Agent-based Networks of Corporate Lending
Grzegorz Halaj
European Central Bank
23/09/2014
Based on research with U. Kocha´
nska (ECB) and Ch. Kok (ECB)
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
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
1 / 23
Motivation
Recent financial crisis: loss of trust on the interbank market; concerns
about failure of one of the key players spreading contagion; small
shocks with detrimental effects
A response from regulators: measures to mitigate the risk ⇒ higher
capital standards + reducing bilateral exposures
I
I
I
I
Large Exposure limits;
Credit Valuation Adjustment to unlock the risk in OTC exposures and
immediately reflect it in the capital
Standard settlement practices (CCP framework)
...but usually only interbank market modelled → a large part of the
network is neglected
Our aim:
I
fill the gap in the literature to improve understanding of:
F
F
linkages between banks and the real economy (non-bank corporate
sector)
risk stemming from interconnectedness
Approach: modelling of banks’ reactions to these measures and to the
changing macroeconomic environment with links to corp sector
(combining risk/return trade-offs, funding conditions...)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
2 / 23
Outline
Modeling framework – agent-based interbank+corporate networks
Four round model – endogenous formation
Interbank augmented by non-bank corporate sector (called: firms)
1
2
3
4
offers of interbank placements based on individual optimisation of
interbank asset structures
funding diversification
negotiation phase: matching offers and preferred funding structure in a
bargaining game
price (i.e. interest rate) adjustment (if demand 6= supply)
Scope for application
stress tests and dynamic balance sheet tool
assessing network effects of credit provision to the real economy
(shocks from corporate sector)
parametrisation of LE and concentration limits (so far only for
interbank)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
3 / 23
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;
Sor amaki 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)
Not explaining how interbank network emerges and how reacts to market conditions
To our knowledge no examples of financial networks incorporating links to the real
economy in a “network fashion”
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
4 / 23
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 (2011);
Bluhm, 2013. Acemoglu (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)
Jackson and Watts (2002) combine stochastic games and matching
problems to study general principles of network formation in
economics
Agent-based approach to address overly complex equilibria – Markose
(2012); Grasselli (2013)
Matching (Chen, 2013); (Duffie and Sun 2012) and price formation
(Eisenschmidt, 2009) ⇒ mechanisms important for us
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
5 / 23
Formation of the lending network – Endogenous networks
The aim of the project is to:
1
understand foundations of the topology of lending networks in the
economy
and (the next steps)
2
analyse sensitivity of the interbank network structures to the
heterogeneity of banks (in terms of size of balance sheet, capital
position, general profitability, counterparty credit risk) and the
changes of market and bank specific risk parameters
3
project the evolution of the lending network (given a macro scenario)
4
assess effectiveness of rule designed to mitigate systemic risk on the
interbank system (esp. pertaining to capital requirements, size and
diversity of interbank exposures)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
6 / 23
4 round model – outline
The following 4 rounds are repeated until 'all interbank assets of a
predefined volume are invested (separate for interbank and bank-firm
network)
1
Firms make loan offers to other banks and firms which are drawn
from a probability map: offers based on optimisation of their
interbank asset structures and corporate lending portfolio
2
Firms formulate their preferred structure of interbank (banks) and
bank (firms) funding from banks drawn in round 1: based on the
diversification of the funding (rollover) risk
3
Firms enter negotiation phase: bargaining game in order to try to
match the preferred allocation of the assets and the preferred
structure of interbank (bank) funding
4
Firms reconsider their pricing offers: firms with open funding gap
incrementally adjust their offers of interest payments on new loan
(optional feature, not used so far in the exposition)
At each step, assets are “matched” with liabilities incrementally
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
7 / 23
Figure 1: The sequential four round procedure of the interbank formation
(formation of bank-firm links separate but analogous)
INITIAL PARAMETERS
Aggregate IB lending / borrowing, capital, RWA, CDS spreads, market interest rates
4 ROUNDS
1) OPTIMISATION
Preferred funding
structure
REPEATED STEPS
Next step
2) OPTIMISATION
Preferred asset
structure
Partial allocation
3) BILATERAL GAMES
Bargaining game
NEW PLACEMENTS
Part of unallocated IB assets
placed in banks as deposits
creating IB linkages
STEPS Repeated
until all IB assets
are allocated
Full allocation
Grzegorz Halaj (ECB)
4) PRICE
Interest rate adjustment
Unallocated IB assets and liabilities
IB Network Completed
Fin. Risk & Network Theory, Cambridge
23/09/2014
8 / 23
Table 1: Overview of data inputs
Item
Banks
Non-financial corporations
Banks
Banks
Banks
Non-financial corporations
Non-financial corporations
Lending relationship
Interest rates on
loans by size and
country
Expected default frequencies
Description
Sources
Coverage
As identified in 2011 EBA Disclosures; 80 banks from EU countries.
