missing&links: a&global&study&onuncovering&financial&...
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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
Anand, K., B. Craig, and G. von Peter (2015): “Filling in the Blanks : Interbank Linkages and Systemic Risk,” Quantitative Finance, 15, 625–636.Baral, P. and J. Fique (2012): “Estimation of Bilateral Connections in a Network: Copula vs. Maximum Entropy,” Mimeo.Musmeci, N., S. Battiston, G. Caldarelli, M. Puliga, A. Gabrielli, (2013) “Bootstrapping Topological Properties and Systemic Risk of Complex Networks Using the Fitness Model”, Journal of Statistical Physics, 151 (3-‐4) 720-‐734.Drehmann, M. and N. Tarashev (2013): “Measuring the Systemic Importance of Interconnected Banks,” Journal of Financial Intermediation, 22, 586–607.Halaj, G. and C. Kok (2013): “Assessing Interbank Contagion using Simulated Networks,” Computational Management Science, 10, 157–186.Mastrandrea, R., T. Squartini, G. Fagiolo, and D. Garlaschelli (2014): “Enhanced Reconstruction ofWeighted Networks from Strengths and Degrees,” New Journal of Physics, 16, 043022.Sarlin, P., T. A. Peltonen, and M. Rancan (2014): “Interconnectedness of the banking sector as avulnerability to crises,” Mimeo Goethe University Frankfurt.Upper, C. (2011): “Simulation methods to assess the danger of contagion in interbank markets,” Journal of Financial Stability, 7, 111–125.