Using agent-based network models to assess financial
contagion
Christoffer Kok
European Central Bank
December 4, 2014
DISCLAIMER: This presentation should not be reported as representing
the views of the European Central Bank (ECB). The views expressed are
those of the authors and do not necessarily reflect those of the ECB
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
1 / 28
Motivation
A key finding in the literature is that the network structure through
which banks are connected to each other is an important factor
driving contagion risk
However, so far, little is known about how financial networks are
formed and about their sensitivity to changes in key bank parameters
and how the many different layers of bank networks affect each other
Basic notion
Interbank network structures, and hence the risk of contagion, are influenced
by banks’ optimising behaviour subject to regulatory (and other) constraints
Understanding how interbank networks emerge can be critical to controlling
and mitigating contagion risk
Changes in macro-prudential policy parameters will affect the contagion risk
because of their impact on banks’ asset allocation and interbank funding
decisions
Implies that well-tailored macro-prudential policy can help reduce interbank
contagion risk by making network structures more resilient
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
2 / 28
Outline
Modeling framework – randomised interbank networks
1
Literature
2
Model descriptions
3
Policy applications
4
Conclusion
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
3 / 28
Literature – general financial networks
Interbank market may (in normal times) act as a shock absorber and peer monitoring
mechanism (see e.g. Bhattacharya and Gale, 1997; Flannery, 1996; Rochet and Tirole,
1996)
But interbank market can also be a source of contagion (Allen and Gale, 2000; Nier et al.,
2008; Allen and Babus, 2009)
Empirical studies using overnight interbank transactions data at national level (Furfine,
1999; Upper and Worms, 2004; Boss et al., 2004; Van Lelyveld and Liedorp,2006;
Soramaki et al., 2007)
But widespread use of entropy measures – too much averaging of the tail risk effects
which may underestimate true contagion risk (Mistrulli, 2005)
Complex network analysis points to robust-yet-fragile character of many networks that
result in knife–edge properties where shocks to particular nodes can have systemic effects
(Nier et al., 2007; Iori et al., 2008; Georg, 2011; Roukny et al., 2013; Battiston and
Caldarelli, 2013)
Not explaining how interbank network emerges and how reacts to market conditions
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
4 / 28
Literature – towards network formation
Networks in other research areas: game theory of Jackson and Wolinsky (1996)
Extensions in finance – exogenous networks: game theory – optimal responses of banks to
shocks to incentives to lend Cohen-Cole et al. (2011); Bluhm et al. 2013. Acemoglu et al.
(2013): dealing with social inefficiency of financial networks; Georg (2011) models
interbank exposures as residuals of banks’ investment activities (but networks simply
drawn from a distribution)
Castiglionesi and Lavarro (2011) propose Nash equilibrium of networks maximising the
investors payoff which depends of the interbank neighbourhood of banks in which an
investor allocated funds (network independent of banks’ decisions)
Agent-based approach to address overly complex equilibria – Markose (2012); Grasselli
(2013)
Matching (Chen and Song, 2013) and price formation (Eisenschmidt and Tapking, 2009)
⇒ mechanisms important for us
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
5 / 28
Model descriptions
Modeling framework – randomised interbank networks
Present three recent papers relying on “agent-based” modelling
frameworks
...which impose certain behavioural assumptions on the banks and
other agents subject to pre-defined budget (and regulatory)
constraints
1
2
3
Halaj and Kok (2014), “Modelling emergence of the interbank
networks”, Quantitative Finance (forthcoming)
Halaj, Kochańska and Kok (2014), “Emergence of EU corporate
lending network”, working paper
Montagna and Kok (2013), “Multi-layered interbank network for
assessing systemic risk”, Kiel Working Papers No. 1873
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
6 / 28
Starting point: Simulated network model
Based on Halaj and Kok (2013), “Assessing interbank contagion
using simulated networks”, Computational Management Science
10(2-3), 157-186
3 steps
1
Probability map that a bank makes an interbank placement to another
bank in the system (mapping using EBA disclosures on exposures)
2
An iterative procedure to generate interbank networks by randomly
picking a link between banks and accepting it with probability taken
from the probability map
3
The algorithm of clearing payments proposed by Eisenberg and Noe
(2001) on the interbank market applying two versions: without and
(modified) with “fire sales” mechanism
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
7 / 28
More advanced approach – Endogenous networks [Halaj
and Kok (2014)]
Network formation modelled in 4 consecutive rounds that are repeated
until (almost) all interbank assets of a predefined volume are invested
4 rounds
1
Banks makes investment offers to other banks drawn from a probability
map: offers based on optimisation of their interbank asset structures
2
Banks formulate their preferred structure of interbank funding from banks
drawn in round 1: based on the diversification of the funding (rollover) risk
3
Banks enter negotiation phase: bargaining game in order to try to match the
preferred allocation of the interbank assets and the preferred structure of
interbank funding
4
Banks reconsider their pricing offers: banks with open funding gap
incrementally adjust their offers of interest payments on new interbank
deposits
At each step, assets are “matched” with liabilities incrementally
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
8 / 28
Prerequisites [HK (2014)]
(banks) N banks in the system
(interbank exposures) Let Lij denotes the interbank placement of
bank j in bank i.
