# Friendship of Stock Market Indices: A Cluster-Based Investigation of Stock Markets

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Methodology

#### 2.2.1. Similarity Matrix

#### 2.2.2. Normalized Modularity

#### 2.2.3. Algorithm

- Constructing the similarity matrix $(W)$.
- Calculating the normalized modularity matrix $({M}_{D})$.
- Based on the spectral gap, determining the number of clusters and optimal k-dimensional representation.
- Appling k-means clustering.

#### 2.2.4. Assessment of Clustering Methods

## 3. Results

#### 3.1. Similarity Metrics

#### 3.2. Comparing Normalized Modularity and Laplacian

#### 3.3. Equity Index Network Structure

#### 3.4. Equity Index Graph

#### 3.5. Risk and Reward

#### 3.6 Time Stability

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Country | Two Clusters | Three Clusters | Five Clusters |
---|---|---|---|

United Arab Emirates | 2 | 3 | 1 |

Saudi Arabia | 2 | 3 | 1 |

Qatar | 2 | 1 | 1 |

Kuwait | 2 | 1 | 1 |

Egypt | 2 | 3 | 1 |

Bahrain | 2 | 1 | 1 |

Vietnam | 2 | 1 | 1 |

Nigeria | 2 | 1 | 1 |

Dow Jones | 1 | 2 | 2 |

Denmark | 1 | 1 | 2 |

Switzerland | 1 | 1 | 2 |

Canada | 1 | 2 | 2 |

Mexico | 1 | 2 | 2 |

Chile | 1 | 2 | 2 |

Argentina | 1 | 2 | 2 |

Hungary | 1 | 1 | 2 |

Morocco | 2 | 1 | 2 |

S&P 500 | 1 | 2 | 2 |

MSCI World | 1 | 2 | 2 |

Czech Republic | 1 | 1 | 2 |

Togo | 2 | 1 | 2 |

Spain | 1 | 1 | 2 |

Norway | 1 | 1 | 2 |

Luxembourg | 1 | 1 | 2 |

France | 1 | 1 | 2 |

South Africa | 1 | 3 | 2 |

Euro Stocks | 1 | 1 | 2 |

Sweden | 1 | 1 | 2 |

UK | 1 | 1 | 2 |

Netherlands | 1 | 1 | 2 |

Finland | 1 | 1 | 2 |

Poland | 1 | 3 | 2 |

Germany | 1 | 1 | 2 |

Belgium | 1 | 1 | 2 |

Italy | 1 | 1 | 2 |

Brazil | 1 | 2 | 2 |

Colombia | 1 | 3 | 2 |

Bangladesh | 2 | 1 | 3 |

Costa Rica | 2 | 1 | 4 |

Zambia | 2 | 1 | 4 |

Malawi | 2 | 1 | 4 |

Venezuela | 2 | 1 | 4 |

South Korea | 2 | 3 | 5 |

Hong Kong | 2 | 3 | 5 |

Thailand | 2 | 3 | 5 |

China | 2 | 3 | 5 |

Kenya | 2 | 3 | 5 |

India | 2 | 3 | 5 |

Namibia | 2 | 3 | 5 |

Turkey | 2 | 3 | 5 |

Indonesia | 2 | 3 | 5 |

Malaysia | 2 | 3 | 5 |

Russia | 2 | 3 | 5 |

Australia | 2 | 3 | 5 |

Taiwan | 2 | 3 | 5 |

Japan | 2 | 3 | 5 |

Ukraine | 2 | 3 | 5 |

Bulgaria | 2 | 1 | 5 |

Romania | 2 | 3 | 5 |

## References

- Barabási, Albert L., and Albert Réka. 1999. Emergence of Scaling in Random Networks. Science 26: 509–12. [Google Scholar] [CrossRef]
- Berlinet, Alain, and Thomas-Agnan Christine. 2011. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Berlin: Springer Science & Business Media, pp. 1–108. ISBN 978-1441990969. [Google Scholar]
- Bolla, Marianna. 2011. Penalized version of Newman-Girvan modularity and their relation to normalized cuts and k-means clustering. Physical Review E 84: 016108. [Google Scholar] [CrossRef] [PubMed]
- Chung, Fan R. G. 1997. Spectral Graph Theory. Providence: American Mathematical Society, No. 92. pp. 14–81. ISBN 978-0821803158. [Google Scholar]
- Engelmann, Bernd, Evelyn Hayden, and Dirk Tasche. 2003. Measuring the Discriminative Power of Rating Systems. Banking and Financial Supervision. Frankfurt: Deutsche Bundesbank. [Google Scholar]
- Erdős, Péter, and Rényi Alfréd. 1960. On the Evolution of Random Graphs. Acta Mathematica Hungarica 5: 17–61. [Google Scholar]
- Erdős, Péter, Mihály Ormos, and Dávid Zibriczky. 2011. Non-parametric and semi-parametric asset pricing. Economic Modelling 28: 1150–62. [Google Scholar] [CrossRef]
- Fama, Eugene, and Kenneth R. French. 1996. Multifactor explanations of asset pricing anomalies. The Journal of Finance 51: 55–84. [Google Scholar] [CrossRef]
- Maurizio, Filippone, Francesco Camastra, Francesco Masulli, and Stefano Rovetta. 2007. A survey of kernel and spectral methods for clustering. Pattern Recognition 41: 176–90. [Google Scholar] [CrossRef]
- Heiberger, Raphael H. 2014. Stock network stability in times of crisis. Physica A: Statistical Mechanics and Its Applications 393: 376–81. [Google Scholar] [CrossRef]
