## _wp1098

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---

### Financial crisis, interconnectedness, and regulatory capital
- Key context and magnitudes:
  - Problems in the U.S. subprime mortgage market in mid-2007 escalated to a liquidity crisis and a full blown financial crisis in 2008-9, producing the worst global recession since 1929.
  - Writedowns in the banking sector through the first half of 2009 amounted to $1.3 trillion and going forward, they may increase to $2.8 trillion.
  - The financial crisis led to a large contraction of the real economy of around1 percent in real terms in 2009.
  - Discretionary government spending among the G20 economies would amount to 2 percent of GDP for 2009 and 1½ percent for 2010 relative to 2007 baselines.
- Externalities motivating policy:
  - Too-Big-to-Fail (TBTF): heavy government support of large financial institutions; could be addressed by limits on scope and scale of activities at the expense of economies of scale.
  - Too-Connected-to-Fail (TCTF): arising from increased interdependence and globalization; illustrated by Herstatt (1974), Long Term Capital Management (1998), Lehman Brothers (September 15, 2008), and the bailout of AIG on September 16, 2008.
- Policy proposal:
  - Introduce regulatory capital charges proportional to an institution’s interconnectedness within the financial system (Too-Connected-to-Fail Capital Charge).
  - Contrast with Basel I/II: traditional charges focus on institution-specific trading and loan-book risks; the TCTF charge penalizes an institution by its incremental contribution to the risk of other institutions and potential government/taxpayer costs.
- Claimed benefits:
  - Induces internalization of TCTF costs and managerial incentives to avoid excessive homogeneity and reliance on a reduced number of counterparties.
  - Complements proposals for more and higher quality capital for systemically important institutions and can complement IMF proposals for bank-funded rescue insurance.
- Methodological lineage:
  - Integrates CoRisk measures, network analysis, one-factor credit portfolio models, and systemic risk indicators based on credit risk models.

### The Too-Connected-to-Fail capital charge: concept and intuition
- Societal loss as the risk metric:
  - Defined flexibly by the systemic regulatory agency depending on dominant features of the financial system (examples: potential deposit losses in deposit-taking systems; losses faced by pension funds if they are major bank creditors).
  - In the U.S. context, societal losses can be roughly proxied by potential losses associated with bank liabilities since government measures effectively guaranteed claims of banks’ senior creditors.
- Core concept:
  - TCTF reflects the incremental contribution of the default of one institution to the potential societal loss from surviving institutions.
  - Calculated by comparing conditional societal loss distributions when the institution survives versus when it defaults.
  - Comparison performed at a predetermined confidence level (consistent with VaR practice) or using Expected Shortfall (ES) as an alternative.

### Two-bank illustrative example (numerical)
- Setup:
  - Two identical banks, A and B; each holds $100 million.
  - Unconditional probability of default for each bank is 5 percent.
  - If one bank defaults, all deposits are lost and the probability of default of the surviving bank increases to 6 percent.
  - Deposits are guaranteed by the government → societal cost equals depositor losses.
- Calculations and implication:
  - In isolation, expected societal loss from default of any bank = $5 million (5 percent × $100 million).
  - If bank A defaults, expected societal loss from subsequent default of bank B = $6 million (6 percent × $100 million).
  - Incremental contribution to societal loss of the default of bank A = $1 million ($6 million − $5 million).
  - If the capital requirement is set equal to expected loss, TCTF capital requirement = $6 million (comprising $5 million unconditional expected loss + $1 million incremental expected loss induced on bank B).
  - Using formula (2) yields an interconnectedness capital charge of $50000 (5 percent × $1 million).

### From societal losses to TCTF capital charges: incremental-portfolio approach
- Incremental-portfolio definition and procedure:
  - For institution J among N banks, define the N−J portfolio as the portfolio of all institutions excluding J.
  - Compute two societal loss distributions for the N−J portfolio:
    - Ex-ante: conditional on J surviving, using PD_I, EAD_I, LGD_I, and dependence parameters.
    - Ex-post: conditional on J defaulting, using PD_I|J, EAD_I|J, LGD_I|J, and conditional dependence parameters.
  - Dependence measures (default correlations) may increase following a failure; empirical studies show higher dependence during downmarkets and recessions.
- Use of a confidence level:
  - Incremental contribution to societal loss of bank J = VaR_J^α(loss distribution when J defaults) − VaR_J^α(loss distribution when J is solvent), evaluated at chosen confidence level 1−α.
  - TCTF capital charge for Bank J = Probability of default of Bank J × Incremental contribution to societal loss of Bank J.
- Interpretation:
  - The TCTF capital charge increases with an institution’s incremental contribution to societal loss and with its probability of default.
  - Imposition provides incentives to change business mix or strengthen solvency.

