Data Mining: Neural Network Applications by Louise Francis CAS Annual Meeting, Nov 11, 2002 Francis...

Data Mining:Neural Network Applications

by Louise Francis

CAS Annual Meeting, Nov 11, 2002Francis Analytics and Actuarial Data Mining, [email protected]

Objectives of PresentationIntroduce insurance professionals to neural networksShow that neural networks are a lot like some conventional statisticsIndicate where use of neural networks might be helpfulShow practical examples of using neural networksShow how to interpret neural network models

A Common Actuarial Application:Loss Development

Sheet2

SUMMARY OUTPUT

Regression Statistics

Multiple R0.9286648277

R Square0.8624183622

Adjusted R Square0.8538195098

Standard Error1525.4802372749

Observations18

ANOVA

dfSSMSFSignificance F

Regression1233394530.147228233394530.147228100.29458883370.0000000269

Residual1637233439.26906092327089.9543163

Total17270627969.416289

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept-367.5107783509690.1485067794-0.53250970590.6016907324-1830.55992478471095.5383680829-1830.55992478471095.5383680829

1989 - Year694.062285709569.304222390210.01471860980.0000000269547.1439305784840.9806408405547.1439305784840.9806408405

RESIDUAL OUTPUTPROBABILITY OUTPUT

ObservationPredicted SeverityResidualsPercentileSeverity

1-367.51077835091363.77943506732.7777777778996.2686567164

2326.5515073586727.87026134898.33333333331054.4217687075

31020.613793068286.806701631613.88888888891307.4204946996

41714.6760787774391.163337280919.44444444441892.1933085502

52408.7383644869-516.5450559367252105.8394160584

63102.8006501964-572.680168268630.55555555562530.1204819277

73796.8629359058-262.436706397636.11111111113534.4262295082

84490.9252216152307.908597626741.66666666674665.8878504673

95184.9875073247-173.558935896147.22222222224798.833819242

105879.0497930342-1213.161942566952.77777777785011.4285714286

116573.1120787436863.314038094958.33333333336169.8113207547

127267.1743644531-923.940041020763.88888888896343.2343234323

137961.2366501625-1791.425329407869.44444444446790.0874635568

148655.298935872-1865.2114723151757436.4261168385

159349.3612215814-799.489098819380.55555555568549.8721227622

1610043.4235072909-1158.896948399486.11111111118884.5265588915

1710737.4857930003688.405388988491.666666666711425.8911819887

1811431.54807870984648.097938989497.222222222216079.6460176991

&A

Page &P

Sheet2

996.26865671640

1054.42176870750

1307.42049469960

2105.83941605840

1892.19330855020

2530.12048192770

3534.42622950820

4798.8338192420

5011.42857142860

4665.88785046730

7436.42611683850

6343.23432343230

6169.81132075470

6790.08746355680

8549.87212276220

8884.52655889150

11425.89118198870

16079.64601769910

&A

Page &P

Severity

Predicted Severity

1989 - Year

Severity

1989 - Year Line Fit Plot

Sheet1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

Sample Percentile

Severity

Normal Probability Plot

Sheet4

SUMMARY OUTPUT



R Square0.9429006601



Observations18

ANOVA


Regression110.887155609910.8871556099264.21339705260

Residual160.65929469020.0412059181

Total1711.5464503001


Intercept7.06377566920.091836593276.916786894806.86909083267.25846050596.86909083267.2584605059

t0.1499030.009222165416.254642323100.13035288710.16945311290.13035288710.1694531129

