 # Financial Networks

Inquiries into the behavior of a financial system often lead to questions about its network structure. Is risk to one institution a risk to the entire system? Network analysis sheds light on such questions.

Networks have been applied in many areas of economics including decentralized markets, bargaining, risk-sharing and diffusion of innovations. However, the literature on applying network analysis to financial systems is still in its early stages. An obstacle is the complexity of modern financial institutions, their services, arrangements, and the instruments they engineer. In response to this complexity, researchers have focused on components of the financial system. For example, lots of research has been devoted to interbank lending networks. In the following we describe our work on a different component, secured lending markets. We analyze the role networks play as conduits of systemic risk.

### Networked repo market

In modern economies, secured lending lies at the heart of the financial system. A case in point is the huge repo (repurchase agreement) market, which supplies trillions of dollars in short-term financing to securities dealers and many other financial institutions. In a repo, a borrower pledges a security as collateral to obtain a short-term loan. Attached is the promise to repay the loan at some later date.

In a simple model of $$n$$ dealer banks, dealer $$i$$ holds $$a_i$$ units of some security and $$c_i$$ units of cash. These holdings change over time according to dynamics observed by the market. Some elementary and empirically motivated laws may be summarized as follows. At each time step, a security is chosen at random and borrower $$j$$ is matched to a lender $$i$$ with conditional probability $$p (i | j)$$. Cash and collateral are exchanged and the process continues.

$p (i | j) = \frac{ c_i}{\sum_{k\in N(j)} c_k}$

Here, $$N(j)$$ denotes the neighborhood of $$j$$ in the network. The process turns out to be a modified Erhenfest urn, a model of particles (securities) mixing among $$n$$ interacting urns (dealer banks). When the network is complete (i.e., $$N(j) = \{1,2, \dots, n\}$$ and $$C = \sum_{k \in N(j)} c_k$$ for all $$j$$) the particles mix with no restriction. In equilibrium (large time) each dealer leverages its initial collateral by a factor of $$A/C$$ where $$A = \sum_{i=1}^n a_i$$. When the network is not complete the equilibrium behavior is more complicated. A simulation of the cash market of 22 primary dealer banks given a randomly generated network. Here, all dealers start with the same endowments. The network plays a major role in equilibrium allocations of cash and securities. Dealers with more central positions in the network are given more opportunity to lend.

Perhaps surprisingly, the process is time reversible for any underlying network structure. Moreover, all dealer banks move assets independently of one another in equilibrium. This is revealed by the following product form for the equilibrium distribution.

$\pi (x) = \frac{1}{Z} \prod_{i=1}^n \binom{e_i}{x_i} \sum_{k\in N(i)} x_k$

Here, $$\pi$$ is a distribution over $$(a_1, \dots, a_n)$$, $$e_i$$ is the initial endowment (cash and collateral) of dealer $$i$$ and $$Z$$ is a normalizing constant.

Equipped with this elementary model of cash flows in a repo network we may ask some pertinent questions. What happens when the dealers can demand some of the loans be repaid at the repo rate? What impact do margins/haircuts have on system stability? Does the network structure play a role in propagating financial distress? It turns out that the model is not always stable and market stability is determined by relationships between the system leverage $$A/C$$, repo rate, haircut and the network. In an upcoming draft we find the precise nature of these relationships.