## 2.     BIS Capital Ratio after a Credit Shock

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---

### Introduction
- Purpose: apply three methodologies to assess financial linkages across Korean banks and quantify risks from interconnectedness:
  - Network approach (bank balance sheet data).
  - Co-risk analysis (CDS spreads, 95th quantile regression).
  - Distress dependence (stock prices and estimated default probabilities).
- Key overall findings:
  - Korean banks are interconnected but no single bank by itself poses systemic threat to the rest of the system; no bank is highly vulnerable to distress in another single bank.
  - Financial risks posed by banks and contagion risk declined significantly after the global financial crisis.
  - Market-based, high-frequency indicators (co-risk and distress dependence) provide forward-looking assessments complementary to traditional FSIs.

### Network approach — Data
- Bilateral interbank exposure data for 18 Korean banks: Busan, Citibank Korea, Daegu, Hana, Industrial Bank of Korea, Jeju, Jeonbuk, KDB, Kookmin, Korea Exchange, Korea Exim, Kwangju, Kyongnam, NongHyup, SC First, Shinhan, Suhyup, and Woori.
- Analysis dates: end-2008, end-2009, and September 2010.
- Bank assets include: bank bonds receivable, loans, call loans, bonds purchased under repurchase agreements, accounts receivable, derivative assets.
- Bank liabilities include: deposits, CD, bank bonds payable, borrowings, call money, bonds sold under repurchase agreements, accounts payable, derivative liabilities.

### Network approach — Methodology
- Simulate losses for an individual bank induced by default of another bank.
- Shock types:
  - Credit shock: write-off of Exposure at Default (EAD) with Loss given Default (LGD) = 100%.
  - Credit-plus-funding shock: direct credit loss plus funding loss = (1 - rollover ratio) * haircut.
    - Rollover rate: Korean Won: 65% (Same April 2009 GFSR); Foreign Currency: 30% (Average rollover ratio during 97 Asian crisis).
    - Haircut (assumed for simulations): 50 percent.
- Default criterion: a bank is in default when its loss > total capital.
- Compare simulated losses to banks’ capital and BIS capital adequacy ratios (Basel II basis).

### Network approach — Results
- Credit shock:
  - Ratio of largest credit-shock-loss-to-capital for Korean banks lies between 4 percent and 33 percent.
  - Mean of these ratios declined from end-2008 through September 2010.
  - No bank’s loss from default of another individual bank exceeds its total capital.
  - BIS capital adequacy ratios (Basel II basis) after largest credit shock: average CAR exceeded 12 percent from 2008 to 2010; no bank’s CAR falls below the 8 percent regulatory minimum.
- Credit-plus-funding shock (assumptions: 50 percent haircut, 65 percent rollover for won, 30 percent rollover for foreign currency):
  - Ratios of largest credit-plus-funding losses relative to capital range from 5 percent to 36 percent.
  - Average ratio declined to less than 11 percent from about 16 percent since end-2008.
  - Even the largest loss for any Korean bank is no more than 40 percent of that bank’s capital.
  - BIS capital adequacy ratios after deducting largest credit-plus-funding shock remain above the 8 percent required minimum for all banks.

### Network approach — Summary
- Simulations indicate Korean banks are well-capitalized and can absorb losses due to default of another single bank; no single-bank knock-on would exhaust another bank’s capital.
- No systemically important financial institution identified that would cause defaults in other banks via single-bank failure; no systemically vulnerable bank identified likely to default due to another single bank’s failure.
- Supporting factors:
  - Direct interbank exposures are not excessively large relative to banks’ capital; households and corporate sector are major funding sources; nondeposit liabilities diversified.
  - Loan-to-deposit (LTD) regulation adopted December 2009 limits wholesale funding (LTD ratio required at or below 100 percent). As of end-September 2010, average LTD of 15 Korean banks subject to regulation = 96.5 percent, down from 115.9 percent in December 2009.
  - Median CAR of Korean banks exceeded 12 percent during the study period; average CAR increased from 12.7 percent as of end-2008 to 14.6 percent as of end-2009.
- Caveat: analysis tests knock-off effect of a single bank; results may differ for combined/multiple bank failures.

