missing&links: a&global&study&onuncovering&financial&...

Missing  Links:A  global  study  on  uncovering  financial  network  structure  from  partial  data

Presented  by  Rod  Garratt,  UCSB

These  views  do  not  necessarily  represent  the  views  of  any  of  the  many  institutions  involved  in  the  project.

Background

• The  working  paper  is  the  final  product  of  the  BCBS  Research  Task  Force  (RTF)  working  group  on  Liquidity  Stress  Testing.

• The  RTF  will  shortly  publish  a  paper  entitled  Making  Supervisory  Stress  Tests  More  Macroprudential:  Considering  Liquidity  and  Solvency  Interactions  and  Systemic  Risk.

• We  also  contributed  two  boxes  for  the  BCBS  Task  Force  on  Coherence  and  Calibration  (CCTF).

Authors

Kartik Anand (DB),  Iman van  Lelyveld (DNB),  Adam  Banai (MNB),  Soeren Friedrich  (BuBa),  Rodney  Garratt  (UCSB),  Gregorz Halaj (ECB),  Bradley  Howell  (BoC),  Ib Hansen  (DB),  SerafinMartinez  Jaramillo  (BoM),  Hwayun Lee  (BoK),  Jose  Luis  Molina-­‐Borboa (BoM),  Stefano  Nobili(BoI),  Sriram Rajan (OFR),  Dilyara Salakhova(BoF),  Thiago Christiano Silva  (BCB),  Laura  Silvestri (BoE),  and  Sergio  Rubens  Stancatode  Souza  (BCB)

Motivation

• Networks  are  ubiquitous  in  finance– Credit  exposures– Funding  structures– Derivative  contracts

• Network  analysis  is  a  big  industry– contemporaneous:  mapping,  reporting,  visualization

– forward-­‐looking:  stress-­‐testing,  risk-­‐analytics  (contagion  analysis  and  systemic  risk)

Motivation

• Network  analysis  often  requires  high-­‐frequency  and  granular  micro-­‐level  data,  which  is  not  always  available– credit  registers  or  supervisory  reports  may  only  cover  exposures  exceeding  a  threshold  that  is  defined  either  in  terms  of  the  absolute  amount  of  the  exposure  or  as  a  fraction  of  the  lender’s  capital.  

– reporting  of  credit  lines  instead  of  actual  exposures– exclusion  of  off-­‐balance  sheet  items

Motivation

• Data  gathering  (by  central  banks  and  banking  supervisors)  was  – at  best  – patchy  in  the  run-­‐up  to  the  financial  crisis

• The  G-­‐20  Data  Gaps  Initiative  has  strengthen  the  reporting  and  collection  of  financial  data  by  member  countries

• Data  are  generally  not  available  beyond  regulatory  perimeters

• Most  empirical  research  focus  on  networks  from  a  single  country  ignoring  international/non-­‐banking  sector  links

Motivation

• Overcoming  data  limitations  – network  reconstruction• Several  algorithms  currently  exist:– maximum  entropy  (most  common)– minimum  density– probability  map

• Sarlin et  al.  (2014)  use  maximum-­‐entropy  to  construct  a  network  of  links  between  EU  banks  and  non-­‐bank  entities

• Several  studies  use  maximum  entropy  to  reconstruct  interbank  exposures  and  subject  them  to  stress-­‐tests  (e.g.,  Upper,  2011)– may  be  unsatisfactory

a)  Real  exposures  network b)  ME  generated  exposures  network

Motivation

Research  questions

• What  is  the  best  method  to  reconstruct  networks?

• What  is  a  good  fit  and  how  can  we  back-­‐test  results?

• Do  different  types  of  networks  need  different  methods?

By  the  numbers

• 17  researchers  conducted  a  horse-­‐race  between  7  network  reconstruction  methods

• Collate  and  present  summary  statistics  for  25  different  financial  networks  spanning  13  jurisdictions

• Rules-­‐of-­‐thumb  to  make  an  informed  choice  on  the  appropriate  network  reconstruction  method  to  use

Methodology

• We  start  with  24  different  domestic  financial  networks  and  one  international  network  for  which  we  have  detailed  granular  data  on  the  full-­‐set  of  bilateral  exposures

• Postulate  we  only  know  the  aggregate  exposures• Reconstruct  the  networks  using  the  aggregate  exposures only