+ 500 randomly generated banks based on TA
Members of the benchmark equity indices in the countries covered
by EBA Disclosures and Halaj and Kok (2014); total 700 firms
EBA, Halaj and Kok (2014)
+ Bankscope
Bloomberg and ECB
Attributes
Total assets, IB assets, securities, securities MtM, equity, CT1 capital, IB liabilities
Loans to non-fin. corporations: calculated by using avg. country
ratio of such loans to TA based on the ECB (MFI) balance sheet
dataset
Economic activity code (NACE), CDS of senior debt with 5 maturity,
and long-term issuer ratings by Moody’s, Fitch and S&P.
Total assets, total equity, total liabilities, NACE code, CDS spreads
of senior debt with 5 maturity, and long term ratings by Moody’s,
Fitch and S&P.
Loans from banks: calculated by using the average country ratio
of loans to total assets of NFCs based on the ECB EA Accounts
dataset.
Lending relations and other supportive variables
Defined as the number of loans with different banks; average figures
by country and NACE sector were applied based on the data provided
through the Working Group on Credit Registers
Avg. interest rates on loans by size of loan and by country based
on the ESCB MIR data; categories of loans as follows: (below 0.25
EUR mn), (equal or above 0.25-1 EUR mn), and (over 1 EUR mn).
Avg. of expected default frequencies for non-financial corp. by
country and NACE.
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
EBA
ECB calculations
Bloomberg
Bloomberg
ECB calculations
ECB calculations
ECB calculations
Moody’s KMV and ECB
23/09/2014
9 / 23
Sampling of the network
Observed nodes (banks + non-bank corporate firms) and +500
generated banks
I
generated banks: based on the total assets and proportional allocation
of other attributes
Lending relationship:
I
{bank}–{firm}: based on aggregate Credit Register data
F
→ 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
+
F
I
I
→ 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 (= 0.01)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
10 / 23
ETT
EQV
EXL
END
FSC
ELI
FIA
EBC
FLG
ELE
FIS
FUM
EFO
DOV
GLA
DIG
HKS
CRA
HON
CTL
HUH
CTH
ILK
CTY
ICP
CNC
IFA
CAV
XNS
CGC
KRA
CPM
KSL
BTH
KES
BIO
KES
BAS
KEL
ALB
CCL
CNA
CPI
CCH
BRB
355
354
CPG
KNE
BNZ
CRH
BT/
ALB
DGE
BSY
KCR
ATR
BLN
EXPEZJ
LAT
BAT
ACG
FRE
BP/BLT
LEM
GFS
ASU
MMO
GKN
APE
BG/
356
353
MAR
GSK
GOGGJFFOE MET
AME
BA/
MHG
GLE
BAB
ALN
NHY
HMS
MET
DNO
AKT
AV/
MEO
HL/
NAS
AKT
AZN
409
410
408
DET
MUN
DJU
DFD
411
407
DSV
DAN
EAC
IMI
DAN
EGE
DAB
ERR
DLH
406405
AHL
EXQ
DNO
ABF
OPE
EXP
COM
NEO
FED
413412
COL
554
556
558
555
557
IMT
BOR
COL
AFE
AKS
FFA
CHR
AHT
414
404
544
546
540
550
549
548
543
553
545
547
541
551
542
552
FLS
NES
CHE
FLU
IHG
AFA
ORK
CBR
534
526
536
530
529
528
539
538
533
527
535
537
531
532
FYN
415
403
NOK
CAR
ARM
GR4
357
362
YLE
CAR
IAG
GAB
416
402
