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July  2022, 18(4): 2783-2803. doi: 10.3934/jimo.2021091

## An online-decision algorithm for the multi-period bank clearing problem

 School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, China

* Corresponding author: Zongwei Chen, czw@163.sufe.edu.cn

Received  February 2021 Revised  March 2021 Published  July 2022 Early access  April 2021

The bank clearing problem (BCP) refers to the problem of designing an optimal clearing algorithm for the interbank payment system. Due to the way in which for the payment system has evolved, the classical BCP model is insufficient for addressing this problem accurately. In particular, delayed settlements are allowed in the now popular high-frequency deferred net settlement (DNS) system. In practice, the characteristics of incoming payment instructions are heavily connected to the time of day, and can be predicted with reasonable precision based on historical data. In this paper, we study the multi-period bank clearing problem (MBCP) by introducing the time dimension and considering future instructions in the decision-making process. We design a new clearing algorithm for MBCP using a model predictive control (MPC) policy, which uses historical data to predict payment instructions in the future. We benchmark the designed algorithm's performance with the classical greedy algorithm on the basis of BCP. Given that the liquidity is regular or relatively low, the numerical results indicate that the designed algorithm significantly improves the quality of clearing decision-making and is robust with respect to forecasting errors and fluctuation of future transactions.

Citation: Zongwei Chen. An online-decision algorithm for the multi-period bank clearing problem. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2783-2803. doi: 10.3934/jimo.2021091
##### References:
 [1] P. Angelini, An analysis of competitive externalities in gross settlement systems, Journal of Banking & Finance, 22 (1998), 1-18.  doi: 10.1016/S0378-4266(97)00043-5. [2] M. L. Bech and R. J. Garratt, Illiquidity in the interbank payment system following wide-scale disruptions, J. Money, Credit Bank, 44 (2012), 903-929. [3] M. L. Bech, C. Preisig and K. Soramaki, Global trends in large-value payments, Economic Policy Review, 14 (2008), 59-81.  doi: 10.2139/ssrn.1141387. [4] D. Bertsimas and J. N. Tsitsiklis, Introduction to linear optimization, 2$^nd$ edition, Cambridge, MA, USA: Athena Scientific, (1997), 139–199. [5] N. Davey and D. Gray, How has the liquidity saving mechanism reduced banks' intraday liquidity costs in CHAPS?, Bank of England Quarterly Bulletin, 54 (2014), 180-189. [6] E. Denbee and B. Norman, The impact of payment splitting on liquidity requirements in RTGS, Working paper of Bank of England, (2010), 35 pp. Available from: https://www.bankofengland.co.uk/working-paper/2010/the-impact-of-payment-splitting-on-liquidity-requirements-in-rtgs. doi: 10.2139/ssrn.1701172. [7] M. Diehl and U. Schollmeyer, Liquidity-Saving Mechanisms: Quantifying the Benefits in TARGET2, Working paper of Deutsche Bundesbank, 2011. Available from: https://www.bis.org/cpmi/publ/d67.pdf. [8] R. Garrat and P. Zimmerman, Centralized netting in financial networks, Journal of Banking & Finance, 112 (2020), Article 105270. doi: 10.1016/j.jbankfin.2017.12.008. [9] M. M. Güntzer, D. Jungnickel and M. Leclerc, Efficient algorithms for the clearing of interbank payments, European Journal of Operational Research, 106 (1998), 212-219. [10] Z. Guo, R. J. Kauffman, M. Lin and D. Ma, Mechanism design for near real-time retail payment and settlement systems, IEEE 2015 48th Hawaii International Conference on System Sciences (HICSS) - HI, USA, (2015), 4824–4833. doi: 10.1109/HICSS.2015.573. [11] M. Hellqvist and T. Laine, (eds.), Diagnostics for the financial markets: Computational studies of payment systems, \emphHelsinki, FIN: Edita Prima Oy, (2012), 267–315 and 341–379. [12] J. McAndrews and S. Rajan, The timing and funding of fedwire funds transfers, Economic Policy Review, 6 (2000), 17-32. [13] O. Merrouche and J. Schanz, Banks' intraday liquidity management during operational outages: Theory and evidence from the UK payment system, Journal of Banking & Finance, 34 (2010), 314-323. [14] J. Nanduri, Y. Jia and A. Oka et al., Microsoft uses machine learning and optimization to reduce E-commerce fraud, Interfaces, 55 (2020), 64-79. [15] T. Nellen, Intraday liquidity facilities, late settlement fee and coordination, Journal of Banking & Finance, 106 (2019), 124-131.  