## _wp0904

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---

### Introduction: purpose, scope, and key choices
- Purpose:
  - Present a method for estimating a set of stability measures of the banking system (BSMs).
  - Demonstrate the approach is reasonable and implementable with limited data.
- Empirical scope:
  - Estimations using publicly available information from 2005 up to the beginning of October 2008.
  - Illustrative implementation: relative changes in stability over time and across U.S. banks’ business lines; cross-region effects between American and European banking groups; extensions to foreign banks’ effects on sovereigns in Latin America, eastern Europe and Asia.
- Key conceptual choices:
  - Conceptualize the banking system as a portfolio of core, systemically important banks.
  - Recover the banking system’s portfolio multivariate density (BSMD) and construct BSMs embedding banks’ distress inter-dependence structure (linear and non-linear), and their changes across the economic cycle.
- Advantages highlighted:
  - BSMs quantify (a) “common” distress in the banks of the system, (b) distress between specific banks, and (c) distress in the system associated with a specific bank.
  - Methodology extensible to NBFIs when data permit.

### Methodology overview: CIMDO framework and steps
- Inputs and steps:
  - Empirical inputs: probabilities of distress (PoDs) for individual banks treated as exogenous inputs.
  - Four modeling steps:
    - Step1: Conceptualize banking system as a portfolio of banks.
    - Step2: Obtain empirical measurements of PoDs for each bank.
    - Step3: Use Consistent Information Multivariate Density Optimizing (CIMDO) methodology with individual PoDs to recover BSMD.
    - Step4: From BSMD estimate proposed banking stability measures (BSMs).
- Core methodological components:
  - CIMDO recovers a posterior multivariate density p by updating a prior q via minimum cross-entropy (Kullback, 1959) subject to constraints matching empirical PoDs.
  - The CIMDO-copula (Segoviano, 2009) is a non-parametric copula recovered from the CIMDO-density that captures time-varying linear and non-linear dependence without imposing parametric copula forms.
  - Evaluation uses the Probability Integral Transformation (PIT) criterion.

### The CIMDO technical approach and copula advantages
- CIMDO technical foundation:
  - Objective functional: C[p,q] = ∫∫ p(x,y) ln( p(x,y) / q(x,y) ) dx dy; constraints force posterior marginals to match empirically estimated PoDs.
  - Optimization yields an exponential-form posterior density where prior q is multiplicatively adjusted by exponential terms involving Lagrange multipliers tied to PoD constraints.
  - CIMDO-recovered distributions outperform common parametric multivariate densities under the PIT criterion (Monte Carlo evidence summarized).
- CIMDO-copula properties and advantages:
  - The copula isolates dependence by transforming marginals to U(0,1) via PIT; the CIMDO-copula is extracted from the inferred CIMDO-density and updates as PoDs change because Lagrange multipliers change.
  - Avoids the Copula Choice Problem (CCP) by not imposing a parametric copula form; addresses the fixed-correlation shortcoming of many parametric copulas by allowing time-varying dependence that reflects changing empirical PoDs.
  - Copulas are invariant to increasing continuous marginal transformations and describe dependence across the entire domain, including tails (critical for distress dependence).

### Banking Stability Measures (BSMs): definitions and components
- Objective: characterize individual-bank PoDs, distress dependence, and changes across the economic cycle from three perspectives:
  - (a) Common distress in banks of the system.
  - (b) Distress between specific banks.
  - (c) Distress in the system associated with a specific bank.
- Measures defined:
  - Joint Probability of Distress (JPoD):
    - JPoD = P(X ∩ Y ∩ R) estimated by integrating BSMD over distress regions; embeds individual PoDs and distress dependence; can rise nonlinearly during crises.
  - Banking Stability Index (BSI):
    - Based on conditional expectation of default probability (Huang (1992)); reflects expected number of banks becoming distressed given at least one bank is distressed.
  - Distress Dependence Matrix (DiDe):
    - Matrix of pairwise conditional probabilities P(row bank distressed | column bank distressed); useful for mapping interlinkages and contagion likelihoods (not causation).
  - Probability that At Least One bank becomes Distressed (PAO) conditional on a specific bank:
    - Quantifies cascade potential and systemic importance; used as a systemic-importance indicator.

### Empirical implementation (2005–beginning October 2008) — inputs and PoD choice
- PoD estimation choices:
  - Structural Approach (SA), Out-of-the-Money option prices (OOM), and CDS-derived PoDs (CDS-PoDs) considered.
  - CDS-PoDs chosen for empirical BSM estimation reported here because CDS spreads often anticipate rating changes and data availability; PoDs treated as exogenous inputs and replaceable if better estimators are available.
- Definition of “distress”:
  - Broader than default, credit, or liquidity risk; intended to capture large losses and possible default as reflected in CDS spreads.

### Key empirical findings — U.S. banks (sample includes Citigroup, Bank of America, JPMorgan, Wachovia, Goldman Sachs, Lehman Brothers, Merrill Lynch, Morgan Stanley, Washington Mutual, AIG)
- Systemic interconnectedness and crisis amplification:
  - U.S. banks are highly interconnected; distress in one bank associated with high probability of distress elsewhere (JPoD and BSI evidence).
  - Distress dependence rises during crisis; systemic risks (JPoD and BSI) rise faster than idiosyncratic risks.
  - Daily percentage changes of the JPoD are larger than daily percentage changes of the individual (average) PoDs.
  - Investment banks (IBs) had larger JPoD and BSI than bank holding companies (BHCs); IBs’ risks were higher at the time of Lehman’s collapse.
- Distress dependence matrix and conditional probabilities (selected precise comparisons):
  - Average conditional probability of other U.S. banks being distressed if any U.S. bank fell into distress:
    - Increased from 27 percent on July 17, 2007 to 41 percent on September 12, 2008.
  - On September 12, 2008 (Lehman-centric observations):
    - Lehman: row-average PoD conditional on any other bank falling into distress reached on average 56 percent.
    - Column-average Lehman: PoD of any other bank conditional on Lehman falling into distress went from 25 percent on July 17, 2007 to 37 percent on September 12, 2008 (an increase of 12 percentage points).
    - Lehman, AIG, WaMu, and Wachovia formed a close cluster; on September 12, 2008:
      - Lehman bankruptcy implied chances of 88 percent for WaMu to fall into distress.
      - Lehman bankruptcy implied chances of 43 percent for AIG to fall into distress.
      - Lehman bankruptcy implied chances of 27 percent for Wachovia to fall into distress.
    - PAO conditional on Lehman becoming distressed reached 97 percent on September 12, 2008.

