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### Introduction and integrated framework
- Presents an integrated framework to examine risks from the macro environment and banks’ interconnectedness and to assess resilience to aggregate and idiosyncratic shocks.
- Framework highlights:
  - Design of scope and methodology of stress tests based on hypothetical data.
  - Assessment of resilience of banks individually and in peer groups by size.
  - Simulation of single-factor shocks: deterioration of entire portfolio and individual sectors, defaults of the largest obligors, interbank contagion, exchange rate and interest rate shocks, and a systemic deposit run.
  - A macro-risk model linking banks’ capital asset ratios (CARs) to expected losses and credit growth, calibrated with a generalized method of moments.
  - Statistical simulations to derive distributions of banks’ credit losses and economic capital.
- Uses CreditRisk+ to estimate banks’ credit portfolio loss distributions, with no assumptions about the cause of default.
- Paper structure (as provided): Section II—macro-risk stress testing model and portfolio loss simulations; Section III—systemic (micro) risk stress tests using quantile regressions; Section IV—design of sensitivity tests; Section V—conclusions.

### Theoretical credit risk model — mechanisms and scenario definitions
- Rationale:
  - Addresses common shortcoming of assuming banks’ assets do not change over the stress horizon by explicitly modeling credit growth effects on RWA and exposures at default.
- Stylized assumptions:
  - Banking system engaged in traditional intermediation; banks rely solely on net interest income to cover operating expenses and credit losses, retaining residual as capital.
  - Credit fully weighted in RWA; operating expenses a fixed fraction of credit; spreads reset at beginning of each period; capital accumulated internally; dividends distributed only if CARs not reduced.
- Core relationships (conceptual):
  - Bank i’s capital at t+1 evolves depending on net interest income (spread × credit), operating expenses, and credit losses.
  - RWA at t+1 evolves depending on credit growth and credit losses.
  - CAR dynamics depend on trade-off between new capital and new RWA; if new capital/new RWA equals current CAR, CAR remains unchanged.
- Scenario definitions:
  - Baseline scenario: banks forecast correctly shocks to credit quality and credit growth and choose a spread that fully absorbs shocks, keeping CARs constant; baseline credit losses equal the mean of the CreditRisk+ loss distribution.
  - Adverse scenario: banks set spreads according to baseline projections and cannot reset spreads until end of period after realization of shocks; adverse scenario losses correspond to the %99 quantile of the CreditRisk+ distribution.
- Multi-period estimation: model can be extended to multiple periods depending on stress test horizon.

### Empirical estimation — dynamic panel and main drivers
- Two dynamic panel equations estimated using quarterly bank-level data (estimation period: 2002Q3–2010Q1):
  - NPL ratio equation (NPL i,t).
  - Credit Growth equation (year-on-year percentage change in total private sector credit for bank i).
- Specification features:
  - Allows endogenous and predetermined variables, up to four lags, individual fixed effects αi and δi.
  - Estimation methods: Difference GMM (Arellano-Bond), System GMM (Arellano-Bover / Blundell-Bond), Hausman-Taylor, Within Group, Pooled OLS; implemented with “xtabond2” and forward orthogonal deviations.
- Key empirical findings:
  - Banks’ credit quality and credit growth are largely driven by real GDP growth and inflation.
  - NPL ratios correlated negatively with GDP growth and positively with inflation.
  - Lending to private sector correlated positively with GDP growth and negatively with inflation.
  - Timing: approximately three quarters for GDP growth to affect NPLs and credit growth; inflation affects credit quality and credit growth in two quarters and one quarter, respectively.
  - Strong inertia: large coefficients on autoregressive terms.

### Modeling credit portfolio losses in CreditRisk+
- Use NPL and credit growth projections from regressions to assess scenario impacts via CreditRisk+.
- Scenario definitions in empirical exercise:
  - Baseline scenario: robust GDP and credit growth, moderate inflation.
  - Inflation scenario: significant increase in inflation.
  - Slowdown scenario: sharp economic slowdown.
- Regression-based projections (system averages under scenarios):
  - Inflation scenario: system’s average NPL ratio increases by around 27 percent and growth of credit to the private sector decelerates to around 14 percent.
  - Slowdown scenario: average NPL ratio increases by nearly 60 percent and credit growth decelerates to roughly 7 percent.
- CreditRisk+ specifics:
  - Models portfolio loss distributions to compute portfolio VaR at the 99 percent confidence level.
  - Losses on individual loans depend on borrower probability of default, exposure at default, and recovery rate.
  - Baseline credit losses = mean of the CreditRisk+ distribution; adverse scenario losses = %99 quantile of the distribution.

### CreditRisk+ implementation details and data proxies
- Data and proxy assumptions:
  - Required provisioning rates used as proxy for default probabilities:
    - performing = 0.03
    - past due = 0.05
    - substandard = 0.25
    - doubtful = 0.5
    - loss = 1
  - Alternative proxies: banks’ internal risk management data, borrowers’ credit risk ratings, or loan classifications.
  - Simulations focus on large loans in banks’ portfolios.
- Simulation steps in CreditRisk+:
  - initial loan default probabilities shocked using projected increase in NPL ratios;
  - exposure at default = loan balances × credit growth projections;
  - 25 percent recovery rate assumed on all loans;
  - generate credit portfolio loss distributions before and after shocks;
  - unexpected losses from simulations and credit growth projections fed into theoretical model to gauge impact on banks’ CARs.
- Modeling features:
  - Portfolio credit risk model generating expected and unexpected losses.
  - Treats credit risk as binary outcomes — a "default mode" model.
  - For many loans with small, independent default probabilities, default frequency approximated by the Poisson distribution.

