless of the order in which the stocks are considered. This property ensures consistency in clustering analyses, where the relationship between stock pairs must be bidirectional and symmetrical.

Analysis of Correlation Coefficients and Clustering of Stocks

Upon loading the data from "returnAstcBiMonthlyEG1.mat" into Octave, the analysis focuses on stocks with bi-monthly returns exceeding 0.009, examining their pairwise correlation coefficients stored in matrix CC. Visualizations such as correlation heatmaps, exemplified by the figure "P5_CC_eg1.pdf," reveal the distribution of correlations among 412 stocks. Typically, such a heatmap illustrates varying degrees of linear association, with strong correlations hinting at potential cluster formations.

Clustering Based on Correlation Thresholds

A common approach in clustering stocks involves applying a threshold to the correlation coefficients—here, a threshold of 0.5. This method assumes that stocks with correlation coefficients greater than 0.5 are sufficiently related to be grouped together. The resulting clustering reveals a distribution characterized by numerous singleton clusters (302 with only one stock), and a relatively small number of multi-stock clusters—28 with 2 stocks, 6 with 3 stocks, and so forth. The predominance of singleton clusters indicates diverse, somewhat uncorrelated stocks, while the larger clusters suggest groups of stocks that move in tandem.

Implications for Portfolio Diversification

The clustering results underline the importance of selecting stocks judiciously for diversification. Clusters with multiple stocks exhibiting high correlation indicate redundancy—investing in several stocks within the same cluster does not substantially diversify risk because their returns tend to move together. Therefore, selecting just one representative stock from each cluster is a strategic approach to constructing a diversified portfolio. This minimizes overlap in stock movements, reduces portfolio risk, and enhances the potential for stable returns, aligning with principles of modern portfolio theory (Markowitz, 1952).

Conclusion

Analyzing correlation coefficients among stocks provides essential insights into their relationships and facilitates effective clustering strategies. The symmetry property of correlation coefficients ensures consistent and reliable clustering outcomes. The observed distribution of stock clusters—with many singleton groups—emphasizes the importance of selecting individual stocks from various clusters to maximize diversification. This approach aims to optimize risk-adjusted returns, an enduring goal in investment management. Future research could expand on dynamic clustering algorithms, incorporate other similarity measures, and explore the impact of different correlation thresholds on portfolio performance.

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