doi: 10.1016/j.jbankfin.2019.06.009. [16] S. J. Qin and T. A. Badgwell, A survey of industrial model predictive control technology, Control Engineering Practice, 11 (2003), 733-764. [17] S. Rosati and S. Secola, Explaining cross-border large-value payment flows: Evidence from TARGET and EURO1 data, Journal of Banking & Finance, 30 (2006), 1753-1782.  doi: 10.1016/j.jbankfin.2005.09.011. [18] Y. M. Shafransky and A. A. Doudkin, An optimization algorithm for the clearing of interbank payments, European Journal of Operational Research, 171 (2006), 743-749.  doi: 10.1016/j.ejor.2004.09.003. [19] K. Soramäki, M. L. Bech, J. Arnold, R. J. Glass and W. E. Beyeler, The topology of interbank payment flows, Physica A Statistical Mechanics & Its Applications, 379 (2007), 317-333. [20] A. Suponenkovs, A. Sisojevs, G. Mosãns et al., Application of image recognition and machine learning technologies for payment data processing review and challenges, 5th IEEE Workshop on Advances in Information, Electronic and Electrical Engineering (AIEEE), (2017), 1–6. doi: 10.1109/AIEEE.2017.8270536. [21] H. Tomura, Payment instruments and collateral in the interbank payment system, J. Econom. Theory, 178 (2018), 82-104.  doi: 10.1016/j.jet.2018.08.008. [22] Y. Wang and S. Boyd, Fast model predictive control using online optimization, IEEE Transactions on Control Systems Technology, 18 (2010), 267-278. [23] M. Willison, Real-time gross settlement and hybrid payment systems: A comparison, Bank of England Quarterly Bulletin, 45 (2005), 53. doi: 10.2139/ssrn.724042. [24] Bank for International Settlements, Statistics on payment, Clearing and Settlement Systems in the CPMI - Figures for 2015, Report of Bank for International Settlements, 2015. Available from: https://www.bis.org/cpmi/publ/d152.pdf. [25] Bank for International Settlements, Real-Time Gross Settlement Systems, Report of Bank for International Settlements, 1997. Available from: https://www.bis.org/cpmi/publ/d22.pdf. [26] Bank for International Settlements, Report on Netting Schemes (Angell Report), Report of Bank for International Settlements, 1989. Available from: https://www.bis.org/cpmi/publ/d02.pdf. [27] Bank for International Settlements, New Developments in Large-Value Payment Systems, 2005, Report of Bank for International Settlements, 2005. Available from: https://www.bis.org/cpmi/publ/d67.pdf.

show all references

##### References:
 [1] P. Angelini, An analysis of competitive externalities in gross settlement systems, Journal of Banking & Finance, 22 (1998), 1-18.  doi: 10.1016/S0378-4266(97)00043-5. [2] M. L. Bech and R. J. Garratt, Illiquidity in the interbank payment system following wide-scale disruptions, J. Money, Credit Bank, 44 (2012), 903-929. [3] M. L. Bech, C. Preisig and K. Soramaki, Global trends in large-value payments, Economic Policy Review, 14 (2008), 59-81.  