### Cross-region results — American and European banking groups
- European sample: Barclays, HSBC, UBS, Credit-Suisse (CSFB), Deutsche.
- Summary findings:
  - European JPoD and BSI moved in tandem with U.S. indicators and coincided with market events.
  - Distress dependence among European banks rose during crisis; daily percentage changes of the European JPoD larger than individual PoDs.
  - European banks’ JPoD and BSI generally lower than U.S. IBs and similar to U.S. BHCs across time; deterioration seen in mid-September 2008.
- DiDe comparisons (July 17, 2007 → September 12, 2008):
  - Average conditional probability of other European banks being distressed if any European bank fell into distress increased from 34 percent to 41 percent.
  - On September 12, 2008:
    - Row-average UBS (PoD conditional on any other bank falling into distress) reached on average 48 percent.
    - Probability of Barclays distress conditional on UBS becoming distressed rose from 18 percent (July 17, 2007) to 31 percent (September 12, 2008).
    - Column-average CSFB: average PoD of European banks conditional on CSFB falling into distress reached 43 percent.
    - Column-average Deutsche: average PoD of American banks conditional on Deutsche falling into distress reached 35 percent.
  - Cross-Atlantic asymmetry on September 12, 2008:
    - Failure of a U.S. bank implied (on average) chances of distress of one European bank of 7 percent (quadrant 3).
    - Average probability of distress of one American bank conditional on a European bank becoming distressed was above 30 percent (quadrant 2).

### Foreign banks’ risks to sovereigns — Latin America, Eastern Europe, Asia
- Method: derive PoDs from CDS spreads on sovereign and bank bonds; estimate cross vulnerabilities.
- Regions, country and bank lists analyzed (as implemented):
  - Latin America: Countries — Mexico, Colombia, Brazil, Chile; Banks — BBVA, Santander, Citigroup, Scotia Bank, HSBC.
  - Eastern Europe: Countries — Bulgaria, Croatia, Hungary, Slovakia; Banks — Intesa, Unicredito, Erste, Societe Generale, Citigroup.
  - Asia: Countries — China, Korea, Thailand, Malaysia, Philippines, Indonesia; Banks — Citigroup, JP Morgan Chase, HSBC, Standard and Chartered, BNP, Deutsche, DBS.
- Aggregate cross-dependence increases (average conditional PoDs, July → September 2008):
  - Quadrant 3 (sovereign distress conditional on bank distress) average conditional PoD increases:
    - Latin America: from 39 to 51 percent.
    - Eastern Europe: from 16 to 46 percent.
    - Asia: from 12 to 34 percent.
  - Quadrant 2 (bank distress conditional on sovereign distress) average conditional PoD increases:
    - Latin America: from 13 to 54 percent.
    - Eastern Europe: from 11 to 45 percent.
    - Asia: from 4 to 30 percent.
- Country- and bank-specific signals and implications:
  - Mexico’s conditional PoD rose significantly on September 16, 2008; Bulgaria and Hungary signaled under significant stress.
  - Indonesia, Korea, and the Philippines showed highest risk among Asian countries analyzed.
  - Geographic presence matters: distress in countries with large foreign-bank presence (e.g., Mexico, Croatia) is highly linked to potential banking distress.
  - Particular foreign banks’ distress maps to greater country risk where they have large presence (examples cited in source: Citigroup → Mexico; Intesa → Hungary; DBS → Indonesia).
  - In Asia, indirect/global effects and liquidity pressures amplify spillovers where direct foreign-bank presence is lower.
- Policy implication highlighted:
  - Limiting systemic risks in advanced-country financial systems would sharply reduce risks to emerging markets.

### Empirical evaluation framework and Monte Carlo evidence
- PIT evaluation approach:
  - If forecast densities coincide with true DGP, the sequence of PITs of observed realizations under those densities are iid U(0,1).
  - For multivariate densities, factorize joint density into marginals and conditionals; apply PIT to each component; for CIMDO re-estimated at each t, evaluation performed cross-sectionally at each t.
- Monte Carlo evaluation procedure (summary of steps):
  - Simulate a bivariate true DGP with marginals matching two empirically observed PoDs; simulate 10,000 observations.
  - Recover CIMDO-density using empirical PoDs.
  - Calibrate competing parametric distributions (multivariate normals, multivariate t, mixture of normals) with matching marginal PoDs.
  - Compute PITs for marginals and conditionals and test for uniformity (K-S tests reported in Segoviano (2006b)).
- Monte Carlo findings (high-level):
  - Given limited information (only PoDs), exact iid U(0,1) PITs are not expected; relative closeness is the focus.
  - CIMDO-distributions outperform parametric distributions (NStd, TCon, etc.) across the domain under the PIT criterion.
  - Improvements are largest in the default region (upper right corner of empirical cdf), critical for estimating extreme portfolio losses.