### Co-risk approach — Data
- Daily financial data from January 1, 2006 through March 31, 2011.
- 11 Korean banks analyzed: Citibank Korea, Hana, Kookmin, Korea Exchange, SC First, Shinhan, Woori (commercial banks); Industrial Bank of Korea, Korea Development, Korea EXIM, Nonghyup (specialty banks).
- Main variable: credit default swap (CDS) spreads for these 11 banks.
- Control variables:
  - TED spread.
  - Spread between one-year KORIBOR and Korean Treasury bond yields.
  - Spread between three-year Korean and U.S. Treasury bond yields.

### Co-risk approach — Methodology
- Use Co-VaR framework via 95th quantile regression (Adrian and Brunnermeier (2009) approach).
- Regression: dependent variable = CDS spread of bank i; independent variables = CDS spreads of remaining banks and common risk factors, estimated at the 95th quantile.
- Construct conditional co-risk matrix where each cell |ij Co-risk| is co-risk of bank j conditional on distress of bank i.
- Indicators:
  - Systemic vulnerability of bank i = average of row i.
  - Systemic importance of bank j = average of column j.
  - Systemic risk = average of all elements in the co-risk matrix.
- Co-risk matrices presented as percentage changes in distress (percent change in CDS spread relative to 95th percentile).

### Co-risk approach — Results
- Conditional co-risk matrices shown for end-September 2010 and October 13, 2008 (Lehman peak).
- Illustrative statistics and dynamics:
  - Pre-crisis average co-risk ~ 5 percent.
  - Average conditional co-risk rose to around 10 to 15 percent during 2007–08.
  - Average conditional co-risk peaked close to 25 percent immediately after the failure of Lehman Brothers.
  - By September 2010 average conditional co-risk remains slightly above pre-crisis levels.
- Comparative observation:
  - Even at the crisis peak, distress among Korean banks was more contained than in other jurisdictions; IMF’s calculation showed conditional co-risk between LCFIs overseas ranged between -6 and 617 percent in March 2008.
- Cross-section finding:
  - Banks with lower financial stability indicators (higher CDS spreads) are more vulnerable to systemic risk and more sensitive to distress from other banks.
- Time-series dynamics:
  - Average conditional co-risk follows major international events: BNP Paribas suspension (Aug. 2007), Bear Stearns’ bankruptcy (Mar. 2008), Lehman Brothers filing for court protection (Sep. 2008), and market volatility due to fiscal concerns in some European countries (May, 2010).
- Example cells from reported tables:
  - Table 2 (end-September 2010): sample cell: bank 1 conditional on bank 2 = 3.2 percent.
  - Table 3 (October 13, 2008): sample cell: bank 1 conditional on bank 5 = 38.9 percent.

### Co-risk approach — Summary
- Co-risk analysis indicates linkage risks were lower before the global financial crisis, rose sharply during major international events (e.g., Bear Stearns, Lehman), and normalized subsequently.
- Korean banks are highly integrated with global financial markets; international events drive conditional default probabilities more than domestic events.
- Banks with higher CDS spreads are more sensitive to distress (may reflect both true vulnerability and investor behavior of selling risky assets first).

### Distress dependence approach — Data
- Daily stock prices and expected default frequencies (EDF) from Moody’s KMV for 11 Korean banks from 2006 to March 31, 2011.
- Banks included: Busan, Cheju, Daegu, Hana, Industrial Bank of Korea, Jeonbuk, Kookmin, Korea Development, Korea Exchange, Shinhan, Woori.