(capital position) total capital e and capital e I ≤ e allocated to the
interbank assets; risk weights ω of interbank exposures.
(CVA) Exposure Lij requires to deduct γi Lij from capital eiI , for γi
being a bank-specific CVA factor, to account for the market based
assessment of the credit risk related with bank i.
(probability map P) of interbank connections drawn from P allowing
for capturing possible customer relationship between banks. Each
bank j draws its counterparties Bjk ⊂ N/{j}, enlarging the set at each
step k: B̄jk+1 = B̄jk ∪ Bjk+1 .
(matching) at each stepPk incremental matching of assets and
liabilities: ājk = ājk−1 − i Lkij , where Lk is a matrix of placements at
step k
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
9 / 28
1st round – Investment of interbank assets [HK (2014)]
General idea of banks’ optimising behaviour
Assumption: each bank maximises return from interbank portfolio adjusted
by risk related to interest rates and counterparts’ defaults (with a
predefined risk aversion parameter) and taking into account customer
relationship, i.e. a drawn sample of banks
Each bank maximises the following function of its interbank exposure
breakdown:
X
>
J(L1j , . . . , LNj ) =
ri Lij − κj (σ ∗ L>
(1)
·j ) Q(σ ∗ L·j )
i∈B̄jk
Outcome: a matrix of exposures LI ,k , whereby optimisation subject to
constraints...
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
10 / 28
...Constraints of the admissible set of strategies [HK
(2014)]
The maximisation is subject to some feasibility and capital constraints.
P
k
0
1 budget constraint –
j|j6=i Lij = āj and Ljj = 0, for aj = aj being
exogenously determined;
2 counterpart’s size constraint – L ≤ l¯k ;
ij
i
P
k
I
>
3 capital constraint –
i|i6=j ωi (Lij + Lij ) ≤ ej − γ (L̄·j + L·j );
4
large exposure limit constraint – Lij ≤ χej .
What if the constraints are too stringent for a bank j? ⇒ bank j reduces
its interbank lending and (technically) the optimisation is solved for ājk
replaced by āik − 2∆āik , āik − 3∆āik ,... until āik − ki ∆āik gives a feasible set
of constraints
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
11 / 28
2nd round – funding diversification [HK (2014)]
Diversification risk gauged by default risk
0
with probability pj
Xj : =
1 with probability 1 − pj
(2)
Remark: pj s are risky (variance based on observed time series of CDS
spreads)
For a covariance matrix D̄X2 of X , the optimised funding risk is measured
F (Lki1 , . . . , LkiN ) = κF [Lki1 . . . LkiN ]D̄X2 [Lki1 . . . LkiN ]>
(3)
Outcome: a matrix of interbank deposits LF ,k , whereby optimisation on
the admissible set:
k
k
k
AFi : = {y ∈ RN
+ |j ∈ B̄j ⇒ yj ≤ āj and j 6∈ B̄j ⇒ yj = 0}.
