Regularization Algorithms for Transition Matrices
by Alexander Kreinin of Algorithmics, and
Abstract: Both estimating portfolio credit risk and pricing credit risky securities require transition matrices for arbitrary time horizons. Simply computing the root of the annual transition matrix is unacceptable because the resulting matrices often contain negative elements. A similar situation exists when taking the logarithm of the annual transition matrix to compute a generator. This paper develops regularization algorithms for obtaining transition matrices and generators that give rise to close approximations of a given annual transition matrix. In our approach, the root or logarithm of the annual transition matrix is transformed, on a row-by-row basis, into a valid transition matrix or generator by projecting each row onto an appropriate set in a Euclidean space. Our methods compare favorably with other known regularization algorithms.