### Scenarios, sensitivity experiments, and key stress-test results
- Selected scenario and sensitivity experiments (percent impacts reported in source):
  - Increase in NPLs by 40 percent (past-due, substandard, doubtful and loss loans increase uniformly by 40 percent).
  - Downgrade of 30 percent of standard loans to substandard.
  - Default of single largest borrower; default of the 2 largest borrowers.
  - Sectoral credit shocks: 20 percent of sectoral exposure downgraded to loss (Agriculture, Construction, Manufacturing, Mining, Trade).
  - Exchange rate risk: 20 percent appreciation; 25 percent depreciation.
  - Interest rate risk: interest rates increase by 250 b.p.; interest rates decrease by 300 b.p.
  - Interbank contagion risk: first round and second round (assumes five medium and small banks with lower CARs simultaneously default on their interbank exposures and the loss given default is 50 percent).
  - Macroeconomic scenarios: High inflation scenario and Slowdown scenario.
- Headline stress-test results:
  - The system’s CAR declines from 9.4 percent to 5.2 percent under the inflation scenario and to 4.5 percent under the slowdown scenarios.
  - The two scenarios have a broadly similar impact on CARs; higher NPLs in the slowdown scenario are mitigated by lower credit growth, while in the inflation scenario increased RWA partly offsets lower credit losses.
  - Ignoring credit growth effects may underestimate the impact of the inflation scenario on banks’ CARs.
- Loss distribution effects:
  - The shock shifts the average bank credit portfolio loss distribution to the right.
  - Expected losses increase by 70 percent.
  - 99th percentile losses increase by 50 percent.
  - Unexpected losses (difference between the 99th percentile loss and the mean of the distribution) exceed capital — "the system’s capital buffers would be insufficient to absorb an extreme shock to credit quality."

### CoStress systemic risk indicator, quantile regression, and MCSR
- CoStress objective: assess level of banking system stress conditional on bank j being in distress; marginal contribution of bank j to systemic risk (MCSR) = difference between conditional stress and median (normal) state.
- Quantile regression approach:
  - Tail focus: model the 90th percentile (τ = 0.90) of conditional quantile function of banks’ NPL ratios using quantile regression (Koenker and Bassett, 1978).
  - Logistic functional form used to accommodate NPL ratios bounded between zero and one.
  - Seven quantile regressions fitted: 0.05, 0.10, 0.25, 0.50, 0.75, 0.90, and 0.95 percentiles.
- MCSR definition:
  - MCSR_j = CoStress_hat(NPL_j | system, τ = 0.90) − CoStress_hat(NPL_j | system, τ = 0.50)
  - MCSR is the percentage point difference between the fitted NPL ratio conditional on bank j being in distress (90.0,=τjNPL) and being in its median state (50.0,=τjNPL).
- Four scenario framework (aggregate vs idiosyncratic; median vs extreme):
  - Scenario A: median aggregate and idiosyncratic shocks (baseline).
  - Scenario B: median aggregate shock and extreme idiosyncratic shock.
  - Scenario C: extreme aggregate shock and median idiosyncratic shock.
  - Scenario D: extreme aggregate and idiosyncratic shocks.
- Main CoStress findings:
  - Aggregate risk factors have greater contribution to systemic risk in the data:
    - CoStress increases by around 75 percent following an extreme shock to commodity prices and the system’s lagged NPLs.
    - Individual banks’ contributions: CoStress increases on average by around 10 percent for an extreme idiosyncratic shock.
  - Scenario D has the largest impact on systemic risk.
  - A simultaneous shock to 5 banks with the largest contributions to systemic risk is broadly similar to that of an extreme aggregate shock.
  - Nonlinearities: marginal impact of extreme shocks to one factor is larger when applied together with extreme shocks to the other factor.
  - Cross-sectional determinants: MCSRs largely driven by interconnectedness (interbank dependence) rather than asset size.

### Selected numeric results from systemic scenarios and marginal contributions (exact entries preserved)
- Systemic risk scenarios — System and banks (percent):
  - Scenario A (median idiosyncratic shock and median aggregate shock): System 26.7; Average 28.3; Bank 1 28.5; Bank 2 26.3; Bank 3 26.9; Bank 4 26.6; Bank 5 26.0; Bank 6 25.0; Bank 7 26.3; Bank 8 26.3; Bank 9 26.7; Bank 10 26.9; Bank 11 26.6
  - Scenario B (extreme idiosyncratic shock and median aggregate shock): System 29.2; Average 29.6; Bank 1 31.3; Bank 2 27.6; Bank 3 30.3; Bank 4 28.6; Bank 5 29.1; Bank 6 31.3; Bank 7 26.7; Bank 8 28.2; Bank 9 28.9; Bank 10 27.6; Bank 11 31.7
  - Scenario C (median idiosyncratic shock and extreme aggregate shock): System 46.3; Average 47.1; Bank 1 48.5; Bank 2 45.5; Bank 3 46.6; Bank 4 44.4; Bank 5 45.4; Bank 6 45.3; Bank 7 45.3; Bank 8 46.3; Bank 9 45.9; Bank 10 45.5; Bank 11 45.9; Bank 12 49.3
  - Scenario D (extreme idiosyncratic shock and extreme aggregate shock): System 49.4; Average 48.7; Bank 1 51.9; Bank 2 47.1; Bank 3 50.7; Bank 4 46.9; Bank 5 50.7; Bank 6 49.2; Bank 7 53.1; Bank 8 46.8; Bank 9 48.4; Bank 10 48.3; Bank 11 46.7; Bank 12 55.4
- Marginal contributions (MC) from Table 3 (percent):
  - MC of extreme idiosyncratic shock when aggregate risk is median (A→B): System 2.5; Average 1.3; Bank 1 2.8; Bank 2 1.3; Bank 3 3.3; Bank 4 2.0; Bank 5 3.1; Bank 6 6.3; Bank 7 0.4; Bank 8 2.0; Bank 9 2.2; Bank 10 0.7; Bank 11 5.1
  - MC of extreme idiosyncratic shock when aggregate risk is extreme (C→D): System 3.1; Average 1.6; Bank 1 3.4; Bank 2 1.6; Bank 3 4.1; Bank 4 2.5; Bank 5 3.9; Bank 6 7.8; Bank 7 0.5; Bank 8 2.5; Bank 9 2.7; Bank 10 0.8; Bank 11 6.2
  - MC of extreme aggregate shock when idiosyncratic risk is median (A→C): System 19.6; Average 18.8; Bank 1 20.0; Bank 2 19.2; Bank 3 19.7; Bank 4 17.8; Bank 5 19.4; Bank 6 20.3; Bank 7 19.9; Bank 8 19.7; Bank 9 18.8; Bank 10 18.9; Bank 11 22.7
  - MC of extreme aggregate shock when idiosyncratic risk is extreme (B→D): System 20.2; Average 19.1; Bank 1 20.5; Bank 2 19.5; Bank 3 20.4; Bank 4 18.3; Bank 5 20.1; Bank 6 21.8; Bank 7 20.1; Bank 8 20.2; Bank 9 19.4; Bank 10 19.1; Bank 11 23.7