RESIDUAL OUTPUT

ObservationPredicted Log(severity)Residuals

17.0637756692-0.1597587123

27.2136786692-0.2529308601

37.3635816692-0.1877702823

47.51348466920.1389847699

57.6633876693-0.1178957533

67.81329066930.0227315326

77.96319366930.2071125846

88.11309666930.3630315438

98.26299966930.2564766281

108.41290266930.0351307475

118.56280566930.3513399825

128.71270866930.0424353937

138.8626116693-0.135188133

149.0125146693-0.1892955676

159.1624176693-0.1087460639

169.3123206693-0.2202542156

179.4622236694-0.1185864538

189.61212666940.0731828593

&A

Page &P

Sheet4

6.90401695690

6.96074780910

7.1758113870

7.65246943910

7.54549191590

7.83602220190

8.17030625380

8.47612821310

8.51947629740

8.44803341680

8.91414565180

8.7551440630

8.72742353630

8.82321910170

9.05367160540

9.09206645370

9.34363721550

9.68530952870

&A

Page &P

Log(severity)

Predicted Log(severity)

t

Log(severity)

t Line Fit Plot

Sheet5

SUMMARY OUTPUTIncremental Severities at Development Ages 1 and 2



R Square0.9075571988



Observations35

ANOVA


Regression220.010192466610.0050962333157.07999968160

Residual322.03821670560.063694272

Total3422.0484091722


Intercept7.35225042680.093245770178.848085200407.16231516947.54218568437.16231516947.5421856843

Dev_age1.01503404530.085458376311.877525512100.84096117641.18910691410.84096117641.1891069141

1980 - Year0.11596479320.008447847713.727140572800.09875710510.13317248140.09875710510.1331724814

RESIDUAL OUTPUT

ObservationPredicted Log(severity)Residuals

17.3522504268-0.44823347

27.4682152201-0.5074674109

37.5841800133-0.4083686263

47.7001448065-0.0476753674

57.8161095998-0.2706176838

67.932074393-0.0960521911

78.04803918620.1222670676

88.16400397940.3121242336

98.27996877270.2395075248

108.39593356590.0520998509

118.51189835910.4022472926

128.62786315240.1272809106

138.7438279456-0.0164044093

148.8597927388-0.0365736371

158.9757575320.0779140734

169.09172232530.0003441285

179.20768711850.135950097

189.32365191170.3616576169

198.36728447210.1575490739

208.48324926530.2642156638

218.59921405860.1975504071

228.71517885180.2235164466

238.8311436450.409329403

248.94710843820.2464248154

259.06307323150.0095231109

269.17903802470.1321893524

279.2950028179-0.0214560898

289.4109676112-0.1603168351

299.5269324044-0.0429633673

309.6428971976-0.1764940579

319.7588619908-0.1398053565

329.8748267841-0.3005411574

339.9907915773-0.2635658539

3410.1067563705-0.1978458829

3510.2227211638-0.3373096723

&A

Page &P

Sheet5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

Dev_age

Residuals

Dev_age Residual Plot

Reported

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

1989 - Year

Residuals

1989 - Year Residual Plot

patholog

6.90401695690

6.96074780910

7.1758113870

7.65246943910

7.54549191590

7.83602220190

8.17030625380

8.47612821310

8.51947629740

8.44803341680

8.91414565180

8.7551440630

8.72742353630

8.82321910170

9.05367160540

9.09206645370

9.34363721550

9.68530952870

8.5248335460

8.74746492910

8.79676446570

8.93869529840

9.2404730480

9.19353325370

9.07259634240

9.31122737710

9.27354672810

9.25065077610

9.48396903710

9.46640313970

9.61905663440

9.57428562670

9.72722572350

9.90891048760

9.88541149140

&A

Page &P

Log(severity)


Dev_age

Log(severity)

Dev_age Line Fit Plot

6.90401695690

6.96074780910

7.1758113870

7.65246943910

7.54549191590

7.83602220190

8.17030625380

8.47612821310

8.51947629740

8.44803341680

8.91414565180

8.7551440630

8.72742353630

8.82321910170

9.05367160540

9.09206645370

9.34363721550

9.68530952870

8.5248335460

8.74746492910

8.79676446570

8.93869529840

9.2404730480

9.19353325370

9.07259634240

9.31122737710

9.27354672810

9.25065077610

9.48396903710

9.46640313970

9.61905663440

9.57428562670

9.72722572350

9.90891048760

9.88541149140

&A

Page &P

Log(severity)


1989 - Year

Log(severity)