### Distress dependence approach — Methodology
- Two-step approach (Segoviano and Goodhart (2009)):
  1. Estimate default probability of each bank using the contingent claims approach (CCA) — equity as a call option on bank assets with strike equal to default-free liabilities (DP); infer asset value A and asset volatility σ_A from observed equity value and equity volatility; distance to default (DD) measured and default probability = N(-DD) under risk-neutral pricing.
  2. Estimate banking stability multivariate densities and quantify distress dependence (joint/default conditional probabilities) to measure distress that one bank or group imposes on another or on the system.
- Advantages: incorporates high-frequency market information (stock prices, EDF), captures nonlinear conditional default dependencies, allows estimation of system-wide distress generated by a bank or group.
- Caveat: distress dependence reflects investors’ perception of risk; market perceptions may not equal true default probabilities.

### Distance-to-default, EDF mapping, and implications
- Distance to default measure is normalized by the volatility of the asset values.
- Probability of default increases as market leverage or volatility of market value of bank assets increase.
- Moody’s KMV EDF maps distance-to-default to a probability of default using an empirically determined function based on historical default data.
- Example conceptual mapping: if historically two out of 100 companies default within a year among firms three standard deviations away from default, the EDF assigns 2 percent to a firm with distance to default = 3.
- EDF advantages over the cumulative normal distribution: EDF assigns more realistic default probabilities; the cumulative normal distribution is typically "on the optimistic side".
- Disadvantage of EDF: the explicit model and parameters used for EDF estimations are not always known.
- This paper relies on EDF statistics rather than the normal distribution as the mapping function.

### Banking Stability Multivariate Density (BSMD) — Methodology
- Banking system modeled as a portfolio of multivariate density functions using the Consistent Information Multivariate Density Optimizing (CIMDO) methodology (Segoviano (2006)).
- CIMDO density obtained by minimizing a conditional default probability function; expected default frequency statistics approximate default probabilities.
- Minimization yields a multivariate distribution where default dependence is estimated from copula functions; marginal probabilities are uniform on [0,1].
- These density functions allow calculation of distress dependence over time reflecting changes in expected default frequency of banks.

### Distress dependence — Results
- Distress dependence statistic: default probability of bank x conditional on the default probability of bank y, P(D_x ≥ d_x | D_y ≥ d_y).
- Distress dependence matrix computed for October 2008 and March 2011 (Table 4):
  - October 2008: highest exposure in the top row was due from conditional default probability of Bank 7.
  - Larger banks (Bank 10 and Bank 11 identified) exhibit the largest conditional default probabilities and are systemically more important.
  - From October 2008 to March 2011, conditional distress statistics for the majority of Korean banks fell, consistent with a decline in global risk aversion.

### Banking Stability Index (BSI) — Construction and Results
- BSI: expected number of bank defaults conditional on the default of at least one other bank in the system.
- Illustrative behavior:
  - Pre-crisis BSI for Korean banking system ≈ 1.4.
  - During elevated distress BSI rose to a peak ≈ 3.
  - After the crisis BSI fell but stayed around 1.9.
- Time path (January 2007 through March 2011): pre-crisis ≈ 1.4 → peak ≈ 3 → post-crisis ≈ 1.9.

### Joint probability of distress — Findings
- Joint probability of distress measures the probability of all banks in the system defaulting (tail-event).
- For Korean banks (January 2007 through March 2011):
  - The joint probability of default is rather small.
  - It spiked during the global financial crisis reflecting nonlinear and larger increases in individual banks’ distress, but it never reached levels that would stress the whole financial system.

### Overall summary of results and interpretation
- All three methodologies indicate:
  - Korean banks are interconnected.
  - The financial risk and contagion risk among Korean banks declined significantly in the aftermath of the global financial crisis.
- Network approach: default of no single Korean bank generates significant distress on other banks; writing off credit-plus-funding risk from banks’ capital still leaves Korean banks with BIS capital ratios higher than the 8 percent required minimum.
- Co-risk model: major international financial events (e.g., Lehman Brothers bankruptcy) were principal drivers of conditional default probability changes for Korean banks; domestic events had relatively smaller impacts.
- Distress dependence: connected banks but limited systemic risk posed by any single bank; distress dependence metrics follow major global financial events mirroring co-risk results.

