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We consider an environment with N risk-neutral financial firms and a continuum of small investors. At any given point in time, each firm has an investment opportunity - a project - which requires an initial payment I and yields a random gross return R at the end of the period. It is assumed that the firm invests all the available funds besides a capital buffer. The resources needed to undertake the project are obtained by issuing liabilities (e.g. deposits or bonds) on which a deterministic rate of return must be paid. The gross return of the project is random, as with some probability q the firm is hit by a negative shock. If no shock hits, the return equals some normal level R. The loss Lb 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 r in order to lend their funds in any given period. Since firms may default, in which case creditors receive a payment equal to zero, the nominal gross rate of return M on the deposits to the firms must be greater or equal than r. Specifically, if we denote by φ the ex ante probability that any given firm defaults (an endogenous variable), we must have: M=r1−ϕ(1)Since, as stated above, default entails a significant cost for a firm, a firm may benefit from entering risk sharing arrangements with other firms and hence diversify risks. Here we consider the case where these arrangements take the form of shares of assets between firms, that is, of claims to the yields of the firms' investments, prior to the realization of the uncertainty. The possibly iterative procedure through which each firm exchanges shares on its whole array of asset holdings can be viewed as a securitization process of the firms' claims. 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 direct exchange of their assets. These exchanges are symmetrical in the analytical model where all N firms are identical ex ante. However in the numerical model this symmetry is broken, supposing that the shares have been issued against a value that is not necessarily present in the system anymore, i.e. we just represent a present state without reconstruction the history that lead to it. We allow for these asset swaps to occur repeatedly. Indirect connections are then also formed. In the analytical model, these asset swaps are allowed to occur, whereby a firm ends up having claims on the returns of projects of firms who swapped assets with the firms it exchanges assets 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 the number of exchange rounds is high enough - in particular, as high as their network distance. 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 propertiesAssign probabilities to each pair of (unconnected) nodes in order to reduce distance between actual and desired properties.Sample link from these probabilitiesA more detailed description is given in Materials S1. Once the links between nodes are established, their firm sizes are sampled and assigned to the nodes taking into account the correlation between firm size and number of links. A fraction of the operational result (fE) 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 aij represents the fraction firm i holds of the investment results of firm j (rj). To calculate the overall exposure we followed [48] and expressed the value Vi of firm i as: Vi=∑kDikrk+∑kaijVj(2)where the matrix D equals zero in all entries but the diagonal ones, that take a value 1-external control (fE). Equation 1 can be written in matrix form and solved for V as: V=(1−A)−1Dr(3)The values of r are obtained from the firms investments, that is their size (Si) minus their capital buffer (Ki) and a risk adjusted return rate q. Some firms are hit by a shock (l) sampled from a Pareto distribution. ri=(Si−Ki)•(q−li)(4)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 (l≤1). A firm defaults if its value is below its level of capitalization. Monte Carlo simulations and analysis: We ran the model for 1,000 parameter sets that were structured as Sobol-sequences [49] to equally cover the sampling space (Table S1). For each parameter set we constructed 10 networks and exposed each to 1000 shocks for each shock distribution, resulting in a total number of model runs of 107. Standardized regression coefficients (SCR) were calculated using the sensitivity package of the R-project [50]. Degree heterogeneity was calculated following [51] as a function of the degrees of the nodes of each link. Modularity expressed the number of links than fall within a given community (cluster) compared to the expected (random) value [52]. Size heterogeneity has been expressed using the Gini coefficient [53], which corresponds to the ratio of the area of the Lorenz curve to the area below the diagonal. Details are given in the Materials S1.
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