## _wp16158

## Source details

**Canonical URL:** [_wp16158](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16158.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16158.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16158.pdf.json)

---

### I. Introduction — concentration risk and limitations of ASRF
- Concentration risk arises from two imperfect diversification types: “name” and sector concentrations (BCBS, 2006b).
- The IRB formula is based on the Asymptotic Single Risk Factor (ASRF) model (Vasicek (2002), Merton (1974)); it is portfolio-invariant and assumes idiosyncratic risk is fully diversified.
- Downside of ASRF: it ignores concentration of exposures; capital charge is the same for portfolios with different concentration risk, other things equal.
- Basel II and Basel III treat concentration risk under Pillar 2 via ICAAP/SREP, leaving large discretion to banks and supervisors for quantitative tools.
- Trade-offs in existing methods:
  - Model-based (analytical) methods: conceptually complex, strongly assumption-dependent.
  - Simulation-based (MC) methods: flexible but computer-intensive; millions of MC iterations often needed.
- Proposed alternative: the partial portfolio approach (PPA):
  - Maintains ASRF assumption for a “granular” portion of the portfolio (small exposures).
  - Applies full MC simulation (systemic and idiosyncratic shocks) only to the “non-granular” largest exposures.
  - Portfolio Credit VaR at confidence level α is the α-percentile of simulated losses minus expected loss for the whole portfolio; the difference from IRB capital is the additional capital for concentration risk.
- Empirical magnitudes reported:
  - Maximum GA for name concentration risk reaches 6.7 percent of the IRB regulatory capital.
  - Sectoral concentration GA (beyond name concentration) ranges from -0.75 percent to 0.8 percent of IRB capital.
- GA estimates comparable to Gordy and Lütkebohmert (2013) and align closely with the Herfindal-Hirschmann Index (HHI).

### II. ASRF essentials and regulatory context
- ASRF model key formulas and structure (as in source equations (1)-(9)):
  - Standardized asset return: Ri = sqrt(rho_i)·X + sqrt(1-rho_i)·epsilon_i, with X ~ N[0,1], epsilon_i ~ N[0,1].
  - Default threshold: α_i = Φ^{-1}(PD_i).
  - Conditional PD given X = x and Conditional Expected Loss (CEL) computed via ASRF equations.
  - Regulatory capital K in ASRF equals CEL at percentile q minus unconditional expected loss.
- Name concentration methods taxonomy:
  - Heuristic: HHI, Gini coefficients; insensitive to PD/LGD changes.
  - Model-based asymptotic: analytical approximations (e.g., Gordy (2004), Emmer and Tasche (2005)); Gordy and Lütkebohmert (2013) focus on m largest exposures.
  - Monte Carlo: GA = simulated Credit VaR − ASRF capital; flexible but computationally intensive.
- Sector concentration approaches:
  - Multi-factor models (Pykthin, Düllmann and Masschelein) and single-factor ad-hoc scaling/mapping techniques.
- Regulatory practice:
  - Concentration risk managed under Pillar 2; many supervisors use HHI and reference tables mapping HHI to additional capital.
  - Large exposure limits (BCBS, 2014): report exposures > 10 percent of Tier 1 capital; single counterparty exposure limit 25 percent of Tier 1 capital (15 percent for G-SIBs).

### III. Partial Portfolio Approach — methodology
- Core idea:
  - Partition portfolio into non-granular sub-portfolio (m largest exposures, m << n) and granular sub-portfolio (remaining positions).
  - Apply MC simulations to the non-granular part (systemic draws X_s and idiosyncratic shocks epsilon_{s,i}).
  - Compute granular-part CEL via ASRF conditional on X_s, avoiding individual idiosyncratic draws for small exposures.
- Implementation steps (S simulation iterations):
  - a) Draw systemic factor X_s ~ N[0,1].
  - b) For each non-granular exposure i, simulate R_{s,i} using epsilon_{s,i}.
  - c) Default test: R_{s,i} ≤ Φ^{-1}(PD_i) ⇒ loss = LGD_i · EAD_i.
  - d) Non-granular loss in iteration s: sum_i LGD_i·EAD_i·I(R_{s,i} ≤ Φ^{-1}(PD_i)).
  - e) Granular loss in iteration s: CEL via ASRF (equation (8)) aggregating groups k with same LGD, PD, and correlations.
  - f) Portfolio loss in iteration s = non-granular loss + granular CEL.
- Final capital:
  - Estimate portfolio loss distribution across S iterations; take relevant quantile (99.9 percent under Basel); subtract expected loss to obtain Credit VaR.
- Flexibility and calibration:
  - Obligor-specific LGDs can be stochastic; asset correlations rho_i can follow Basel formula, historical estimates, or market-implied; choice of m can be fixed, fractional, or threshold-based.
  - Residual pairwise correlations among idiosyncratic shocks can be modeled via a partial correlation matrix.

