[PPT] - Network analysis of the e-Mid interbank e-MID market: implications PowerPoint Presentation

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

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

Vasilis Hatzopoulos and Giulia Iori City University London

The research leading to these results has received funding from the European Union, Seventh Framework Programme FP7/2007-2013 under 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 risk September 11-14, 2012 1 / 54

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Introduction

In normal times, interbank markets are among the most

liquid in the financial sector and the financial literature has historically devoted a relatively low consideration to the interbank market due to the short term nature of the exchanged deposits.

During the 2007-2008 financial crisis though liquidity in the

interbank market has considerably dried up, even at short maturities, and an increasing dispersion in the credit conditions of different banks has emerged.

These events have triggered a new interest in interbank

markets.

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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 D09 0.5 1 1.5 2 2.5 3 x 10

4

Month Daily volume / 106 Euros Daily volumes avaraged within a month Aggressor to quoter volumes Quoter to aggressor volumes J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 D09 100 200 300 400 500 600 700 Month Daily trades Daily trades avaraged within a month Banks proposing to borrow Banks proposing to lend

Figure: Left: monthly average of daily volumes. Left: monthly average

f daily trades. In both cases trades have been separated into borrow

initiate deals and lending initiated deals).

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Effect of the crisis: dispertion of credit spreads

0.00 0.01 0.02 0.03 0.04 month mean of cross−sectional variance of daily spread J99 J01 J03 J05 J07 J09

borrowers

0.00 0.01 0.02 0.03 0.04 month J99 J01 J03 J05 J07 J09

lenders

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

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The literature has presented two main explanations for the volume collapse (or freeze) in the money market and for the raise in spreads during the recent turmoil:

liquidity hoarding: banks were hoarding liquidity in order

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

Trust evaporation: banks, rationally or irrationally,

perceive an increase in the counterparty-risk and became reluctant to lend. Our analysis is an attempt to quantify the second effect.

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Randomness in matching and Trust Evaporation

The question we address is whether banks behaviour

regarding the choice of counter parties in a trade changed before and during the subprime crisis.

In particular we try and quantify the level of randomness in

the weights distribution across the links of the credit network.

We interpret this randomness as a proxy of the level of

trust among credit institution.

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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 screen

based fully electronic interbank market. Outside Italy interbank trades are largely bilateral or undertaken via voice brokers.

The central system is located in the office of the SIA and

the peripherals on the premises of the member participants.

The names of quoting banks are visible next to their

quotes to facilitate credit line checking. A transaction is finalized if the ordering bank accepts a listed bid/offer.

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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 the US, 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 increased systemically until the beginning of the financial crisis, with an average of 450 transactions each day and an average exposure 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 money market in the Euro Area. In the last report on money markets (European Central Bank, 2010), it recorded 12% of the total

vernight turnovers.

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e-MID Dataset Market composition reshuffling 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 the subsequent Business Day;

Tomorrow next (T/N): Trades for a transfer of funds on the

first Business Day following the day of the trade and to return on the second Business Day following that of the trade;

Spot next (S/N): Trades for a transfer of funds on the

second Business Day following the day of the trade and to return on the third Business Day following that of the trade;

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

e-MID

Time Deposits: Trades for an initial transfer of funds and to

return at a predetermined maturity (from 1 week to 12 months);

Broken Date Deposit: Trades with freely agreed Initial

Value Date and Final Value Date between parties without standardization obligations provided that both dates do not coincide with the previous ones and that the two days are not separated by a period superior to a calendar year We only look at ON and ONL transactions!!

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Dataset

The data base is composed by the records of all

transactions registered in the period 01/1999–12/2009 for a total of 1.523.510 transactions.

For each contract we have information about the date and

time of the trade, the quantity, the interest rate and the encoded name of the quoting and ordering bank.