### Stepwise procedure and implementation steps (summary)
- Steps to calculate interconnectedness capital charge for institution J:
  1. For each institution other than J, specify the probability of default of the remaining institutions in the events that J survives or defaults.
  2. For each institution, determine the societal exposure at default and the societal loss given default for each of the two events (typical choices: nominal amount of potential losses incurred by the government in case of default).
  3. Construct the societal loss distributions for the incremental portfolio (all institutions excluding J) under the events that J survives (ex-ante) or defaults (ex-post).
  4–7. Pick confidence level, compute VaR/ES, compute incremental contribution, and compute TCTF charge as PD_J × incremental contribution.
- Confidence levels and tail measures:
  - Typical confidence levels used for VaR: 95 percent, 99 percent, and 99.5 percent; alternatively, Expected Shortfall (ES) can be used.
- Key implementation challenges (steps 1–3):
  - Step 1: Specification of PD — likely one of the most difficult elements.
  - Step 2: Determination of EAD and LGD — particularly challenging; may use rough estimates based on liabilities potentially guaranteed by the government (at minimum, include deposits; may justifiably include senior creditor claims).
  - Step 3: Specification of a loss-distribution model and degree of association (comovement) between PDs.
- Iteration and model-feedback:
  - A rigorous calculation requires an iterative process or finding a fixed point: the capital charge depends on PD; adding the charge increases capital, lowers leverage and asset volatility, which reduces PD and thus the charge.
  - One regulatory iteration errs on the conservative side and is a second-best solution when data quality and model risk constrain full optimization.

### Comparisons with other systemic-risk capital approaches
- Measurement of contribution to systemic risk:
  - TCTF: incremental contribution measured as change in risk metric of a reference portfolio (VaR or ES) induced by failure of the institution; the reference portfolio excludes the failed institution both ex-ante and ex-post.
  - Tarashev et al. and Gauthier et al.: capital charge based on marginal contribution to risk from adding one institution to an ex-ante portfolio; their portfolios are conditional on survival and cannot account for impact of the institution’s failure.
- Treatment of PDs and parameter changes:
  - TCTF factors in increases in default risk triggered by failure and can account for changes in EAD, LGD, and association parameters.
  - Tarashev et al. and Gauthier et al. do not factor in increases in default risk triggered by an institution’s failure, nor changes in EAD or association parameters.
- Generalization:
  - TCTF methodology can be generalized to simultaneous default of more than one institution.

### Toolkit and data inputs
- Main data requirement: how an institution’s survival or default affects PDs of other institutions (from market sources like Moody’s KMV, CDS and bond spreads, equity prices, or derived via reduced-form/structural models).
- Two methodologies described for PDs:
  - CoRisk analysis (market-based, quantile-regression approach).
  - Network analysis (balance-sheet direct-exposure approach).
- Calculation of societal EAD and LGD is more complex and less developed; pragmatic approach: rough estimates based on potentially government-guaranteed liabilities.

### CoRisk Analysis (PD estimation approach)
- Purpose: estimate PDs conditional on failure of a specific institution while accounting for market interaction from direct or indirect exposures.
- Method: CoRisk extends CoVaR (Adrian and Brunnermeier (2009)); estimation via quantile regression (Koenker and Basset (1978)).
- Practical considerations:
  - Choice of quantile τ: 50th percentile for normal periods; upper quantiles such as the 95th or 99th percentiles for a more cautious approach (tradeoff: higher τ → higher TCTF capital → reduced credit supply).
  - Value for PD of the institution assumed to fail: obvious choice is one; alternatives include a high historical percentile (e.g., 95th percentile) or an intervention-associated PD under prompt corrective action.
  - Use a Cox transformation for PDs before estimating regressions because PDs are bounded between 0 and 1.

### Network Analysis (balance-sheet approach)
- Purpose: use direct exposure data to model contagion via balance-sheet linkages (interbank exposures, assets = liabilities of counterparties).
- Mechanism:
  - Institutions directly exposed that fail in contagion rounds have conditional PD set equal to one and their societal losses added to the loss distribution.
  - Survivors’ conditional PDs computed using balance-sheet default-risk models (structural Merton (1974) or financial-ratio Altman (1968)).
- Merton-style relations and conversions:
  - Distance-to-default (DD) formula and Φ-based conversion to risk-neutral conditional PDs, with further conversion from risk-neutral to physical PDs via Bohn (1999) approximation.
- Additional features:
  - Captures second-round effects from forced asset sales and price declines.
- Data caution: network analysis is data- and resource-intensive.

