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Probability of Default as a Function of Correlation: The problem of non-uniqueness

by Toby Daglish of the University of Iowa, and
Wei Li of the University of Iowa

September 26, 2005

Abstract: It is common practice in industry for traders to use copula models, combined with observed market prices, to calculate implied correlations for firm defaults. The actual feasibility of this calculation depends on the assumption that there is a one-to-one mapping between values of CDO tranches, and the correlation implicit in the copula. This paper presents several proofs which demonstrate that, for sufficiently large portfolios of underlying credits, the probability of certain number of default are hump shaped as a function of the correlation. We follow our analytical results with some numerical examples of pricing CDOs, demonstrating the nonuniqueness problem of implied correlations.

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