University of Hertfordshire

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Mathematical Foundations for Balancing the Payment System in the Trade Credit Market

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  • Tomaž Fleischman
  • Paolo Dini
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Original languageEnglish
Article numbere452
JournalJournal of Risk and Financial Management
Early online date21 Sep 2021
Publication statusE-pub ahead of print - 21 Sep 2021


The increasingly complex economic and financial environment in which we live makes the management of liquidity in payment systems and the economy in general a persistent challenge. New technologies make it possible to address this challenge through alternative solutions that complement and strengthen existing payment systems. For example, interbank balancing and clearing methods (such as real-time gross settlement) can also be applied to private payments, complementary currencies, and trade credit clearing to provide better liquidity and risk management. The paper defines the concept of a balanced payment system mathematically and demonstrates the effects of balancing on a few small examples. It then derives the construction of a balanced payment subsystem that can be settled in full and therefore that can be removed in toto to achieve debt reduction and payment gridlock resolution. Using well-known results from graph theory, the main output of the paper is the proof—for the general formulation of a payment system with an arbitrary number of liquidity sources—that the amount of liquidity saved is maximum, along with a detailed discussion of the practical steps that a lending institution can take to provide different levels of service subject to the constraints of available liquidity and its own cap on total overdraft exposure. From an applied mathematics point of view, the original contribution of the paper is two-fold: (1) the introduction of a liquidity node with a store of value function in obligation-clearing; and (2) the demonstration that the case with one or more liquidity sources can be solved with the same mathematical machinery that is used for obligation-clearing without liquidity. The clearing and balancing methods presented are based on the experience of a specific application (Tetris Core Technologies), whose wider adoption in the trade credit market could contribute to the financial stability of the whole economy and a better management of liquidity and risk overall.

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