*Source: IMF staff calculations and analysis as presented in the provided document excerpt.*

### 2.     BIS Capital Ratio after a Credit Shock ..........................................................................

### _wp11201 - 2.     BIS Capital Ratio after a Credit Shock ..........................................................................

### Sections included in this content unit
- 2.     BIS Capital Ratio after a Credit Shock .................................................................................. 8
- 3.     Credit-Plus-Funding Shock on Bank Capital ..........................................................................  9
- 4.     BIS Capital Ratio after a Credit-Plus-Funding Shock ............................................................. 9
- 5.     Changes in the Average Conditional Co-risk ................................................................. 16
- 6.     Distribution of Bank Asset Over Time ........................................................................... 21
- 7.     Stock Price and Default Probabilities of Korean Banks ................................................. 22
- 8.     Viewing Banking System Stability as a Portfolio of Banks ........................................... 23
- 9.     Expected Number of Bank Defaults, 2007–11 ............................................................... 26
- 10.   Joint Probability of Default, 2007–11 ............................................................................. 27

### Tables listed in this content unit
- 1.     Financial Shock Scenarios ................................................................................................ 6
- 2.     Conditional Co-risk Estimates, as of end of September 2010 ........................................ 14
- 3.     Conditional Co-risk Estimates, as of October 13, 2008 .................................................. 15
- 4.     Distress Dependence Matrix (October 2008 and March 2011) ...................................... 25

*Content unit: _wp11201 - 2.     BIS Capital Ratio after a Credit Shock (PDF chapter/section).*

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

### _wp11201 - References .............................................................................................................

### Introduction
- Purpose: apply three methodologies to assess financial linkages across Korean banks and quantify risks from interconnectedness: (i) network approach (bank balance sheet data), (ii) co-risk analysis (CDS spreads, 95th quantile regression), and (iii) distress dependence (stock prices and estimated default probabilities).
- Key overall findings:
  - Korean banks are interconnected but no single bank by itself poses systemic threat to the rest of the system; no bank is highly vulnerable to distress in another single bank.
  - Financial risks posed by banks and contagion risk declined significantly after the global financial crisis.
  - Market-based, high-frequency indicators (co-risk and distress dependence) provide forward-looking assessments complementary to traditional FSIs.

### Network approach

#### Data
- Bilateral interbank exposure data for 18 Korean banks (covering the bulk of the banking system): Busan, Citibank Korea, Daegu, Hana, Industrial Bank of Korea, Jeju, Jeonbuk, KDB, Kookmin, Korea Exchange, Korea Exim, Kwangju, Kyongnam, NongHyup, SC First, Shinhan, Suhyup, and Woori.
- Analysis dates: end-2008, end-2009, and September 2010.
- Bank assets include: bank bonds receivable, loans, call loans, bonds purchased under repurchase agreements, accounts receivable, derivative assets.
- Bank liabilities include: deposits, CD, bank bonds payable, borrowings, call money, bonds sold under repurchase agreements, accounts payable, derivative liabilities.

#### Methodology
- Simulate losses for an individual bank induced by default of another bank.
- Two shock types:
  - Credit shock: write-off of Exposure at Default (EAD) with Loss given Default (LGD) = 100%.
  - Credit-plus-funding shock: direct credit loss plus funding loss = (1 - rollover ratio) * haircut.
    - Rollover rate: Korean Won: 65% (Same April 2009 GFSR); Foreign Currency: 30% (Average rollover ratio during 97 Asian crisis).
    - Haircut (assumed for simulations): 50 percent (for funding loss due to fire sales).
- Default criterion: a bank is in default when its loss > total capital.
- Compare simulated losses to banks’ capital and BIS capital adequacy ratios (Basel II basis).