### One-factor credit portfolio risk model (benchmark)
- Role: construct loss distribution for a portfolio of risky assets; extendable to several factors.
- Model elements:
  - xi = F_i^{-1}(PD_i) sets default threshold.
  - Factor loading ai between 0 and 1; asset correlation between i and j equals ai·aj.
  - Conditional PD given common factor realization computed via model equations.
- Flexibility:
  - Can accommodate skewness, fat tails, extreme-value and empirical distributions by choosing Fi, G, H accordingly.
- Estimation approaches: infer normalized asset values, extract common factors (factor analysis/PCA), compute loss distribution via Monte Carlo or analytical algorithms.
- Empirical observation: correlations during stressful events are higher than during normal periods; model this via factor-loading dependence on common shocks or PD.

### Illustrative empirical example: data and assumptions
- Data and balance-sheet aggregates:
  - Weekly EDFs from Moody’s KMV for 26 global financial institutions for sample period May 2, 2003 – February 27, 2009.
  - Aggregate balance-sheet figures as of end-2008: institutions held together US$ 36 trillion in assets and US$ 9¼ trillion in total debt.
- Methodological choices in example:
  - CoRisk analysis used to calculate impact of single-institution failure on PDs of remaining institutions.
  - Equation (3) estimated for the 95th quantile regression for all institutions pairwise.
  - Common macro and financial risk factors proxied by the first two principal components of PDs.
  - Probabilities of default conditional on survival are set equal to values observed in the last week of February 2009; conditional PDs given default calculated by adding CoRisk percent increase to survival PDs.
  - Societal losses set equal to total debt reported in end-2008 financial statements; LGD assumed equal to 100 percent.
  - Factor loading calculated according to Proposal II of the Basle Committee on Banking Supervision (BCBS) (2004); BCBS formula assigns lower correlations for higher PDs and imposes an upper bound of 0.24.
  - Monte Carlo simulation: number of simulations = 5000.
  - Capital charges reported in Table 2 are given in billions of U.S. dollars and as percent of the institutions’ total assets.

### Key empirical findings and statistics from the example
- Citigroup:
  - Stands out with the highest TCTF capital charge at around 15 percent of total assets.
  - At the 95 percent confidence level, the VaR of Citigroup’s incremental portfolio almost doubles to 32 percent of the nominal amount of the portfolio in case Citigroup defaults.
- Other institutions:
  - Spanish banks BBVA and Santander, and French banks Credit Agricole and Societe Generale, have incremental portfolios roughly as large as Citigroup, but their PDs are one or two orders of magnitude lower than Citigroup’s PD, producing substantially lower TCTF capital requirements.
- Caveats on parameter choices:
  - Normal-distribution assumptions in the one-factor model can understate tail risk and bias down TCTF capital charges.
  - BCBS correlation formula cap (0.24) and lower correlations for higher PDs can understate correlations in tail scenarios.
- Suggested adjustments:
  - Use structural models with a constant, high correlation consistent with tail risk scenarios, or specify correlation as an increasing function of PD.

### Interpretation, policy implications, and recommendations
- Determinants of TCTF charges:
  - Depend on both the institution’s own risk (PD) and the incremental contribution its default poses to other institutions (incremental societal loss).
- Regulatory perimeter and application:
  - Methodology can help define regulatory perimeter by including institutions whose incremental contribution to societal loss exceeds a predetermined threshold; allows a dynamic perimeter.
  - Extend TCTF charges beyond banks to entities whose failure would have macro-financial linkages (e.g., trade-finance backstops).
- Implementation and governance:
  - Use TCTF methodology as input to systemic-institution scorecards.
  - To avoid conflicts of interest, calculation could be performed by a neutral multilateral institution such as the BIS or the IMF.
- Design alternatives and distribution:
  - Though charges are discussed as applied to the failing institution, alternative allocation rules exist (e.g., summing individual TCTF charges and distributing proportionally to each institution’s PD).
- Procyclicality and mitigation:
  - TCTF charges are procyclical as PDs rise during downturns; mitigate by using PD estimates conditional on extreme events or implement within a prompt corrective action (PCA) framework using intervention-associated PD ceilings.
- Harmonization:
  - Harmonize measurement practices across jurisdictions; use TCTF charges as input for cross-jurisdictional systemic institution scorecards.

