e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Network analysis of the e-Mid interbankmarket: implications for systemic risk

Vasilis Hatzopoulos and Giulia IoriCity University London

The research leading to these results has received funding from theEuropean Union, Seventh Framework Programme FP7/2007-2013under grant agreement FET Open Project FOC, Nr. 255987.

Latsis Symposium 2012, Zurich

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 1 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Introduction

• In normal times, interbank markets are among the mostliquid in the financial sector and the financial literature hashistorically devoted a relatively low consideration to theinterbank market due to the short term nature of theexchanged deposits.

• During the 2007-2008 financial crisis though liquidity in theinterbank market has considerably dried up, even at shortmaturities, and an increasing dispersion in the creditconditions of different banks has emerged.

• These events have triggered a new interest in interbankmarkets.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 2 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Effect of the crisis: market freeze

J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 D090

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Daily volumes avaraged within a month

Aggressor to quoter volumes

Quoter to aggressor volumes

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ily t

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Daily trades avaraged within a month

Banks proposing to borrow

Banks proposing to lend

Figure: Left: monthly average of daily volumes. Left: monthly averageof daily trades. In both cases trades have been separated into borrowinitiate deals and lending initiated deals).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 3 / 54

e-MID

Dataset

Market composition

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Network Metrics

Effect of the crisis: dispertion of credit spreads

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Figure: Montly average of the cross-sectional variance of spreads forborrower (left) and lender(right).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 4 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

The literature has presented two main explanations for thevolume collapse (or freeze) in the money market and for theraise in spreads during the recent turmoil:• liquidity hoarding: banks were hoarding liquidity in order

to anticipate additional money demand, both for internalneeds, and from external operators.

• Trust evaporation: banks, rationally or irrationally,perceive an increase in the counterparty-risk and becamereluctant to lend.

Our analysis is an attempt to quantify the second effect.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 5 / 54

e-MID

Dataset

Market composition

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Randomness in matching and Trust Evaporation

• The question we address is whether banks behaviourregarding the choice of counter parties in a trade changedbefore and during the subprime crisis.

• In particular we try and quantify the level of randomness inthe weights distribution across the links of the creditnetwork.

• We interpret this randomness as a proxy of the level oftrust among credit institution.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 6 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

e-MID: electronic Market for Interbank Deposits

• This market is unique in the Euro area in being a screenbased fully electronic interbank market. Outside Italyinterbank trades are largely bilateral or undertaken viavoice brokers.

• The central system is located in the office of the SIA andthe peripherals on the premises of the memberparticipants.

• The names of quoting banks are visible next to theirquotes to facilitate credit line checking. A transaction isfinalized if the ordering bank accepts a listed bid/offer.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 7 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

e-MID

Both italian banks and foreign banks can exchange funds.Market players are 246 members from 29 EU countries and theUS, of which:• 30 central banks acting as market observers• 2 Ministries of Finance• 108 domestic banks• 106 international banks

The number of transactions and the volume increasedsystemically until the beginning of the financial crisis, with anaverage of 450 transactions each day and an averageexposure of about 5.5 million euros per transaction.According to European Central Bank (2007) e-MID accounted,before the crisis, for 17% of total turnover in unsecured moneymarket in the Euro Area. In the last report on money markets(European Central Bank, 2010), it recorded 12% of the totalovernight turnovers.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 8 / 54

e-MID

Dataset

Market composition

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Entropy

Network Metrics

e-MID

Types of trade:• Overnight (O/N): Trades for a transfer of funds to be

effected on the day of the trade and to return on thesubsequent Business Day;

• Tomorrow next (T/N): Trades for a transfer of funds on thefirst Business Day following the day of the trade and toreturn on the second Business Day following that of thetrade;

• Spot next (S/N): Trades for a transfer of funds on thesecond Business Day following the day of the trade and toreturn on the third Business Day following that of the trade;

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 9 / 54

e-MID

Dataset

Market composition

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Entropy

Network Metrics

e-MID

• Time Deposits: Trades for an initial transfer of funds and toreturn at a predetermined maturity (from 1 week to 12months);

• Broken Date Deposit: Trades with freely agreed InitialValue Date and Final Value Date between parties withoutstandardization obligations provided that both dates do notcoincide with the previous ones and that the two days arenot separated by a period superior to a calendar year

We only look at ON and ONL transactions!!