• Compare  reconstructions  with  the  actual  networks

Methodology

Step  1   Original  network

Step  2  Compute  the  totals  or  marginals

Step  3  “Forget”  actual  network

Step  4  Reconstruct  network  (7  algorithms)

Step  5    Compare  reconstruction  with  actual  network

Collection  of  networksType Number CountriesInterbank 13 Brazil,  BIS,  Canada,  Denmark,  France,  Germany,  Hungary,  

Italy,  Mexico  (x3),  NetherlandsPayment 5 Brazil,  Mexico  (x3),  USRepo 2 Denmark,  MexicoFXS 1 HungaryCDS 2 UK,  USEquity 1 MexicoDerivatives 1 Mexico

Summary  of  interbank  networks  I

Motivation Research questions Methodology Data Results Conclusion References

Summary of interbank networks

InterbankBIS03 BR02 CA01 DE01 DK01 FR01 HU06 IT01 KR01 MX0103 MX0303 MX0602 NL01

Num of links 812 418 29 10675 77 46 131 3084 263 420 129 50 576Density 87.3 4.1 96.7 3.3 42.3 41.8 14.1 1 85.9 23.3 7.1 2.8 2.4Avg Degree 26.2 4.1 4.8 18.9 5.5 4.2 4.2 5.6 14.6 9.8 3 1.2 3.7Med Degree 28 1.5 5 14 5 4 3 3 15 10 2 1 1Assortativity -.12 -.36 -.63 -.3 -.42 -.44 -.43 -.17 -.26 -.36 -.2 -.48Clustering 22.4 3.1 17.8 41.3 21.5 10.8 23 19.7 21.2 16.2 6.5 3.4 7.5Lender Dep 28.8 63.9 54.4 42.9 37.5 38.7 51.1 74.9 31.6 56.6 76.3 84.4 79.4Borrower Dep 31.6 60.2 46.3 71 39.6 39 44.9 88.2 24.9 53.4 63.6 78.8 73.1Mean HHI Assets .17 .41 .41 .29 .27 .18 .41 .67 .19 .41 .51 .44 .62Median HHI Assets .15 .31 .42 .22 .25 .22 .29 .71 .17 .33 .47 .46 .73Mean HHI Liabilities .19 .42 .35 .6 .25 .29 .2 .82 .15 .38 .33 .32 .42Median HHI Liabilities .15 .35 .28 .6 .23 .26 .08 1 .14 .28 .28 0 .35Core Size (% banks) 77.4 9.8 66.7 6.5 42.9 36.4 22.6 3.2 77.8 32.6 16.3 7 7Error score (% links) 4.4 46.7 3.4 11.2 14.3 10.9 17.6 19.9 3.4 23.6 38 76 26.9

Summary of other networks

Summary  of  interbank  networks  II

Appendix

Summary of other networks

Payments CDS Repo OtherBR04 MX0703 MX0803 MX0903 US02 MX0503 OFR02 UK01 DK02 MX0203 HU07 MX0403

Num of links 1396 800 149 302 169027 70 3267 2004 18 74 221 99Density 13.3 43.3 8.3 16.3 .5 3.9 .6 1.8 13.6 4.1 2.3 5.5Avg Degree 13.6 18.6 3.5 7 29.4 1.6 4.4 6 1.5 1.7 2.2 2.3Med Degree 6 18 0 0 10 1 2 1 .5 1 1 1Assortativity -.52 -.45 -.37 -.26 -.27 -.15 -.79 -.72 -.73 -.16 -.57 -.31Clustering 12.2 17.9 5.8 9.5 14.7 2.8 18.2 13.4 3.5 3.2 1.3 5.9Lender Dep 62.9 52.2 63.9 39.2 60.8 74.4 69.4 59.7 71.4 75.2 77 67.1Borrower Dep 59.3 53.5 63.7 49.1 61.4 71 69.1 63.4 95.1 69.7 74.1 60Mean HHI Assets .5 .38 .27 .12 .46 .44 .51 .36 .32 .45 .39 .31Median HHI Assets .41 .3 0 0 .38 .38 .44 .2 .11 .45 .12 .12Mean HHI Liabilities .47 .41 .24 .17 .49 .3 .31 .51 .84 .3 .47 .21Median HHI Liabilities .39 .3 0 0 .42 0 0 .46 1 0 .35 0Core Size (% banks) 20.4 44.2 18.6 32.6 2.6 9.3 2 5.4 16.7 9.3 7.1 11.6Error score (% links) 10.7 4 12.1 2.6 27.4 61.4 5.2 .9 22.2 55.4 33.5 56.6

