Using data from world stock exchange indices prior to and during periods of global financial crises, clusters and networks of indices are built for different thresholds and diverse periods of time, so that it is then possible to analyze how clusters are formed according to correlations among indices and how they evolve in time, particularly during times of financial crises. Further analysis is made on the eigenvectors corresponding to the second highest eigenvalues of the correlation matrices, revealing a structure peculiar to markets that operate in different time zones.
This work uses a distance measure based on Spearman's rank correlation in order to build three dimensional networks based on hierarchies. This approach makes it possible to show how clusters form at different levels of correlations, and how the formations of those networks evolve in time.
The data used are the time series of international stock exchange indices taken from many countries in the world so as to represent both evolved and developing economies in a variety of continents. The series were taken both prior to and during some of the severest financial crises of the past decades, namely the 1987 Black Monday, the 1997 Asian Financial Crisis, the 1998 Russian Crisis, the burst of the dot-com bubble in 2001, and the crisis after September, 11,2001, so that different regimes of volatility are represented. The idea is to be able to study how clusters behave when going from periods of low volatility to more turbulent times. The Subprime Mortgage crisis of 2008 is analyzed in a companion article, along with the years ranging from 2007 to 2010.
The data consists on some of the benchmark indices of a diversity of stock exchanges around the world. The number goes from 16 indices (1986) to 79 indices (2001). The choice of indices was mainly based on availability of data, since some of the stock exchanges being studied didn’t develope or had no recorded indices until quite recently. As there are differences between some of the days certain stock markets operate, some of the operation days had to be deleted and others duplicated. The rule was the following: when more than 30% of markets didn’t operate on a certain day, that day was deleted. When that number was bellow 30%, we repeated the value of the index for the previous day for the markets that didn’t open. Special care had to be given to markets whose weekends didn’t correspond to the usual ocidental days, like some Arab countries. For those markets, we adjusted the weekends in order to match those of the majority of markets. All that was done in order to certify that we minimized the number of days with missing data, so that we could measure the log-returns of two consecutive days of a market’s operation whenever that was possible.
The correlation matrix of the time series of financial data encodes a large amount of information, and an even greater amount of noise. That information and noise must be filtered if one is to try to understand how the elements (in our case, indices) relate to each other and how that relation evolves in time. One of the most common filtering procedures is to represent those relations using a Minimum Spanning Tree [1]- [15], which is a graph containing all indices, connected by at least one edge, so that the sum of the edges is minimum, and which presents no loops. Another type of representation is that of Maximally Planar Filtered Graphs [16]- [21], which admits loops but must be representable in two dimensional graphs without crossings.
Yet another type of representation is obtained by establishing a number which defines how many connections (edges) are to be represented in a graph of the correlations between nodes. There is no limitation with respect to crossing of edges or to the formation of loops, and if the number is high enough, then one has a graph where all nodes are connected to one another. These are usualy called Asset Trees, or Asset Graphs [22]- [29], for they are no trees in the network sense. Another way to build asset graphs is to establish a value (threshold) such that distances above it are not considered. This eliminates connections (edges) as well as indices (nodes), but also turn the diagrams more understandable by filtering both information and noise. Some previous works using graphic representations of correlations between international assets (indices or otherwise) can be found in [5], [20], and [28].
In the present work, we shall build asset trees of indices based on threshold values for a distance measure between them that is based on the Spearman rank correlation among indices. By establishing increasing values for the thresholds, we shall be able to devise the formation of clusters among indices, and how they evolve in time. Details of the way that is done are given in Section 2.
A second topic is pursued in Section 3, where an analysis is made of the eigenvector corresponding to the second highest eigenvalue of the correlation matrix of each group of data being studied. It is well-known ( [30] and references therein) that the eigenvector corresponding to the highest eigenvalue of a correlation matrix of financial assets corresponds to a “market mode” that resembles the common movement of the market as a whole, and that often the eigenvectors corresponding to the second and sometimes the third highest eigenvalues also carry some information on the internal structure of the market being studied. What is implied in Section 3 is that this internal structure reflects the fact that stock markets ope
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