## _wp07239

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### I. Introduction and objective
- Context:
  - New Basel Capital Accord (Basel II) implementation guidelines issued end of 2006; AMA requires one-year holding period at a 99.9 percent statistical confidence level.
  - Operational risk defined as risk of loss from internal processes, people, systems, or external events.
- Main objective:
  - Optimize generalized parametric distributions (GEV and GPD) under Extreme Value Theory (EVT) and assess the g-and-h distribution for measurement of extreme operational losses, addressing parameter instability and threshold selection to produce stable and robust capital estimates.

### II. Methods and model specifications
- EVT families and alternatives:
  - GEV (Generalized Extreme Value) for normalized maxima; parameters μ, σ > 0, ξ (shape); tail index α = 1/ξ.
  - GPD (Generalized Pareto Distribution) via POT (Peak-over-Threshold); estimated quantile function x̂_{t,p,n,k} = t + (σ̂_t/ξ̂) [ (p)^{-ξ̂} − 1 ].
  - g-and-h distribution: F_{g,h}(z) = μ + σ [ (exp(gz) − 1)/g ] exp( (h z^2)/2 ).
- Estimation algorithms and diagnostics:
  - Estimators: Moment (Dekkers, Einmahl and de Haan, 1989), Maximum Likelihood (MLE), Pickands (1975), Drees-Pickands (Drees, 1995), Hill (1975), Phillips-Loretan (PL) adjustment for ξ.
  - Threshold diagnostics: mean excess function (MEF), stability of ξ across thresholds, goodness-of-fit of upper-tail quantiles, residuals R_i = log[1 + ξ̂ (X_i − t)/β̂] should be i.i.d. unit exponential.
  - Introduced “threshold-quantile surface”; simplifications: single point estimate graphs and single threshold upper tail graphs.

### III. Data, simulation design, and scaling
- LDCE base and augmentation:
  - LDCE coverage: 55,766 operational loss events > US$10,000 over 2000–2004; aggregate total loss exposure US$25.9 billion.
  - LDCE sample: 23 participating banks; 10 firms submitted comprehensive internal loss data.
  - Total simulated observations: 9,996 (scaling factor applied to reported loss frequency: 2.91× to reach about 10,000 observations).
- Simulation mechanics:
  - Five-year horizon; daily events randomly disseminated; left endpoint rescaled to zero and first two moments matched to LDCE statistics.
  - Severity families: EV0, EV1 (Fréchet sub-model chosen), EV2; calibrated ξ ∈ {−1.0; 0; 0.85; 1.0; 1.15} (as specified).
  - Fundamental-value adjustment and bank-specific scaling using quarterly total assets (TA), gross income (GI), Tier 1 capital (T1); main analysis scaled by GI.
- Bank sample for scaling: Bank 1—Bank of America (BoA); Bank 2—J.P. Morgan Chase (JPMC); Bank 3—Wachovia; Bank 4—Washington Mutual (WaMu).

### IV. Key descriptive statistics (simulated losses)
- LDCE and simulation summary:
  - LDCE aggregate: 55,766 events; total loss US$25.92 billion; average loss (EL) per bank US$465 (’000).
  - Total simulated observations: 9,996.
- Selected simulated descriptive statistics (five-year horizon, Table 4, US$ ’000):
  - EV0 (ξ=0): mean 670.2; median 437.4; maximum 5,957.1; std. dev. 685.4; coeff. of variance 1.0; 99.900 percentile 4,337.8; 99.999 percentile 5,896.4.
  - EV1 (ξ=0.85): median 39.5; maximum 524,400.0; std. dev. 8,390.3; coeff. of variance 12.5; 99.900 percentile 83,774.7; 99.999 percentile 512,433.9.
  - EV1 (ξ=1.0): median 61.0; maximum 575,708.3; std. dev. 8,740.2; coeff. of variance 13.0; 99.900 percentile 78,948.6; 99.999 percentile 562,326.7.
  - EV1 (ξ=1.15): median 126.5; maximum 415,569.5; std. dev. 7,078.8; coeff. of variance 10.6; 99.900 percentile 75,444.5; 99.999 percentile 401,801.6.
  - EV2 (ξ=−1.0): median 471.8; maximum 1,932.6; std. dev. 591.8; coeff. of variance 0.9; 99.900 percentile 1,931.9; 99.999 percentile 1,932.6.
- One-year horizon (Table 5, US$ ’000):
  - EV0 (ξ=0): mean 447.8; median 413.7; std. dev. 211.0; 99.900 percentile 1,360.9; 99.999 percentile 1,418.7.
  - EV1 (ξ=0.85): median 47.7; maximum 83,730.9; std. dev. 2,682.8; 99.900 percentile 41,828.0; 99.999 percentile 82,066.9.
  - EV1 (ξ=1.0): median 59.1; maximum 183,839.0; std. dev. 8,540.5; 99.900 percentile 57,736.5; 99.999 percentile 183,501.8.
  - EV1 (ξ=1.15): median 137.3; maximum 97,869.2; std. dev. 2,616.1; 99.900 percentile 25,760.3; 99.999 percentile 96,654.9.
  - EV2 (ξ=−1.0): median 467.7; maximum 510.9; std. dev. 64.1; 99.900 percentile 680.8; 99.999 percentile 680.9.
- Rule of thumb: annualization factor for T=5 reduces five-year percentile values by factor 1/T×, i.e., 44.7 percent for T=5 (annual values ~55 percent of five-year values).

### V. Estimation results, tail behavior and model comparisons
- GEV/GPD/g-and-h comparisons:
  - GEV tends to overstate actual losses at high quantiles.
  - GPD POT method accurately represents residual upper-tail risk when threshold diagnostics satisfied.
  - g-and-h offers more consistent upper-tail fit for very extreme quantiles (≥ 99.99 percent) and is convenient for Monte Carlo simulation.
- Representative point estimates (Bank 1, five-year horizon, Table 8, percent of GI):
  - 99.700: empirical 0.17; normal 21.89; GEV exact (LRS) 0.33; Moment GP 0.17; MLE GP 0.28; g-and-h 0.17.
  - 99.900: empirical 0.50; normal 24.55; GEV exact (LRS) 1.17; Moment GP 0.47; MLE GP 0.87; g-and-h 0.53.
  - 99.999: empirical 4.03; normal 33.68; GEV exact (LRS) 227.75; Moment GP 35.94; MLE GP 112.53; g-and-h 101.39.
- Relative deviations (Bank 1, Table 9, percent deviation from empirical):
  - 99.900: normal distr. deviation 4,764.9; GEV exact (LRS) 131.0; Moment GP −7.79; MLE GP 69.30; g-and-h 3.20.
  - 99.999: GEV exact (LRS) 5,556.6; Moment GP 792.7; MLE GP 2,694.9; g-and-h 805.2.
- Estimated parameters (Bank 1, Table 11, scaled by GI):
  - GEV (LRS): μ 0.000003; σ 0.000005; ξ 1.145300.
  - GPD Moment: μ 0.000118; β 0.000006; ξ 0.949170.
  - GPD MLE: μ 0.000141; β 0.000004; ξ 1.044900.
  - Pickands: μ 0.000268; β 0.000002; ξ 1.156000.
  - Drees-Pickands: μ 0.000265; β 0.000002; ξ 1.150300.
  - g-and-h: μ 0.000005; σ 0.000010; g 1.985492; h −0.031397.
- Cross-bank parameter notes:
  - Shape ξ estimates generally cluster around 0.9–1.3 across GEV/GPD methods and banks.
  - g-and-h implied higher moments (skewness, kurtosis) extremely large (e.g., skewness 1.536E+06; kurtosis 1.473E+12 for Bank 1).

### VI. Threshold diagnostics, sensitivity, and optimization
- Threshold-quantile surface properties:
  - Constructed on quantile range 98.0th–99.9th at 0.05 percent increments; x-axis: number of exceedances (threshold); z-axis: percentile level.
  - Higher percentile levels → higher volatility of point estimates across threshold values.
  - In- vs out-of-sample point estimates similar up to 99.0th percentile; beyond 99.7th percentile estimation uncertainty rises substantially.
  - For reliable point estimates at confidence ≤ 99.0 percent, at least 1.5 percent of events should be classified as extremes; estimation uncertainty increases significantly when confidence > 99.7 percent or exceedances < 0.5 percent.
- Local optimization (p = 99.9 percent) findings:
  - Optimal local threshold quantiles *lq range from 97.52 percent to 98.78 percent depending on bank and method.
  - Unconditional local threshold (*,luq) around 98.0th percentile; conditional (*,lcq) often corresponds to q = 99.0 with optimal local percentile * ,lcp.
- Global optimization findings:
  - Optimal global threshold *,gcq and percentile *,gcp reported (Tables):
    - Bank 1: *,gcq 98.00 [Moment; 2.95]; *,gcp 98.30 [Moment; 0.29].
    - Bank 2: *,gcq 98.10 [Pickands; 0.15]; *,gcp 98.75 [Pickands; 0.17].
    - Bank 3: *,gcq 98.30 [Moment; 0.00]; *,gcp 98.95 [Moment; 0.00].
    - Bank 4: *,gcq 98.75 [Moment; 0.22]; *,gcp 98.35 [Moment; 0.00].
- Single-point and single-threshold simplifications:
  - Moment and MLE (with PL) most consistent in threshold quantile “comfort zone” between 98.5th and 99.0th percentile.
  - Upper tail convexity declines at higher thresholds; a threshold ~99.0th percentile offers reliable results for many methods.
- Sensitivity summaries:
  - Sensitivity of upper-tail fit to number of exceedances marginal between 98.0th and 99.7th percentiles; higher thresholds increase out-of-sample parameter uncertainty.
  - Point estimates more sensitive to desired confidence level than to threshold choice within a high-threshold range; selecting the percentile level is often more crucial than fine-tuning threshold if threshold is already high.