Return

Summary  of  interbank  networks  III

Abbreviations:  Brazil  (BR),  BIS  cross  border  banking  system  exposure  data  (BIS),  Canada (CA),  Denmark  (DK),  France  (FR),  Germany  (DE),  Hungary  (HU),  Italy  (IT),  South  Korea  (KR),  Mexico (MX),  the  Netherlands  (NL),  the  United  Kingdom  (UK),  and  the  United  States  (US).

Interbank Payment Repo CDS OtherCountries  CA  BR  BIS  DK  FR  HU  IT  KR  

MX  BR  MX  US  DK  MX  MX  US  UK  HU  MX

Number  of  links 30  (587)  3084 149  (34335)  169027 18  (46)  74 70  (1317)  3267 99  (160)  221Density  (%) 1  (44)  100 1  (16)  43 4  (9)  14 1  (3)  4 2  (4)  5Ave.  Degree 4  (9)  26 3  (14)  29 2  (2)  2 2  (4)  5 2  (2)  2Med.  Degre 2  (8)  28 0  (7)  18 1  (1)  1 1  (1)  2 1  (1)  1Assortativity -­‐.44  (-­‐.31)  -­‐.12 -­‐.52  (-­‐.38)  -­‐.26 -­‐.73  (-­‐.44)  -­‐.16 -­‐.79  (-­‐.52)  -­‐.15 -­‐.57  (-­‐.44)  -­‐.31Clustering 3  (26)  100 6  (12)  18 3  (3)  3 3  (11)  18 1  (4)  6Core  (%) 3  (43)  83 3  (24)  44 9  (13)  17 2  (8)  13 7  (9)  12Error  score  (%) 0  (16)  47 3  (11)  27 22  (39)  55 2  (23)  61 33  (45)  57

Types  of  Networks

Reconstruction  methodsAuthors Code Short  DescriptionUpper  (2011) Maxe Maximum  entropy  methodDrehmann and  Tarashev (2013)

Dreh Perturbed  maximum  entropy  method

Baral and  Fique (2012) Bara Copula  to  allocate  the  marginalsAnand  et  al.  (2015) Anan Minimum  Density  method  which  minimizes  the  number  

of  links  required  for  distributing  a  given  volume  of  loansHalaj and  Kok (2013) Hala Probability  map  for  the  likelihood  of  linksMusmeci et  al.  (2013) Batt Fitness  model  that  determines  the  likelihood   of

linkageMastrandrea et  al.  (2014)

Mast Weighted  configuration  model  for  the  distribution  of  the  reconstructed  networks;  constraints  are  onlysatisfied  on  average

Maxe

“A  source  of  information  that  is  widely  available  is  banks’  balance  sheets.  In  contrast  to  credit  registers  they  do  not  identify  point-­‐to-­‐point  exposures  …,  but  only  contain  information  on  total  interbank  lending  and  borrowing  of  the  reporting  institution.  Nevertheless,  this  data  can  still  be  used  to  draw  inferences  on  bilateral  exposures,  although  the  researcher  has  to  make  assumptions  on  how  banks  spread  their  interbank  lending.”  (Upper  2011)

Maxe• Assume  banks  spread  their  lending  as  evenly  as  possible  

given  the  asset  and  liability  positions  reported  in  the  balance  sheets  of  all  other  banks.  

• In  technical  terms,  this  corresponds  to  maximizing  the  entropy  (ME)  of  interbank  linkages

• Upper  and  Worms  (2004)  and  Elsinger,  Lehar  and  Summer  (2006a)  extended  ME  to  handle  zero  entries  on  the  diagonal  of  the  matrix

• Corresponds  to  the  most  likely  structure  of  lending  given  the  row  and  column  sums  of  the  interbank  matrix  as  well  as  any  other  pieces  of  information  that  has  been  incorporated  in  the  estimation  program  as  a  constraint.  

• From  a  practical  point  of  view,  ME  yields  a  unique  estimate.

