Large portfolio losses in a turbulent market

Abstract

Consider a large credit portfolio of defaultable obligors in a turbulent market. Accordingly, the credit quality process of each obligor is described by a stochastic differential equation consisting of a drift term reflecting the trend, an individual volatility term reflecting the idiosyncratic risk, and a common volatility term reflecting the systematic risk. Moreover, for each obligor a market beta is used to measure its loading on the systematic risk. The obligor defaults at the first passage time of the credit quality process. We approximate the portfolio loss as the portfolio size becomes large. For the usual case where the individual defaults do not become rare, we establish a limit theorem for the portfolio loss, while for the other case where the individual defaults become rare, which is due to portfolio effect, we establish an asymptotic estimate for its tail probability. Both results show that the portfolio loss is driven by the systematic risk, while this driving force is amplified by the market beta. As an application, we derive asymptotic estimates for the value at risk and expected shortfall of the portfolio loss. Moreover, we implement intensive numerical studies to examine the accuracy of the obtained approximations and conduct some sensitivity analysis.