The Rearrangement Algorithm of Puccetti and Rüschendorf: Proving the Convergence

In 2012 Puccetti and Rüschendorf [J. Comp. Appl. Math., 236 (2012)] proposed a new algorithm to compute the upper Value-at-Risk (VaR), at a given level of confidence, of a portfolio of risky positions, whose mutual dependence is unknown. The algorithm was called Rearrangement, as it consists precisely in rearranging the columns of a matrix, whose entries are quantiles of the marginal distributions. In the following years the algorithm has performed quite well in several practical situations, but the convergence has remained an open problem. In the present paper we show that the rearrangement algorithm converges, once the deterministic procedure has been precisely defined and an initial optimality condition is satisfied.