We analyze the size dependence and temporal stability of firm bankruptcy risk
in the US economy by applying Zipf scaling techniques. We focus on a single
risk factor-the debt-to-asset ratio R-in order to study the stability of the
Zipf distribution of R over time. We find that the Zipf exponent increases
during market crashes, implying that firms go bankrupt with larger values of R.
Based on the Zipf analysis, we employ Bayes's theorem and relate the
conditional probability that a bankrupt firm has a ratio R with the conditional
probability of bankruptcy for a firm with a given R value.

Public debt is one of the important economic variables that quantitatively
describes a nation's economy. Because bankruptcy is a risk faced even by
institutions as large as governments (e.g. Iceland), national debt should be
strictly controlled with respect to national wealth. Also, the problem of
eliminating extreme poverty in the world is closely connected to the study of
extremely poor debtor nations. We analyze the time evolution of national public
debt and find "convergence": initially less-indebted countries increase their
debt more quickly than initially more-indebted countries.