Higher-order saddlepoint approximations in the Vasicek portfolio credit loss model

Abstract

This paper utilizes the saddlepoint approximation as an efficient tool to estimate the portfolio credit loss distribution in the Vasicek model. Value-atrisk (VaR), the risk measure chosen in the Basel II Accord for the evaluation of capital requirement, can then be found by inverting the loss distribution. The VaR contribution (VaRC), expected shortfall (ES) and ES contribution (ESC) can all be calculated accurately. Saddlepoint approximation is well known to provide good approximations to very small tail probabilities, which makes it a very suitable technique in the context of portfolio credit loss. The portfolio credit model we employ is the Vasicek one-factor model, which has an analytical solution if the portfolio is well diversified. The Vasicek asymptotic formula fails, however, when the portfolio is dominated by a few loans much larger than the rest. We show that saddlepoint approximation is able to handle such exposure concentration. We also point out that the saddlepoint approximation technique can be readily applied to more general Bernoulli mixture models (possibly multi-factor). It can further handle portfolios with random loss given default (LGD).