s . so c - ph ] 6 M ar 2 00 6 Evaluation of Tranche in Securitization and Long - range Ising Model

This econophysics work studies the long-range Ising model of a finite system with N spins and the exchange interaction J N and the external field H as a model for homogeneous credit portfolio of assets with default probability P d and default correlation ρ d. Based on the discussion on the (J, H) phase diagram, we develop a perturbative calculation method for the model and obtain explicit expressions for P d , ρ d and the normalization factor Z in terms of the model parameters N and J, H. The effect of the default correlation ρ d on the probabilities P (N d , ρ d) for N d defaults and on the cumulative distribution function D(i, ρ d) are discussed. The latter means the average loss rate of the " tranche " (layered structure) of the securities (e.g. CDO), which are synthesized from a pool of many assets. We show that the expected loss rate of the subordinated tranche decreases with ρ d and that of the senior tranche increases linearly, which are important in their pricing and ratings.