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Operational Risk Capital: An Analysis
Kabir Dutta ARIA Conference, Washington DC
August 7, 2006
The views expressed in this presentation do not necessarily reflect those of the Federal Reserve System.
Agenda
• Characteristics of Data• Capital Estimation• Results
Data
Background• Under the AMA, Banks must use a combination of the
following four elements in quantifying operational risk exposure:
– internal loss event data
– external loss event data
– scenario analysis
– business environment and internal control factor assessments
• These elements must be combined in a manner that most effectively enables quantification of operational risk exposure.
Data Observations•Results from QIS-4 and the Benchmarking Exercise suggest the following:
–Institutions have made considerable progress in developing internal loss data collection systems.
–Many institutions have acquired external databases, but use of external data varies considerably.
Data Observations (Continued)–Institutions have begun using scenario analysis, but significant work remains in this area.
–Many institutions are using some form of tools to assess Business Environment and Internal Control Factors (BE&ICF).
Unit of measure •The level of granularity seen in QIS-4 varied significantly, with the number of units of measure ranging from 1 to over 100.
•Several banks submitted only ‘Enterprise Level’ capital computations.
•The others computed capital at business level or loss event type level, or some combination of the two.
Operational Loss Frequency (LDCE)
Table 1Number of Losses, Annualized
By Business Line and Event Type
Internal Fraud
External Fraud
Employ-ment
Practices & Workplace
Safety
Clients, Products &
Business Practices
Damage to Physical Assets
Business Disruption & System Failures
Execution, Delivery &
Process Mgmt
Other Fraud TotalPercent of
Total
Corporate Finance 1.4 2.1 10.9 14.1 0.4 22.6 6.0 1.6 59.1 0.3%2.4% 3.6% 18.4% 23.9% 0.7% 38.2% 10.2% 2.7%
Trading & Sales 4.1 2.5 30.4 35.7 5.8 43.7 1,204.2 8.5 1,334.9 7.3%0.3% 0.2% 2.3% 2.7% 0.4% 3.3% 90.2% 0.6%
Retail Banking 419.8 6,218.3 690.2 810.5 103.1 39.2 2,256.7 126.3 385.0 11,049.1 60.1%3.8% 56.3% 6.2% 7.3% 0.9% 0.4% 20.4% 1.1% 3.5%
Commercial Banking 8.5 484.1 31.5 65.4 1.4 5.2 254.2 4.0 80.6 934.9 5.1%0.9% 51.8% 3.4% 7.0% 0.1% 0.6% 27.2% 0.4% 8.6%
96.3 81.4 32.4 6.8 1.8 9.5 549.3 1.0 41.7 820.3 4.5%11.7% 9.9% 3.9% 0.8% 0.2% 1.2% 67.0% 0.1% 5.1%
Agency Services 1.4 6.1 7.0 57.1 1.8 25.8 829.6 928.7 5.1%0.2% 0.7% 0.8% 6.1% 0.2% 2.8% 89.3%
Asset Management 0.3 47.0 19.2 24.7 0.2 6.6 335.1 16.2 449.3 2.4%0.1% 10.5% 4.3% 5.5% 0.0% 1.5% 74.6% 3.6%
Retail Brokerage 11.0 19.0 254.4 606.2 1.8 404.1 36.4 1,333.1 7.3%0.8% 1.4% 19.1% 45.5% 0.1% 30.3% 2.7%
Other 76.3 304.5 321.6 72.6 22.0 3.9 633.9 13.0 13.9 1,461.8 8.0%5.2% 20.8% 22.0% 5.0% 1.5% 0.3% 43.4% 0.9% 1.0%
Total 619.2 7,164.9 1,397.7 1,693.2 136.5 135.6 6,489.7 150.4 583.9 18,371.1 100.0%3.4% 39.0% 7.6% 9.2% 0.7% 0.7% 35.3% 0.8% 3.2% 100.0%
Sample 1: Losses ≥ $10,000 Occurring in Years When Data Capture Appears Stable
Payment & Settlement
Operational Loss Severity (LDCE)
Figure 1. Loss Severity by Business Line Across Three LDCEs Figure 2. Loss Severity by Event Type Across Three LDCEsa
a) The following abbreviations are used: EPWS denotes Employment Practices and Workplace Safety; CPBP denotes Clients, Products and Business Practices; DPA denotesDamage to Physical Assets; BDSF denotes Business Disruption and System Failures; and EDPM denotes Execution, Delivery and Process Management.
0% 20% 40% 60% 80%
Corporate Finance
Trading & Sales
Retail Banking
Cmcl. Banking
Pmt. & Settlement
Agency Services
Asset Mgmt.
