Correlation matrix with block structure and efficient sampling methods

Abstract

Random sampling from a multivariate normal distribution is essential for Monte Carlo simulations in many credit risk models. For a portfolio of N obligors, standard methods usually require O(N) calculations to get one random sample. In many applications, the correlation matrix has a block structure that, as we show, can be converted to a “quasi-factor” model. As a result, the cost to get one sample can be reduced to O(N). Such a conversion also enables us to check whether a user-defined “correlation” matrix is positive semidefinite and “fix” it if necessary in an efficient manner. Disclaimer: The models and analyses presented here are exclusively part of a quantitative research effort intended to improve the computation time of Monte Carlo simulations when we deal with a correlation matrix that has a block structure. The views expressed in this paper are the authors’ own and do not necessarily represent the views of Standard & Poor’s. Furthermore, no inferences should be made with regard to Standard & Poor’s credit ratings or any current or future criteria or models used in the ratings process for credit portfolios or any type of financial security.