### IV. Testing the approach — IACPM-ISDA synthetic portfolio results
- Portfolio summary (Table 1):
  - Total size: $100 Billion
  - Number of Exposures: 6000 (2 per obligor)
  - Number of Obligors: 3000
  - Credit Rating: 8 rating buckets, average = BBB
  - Number of Industries: 61 (M-KMV classification)
  - Number of Countries: 7
  - LGD: 22% to 58%  (average = 41%)
  - Mean Size (standard deviation): $16.7 million ($101.7 million)
  - Smallest Exposure/Largest Exposure: $1 million / $1,250 million
- Methodology and simulation assumptions:
  - Shortest Maturity/Longest Maturity: 6 months / 7 years
  - Correlation (R^2) Average = 20%
  - All exposures assumed 1 year maturity for capital charge calculations.
  - Credit VaR (CVaR) for PPA estimated by 100,000 simulations with importance sampling.
  - Basel correlation formula applied in portfolio simulations to focus on name concentration.
- Partial Portfolio versus IRB (Table 2 — PPA results):
  - Partial portfolio with threshold=0%:
    - m: 3000
    - Credit VaR ($ millions): 3,347
    - Percent difference compared to IRB: 6.7
    - Computing time (sec): 145
  - Partial portfolio with threshold=0.05%:
    - m: 291
    - Credit VaR ($ millions): 3,331
    - Percent difference compared to IRB: 6.3
    - Computing time (sec): 128
  - Partial portfolio with threshold=0.5%:
    - m: 19
    - Credit VaR ($ millions): 3,322
    - Percent difference compared to IRB: 3.1
    - Computing time (sec): 127
  - IRB approach:
    - m: 0
    - Credit VaR ($ millions): 3,230
- Observations:
  - CVaR estimated using the PPA is higher than IRB regulatory capital because it accounts for positive idiosyncratic risk.
  - Percent difference relative to IRB increases with the number of obligors in the non-granular portfolio, reaching 6.7 percent when m=3000.
  - CVaR estimated by PPA converges toward IRB capital around a non-granular portfolio size threshold of one percent (at that threshold there are four obligors in the non-granular portfolio).
- Comparison with Gordy and Lütkebohmert (2013):
  - Benchmark GA uses G&L parameterization σ =0.25 and θ =0.25.
  - The GA based on PPA is slightly higher than the G&L GA; overall very similar results, with PPA retaining simulation flexibility.

### V. Sensitivity analyses and LGD assumptions
- LGD specifications tested:
  - i) Fixed LGD = 0.45 (senior unsecured IRB level).
  - ii) LGDs drawn from a beta distribution with exposure-specific parameters from IACPM-ISDA.
- Findings:
  - GA is sensitive to LGD parameterization; deterministic versus stochastic LGD assumptions materially affect CVaR.
  - Introducing randomness in LGD (beta draws) increases CVaR and hence increases GA compared to deterministic LGD=0.45.

### VI. Application to semi-hypothetical GCC bank portfolios — data and assumptions
- Sample: 31 large banks in Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, and UAE.
- Country firm counts, EDFs and average asset correlations (Table 3):
  - Bahrain: Number of firms 29; Average EDF by end 2013 (percent) 0.52; Long-term average EDF* (percent) 1.10; Average asset correlation 0.12
  - Kuwait: Number of firms 166; Average EDF by end 2013 (percent) 1.12; Long-term average EDF* (percent) 2.93; Average asset correlation 0.15
  - Oman: Number of firms 75; Average EDF by end 2013 (percent) 1.78; Long-term average EDF* (percent) ---; Average asset correlation 0.17
  - Qatar: Number of firms 42; Average EDF by end 2013 (percent) 0.10; Long-term average EDF* (percent) 0.54; Average asset correlation 0.27
  - Saudi Arabia: Number of firms 155; Average EDF by end 2013 (percent) 0.16; Long-term average EDF* (percent) 0.46; Average asset correlation 0.30
  - UAE: Number of firms 87; Average EDF by end 2013 (percent) 0.61; Long-term average EDF* (percent) 1.98; Average asset correlation 0.26
  - Note on long-term EDF*: 2006-2014 for Saudi Arabia, 2009-2014 for other countries (except Oman, insufficient data so EDFs scaled up by factor of 3.25).
- Assumed share of listed companies’ debt financed by domestic banks (percent):
  - Bahrain 75; Kuwait 90; Oman 50; Qatar 75; Saudi Arabia 70; UAE 50
- LGD set at 45 percent for semi-hypothetical portfolios.
- Non-granular part: obligors identified by EDFs and asset correlations from Moody’s KMV; remaining portfolio treated as granular.
- Simulations: PPA with 100,000 Monte Carlo iterations and importance sampling.