Retail Brokerage
No Info. / Other
2000 LDCE 2001 LDCE 2004 LDCE
0% 20% 40% 60% 80%
Internal Fraud
External Fraud
EPWS
CPBP
DPA
BDSF
EDPM
No Information
2000 LDCE 2001 LDCE 2004 LDCE
LDCE Sample Descriptive Details
Number of Individual Losses Reported by 2004 LDCE Participants
Number
of Losses of 10,000 or
More
Total Number of Losses
Total Loss Amount
($M)
Comprehensiveness of Loss Data1
Number of Losses of $10,000 or More
Number of Participants
Fully Comprehensive
Partially Comprehensive
No Information
Provided
0-250 6 640 134,679 212 2 2 2
250-1,000 5 2,253 6,125 283 2 1 2
1,000-2,500 8 13,404 43,814 8,151 5 1 2
2,500+ 4 39,469 1,342,147 17,275 1 3 0
Total 23 55,766 1,526,765 25,920 10 7 6
Data Characteristics and Challenges
• Internal Data– Not too old
• Data quality appears to vary across institutions
– Loss thresholds– Rounding of loss amounts– Length of time series available– Accuracy of loss timestamps
References (continued)
•AMA Benchmarking Exercise, QIS-4, and LDCE results:–http://www.bos.frb.org/bankinfo/qau/pd051205.pdf
•Summary findings of QIS-4:–http://www.federalreserve.gov/boarddocs/press/bcreg/2006/20060224/default.htm
Capital Calculation
Overview
• We found some degree of central tendency among number of institutions using an AMA along some important dimensions.– Capital estimates vs. total assets and other
exposure indicators.– Capital estimates in QIS4 vs. the number of
losses reported in LDCE.
• Use of Loss Distribution Approach
Overview (CONT)
• There is significant variation across all the institutions, with outliers identified along many different dimensions.
• This variation could arise from several different sources.– Cross-firm differences in risk profile.– Differences in data completeness.– Differences in methodology, including use of
the four elements.
Reference
• A Tale of Tails: An Empirical Analysis of Loss Distribution Models for Estimating Operational Risk Capital. White Paper of the Federal Reserve Board, July 2006.
– http://www.federalreserve.gov/generalinfo/basel2/whitepapers.htm
Important Questions
• Operational Risk Characteristics – Which techniques fit the loss data and result
in meaningful capital estimates?– Which commonly used techniques do not fit
the loss data?– Is there a single model that can be used in
all cases? • consistently in some cases
– How do the capital estimates vary with respect to the model assumptions across different institutions classified by assets size, income, and other criteria?
Some Believe and Suggestions:• Operation Loss Data will be impossible to model• Data Contamination and Outliers• Ignore the Outliers• Truncate Severity Distribution• Impossible to Measure the Risk at 99.9% level• The Operational Risk has to be an application of
Extreme Value Theory (EVT)• Body and tail can’t be fitted using same distribution
The Problem:Modeling skewness and kurtosis
• Finding appropriate leptokurtic behavior in the loss data
• Constructing and calibrating models to reflect the observed leptokurtic behavior
• Testing of model behavior
Exploratory Data Analysis
• Various experiments were performed• Skewness and kurtosis are not absolute
concepts– They are relative
• LDCE data vary with many types of kurtosis values but similar skewness
– Heavy-tailed loss severity
• Distributions that can’t model the kurtosis variability will not be able to model the data.
Performance Measures
• We use these criteria to measure the performance of our models:– Good Fit– Realistic– Well Specified– Flexible– Simple
• Model performance measured at the enterprise, business line, and event type levels
Hoaglin, Mosteller, and Tukey (1985)
Using Quantiles to Study Shapes (Chapter 10).
Summarizing the Shape Numerically: The g-and-h Distribution
(Chapter 11) .
