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

• Published in: Physica A Vol. 392; no. 20; pp. 4989 - 4996
• Main Authors: Katz, Yuri A., Tian, Li
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.10.2013
• Subjects: [formula omitted]-Gaussian distribution; Default risk; First stopping time; First stopping time; q-Gaussian distribution; Default risk
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2013.06.035

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. •Probability of default.•Credit risk.•Superstatistics of leverage returns.•q-Gaussian distributions.