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 10 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Dataset

• The data base is composed by the records of alltransactions registered in the period 01/1999–12/2009 fora total of 1.523.510 transactions.

• For each contract we have information about the date andtime of the trade, the quantity, the interest rate and theencoded name of the quoting and ordering bank.

• The banks are reported together with a code representingtheir country and, when the bank is Italian, a final labelthat indicates the class of capitalization (major, large,medium, small, minor)

• The aggregate characteristics of the entire set oftransactions can thus be studied in terms of the statisticaland topological properties of this HIGHLYHETEROGENEOUS network.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 11 / 54

e-MID

Dataset

Market composition

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Entropy

Network Metrics

System Heterogeneity: number of banks per group

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009MA 6 7 7 6 6 6 6 6 6 4 4GR 12 12 8 7 9 7 7 9 9 6 5ME 26 26 26 23 17 14 13 13 12 12 10PI 68 64 74 61 57 52 53 58 54 53 51MI 76 59 33 31 28 26 26 14 16 20 19FB 2 13 20 31 48 54 60 59 62 58 39

Table: Banks active as borrowers per group per year.

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009MA 6 7 7 6 6 6 6 6 6 4 4GR 12 12 8 7 9 7 7 9 9 6 4ME 26 26 26 23 18 14 13 13 11 11 10PI 75 65 76 66 60 58 55 62 58 54 55MI 91 70 44 39 34 35 33 17 20 22 22FB 3 11 21 32 48 57 61 66 70 70 45

Table: Banks active as lenders per group per year.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 12 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

System Heterogeneity: volume per group

Month

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Figure: Average daily volume per maintenance period per group aslender (dashed line) and borrower (continuous line).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 13 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

System Heterogeneity: net volume per group

Month

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Figure: Net percentage traded volume per group as lender andborrower.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 14 / 54

e-MID

Dataset

Market composition

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Entropy

Network Metrics

System Heterogeneity: intratrade time

group ∆tp1 ∆tp2 ∆tp3

MA 336 483 190GR 271 606 951ME 777 1627 730PI 7594 3639 7703MI 13623 9961 7240FB 6398 7854 3881

Table: Borrower Intratrade time (in seconds) per group in p1, p2, p3.

start endpre-crisis (p1) 2006-07-05 2007-08-08subprime (p2) 2007-08-09 2008-09-14

after Lehman (p3) 2008-09-15 2009-10-21

Table: The three periods in yyyy-mm-dd format.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 15 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

System Heterogeneity: intratrade time

group ∆tp1 ∆tp2 ∆tp3

MA 179 175 1068GR 1043 585 1378ME 1163 1422 2704PI 1381 1115 1296MI 665 996 1949FB 7167 5528 6812

Table: Lenders Intratrade time (in seconds) per group in p1, p2, p3.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 16 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

To summarize:• larger banks overall trade larger volumes• larger banks trade more often than smaller banks• some banks tend to trade predominantly on one side of

the market• individual trades have comparable volumes

We choose the number of exchanges between two parties asthe weight of the edge.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 17 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Random reshuffling

In order to compare the values of various network metrics withrandom null hypothesis that preserve the system’sheterogeneity

• We use the edge swap algorithm to generate syntheticnetworks to use as null models.

• An edge swap selects two ordered pairs (x , y),(u, v) andswaps the endpoints (target nodes) while keeping thesources fixed. This procedure preserves the degree ofeach node. Not all edges swaps are accepted during arewiring process as some swaps can produce graphs thatcontain self loops or parallel edges. Such sampling bias isreduced in the limit of large or sparse graphs.

• We perform 4E swaps per randomisation and averageover 100 randomizations for each of the 133 empiricalnetwork.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 18 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

For directed and weighted representations we can construct arandomisation using the edge swap procedure (that nowconserves the vertex in-out strength sequence but not thein-out degree sequence) in the following way.• We define the number of transactions between to edges

as the weight of the edge.• Each weighted directed edge with weight wuv is further

inserted wuv − 1 times in the network and all edges havetheir weights set to 1.

