## _wp10107

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### Too-Connected-to-Fail (TCTF) risk: characterization and implications
- Concept:
  - The failure of one institution could lead to successive rounds of failures of other institutions via direct and indirect linkages.
  - Direct linkages arise from balance sheet claims or reliance on credit lines; indirect linkages arise through derivatives and securities with market values linked to another institution’s failure.
- Regulatory relevance:
  - Prudential regulation and systemic risk surveillance require methods to assess TCTF risk as emphasized by BIS, FSB, and IMF (2009a, b).
  - Balance sheet-based network analysis is a practical approach suitable for data available to supervisors and cross-country banking data (BIS); it complements market-based methods when secondary market prices are available.
- Distinctions among systemic concepts:
  - TCTF is not necessarily Too-Big-to-Fail (TBTF).
  - TCTF differs from Too-Many-to-Fail (TMTF); in TMTF simultaneous failures arise from herd behavior, not cascading failures caused by a single institution.

### Key cross-country findings from balance sheet-based network analysis
- Global (BIS cross-country bank claim data for reporting jurisdictions):
  - Main sources of risk in the global banking system are shocks that could impair the solvency of banks based in the United States and the United Kingdom.
  - Among BIS reporting jurisdictions in Latin America:
    - The banking system in Brazil appears to be the most resilient to global shocks.
    - The banking system in Chile may be more exposed to funding risk from Spanish banks.
- Country-specific (Chile) summary:
  - Main domestic sources of risk are shocks affecting banks’ claims on households and domestic corporations as publicly reported.
  - To a lesser extent, exposure to shocks affecting banks domiciled in Spain and the United States.
  - On balance sheet data, no bank operating in Chile appears to be a TCTF institution.
  - Surveillance should focus on linkages with foreign banks and the creditworthiness of claims on the non-bank financial sector.

### Graph-theoretic representation and analytical approach
- Representation:
  - Financial system mapped as a directed graph (digraph): institutions as vertices; directed edges (arcs) represent claims.
  - Example depiction: a stylized digraph of seven banks with inbound arcs representing claims (e.g., bank 7 has claims on banks 4 and 6).
- Practical utility:
  - Digraph mapping visualizes linkages and can guide regulatory perimeter design.
  - Balance sheet-based network analysis builds on simple accounting identities and simulations to analyze credit, funding, and OBS shocks.
- Modelling scope:
  - The paper describes simple tools (balance sheet-based network analysis) and notes that more sophisticated graph theory–based models are discussed in the appendix.

### A. The Basic Accounting Identity
- Framework:
  - Equality between asset side and liability side of a bank’s balance sheet underpins the analysis.
  - Asset side includes claims on financial institutions, corporations, households (loans, receivables, equity, debt securities).
  - Liability side includes shareholders’ equity (bank capital) and creditors including depositors.
- Capital as cushion:
  - Bank capital is the first cushion; if asset declines exceed bank capital, the bank defaults.
- Default propagation mechanisms:
  - Direct losses on claims owed by the defaulted bank reduce surviving banks’ capital.
  - Funding withdrawals can force fire-sales, generating mark-to-market losses across institutions.
- Regulatory note:
  - A “default” can be treated as capital falling below the regulatory minimum by reducing effective capital by the minimum regulatory capital.

### B. Credit Shocks
- Definition: losses on the asset side due to defaults of counterparties; loss given default > 0 (recovery ratio < 100 percent).
- Default condition (as presented):
  - Capital − ∑ Claims on defaulted banks × Loss Given Default ≤ 0
- Simulation procedure:
  - Set an initial bank default; apply the equation to other banks; iterate until no new defaults occur.
- Assumption: surviving banks can replace funding from defaulted banks without major difficulty for pure credit-shock analysis.

### C. Funding Shocks
- Definition: liability shock from sudden withdrawal of funding.
- Under normal conditions:
  - Assets sold equal funding loss; capital unaffected; leverage declines.
- Under stressed/fire-sale conditions:
  - Forced asset liquidations at depressed prices generate capital losses and can cause defaults; collusive markdowns may accentuate losses.
- Default condition with funding and credit effects (as presented):
  - Capital − ∑ Claims on defaulted banks × Loss Given Default − ∑ funding withdrawal × (1 − replacement ratio) × fire sale loss ratio ≤ 0
- Use: simulate resilience to funding shocks and contagion from initial withdrawals.