NREYAR
BIF
ANT
GEN
PGS
YTY
HAR
GER
ITR
NDA
417
401
KLE
TGS
GJ
WUF
AAL
BOC
NLG
GN
ITV
PRS
BLV
418
400
WRT
GES
AME
OKM
BIO
TEL
GRI
VIK
SBR
BAV
419
399
REC
OLV
GRL
AGK
288287
BNO
VAL
GYL
JMA
ORA
BO
RCLSCH STBSUB
420
398
VAI
ADM LUN
GYL
291290289 286285284
AUR
ORE
KGF
STL
292
VAC
S
DR
ATL
421
397
283
ADN
OKD
358
361
HH
WAT
293
FOA
LAN
OKD
HAR
282
422
448
WPP HOE
UNR
OAH
294
ORN
LGE
MAE
UPM
281
HOE
WOS
ORN
423
447
295
MAE
TUT
LSE
OUT
280
AMB
TUL
MRW HVI
IC
424
446
296
OTE
359
360
MKS
AMB
TRH
279
PNA
WMHIMA
ALM
TIK
425
445
297
PKC
MGG
JMI
TIE
ALM
WTB JDA
278
PKK
TLS
426
444
MRO
ALK
PON
298
TLT
WEI JUT
POY
ELI
277
427
443
TEM
MND
QPR
TPS
AAB
299
VOD KBH
RAI
428
442
TLV
NG/
RMR
323
ZEA
TTM
LAS
UU/ KRE
RAP
429
441
TAM
300
WDH
RTR
NXT
SUY
RUT
430
440
ULV LLB
SSK
322
VIP
RES
STE
ELUBOLATCATC
OML
LLA
REG
431432
439
STE
301
SAG
VII
STC
TLW
SAM
STC
GET
AZN
438
SAA
LOL
SSH
SCL
SRV
321
PSO
SCI
SDA
STQ
433434435436437
SOS
SOP
VIB
TT/
LUX
HMB
ASS
302
VJB
PSN
MAT
320
TPK
TVE
VWS
PFC
INV 568567566565 ALF
TPD
MRK
303
MOL
TSC
VEL
BLT
JRV
319
PRU
MTB
564 ABB
TAT
UIE
LUP 569
304
RRS
EEG
ARC
MNB
SL/
318
TRY
RB/
NEU
305
STA
TRI
MTG 570
576 VOL
HAE
TKM
REL
317
NEW
SSE
TOT
306
RSL
NKT
SPD
NCN
TAL
316
REX
TOR
NOK
DGO
BAL
KCM
DCP
FOI
307308
NDA
575 TLS
DLE
RIO
GZE
KA1
FBD
SN/SMI
CRH
TPS
OEG SFG
SKN
RR/
FDP
LJM
GRZ
NRD
SHP
CPL
RDS
SVT
FFY
TOP
RDS
SAN 571572573574 ERI
SDR
309310311312313314315
RMG
CGN
SGE
NOR
RSA
SAB
PKG
LSC
GRD PRF
GLB
TOP
NRD
GWM
TKD
LTT
DPK
SCV
TEL
461 460GCC
NOR
YZA
TIV
GRN
NRS
SECSKA SCA
SWM
AEX
TDC
ZOV
LOK
NOR
HBR
MIG
AER
NOV
462
SKFSSA
STR
IFP
LAP
BRV
NZY
ROV
ABB
NTR
SPN
INM
NUN
SPG
466 ZMN
NKA
VNF
OSS
SOL
IR5
OJB
SMA
UTV
PND
SKJ
OLF
RRR
PAR
KDR
SKA
COR
EUR
463
PAA
BZW
TVC
SIM
PRI
GTC
SIF
REC
KMR
SCH
GTOFUR ASM
MT
REA
RAR
VEF
SBS
PEO
TOT
REL
SAS
RIA
SAN
RIL
SAL
RBL
KYG
RTX
ROC
RBR
ROC
LTS
HEI
AKZ
SKG
464 465 RYA
RER
VSS
KSP
BHW
MER
FRM
TKB
AH 514513512511
PRP
JSW
MIO
510 AF
RJRRKB
TMA
OGN
ACP
ORM
SCM
PPTR
CIPAC
OVG
DSM 515
IPM
SAF
SMA
PWL
509 AGN
KER
KPN 516
525 ZIG
559 TPE
517
KGH
524 WKL
PHI 518
SNS
523 UNA
LWB
PNL 519520521522
PZU
RAN
UL
MBK
BEL
BSL
BEF
COF
ABI
RENRDSIMSBM
TNT
COL
OPLPGEPKNPGNIRB SRA
2827 AGS
DIE
ACK
DEL29 26
34UMI
VUB
DAI CON BEI
DL30 33
UCB
SES
ELI
DB1
BMW
3132 THR
GSZ
TNE
GBL
SOL
SLN TMR
BEK
LHA
BAY
BIP
MT
FOY APA
DANPANAPP
DPW
BAS
96
97
95
98
94
99
93
100
92
101
91
102
90
103
89
104
88
105
87
106
86
107
85
108
84
109
83
110
82
KBC
FHB456455ANY
111
81
112
80
113
79
114
78
115
77
116
76
117
75
118
74
SFN
SYN
119
73
120
72
121
71
122
70
123
ENG EBR DIA MTS
69
124
68
LXM 507
MTE457458459
125
DTE
ALV
67
126
66
127
65
128
64
129
SOC
63
130
62
131
FER
61
132
60
REIRTLSES
MOLPANRABRIC
133
59
134
58
135
57
136
AMS
56
137
55
138
54
139
53
140
52
141
51
FCC
EOA
ADS
142
50
143
49
144
48
145
47
146
46
147
ACS
45
148
44
149
43
150
42
151
41
152
40
332331330329328
