## Credit Loss in Translation: Informing Bank Provisions and Capital Buffer Requirements with Forward-Looking Credit Loss Distributions — Working Paper No. WP/2025/228

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### Introduction: purpose, scope, and motivation
- Purpose of the model framework:
  - Assessing the adequacy of provisions at the bank-portfolio level.
  - Conducting stress tests conditional on macro-financial scenarios.
  - Informing the sufficiency of capital requirements, including a countercyclical capital buffer (CCyB) and a positive neutral version thereof (PN-CCyB).
- Operationalization:
  - Designed to be “operationalizable” by central authorities using bank data available to them.
  - Model was set up and operationalized with Colombian banking system data in IMF technical assistance missions in November 2023 and May 2024.
- Motivation and context:
  - Revisions to accounting and prudential frameworks over the past 10 to 15 years increased relevance of expected credit loss (ECL) models.
  - IFRS 9 (IASB, 2014) and CECL (FASB, 2016) frameworks highlighted; IFRS 9 is relevant for more than 90 percent of the countries worldwide.
  - Macroprudential capital requirements prominence increased with Basel III and tools such as the CCyB and PN-CCyB (BCBS, 2022).
- Methodological contribution:
  - Introduces an integrated, top-down, simulation-based model framework to generate credit loss distributions by stochastically simulating a large number of macro-financial scenarios (instead of handpicked scenarios and weights).

### Conceptual framework: PiT and TTC distributions and uses
- Point-in-time (PiT) credit loss distribution:
  - Definition: forward-looking probability distribution of credit losses conditional on the current position in the business and financial cycle and bank-portfolio risk parameters.
  - Under IFRS 9:
    - Stage 1: credit losses computed over a 1-year horizon.
    - Stages 2 and 3: lifetime horizon.
  - Under CECL: lifetime horizon for all asset quality classes.
  - Statistic of interest: the mean of the PiT credit loss distribution = expected loss (ECL_PiT) for provisioning.
  - Distributional characteristics: bounded at zero; compressed toward zero in booms; expanded to the right in recessions.
- PiT distributions and capital requirements:
  - Provisions buffer expected losses; capital covers unexpected losses (tail metrics).
  - Example benchmark: Basel Committee (2006) aims for Pillar 1 requirements to cover the 99.9th percentile under the IRB approach.
  - Benchmarking compares sum of provisions and minimum capital requirements to PiT loss distribution to determine fraction of scenarios where losses exceed combined buffers.
- Through-the-cycle (TTC) distribution and CCyB calibration:
  - PiT distributions alone do not indicate the system’s cyclical position; a TTC reference is required.
  - Proposed TTC approach: build a TTC distribution of historical tail credit losses from one specific quantile of historical PiT distributions across time (example uses 25 years, quarterly snapshots → 100 observations; example quantile used = 95th percentile denoted C_Cy_Q–).
  - CCyB calibration: set buffer to fill gap between sum of provisions plus capital requirements and chosen TTC quantile; allows zero buffer in recession and gradual build in neutral/boom phases; Annex A3. derives calibration mathematically.
  - Caveat: CCyB applies to total RWA; calibration should consider credit loss dynamics and all other components of banks’ profit and loss distributions.

### Model overall structure and simulation workflow
- Four modules:
  - Module A: macro-financial model component.
  - Module B: transition matrix model component.
  - Module C: LGD module.
  - Module D: combines outputs to obtain a credit loss distribution for all bank-portfolios.
- Modules A, B, C split into estimation and simulation sub-components (A1/A2, B1/B2, C1/C2).
- Simulation workflow:
  - Simulate a large number of macro-financial scenarios (example: 5,000 multivariate paths, 30 years forward, quarterly).
  - Feed simulated macro-financial density forecasts through Modules B and C to produce portfolio-level credit loss distributions aggregated in Module D.
- Rationale versus IFRS 9 small-scenario approach:
  - Simulation-based approach avoids hand-picked scenarios and subjective weights, covers full distribution, accounts for nonlinearities, and yields robust estimates of means and tail percentiles.

### Required data inputs (dimensions preserved exactly)
- Exposure stocks: B x P x T x S5 x C — Historical end-of-period t stocks in currency units, by bank and portfolio.
- Transition flows: B x P x T x S3 x S3 — Historical cross-flows between Stages 1/2/3 through period t in % of the end-of-previous-period stock, by bank and portfolio.
- Interest rates: B x P — Annualized current interest rates for outstanding stock of exposures, by bank and portfolio.
- Repayments: B x P x S3 x H — Principal repayment distribution of initial Stage 1 and 2 exposures, by bank and portfolio, from current position up to H quarters into the future.
- Unsecured LGDs: (B x) P — Unsecured LGDs for recently defaulted loans.
- Historical macro data: K x T — History of relevant macro-financial variables.
- Collateralization ratios: B x P x C x S5 — Fitted parameters of the log-normal distribution of portfolio-level collateralization ratios.
- Note on dimensions: B = banks; P = portfolios; T = historical time; S3 = Stages 1 to 3; S5 = Stage 1, Stage 2 and the three time-since-default buckets of Stage 3; H = future time horizon; K = macro-financial variables; C = collateral type.

### Module A — macro-financial model details
- Model choice:
  - Conventional VAR or any macro-financial model; paper uses two VAR models:
    - First VAR: endogenous domestic macro-financial variables + exogenous international variables.
    - Second auxiliary VAR: endogenizes relationships between international variables.
- Density forecast generation:
  - Resampling to account for residual and coefficient uncertainty:
    - Draw from coefficient vector mean and covariance (parametric bootstrap assuming multivariate normality) for coefficient uncertainty.
    - Add noise by drawing from VAR residuals (parametric or nonparametric) for residual uncertainty.
  - Width of final credit loss distributions examined for dependence on treatment of uncertainties.
- Choice of macro-financial variables (inputs to follow-on modules): GDP growth, price inflation, unemployment rate, bilateral/effective exchange rate, wage growth, house price growth, short-term sovereign bond yield, and a term spread; exogenous variables (e.g., Fed Funds) included in VARX.

