## Introduction

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---

### Purpose and objectives
- Three objectives:
  - Provide haircut models depending on the type of eligible assets and the availability of data.
  - Present a framework for quantifying expected uncollateralized exposure under haircut models.
  - Simulate the uncollateralized exposure and validate haircut models using a Machine Learning technique (Variational Autoencoders, VAEs).

### Core principles for setting haircuts
- Non-procyclicality: Haircuts should not be procyclical to avoid amplifying fire sales during crises.
- Data-drivenness: Haircuts should be informed by datasets that capture risks embedded in target assets.
- Conservatism: Haircuts should protect the central bank by benchmarking against losses during historical stress episodes.
- No arbitrage gap: Avoid overly lenient collateral risk management that could create adverse selection.

### Classification of modeler situations (2×2 framework)
- Dimensions:
  - Type of collateral: marketable vs non-marketable.
  - Data availability: data-rich vs data-limited.
- Model recommendations:
  - Marketable, data-rich: Duration Approximation with Stressed Volatility (DASV) model.
  - Marketable, data-limited: Duration Approximation with Cross-country Regression (DACR) model.
  - Non-marketable, data-rich: Asymptotic Single Risk Factor with Adjustments (ASRFA) model.
  - Non-marketable, data-limited: cross-country regression, Merton (1974) structural model if balance sheet data exists, or satellite/extrapolation methods for short samples.

### Tail-risk approach to haircuts
- Tail-risk measures: Value-at-Risk (VaR) and Expected Shortfall (ES).
- VaR-based haircut condition (notation preserved):
  - choose h such that VaR_α = 0 where VaR_α = inf{L*: Pr(Loss_{t+Δt} ≥ L* | t = τ_D) ≥ 1−α}.
- ES-based haircut condition:
  - choose h such that E[Loss_{t+Δt} | Loss_{t+Δt} ≥ VaR_α, t = τ_D] = 0.
- Key parameters central banks must set: VaR percentile(s), time to liquidation Δt, VaR percentile for parameter uncertainty, and risk-free interest rate volatility.

### Illustrative toy model (geometric Brownian motion collateral)
- Collateral A_t follows geometric Brownian motion with drift μ_A and volatility σ_A.
- Exact VaR-based haircut formula:
  - A_{t+Δt} = A_t exp((μ_A − σ_A^2/2)Δt − Φ(α) σ_A √Δt).
  - h = 1 − exp((μ_A − σ_A^2/2)Δt − Φ(α) σ_A √Δt).
- Linear approximation (when exponent is small):
  - h = Φ(α) σ_A √Δt.
- Baseline illustrative parameters used later:
  - μ = 0%, σ = 10%, Δt = 10/252, VaR percentile α = 0.99.

### Dependence between collateral volatility and counterparty default risk (wrong-way risk)
- Volatility of collateral may be correlated with counterparty credit quality; structural Merton-like setup.
- Dynamic specifications:
  - d log σ_t^2 = κ(log σ̅^2 − log σ_t^2) dt + ω dW_{σ,t}.
  - dX_t = X_t(μ_X dt + σ_X dW_{X,t}).
- Correlation parameter:
  - ρ = dW_{σ,t} dW_{X,t} / dt.
- Numerical illustrative parameter set:
  - σ̅ = σ_0 = 10%, κ = 0.01, ω = 0.01, X_0 = 1, D = 0.9, μ_X = 0, σ_X = 20%.
- Finding: As correlation becomes more negative, volatility conditional on default increases and haircuts rise.

### Central bank residual uncollateralized exposure framework
- Transaction-level residual exposure:
  - E_{t+Δt}^U = L − C_{t+Δt} = L(1 − (1/(1−h)) A_{t+Δt}/A_t).
- Expected uncollateralized exposure conditional on counterparty default:
  - E[max(E_{t+Δt}^U, 0) | t = τ_D] = L/(1−h) ⋅ E[max(1 − h − A_{t+Δt}/A_t, 0) | t = τ_D].
  - Interpretation: expected uncollateralized exposure is equivalent to a put option with strike 1−h and maturity Δt.
- Portfolio-level exposure definitions:
  - PEE_{i,t} = E[max(∑_{k=1}^{K_i} E_{i,k,t+Δt}^U, 0) | t = t_i^D].
  - Aggregated PEE_t = ∑_{i=1}^N (1 − R_i) ∫ dt Pr(t = t_i^D) E[max(∑_{k=1}^{K_i} E_{i,k,t+Δt}^U, 0) | t = t_i^D]_{t_i}^{T_i}.