153
GAM
39
154
38
155
37
156
FME
VOW
333
157
276
158
275
159
327
274
ANA
160
273
161
272
162
334
271
163
IIA
EVN
270
164
269
165
326
268
166
LNZ
CWI
267
GAS
167
266
168
265
335
169
264
170
263
171
FRE
TKA MMK 7 6 5 CAI
262
172
261
325
173
260
174
ABE
259
175
8
258
176
43
257
177
336
256
178
255
179
254
180
253
181
109
POS 11
252
182
324
251
183
GRF
21 AND
250
184
249
185
248
186
247
187
246
HEI
SIE
337
188
245
189
244
190
243
191
12
242
192
241
193
240
194
352
VIS
OMV 13
25 ZAG
239
195
238
196
237
197
236
198
235
199
234
200
233
201
232
202
231
24
203
230
338
204
229
205
228
206
227
207
226
208
225
209
224
210
223
211
222
212
221
213
220
214
219
215
218
216
217
IBE
HEN
SAP
351
RHI 14
1516
2223 WIE
339
21
17
TEF
350
SBO 181920 VOE
IFX
RWE
340
ITX
STR
VIG
349
TKAVER
SDF
MUV
CACAP
341
TRE
348
MLSGO ENCSMT
LXS
MRK
342
LIN
BN
IDR
ALO
343344345346347
EDF
ALU
371
370
SCY
372
369
EI
373
368
AIR
374
367
IAG
GSZ 376
375
366
REP
365 AI
GTO 377
JAZ
364 AC
378
363
KER 379
MAP TL5 OHL REE
396 VIV
OR 380
395 DG
381
394
LG 382
393 VIE
392
LR 383
384
391
VK
KRK GRV
385
390
MC
386
389
387
388
UL
579
578
ESO
EDP
RAM
EDP
MEL580
ORA
577
FCP
CTT
FP
GAL
COR
RIPUB
581 586
ZVT
GLI
COM
TEC
MCP
CFN
585
RNOSAFSANSUSOL
PET582
BZU
PMI
CNH
583
584
BPE
SCO
CPR
CPR
560
AZM
EGP
AGL
IBS
MPI
POS TLS
ENE
ATL
476
GPA
BAN
477
475
ENI
478
474
479
473
GI
480
472
EXO 482
481
471
IPR
BES
470
STS
483
469
F
I
INA
ALT
484
468
A2A
485
467
FNC 486
JMT 561
563 ZON
506 YOO
GTK
487
505
504 WDF
LIG
VAF
GO
FFG
FOY
EUP
LUX 488
FRI
489
503
ELB
GEK
ELT
XAK
US
502
ELL
EXA
MAR
VAF
DRO
491
501
MS 490
HSBGHM GCLFIM
ELP
HYG
492
500
TOD
HTO
493
499
EEE
494
498
MB
495
497
KLM
EGL
SCT
496
TRN
IHI
CW
IAS
MED
OTO
450 449 CEN
IKT
NBA
TDS
TEN
GAPERMAPE
TAT
PCPRY
INT
TIT
562
EYD
IHG
6PM
PTI
SUC
STM
SPM
INK
SRG
SFE
HB
TSH
IAT
INL
PTC
SCP
AVA
RED
SLB
LOM
508 TML
451
454 AEG
TRA
LOG35 36FWW
BEL
REN
SNC
TIT
SVA
SON
KLE
SEM
SON
PLA
ORE
SON
LHH
VCW
KOR
MLT
SFC
EYA
KRI
OLT
LPLLUIMIT
LAM
452 453 OLY
TEN
KYL
MIA
RS2
MIG
SID
MET
EPI
MLS
MTPMDS MDIPZC
SAR
MOH
REV
MYT
QUE
NIR
PPC
OPA
PLA
PAP
PPA
MSI
Figure 2: Network of non-bank corporate borrowing
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
11 / 23
Applications – policy implications
Event-driven contagion (realised)
Deterioration of credit quality in a given sector (NACE) – corporate loan losses
trigger contagion
Plan: realised for pure interbank network (Halaj and Kok, 2014)
Large Exposure limits – compactness of the networks (planned)
lower bilateral exposures allowed ⇒ more connections
Network reactions to adverse market conditions (planned)
passing macro scenarios via dynamic BS model (Halaj, 2013):
baseline macro scenario ⇒ optimising behaviour of banks ⇒ change in banks’ preferred
aggregate interbank lending and borrowing ⇒ endogenous formation of the interbank under
