Probability of Default as a Function of Correlation: The Problem of Non-uniqueness

It is common practise 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 non-uniqueness problem of implied correlations.