Line Fit Plot

SUMMARY OUTPUT



R Square0.9276180493



Observations35

ANOVA


Regression218.60875949369.3043797468205.04958277710

Residual321.45203978410.0453762433

Total3420.0607992777


Intercept7.26890485450.078703350292.358264748807.10859151197.4292181977.10859151197.4292181971434.9780823539

Dev_age0.77960347460.072130462510.808241733700.6326786550.92652829430.6326786550.92652829432.1806074279

1980 - Year0.12577015470.007130338617.638735131700.11124614250.14029416690.11124614250.14029416691.1340214888

RESIDUAL OUTPUT

ObservationPredicted Log(Cum Sev)Residuals

17.2689048545-0.3648878976

27.3946750091-0.4339272

37.5204451638-0.3446337769

47.64621531850.0062541206

57.7719854732-0.2264935573

67.8977556279-0.061733426

78.02352578260.1467804713

88.14929593730.3268322758

98.27506609190.2444102055

108.40083624660.0471971702

118.52660640130.3875392505

128.6523765560.102767507

138.7781467107-0.0507231744

148.9039168654-0.0806977637

159.02968702010.0239845854

169.1554571747-0.063390721

179.28122732940.0624098861

189.40699748410.2783120445

198.04850832910.0390418362

208.17427848380.1359207985

218.30004863850.0602903257

228.42581879310.1837227576

238.55158894780.2827860969

248.67735910250.2419212543

258.80312925720.0906300645

268.92889941190.1760804444

279.05466956660.0572640487

289.1804397213-0.137920344

299.30620987590.0431993176

309.4319800306-0.1328898668

319.5577501853-0.1345176809

329.68352034-0.243880162

339.8092904947-0.2038719059

349.9350606494-0.1591313434

3510.0608308041-0.2986456414

&A

Page &P

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

Dev_age

Residuals

Dev_age Residual Plot

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

1980 - Year

Residuals

1980 - Year Residual Plot

6.90401695690

6.96074780910

7.1758113870

7.65246943910

7.54549191590

7.83602220190

8.17030625380

8.47612821310

8.51947629740

8.44803341680

8.91414565180

8.7551440630

8.72742353630

8.82321910170

9.05367160540

9.09206645370

9.34363721550

9.68530952870

8.08755016530

8.31019928230

8.36033896410

8.60954155080

8.83437504470

8.91928035690

8.89375932170

9.10497985630

9.11193361530

9.04251937730

9.34940919350

9.29909016380

9.42323250450

9.4396401780

9.60541858880

9.77592930590

9.76218516270

&A

Page &P

Log(Cum Sev)

Predicted Log(Cum Sev)

Dev_age

Log(Cum Sev)

Dev_age Line Fit Plot

6.90401695690

6.96074780910

7.1758113870

7.65246943910

7.54549191590

7.83602220190

8.17030625380

8.47612821310

8.51947629740

8.44803341680

8.91414565180

8.7551440630

8.72742353630

8.82321910170

9.05367160540

9.09206645370

9.34363721550

9.68530952870

8.08755016530

8.31019928230

8.36033896410

8.60954155080

8.83437504470

8.91928035690

8.89375932170

9.10497985630

9.11193361530

9.04251937730

9.34940919350

9.29909016380

9.42323250450

9.4396401780

9.60541858880

9.77592930590

9.76218516270

&A

Page &P

Log(Cum Sev)

Predicted Log(Cum Sev)

1980 - Year

Log(Cum Sev)