### Duration Approximation with Stressed Volatility (DASV) — marketable assets, data-rich
- Duration-based haircut formula:
  - h_a^α = T ⋅ Δy_{α,T}.
- Δy_{α,T} computed under normal assumption:
  - VaR: Δy_{α,T} = Φ^{-1}(α) ⋅ σ_α ⋅ √Δt.
  - ES: Δy_{α,T} = φ(Φ^{-1}(α)) / (1−α) ⋅ σ_α ⋅ √Δt.
- Duration T computed via:
  - T = −(1/P) dP/dy (or numerical differentiation, modified duration, or remaining maturity as conservative proxy).
- Stressed volatility approaches:
  - Historical volatility (rolling standard deviation over d days, often d = Δt = 10 business days).
  - EGARCH model (parameters: autoregressive coefficient ρ, mean-reverting level σ̅, volatility of volatility ν, asymmetry γ). EGARCH equations preserved.
- Time to liquidation guidance:
  - Benchmark Δt = 10 business days for liquid collateral (Basel reference); longer for illiquid assets; illiquidity may be captured by λ>1 in extended formulas.
- Consistency across related markets:
  - For less liquid domestic bonds vs liquid Eurobonds, extend stressed yield change with spread volatility σ_spr and illiquidity adjustment λ:
    - VaR: Δy_{α,T} = Φ^{-1}(α) ⋅ (σ_α + σ_spr) ⋅ √(λ Δt).

### Duration Approximation with Cross-country Regression (DACR) — marketable assets, data-limited
- Purpose: construct hypothetical sovereign spreads where secondary markets or ratings are unavailable.
- Cross-country regression specifications:
  - Basic: Spread_{i,t} = α_{i,t} + β_{Debt,t} Debt_{i,t} + β_{GDP,t} GDP_{i,t} + ε_{i,t}.
  - Extended: includes controls Z_{i,t} = (Trade_{i,t}, PerGDP_{i,t}, FX_{i,t}, π_{i,t})'.
- Use five-year averages of estimated coefficients; adjust coefficients for parameter uncertainty:
  - β_{Debt}^{adj} = β̅_{Debt} + Φ^{-1}(θ) ⋅ σ_{Debt}.
  - β_{GDP}^{adj} = β̅_{GDP} − Φ^{-1}(θ) ⋅ σ_{GDP}.
  - θ = risk tolerance for parameter uncertainty.
- Compute model-based spreads for target country:
  - Spread_{target,t} = α̅ + β_{Debt}^{adj} Debt_{target,t} + β_{GDP}^{adj} GDP_{target,t}.
- Compute spread-change volatility (five-year window recommended) to feed into haircut formulas.
- Illustrative hypothetical-country numeric example:
  - Spread volatility = 0.0274.
  - Risk-free rate volatility assumed 0.01.
  - Time to liquidation assumed one month (1/12 annualized).
  - Duration-based linear formula: h_α = T ⋅ Φ^{-1}(α) ⋅ (0.01 + 0.0274) ⋅ √(1/12).
  - Nonlinear formula: h_α^{nonlinear} = 1 − exp(−T ⋅ Φ^{-1}(α) ⋅ (0.01 + 0.0274) ⋅ √(1/12)).
  - Observation: higher VaR percentile → higher haircuts; duration approximation tends to produce higher haircuts than nonlinear formula.

### Asymptotic Single Risk Factor with Adjustments (ASRFA) — non-marketable assets, data-rich
- ASRF model basics:
  - X_{i,t} = √ρ_i M_t + √(1−ρ_i) ε_{i,t}.
- Closed-form stressed PD (VaR of PD):
  - PD_α = Φ(Φ^{-1}(PD̅) − √ρ Φ^{-1}(1−α) / √(1−ρ)).
- ES for PD:
  - ES_α = 1/(1−α) ⋅ Φ_2(Φ^{-1}(PD̅), Φ^{-1}(1−α), √ρ).
- Inputs: unconditional PD, LGD, percentile α, correlation parameter ρ.
- Parameter uncertainty for correlation:
  - ρ^* = ρ̂ + M_ρ σ_ρ, where σ_ρ = std(ρ̂) and M_ρ can be set to Φ^{-1}(α) or calibrated.
- Additional adjustments: granularity adjustment; LGD-PD dependence adjustments if material.