specified regulatory regime ⇒ adverse macro shock ⇒ banks defaults ⇒ contagion
CVA – crowding out bad quality borrowers (planned)
supposedly, banks would shift towards lending to high quality borrowers
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
12 / 23
Figure 3: Contagion simulation
BE
AT
CY
HU
DE
DK
ES
FI FR
GB
GR IE
LU
IT
NL
PT
NO
SE SI
Consumer, Non-cyclical [7]
Consumer, Cyclical [8]
Non-bank financial [11]
Consumer, Non-cyclical [1]
Energy, Basic materials [2]
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)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
13 / 23
Figure 4: Contagion simulation for different deterioration of credit quality
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
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
Contagion mechanism – cascade triggered by a deterioration of credit
quality of loan portfolios to companies in a given NACE sector for
(y-axis) PD ∈ {5%, 10%, . . . , 100%} and 50% LGD
“Spectral” graph of contagion losses of 500+ banks (the darker the
bar, the higher the fraction of capital wiped out by contagion)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
14 / 23
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
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 (counterfactual example!) Germany
Each bar indicates a defaulting bank
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
15 / 23
Conclusions
Endogenous interbank networks give an important insight into the
role of banks’ investment and funding strategies in shaping the
interbank market and non-bank firms’ funding channels. The simple,
mechanistic cascade models are too simplistic in assuming that banks
do not react to actions of other interbank participants and market
conditions.
It is easier to introduce heterogeneity of agents if the network
approach is taken rather than macroeconomic (e.g. general
equilibrium) framework.
In the proposed framework, 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.
More stability and robustness checks must be performed in order to
understand the complexity of the relationship between market
parameters and network topologies.
The model needs to be calibrated to the observed interbank / lending
networks. How far are we from the truth?
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
16 / 23
APPENDIX
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
17 / 23
Prerequisites
(nodes) N banks and M non-bank firms: capital and bank borrowing
+ out-degree distribution within (NACE) sectors
(exposures) Let Lij denotes the interbank (bank) placement (loan) of
bank j in bank (firm) i.
(capital position – constraint for risk-taking) total capital e and
capital e I ≤ e allocated to the interbank assets, e C ≤ e allocated to
non-bank firms; risk weights ω of exposures.