Line Fit Plot

AccYrCumClaimsDevAgeExposureCumFrequencyFrequencylnFreq

197419121110000.17381820.1738182-1.7497449

197428542110000.25945450.0856364-2.4576441

197433503110000.30454550.0450909-3.0990724

197439454110000.35863640.0540909-2.9170873

197440575110000.36881820.0101818-4.5871419

197441046110000.37309090.0042727-5.4554795

197441497110000.37718180.0040909-5.4989636

197441558110000.37772730.0005455-7.5137078

197441649110000.37854550.0008182-7.1083038

1974416710110000.37881820.0002727-8.2066717

1974416911110000.37900000.0001818-8.6119535

1974416912110000.37900000.0000000-16.1180957

1974416913110000.37900000.0000000-16.1180957

1974417014110000.37909090.0000909-9.3045512

19752,2191110000.20172730.2017273-1.6008381

19753,3022110000.30018180.0984545-2.3181593

19753,9153110000.35590910.0557273-2.8872838

19754,4624110000.40563640.0497273-3.0011997

19754,6185110000.41981820.0141818-4.2557875

19754,6736110000.42481820.0050000-5.2982974

19754,6967110000.42690910.0020909-6.1701085

19754,7048110000.42763640.0007273-7.2260715

19754,7089110000.42800000.0003636-7.9190812

19754,71110110000.42827270.0002727-8.2066717

19754,71211110000.42836360.0000909-9.3045512

19754,71612110000.42872730.0003636-7.9190812

19754,71613110000.42872730.0000000-16.1180957

19754,71614110000.42872730.0000000-16.1180957

19754,71615110000.42872730.0000000-16.1180957

19754,71616110000.42872730.0000000-16.1180957

19754,71717110000.42881820.0000909-9.3045512

19762,3471110000.21336360.2133636-1.5447569

19763,7022110000.33654550.1231818-2.0940930

19764,2783110000.38890910.0523636-2.9495410

19764,7684110000.43345450.0445455-3.1112429

19764,9155110000.44681820.0133636-4.3152105

19764,9836110000.45300000.0061818-5.0861267

19765,0037110000.45481820.0018182-6.3098633

19765,0078110000.45518180.0003636-7.9190812

19765,0129110000.45563640.0004545-7.6959927

19765,01210110000.45563640.0000000-16.1180957

19765,01311110000.45572730.0000909-9.3045512

19765,01412110000.45581820.0000909-9.3045512

19765,01513110000.45590910.0000909-9.3045512

19765,01514110000.45590910.0000000-16.1180957

19772,9831120000.24858330.2485833-1.3919767

19774,3462120000.36216670.1135833-2.1752176

19775,0553120000.42125000.0590833-2.8288047

19775,6964120000.47466670.0534167-2.9296306

19775,8185120000.48483330.0101667-4.5886310

19775,8616120000.48841670.0035833-5.6314339

19775,8847120000.49033330.0019167-6.2571155

19775,8928120000.49100000.0006667-7.3130704

19775,8969120000.49133330.0003333-8.0060676

19775,89710120000.49141670.0000833-9.3914626

19775,90011120000.49166670.0002500-8.2936497

19775,90012120000.49166670.0000000-16.1180957

19782,5381120000.21150000.2115000-1.5535298

19783,9062120000.32550000.1140000-2.1715560

19784,6333120000.38608330.0605833-2.8037338

19785,1234120000.42691670.0408333-3.1982541

19785,2425120000.43683330.0099167-4.6135284

19785,2756120000.43958330.0027500-5.8961180

19785,2867120000.44050000.0009167-6.9946576

19785,2928120000.44100000.0005000-7.6007025

19785,2989120000.44150000.0005000-7.6007025

19785,30210120000.44183330.0003333-8.0060676

19785,30411120000.44200000.0001667-8.6989149

19785,30412120000.44200000.0000000-16.1180957

19785,30613120000.44216670.0001667-8.6989149

19793,5481120000.29566670.2956667-1.2185222

19795,1902120000.43250000.1368333-1.9889909

19795,7793120000.48158330.0490833-3.0142337

19796,2064120000.51716670.0355833-3.3358751

19796,3135120000.52608330.0089167-4.7198219

19796,3296120000.52741670.0013333-6.6199982

19796,3397120000.52825000.0008333-7.0899568

19796,3438120000.52858330.0003333-8.0060676

19796,3479120000.52891670.0003333-8.0060676