### Non-marketable assets, data-limited — three representative cases
- Case I: balance sheet data available → use Merton (1974) structural model:
  - PD = Φ((log(X_0/D) + (μ_X − 0.5 σ_X^2) T) / (σ_X √T)).
  - Stress by increasing σ_X or reducing μ_X, or use estimated PD as unconditional PD in ASRFA.
- Case II: balance sheet data not available → cross-country sector-level regression:
  - Spread_{i,j,t} = α_{i,t} + β_{M,i,t} Macro_{i,t} + β_{S,i,j,t} Sector_{i,j,t} + β_z Z_{i,j,t} + ε_{i,t}.
  - Use model-based sector spreads + assumed LGD to infer PDs.
- Case III: short sample missing crises → satellite/extrapolation models (e.g., logistic extrapolation) to estimate steady-state PD and derive stressed PD.

### Simulation of counterparty risk exposures (Nelson-Siegel and portfolio cases)
- Portfolios considered:
  - (i) equal allocation across maturities;
  - (ii) larger allocation to short-term bonds;
  - (iii) larger allocation to long-term bonds.
- Nelson-Siegel model:
  - y_{model,t}(τ, λ) = L_t + (1 − exp(−λ τ))/(λ τ) S_t + ((1 − exp(−λ τ))/(λ τ) − exp(−λ τ)) C_t.
- Monte Carlo simulations with NS factor shocks (level and slope shocks normal; curvature fixed).
- Findings:
  - Expected uncollateralized exposure increases with NS level factor volatility; strongest for long-term concentrated portfolios.
  - Expected uncollateralized exposure increases with NS slope factor volatility; strongest for short-term concentrated portfolios.

### Machine Learning validation: Variational Autoencoders (VAEs)
- VAE architecture: Encoder → latent multivariate normal p(z) → Decoder generating p*(x | z).
- VAE use:
  - Generate stress scenarios for bond yields by learning latent stochastic factors from historical yield curves.
  - Produce diverse, non-normal yield distributions for stress testing and model validation.
- Empirical application:
  - VAE trained on U.S. Treasury yield curve data; simulated 10,000 yield-curve paths.
  - Simulated distributions for 1-year and 20-year yields show non-normal features.
  - Latent variables interpreted as overlapping level/slope/curvature drivers.
  - Realized VaR percentiles under VAE-generated shocks were slightly higher beyond the 93 percentile, suggesting small underestimation of haircuts but overall DASV haircuts were sufficiently conservative under VAE validation.
- Caveat: VAEs useful for validation and identifying model risk add-ons; direct policy-setting use constrained by interpretability and accountability concerns.

### Empirical examples and parameter guidance
- Time to liquidation guidance:
  - Δt = 10 business days for liquid collateral (Basel reference).
  - Δt = 20 business days for illiquid assets reference.
- Baseline illustrative parameters:
  - μ = 0%, σ = 10%, Δt = 10/252, VaR percentile α = 0.99.
  - Risk-free interest rate volatility example: 0.01.
  - Spread volatility example (hypothetical country): 0.0274.
  - Time to liquidation in DACR example: 1/12.
- Empirical DASV application:
  - U.S. Treasury yield curve daily data from June 1, 2005 to May 28, 2025.
  - EGARCH estimated for multiple maturities with autoregressive coefficient close to one and volatility-of-volatility declining with maturity (except 20-year).

### Model governance, adjustments, and operational notes
- Parameter uncertainty adjustments (VaR percentile for parameters) recommended, distinct from VaR percentile α for market shocks.
- Duration buckets: balance operational efficiency vs accuracy; conservative practice is to assign duration equal to the upper bound of each bucket.
- Combining market data and credit ratings: watch for double-counting of credit risk, data gaps in ratings, procyclicality from rating changes, and credibility concerns.
- Recommended robustness checks:
  - Use nonlinear formula when duration approximation produces excessively conservative haircuts.
  - Apply granularity and LGD-PD correlation adjustments when needed.