### D. Risk Transfer Shocks and Off-Balance Sheet (OBS) Exposures
- Definition: losses from OBS exposures; requires data on OBS exposure size and provisions.
- Extended balance-sheet treatment includes OBS exposures and provisions.
- Default condition under OBS shock (as presented):
  - Capital − Off‑balance sheet exposure + provisions ≤ 0
- OBS-induced defaults can trigger subsequent credit and funding shocks.
- Practical note: OBS analysis requires additional data not always publicly available.

### III. BALANCE SHEET-BASED NETWORK ANALYSIS IN PRACTICE — scenarios and parameter assumptions
- Standard parameter assumptions used in simulations:
  - Loss given default in credit shock scenario set equal to 70 percent.
  - In funding shocks: only 40 percent of the funding can be replaced.
  - Forced sales at fire sale prices 50 percent below book value.
- Risk transfer shocks excluded from global analysis for aggregate data limitations; in Chile, credit risk transfers between institutions are negligible.
- Caveat: calculations assume extreme adverse scenarios (bank/system fails or sudden funding withdrawal — interpreted as very low probability events).

### A. TCTF Risk Analysis in Mature and Emerging Market Countries
- Data source: BIS consolidated banking statistics (immediate borrower and ultimate borrower bases).
  - Consolidated statistics capture a country’s aggregate banking system claims on the rest of the world.
  - Immediate borrower basis: contractual claims by head office and branches assigned to home country.
  - Ultimate borrower basis: accounts for risk transfers.
- Limitations:
  - Aggregate BIS data approximate interbank exposures and include claims on non-bank financial institutions, non-financial institutions, and households.
  - The analysis avoids inferring risk transfers from immediate vs ultimate borrower differences due to aggregation concerns.
- Country sample: 20 countries using BIS consolidated claims on an immediate borrower basis for Q1 and Q3 2009 (including Brazil, Chile, Mexico; Austria, Belgium, Canada, Denmark, France, Germany, Greece, Ireland, Italy, Japan, Netherlands, Portugal, Spain, Sweden, Switzerland, United Kingdom, United States).
- Failure criterion for national banking system: losses from shocks exceed aggregate capital in the system.
- Key empirical findings (2009, simulations using stated assumptions):
  - Shocks originating in the UK and the US may cause failures in most countries; banks headquartered in the UK and the US could be TCTF globally.
  - Systemic risk from German banking system increased two-fold between March 2009 and September 2009.
  - French-origin shocks could heavily impact Belgium and Netherlands; Swedish banks could be exposed to shocks to Danish banks.
  - Chile’s banking system was most exposed among analyzed Latin American countries to funding risks from Spanish banks.
  - Brazilian banking system appears robust to adverse shocks from major banking systems.
- TCTF measures described:
  - Vulnerability measure: average capital loss a national banking system suffers due to shocks in other countries.
  - Risk measure: average capital loss induced by the failure of a country’s banking system on other countries.
  - Example: Austria loses on average 23 percent of its capital due to shocks in other banking systems; shocks to Austria induce average losses of around 5 percent in other countries.
- Overall 2009 pattern: both TCTF risks and vulnerabilities increased during 2009.
- Contagion dynamics: absence of direct bilateral exposure does not preclude simultaneous failures if both systems connect to a common failing system.