doi: 10.2139/ssrn.1141387. [4] D. Bertsimas and J. N. Tsitsiklis, Introduction to linear optimization, 2$^nd$ edition, Cambridge, MA, USA: Athena Scientific, (1997), 139–199. [5] N. Davey and D. Gray, How has the liquidity saving mechanism reduced banks' intraday liquidity costs in CHAPS?, Bank of England Quarterly Bulletin, 54 (2014), 180-189. [6] E. Denbee and B. Norman, The impact of payment splitting on liquidity requirements in RTGS, Working paper of Bank of England, (2010), 35 pp. Available from: https://www.bankofengland.co.uk/working-paper/2010/the-impact-of-payment-splitting-on-liquidity-requirements-in-rtgs. doi: 10.2139/ssrn.1701172. [7] M. Diehl and U. Schollmeyer, Liquidity-Saving Mechanisms: Quantifying the Benefits in TARGET2, Working paper of Deutsche Bundesbank, 2011. Available from: https://www.bis.org/cpmi/publ/d67.pdf. [8] R. Garrat and P. Zimmerman, Centralized netting in financial networks, Journal of Banking & Finance, 112 (2020), Article 105270. doi: 10.1016/j.jbankfin.2017.12.008. [9] M. M. Güntzer, D. Jungnickel and M. Leclerc, Efficient algorithms for the clearing of interbank payments, European Journal of Operational Research, 106 (1998), 212-219. [10] Z. Guo, R. J. Kauffman, M. Lin and D. Ma, Mechanism design for near real-time retail payment and settlement systems, IEEE 2015 48th Hawaii International Conference on System Sciences (HICSS) - HI, USA, (2015), 4824–4833. doi: 10.1109/HICSS.2015.573. [11] M. Hellqvist and T. Laine, (eds.), Diagnostics for the financial markets: Computational studies of payment systems, \emphHelsinki, FIN: Edita Prima Oy, (2012), 267–315 and 341–379. [12] J. McAndrews and S. Rajan, The timing and funding of fedwire funds transfers, Economic Policy Review, 6 (2000), 17-32. [13] O. Merrouche and J. Schanz, Banks' intraday liquidity management during operational outages: Theory and evidence from the UK payment system, Journal of Banking & Finance, 34 (2010), 314-323. [14] J. Nanduri, Y. Jia and A. Oka et al., Microsoft uses machine learning and optimization to reduce E-commerce fraud, Interfaces, 55 (2020), 64-79. [15] T. Nellen, Intraday liquidity facilities, late settlement fee and coordination, Journal of Banking & Finance, 106 (2019), 124-131.  doi: 10.1016/j.jbankfin.2019.06.009. [16] S. J. Qin and T. A. Badgwell, A survey of industrial model predictive control technology, Control Engineering Practice, 11 (2003), 733-764. [17] S. Rosati and S. Secola, Explaining cross-border large-value payment flows: Evidence from TARGET and EURO1 data, Journal of Banking & Finance, 30 (2006), 1753-1782.  doi: 10.1016/j.jbankfin.2005.09.011. [18] Y. M. Shafransky and A. A. Doudkin, An optimization algorithm for the clearing of interbank payments, European Journal of Operational Research, 171 (2006), 743-749.  doi: 10.1016/j.ejor.2004.09.003. [19] K. Soramäki, M. L. Bech, J. Arnold, R. J. Glass and W. E. Beyeler, The topology of interbank payment flows, Physica A Statistical Mechanics & Its Applications, 379 (2007), 317-333. [20] A. Suponenkovs, A. Sisojevs, G. Mosãns et al., Application of image recognition and machine learning technologies for payment data processing review and challenges, 5th IEEE Workshop on Advances in Information, Electronic and Electrical Engineering (AIEEE), (2017), 1–6. doi: 10.1109/AIEEE.2017.8270536. [21] H. Tomura, Payment instruments and collateral in the interbank payment system, J. Econom. Theory, 178 (2018), 82-104.  doi: 10.1016/j.jet.2018.08.008. [22] Y. Wang and S. Boyd, Fast model predictive control using online optimization, IEEE Transactions on Control Systems Technology, 18 (2010), 267-278. [23] M. Willison, Real-time gross settlement and hybrid payment systems: A comparison, Bank of England Quarterly Bulletin, 45 (2005), 53. doi: 10.2139/ssrn.724042. [24] Bank for International Settlements, Statistics on payment, Clearing and Settlement Systems in the CPMI - Figures for 2015, Report of Bank for International Settlements, 2015. Available from: https://www.bis.org/cpmi/publ/d152.pdf. [25] Bank for International Settlements, Real-Time Gross Settlement Systems, Report of Bank for International Settlements, 1997. Available from: https://www.bis.org/cpmi/publ/d22.pdf. [26] Bank for International Settlements, Report on Netting Schemes (Angell Report), Report of Bank for International Settlements, 1989. Available from: https://www.bis.org/cpmi/publ/d02.pdf. [27] Bank for International Settlements, New Developments in Large-Value Payment Systems, 2005, Report of Bank for International Settlements, 2005. Available from: https://www.bis.org/cpmi/publ/d67.pdf.