### VII. Robustness, reliability, and methodological caveats
- Data and modeling caveats:
  - Analysis assumes i.i.d. loss events; excludes joint dependence across business lines or event types and multivariate EVT extensions.
  - LDCE left-truncation and minimum reporting thresholds may bias EL upward and affect tail behavior.
  - Aggregation of i.i.d. extremes is not equivalent to multivariate joint extremes; multivariate EVT inconclusive without ordering.
- Estimation caveats:
  - Scarcity of historical loss data undermines back-testing at extreme quantiles (e.g., 99.9th and above) and produces parameter risk.
  - EVT-based VaR requires treatment of second-order effects and slow convergence; unchecked application yields unstable out-of-sample estimates.
  - g-and-h avoids discrete threshold choice but depends on percentile selection and spacing—introduces its own estimation risk.

### VIII. Regulatory assessment and policy-relevant findings
- Comparative outcomes vs standardized approaches:
  - Simulated cumulative operational losses rarely exceed 2 percent of quarterly gross income; standard BIA fixed charge 15 percent of average gross income appears indiscriminate relative to simulated exposure.
  - Parametric AMA internal models can imply capital savings up to almost 97 percent relative to standardized volume-based charges.
  - To match a uniform 15 percent GI charge, AMA would need statistical confidence levels at least 99.99 percent, which materially increases estimation risk.
- Practical recommendations:
  - Use enterprise-level “full-data” aggregation to maximize loss data for parametric estimation, acknowledging trade-offs.
  - Prefer Moment and MLE algorithms for GPD tail estimation under varying tail shapes; consider PL-adjusted shape for improved 99.9th percentile estimates.
  - Consider lowering regulatory confidence target from 99.9 percent toward 99.7 percent (e.g., reduction by 0.2 percentage points) to materially reduce UL and threshold/estimation uncertainty and required sample sizes.
  - Prioritize selection of percentile level and comprehensive threshold-quantile surface diagnostics over solely raising threshold quantiles.
  - Employ threshold-quantile surface optimization (local/global) and sensitivity analysis before fixing thresholds and percentile levels.
- Regulatory interpretation:
  - Large disagreement between AMA and standardized charges indicates prima facie inconsistency in operational risk measurement under current framework if loss data are empirically valid.
  - Scarcity of data suggests combining internal measurement models with capital buffers and qualitative overlays; qualitative overlay recommended to address one-off events and reporting heterogeneity.

### IX. Conclusions and future research directions
- Empirical conclusions:
  - AMA-consistent parametric EVT (GPD) and g-and-h are complementary: GPD preferred for in-sample estimation up to ~99.0–99.9th percentile with adequate exceedances; g-and-h more consistent for ultra-extreme quantiles (e.g., ≥ 99.99 percent) or small samples.
  - Estimation uncertainty increases sharply beyond the 99.7th percentile or when exceedances represent < 0.5 percent of observations.
  - Most reliable GPD specification found using Moment and MLE for thresholds beyond about the 98.5th percentile and confidence no higher than 99.0 percent.
- Suggested future work:
  - Multivariate EVT to assess joint dependence of extremes across business lines/event types.
  - Further application and testing of g-and-h across diverse bank loss profiles.
  - Development of an optimization algorithm for the threshold-quantile surface akin to Danielsson et al. (2001) bootstrap to choose threshold objectively on bias-variance trade-off.
  - Detailed assessment of reporting modes and granularity to inform supervisory guidance and self-assessment controls.

*Source: Content from _wp07239 (PDF chapter/section).*

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

### _wp07239 - References

### Tables and Figures (document inventory)
- Tables listed include:
  - Table 1. Overview of Operational Risk Measures According to the Basel Committee on Banking Supervision
  - Table 2. Aggregate Operational Risk Losses of U.S. Commercial Banks (Longitudinal)
  - Table 3. Aggregate Operational Risk Losses of U.S. Commercial Banks (Cross-Sectional)
  - Table 4. Descriptive Statistics of Simulated Daily Operational Risk Losses (Five-Year Horizon)
  - Table 5. Descriptive Statistics of Simulated Daily Operational Risk Losses (One-Year Horizon)
  - Table 6. Descriptives of Simulated Loss Series Based on the Aggregate Loss Statistics (2000–2004)
  - Table 7. Descriptives of Quarterly Aggregates of Simulated Loss Series
  - Table 8. Point Estimates of the Empirical, Normal and EVT-Based Distribution (Five-Year Risk Horizon)
  - Table 9. Relative Deviation of Point Estimates of the Empirical, Normal and EVT-Based Distribution (Five-Year Risk Horizon)
  - Table 10. Point Estimates of the Empirical, Normal and EVT-Based Distribution (One-Year-Risk Horizon)
  - Table 11. Estimated Parameter Values from the Numerical Evaluation of the Tail Behavior
  - Table 12. Summary Statistics of the Deviation of Point Estimates from the Actual Upper Tail (Variable Threshold) Choice
  - Table 13. Summary Statistics of the Deviation of Point Estimates Around the Actual Upper Tail Quantile (Variable Percentile)
  - Table 14. Summary Statistics of Quantile (Point) Estimates of the EVT-Based Distribution
- Figures listed include:
  - Figure 1. Loss Distribution Approach (LDA) for AMA of Operational Risk
  - Figure 2. Upper Tail Graphs of Estimated Shape Parameter for a Continuous Threshold Choice
  - Figure 3. Threshold-Quantile Surface of Operational Risk Event
  - Figure 4. Aggregate Sensitivity of Estimated Upper Tail to the Threshold Choice
  - Figure 5. Aggregate Sensitivity of Individual Point Estimates to the Percentile Level of Statistical Confidence
  - Figure 6. Single Point Estimate Graphs of GDP-Fitted Distribution (Variable Threshold)
  - Figure 7. Relative Deviation of Point Estimates (Point Estimate Residuals) from the Actual Quantile Level
  - Figure 8. Single Point Estimate Graphs of GPD-Fitted Distribution Threshold Upper Tail Graphs
  - Figure 9. Relative Deviation Of Point Estimates (Point Estimate Residuals) from Actual Quantile Value

### I. INTRODUCTION — context and regulatory change
- After more than seven years in the making, the New Basel Capital Accord (Basel II) "finally took effect" with final draft implementation guidelines issued at the end of 2006 (Basel Committee, 2004, 2005, and 2006b).
- The revised regulatory framework moves away from rigid controls and aligns minimum capital requirements closer to asset risk by allowing advanced risk measurement techniques.
- The revised Accord is described as the second round of regulatory changes since the original Basel Accord of 1988.
- Operational risk and corporate governance are highlighted as salient elements of the new banking regulation in response to accounting scandals and complexity of intermediation.
- The paper presents a generalized parametric estimation framework for operational risk of U.S. commercial banks compliant with AMA risk measurement standards under Basel II, in the context of new U.S. supervisory implementation guidelines published on February 28, 2007 (Federal Information & News Dispatch, 2007).

### A. Definition of operational risk, measurement concepts, and Basel II regulatory framework
- Common operational risk definition cited: risk of loss from acts undertaken (or neglected) in carrying out business activities, including inadequate or failed internal processes and information systems, misconduct by people, or external events (Basel Committee, 2004, 2005, and 2006b; Coleman and Cruz, 1999).
- Operational risk characteristics:
  - Diverse origins (internal/external disruptions), unpredictable overall financial impact.
  - Focuses on tail events rather than central tendencies; loss profiles contain occasional extreme losses among frequent low-severity events.
  - Banks partition losses into expected losses (EL) absorbed by net profit and unexpected losses (UL) covered by risk reserves/core capital or hedging.
- Three major measurement concepts:
  - Volume-based approach: exposure as a function of business complexity.
  - Qualitative self-assessment: subjective judgment, comprehensive process review.
  - Quantitative techniques: joint and marginal distributions of severity and frequency, scenario testing, Value-at-Risk (VaR) for extreme quantiles (Mats, 2002 and 2003).
- Basel II measurement approaches (Operational Risk Subgroup of AIGOR):
  - Basic Indicator Approach (BIA): fixed percentage of gross income as a volume-based metric; BIA requires banks to provision a fixed percentage (15 percent) of their average gross income over the previous three years for operational risk losses.
  - Standardized Approach (TSA): sets regulatory capital to at least the three-year average of the summation of different regulatory capital charges (as percentages of gross income) across business lines in each year.
  - Advanced Measurement Approaches (AMA): internal risk measurement systems and scenario analysis to calculate regulatory capital that cover total operational risk exposure (EL and UL) over a one-year holding period at a 99.9 percent statistical confidence level.
- Loss Distribution Approach (LDA) identified as the most common AMA form: annual distribution of number and total loss amount of events, combining frequency and severity.
- Basel II foundational pillars quoted:
  - (i) "Minimum Capital Requirements" (Pillar 1)
  - (ii) "Supervisory Review" (Pillar 2)
  - (iii) "Market Discipline" (Pillar 3)
- Basel Committee specifies eight business lines (BL) and seven event types (ET) for operational risk reporting; a unit of measure could be a business line, an event type category, or both.
- Note: negative or zero gross income periods are excluded in BIA and TSA.

