Lucas, André, Pieter Klaassen, Peter Spreij, and Stefan Straetmans, "An Analytic Approach to Credit Risk of Large Corporate Bond and Loan Portfolios", Journal of Banking & Finance, Vol. 25, No. 9, (September 2001), pp. 1635-1664.
Abstract: We derive an analytic approximation to the credit loss distribution of large portfolios by letting the number of exposures tend to infinity. Defaults and rating migrations for individual exposures are driven by a factor model in order to capture co-movements in changing credit quality. The limiting credit loss distribution obeys the empirical stylized facts of skewness and heavy tails. We show how portfolio features like the degree of systematic risk, credit quality and term to maturity affect the distributional shape of portfolio credit losses. Using empirical data, it appears that the Basle 8% rule corresponds to quantiles with confidence levels exceeding 98%. The limit law's relevance for credit risk management is investigated further by checking its applicability to portfolios with a finite number of exposures. Relatively homogeneous portfolios of 300 exposures can be well approximated by the limit law. A minimum of 800 exposures is required if portfolios are relatively heterogeneous. Realistic loan portfolios often contain thousands of exposures implying that our analytic approach can be a fast and accurate alternative to the standard Monte-Carlo simulation techniques adopted in much of the literature.
Keywords: Credit risk, Factor model, Fat tails, Skewness, Asymptotic analysis, Nonnormal factors.
Additional proofs: here.