### B. Chile: TCTF Risk Analysis in an Advanced Emerging Market Economy
- Data source: Monthly Report on Financial Institutions (SBIF) — claims and liabilities vis-à-vis central bank, foreign banks, corporate, nonbank financial and household sectors for twenty domestic banks, foreign subsidiaries, and five foreign branches (one- to two-month lag).
- Construction of interbank exposure matrix:
  - In absence of detailed counterparty data, amounts owed are divided equally among all banks reported as having claims on other banks (assumes maximum diversification).
  - Alternatives noted: proportional assignment by claimant size, entropy techniques.
- Modeling assumptions:
  - Foreign banks modeled as a single sector (conservative assumption implying simultaneous shocks across all foreign banks).
  - Network augmented to include BCCH, non-bank financial institutions, corporate sector, household sector.
    - Non-bank financial institutions included only as creditors.
    - Corporate sector included only as debtor.
    - Households included as both depositors and borrowers.
- Empirical outcomes (January, July, December 2009):
  - Domestic interbank exposure relatively unimportant; a single bank default would not lead to further defaults in most cases.
  - Example average capital loss changes between January and December 2009:
    - Bank 5: declined from 10.2 percent to 1.1 percent.
    - Bank 3: declined from 7.4 percent to 0.7 percent.
    - Bank 4: increased from 0.3 percent to 6.9 percent; capital losses concentrated in Bank 12 (35 percent), Bank 21 (54 percent), Bank 25 (47 percent).
  - Trend: domestic banks reduced claim exposure to corporate and household sectors during 2009:
    - Number of banks that could default after system-wide corporate sector collapse declined to 14 from 17.
    - Average capital losses from corporate sector defaults declined to 76 percent from 82 percent.
    - Average capital losses from household loan defaults fell to 59 percent from 68 percent.
  - Funding reliance shifts in 2009:
    - Increased reliance on wholesale funding from non-bank financial institutions: a sudden withdrawal of non-bank financial deposits in December 2009 could cause problems in Banks 16, 17, 18, and 20 (up from 2 banks in January 2009).
    - Central bank funding importance: average capital losses would be approximately 3 percent if central bank funding unavailable; concentrated impacts in December 2009 on Bank 4 (32 percent), Bank 11 (24 percent), Bank 13 (22 percent).
- TCTF measures for Chile:
  - TCTF vulnerability: average capital loss a bank suffers from shocks to other banks.
  - TCTF risk: average capital loss a bank induces on other banks.
  - Excess capital losses: banking-system capital losses excluding losses of the triggering bank; used as a TCTF risk metric (incremental/marginal contribution).
  - Example: January 2009, Bank 3 accounted for 6.7 percent of total system capital; a hypothetical credit shock to Bank 3 would cause 7.9 percent total system capital loss, of which 1.2 percent is excess capital loss attributable to exposures to Bank 3 (TCTF risk equivalence 1.2 percent of system capital).
  - By December 2009, excess capital losses were relatively small, at most equal to 1½ percent for shocks from Bank 4.
- Bottom-line empirical conclusion for Chile:
  - TCTF risk in the domestic banking system is relatively small.
  - Financial surveillance should focus on linkages with foreign banks and non-bank financial institutions.

### IV. CONCLUSIONS (Key takeaways)
- Interconnectedness and TCTF externalities amplified and transmitted shocks during the 2008/9 global financial crisis.
- Balance sheet-based network analysis:
  - Captures interconnectedness and TCTF risk using publicly available balance sheet data.
  - Uses simple accounting identities rather than complex graph-theory mathematics.
  - Is easy to implement where balance sheet data are available and provides results with clear economic interpretation.
- Empirical highlights:
  - Global analysis (BIS cross-country claims) indicates shocks from UK and US banks (and to a lesser extent Germany) as main global risk sources in 2009 — interpret cautiously due to data aggregation.
  - For Chile, bank-level data suggests low domestic TCTF risk; surveillance should focus on foreign bank and non-bank financial institution linkages.
- Limitations and complementarities:
  - Balance sheet data disadvantages: potential manipulation, reporting lags, and timeliness issues when exposures change rapidly.
  - Complement with market-based measures (security prices, common economic factors) to address some disadvantages; note market prices may reflect liquidity and technical drivers beyond fundamentals.

### APPENDIX I — Graph theory framework, properties, and policy considerations
- Framework:
  - Graph theory is the natural analytical framework once the system is mapped into a digraph: banks as vertices; bilateral exposures as arcs.
- Topological properties and metrics:
  - size: the number of institutions in the system;
  - connectivity: the relative number of existing linkages to the maximum number of linkages;
  - clustering coefficient: probability that two institutions “close” to a third are also “close” to each other.
- Applications and challenges:
  - Graph-theoretic models have been applied extensively to payment and settlement systems; extending to complex banking systems is difficult due to heterogeneity of cross-claims.
  - Recent stylized models (Erdős–Rényi, small-world, BIS-based topological analyses) explore contagion implications.
  - Calibrating graph theory–based models with real data is a major challenge for operational surveillance; hence, applied policy work often uses simpler balance sheet approaches described above.

