Comptroller of the Currency

Comptroller of the CurrencyAdministrator of National Banks

Managing Model Risk in Retail Scoring

Dennis Glennon

Credit Risk Analysis DivisionOffice of the Comptroller of the Currency

September 28, 2012

The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the Office of the Comptroller of the Currency. All errors are the responsibilities of the authors.

FilenameFilename2 Agenda Introduction to Model Risk What is it? Why is it relevant?

Managing Model RiskOverview of Sound Model Development and Validation Procedures

Emerging Issues Related to Model Risk2Filename3 Models Risk: What is it?Model Risk Potential for adverse consequences from decisions based on incorrect or misused model outputsModel errors that produce inaccurate outputsModel may be used incorrectly or inappropriately (i.e., using a model outside the environment for which it was designed).

Model risk emerges from the process used to develop models for measuring credit risk.

The process introduces a secondary loss exposure beyond that of credit risk alone

e.g., poor underwriting decisions based on erroneous models or overly broad interpretations of model results.

3Filename4 Model Risk: What is it?Credit Risk: The risk to earnings or capital from an obligor's failure to meet the terms of any contract with the bank or otherwise fails to perform as agreed.

A conceptually distinct exposure to loss.

There are many reasons for poor model-based results including:

Poor modeling (i.e., inadequate understanding of the business)Poor model selection (i.e., overfitting)Inadequate understanding of model useChanging conditions in the market

4Filename5Managing Model RiskThe goal of model-risk analysis is to isolate the effect of a bank's choice of risk-management strategies from those associated with incorrect or misused model output.

Model Validation is an essential component of a sound model-risk management process. Validate at time of model development/implementationOngoing monitoringRe-validate

5Filename6 Model RiskModel validation can be costly.

However, using unvalidated models to underwrite, price, and/or manage risk is potentially an unsafe and unsound practice.

The best defense against model risk is the implementation of formal, prudent, and comprehensive model-validation procedures.

6Filename7Sound modeling practices

In many cases, there are generally accepted methods of building and validating models.

These methods incorporate procedures developed in the finance, statistics, econometrics, and information theory literature.

Although these methods are valid, they may not be appropriate in all applications.

A model selected for its ability to discriminate between high and low risk may perform poorly at predicting the likelihood of default.7Model Risk: Sound Modeling PracticesFilename8Two primary modeling objectives

Classification: The model is used to rank credits by their expected relative performance

Prediction: The model is used to accurately predict the probability of the outcome

Modelers typically have one of these objectives in mind when developing and validating their models 8Models as Decision ToolsFilename9 9 01y Score (quintiles)y1020408060010010305070900102040608010030705090 Score (quintiles)9753111652117411951543[0.1][0.08][0.45][0.44][0.67][0.92][0.3][0.5][0.7][0.9][#B / (#G + #B)][bad rate][bad rate]obs. bad (B) - y=1obs. good (G) - y=0 Model 2 Model 1Model Selection: Which model is better?Filename 10A comparison of models: visual summary

Reliable and AccurateReliable, but not AccurateModels as Decision ToolsOdds: 33:1Bad %: 3.0%Score: 253Odds: 12.2:1Bad %: 7.6%Odds: 33:1Bad %: 3.0%Filename1011Illustrative Example

ln(20/1) = 3.0bad rate = 5%ln(4/1) = 1.4bad rate = 20%Filename1112The model design should reflect how the model will be used.

As such, the choices of:

sample designmodeling techniquevalidation procedures

should reflect the intended purpose for which the model will ultimately be used.

To effectively manage model risk, the right tools must be used.

12Models as Decision ToolsFilename13Models are developed for different purposes i.e., classification or prediction. As such, the choices of:

sample designmodeling techniquevalidation procedures

are driven by the intended purpose for which the model will ultimately be used. 13Models as Decision ToolsFilename14 Model Validation The classification objective is the weaker of two conditions.

There are well-developed methods outlined in the literature and accepted by the industry that are used to assess the validity of models developed under that objective.

In practice, we see:DevelopmentKS / Gini used as the primary model selection toolThese evaluated on the development, hold out, and out-of-time samplesValidation KS / KSStability test (e.g., PSI, characteristic analysis, etc.)Backtesting analysis14Filename15 Model ValidationAlmost all scoring models generate KS values that reject the null that the distribution of good accounts is equal to the distribution of bads.

