Journal of Risk Model Validation (69–76) Volume 1/Number 3, Fall 2007 Multiple hypotheses testing of transition matrices Victor de la Pena Department of Statistics, Columbia University, Room 1027, Mail Code 4690, 1255 Amsterdam Avenue, New York, NY 10027, USA; email: vp@stat.columbia.edu Adrian Hernandez-del-Valle Department of Statistics, Columbia University, 2700 Broadway Apt. 3B, New York, NY 10025, USA; email: ah2488@columbia.edu or ahdv@hotmail.com Ricardo Rivera State of New York Banking Department and New York University, One State Street, New York, NY 10025, USA; email: rar7@nyu.edu We propose a multiple hypotheses test of equality between transition matrices and apply our procedure to solve the problem of “accuracy” of credit rating transition matrices. 1 INTRODUCTION A transition probability (TP) is the probability that a company will go from a given credit rating to another in a specific period of time, eg, dropping from AAA to AA in a year, and a transition matrix (TM) is a probability matrix that portrays all possible transitions. Table 1 is an example of a Moody’s TM. The comparison of TMs is an important problem ahead of international imple- mentation of the Basel II regulations on bank capital, particularly for backtesting – “validating” rating estimates against realized ratings (see Basel Committee on Banking Supervision (2005, p. 9)). The logic is threefold. (i) Regulatory and economic capital: a credit rating is linked to the borrowers capacity of repaying debt, and thus determines the ability of a company or government to obtain funding, as well as the cost of the resources. A borrower with a strong credit rating, eg, AAA, can obtain resources faster and cheaper than one with a high probability of default, a D rating. (ii) Banks use TMs for portfolio management and trading: an AAA rate is not worth much if the probability of falling to a lower rate in a short period of time is high, eg, imagine a TP of 90% droppingfrom AAA to D in a year. Would this borrower really be qualified as AAA? Some investors are required to sell their bond holdings automatically if the bonds are downgraded. Would these investors buy paper knowing they may have to sell it off in a rush in the immediate future? The first author acknowledges partial support from NSF grant DMS-05-0594. 69