Within the framework of maximum entropy principle we show that the finite-size long-range Ising model is the adequate model for the description of homogeneous credit portfolios and the computation of credit risk when default correlations between the borrowers are included. The exact analysis of the model suggest that when the correlation increases a first-order-like transition may occur inducing a sudden risk increase. Such a feature is not reproduced by the standard models used in credit risk modeling.

The purpose of the present paper is twofold. In the first part, we provide an algebraic characterization of several families of ϭ1/2 n nр5 Bogomol'nyi-Prasad-Sommerfield ͑BPS͒ states in M theory, at threshold and non-threshold, by an analysis of the BPS bound derived from the Nϭ1 Dϭ11 super Poincaré algebra. We determine their BPS masses and their supersymmetry projection conditions, explicitly. In the second part, we develop an algebraic formulation to study the way BPS states transform under GL(32,R) transformations, the group of automorphisms of the corresponding super Poincaré algebra. We prove that all ϭ 1 2 non-threshold bound states are SO(32) related with ϭ 1 2 BPS states at threshold having the same mass. We provide further examples of this phenomena for less supersymmetric ϭ 1 4 , 1 8 non-threshold bound states.

Abstract. This paper develops the Jungle model in a credit portfolio framework. The Jungle model is able to model credit contagion, produce doubly-peaked probability distributions for the total default loss and endogenously generate quasi phase transitions, potentially leading to systemic credit events which happen unexpectedly and without an underlying single cause. We show the Jungle model provides the optimal probability distribution for credit losses, under some reasonable empirical constraints. The Dandelion model, a particular case of the Jungle model, is presented, motivated and exactly solved. The Dandelion model provides an explicit example of doubly-peaked probability distribution for the credit losses. The Diamond model, another instance of the Jungle model, experiences the so called quasi phase transitions; in particular, both the U.S. subprime and the European sovereign crises are shown to be potential examples of quasi phase transitions. We suggest how the Jungle model is able to explain a series of empirical stylized facts in credit portfolios, hard to reconcile by some standard credit portfolio models. We look at model risk in a credit risk framework under the Jungle model, especially in relation to systemic risks posed by doubly-peaked distributions and quasi phase transitions.