### Module B — transition matrices, Z-scores, and bridge equations
- Module B1:
  - Construct Z-scores from historical transition matrices using Belkin et al. (1998) methodology.
  - Z-scores compress a time series of transition matrices to one number per point in time:
    - Z-Score = 0 → transition matrix equals long-run average.
    - Positive Z-Score → more favorable transitions.
    - Negative Z-Score → more unfavorable transitions.
  - Data should cover years predating IFRS 9 (2018) where possible; new business inflows into Stage 1 are excluded since model quantifies ECLs for currently outstanding exposures.
  - IFRS 9 staging indicators examples provided (Stage 1 → Stage 2 triggers; Stage 2 → Stage 3 triggers).
- Module B2:
  - Model Z-score time series as functions of macro-financial drivers using Bayesian Model Averaging (BMA) to account for model uncertainty.
  - Use separate bridge equations rather than include Z-scores directly in the VAR.
  - BMA allows sign constraints on long-run multipliers and assigns empirical weights to equations.
  - Density forecast generation for Z-scores:
    1. Draw coefficients from estimated means and covariance matrices.
    2. Draw from Z-score residuals (bank-portfolio specific).
    3. Use macro-financial density forecasts from VARX as input.
  - Z-score density forecasts decomposed into coefficient, model, and residual uncertainty.

### Module C — Loss Given Default (LGD) module
- Two structural LGD methods:
  1. Real estate-collateralized portfolios: collateral value indexed with house price trajectories simulated by macro model (indexation approach).
  2. Other portfolios: Frye & Jacobs (2012) methodology (FJ), establishing structural link between PDs and LGDs.
- Time-since-default and collateralization:
  - Defaults split into three time-since-default buckets; for loans in long default, LGD assumed 100 percent.
  - Loan-level collateralization ratio (CR = available collateral / loan exposure) fitted with lognormal distribution parameters.
  - If no collateral, apply unsecured LGD informed by historical recoveries.
- Portfolio-level LGD logic:
  - Fitted CR distribution computes fraction overcollateralized (LGD = 0 percent) and remainder where unsecured LGD applies.
  - Equation (1) expresses LGD as integral using lognormal CR distribution (as provided in source).
- Real-estate collateral specifics:
  - Adjust CR distribution for uncertainty in time-to-foreclosure (TTF) and future sales price (SP):
    - Multiply CR distribution by lognormal distributions for TTF and SP uncertainty.
    - CR translation: CR ∼ LN(μ,σ^2) −→ CR ∼ LN(μ+μ_TTF+μ_SP, σ^2+σ^2_TTF+σ^2_SP).
  - Loan amortization increases CR over time if collateral values unchanged.

### LGD — Frye-Jacobs (FJ) methodology (Vašíček distribution assumption)
- FJ LGD formula:
  - LGD_{t0+h} = Φ( Φ^{-1}(PD_{t0+h}) − k ) · PD_{t0+h}
  - k = [ Φ^{-1}(PD̄) − Φ^{-1}(PD̄ · LGD̄) ] / sqrt(1 − ρ)  (ρ = default correlation parameter)
- Use cases (paper embeds Case 2):
  - Case 1: Given PD_TTC (T0), LGD_TTC (T0), PD_PiT (T0 & horizon) → derive LGD_PiT path.
  - Case 2: Given PD_TTC (T0), LGD_PiT (T0), PD_PiT (T0 & horizon) → impute LGD_TTC to match LGD_PiT and derive LGD_PiT path.
  - Case 3 and Case 4 described for alternative imputations and implications.

### Module D — Expected Credit Loss (ECL) calculation (Stages 1–3) (formulae preserved)
- Stage 1 (quarterly discounting, losses mid-quarter):
  - ECL_S1 = sum_{X=1..2} sum_{q=1..4} EXP_{S_X,t=q−1} · TR_{X→3,t=q} · LGD_{S_X,t=q} / (1 + r)^{q−0.5}
- Stage 2 (lifetime up to quarter M; r is quarterly):
  - ECL_S2 = sum_{X=1..2} sum_{q=1..M} EXP_{S_X,t=q−1} · TR_{X→3,t=q} · LGD_{S_X,t=q} / (1 + r)^{q−0.5}
- Stage 3 (time-since-default buckets b):
  - ECL_S3 = sum_{b=1..3} (1 − TRC_{3→1,b} − TRC_{3→2,b}) · EXP_{S3,b,t=0} · LGD_{b,t=1}
- Cure rates TRC_{3→X,b} computed from scenario-specific simulated transition matrices:
  - First bucket cure computed over a three-year horizon.
  - Second bucket cure computed over a one-year horizon.
  - Longest time-since-default bucket: cure rate set to 0.

### Collateralization ratios, amortization, and real-estate foreclosure specifics (preserved formulae and parameters)
- Collateralization ratio mapping:
  - CR ∼ LN(μ, σ^2); loss(CR) = LGD_unsec · (1 − CR) if CR < 1; = 0 if CR ≥ 1.
  - Expected LGD (closed form): LGD = LGD_unsec · [ Φ_{LN}(1) − exp(μ + σ^2/2) · Φ_{N}( (−μ − σ − σ^2) / σ ) ].
- Loan amortization effect on CR:
  - If fraction α repaid, E → E(1 − α) and CR ∼ LN( μ − ln(1 − α), σ^2 ).
  - Model assumption: half of amortization linked to repayments and half to maturing loans.
- Time-to-foreclosure (TTF) and house prices (Colombia example):
  - Colombia average sale (TTF) = four years with a standard deviation of about five years.
  - Product of independent lognormals yields CR ∼ LN( μ + μ_{TTF} + μ_{SP} + μ_{HP}, σ^2 + σ^2_{TTF} + σ^2_{SP} ).