### B. Literature review — empirical and theoretical background
- Historical loss experience is a primary indicator for reserves, but data paucity and inconsistent recording complicate measurement.
- Regulatory researchers (Federal Reserve Bank of Boston; Basel Committee) examined data limitations and statistical patterns of operational risk; academics and industry developed quantitative methods to address measurement gaps (multiple cited works).
- Theoretical models examine economic incentives for ORM:
  - ORM as profitability optimization (Leippold and Vanini, 2003).
  - Operational risk effects modeled as insurance problems affecting utility of depositors and shareholders (Banerjee and Banipal, 2005; Crouhy et al., 2004).
  - Crouhy et al. (2004): partial recognition of insurance could make some banks better off if self-insured depending on capital structure and other factors.
  - Banerjee and Banipal (2005): allowing banks to choose insurance/regulatory capital may be optimal as banking volume increases; blanket regulation can be welfare reducing.
- Empirical alignment:
  - The paper extends parametric specification of tail convergence of simulated loss data (Mignola and Ugoccioni, 2005).
  - Objective to optimize generalized parametric distributions (GEV and GPD) under EVT for measurement of high-impact, low-probability operational losses.
  - Acknowledges parameter instability of estimated asymptotic tail shape at extreme quantiles and investigates effects of parameterization and statistical power on risk estimates.
  - Proposes a threshold diagnostic and model specification for stable and robust capital estimates consistent with a fair value–based insurance premium of operational risk (Banerjee and Banipal, 2005).
  - Considers gross income as exposure factor to evaluate coherence of current regulatory framework independent of uniform capital charges under standardized approaches (Mignola and Ugoccioni, 2006).
- Notes on alternative distributions:
  - Prior studies suggest weak convergence to GPD at high empirical quantiles due to second order behavior of a slowly varying loss severity function.
  - Degen et al. (2006) propose the g-and-h distribution as a suitable candidate for operational risk loss density; the paper examines g-and-h as accommodating diverse loss severity.

### C. Objective (as stated in the supplied content)
- The main objective described is the optimization of generalized parametric distributions (GEV and GPD) as the most commonly used statistical techniques for measurement of operational risk exposure as extreme loss under extreme value theory (EVT), while addressing parameter instability and threshold selection issues to produce stable and robust capital estimates.

*Source: _wp07239 - References (excerpt provided).*

### 10.      In this paper, we investigate the concept of EVT and the g-and-h distribution as

### _wp07239 - 10.      In this paper, we investigate the concept of EVT and the g-and-h distribution as

### Overview and objectives
- Investigates EVT and the g-and-h distribution as measurement methods of operational risk in compliance with the regulatory capital standards of the New Basel Capital Accord.
- Establishes an analytical framework to assess the parametric fit of a limiting distribution to a sequence of i.i.d. normalized extremes of operational risk losses.
- Uses simulated aggregate loss profiles based on the aggregate statistics of the historical loss experience of U.S. commercial banks that participated in a recent operational risk survey.
- Aims to quantify the potential impact of operational risk on income performance of selected banks under the aggregate historical loss experience of the largest U.S. commercial banks over the last five years.

### Methods: EVT, GEV, GPD, and the g-and-h distribution
- Two generalized EVT parametric families fitted to empirical quantile functions using a “full-data” approach (aggregate enterprise level):
  - GEV (Generalized Extreme Value) — models asymptotic tail behavior of order statistics of normalized maxima (or minima).
  - GPD (Generalized Pareto Distribution) — used as an exceedance function (POT method) to model residual risk beyond a high threshold where linear conditional mean excess holds.
- g-and-h distribution — alternative generalized parametric specification that models heavy-tailed behavior by transforming a standard normal variable and relies on order statistics of extremes; does not require a discrete threshold choice as EVT does.
- Estimation algorithms and diagnostics considered: Moment, Maximum Likelihood (MLE), Pickands, Drees-Pickands, Hill.
- Introduces the concept of a “threshold-quantile surface” and its simplifications (“point estimate graphs” and “single threshold upper tail graphs”) to illustrate sensitivity of point estimates to threshold choice, estimation method, and confidence level.

### Key findings and comparisons
- Standard volume-based metric (e.g., BIA-style metrics) is simplistic and masks substantial heterogeneity of relative operational risk losses across banks.
- Simulation results:
  - Cumulative operational risk losses from consolidated banking activities hardly surpass 2 percent of quarterly reported gross income.
  - These simulated losses clearly undershoot a fixed regulatory benchmark in the range between 12 and 18 percent.
- Internal (AMA-consistent) generalized parametric models:
  - Yield closer analytical risk estimates of unexpected loss (UL) than standardized approaches.
  - Can imply substantial capital savings of up to almost 97 percent relative to standardized volume-based charges.
- Sensitivity and estimation uncertainty:
  - Regulatory 99.9th percentile requirement restricts methods to those whose estimates are notoriously sensitive to calibration method, threshold choice, and empirical loss profile heterogeneity.
  - Scarcity of historical loss data defies back-testing and produces considerable parameter risk for high percentile residual risk estimation.
  - For reliable point estimates at a statistical confidence level of no higher than 99.0 percent, at least 1.5 percent of all operational risk events would need to be considered extremes by historical standards.
  - Estimation uncertainty increases significantly when confidence exceeds 99.7 percent or when fewer than 0.5 percent of all loss events can be classified as exceedances for parametric upper tail fit.
  - A marginal reduction of the percentile level by 0.2 percentage points from 99.9 percent to 99.7 percent:
    - Reduces both UL and the optimal threshold at lower estimation uncertainty.
    - Requires a far smaller sample size of essential loss data.
- Model performance:
  - For GPD, the Moment and MLE estimation algorithms generate the most accurate point estimates for approximation of asymptotic tail behavior, regardless of varying tail shape properties.
  - The g-and-h distribution offers a more consistent upper tail fit than GPD and generates more accurate and realistic point estimates only beyond the critical 99.9th percentile level.

### Regulatory assessment and implications
- Compares AMA internal-model (EVT and g-and-h based) results to standardized measurement methodologies (e.g., BIA) to evaluate empirically valid levels of regulatory capital coverage for operational risk.
- Notes limitation: analysis uses simulated aggregate losses from a generic loss severity function based on descriptive statistics of the 2004 Loss Data Collection Exercise (LDCE); cannot investigate robustness of internal models to accommodate different loss distributions across banks (internally consistent cross-sectional differences).
- Discusses the suitability of volume-based measurement (BIA) testing either by:
  - Contrasting the analytical loss estimate at the 99.9th percentile (adjusted by gross income) and a standard capital charge of 15 percent of average gross income over three years, or
  - Estimating the level of statistical confidence at which the analytical point estimate concurs with a capital charge of 15 percent.

### Estimation diagnostics, recommendations, and operational guidance
- For EVT applicability, two core determinations are necessary:
  - Whether GEV of normalized i.i.d. maxima correctly identifies asymptotic tail convergence of designated extremes.
  - Whether fitting GPD as an exceedance function to order statistics (conditional mean excess over a sufficiently high threshold) yields a reliable, non-degenerate limit law satisfying the external types theorem.
- g-and-h calibration avoids discrete threshold choice uncertainty but depends on specification of number and spacing of percentiles in the upper tail; quantile spacing affects parameterization of higher moments and introduces estimation risk.
- Recommendations implicit in analysis:
  - Use enterprise-level “full-data” aggregation to maximize available loss data for parametric estimation, acknowledging the trade-off of grouping similar losses.
  - Prefer Moment and MLE algorithms for GPD tail estimation under varying tail shapes.
  - Consider relaxing the regulatory confidence level (e.g., from 99.9 percent toward 99.7 percent) to materially reduce estimation uncertainty and required extreme-observation sample sizes.
  - Employ threshold-quantile surface diagnostics to assess robustness of point estimates across thresholds and confidence levels.

### Technical and methodological caveats
- Analysis assumes i.i.d. loss events and excludes joint patterns of extreme behavior across business lines (BLs) or event types (ETs).
- Aggregation of i.i.d. random extremes is not equivalent to joint distribution of marginal extremes; multivariate EVT extension is inconclusive without principled ordering in vector space.
- EVT-based VaR (EVT-VaR) requires careful treatment of second-order effects and slow convergence of extreme distributions; unchecked application can yield unstable out-of-sample parameter estimates.

*Italic: Source — Content from _wp07239 - 10.      In this paper, we investigate the concept of EVT and the g-and-h distribution as (PDF chapter/section).*

### 23.      Let

### _wp07239 - 23.      Let

### Extreme Value Theory and the Generalized Extreme Value (GEV) distribution
- Define Y(1), ..., Y(n) as the sample maximum (or minimum) of i.i.d. random variables X1, ..., Xn with common c.d.f. F and density f; ascending kth order statistics X(1) ≤ ... ≤ X(n).
- GEV convergence statement (equation (1)):
  - There exist normalizing constants an > 0 and bn such that
    - lim_{n→∞} Pr[(Y_n − b_n)/a_n ≤ x] = G(x)
  - If F^n(a_n x + b_n) ≈ G(x), normalized maxima fall within the maximum domain of attraction (MDA) of G, i.e., F ∈ D(G).
- GEV density/unified form (equations (2) and (3)):
  - G(x) =
    - { exp[−(1+ξ (x−μ)/σ)^{-1/ξ}], for ξ ≠ 0 with support 1+ξ (x−μ)/σ > 0
    - exp[−exp(−(x−μ)/σ)], for ξ = 0
  - Re-parameterization: tail index α = 1/ξ; σ > 0 (scale), μ (location), ξ (shape/tail parameter).
- Interpretation of ξ:
  - ξ indicates the velocity of asymptotic tail decay (“tail thickness”).
  - The smaller the absolute value of the tail index (or the higher the shape parameter), the larger the weight of the tail and the slower the speed at which the tail approaches its peak at y = 0.
  - Example: if 1/2 ξ = ?, the text states: "if 12ξ=, the first moment (mean) and the second moment (variance) exist, but higher moments have a infinite value." (textual presentation preserved).

### Estimation of GEV parameters
- Two assumptions for estimation:
  - (i) Distribution of normalized maxima converges perfectly to GEV: estimate μ, σ, ξ concurrently by numerical iteration to obtain point estimates.
    - Implicit MLE/log-likelihood form referenced in equation (4).
  - (ii) Common distribution only satisfies F ∈ D(G): semi-parametric estimation using tail approximation F(x) ≈ x^{-1/ξ} L(x) for 0 < λ and slowly varying L(x).
    - As n→∞ and for threshold t_n = b_n + a_n x, approximation leads to semi-parametric point estimate forms (equations (5) etc.).
- When empirical extremes are insufficient:
  - Define threshold t equal to order statistic X_{n,k} of k exceedances out of n.
  - Out-of-sample point estimate of quantile (equation (6)):
    - x̂_{p,n,k} = X_{n,k} p^{-ξ}
    - where X̂_k > x_p and L̂_{n,k} ≈ L.

