Fig 1.
The network of portfolio holdings can be represented as a bipartite graph. The two vertex classes are the funds {i, j, k, …} and the assets {α, β, γ, …} in their portfolios. Each edge (i, α) represents a specific holding relationship. The edge weight Wiα is equal to the total market value of security α owned by fund i in its portfolio.
Fig 2.
Three case studies, with the same values of the portfolios (Si = Sj = 3) but different values of the inverse Herfindahl-Hirschmann index and of the cosine similarity between them.
Table 1.
Summary statistics of the bipartite network of holdings.
Fig 3.
Probability density functions of the inverse Herfindahl–Hirschmann index h (A), asset degree k (B) and cosine similarity s (C) for three reference quarters. The network is heterogeneous and all the distributions are broad and fat tailed. Most portfolios have few dominant assets but there exist some that have thousands; most assets belong to few funds, but some are extremely popular. Similarities extend over more than 10 orders of magnitude and there exist pairs of almost identical portfolios. Only values s > 0 have been considered to compute the histogram.
Fig 4.
Similarity and systemic risk: Comparison to the null models.
Complementary cumulative distribution functions of the similarity for a snapshot of the real network (2006Q3) and the random benchmarks RH and DPR (A); the same comparison is performed for the system’s riskiness measured by the total percentage loss ensuing from the propagation of a negative shock of -30% in the value of the 10 most common stocks in the network (B). Large values of the similarity between portfolios are more likely in the real world than in the randomized networks even when the degree sequence of assets is preserved. The real network appears as the most fragile. Such fragility is not explained by the role of very popular assets, as shown by the comparison with the the DPR case.
Fig 5.
Similarity and systemic damage over time.
Median of the similarity s (A) and systemic damage after T = 4 periods (B). The similarity across portfolios and the systemic riskiness in the real network are strongly correlated. Both quantities have reduced and the network is less risky after the crisis.