- Gregory, Leibon, Scott Pauls, Daniel Rockmore, and Robert Savell. 2008. Topological Structures in the Equities Market Network. PNAS 105: 20589–94. [Google Scholar] [CrossRef]
- Von Luxburg, Ulrike. 2007. Tutorial on Spectral Clustering. Statistics and Computing 17: 395–416. [Google Scholar] [CrossRef]
- Maldonado, Rita, and Saunders Anthony. 1981. International portfolio diversification and the inter-temporal stability of international stock market relationships, 1957–1978. Financial Management 10: 54–63. [Google Scholar] [CrossRef]
- MSCI. 2018. Market Classification. Available online: https://www.msci.com/market-classification (accessed on 3 November 2018).
- Ormos, Mihály, and Dávid Zibriczky. 2014. Entropy-Based Financial Asset Pricing. PLoS ONE 9: E115742. [Google Scholar] [CrossRef] [PubMed]
- Shi, Jianbo, and Jitendra Malik. 2000. Normalized cuts and image segmentation. IEEE Pattern Analysis and Machine Intelligence 22: 888–905. [Google Scholar] [CrossRef] [Green Version]
- Sharpe, William F. 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19: 425–42. [Google Scholar]
- Song, Dong-Ming, Michele Tumminello, Wei-Xing Zhou, and Rosario N. Mantegna. 2011. Evolution of worldwide stock markets, correlation structure, and correlation-based graphs. Physical Review E 84: 026108. [Google Scholar] [CrossRef] [PubMed]
- Takumasa, Sakakibara, Tohgoroh Matsuib, Atsuko Mutoha, and Nobuhiro Inuzuka. 2015. Clustering mutual funds based on investment similarity. Procedia Computer Science 60: 881–90. [Google Scholar] [CrossRef]
- Yalamova, Rossitsa. 2009. Correlations in Financial Time Series during Extreme Events-Spectral Clustering and Partition Decoupling Method. Paper presented at World Congress on Engineering, London, UK, July 1–3; Volume 2, pp. 1376–78. [Google Scholar]
- Zhao, Yanchang. 2012. R and Data Mining: Examples and Case Study. Cambridge: Academic Press, pp. 49–59. [Google Scholar]

**Figure 4.**Largest eigenvalues of Gaussian- and relative entropy-based normalized modularity matrices.

**Figure 6.**Histogram of 10,000 Gaussian similarities which are generated from i.i.d. 250 dim. standard normal samples.

**Figure 7.**Explained percentage variance of Gaussian-kernel based clusters after zero out similarities less than 0.2.

**Figure 10.**Three Gaussian-kernel based normalized modularity clusters, edges with weights stronger than 0.5.

**Figure 12.**Histogram of vertex weights, five Gaussian cluster, two nodes are connected if their Gaussian similarity is stronger than 0.2.

**Figure 13.**Histogram of vertex count-weights, five Gaussian cluster, two nodes are connected if their similarity is stronger than 0.2.

Index | Mean | Variance | Skewness |
---|---|---|---|

.CSI300 | 0.018 | 0.056 | −0.336 |

.XU100 | 0 | 0.026 | −0.809 |

.DJI | 0.012 | 0.009 | −0.819 |

.UAX | −0.034 | 0.037 | −0.721 |

.WORLD | 0.004 | 0.002 | −1.889 |

Method | Coeff. of Cluster | p-Value |
---|---|---|

Geographical | −0.000036 | 0.394 |

MSCI | −0.000041 | 0.293 |

Spectral | −0.000112 | 0.027 |

Clusters | p-Value of Intercept | p-Value of s.d. | ${\mathit{R}}^{2}$ |
---|---|---|---|

Total Sample | 0.62 | 0.12 | 0.05 |

First cluster | 0.62 | 0.02 | 0.68 |

Second cluster | 0.29 | 0.00 | 0.59 |

Fifth cluster | 0.93 | 0.71 | 0.01 |

ADF t-Value | ADF p-Value |
---|---|

−2.67 | 0.32 |

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## Share and Cite

**MDPI and ACS Style**

Nagy, L.; Ormos, M.
Friendship of Stock Market Indices: A Cluster-Based Investigation of Stock Markets. *J. Risk Financial Manag.* **2018**, *11*, 88.
https://doi.org/10.3390/jrfm11040088

**AMA Style**

Nagy L, Ormos M.
Friendship of Stock Market Indices: A Cluster-Based Investigation of Stock Markets. *Journal of Risk and Financial Management*. 2018; 11(4):88.
https://doi.org/10.3390/jrfm11040088

**Chicago/Turabian Style**

Nagy, László, and Mihály Ormos.
2018. "Friendship of Stock Market Indices: A Cluster-Based Investigation of Stock Markets" *Journal of Risk and Financial Management* 11, no. 4: 88.
https://doi.org/10.3390/jrfm11040088