19796,34710120000.52891670.0000000-16.1180957

19796,34811120000.52900000.0000833-9.3914626

19796,34812120000.52900000.0000000-16.1180957

19796,34813120000.52900000.0000000-16.1180957

19804,5831120000.38191670.3819167-0.9625526

19806,1062120000.50883330.1269167-2.0642238

19806,6563120000.55466670.0458333-3.0827415

19807,0324120000.58600000.0313333-3.4630696

19807,1285120000.59400000.0080000-4.8283012

19807,1396120000.59491670.0009167-6.9946576

19807,1477120000.59558330.0006667-7.3130704

19807,1508120000.59583330.0002500-8.2936497

19807,1519120000.59591670.0000833-9.3914626

19807,15310120000.59608330.0001667-8.6989149

19807,15411120000.59616670.0000833-9.3914626

19807,15412120000.59616670.0000000-16.1180957

19814,4301120000.36916670.3691667-0.9965068

19815,9672120000.49725000.1280833-2.0550734

19816,5103120000.54250000.0452500-3.0955504

19816,7754120000.56458330.0220833-3.8129276

19816,8545120000.57116670.0065833-5.0231989

19816,8736120000.57275000.0015833-6.4481598

19816,8837120000.57358330.0008333-7.0899568

19816,8898120000.57408330.0005000-7.6007025

19816,8929120000.57433330.0002500-8.2936497

19816,89410120000.57450000.0001667-8.6989149

19816,89511120000.57458330.0000833-9.3914626

19824,4081110000.40072730.4007273-0.9144740

19825,8492110000.53172730.1310000-2.0325572

19826,2643110000.56945450.0377273-3.2773694

19826,5264110000.59327270.0238182-3.7373018

19826,5715110000.59736360.0040909-5.4989636

19826,5896110000.59900000.0016364-6.4152177

19826,5947110000.59945450.0004545-7.6959927

19826,5968110000.59963640.0001818-8.6119535

19826,6009110000.60000000.0003636-7.9190812

19826,60210110000.60018180.0001818-8.6119535

19834,8611110000.44190910.4419091-0.8166509

19836,4372110000.58518180.1432727-1.9430046

19836,8693110000.62445450.0392727-3.2372224

19837,1344110000.64854550.0240909-3.7259166

19837,1965110000.65418180.0056364-5.1784984

19837,2056110000.65500000.0008182-7.1083038

19837,2117110000.65554550.0005455-7.5137078

19837,2128110000.65563640.0000909-9.3045512

19837,2149110000.65581820.0001818-8.6119535

19844,2291110000.38445450.3844545-0.9559295

19845,6452110000.51318180.1287273-2.0500585

19846,0533110000.55027270.0370909-3.2943807

19846,4194110000.58354550.0332727-3.4030142

19846,5065110000.59145450.0079091-4.8397298

19846,5236110000.59300000.0015455-6.4723725

19846,5297110000.59354550.0005455-7.5137078

19846,5318110000.59372730.0001818-8.6119535

19853,7271110000.33881820.3388182-1.0822914

19854,8302110000.43909090.1002727-2.2998605

19855,3213110000.48372730.0446364-3.1092042

19855,7174110000.51972730.0360000-3.3242336

19855,7775110000.52518180.0054545-5.2112877

19855,7986110000.52709090.0019091-6.2610757

19855,8027110000.52745450.0003636-7.9190812

19863,5611120000.29675000.2967500-1.2148649

19865,0452120000.42041670.1236667-2.0901647

19865,6563120000.47133330.0509167-2.9775630

19866,0404120000.50333330.0320000-3.4420163

19866,0965120000.50800000.0046667-5.3672888

19866,1116120000.50925000.0012500-6.6845317

19874,2591130000.32761540.3276154-1.1159147

19876,0492130000.46530770.1376923-1.9827330

19876,7673130000.52053850.0552308-2.8962333

19877,2064130000.55430770.0337692-3.3882023

19877,2825130000.56015380.0058462-5.1419542

19884,4241140000.31600000.3160000-1.1520127

19886,7002140000.47857140.1625714-1.8166372

19887,5483140000.53914290.0605714-2.8039303

19888,1054140000.57892860.0397857-3.2242449

19895,0051140000.35750000.3575000-1.0286196

19897,4072140000.52907140.1715714-1.7627550

19898,2873140000.59192860.0628571-2.7668891

Table 3: Cumulative Paid Losses (000s)

