Abstract—In this paper an application by means real financial data is made in the field of migration models, i.e. the models that analyzes default studies and rating transitions. Data refers to long- term ratings from Standard &Poor’s historical file from 1975 to 2007. The mathematical tools used are Markov, semi-Markov and backward recurrence time processes. Keywords— Migration models, Rating transitions, Default studies, Markov chains, Semi-Markov chains, Backward. I. INTRODUCTION NTERNATIONAL organizations such as, Fitch, Moody’s and Standard & Poor’s, evaluate credit risk by giving different ratings to firms agreeing to be inspected. This represents a measure of the creditworthiness of bonds issued. Obviously, the lower the rating, the higher the interest rate that the firm should pay. The rating scale goes from AAA (the highest rating), i.e. extremely strong capacity to meet financial commitments to C, i.e. a bankruptcy petition has been filed or similar action taken, but payments of financial commitments are continued. The rating D means the payment default on financial commitments. The ratings change with time and one way of following their evolution is by means of Markov processes, see. [2], [8], [9], [11]. II. PROBLEM FORMULATION Although in the industry almost models make use of Markov chains the problem of the poor fitting of Markov process in the credit risk environment has been outlined, [1], [3], [12], [13], [16]. Some of these problems include: the duration inside a state. Actually the probability of changing rating depends on the time that a firm remains in the same rating, see in particular [3]. Under the Markov G. D’Amico is with the Department of Drug Sciences, University “G. D’Annunzio” ITALY (e-mail: g.damico@unich.it). G. Di Biase is with the Department of Drug Sciences, University “G. D’Annunzio” ITALY (phone: 0039-0871-3554608; fax 0039-0871-3554622; e-mail: dibiase@unich.it). J. Janssen is with the JACAN & EURIA, University of Bretagne Occidental FRANCE (e-mail: janssenja@wanadoo.fr). R. Manca is with the Department of Mathematics for Actuarial,Economic and Financial Decisions, University “La Sapienza” ITALY (e-mail: raimondo.manca@iniroma1.it). assumption, this probability depends only on the rank own at previous transition; the dependence of the rating evaluation from the epoch of the assessment. This means that, in general, the rating evaluation depends on when it is done and, in particular, on the business cycle; the dependence of the new rating from all history of the firm’s rank evolution, not only from the last evaluation. Actually the effect exists only in the downward cases but not in the case of upward ratings in the sense that if a firm gets a lower rating (for almost all rating classes ) then there is a higher probability that the next rating will be lower than the preceding one. Recently semi-Markov models applied to rating evolution have been appeared in literature, see i.e. [4], [5], [10], [17], [6]. In the last quoted paper also a backward process has been introduced in the model. In this paper a real data application of the semi-Markov model with backward is presented III. PROBLEM SOLUTION In order to consider dependence of the rating evaluation from the laps of time in which a firm remains in the same rating a homogeneous semi-Markov process is introduced. Given a probability space let us consider two random variables: , n n J T n ∈ ` respectively with state space I, that represents the rating at the n-th transition, and ` that represents the time of the n-th transition: : n J I Ω → : n T Ω→ ` Suppose that the process (J n , T n ) is a homogeneous Markov renewal process (MRP). The kernel Q =[Q ij (t)] associated to the MRP expresses the following probability: [ ] 1 1 () , | . ij n n n n Q t PJ jT T tJ i + + = = − ≤ = The probability to make next transition at time t in state j given that the process entered in state i at the starting time is given by: Semi-Markov Backward Credit Risk Migration Models Compared with Markov Models G. D’Amico, G. Di Biase, J. Janssen, R. Manca I PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON APPLIED MATHEMATICS, SIMULATION, MODELLING (ASM'09) PROCEEDINGS OF THE 3RD INTERNATIONAL CONFERENCE ON CIRCUITS, SYSTEMS AND SIGNALS (CSS'09) ISBN: 978-960-474-147-2 112 ISSN: 1790-5117