### Conclusion (summary of contributions and policy implications)
- Contributions:
  - Suite of quantitative models tailored to asset type and data availability.
  - Novel model applicable under data-limited environment.
  - Models promote internal consistency, reduce subjectivity, and enhance public accountability.
- Monitoring residual uncollateralized exposures emphasized; quantification via traditional models and Machine Learning.
- Policy implications:
  - Framework strengthens central bank collateral and counterparty risk management.
  - Contributes to financial stability and effective liquidity provision.
- Directions for future research:
  - Enhance cross-country model using AI.
  - Separate working paper planned on a “Super Economist” model leveraging an LLM to transform Article IV reports into forward-looking quantitative indicators for inclusion in cross-country spread construction.

_Italic source: wpiea2025225-source-pdf - Introduction — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025225-source-pdf.pdf_

### Introduction ...........................................................................................................

### Introduction

### Major themes covered
- Tail risk measures for setting haircuts and protecting central bank balance sheets.
- Comparison of Value-at-Risk and Expected Shortfall.
- Mathematical definition of haircuts and determining VaR percentile.
- External factors not accounted for within the modeling framework.
- Dependence between collateral volatility and counterparty default risk.
- Central bank’s uncollateralized exposure and classification of the modeler’s situation.
- Practical model choices for different data environments (data-rich and data-limited).
- Model overviews: DASV, DACR, ASRFA, ASRF adjustments, and machine learning-based simulation approaches (VAE, Nelson-Siegel).
- Operational considerations: duration buckets, duration approximation, stressing volatility, time to liquidation, consistency across markets, combining with credit rating data.
- Applications to real-world and hypothetical data, and comparative model illustrations.

### Models and methods introduced (names preserved exactly)
- DASV Model
- DACR Model
- ASRFA Model
- Original ASRF Model (and Adjusting Parameters in ASRF Model)
- Nelson-Siegel Model
- VAE (Variational Auto-Encoder) for simulating counterparty risk exposure
- EGARCH (parameter estimates table referenced)

### Key modeling and implementation topics (section headings preserved)
- Setting Haircuts with a Tail Risk Measure
- Protecting Central Bank’s Balance Sheet with Tail Risk Measures
- Value-at-Risk vs. Expected Shortfall
- Mathematical Definition of Haircuts
- Determining VaR Percentile
- External Factors Not Accounted for Within the Modeling Framework
- An Illustrative Toy Model
- Dependance Between Collateral Volatility and Counterparty Default Risk
- Central Bank’s Uncollateralized Exposure
- Classifying the Modeler’s Situation
- Setting Duration Bucket
- Marketable Assets in a Data-Rich Environment
- Duration Approximation
- Stressing Volatility
- Time to Liquidation
- Ensuring Consistency in Haircuts Models for Relevant Markets
- Combining with Credit Rating Data
- Application to Real-World Data
- Marketable Assets in a Data-Limited Environment
- Constructing Sovereign Spreads with Cross-Country Regression
- Application to a Hypothetical Country
- Non-Marketable Assets in a Data-Rich Environment
- Description of Original ASRF Model
- Non-Marketable Assets in a Data-Limited Environment (Case I, Case II, Case III)
- Simulating Counterparty Risk Exposure with ML
- Three Cases of Portfolios of Bonds Posted as Collateral
- Simulate Counterparty Risk Exposure with VAE

### Figures and tables (items and identifiers preserved exactly)
- Figures include: 1. Potential Losses, Value-at-Risk and Expected Shortfall; 2. The Impact of Corelation Between Collateral Volatility and Counterparty Default Risk; 3. Expected Uncollateralized Exposure and Volatility; 4. Expected Uncollateralized Exposure and Time to Liquidation; 5. Understanding the Relation Between Time and Liquidation and Market Risk; 6. Historical Data of the US Treasury Bond Yields; 7. DASV Model with VaR; 8. Overview of DACR Model; 9. Applying DACR Model to a Hypothetical Country; 10. Model-based Sovereign Spreads; 11. Conditional Distribution under ASRF Model; 12-a. High Correlation ρ=0.8; 12-b. Low Correlation ρ=0.2; 13. VaR of PD as a Function of the Correlation Parameter; 14. Default in Structural Credit Risk Model; 15. Three Hypothetical Compositions of Bond Portfolio; 16. Relation Between Level Favtor Volatility and Expected Uncollateralized Exposure; 17. Relation Between Slope Factor Volatility and Expected Uncollateraled Exposure; 18. Architecute of VAE; 19-a. Simulated Distribution of One-year Bond; 19-b. Simulated Distribution of 20-year Bond; 20-a. The Impact of the First Late Variable on Yield Curve; 20-b. The Impact of the Second Late Variable on Yield Curve; 21. VaR Percentile for Setting Haircuts and VaR Percentile Under VAE-generated Shocks.
- Tables include: 1. Pros and Cons of Value-at-Risk and Expected Shortfall; 2. Model Choice Given Data Availability Asset Type; 3. EGARCH Model Parameter Estimates; 4. Haircut Schedule under Different VaR Percentiles for Spreads; 5. Parameter Values and Configuration of VAE.

