Characterization of Lower-bound and Upper-bound of Required Settlement Fund under Real-Time Gross Settlement. Hitoshi Hayakawa. Hokkaido University
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1 Characterization of Lower-bound and Upper-bound of Required Settlement Fund under Real-Time Gross Settlement Hitoshi Hayakawa Hokkaido University September 9, 2015 Financial Risk & Networks Theory Conference 2015 H Hayakawa (Hokkaido Univ) September 9, / 15
2 Interbank Settlement Systems Interbank Settlement System: TARGET2, CHAPS, BOJ-Net, Fedwire real-time gross settlement (RTGS) system - small risk, large requirement of settlement fund designated-time net settlement (DTNS) system - large risk, small requirement of settlement fund H Hayakawa (Hokkaido Univ) September 9, / 15
3 Research Question - - How much settlement fund is to be required under RTGS? - in the face of complex network of obligations (about 500 participants, 50, 000 daily transactions in the BOJ-NET) which and how network structure matters? - theoretical approach formalize as network problem (, omitting incentive) characterize the problem with network factors H Hayakawa (Hokkaido Univ) September 9, / 15
4 What this paper does settlement in interbank settlement systems banks settle obligations on behalf of non-financial customers - given obligations in the morning, all settled within a day - zero balance in the morning - intraday lending by central bank - borrowed amount returned at the end of the day required amount of settlement fund under RTGS - minimum required amount for each realized order of settlement no redundant lending, no reserve H Hayakawa (Hokkaido Univ) September 9, / 15
5 What this paper does example Formalize a pair of minimization/maximization problem - for given distribution of obligations with settlement unit constraint, - evaluate lower-bound and upper-bound of required settlement fund - in relation to physically available order of settlement Characterize the problems in terms of network factors - Propose original concepts on twisted nature of payment network : arrow-twisted, vertex-twisted H Hayakawa (Hokkaido Univ) September 9, / 15
6 Framework base elements V, A < V, A > : directed graph (vertices, arrows) additional elements f, s, p f : A R + (flow) : amount of obligation s : A {1, 2,, A } (sequence) : order of settlement p : V R 0+ (potential) : allocation of settlement fund Networks f-network : < V, A, f > : distribution of obligations fsp-network : < V, A, f, s, p > : settlement procedure (for < V, A, f >) Example H Hayakawa (Hokkaido Univ) September 9, / 15
7 Basic Properties for Networks closed < V, A, f > : balanced distribution of obligations v V, sum of payments indicated by f are balanced e-( exact- )covered < V, A, f, s, p > : proper settlement procedure (feasible) For payments of < V, A, f >, all the payments are executable at any point under the order indicated by s, and also (no redundancy) there exists no other feasible < V, A, f, s, p > such that p(v) p(v) for every v V, and v V, p (v) < p(v) H Hayakawa (Hokkaido Univ) September 9, / 15
8 Formalization of our problem Given a closed f-network (balanced distribution of obligations) N f =< V, A, f >, min s,p v V p(v) max s,p v V p(v) st, < V, A, f, s, p > is e-covered (proper settlement procedure) H Hayakawa (Hokkaido Univ) September 9, / 15
9 Approach: Closed Cycle Decomposition Theorem (Closed Cycle Decomposition (fulkerson, 1962)) For any closed f-network N f =< V, A, f >, there always exists a closed cycle decomposition N f = c C < V c, c, f c > = (1 cycle closed f-network) Example for closed cycle decomposition H Hayakawa (Hokkaido Univ) September 9, / 15
10 Problem rewritten with Closed Cycle Decomposition Given a closed f-network N f, with some closed cycle decomposition N f = c C < V c, c, f c >= c C < V c, c, f c >, x min (N f ) = c C f c + R min (N f, C, { f c} c C ) x max (N f ) = c C f c ( c 1) + R max (N f, C, { f c} c C ) (cycle oriented amount) + (residual) Example for R min (N f, C, { f c} c C ) = Rmax (N f, C, { f c} c C ) = 0 H Hayakawa (Hokkaido Univ) September 9, / 15
11 Major Network Factors : arrow-twisted Theorem (arrow-twisted) Given a closed f-network N f, there exist arrow-twisted cycles (C, { f c} c C ), Rmin (N f, C, { f c} c C ) > 0, Example for R min (N f, C, {f c } c C ) > 0 Economic interpretation: synchronization inconsistency regarding each payment H Hayakawa (Hokkaido Univ) September 9, / 15
12 Major Network Factors : vertex-twisted Theorem (vertex-twisted) Given a closed f-network N f, there exist vertex-twisted cycles (C, { f c} c C ), Rmax (N f, C, {f c } c C ) < 0 Example for R max (N f, C, { f c} c C ) > 0 Economic interpretation: synchronization inconsistency regarding each subject H Hayakawa (Hokkaido Univ) September 9, / 15
13 arrow-twisted and vertex-twisted Theorem (arrow-twisted and vertex-twisted) Given a set of cycles C, C is arrow-twisted C is vertex-twisted arrow-twisted, so vertex-twisted vertex-twisted, but not arrow-twisted Economic interpretation: scenario dependent feature of arrow-twisted H Hayakawa (Hokkaido Univ) September 9, / 15
14 Summary Introduced a mathematical framework for RTGS order of settlement - timing of payment in addition to balance-sheet linkage Provided a general analysis on which and how network structure matters for required amount of settlement fund arrow-twisted - synchronization inconsistency regarding each payment vertex-twisted - synchronization inconsistency regarding each subject relation between arrow-twisted and vertex-twisted - scenario dependent feature H Hayakawa (Hokkaido Univ) September 9, / 15
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