### VII. Semi-hypothetical portfolio results — CVaR variants and GA measures
- Two Credit VaR variants estimated:
  - CVaR-IRB: asset correlations per regulatory IRB formula (isolates name concentration effect).
  - CVaR-MKMV: asset correlations from Moody’s KMV (captures sector concentration when compared to CVaR-IRB).
- Additional variant for subset (Saudi Arabia and UAE): CVaR-MKMV-CD where non-granular obligor shocks are drawn from a multivariate normal with pairwise correlations equal to the partial correlation matrix (filters out the common systematic factor).
- Interpretation caveats:
  - Sector concentration assessment ideally requires multi-factor models; single-factor estimates here may over- or under-estimate CVaR.
  - Using IRB asset correlations for the granular portion without SME size adjustments likely over-estimates CVaR.
- Key findings:
  - CVaR-IRB consistently lies on or above the IRB line.
  - IRB underestimation of credit risk (without concentration adjustments) can be sizable: almost 12 percent on average for the 31 banks, with a peak of over 50 percent in one case (IRB capital less than half CVaR-IRB).
  - CVaR-MKMV variants mostly larger than IRB with exceptions.
- Granularity Adjustments (GAs), expressed as percent of total portfolio exposure (semi-hypothetical portfolios):
  - Range: 0.02 to 4.12 percent
  - Average: 0.62
  - Median: 0.14
  - (Standard deviation: value not provided in supplied content)

### VIII. Comparison PPA GA versus Gordy & Lütkebohmert and relationship with HHI
- PPA GA vs G&L:
  - Generally closely aligned; PPA GA most often slightly below G&L GA.
  - Largest observed difference: 4.12 versus 2.26 percent for the most concentrated portfolio (HHI: 3.35 percent).
  - PPA can be used to calibrate G&L parameters, but calibration likely country-specific.
- PPA GA and HHI:
  - HHI computed on exposure shares, expected losses, and IRB capital shares (practically summing squared shares for non-granular portion relative to overall portfolio).
  - Linear regression of PPA GA on HHI (exposure shares) yields:
    - R^2 = 0.96
    - GA ≈ 1.3 × HHI (coefficient ≈ 1.3)
    - Intercept not statistically significant at a 95 percent confidence level.
  - Using HHIs based on expected losses or IRB capital does not improve fit.
  - Policy implication: strong linear relationship supports mapping HHI to GA in supervisory benchmark models, subject to calibration.

### IX. Sectoral adjustment (PPA-SA), correlated draws (PPA-SA-CD), and limitations
- PPA-SA defined as CVaR-MKMV − CVaR-IRB, expressed as percent of total exposure.
- Sectoral adjustments can be positive or negative; in semi-hypothetical portfolios they are nearly evenly split with average and median close to zero.
- Relationship between PPA-SA and weighted average difference between market-based and IRB-based asset correlations:
  - Positive relationship with moderate fit: R^2 = 0.49.
  - Conclusion: simple linear rules for sectoral concentration are harder to derive than for name concentration.
- PPA-SA-CD (correlated draws for non-granular obligors) on 11 portfolios:
  - Generally does not dramatically change sectoral adjustment size.
  - Exceptions where changes are significant, including sign reversals in some portfolios (e.g., portfolios 8 and 11).
  - Largest changes occur for portfolios with the largest share of non-granular exposures.
- Caveat: modeling more realistic dependence across the whole portfolio could yield larger effects; outside this paper’s scope.

