Formula omitted]-Gaussian distributions of leverage returns, first stopping times, and default risk valuations

Abstract

We study the probability distributions of daily leverage returns of 520 North American industrial companies that survive de-listing during the financial crisis, 2006–2012. We provide evidence that distributions of unbiased leverage returns of all individual firms belong to the class of q-Gaussian distributions with the Tsallis entropic parameter within the interval 1<q<2. The fat tails of the observed distributions imply a much higher probability of extreme movements in a company’s leverage ratio than forecasted by the normal distribution (q=1). Motivated by these findings, we develop a q-Gaussian generalization of traditional structural models of default. Derived exact analytical expressions for the probability distribution of a first stopping time and its intensity forecast significantly higher probability of default and much wider credit spreads at short time-horizons. Our findings are broadly consistent with the results of empirical studies in equity markets and are essential for single-name default forecasting as well as valuations of portfolio credit risk and economic capital, which might be underestimated by a classic theory of diversified portfolio optimization.