In Exploring Data Tables Trend and Shapes
g-and-h distribution is a functional transformation of the standard normal variable:
g
hZgZeBAZhgX
)2/2exp()1()(
, = hgBYA ,
g = 0 is a h-distribution (no skewness)
h= 0 is a g-distribution (no kurtosis)
)2/2exp()(,0
hZBZAZhX = hBYA ,0
g
gZeBAZ
gX
)1()(
0,
Q-Q Plot – Body and Lower Tail
O bserved Loss
g-and-h (4-parameter) LoglogisticEVT 5% 45 degree line
Observed Loss
Exp
ecte
d L
oss
85%0% Observed Loss
Exp
ecte
d L
oss
85% 97%
Q-Q Plot – Upper Tail
O bserved Loss
g-and-h (4-parameter) LoglogisticEVT 5% 45 degree line
Observed Loss
Exp
ecte
d L
oss
97% 99.9%
Q-Q Plot – Extreme Tail
O bserved Loss
g-and-h (4-parameter) LoglogisticEVT 5% 45 degree line
Observed Loss
Exp
ecte
d L
oss
99.9%
Tail Shapes for Various Distributions
Enterprise Level Capital
g-and-h Emp Exp Gamma WeibullEVT5%
EVT10% GPD
Log-logistic
Truncated Lognormal GB2
# Modeled 7 7 7 7 7 7 7 7 7 7 7# that Fit 7 7 0 0 0 6 6 5 4 5 5
25th 0.37 0.15 0.08 0.08 0.03 10.92 2.67 4.15 3.86 0.28 4.11Med 0.79 0.27 0.10 0.11 0.04 43.31 7.85 4.98 4.84 0.90 5.9375th 1.04 1.39 0.22 0.26 0.04 138.34 38.59 10.84 7.31 18.15 7.81
0-1.5% 7 5 - - - 1 2 - - 2 11.5-3% - 2 - - - 1 - - - - 13-20% - - - - - 1 2 4 4 1 220-100% - - - - - 1 2 - - 1 1100% + - - - - - 2 - 1 - 1 -
25th 6.45 1.20 1.29 0.57 2.30 147.50 38.13 71.08 62.61 4.54 61.94Med 16.79 2.28 2.37 0.60 6.09 648.50 117.52 90.90 63.44 16.27 97.2275th 18.65 4.64 5.26 0.78 27.25 2763.62 764.60 192.29 137.34 418.69 160.85
0-50% 7 7 - - - 2 3 - - 2 250-100% - - - - - - - 2 2 - 1100-200% - - - - - - 1 1 1 1 1200-1000% - - - - - 2 1 1 1 1 -1000% + - - - - - 2 1 1 - 1 1
Reasonable Results Rarely Fit the Data
Panel A: Summary Statistics of Capital Estimates as a Percentage of Assets for All Models
Generally Yielded Unreasonable Capital Estimates
Panel D: Capital Estimates as a Percentage of Gross Income for Models that Fit (Frequency)
Panel C: Summary Statistics of Capital Estimates as a Percentage of Gross Income for All Models
Panel B: Capital Estimates as a Percentage of Assets for Models that Fit (Frequency)
Enterprise Level Capital
g-and-h Emp Exp Gamma WeibullEVT5%
EVT10% GPD
Log-logistic
Truncated Lognormal GB2
# Modeled 7 7 7 7 7 7 7 7 7 7 7# that Fit 7 7 0 0 0 6 6 5 4 5 5
25th 0.37 0.15 0.08 0.08 0.03 10.92 2.67 4.15 3.86 0.28 4.11Med 0.79 0.27 0.10 0.11 0.04 43.31 7.85 4.98 4.84 0.90 5.9375th 1.04 1.39 0.22 0.26 0.04 138.34 38.59 10.84 7.31 18.15 7.81
0-1.5% 7 5 - - - 1 2 - - 2 11.5-3% - 2 - - - 1 - - - - 13-20% - - - - - 1 2 4 4 1 220-100% - - - - - 1 2 - - 1 1100% + - - - - - 2 - 1 - 1 -
25th 6.45 1.20 1.29 0.57 2.30 147.50 38.13 71.08 62.61 4.54 61.94Med 16.79 2.28 2.37 0.60 6.09 648.50 117.52 90.90 63.44 16.27 97.2275th 18.65 4.64 5.26 0.78 27.25 2763.62 764.60 192.29 137.34 418.69 160.85
0-50% 7 7 - - - 2 3 - - 2 250-100% - - - - - - - 2 2 - 1100-200% - - - - - - 1 1 1 1 1200-1000% - - - - - 2 1 1 1 1 -1000% + - - - - - 2 1 1 - 1 1
Reasonable Results Rarely Fit the Data
Panel A: Summary Statistics of Capital Estimates as a Percentage of Assets for All Models
Generally Yielded Unreasonable Capital Estimates
Panel D: Capital Estimates as a Percentage of Gross Income for Models that Fit (Frequency)
Panel C: Summary Statistics of Capital Estimates as a Percentage of Gross Income for All Models
Panel B: Capital Estimates as a Percentage of Assets for Models that Fit (Frequency)