### X. Conclusions, practical implications, and further work
- Aim: adjust capital requirements to reflect credit concentration risk since Pillar 1 (standardized and IRB) does not capture concentration risk.
- Pillar 2 role: concentration risk typically addressed via ICAAP and supervisory challenge; benchmarking tools are used but may be difficult for some authorities.
- Trade-offs:
  - Model-based approaches: good for uniform rules, but complex to extend and calibrate.
  - Simulation-based approaches: flexible, useful for calibration, but computationally intensive.
- Partial portfolio approach advantages:
  - Combines benefits of model-based and simulation-based methods.
  - Limits computational burden and data needs.
  - For name-concentration adjustments, minimum data: exposures in non-granular portion, corresponding PDs, and average PDs in granular clusters.
  - Dependence-structure data (asset value correlations among firms and sectors) improve calibration for name concentration and are indispensable for sector concentration assessment.
- Uses:
  - Static cross-sectional analysis, tracking concentration risk over time, bilateral surveillance, solvency stress testing, technical assistance to supervisors, and policy tool for gauging concentration risk across banking systems.
- Further work:
  - Apply PPA to real-world bank portfolios with bank-specific granular data (suggested for supervisors with access to such data).

*Source — IMF working paper chapter "4. Bank Funding of Domestic Companies in Semi-Hypothetical Portfolios" (source PDF content provided).*

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

### _wp16158 - References .............................................................................................................

### Figures
- 1. Partial Portfolio Approach: Credit VaR for Varying m ...................................................... 16
- 2. Granularity Adjustment of Partial Portfolio and G&L (2013) Methods............................. 17
- 3. Partial Portfolio Approach: Credit VaR for Varying m and Different LGD Assumptions  18
- 4. Share of Non-Granular Part of Loan Portfolios .................................................................. 20
- 5. Partial Portfolio Approach for Semi-Hypothetical Portfolios ............................................ 23
- 6. Granularity Adjustment: Partial Portfolio Approach and Gordy and Lütkebohmert ......... 24
- 7. Granularity Adjustment: Partial Portfolio Approach and HHI ........................................... 25
- 8. Distribution of Sectoral Adjustments Across the Semi-Hypothetical Portfolios................ 26
- 9. Sectoral Adjustment and Weighted Average Difference between MKMV and IRB-Based 
  Asset Correlation ............................................................................................................ 27
- 10. Sectoral Adjustments with and without Correlated Draws in the Simulation .................. 27

### Tables
- 1. Characteristics of the IACPM-ISDA Portfolio ................................................................... 14
- 2. Regulatory Capital: Partial Portfolio Method versus IRB Model ....................................... 15
- 3. Characteristics of Semi-Hypothetical Portfolios ................................................................ 19

*Document: _wp16158 - References.*

### 4. Bank Funding of Domestic Companies in Semi-Hypothetical Portfolios .......................... 19

### 4. Bank Funding of Domestic Companies in Semi-Hypothetical Portfolios .......................... 19

### I. INTRODUCTION — concentration risk and limitations of ASRF
- Concentration risk arises from two imperfect diversification types: “name” and sector concentrations (BCBS, 2006b).
- The IRB formula is based on the Asymptotic Single Risk Factor (ASRF) model (Vasicek (2002), Merton (1974)); it is portfolio-invariant and assumes idiosyncratic risk is fully diversified.
- Downside of ASRF: it ignores concentration of exposures; capital charge is the same for portfolios with different concentration risk, other things equal.
- Basel II and Basel III treat concentration risk under Pillar 2 via ICAAP/SREP, leaving large discretion to banks and supervisors for quantitative tools.
- Trade-offs in existing methods:
  - Model-based (analytical) methods: conceptually complex, strongly assumption-dependent.
  - Simulation-based (MC) methods: flexible but computer-intensive; millions of MC iterations often needed.
- Proposed alternative: a “partial portfolio” approach that:
  - Maintains ASRF assumption for a “granular” portion of the portfolio (small exposures).
  - Applies full MC simulation (systemic and idiosyncratic shocks) only to the “non-granular” largest exposures.
  - Portfolio Credit VaR at confidence level α is the α-percentile of simulated losses minus expected loss for the whole portfolio; the difference from IRB capital is the additional capital for concentration risk.
- Empirical magnitudes reported:
  - Maximum GA for name concentration risk reaches 6.7 percent of the IRB regulatory capital.
  - Sectoral concentration GA (beyond name concentration) ranges from -0.75 percent to 0.8 percent of IRB capital.
- Findings regarding parameter assumptions:
  - GA size depends on parameter assumptions, particularly whether parameters are stochastic or deterministic.
  - GA estimates comparable to Gordy and Lütkebohmert (2013) and align closely with the Herfindal-Hirschmann Index (HHI).

