^{1}

^{*}

^{2}

^{1}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: LL AC AS. Performed the experiments: LL AC AS. Analyzed the data: LL AC AS. Contributed reagents/materials/analysis tools: LL AC AS. Wrote the manuscript: LL AC AS. Designed the software used in analysis: LL.

The 2007-2008 financial crisis solidified the consensus among policymakers that a macro-prudential approach to regulation and supervision should be adopted. The currently preferred policy option is the regulation of capital requirements, with the main focus on combating procyclicality and on identifying the banks that have a high systemic importance, those that are “too big to fail”. Here we argue that the concept of systemic risk should include the analysis of the system as a whole and we explore systematically the most important properties for policy purposes of networks topology on resistance to shocks. In a thorough study going from analytical models to empirical data, we show two sharp transitions from safe to risky regimes: 1) diversification becomes harmful with just a small fraction (~2%) of the shocks sampled from a fat tailed shock distributions and 2) when large shocks are present a critical link density exists where an effective giant cluster forms and most firms become vulnerable. This threshold depends on the network topology, especially on modularity. Firm size heterogeneity has important but diverse effects that are heavily dependent on shock characteristics. Similarly, degree heterogeneity increases vulnerability only when shocks are directed at the most connected firms. Furthermore, by studying the structure of the core of the transnational corporation network from real data, we show that its stability could be clearly increased by removing some of the links with highest centrality betweeness. Our results provide a novel insight and arguments for policy makers to focus surveillance on the connections between firms, in addition to capital requirements directed at the nodes.

The increasing complexity and globalization of financial markets, together with excessive leverage, have been singled out jointly by the Financial Service Authority and the Financial Stability Board as major contributors to the financial crisis of 2007-2009 [

The currently preferred policy option is the regulation of capital requirements. The Basel III accords establish higher overall equity requirements and combat pro-cyclicality by forcing banks to build up an extra capital conservation buffer of 2.5% during good times. In the cross-sectional dimension, much effort is currently concentrated in the “too big to fail” (or better, “too central to fail” [

A related research strand focus on the effects of the network topology as a whole. The baseline discussion here is on the effects of network connectivity on its shock resistance. On one hand, higher interconnectedness can reduce the probability of default, as it allows adverse shocks to dissipate quicker [

On the other hand, most theoretical models of financial networks rely on extreme examples, such as complete networks. The most often used counterparts used are ring and star networks, assuming complete degree homogeneity or heterogeneity, respectively. While these are important cornerstones, real networks have intermediate values with corresponding resistances that are still to be explored. Numerical simulations have been mostly based on Erdös-Renyi random graphs, whereas most real networks have much more heterogeneous and often scale-free degree distributions [

Taking into account its implications on society, the available information on the effects of network topology on its shock-resistance is surprisingly scarce and fragmented. Furthermore, most of these studies have been carried out either categorically or varying one parameter at a time (OAT). However, OAT sensitivity analysis should only be used when the model is proven linear [

Here we present a thorough study on contagion resulting from overlapping risk exposure, aimed to provide insights on the interplay of different network features. Our research starts from an analytical foundation and then explores the hyperspace of network topology features with numerical simulations. Finally, in order to make the connection with real world systems, we collect empirical data examples from the literature and assess the robustness of the core-group of the transnational corporate ownership network [

We consider an environment with

Node size represents the asset volume of a firm, node color its level of capitalization (green: healthy, yellow: critical, red: default). Arrows thickness represents the amount of direct exposure. A) Firm 1 is hit by a relatively large shock and defaults. B) Firms 2 and 3 default due to their direct exposure to firm 1, the capitalization level of firm 4 drops to a critical level. C) Including the effect of indirect exposure, firm 4 defaults and the shock spreads until firm 8. Note that firm 5 and 7 can propagate the shock without having to default themselves. Nodes 9 and 10 are in an isolated cluster and are not affected.

A firm may benefit from entering risk sharing arrangements with other firms which allow it to diversify risks. The specific pattern of exchanges among firms is represented as a network, where a direct linkage between two firms reflects the fact that each firm holds a part of the asset of the other. Indirect exposure results from taking into account that a firm ends up having claims on the returns of projects of firms who hold assets of the firms it trades with, and so on. As a consequence a pair of firms lying at a certain distance in the network will have some reciprocal exposure to the yields of each other’s projects provided they are linked by a path through the network.