*Source: Appendix I. Graph Theory-Based Network Models.*

### Appendix I.  Graph Theory-Based Network Models .............................................................. 22

### Appendix I.  Graph Theory-Based Network Models

### Too-Connected-to-Fail (TCTF) risk: characterization and implications
- The recent financial crisis experienced in 2008/9 raised concerns on adverse consequences associated with externalities inherent in the financial system, in particular interconnectedness risk, or the Too-Connected-to-Fail (TCTF) problem.
- Simplistic characterization: the failure of one institution could lead to successive rounds of failures of other institutions in the system via direct and indirect linkages.
  - Direct linkages arise from balance sheet claims that expose one institution to the default of other institutions, or from reliance on credit lines that can be withdrawn abruptly.
  - Indirect linkages can arise through derivatives contracts and securities with market values linked to the failure of an otherwise unrelated institution.
- Regulatory and surveillance relevance:
  - Prudential regulation and systemic risk surveillance call for methods to assess TCTF risk as emphasized by BIS, FSB, and IMF (2009a, b).
  - Balance sheet-based network analysis offers a practical approach suitable for data typically available to financial supervisors and cross-country banking data from institutions such as the BIS.
  - When secondary market prices are available, network analysis complements market-based methods.
- Distinctions among systemic concepts:
  - TCTF is not necessarily the same as Too-Big-to-Fail (TBTF). Example: a bank that is TBTF due to large retail deposit share may not be TCTF if interbank exposures are negligible.
  - The Too-Many-to-Fail (TMTF) problem is associated with simultaneous failure of many institutions owing to herd behavior (Acharya and Yorulmazer, 2007); TMTF differs from TCTF because in TMTF the failure of one institution does not cause other institutions’ failures.

### Key cross-country findings from balance sheet-based network analysis
- Global findings (based on BIS cross-country bank claim data for reporting jurisdictions):
  - Main sources of risk in the global banking system are shocks that could impair the solvency of banks based in the United States and the United Kingdom.
  - Among BIS reporting jurisdictions in Latin America:
    - The banking system in Brazil appears to be the most resilient to global shocks.
    - The banking system in Chile may be more exposed to funding risk from Spanish banks.
- Country-specific (Chile) findings using bank-specific disaggregated data on claims vis-à-vis domestic banks, foreign banks, and other non-bank financial institutions:
  - Main domestic sources of risk in the Chilean banking system are shocks that affect banks’ claims on households and domestic corporations as publicly reported by the supervisory agency.
  - To a lesser extent, the banking system is exposed to shocks affecting banks domiciled in Spain and the United States.
  - On the basis of balance sheet data, no bank operating in Chile appears to be a TCTF institution.
  - From a financial stability perspective, the focus is better placed on linkages with foreign banks and the creditworthiness of claims on the non-bank financial sector.

### Graph-theoretic representation and analytical approach
- A financial or banking system can be visualized as a network of institutions (vertices) linked through directed edges (arcs); the combination constitutes a directed graph or digraph.
  - Figure 1 depicts a stylized digraph of seven banks, with claims represented by inbound arcs (example: bank 7 has claims on banks 4 and 6).
- Practical utility:
  - Mapping a financial system into a digraph provides visualization of linkages across institutions that can guide the design of the perimeter of regulation.
  - Balance sheet-based network analysis builds on simple balance sheet accounting identities and simulations to analyze different types of shocks affecting the banking system.
- Modeling scope:
  - The paper describes a simple set of tools (balance sheet-based network analysis) and notes that more sophisticated graph theory-based models are described in some detail elsewhere in the appendix.

