Entropy in Financial Contagion Research Prof. Michael Stutzer Dept. of Finance University of Colorado

Chain Reaction ?? Banks owe payments to each other. Banks depend on payments received to make payments owed. Thus shocks impeding some banks’ ability to pay may impede other banks’ ability to pay: cascading difficulties

Some References Upper and Worms, EuroEconRev., 2004 Estimates payments matrix, uses it in stress-scenario simulation Upper, J. Financial Stability, 2011 Surveys subsequent simulation studies of others Golan, Judge and Robinson, RevEconStat, 1994 Foundation for interbank payment matrix estimation

Basic Concept Snapshot of interbank payments matrix X A bank fails if total losses – direct and indirect from cascades – exceed the bank’s capital Modeling goals: Estimate frequency and severity of bank failures Estimate a bank’s probability of causing others to fail Identify banks that are most important to this process Raise capital requirements for those critical banks, or regulate them more closely. Network notion of “centrality” is important here. Has the potential to model other systemic risks, e.g. temporary systemic snafus for the daylight overdraft mechanism.

Interbank Payments Matrix X

Suppose this is all we know. Info-Metrics Rescue #1 Suppose all you know are: Row sums ai : each bank’s interbank assets Column sums lj : each bank’s interbank liabilities Suppose this is all we know. Info-Metrics to the Rescue !

The solution is as-if we assumed independence: Unless we allow banks to owe money to themselves, we: Add new constraints Minimize the Mutual Information : Solution by Duality: Has canonical form:

Toy Example Suppose the unknown X is: Only Fractions A and L are Known The estimated matrix is: Estimator Spreads Flows Across Banks: Bold Cells Aren’t Zero

A Drawback of This Estimator Theory and empirical work show (Memmel & Sachs): Mutual Information System Stability Hence minimizing mutual information may overestimate system stability, i.e. underestimate probability and severity of failure cascades This is due to tendency of minimum mutual information estimator to spread known column and row sums too evenly across unknown cells Potential fixes to this include: Acquire more data about unknown cells, to avoid need for estimating them Use additional constraints to incorporate known qualitative information

Elaboration Known Cell Values Are Easily Incorporated Plug ‘em in, and then re-normalize remaining unknown cells to be joint probabilities. Minimize mutual information as before The size of this problem is smaller. Accommodate Typical 2-Tier Structure

A Component: 3 small banks, 1 Big Link Components Like this Together. Minimize Resulting Mutual Information.

Incorporate Measurement Error Flow data is subject to measurement error Even if no point-in-time error, flows are changing over time Info-Metrics Rescue #2 Use Golan & Judge Support Set Methods to obtain probability distributions for uncertain cells. Use to construct interval estimates for those cells.