• The resulting multigraph is then rewired as a directedunweighted graph where each edge now indicates a singletransactions and the number of edges between u and vcorrespond to their number of transactions.

• The rewired multigraph is then collapsed to a directedweighted graph via the reverse procedure (i.e. all mdirected and unweighted edges between u and v arecollapsed into a single edge with weight m).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 19 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

• Quantitites that are preserved in the randomisedensemble after the degree-preserving randomisation canbe then traced back/explained as consequences of the inand out degrees distributions and thus the degreedistribution assessed in terms of its information content inthe context of the real-world network.

• Quantitites that are preserved in the randomisedensemble after the strength preserving randomisation canbe then traced back/explained as consequences of thestrength distributions (or heterogeneity in size).

• If system heterogeneity cannot fully explain the dynamicsof the various metrics then we cannot say that bankbehaviour is random.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 20 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Choice of timescale:

Two natural timescales in the system

• daily:maturity of the interbank loans• monthly: around 23 business days-known as a

maintenance period

We perform the analysis at the monthly time scale as want tomonitor the frequency of exchanges between counter partiesand compare it with a random null hypothesis that preservesbank’s heterogeneity in strengths (number of trades).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 21 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Example Monthly Networks in p1

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Figure: eMid bank Network in maintenance period 14-Feb-2007 to13-Mar-2007

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 22 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Example Monthly Networks in p2

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Figure: eMid bank Network in maintenance period 11-Jun-2008 to08-Jul-2008

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 23 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Example Monthly Networks in p2

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Figure: eMid bank Network in maintenance period 08-Oct-2008 to11-Nov-2008

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 24 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Basic network metrics

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Figure: Number of nodes(top left), number of edges(top right),average degree (bottom left) and edge density(bottom right ) for theset of networks defined on non-overlapping intervals of δt = 1maintenance period.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 25 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Information-theoretic description of networks

Letw̃ij =

wij∑i

∑j

wij

be the normalised weight, or flux, from node i to node j .The probability of observing lending activity from i or borrowingactivity from j are respectively

p(li )→ p(w̃i ) =∑

j

w̃ij

p(bj )→ p(w̃j ) =∑

i

w̃ij

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 26 / 54

e-MID

Dataset

Market composition

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Entropy

Network Metrics

The expressionsp(li |bj ) = w̃ij/w̃j

p(bj |li ) = w̃ij/w̃i

are respectively the conditional probabilities of observing atransaction with i being the source given that j is the sink and jbeing the sink given that i is the source.Finally the joint probability of observing a transaction with ibeing the source and j begin the sink is

p(li ,bj )→ p(w̃i , w̃j ) = w̃ij

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 27 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Following T. Wilhelm and J.Holunder1, we define

Lender Entropy:

H(L) = −∑

i

∑j

w̃ij log∑

j

w̃ij (1)

Borrower Entropy:

H(B) = −∑

j

∑i

w̃ij log∑

i

w̃ij (2)

Lender Entropy given the borrower is known:

H(L|B) =∑

j

p(bj )H(L|bj ) = −∑

i

∑j

w̃ij logw̃ij∑k w̃kj

(3)

1Information theoretic description of networks. Phys A, (385), 2007Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 28 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Borrower Entropy given the lender is known:

H(B|L) =∑

i

p(li )H(B|li ) = −∑

i

∑j

w̃ij logw̃ij∑k .w̃ik

(4)

Joint Entropy:

H(L,B) = −∑

i

∑j

w̃ij log w̃ij (5)

Mutual Information:

I(L,B) = H(L,B)− H(L|B)− H(B|L) (6)

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 29 / 54

e-MID

Dataset

Market composition

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Entropy

Network Metrics

• The minimum value (no uncertainty) is achieved when aborrower only trades with one lender or viceversa H = 0.

• The maximum values for all of the above quantities areHmax = log(N) except for H(L,B)max = 2 log(N).