### Estimation of individual banks’ PoDs — approaches, trade-offs, and implications
- Structural Approach (SA):
  - Premise: firm asset value evolution with default triggered when asset value falls below a threshold tied to leverage.
  - Data inputs: bank equity prices and balance sheets.
  - Advantages: explicit default definition; links market perceptions and balance-sheet structure; allows decomposition of default determinants.
  - Challenges: asset-value proxying by stock prices; asset-return non-normality; sensitivity to volatility estimates; calibration difficulties.
- CDS-derived PoDs:
  - Advantages: avoid explicit modeling of asset distributions and volatilities.
  - Caveats: affected by CDS-market liquidity and general risk aversion.
- Out-of-the-Money (OOM) option-based PoDs:
  - Theoretical basis: under fundamental theorem of asset pricing, PoDs inferred from OOM option prices avoid explicit estimation of asset-price distributions, volatilities, recovery rates, and other inputs in principle.
  - Practical advantages: where liquid option markets exist, broader coverage possible.
  - Practical caveats: requires availability of OOM option prices across strikes and involves calibration efforts.
- Comparative assessment and practical note:
  - PoDs are exogenous inputs to the CIMDO estimation; CIMDO can be implemented with PoDs derived by any method deemed appropriate in different countries.
  - Authors conducting research to empirically determine most appropriate PoD methodology.

### Conclusions, implications, and research agenda
- The CIMDO/BSMD framework:
  - Provides complementary measures of banking stability across three perspectives (common distress, pairwise distress dependence, system distress associated with a specific bank).
  - Is implementable with limited data (individual-bank PoDs) and captures linear and non-linear default interdependence.
  - Quantifies time variation in default interdependence, relaxing fixed-correlation assumptions common in standard risk models.
- Suggested extensions and research agenda:
  - Attempt to predict future movements of BSMs as early-warning indicators.
  - Explore macroeconomic and financial factors and shocks influencing BSMs to identify macro-financial linkages.
  - Investigate factors and instruments that can limit and reverse tendencies toward instability to inform policy interventions.
  - Extend the framework to include NBFIs and broader cross-border surveillance as data permit.

*Source: _wp0904 - IMF working paper content (Sections III, Box 1, Appendixes I–IV, and empirical results summarized).*

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

### References

### Tables and Figures Inventory
- Tables
  - 1. Distress Dependence Matrix ................................................................................................18
  - 2. Distress Dependence Matrix: American and European Banks ............................................29
  - 3. Distress Dependence Matrix: Latin America. Sovereigns and Banks .................................33
  - 4. Distress Dependence Matrix: Eastern Europe. Sovereigns and Banks................................34
  - 5. Distress Dependence Matrix: Asia. Sovereigns and Banks .................................................35
- Figures
  - 1. The Probability of Distress ....................................................................................................8
  - 2. The Banking System’s Multivariate Density .........................................................................9
  - 3. Probability That At Least One Bank Becomes Distressed ..................................................19
  - 4. Joint Probability of Distress.................................................................................................23
  - 5. Banking Stability Index .......................................................................................................24
  - 6. Daily Percentage Increase: Joint and Average Probability of Distress................................25
  - 7. PAO: Lehman ......................................................................................................................26
  - 8. Foreign-Bank and Sovereign Risks .....................................................................................32
- Box
  - 1. Drawbacks to the Characterization of Distress Dependence of Financial Returns with Correlations..............................................................................................................................14
- Appendixes
  - I. Copula Functions ..................................................................................................................37
  - II. CIMDO-copula....................................................................................................................39
  - III. CIMDO-density and CIMDO-copula Evaluation Framework ..........................................40
  - IV. Estimation of Probabilities of Distress of Individual Banks .............................................47

### Introduction: Purpose and scope
- Purpose
  - Present a method for estimating a set of stability measures of the banking system (BSMs).
  - Demonstrate that the approach is reasonable and implementable with limited data.
- Scope of empirical implementation
  - Estimations using publicly available information from 2005 up to the beginning of October 2008.
  - Illustrative methodology: examine relative changes in stability over time and among different banks’ business lines in the US banking system.
  - Analyze cross-region effects between American and European banking groups.
  - Extend technique to incorporate effect of foreign banks on sovereigns in Latin America, eastern Europe and Asia.
- Key conceptual choices
  - Conceptualize the banking system as a portfolio of core, systemically important banks.
  - Infer the banking system’s portfolio multivariate density (BSMD) and construct BSMs that embed banks’ distress inter-dependence structure, capturing linear and non-linear dependencies and their changes across the economic cycle.
- Advantages emphasized
  - BSMs allow quantification of (a) “common” distress in the banks of the system, (b) distress between specific banks, and (c) distress in the system associated with a specific bank.
  - Methodology can be extended to include NBFIs (insurance companies, hedge funds etc.) when data are available.

### Methodology overview and steps
- Empirical inputs
  - Use empirical measurements of probabilities of distress (PoDs) for individual banks.
  - Data requirements described as minimal and generally available across countries.
- Core methodological components
  - Use Consistent Information Multivariate Density Optimizing (CIMDO) methodology to recover the banking system’s multivariate density (BSMD) from individual banks’ PoDs.
  - Employ a novel non-parametric copula approach: the CIMDO-copula (Segoviano, 2009) to capture full dependence structure (linear and non-linear) without explicitly choosing parametric copula forms.
  - PIT criterion used for multivariate density evaluation (Diebold et al (1999)).
- Four modeling steps (Figure 2)
  - Step1: Conceptualize the banking system as a portfolio of banks.
  - Step2: For each bank, obtain empirical measurements of probabilities of distress (PoDs).
  - Step3: Use CIMDO methodology with individual banks’ PoDs to recover the BSMD.
  - Step4: Based on the BSMD, estimate the proposed banking stability measures (BSMs).

### Distress dependence, economic cycle, and stability implications
- Nature of distress dependence
  - Banks are linked directly (inter-bank deposit market, participations in syndicated loans) and indirectly (lending to common sectors, proprietary trades).
  - Distress dependence varies across the economic cycle and tends to rise in times of distress due to contagion after idiosyncratic shocks or negative systemic shocks.
- Implications for system-wide measures
  - Joint probability of distress (JPoD): probability that all banks in the system experience large losses simultaneously; embeds banks’ distress dependence and may exhibit larger and nonlinear increases than individual banks’ PoDs during stressed periods.
- Limitations of standard approaches
  - Correlation coefficient is inadequate for capturing distress (tail) dependence because distress is a tail event in the distribution of implied asset price movements.
  - Parametric copula modeling is difficult to implement under data constraints and for tail dependence.
- Methodological response
  - Adopt a reduced-form or non-parametric copula (CIMDO-copula) that captures default dependence and its changes across points in the economic cycle, is implementable under data constraints, and produces robust estimates under the PIT criterion.