Results of the greedy policy
Results of a clearing policy considering future payments
Aggregated settlement interval
The empirical cumulative distribution function of $\eta(x)$ with different reference periods $(P)$
Performance under the three policies during a working day
Comparing Different Policies
The empirical cumulative distribution function of $\eta $$(x) with different interval lengths Data sample  Sending Bank ID Receiving Bank ID Payment Amount Submit Time 12 4 56000.00 Day1 9:40:06 4 11 532200.00 Day2 8:30:21 13 11 367506.68 Day3 8:30:46  Sending Bank ID Receiving Bank ID Payment Amount Submit Time 12 4 56000.00 Day1 9:40:06 4 11 532200.00 Day2 8:30:21 13 11 367506.68 Day3 8:30:46 payment_i of banks (in billion CNY)  BankID 1 2 3 4 5 6 7 8 9 10 payment_i 61.93 17.22 10.99 46.82 44.34 0.71 18.86 6.78 14.05 26.11 Bank ID 11 12 13 14 15 16 17 18 19 20 payment_i 16.64 23.39 7.9 3.19 7.36 10.87 6.1 3.47 1.76 1.91  BankID 1 2 3 4 5 6 7 8 9 10 payment_i 61.93 17.22 10.99 46.82 44.34 0.71 18.86 6.78 14.05 26.11 Bank ID 11 12 13 14 15 16 17 18 19 20 payment_i 16.64 23.39 7.9 3.19 7.36 10.87 6.1 3.47 1.76 1.91 Descriptive statistics of the MPC policy's \eta$$ (x)$ under different values of $P$
 Reference Period$(P)$ Mean S.D. Upper 5% Lower 5% 20 days 1.22 0.26 2.30 0.90 10 days 1.19 0.25 2.28 0.86 5 days 1.19 0.28 2.00 0.78 2 days 1.15 0.31 1.98 0.73 1 day 1.08 0.29 2.02 0.68
 Reference Period$(P)$ Mean S.D. Upper 5% Lower 5% 20 days 1.22 0.26 2.30 0.90 10 days 1.19 0.25 2.28 0.86 5 days 1.19 0.28 2.00 0.78 2 days 1.15 0.31 1.98 0.73 1 day 1.08 0.29 2.02 0.68
Mean, standard deviation, and lower $5$th percentile of $\eta $$(x) for different values of L and different R  R 20\% 10\% 8\% 6\% 4\% 2\% Mean 1.35 1.33 1.34 1.23 1.17 1.16 L=600 S.D. 0.94 0.82 0.81 0.33 0.24 0.15 Lower 5% 0.23 0.61 0.78 0.84 0.84 0.94 Mean 1.23 1.38 1.30 1.21 1.24 1.18 L=300 S.D. 0.55 0.89 0.68 0.27 0.23 0.15 Lower 5% 0.40 0.68 0.80 0.84 0.91 0.97 Mean 1.24 1.31 1.31 1.23 1.21 1.21 L=120 S.D. 0.50 0.66 0.61 0.26 0.21 0.16 Lower 5% 0.62 0.76 0.91 0.90 0.95 0.97 Mean 1.22 1.25 1.27 1.22 1.22 1.21 L=60 S.D. 0.54 0.51 0.49 0.22 0.21 0.16 Lower 5% 0.66 0.74 0.83 0.94 0.95 1.01 Mean 1.23 1.29 1.25 1.22 1.21 1.23 L=30 S.D. 0.49 0.66 0.45 0.23 0.21 0.16 Lower 5% 0.68 0.73 0.83 0.95 0.96 1.00  R 20\% 10\% 8\% 6\% 4\% 2\% Mean 1.35 1.33 1.34 1.23 1.17 1.16 L=600 S.D. 0.94 