• Intuitively H(X|Y) is the amount of uncertainty remainingabout X after Y is known, it tells us how much Y spreadsits trades (high entropy) or concentrate them with fewcounter parties (low entropy).

• The intuitive meaning of mutual information is the amountof information that knowing either variable provides aboutthe other, is a quantity that measures the mutualdependence of the two random variables, it tells us howmuch banks forms exclusive trading partnerships.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 30 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

An an individual bank level the following conditional entropiescan be defined:Lender entropy given that the borrower is node j:

H(L|bj ) = −∑

i

w̃ij∑k w̃kj

logw̃ij∑k w̃kj

(7)

Borrower Entropy given that the lender is node i:

H(B|li ) = −∑

j

w̃ij∑k w̃ik

logw̃ij∑k w̃ik

(8)

For individual lenders/borrowers the maximum entropy is equalto the Log of their out/in degrees.

An alternative approach to quantify the randomness ofindividual links in a network has been proposed recently byMantegna’s group.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 31 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics0.3

00.3

50.4

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H(X

|Y)

/ H

(X|Y

) m

ax

J06 J07 J08 J09 J10

H(B/L)

H(B/L) random

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H(Y

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) m

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H(L/B)

H(L/B) random

Figure: Conditional entropies. Left panel: entropy of borrower givenlender on empirical networks and randomised sample(squares) bothnormalised by their maximum possible values. Right panel: same asleft for entropy of lender given borrower.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 32 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics0.0

50.1

00.1

50.2

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Month

I(X

,Y)

/ I(

X,Y

) m

ax

J06 J07 J08 J09 J10

I(L,B)

I(L,B) random

Figure: Mutual information, squares(empirical), triangles (randomizedsample). Information about sender if receiver is known and vice versa.I(A,B) = H(A)− H(A|B) = H(B)− H(B|A)

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 33 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Matching probabilities: are partners chosen randomly?(withSalvatore Micciché and Rosario Mantegna)

90 100 110 120 130

months0

0.05

0.1

0.15

0.2

frac

tion

of li

nks

bonferroniFDR

Lender Initiated

90 100 110 120 130

months0

0.05

0.1

0.15

0.2

frac

tion

of li

nks

bonferroniFDR

Borrower Initiated

Figure: Fraction of non random matches: Lenders as aggressor (Left),Borrowers as aggressor(Right).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 34 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Relationship Lending: Results from the literature.

The empirical literature highlighted the presence ofRelationship Lending in OTC interbank transactions.

• Banks develop long-lasting trading relationships onthe interbank market on different maturities (Cocco etal. 2009)

• Relationships are influenced by banks characteristics(Size, Non-performing loans, etc.).

• Relationships reinforce over time (Affinito 2011)• The interest rate on interbank transactions is

influenced by the strength of the relationship.• Relationships lasted after the crisis (Affinito 2011, and

Brauning 2011).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 35 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Interpretations

Main interpretations:• Relationships as liquidity insurance (high reserve

imbalances, and volatile liquidity shocks increase theuse of relationship lending) (Cocco et al. 2009).

• ON Relationships as a strategic relationship (Bankstransacting outside ON have higher relationships).

• Banks continuously monitor each other (bankscharacteristics are always important) (Furfine 2001,Affinito 2011).

• Before the crisis banks paid a premium forrelationship. The opposite holds after the crisis (roleof information?). (Brauning 2011)

Work in progress: Empirical investigation of Relationshiplending in the e-Mid ON market (with Asena Temizsoy, GabrielMontes-Rojas and Mantegna’s group)

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 36 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Network Metrics

• Affinity

Affinity is a measure of similarity among nodes and is definedas

knn,i =1ki

∑j∈V(i)

kj (9)

If knn(k) is an increasing function of k , vertices with highdegree have a larger probability to be connected with largedegree vertices: assortative mixing. A decreasing behavior ofknn(k) defines disassortative mixing, in the sense that highdegree vertices have a majority of neighbors with low degree,while the opposite holds for low degree vertices.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 37 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Similarly we can define the following four measues

k in−outnn,i =

1k in

i

∑j∈V(i)

koutj (10)

k in−innn,i =

1k in

i

∑j∈V(i)

k inj (11)

kout−outnn,i =

1kout

i

∑j∈V(i)

koutj (12)

kout−innn,i =

1kout

i

∑j∈V(i)

k inj (13)

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 38 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

• kout−outnn,i tell us if the borrowers of a lending bank are

lenders themselves and to how many banks they lend.• k in−in

nn,i tells us if the lenders of a borrowing bank and areborrower themselves and from how many banks theyborrow.