### II. CONCENTRATION RISK IN THE BASEL CAPITAL FRAMEWORK
- ASRF model essentials:
  - For obligor i, standardized asset return Ri = sqrt(rho_i)·X + sqrt(1-rho_i)·epsilon_i, with X ~ N[0,1], epsilon_i ~ N[0,1], covariances as specified (equations (1)-(9)).
  - Default threshold α_i = Φ^{-1}(PD_i).
  - Conditional PD given X = x expressed by equation (7).
  - Conditional expected loss (CEL) formula given by equation (8).
  - Regulatory capital K in ASRF equals total CEL at percentile q minus unconditional expected loss (equation (9)).
- Name concentration methods classified:
  - Heuristic: HHI, Gini coefficients; simple linear mappings to GA; insensitive to PD/LGD changes.
  - Model-based asymptotic: analytical approximations (e.g., Gordy (2004), Emmer and Tasche (2005)); precise but complex; Gordy and Lütkebohmert (2013) simplify by focusing on m largest exposures.
  - Monte Carlo: flexible, GA = simulated Credit VaR − ASRF capital; computationally intensive.
- Sector concentration methods classified:
  - Multi-factor models: typically no closed-form solutions; simplifications (Pykhtin (2004), Düllmann and Masschelein (2006)) introduce tractability at cost of strong assumptions.
  - Single-factor adjustments: ad-hoc scaling, mapping techniques (BET by Moody’s, Garcia Cespedes et al. (2005)).
- Regulatory treatment under Basel II/III:
  - Concentration risk addressed under Pillar 2 (ICAAP/SREP); methods and measurement left to national supervisors.
  - Supervisory practices cited:
    - Many regulators use HHI to evaluate concentration and adjust required capital (EBA, 2014).
    - Spanish Central Bank and PRA provide reference tables mapping HHI intervals to additional capital.
    - Sweden (Finansinspektionen, 2015) proposes continuous formulas based on HHI and Gordy & Lütkebohmert (2013) for IRB banks.
    - Other authorities require stress tests, parameter evaluation, or additional capital for significant exposures.
  - Large exposure limits (BCBS, 2014):
    - Report all exposures exceeding 10 percent of Tier 1 capital.
    - Single counterparty exposure cannot exceed 25 percent of Tier 1 capital (15 percent for G-SIBs).

### III. A PARTIAL PORTFOLIO APPROACH TO CONCENTRATION RISK — methodology
- Core idea:
  - Partition portfolio into non-granular sub-portfolio (m largest exposures, m << n) and granular sub-portfolio (remaining positions).
  - Apply MC simulations to the non-granular part using draws of the systemic factor and idiosyncratic shocks for each of the m exposures.
  - Compute CEL via ASRF formula (equation (8)) for the granular part conditional on the systemic draw, avoiding full simulation of idiosyncratic shocks for small exposures.
- Implementation steps (for S simulation iterations):
  - a) Draw systemic factor X_s ~ N[0,1].
  - b) For each exposure in the non-granular part, simulate standardized asset return R_{s,i} by generating idiosyncratic shock epsilon_{s,i}.
  - c) Apply default threshold test using equation (6): default if R_{s,i} ≤ Φ^{-1}(PD_i); loss if default = LGD_i · EAD_i.
  - d) Total loss for non-granular portfolio in iteration s: sum over i of LGD_i·EAD_i·I(R_{s,i} ≤ Φ^{-1}(PD_i)).
  - e) Loss for granular portfolio in iteration s: compute CEL via equation (8) aggregating groups k with same LGD, PD, and correlations.
  - f) Total portfolio loss in iteration s = non-granular loss + granular CEL.
- Final capital calculation:
  - Compute the portfolio loss distribution across S iterations, take the relevant quantile (99.9 percent under Basel), subtract expected loss to obtain Credit VaR.
- Flexibility and calibration options:
  - Obligor-specific LGDs in non-granular part can be stochastic draws; asset correlations rho_i can be Basel formulas, historical estimates, or market-implied.
  - Choice of m can be fixed number, fraction of portfolio, or based on size threshold.
  - Residual pairwise correlations among idiosyncratic shocks can be introduced via a partial correlation matrix.