KS is also used to identify a specific model with the maximum separation across alternative models.

In practice, however, the difference between the maxKS and those of alternative models is never tested using statistical methods (although there are tests outlined in the literature e.g., Krzanowski and Hand, 2011).

More importantly, once a model is selected, few modelers apply a statistical test to determine if the KS has change significantly over time to conclude the model is no longer working as expected.

15Filename16 Model ValidationThe test that have been developed, however, tend to be sensitive to sample size. Given the size of development and validation samples, very small changes may be statistically significant.

OPEN ISSUE 1: Are there tests banks can use to test for statistical significance that are not overly sensitive to sample size.

16Filename17 Model ValidationPredictive models are developed under a model accuracy objective.

As a result, a goodness-of-fit test is required for model selection.

Common performance measures used to evaluate predictive models:

Interval TestChi-Square TestHosmer-Lemeshow (H-L) Test

Unfortunately, the goodness-of-fit tests assume defaults are independent events. If the events are dependent, the tests will reject the null too frequently.17Filename18 Model ValidationThe Vasicek Test is an alternative test of accuracy that allows for dependence.

The Vasicek Test is designed to capture the effect of dependence on the size of the confidence bands.

Formula used to derive the confidence bands

where Vint is the width of the interval; ~ N(0,1); Z.95=1.64; and correlation.

18

Filename 19Vasicek Test: An Example Vasicek Test AnalysisSegmentAccountsEstimated PDActual PDVasicek Upper Bound 95% CI = 0.15 = 0.05 = 0.015110000.000000.002000.0000030.000000.000000.00005210000.000010.000000.0000580.000040.000030.00024310000.000080.000000.0003230.000230.000150.00062410000.000310.001000.0012720.000870.000590.00141510000.001020.004000.0039570.002650.001830.00299610000.003130.008000.0114660.007600.005360.00659710000.010030.019000.0335410.022300.016180.01620810000.037670.063000.1078770.073920.056050.04948910000.187980.267000.3938360.297710.245380.212201010000.759280.549000.9271030.864250.819190.78578Filename1920 Model Validation: Vasicek TestIf is too high the bands are too wide: too many models would pass the test

is not known and has to be estimated. For point-in-time based models, can be very smallFor through-the-cycle based models, can be large

In practice, we often see models fail the interval/Chi-square test, but pass the Vasicek test (especially when samples are large).

Open Issue 2: How do we resolve the inconsistency? 20Filename21Sensitivity of Validation Test to Sample SizeAccuracy tests tend to reject models thatdiscriminate well consistent with the expectations of the LOB

Measurement can be so precise that even a small, non-relevant difference in point estimates can be considered statistically significant.

21Filename22 Illustrative Example22

Filename23 Illustrative Example23

Filename24 Illustrative Example24

Filename25 Interval Tests with Large SamplesConclusion:Statistical difference: significantEconomic difference: insignificant

Solutions?Reduce the number observations using a sample: less powerful testRedefine the testInterval testFocus on capital25Filename26 Interval Tests with Large Samples26(5)(4)-1%+1%0 (1)(2)(3)Filename27 Interval TestRestate the null as an interval defined over an economically acceptable range

If the CI1- around the point estimate is within the in interval, conclude no economically significant difference

May want to reformulate the interval test in terms of an acceptable economic bias in the calculation of regulatory capital

Open Issue 3: How do we reconcile business and statistical significance?27Filename28ConclusionActive management of model risk

Sound model development, implementation, and use of models are vital elements, andRigorous model validation is critical to effective model risk management.

Model Risk should be managed like other risksIdentify the sourceManage it properly

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scorelog(odds)development log oddsactual log odds

