Analytical Approximations for Loan and Credit Derivatives Portfolios

Abstract

Banks often seek to reduce the default risk exposure associated with their corporate loan portfolios by entering into credit derivative positions. They can, for example, buy default protection on selected borrowers, or diversify the portfolio by selling protection on other names. The design of suitable credit derivative positions and the estimation of risk capital reserves supporting a portfolio of loans and credit derivatives requires the computation of the probability distribution of the future portfolio value. The distribution of portfolio value takes a complicated form because it takes account of losses from default, interest income, protection and default risk premia, and the mark-to-market volatility of the derivative positions. This paper develops an analytical approximation for this distribution. The approximation is based on a small-time expansion of a transform of the portfolio value. It applies to many standard intensity-based models of firm-by-firm default timing. Numerical results illustrate the approximation for the value at risk of a portfolio of loans with hedging positions in credit swaps.

Keywords: profit and loss distribution, correlated defaults, mark-to-market, short-term approximation