### IV. TESTING THE APPROACH — overview and synthetic portfolio application
- Tests performed:
  - Application to two sets of portfolios: a synthetic IACPM-ISDA portfolio and a set of semi-hypothetical portfolios; comparisons made with IRB regulatory capital and Gordy & Lütkebohmert (2013) GA.
- Application to the IACPM-ISDA 2006 synthetic portfolio — Data (Table 1 summary):
  - Portfolio Summary:
    - $100 Billion
    - Number of Exposures: 6000 (2 per obligor)
    - Number of Obligors: 3000
    - Credit Rating: 8 rating buckets, average = BBB
    - Number of Industries: 61 (M-KMV classification)
    - Number of Countries: 7
  - Exposures Characteristics:
    - LGD: 22% to 58%  (average = 41%)
    - Mean Size (standard deviation): $16.7 million ($101.7 million)
    - Smallest Exposure/Largest Exposure: $1 million / $1,250 million

*Italic: Source — IMF working paper chapter "4. Bank Funding of Domestic Companies in Semi-Hypothetical Portfolios" (source PDF content provided).*

### 2.5 years

### _wp16158 - 2.5 years

### Methodology and Assumptions
- Shortest Maturity/Longest Maturity: 6 months / 7 years
- Correlation (R^2) Average = 20%
- Capital charges derived using both the PPA (partial portfolio approach) and the IRB formula, using LGDs and (annualized) PDs from the IACPM-ISDA dataset.
- The exposure-specific correlation with the systemic risk factor (ݓ
௜
) is present in the dataset but the Basel correlation formula is applied in portfolio simulations to focus on name concentration.
- All exposures are assumed to have a maturity of one year for capital charge calculations.
- Credit VaR (CVaR) for the partial portfolio approach is estimated by 100,000 simulations of portfolio losses with importance sampling.

### Partial Portfolio Versus IRB: Main Results (Table 2)
- Partial portfolio with threshold=0%:
  - Number of obligors in the non-granular portfolio (m): 3000
  - Credit VaR ($ millions): 3,347
  - Percent difference compared to IRB: 6.7
  - Computing time (sec): 145
- Partial portfolio with threshold=0.05%:
  - m: 291
  - Credit VaR ($ millions): 3,331
  - Percent difference compared to IRB: 6.3
  - Computing time (sec): 128
- Partial portfolio with threshold=0.5%:
  - m: 19
  - Credit VaR ($ millions): 3,322
  - Percent difference compared to IRB: 3.1
  - Computing time (sec): 127
- IRB approach:
  - m: 0
  - Credit VaR ($ millions): 3,230

- Observations:
  - CVaR estimated using the PPA is higher than IRB regulatory capital because it accounts for positive idiosyncratic risk.
  - The percent difference relative to IRB increases with the number of obligors in the non-granular portfolio, reaching 6.7 percent when m=3000.
  - CVaR estimated by PPA converges toward IRB capital around a non-granular portfolio size threshold of one percent (at that threshold there are four obligors in the non-granular portfolio).

### Comparison with Gordy and Lütkebohmert (2013) Granularity Adjustment (GA)
- Benchmarking approach: apply the simplified analytical expression for the GA (Gordy and Lütkebohmert, 2013) to the non-granular portfolio, using parameterization ߛ =0.25 and ߦ =0.25.
- Result: The GA based on PPA is slightly higher than the GA proposed by Gordy and Lütkebohmert (2013).
- Conclusion: The two approaches produce very similar results; PPA retains simulation-based flexibility lacking in the Gordy and Lütkebohmert (2013) method.

### Sensitivity Analysis (LGD assumptions)
- Alternative LGD assumptions tested:
  - i) Fixed LGD level of 0.45 (corresponding to LGD for senior, unsecured claims under IRB).
  - ii) Draws of LGDs from a beta distribution with exposure-specific parameters from IACPM-ISDA.
- Findings:
  - GA is sensitive to LGD parameterization; deterministic versus stochastic LGD assumptions materially affect CVaR.
  - Introducing randomness in LGD (beta draws) increases CVaR and hence increases GA compared to deterministic LGD=0.45.

