Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem

Abstract

The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often constructed using output from disparate modeling sources and can be subjectively adjusted, for example, increasing the estimated correlation between two risk sources to confer reserving prudence. Hence, while individual elements still obey the assumptions of correlation values, the overall matrix is often not mathematically valid (not positive semidefinite). This can prove problematic in using the matrix in statistical models. The first objective of this article is to review existing techniques that address the nearest positive semidefinite matrix problem in a very general setting. The chief approaches studied are Semidefinite Programming (SDP) and the Alternating Projections Method (APM). The second objective is to finesse the original problem specification to consider imposition of a block structure on the initial risk correlation matrix. This commonly employed technique identifies off-diagonal subsets of the matrix where values can or should be set equal to some constant. This may be due to similarity of the underlying risks and/or with the goal of increasing computational efficiency for processes involving large matrices. Implementation of further linear constraints of this nature requires adaptation of the standard SDP and APM algorithms. In addition, a new Shrinking Method is proposed to provide an alternative solution in the context of this increased complexity. “Nearness” is primarily considered in terms of two summary measures for differences between matrices: the Chebychev Norm (maximum element distance) and the Frobenius Norm (sum of squared element distances). Among the existing methods, adapted to function appropriately for actuarial risk matrices, APM is extremely efficient in producing solutions that are optimal in the Frobenius norm. An efficient algorithm that would return a positive semidefinite matrix that is optimal in Chebychev norm is currently unknown. However, APM is used to highlight the existence of matrices close to such an optimum and exploited, via the Shrinking Method, to find high-quality solutions. All methods are shown to work well both on artificial and real actuarial risk matrices provided under collaboration with Tokio Marine Kiln (TMK). Convergence speeds are calculated and compared and sample data and MATLAB code is provided. Ultimately the APM is identified as being superior in Frobenius distance and convergence speed. The Shrinking Method, building on the output of the APM algorithm, is demonstrated to provide excellent results at low computational cost for minimizing Chebychev distance.