### Scenarios, cases, and classification structure
- Three cases for non-marketable assets in a data-limited environment:
  - Case I: Balance Sheet Data is Available
  - Case II: Balance Sheet Data is Not Available
  - Case III: Insufficiently Long Sample Period
- Three cases of portfolios of bonds posted as collateral (figure and section listed).
- Classification of the modeler’s situation (section present to guide model choice).

### Implementation-relevant parameters and items explicitly listed
- VaR percentile determination (section dedicated to this topic).
- Time to liquidation (section dedicated; appears also in figures 4 and 5).
- Duration bucket and duration approximation (sections dedicated).
- Stressing volatility (section dedicated).
- Combining with credit rating data (section dedicated).
- Model choice guidance by data availability and asset type (Table 2 referenced).

*IMF Working Papers — A Quantitative Approach to Central Bank Haircuts and Counterparty Risk Management (Introduction and table of contents as provided).*

### Introduction

### wpiea2025225-source-pdf - Introduction

### Purpose and objectives
- Three objectives:
  - Provide haircut models depending on the type of eligible assets and the availability of data.
  - Present a framework for quantifying expected uncollateralized exposure under haircut models.
  - Simulate the uncollateralized exposure and validate haircut models using a Machine Learning technique (Variational Autoencoders, VAEs).

### Core principles for setting haircuts
- Non-procyclicality: Haircuts should not be procyclical to avoid amplifying fire sales during crises.
- Data-drivenness: Haircuts should be informed by datasets that capture risks embedded in target assets.
- Conservatism: Haircuts should protect the central bank by benchmarking against losses during historical stress episodes.
- No arbitrage gap: Avoid overly lenient collateral risk management that could create adverse selection.

### Classification of modeler situations (2×2 framework)
- Dimensions:
  - Type of collateral: marketable (e.g., securities) vs non-marketable (primarily credit claims such as loan portfolios).
  - Data availability: data-rich vs data-limited.
- Model recommendations:
  - Marketable, data-rich: Duration Approximation with Stressed Volatility (DASV) model.
  - Marketable, data-limited: Duration Approximation with Cross-country Regression (DACR) model.
  - Non-marketable, data-rich: Asymptotic Single Risk Factor with Adjustments (ASRFA) model.
  - Non-marketable, data-limited: cross-country regression, Merton (1974) structural model if balance sheet data exists, or satellite/extrapolation methods for short samples.

### Tail-risk approach to haircuts
- Tail-risk measures: Value-at-Risk (VaR) and Expected Shortfall (ES).
- VaR and ES used to define haircuts such that conditional loss measures equal zero:
  - VaR-based haircut condition (notation preserved from source): choose h such that VaR_α = 0 where VaR_α = inf{L*: Pr(Loss_{t+Δt} ≥ L* | t = τ_D) ≥ 1−α}.
  - ES-based haircut condition: choose h such that E[Loss_{t+Δt} | Loss_{t+Δt} ≥ VaR_α, t = τ_D] = 0.
- Key parameters central banks must set: VaR percentile(s), time to liquidation Δt, VaR percentile for parameter uncertainty, and risk-free interest rate volatility.