### Caveats and limitations
- The numerical example is illustrative and uses simplifying assumptions and approximations; results are not definitive.
- Normal latent-asset distributions may neglect tail risk; fat-tail distributions (Student-t, Gumbel) may be more appropriate.
- BCBS correlation cap and inversion of correlation with PD may understate correlations in tails; structural models or alternative correlation specifications are preferable.
- During stress periods authorities may still need to support institutions not deemed TBTF or TCTF to avoid destabilizing runs or investor-sentiment spirals.

*Source: _wp1098 - References and Section Excerpts.*

### References..............................................................................................................

### _wp1098 - References..............................................................................................................

### Financial crisis, interconnectedness, and regulatory capital
- Problems in the U.S. subprime mortgage market in mid-2007 escalated to a liquidity crisis and a full blown financial crisis in 2008-9, producing the worst global recession since 1929.
- Writedowns in the banking sector through the first half of 2009 amounted to $1.3 trillion and going forward, they may increase to $2.8 trillion.
- The financial crisis led to a large contraction of the real economy of around1 percent in real terms in 2009.
- Discretionary government spending among the G20 economies would amount to 2 percent of GDP for 2009 and 1½ percent for 2010 relative to 2007 baselines.
- Two externalities highlighted:
  - Too-Big-to-Fail (TBTF): heavy government support of large financial institutions; could be addressed by limits on scope and scale of activities at the expense of economies of scale.
  - Too-Connected-to-Fail (TCTF): arising from increased interdependence and globalization; illustrated by Herstatt (1974), Long Term Capital Management (1998), Lehman Brothers (September 15, 2008), and the bailout of AIG on September 16, 2008.
- Proposal: introduce regulatory capital charges proportional to an institution’s interconnectedness within the financial system or to other systemic institutions (Too-Connected-to-Fail Capital Charge).
  - Traditional Basel I and Basel II charges focus on individual institution trading and loan book risks.
  - TCTF capital charge penalizes institutions by their incremental contribution to the risk of other institutions and the potential costs borne by governments/taxpayers.
- Benefits:
  - Induces institutions to internalize costs associated with TCTF.
  - Provides managerial incentives to avoid excessive homogeneity and reliance on a reduced number of counterparties.
  - Complements proposals for more and higher quality capital for systemically important institutions.
  - Can complement an IMF proposal to make banks pay insurance fees to fund future rescues.
- Methodological lineage: extends and integrates CoRisk measures, network analysis, one-factor credit risk portfolio models, and systemic risk indicators based on credit risk models.

### The Too-Connected-to-Fail capital charge: concept and intuition
- Goal: measure systemic impact using societal loss as the risk metric.
- Societal loss:
  - Defined flexibly by the systemic regulatory agency depending on dominant features of the financial system.
  - Examples: potential deposit losses in deposit-taking systems; losses faced by pension funds if they are major bank creditors.
  - In the U.S. context, societal losses can be roughly proxied by potential losses associated with bank liabilities since government measures effectively guaranteed claims of banks’ senior creditors.
- TCTF capital charge concept:
  - Reflects the incremental contribution of the default of one institution to the potential societal loss from surviving institutions.
  - Calculated by comparing conditional societal loss distributions when the institution survives versus when it defaults.
  - Comparison performed at a predetermined confidence level (consistent with VaR practice) or using Expected Shortfall (ES) as an alternative.

### Two-bank illustrative example
- Setup:
  - Two identical banks, A and B.
  - Each bank holds $100 million.
  - Unconditional probability of default for each bank is 5 percent.
  - If one bank defaults, all deposits are lost and the probability of default of the surviving bank increases to 6 percent.
  - Deposits are guaranteed by the government → societal cost equals depositor losses.
- Calculations:
  - In isolation, expected societal loss from default of any bank = $5 million (5 percent × $100 million).
  - If bank A defaults, expected societal loss from subsequent default of bank B = $6 million (6 percent × $100 million).
  - Incremental contribution to societal loss of the default of bank A = $1 million ($6 million − $5 million).
  - TCTF capital charge should be proportional to the incremental contribution: in the example, capital requirement = $6 million (expected loss) if set equal to expected loss, comprising $5 million unconditional expected loss + $1 million incremental expected loss induced on bank B.
  - Using formula (2) below yields an interconnectedness capital charge of $50000 (5 percent × $1 million).