*Source: Appendix I. Graph Theory-Based Network Models.*

### Appendix 1.

### Appendix 1.

### A. The Basic Accounting Identity
- Balance sheet-based network analysis starts from the equality between the asset side and the liability side of a bank’s balance sheet.
- Asset side: claims on other financial institutions, corporations, and households (loans, receivables, equity shares, debt securities).
- Liability side: shareholders’ equity (bank capital) and different creditors, including depositors.
- Bank capital is the first cushion against declines in asset values; if asset declines exceed bank capital, the bank defaults.
- Default propagation mechanisms:
  - Direct losses on claims owed by the defaulted bank reduce surviving banks’ capital.
  - Funding withdrawals by or from the defaulted bank can force fire-sales, generating mark-to-market losses across institutions.
- Note on regulatory perspective: the analysis can be adapted to treat a “default” as occurring when capital falls below the regulatory minimum by reducing the effective capital by the minimum regulatory capital.

### B. Credit Shocks
- Definition: losses on the asset side due to defaults of counterparties; claims on other banks are reduced because loss given default > 0 (recovery ratio < 100 percent).
- Default condition under credit shocks (Equation (1) as presented):
  - Capital − ∑ Claims on defaulted banks × Loss Given Default ≤ 0
- Simulation procedure:
  - Set an initial bank default.
  - Apply equation to other banks to determine whether losses trigger further defaults.
  - Iterate until no new defaults occur.
- Assumption for pure credit-shock analysis: surviving banks can replace funding from defaulted banks without major difficulty.

### C. Funding Shocks
- Definition: liability shock from sudden withdrawal of funding.
- Under normal market conditions:
  - Assets sold equal funding loss; capital unaffected; leverage declines.
- Under stressed/ fire-sale conditions:
  - Forced asset liquidations at depressed prices generate capital losses and can cause defaults.
  - Collusive mark-downs by other market participants may accentuate losses.
- Default condition with funding and credit effects (Equation (2) as presented):
  - Capital − ∑ Claims on defaulted banks × Loss Given Default − ∑ funding withdrawal × (1 − replacement ratio) × fire sale loss ratio ≤ 0
- Simulation use: determine resilience to funding shocks and contagion from initial funding withdrawals.

### D. Risk Transfer Shocks and Off-Balance Sheet (OBS) Exposures
- Definition: losses from OBS exposures; requires data on OBS exposure size and provisions.
- Extended balance-sheet treatment includes OBS exposures and provisions.
- Default condition under OBS shock (Equation (3) as presented):
  - Capital − Off‑balance sheet exposure + provisions ≤ 0
- OBS-induced defaults can trigger subsequent credit and funding shocks across the system.
- Practical note: OBS analysis requires additional data not always publicly available.

### III. BALANCE SHEET-BASED NETWORK ANALYSIS IN PRACTICE
- Application domains illustrated: global banking system TCTF/interconnectedness risk, and Chile’s domestic banking system.
- Scenarios and parameter assumptions used in practice:
  - Loss given default in credit shock scenario set equal to 70 percent.
  - In funding shocks: only 40 percent of the funding can be replaced; forced sales at fire sale prices 50 percent below book value.
- Risk transfer shocks not considered in global analysis due to aggregate data limitations; in Chile, credit risk transfers between institutions are negligible.
- Important caveat: calculations assume either a bank/system fails or a sudden funding withdrawal — interpreted as extreme adverse scenarios (very low probability events).

