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Abstract: Interdependencies between firms that hold each other's shares (or other obligations) are a channel through which shocks to one firm can be transmitted to others. Thus, a small yet sufficient number of failing firms may bring down the entire network. The key question is what is the impact of firms' cash reserves and structure of the cross holdings network upon the system's exposure to risk.

We study a financial network of equity share cross-holdings. In the model, a firm's value depends on its cash endowment and the shares it owns in other firms. If a firm's value falls below a failure threshold, it discontinuously imposes losses on its counter-parties (i.e., default costs). We find that the model's equilibria---understood as consistent combinations of firm market values---are distributed non-uniformly in space: while there is concentration of equilibria around the "worst" and the "best" equilibria, generally, there are many equilibria "in-between", suggesting that the extreme equilibria---focused on in other studies of similar models---are by no means representative.

If we let firm market values change according to a natural dynamic, we find that, in many regimes when firms' cash endowments are non-extreme, a large fraction of the model's state space is risk-free in that, if the firms start with their market values being in this safe zone, the initially solvent firms remain solvent or, potentially, firms switch from insolvency to solvency. We also provide a formal characterization of the risk zone, with all the firms whose market values started in the risk zone ending up insolvent. The size of the risk zone is determined by the firms' cash reserves, shock magnitude and occurrence threshold, and the firms cross holdings.

Joint work with Santosh Venkatesh and Rakesh Vohra.