(probability map P) of interbank and bank-firm connections drawn
from P allowing for capturing possible customer relationship between
banks and firms. Each bank j draws its counterparties Bjk ⊂ N/{j},
enlarging the set at each step k: B¯jk+1 = B¯jk ∪ Bjk+1 ;
In addition, firms choose max number (mj ) of banks granting loans
based on out-degree distribution, i.e. #Bjk+1 ≤ mj
(matching) atPstep k incremental matching of assets and liabilities:
a¯jk = a¯jk−1 − i Lkij , where Lk is a matrix of placements at step k
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
18 / 23
1st round – Criteria for investment of interbank assets
General idea of banks’ optimising behaviour
Assumption (i): each bank maximises return from loan 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 and firms
Assumption (ii): optimisation of interbank portfolio separate from
optimisation of non-bank corporate loan portfolio
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∈B¯jk
Outcome: a matrix of exposures LI ,k , whereby optimisation subject to
constraints...
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
19 / 23
...Constraints of the admissible set of strategies
The maximisation is subject to some feasibility and capital constraints.
P
1 budget constraint –
¯jk and Ljj = 0, for aj0 = aj being
j|j6=i Lij = a
exogenously determined;
2 counterpart’s size constraint – L ≤ l¯k ;
ij
i
P
k
3 capital constraint –
ω
(L
+
L
) ≤ ejI − γ > (L¯·j + L·j );
i
ij
i|i6=j
ij
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 a¯jk
aik gives a feasible set
aik , a¯ik − 3∆¯
aik ,... until a¯ik − ki ∆¯
replaced by a¯ik − 2∆¯
of constraints
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
20 / 23
2nd round – funding diversification
Diversification risk gauged by default risk
0
with probability pj
Xj : =
1 with probability 1 − pj
(2)
Assumption: pj s are risky (variance based on time series of CDS spreads)
For a covariance matrix D¯X2 of X , the optimised funding risk is measured
F (Lki1 , . . . , LkiN ) = κF [Lki1 . . . LkiN ]D¯X2 [Lki1 . . . LkiN ]>
(3)
Outcome: a matrix of interbank deposits LF ,k , whereby optimisation on
the admissible set:
¯k
¯jk and j 6∈ B¯jk ⇒ yj = 0}.
AFi : = {y ∈ RN
+ |j ∈ Bj ⇒ yj ≤ a
REMARK: inclusion of non-bank corporate sector implies that (3) is also
solved by non-bank firms (⇒ LF ,k is (N + M) × (N + M) matrix)
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
21 / 23
3rd round – the game
Assumption: banks negotiate loans in pairs simultaneously (pair (i 0 , j)
knows the outcome of (i 00 , j) after both games are completed). Case
LIij,k > LFij ,k
h
ih
i
Gijk (x) = Uijl,k∗ − sijl,k · (x − LFij ,k ) Uija,k∗ − sija,k · (LIij,k − x)
(4)
where sijl,k is a measure of how much bank i is willing to deviate from his
optimal funding strategy, i.e.
!
Uijl,k (LFij ,k ) − Uijl,k (LIij,k )
l,k
,0 ,
sij = max
|LIij,k − LFij ,k |
,k
,k
where Uijl,k (x) = −F (LFi1,k , . . . , LFij−1
, x, LFij+1
, . . . , LFiN,k )
(for sija,k analogously,... and for LIij,k < LFij ,k similar)
Goal of the game: maximisation of Gijk
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
22 / 23
4th round – price adjustment [optional]
After the first 3 rounds of a step k some banks may still have a gap in
the interbank funding ⇒ adjustment to the offered interest rate on
new interbank deposits to increase a chance to obtain funding in step
k +1
P k+1
If at the step k + 1 the gap amounts to g k+1 : = li −
L¯
then
i
j
ij
the adjusted offered rate satisfies rik+1 = rik exp(αgik+1 /li ).
REMARK: in the baseline case we assume α = 0
Grzegorz Halaj (ECB)
Fin. Risk & Network Theory, Cambridge
23/09/2014
23 / 23
© Copyright 2024 ExpyDoc