#### Results
- Credit shock:
  - Ratio of largest credit-shock-loss-to-capital for Korean banks lies between 4 percent and 33 percent.
  - Mean of these ratios declined from end-2008 through September 2010.
  - No bank’s loss from default of another individual bank exceeds its total capital.
  - BIS capital adequacy ratios (Basel II basis) after largest credit shock: average CAR exceeded 12 percent from 2008 to 2010; no bank’s CAR falls below the 8 percent regulatory minimum.
- Credit-plus-funding shock:
  - Assumptions: 50 percent haircut, 65 percent rollover for won, 30 percent rollover for foreign currency.
  - Ratios of largest credit-plus-funding losses relative to capital range from 5 percent to 36 percent.
  - Average ratio declined to less than 11 percent from about 16 percent since end-2008.
  - Even the largest loss for any Korean bank is no more than 40 percent of that bank’s capital.
  - BIS capital adequacy ratios after deducting largest credit-plus-funding shock remain above the 8 percent required minimum for all banks.

#### Summary (Network approach)
- Simulations indicate Korean banks are well-capitalized and can absorb losses due to default of another single bank; no single-bank knock-on would exhaust another bank’s capital.
- No systemically important financial institution identified that would cause defaults in other banks via single-bank failure; no systemically vulnerable bank identified likely to default due to another single bank’s failure.
- Supporting factors:
  - Direct interbank exposures are not excessively large relative to banks’ capital; households and corporate sector are major funding sources; nondeposit liabilities diversified.
  - Loan-to-deposit (LTD) regulation adopted December 2009 limits wholesale funding (LTD ratio required at or below 100 percent). As of end-September 2010, average LTD of 15 Korean banks subject to regulation = 96.5 percent, down from 115.9 percent in December 2009.
  - Median CAR of Korean banks exceeded 12 percent during the study period; average CAR increased from 12.7 percent as of end-2008 to 14.6 percent as of end-2009.
- Caveat: analysis tests knock-off effect of a single bank; results may differ for combined/multiple bank failures.

### Co-risk approach

#### Data
- Daily financial data from January 1, 2006 through March 31, 2011.
- 11 Korean banks analyzed: Citibank Korea, Hana, Kookmin, Korea Exchange, SC First, Shinhan, Woori (commercial banks); Industrial Bank of Korea, Korea Development, Korea EXIM, Nonghyup (specialty banks).
- Main variable: credit default swap (CDS) spreads for these 11 banks.
- Control variables:
  - TED spread (measures risk aversion in international financial markets).
  - Spread between one-year KORIBOR and Korean Treasury bond yields (domestic inter-bank credit risk).
  - Spread between three-year Korean and U.S. Treasury bond yields (domestic vs international interest rate difference).

#### Methodology
- Measure increase in default risk of bank i conditional on default of bank j using Co-VaR framework via 95th quantile regression (Adrian and Brunnermeier (2009) approach).
- Regression specification: dependent variable = CDS spread of bank i; independent variables = CDS spreads of remaining banks and common risk factors, estimated at the 95th quantile.
- Construct conditional co-risk matrix where each cell |ij Co-risk| is co-risk of bank j conditional on distress of bank i.
- Indicators derived from matrix:
  - Systemic vulnerability of bank i = average of row i.
  - Systemic importance of bank j = average of column j.
  - Systemic risk = average of all elements in the co-risk matrix.
- Co-risk matrices presented as percentage changes in distress (percent change in CDS spread relative to 95th percentile).

#### Results
- Conditional co-risk matrices shown for end-September 2010 and October 13, 2008 (peak after Lehman collapse).
- Illustrative statistics and dynamics:
  - Pre-crisis average co-risk ~ 5 percent.
  - Average conditional co-risk rose to around 10 to 15 percent during 2007–08.
  - Average conditional co-risk peaked close to 25 percent immediately after the failure of Lehman Brothers (note: referenced as “peaked close to 25 percent”).
  - By September 2010 average conditional co-risk remains slightly above pre-crisis levels.
- Comparative observations:
  - Even at the crisis peak, distress among Korean banks was more contained than in other jurisdictions; IMF’s calculation showed conditional co-risk between LCFIs overseas ranged between -6 and 617 percent in March 2008.
- Cross-section finding:
  - Banks with lower financial stability indicators (higher CDS spreads) are more vulnerable to systemic risk and more sensitive to distress from other banks.
- Time-series dynamics:
  - Average conditional co-risk follows major international events: BNP Paribas suspension (Aug. 2007), Bear Stearns’ bankruptcy (Mar. 2008), Lehman Brothers filing for court protection (Sep. 2008), and market volatility due to fiscal concerns in some European countries (May, 2010).
- Tables (reported in source) provide detailed conditional co-risk percentages for each bank pair at the two dates; example values:
  - Table 2 (end-September 2010): sample cell: bank 1 conditional on bank 2 = 3.2 percent.
  - Table 3 (October 13, 2008): sample cell: bank 1 conditional on bank 5 = 38.9 percent; importance and vulnerability aggregates reported per bank.