• Thus kout−outnn,i and kout−out

nn,i can be used to monitorsystemic risk. Potentially dangerous situation for the allsystem, in case of defaults, are those in which banks witha large k in

i also have a large k in−innn,i , or banks with a large

kouti also have a large kout−out

nn,i .

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 39 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

kout−innn,i allows us to identify banks that may pose a serious

liquidity problem to their neighbours if they exit the market.

Such banks are those with a high kouti and low kout−in

nn,i , that isbanks who lend to several counter parties who in turn borrowfrom very few other banks.

If one of these high kouti and low kout−in

nn,i bank stops lending itscounter parties may find it difficult to satisfy their liquidity needsfrom the few remaining lenders (unless they can find newlenders).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 40 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Similarly k in−outnn,i allows us to identify banks that may be unable

to reallocate efficiently their liquidity if of their neighbours exitthe market.

Such banks are those with a high k ini and low k in−out

nn,i , that isbanks who borrow from several counter parties who in turnlend to very few other banks.

If one of these high k ini and low k in−out

nn,i bank stops borrowing itscounter parties may find it difficult to reallocate their liquiditysurplus to their few remaining borrowers (unless they arewilling to find new borrowers).

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 41 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Degree correlations

For a directed network four Pearson degree correlationcoefficients can be defined. r(α, β) where α, β ∈ {kin, kout}measures the tendencies of edges to have high α as sourcesand high β as targets and is defined as.

r(α, β) =

E−1∑

i

[(jαi − jα)(hβi − hβ)]

σασβ

where E is the number of edges in the network (tradingpartnerships taking into account the flow of liquidity) and jαi , hβithe α and β-degree of the source j and target h node for edge i .

For a directed weighted coefficient it suffices to replace degreeby strength.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 42 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Strenght correlations

−0.3

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r(S

in,S

in)

samplerw

pearson degree correlation coefficient

J99 J01 J03 J05 J07 J09

−0.3

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r(S

in,S

out)

J99 J01 J03 J05 J07 J09

samplerw

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r(S

out,S

in)

J99 J01 J03 J05 J07 J09

samplerw

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−0.1

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0.1

0.2

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Month

r(S

out,S

out)

J99 J01 J03 J05 J07 J09

samplerw

Figure: The four directed Pearson correlation coefficients. Verticallines correspond to the periods p1, p2, p3.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 43 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Degree correlations

−0.3

−0.1

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Month

r(S

in,S

in)

samplerw

pearson strength correlation coefficient

J99 J01 J03 J05 J07 J09

−0.3

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r(S

in,S

out)

J99 J01 J03 J05 J07 J09

samplerw

−0.3

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r(S

out,S

in)

J99 J01 J03 J05 J07 J09

samplerw

−0.3

−0.1

0.0

0.1

0.2

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Month

r(S

out,S

out)

J99 J01 J03 J05 J07 J09

samplerw

Figure: The four weighted directed Pearson correlation coefficients.Vertical lines correspond to the periods p1, p2, p3.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 44 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Clustering

From the point of view of a vertex i four directed clusteringcoefficients,in, out, middle, cycle can be distinguisheddepending on the direction of edges participating in a triangle:• in: when node i holds two inward edges.• out : when node i holds two outward edges.• middle: when one of the neighbours of i holds two outward

edges and the other holds two inward edges.• cycle: when there is a cyclical relation between i and its

neighbours.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 45 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

• The in coefficient represents high risk as if the bank i whohas borrowed from two other banks j and k fails to repayits loans then j and k may also fail to settle their obligationthat completes the triangle.

