Asymptotic Loan Portfolio Loss with Multi-Factors and Skew Elliptical Distributions

This paper extends the Vasicek asymptotic result for the Gaussian single factor portfolio loss distribution to multi-factors with skew elliptical distributions. In particular, we derive new asymptotic results and analytical solutions for portfolio loss distribution when asset returns follow skew normal, Student’s t and skew t distributions. The sensitivities of portfolio loss value-at-risk with respect to model parameters are investigated under the new setting. Generally, a higher default threshold, a higher correlation, a more negative skewness and a fatter tail can all lead to a bigger value-at-risk of portfolio loss. An exception is noted for portfolio with small default probability; the value-at-risk decreases as correlation increases beyond a threshold. Empirical tests based on the U.S. sector charge-off ratios, published by the Federal Reserve for the period Q1:1985 to Q4:2011, show that the performance of our multi-factor model dominates that of the single factor model. While the Basel II single factor captures market wide systematic factor for defaults, the multi-factor model uncovers an additional and significant systemic risk factor that explains sector defaults. The loan portfolio loss estimated using the single factor model currently adopted by the regulatory framework is downward biased when compared with the estimates from the multi-factor model.