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IL-UPNYinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratebad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-2032028656.21.751.7136.6354.50.310.3353.2310439108.80.941.281.4250.20.420.4249.230069368.31.4101.352.8176.60.640.517629091740.72.4221.853.6124.60.870.6124.2280101326.43.6372.334.2881.1110.887.6270102826.83.6372.618.262.11.6160.961.8260108817.15.5603.212.243.82.2241.243.62509739.19.9964.210.830.93.1311.530.82407556.114.11065.321.84.4331.82306915.415.61086.215.46.1422.22253003.920.4616.712.97.2222.42206420.34.736.710.98.452.4210639.59.566.77.711.572.5200474.617.986.85.415.672.5190454.617.986.83.820.892.61802511.58.026.82.727.072.7170161.737.066.91.934.562.81608712.516.91.343.532.8below12150.066.9150.062.9odds atodds atcutoff cum13.9620726571cutoff cum41.388659851scoreactualdevelopmentrangelog oddslog odds3204.02891675695.8707083463104.68951133445.5222605983004.22390976665.17388728822903.70622809244.82510860642803.27336401024.47733681452703.28840188754.12874598892602.83907846353.77963381742502.20827441353.43075618392401.80828877123.08190996982301.68639895362.73436750942251.36097655312.55722731142203.0106208862.38876278922102.25129179862.04122032892001.52605630351.68639895361901.52605630351.33500106671802.44234703540.9932517731700.53062825110.64185388621601.94591014910.2623642645below00

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IL-LMIinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratebad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-20320000.000.0000.000.00310000.000.0000.000.00300000.000.0000.000.0166.2290000.000.02500.000.011828020100.02100.013.283.11.201.283.82702712.57.4213.813.8591.701.659.42601516.613.22013.31141.92.342.240.82503726.912.74712.95.429.73.3122.9302404785.515.47414.020.44.7223.72308092.727.021919.7156.3514.82255882.429.417322.112.67.4435.4220825.814.71221.810.68.675.62101315.615.22021.57.511.8155.9200863.820.81821.55.415.6136.2190401.934.51421.73.820.886.418016150.0821.82.727.046.51706516.7121.81.934.526.616010100.0121.91.343.506.6below20100.0221.90.952.616.6odds atodds atcutoff cum3.5272644684cutoff cum17.3522865973scoreactualdevelopmentrangelog oddslog odds3200031000300002900028004.42004470192702.52572864434.07753744392601.8870696493.73528582692501.93152141163.39114704582401.70474809223.01553490092300.9932517732.70805020112250.87546873742.5336968142201.75785791762.36085400112101.72276659772.01490302052001.33500106671.68639895361900.64185388621.335001066718000.9932517731701.60943791240.641853886216000.2623642645below0-0.1053605157

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Chart54.68213122715.14749447685.08759633524.80073696954.56017282014.45434729634.19569705654.1075897893.7977338593.76120011573.40119738173.41444260842.94968833513.06805293512.41591377832.72129542792.64617479742.37024374152.18605127672.02814824731.93152141161.8562979904


bc_otherBC-OTHERinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratecumulbad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-203092181080.920.9191.21720.610.61722998151620.650.7132.8121.60.870.8121.628977395.61.080.889.2861.190.98627967466.41.5101.06060.81.6111.160.826963944.62.2141.338.2432.3151.443259650303.2211.622.430.43.2211.730.424966219.15.0332.128.221.54.4292.121.423966911.28.2552.917.815.26.2412.615.222983214.16.6553.410.78.5713.52199028.910.1914.37.611.61054.52094536.912.7574.86.413.5615.12042210000.104.85.415.635.119915146.714.83.820.835.21891310000.104.82.727.045.2179510000.104.81.934.525.2169710000.104.81.343.535.2159210000.104.80.952.615.3149010000.004.80.70.005.3below110000.104.80.566.715.3odds atNOTE: these accounts are active accounts only, the bank usescutoff cum19.742566211all booked (non-active and active) for their analysisbad rate cu4.8210042568accountsever 60 dpdcharge-offoverall137394.92.90.59scoreactualdevelopment95 2q368630.50rangelog oddslog odds3q14408.75.80.673094.68213122715.14749447684q12529.56.70.712995.08759633524.800736969596 1q11417.94.80.612894.56017282014.45434729632q25095.33.30.622794.19569705654.1075897893q17565.22.80.542693.7977338593.76120011574q9364.21.50.362593.40119738173.414442608497 1q12882.70.80.302492.94968833513.06805293512q11141.40.40.292392.41591377832.72129542793q11580.300.002292.64617479742.37024374154q777000.002192.18605127672.02814824732091.93152141161.85629799040.522046.9077552791.6863989536199663423.16576789661992.63905732961.3350010667charge-off rate1896.9077552790.993251773for 1996 vintage1796.9077552790.6418538862in 12/971696.9077552790.2623642645excluding non-active accounts1596.907755279-0.10536051571496.907755279-0.3566749439below6.907755279-0.6931471806