0.82 0.81 0.33 0.24 0.15 Lower 5% 0.23 0.61 0.78 0.84 0.84 0.94 Mean 1.23 1.38 1.30 1.21 1.24 1.18 L=300 S.D. 0.55 0.89 0.68 0.27 0.23 0.15 Lower 5% 0.40 0.68 0.80 0.84 0.91 0.97 Mean 1.24 1.31 1.31 1.23 1.21 1.21 L=120 S.D. 0.50 0.66 0.61 0.26 0.21 0.16 Lower 5% 0.62 0.76 0.91 0.90 0.95 0.97 Mean 1.22 1.25 1.27 1.22 1.22 1.21 L=60 S.D. 0.54 0.51 0.49 0.22 0.21 0.16 Lower 5% 0.66 0.74 0.83 0.94 0.95 1.01 Mean 1.23 1.29 1.25 1.22 1.21 1.23 L=30 S.D. 0.49 0.66 0.45 0.23 0.21 0.16 Lower 5% 0.68 0.73 0.83 0.95 0.96 1.00 Computation time (upper 99\% ) at different values of g  g=1 g=2 g=5 g=10 g=20 g=40 L =600 0.73 0.52 0.13 0.06 0.03 0.01 L =300 3.20 1.11 0.41 0.16 0.13 0.05 L =120 28.68 5.95 1.38 0.89 0.42 0.39 L =60 >60.00 28.44 3.78 1.42 1.39 1.39 L =30 >30.00 >30.00 14.42 3.80 1.83 1.56  g=1 g=2 g=5 g=10 g=20 g=40 L =600 0.73 0.52 0.13 0.06 0.03 0.01 L =300 3.20 1.11 0.41 0.16 0.13 0.05 L =120 28.68 5.95 1.38 0.89 0.42 0.39 L =60 >60.00 28.44 3.78 1.42 1.39 1.39 L =30 >30.00 >30.00 14.42 3.80 1.83 1.56 Mean, standard deviation, and lower 5 th percentile of \eta$$ (x)$ for different values of $L$ and $g$ with $R$ = $6\%$
 $g=1$ $g=2$ $g=5$ $g=10$ $g=20$ $g=40$ mean 1.27 1.25 1.22 1.11 1.08 1.03 $L$=600 S.D. 0.37 0.29 0.33 0.23 0.24 0.24 lower 5% 0.79 0.89 0.81 0.76 0.67 0.68 mean 1.24 1.24 1.21 1.17 1.09 1.06 $L$=300 S.D. 0.26 0.29 0.27 0.26 0.20 0.21 lower 5% 0.85 0.88 0.84 0.82 0.75 0.72 mean 1.22 1.24 1.23 1.22 1.18 1.12 $L$=120 S.D. 0.26 0.26 0.27 0.24 0.22 0.20 lower 5% 0.91 0.93 0.89 0.92 0.88 0.81 mean - 1.21 1.22 1.20 1.19 1.16 $L$=60 S.D. - 0.25 0.23 0.21 0.21 0.19 lower 5% - 0.93 0.93 0.93 0.95 0.90 mean - - 1.22 1.21 1.21 1.19 $L$=30 S.D. - - 0.23 0.22 0.22 0.19 lower 5% - - 0.96 0.89 0.91 0.94
 $g=1$ $g=2$ $g=5$ $g=10$ $g=20$ $g=40$ mean 1.27 1.25 1.22 1.11 1.08 1.03 $L$=600 S.D. 0.37 0.29 0.33 0.23 0.24 0.24 lower 5% 0.79 0.89 0.81 0.76 0.67 0.68 mean 1.24 1.24 1.21 1.17 1.09 1.06 $L$=300 S.D. 0.26 0.29 0.27 0.26 0.20 0.21 lower 5% 0.85 0.88 0.84 0.82 0.75 0.72 mean 1.22 1.24 1.23 1.22 1.18 1.12 $L$=120 S.D. 0.26 0.26 0.27 0.24 0.22 0.20 lower 5% 0.91 0.93 0.89 0.92 0.88 0.81 mean - 1.21 1.22 1.20 1.19 1.16 $L$=60 S.D. - 0.25 0.23 0.21 0.21 0.19 lower 5% - 0.93 0.93 0.93 0.95 0.90 mean - - 1.22 1.21 1.21 1.19 $L$=30 S.D. - - 0.23 0.22 0.22 0.19 lower 5% - - 0.96 0.89 0.91 0.94
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