#### A. TCTF Risk Analysis in Mature and Emerging Market Countries
- Data source for cross-country claims: BIS consolidated banking statistics (immediate borrower and ultimate borrower bases).
  - Consolidated statistics capture a country’s aggregate banking system financial claims on the rest of the world.
  - Immediate borrower basis: contractual claims by head office and branches assigned to home country.
  - Ultimate borrower basis: account for impact of risk transfers.
- Limitations:
  - Aggregate BIS data approximate interbank exposures and include claims on non-bank financial institutions, non-financial institutions, and households.
  - The analysis avoids inferring risk transfers from immediate vs ultimate borrower differences due to aggregation concerns.
- Country sample: 20 countries using BIS consolidated claims on an immediate borrower basis for Q1 and Q3 2009 (included emerging markets Brazil, Chile, Mexico; mature markets Austria, Belgium, Canada, Denmark, France, Germany, Greece, Ireland, Italy, Japan, Netherlands, Portugal, Spain, Sweden, Switzerland, United Kingdom, United States).
- Failure criterion for national banking system: losses from shocks exceed aggregate capital in the system.
- Key empirical findings (2009, based on simulations using the stated assumptions):
  - Shocks originating in the UK and the US may cause failures in most countries; banks headquartered in the UK and the US could be Too-Connected-to-Fail (TCTF) globally.
  - Systemic risk from German banking system increased two-fold between March 2009 and September 2009.
  - French-origin shocks could heavily impact Belgium and Netherlands; Swedish banks could be exposed to shocks to Danish banks.
  - Chile’s banking system was most exposed among analyzed Latin American countries to funding risks from Spanish banks.
  - Brazilian banking system appears robust to adverse shocks from major banking systems.
- TCTF vulnerability and risk measures (concepts and example):
  - Vulnerability measure: average capital loss a national banking system suffers due to shocks in other countries.
  - Risk measure: average capital loss induced by the failure of a country’s banking system on other countries.
  - Example: Austria loses on average 23 percent of its capital due to shocks in other banking systems; shocks to Austria induce average losses of around 5 percent in other countries.
- Overall 2009 pattern: both TCTF risks and vulnerabilities increased during 2009.
- Contagion dynamics note: absence of direct bilateral exposure does not preclude simultaneous failures if both systems connect to a common failing system; example cascade from UK shock across multiple European systems.

#### B. Chile: TCTF Risk Analysis in an Advanced Emerging Market Economy
- Data: Monthly Report on Financial Institutions (SBIF) providing claims and liabilities vis-à-vis central bank, foreign banks, corporate, nonbank financial and household sectors for twenty domestic banks, foreign subsidiaries, and five foreign branches (one- to two-month lag).
- Construction of interbank exposure matrix:
  - In absence of detailed counterparty data, the amount a bank owes in the system is divided equally among all banks reported as having claims on other banks (assumes maximum diversification).
  - Alternatives acknowledged: proportional assignment by claimant size, entropy techniques.
- Foreign banks modeled as a single sector (conservative assumption implying simultaneous shocks across all foreign banks).
- Network augmented to include: Central Bank of Chile (BCCH), non-bank financial institutions, corporate sector, household sector.
  - Non-bank financial institutions included only as creditors (SBIF reports their deposits on individual banks but not their obligations).
  - Corporate sector included only as debtor (constructed using corporate loans and receivables).
  - Households included as both depositors and borrowers via deposits and consumer loans.
- Empirical outcomes (January, July, December 2009 reported):
  - Domestic interbank exposure relatively unimportant; a single bank default would not lead to further defaults in most cases.
  - Example average capital loss changes between January and December 2009:
    - Bank 5: average capital loss triggered by its hypothetical default declined from 10.2 percent to 1.1 percent.
    - Bank 3: declined from 7.4 percent to 0.7 percent.
    - Bank 4: increased from 0.3 percent to 6.9 percent; capital losses concentrated in Bank 12 (35 percent), Bank 21 (54 percent), Bank 25 (47 percent).
  - Trend: domestic banks reduced claim exposure to corporate and household sectors during 2009:
    - Number of banks that could default after system-wide corporate sector collapse declined to 14 from 17.
    - Average capital losses from corporate sector defaults declined to 76 percent from 82 percent.
    - Average capital losses from household loan defaults fell to 59 percent from 68 percent.
  - Funding reliance shifts in 2009:
    - Increased reliance on wholesale funding from non-bank financial institutions: a sudden withdrawal of non-bank financial deposits in December 2009 could cause problems in Banks 16, 17, 18, and 20 (up from 2 banks in January 2009).
    - Central bank funding importance: average capital losses would be approximately 3 percent if central bank funding unavailable; concentrated impacts in December 2009 on Bank 4 (32 percent), Bank 11 (24 percent), Bank 13 (22 percent).
- TCTF measures for Chile (Table 6 and Table 7 summarized concepts and outcomes):
  - TCTF vulnerability: average capital loss a bank suffers from shocks to other banks.
  - TCTF risk: average capital loss a bank induces on other banks.
  - Excess capital losses: banking-system capital losses excluding losses of the triggering bank; used as a TCTF risk metric (related to incremental/marginal contribution to risk).
  - Example: January 2009, Bank 3 accounted for 6.7 percent of total system capital; a hypothetical credit shock to Bank 3 would cause 7.9 percent total system capital loss, of which 1.2 percent is excess capital loss attributable to exposures to Bank 3 (TCTF risk equivalence 1.2 percent of system capital).
  - By December 2009, excess capital losses were relatively small, at most equal to 1½ percent for shocks from Bank 4.
- Bottom-line empirical conclusion for Chile: TCTF risk in the domestic banking system is relatively small; focus for financial surveillance should be linkages with foreign banks and non-bank financial institutions.

