Computing Optimal Recovery Policies for Financial Markets

The current financial crisis motivates the study of correlated defaults in financial systems. In this paper we focus on such a model which is based on Markov random fields. This is a probabilistic model where uncertainty in default probabilities incorporates expert's opinions on the default risk (based on various credit ratings). We consider a bilevel optimization model for finding an optimal recovery policy: which companies should be supported given a fixed budget. This is closely linked to the problem of finding a maximum likelihood estimator of the defaulting set of agents, and we show how to compute this solution efficiently using combinatorial methods. We also prove properties of such optimal solutions. A practical procedure for estimation of model parameters is also given. Computational examples are presented and  experiments indicate that our methods can find optimal recovery policies for up to about 100 companies.  The overall approach is evaluated on a real-world problem concerning the major banks in Scandinavia and public loans. To our knowledge  this is a first attempt to apply combinatorial optimization techniques to this important, and expanding,  area of default risk analysis.