Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

Correlation stress testing is employed in several financial models for determining the value-at-risk (VaR) of a financial institution’s portfolio. The possible lack of mathematical consistence in the target correlation matrix, which must be positive semidefinite, often causes breakdown of these models. The target matrix is obtained by fixing some of the correlations (often contained in blocks of submatrices) in the current correlation matrix while stressing the remaining to a certain level to reflect various stressing scenarios. The combination of fixing and stressing effects often leads to mathematical inconsistence of the target matrix. It is then naturally to find the nearest correlation matrix to the target matrix with the fixed correlations unaltered. However, the number of fixed correlations could be potentially very large, posing a computational challenge to existing methods. In this paper, we propose an unconstrained convex optimization approach by solving one or a sequence of continuously differentiable (but not twice continuously differentiable) convex optimization problems, depending on different stress patterns. This research fully takes advantage of the recently developed theory of strongly semismooth matrix valued functions, which makes fast convergent numerical methods applicable to the underlying unconstrained optimization problem. Promising numerical results on practical data (RiskMetrics database) and randomly generated problems of larger sizes are reported. Research of H. Qi was partially supported by EPSRC Grant EP/D502535/1. Research of D. Sun was partially supported by Grant R146-000-104-112 of the National University of Singapore. H. Qi School of Mathematics, The University of Southampton, Highfield, Southampton SO17 1BJ, UK e-mail: [email protected] D.F. Sun ( ) Department of Mathematics and Risk Management Institute, National University of Singapore, Singapore 117543, Singapore e-mail: [email protected]