### Generalized Pareto Distribution (GPD): definition and parametric specification
- GPD approximates GEV via the Pickands-Balkema-de Haan theorem when sample mean excess is positive linear and Fisher-Tippett theorem holds.
- Conditional mean excess distribution definition: Pr[X − t ≤ y | X > t] for exceedance values Y(1), ..., Y(n).
- Limit representation (equation (7)) and GPD form (equation (8)):
  - GPD unifies GP0 (exponential), GP1 (Pareto), GP2 (beta) with shape parameter ξ (continuity at ξ = 0).
  - Support: x ≥ 0 when ξ ≥ 0; 0 ≤ x ≤ −β/ξ when ξ < 0.
- Peak-over-Threshold (POT) method:
  - Fits GPD to order statistics X_{n−k+1:n}, ..., X_{n:n} beyond threshold t ≥ 0.
  - Trade-off in threshold selection: sufficiently high to support parametric tail estimation vs. enough exceedances to estimate parameters and maintain linear mean excess.
  - Threshold can also be selected by timing of occurrence of extremes.
- GPD fitting and parametrization (equations (9) and (10)):
  - Locally estimated GPD W_{t,k}(·; ξ, σ̂) for exceedances beyond threshold fitted to full sample distribution W_{t,n}(·; ξ, μ̂, σ̂) by selecting μ̂ and σ̂ such that W_{t,k} = W_{t,n}.
  - Reparametrizations: σ̂_{k,n} = σ̂% and μ̂_t = μ̂ − σ̂%/(1 − ξ̂)
  - Estimated GPD quantile function (equation (10)):
    - x̂_{t,p,n,k} = t + (σ̂_t/ξ̂) [ (p)^{-ξ̂} − 1 ]  (preserved symbolic form from text)

### Estimation methods and threshold diagnostics for GPD
- Four major estimation types for GPD tail index (shape parameter ξ):
  - (I) Moment estimator (Dekkers, Einmahl and de Haan, 1989)
  - (ii) Maximum likelihood estimator (MLE)
  - (iii) Pickands (1975) estimator
  - (iv) Drees-Pickands (Drees, 1995) estimator
  - Hill (1975) estimator is also mentioned as predicated on calibrating ξ to exceedance volatility.
- Threshold diagnostics:
  - Goal: identify highest possible threshold such that mean excess has positive linear slope (Pickands-Balkema-de Haan requirement).
  - Common complementary diagnostics:
    - (I) Mean excess function (MEF) and distribution of GPD residuals.
    - (ii) Stability of estimated tail (index) parameter across thresholds.
    - (iii) Goodness of fit of point estimates of upper-tail quantiles.
- Empirical vs. analytical MEF and residuals:
  - Sample (empirical) MEF (equation (11)):
    - e_n(t) = (1/n_t) ∑_{i=1}^{n_t} (X_{(i)} − t) , with notation preserved.
  - Analytical MEF: constant exponential MEF W_0^{-1}(t) and positive linear analytical MEF for GPD:
    - W_0^{-1}(t) and W_{ξ}^{−1}(t) forms preserved in text.
  - Diagnostic rules:
    - Correct parametric specification and convergence if empirical MEF lies between the exponential constant analytical MEF and the positive linear analytical MEF of GPD.
    - The threshold that yields the smallest number of ordered maxima still supporting a linear MEF is preferred.
  - Q-Q plot of order statistics and empirical mean excess values used to identify threshold inflection points from constant to positively sloped MEF.
  - Residuals R_i = log[1 + ξ̂ (X_i − t)/β̂] should be i.i.d. unit exponentially distributed (with constant drift) if threshold is sufficiently high.
- Reliability of tail estimation and goodness-of-fit:
  - Select highest threshold that generates stable tail index estimates and linear mean excess values.
  - Coincidence of stable parameter estimates and linear MEF is imperative for reliable numerical evaluation of point estimates in the tail.

### g-and-h distribution: definition and parametric estimation
- g-and-h distribution as alternative to estimate residual risk of extreme losses and model heavy skew/kurtosis in loss data:
  - Transformation of standard normal z (equation (12)):
    - F_{g,h}(z) = μ + σ [ (exp(gz) − 1)/g ] exp( (h z^2)/2 )  (preserved transformation structure as in text)
  - Parameters: μ, σ ≥ 0 (location, scale), g (skewness), h (kurtosis); domain of attraction includes all real numbers.
  - g and h can be constants or real-valued functions of z^2, but here specified as constants.
- Practical notes:
  - Martinez and Iglewicz (1984) show wide approximation capability, including approximating GEV and GPD shapes.
  - Quantile-based estimation methods (McCulloch, 1996; Hoaglin, 1985) are typical and can deliver more accurate empirical tail fit than moments or MLE.
  - g-and-h is convenient for Monte Carlo simulation since it is a monotonic transformation of the standard normal.

### Data description and simulation context
- Operational loss data challenges:
  - High-quality operational loss-event data are scarce due to measurement differences, data availability variations across banks, variability in operational risk management, banking activities, and reporting standards.
  - One-off significant operational risk events and idiosyncratic loss data make analytical modeling and comparative analysis difficult.
- Implication:
  - Simulation of quasi-empirical distributions and Monte Carlo methods are pertinent given data limitations (context for use of g-and-h and EVT methods).

*Source: _wp07239 - 23.      Let (PDF chapter/section) — content extracted from the supplied source document.*

### 36.      Given the diverse characteristics of operational risk across banks, we redress the

### _wp07239 - 36.      Given the diverse characteristics of operational risk across banks, we redress the

### Data and simulation design
- Simulation base: Loss Data Collection Exercise (LDCE) average probability and loss severity of operational risk events reported by selected U.S. banks over a five-year period from 2000 to 2004.
- LDCE coverage: 55,766 operational risk loss events of more than US$10,000 over 2000–2004, amounting to an aggregate total loss exposure of US$25.9 billion—just shy of 56 percent of aggregate (seasonally-adjusted) reserves of U.S. depository institutions at year-end 2004.
- LDCE sample composition: Out of 23 participating banks, only the 10 firms that submitted comprehensive internal loss data.
- Annualization: LDCE covers losses only until the end of the second quarter of 2004; total number of loss events for 2004 was annualized.
- Sample-size augmentation: Average number of operational risk events (3,350) deemed insufficient for GPD estimation; annual loss frequency rescaled by factor 2.91 to increase original sample size to a total of 10,000 observations per bank.

### Loss generation and severity modeling
- Two-step loss data generation: (1) Monte Carlo-based loss simulation; (2) fundamental value adjustment.
- Aggregate loss definition: Aggregate loss S_t defined as the N_t-fold convolution of individual loss amounts over a five-year risk horizon, following the Loss Distribution Approach (LDA).
- Severity distribution specification:
  - Considered families: three sub-classes EV0, EV1 and EV2 of GEV.
  - Chosen sub-model: Fréchet (EV1) sub-model distribution of GEV.
  - Calibrated shape parameters: {}1.0; 0; 0.85;1.0;1.15ξ∈− (as specified in source).
  - Rationale: accommodates the full spectrum of asymptotic tail behavior in line with LDCE sample statistics.
- Simulation mechanics:
  - Generate series of N_t random draws from f_t(x) to simulate i.i.d. loss events L_{t,i} over the first year t=1.
  - Loss events are randomly disseminated as daily events over each year.
  - Left endpoint of density function rescaled to zero; first two moments matched so average aggregate loss S_t is of same order as LDCE aggregate statistics.
  - Multi-year simulation: repeat generation for years 2–5 with time-varying frequency {2;3;4;5}_t N ∈ and severity f_t(x) ∈ changing each year according to LDCE.
- Alternative aggregation methods noted: fast Fourier transform or analytical approximation (per Klugman et al. (2004)).

### Fundamental-value adjustment and bank-specific scaling
- Sample banks for fundamental adjustment:
  - Bank 1—Bank of America (BoA)
  - Bank 2—J.P. Morgan Chase (JPMC)
  - Bank 3—Wachovia
  - Bank 4—Washington Mutual (WaMu)
- Fundamental controls used to normalize losses: quarterly total assets (TA), gross income (GI) and Tier 1 capital (T1).
- Quarterly procedure:
  - Split each annual series of generic loss events L_{t,i} into four equally sized batches matching quarterly loss frequency 4_t N.
  - Adjust each L_{t,i} by reported quarterly TA, GI and T1 to obtain loss ratios reflecting financial impact at time of occurrence.
- Data sources for fundamentals: Federal Reserve Board National Information Center (FRBNIC) and SNL Data Source of the SEC for consolidated banking operations.
- Choice of scaling factor for main analysis: gross income (GI), in line with quantitative criteria of New Basel Capital Accord.

### Key descriptive outcomes and comparative observations
- Distributional features:
  - All bank loss distributions are fat-tailed; skewness particularly pronounced for Banks 1 and 2.
  - High-impact events generate relatively small amounts of exposure relative to quarterly fundamental values.
- Maximum relative loss observations:
  - Aggregating daily losses on a quarterly basis over the five-year sample, Bank 4 (lowest reported gross income) would have recorded an amount of quarterly gross income lost due to operational risk events of 14.5 percent—just shy of the indiscriminate 15 percent rate in line with BIA.
  - Conclusion: a blanket regulatory capital charge based on a standard metric of unexpected operational risk appears no least too high relative to actual operational risk exposure, and at best, too inflexible to account for cross-sectional variation of loss profiles.
- Effect of scaling choice: multiples of quarterly aggregate losses hardly vary by type of fundamental adjustment (TA, GI, T1), indicating limited sensitivity of relative operational risk exposure quantification to the choice among these fundamentals.
- Advantages of the simulated approach:
  - Simulated loss data are generic to all sample banks, enabling ruling out reporting bias of operational risk events from distinctive patterns of loss frequency.
  - Simulated data avoid some common limitations of historical operational risk losses (heterogeneous definitions, short time periods, dominance by few high-severity losses).