Summed ET Capital Estimates
g-and-h Emp Exp Gamma WeibullEVT5%
EVT10% GPD
Log-logistic
Truncated Lognormal
25th 0.65 0.21 0.10 0.11 0.01 1.00 0.85 4.17 1.58 1.21Med 0.93 0.36 0.12 0.15 0.01 2.78 2.66 11.37 3.38 3.0275th 1.36 1.56 0.34 0.43 0.13 239.97 5.70 168.11 4.46 30.46
0-1.5% 5 5 6 6 6 3 3 - 2 21.5-3% 2 2 1 1 1 1 2 2 1 13-20% - - - - - - 1 2 3 220-100% - - - - - - - - 1 1100% + - - - - - 3 1 3 - 1
25th 10.69 3.33 1.65 1.91 0.22 24.20 20.83 79.94 32.38 21.75Med 20.83 8.12 2.56 2.90 0.25 41.57 34.27 170.23 52.90 45.1875th 26.36 32.72 7.26 9.47 2.58 4813.44 96.95 3507.88 87.20 660.85
0-50% 7 7 7 7 7 4 5 2 2 450-100% - - - - - - - - 3 1100-200% - - - - - - 1 2 1 -200-1000% - - - - - - - - 1 -1000% + - - - - - 3 1 3 - 2
Panel D: Capital Estimates as a Percentage of Gross Income for Models that Fit (Frequency)
Reasonable Results Rarely Fit the Data Generally Yielded Unreasonable Capital Estimates
Panel B: Capital Estimates as a Percentage of Assets for All Models (Frequency)
Panel A: Summary Statistics of Capital Estimates as a Percentage of Assets for All Models
Panel C: Summary Statistics of Capital Estimates as a Percentage of Gross Income for All Models
Summed BL Capital Estimates
g-and-h Emp Exp Gamma WeibullEVT5%
EVT10% GPD
Log-logistic
Truncated Lognormal
25th 0.70 0.21 0.12 0.13 0.11 2.71 0.48 9.49 4.98 1.68Med 1.55 0.51 0.23 0.29 0.22 7.32 1.91 22.81 5.87 2.3575th 2.16 1.46 1.39 1.39 1.38 61.73 5.19 35.52 10.16 4.95
0-1.5% 3 5 5 5 5 2 3 - - 21.5-3% 4 2 2 2 2 - 1 - - 33-20% - - - - - 2 3 3 6 120-100% - - - - - 2 - 3 - 1100% + - - - - - 1 - 1 1 -
25th 12.97 3.25 1.86 2.03 1.74 44.03 7.60 166.59 86.34 32.55Med 27.32 11.39 5.25 6.58 4.92 195.94 28.60 341.49 102.92 41.5875th 42.42 29.04 27.25 27.39 26.99 1129.39 123.98 775.11 232.36 72.84
0-50% 5 7 7 7 7 2 4 - - 450-100% 2 - - - - 1 1 1 3 2100-200% - - - - - 1 1 1 2 -200-1000% - - - - - 1 1 4 1 -1000% + - - - - - 2 - 1 1 1
Panel D: Capital Estimates as a Percentage of Gross Income for Models that Fit (Frequency)
Panel C: Summary Statistics of Capital Estimates as a Percentage of Gross Income for All Models
Panel B: Capital Estimates as a Percentage of Assets for All Models (Frequency)
Panel A: Summary Statistics of Capital Estimates as a Percentage of Assets for All Models
Reasonable Results Rarely Fit the Data Generally Yielded Unreasonable Capital Estimates
Enterprise Distribution Rankings
EVT EVT
10% 5%A 1 2 3 5 4 11 6 9 7 8 10B 1 2 3 4 6 5 10 8 7 9 11C 1 2 3 4 5 9 6 10 8 7 11D 1 2 3 5 4 6 7 8 9 10 11E 1 3 4 6 5 2 8 10 9 7 11F 1 2 3 4 6 5 10 7 9 11 8G 1 2 3 4 5 6 10 7 9 11 8Mean 1.0 2.1 3.1 4.6 5.0 6.3 8.1 8.4 8.3 9.0 10.0Med 1.0 2.0 3.0 4.0 5.0 6.0 8.0 8.0 9.0 9.0 11.0SD 0.0 0.4 0.4 0.8 0.8 2.9 1.9 1.3 1.0 1.7 1.4
Model
Weibull Exp g-and-hTruncated Lognormal
Log-logistic GB2 GPDGamma EmpBank
Summary of 99.9% Capital Estimates Summary for g-and-h
Mean 0.70 1.47 1.21 0.98 0.90Std Dev 0.40 0.98 0.82 0.61 0.6025th 0.37 0.70 0.52 0.65 0.55Med 0.79 1.55 1.49 0.93 0.8675th 1.04 2.16 1.70 1.36 1.28
Enterprise Level One Correlation Zero Correlation
Summed by Business LinesOne Correlation Zero Correlation
Summed by Event Types
Conclusion
• Flexibility in terms of skewness-Kurtosis is needed to model oprisk data
• Oprisk data can be modeled using LDA and at 99.9% and at all levels
• Our analysis can be used for product development and securitization in oprisk and other insurance areas