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
12 / 28
Data
Sample of 80 EU banks (based on EBA stress test disclosures)
Bankscope data on TAi , interbank borrowing and lending (l and a),
customer loans Li , securities Si and capital ei
Probability map based on 2011 EBA disclosures on geographical
breakdown of banks’ portfolios
Risk weights for interbank lending are set to 20%, while risk weights
for loans and securities are based on average figures across banks
(due to data gap)
3-month money market rates and bank individual CDS spread
(country averages when not available) were used to approximate the
expected return and riskiness of interbank placements (Note: internal
capital allocated to IB portfolio based on Credit Valuation
Adjustment concept)
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
13 / 28
Halaj, Kochańska and Kok (2014): Extension of HK
(2014) to cover bank-firm linkages
Sampling of the bank-firm network:
Observed nodes (banks + non-bank corporate firms) incl. +500
generated banks and 900 publicly listed firms
I
generated banks: based on the total assets and proportional allocation
of other attributes based on country averages
Lending relationship:
I
{bank}–{firm}: based on national Credit Register data
F
F
I
I
→ out-degree distribution (for each NACE sector) → the cardinal
number of set Bjk of firms k to which a bank j grants loans is
constrained by a number mj drawn from the out-degree distribution,
i.e. #Bjk ≤ mj
→ probability that a bank in a given country lends to a firm from a
given country and a given (NACE) sector
{EBA sample bank}–{EBA sample bank}: EBA disclosures
{small bank}–{EBA sample bank}: arbitrary [small] probability of
connection to other domestic banks (= 0.01)
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
14 / 28
The bank-firm network [HKK (2014)]
Network of non-bank corporate borrowing
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Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
15 / 28
Multi-Layered Financial System [Montagna and Kok
(2013)]
We include 3 layers:
1
Layer l1 : long-term bilateral
exposures (interbank
counterparty risk)
2
Layer l2 : short-term bilateral
exposures (funding risk)
3
Layer l3 : portfolios overlap
(liquidity risk)
Each layer characterized by: →
its own time-scale;
→ its own contagion mechanism;
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
16 / 28
The nodes : Banks Balance Sheets [MK (2013)]
Banks Balance sheets:
ai = ei + lil + lis + ci + oia
li = bil + bis + di + oil
ai = li + eqi
Banks have to fulfill (in each period t = 0, 1, . . . ) two constraints:
γi ≥ γ̄
ci ≥ β(di + bis )
Banks have two ways to increase their liquidity and risk-weighted CAR:
Withdrawing liquidity from the short-term interbank market
Selling securities
Since the first way is cheapest (price adjustments will reduce the capital),
we assume it is the best choice for the banks
Securities prices (µ = 1, 2, · · · , M):
(
)
PN
i
−α
·
sell
µ
µ
i
pµ = pµ0 · exp
PN i
s
i µ
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
17 / 28
Network Generator [MK (2013)]
Layers l1 and l2 : we generate interbank networks according to a
probability matrix P (based on Halaj and Kok (2013)):
1 Extract a bank i from the sample, i = 1, 2, · · · , N;
2 Generate link according to P: P
ij is the probability for the link i → j
to exist;
3 A random number ∈ [0, 1] indicates the percentage of bank i IBA is
deposited in bank j IBL (NOTE: values are truncated to limit the
exposures);
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
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Network Generator [MK (2013)]
Layer l3 : we generate random securities portfolios1 :
1
We decide the number M of securities in the system;
2
We generate a random bipartite network between N banks and M
securities: each link has the same probability p to exist;
3
All the out-coming links have the same weight;
NOTE: given the bipartite network, it is easy to build up the network of
the overlapping portfolios:
"
"
##
M
X
sjµ
siµ
3
· max 1; µ
Wij =
sjtot
sj
µ=1
1
No data available for proper calibration.