bc_other00000000000000000000000000000000000000


bc-nyct aff0000000000000000000000


bc-nyct crBC-NYCTaffinityinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratecumulbad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-202991110000.10.00.1258.61140.900.9118.82899918.55.154.615983.71.211.184.4279521129.30.841.4200.659.41.791.659.8269120779.51.2151.311942.22.3282.142.42591317100.31.0131.266.629.93.2432.530.2249157359.51.7261.347.621.24.5713.221.4239144033.32.9421.72715.16.2893.915.2229126523.84.0512.120.810.78.51084.712.8219120313.56.9832.87.611.61405.720959110.48.8523.26.413.5806.2204310.19.003.25.415.606.2199013.30.003.23.80.006.218906.80.003.22.70.006.2179210000.103.21.934.516.21690200.003.21.40.006.215905.50.003.210.006.2149110000.103.20.758.816.2139110000.103.20.566.716.2below020.003.20.30.006.2odds atNOTE: these accounts are active accounts only, the bank usescutoff cum30.7019135104all booked (non-active and active) for their analysiscum bad r3.1543837241accountsever 60 dpdcharge-offoverall107573.82.40.63scoreactualdevelopment95 2q5503.82.90.76rangelog oddslog odds3q19917.44.80.652996.9077552794.73619844844q7334.130.732892.91777073214.427238977596 1q11053.42.40.712794.86213528584.08429422642q14133.82.40.632694.37575702173.7424202213q7602.41.20.502594.6081656953.39785848044q19724.32.70.632494.08597631263.054001181797 1q6520.90.30.332393.5055573972.71469474382q5161.40.20.142293.16968558072.37024374153q626000.002192.60268968542.02814824734q4390.100.002092.34180580611.85629799042042.31253542381.68639895360.571992.58776403521.3350010667199652502.33897142861891.91692261220.993251773charge-off rate1796.9077552790.6418538862for 1996 vintage1692.99573227360.3364722366in 12/971591.70474809220excluding non-active accounts1496.907755279-0.35667494391696.907755279-0.6931471806below0.6931471806-1.2039728043

bc-nyct cr00000000000000000000000000000000000000


bc-nyct0000000000000000000000


il-otherBC-NYCTcash reservesinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratecumulbad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-202991581550.61.00.6781140.910.9118.8289370911.141.072.483.71.241.184.4279944392.5241.949.859.41.7161.559.8269133736.22.7362.34242.22.3311.942.4259138024.93.9532.827.829.93.2452.330.22491374214.5623.23021.24.5622.921.4239135413.96.7913.92015.16.2843.515.22291252156.3784.31610.78.51074.312.82191063109.1974.87.611.61245.1209545811.1615.26.413.5745.62048410.19.085.25.415.6135.719911913.37.085.23.820.8255.9189716.812.895.32.727.0196.0179484.817.285.31.934.5176.116923204.815.31.441.7106.2159135.515.425.4150.076.314912420.025.40.758.876.3139310000.105.40.566.726.4below4233.315.40.376.936.4odds atNOTE: these accounts are active accounts only, the bank usescutoff cum18.177594555all booked (non-active and active) for their analysiscum bad r5.2144183001accountsever 60 dpdcharge-offoverall199814.50.10.02scoreactualdevelopment95 2q6286.50.20.03rangelog oddslog odds3q17.68.10.20.022995.04342511694.73619844844q18908.20.20.022894.51085950654.427238977596 1q169460.10.022793.66356164614.08429422642q17065.10.10.022693.58905911883.7424202213q19924.400.002593.21486780353.39785848044q22115.20.10.022493.04452243773.054001181797 1q23193.800.002392.63188884012.71469474382q22982.900.002292.70805020112.37024374153q22931.200.002192.3025850932.02814824734q12440.100.002092.07944154171.85629799042042.31253542381.68639895360.021992.58776403521.3350010667199676030.07379981591891.91692261220.993251773charge-off rate1791.56861591790.6418538862for 1996 vintage1692.99573227360.3364722366in 12/971591.70474809220excluding non-active accounts1491.3862943611-0.35667494391696.907755279-0.6931471806below0.6931471806-1.2039728043