### Time-since-default bucketing limits (Table 4 preserved)
- Limits (days since default) by portfolio and collateral type:
  - Commercial/Micro, None: bucket1–2 limit = 210, bucket2–3 limit = 420
  - Commercial/Micro, Real Estate: 540, 1080
  - Commercial/Micro, Financial: 360, 720
  - Commercial/Micro, Other: 360, 720
  - Consumer/Mortgage, None: 30, 90
  - Consumer/Mortgage, Real Estate: 360, 720
  - Consumer/Mortgage, Financial: 360, 720
  - Consumer/Mortgage, Other: 270, 540

### Point-in-time versus through-the-cycle outputs and uses
- PiT outputs:
  - PiT credit loss distributions generated from current cycle starting point; mean = ECL_PiT used for provisioning.
- TTC outputs:
  - TTC distribution constructed from historical tail metrics of PiT distributions (section 2.3); used as comparator/anchor for CCyB calibration.

### Model extensions (examples)
- Split portfolios by fixed vs variable rate; estimate Z-scores separately.
- Endogenize discounting of recovery values and expected losses using VARX interest rate paths (current uses loan T0 interest rates).
- Estimate Z-scores for each row of transition matrices rather than full-matrix Z-scores.
- Use nonlinear macro models for scenario simulations (e.g., regime-switching VAR(X)).

### Illustrative application: Colombian banking system (key dataset and selected outcomes)
- Bank sample and system coverage:
  - 55 Colombian banks (close to 100 percent banking system coverage).
  - Banking system = 71 percent of financial system assets at end-2023 and represents about 64 percent of nominal GDP in 2023.
  - Largest 3, 8, and 13 of 55 banks represent 50, 75, and 90 percent share of banking system assets, respectively.
- Portfolio segmentation and shares at end-2023:
  - Commercial 50 percent, consumer 30 percent, mortgages 16 percent, microcredit 4 percent.
- Domestic vs foreign lending at end-2023:
  - Domestic loan book = 89 percent, foreign lending through branches = 1 percent, foreign subsidiary loan book = 10 percent.
- Provision coverage comparison (percent of end-2023 loan book exposures):
  - Commercial portfolio: model estimate = 4.4 percent; banks = 4.5 percent.
- Stage 1 lifetime-horizon sensitivity (CECL-style):
  - Multiples (lifetime vs 1-year) = 1.7x (commercial), 1.5x (consumer), 1.2x (microcredit), 2.2x (mortgages).
  - Note: mortgage multiple 2.2x reflects mortgages’ longest average duration.
- Z-score model implementation detail:
  - Panel BMA with banks as cross-section; pandemic period excluded from estimation: 2020Q2—2022Q2 (nine quarters).

### Sensitivity and counterfactual experiment findings (selected)
- Sensitivity 1: Excluding pandemic period from VAR(X) estimation → tighter credit loss distributions; means not overly sensitive but upper-tail provisioning more sensitive.
- Sensitivity 2a: Switching residual uncertainty off (domestic VARX, auxiliary foreign VAR, Z-score equations) → noticeably tighter credit loss distributions and left-shift of means.
- Sensitivity 2b: Switching model uncertainty component of BMA off → in this Colombia application aggregate system credit loss distributions did not change notably.
- Sensitivity 3: Re-anchoring macro-financial baseline paths to IMF WEO October 2023 → re-centered macro densities shift PiT credit loss distributions left.
- Sensitivity 4: Stage 1 horizon change from 1-year to lifetime → provisioning needs increase for all portfolios (most pronounced for mortgages).

### CCyP versus CCyB and CCyB calibration equations
- Objectives and transmission:
  - Both CCyP and CCyB aim to accumulate releasable buffers in booms and lean against cyclical imbalances.
  - CCyP: mainly through provision stock build-up (directly affects capital ratios).
  - CCyB: mainly through dynamic capital requirements (affects capital ratios secondarily).
- Colombia CCyP design specifics:
  - Release triggered when four threshold-based indicators jointly signal downturn.
  - During release, banks can use up to 70 percent of their countercyclical provision stock to buffer periodic losses.
  - After release phase (up to six months), normal-times CCyP coverage to be reached after no more than 18 subsequent months.
- CCyB calibration within model (key expressions preserved):
  - K / RWA ≥ ρ^*  where ρ^* = ρ + CCyB
  - Focus on credit-risk component: k_CR(p) ≥ (ρ + CCyB) · d_RW · b_a_CR(p)  (normalized by total credit risk assets A_CR)
  - Minimum capital requirement (fraction of credit risk exposures): k_rmin_CR = ρ · d_RW · b_a_CR(p)
  - CCyB derived from TTC quantile CCyQ and PiT expected losses ECL_PiT:
    - CCyB = max{ 0, CCyQ / (d_RW · b_a_CR) − p / b_A_CR − ρ }
    - Interpretation: CCyB positive when sum of provisions and minimum capital requirements insufficient to cover chosen TTC quantile.

*Source: wpiea2025228-source-pdf — IMF Working Paper No. WP/2025/228.*

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

### References

### Introduction
- Purpose of the model framework:
  - Assessing the adequacy of provisions at the bank-portfolio level.
  - Conducting stress tests conditional on macro-financial scenarios.
  - Informing the sufficiency of capital requirements, including a countercyclical capital buffer (CCyB) and a positive neutral version thereof (PN-CCyB).
- Operationalization:
  - Designed to be “operationalizable” by central authorities (central banks, micro- and macroprudential oversight institutions) using bank data available to them.
- Motivation and context:
  - Revisions to accounting and prudential frameworks over the past 10 to 15 years increased relevance of expected credit loss (ECL) models.
  - IFRS 9 framework (IASB, 2014) and CECL framework (FASB, 2016) highlighted; IFRS 9 is relevant for more than 90 percent of the countries worldwide.
  - Macroprudential capital requirements prominence increased with Basel III and tools such as the CCyB and PN-CCyB (BCBS, 2022).
  - General discussions on provisioning and capital requirements: Borio & Lowe (2001) and Gaston & Song (2014).
- Supervisory practice and risks:
  - Supervisory authorities in most countries mostly do not examine the adequacy of accounting provisions; provisioning model development often left to banks and auditors.
  - Risks include potentially inadequate calibration leading to underestimation of risk, balance sheet misrepresentation, or tailoring provisions to offset profits and reduce taxes.
- Compatibility and flexibility:
  - Model is not specific to any accounting regime; IFRS 9 is an example (primer in Annex A1).
  - To switch from IFRS-9 to CECL, change the Stage 1 horizon from 12 months to lifetime (counterfactual exercise for IFRS 9 regimes).
- Methodological contribution:
  - No integrated, top-down, simulation-based model framework for generating credit loss distributions (for provisioning and CCyB calibration) was known to exist prior to this development.
  - Instead of handpicked scenarios and scenario weights, the model stochastically simulates a large number of scenarios from a macro-financial model engine to obtain credit loss distributions, accounting for nonlinearities and improving estimation of distribution moments (means and tail percentiles).
- Use for macroprudential policy:
  - Model can be added to a suite of indicators and models for judging cyclical position to inform capital-based policies (guidance: BCBS (2010) and BIS (2015)).
  - Related approaches: regime switching model for cyclical position (Brave & Lopez (2021)); using cyclical swings in profitability to build buffers (Pfeifer & Hodula (2021)).
  - Model generates simulated point-in-time and through-the-cycle distributions of losses and capital needs to inform CCyB and PN-CCyB.
- Stress-testing applicability:
  - Model computes ECLs for currently outstanding exposures (not future originations) and implied provisioning requirements conditional on future scenarios.
  - Not originally a stress testing model year-by-year, but can be adapted for stress test purposes.
- Paper structure (as presented):
  - Section 2: general model framework.
  - Section 3: model structure and module details.
  - Section 4: quantitative results using Colombian banking system data.
  - Section 5: conclusion.
- Operational note:
  - Model was set up and operationalized with Colombian banking system data in an IMF technical assistance mission in November 2023 and May 2024; related TA report title given in the source.

