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An agent based model of payment systems - Talk at Norges Bank 24 March 2011
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An agent based model of payment systems - Talk at Norges Bank 24 March 2011


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The paper builds an agent based model where banks choose the amount of liquidity to settle a given flow of payments. The paper pays special attention to a realistic settlement process with complex …

The paper builds an agent based model where banks choose the amount of liquidity to settle a given flow of payments. The paper pays special attention to a realistic settlement process with complex dynamics and studies the equilibrium level of liquidity that is a result of the game between the banks. The paper investigates liquidity usage with various system sizes and volumes, and under different liquidity cost parameters.

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  • 1. An agent-based model of payment systems
    Marco GalbiatiKimmo SoramäkiNorges Bank, Oslo 24 March 2011
  • 2. Interbank Payment Systems
    Source: Bech, Preisig, Soramäki (2008), FRBNY Economic Policy Review / September 2008
  • 3. The values are large
    ~1939 tr
    ~194 tr
    ~120 tr
    ~5 tr
  • 4. The liquidity is costly
    Luckily central banks provide free intraday credit against collateral
    But collateral has a cost (explicit or opportunity)
    The cost is perhaps around 8-16bp (research on eMid data, banks internal pricing)
    Higher during crisis: Baglioniand Monticini (2010) say it was 10 times as high in 2007
    And liquidity can be recycled during the day
    Provide a free source of liquidity, amount to 90% of liquidity usage
    Timing is uncertain, depends on other banks
    Banks manage intraday liquidity
    Synchronize payments with incoming liquidity -> to save on external liquidity
    Delaying payments is also costly: sanctions, service levels, agreements, processing schedules
    But it’s a hard problem…
  • 5. … payment arrival uncertain
    Summed over the network, instructions arrive at a steady rate
    When liquidity is high payments are submitted promptly and banks process payments independently of each other
    Source: Beyeler, Glass, Bech and Soramäki (2007), Physica A, 384-2, pp 693-718.
  • 6. … liquidity dynamics are complex
    Reducing liquidity leads to episodes of congestion when queues build, and cascades of settlement activity when incoming payments allow banks to work off queues. Payment processing becomes coupled across the network
  • 7. … and playing a game
    In the collateralized credit game, early settlement (morning, morning) is a unique equilibrium if the opportunity cost of collateral is less than the cost of delaying(C < D). This outcome is efficient.
    Conversely, late settlement (afternoon, afternoon) is a unique equilibrium if C > D, and the game is a prisoner’s dilemma. Late settlement is inefficient.
    Source: Bech and Garratt (2006), Journal of Economic Theory and Bech (2008), FRBNY Economic Policy Review
  • 8. Objective and approach
    To study liquidity demand in a payment system
    The paper draws on:
    Game theoretic models that incorporate behaviour: Angelini (1998), Chakravorti (2000), Bech and Garratt (2006)
    Payment system simulations that bring more realism to complex dynamics and interactions in the settlement process: Koponen and Soramäki (1998), Leinonen, ed. (2005, 2007), Devriese and Mitchell (2005)
    Agent based models for interaction between banks in a continuous game: Tesfatsion (2001), Axelrod and Tesfatsion(2006), Herbert (2007)
  • 9. Model
    Consists of a sequence of days
    At the beginning of each day
    Identical banks simultaneously choose funds to commit that maximize expected payoff
    Payoff depends on own and others’ choices
    Consists of liquidity and delay cost component:
    Payment simulation model (RTGS),
    Poisson arrival of unit size payments, complete network
    Banks follow simple rule: settle payment if liquidity, queue if not
    At end
    Banks receive payoff
    Banks update beliefs on others’ choices
    Banks update payoff for own/others liquidity combinations
    After enough days banks converge to stable choices and beliefs
  • 10. Aggregationproperty
    Turns out, we can simplify the problem
    In the long run (within a day), the distribution of liquidity does not matter, only its total level does
    For the short run, bank’sown liquidity mattersmore than total level
    Banks maintain beliefs ontotal liquidity, and a payoffmatrix on own liquidity and total liquidity
    Equal distribution
    Highly skewed
  • 11. Whyis this?
    The system can be understood as a Markov process
    Transition probabilities for moving liquidity from i to j
    Converges to a steady state (assuming no liquidity constraint)
    Holds better
    When convergence is fast (symmetric and complete network fastest)
    The ratio of payments to liquidity is high
  • 12. Convexity property of delays
    Delays are a convex function of own and others’ liquidity
    A upward shift of the curve (more liquidity) reduces less delays (red area) the higher the curve is
  • 13. … same “empirically”
    Delays as a function of own and others’ liquidity (N=13)
  • 14. Costs
    Recall, costs depend on own liquidity and others liquidity -> which jointly determine delays
    Red = high price for liquidity, Blue = low price for liquidity
  • 15. Model parameters
    2-30 banks, 400-6000 payments
    Base case (15/3000) “looks like” CHAPS
    Enough payments for property 1 to hold
    Combinations explore different aspects of system size
    Liquidity choices (for each bank)
    From 0 (nothing settles)
    To 50 (virtually no delays)
    Price of liquidity
    From free
    Until equilibrium liquidity is 0
    Price of delays is normalized to 1
  • 16. Liquidity demand
    With “CHAPS” inferred liquidity price
    -> Banks provide 3-10% less than planner
    -> The cost might be ~35-75bp
  • 17. More banks of same size
    In a larger system liquidity gets lost…
    Variance in cascade length (and bank’s incoming payments) is increased
    Due to convexity, increasing variance increases expected value
    Delay per payment
  • 18. … increase liquidity consumption
  • 19. … and increase costs
  • 20. Concentration increases efficiency
    Same volume distributed across different number of banks
    “Pooling effect”, economies of scale
    Argument for tiered structures
  • 21. … and reduces costs
  • 22. Summing up
    The paper put together “realistic” liquidity dynamics with bank behavior
    Derived liquidity demand function
    Banks underprovide liquidity
    Explored system size
    Concentrating payments among a few banks is more efficient -> Tiering
    Economies of scale in “pooling effect” but not in “open access” sense
  • 23. Liquidity saving mechanisms
    Galbiati and Soramäki (2010), BoE Working Paper No. 400
    Extends the model with
    Payment urgency: [0,1]
    Two streams of settlement:
    RTGS – as in Galbiati and Soramäki (2008)
    LSM – a partial netting algorithm as in Bech and Soramäki (2002)
    An additional choice variable: urgency theshold above which payment is submitted to RTGS
    Instead of “Fictitious play” uses a pre-calculated pay-off matrix and standard methods to solve it
    Focus on symmetric equilibria
  • 24. Withholding vs two stream LSM
    Withholding low priority payments
    Submitting low priority payments to LSM stream
    Priority threshold
  • 25. Dynamics
    Other banks use mostly LSM
    Other banks use mostly RTGS
    Other banks post little liquidity
    Other banks post much liquidity
  • 26. Equilibria
    Low price of liquidity ->
    -> High price of liquidity
  • 27. Mainresults
    Planner virtually never uses LSM stream
    Banks mix RTGS with LSM for a wide liq. price range
    Banks generally underprovide liquidity
    Banks are better off with LSM than without (less far from planner’s choice)
    Bad equilibria are possible -> co-ordination may be needed when introducing them
  • 28. Thank you!
    More information at