#### Summary (Co-risk)
- Co-risk analysis indicates linkage risks were lower before the global financial crisis, rose sharply during major international events (e.g., Bear Stearns, Lehman), and normalized subsequently.
- Korean banks are highly integrated with global financial markets; international events drive conditional default probabilities more than domestic events.
- Banks with higher CDS spreads are more sensitive to distress (may reflect both true vulnerability and investor behavior of selling risky assets first).

### Distress dependence approach

#### Data
- Daily stock prices and expected default frequencies (EDF) from Moody’s KMV for 11 Korean banks from 2006 to March 31, 2011.
- Banks included: Busan, Cheju, Daegu, Hana, Industrial Bank of Korea, Jeonbuk, Kookmin, Korea Development, Korea Exchange, Shinhan, Woori.

#### Methodology
- Follow Segoviano and Goodhart (2009) two-step approach:
  1. Estimate default probability of each bank using the contingent claims approach (CCA) based on option-pricing linking market value of bank assets to equity value.
     - Equity modeled as a call option on underlying bank assets with strike equal to default-free liabilities (DP).
     - Asset dynamics assumed to follow geometric Brownian motion with drift μ_A and volatility σ_A.
     - Use option-pricing relations to infer implied asset value A and asset volatility σ_A from observed equity value and equity volatility.
     - Distance to default (DD) measured (function of A, DP, σ_A, T); default probability measured as N(-DD) under risk-neutral pricing.
  2. Estimate banking stability multivariate densities and quantify distress dependence (joint/default conditional probabilities) to measure distress that one bank or group imposes on another or on the system.
- Advantages: incorporates high-frequency market information (stock prices, EDF), captures nonlinear conditional default dependencies, and allows estimation of system-wide distress generated by a bank or group.
- Caveat: distress dependence reflects investors’ perception of risk; market perceptions may not equal true default probabilities.

#### Summary (Distress dependence)
- Distress dependence captures the level of distress a bank or group creates on another bank or system by using market data (stock prices, EDF) and CCA-implied default probabilities.
- Like co-risk results, distress dependence shows strong inter-linkages in Korea but limited systemic risk posed by any single bank; system-level distress has declined significantly after the global financial crisis.
- Limitation: measures the market-perceived distress and may not always reflect actual default probabilities (markets can misprice tail events).

*Source: IMF staff summary of the content unit "_wp11201 - References ............................................................................................................."*

### 2.28 percent since N(-2) = 0.0228. The probability of default

### _wp11201 - 2.28 percent since N(-2) = 0.0228. The probability of default

### Distance-to-default, EDF mapping, and implications
- Distance to default measure is normalized by the volatility of the asset values and thereby mitigates problems arising if the probability distribution of asset values is not lognormal.
- The probability of default increases as the market leverage of a bank or the volatility of the market value of bank assets increase.
- Moody’s KMV EDF (expected default frequency) maps distance-to-default to a probability of default using an empirically determined function based on historical default data. Example conceptual mapping:
  - If historically two out of 100 companies default within a year among firms three standard deviations away from default, the EDF assigns 2 percent to a firm with distance to default = 3.
- Advantages of EDF over the cumulative normal distribution:
  - EDF assigns more realistic default probabilities; the cumulative normal distribution is typically "on the optimistic side" and assigns lower default probabilities for firms three or more standard deviations away from default.
- Disadvantages of EDF:
  - The explicit model and parameters used for EDF estimations are not always known.
- This paper relies on EDF statistics rather than the normal distribution as the mapping function.