### Conceptual Framework
- Overall use cases of model-derived credit loss distribution:
  - Inform provisions (section 2.1).
  - Challenge microprudential capital requirements (section 2.2).
  - Inform calibration of the Countercyclical Capital Buffer (CCyB, section 2.3).

#### 2.1 PiT Credit Loss Distribution and Provisioning
- Definition:
  - The point-in-time (PiT) credit loss distribution is the forward-looking probability distribution of credit losses obtained at a certain point in time, conditional on the specific position in the business and financial cycle and bank-portfolio risk parameters at that moment.
- Accounting-regime-specific usage:
  - Under IFRS 9:
    - Provisioning is specific to three asset quality stages (see Annex A1).
    - Stage 1: credit losses computed over a 1-year horizon and loss distributions reflect this.
    - Stages 2 and 3: lifetime horizon for loss distributions.
  - Under CECL (U.S.):
    - Provisioning based on expected credit losses with a lifetime horizon for all asset quality classes.
  - Under previous IAS 39 and many national frameworks:
    - Stage 1 and 2 exposures were not provisioned for.
- Statistic of interest for provisioning:
  - The mean of the point-in-time credit loss distributions corresponds to the expected loss (ECL_PiT) for which provisions are to be held.
- Distributional characteristics:
  - Credit loss distributions are bounded at zero (losses cannot be negative).
  - In boom times: expected losses lower and distribution compressed toward zero.
  - In recession times: expected losses higher and distribution expanded to the right.

#### 2.2 PiT Credit Loss Distribution and Capital Requirements
- Role distinction:
  - Loan loss provisions buffer expected credit losses.
  - Capital requirements protect against “unexpected losses,” i.e., tail metrics of the loss distribution.
  - Example: Basel Committee on Banking Supervision (2006) aims for Pillar 1 requirements to cover the 99.9th percentile under the IRB approach.
- Benchmarking procedure using PiT distributions:
  - Minimum capital requirements (sum of Pillar 1 and Pillar 2 requirements plus the capital conservation buffer and possible buffers for systemically important banks) are expressed as a share of credit exposure.
  - The sum of provisions and minimum capital requirements can be compared to the point-in-time credit loss distribution to determine the fraction of forward-looking loss scenarios where losses exceed the combined buffers (illustrated in Figure 1).
- Application levels:
  - Benchmarking can be performed at the level of each bank-portfolio, each bank, and the entire banking system.

*Source: IMF Working Paper excerpt (pages labeled INTERNATIONAL MONETARY FUND4–7 and Figures as described in the provided content).*

### Annex A3.  derives the benchmarking mathematically.

### Annex A3.  derives the benchmarking mathematically.

### Benchmark interpretation and scope
- The benchmark reveals which quantile of the forward-looking PiT loss distribution is covered by provisions and capital requirements.
- The benchmark is not normative; it does not indicate whether capital requirements "pass" or "fail" a test.
- The Basel framework makes no quantitative claim as to the quantile of unexpected losses covered by capital requirements under the standardized approach.
- Under the IRB approach, loan losses are derived using a through-the-cycle (TTC) PD and downturn (conservative) LGDs.
- TTC PDs can be higher (lower) than PiT PDs in boom (recession) times.

### TTC losses and the Countercyclical Capital Buffer (CCyB)
- Purpose of the CCyB:
  - Intended to absorb losses arising in a sudden downturn.
  - Intended to lean against the cycle and dampen it during boom times.
  - Accumulated during periods of excessive credit growth, when realized point-in-time credit losses tend to be small.
- PiT credit loss distributions alone cannot serve as the sole source for CCyB calibration because they do not convey a banking system’s relative position in the credit cycle; a through-the-cycle (TTC) reference metric is required.
- Alternative approach proposed:
  - Use a TTC distribution of historical tail credit losses built from one specific quantile of all historical PiT distributions over a longer time period spanning boom and downturn episodes.
  - Illustration: quarterly snapshots over 25 years lead to a distribution with 100 observations.
  - A quantile of that distribution (Figure 1 uses the 95th percentile, denoted as C_Cy_Q–, the dashed yellow vertical line) serves as the anchor point for calibration.
  - Given its definition, this point is expected to be stable when more (quarterly) data accumulates over time.
- CCyB calibration mechanism:
  - The CCyB is informed by the amount of capital required to fill the gap between the sum of provisions and capital requirements on one hand and the quantile of the TTC loss distribution on the other.
  - The mechanism allows setting the buffer to zero in a recession and building it up gradually in neutral or boom phases, including the possibility of a positive neutral CCyB.
  - Annex A3. derives the calibration mathematically.
- Caveat:
  - Credit losses form only one component of overall risk.
  - Since the CCyB applies to total risk-weighted assets (RWA), calibration should consider credit loss dynamics and all other components of a profit and loss distribution confronting banks, including interest income and expenses, fee and commission income and expenses, and market risk-related valuation gains and losses.