### Research agenda and intended extensions
- Immediate research aims
  - i. Examine relative changes in the BSMs over time and between countries to identify occasions and determinants of changes in banking system riskiness.
  - ii. Attempt to predict future movements of the BSMs for use as an early-warning mechanism.
  - iii. Explore significant macroeconomic and financial factors and shocks influencing the BSMs to identify macro-financial linkages.
  - iv. Explore factors that can limit and reverse tendencies towards instability to identify instruments to control instability.
- Relation to prior work
  - Builds on earlier research on portfolio credit risk, procyclicality of banking regulation, macroeconomic stress testing, and systemic risk measures (Segoviano (1998), Segoviano and Lowe (2002), Goodhart and Segoviano (2004), Segoviano and Padilla (2006), Goodhart, Hofmann and Segoviano (2004, 2006), Aspachs et al. (2006)).
  - Advances prior models by explicitly incorporating banks’ distress dependencies and their changes across the economic cycle.

*Source: _wp0904 - References..............................................................................................................*

### Section III describes the procedure to recover the BSMD.

### III.   BANKING SYSTEM MULTIVARIATE DENSITY

### A. The CIMDO Approach: Modeling the Banking System Multivariate Density
- Purpose:
  - The Banking System Multivariate Density (BSMD) characterizes both the individual and joint asset value movements of the portfolio of banks representing the banking system.
  - The BSMD is recovered using the Consistent Information Multivariate Density Optimizing (CIMDO) methodology (Segoviano, 2006b).
  - Individual banks’ probabilities of distress (PoDs) are treated as exogenous inputs to the CIMDO framework, allowing flexibility in PoD estimation methods (Structural Approach (SA); Credit Default Swaps (CDS); Out of the Money Option Prices (OOM) — discussed further in Section V).
- Key technical advantages over traditional risk models:
  - CIMDO embeds the banks’ distress dependence structure—characterized by the CIMDO-copula function (Segoviano, 2009)—capturing linear and non-linear distress dependencies among banks.
  - Dependence is allowed to change throughout the economic cycle, reflecting that dependence increases in periods of distress. This contrasts with traditional models that usually account only for linear dependence (correlations) assumed constant over the cycle or a fixed period.
- Methodological foundation:
  - CIMDO is based on the minimum cross-entropy approach (Kullback, 1959). A posterior multivariate distribution p—the CIMDO-density—is recovered by updating a prior density q with empirical information via a set of constraints (empirically estimated PoDs).
  - The CIMDO-objective (cross-entropy) functional is defined as:
    - C[p,q] = ∫ ∫ p(x,y) ln( p(x,y) / q(x,y) ) dx dy, where q(x,y) and p(x,y) ∈ L^2(R).
  - The prior distribution q follows a parametric form consistent with economic intuition (e.g., defaults triggered by asset value drops below a threshold) and theoretical models (structural approach). However, q is usually inconsistent with empirically observed distress measures, motivating the constraint-based update.
- Consistency constraints and optimization:
  - Constraints are imposed on the marginal densities of the posterior multivariate density to match empirically estimated bank PoDs. In formal terms (for banks X and Y):
    - ∫∫_{x_d^x}^{∞} ∫_{y_d^y}^{∞} p(x,y) dx dy = PoD_x^{t}
    - ∫∫_{x_d^x}^{∞} ∫_{y_d^y}^{∞} p(x,y) dx dy = PoD_y^{t}
    - Indicator functions χ_x(·), χ_y(·) are defined with distress thresholds d_x, d_y estimated for each bank.
  - The posterior p(x,y) must satisfy p(x,y) ≥ 0 and ∫∫ p(x,y) dx dy = 1.
  - The constrained minimization functional L[p] (including Lagrange multipliers λ_1, λ_2 for PoD constraints and μ for additivity) is minimized via calculus of variations.
  - The optimal posterior multivariate density solution takes the exponential form (displayed in the source as equation (3)), where the prior q(x,y) is adjusted multiplicatively by exponential terms involving the Lagrange multipliers and indicator functions.
- Performance and evaluation:
  - CIMDO-recovered distributions outperform commonly used parametric multivariate densities under the Probability Integral Transformation (PIT) criterion.6
  - The rationale: CIMDO uses empirical information embedded in the constraint set to adjust the multivariate density “shape” via optimization, whereas parametric approaches adjust shape via fixed parameter sets.
  - Appendix 3 summarizes the PIT evaluation criterion and Monte Carlo study results (detailed development in Segoviano (2006b)).

### B. The CIMDO-copula: Distress Dependence among Banks in the System
- Role of the copula in BSMD:
  - The BSMD embeds both linear and nonlinear default dependence among banks; this dependence structure is characterized by the copula of the BSMD—the CIMDO-copula—which changes over time consistent with changes in empirically observed PoDs.
- Copula approach overview:
  - Any multivariate density can be decomposed into (i) marginal distributions for each variable and (ii) the dependence structure among variables (the copula).
  - Marginal information is “sterilized” by transforming marginals into uniform U(0,1) variables; the joint distribution of these uniforms is the copula, which contains only dependence information.
  - For random variables x and y with marginals F_x and H_y and joint G_xy, define u = F_x(x), v = H_y(y); u and v ∼ U(0,1). The copula density c(u,v) relates to the original densities by:
    - c(u,v) = g(F_x^{-1}(u), H_y^{-1}(v)) / ( f_x(F_x^{-1}(u)) h_y(H_y^{-1}(v)) )
    - Rewritten in x,y terms: g(x,y) = c(F_x(x), H_y(y)) f_x(x) h_y(y).
  - If variables are independent, the ratio in equation (5) equals one.
- Advantages of copulas vs. correlations:
  - Copula functions describe linear and non-linear dependencies of any type of multivariate densities, across the entire domain.
  - Copulas are invariant under increasing and continuous transformations of marginals.
- Shortcomings of standard parametric copula approaches:
  - (i) Copula choice problem (CCP): choosing, specifying, and calibrating a parametric copula is challenging and results are sensitive to functional form and parameter values; joint distributions of distress required for correct specification are generally not available.
  - (ii) Parametric copulas commonly require correlation parameters that are typically held fixed through time (see Appendix 1). Even dynamically-updated correlations (e.g., rolling windows) remain fixed within windows and the choice of window length is often subjective.8
- CIMDO-copula advantages (implied by exposition):
  - The CIMDO-copula embeds a time-varying dependence structure that updates consistently with empirically observed PoDs, thereby addressing the CCP and the fixed-dependence shortcomings of standard parametric copulas.