### Application to Semi-Hypothetical Portfolios (GCC banks)
- Data:
  - Group of 31 large banks in Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, and UAE.
  - Number of firms, Average EDF by end 2013, Long-term average EDF*, Average asset correlation (Table 3):
    - Bahrain: Number of firms 29; Average EDF by end 2013 (percent) 0.52; Long-term average EDF* (percent) 1.10; Average asset correlation 0.12
    - Kuwait: Number of firms 166; Average EDF by end 2013 (percent) 1.12; Long-term average EDF* (percent) 2.93; Average asset correlation 0.15
    - Oman: Number of firms 75; Average EDF by end 2013 (percent) 1.78; Long-term average EDF* (percent) ---; Average asset correlation 0.17
    - Qatar: Number of firms 42; Average EDF by end 2013 (percent) 0.10; Long-term average EDF* (percent) 0.54; Average asset correlation 0.27
    - Saudi Arabia: Number of firms 155; Average EDF by end 2013 (percent) 0.16; Long-term average EDF* (percent) 0.46; Average asset correlation 0.30
    - UAE: Number of firms 87; Average EDF by end 2013 (percent) 0.61; Long-term average EDF* (percent) 1.98; Average asset correlation 0.26
  - Notes:
    - (*): 2006-2014 for Saudi Arabia, 2009-2014 for the other countries (except Oman, insufficient data so EDFs of Omani firms scaled up by factor of 3.25).
  - Assumed share of listed companies’ debt financed by domestic banks (percent) (Table 4):
    - Bahrain 75
    - Kuwait 90
    - Oman 50
    - Qatar 75
    - Saudi Arabia 70
    - UAE 50
  - LGD set at 45 percent for semi-hypothetical portfolios.
  - All exposures assumed 1 year maturity.
  - Non-granular part: obligors individually identified by EDFs and asset correlations with the systematic factor estimated from Moody’s KMV.
  - Remaining portfolio treated as granular.

### Estimation for Semi-Hypothetical Portfolios
- PPA results based on Monte Carlo simulations with 100,000 iterations and importance sampling.
- Two kinds of Credit VaR estimated:
  - CVaR-IRB: asset correlations per regulatory IRB formula (isolates effect of name concentration).
  - CVaR-MKMV: asset correlations estimated from Moody’s KMV database (captures sector concentration when compared to CVaR-IRB).
- Additional measure (for subset of banks): CVaR-MKMV-CD where shocks to firms’ asset values in the non-granular portion are drawn from a multi-normal distribution with pairwise correlations equal to the partial correlation matrix among firms’ asset values (filtering out the common systematic factor). This was applied for semi-hypothetical portfolios of Saudi Arabia and UAE banks where asset correlations could be estimated.

### Interpretation Caveats on Sector Concentration
- Sector concentration measures require multi-factor models for proper assessment; the single-factor model used here may over- or under-estimate CVaR depending on intra- and inter-sector correlation structure.
- The IRB asset correlations used for the granular portion, without SME size adjustments, likely lead to over-estimation of CVaR.

### Results for Semi-Hypothetical Portfolios
- Comparison of IRB capital and partial portfolio CVaR:
  - CVaR-IRB consistently lies on or above the IRB line.
  - Underestimation of credit risk by IRB (without concentration adjustments) can be sizable: almost 12 percent on average for the 31 banks, with a peak of over 50 percent in one case (IRB capital less than half CVaR-IRB).
  - CVaR-MKMV variants are mostly larger than IRB with some exceptions.
- Granularity Adjustments (GAs), expressed as percent of total portfolio exposure:
  - Range: 0.02 to 4.12 percent
  - Average: 0.62
  - Median: 0.14
  - Standard deviation: (value not provided in supplied content; reported as "standard deviation of" with no number)

*Source: _wp16158 - 2.5 years (PDF chapter/section)*

### 0.99 percent. The range looks very close to that of 0.02 to 3.81 percent reported by Gordy

### _wp16158 - 0.99 percent. The range looks very close to that of 0.02 to 3.81 percent reported by Gordy

### Comparison of Granularity Adjustments: Partial Portfolio Approach (PPA) vs Gordy & Lütkebohmert (G&L)
- The paper compares the PPA GA (based on the CVaR-IRB capital measure) with the G&L GA calculated on the semi-hypothetical portfolios using the same parameterization as in section IV.A.
- Overall alignment:
  - "The two measures are generally closely aligned, with the PPA GA lying most of the times slightly below G&L GA."
  - A more direct comparison is visualized by plotting G&L GA on a 45 degree line against the PPA GA.
- Largest observed difference:
  - 4.12 versus 2.26 percent (marked with the full red dot in Figure 6) — obtained for the portfolio with the highest concentration among those considered, as measured by HHI: 3.35 percent.
- Calibration note:
  - The PPA approach could be used to calibrate parameters in the G&L GA approach, but calibration would need to be performed separately for each country (variance of the systematic factor likely differs across countries). This was not feasible given the limited sample size.

### Relationship between PPA GA and Herfindahl-Hirschman Index (HHI)
- HHI calculation approach:
  - HHI is calculated with respect to shares of exposures, shares of expected losses, and IRB capital charges, always assuming infinitely granular exposures in the granular portion of the portfolio.
  - Practically, this translates into calculating and summing squared shares only for the non-granular portion, with shares measured relative to the overall portfolio.
- Empirical fit:
  - A linear regression of the PPA GA on the HHI based on exposure shares yields a very good fit:
    - R^2 of 0.96.
    - The GA is linear in HHI with a coefficient of approximately 1.3.
  - Statistical note: "The intercept is not statistically significant at a 95 percent confidence level."
  - Using HHIs calculated on shares of expected losses or IRB capital charges does not lead to improvements in fit.
- Policy implication:
  - The strong linear relationship supports the practice adopted by several supervisory authorities of creating benchmark models that map a bank’s HHI to a granularity adjustment, subject to proper calibration.