### The model — overall structure
- The model is composed of four modules:
  - Module A: macro-financial model component.
  - Module B: transition matrix model component.
  - Module C: LGD module.
  - Module D: combines outputs to obtain a credit loss distribution for all bank-portfolios.
- Modules A, B, and C are split into estimation and simulation sub-components (A1/A2, B1/B2, C1/C2).
- Simulation workflow:
  - Simulated macro-financial density forecasts are fed through the model suite.
  - A large number of macro-financial scenarios (e.g., 5,000 multivariate paths, 30 years forward in time, with quarterly frequency) are simulated from Module A2 and fed consistently through subsequent modules to produce credit loss distributions for all banks and portfolios in Module D.
- Rationale for simulation-based approach versus IFRS 9 small-scenario approach:
  - IFRS 9 requires a small number of scenarios designed by banks with self-set scenario weights; this is subject to ad hoc choices and uncertain proximity to the “true” ECL.
  - The simulation-based approach:
    - Avoids hand-picked scenarios and subjective weights.
    - Covers the whole distribution with many simulated scenarios.
    - More properly accounts for all nonlinearities determining credit loss distributions (assuming the model captures them).
    - Provides more robust estimates of the credit loss distributions’ mean and other statistical moments.

### Module interconnections and detailed components (as illustrated)
- Key functional elements shown (conceptual, as in Figure 3):
  - VAR model estimation and simulation feeding Z-score conditional forecasting.
  - TM (transition matrix) density forecasts and augmentation from 3x3 TMs to obtain 4x4 TMs.
  - Repayment profiles projecting portfolio balances (flow → stock) and portfolio-level balance density forecasts.
  - LGD forecasting produces LGD density forecasts and parameters for distributions of collateralization ratios, sales ratio, and time-to-liquidation; LGD unsecured.
  - ECL calculation aggregates to produce an ECL distribution.
  - Scale-up on-balance sheet portfolios and rescaling steps for off-balance sheet exposure ratios.

### Required data inputs (Table 1 summary)
- The model requires historical macro-financial data and bank-portfolio level micro data.
- Data input items, dimensions, and comments (preserve dimensions and descriptions exactly):
  - Exposure stocks: B x P x T x S5 x C — Historical end-of-period t stocks in currency units, by bank and portfolio.
  - Transition flows: B x P x T x S3 x S3 — Historical cross-flows between Stages 1/2/3 through period t in % of the end-of-previous-period stock, by bank and portfolio.
  - Interest rates: B x P — Annualized current interest rates for outstanding stock of exposures, by bank and portfolio.
  - Repayments: B x P x S3 x H — Principal repayment distribution of initial Stage 1 and 2 exposures, by bank and portfolio, from current position up to H quarters into the future.
  - Unsecured LGDs: (B x) P — Unsecured LGDs for recently defaulted loans.
  - Historical macro data: K x T — History of relevant macro-financial variables.
  - Collateralization ratios: B x P x C x S5 — Fitted parameters of the log-normal distribution of portfolio-level collateralization ratios.
- Note text (preserved):
  - The dimensions denote the: B = banks; P = portfolios; T = historical time; S3 = Stages 1 to 3; S5 = Stage 1, Stage 2 and the three time-since-default buckets of Stage 3; H = future time horizon; K = macro-financial variables; C = collateral type.
- Model flexibility and portfolio segmentation guidance:
  - The model allows different portfolio segmentations; segmentation should be chosen per economic and data availability considerations.
  - Exposures with similar characteristics (similar sensitivities to the macro-financial environment and unsecured loss rates) should be included in the same portfolio type.
  - A sufficiently high number of contract-level exposures should be included in each portfolio to observe meaningful historical transition matrices and estimate robust macro-financial relationships.
  - Possible portfolio definitions include counterparts (large non-financial corporates, SMEs, financial firms, public sector and local government, households), lending purpose (consumer finance, mortgage lending, project finance) and instrument type (micro-finance, loans, bonds).

### Module A: Macro-financial model details
- Model choice and purpose:
  - A conventional vector autoregressive (VAR) model may form the starting point, though any macro-financial model can serve the same purpose.
  - Purpose: capture historical dependencies of relevant macro-financial drivers and produce multivariate, multi-period density forecasts.
- VAR implementation specifics:
  - Macro module set up to contain two VAR models for the paper’s illustration:
    - First VAR: endogenous domestic macro-financial variables (such as GDP growth, unemployment, and interest rates) plus exogenous international variables (such as the oil price or U.S. interest rates).
    - Second auxiliary VAR: endogenizes relationships between the international variables.
- Density forecast generation:
  - Use resampling methods to account for residual and coefficient uncertainty.
  - Draw from the coefficient vector mean and covariance matrix estimates to capture coefficient uncertainty (parametric bootstrap assuming multivariate normality).
  - Add noise by drawing from the VAR models’ residuals to capture residual uncertainty (residual drawing can be nonparametric or parametric).
  - The width of final credit loss distributions can be examined for dependence on:
    - accounting for only coefficient uncertainty vs coefficient and residual uncertainty combined;
    - parametric (normal) versus nonparametric treatment of residuals.
- Example:
  - Figure 4 illustrates density forecasts for selected macro variables for Colombia, displaying historical variables up until the dashed vertical line in 2023Q3 and the scenario-conditional density for the ten-year period starting in 2023Q4. Shaded bands show the width of the distribution from the 25th to the 75th percentile.
- Choice of macro-financial variables:
  - Include variables required by subsequent modules (B, C, and D), usually: GDP growth, price inflation, the unemployment rate, the relevant bilateral exchange rate (or an effective, i.e., trade weighted, exchange rate), wage growth, house price growth, a short-term sovereign bond yield, and a term spread (e.g., the difference between a long- and short-term sovereign bond yield).
  - Exogenous variables can be included in a VARX model, such as the U.S. Fed Funds or other major central bank policy rates, GDP growth rates of major trading partners, and import or export-related commodity prices such as for oil.
- All variables will serve as potential inputs to the transition matrix (Z-score) models.