### Sensitivity analysis — shocks, methodology, and principal findings
- Shocks used in sensitivity tests:
  - Credit and concentration shocks:
    - (i) 40 percent uniform increase in adversely classified loans across loan categories.
    - (ii) 30 percent downgrade of performing loans to substandard.
    - (iii) simultaneous default of the single largest and two largest obligors.
    - (iv) 20 percent default rate on banks’ lending to agriculture, mining, construction, trade and manufacturing sectors.
  - Market, liquidity and interbank contagion shocks:
    - (i) 25 percent (depreciation) and 20 percent (appreciation) simultaneous movement against all currencies.
    - (ii) parallel upward (downward) shift in the yield curve of 250 basis points (300 basis points).
    - (iii) five day systemic deposit run resulting in a cumulative outflow of 20 percent of total deposits in domestic currency.
    - (iv) simultaneous default on interbank exposures by five banks with lowest CARs.
- Methodology and key assumptions:
  - Credit risk: simulated downgrades to loan rating transition matrix; impact on CARs computed by deducting increase in required provisions from capital.
  - Default risk: Z-scores estimated over a 3-year rolling window with formula: Z-score = (ROA + Capital/assets) / σ(ROA).
  - Market risk: interest rate risk assessed via re-pricing gap model; FX risk via sensitivity of net open FX position.
  - Liquidity risk: five day deposit run with cumulative outflow around 20 percent of deposits in domestic currency; assumed banks have 70 percent of their liquid assets in domestic currency readily available to meet daily withdrawals.
  - Interbank contagion: assumed five banks with low capital buffers default simultaneously on interbank obligations and measured systemic impact including second-round effects.
- Main sensitivity findings:
  - Credit risk is a major vulnerability:
    - A 40 percent increase in adversely classified loans would erode the capital buffers of banks accounting for around 15 percent of the system’s assets.
    - Severe shocks (30 percent downgrade of performing loans to substandard, or simultaneous default of the two largest obligors) would practically deplete the system’s Tier 1 capital.
    - A simultaneous default of five banks with weak capital buffers on interbank exposures would undermine the solvency of two other banks, including a systemically important one.
    - Z-scores indicate high default risk, particularly for medium and small banks whose Z-scores are well below pre-crisis levels.
  - Direct FX and interest rate exposures are small; a 20 percent appreciation will have only a modest impact on capital adequacy, though FX and interest rate risk can indirectly increase credit risk.
  - Liquidity: system can withstand a hypothetical five day deposit run, but liquid assets fall to around 7 percent of total assets; medium-sized banks are vulnerable and two banks (around 5 percent of system assets) would require liquidity assistance.

### Selected numeric entries from sensitivity tables (preserved exactly)
- Table 5 — Selected entries (In percent, unless indicated otherwise):
  - Reported capital adequacy ratio: 0 41 00.0 34.4 65.6 0.1
  - Adjusted capital adequacy ratio (adjusted for underprovisioning): 35 65.2 58.0 36.9 2.3
  - Increase in NPLs by 40 percent: 3 54 5 15.2 76.8 8.0 2.7
  - Downgrade of 30 percent of standard loans to substandard: 4 64 43 9.4 56.4 4.2 4.1
  - Default of the 2 largest borrowers: 6 44 39 4.5 64.4 4.2 4.1
  - Exchange rate risk — 20 percent appreciation: 4 55 6.9 85.1 8.0 2.3
  - Interest rates increase by 250 b.p.: 2 66 4.6 58.5 36.9 2.2
  - Macro scenarios — High inflation scenario: 5 54 15.2 80.6 4.2 3.8
  - Notes:
    - 1/ Additional capital needed by banks to reach CAR of 14 percent; in percent of 2010 GDP.
    - 2/ Adjusted for underprovisioning.
    - 3/ Past-due, substandard, doubtful and loss loans increase uniformly by 40 percent.
    - 4/ 30 percent of standard (performing) loans are downgraded to substandard.
    - 5/ 20 percent of sectoral exposure is downgraded to loss.
- Table 6 — Selected liquidity and Z-score entries (In percent, unless indicated otherwise):
  - Initial position — Liquid assets to total assets (system / Large banks / Medium banks / Small banks): 1/18.6 20.8 13.0 21.4
  - Initial position — Liquid assets to total deposits: 1/42.5 45.0 33.8 111.2
  - Liquidity scenario (5-day deposit run total outflow 22.5 percent of deposits in domestic currency) — Liquid assets to total assets: 7.2 8.2 4.4 15.4
  - Liquidity scenario — Liquid assets to total deposits: 19.2 20.7 13.1 86.0
  - Z-Score Index (Dec-07 / Dec-09 / Jun-10): Dec-07 37.1 20.8 71.8 48.6; Dec-09 7.7 15.2 3.4 2.9; Jun-10 10.1 19.1 5.0 4.5
  - Footnotes:
    - 1/ Liquid assets in domestic currency.
    - 2/ 5-day deposit run resulting in a total outflow of 22.5 percent of deposits in domestic currency.
    - 3/ Financial fragility index defined as the sum of ROA and the capital to asset ratio divided by the standard deviation of ROA. Lower Z-score implies higher default risk.

### Appendix I — CreditRisk+ mechanics and limitations
- Two-stage approach:
  - Stage 1: Estimates the frequency of defaults and the severity of losses.
  - Stage 2: Derives the loss distribution combining frequency and severity.
- Modeling steps and approximations:
  - Exposures net of recoveries grouped into discrete loan bands; exposure in each band approximated by a common average; derive loss distributions for each band and aggregate.
  - Probability generating function for portfolio defaults:
    - F(z) = sum_{n=0}^{∞} p(n defaults) z^n
    - Individual obligor: F_A(z) = p_A z + (1 − p_A)
    - Portfolio: F(z) = ∏_A ( (1 − p_A) + p_A z )
    - For small p_A, linearization yields F(z) = exp( μ (z − 1) ) where μ = sum_A p_A
  - Poisson approximation:
    - Pr(n defaults) = e^{−μ} μ^n / n!
    - Poisson assumption implies mean equals variance.
- Limitations and extensions:
  - Historical data show variance of default rates > Poisson implies; Poisson can underestimate tail risk.
  - Extensions (e.g., CSFB) allow for pair-wise default correlations to capture fatter tails.