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il-upnyBC-NYCTinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratecumulbad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-20299000.000.0216.41140.000.0118.82893173160.310.3163.283.71.241.284.4279655108.20.960.7138.859.41.7111.559.8269173481.61.2211.078.442.22.3402.042.4259232369.41.4331.243.629.93.2752.630.2249309639.22.5771.729.621.24.51393.321.4239344221.84.41512.51915.16.22144.215.2229346114.86.32193.41510.78.52965.212.821930269.59.52884.47.611.63526.320913217.511.81554.96.413.51796.820423224.314.95.415.646.819936352.814.93.820.886.8189211000000.004.92.727.066.8179205.714.934.91.934.576.816911109.114.91.441.756.91594325.014.9150.026.91497614.314.90.758.846.91693233.314.90.566.726.9below70.855.644.90.376.956.9odds atNOTE: these accounts are active accounts only, the bank usescutoff cum19.3627723712all booked (non-active and active) for their analysiscum bad r4.9109226473accountsever 60 dpdcharge-offoverall243756.34.20.67scoreactualdevelopment95 2q58712.17.70.64rangelog oddslog odds3q21309.47.30.7829904.73619844844q136011.89.10.772895.75574221364.427238977596 1q251810.780.752794.68398136644.08429422642q38207.45.30.722694.4018292623.7424202213q57045.93.60.612594.23988686753.39785848044q15075.53.10.562493.66867674683.054001181797 1q21374.21.80.432393.08190996982.71469474382q18302.60.50.192292.69462718082.37024374153q16900.20.10.502192.25129179862.02814824734q1092000.002092.01490302051.85629799042043.09104245341.68639895360.611993.55534806151.33500106671996135494.841397889118911.5129254650.993251773charge-off rate1791.74046617480.6418538862for 1996 vintage1692.3025850930.3364722366in 12/971591.09861228870excluding non-active accounts1491.7917594692-0.35667494391690.6931471806-0.6931471806below-0.2231435513-1.2039728043

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il-lmi00000000000000000000


IL-OTHERinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratecumulbad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-20320227112.50.920.9132.4327.40.310.3329.6310460152.30.730.7112.6232.30.420.4239.2300114366.21.5171.290164.80.670.5166290160556.31.7281.563.6119.60.8130.7117.82802115452.2461.736.2831.2250.983.6270256031.83.0782.121.658.91.7431.159.2260278518.15.21462.915.241.82.3651.442250251910.88.52134.01029.63.3821.829.824020407.611.62375.0214.5932.12302115516.73536.414.96.31332.62259463.920.41937.112.67.4702.92201708.410.6187.110.68.6152.92102557.811.4297.27.511.8303.12001847.811.4217.25.315.9293.21901343.323.3317.33.820.8283.3180715.515.4117.42.727.0193.4170292.231.397.41.934.5103.41606516.717.41.343.533.4below81.343.537.4150.043.5odds atcutoff cum13.0680402243accountsever 60 dpdcharge-offoverall552015.73.40.60scoreactualdevelopment95 2q254414.310.30.72rangelog oddslog odds3q457110.87.20.673204.72295322165.79118266494q398410.270.693105.02585225995.448029639896 1q43229.15.80.643004.19268046295.10473261752q57367.34.30.592904.03069453514.78415284153q53946.53.80.582803.80666248984.41884060784q47115.83.50.602703.45946628984.075841090797 1q44144.32.10.492602.89591193833.73289633952q65681.60.90.562502.37954613413.38777436133q71491.20.10.082402.02814824733.04452243774q5808000.002301.60943791242.7013612132251.36097655312.5336968140.612202.12823170582.36085400112102.05412373372.01490302052002.05412373371.66770682061901.19392246851.33500106671801.70474809220.9932517731700.78845736040.64185388621601.60943791240.2623642645below0.26236426450

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interval and cumulative known odds actual interval oddsinterval and cumulative known odds actual bad rate intervalinterval and cumulative known odds actual bad rate badsinterval and cumulative known odds actual bad rate cumulativeinterval and cumulative known odds double of odd using score-20interval and cumulative known odds development interval oddsinterval and cumulative known odds development bad rate intervalinterval and cumulative known odds development cumul badsinterval and cumulative known odds development bad rate cumulative

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development log oddsactual log odds

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00000000000000000000000000000000000000