*Source: IMF Working Papers — Informing Bank Provisions and Capital Requirements with Forward-Looking Credit Loss Distributions (Annex A3).*

### Section  3.5.  House  prices  should  be  included  in  the  model  as  this  variable  will  serve  as  an  important

### Section 3.5. House prices should be included in the model as this variable will serve as an important direct input to the LGD module as well, for its portion that is responsible for delivering LGD estimates for the real estate-collateralized part of the banks’ loan portfolios.

### Role of house prices and international variables
- House prices are an explicit input to the LGD module for real estate-collateralized portfolios.
- Exogenous international variables are endogenized using a second VAR model to capture variation and dependencies across international variables, which feed into uncertainty (the density forecasts) of domestic macro-financial variables.
- Density forecasts are generated in two stages:
  - Density forecasts from the “international VAR” are simulated first.
  - These international VAR density forecasts are then used as input to the density forecast simulation with the “domestic VARX.”

### Module B1: Constructing Z-Scores from transition data
- Methodological choice:
  - The Z-score methodology of Belkin et al. (1998) is employed for rating transitions and applied to credit quality class transitions relevant under IFRS 9, as suggested in Gross et al. (2020).
  - Z-scores are conditioned on macro-financial factors using a Bayesian Model Averaging Method (BMA); see Gross & Población (2017).
- Purpose and data preparation:
  - Historical transition flow data are collected as transition matrices after defining stage migration criteria.
  - Stage balances (Stages 1, 2, and 3) evolve through stage migration flows and other factors (new business, repayment of principal, write-offs).
  - New business inflows into Stage 1 are not considered because the model quantifies ECLs and implied provision needs for currently outstanding bank portfolios (including off-balance sheet assets).
  - Data generation should ideally cover years predating the introduction of IFRS 9 in 2018.
- IFRS 9 staging indicators (examples):
  - Move from Stage 1 to Stage 2: minimal absolute or relative changes in PDs since origination; observed or expected significant changes in operating profits of firm borrowers; rating downgrades; material rises in market-based CDS spreads or credit spreads of corporate bonds; modification and restructuring (interest payment holidays, interest rate step-ups, call for additional collateral or guarantees); a rebuttable presumption of reaching 30 days past due.
  - Move from Stage 2 to Stage 3: outright bankruptcy; a rebuttable presumption of delinquent payments surpassing 90 days past due.
- Practical choices in staging:
  - Two possible avenues to define staging criteria when preparing transition flow data:
    - Use banks’ own historical stage classifications to derive historical transition matrices (reflects banks’ staging assumptions but may hamper comparability).
    - Define a set of criteria in a “top-down” manner to ensure comparability across banks.
- Interpretation of Z-scores:
  - The Z-score compresses a time series of transition matrices into one number per point in time, reducing dimensionality for econometric modeling.
  - For any bank-portfolio:
    - Z-Score = 0 corresponds to the transition matrix equaling the long-run average transition matrix.
    - Positive Z-Score indicates a more “favorable” transition matrix (fewer defaults, more migrations to better stages).
    - Negative Z-Score indicates more “unfavorable” transitions.

### Module B2: Econometric models for Z-scores to relate them to macro-financial variables
- Modeling approach:
  - Z-score time series estimates are modeled as a function of macro-financial drivers using Bayesian Model Averaging (BMA) (Gross & Población, 2019) to account for model uncertainty and avoid hand-picking equations.
  - Separate bridge equations are used rather than including Z-scores directly in the VAR because there are typically several dozens up to hundreds of Z-score time series, which would render the VAR too high-dimensional and not estimable.
  - Separate bridge equations allow consideration of time contemporaneous macro–Z-score relationships on top of lagged relationships.
  - The BMA implementation allows imposition of sign constraints on long-run multipliers (LRMs) of the Z-score models’ right-hand side variables.
- Model selection and inclusion:
  - Macro-financial variables are retained in individual equations comprised by the “model space” only when they have the predefined “right” signs.
  - Empirical weights are assigned to each equation in the model space using in-sample or out-of-sample criteria.
  - Right-hand side variables found relevant in Z-score models should be included in the macro module (Module A).
- Density forecast generation for Z-scores:
  - Required steps:
    1. Draw coefficients from the Z-score models using estimated coefficient means and coefficient covariance matrices.
    2. Draw from the Z-score model residuals (bank portfolio specific).
    3. Use macro-financial density forecasts simulated from the VARX model as input.
  - Z-score density forecasts can be decomposed into coefficient, model, and residual uncertainty (model uncertainty is assessable because of BMA).
  - VARX model density forecasts decompose into coefficient and residual uncertainty only (no BMA involved).

