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- 1. An agent-based model of payment systems Marco Galbiati Bank of England Kimmo Soramäki ECB, www.soramaki.net ECB-BoE Conference Payments and Monetary and Financial Stability 12-13 November 2007
- 2. Motivation, related work Model Results Conclusions Overview of the presentation Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 3. Motivation, related work Model Results Conclusions Overview of the presentation Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 4. Motivation, related work Model Results Conclusions Overview of the presentation Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 5. Motivation, related work Model Results Conclusions Overview of the presentation Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 6. Motivation, related work Model Results Conclusions Overview of the presentation Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 7. Liquidity in payment systems Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 8. Liquidity in payment systems Deferred Net Settlement vs Real Time Gross Settlement Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 9. Liquidity in payment systems Deferred Net Settlement vs Real Time Gross Settlement Liquidity risk (and operational risk) Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 10. Liquidity in payment systems Deferred Net Settlement vs Real Time Gross Settlement Liquidity risk (and operational risk) Liquidity as a common good Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 11. Liquidity is costly: tradeoff cost-of-liquidity / cost-of-delay Liquidity in payment systems Deferred Net Settlement vs Real Time Gross Settlement Liquidity risk (and operational risk) Liquidity as a common good Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 12. Related literature Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 13. Related literature <ul><li>Simulations </li></ul><ul><ul><li>Koponen-Soramaki (1998), Leinonen, ed. (2005, 2007) </li></ul></ul><ul><ul><li>Work at BoE, BoFr, US Fed, BoC, BoFin, others. </li></ul></ul><ul><ul><li>Use actual payment data and investigate alternative scenarios: </li></ul></ul><ul><ul><li>effect on payment delays, liquidity needs, and risks </li></ul></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 14. Related literature <ul><li>Simulations </li></ul><ul><ul><li>Koponen-Soramaki (1998), Leinonen, ed. (2005, 2007) </li></ul></ul><ul><ul><li>Work at BoE, BoFr, US Fed, BoC, BoFin, others. </li></ul></ul><ul><ul><li>Use actual payment data and investigate alternative scenarios: </li></ul></ul><ul><ul><li>effect on payment delays, liquidity needs, and risks </li></ul></ul><ul><li>Game theoretic models </li></ul><ul><ul><li>Angelini (1998), Bech-Garrat (2003), McAndrews (2007)… </li></ul></ul><ul><ul><li>Investigate "liquidity management games" to analyze intraday liquidity management behavior of banks in a RTGS (and DNS) environment </li></ul></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 15. Model overview RTGS á la UK CHAPS: banks choose an opening balance at the beginning of each day, used to settle payments during the day. Banks face a random stream of payment orders, to be settled out of their liquidity. Beside funding costs, banks (may) experience delay costs Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 16. Model overview RTGS á la UK CHAPS: banks choose an opening balance at the beginning of each day, used to settle payments during the day. Banks face a random stream of payment orders, to be settled out of their liquidity. Beside funding costs, banks (may) experience delay costs Banks adapt their opening balances over time, learning from experience, until equilibrium is reached We look at properties of equilibrium liquidity Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 17. Model overview We consider two scenarios: “ normal conditions” and “operational failures” RTGS á la UK CHAPS: banks choose an opening balance at the beginning of each day, used to settle payments during the day. Banks face a random stream of payment orders, to be settled out of their liquidity. Beside funding costs, banks (may) experience delay costs Banks adapt their opening balances over time, learning from experience, until equilibrium is reached We look at properties of equilibrium liquidity Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 18. Settlement algorithm i receives order to pay to i time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 19. Settlement algorithm i receives order to pay to i if i has funds the order is settled : j receives funds else, the order is queued time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 20. Settlement algorithm i receives order to pay to i if i has funds the order is settled : j receives funds if j has queued payments, the first one (say to k ) is settled else, the order is queued time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 21. Settlement algorithm i receives order to pay to i if i has funds the order is settled : j receives funds if j has queued payments, the first one (say to k ) is settled if k has queued payments, the first one (to ...) is settled else, the order is queued time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 22. Settlement algorithm i receives order to pay to i if i has funds the order is settled : j receives funds if j has queued payments, the first one (say to k ) is settled if k has queued payments, the first one (to ...) is settled ... cascade ends when the recipient of the payment has no queued payments else, the order is queued time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 23. Settlement algorithm i receives order to pay to i if i has funds the order is settled : j receives funds if j has queued payments, the first one (say to k ) is settled if k has queued payments, the first one (to ...) is settled ... cascade ends when the recipient of the payment has no queued payments else, the order is queued the algorithm is run 30 million times, for different liquidity levels k receives order to pay to z time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 24. Settlement algorithm i receives order to pay to i if i has funds the order is settled : j receives funds if j has queued payments, the first one (say to k ) is settled if k has queued payments, the first one (to ...) is settled ... cascade ends when the recipient of the payment has no queued payments else, the order is queued the algorithm is run 30 million times, for different liquidity levels k receives order to pay to z Payment orders arrive according to a Poisson process. Each bank equally likely as sender/ recipient complete symmetric network time Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 25. Settlement