0000000000000000000000


IL-LMIinterval and cumulative known oddsactualactualdouble ofdevelopmentdouble ofscore rangetotat loansintervalbad rateodd usingintervalbad ratebad rateodd usingoddsintervalbadscumulativescore-20oddsintervalbadscumulativescore-20320000.000.0000.000.00310000.000.0000.000.00300000.000.0000.000.0166.2290000.000.02500.000.011828020100.02100.013.283.11.201.283.82702712.57.4213.813.8591.701.659.42601516.613.22013.31141.92.342.240.82503726.912.74712.95.429.73.3122.9302404785.515.47414.020.44.7223.72308092.727.021919.7156.3514.82255882.429.417322.112.67.4435.4220825.814.71221.810.68.675.62101315.615.22021.57.511.8155.9200863.820.81821.55.415.6136.2190401.934.51421.73.820.886.418016150.0821.82.727.046.51706516.7121.81.934.526.616010100.0121.91.343.506.6below20100.0221.90.952.616.6odds atodds atcutoff cum3.5272644684cutoff cum17.3522865973scoreactualdevelopmentrangelog oddslog odds3200031000300002900028004.42004470192702.52572864434.07753744392601.8870696493.73528582692501.93152141163.39114704582401.70474809223.01553490092300.9932517732.70805020112250.87546873742.5336968142201.75785791762.36085400112101.72276659772.01490302052001.33500106671.68639895361900.64185388621.335001066718000.9932517731701.60943791240.641853886216000.2623642645below0-0.1053605157

0000000004.42004470192.52572864434.07753744391.8870696493.73528582691.93152141163.39114704581.70474809223.01553490090.9932517732.70805020110.87546873742.5336968141.75785791762.36085400111.72276659772.01490302051.33500106671.68639895360.64185388621.335001066700.9932517731.60943791240.641853886200.26236426450-0.1053605157


0000000004.42004470192.52572864434.07753744391.8870696493.73528582691.93152141163.39114704581.70474809223.01553490090.9932517732.70805020110.87546873742.533696814


Chart12.54192026691.08180517042.92176975811.25276296853.11613498791.43508452533.34926041611.60943791243.65272015411.82454929213.91876181442.09186406174.266699242.44234703544.63427220112.72785282845.11386647253.18635263325.90333659713.46260600986.596625853.9239515763

Development (K-S = 32.1)Validation (K-S = 34.3)Score Bandsln(good/bad)Risk-Rating Model

AHousehold Scoring modelp:\wp\cscore\bank\hhold98\hhold1gm.wk4GM risk model - Wave 2 (development) and Wave 4 (validation) solicitationnote that the reported data for Wave 2 is based onUSED COLS E AND Fcum percentage badfrom the bank'smeans at upper score band used for Wave 4 thus IDevelopment sampleFrom validation reportcluster 1WAVE 2reportscorrected using col g and h, and splitting .5 accross bandsodds by validationWave 4Wave 2alignedoddsoddsnumber ofnumber ofnumber ofnumber ofscore rangesgoodsbadscum perccum percK-Sclus 1from the validationraw scrscoreWave 2WAVE 4goods **bads **goodsbadsWave 2goodbadtablereport20.0764410.992.951665513111058596364412.70404271551665513110.05997911260.2027842227-14.2805110092.16.640.05997911260.202784222726.4965317.43.512277.56611214069665318.574130105912277.56610.10419367620.3050270688-20.08333926587.1323.050.04421456350.102242846130.7365919.8312550587.51241562600.0451959090.090873936634.6166523.14.212772.55351268554966522.559020044525322.51122.50.19538677610.4786542923-28.326751619814.740.120.0459971910.082753286938.2967024.68130854871286052100.04712258720.07532869341.8667529.38513317.54401331045367528.481661272926402.59270.29046924520.6220417633-33.157251816725.955.550.04795988190.06805877845.368031.213575381.51332542700.04888720830.059010054148.5768441.146.213855329.51382533668438.5794655415274307110.38925201670.7320185615-34.276654477541.2669.190.04989556320.05096674451.7168842.98141002991388532300.05077787380.046249033354.9569352.058.1141902631431527569350.3380782918282905620.49113187840.8189481825-32.7816304157.479.070.05110198790.040680587858.0469756.0314195235.51406525100.05111999420.036426914261.170165.1114515199.51432522000.05227239990.030858468764.1770682.1511.514667.5173.51470517970671.285949055143377.5608.50.64734586570.913070379-26.572451321972.0588.280.05282159320.026836813667.471087.0814695158.51463016800.05292062810.02451662870.7371599.0615.314852.5128.514760149715102.952961672529547.52870.75375432150.9574632637-20.370894220784.294.050.05348782770.019876256874.34720138.37149951011494510800.05400100840.015622583178.2725160.124.215107.5801504594725166.312154696130102.51810.86216148080.9854601701-12.329868930691.697.260.0544061510.012374323382.62731229.815237.553151706600.05487431580.008197989288.37739382.6231.91534530.51530540739366.257485029930582.583.50.97229724860.9983758701-2.607862143896.499.020.0552614520.004717710898.18753732.650.67692.510.51538521753732.6190476197692.510.51100.02770275140.00162412992776806465277680646527768064651Development sampleodds by validationlog oddslog odsscoreoddsscore rangeswave 4wave 2rangeWAVE 4Wave 21.08180517042.54192026696442.9512.70404271551.25276296852.92176975816533.518.57413010591.43508452533.11613498796654.222.55902004451.60943791243.3492604161675528.48166127291.82454929213.65272015416846.238.57946554152.09186406173.91876181446938.150.33807829182.44234703544.2666992470611.571.28594905512.72785282844.634272201171515.3102.95296167253.18635263325.113866472572524.2166.31215469613.46260600985.903336597173931.9366.25748502993.92395157636.5966258575350.6732.619047619Development sampleodds by validationscoreWAVE 2cum bad ratescore rangesWAVE 4cum bad raterangegoodsbadsWave 2Wave 2goodsbadsWave 4odds6441665513112.27524679312.70404271553631237.3081710842.951219512265312277.56611.813862640618.574130105910873087.15951802573.529220779266525322.51122.51.581234932922.559020044519424626.78728109934.203463203567526402.59271.186190149428.481661272933386716.22892570974.9746646796684274307110.859948265938.579465541557939315.41798097726.2223415682693282905620.609723908650.338078291878809734.29281026798.09866392670643377.5608.50.411937567171.2859490551102658893.116879977811.546681664871529547.52870.1977863415102.9529616725111977332.042468849315.27557980972530102.51810.0967815728166.3121546961108264471.15659330724.219239373673930582.583.50.0330817012366.257485029992912910.616366339131.92783505157537692.510.50.0036952964732.61904761972821440.264674957450.56944444447432750.090642108799.093333333327768064652.2752467937669660477.308171084