### Module C: Loss Given Default (LGD)
- Literature and approach:
  - LGD literature notes negative empirical relationship between default and recovery rates on corporate bonds and macro drivers (Altman et al., 2005).
  - Moody’s KMV’s LossCalc model has endogenous LGDs driven by collateral, debt type, firm-level financials, industry factors, and macro-financial factors (Gupton & Stein, 2005).
  - Evidence exists that PDs and LGDs are positively correlated (Bellotti & Crook, 2012).
- LGD methods employed:
  - Two structural LGD methods are used:
    1. For real estate-collateralized portfolios: collateral value is indexed with house price trajectories simulated by a macroeconomic model (indexation approach also considered in Gross & Población (2017) and Gross et al. (2020)).
    2. For all other portfolios: Frye & Jacobs (2012) methodology (FJ), which establishes a structural link between PDs and LGDs.
- Treatment by time-since-default and collateralization:
  - Loans in default are split into three buckets based on time-since-default; thresholds to move between buckets can be defined by portfolio and collateral type. For loans in long default, an LGD of 100 percent is assumed.
  - For each portfolio, the loan-level collateralization ratio (CR) — the ratio of available collateral to loan exposure — is fitted using a lognormal distribution to capture available collateral with two parameters efficiently.
  - If no collateral is available, an unsecured LGD (informed by historical recoveries on unsecured parts of loan exposures) is applied.
  - For uncollateralized portfolios no CR distribution is fitted.
- Portfolio-level LGD computation logic:
  - The fitted CR distribution allows computing the fraction of overcollateralized loans (LGD = 0 percent) and the remainder where unsecured LGD rates apply to the uncollateralized part of each loan.
  - The framework logic is captured by equation (1) and the resulting LGD derivation is provided in Annex A4..2.
  - Equation (1) (as shown in source) expresses LGD in integral form and relates LGD to LGD_unsec and the lognormal CR distribution components.
- Real estate collateral specifics:
  - For real estate collateral, the CR distribution is adjusted to account for uncertainty in time to collateral liquidation (TTF) and future sales price (SP):
    - Step 1: Multiply CR distribution by a lognormal distribution reflecting uncertainty about time from default to foreclosure and simulated change in average house prices along the scenario horizon. The distribution of TTF is informed by empirical moments. House price index fluctuations are captured by the change in the house price index from the macroeconomic model, discounted back to the time of default, and fitted with a lognormal distribution.
    - Step 2: Multiply the CR distribution by a lognormal distribution reflecting uncertainty about the precise sale price relative to current collateral valuation.
    - Loan amortization is considered to project future CR distributions.
  - The CR distribution translation:
    - C R ∼ LN(μ,σ^2) −→ C R ∼ LN(μ+μ_TTF+μ_SP, σ^2+σ^2_TTF+σ^2_SP) (equation (2) as in the source).
- Projection dynamics:
  - For defaults occurring after the first year in simulation, unsecured LGDs are projected using the FJ methodology and CR distributions are shifted according to portfolio amortization (reflecting higher credit risk when exposures decrease).
  - As principal is repaid over time and assuming collateral values remain unchanged, collateralization ratios increase.

*Source: IMF Working Paper — Section content as provided in the supplied PDF excerpt.*

### Annex A4..3 derives how the collateralization ratio distributions are affected by this.

### wpiea2025228-source-pdf - Annex A4..3 derives how the collateralization ratio distributions are affected by this.

### Module D: Expected Loss (ECL) calculation (Stages 1–3)
- Stage 1 ECL formula (quarterly discounting, losses assumed to occur mid-quarter):
  - ECL_S1 = sum_{X=1..2} sum_{q=1..4} EXP_{S_X,t=q−1} · TR_{X→3,t=q} · LGD_{S_X,t=q} / (1 + r)^{q−0.5}
- Stage 2 ECL formula (lifetime up to quarter M; interest rate r is quarterly):
  - ECL_S2 = sum_{X=1..2} sum_{q=1..M} EXP_{S_X,t=q−1} · TR_{X→3,t=q} · LGD_{S_X,t=q} / (1 + r)^{q−0.5}
- Stage 3 ECL formula (time-since-default buckets b):
  - ECL_S3 = sum_{b=1..3} (1 − TRC_{3→1,b} − TRC_{3→2,b}) · EXP_{S3,b,t=0} · LGD_{b,t=1}
- Cure rates (TRC_{3→X,b}) computed from scenario-specific simulated transition matrices:
  - First bucket cure computed over a three-year horizon
  - Second bucket cure computed over a one-year horizon
  - Longest time-since-default bucket: cure rate set to 0

### LGD module — Frye-Jacobs (FJ) methodology and application cases
- FJ LGD formula (Vašíček distribution assumption):
  - LGD_{t0+h} = Φ( Φ^{-1}(PD_{t0+h}) − k ) · PD_{t0+h}
  - k = [ Φ^{-1}(PD̄) − Φ^{-1}(PD̄ · LGD̄) ] / sqrt(1 − ρ)  (ρ = default correlation parameter)
- Use cases (data/inputs → outputs):
  - Case 1: Given PD_TTC (T0), LGD_TTC (T0), PD_PiT (T0 & horizon) → derive LGD_PiT path
  - Case 2: Given PD_TTC (T0), LGD_PiT (T0), PD_PiT (T0 & horizon) → impute LGD_TTC to match LGD_PiT and derive LGD_PiT path (this paper embeds Case 2)
  - Case 3: Given LGD_TTC (T0), LGD_PiT (T0), PD_PiT (T0 & horizon) → impute PD_TTC then derive LGD_PiT path
  - Case 4: Given PiT loss-rate path and PD_TTC & LGD_TTC → imply PD_PiT and LGD_PiT

### Collateralization ratios → loss rates (lognormal mapping)
- Portfolio-level collateralization ratio CR = C/E modeled as CR ∼ LN(μ, σ^2).
- Loss function given collateralization:
  - loss(CR) = LGD_unsec · (1 − CR), if CR < 1; = 0, if CR ≥ 1
- Expected LGD (integral formulation):
  - LGD = LGD_unsec · ∫_{0}^{1} (1 − CR) · LN_pdf(CR) dCR
- Closed-form expression using lognormal/normal CDFs:
  - LGD = LGD_unsec · [ Φ_{LN}(1) − exp(μ + σ^2/2) · Φ_{N}( (−μ − σ − σ^2) / σ ) ]
  - (notation: Φ_{LN} is lognormal CDF; Φ_{N} is normal CDF)

### Loan amortization effects on CR distributions (A4..3)
- Principal amortization reduces exposures E; if fraction α is repaid, E → E(1 − α).
- Under amortization, CR follows:
  - CR = C / (E (1 − α)) ∼ LN( μ − ln(1 − α), σ^2 )
- Portfolio-level amortization implementation:
  - Amortization arises from principal repayments (which increase CR) and loan maturities (collateral leaves portfolio)
  - Model assumption: half of amortization is linked to repayments and half to maturing loans

### Real-estate collateral: time-to-foreclosure and future house prices (A4..4)
- Baseline CR distribution remains lognormal: CR ∼ LN(μ, σ^2)
- Time-to-foreclosure (TTF):
  - In Colombia average sale (time-to-foreclosure) = four years with a standard deviation of about five years
  - Simulate TTF distribution; combine with house price distribution N years forward to obtain foreclosure price; discount back to default time
  - Sale price uncertainty modeled as lognormal with parameters μ_{SP}, σ_{SP}
- Future house price shifts accounted via μ_{HP}
- Product of independent lognormals is lognormal; resulting CR distribution parameters:
  - CR ∼ LN( μ + μ_{TTF} + μ_{SP} + μ_{HP}, σ^2 + σ^2_{TTF} + σ^2_{SP} )