### Conclusion — framework utility and main tools
- The paper presents an integrated framework for systemic risk analysis that considers both the macroeconomic environment and banks’ interconnectedness.
- Key elements:
  - Banks’ CARs modeled to consider simultaneous impact of future credit losses, credit growth, and the credit spread.
  - Analysis focuses on economic measures of solvency and uses a generalized method of moments to calibrate shocks to NPLs and credit growth.
  - Uncertainty about banks’ future losses modeled in CreditRisk+, which relies on analytical techniques to find banks’ credit portfolio loss distributions.
  - A simple systemic risk indicator (CoStress) proposed to measure tail risk comovements among banks.
  - Quantile regressions and CreditRisk+ used to model banks’ conditional VaRs.

*Source: _wp1253 - References (content excerpt).*

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

### _wp1253 - References

### Introduction
- Presents an integrated framework to examine risks from the macro environment and banks’ interconnectedness and to assess resilience to aggregate and idiosyncratic shocks.
- Framework highlights:
  - Design of scope and methodology of stress tests based on hypothetical data.
  - Assessment of resilience of banks individually and in peer groups by size.
  - Simulation of single-factor shocks: deterioration of entire portfolio and individual sectors, defaults of the largest obligors, interbank contagion, exchange rate and interest rate shocks, and a systemic deposit run.
  - A macro-risk model linking banks’ capital asset ratios (CARs) to expected losses and credit growth, calibrated with a generalized method of moments.
  - Statistical simulations to derive distributions of banks’ credit losses and economic capital.
- Uses CreditRisk+ to estimate banks’ credit portfolio loss distributions, with no assumptions about the cause of default.
- Paper structure (as provided): Section II—macro-risk stress testing model and portfolio loss simulations; Section III—systemic (micro) risk stress tests using quantile regressions; Section IV—design of sensitivity tests; Section V—conclusions.

### Credit Risk Stress Testing — Overview and Literature
- Purpose: identify common vulnerabilities across institutions that can lead to systemic failure.
- Notes on frameworks:
  - Early Excel-based tests (e.g., Čihák (2007)) are easy to implement but determine shocks ad hoc and often do not model link between macro environment and banks’ losses explicitly.
  - Second-generation frameworks (e.g., Schmieder (2011)) allow RWA to change under stress.
  - Reduced-form “satellite” approaches map exogenous macro shocks into credit quality indicators (Foglia (2009), Sorge (2004), Chan-Lau (2006), Quagliariello (2009), Borio et al (2012)).
  - Integrated frameworks exist (Boss et al., 2006; Aikman et al., 2009—RAMSI), combining macro scenarios, satellite credit and market risk models, and interbank/network elements.
- Contribution of this paper:
  - Proposes a simple method to construct integrated systemic risk scenarios assessing marginal contributions of macroeconomic and idiosyncratic factors.
  - Derives a stylized theoretical model measuring impact of future credit losses, credit growth and net interest income on banks’ CARs, including direct effects of credit growth on RWA and indirect effects on future credit losses.
- Macro-financial stress testing framework steps (as in Figure 1):
  - Model econometrically banks’ credit quality (NPL) and credit growth as functions of risk factors and project under scenarios.
  - Use projections to model credit portfolio loss distributions in CreditRisk+ and measure expected losses and economic capital.
  - Insert credit loss and credit growth projections into theoretical model to estimate scenario impacts on CARs.

### Theoretical Credit Risk Model — Key Mechanisms and Baseline vs Adverse Scenarios
- Rationale:
  - Common shortcoming: assuming banks’ assets do not change over the stress horizon; ignores credit growth effects which can raise RWA and exposures at default.
- Stylized assumptions:
  - Banking system engaged in traditional intermediation; banks rely solely on net interest income to cover operating expenses and credit losses, retaining residual as capital.
  - Credit fully weighted in RWA; operating expenses a fixed fraction of credit; spreads reset at beginning of each period; capital accumulated internally; dividends distributed only if CARs not reduced.
- Core relationships (conceptual summary based on model equations in text):
  - Bank i’s capital at t+1 evolves depending on net interest income (spread × credit), operating expenses, and credit losses.
  - RWA at t+1 evolves depending on credit growth and credit losses.
  - CAR dynamics depend on trade-off between new capital and new RWA; if new capital/new RWA equals current CAR, CAR remains unchanged.
- Baseline scenario:
  - Banks forecast correctly shocks to credit quality and credit growth and choose a spread that fully absorbs shocks, keeping CARs constant (baseline condition defined mathematically in the paper).
  - Baseline credit losses equal the mean of the CreditRisk+ loss distribution.
- Adverse scenario:
  - Banks set spreads according to baseline projections and cannot reset spreads until end of period after realization of shocks.
  - Adverse scenarios defined where credit losses correspond to the 99th percentile of the distribution (i.e., %99 quantile).
  - Under adverse scenario, CARs become functions of the magnitudes of credit quality and credit growth surprises; higher-than-expected credit losses and credit growth reduce CARs.
- Multi-period estimation: model can be extended to multiple periods depending on stress test horizon.

### Empirical Credit Risk Model — Estimation and Key Findings
- Two dynamic panel equations estimated using quarterly bank-level data:
  - NPL ratio equation (NPL i,t).
  - Credit Growth equation (year-on-year percentage change in total private sector credit for bank i).
- Specification allows endogenous and predetermined variables, up to four lags, individual fixed effects αi and δi.
- Estimation challenges and methods:
  - OLS levels estimator is inconsistent in dynamic panels; Within Groups (WG) estimator biased downward due to induced endogeneity.
  - Use of GMM estimators: Arellano and Bond (1991) “difference” GMM and Arellano and Bover (1995)/Blundell and Bond (1998) “system” GMM; system GMM better when series are highly autoregressive.
  - Implementation: “xtabond2” with forward orthogonal deviations transform used due to small unbalanced panel; also exploit Hausman and Taylor (1981) as alternative and WG and Pooled OLS as lower and upper bounds.
- Empirical results summary:
  - Banks’ credit quality and credit growth are largely driven by real GDP growth and inflation (Table 1 referenced).
  - Signs and robustness:
    - NPL ratios correlated negatively with GDP growth and positively with inflation.
    - Lending to private sector correlated positively with GDP growth and negatively with inflation.
  - Timing (lags):
    - It takes approximately three quarters for GDP growth to affect NPLs and credit growth.
    - Inflation affects credit quality and credit growth in two quarters and one quarter, respectively.
  - Strong inertia: large coefficients on autoregressive terms.

