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Banks on the verge of a crisis: phase transitions and hysteresis in banking systems

Tomaso Aste1,2

1 Department of Computer Science, University College London, 2 Systemic Risk Centre, London School of Economics and Political Sciences References: Annika Birch and TA "Systemic Losses Due to Counter Party Risk in a Stylized Banking System" Journal of Statistical Physics 156 (2014) 998 - 1024 Annika Birch, Zijun Liu & TA "A counterparty risk study for the UK banking system" ssrn.com/abstract=2599891, under submission T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 1 / 24

Phase transitions

Phase transition

Change in the system state

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 2 / 24

Phase transitions

Phase transition

Change in financial market structure across the banking crisis

1 8 / 1 2 / 2 2 4 / 1 2 / 2 2 2 7 / 1 2 / 2 4 2 6 / 1 2 / 2 6 2 6 / 1 2 / 2 8 2 8 / 1 2 / 2 1 2 8 / 1 2 / 2 1 2

ES(Ta, Tb): NYSE data set

18/12/2000 24/12/2002 27/12/2004 26/12/2006 26/12/2008 28/12/2010 28/12/2012 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

N Musmeci, TA, Tiziana Di Matteo, arXiv:1605.08908 (2016) T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 3 / 24

Phase transitions

What is a phase transition?

in Physics

• Change in the system state (liquid to solid) as consequence of a change

in the parameters (temperature, pressure)

• Change in the internal energy
• Change in the system entropy
• Emergence of collective properties
• Order parameter becomes finite (symmetry breaking)
• Appearence of long-range correlations near the transition point
• Appearance of “soft modes”

in General

• Change in the system state
• ?

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 4 / 24

Tipping Points

Tipping Point

the point at which small changes or incidents can cause large changes

• An object at a point of unstable equilibrium
• A rare phenomenon becoming rapidly more common
• The point at which a technology becomes dominant and the “winner

takes all”

• A change happening as a consequence of a small cause that cannot

be easly reverted

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 5 / 24

Tipping Points

Tipping Point

the point at which small changes or incidents can cause large changes

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 6 / 24

Tipping Points

Tipping Point

the point at which small changes or incidents can cause large changes

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 7 / 24

Tipping Points

Tipping Point

the point at which small changes or incidents can cause large changes

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 8 / 24

Simple stylized banking model

UK banking system data from BoE

We use form supervisory reports from Bank of England (BoE) for the years 2011, 2012 and 2013. Lending (unsecured, secured and undrawn) Holdings of equity and fixed-income securities (marketable securities) issued by banks; Credit default swaps (CDS) bought and sold Securities lending and borrowing (gross and net of collateral); Repo and reverse repo (gross and net of collateral); Derivatives exposures (with breakdown by type of derivative) UK banks have to report their 20 largest counterparties to the BoE semi-annually. If the top 20 does not have at least six UK-based counterparties, banks report exposures to up to six UK-based counterparties in addition to the top 20. Branches of foreign banking groups in the UK are not included. There are 176 UK banks reporting to BoE.

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 9 / 24

Simple stylized banking model

UK Banking Network

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 10 / 24

Simple stylized banking model

UK Banking Network

(a) In-degree (b) In-weight (c) Out-degree (d) Out-weight

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 11 / 24

Simple stylized banking model

Phase transition in a Stylized Banking System

We propose a model combing the balance sheet based model12, with the contagion model3 creating a stylized banking system4. We distinguish between normally operating banks and distressed banks: Si(t) =

I

1 if bank i is operating normally if bank i is distressed . We consider a system of N banks that borrow and deposit money into each-other though an interbank network

• 1P. Gai et al. (2007). In: Journal of Risk Finance 8.2, pp. 156–165.
• 2E. Nier et al. (2007). In: Journal of Economic Dynamics and Control 31.6,
• pp. 2033–2060.
• 3J. P. Solorzano-Margain et al. (2013). In: Computational Management Science,
• pp. 1–31.
• 4S. Heise et al. (2012). In: The European Physical Journal B 85.4, pp. 1–19.

