10 Correlations, Hierarchies, Networks and Clustering
265
• The Financial sector is strongly connected within, but also to others.
• The assets of the classic Markowitz portfolio are always located on the outer leaves
of the tree [108, 116, 119].
• The maximum eigenvalue of the correlation matrix, which carries most of the
correlations, is very large during market crashes [39] (increased value of the mean
correlation).
• The MST shrinks during market crashes [108] and contains a low number of
clusters [97].
• The MST provides a taxonomy which is well compatible with the sector classification provided by an outside institution [90, 108].
• Scale free structure of the MST (i.e. the degree of vertices is power law distributed
f (n) ∼ n
−α ) [16, 73, 108, 150], but the scaling exponent depends on market
period and window width [106].
• The MST obtained with the one-factor model is very different from the one
obtained using real data [16]. This invalidates the Capital Asset Pricing Model
which is based on the one-factor model r i (t) = α i + β i r M (t) + i (t).
• Stocks compose a hierarchical system progressively structuring as the sampling
time horizon increases [17, 145].
• The correlation among market indices presents both a fast and a slow dynamics.
The slow dynamics is a gradual growth associated with the development and
consolidation of globalization. The fast dynamics is associated with events that
originate in a specific part of the world and rapidly (in less than 3 months) affect
the global system [98, 130].
• Removing the dynamics of the center of mass decreases the level of correlations,
but also makes the cluster structure more evident [17].
• Scale invariance of correlation structure (by subtraction of the market mode) might
have important implications for risk management, because it suggests that correlations on short time scales might be used as a proxy for correlations on longer
time-horizons [17].
• The MST is star-like in low-volatility segments, and chain-like in high-volatility
segments [155].
• Volatility shocks always start at the fringe and propagate inwards [155].
• The “post-subprime” regime correlation matrix shows markedly higher absolute
correlations than the others [115].
• In [115], authors find far less asset class separation in the post-subprime period.
• One can distinguish three types of topological configurations for the companies:
(i) important nodes, (ii) links and (iii) dangling ends [150].
• A node keeps the majority of its neighbours. The non-randomness of the stock
market topology is thus a robust property [150].
• The largest eigenvector of the correlation matrix is strongly non-Gaussian, tending
to uniform - suggesting that all companies participate. Authors find indeed that
all components participate approximately equally to the largest eigenvector. This
implies that every company is connected with every other company. In the stock
market problem, this eigenvector conveys the fact that the whole market “moves”
265
• The Financial sector is strongly connected within, but also to others.
• The assets of the classic Markowitz portfolio are always located on the outer leaves
of the tree [108, 116, 119].
• The maximum eigenvalue of the correlation matrix, which carries most of the
correlations, is very large during market crashes [39] (increased value of the mean
correlation).
• The MST shrinks during market crashes [108] and contains a low number of
clusters [97].
• The MST provides a taxonomy which is well compatible with the sector classification provided by an outside institution [90, 108].
• Scale free structure of the MST (i.e. the degree of vertices is power law distributed
f (n) ∼ n
−α ) [16, 73, 108, 150], but the scaling exponent depends on market
period and window width [106].
• The MST obtained with the one-factor model is very different from the one
obtained using real data [16]. This invalidates the Capital Asset Pricing Model
which is based on the one-factor model r i (t) = α i + β i r M (t) + i (t).
• Stocks compose a hierarchical system progressively structuring as the sampling
time horizon increases [17, 145].
• The correlation among market indices presents both a fast and a slow dynamics.
The slow dynamics is a gradual growth associated with the development and
consolidation of globalization. The fast dynamics is associated with events that
originate in a specific part of the world and rapidly (in less than 3 months) affect
the global system [98, 130].
• Removing the dynamics of the center of mass decreases the level of correlations,
but also makes the cluster structure more evident [17].
• Scale invariance of correlation structure (by subtraction of the market mode) might
have important implications for risk management, because it suggests that correlations on short time scales might be used as a proxy for correlations on longer
time-horizons [17].
• The MST is star-like in low-volatility segments, and chain-like in high-volatility
segments [155].
• Volatility shocks always start at the fringe and propagate inwards [155].
• The “post-subprime” regime correlation matrix shows markedly higher absolute
correlations than the others [115].
• In [115], authors find far less asset class separation in the post-subprime period.
• One can distinguish three types of topological configurations for the companies:
(i) important nodes, (ii) links and (iii) dangling ends [150].
• A node keeps the majority of its neighbours. The non-randomness of the stock
market topology is thus a robust property [150].
• The largest eigenvector of the correlation matrix is strongly non-Gaussian, tending
to uniform - suggesting that all companies participate. Authors find indeed that
all components participate approximately equally to the largest eigenvector. This
implies that every company is connected with every other company. In the stock
market problem, this eigenvector conveys the fact that the whole market “moves”