Higher-order Saddlepoint Approximations in the Vasicek Portfolio Credit Loss Model

by Xinzheng Huang of Delft University of Technology,
Cornelis W. Oosterlee of Delft University of Technology, and
Hans van der Weide of Delft University of Technology

Fall 2007

Abstract: This paper utilizes the saddlepoint approximation as an efficient tool to estimate the portfolio credit loss distribution in the Vasicek model. Value at Risk (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. 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 however fails when the portfolio is dominated by a few loans. 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 LGD.

JEL Classification: C63, G11, G21.

Keywords: Portfolio credit risk, Value at Risk, Expected Shortfall, VaR contribution, saddlepoint approximation.

Published in: Journal of Computational Finance, Vol. 11, No. 1, (Fall 2007), pp. 93-113.

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