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 12 / 24

Simple stylized banking model

Balance Sheet

Liabili&es Li(t) Assets Ai(t) Deposits

ˆ Li(t)

Interbank Borrowing  Li(t) = g j,i(t)

j=1 N

∑

Non Interbank Assets

(stocks + external assets)

ˆ A

i(t)

Capital: Ei(t)=Ai(t)-Li(t) Interbank Lending gi, j(t)

j=1 N

∑

S j(t)

aσ bσp(t)

Liabilities: Li(t) = sum of the bank’s customer deposits, ˆ Li(t), and interbank bor- rowings q

j gj,i.

Assets: Ai(t) = sum of non-interbank assets, ˆ Ai(t), interbank lending to non distressed banks q

j gi,j(t)Sj(t)

Bank’s capital: Ei(t) = Ai(t) − Li(t) (> 0)

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 13 / 24

Simple stylized banking model

Systemic risk

Liabili&es Li(t) Assets Ai(t) Deposits

ˆ Li(t)

Interbank Borrowing  Li(t) = g j,i(t)

j=1 N

∑

Non Interbank Assets

(stocks + external assets)

ˆ A

i(t)

Capital: Ei(t)=Ai(t)-Li(t) Interbank Lending gi, j(t)

j=1 N

∑

S j(t)

aσ bσp(t)

gi, jS j Si S j

Annika Birch and TA "Systemic Losses Due to Counter Party Risk in a Stylized Banking System" Journal of Statistical Physics 156 (2014) 998 - 1024 Annika Birch, Zijun Liu & TA "A counterparty risk study for the UK banking system" ssrn.com/abstract=2599891, under submission T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 14 / 24

Simple stylized banking model

Dynamics

Balance Sheet Equation: a bank operate normally if Ai(t) ≥ Li(t), it is in distress otherwise. The state of a bank at time t + 1 is Si(t + 1) =

I

1 if Ai(t) − Li(t) ≥ 0 if Ai(t) − Li(t) < 0 . The state of the system is associated with two main quantities: Non-Counterparty-dependent balance sheet Li(t)

¸ ˚˙ ˝ Liabilities

− ˆ Ai(t)

¸ ˚˙ ˝ Non-Counterparty-dependent assets

Counterparty-dependent assets:

q

j gi,jSj(t)

mean: aσ; Variance: σ2 mean: bσpt Homogeneous system: all banks same size and balance sheet quantities are randomly distributed with

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 15 / 24

Simple stylized banking model

Dynamics

Balance Sheet Equation: a bank operate normally if Ai(t) ≥ Li(t), it is in distress otherwise. The state of a bank at time t + 1 is Si(t + 1) =

I

1 if Ai(t) − Li(t) ≥ 0 if Ai(t) − Li(t) < 0 . The state of the system is associated with two main quantities: Non-Counterparty-dependent balance sheet Li(t)

¸ ˚˙ ˝ Liabilities

− ˆ Ai(t)

¸ ˚˙ ˝ Non-Counterparty-dependent assets

Counterparty-dependent assets:

q

j gi,jSj(t)

mean: aσ; Variance: σ2 mean: bσpt Homogeneous system: all banks same size and balance sheet quantities are randomly distributed with

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 15 / 24

Onset of a fragile state with irreversible dynamics

Fixed Point Solutions (Normal distribution)

pt+1 = Φ(bpt − a) p = Φ(bp − a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b= 7 a2= 1.96 a1= 5.04 p Φ(bp − a)

Small a (large non-interbank assets):

• nly one solution p = 1
• all banks functioning normally

Large a (small non-interbank assets):

• nly one solution p = 0
• all banks in distress

Intermediate a: three solutions

• one unstable and 2 stable solutions

The central fixed point is unstable and forms a barrier. The dynamics becomes irreversible.