### Methodological caveats and differences from prior literature
- Left truncation and threshold issues:
  - LDCE truncation (minimum reporting threshold) may explain high average operational risk loss of US$464,800.
  - Some LDCE-participating banks collect only losses exceeding a minimum threshold, which curtails empirical loss distribution and increases EL; limiting variation of losses implies a higher probability of extreme losses.
- Differences from Mignola and Ugoccioni (2006):
  - This paper parameterizes time-varying severity and frequency using aggregate LDCE information rather than compounding a known severity function over multiple periods.
  - Allows more extreme events and abstains from matching the regulatory 99.9 percent quantile of aggregated losses to a pre-defined capital-gross income ratio.
  - Critique: calibrating to a pre-specified 99.9th percentile and regulatory benchmarks may induce biased empirical upper-tail fit and sensitivity to normative assumptions.
- Aggregation effects:
  - Loss aggregation critical to detecting dissimilar changes of loss exposure relative to fundamentals and to avoid understatement of EL and UL.
  - Aggregating scaled daily losses rather than scaling quarterly aggregates would downward bias the aggregate measure of quarterly operational risk exposure; bias increases with number of observations in aggregation period.

### Estimation procedure overview
- Distributions estimated: GEV, GPD and g-and-h to assess asymptotic tail behavior of simulated operational risk losses for four U.S. commercial banks based on LDA in accordance with AMA standards.
- Primary focus: fitting GPD via the POT procedure with threshold diagnostics for tail fit.
- Full-data approach: fit all annually generated loss data (one-year risk horizon) adjusted by quarterly reported gross income to a parametric loss distribution across five years, without BL or ET granularity.
- Outcome targets: point estimates of Unexpected Loss (UL) for specific percentile levels of statistical confidence, followed by robustness tests.
- Analytical scope: threshold diagnostics and robustness tests presented for Bank 1 as pars pro toto; summary statistics provided for all sample banks.

*Source: _wp07239 - 36.      Given the diverse characteristics of operational risk across banks, we redress the*

### 47.      Conventional threshold diagnostics (based on mean excess properties and parameter

### _wp07239 - 47.      Conventional threshold diagnostics (based on mean excess properties and parameter

### Conventional threshold diagnostics and mean excess function (MEF)
- Conventional threshold diagnostics (based on mean excess properties and parameter stability) identify the range of eligible threshold quantiles to guide qualitative assessment of how threshold for GPD should be selected from a sample of loss data.
- Analyze the slope of the MEF to determine whether the empirical distribution satisfies the external types theorem of asymptotic tail behavior under GEV as the fundamental requisite of GPD.
- Sensitivity criteria for eligible thresholds:
  - Sensitivity of the estimated GPD tail index parameter to a variable threshold choice.
  - Goodness of POT-based upper tail fit of GPD-based point estimates.
- Optimal threshold objective:
  - Support a stable parametric approximation of linear conditional mean excess of extremes.
  - Allow in-sample point estimation over a maximum range of percentiles.

### Sample period, data availability, and exceedance selection
- Limited availability of extreme observations frequently precludes annualization of risk estimates derived from a LDA-based implementation of AMA.
- Use of a five-year time period of loss data is consistent with empirical loss experience of banks in QIS and U.S. federal bank regulator guidance:
  - U.S. federal regulators specify five years of internal operational risk loss data and permit external data for regulatory capital calculation.
  - Basel Committee provides only three years of data after initial adoption of AMA and then five years.
- Empirical findings on exceedances:
  - Empirical MEFs of exceedances within the top docile (from k=10 to 10 percent of the total sample size) are positive linear and indicate very extreme observations far removed from average loss experience.
  - Candidate threshold quantiles based on relative incidence of exceedances k=50, 100, 150, and 200.
  - For Bank 1, the most suitable threshold quantile designates about k=100 exceedances (see Table 9).
    - This threshold relegates out-of-sample estimation of point estimates to quantiles only beyond the 99.0th percentile of the empirical quantile function while mitigating parameter uncertainty and possible estimation bias.
  - To control bias from clustered extremes, select exceedances from equal number of non-overlapping blocks of observations over the sample period.

### Point estimation: methods and assessment
- Point estimates derived from GPD and the g-and-h distribution given asymptotic tail behavior of i.i.d. loss events over a five-year period.
- Goodness of upper tail fit assessed by convergence between actual quantile values and analytical point estimates of the GPD limit distribution.
- With k=100 exceedances:
  - Moment and Hill estimation methods offer the most consistent upper tail fit across different loss profiles, especially within the quantile range from the 99.0 to the 99.97th percentile.
  - Hill estimation yields more accurate results beyond the 99.9 percent quantile but understates loss severity for extreme quantiles of 99.99 percent or higher (similar to Moment method).
  - MLE, Pickands and Drees-Pickands overestimate actual losses but better specify conditional mean excess within the 99.5th to 99.97th percentile.
  - Philips-Loretan (PL) (1990) estimation of the shape parameter tends to improve analytical point estimates of the Moment (and MLE) methods at the 99.9th percentile at the expense of less accurate upper tail fit.

### Shape parameter behavior and cross-bank variation
- Expectation: original GEV shape parameter 1.0ξ= largely unchanged for GPD given persistence of asymptotic tail convergence.
- Observed significant differences of shape parameter estimates across banks and estimation methods (see Table 11):
  - Example: Bank 3 shows extremely large losses coinciding with quarters of below-average gross income, amplifying relative operational risk exposure.
  - Pickands and Drees-Pickands methods generate shape parameter values around 1.15, well above Moment and MLE results, because they assign more tail weight to most extreme observations and support a high shape parameter value (or small tail index) with lower velocity of asymptotic tail decay.

### Estimation results and model comparisons
- Summary of quantile point estimation based on GEV and GPD (“EVT-VaR”) and g-and-h distribution fitted simulated loss data over a five-year risk horizon (see Tables 8 to 10).
- Key observations:
  - GEV tends to overstate actual losses at high quantiles.
  - POT method of GPD for normalized i.i.d. maxima in excess of designated sample threshold accurately represents residual risk in the upper tail.
  - Empirical quantile values increase precipitously over percentile range from 95.0 to 99.999 percent; rate of change varies by bank and depends on loss variance increasing from Bank 1 to Bank 4.
  - Moment and MLE methods provide most consistent and stable GPD estimation at the 95th percentile or higher.
  - For AMA-required probability level of 99.9 percent, inclusion of the PL method for the shape parameter (combined with Moment or MLE) generates best analytical point estimates of historical operational risk exposure:
    - Historical operational risk exposure at 99.9th percentile ranges between 0.61 percent (Bank 1) and 2.39 percent (Bank 4) of gross income (per quarter).
- Effect of risk horizon:
  - Accumulation of losses over multiple years masks considerable variability of estimated operational risk exposure.
  - Risk horizon amplifies sensitivity of goodness of upper tail fit to percentile level and increases estimation risk and parameter uncertainty due to intermittent annual loss frequency and variable loss skewness.
  - The final year of simulated losses offers the most robust and consistent GPD estimates of any sample year.
  - Recommendation: In view of data limitations, a risk-horizon of at least three years appears sensible based on average historical loss experience of U.S. commercial banks.
- Model complementarity:
  - GPD supports closest approximation of actual quantiles around the 99.0th percentile.
  - g-and-h distribution outperforms GPD for very extreme quantiles of 99.99 percent and higher; g-and-h is particularly stable and accurate around the 97.5th percentile.
  - Around the 99.9th percentile, GPD estimates are superior to g-and-h.
  - Conclusion: GPD and the g-and-h distribution are complementary—GPD for sufficiently large sample sizes and in-sample estimation near 99.0th percentile; g-and-h for very extreme out-of-sample quantiles or small samples/shorter risk horizons.

### Robustness of GPD point estimates to threshold choice
- Identified limitation: Global threshold choice may be inadequate for optimal specification of GPD at specific percentiles of interest.
- Proposed measurement approach: the “threshold-quantile surface”.
  - The threshold-quantile surface represents all point estimates for any combination of threshold value and percentile level over a pre-specified quantile range for a given GPD parameterization.
  - Constructed on an equidistant grid over quantile range from the 98.0th to the 99.9th percentile at 0.05 percent increments:
    - x-axis: number of exceedances (designated by threshold quantile).
    - z-axis: level of statistical confidence (percentile level).
  - The 45-degree line from [0.980; 0.980] to [0.999; 0.999] identifies point estimates whose percentile level coincides with threshold quantile (in- vs out-of-sample estimation).
- Observed properties of threshold-quantile surface:
  - Higher percentile levels entail generally higher volatility of point estimates across eligible threshold values.
  - In- and out-of-sample point estimates do not differ greatly up to the 99.0th percentile.
  - Some estimation methods (Pickands, Hill) generate less robust point estimates and a disturbed surface for different threshold quantiles.
  - Number of exceedances matters less for lower percentiles unless threshold is too restrictive leaving very few exceedances.
  - Point estimates beyond the 99.7th percentile involve significant parameter uncertainty and estimation risk irrespective of threshold choice.
  - Lower threshold values:
    - More “forgiving” but tend to slightly overstate loss severity.
    - Lower quantiles involve less estimation risk if at least 0.2 percent of all observations are classified as exceedances.
  - All threshold-quantile surface estimations indicate most consistent estimation results around the 99.0th percentile.