*Source: _wp0904 - Section III describes the procedure to recover the BSMD.*

### Box 1. Drawbacks to the Characterization of Distress Dependence of Financial Returns

### Box 1. Drawbacks to the Characterization of Distress Dependence of Financial Returns with Correlations

### Drawbacks of Using Correlation to Characterize Financial-returns Dependence
- Linear correlation is natural for multivariate normal distributions but financial-asset returns are seldom multivariate normal (empirical support cited).
- For heavy-tailed distributions that usually characterize financial asset returns, variances might not be finite; hence correlation becomes undefined.
- Zero linear correlation does not imply independence; strong dependence can coexist with zero covariance (example referencing moment properties of the standard normal).
- Linear correlation is not invariant under nonlinear strictly increasing transformations; ρ(X,Y) generally differs from ρ(T(X),T(Y)).
- Correlation measures center-of-distribution dependence and gives little weight to tail events; therefore it is ill-suited to characterize distress dependence (tail risk) when marginal distributions are non-normal.

### The CIMDO-copula: Method and Advantages
- Method summary:
  - Infer the CIMDO-density (Section III.A).
  - Extract the copula function (CIMDO-copula) from the multivariate CIMDO-density using estimated marginal densities and Sklar’s theorem.
- Benefits preserved relative to standard copula approach:
  - Describes linear and non-linear dependencies; dependence structure invariant under increasing, continuous transformations of marginals.
  - Characterizes dependence along the entire domain of the CIMDO-density and is more robust in the tail where distress dependence is of primary interest.
- How CIMDO-copula avoids standard copula drawbacks:
  - Circumvents the Copula Choice Problem: the CIMDO-copula is recovered from the inferred CIMDO-density without imposing explicit parametric copula forms; under limited information (only marginal probabilities of distress), the CIMDO-copula is easily implementable and outperforms common parametric copulas under the PIT criterion, especially in the tail (see Appendix 3 referenced).
  - Avoids imposing constant correlation parameters: the CIMDO-copula “automatically” updates when empirical PoDs change because the Lagrange multipliers (λ12, λ1, λ2 and μ) of the CIMDO functional vary with constraints, so the copula and default dependence change over time and across the economic cycle.
- Formal representation:
  - The CIMDO-copula c_c(u,v) is given by an integral expression (equation (6)) that is a nonlinear function of λ12, λ1, λ2, and μ; changes in empirical PoDs change Lagrange multipliers and thus alter the CIMDO-copula and implied default dependence.

### Banking Stability Measures (BSMs): Definitions and Components
- Objective: characterize individual-bank PoDs, distress dependence, and changes across the economic cycle; quantify stability from three complementary perspectives:
  - (a) Common distress in banks of the system.
  - (b) Distress between specific banks.
  - (c) Distress in the system associated with a specific bank.

- Illustrative CIMDO-density for a three-bank system (x, y, r) given by equation (7): p(x,y,r) ∝ exp{ q(x,y,r) + μ + λ1χ1 + λ2χ2 + λ3χ3 } (structure preserved as in source).

A. Common Distress: JPoD and BSI
- Joint Probability of Distress (JPoD):
  - JPoD = P(X ∩ Y ∩ R) estimated by integrating the BSMD over distress regions (equation (8)).
  - JPoD embeds individual PoDs and distress dependence; distress dependence can rise in crisis causing larger, nonlinear increases in JPoD relative to average individual PoDs.
- Banking Stability Index (BSI):
  - Based on conditional expectation of default probability (Huang (1992)).
  - Reflects expected number of banks becoming distressed given at least one bank is distressed; higher number → increased instability.
  - Example for two banks given by equation (9).

B. Distress Between Specific Banks: Distress Dependence Matrix (DiDe)
- DiDe contains pairwise conditional probabilities: probability of distress of row bank given column bank is distressed.
- Useful for insights into interlinkages and likelihood of contagion (does not imply causation).
- Example estimation formula (equation (10)) for P(X|Y).

C. Distress in the System Associated with a Specific Bank: PAO
- Probability that At Least One bank becomes Distressed (PAO) given a specific bank is distressed quantifies cascade potential and systemic importance of that bank.
- Example formula for a four-bank system given by equation (11).
- PAO used as indicator of systemic importance (Venn-diagram illustrated in Figure 3).

### Empirical Implementation and Results (2005–October 2008; CDS-PoDs used)
- PoD estimation choice and rationale:
  - Alternatives: structural approach (SA), out-of-the-money option (OOM), CDS-derived PoDs (CDS-PoDs).
  - SA and OOM had parametrization/data issues; CDS-PoDs have drawbacks (liquidity, risk aversion effects) but often anticipate rating changes and were chosen for empirical BSM estimation.
  - PoDs are exogenous inputs in the CIMDO framework and can be replaced if better estimators are found.
- Definition of “distress” risk: broader than default, credit, or liquidity risk; captures large losses and possible default as reflected in CDS spreads.

