q-Gaussian distributions of leverage returns, first stopping times, and default risk valuations

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.