### Consolidation and sensitivity summaries
- Collapse of one dimension of the threshold-quantile surface into descriptive statistics:
  - Consolidate z-axis (percentile levels) into a descriptive statistic of upper tail shape to show aggregate sensitivity of point estimates to threshold quantiles.
  - Consolidate x-axis (threshold quantiles) to show aggregate sensitivity of point estimates to threshold choice at each percentile level.
  - For each approach calculate:
    - (I) relative deviation of estimated from actual upper tail shape at each threshold quantile (see Figure 4).
    - (II) relative deviation of point estimates from actual quantile values across percentile range (see Figure 5).
- Findings on sensitivity:
  - Sensitivity of upper tail fit to number of exceedances is marginal for threshold quantiles between the 98.0th and the 99.7th percentile; higher thresholds increase parameter uncertainty from out-of-sample estimation.
  - Positive correlation between threshold quantile and deviation of estimated from actual median of the upper tail: fewer exceedances → greater variation of point estimates (see Table 12).
  - Point estimates more sensitive to changes in desired level of statistical confidence if threshold quantile is flexible (see Figure 5).
  - Positive correlation between deviation of median point estimate from actual quantile and threshold quantile: greater estimation risk as one traverses the upper tail (see Table 13).
  - Ignoring extreme results for 99.9th threshold quantile (insufficient exceedances), selection of the right percentile level rather than threshold choice (within a high threshold quantile range) matters most for robust point estimates and optimal parametric GPD-based upper tail fit.

*Source: _wp07239 - excerpt from provided IMF PDF content.*

### 59.      As an alternative simplification, we disregard one dimension of the threshold-quantile

### _wp07239 - 59.      As an alternative simplification, we disregard one dimension of the threshold-quantile

### Single-point and single-threshold graph simplifications
- Two-dimensional renditions of the threshold-quantile surface:
  - Single point estimate graph: fixed percentile level over a continuous range of thresholds (a “horizontal slice”).
  - Single threshold upper tail graph: fixed loss threshold over a percentile range (a “vertical slice”).
- Comparative findings across estimation methods:
  - Moment and MLE estimation methods with Philips-Loretan (PL) produce the most consistent point estimates within a “comfort zone” of threshold quantiles between the 98.5th and the 99.0th percentile.
  - Individual sensitivity of point estimates to increasing threshold differs by estimation method, but the general pattern of variation hardly varies across different percentile levels.
- Upper tail convexity and threshold effects:
  - Upper tail fit shape varies by estimation method and exhibits distinct convexity, which declines at higher thresholds.
  - A threshold quantile around the 99.0th percentile offers the most reliable results, particularly for the Moment and the MLE estimation method with and without PL estimated tail shape.

### Backward induction for optimal threshold choice (overview)
- Two optimization approaches using the threshold-quantile surface:
  - Global optimization: selects combinations of threshold quantiles and estimation methods that engender the most consistent upper tail fit of point estimates (consolidated z axis).
  - Local optimization: identifies a threshold quantile that minimizes deviation of point estimates from actual values at a specific percentile level (single point estimate graph).
- Sensitivity analyses complement optimization:
  - Sensitivity of individual point estimates to flexible threshold choice can be analyzed at each percentile level within the upper tail or at a specific percentile level via single threshold upper tail graphs.
- Tables and figures referenced provide diagnostics and summarization (Table 8, Table 11–14; Figures 2–9).

### Local threshold optimization (detailed procedure and findings)
- Local optimization procedure for a given percentile p (focus on p=99.9th percentile):
  1. Choose the most consistent GPD estimation method that generates point estimates with the lowest sensitivity to a variable threshold between the 97.0th and the 99.5th percentile (see Table 14(5)).
  2. Select the local threshold quantile *lq that engenders the smallest deviation of the point estimate from the actual quantile value (see Table 14(6)).
  3. Qualify *lq on the smallest joint divergence of point estimates over different estimation methods (see Table 14(7)) to reduce estimator-specific bias.
  4. Complement with sensitivity analysis of upper tail fit over a percentile range to a threshold q close to *lq; the single threshold upper tail graph identifies the optimal local percentile level (*lp).
- Key quantitative findings:
  - Risk estimates at the critical percentile level of p=99.9 percent of AMA are most accurate for threshold quantiles *lq ranging from 97.52 percent to 98.78 percent, depending on the estimation method and sample bank.
  - There are decreasing marginal benefits from higher thresholds to support point estimates at high levels of statistical confidence according to AMA standards.
  - Higher percentiles require disproportionately more loss data to support a higher loss threshold that maintains linear ME of exceedances.
- Conditional vs. unconditional local optimization:
  - Unconditional local threshold choice (*,luq) supports the closest alignment of a single point estimate to the actual value across all estimation methods (see Table 14(8)); without restricting by estimation method, the optimal threshold declines significantly to around the 98.0th percentile.
  - Percentile levels remain largely unchanged between conditional and unconditional optimization, with unconditional generating more accurate point estimates.
  - Conditional optimization (*,lcq) based on the estimation method with best overall upper tail fit produces a more balanced trade-off between threshold and percentile levels; conditional optimization at p=99.9 percent identifies * ,lcq that corresponds to the deterministic threshold quantile q=99.0 percent for the optimal local percentile * ,lcp.

### Global threshold optimization (detailed procedure and findings)
- Purpose and approach:
  - Global optimization considers joint sensitivity of point estimates to both threshold and percentile level.
  - Global threshold * ,gcq minimizes the difference between the median point estimate and the actual median value of the upper tail based on the estimation method for the best original upper tail fit (see Table 14(15); Figure 7; Table 12).
  - Global percentile level * ,gcp minimizes the difference between the actual percentile value and the dispersion of point estimates over the entire range of thresholds at a certain level of statistical confidence (see Table 14(16); Figure 9).
- Key comparative findings:
  - * ,gcq almost always concurs with * ,lcq (see Table 14(9)).
  - Without restrictions on estimation method, * ,gcq is much higher than * ,luq (see Table 14(8)).
  - Point estimates below the 99.0th percentile are least sensitive to the threshold choice (consistent with Table 14(4) and Table 13).
  - The most reliable estimation of quasi-empirical losses for the best overall upper tail fit requires in-sample estimation for all but one sample bank.
  - Insufficient observations of extremes preclude thresholds that support reliable, in-sample upper tail fit as the level of statistical confidence increases.

### Conclusion — summary of empirical and policy-relevant findings
- Methods and objective:
  - Investigated GEV, GPD, and the g-and-h distribution to assess fat-tailed limiting behavior of large operational risk losses and quantified hypothetical impact of unexpected operational loss (UL) on selected U.S. commercial banks.
  - Focused primarily on optimal GPD-based specification of tail dependence with comprehensive threshold diagnostics (mean excess, tail index stability, goodness of upper tail fit); also derived in- and out-of-sample point estimates for GEV and g-and-h.
- Main empirical findings:
  - Based on simulated operational loss data adjusted by quarterly reported gross income, impact at the 99.9th percentile required by AMA appears minimal: the bank with the most distinct low-frequency, high-severity profile would have lost only 1.75 percent of gross income over a five year horizon, whereas standardized approaches would have required a fixed capital charge of 15 percent of three-year average gross income.
  - Parametric risk estimates of i.i.d. normalized maxima at the required 99.9th percentile implied capital savings of up to almost 97 percent compared to a uniform measure of operational risk exposure.
  - The g-and-h distribution (and its power law variant) outperformed GEV and GPD in terms of goodness of upper tail fit but underestimated actual losses in all but the most extreme quantiles of 99.95 percent and higher; EVT-based estimates overstated excess elongation in asymptotic tail decay at those extreme quantiles.
  - Parameter instability at high percentile levels is largely caused by lack of sufficient empirical loss data; GPD is useful for out-of-sample estimates of asymptotic tail behavior after comprehensive sensitivity analysis.
  - Estimation uncertainty increases significantly at statistical confidence levels beyond the 99.7th percentile or for threshold quantiles that classify less than 0.5 percent of all losses as exceedances under parametric GPD upper tail fit.
  - Most reliable parametric specification under GPD found using Moment and MLE estimation methods for thresholds beyond about the 98.5th percentile and statistical confidence no higher than 99.0 percent, regardless of varying tail shapes.
- Regulatory implication and interpretation:
  - Large differences between AMA risk estimates and standardized capital charges (BIA and TSA) indicate prima facie inconsistency of operational risk measurement under the current regulatory framework if loss data are empirically valid.
  - At a uniform capital charge of 15 percent of average annual gross income, AMA would require much higher levels of statistical confidence (at least 99.99 percent) to match standardized measures, increasing estimation risk and parameter uncertainty.
  - Scarcity of historical loss data constrains reliable back-testing at very extreme quantiles and limits comparability of point estimates across distributions at the 99.9th percentile.
  - AMA criteria appear set beyond historical loss experience of banks, encouraging combination of internal measurement models and creation of capital buffers.
- Practical recommendation distilled from analysis:
  - Prioritize selection of percentile level and careful sensitivity diagnostics over solely raising threshold quantiles.
  - Use Moment and MLE estimation methods (with and without PL) for GPD-based upper tail approximation when threshold quantiles are around or above the 98.5th percentile but limit desired statistical confidence to no higher than 99.0 percent to contain estimation uncertainty.
  - Recognize that lowering the regulatory target from p=99.9 percent to p=99.7 percent yields disproportionate reductions in UL and in required sample sizes, potentially improving estimation reliability given scarce data.