### Modeling Banks’ Credit Portfolio Losses in CreditRisk+
- Use NPL and credit growth projections (from regression model) to assess impact of scenarios on banks’ credit portfolio losses via CreditRisk+.
- Scenario definitions in the empirical exercise:
  - Baseline scenario: robust GDP and credit growth, moderate inflation.
  - Inflation scenario: significant increase in inflation.
  - Slowdown scenario: sharp economic slowdown.
- Regression-based projections for system averages under scenarios (as reported):
  - Inflation scenario: system’s average NPL ratio increases by around 27 percent and growth of credit to the private sector decelerates to around 14 percent.
  - Slowdown scenario: average NPL ratio increases by nearly 60 percent and credit growth decelerates to roughly 7 percent.
- CreditRisk+ use:
  - Models portfolio loss distributions to compute portfolio VaR at the 99 percent confidence level.
  - Losses on individual loans depend on borrower probability of default, exposure at default, and recovery rate.
  - The baseline credit losses used in the theoretical model equal the mean of the CreditRisk+ distribution; adverse scenario losses correspond to the %99 quantile of the distribution.

*Source: _wp1253 - References (content excerpt).*

### Box 1 and Appendix I. The data requirements for the simulations include data on banks’

### _wp1253 - Box 1 and Appendix I. The data requirements for the simulations include data on banks’

### Data requirements and proxies for default probabilities
- Default probabilities are not directly observable; required provisioning rates are used as a proxy and "reflect the expected impairment on loans in that classification":
  - performing = 0.03
  - past due = 0.05
  - substandard = 0.25
  - doubtful = 0.5
  - loss = 1
- Alternative proxies: banks’ internal risk management data systems, borrowers’ credit risk ratings, or loan classifications.
- Simulations focus on large loans in banks’ portfolios.

### Simulation setup and assumptions (CreditRisk+ implementation)
- Steps used to simulate the macro scenario in CreditRisk+:
  - initial loan default probabilities were shocked by using the projected increase in NPL ratios;
  - exposure at default is found by multiplying the loan balances by the credit growth projections;
  - a 25 percent recovery rate is assumed on all loans;
  - under the previous assumptions, banks’ credit portfolio loss distributions are generated before and after the shocks; and
  - unexpected losses from the simulations and the credit growth projections from the regression analysis are fed into the theoretical model to gauge the impact of the scenario on banks’ CARs.
- CreditRisk+ modeling features (Box 1):
  - portfolio credit risk model generating expected and unexpected losses;
  - treats credit risk as binary outcomes (default or no default) — a "default mode" model;
  - models default as a continuous random variable; under assumptions of many loans with small, independent default probabilities, the frequency of defaults is closely approximated by the Poisson distribution (see Appendix I).

### Key empirical findings from regressions and stress tests
- Regression estimation period: 2002Q3–2010Q1 (Table 1: Macro Determinants of Credit Risk).
  - Credit quality equation (Dependent variable: NPL ratio) and Credit growth equation (Dependent variable: Private Sector Credit Growth) estimated with panel methods (Difference GMM, System GMM, Hausman-Taylor, Within Group, Pooled OLS).
  - Notation and estimation details preserved in source (Arellano-Bond, Blundell-Bond, Hausman-Taylor, fixed effects, p-values in italic; * denotes significance at the 10 percent level; ** at the 5 percent level; *** at the 1 percent level).
- Stress test headline results (Table 2 and text):
  - The system’s CAR declines from 9.4 percent to 5.2 percent under the inflation scenario and to 4.5 percent under the slowdown scenarios.
  - The two scenarios have a broadly similar impact on CARs; higher NPLs in the slowdown scenario are mitigated by lower credit growth, while in the inflation scenario increased RWA partly offsets lower credit losses.
  - Ignoring credit growth effects may underestimate the impact of the inflation scenario on banks’ CARs.
- Loss distribution effects (Figure 2 summary):
  - The shock shifts the average bank credit portfolio loss distribution to the right.
  - Expected losses increase by 70 percent.
  - 99th percentile losses increase by 50 percent.
  - Unexpected losses (defined as the difference between the 99th percentile loss and the mean of the distribution) exceed capital — "the system’s capital buffers would be insufficient to absorb an extreme shock to credit quality."

### Scenario and sensitivity experiments (selected results from Table 2)
- Reported capital adequacy ratio and adjusted capital adequacy ratio shown for system and bank size cohorts; adjusted for underprovisioning.
- Selected sensitivity and scenario experiments (percent impacts reported in source):
  - Increase in NPLs by 40 percent (past-due, substandard, doubtful and loss loans increase uniformly by 40 percent).
  - Downgrade of 30 percent of standard loans to substandard (30 percent of standard (performing) loans are downgraded to substandard).
  - Credit concentration and counterparty defaults (default of single largest borrower; default of the 2 largest borrowers).
  - Sectoral credit shocks: 20 percent of the sectoral exposure downgraded to loss (Agriculture, Construction, Manufacturing, Mining, Trade).
  - Exchange rate risk: 20 percent appreciation; 25 percent depreciation.
  - Interest rate risk: interest rates increase by 250 b.p.; interest rates decrease by 300 b.p.
  - Interbank contagion risk: first round and second round (assumes five medium and small banks with lower CARs simultaneously default on their interbank exposures and the loss given default is 50 percent).
  - Macroeconomic scenarios: High inflation scenario and Slowdown scenario (reported impacts on CAR in Table 2; system CAR under scenarios reported above).

### Systemic risk indicator (CoStress) and tail comovement methodology
- Objective: assess risk of systemic spillovers as the level of banking system stress conditional on bank i being in distress; marginal contribution of bank i to systemic risk is the difference between conditional stress and median (normal) state.
- Tail focus: model the 90th percentile of the conditional quantile function of banks’ NPL ratios using quantile regression.
  - Quantile regression (Koenker and Bassett, 1978) models conditional quantile function rather than conditional mean; suited to capture nonlinear effects at higher levels of stress.
- Drivers considered:
  - Common risk factors (e.g., year-on-year change in a monthly commodity price index as a common factor).
  - Idiosyncratic factors captured by individual banks’ NPL ratios.
- Functional form and estimation:
  - Logistic functional form (Wilson, 1997a, 1997b) used to accommodate NPL ratios bounded between zero and one.
  - Quantile regression model (90th percentile) estimated using monthly data:
    - ttjj
      N
      n
      t n n t t
      NPLCRFNPLaNPL
      ,,,,,1,ττττττ
      εβββ++++=
      ∑
      −
    - where tNPL,τ is the 90th percentile (τ = 0.90) of the conditional quantile distribution of the logit transformation of the pooled NPL ratios excluding bank j; 1−tNPL is the one-quarter lag; CRFn denotes the n-th common risk factor; jNPL is the NPL ratio of the explanatory bank j; τα is a constant; t,τε is the disturbance.
- Implementation notes:
  - Tail comovement may arise from direct financial linkages (interbank exposures, ownership linkages) or from exposure to common risk factors (e.g., sectoral exposures).
  - Market-based indicators (CDS spreads, default probabilities, equity prices) are alternative forward-looking proxies but were not used here.
  - Multicollinearity caveat: common factor and the explanatory bank NPL may be related; principal component analysis and orthogonalization could address this but would complicate interpretation.

