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