One of the trickier questions in the business of calculating default risk — the likelihood of a firm missing payments on a credit note — is in the feedback dynamics of the thing. Sure, you can estimate interest coverage, capital requirements, etc., but all those things presume that everything else stays more or less static, which is rarely the case, and even less likely during extreme market duress, such as that faced by Bear Stears.
One way of asking the question is as follows: How does your default risk change based on the number of times you have been perceived to have default risk? What is the feedback?
A recent paper looks at this subject using a stochastic urn moel, one where you pull a ball from an imaginary urn, and then, based on the color of the ball take a particular action. Specifically, the ball color tells you the likelihood of default — none, risky, or default in this simplified three-level model — and then there is a reinforcement mechanism that kicks in based on the number of times you have been in the risky state.
Got it? It’s fairly straightforward, at least conceptually, even if the mathematics of these Polya urn processes can be a bear (no pun intended). Analytics aside, the results are interesting, with the stochastic urn model doing a better job than more traditional default models in matching actual defaults. The following graph shows that nicely, with the traditional Z-score model not doing nearly as nice a job UbGesm (the urn model) in tracking against actual defaults.