### Illustrative toy model (geometric Brownian motion collateral)
- Collateral A_t follows geometric Brownian motion with drift μ_A and volatility σ_A.
- Exact VaR-based haircut formula preserved from source:
  - A_{t+Δt} = A_t exp((μ_A − σ_A^2/2)Δt − Φ(α) σ_A √Δt).
  - h = 1 − exp((μ_A − σ_A^2/2)Δt − Φ(α) σ_A √Δt).
- Linear approximation (when exponent is small):
  - h = Φ(α) σ_A √Δt.
- Baseline illustrative parameters used later: μ = 0%, σ = 10%, Δt = 10/252, VaR percentile α = 0.99.

### Dependence between collateral volatility and counterparty default risk (wrong-way risk)
- Volatility of collateral may be correlated with counterparty credit quality (structural Merton-like setup).
- Example dynamic specifications preserved:
  - d log σ_t^2 = κ(log σ̅^2 − log σ_t^2) dt + ω dW_{σ,t}.
  - dX_t = X_t(μ_X dt + σ_X dW_{X,t}).
- Key correlation parameter ρ = dW_{σ,t} dW_{X,t} / dt.
- Numerical illustrative parameter set: σ̅ = σ_0 = 10%, κ = 0.01, ω = 0.01, X_0 = 1, D = 0.9, μ_X = 0, σ_X = 20%.
- Finding: As correlation becomes more negative, volatility conditional on default increases and haircuts rise.

### Central bank residual uncollateralized exposure framework
- Transaction-level residual exposure:
  - E_{t+Δt}^U = L − C_{t+Δt} = L(1 − (1/(1−h)) A_{t+Δt}/A_t).
- Expected uncollateralized exposure conditional on counterparty default:
  - E[max(E_{t+Δt}^U, 0) | t = τ_D] = L/(1−h) ⋅ E[max(1 − h − A_{t+Δt}/A_t, 0) | t = τ_D].
  - Interpretation: expected uncollateralized exposure is equivalent to a put option with strike 1−h and maturity Δt.
- Portfolio-level exposure definitions:
  - PEE_{i,t} = E[max(∑_{k=1}^{K_i} E_{i,k,t+Δt}^U, 0) | t = t_i^D].
  - Aggregated PEE_t = ∑_{i=1}^N (1 − R_i) ∫ dt Pr(t = t_i^D) E[max(∑_{k=1}^{K_i} E_{i,k,t+Δt}^U, 0) | t = t_i^D]_{t_i}^{T_i}.

### Duration Approximation with Stressed Volatility (DASV) — marketable assets, data-rich
- Duration-based haircut formula:
  - h_a^α = T ⋅ Δy_{α,T}
  - Δy_{α,T} computed for VaR and ES under normal assumption:
    - VaR: Δy_{α,T} = Φ^{-1}(α) ⋅ σ_α ⋅ √Δt.
    - ES: Δy_{α,T} = φ(Φ^{-1}(α)) / (1−α) ⋅ σ_α ⋅ √Δt.
- Duration T computed via:
  - T = −(1/P) dP/dy (or numerical differentiation, modified duration, or remaining maturity as conservative proxy).
- Stressed volatility approaches:
  - Historical volatility (rolling standard deviation over d days, often d = Δt = 10 business days).
  - EGARCH model (parameters: autoregressive coefficient ρ, mean-reverting level σ̅, volatility of volatility ν, asymmetry γ). EGARCH equations preserved from source.
- Time to liquidation guidance:
  - Benchmark Δt = 10 business days for liquid collateral (Basel reference), longer for illiquid assets; illiquidity may be captured by λ>1 in extended formulas.
- Consistency across related markets:
  - For less liquid domestic bonds vs liquid Eurobonds, extend stressed yield change with spread volatility σ_spr and illiquidity adjustment λ:
    - VaR: Δy_{α,T} = Φ^{-1}(α) ⋅ (σ_α + σ_spr) ⋅ √(λ Δt).