algorithm <ul><li>Distribution of others’ liquidity does not matter (much), only total level does </li></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 26. Settlement algorithm <ul><li>Distribution of others’ liquidity does not matter (much), only total level does </li></ul>funds committed by i Delays Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 27. Settlement algorithm <ul><li>Distribution of others’ liquidity does not matter (much), only total level does </li></ul>funds committed by i Delays Costs Costs funds committed by i Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 28. The liquidity game <ul><li>Costs minimized at different liquidity levels, depending on others’ funds each bank plays a “two-player game“ </li></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 29. The liquidity game <ul><li>Costs minimized at different liquidity levels, depending on others’ funds each bank plays a “two-player game“ </li></ul>Best reply <ul><ul><li>if others post 1, I should post 24 </li></ul></ul><ul><ul><li>if others post 5, I should post 15 </li></ul></ul><ul><ul><li>if others post 50, I should post 10…. </li></ul></ul>funds committed by others Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 30. Learning the equilibrium Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 31. Learning the equilibrium <ul><ul><li>Banks adapt actions over time, </li></ul></ul><ul><ul><li>on the basis of experience (“Fictitious play”) </li></ul></ul><ul><ul><li>up to equilibrium. </li></ul></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 32. Learning the equilibrium <ul><ul><li>Banks adapt actions over time, </li></ul></ul><ul><ul><li>on the basis of experience (“Fictitious play”) </li></ul></ul><ul><ul><li>up to equilibrium. </li></ul></ul><ul><ul><li>Property of Fictitious Play: </li></ul></ul><ul><ul><li>IF actions converge to a pure profile (or to a distribution) </li></ul></ul><ul><ul><li>THEN that is a Nash equilibrium of the game </li></ul></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 33. Learning the equilibrium <ul><ul><li>Banks adapt actions over time, </li></ul></ul><ul><ul><li>on the basis of experience (“Fictitious play”) </li></ul></ul><ul><ul><li>up to equilibrium. </li></ul></ul><ul><li>In our case, actions do converge </li></ul><ul><ul><li>We show (analytically) that given cost parameters, </li></ul></ul><ul><ul><li>all possible equilibria feature same average liquidity </li></ul></ul><ul><ul><li>So we can speak about “ the equilibrium liquidity demand” </li></ul></ul><ul><ul><li>Property of Fictitious Play: </li></ul></ul><ul><ul><li>IF actions converge to a pure profile (or to a distribution) </li></ul></ul><ul><ul><li>THEN that is a Nash equilibrium of the game </li></ul></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 34. Total costs price of delays = 1 price of delays = 2 price of delays = 5 price of delays = 20 cost, i cost, i cost, i cost, i funds committed by i funds committed by i funds committed by i funds committed by i funds committed by <j> funds committed by others funds committed by others funds committed by others funds committed by others Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 35. Base case results (15 banks) <ul><li>Banks (naturally) use more liquidity when delay price is high </li></ul><ul><li>At price parity, banks commit exactly 1 unit </li></ul><ul><li>The amount used increases by 13-15 units per bank (200-230 for the system) for each 10-fold increase in the price of delays </li></ul><ul><li>Banks will practically not commit over 49 units </li></ul>price of delays funds committed delays funds committed by i 10-fold decrease for each ~20 units of liquidity Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 36. Efficiency <ul><li>The outcome is not efficient </li></ul><ul><li>Higher levels of liquidity would yield overall lower costs </li></ul>price of delays price of delays liquidity costs orange = best non-equilibrium common action Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 37. System size, fixed turnover by bank <ul><li>Banks post more liquidity for a given payment volume, the more other banks there are in the network </li></ul><ul><li>Due to higher variation in the time to receive your own funds back </li></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 38. System size, fixed total turnover <ul><li>Concentrated systems are more liquidity efficient </li></ul><ul><li>Smaller number of banks -> higher value of payments per bank -> economies of scale </li></ul>Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 39. <ul><li>One bank can receive, but cannot send for first half of the day (liquidity sink) </li></ul><ul><li>Delays of “non-incident” banks are increased </li></ul><ul><li>More so, when liquidity is scarce </li></ul><ul><li>We expect banks to choose in equilibrium a higher level of liquidity </li></ul><ul><ul><li>e.g. (with delay cost 4) </li></ul></ul><ul><ul><li>if others choose 14, in normal circumstances I should choose 10, in case of an incident 14 </li></ul></ul>increase in delays example of changed behavior cost, i funds committed by i funds committed by <j> funds committed by i increase in delays for i (0,1) Operational incident 1 Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 40. Operational incident 2 <ul><li>With low delay cost, only small difference </li></ul><ul><li>As delays get costlier, more liquidity is used </li></ul><ul><li>At extremely high delay cost, adding funds does not help </li></ul>price of delays funds committed by i Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1
- 41. <ul><li>We developed a model with endogenous decisions by banks on their level of funding </li></ul><ul><li>We investigated the game with more “realistic” costs from settlement than analytical game theoretic models </li></ul><ul><li>The game collapses to “me vs. others” as only the aggregate behavior of others is relevant. The type of the game depends on model parameters (system size and delay cost) </li></ul><ul><li>Equilibrium </li></ul><ul><ul><li>not a social optimum </li></ul></ul><ul><ul><li>more participants, fewer payments par bank and higher delay costs more liquidity </li></ul></ul><ul><li>Operational incident impact on liquidity holdings is ambiguous </li></ul><ul><ul><li>payoffs are not improved in equilibrium </li></ul></ul><ul><li>Model being extended to account for alternative delay cost specifications, and heterogeneous banks -> policy purposes of Bank of England </li></ul>Conclusions Social Optimum 11 Size I 12 Size II 13 Conclusions 16 Incident II 15 Incident I 14 Base case 10 Total costs 9 Learning 8 The game 7 Algorithm II 6 Algorithm 5 Model 4 Literature 3 Liquidity 2 Overview 1

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