### Time-since-default bucketing (Table 4)
- Limits (days since default) by portfolio and collateral type (as provided in Table 4):
  - Commercial/Micro, None: bucket1–2 limit = 210, bucket2–3 limit = 420
  - Commercial/Micro, Real Estate: 540, 1080
  - Commercial/Micro, Financial: 360, 720
  - Commercial/Micro, Other: 360, 720
  - Consumer/Mortgage, None: 30, 90
  - Consumer/Mortgage, Real Estate: 360, 720
  - Consumer/Mortgage, Financial: 360, 720
  - Consumer/Mortgage, Other: 270, 540

### Point-in-time (PiT) vs Through-the-cycle (TTC) distributions
- Model generates PiT credit loss distributions using current cycle starting point; mean = expected credit loss (informs provisions)
- Through-the-cycle distribution constructed as accumulation of historical tail metrics of PiT distributions (see section 2.3 referenced)
- PiT inputs: last observations of macro-financial variables and portfolio parameters (Z-scores, observed Stage composition)

### Model extensions (examples listed)
- Split portfolios by fixed vs. variable rate components; estimate Z-scores and Z-score bridge equations separately
- Endogenize discounting of recovery values and expected losses using interest rate paths from VARX component (current implementation uses loan T0 interest rates)
- Estimate Z-scores for each row of bank-portfolio transition matrices instead of full-matrix Z-scores
- Employ nonlinear macro models for scenario simulations (e.g., regime-switching VAR(X))

### Illustrative application: Colombia (key dataset and outcomes)
- Bank sample: 55 Colombian banks (close to 100 percent banking system coverage); banking system = 71 percent of financial system assets at end-2023 and represents about 64 percent of nominal GDP in 2023
- Concentration: largest 3, 8, and 13 of 55 banks represent 50, 75, and 90 percent share of banking system assets, respectively
- Four bank portfolio types: commercial, microcredit, consumer lending, mortgage credit
  - Portfolio shares of loan book at end-2023: commercial 50 percent, consumer 30 percent, mortgages 16 percent, microcredit 4 percent
- Domestic vs foreign lending scope in model: domestic loan book = 89 percent, foreign lending through branches = 1 percent, foreign subsidiary loan book = 10 percent (end-2023)
- Solo banks’ domestic bond exposures at end-2023: 13 percent of total (split into 9 and 4 percent for public and private corporate bond exposures); foreign bonds 8 percent excluded from scope
- Z-score model: panel BMA with banks as cross-section; pandemic period excluded from estimation sample: 2020Q2—2022Q2 (nine quarters)
- Provision coverage comparison (expressed as percent of end-2023 loan book exposures):
  - Commercial portfolio: model estimate = 4.4 percent; banks = 4.5 percent
- Stage 1 lifetime-horizon sensitivity (CECL-style): switching Stage 1 horizon from 1-year to lifetime increases provision coverage multiples:
  - Multiples (lifetime vs 1-year) = 1.7x (commercial), 1.5x (consumer), 1.2x (microcredit), 2.2x (mortgages)
  - Note: mortgage multiple 2.2x reflects mortgages’ longest average duration

### Sensitivity and counterfactual experiments (selected findings)
- Sensitivity 1: Excluding pandemic period from VAR(X) estimation → tighter credit loss distributions; means of PiT distributions not overly sensitive but upper-tail provisioning more sensitive
- Sensitivity 2a: Switching residual uncertainty off (domestic VARX, auxiliary foreign VAR, Z-score equations) → noticeably tighter credit loss distributions and left-shift of means
- Sensitivity 2b: Switching model uncertainty component of BMA off → in this Colombia application, credit loss distributions at aggregate system level did not change notably (but this may differ across applications)
- Sensitivity 3: Re-anchoring macro-financial baseline paths (example: re-anchor real GDP growth, unemployment, CPI from domestic VARX to IMF WEO October 2023 baseline) → re-centered macro densities shift PiT credit loss distributions left (lower PiT credit losses and provisioning)
- Sensitivity 4: Stage 1 ECL horizon change from 1-year to lifetime → provisioning needs increase for all portfolios (most pronounced relative impact for mortgages)

### CCyP vs CCyB (comparison and CCyB calibration approach)
- Objectives: CCyP and CCyB both aim to accumulate releasable buffers in booms and lean against cyclical imbalances
- Transmission:
  - CCyP: primarily through build-up of provision stocks (directly affects actual capital ratios)
  - CCyB: primarily through dynamic capital requirements (affects actual capital ratios secondarily)
- Colombia CCyP design specifics:
  - Release triggered when four threshold-based indicators jointly signal downturn
  - During release, bank can use up to 70 percent of its countercyclical provision stock to buffer periodic losses
  - After release phase (up to six months), normal-times CCyP coverage to be reached after no more than 18 subsequent months
- CCyB calibration within model (key equations):
  - K / RWA ≥ ρ^*  where ρ^* = ρ + CCyB
  - Focus on credit-risk component: k_CR(p) ≥ (ρ + CCyB) · d_RW · b_a_CR(p)  (normalized by total credit risk assets A_CR)
  - Minimum capital requirement (fraction of credit risk exposures): k_rmin_CR = ρ · d_RW · b_a_CR(p)
  - CCyB derived from TTC quantile CCyQ and PiT expected losses ECL_PiT:
    - CCyB = max{ 0, CCyQ / (d_RW · b_a_CR) − p / b_A_CR − ρ }
    - Interpretation: CCyB positive when sum of provisions and minimum capital requirements insufficient to cover chosen TTC quantile

*Source: wpiea2025228-source-pdf - Annex A4..3 derives how the collateralization ratio distributions are affected by this.*

### References

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*Credit Loss in Translation: Informing Bank Provisions and Capital Buffer Requirements with Forward-Looking Credit Loss Distributions — Working Paper No. WP/2025/228*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025228-source-pdf.pdf_