### Empirical patterns for Korean banks (stock prices and EDF)
- Larger banks have higher book value and lower default frequency, reflecting market perception of smaller default probabilities for larger banks (Figure 7 summary).
- EDF statistics used are derived from Moody’s KMV and stock-price-derived implicit asset values.

### Banking Stability Multivariate Density (BSMD) — methodology
- The banking system is modeled as a portfolio of multivariate density functions using the Consistent Information Multivariate Density Optimizing (CIMDO) methodology (Segoviano (2006)).
- CIMDO density function is obtained by minimizing a conditional default probability function; the expected default frequency statistics are used to approximate default probabilities.
- Minimization yields a multivariate distribution where default dependence is estimated from copula functions; marginal probabilities are uniform on [0,1].
- These density functions allow calculation of distress dependence over time reflecting changes in expected default frequency of banks.

### Distress dependence: definition and empirical findings
- Distress dependence statistic: default probability of bank x conditional on the default probability of bank y, represented conceptually as P(D_x ≥ d_x | D_y ≥ d_y).
- Distress dependence matrix computed for two dates: October 2008 and March 2011 (Table 4).
  - October 2008: the highest level of exposure in the top row was due from the conditional default probability of Bank 7.
  - In general, larger banks (Bank 10 and Bank 11 identified) exhibit the largest conditional default probabilities and are systemically more important.
  - From October 2008 to March 2011, conditional distress statistics for the majority of Korean banks fell, consistent with a decline in global risk aversion.

### Banking Stability Index (BSI) — construction and results
- BSI: expected number of bank defaults conditional on the default of at least one other bank in the system.
- Illustrative behavior:
  - The BSI index for the Korean banking system before the outbreak of the global financial crisis was around 1.4 (similar to major U.S. and European banks).
  - During elevated distress in global financial markets the BSI rose to a peak at around 3, indicating that three banks may default conditional on the distress of another bank.
  - For major U.S. and European banks during the same period, this ratio rose to more than 3.5.
  - After the crisis, the BSI fell but stayed around 1.9, which is still higher than pre-crisis levels.
- Figure 9 (January 2007 through March 2011) shows the time path: pre-crisis ≈ 1.4 → peak ≈ 3 → post-crisis ≈ 1.9.

### Joint probability of distress — tail-event assessment
- Joint probability of distress measures the probability of all banks in the system defaulting (tail-event).
- For a two-bank system, joint probability expressed conceptually as a double integral over joint tail densities.
- Empirical finding for Korean banks (January 2007 through March 2011):
  - The joint probability of default is rather small.
  - It spiked during the global financial crisis reflecting nonlinear and larger increases in individual banks’ distress, but it never reached levels that would stress the whole financial system.

### Summary of results and interpretation
- All three methodologies applied — network, co-risk, and distress dependence approaches — indicate that:
  - Korean banks are interconnected.
  - The financial risk and contagion risk among Korean banks declined significantly in the aftermath of the global financial crisis.
- Network approach findings:
  - Default of no single Korean bank generates significant distress on other banks in the system.
  - Writing off credit-plus-funding risk from banks’ capital still leaves Korean banks with BIS capital ratios higher than the 8 percent required minimum (textual finding).
- Co-risk model findings:
  - Major international financial events (e.g., Lehman Brothers bankruptcy) were principal drivers of conditional default probability changes for Korean banks.
  - Domestic events had relatively smaller impacts, indicating high integration of Korean banks with global financial markets as an important source of distress.
- Distress dependence findings:
  - Korean banks are connected, but the systemic risk a single bank can pose to the whole system is limited.
  - Distress dependence metrics follow major global financial events, mirroring co-risk results.

*Source: IMF staff calculations and analysis as presented in the provided document excerpt.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2011/_wp11201.pdf_