Maxe• Three  reasons  why  ME  might  not  be  a  particularly  good  description  of  reality.  1. First,  fixed  costs  for  screening  of  potential  borrowers  and  

monitoring  loans  may  render  small  exposures  unavialable

2. relationship  lending  may  limit  the  number  of  counterparties  of  any  one  bank  and  could  thus  lead  to  a  higher  degree  of  market  concentration  than  suggested  by  ME

3. ME  results  in  all  banks  holding  essentially  the  same  portfolio  of  interbank  assets  and  liabilities,  differing  only  by  size  and  by  the  fact  that  no  bank  has  any  claims  on  itself  – too  homogeneous,  misleading  results  on  contagion

Modifications  of  ME  approach

• Formally,  maximum  entropy  involves  maximizing  the  entropy  function

Max  –Σi,jXijlog(Xij/Qij)subject  to  constraints  (total  assets  and  liabilities),  where  (Qij)  is  the  prior  information  on  bilateral  exposures– product  of  marginals in  simplest  case

Dreh,  Bara  

• Modifications  that  assign  off-­‐diagonal  elements  of  Qij to  be  zero  (in  addition  to  diagonal  elements)  – Dreh =  randomly– Bara  =  impose  core-­‐periphery  structure– Uses  RAS  algorithm• an  iterative  algorithm  for  estimating  cell  values  of  a  contingency  table  such  that  the  marginal  totals  remain  fixed  and  the  estimated  table  decomposes  into  an  outer  product.

Anan• Anand,  K.,  B.  Craig,  G.  von  Peter  (2014)• Benchmark  model  which  minimizes  number  of  necessary  links  for  transmitting  a  certain  volume  of  interbank  loans:  a  “minimum  density”  (MD)  approach.

• Most  probable  links  are  identified  and  loaded  with  the  largest  possible  exposures,  maintaining  consistency  with  the  aggregate  lending  and  borrowing  for  each  bank.

Anan• Important:  establishing  and  maintaining  links  is  costly,  banks  do  not  spread  lending  and  borrowing  evenly  across  all  possible  counterparts  because  of  monitoring  costs.  

• Real  interbank  networks  are  sparse  and  present  the  phenomena  of  disassortativemixing.  

• The  MD  approach  is  formulated  as  a  constrained  optimization  problem.

Hala

• Halaj and  Kok (2013)  1. a  random  pair  of  banks  is  chosen  with  the  same  

probability  and  the  link  is  created  with  a  probability  given  by  a  probability  map  (likelihood  of  connection  based  on  aggregate  data).  

2. If  the  link  is  created  then  a  random  (uniform  [0,1])  number  is  generated  which  indicates  the  percentage  of  interbank  liabilities    of  the  first  bank  from  the  pair  coming  from  the  second  bank  in  such  pair,  this  assignment  is  consistent  with  the  asset  side  of  the  second  bank.  

3. This  process  is  repeated  until  no  more  liabilities  need  to  be  assigned.  

Mast

• Mastrandreaet  al  (2014)• Weighted  configuration  model• Connects  nodes  to  allocate  (match  together)  node  strengths

• Uses  maximum  entropy  to  allocate  strengths  in  unbiased  fashion

Batt

• Musmeci et  al  (2013)• Probability  of  link  depends  on  fitness  of  nodes• A  known  non-­‐topological  property  is  assumed  to  be  proportional  to  fitness  used  (in  case  of  Fedwiremax  net  debit  caps  were  used)

• Using  computed  probabilities  a  series  of  adjacency  matrices  are  sampled

Similarity  measuresMeasure Short  DescriptionHamming Sum  of  the  difference  between  the  original  and  estimated  adjacency  

matrices.  This  measure  captures  the  number  of  instances  where  the  original  matrix  had  a  link,  but  the  estimated  did  not  (false  negative)  and  where  the  original  matrix  did  not  have  a  link,  but  the  estimated  matrix  did  (false  positive);  range:  positive  scalar

Jaccard The  number  of  links  belonging   to  both  the  original  and  estimated  networks  divided  by  the  number  of  links  that  belong  to  at  least  one  network;  range:  [0,  1].

Cosine Cosine  of  angle  between  the  vectorized original  network  and  estimated  network;  range:  [0,  1]  since  no  negative  values.

Jensen Jensen-­‐Shannon  divergence  between  networks,  where  we  normalize  all  entries  in  the  networks  to  sum  up  to  one;  range:  [0,1).  

Accuracy The  percentage  of  true  positive  and  true  negative  links.  range:  [0,1).  