### Duration Approximation with Cross-country Regression (DACR) — marketable assets, data-limited
- Purpose: construct hypothetical sovereign spreads where secondary markets or ratings are unavailable.
- Cross-country regression specifications:
  - Basic: Spread_{i,t} = α_{i,t} + β_{Debt,t} Debt_{i,t} + β_{GDP,t} GDP_{i,t} + ε_{i,t}.
  - Extended: includes controls Z_{i,t} = (Trade_{i,t}, PerGDP_{i,t}, FX_{i,t}, π_{i,t})'.
- Use five-year averages of estimated coefficients to stabilize inputs; adjust coefficients for parameter uncertainty:
  - β_{Debt}^{adj} = β̅_{Debt} + Φ^{-1}(θ) ⋅ σ_{Debt}.
  - β_{GDP}^{adj} = β̅_{GDP} − Φ^{-1}(θ) ⋅ σ_{GDP}.
  - θ = risk tolerance for parameter uncertainty (distinct from α).
- Compute model-based spreads for target country:
  - Spread_{target,t} = α̅ + β_{Debt}^{adj} Debt_{target,t} + β_{GDP}^{adj} GDP_{target,t}.
- Compute spread-change volatility (five-year window recommended) to feed into haircut formulas.
- Illustrative hypothetical-country numeric example:
  - Spread volatility = 0.0274 (time-average volatility between 2019 and 2023).
  - Risk-free rate volatility assumed 0.01.
  - Time to liquidation assumed one month (1/12 annualized).
  - Duration-based linear formula used: h_α = T ⋅ Φ^{-1}(α) ⋅ (0.01 + 0.0274) ⋅ √(1/12).
  - Nonlinear formula used: h_α^{nonlinear} = 1 − exp(−T ⋅ Φ^{-1}(α) ⋅ (0.01 + 0.0274) ⋅ √(1/12)).
  - Observations: higher VaR percentile → higher haircuts; duration approximation tends to produce higher (more conservative) haircuts than nonlinear formula.

### Asymptotic Single Risk Factor with Adjustments (ASRFA) — non-marketable assets, data-rich
- ASRF model basics (Vasicek/Gordy foundation):
  - Firm credit quality: X_{i,t} = √ρ_i M_t + √(1−ρ_i) ε_{i,t}, with normals M and ε.
  - Closed-form stressed PD (VaR of PD) preserved:
    - PD_α = Φ(Φ^{-1}(PD̅) − √ρ Φ^{-1}(1−α) / √(1−ρ)).
  - ES for PD:
    - ES_α = 1/(1−α) ⋅ Φ_2(Φ^{-1}(PD̅), Φ^{-1}(1−α), √ρ).
- Inputs: unconditional PD, LGD, percentile α (risk tolerance), correlation parameter ρ.
- Parameter uncertainty for correlation addressed by adjustment:
  - ρ^* = ρ̂ + M_ρ σ_ρ, where σ_ρ = std(ρ̂) and M_ρ can be set to Φ^{-1}(α) or calibrated.
  - Use ρ^* in PD_α computation to produce conservative stressed PD.
- Additional adjustments: granularity adjustment if portfolio is not perfectly granular; adjustments for LGD-PD dependence if material.

### Non-marketable assets, data-limited — three representative cases
- Case I: balance sheet data available → use Merton (1974) structural model to estimate PD proxy:
  - PD = Φ((log(X_0/D) + (μ_X − 0.5 σ_X^2) T) / (σ_X √T)).
  - Stress by increasing σ_X or reducing μ_X, or use estimated PD as unconditional PD in ASRFA.
- Case II: balance sheet data not available → cross-country sector-level regression for spreads:
  - Spread_{i,j,t} = α_{i,t} + β_{M,i,t} Macro_{i,t} + β_{S,i,j,t} Sector_{i,j,t} + β_z Z_{i,j,t} + ε_{i,t}.
  - Use model-based sector spreads + assumed LGD to infer PDs.
- Case III: short sample missing crises → satellite/extrapolation models (e.g., logistic extrapolation) to estimate steady-state PD and derive stressed PD; reliance on satellite models should diminish once sample includes a business cycle.

### Simulation of counterparty risk exposures
- Portfolios considered: (i) equal allocation across maturities; (ii) larger allocation to short-term bonds; (iii) larger allocation to long-term bonds.
- Nelson-Siegel (NS) model used for yield-curve simulation:
  - y_{model,t}(τ, λ) = L_t + (1 − exp(−λ τ))/(λ τ) S_t + ((1 − exp(−λ τ))/(λ τ) − exp(−λ τ)) C_t.
  - Monte Carlo simulations with NS factor shocks (level and slope shocks normal; curvature fixed).
- Findings from NS simulations:
  - Expected uncollateralized exposure increases with NS level factor volatility; effect strongest for portfolios concentrated in long-term bonds.
  - Expected uncollateralized exposure increases with NS slope factor volatility; effect strongest for portfolios concentrated in short-term bonds.