### From societal losses to TCTF capital charges: incremental portfolio approach
- Core idea:
  - For institution J among N banks, define the N−J portfolio as the portfolio of all institutions excluding J (the incremental portfolio).
  - Compute two societal loss distributions for the N−J portfolio:
    - Ex-ante (conditional on J surviving) using PD_I, EAD_I, LGD_I, and dependence parameters (e.g., asset correlation in the one-factor Gaussian model).
    - Ex-post (conditional on J defaulting) using PD_I|J, EAD_I|J, LGD_I|J, and conditional dependence parameters.
  - Dependence measures (default correlations) may increase following a failure; empirical studies show higher dependence during downmarkets and recessions.
- Use of a confidence level:
  - The incremental contribution to societal loss of bank J is the difference between the VaRα values of the conditional (J defaults) and unconditional (J solvent) loss distributions at confidence level 1−α:
    - Incremental contribution to societal loss of Bank J = VaR_J^α (Loss distribution when J defaults) − VaR_J^α (Loss distribution when J is solvent).
  - The TCTF capital charge for Bank J = Probability of default of Bank J × Incremental contribution to societal loss of Bank J.
- Interpretation:
  - The TCTF capital charge increases with an institution’s incremental contribution to societal loss and with its probability of default.
  - Imposition of this charge provides incentives for banks to change business activities or strengthen solvency to reduce systemic impact.

### Implementation steps (summary)
- For institution J in relation to a group of connected institutions, the steps for calculating the interconnectedness capital charge are:
  1. For each institution other than J, specify the probability of default of the remaining institutions in the events that institution J survives or defaults.
  2. For each institution in step 1, determine the societal exposure at default and the societal loss given default for each of the two events (typical choices: nominal amount of potential losses incurred by the government in case of default).
  3. Construct the societal loss distributions for the incremental portfolio that comprises all institutions excluding J under the events that J survives (ex-ante loss distribution) or defaults (ex-post loss distribution).

*Source: _wp1098 - References..............................................................................................................*

### 4.         Pick up a given confidence level. Typical values correspond to those used for

### _wp1098 - 4.         Pick up a given confidence level. Typical values correspond to those used for

### Stepwise procedure for calculating the TCTF capital charge
- Pick up a given confidence level. Typical values correspond to those used for calculating Value-at-Risk (VaR), i.e. 95 percent, 99 percent, and 99.5 percent (alternatively, a tail-risk measure such as Expected Shortfall (ES) can be used).
- Calculate the VaR in the ex-ante and ex-post societal loss distributions at the specified confidence level.
- Calculate the incremental contribution to societal loss as the difference between the VaR conditional on the default of institution J (VaR of ex-ante loss distribution) and the VaR conditional on the survival of institution J (VaR of ex-post loss distribution).
- Calculate the TCTF capital charge as the product of the probability of default of institution J and its incremental contribution to societal loss.

### Key implementation challenges (steps 1–3)
- Step 1: Specification of the probability of default — likely one of the most difficult elements.
- Step 2: Determination of the exposure at default and loss given default — particularly challenging; academic and practitioner advances are still evolving and at a relatively primitive stage compared with probability-of-default methods.
  - Under data/model limitations, it may be appropriate to use rough estimates of exposure at default and loss given default based on liabilities that could potentially be guaranteed by the government if a financial institution were to default.
  - These liabilities should include at least deposits; policy responses during the recent crisis suggest it would be justifiable to also include senior creditor claims.
- Step 3: Specification of a loss distribution model and its associated characteristics.
  - Foremost among model choices is the specification of the degree of association or dependence (i.e., comovement) between the probabilities of default of individual institutions.
- Steps 4 to 7 are relatively straightforward once steps 1 to 3 are specified.

### Iteration and model feedback
- A rigorous calculation of the TCTF capital charge requires an iterative process or finding a fixed point (as emphasized in Gauthier, Lehar and Souissi (2009)).
  - Equation (2) (referenced) indicates the capital charge depends on the probability of default.
  - Adding the TCTF capital charge increases institution capital and reduces leverage, which usually lowers asset volatility, which in turn reduces the probability of default and thus reduces the TCTF capital charge.
- From a regulatory perspective, one iteration of the TCTF capital charge errs on the conservative side and appears to be a second-best solution where optimal solutions are unfeasible due to data quality and model risk.

### Differences with other systemic-risk capital approaches
- Two important differences between the TCTF capital charge and approaches by Tarashev, Borio, and Tsatsaronis (2009), and Gauthier, Lehar, and Souissi (2009):
  - Measurement of an institution’s contribution to systemic risk:
    - TCTF: Incremental contribution is proportional to the change in the risk metrics of a reference portfolio (VaR or expected shortfall) induced by the failure of that institution. The reference portfolio in the TCTF capital charge excludes the failed institution both ex-ante and ex-post the failure event.
    - Tarashev et al. and Gauthiers et al.: Capital charge is calculated in reference to two different portfolios: the ex-ante portfolio does not include the institution while the ex-post portfolio does. In these approaches, the capital charge is based on the marginal contribution to risk from adding one institution to the ex-ante portfolio; these portfolios are conditional on the institution surviving and therefore cannot account for the impact of the institution’s failure.
  - Treatment of probabilities of default and parameter changes triggered by failure:
    - TCTF factors in the increase in default risk triggered by the failure of one institution and can account for changes in other parameters such as societal exposure at default and the degree of association (correlation) between institutions.
    - Tarashev et al. and Gauthiers et al. do not factor in the increase in default risk triggered by an institution’s failure, nor changes in societal exposure at default or association parameters.
- Generalization:
  - The TCTF methodology can be generalized to accommodate the simultaneous default of more than one institution; this is not a limitation in the CoRisk and network analysis methods described elsewhere.