A. Relative Changes of Stability Over Time (U.S. banks, to October 2008)
- Sample: BHCs — Citigroup, Bank of America, JPMorgan, Wachovia; IBs — Goldman Sachs, Lehman Brothers, Merrill Lynch, Morgan Stanley; also Washington Mutual (WaMu) and AIG.
- Key empirical findings (perspective 1: JPoD and BSI):
  - U.S. banks are highly interconnected; distress in one bank associated with high probability of distress elsewhere as shown by JPoD and BSI.
  - Distress dependence across banks rises during crisis; systemic risks (JPoD and BSI) rise faster than idiosyncratic risks.
  - Daily percentage changes of the JPoD are larger than daily percentage changes of the individual (average) PoDs (Figure 6 evidence).
  - Investment banks (IBs) show larger JPoD and BSI than bank holding companies (BHCs); IBs’ risks were higher at time of Lehman’s collapse.

- Key empirical findings (perspective 2: DiDe; examples comparing July 1, 2007 and September 12, 2008):
  - Average conditional probability of other U.S. banks being distressed if any U.S. bank fell into distress increased from 27 percent on July 17, 2007 to 41 percent on September 12, 2008.
  - On September 12, 2008:
    - Lehman: large PoD conditional on any other bank falling into distress reached on average 56 percent (row-average Lehman).
    - Lehman default estimated to raise chances of default elsewhere by 46 percent; the PoD of any other bank conditional on Lehman falling into distress went from 25 percent on July 17, 2007 to 37 percent on September 12, 2008 (column-average Lehman).
    - Lehman, AIG, WaMu, and Wachovia formed a cluster of particularly close links; on September 12, a Lehman bankruptcy implied chances of 88 percent for WaMu, 43 percent for AIG, and 27 percent for Wachovia to fall into distress.
  - On September 12, 2008 the probability that one or more banks would become distressed given Lehman became distressed (PAO) reached 97 percent (perspective 3).

B. Cross-Region Effects Between American and European Banking Groups
- Sample included five major European banks: Barclays, HSBC, UBS, Credit-Suisse (CSFB), Deutsche.
- Findings:
  - European banks’ JPoD and BSI move in tandem with U.S. indicators and coincide with relevant market events.
  - Distress dependence among European banks rose during crisis; daily percentage changes of the European JPoD larger than individual PoDs.
  - Risks for European banks (JPoD and BSI) generally lower than U.S. IBs and similar to U.S. BHCs across time; deterioration noted in mid-September.
  - From DiDe comparisons (July 17, 2007 → September 12, 2008):
    - Average conditional probability of other European banks being distressed if any European bank fell into distress increased from 34 percent to 41 percent.
    - On September 12, 2008:
      - UBS showed largest PoD conditional on any other bank falling into distress, reaching on average 48 percent (row-average UBS).
      - Probability of Barclays distress conditional on UBS becoming distressed rose from 18 percent (July 17, 2007) to 31 percent (September 12, 2008).
      - Column-average CSFB implied (average) PoD of European banks conditional on CSFB falling into distress reached 43 percent.
      - Column-average Deutsche: (average) PoD of American banks conditional on Deutsche falling into distress reached 35 percent.
    - Cross Atlantic link asymmetry: on September 12, 2008, failure of a U.S. bank implied (on average) chances of distress of one European bank of 7 percent (quadrant 3), while the (average) probability of distress of one American bank conditional on a European bank becoming distressed was above 30 percent (quadrant 2).

C. Foreign Banks’ Risks to Sovereigns (Latin America, Eastern Europe, Asia)
- Method: use CDS spreads on sovereign and bank bonds to derive PoDs and estimate cross vulnerabilities.
- Regions and entities analyzed include:
  - Latin America: Countries — Mexico, Colombia, Brazil, Chile; Banks — BBVA, Santander, Citigroup, Scotia Bank, HSBC.
  - Eastern Europe: Countries — Bulgaria, Croatia, Hungary, Slovakia; Banks — Intesa, Unicredito, Erste, Societe Generale, Citigroup.
  - Asia: Countries — China, Korea, Thailand, Malaysia, Philippines, Indonesia; Banks — Citigroup, JP Morgan Chase, HSBC, Standard and Chartered, BNP, Deutsche, DBS.
- Key observations:
  - Cross dependencies rose sharply between July 2007 and September 2008, implying systemic risks leapt.
  - Quadrant 3 (sovereign distress conditional on bank distress) average conditional PoD increases (July → September 2008):
    - Latin America: from 39 to 51 percent.
    - Eastern Europe: from 16 to 46 percent.
    - Asia: from 12 to 34 percent.
  - Quadrant 2 (bank distress conditional on sovereign distress) average conditional PoD increases:
    - Latin America: from 13 to 54 percent.
    - Eastern Europe: from 11 to 45 percent.
    - Asia: from 4 to 30 percent.
  - Country- and bank-specific signals: e.g., on September 16, 2008 Mexico’s conditional PoD rose significantly; Bulgaria and Hungary signaled under significant stress; Indonesia, Korea, and the Philippines at highest risk in Asia.
  - Geographic presence matters: distress in countries with large foreign-bank presence (Mexico, Croatia) is highly linked with potential banking distress; distress of particular foreign banks maps to greater country risk in the regions where they have large presence (examples: Citigroup → Mexico; Intesa → Hungary; DBS → Indonesia).
  - Indirect/global effects are important for Asia where direct foreign-bank presence is lower; systemic spillovers and liquidity pressures amplify indirect links.
  - Policy implication: limiting systemic risks in advanced-country financial systems would sharply reduce risks to emerging markets.

### Conclusions and Implications
- The proposed CIMDO/BSMD framework:
  - Provides complementary measures of banking stability across three perspectives.
  - Can be constructed from a limited data set (empirical default probabilities of individual banks) and is implementable in many countries.
  - Embeds banks’ default interdependence structure (copula), capturing linear and non-linear dependencies.
  - Quantifies changes in default interdependence over time, relaxing fixed-correlation assumptions common in risk models.
- Empirical application up to October 2008 demonstrates implementation flexibility and relevance for cross-border banking stability surveillance.
- Suggested research agenda and applications:
  - Attempt to predict future movements of the BSMs as early-warning indicators.
  - Explore macroeconomic and financial factors and shocks influencing BSMs to identify macro-financial linkages.
  - Investigate factors and instruments that can limit and reverse tendencies toward instability to inform policy interventions.