### Machine Learning validation: Variational Autoencoders (VAEs)
- VAEs used to generate stress scenarios for bond yields by learning latent stochastic factors from historical yield curves.
- VAE architecture: Encoder → latent multivariate normal p(z) → Decoder generating p*(x | z).
- VAE advantages: interpretable latent space, ability to generate diverse, non-normal yield distributions for stress testing and model validation.
- Empirical application:
  - VAE trained on U.S. Treasury yield curve data; simulated 10,000 yield-curve paths.
  - Simulated distributions for 1-year and 20-year yields show non-normal features.
  - Latent variables interpreted as overlapping level/slope/curvature drivers (Figure examples in source).
  - Realized VaR percentiles under VAE-generated shocks were slightly higher beyond the 93 percentile, suggesting small underestimation of haircuts but overall DASV haircuts were sufficiently conservative under VAE validation.
- Caveat: ML models like VAEs are useful for validation and identifying potential add-ons for model risk, but direct use for policy-setting is constrained by interpretability and accountability concerns.

### Empirical examples and parameter guidance
- Time to liquidation guidance: Δt = 10 business days for liquid collateral (Basel reference), 20 business days for illiquid assets reference; can be longer depending on local liquidity conditions.
- Baseline illustrative parameters used in figures and examples:
  - μ = 0%, σ = 10%, Δt = 10/252, VaR percentile α = 0.99.
  - Risk-free interest rate volatility example: 0.01.
  - Spread volatility example (hypothetical country): 0.0274 (274 basis points).
  - Time to liquidation in DACR example: 1/12 (one month).
- Empirical DASV application: U.S. Treasury yield curve daily data from June 1, 2005 to May 28, 2025; EGARCH estimated for multiple maturities with autoregressive coefficient close to one and volatility-of-volatility declining with maturity (except 20-year).

### Model governance, adjustments, and operational notes
- Parameter uncertainty adjustments (VaR percentile for parameters) recommended, distinct from VaR percentile α for market shocks.
- Duration buckets: balance operational efficiency vs accuracy; conservative practice is to assign duration equal to the upper bound of each bucket.
- Combining market data and credit ratings: watch for double-counting of credit risk, data gaps in ratings, procyclicality from rating changes, and credibility concerns.
- Recommended robustness checks: use nonlinear formula when duration approximation produces excessively conservative haircuts; apply granularity and LGD-PD correlation adjustments when needed.

_Italic source: wpiea2025225-source-pdf - Introduction — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025225-source-pdf.pdf_

### Conclusion

### Conclusion

### Summary of contributions
- Proposed a suite of quantitative models tailored to the nature of eligible assets—marketable or non-marketable—and the availability of relevant data.
- Developed a novel model applicable under data-limited environment.
- Models aim to promote internal consistency, reduce subjectivity, and enhance public accountability.

### Haircut calibration principles
- Key principles guiding haircut calibration include:
  - non-procyclicality
  - data-drivenness
  - conservatism
  - the elimination of arbitrage opportunities

### Monitoring residual uncollateralized exposures
- Emphasized the importance of monitoring residual uncollateralized exposures, which can pose hidden risks during stress periods.
- Demonstrated how such exposures can be quantified using:
  - traditional financial models
  - advanced techniques like Machine Learning

### Machine Learning applications and validation
- Applied Variational Autoencoders (VAEs) to simulate stress scenarios for bond yields and validate haircut models.
- Illustrated VAEs’ potential to enhance risk assessment.

### Policy implications and framework impact
- Provides a robust framework for central banks to strengthen their collateral and counterparty risk management practices.
- Framework contributes to financial stability and effective liquidity provision.

### Directions for future research
- Plan to enhance the cross-country model using AI.
- Intend to write a separate working paper on a “Super Economist” model that:
  - leverages a Large Language Model (LLM) to transform Article IV reports into forward-looking quantitative indicators
  - includes those indicators as a regressor in the cross-country model for constructing sovereign bond spreads for countries with limited data availability

*IMF Working Paper No. WP/2025/225 — A Quantitative Approach to Central Bank Haircuts and Counterparty Risk Management, Conclusion.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025225-source-pdf.pdf_