The banks are reported together with a code representing

their country and, when the bank is Italian, a final label that indicates the class of capitalization (major, large, medium, small, minor)

The aggregate characteristics of the entire set of

transactions can thus be studied in terms of the statistical and topological properties of this HIGHLY HETEROGENEOUS network.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

System Heterogeneity: number of banks per group

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 MA 6 7 7 6 6 6 6 6 6 4 4 GR 12 12 8 7 9 7 7 9 9 6 5 ME 26 26 26 23 17 14 13 13 12 12 10 PI 68 64 74 61 57 52 53 58 54 53 51 MI 76 59 33 31 28 26 26 14 16 20 19 FB 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 2009 MA 6 7 7 6 6 6 6 6 6 4 4 GR 12 12 8 7 9 7 7 9 9 6 4 ME 26 26 26 23 18 14 13 13 11 11 10 PI 75 65 76 66 60 58 55 62 58 54 55 MI 91 70 44 39 34 35 33 17 20 22 22 FB 3 11 21 32 48 57 61 66 70 70 45

Table: Banks active as lenders per group per year.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

System Heterogeneity: volume per group

Month Volume J99 J01 J03 J05 J07 J09 1e+05 2e+05 3e+05 4e+05 MA b MA l

Month Volume J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 1e+05 2e+05 3e+05 4e+05 GR b GR l Month Volume J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 1e+05 2e+05 3e+05 4e+05 ME b ME l

Month Volume J99 J01 J03 J05 J07 J09 1e+05 2e+05 3e+05 4e+05 PI b PI l

Month Volume J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 1e+05 2e+05 3e+05 4e+05 MI b MI l Month Volume J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 1e+05 2e+05 3e+05 4e+05 FB b FB l

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

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

System Heterogeneity: net volume per group

Month Volume MA J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

Month Volume GR J99 J01 J03 J05 J07 J09 −1.0 −0.5 0.0 0.5 1.0 Month Volume ME J99 J01 J03 J05 J07 J09 −1.0 −0.5 0.0 0.5 1.0 Month Volume PI J99 J01 J03 J05 J07 J09 −1.0 −0.5 0.0 0.5 1.0

Month Volume MI J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00

Month Volume FB J99 J01 J03 J05 J07 J09 −1.0 −0.5 0.0 0.5 1.0

Figure: Net percentage traded volume per group as lender and borrower.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

System Heterogeneity: intratrade time group ∆tp1 ∆tp2 ∆tp3 MA 336 483 190 GR 271 606 951 ME 777 1627 730 PI 7594 3639 7703 MI 13623 9961 7240 FB 6398 7854 3881

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

start end pre-crisis (p1) 2006-07-05 2007-08-08 subprime (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.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

System Heterogeneity: intratrade time group ∆tp1 ∆tp2 ∆tp3 MA 179 175 1068 GR 1043 585 1378 ME 1163 1422 2704 PI 1381 1115 1296 MI 665 996 1949 FB 7167 5528 6812

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

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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 as the weight of the edge.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Random reshuffling In order to compare the values of various network metrics with random null hypothesis that preserve the system’s heterogeneity

We use the edge swap algorithm to generate synthetic

networks to use as null models.

An edge swap selects two ordered pairs (x, y),(u, v) and

swaps the endpoints (target nodes) while keeping the sources fixed. This procedure preserves the degree of each node. Not all edges swaps are accepted during a rewiring process as some swaps can produce graphs that contain self loops or parallel edges. Such sampling bias is reduced in the limit of large or sparse graphs.

We perform 4E swaps per randomisation and average
ver 100 randomizations for each of the 133 empirical

network.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

For directed and weighted representations we can construct a randomisation using the edge swap procedure (that now conserves the vertex in-out strength sequence but not the in-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 have their weights set to 1.

The resulting multigraph is then rewired as a directed

unweighted graph where each edge now indicates a single transactions and the number of edges between u and v correspond to their number of transactions.

The rewired multigraph is then collapsed to a directed

weighted graph via the reverse procedure (i.e. all m directed and unweighted edges between u and v are collapsed into a single edge with weight m).

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Quantitites that are preserved in the randomised

ensemble after the degree-preserving randomisation can be then traced back/explained as consequences of the in and out degrees distributions and thus the degree distribution assessed in terms of its information content in the context of the real-world network.

Quantitites that are preserved in the randomised

ensemble after the strength preserving randomisation can be then traced back/explained as consequences of the strength distributions (or heterogeneity in size).

If system heterogeneity cannot fully explain the dynamics
f the various metrics then we cannot say that bank

behaviour is random.

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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 to monitor the frequency of exchanges between counter parties and compare it with a random null hypothesis that preserves bank’s heterogeneity in strengths (number of trades).