*Source: Box 1. Drawbacks to the Characterization of Distress Dependence of Financial Returns with Correlations (excerpted content).*

### APPENDIX I. COPULA FUNCTIONS

### APPENDIX I. COPULA FUNCTIONS

### Definition and purpose of copulas
- A copula isolates the dependence structure between two random variables x and y by sterilizing marginal information via the Probability Integral Transformation (PIT), mapping each marginal to U(0,1).
- Under the PIT two new variables are defined as u = F(x) and v = H(y), both distributed as U(0,1) with joint density c(u,v).
- Under the distribution of transformations of random variables (Cassella and Berger, 1990) the copula density is defined (equation (12)) as:
  - c(u,v) = g(F^{-1}(u), H^{-1}(v)) / [ f(F^{-1}(u)) h(H^{-1}(v)) ]  (presented in the source as equation (12) with g, f, h defined densities).
- From equation (13) the joint density of x and y can be written as:
  - g(x,y) = c(F(x), H(y)) f(x) h(y)  (equation (13)).
- If x and y are independent then c(F(x),H(y)) = 1.

### Sklar’s Theorem and corollary
- Sklar’s Theorem (Sklar, 1959) (equation (14)):
  - G(x,y) = C( F(x), H(y) )
- If F and H are continuous then C is unique; otherwise C is uniquely determined on RanF x RanH.
- Converse: Given a copula C and marginals F and H, the multivariate function G defined by equation (14) is a joint distribution with margins F and H.
- Corollary (Nelsen, 1999): for continuous marginals F and H, with quasi-inverses F^{-1}, H^{-1}, there exists a unique copula C: [0,1] x [0,1] → [0,1] such that G(x,y)=C(F(x),H(y)).
- Cross partial derivatives lead to (equation (15)):
  - g(x,y) = c(F(x),H(y)) f(x) h(y)

### Common parametric copulas used in finance
- Gaussian-copula (equation (16)):
  - C^{Ga}_{\rho}(u,v) = \int_{-\infty}^{\Phi^{-1}(u)} \int_{-\infty}^{\Phi^{-1}(v)} (1/(2\pi\sqrt{1-\rho^2})) exp[ - (s^2 - 2\rho s t + t^2) / (2(1-\rho^2)) ] ds dt
  - Where ρ is the linear correlation coefficient and Φ^{-1} denotes the inverse cdf of the standard normal.
- t-copula (equation (17)):
  - C^{t}_{\rho,\nu}(u,v) = \int_{-\infty}^{t^{-1}_{\nu}(u)} \int_{-\infty}^{t^{-1}_{\nu}(v)} [ multivariate t density with ν degrees of freedom and correlation ρ ] ds dt
  - Where t^{-1}_{\nu} denotes the inverse cdf of the standard univariate t-distribution with ν degrees of freedom.
- Both copulas depend only on ρ (and ν for the t-copula).

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### APPENDIX II. CIMDO-COPULA

### Derivation of the CIMDO-copula
- Starting from the general copula definition (as in Appendix I), the copula of a prior with density q(x,y) is (equation (18)):
  - c^{q}(u,v) = q(F^{-1}(u), H^{-1}(v)) / [ ∫_{-\infty}^{+\infty} q(x,y) dy ∫_{-\infty}^{+\infty} q(x,y) dx ] (expressed in integral form in the source).
- The CIMDO distribution with prior q(x,y) is specified in exponential form (source presents a CIMDO density expression involving μ, λ, χ; see equation (19) for the CIMDO-copula).
- The CIMDO-copula (equation (19)) is a nonlinear function of parameters μ, λ_1, λ_2, χ (presented in source as 12 , , μ λ λ χ notation), which change as the PoDs of the banks under analysis change.
- Therefore, the CIMDO-copula captures changes in PoDs as these change across economic cycles.

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### APPENDIX III. CIMDO-DENSITY AND CIMDO-COPULA EVALUATION FRAMEWORK

### Probability Integral Transformation (PIT) approach for density evaluation
- Diebold et al. (1998, 1999) show that if forecast densities coincide with the true DGP, the sequence of PITs of observed realizations under those forecast densities are iid U(0,1).
- For multivariate (Mth) densities, factorize the joint density into product of one marginal and M-1 conditionals; the PITs of the M series will be iid U(0,1) individually and jointly if forecasts are correct.
- For CIMDO (which is re-estimated at each time t), evaluation is performed cross-sectionally at each t: the product of conditionals and marginal PITs at each t should be iid U(0,1). A proof and Monte Carlo development are in Segoviano (2006b).

### Theoretical proofs summarized
- PIT Theorem (presented): If U = F(X) where F is the cdf of an absolutely continuous random vector X, then U ∼ U(0,1).
- Independence of PITs for decomposed marginals/conditionals: Using the distribution-of-transformations result (Cassella and Berger, 1990) and the structure of the conditional/marginal cdfs, the transformed variables u and v are shown to be independent at a given period of time.

### Monte Carlo evaluation procedure (stepwise)
1. Take values of two empirically observed PoDs of different financial assets.
2. Simulate the DGP of a bivariate density whose marginals match those PoDs; simulate 10,000 observations under this true DGP.
3. Recover the CIMDO-density using the empirically observed PoDs.
4. Calibrate common parametric distributions (multivariate normals, multivariate t-distributions, mixture of normals) ensuring their PoDs match empirically observed PoDs.
5. Decompose competing distributions into product of marginals and conditionals.
6. Compute PITs (z-variables) of realizations under the conditional and marginal distributions:
   - P_{x|y}(x|y), P_{y|x}(y|x), P_x(x)
   - The evaluated distributions include: NStd, NCon, TCon, NMix and CIMDO (as named in the source).
7. Test whether the series of z-variables are iid U(0,1):
   - Independence test: not necessary at a given period t because PITs are independent by the proof.
   - Uniformity test: compare empirical cdf of z-variables to the 45° line (or perform Kolmogorov-Smirnov tests; K-S results are reported in Segoviano (2006b)).

