An agent-based model of payment systems
Marco GalbiatiKimmo Soramäki
Norges Bank, Oslo 24 March 2011
Interbank Payment Systems
Source: Bech, Preisig, Soramäki (2008), FRBNY Economic Policy Review / September 2008
The values are large
~1939 tr
~194 tr ~120 tr
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: Baglioni and 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…
0
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 0 0 0
1 2 0 0 0
1 4 0 0 0
1 6 0 0 0
1 8 0 0 0
2 0 0 0 0
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 1 8 0 0 2 00 0
Time
0
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 00 0
1 2 00 0
1 4 00 0
1 6 00 0
1 8 00 0
2 0 00 0
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 00 1 4 00 1 6 00 1 8 00 2 0 0 0
Time
PaymentSystem
When liquidity is high payments are submitted promptly and banks process payments independently of each other
Instructions Payments
Summed over the network, instructions arrive at a steady rate
… payment arrival uncertain
Liquidity
0
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 0 0 0
1 2 0 0 0
1 4 0 0 0
5 5 0 0 5 7 0 0 5 9 0 0 6 1 0 0
Instructions
Paym
ents
5 5 0 0
5 6 0 0
5 7 0 0
5 8 0 0
5 9 0 0
6 0 0 0
6 1 0 0
5 5 0 0 5 7 0 0 5 9 0 0 6 1 0 0
Instructions
Paym
ents
Source: Beyeler, Glass, Bech and Soramäki (2007), Physica A, 384-2, pp 693-718.
0
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 0 0 0
1 2 0 0 0
1 4 0 0 0
5 5 0 0 5 7 0 0 5 9 0 0 6 1 0 0
Instructions
Paym
ents
0
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 0 0 0
1 2 0 0 0
1 4 0 0 0
1 6 0 0 0
1 8 0 0 0
2 0 0 0 0
0 2 00 4 0 0 6 00 8 00 1 0 0 0 1 2 0 0 1 4 0 0 1 6 00 1 8 0 0 2 0 0 0
Time
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
PaymentSystem
Instructions Payments
… liquidity dynamics are complex
0
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 00 0
1 2 00 0
1 4 00 0
1 6 00 0
1 8 00 0
2 0 00 0
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 00 1 4 00 1 6 00 1 8 00 2 0 0 0
Time
1 E -0 4
0 .0 0 1
0 .0 1
0 .1
1
1 1 0 1 0 0 1 0 0 0 1 0 0 0 0
Cascade Length
Freq
uenc
y1
1
Liquidity
… and playing a game
Source: Bech and Garratt (2006), Journal of Economic Theory and Bech (2008), FRBNY Economic Policy Review
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.
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)
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:
• During– 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
Aggregation property
• 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
Why is 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
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
time
bala
nce
… same “empirically”
Delays as a function of own and others’ liquidity (N=13)
Costs
• Recall, costs depend on own liquidity and others liquidity -> which jointly determine delays
• Red = high price for liquidity, Blue = low price for liquidity
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
Liquidity demand
With “CHAPS” inferred liquidity price
-> Banks provide 3-10% less than planner
-> The cost might be ~35-75bp
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
Dela
y pe
r pay
men
t
… increase liquidity consumption
… and increase costs
Concentration increases efficiency
Same volume distributed across different number of banks
“Pooling effect”, economies of scale
Argument for tiered structures
… and reduces costs
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
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
Withholding vs two stream LSM
Withholding low priority payments
Submitting low priority payments to LSM stream
Priority threshold
Dynamics
Other banks post little liquidity
Other banks post much
liquidity
Other banks use mostly RTGS
Other banks use mostly LSM
-> High price of liquidity
Equilibria
threshold
liqui
dity
Low price of liquidity ->
Main results• 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
Thank you!
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