In this paper, we develop a model supporting the so-called square-root formula used in Solvency II to aggregate the modular life SCR. Describing the insurance policy by a Markov jump process, we can obtain expressions similar to the square-root formula in Solvency II by means of limited expansions around the best estimate. Numerical illustrations are given, based on German population data. Even...

To prove Eq. (S1) we recall that the square root of the positive definite Fisher matrix is given by the matrix with the same (orthonormal) eigenvectors as F and the square root of its eigenvalues. We indicate as fi > 0 and vi (with v> i vj = δi,j) the eigenvalues and eigenvectors of F, respectively (Fvi = fivi). Notice that F is real and symmetric and thus diagonalize. In addition, F is positiv...

Using the block vec matrix, I give a necessary and sufficient condition for factorization of a matrix into the Kronecker product of two other matrices. As a consequence, I obtain an elementary algorithmic procedure to decide whether a matrix has a square root for the Kronecker product. Introduction My statistician colleague, J.E. Chacón, asked me how to decide if a real given matrix A has a squ...

in this paper, we show that a matrix a in mn(c) that has n coneigenvectors, where coneigenvaluesassociated with them are distinct, is condiagonalizable. and also show that if allconeigenvalues of conjugate-normal matrix a be real, then it is symmetric.

We present efficient implement at ions of the balanceand-truncate model reduction technique for large-scale systems. The key observation that distinguishes our approach is that Krylov subspace methods (Arnoldi and Lanczos) directly yield approximate low-rank square roots1 of the system Gramians; the balancing transformation can be then constructed from these square roots, obviating the need for...