3 leonidov

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


Balance sheet


INTERBANK NETWORKS: CONTAGION MODELLING



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





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

]





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





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!





Benchmark case. Contagion of more than 5 % of the networkresulting form the default of a single bank





Comparison with analytic calculation


Russian interbank market


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.


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


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.


Interbank network: bow-tie structure, nodes and weights


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


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


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


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


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


Contagion tree Out → In

Li ,j(y) =∞∑n,m

POut/In(n,m|i , j)[(1− v

Out/Inm ) + v

Out/Inm y

]


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

]


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)

]


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

]


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


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


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


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


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.


Date post:	16-Jun-2015
Category:	Technology
Upload:	yandex
View:	175 times
Download:	0 times