ResultsMotivation Research questions Methodology Data Results Conclusion References

Results

Country Type Code Hamming Jaccard Cosine Jensen Accuracy

Brazil Interbank BR02 Anan* Batt* Maxe Batt Anan*Payment BR04 Anan* Batt Bara Maxe Anan*

BIS Interbank BIS03 Dreh* Bara, Dreh,Maxe

Bara, Maxe Bara, Maxe Bara, Dreh,Maxe

Canada Interbank CA01 Bara, Dreh,Maxe

Bara, Dreh,Maxe

Bara, Maxe Bara, Maxe Bara, Maxe

Denmark Interbank DK01 Anan* Bara, Dreh,Maxe

Bara, Maxe Batt Anan*

Repo DK02 Anan Anan Bara, Maxe Anan AnanFrance Interbank FR01 Bara, Dreh,

MaxeBara, Dreh,Maxe

Bara, Maxe Bara Bara, Dreh,Maxe

Germany Interbank DE01 Anan Batt Anan* Batt Anan

Hungary Interbank HU06 Anan Hala* Bara, Maxe Batt AnanFX Swaps HU07 Hala Bara, Dreh,

MaxeBara, Maxe Bara Hala

Italy Interbank IT01 Hala Anan* Bara, Maxe Batt HalaKorea Interbank KR01 Bara, Dreh,

MaxeBara, Dreh,Maxe

Bara, Maxe Bara Bara, Dreh,Maxe

Results

Motivation Research questions Methodology Data Results Conclusion References

ResultsCountry Type Code Hamming Jaccard Cosine Jensen Accuracy

Mexico

InterbankMX0103 Anan* Bara, Dreh,

MaxeBara Batt Anan*

MX0303 Anan Bara, Dreh,Maxe

Hala Batt Anan

MX0603 Anan Bara, Dreh,Maxe

Bara, Maxe Batt Anan

Repo MX0203 Anan* Bara, Dreh,Maxe

Bara Batt Anan*

Equity MX0403 Anan Hala Bara, Maxe Batt AnanDerivatives MX0503 Anan Bara, Dreh,

MaxeMaxe Batt Anan*

PaymentsMX0703 Batt Bara, Dreh,

MaxeBara, Maxe Bara, Maxe Anan*

MX0803 Anan* Bara, Dreh,Maxe

Maxe Batt Anan

MX0903 Bara, Dreh,Maxe

Bara, Dreh,Maxe

Bara, Maxe Maxe Bara, Maxe

Netherlands Interbank NL01 Anan Anan Dreh* Batt Anan*UnitedKingdom

CDS UK02 Batt, Hala Anan Batt*

United States

Payments US02 Anan Anan Bara Bara Anan

CDS OFR03 Hala Batt Maxe Bara* Hala

Figure 1: CA01

20

Figure 21: MX0903

40

Figure 22: NL01

41

US02• Fedwire Funds  transactions  data• Very  large,  but  very  sparse  network

– Roughly  7000  participants,  (we  only  included  banks  that  made  or  received  at  least  one  payment  during  the  sample  period)  so  this  reduced  #  to  ~6000.

– allows  for  60002 =  ~36  million  links  – Less  than  200,000  links  in  original  matrix– Max  entropy  methods  (Maxe,  bara,  dreh)  do  terribly  on  similarity  measures  based  on  correct  links:  Hammand~30  million;  Jaccard near  0

– Anan  does  quite  well:  same  order  of  magnitude  for  Hammand• Mast  and  Batt did  not  converge

Results

• The  “winner”  depends  on  the  similarity  measure– Accuracy:  Anan– Jaccard:  Maxe,  Dreh and  Bara– Jensen:  Batt

• Results  also  depend  on  network  sparcity/rules-­‐of-­‐thumb– Sparse  networks:  Anan,  Batt and  Hala– Dense  networks:  Maxe,  Dreh and  Bara

• Mast  is  also  strong  performer  under  the  Accuracy  and  Hamming  metrics

Conclusion

• Horse  race  of  network  reconstruction  methods

• Unique  dataset  of  25  networks  from  13  jurisdictions

• No  universal  winner• Winners  depend  on  the  similarity  measure  and  the  structure  of  underlying  network

• No  free  lunch

More  to  be  done…

• What  errors  matter?• Identification  of  “worse-­‐case  scenario”  re-­‐constructions

Thank  You!

References

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