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Improved block rearrangement algorithm
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DOI：10.1007/s10479-025-06682-w.png)
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Abstract
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In the context of finding risk bounds for portfolios of risks, Puccetti and Rüschendorf (J Comput Appl Math 236(7):1833–1840, 2012) introduce the rearrangement algorithm (RA) as a tool for (optimally) rearranging matrices by permuting, in each step, the elements of a given column. The RA also has applications in finance and operations research. Bernard and McLeish (Asia-Pac J Oper Res 33(05):1650040, 2016) and Bernard et al. (J Risk Insur 84(3):923–959, 2017) show that, in principle, better results can be expected by permuting the rows of randomly chosen blocks of the matrix. They label such an algorithm the block rearrangement algorithm (BRA). Various versions of BRA exist, and they mainly differ with respect to the manner in which the blocks (i.e., the submatrices) are chosen in each step. In this paper, we aim to develop an improved version of BRA based on a dynamic choice of block sizes. That is, we seek to find the optimal sequence of block (submatrix) sizes. To achieve this, we refine the BRA by sampling the block size, $$r_t,$$ at the t-th step from a Beta distribution with two parameters that evolve over the different steps. The proposed BRA Beta is designed to select large block sizes initially (similar to BRA Binomial), and then transition to smaller sizes (resembling the RA). A numerical study demonstrates that BRA Beta outperforms other variants of the RA available in the literature. For example, the improvement in variance reduction achieved by BRA Beta is double that of other BRA-based algorithms when the risks are heterogeneous and the portfolio is large (see Sect. 5).
Keywords:
Rearrangement algorithm
Risk bounds
Aggregation risk
Variance of the sum
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