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Workshop on random graphs 24—26 oct Доклад Андрея Леонидова
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Systemic Interbank Network Risks in Russia
A. Leonidov(1,2), E. Rumyantsev(2,3)
(1) P.N. Lebedev Physical Institute(2) Moscow Institute of Physics and Technology
(3) Central Bank of Russian Federation
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 1 / 28
BOE view of an interbank network
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 2 / 28
Balance sheet
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 3 / 28
INTERBANK NETWORKS: CONTAGION MODELLING
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
Solvency condition for a bank i :
(1− φ)AIBi + qAM
i − LIBi − Di > 0
Definition of variables:
AIB: liquid interbank assetsAM: illiquid assetsLIB: interbank liabilitiesφ: Fraction of contacts going bankruptD: DepositsK = AIB + AM − LIB − D: capital buffer
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 4 / 28
INTERBANK NETWORKS: CONTAGION MODELLING
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
Vulnerable banks are those that default because of a default of one of theircounterparties (i.e. φ = 1/j)
A bank i with j incoming links is thus vulnerable in this sense if
Ki − (1− q)AMi <
AIBi
j
Because of the randomness of the capital buffer Ki (e.g. due to that of thedeposits) contagion is probabilistic with a probability vj :
vj = Prob[Ki − (1− q)AM
i <AIB
i
j
]
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 5 / 28
INTERBANK NETWORKS: CONTAGION MODELLING
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
H1(y) - generating function for the probability of reaching an outgoingvulnerable cluster of given size by following a random outgoing link form avulnerable bank
H1(y) = Pr[reach safe bank] + y∑j,l
vj · rjk [H1(y)]k
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 6 / 28
INTERBANK NETWORKS: CONTAGION MODELLING
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
Generating function for the size of a vulnerable cluster
H0(y) = 1− G0(1) + yG0 [H1(y)]
Size of the vulnerable cluster S :
S = G0(1) +G ′
0(1)G1(1)
1− G ′1(1)
Phase transition:
G ′1(1) = 1 ⇔
∑j,k
j · k · vj · pjk = z
In this case: opposing effects from j · k and vj lead to two phase transitionsand a contagion window in z!
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 7 / 28
INTERBANK NETWORKS: CONTAGION MODELLING
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
Benchmark case. Contagion of more than 5 % of the networkresulting form the default of a single bank
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 8 / 28
INTERBANK NETWORKS: CONTAGION MODELLING
P. Gai, S. Kapadia, ”Contagion in financial networks”
Bank of England Working Paper No.383
Comparison with analytic calculation
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 9 / 28
Russian interbank market
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 10 / 28
Russian interbank network: data description
Uncollaterized interbank rouble deposits of all maturities in the periodfrom January 11, 2011 till December 30, 2011 are considered.
Interbank network for N banks is fully characterized by an orientedweighted graph GW = (N,W ), where W = {wij} is an N × N matrixof wij > 0 of liabilities of the bank i with respect to the bank j .
By definition the outgoing links correspond to liabilities, the incomingones - to claims.
The interbank network graph is scale-free in both in- and out- degreesand is characterized by significant clustering.
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 11 / 28
Definition of vulnerability
Solvency coefficient H1 as defined by CBRF:
H1 =K∑
i AiKpi + PP + OP + others
Here
K is capital
Kpi - risk coefficients
All instruments are divided into 5 groups i = 1, · · · 5 and Kp1 = 0,Kp2 = 20 %, etc. For the interbank market the risk coefficient is 20 %
PP - market risk
OP - operational risk
others - other contributions
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 12 / 28
Definition of vulnerability
A default condition is
H1 =K∑
i AiKpi + PP + OP + others< H1∗
where for banks H1∗ = 10 %, for others - H1∗ = 12 %
Calculation using H1:
H1 ⇒ K − P∑i AiKpi + PP + OP + others
where P is a reserve kept for the case when one or severalcounteragents default.
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 13 / 28
Interbank network: bow-tie structure, nodes and weights
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 14 / 28
In- and Out- degree distributions
0 1 2 3 4
−12
−10
−8
−6
−4
−2
log(kIn)
log(
p)
log(p) = − 3.77 log(kIn) + 5.19
P(kin) ∼1
k3.77in
0 1 2 3 4
−12
−10
−8
−6
−4
log(kOut)lo
g(p)
log(p) = − 8.7 log(kOut) + 27.34
P(kout) ∼1
k8.7out
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 15 / 28
Disassortativity
0 10 20 30 40 50 60 70
1015
2025
30
Number of incoming links
Ave
rage
num
ber
of c
ount
erpa
rtie
s’s
outg
oing
link
s
0 20 40 60 80 1000
510
15
Number of outgoing links
Ave
rage
num
ber
of c
ount
erpa
rtie
s’s
outg
oing
link
s
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 16 / 28
Empirical default distributions
Probability that at least one incoming link is vulnerable:
Out → In-Out
5 10 15 20
0.00
20.
