  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. For 2,737 bankrupt
firms, we demonstrate size dependence in assets change during the bankruptcy
proceedings. Prepetition firm assets and petition firm assets follow Zipf
distributions but with different exponents, meaning that firms with smaller
assets adjust their assets more than firms with larger assets during the
bankruptcy process. We compare bankrupt firms with nonbankrupt firms by
analyzing the assets and liabilities of two large subsets of the US economy:
2,545 Nasdaq members and 1,680 New York Stock Exchange (NYSE) members. We find
that both assets and liabilities follow a Pareto distribution. The finding is
not a trivial consequence of the Zipf scaling relationship of firm size
quantified by employees-although the market capitalization of Nasdaq stocks
follows a Pareto distribution, the same distribution does not describe NYSE
stocks. We propose a coupled Simon model that simultaneously evolves both
assets and debt with the possibility of bankruptcy, and we also consider the
possibility of firm mergers.