### IV. CONCLUSIONS (Key takeaways)
- Interconnectedness and Too-Connected-to-Fail (TCTF) externalities amplified and transmitted shocks during the 2008/9 global financial crisis.
- Balance sheet-based network analysis:
  - Captures interconnectedness and TCTF risk using publicly available balance sheet data.
  - Uses simple accounting identities rather than complex graph-theory mathematics.
  - Is easy to implement where balance sheet data are available and provides results with clear economic interpretation.
- Empirical highlights:
  - Global analysis (BIS cross-country claims) indicates shocks from UK and US banks (and to a lesser extent Germany) as main global risk sources in 2009 — interpret cautiously due to data aggregation.
  - For Chile, bank-level data suggests low domestic TCTF risk; surveillance should focus on foreign bank and non-bank financial institution linkages.
- Limitations and complementarities:
  - Balance sheet data disadvantages: potential manipulation, reporting lags, and timeliness issues especially when exposures change rapidly.
  - Complementing balance-sheet analysis with market-based measures (security prices, common economic factors) can help address some disadvantages; caution that market prices may reflect liquidity and technical drivers beyond fundamentals.

*Source: Appendix 1. (content provided in the supplied PDF chapter).*

### APPENDIX I.  GRAPH THEORY-BASED NETWORK MODELS

### APPENDIX I.  GRAPH THEORY-BASED NETWORK MODELS

### Graph theory as analytical framework
- Graph theory provides the natural analytical framework for analyzing the properties of a financial system once it has been mapped into a digraph.
- Banks are represented as vertices and bilateral exposures or linkages as arcs; this mapping enables classification of banks into clusters and evaluation of system changes following removal of selected vertices and their arcs.
- Random graph theory can introduce randomness to:
  - the number of banks (vertices) in the digraph, and/or
  - the existence of linkages (arcs) between banks.
- Random graph theory can accommodate the empirical observation that both the number of participants in a connected financial system and the nature of their cross-linkages change over time.

### Topological properties and metrics
- Key topological properties used in analysis include:
  - size: the number of institutions in the system;
  - connectivity: the relative number of existing linkages to the maximum number of linkages;
  - clustering coefficient: the probability that two institutions “close” to a third one are also “close” to each other.
- These metrics have been applied extensively to payment and settlement systems to characterize network structure and centrality.

### Applications to payment and settlement systems vs. complex banking systems
- Graph theory has been used extensively to analyze payment and settlement systems; the literature has focused mostly on topological properties.
- Translating insights from payment/settlement systems to more complex systems (interbank market, domestic banking systems, global financial system) is difficult because:
  - cross-claims across financial institutions exhibit greater complexity and heterogeneity than the homogeneous transactions in payment and settlement systems.
- Despite these difficulties, recent advances extend graph-theoretic approaches to stylized models of complex banking systems:
  - Hattori and Suda use BIS banking data to analyze topological properties of cross-border banking networks and implications for banking stability.
  - Nier et al. use Erdős–Rényi graphs to explore how different topological properties affect propagation of defaults.
  - Gai and Kapadia apply small-world models to analyze contagion in stylized networks via numerical simulations.

### Practical challenges and policy-use considerations
- Calibrating graph theory–based models with real data remains a major challenge for operational use in surveillance and supervisory contexts.
- Because of calibration difficulties, applied research in policy-making institutions has often opted for simpler approaches based on balance sheet data (described in the main text of the source) as practical alternatives for systemic risk analysis and policy work.

*Source: APPENDIX I.  GRAPH THEORY-BASED NETWORK MODELS*

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