### Basic toolkit and data inputs (overview)
- The first step requires knowing how an institution’s survival or default affects the probability of default of other institutions.
  - For systemic institutions, survival/default effects may be straightforward because probability-of-default estimates may be available from risk data providers such as Moody’s KMV, or derived from market prices (CDS and bond spreads, equity prices) using reduced-form and/or structural models of default risk and adjusting for the market price of risk.
- Section III (referenced) describes two methodologies for calculating probabilities of default: CoRisk analysis and network analysis.
- Calculation of societal exposure at default and corresponding loss given default is more complicated and currently less developed; rough estimates based on potentially government-guaranteed liabilities are a pragmatic approach under current limitations.

*Source: IMF working paper excerpt (_wp1098 - 4.         Pick up a given confidence level. Typical values correspond to those used for calculating VaR)*

### references.

### references.

### CoRisk Analysis
- Purpose: estimate probabilities of default conditional on the failure of a specific institution while accounting for market interaction from direct or indirect exposures.
- Method: CoRisk, based on CoVaR (Adrian and Brunnermeier (2009)) and extended in Chan-Lau (2010), measures risk codependence (how the risk of one institution reacts to changes in the risk of another after controlling for common risk factors).
- Estimation technique: quantile regression (Koenker and Basset (1978)).
- Interpretation:
  - Each quantile regression identifies different risk regimes (low risk to high risk).
  - Including the probability of default of another institution among covariates yields CoRisk estimates at different quantiles.
- Illustration (Figure 2): relationship between Lehman Brothers and AIG CDS spreads for July 1, 2003 – September 12, 2008; quantile regressions at the 5th, 50th, and 95th quantiles show steeper slopes at higher quantiles, indicating higher CoRisk in high-risk regimes.
- Relevant quantile regression for TCTF capital charge (equation (3)):
  - ,,
    K
    ikkjj
    k
    PDRPD
    
       
    
    ,
  - PD denotes probability of default, Rk denotes common factors, τ the quantile, and β the coefficients associated with covariates.
- Practical considerations affecting TCTF capital charge:
  - Choice of quantile τ:
    - 50th percentile could be used in normal periods.
    - Upper quantiles such as the 95th or 99th percentiles represent a more cautious approach.
    - Tradeoff: higher quantile → higher TCTF capital requirement → reduced supply of credit from affected bank.
    - Use of upper quantiles is less vulnerable to procyclicality and aligns with proposals to reduce procyclical capital requirements (Brunnermeier et al, 2009; IMF, 2009c).
  - Value for the probability of default of the institution assumed to fail:
    - Obvious choice: value of one (institution has already failed).
    - Alternative: use a high historical percentile (e.g., 95th percentile) of the PD distribution if data covers at least one business cycle or major crisis.
    - Another alternative: use a threshold “intervention” probability of default under prompt corrective action (regulator intervenes before default); use that intervention-associated PD in equation (3).
- Implementation note: since PDs are bounded between 0 and 1, use a Cox transformation before estimating CoRisk quantile regressions.

### Network Analysis
- Purpose: complementary approach to CoRisk when direct exposure data across institutions is available; useful where market data is not available or reliable.
- Core idea: balance sheets are linked by direct exposures (e.g., interbank exposures); liabilities of one institution are assets of another. Default of one reduces asset value of counterparties, raising their PDs and potentially triggering cascading failures.
- Mechanism:
  - Directly exposed institutions that fail in contagion rounds: conditional PD set equal to one; their societal losses are added to the loss distribution.
  - For banks that survive the initial trigger, conditional PDs are computed using balance-sheet models of default risk (structural models like Merton (1974) or financial-ratio models like Altman (1968)).
- Merton-style relation (distance-to-default, DD) (equation (4)):
  - 2
    ln
    2
    A
    rT
    L
    DD
    T
    
    
    
      
      
    
      
      
    