*Source: Authors' estimates (excerpts from Box 1 and Appendix I of the supplied content).*

### Box 2. Quantile Regression

### Box 2. Quantile Regression

### Quantile regression method
- Quantile regression estimates the conditional quantile function by solving an optimization problem.
- While OLS regression minimizes the sum of squared residuals to obtain an estimate of the conditional mean function, quantile regressions minimize the sum of weighted absolute residuals to obtain an estimate of the conditional quantile function (further details on the minimization problem are presented in Appendix II).
- In the application, the NPL ratio of one bank is fitted against the NPL ratios of the other banks using two alternative methods: quantile regression and OLS.
- Seven quantile regressions are fitted corresponding to the 0.05, 0.10, 0.25, 0.50, 0.75, 0.90, and 0.95 percentiles.
- The quantile regression estimate of the conditional median function is identified as the 0.50 quantile (red solid line in the figure), and the OLS estimate is the conditional mean (dashed yellow line).
- The two regression estimates diverge significantly due to asymmetry of the conditional density and sensitivity of OLS to outliers.
- Quantile regressions are better suited to model heterogeneous conditional distributions (Koenker and Hallock, 2001) because:
  - dispersion of banks’ NPL ratios tends to increase with the distress level of the “explanatory” bank;
  - narrow spacing in the lower quantiles reveals higher density and shorter lower tail;
  - wide spacing in the upper quantiles indicates lower density and longer upper tail;
  - spacing decreases sharply between the 90th and 95th percentiles since the upper tail fattens at the extreme end of the distribution.

### CoStress measure and MCSR (marginal contribution to systemic risk)
- For each bank j the prediction from the quantile regression is referred to as the CoStress measure of bank j — the level of systemic stress conditional on distress at bank j.
- Following Adrian and Brunnermeier (2009), banks’ marginal contribution to systemic risk (MCSR) is defined as the percentage point difference between the systemic risk indicator (CoStress) conditional on an extreme shock to bank j and conditional on its median state:
  - MCSR_j = CoStress_hat(NPL_j | system, τ = 0.90) − CoStress_hat(NPL_j | system, τ = 0.50)
  - Notation in the source: ),ˆˆ( 5.0NPLj, |system9.0NPLj, |system ,5.0,9.0,9.0 == − = ττ τττ β macromacroj PLNPLNMCSR (equation (18))
- MCSR is the difference between the average fitted NPL ratio conditional on bank j being in distress (90.0,=τjNPL) and being in its median state (50.0,=τjNPL).
- Shocks are calibrated with historical data.
- To find the marginal contribution of bank j to systemic risk other factors (macro environment, state of banks in previous period) are held constant.
- The estimation procedure enables ranking of individual banks by systemic importance proxied by MCSR.

### Systemic risk scenarios (four scenario framework)
- Two types of risk factors: aggregate and idiosyncratic.
- Two shock magnitudes: median shock (50th percentile) and extreme shock (90th percentile).
- Four scenarios:
  - Scenario A: median aggregate and idiosyncratic shocks (baseline).
  - Scenario B: median aggregate shock and extreme idiosyncratic shock.
  - Scenario C: extreme aggregate shock and median idiosyncratic shock.
  - Scenario D: extreme aggregate and idiosyncratic shocks.
- “Shocked” variables assume their median historical value if the shock is “normal”, and the 90th percentile of the historical distribution if the shock is extreme.
- Marginal contributions are calculated as differences between fitted CoStress indicators across scenarios:
  - Marginal contribution of an extreme idiosyncratic shock when aggregate risk is median: CoStress_hat_B − CoStress_hat_A
  - Marginal contribution of an extreme aggregate shock when idiosyncratic risk is median: CoStress_hat_C − CoStress_hat_A
  - Marginal contribution of an extreme idiosyncratic shock when aggregate risk is extreme: CoStress_hat_D − CoStress_hat_C
  - Marginal contribution of an extreme aggregate shock when idiosyncratic risk is extreme: CoStress_hat_D − CoStress_hat_B
- CoStress (PLN̂) are the fitted values and subscripts indicate respective scenarios.

### Main findings (systemic risk results and interpretation)
- Fitted CoStress values and MCSR of aggregate and idiosyncratic factors are presented in Table 3 (system-level and bank-level entries).
- Aggregate risk factors have a greater contribution to systemic risk in the data:
  - CoStress increases by around 75 percent following an extreme shock to commodity prices and the system’s lagged NPLs.
  - Individual banks’ contributions are smaller: CoStress increases on average by around 10 percent for an extreme idiosyncratic shock.
- Scenario D (both aggregate and idiosyncratic shocks extreme) has the largest impact on systemic risk.
- A simultaneous shock to several banks can be significant:
  - Impact of a simultaneous shock to 5 banks with the largest contributions to systemic risk is broadly similar to that of an extreme aggregate shock.
- Nonlinearities: marginal impact of extreme shocks to one factor is larger when applied together with extreme shocks to the other factor (e.g., extreme idiosyncratic shock impact is bigger when aggregate factor is also extreme).
- Cross-sectional determinants of MCSR (limited by small sample and outliers):
  - Plots of MCSR against bank-specific factors (NPL ratios, share of system’s NPLs, share of system’s total assets, interbank borrowing as share of total and of banks’ capital) suggest MCSRs are largely driven by interconnectedness (dependence on interbank market) rather than asset size.
  - Results are broadly similar to other studies (e.g., Adrian and Brunnermeier, 2009) that find a loose cross-sectional relationship between unconditional VaRs and MCSR.