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 16 / 24

Efficiency, Resistance, Failure and Recovery Cost

Efficiency, Resistance, Failure and Recovery

Let us consider a banking system put to stress by decreasing non-interbank assets (i.e. increasing a ) with interbank assets unchanged (i.e. b constant)

−10 −5 5 10 15 0.2 0.4 0.6 0.8 1

a

Fraction of surviving banks, p a = a2 ≈ 5.04 cost b = 15 b = 7 b = 0 a = a1 ≈ 1.96 Fraction of surviving banks as a function of a given b. Solid lines indicate stable fixed points, dotted lines unstable fixed points. For b = 7, starting from fully oper- ating banks (i.e. p0 = 1) an infinite avalanche can bring down the system when a2 ≈ 5.04. Once all banks are distressed, a needs to be lowered to a1 ≈ 1.96, in order for the system to return to a stable state. The differ- ence between a2 and a1 is the capi- tal to be injected into the system to reverse the state

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 17 / 24

Efficiency, Resistance, Failure and Recovery Cost

Efficiency, Resistance, Failure and Recovery

Let us consider a banking system put to stress by decreasing non-interbank assets (i.e. increasing a ) with interbank assets unchanged (i.e. b constant)

−10 −5 5 10 15 0.2 0.4 0.6 0.8 1

a

Fraction of surviving banks, p a = a2 ≈ 5.04 cost b = 15 b = 7 b = 0 a = a1 ≈ 1.96

Tipping Tipping Point Point!

Fraction of surviving banks as a function of a given b. Solid lines indicate stable fixed points, dotted lines unstable fixed points. For b = 7, starting from fully oper- ating banks (i.e. p0 = 1) an infinite avalanche can bring down the system when a2 ≈ 5.04. Once all banks are distressed, a needs to be lowered to a1 ≈ 1.96, in order for the system to return to a stable state. The differ- ence between a2 and a1 is the capi- tal to be injected into the system to reverse the state

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 17 / 24

Numerical results: robustness of analytical results

Interbank network effect: simulation results

−5 −4 −3 −2 −1 1 2 3 0.5 1

Normal Distribution

−5 −4 −3 −2 −1 1 2 3 0.5 1

Fraction of surivivng banks, p (µL − µA)/(σ2

A + σ2 L)1/2

Student t Distribution, Degrees of Freedom ν = 2 θ = 0.3 θ = 0.1 θ = 0.3 (B) (A) θ = 0.1 θ = 0.0 θ = 0.0

−5 −4 −3 −2 −1 1 2 3 0.5 1

Small World Network, c = 12, β = 0.1

−5 −4 −3 −2 −1 1 2 3 0.5 1

Fraction of surivivng banks, p (µL − µA)/(σ2

A + σ2 L)1/2

Core-Periphery Network, α = 0.75, m = 15 θ = 0.3 θ = 0.1 θ = 0.0 θ = 0.3 θ = 0.1 θ = 0.0 (A) (B) −4.2 −3.2 −2.2 −1.2 −0.2 0.8 1.8 0.01 0.02 0.03 0.04 (µL − µA)/(σ2

L + σ2 A)1/2

Connection probability, α 0.2 0.4 0.6 0.8

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 18 / 24

Heterogeneous System Calibrated with UK data

Heterogeneous system simulations calibration

The system consists of N = 175 banks. Assets Ai(0) for each bank are drawn from a distribution with mean µAi retrieved from the data and variance f AµAi with fA = 0.001 Liabilities Li(0) for each bank are drawn from a distribution with mean µLi retrieved from the data but multiplied by a factor f L µLi, variance is assumed f AµAi Normal distributions are assumed Interbank assets are retrieved from the data Interbank lending structure is retrieved from the data recovery rate is q = 0

