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

Annual value (euros) Liquidity need Age of the universe (hours)0.00E+00

5.00E+14

1.00E+15

1.50E+15

2.00E+15

2.50E+15

Annual value (euros) Liquidity need0.00E+00

5.00E+14

1.00E+15

1.50E+15

2.00E+15

2.50E+15

Annual value (euros)0.00E+00

5.00E+14

1.00E+15

1.50E+15

2.00E+15

2.50E+15

Annual value (euros) Liquidity need Age of the universe (days)0.00E+00

5.00E+14

1.00E+15

1.50E+15

2.00E+15

2.50E+15

~1939 tr

~194 tr ~120 tr ~5 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 0 0 0

Time

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 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

Pay

men

ts

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

Pay

men

ts

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

Pay

men

ts

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 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 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 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

Fre

qu

ency

1

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

Del

ay p

er p

aym

ent

… 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!

More information at

www.financialnetworkanalysis.com