004
0.00
60.
008
0.01
0
In degree
Vul
nera
ble
prob
abili
ty
In-Out → In-Out
5 10 15 20 25
0.00
00.
005
0.01
00.
015
0.02
00.
025
0.03
0
In degree
Vul
nera
ble
prob
abili
ty
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 17 / 28
Empirical default distributions
Probability that at least one incoming link is vulnerable:
Out → In
5 10 15 20
0.00
0.02
0.04
0.06
0.08
In degree
Vul
nera
ble
prob
abili
ty
In-Out → In
5 10 15 20 25
0.00
0.02
0.04
0.06
0.08
0.10
0.12
In degree
Vul
nera
ble
prob
abili
ty
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 18 / 28
Strongly connected component
There exists a strongly connected componentThe weight of this component did significantly increase
Jan Mar May Jul Sep Nov Jan
020
4060
8010
0
Date
Out
stan
ding
, %
050
100
150
200
Coun
terp
artie
s
Total outstandingCounterparties
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 19 / 28
Contagion tree Out → In
Li ,j(y) =∞∑n,m
POut/In(n,m|i , j)[(1− v
Out/Inm ) + v
Out/Inm y
]
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 20 / 28
Contagion tree In-Out → In
Nk,l(y) =∞∑s,r
PIn−Out/In(s, r |k, l)[(1− v
In−Out/Inr ) + v
In−Out/Inr y
]
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 21 / 28
Contagion tree In-Out → In-Out & In
Mk,l(x ,N(y)) =∞∑
u,t,s,r
PIn−Out/In−Out(u, t, s, r |k, l)
∗[(1− v
In−Out/In−Outr ) + xv
In−Out/In−Outr Mu
u,t(x ,N(y))Ntu,t(y)
]
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 22 / 28
Contagion tree Out → In-Out → In-Out & In
Ki ,j(x , y) =∞∑
k,l ,n,m
POut/In−Out(k, l , n,m|i , j)
∗[(1− v
Out/In−Outm ) + xv
Out/In−Outm [Mk,l(x , y)]k [Nk,l(y)]l
]
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 23 / 28
Contagion clusters In-Out → In-Out & In
Let F (x , y) be the generation function for the probability for a bankfrom In-Out being linked with In-Out and In components:
F (x , y) =∞∑
i ,j=0
pInOutij x iy j
The generation function for default cluster originating in In-Out thenreads:
GInOut(x , y) = F (M(x ,N(y)),N(y))
The average size of default clusters is given by
dGInOut(x , x)
dx
∣∣∣∣x=1
= 1
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 24 / 28
Contagion clusters Out → In-Out & In
Let G (x , y) be the generation function for the probability for a bankfrom Out being linked with In-Out and In components:
G (x , y) =∞∑
i ,j=0
pijxiy j
The generation function for default cluster originating in Out thenreads:
GOut(x , y) = G (K (M(x ,N(y)),N(y)), L(y))
The average size of default clusters is given by
dGOut(x , x)
dx
∣∣∣∣x=1
= 1
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 25 / 28
Simulation: dependence upon out-degree
0 20 40 60 80 100
05
1015
2025
Out degree
Aver
age
defa
ult c
lust
er s
ize
In−Out clusterOut cluster
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 26 / 28
Simulation: default cluster size distribution
5 10 15 20 25
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Default cluster size
Prob
abilit
y
In−Out clusterOut cluster
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 27 / 28
Conclusions
Taking into account the bow-tie structure of the interbank network isvery essential.
Despite of the complicated topology of the original graph, the defaultclusters are (almost) always tree-like.
This allows to describe default clusters in terms of generatingfunctions taking into account the bow-tie structure of the originalinterbank network graph.
The realistic contagion in the RF interbank market is a relativelysmall effect, nothing dramatic.
A. Leonidov(1,2), E. Rumyantsev(2,3) (LPI, MIPT, CBRF)Random Graphs and their Aplications 24.10.2013 28 / 28