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Example Monthly Networks in p1

1 2 3 5 7 8 12 13 14 17 19 25 29 32 33 34 36 37 38 46 48 49 52 53 56 58 62 65 66 67 68 71 72 76 77 82 85 87 89 100 104 106 110 111 114 116 119 122 123 126 128 129 132 134 135 136 148 149 150 152 153 157 159 160 168 170 172 173 175 177 181 188 191 193 197 199 201 202 205 208 209 210 211 212 213 215 219 221 222 224 226 227 228 229 230 231 233 234 236 237 239 240 241 244 247 248 251 252 257 258 259 260 261 262 263 264 265 266 267 268 271 273 275 276 278 279 280 281 282 283 284 285 286 287 288 290 291 292 293 294 296 297 298 299 301 302 306 307 308 309 310 312 313 314 315 316 317 318 319 320 324 326 327 328 329 330 332

Figure: eMid bank Network in maintenance period 14-Feb-2007 to 13-Mar-2007

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Example Monthly Networks in p2

2 3 5 7 8 12 13 14 17 18 19 29 32 33 34 36 37 38 46 48 49 52 53 56 62 65 66 67 68 71 72 76 77 82 85 100 104 106 110 111 114 116 119 122 123 128 129 132 135 136 148 150 153 157 159 160 168 170 172 173 175 181 191 193 197 199 201 202 205 208 209 210 211 212 213 215 219 221 224 226 227 228 229 230 231 233 234 236 237 239 240 244 247 248 251 252 253 257 258 259 261 263 264 265 266 267 268 271 273 275 276 278 279 280 281 282 283 284 286 287 288 290 291 292 293 294 295 296 297 298 299 302 306 308 310 311 312 313 314 315 316 318 319 320 326 328 329 333 336 337 338 339 340 341 342 344 345

Figure: eMid bank Network in maintenance period 11-Jun-2008 to 08-Jul-2008

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Example Monthly Networks in p2

2

3 5 7 8 12 13 14 17 19 29 32 33 34 36 37 38 46 48 49 52 53 56 62 65 66 67 68 71 72 76 77 82 85 100 104 106 110 111 114 116 119 122 123 128 129 132 135 136 148 150 153 157 159 160 168 170 172 173 175 177 181 191 193 197 199 201 202 205 208 209 210 211 212 213 215 219 221 226 227 229 233 234 236 237 239 244 247 248 251 252 257 258 259 260 261 263 265 266 268 271 278 279 281 282 283 284 288 290 291 292 293 294 296 298 299 306 308 310 312 313 314 315 316 318 319 320 328 329 333 336 337 338 339 341 342 345 346

Figure: eMid bank Network in maintenance period 08-Oct-2008 to 11-Nov-2008

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Basic network metrics

120 140 160 180 200 Month N J99 J01 J03 J05 J07 J09 1000 2000 3000 4000 Month E J99 J01 J03 J05 J07 J09 10 15 20 25 Month <k> J99 J01 J03 J05 J07 J09 0.06 0.08 0.10 0.12 Month E/N(N−1) J99 J01 J03 J05 J07 J09

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

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Information-theoretic description of networks Let ˜ wij = 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 borrowing activity from j are respectively p(li) → p( ˜ wi) =

j

˜ wij p(bj) → p( ˜ wj) =

i

˜ wij

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The expressions p(li|bj) = ˜ wij/ ˜ wj p(bj|li) = ˜ wij/ ˜ wi are respectively the conditional probabilities of observing a transaction with i being the source given that j is the sink and j being the sink given that i is the source. Finally the joint probability of observing a transaction with i being the source and j begin the sink is p(li, bj) → p( ˜ wi, ˜ wj) = ˜ wij

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Following T. Wilhelm and J.Holunder1, we define Lender Entropy: H(L) = −

i
j

˜ wij log

j

˜ wij (1) Borrower Entropy: H(B) = −

j
i

˜ wij log

i

˜ wij (2) Lender Entropy given the borrower is known: H(L|B) =

j

p(bj)H(L|bj) = −

i
j

˜ wij log ˜ wij

k ˜

wkj (3)

1Information theoretic description of networks. Phys A, (385), 2007

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Borrower Entropy given the lender is known: H(B|L) =

i

p(li)H(B|li) = −

i
j

˜ wij log ˜ wij

k . ˜

wik (4) Joint Entropy: H(L, B) = −

i
j

˜ wij log ˜ wij (5) Mutual Information: I(L, B) = H(L, B) − H(L|B) − H(B|L) (6)

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The minimum value (no uncertainty) is achieved when a

borrower only trades with one lender or viceversa H = 0.