### Monte Carlo results (summary of findings)
- Given limited data (only PoDs), achieving exact iid U(0,1) PIT series is not expected; focus is on relative closeness.
- CIMDO-distributions outperform parametric distributions (NStd, TCon, etc.) across the whole domain of the distribution under the PIT criterion.
- Improvements are largest in the region of default (upper right corner of the empirical cdf), which is critical for estimating extreme portfolio losses because it represents probabilities of default for portfolio assets.
- Empirical cdf plots provided in the source show CIMDO closer to the 45° line (True DGP) than NStd and TCon for z_{x|y} and z_{y} variables.

### Copula evaluation under the PIT framework
- Copula functions, being bivariate densities of U and V with uniform marginals on [0,1], can be decomposed as c(u,v) = c_v(v) c_{u|v}(u|v) or c(u,v) = c_u(u) c_{v|u}(v|u).
- Because the marginal density of V is the constant 1 on [0,1], the conditional density of U given v is c_{u|v}(u|v)=c(u,v).
- Integrating shows the distribution of U given fixed v is uniform (see equation (22) in the source).
- Once decomposed, marginal and conditional copulas are evaluated under the same PIT criterion; a Monte Carlo experiment similar to the density experiment was carried out.

*Source: _wp0904 - APPENDIX I. COPULA FUNCTIONS*

### APPENDIX IV. ESTIMATION OF PROBABILITIES OF DISTRESS OF INDIVIDUAL BANKS

### APPENDIX IV. ESTIMATION OF PROBABILITIES OF DISTRESS OF INDIVIDUAL BANKS

### Structural Approach
- Basic premise: a firm’s underlying asset value evolves stochastically and default is triggered when it falls below a pre-specified barrier (the default-threshold), modeled as a function of the firm's leverage structure.
- Determinants of a firm’s PoD under the SA:
  - (a) the market value of its assets, A_V;
  - (b) the uncertainty or volatility of the asset value, A_σ;
  - (c) the degree of leverage or the extent of the firm’s contractual liabilities, measured as the book value of liabilities at time t, t_D (with maturity T).
- Implementation: under the most basic and popular version, A_V and A_σ are modeled following the Merton (1974) framework using information concerning equity prices.
- Data inputs required: bank equity prices and balance sheets of each bank under analysis—variables usually available for the largest banks across developed and developing countries.
- Advantages:
  - Explicit definition of default via a threshold tied to leverage.
  - PoD summarizes market perceptions (via equity prices) and financial conditions (via default threshold).
  - Allows separate analysis of factors determining default, improving understanding relative to approaches that only analyze changes in asset returns without an explicit default threshold.
- Challenges and disadvantages:
  - Proper specification and calibration of the distribution of firms’ asset values is difficult because stock prices are only a proxy for asset values.22
  - Financial firms’ logarithmic asset values empirically do not follow normal distributions; different parametric distributions have been proposed to improve modeling.
  - Availability of proxies indicating the evolution of underlying asset values is crucial for calibration and implementation.
  - PoDs are very sensitive to volatility values; numerous methods to estimate realized and implied volatilities can produce significantly different volatility values.
  - Sensitivity implies PoDs may reflect factors affecting aggregate equity market level and variance rather than the particular robustness or fragility of the banking system.
- Footnotes cited in source:
  - 22: The CreditMetrics methodology (Gupton, Finger, and Bhatia, 1997) and Moody’s-KMV methodology (Crosbie, 1998) are applications built upon this approach.
  - 23: Servigny and Renault (2002) and Schonbucher (2003) document that equity return second moments tend to be significantly higher than asset return ones.

### CDS Spreads
- Estimation characteristics:
  - PoDs inferred from CDS spreads are not subject to modeling the distribution of asset prices or explicit estimation of CDS volatilities (see Box 2 in source).
- Advantages relative to SA:
  - Avoids explicit estimation of asset distributions and volatilities.
- Disadvantages and caveats:
  - PoDs inferred from CDS spreads are affected by liquidity in the particular CDS market.
  - PoDs are affected by generalized risk aversion in the financial system.

### Out-of-the-Money Option Prices
- Theoretical basis:
  - Under the fundamental theorem of asset pricing, estimation of PoDs from out-of-the-money (OOM) calls or puts is not subject to explicit estimation of the distribution of asset prices, volatilities, recovery rates (RR), price of delivery options, or convenience.24
- Practical advantages:
  - For some banking sectors, liquid markets of plain vanilla options might be more extensive than liquid CDS markets, potentially allowing estimation of PoDs for a larger set of banks in developed and developing economies.
- Practical caveats:
  - CDS markets are growing and becoming liquid rapidly; option market liquidity may be overtaken by CDS liquidity in some cases.
  - Methodology depends on:
    - (i) availability of information of OOM option prices at different strike prices;
    - (ii) model calibrations, which in some cases can become cumbersome.
- Footnote cited in source:
  - 24: For an excellent presentation, we refer readers to Neftci (2004).

### Comparative assessment and methodological implications
- Summary of trade-offs:
  - Structural Approach: explicit default definition, data-availability advantage for large banks, but high sensitivity to volatility and strong modeling/calibration demands.
  - CDS-based PoDs: avoid asset-distribution modeling but are sensitive to market liquidity and risk aversion.
  - OOM option-based PoDs: avoid many explicit model inputs under no-arbitrage pricing, potentially broader coverage where option markets are liquid, but require OOM price availability and calibration.
- Practical note from source:
  - The authors are conducting research to empirically determine the most appropriate methodology to estimate individual banks’ PoDs.
  - PoDs are exogenous variables to the estimation of the banking system multivariate density; the CIMDO methodology can be implemented with PoDs derived by any method considered most appropriate or available in different countries.

*Source: APPENDIX IV. ESTIMATION OF PROBABILITIES OF DISTRESS OF INDIVIDUAL BANKS (IMF working paper content).*

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