*Source: Excerpt from _wp07239 (PDF chapter/section)._

### 72.      The cross-sectional variation of loss profiles and the trade-off between reliability and

### _wp07239 - 72.      The cross-sectional variation of loss profiles and the trade-off between reliability and accuracy of risk estimates

### Cross-sectional variation, model reliability, and accuracy
- AMA yields more accurate estimates of unexpected operational risk and appeal to the usage of economic capital as determinant of banking performance, but it fails to address structural and systemic effects of heterogeneous loss data.
- The cross-sectional variation of loss profiles and the trade-off between reliability and accuracy of risk estimates underscore an inherent joint hypothesis problem of operational risk measurement—either the model is correct and the testable data is not or vice-versa.
- Standardization of units of measure notwithstanding, consistent AMA estimation might be compromised by:
  - cross-sectional variation of reported loss frequency; and
  - incentives of loss fragmentation.
- Normative assumptions eschew such measurement bias in favor of greater reliability—but they do so at the expense of:
  - less discriminatory power; and
  - higher capital charges.

### Revised regulatory framework and measurement practice
- A revised regulatory framework for operational risk would benefit from a comparative assessment of the time-varying impact of different loss profiles of banks under different measurement approaches.
- Analytical benefits of upper tail specification based on order statistics are noted, but:
  - the most common measurement models used by banks do not identify the incidence of extremes as a dynamic process.
- One-off operational risk events elude purely quantitative models; therefore:
  - sound risk measurement would require a qualitative overlay; and
  - the qualitative overlay's prominence needs careful balancing with a considerable degree of judgment and mindful interpretation of historical precedence.

### Future research directions and methodological extensions
- Future terrain to be covered includes:
  - a detailed analysis of joint dependence of extremes based on multivariate EVT approaches to assess alleged correlation effects across different units of measure;
  - demonstrating that loss estimates exhibit granularity to capture risk drivers in a structural model specification;
  - circumspect investigation of different modes of reporting operational risk losses in terms of timing and organizational detail to help derive regulatory controls of self-assessment that promote consistent sensitivity measures of loss severity and loss frequency;
  - more extensive application of the g-and-h distribution across different loss profiles of banks (Dutta and Perry, 2006; Degen et al., 2006).
- Methodological development recommendation:
  - develop d an optimization algorithm for the “threshold-quantile surface” in the spirit of the Danielsson et al. (2001) bootstrap method to objectively choose a threshold level based on a bias-variance trade-off.

*Source: _wp07239, pages 72–74 (page 38 in document).*

### REFERENCES

### _wp07239 - REFERENCES

### Operational risk measurement and regulatory context
- Basel Committee guidance and standards cited include:
  - Basel Committee on Banking Supervision, 2006a, Observed Range of Practice in Key Elements of Advanced Measurement Approaches (AMA). BCBS Publications No. 131.
  - Basel Committee on Banking Supervision, 2006b, Basel II: International Convergence of Capital Measurement and Capital Standards: A Revised Framework—Comprehensive Version. BCBS Publications No. 128.
  - Basel Committee on Banking Supervision, 2005 and 2004 publications on the Revised Framework and Operational Risk.
- AMA quantitative criteria referenced:
  - one-year holding period and a 99.9th percentile confidence interval as the AMA soundness standard.
  - Banks allowed to adjust total operational risk up to 20 percent of the total operational risk capital charge (capital deduction up to 20%).

### Key empirical results and aggregate statistics (LDCE and simulated calibrations)
- LDCE aggregate operational risk losses (Table 2, longitudinal, U.S. commercial banks, 1999–2004):
  - Total (1999-2004): 39.92 (U.S.$bn) total loss; total loss per bank 2.10 (U.S.$bn); total number of loss events 3,650; total loss per bank 54,934 (U.S.$ ‘000); average loss per bank 3,597 (U.S.$ ‘000).
  - Average (1999-2004): 15 participating banks; mean total loss 6.65 (U.S.$bn); total loss per bank 0.35 (U.S.$bn); average loss per bank (EL) 727 (U.S.$ ‘000); average loss per bank (all banks) 9,156 (U.S.$ ‘000); average loss per bank (per bank) 600 (U.S.$ ‘000).
  - Total (1999-2004* annualized): 45.92 (U.S.$bn) total loss; total loss per bank 2.40 (U.S.$bn); total number of loss events 3,650; total loss per bank 58,378 (U.S.$ ‘000); average loss per bank 3,769 (U.S.$ ‘000).
  - Note: 2004 annualized (*) because LDCE submissions were until June 30, 2004.
- Cross-sectional LDCE (Table 3):
  - Total loss events of all banks: 55,766; no. of loss events per bank: 2,425; total loss (in US$bn.) 25.92; total loss per bank (in US$bn.) 1.13; average loss (EL) per bank (in US$ ‘000) 465.
  - Frequency intervals and corresponding average loss (EL) per bank (in US$ ‘000):
    - 0-250: 331
    - 251-1,000: 126
    - 1,001-2,500: 608
    - 2,501+: 438

### Simulated operational risk exposure (calibrated to LDCE aggregate statistics)
- Simulation setup (Tables 4 and 5 notes):
  - Total simulated observations: 9,996.
  - Scaling factor applied to reported loss frequency: 2.91× to reach about 10,000 observations.
  - Simulations performed on quarterly basis; left endpoint rescaled to zero before mean calibration.
- Descriptive statistics for simulated daily operational risk exposure over a five-year horizon (Table 4, in US$ ’000) for random sampling from GEV with various shape parameter ξ:
  - EV0 (ξ=0):
    - mean: 670.2
    - median: 437.4
    - maximum: 5,957.1
    - std. dev.: 685.4
    - coeff. of variance: 1.0
    - JB statistic: 1.2E+04; p-value: 0.00
    - 99.900 percentile: 4,337.8
    - 99.999 percentile: 5,896.4
  - EV1 (ξ=0.85):
    - median: 39.5
    - maximum: 524,400.0
    - std. dev.: 8,390.3
    - coeff. of variance: 12.5
    - JB statistic: 2.0E+09; p-value: 0.00
    - 99.900 percentile: 83,774.7
    - 99.999 percentile: 512,433.9
  - EV1 (ξ=1.0):
    - median: 61.0
    - maximum: 575,708.3
    - std. dev.: 8,740.2
    - coeff. of variance: 13.0
    - JB statistic: 2.8E+09; p-value: 0.00
    - 99.900 percentile: 78,948.6
    - 99.999 percentile: 562,326.7
  - EV1 (ξ=1.15):
    - median: 126.5
    - maximum: 415,569.5
    - std. dev.: 7,078.8
    - coeff. of variance: 10.6
    - JB statistic: 1.1E+09; p-value: 0.00
    - 99.900 percentile: 75,444.5
    - 99.999 percentile: 401,801.6
  - EV2 (ξ=-1.0):
    - median: 471.8
    - maximum: 1,932.6
    - std. dev.: 591.8
    - coeff. of variance: 0.9
    - JB statistic: 1.8E+03; p-value: 0.00
    - 99.900 percentile: 1,931.9
    - 99.999 percentile: 1,932.6
- Descriptive statistics for simulated median daily operational risk exposure over a one-year horizon (Table 5, in US$ ’000):
  - EV0 (ξ=0): mean 447.8; median 413.7; std. dev. 211.0; 99.900 percentile 1,360.9; 99.999 percentile 1,418.7.
  - EV1 (ξ=0.85): median 47.7; maximum 83,730.9; std. dev. 2,682.8; 99.900 percentile 41,828.0; 99.999 percentile 82,066.9.
  - EV1 (ξ=1.0): median 59.1; maximum 183,839.0; std. dev. 8,540.5; 99.900 percentile 57,736.5; 99.999 percentile 183,501.8.
  - EV1 (ξ=1.15): median 137.3; maximum 97,869.2; std. dev. 2,616.1; 99.900 percentile 25,760.3; 99.999 percentile 96,654.9.
  - EV2 (ξ=-1.0): median 467.7; maximum 510.9; std. dev. 64.1; 99.900 percentile 680.8; 99.999 percentile 680.9.
  - Relative rule of thumb: annualization of operational risk exposure suggests a decrease of percentile values over five years by factor 1T×, i.e., 44.7 percent for T=5 (so annual values ~55 percent of five-year values).