AccidentMonths of Development

Year1224364860728496108120132144156168180192204216122436486072SquaredSquared

19802671,9754,5877,37510,66115,23217,88818,54118,93719,13019,18919,20919,23419,23419,24619,24619,24619,246$19,246SeverityPredictedPredictedResidualResidual

19813102,8095,6869,38614,88420,65422,01722,52922,77222,82123,04223,06023,12723,12723,12723,12723,15923,159Year1980 - YearSeverityLog(severity)tYearActualLinearExponentialLinearExponential

19823702,7447,28113,28719,77323,88825,17425,81926,04926,18026,26826,36426,37126,37926,39726,39726,39719800996.26865671646.904016956901980$996$(368)$1,169$1,859,894$29,7843253.7067545305

19835773,8779,61216,96223,76426,71228,39329,65629,83929,94429,99729,99929,99930,04930,04930,049198111054.42176870756.960747809111981$1,054$327$1,358$529,795$92,0864065.1230101302

19845094,51812,06721,21827,19429,61730,85431,24031,59831,88932,00231,94731,96531,98631,986198221307.42049469967.17581138721982$1,307$1,021$1,577$82,258$72,9304274.1433021807

19856305,76316,37224,10529,09132,53133,87834,18534,29034,42034,47934,49834,52434,524198332105.83941605847.652469439131983$2,106$1,715$1,833$153,009$74,6665483.7340876945

19861,0788,06617,51826,09131,80733,88334,82035,48235,60735,93735,95735,96235,962198441892.19330855027.545491915941984$1,892$2,409$2,129$266,819$56,0576866.2613981763

19871,6469,37818,03426,65231,25333,37634,28734,98535,12235,16135,17235,172198552530.12048192777.836022201951985$2,530$3,103$2,473$327,963$3,2347474.7081712062

19881,75411,25620,62427,85731,36033,33134,06134,22734,31734,37834,378198663534.42622950828.170306253861986$3,534$3,797$2,873$68,873$437,1757286.359530262

19891,99710,62821,01529,01433,78836,32937,44637,57137,68137,681198774798.8338192428.476128213171987$4,799$4,491$3,338$94,808$2,134,3329000

19902,16411,53821,54929,16734,44036,52836,95037,09937,099198885011.42857142868.519476297481988$5,011$5,185$3,878$30,123$1,285,3219062.8019323672

19911,92210,93921,35728,48832,98235,33036,05936,059198994665.88785046738.448033416891989$4,666$5,879$4,505$1,471,762$25,9448455.0517104216

19921,96213,05327,86938,56044,46145,98845,9881990107436.42611683858.9141456518101990$7,436$6,573$5,233$745,311$4,853,57211492.03187251

19932,32918,08638,09951,95358,02958,0291991116343.23432343238.755144063111991$6,343$7,267$6,080$853,665$69,45710928.0719280719

19943,34324,80652,05466,20366,2031992126169.81132075478.7274235363121992$6,170$7,961$7,063$3,209,205$797,61612372.5118483412

19953,84734,17159,23259,2321993136790.08746355688.8232191017131993$6,790$8,655$8,205$3,479,014$2,002,34112577.19054242

19966,09033,39233,3921994148549.87212276229.0536716054141994$8,550$9,349$9,532$639,183$964,72214845.0029922202

19975,4515,4511995158884.52655889159.0920664537151995$8,885$10,043$11,074$1,343,042$4,792,11517604.8428645028

19961611425.89118198879.3436372155161996$11,426$10,737$12,864$473,902$2,069,47717364.5345813833

Claims Closed with Payment19971716079.64601769919.6853095287171997$16,080$11,432$14,945$21,604,814$1,287,605