    ,
  - where A is asset value, L is face value of liabilities, r is the risk-free rate, σ is volatility of asset value, and T is the time horizon.
- Risk-neutral conditional PD for institution I given failure of J (equation (5)):
  - *1
    (|   )
    I
    PDDD   J
    
     ,
  - where Φ is the cumulative normal distribution and |DD   J is DD of I after accounting for J's failure.
- Converting risk-neutral to physical PDs (Bohn (1999) approximation) (equation (6)):
  - *
    1
    ,
    2
    2
    I
    II  M
    PD
    PDT  
    
    
        
      
    
        
        
    
  - where T is time horizon, ,I  M
     is correlation between asset and market returns, and β is market price of risk (market return in excess of risk-free rate per unit of market volatility).
- Additional features:
  - Network analysis can capture second-round effects from forced asset sales (margin calls or rebuilding capital ratios), depressing prices and further reducing asset values of survivors.
- Data requirement caution: implementing network analysis can be data- and resource-intensive.

### The One-Factor Credit Portfolio Risk Model
- Role: benchmark for constructing loss distribution of a portfolio of risky assets (e.g., systemic institutions for TCTF capital charge).
- Features:
  - Captures joint behavior of default events; extendable to several common factors (Hull and White, 2004).
  - Loss distribution computable via Monte Carlo simulation or numerical algorithms (Andersen, Sidenius, and Basu, 2003).
- Model specification (equation (7)):
  - 2
    1,
    iiii
    xa Ma  Z      
  - xi, M, and Zi are random variables with zero mean and unit variance and cumulative distribution functions Fi, G, and H respectively.
  - Factor loading ai restricted to values between 0 and 1.
- Default threshold:
  - xi determined by current PDi via xi = F_i^{-1}(PD_i).
- Conditional PD given common factor realization (equation (8)):
  - 
    1
    2
    Prob|(  |).
    1
    ii
    iii
    i
    xa M
    xx   MPD  t   MH
    a
    
    
    
    
    
    
    
    
- Flexibility:
  - Contrary to fixation on the Gaussian case, the one-factor model can accommodate skewness, fat tails, extreme value, and empirical distributions by choosing appropriate distribution functions for asset value and shocks.
- Implication for asset correlation:
  - Asset correlation between institutions i and j equals ai·aj.
  - Practical estimation reduces to: infer normalized asset values for each institution, identify the common factor, estimate factor loadings.
- Estimation approaches:
  - Use structural models to infer normalized asset values from estimated PDs.
  - Use factor analysis or principal component analysis to extract common factor(s).
  - Compute loss distribution via Monte Carlo or analytical algorithms.
- Empirical observation note: correlations during stressful events are higher than during normal periods; this asymmetry can be modeled by allowing factor loading to increase with common shock M or PD.

### An Illustrative Example
- Data: weekly time series of EDFs from Moody’s KMV for 26 global financial institutions for sample period May 2, 2003 – February 27, 2009.
- Aggregate balance-sheet figures as of end-2008: institutions held together US$ 36 trillion in assets and US$ 9¼ trillion in total debt (Primark Datastream and Google Finance).
- Implementation choices in example:
  - CoRisk analysis used to calculate impact of single institution failure on PDs of remaining institutions.
  - Equation (3) estimated for the 95th quantile regression for all institutions pairwise.
  - Approximation: common macro and financial risk factors proxied by the first two principal components of PDs.
- Findings from estimation (Table 1 summary):
  - On average, U.S. banks are the most vulnerable to shocks within the U.S. and outside the U.S., attributed mostly to vulnerabilities of the three investment banks in the sample.
  - Spanish banks and Canadian banks are the least vulnerable on average.
  - Failures of U.S. banks, on average, are most damaging to the financial system; driven mostly by the largest U.S. commercial bank in the sample.
  - Systemic risk created by failures of banks in countries other than the U.S. is roughly the same, and slightly lower for German and Canadian institutions.
- Next step referenced: calculation of the Too-Connected-to-Fail capital charge follows steps outlined earlier.

*Source: _wp1098 - references.*

### section II.C and uses the one-factor Gaussian credit risk model described in section III.C.

### _wp1098 - section II.C and uses the one-factor Gaussian credit risk model described in section III.C.

### Methodology and model assumptions
- Probabilities of default conditional on survival are set equal to the values observed in the last week of February 2009.
- Probabilities of default conditional on the default of an institution are calculated by adding the CoRisk percent increase caused by that institution to the probabilities corresponding to the survival event.
- Societal losses are set equal to the total debt as reported by the financial institutions in their end-2008 financial statements (used as a short-cut and for illustration).
- Loss-given-default is assumed equal to 100 percent.
- The factor loading (correlation of the institution’s asset returns with the common factor) is calculated according to Proposal II of the Basle Committee on Banking Supervision (BCBS) (2004).
- Monte Carlo simulation: number of simulations set to 5000.
- Capital charges reported in Table 2 are given in billions of U.S. dollars and as percent of the institutions’ total assets.
- The BCBS formula assigns lower correlations for higher probabilities of default and imposes an upper bound of 0.24.