The maximum values for all of the above quantities are

Hmax = log(N) except for H(L, B)max = 2 log(N).

Intuitively H(X|Y) is the amount of uncertainty remaining

about X after Y is known, it tells us how much Y spreads its trades (high entropy) or concentrate them with few counter parties (low entropy).

The intuitive meaning of mutual information is the amount
f information that knowing either variable provides about

the other, is a quantity that measures the mutual dependence of the two random variables, it tells us how much banks forms exclusive trading partnerships.

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An an individual bank level the following conditional entropies can be defined: Lender entropy given that the borrower is node j: H(L|bj) = −

i

˜ wij

k ˜

wkj log ˜ wij

k ˜

wkj (7) Borrower Entropy given that the lender is node i: H(B|li) = −

j

˜ wij

k ˜

wik log ˜ wij

k ˜

wik (8) For individual lenders/borrowers the maximum entropy is equal to the Log of their out/in degrees. An alternative approach to quantify the randomness of individual links in a network has been proposed recently by Mantegna’s group.

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0.30 0.35 0.40 0.45 0.50 0.55 Month H(X|Y) / H(X|Y) max J06 J07 J08 J09 J10 H(B/L) H(B/L) random 0.30 0.35 0.40 0.45 0.50 0.55 Month H(Y|X) / H(Y|X) max J06 J07 J08 J09 J10 H(L/B) H(L/B) random

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

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0.05 0.10 0.15 0.20 Month I(X,Y) / I(X,Y) max J06 J07 J08 J09 J10 I(L,B) I(L,B) random

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

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

90 100 110 120 130

months

0.05 0.1 0.15 0.2

fraction of links

bonferroni FDR

Lender Initiated

90 100 110 120 130

months

0.05 0.1 0.15 0.2

fraction of links

bonferroni FDR

Borrower Initiated

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

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Relationship Lending: Results from the literature. The empirical literature highlighted the presence of Relationship Lending in OTC interbank transactions.

Banks develop long-lasting trading relationships on

the interbank market on different maturities (Cocco et

al. 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).

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Interpretations Main interpretations:

Relationships as liquidity insurance (high reserve

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

ON Relationships as a strategic relationship (Banks

transacting outside ON have higher relationships).

Banks continuously monitor each other (banks

characteristics are always important) (Furfine 2001, Affinito 2011).

Before the crisis banks paid a premium for
relationship. The opposite holds after the crisis (role
f information?). (Brauning 2011)

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

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Network Metrics

Affinity

Affinity is a measure of similarity among nodes and is defined as knn,i = 1 ki

j∈V(i)

kj (9) If knn(k) is an increasing function of k, vertices with high degree have a larger probability to be connected with large degree vertices: assortative mixing. A decreasing behavior of knn(k) defines disassortative mixing, in the sense that high degree vertices have a majority of neighbors with low degree, while the opposite holds for low degree vertices.

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Similarly we can define the following four measues kin−out

nn,i

= 1 kin

i

j∈V(i)

kout

j

(10) kin−in

nn,i

= 1 kin

i

j∈V(i)

kin

j

(11) kout−out

nn,i

= 1 kout

i

j∈V(i)

kout

j

(12) kout−in

nn,i

= 1 kout

i

j∈V(i)

kin

j

(13)

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kout−out

nn,i

tell us if the borrowers of a lending bank are lenders themselves and to how many banks they lend.

kin−in

nn,i

tells us if the lenders of a borrowing bank and are borrower themselves and from how many banks they borrow.

Thus kout−out

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Strenght correlations

−0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sin,Sin) sample rw

pearson degree correlation coefficient

J99 J01 J03 J05 J07 J09 −0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sin,Sout) J99 J01 J03 J05 J07 J09 sample rw −0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sout,Sin) J99 J01 J03 J05 J07 J09 sample rw −0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sout,Sout) J99 J01 J03 J05 J07 J09 sample rw

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

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Degree correlations

−0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sin,Sin) sample rw

pearson strength correlation coefficient

J99 J01 J03 J05 J07 J09 −0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sin,Sout) J99 J01 J03 J05 J07 J09 sample rw −0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sout,Sin) J99 J01 J03 J05 J07 J09 sample rw −0.3 −0.1 0.0 0.1 0.2 0.3 Month r(Sout,Sout) J99 J01 J03 J05 J07 J09 sample rw

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

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

Clustering From the point of view of a vertex i four directed clustering coefficients,in, out, middle, cycle can be distinguished depending 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 risk September 11-14, 2012 45 / 54

SLIDE 46

e-MID Dataset Market composition reshuffling Entropy Network Metrics

The in coefficient represents high risk as if the bank i who

has borrowed from two other banks j and k fails to repay its loans then j and k may also fail to settle their obligation that completes the triangle.