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
19 / 28
Model dynamics [MK (2013)]
Simulations:
Generate the multi-layered financial structure;
Shock the system;
Banks suffer losses from LT IB exposures;
Banks decide the amount to roll-over on the ST IB market;
Banks sell securities to (i) fulfill requirements and (ii) pay back ST
creditors;
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
20 / 28
Applications – policy implications [HK (2014)]
x-axis: Large exposure (LE)
thresholds (in per cent)
Figure 1: Topological measures of networks
emerging under different large exposure thresholds
y-axis: Topological network
measures
More stringent LE limits
could lead to adjustments
in banks’ network
connections
On average, the number of
network connections
increase when LE limits are
lowered
...and lowers the degree of
concentration
(“betweenness”)
average number of links (left-hand scale)
betweenness (right-hand scale)
23.0%
1.5%
22.5%
1.4%
22.0%
1.3%
21.5%
1.2%
21.0%
1.1%
20.5%
1.0%
10%
15%
20%
25%
Source: own calculations
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
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Applications – policy implications [HK (2014)]
x-axis: CDS spread (in bps)
y-axis: difference of CAR
after adverse stress testing
shock between LE=15%
and LE=25% regime (in
pp, positive number means
that by lowering LE limit
contagion losses rise)
Figure 2: Counterparty credit quality and the
impact of LE limits on the losses incurred due to
contagion
1
0.5
0
The size of a circle is
proportional to bank’s total
assets
−0.5
Stringent LE limits overall
tend to lower contagion
risk
−1.5
Benefits for the safest part
of the banking system
−1
−2
−2.5
−3
0
500
1000
1500
2000
2500
Source: own calculations
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
22 / 28
Applications – policy implications [HKK (2014)]
Figure 3: Contagion impact on banks: uniform shock to PDs of all sectors
A
B
C
D
E
F
G
H
I
J
K
L
AT D E D E D E D E D E D E D E D E D E D E D E D E D E D E D K D K ES ES FI FR G B G B G B G R IT IT IT N L N O PL SI
Contagion mechanism – cascade triggered by a deterioration of credit quality
of loan portfolios to companies in a given NACE sector imposing 5% PD
and 50% LGD
“Spectral” graph shows impact of the contagion losses of 500+ banks (the
darker the bar, the higher the fraction of capital wiped out by contagion)
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
23 / 28
Applications – policy implications [HKK (2014)]
Figure 4: Feedback to the non-bank corporate sector
50%
60%
70%
80%
90%
BE_D
DE_B
DE_C
DE_D
DE_H
DE_J
DK_B
DK_C
DK_D
DK_F
DK_G
DK_H
DK_J
DK_L
EE_C
EE_E
EE_H
FI_B
FI_C
FI_G
FI_H
FI_I
FI_J
GB_B
GB_C
GB_L
GR_F
GR_H
HU_C
IE_B
LU_C
NO_B
PL_B
SE_C
SI_C
100%
Defaults of banks triggered by banks failing to pay back their obligations as
a result of losses related to decreasing credit quality of manufacturing loan
portfolio in Germany
IB contagion may feedback to the corporate sector via lending relationships;
i.e. banks may cut back on lending when hit by shock (e.g. losing +30 p.c.
of initial capital)
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
24 / 28
Applications – policy implications [HKK (2014)]
Figure 5: Second round defaults of banks in the cascade of contagion spreading
triggered by losses in the portfolio of loans to the manufacturing sector in DE
3%
5%
7%
9%
11%
13%
15%
17%
19%
21%
23%
25%
AT DE DE DE DE DE DE DE DE DE DE DE DE DE DE DK DK ES ES FI FR GB GB GB GR IT IT IT NL NO PL SI
LE limits (or risk weights on capital allocated to IB lending) can mitigate
contagion losses
Example shows impact on banks’ capital ratio from default on loans to
German manufacturing firms for different LE limits
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
25 / 28
Applications – policy implications [MK (2013)]
Illustration of propagation mechanism across different layers: blue links →
LT exposures; green links → ST exposures; cyan links → common
exposures
NON LINEARITIES: multidimensional paths amplify the range of
propagation of the initial shock!
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
26 / 28
Applications – policy implications [MK (2013)]
( N = 50, M = 30, γ = 8%, β = 5% )
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
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Conclusions
Endogenous interbank networks give an important insight into the
role of banks’ investment and funding strategies in shaping the
interbank market.
The simple, mechanistic cascade models are too simplistic in
assuming that banks do not react to actions of other interbank
participants and market conditions.
In the proposed frameworks, we are able to analyse different policy
measures addressing the systemic risk – their ultimate impact on the
market structure and efficiency in reducing the contagion risk
Illustrate clear non-linearities in contagion effects when taking into
account banks’ endogenous responses and multi-layered interlinkages
Christoffer Kok (ECB)
Agent-based network models
December 4, 2014
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