### Key empirical findings and statistics
- Citigroup stands out with the highest TCTF capital charge at around 15 percent of total assets.
- At the 95 percent confidence level, the VaR of Citigroup’s incremental portfolio almost doubles to 32 percent of the nominal amount of the portfolio in case Citigroup defaults.
- Spanish banks BBVA and Santander, and the French banks Credit Agricole and Societe Generale, have incremental portfolios roughly as large as Citigroup, but:
  - Their probabilities of default are one or two orders of magnitude lower than the probability of default of Citigroup, producing substantially lower TCTF capital requirements.
- The numerical example’s parameter choices and simplifying assumptions (e.g., normal distributions in the one-factor credit portfolio risk model) could understate tail risk and thereby bias down the TCTF capital charges reported in Table 2.
- Two assumptions noted to bias down reported TCTF charges:
  - Assuming normal distributions (neglecting fat tails; alternatives: Student-t or Gumbel).
  - Using the BCBS correlation formula that caps correlation at 0.24 and assigns lower correlations to higher PDs.
- Suggestions to address these shortcomings:
  - Use structural models (section III.C) with a constant, high correlation value consistent with tail risk scenarios, or specify correlation as an increasing function of the probability of default.

### Interpretation and implications
- TCTF capital charges depend on both:
  - The risk of the individual institution (its probability of default), and
  - The risk its default poses to other institutions (incremental contribution to societal losses).
- The results support prudential measures aimed at reducing the risk of individual institutions deemed systemic (example: Chile requiring Banco Santander to hold higher capital charges due to dominance).
- The methodology can help define the regulatory perimeter:
  - Include institutions (financial or non-financial) whose incremental contribution to societal loss exceeds a predetermined threshold.
  - Allows a dynamic perimeter: some currently unregulated institutions may be added; some currently regulated institutions may be dropped if their systemic contribution declines.
- Macro-financial linkage application:
  - TCTF capital charges should account for impacts beyond banks (e.g., banks that underpin trade financing or export-import sectors).
  - CoRisk analysis can be used when direct exposures are not observable.

### Policy recommendations and design considerations
- The proposed TCTF capital charge methodology:
  - Should be proportional to the incremental contribution to societal losses due to an institution’s failure.
  - Is aligned with concepts such as Value-at-Risk and Expected Shortfall, facilitating adoption consistent with the spirit of Basel II.
  - Can be implemented using a toolkit of CoRisk analysis, network analysis, and portfolio credit risk models.
- Practical implementation points:
  - Use the methodology as input to construct a scorecard for systemic institutions.
  - To avoid conflicts of interest, calculation of the charge could be performed by a neutral multilateral institution such as the BIS or the IMF.
- Design alternatives and distribution issues:
  - Charges are discussed as applied to the failing institution, but there are arguments for applying charges to institutions most affected by the failure.
  - One mitigation: add up individual TCTF charges and distribute them proportionally to each institution’s probability of default.
- Addressing procyclicality:
  - TCTF capital charges are procyclical as PDs rise during downturns.
  - Procyclicality could be reduced by using PD estimates conditional on extreme events.
  - Implement TCTF charges within a prompt corrective action (PCA) framework where thresholds trigger remedial actions—this can place a ceiling on maximum PD for given institutions.
- Harmonization:
  - Harmonize regulatory practices for measuring interconnectedness risk across jurisdictions.
  - An intermediate step: use the TCTF charge as an input into cross-jurisdictional systemic institution scorecards.

### Caveats and limitations
- The numerical example is illustrative and uses rough approximations and simplifying assumptions; results should not be taken at full face value.
- Assuming normal latent asset distributions can neglect tail risk; fat-tail distributions may be more appropriate.
- The BCBS correlation cap and inversion of correlation with PD may understate correlation in tail scenarios; structural models or alternative correlation specifications may be preferable.
- During stress periods authorities may need to support institutions not deemed too-big-to-fail or too-connected-to-fail to avoid destabilizing runs or spirals in investor sentiment.

*Source: Moody’s KMV and author’s calculations; numerical example and discussion as presented in the specified section of the source PDF.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1098.pdf_