### Conditional VaR modeling using CreditRisk+
- Regression estimates were used to model individual banks’ VaRs conditional on shocks to other banks via CreditRisk+.
- To estimate a bank’s conditional VaRs: project bank’s NPL ratio conditional on an extreme (90th percentile) shock to the “explanatory” bank using the quantile regression line, then simulate the bank’s conditional credit portfolio loss distribution in CreditRisk+.
- Table 4 shows conditional VaRs for the five banks with the largest MCSR as a percentage of their capital; Bank 2 emerges as the biggest contributor to systemic risk within the group.
- CreditRisk+ can also derive banks’ VaRs conditional on integrated systemic risk scenarios including shocks to individual banks, aggregate risk factors, and credit growth.

### Key numeric results from Table 3 (selected entries preserved exactly)
- Systemic risk scenarios — System and banks (percent):
  - Scenario A (median idiosyncratic shock and median aggregate shock): System 26.7; Average 28.3; Bank 1 28.5; Bank 2 26.3; Bank 3 26.9; Bank 4 26.6; Bank 5 26.0; Bank 6 25.0; Bank 7 26.3; Bank 8 26.3; Bank 9 26.7; Bank 10 26.9; Bank 11 26.6
  - Scenario B (extreme idiosyncratic shock and median aggregate shock): System 29.2; Average 29.6; Bank 1 31.3; Bank 2 27.6; Bank 3 30.3; Bank 4 28.6; Bank 5 29.1; Bank 6 31.3; Bank 7 26.7; Bank 8 28.2; Bank 9 28.9; Bank 10 27.6; Bank 11 31.7
  - Scenario C (median idiosyncratic shock and extreme aggregate shock): System 46.3; Average 47.1; Bank 1 48.5; Bank 2 45.5; Bank 3 46.6; Bank 4 44.4; Bank 5 45.4; Bank 6 45.3; Bank 7 45.3; Bank 8 46.3; Bank 9 45.9; Bank 10 45.5; Bank 11 45.9; Bank 12 49.3
  - Scenario D (extreme idiosyncratic shock and extreme aggregate shock): System 49.4; Average 48.7; Bank 1 51.9; Bank 2 47.1; Bank 3 50.7; Bank 4 46.9; Bank 5 50.7; Bank 6 49.2; Bank 7 53.1; Bank 8 46.8; Bank 9 48.4; Bank 10 48.3; Bank 11 46.7; Bank 12 55.4
- Marginal contributions (MC) from Table 3 (percent, transitions indicated):
  - MC of extreme idiosyncratic shock when aggregate risk is median (A→B): System 2.5; Average 1.3; Bank 1 2.8; Bank 2 1.3; Bank 3 3.3; Bank 4 2.0; Bank 5 3.1; Bank 6 6.3; Bank 7 0.4; Bank 8 2.0; Bank 9 2.2; Bank 10 0.7; Bank 11 5.1
  - MC of extreme idiosyncratic shock when aggregate risk is extreme (C→D): System 3.1; Average 1.6; Bank 1 3.4; Bank 2 1.6; Bank 3 4.1; Bank 4 2.5; Bank 5 3.9; Bank 6 7.8; Bank 7 0.5; Bank 8 2.5; Bank 9 2.7; Bank 10 0.8; Bank 11 6.2
  - MC of extreme aggregate shock when idiosyncratic risk is median (A→C): System 19.6; Average 18.8; Bank 1 20.0; Bank 2 19.2; Bank 3 19.7; Bank 4 17.8; Bank 5 19.4; Bank 6 20.3; Bank 7 19.9; Bank 8 19.7; Bank 9 18.8; Bank 10 18.9; Bank 11 22.7
  - MC of extreme aggregate shock when idiosyncratic risk is extreme (B→D): System 20.2; Average 19.1; Bank 1 20.5; Bank 2 19.5; Bank 3 20.4; Bank 4 18.3; Bank 5 20.1; Bank 6 21.8; Bank 7 20.1; Bank 8 20.2; Bank 9 19.4; Bank 10 19.1; Bank 11 23.7
- Note from Table 3: "The Co-Stress measure is defined as the 90th percentile of the' NPL ratios of the banks in the system. Exteme shocks are defined as the 90th percentile of the variables, while median shocks as the 50th percentile."

### Sensitivity analysis — shocks, methodology, and key findings
- Shocks used in sensitivity tests:
  - Credit and concentration shocks:
    - (i) 40 percent uniform increase in adversely classified loans across loan categories (past due, substandard, doubtful, and loss).
    - (ii) 30 percent downgrade of performing loans to substandard.
    - (iii) simultaneous default of the single largest and two largest obligors.
    - (iv) 20 percent default rate on banks’ lending to the agriculture, mining, construction, trade and manufacturing sectors.
  - Market, liquidity and interbank contagion shocks:
    - (i) 25 percent (depreciation) and 20 percent (appreciation) simultaneous movement against all currencies.
    - (ii) parallel upward (downward) shift in the yield curve of 250 basis points (300 basis points).
    - (iii) five day systemic deposit run resulting in a cumulative outflow of 20 percent of total deposits in domestic currency.
    - (iv) simultaneous default on interbank exposures by five banks with lowest CARs.
- Methodology and assumptions:
  - Credit risk: simulated downgrades to loan rating transition matrix; impact on CARs computed by deducting increase in required provisions from capital.
  - Default risk assessed using Z-scores estimated over a 3-year rolling window with formula: Z-score = (ROA + Capital/assets) / σ(ROA).
  - Market risk: interest rate risk assessed via re-pricing gap model; FX risk via sensitivity of net open FX position.
  - Liquidity risk: five day deposit run with cumulative outflow around 20 percent of deposits in domestic currency; assumed banks have 70 percent of their liquid assets in domestic currency readily available to meet daily withdrawals.
  - Interbank contagion: assumed five banks with low capital buffers default simultaneously on interbank obligations and measured systemic impact including second-round effects.
- Main findings from sensitivity analysis:
  - Credit risk is a major vulnerability:
    - A 40 percent increase in adversely classified loans would erode the capital buffers of banks accounting for around 15 percent of the system’s assets (Table 5).
    - Severe shocks (30 percent downgrade of performing loans to substandard, or simultaneous default of the two largest obligors) would practically deplete the system’s Tier 1 capital.
    - A simultaneous default of five banks with weak capital buffers on interbank exposures would undermine the solvency of two other banks, including a systemically important one.
    - Z-scores indicate high default risk, particularly for medium and small banks whose Z-scores are well below pre-crisis levels (Table 6).
  - Direct FX and interest rate exposures are small; a 20 percent appreciation will have only a modest impact on capital adequacy, though FX and interest rate risk can indirectly increase credit risk.
  - Liquidity: system can withstand a hypothetical five day deposit run, but liquid assets fall to around 7 percent of total assets; medium-sized banks are vulnerable and two banks (around 5 percent of system assets) would require liquidity assistance.

