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

2. 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

3. 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

4. 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.

5. Interbank Payments Matrix X

6. 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 !

7. 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:

8. 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

9. 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

10. 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

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

12. 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.