A12.70404271552.9518.57413010593.522.55902004454.228.4816612729538.57946554156.250.33807829188.171.285949055111.5102.952961672515.3166.312154696124.2366.257485029931.9732.61904761950.6

Wave 2 (l-axis): developmentWave 4 (4-axis): validationX-AxisY-AxisTitle

12.70404271552.9518.57413010593.522.55902004454.228.4816612729538.57946554156.250.33807829188.171.285949055111.5102.952961672515.3166.312154696124.273931.975350.6

Wave 2Wave 4X-AxisY-AxisOdds (GM Risk Model: Cluster 1)

2.54192026691.08180517042.92176975811.25276296853.11613498791.43508452533.34926041611.60943791243.65272015411.82454929213.91876181442.09186406174.266699242.44234703544.63427220112.72785282845.11386647253.18635263325.90333659713.46260600986.596625853.9239515763

Wave 2Wave 4X-AxisY-Axisln odds (GM Risk Model: Cluster 1)

0000000000000000000000

Development (K-S = 32.1)Validation (K-S = 34.3)Score Bandsln(good/bad)Log Odds Curve

0.05997911260.20278422270.04421456350.10224284610.0451959090.09087393660.0459971910.08275328690.04712258720.0753286930.04795988190.0680587780.04888720830.05901005410.04989556320.0509667440.05077787380.04624903330.05110198790.04068058780.05111999420.03642691420.05227239990.03085846870.05282159320.02683681360.05292062810.0245166280.05348782770.01987625680.05400100840.01562258310.0544061510.01237432330.05487431580.00819798920.0552614520.00471771080.02770275140.0016241299