The middle coefficient represents the case where the

counter parties j, k of bank i are either borrowing or lending from the other two banks, thus a collapse of i will trigger a possible collapse of the whole triangle.

By contrast out does not increase systemic risk as the

collapse of a bank with two outward (lending) edges will not render the other two banks unable to fulfill the

bligation between them.
Finally cycle does not represent high systemic risk due to

the cyclical relation of liquidity flow.

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SLIDE 47

e-MID Dataset Market composition reshuffling Entropy Network Metrics

Clustering

0.0 0.1 0.2 0.3 0.4 0.5 Month cycle sample rw

clustering coefficients

J99 J01 J03 J05 J07 J09 0.0 0.1 0.2 0.3 0.4 0.5 Month midle J99 J01 J03 J05 J07 J09 sample rw 0.0 0.1 0.2 0.3 0.4 0.5 Month in J99 J01 J03 J05 J07 J09 sample rw 0.0 0.1 0.2 0.3 0.4 0.5 Month

ut

J99 J01 J03 J05 J07 J09 sample rw

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

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SLIDE 48

e-MID Dataset Market composition reshuffling Entropy Network Metrics

Clustering

0.000 0.002 0.004 Month cycle sample rw

clustering coefficients

J99 J01 J03 J05 J07 J09 0.00 0.01 0.02 0.03 0.04 Month midle J99 J01 J03 J05 J07 J09 sample rw 0.000 0.004 0.008 Month in J99 J01 J03 J05 J07 J09 sample rw 0.000 0.005 0.010 0.015 0.020 Month

ut

J99 J01 J03 J05 J07 J09 sample rw

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

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SLIDE 49

e-MID Dataset Market composition reshuffling Entropy Network Metrics

Path lengths

Since none of the maintenance period networks are

strongly connected, i.e. there aren’t directed paths between all pairs in both directions, it becomes problematic to calculates quantities such as the average shortest path length, diameter, etc.

However we can look at the number of pairs in the network

for which a path exists (in both directions).

Let nr be the fraction of vertex pairs that can be reached

via a directed path (of any length). Then fr =

nr N(N−1)is the

fraction of all pairs that can be reached in the graph.

We can further divide fr by edge density Eρ =

E N(N−1) to

get the quantity nr

E which is the number of reachable pairs

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

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

0.4 0.5 0.6 0.7 Month reachable pairs / all pairs J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 5 6 7 8 9 Month (reachable pairs / all 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 the number of pairs reachable by a directed path in units of the number of edges in the system.

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SLIDE 51

e-MID Dataset Market composition reshuffling Entropy Network Metrics

Centralities Centrality measures the importance of a vertex in a network in terms 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 all

ther vertices .
The in and out centralities measure the in reps. out degree
f a vertex normalised by the number of all vertices.

It is also interesting to calculate the centralities for each capitalization group separately. In this case the centrality of a group i is defined as the average centrality over the group members.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

0.000 0.004 0.008 Month betweenness sample rw

centralities

J99 J01 J03 J05 J07 J09 0.0 0.1 0.2 0.3 0.4 Month closeness sample rw J99 J01 J03 J05 J07 J09 0.00 0.05 0.10 0.15 0.20 Month in centrality sample rw J99 J01 J03 J05 J07 J09 0.00 0.05 0.10 0.15 0.20 Month

ut centrality

sample rw J99 J01 J03 J05 J07 J09

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

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

0.00 0.01 0.02 0.03 0.04 0.05 0.06 month betweeness centrality J99 J00 J01 J02 J03 J04 J05 J06 J07 J08 J09 MA GR ME PI MI FB

Figure: The unweighted betweenness centrality averaged over banks

f each group.

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e-MID Dataset Market composition reshuffling Entropy Network Metrics

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 month betweeness centrality 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 of each group.

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