### Tail estimation methods, point estimates, and model comparisons
- Estimation methods applied (multiple tables and notes):
  - GEV estimation via Linear Combination of Ratios of Spacings (LRS) for μ[location], σ[scale], ξ[shape].
  - GPD (Peak-over-Threshold) estimation with Moment GP, Maximum Likelihood (MLE) GP (initialised with Moment GP), Pickands GP, Drees-Pickands (DP) GP, and Hill GP1 (Pareto).
  - Phillips-Loretan (PL) alternative shape parameter used in some Moment and MLE GP computations.
  - g-and-h distribution fitted using 40 equidistant percentile points for μ, σ, g, h.
- Point estimates for VaR quantiles (Table 8, five-year risk horizon, scaled by gross income, Bank 1) — selected percentiles (in percent):
  - Empirical distribution (emp. distr.) and normal distribution (normal distr.) comparisons shown alongside EVT (GEV exact LRS, GEV out-of-sample LRS) and multiple GPD methods.
  - Examples of point estimates (percent) at selected percentiles (Bank 1):
    - 99.700: emp. distr. 0.17; normal distr. 21.89; GEV exact (LRS) 0.33; GEV out-of-sample (LRS) 0.25; Moment GP 0.17; MLE GP 0.28; Pickands 0.18; DP 0.20; Hill GP1 0.17; g-and-h 0.17.
    - 99.900 (AMA criterion): emp. distr. 0.50; normal distr. 24.55; GEV exact (LRS) 1.17; GEV out-of-sample (LRS) 0.87; Moment GP 0.47; MLE GP 0.87; Pickands 0.55; DP 0.59; Hill GP1 0.54; g-and-h 0.53.
    - 99.999: emp. distr. 4.03; normal distr. 33.68; GEV exact (LRS) 227.75; GEV out-of-sample (LRS) 168.98; Moment GP 35.94; MLE GP 112.53; Picklands 66.14; DP 75.75; Hill GP1 104.60; g-and-h 101.39.
  - The 99.9th percentile is highlighted as the AMA quantitative criterion.
- Relative deviations of analytical point estimates from empirical quantiles (Table 9, five-year horizon, Bank 1):
  - Example deviations (in percent) at key percentiles:
    - 99.900: normal distr. deviation 4,764.9; GEV exact (LRS) 131.0; GEV out-of-sample (LRS) 71.5; Moment GP -7.79; Moment GP (PL) 73.16; MLE GP 69.30; Pickands 17.78; DP 6.31; Hill GP1 5.76; g-and-h 3.20.
    - 99.999: GEV exact (LRS) 5,556.6; GEV out-of-sample (LRS) 4,096.9; Moment GP 792.7; MLE GP 2,694.9; Pickands 1,542.7; DP 1,781.5; Hill GP1 2,497.9; g-and-h 805.2.
  - Relative deviation values vary widely across methods and extreme percentiles, often extremely large.
- Estimated parameter values (Table 11) for GEV and GPD (scaled by gross income, Banks 1–4):
  - Panel A: Bank 1 (BoA)
    - GEV (LRS): μ [location] 0.000003; σ [scale] 0.000005; ξ [shape] 1.145300.
    - GPD Moment: μ 0.000118; β [scale] 0.000006; ξ 0.949170.
    - GPD MLE: μ 0.000141; β 0.000004; ξ 1.044900.
    - Pickands: μ 0.000268; β 0.000002; ξ 1.156000.
    - Drees-Pickands: μ 0.000265; β 0.000002; ξ 1.150300.
    - Hill GP1 tail index α = 0.949170 (implied by ξ).
    - g-and-h distribution: μ 0.000005; σ 0.000010; g 1.985492; h -0.031397.
  - Panel B: Bank 2 (JPMC)
    - GEV (LRS): μ 0.000004; σ 0.000006; ξ 1.161500.
    - GPD Moment: μ -0.000034; β 0.000009; ξ 0.944180.
    - GPD MLE: μ -0.000072; β 0.000010; ξ 0.935230.
    - Pickands: μ -0.000321; β 0.000027; ξ 0.730800.
    - Drees-Pickands: μ 0.000048; β 0.000007; ξ 1.003600.
    - Hill GP1 α = 1.010200 (implied).
    - g-and-h: μ 0.000006; σ 0.000012; g 2.009666; h -0.031397.
  - Panel C: Bank 3 (Wachovia) and Panel D: Bank 4 (WaMu) analogous parameter sets reported; shape ξ estimates generally around or above 0.9–1.3 for GEV/GPD methods.
  - Note: g-and-h implied higher moments (skewness, kurtosis) are extremely large (e.g., Bank 1: skewness 1.536E+06; kurtosis 1.473E+12).
- Summary of GPD/GPD-based estimation sensitivity and best-fit choices (Table 14, Bank 1–4):
  - Discrete threshold choice q=99.0 for entire upper tail:
    - Best GPD estimation method for upper tail fit:
      - Bank 1: Moment
      - Bank 2: Pickands
      - Bank 3: Moment
      - Bank 4: Moment
    - Best point estimate at q=99.9 percent and dev. from actual:
      - Bank 1: 0.52 [Hill; 3.20]; actual 0.50.
      - Bank 2: 0.54 [Pickands; -8.41]; actual 0.59.
      - Bank 3: 1.33 [MLE; 0.40]; actual 1.32.
      - Bank 4: 1.75 [MLE; 0.32]; actual 1.75.
  - Variable threshold choice (q=97.0–99.5) for specific percentile p=99.9 (AMA):
    - Most consistent GPD method (std. dev. of point estimate) across banks: MLE with small std. dev. values:
      - Bank 1: MLE [0.01]; best threshold quantile 98.60 [dev. from actual value 6.21].
      - Bank 2: MLE [0.03]; best threshold quantile 98.24 [dev. from actual value 11.26].
      - Bank 3: MLE [0.09]; best threshold quantile 97.57 [dev. from actual value -4.92].
      - Bank 4: MLE [0.10]; best threshold quantile 97.52 [dev. from actual value -3.57].
    - Optimal local threshold quantiles (unconditional) and methods reported (e.g., Bank 1: 98.50 [Drees-Pickands; 2.29]).
  - From single threshold upper tail graph (variable percentile p=97.5–99.99 for q=99.0):
    - Most consistent GPD method across percentile variation: Moment with std. dev. values 0.32 (Bank 1), 0.45 (Bank 2), 0.79 (Bank 3), 1.03 (Bank 4).

### Findings on tail behavior, sensitivity, and estimation uncertainty
- Tail behavior is highly sensitive to:
  - Choice of shape parameter ξ.
  - Choice of threshold quantile (threshold-quantile surface, Figures 3 and 4).
  - Choice of tail estimation method (Moment, MLE, Pickands, Drees-Pickands, Hill GP1).
- Aggregate sensitivity diagnostics (Figure 4; Table 12 summary):
  - Absolute relative deviations of estimates from empirical upper-tail values vary widely across banks and estimation methods.
  - Example Bank 1 (Moment): mean deviation 38.45 percent, median deviation 5.47 percent; minimum deviation 1.53 percent; maximum 491.47 percent.
  - Correlation ρ between deviation and threshold ranges reported e.g., Bank 1 mean deviation correl. ρ 0.54; median correl. 0.61.
- Point-estimate dispersion across percentile levels (Figure 5; Table 13 summary):
  - For Bank 1 (Moment): mean absolute deviation across threshold range 29.10 percent, median 26.64 percent; actual upper-tail value 0.10 percent (reference).
  - Deviations show negative correlation with percentile (e.g., correl. ρ -0.95 reported in Table 13 for one series), indicating systematic patterns depending on percentile selection.
- Single-point estimate threshold sensitivity (Figures 6–9):
  - Deviations at specific percentile targets (e.g., 99.9th) can range from large negative to very large positive percentages depending on threshold and estimation method.
  - Hill GP1 often scales differently (plotted on a separate axis in figures) and can produce outlying behavior versus other GPD methods.
- Overall implication: EVT-based quantile estimation for operational risk capital (especially at extreme percentiles such as 99.9th and above) is subject to large model, threshold, and estimation-method uncertainty; careful threshold selection and sensitivity analysis are essential.

*Italic line: Source: _wp07239 - REFERENCES (PDF chapter/section).*

### 99.5 percent in order to avoid high parameter uncertainty for an overly restrictive designation of exceedances.

### 99.5 percent in order to avoid high parameter uncertainty for an overly restrictive designation of exceedances.

### Estimation approach and threshold choice
- Large sample size of quasi-empirical losses permits a high threshold choice to support estimation of conditional mean excess; threshold quantile range chosen: 98.0–99.9 percent with smaller intervals for global optimization of the threshold choice.
- Tables 12 and 13 report only results for the estimation method that provides the best original upper tail fit (see Tables 8–10).
- Variable percentile range for discrete threshold choice considered: p = 97.5–99.99.

### Summary analysis of GPD estimation results — overview
- Simulated losses over quarterly gross income reported for four banks: Bank 1 (BoA), Bank 2 (JPMC), Bank 3 (Wach), Bank 4 (WaMu).

### From single threshold upper tail graph (Figure 8)
- Variable percentile (p=97.5–99.99) for discrete threshold choice q = 99.0:
  - (11) Best percentile for (10)
    - Bank 1 (BoA): 99.92 [-38.89]
    - Bank 2 (JPMC): 99.91 [-30.44]
    - Bank 3 (Wach): 99.92 [-35.20]
    - Bank 4 (WaMu): 99.94 [-32.39]
  - (12) Best percentile *l_p across all GPD methods [avg. dev. from actual value]
    - Bank 1 (BoA): 99.92 [-0.08]
    - Bank 2 (JPMC): 99.91 [-12.59]
    - Bank 3 (Wach): 99.92 [2.60]
    - Bank 4 (WaMu): 99.94 [12.75]
  - (13) Optimal local percentile *,lu_p (unconditional)3 [GPD method; dev. from actual value]
    - Bank 1 (BoA): 99.92 [MLE PL; -14.04]
    - Bank 2 (JPMC): 99.91 [MLE PL; -1.31]
    - Bank 3 (Wach): 99.82 [Pickands; -13.96]
    - Bank 4 (WaMu): 99.82 [Pickands; -13.49]
  - (14) Optimal local percentile *,lc_p for (1) [method; std. dev. from actual value]
    - Bank 1 (BoA): 99.92 [Moment; -38.89]
    - Bank 2 (JPMC): 99.91 [Pickands; -26.48]
    - Bank 3 (Wach): 99.92 [Moment; -35.20]
    - Bank 4 (WaMu): 99.94 [Moment; -32.39]

### From Tables 11 & 12 (threshold-quantile surface)
- Variable threshold choice (q = 98.0–99.9) for entire upper tail (p = 98.0–99.9):
  - (15) Optimal global threshold quantile *,gc_q for (1)2 [method; abs. dev. of median estimate]
    - Bank 1 (BoA): 98.00 [Moment; 2.95]
    - Bank 2 (JPMC): 98.10 [Pickands; 0.15]
    - Bank 3 (Wach): 98.30 [Moment; 0.00]
    - Bank 4 (WaMu): 98.75 [Moment; 0.22]
  - (16) Optimal global percentile *,gc_p for (1) [method; abs. dev. of median estimate]
    - Bank 1 (BoA): 98.30 [Moment; 0.29]
    - Bank 2 (JPMC): 98.75 [Pickands; 0.17]
    - Bank 3 (Wach): 98.95 [Moment; 0.00]
    - Bank 4 (WaMu): 98.35 [Moment; 0.00]

*Source: _wp07239 - 99.5 percent in order to avoid high parameter uncertainty for an overly restrictive designation of exceedances.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07239.pdf_