### Sectoral Adjustment (PPA-SA) and Its Drivers
- Definition:
  - Sectoral adjustment (PPA-SA) is defined as the difference between CVaR-MKMV (calculated with firm-by-firm market-based asset correlations) and CVaR-IRB, expressed as a percentage of total exposure.
- Theoretical context:
  - Unlike name concentration (where the infinitely granular portfolio is a natural floor), sector concentration can move Credit VaR either above or below the IRB capital charge; hence sectoral adjustments can be positive or negative.
- Empirical distribution:
  - In the semi-hypothetical portfolios analyzed, sectoral adjustments are almost equally split between positive and negative values, with average and median close to zero.
- Relationship to asset-correlation differences:
  - The paper examines the relationship between PPA-SA and the weighted average of differences between market-based and IRB-based asset correlations (weighted by sector exposures).
  - Empirical fit:
    - There is a clear positive relationship, but the linear fit is moderate:
      - R^2 is 0.49.
    - Conclusion: "Finding a simple, linear relationship for sectoral concentration, amenable to a simple rule with straight calculations also for small and unsophisticated banks, is far less easy than for name concentration."

### Sectoral Adjustment with Correlated Draws (PPA-SA-CD)
- Additional experiment:
  - For a limited sample of 11 bank portfolios, sectoral adjustment is also computed as the difference between CVaR-MKMV-CD (market-based correlations and correlated draws for non-granular obligors) and CVaR-IRB, expressed as a percentage of total exposure and labeled "PPA-SA-CD."
- Results summary:
  - Most of the time, introducing correlated draws for the non-granular part does not dramatically change the size of the sectoral adjustment.
  - Exceptions:
    - In some portfolios (e.g., portfolios 5 and 8) changes appear significant; in portfolios 8 and 11 there is a sign reversal.
  - Largest changes occur for portfolios with the largest share of non-granular exposures because the sectoral adjustment is obtained by applying market-based correlations only to these exposures while keeping the granular part’s correlation structure fixed.
- Caveat:
  - Introducing a more realistic dependence structure for the whole portfolio could lead to more significant changes; this extension is beyond the scope of the working paper.

### Conclusions and Practical Implications
- Problem addressed:
  - How to adjust banks’ capital requirements to reflect credit concentration risk, given that Pillar 1 approaches (standardized and IRB) do not capture concentration risk.
- Role of Pillar 2:
  - Concentration risk is expected to be addressed under Pillar 2 via banks’ ICAAPs and supervisory challenge; supervisors sometimes use benchmarking tools, but they may be difficult for many authorities to implement.
- Methodological trade-offs:
  - Model-based approaches: good for comprehensive rules and applicability across banks, but hard to incorporate advanced features and require calibration.
  - Simulation-based approaches: flexible and useful for calibrating models, but computationally intensive.
- Partial portfolio approach (proposed method):
  - Splits portfolios into "granular" and "non-granular" portions.
  - Treats granular portion with the Asymptotic Single Risk Factor (ASRF) framework and non-granular portion with Monte Carlo simulations.
  - Measures Credit VaR minus IRB capital charge and decomposes the difference into name concentration and sector concentration within a single-factor framework.
- Advantages:
  - Combines benefits of model-based and simulation-based methods.
  - Limits computational burden and data requirements.
  - For name-concentration adjustments, minimum data needs: exposures in the non-granular portion, corresponding probabilities of default, and average probabilities of default in clusters of the granular portion.
  - Dependence-structure data (asset value correlations among firms and sectors) improve calibration for name concentration and are indispensable for sector concentration.
- Uses:
  - Suitable for static cross-sectional analysis and for tracking evolution of concentration risk over time.
  - Useful in bilateral surveillance, solvency stress testing, technical assistance to supervisors, and as a policy tool to gauge concentration risk across banking systems.
- Areas for further work:
  - Application of the partial portfolio approach to real-world bank portfolios (e.g., by individual regulators with access to more granular, bank-specific data).

*Source: IMF working paper excerpt (figures and appendix referenced).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16158.pdf_