The contagion process derives from the exposure of common assets losses; it is not a default-cascade with self-enhancing mechanism such as those described in {{607 Upper, Christian 2011}}. However, unlike most classical common exposure models, it does include indirect exposure, exemplified by this simply case: Firm A owns 50% of firm B who again owns 50% of firm C. If firm C is hit by a shock that reduces its value to cero, firm A would lose 25% of the original value of C. The process of asset exchange in our model is analogous to the transmission of pathogens in disease contagion. A “bad” asset (think of an eventually unpaid mortgage) will expose all common owners to a shock. This is transmitted from the originator bank to others who purchase its mortgage-backed securities, and further down the line to the bondholders of the companies which purchase those securities.

In an earlier theoretical paper [

Throughout the whole paper, the term “optimal” is referring to the least number of mean defaults.

For the sake of completeness, we find it important to give a brief summary here of the main analytical findings about our model. The reader is referred to [

Regarding heterogeneity, the main finding is that heterogeneity tends to favor assortative matching, that is, firms facing similar shock distributions should band together. This does not necessarily mean that they differ because one type of distribution is necessarily more “risky” or more correlated than another one. They could be different without being ordered in a first or even second order stochastic sense. But the fact that they are different means that the optimal structure to deal with shocks could be different between the two of them, and mixing firms of different types would lead to inferior risk-sharing properties. That means, in practice, that some activities should be isolated from others, for instance by separating the banks’ retail and investment activities. In this respect, it was also found that the symmetric structure is optimal when the shocks are not too large (because this maximizes risk-sharing possibilities) while the star structure is optimal for larger shocks.

Let us now present the main results of our paper, namely those obtained from our numerical simulation program (see Materials and Methods below for details on our procedure and analysis).

Shock sizes are sampled from a Pareto distribution with a scale parameter of either 0.5 (“fat”), 1.5 (“small”) or with 5% probability 0.5 and 1.5 with probability of 95% (“mix”). Shocks are either directed at the largest firm (“Size”), the most connected firm (“Degree”) or at a random firm (“Random”). Significance in linear regression is indicated by the border line, solid: p<0.01, dashed: p<0.05, dotted: p<0.1.

The upper panel shows the interaction between connectivity and modularity, the lower panel the interaction between connectivity and level of capitalization. Blue contour lines represent the interpolation using a loess-function. Note that the number of mean defaults has been normalized for the range of each panel, and hence quantitative comparisons between panels are not possible.

The trade-off between diversification and contagion found in the analytical results could be confirmed and extended to other parameters. When shocks are sampled from a small tailed distribution, the optimal structure is well connected and uniform: default probability increases with the number of connections and decreases with modularity and size heterogeneity. But when shocks are sampled from a fat tailed distribution, the optimal structure is the opposite, sparse and heterogeneous (

All response curves present high concavity. Once a given degree of density, modularity and heterogeneity is reached, further changes of these parameters result in little variation of default probability (

The line colors represent different levels of mean degree (<k>).Continuous lines represent the trends for random graphs without capital buffer, Dashed lines represent the effect of the applied modifying factor: a 10% capitalization level (top), the highest possible degree heterogeneity (middle) and the highest possible modularity (bottom).

With constant network topology, default probability could be predicted from the fraction of shocks sampled from a fat-tailed distribution. The relationship was linear and the slope increased with density (

After completing the numerical study summarized above, we moved towards a closer connection with real systems by considering the core-group of the transnational corporate ownership network [

Removal was done randomly (black squares), or preferentially at the edges with the highest betweeness centrality (red circles) or the highest degree heterogeneity (grey diamonds). Shock were sampled from a small-tailed Pareto distribution (σ=1.5, lower panel) or a fat-tailed Pareto distribution (σ=0.5, upper panel).