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 19 / 24

Heterogeneous System Calibrated with UK data

Heterogeneous system simulations calibration

The system consists of N = 175 banks. Assets Ai(0) for each bank are drawn from a distribution with mean µAi retrieved from the data and variance f AµAi with fA = 0.001 Liabilities Li(0) for each bank are drawn from a distribution with mean µLi retrieved from the data but multiplied by a factor f L µLi, variance is assumed f AµAi Normal distributions are assumed Interbank assets are retrieved from the data Interbank lending structure is retrieved from the data recovery rate is q = 0

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 19 / 24

Heterogeneous System Calibrated with UK data

Results

Fraction of surviving banks as function of mean liabilities. Com- parison with homogeneous system model (mean field, MF) and sys- tem with no contagion (null, N)

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1

p 2011 H2

pS pN pMF 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.05 0.1 0.15

fL STD(pS)

MF:fL=1.0383 fL(Max(STD(pS)))=1.038

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1

p 2012 H2

p − simulation p − Null p − mean field 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.1 0.2

fL STD(p)

MF:fL=1.0495 fL(Max(STD(pS)))=1.042

2013

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1

p 2013 H1

p − simulation p − Null p − mean field 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.1 0.2

fL STD(p)

MF:fL=1.0498 fL(Max(STD(pS)))=1.042

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 20 / 24

Heterogeneous System Calibrated with UK data

Fraction of surviving banks as function of mean liabilities Results for different bank types: “Large Banks" (LB), “Building Soci- eties" (BS), “Investment Banks" (IB), “Oversea Banks" (OB) and “Other Commercial Banks" (CB)

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1

p 2011 H2

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.1 0.2

fL STD(p)

LB BS IB OB CB

fL(Max(STD(pS

LB)))=1.0390

fL(Max(STD(pS

BS)))=1.0380

fL(Max(STD(pS

IB)))=1.0130

fL(Max(STD(pS

OB)))=1.0380

fL(Max(STD(pS

CB)))=1.0330

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1

p 2012 H2

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.1 0.2

fL STD(p)

LB BS IB OB CB

fL(Max(STD(pS

CB)))=1.0420

fL(Max(STD(pS

OB)))=1.0430

fL(Max(STD(pS

IB)))=1.0420

fL(Max(STD(pS

BS)))=1.0420

fL(Max(STD(pS

LB)))=1.0420

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.5 1

p 2013 H1

0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.2 0.4

fL STD(p)

LB BS IB OB CB

fL(Max(STD(pS

LB)))=1.0430

fL(Max(STD(pS

BS)))=1.0420

fL(Max(STD(pS

IB)))=1.0420

fL(Max(STD(pS

CB)))=1.0430

fL(Max(STD(pS

OB)))=1.0420

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 21 / 24

Heterogeneous System Calibrated with UK data

Emergence of systemic failure

Fraction of surviving banks against the fraction of capital

blue crosses - 2007; black circles - 2012 θ fraction of interbank assets to total assets σ amplitude of fluctuations p fraction of surviving banks T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 22 / 24

Conclusions

Conclusions

a simple model from physics for a banking system where systemic fragility emerges in the fragile state a failure can trigger an avalanche that brings down the entire system Once the system fails it cannot go back to operating state without a recovery cost Numerical simulations show that the prediction of the homogeneous model are reproducing well the behavior of more realistic heterogeneous systems calibrated on BoE UK data tipping points phase transitions

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 23 / 24

Conclusions

Thank You

http://fincomp.cs.ucl.ac.uk/

References:

Annika Birch and TA "Systemic Losses Due to Counter Party Risk in a Stylized Banking System" Journal of Statistical Physics 156 (2014) 998 - 1024 Annika Birch, Zijun Liu∗ & TA "A counterparty risk study for the UK banking system" To be submitted

T Aste (UCL, SRC) Phase Transitions in Banking Systems CoSyDy, QMUL, 06/07/16 24 / 24