Data AData BX-AxisY-AxisTitle

Chart334.106885745734.556885745722.75285171122.55285171116.180464301416.590464301413.626424214513.266424214511.442721230311.48272123039.8435972639.8635972639.06885047018.64885047017.60499432467.90499432467.35225686377.16225686376.8328840976.5428840976.36527527395.96527527395.71822780935.49822780935.40771310665.03771310664.48828280154.63828280154.40829504534.13829504533.57352079673.76352079673.43835419363.37835419362.93171254922.96171254922.26584525432.45584525431.14373694051.4337369405

actualpredictedscore rangedefault rateDefault Rates

Sheet25lgd * ead$10calculated fromsd = sqrt[p(1-p)/n]Probability RangesinitialEFGHIJrow 10LMNt= [act - pred]/sdQVasicek TestLowerUpperact - predbadgoodtotalactual bad rateactual-predictedpredictedobssdt-testreject H0p-valuesprobprobdifferenceDefaultNon-DefaultrangeKKH0: act=pred at .05139100-0.451862.84428728723,137500037.2610693.68-2.734.5610626.5350000.0067253405-4.01466663570.0000645.69823223390.21067.64258555133,932500021.359385.521.222.559569.6250000.00591042142.03031208890.0423832.1266031723-0.41952.52321506794,047500019.058987.99-2.4616.598500.3150000.005260804-4.67609135520.0000024.84379414170.36555.32121072624,445500011.117259.482.1613.277758.3750000.0047971754.50264997560.0000120.6415451214-0.04586.13606151444,414500011.727400.33-0.2411.487345.2050000.004508701-0.53230409360.5945418.3673261012-0.02499.17986314764,50150009.987011.60-0.129.866986.0550000.0042167978-0.28457613170.7759816.3091179100.42306.44252350624,69450006.136348.072.528.656741.9250000.00397512886.33941729370.0000014.7779088.19.1-0.3485.24971623164,51550009.706952.63-1.87.906608.4150000.0038157843-4.71724775590.0000013.8478397.48.10.19301.11284318294,69950006.026334.901.147.166489.4550000.00364671843.12609822760.0017812.92470106.77.40.29240.14420485184,76050004.806192.861.746.546401.5550000.00349708194.97557697750.0000012.15771116.16.70.4178.26376369444,82250003.576013.432.45.976327.9450000.00334945677.16534111540.0000011.44239125.66.10.22208.91139046454,79150004.186113.671.325.506272.8250000.00322363854.09475188840.0000410.86152135.15.60.37140.88565532944,85950002.825823.202.225.046220.1150000.0030931957.17704515480.0000010.28374144.75.1-0.15276.91414007684,72350005.546277.45-0.94.646173.2250000.0029742717-3.02595083820.002499.77569154.24.70.27125.91475226624,87450002.525710.481.624.146108.0350000.002816755.75130920150.000009.12553163.84.2-0.19245.1760398364,75550004.906204.62-1.143.766049.5850000.0026914234-4.23567695780.000028.62314173.33.80.06150.91770968214,84950003.025884.980.363.385974.9350000.00255508181.40895686540.158918.08802182.83.3-0.03157.0856274584,84350003.145918.50-0.182.965868.5050000.0023974967-0.7507831080.452827.48019192.22.8-0.19179.79226271274,82050003.606019.29-1.142.465683.2650000.0021888506-5.20821290910.000006.68619200.12.2-0.29158.68684702524,84150003.175926.71-1.741.434942.0550000.0016811787-10.34988125640.000004.772801000008679.144699612391320.8553003877100000regulatorycapital (K)$136,459$136,648if lgd=.33, then ead = $30 for total dollar value of$3,000,000cap ratio4.554.55test of h-l calculationORIGINAL DATAactualactualcalculatedfrom logitgroupbadgoodtotalactualpredictedactual-pred sqrdcol m / coln(b)(d)col b/col dpred probbadbadn x pred_r x (1-pred_r)11863313750000.3730.3461863172818225113116.11821068393250000.2140.2261068112836008734.1223953404750000.1910.1669538301512969221.8664555444550000.1110.1335556631166457520.2745586441450000.1170.1155865741445080.2836499450150000.1000.099499493364450.0817306469450000.0610.0863064321587639540.1888485451550000.0970.079485395810036422.2529301469950000.0600.07230135832493329.77210240476050000.0480.065240327756930624.75611178482250000.0360.0601782981440028051.34212209479150000.0420.055209275435626016.76713141485950000.0280.0501412521232123951.51014277472350000.0550.04627723220252219.15615126487450000.0250.041126207656119833.07816245475550000.0490.038245188324918117.94117151484950000.0300.0341511693241631.98518157484350000.0310.030157148811440.56419180482050000.0360.025180123324912027.12520159484150000.0320.01415972756971107.120867991321100000hl-stat476.302100000

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Comptroller of the Currency