The numerical results obtained here support and extend significantly the underlying analytical findings of [

As interconnectedness allows for higher risk diversification, many studies e.g. [

Considering the risk exposures that result from indirect neighbors greatly enhances the “effective” connectivity of the network. This accentuates the effects of the parameters that influence segmentation. An infinitely big shock could bring to bankruptcy every firm that can be reached by a path through the network. Real world shocks are not infinite but neither is network size and shocks can be several orders larger than the level of capitalization of many firms. This leads to a percolation phenomenon: when a giant cluster forms, almost all firms are susceptible to large shocks. Higher modularity increases the density at which the giant cluster forms and therefore decreases systemic risk.

Heterogeneity in degree distribution has received much attention in the literature during the last decade. In line with the seminal results of [

Higher size heterogeneity can be considered as a different way of isolation since it leads to a more unequal distribution of the number of defaults. Consider the extreme case where nearly all of the investments are realized by one single firm connected (directly or indirectly) to many very small firms; In a regular network a large enough shock would result in the default of most firms, independent of which firm was hit. In a heterogeneous network no firm defaults when a small firm is hit which is much more likely given their larger number. Therefore size heterogeneity provides safety in the presence of large shocks. On the other hand, small shocks that would not result in any default under a regular size distribution can lead to a default of all firms when the biggest firm is hit. However, this effect is important only for very extreme heterogeneity and is only socially beneficial under the assumption of linearity between number of defaults and system cost [

Our results confirm that capital requirements are an important tool (at least when the requirement are actually fulfilled and not eluded with accounting tricks), however the level of capital requested should depend on macro-prudential criteria. In that sense the Basel II and III accords establish higher capital requirements for SIFIs. The analysis of different national banking systems has shown that the SIFIs not necessarily are the biggest firms. Also, local measurements, such as number of links of a financial institution, are insufficient [

We consider an environment with

The gross return of the project is random, as with some probability _{b} is a random variable, with a Pareto distribution function.

Since the return on a firm's investment is subject to shocks, while the return promised to its creditors is deterministic, when the firm is hit by a shock it may be unable to meet the required payments on its liabilities, in which case it must default. Default costs are assumed to be substantial, so that the value of a firm is maximized when its probability of default at any point in time is minimized.

There is a large set of investors, who are the source of the supply of funds to firms. Investors are risk neutral and require an expected gross rate of return equal to

Since, as stated above, default entails a significant cost for a firm, a firm may benefit from entering

More precisely, let us posit that each firm exchanges a fraction 1-ϕ of its standing shares, giving rights to the return on its investments, for shares held by other firms. The specific pattern of exchanges among firms is formalized by a network, where a direct linkage between two firms reflects the fact that they undertake a

In the numerical model, we represented the financial network of exposures as a directed weighted graph that was constructed by a variant of the preferential attachment algorithm. We started from a given set of nodes (firms) connected with a few (~1%) random edges between them. Subsequent links were added until obtaining the desired network characteristics repeating the following algorithm for each link:

Calculate current network properties (modularity and heterogeneity)

Compare these results to the desired network properties

Assign probabilities to each pair of (unconnected) nodes in order to reduce distance between actual and desired properties.

Sample link from these probabilities

A more detailed description is given in _{E}) is assigned to an external entity, emulating private investors whose default would not affect systemic risk. The rest is split evenly between the original node and all its neighbors. The result is an adjacency matrix A, where each entry _{ij} represents the fraction firm _{j}).

To calculate the overall exposure we followed [_{i} of firm

where the matrix _{E}

The values of _{i}) minus their capital buffer (_{i}) and a risk adjusted return rate

The Pareto distribution has a fat-tail when its shape parameter (γ) is <1 and a small tail otherwise. The shock can affect either one single or several firms and be directed either at the most connected, biggest or randomly chosen firms. The shock is limited to the total size of the affected firm (

We ran the model for 1,000 parameter sets that were structured as Sobol-sequences [^{7}. Standardized regression coefficients (SCR) were calculated using the sensitivity package of the R-project [

Degree heterogeneity was calculated following [

(TIF)

(TIF)

(TIF)

(TIF)

(TIF)

(TIF)

(TIF)

(TIF)

(DOCX)

We thank Stefania Vitali and Stefano Battiston for sharing the Orbis data on corporate ownership.