• The middle coefficient represents the case where thecounter parties j , k of bank i are either borrowing orlending from the other two banks, thus a collapse of i willtrigger a possible collapse of the whole triangle.

• By contrast out does not increase systemic risk as thecollapse of a bank with two outward (lending) edges willnot render the other two banks unable to fulfill theobligation between them.

• Finally cycle does not represent high systemic risk due tothe cyclical relation of liquidity flow.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 46 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Clustering

0.0

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cycle

sample

rw

clustering coefficients

J99 J01 J03 J05 J07 J09

0.0

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mid

le

J99 J01 J03 J05 J07 J09

sample

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in

J99 J01 J03 J05 J07 J09

sample

rw

0.0

0.1

0.2

0.3

0.4

0.5

Month

out

J99 J01 J03 J05 J07 J09

sample

rw

Figure: The four directed unweighted clustering coefficients. Verticallines correspond to the periods p1, p2, p3.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 47 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Clustering

0.0

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cycle

sample

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clustering coefficients

J99 J01 J03 J05 J07 J09

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mid

le

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in

J99 J01 J03 J05 J07 J09

sample

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0.0

00

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0.0

10

0.0

15

0.0

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Month

out

J99 J01 J03 J05 J07 J09

sample

rw

Figure: The four directed weighted clustering coefficients. Verticallines correspond to the periods p1, p2, p3.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 48 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Path lengths

• Since none of the maintenance period networks arestrongly connected, i.e. there aren’t directed pathsbetween all pairs in both directions, it becomesproblematic to calculates quantities such as the averageshortest path length, diameter, etc.

• However we can look at the number of pairs in the networkfor which a path exists (in both directions).

• Let nr be the fraction of vertex pairs that can be reachedvia a directed path (of any length). Then fr = nr

N(N−1) is thefraction of all pairs that can be reached in the graph.

• We can further divide fr by edge density Eρ = EN(N−1) to

get the quantity nrE which is the number of reachable pairs

in units of the number of edges in the system..

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 49 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics0.4

0.5

0.6

0.7

Month

reachable

pairs / a

ll pairs

J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09

56

78

9

Month

(re

achable

pairs / a

ll pairs)

/ density

J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09

Figure: The fraction of pairs reachable by a directed path(top) and thenumber of pairs reachable by a directed path in units of the number ofedges in the system.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 50 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

Centralities

Centrality measures the importance of a vertex in a network interms of channelling the flow of a quantity, in our case liquidity.• The betweenness centrality of a vertex v is the sum of

shortest paths between all nodes that pass through v .• The closeness centrality of vertex v is the inverse of the

sum of the distance (sum of shortest paths) of v to allother vertices .

• The in and out centralities measure the in reps. out degreeof a vertex normalised by the number of all vertices.

It is also interesting to calculate the centralities for eachcapitalization group separately. In this case the centrality of agroup i is defined as the average centrality over the groupmembers.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 51 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics

0.0

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betw

eenness

sample

rw

centralities

J99 J01 J03 J05 J07 J09

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clo

seness

sample

rw

J99 J01 J03 J05 J07 J09

0.0

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in c

entr

alit

y

sample

rw

J99 J01 J03 J05 J07 J09

0.0

00.0

50.1

00.1

50.2

0Month

out centr

alit

y

sample

rw

J99 J01 J03 J05 J07 J09

Figure: The four directed unweighted centrality measures. Verticallines correspond to the periods p1, p2, p3.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 52 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics0.0

00.0

10.0

20.0

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40.0

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month

betw

eeness c

entr

alit

y

J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09

MA

GR

ME

PI

MI

FB

Figure: The unweighted betweenness centrality averaged over banksof each group.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 53 / 54

e-MID

Dataset

Market composition

reshuffling

Entropy

Network Metrics0.0

00.0

10.0

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betw

eeness c

entr

alit

y

J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09

MA

GR

ME

PI

MI

FB

Figure: The weighted betweenness centrality averaged over banks ofeach group.

Giulia Iori Network analysis of the e-Mid interbank market: implications for systemic riskSeptember 11-14, 2012 54 / 54