### Selected results from sensitivity and stress tables (preserved exactly)
- Table 5 — Selected entries (In percent, unless indicated otherwise):
  - Reported capital adequacy ratio: 0 41 00.0 34.4 65.6 0.1  (entries presented in table format in source)
  - Adjusted capital adequacy ratio (adjusted for underprovisioning): 35 65.2 58.0 36.9 2.3
  - Increase in NPLs by 40 percent: 3 54 5 15.2 76.8 8.0 2.7
  - Downgrade of 30 percent of standard loans to substandard: 4 64 43 9.4 56.4 4.2 4.1
  - Default of the 2 largest borrowers: 6 44 39 4.5 64.4 4.2 4.1
  - Exchange rate risk — 20 percent appreciation: 4 55 6.9 85.1 8.0 2.3
  - Interest rates increase by 250 b.p.: 2 66 4.6 58.5 36.9 2.2
  - Macro scenarios — High inflation scenario: 5 54 15.2 80.6 4.2 3.8
  - Note footnotes from Table 5:
    - 1/ Additional capital needed by banks to reach CAR of 14 percent; in percent of 2010 GDP.
    - 2/ Adjusted for underprovisioning.
    - 3/ Past-due, substandard, doubtful and loss loans increase uniformly by 40 percent.
    - 4/ 30 percent of standard (performing) loans are downgraded to substandard.
    - 5/ 20 percent of sectoral exposure is downgraded to loss.
- Table 6 — Selected liquidity and Z-score entries (In percent, unless indicated otherwise):
  - Initial position — Liquid assets to total assets (system / Large banks / Medium banks / Small banks): 1/18.6 20.8 13.0 21.4
  - Initial position — Liquid assets to total deposits: 1/42.5 45.0 33.8 111.2
  - Liquidity scenario (5-day deposit run total outflow 22.5 percent of deposits in domestic currency) — Liquid assets to total assets: 7.2 8.2 4.4 15.4
  - Liquidity scenario — Liquid assets to total deposits: 19.2 20.7 13.1 86.0
  - Z-Score Index (Dec-07 / Dec-09 / Jun-10): Dec-07 37.1 20.8 71.8 48.6; Dec-09 7.7 15.2 3.4 2.9; Jun-10 10.1 19.1 5.0 4.5
  - Footnotes:
    - 1/ Liquid assets in domestic currency.
    - 2/ 5-day deposit run resulting in a total outflow of 22.5 percent of deposits in domestic currency.
    - 3/ Financial fragility index defined as the sum of ROA and the capital to asset ratio divided by the standard deviation of ROA. Lower Z-score implies higher default risk.

### Conclusion — framework and tools
- The paper presents an integrated framework for systemic risk analysis that considers both the macroeconomic environment and banks’ interconnectedness.
- Key elements of the framework:
  - banks’ CARs are modeled to consider simultaneous impact of future credit losses, credit growth, and the credit spread;
  - analysis focuses on economic measures of solvency and uses a generalized method of moments to calibrate shocks to NPLs and credit growth;
  - uncertainty about banks’ future losses is modeled in CreditRisk+, which relies on analytical techniques to find banks’ credit portfolio loss distributions;
  - a simple systemic risk indicator (CoStress) is proposed to measure tail risk comovements among banks in the system;
  - quantile regressions and CreditRisk+ are used to model banks’ conditional VaRs.

*Source: Authors' estimates (Box 2. Quantile Regression, _wp1253).*

### Appendix I. Default Risk Modeling in CreditRisk+

### Appendix I. Default Risk Modeling in CreditRisk+

### Overview of the two-stage approach
- CreditRisk+ derives banks’ portfolio loss distributions in a two-stage process:
  - Stage 1: Estimates the frequency of defaults and the severity of losses.
  - Stage 2: Derives the loss distribution combining frequency and severity.
- The amount lost in a given default equals the exposure to the obligor less a recovery amount.
- Exposures, net of recoveries, are grouped into discrete loan bands; exposure level for each band is approximated by a common average. Loss distributions are derived for each exposure band and accumulated across bands to generate an overall distribution.
- Basic inputs required for the estimation of the model:
  - Loan exposures
  - Their default rates
  - Mean recovery rates
- Default rates can be approximated by mapping obligors’ credit ratings or using proxies such as NPLs, required provisioning rates, etc.
- Given its simplicity, the model has parsimonious data requirements.

### Probability generating function and portfolio aggregation
- For a portfolio with N obligors each with annual probability of default p_A, CreditRisk+ introduces a probability generating function defined using auxiliary variable z:
  - F(z) = sum_{n=0}^{∞} p(n defaults) z^n  (equation (20))
- Individual obligor probability generating function (two states: default or no default):
  - F_A(z) = p_A z + (1 − p_A)  (equation (21))
- Assuming individual default events are independent, the portfolio generating function is the product of individual generating functions:
  - F(z) = ∏_A ( (1 − p_A) + p_A z )  (equation (22))
- Taking logarithms:
  - log F(z) = sum_A log( (1 − p_A) + p_A z )  (equation (23))
- For small individual default probabilities the logarithm can be linearized:
  - log(1 − p_A + p_A z) = −(1 − z) p_A + higher-order terms  (equation (24) in source form)
- In the limit this yields:
  - F(z) = exp( μ (z − 1) )  (equation (25))
  - where the expected number of defaults in the portfolio is μ = sum_A p_A  (equation (26))

### Poisson approximation for default counts
- The probability distribution corresponding to the probability generating function is obtained by Taylor expansion:
  - F(z) = exp(−μ) sum_{n=0}^{∞} μ^n z^n / n!  (equation (27))
- If individual default probabilities are small, the probability of having n default events in the portfolio in one year is:
  - Pr(n defaults) = e^{−μ} μ^n / n!  (equation (28))
- This is the Poisson distribution with single parameter μ (the expected number of defaults). The Poisson assumption implies the mean equals the variance.

### Limitations and extension
- Historical data show the variance of default rates is significantly higher than the Poisson assumption implies, especially for lower quality exposures. Thus the Poisson assumption can underestimate actual default probabilities and understate tail risk.
- To capture fatter tails of observed loss distributions, Credit Suisse First Boston (CSFB) extend the basic model to allow for pair-wise default correlations among obligors.

*Source: Appendix I. Default Risk Modeling in CreditRisk+*

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