AccidentMonths of Development270627969.416289$37,233,439$21,048,434

Year122436486072000000000000% Unexplained Variance0.1380.078

19742686078581,0901,3331,7432,0002,0762,1132,1292,1372,1412,1432,1432,1452,1452,1452,145% Explained Variance

19752946919131,1951,6202,0762,2342,2932,3202,3312,3392,3412,3432,3432,3432,3432,344Dev Age 1 and 2, Incremental Data

19762836429611,4071,9942,3752,5042,5492,5802,5902,5962,6002,6022,6032,6032,603YearSeverityLog(severity)Dev_age1980 - Year

19772747071,1761,6882,2952,5452,6892,7772,8092,8172,8242,8252,8252,8262,8261980996.26865671646.904016956900

19782696581,2281,8192,2172,4752,6132,6712,6912,7062,7102,7112,7142,71719811054.42176870756.960747809101

19792497711,5812,1012,5282,8162,9302,9612,9732,9792,9862,9882,99219821307.42049469967.17581138702

19803051,1071,7132,3162,7482,9423,0253,0493,0633,0773,0793,08019832105.83941605847.652469439103

19813431,0421,6082,2602,5962,7342,8012,8352,8542,8592,86019841892.19330855027.545491915904

19823501,2421,9222,4072,6612,8342,8872,9022,9112,91519852530.12048192777.836022201905

19834281,2571,8412,3452,6832,8532,9082,9202,92519863534.42622950828.170306253806

19842911,0041,5772,0542,4062,5832,6222,63619874798.8338192428.476128213107

19853031,0011,5752,0802,4442,5862,61719885011.42857142868.519476297408

19863181,0551,9062,5242,8742,95819894665.88785046738.448033416809

19873431,4382,3843,1723,55919907436.42611683858.9141456518010

19883911,6713,0823,77119916343.23432343238.755144063011

19894331,9413,24119926169.81132075478.7274235363012

19905331,92319936790.08746355688.8232191017013

199133919948549.87212276229.0536716054014

19958884.52655889159.0920664537015

Cumulative Reported Claims199611425.89118198879.3436372155016

199716079.64601769919.6853095287017

AccidentMonths of Development198050388.52483354610

Year122436486072000000000000198162958.747464929111

19741,9122,8543,3503,9454,0574,1044,1494,1554,1644,1674,1694,1694,1694,1704,1704,1704,1704,170198266138.796764465712

19752,2193,3023,9154,4624,6184,6734,6964,7044,7084,7114,7124,7164,7164,7164,7164,7164,717198376218.938695298413

19762,3473,7024,2784,7684,9154,9835,0035,0075,0125,0125,0135,0145,0155,0155,0155,0151984103069.24047304814

19772,9834,3465,0555,6965,8185,8615,8845,8925,8965,8975,9005,9005,9005,9005,900198598339.193533253715

19782,5383,9064,6335,1235,2425,2755,2865,2925,2985,3025,3045,3045,3065,306198687139.072596342416

19793,5485,1905,7796,2066,3136,3296,3396,3436,3476,3476,3486,3486,3481987110629.311227377117

19804,5836,1066,6567,0327,1287,1397,1477,1507,1517,1537,1547,1541988106529.273546728118

19814,4305,9676,5106,7756,8546,8736,8836,8896,8926,8946,8951989104119.250650776119

19824,4085,8496,2646,5266,5716,5896,5946,5966,6006,6021990131479.4839690371110

19834,8616,4376,8697,1347,1967,2057,2117,2127,2141991129189.4664031397111

19844,2295,6456,0536,4196,5066,5236,5296,5311992150499.6190566344112

19853,7274,8305,3215,7175,7775,7985,8021993143909.5742856267113

19863,5615,0455,6566,0406,0966,1111994167689.7272257235114

19874,2596,0496,7677,2067,2821995201099.9089104876115

19884,4246,7007,5488,1051996196429.8854114914116

19895,0057,4078,287

19904,8897,314

19914,044Dev Age 1 and 2, Cumulative Data

YearCum SeverityLog(Cum Sev)Dev_age1980 - Year

Outstanding Claims1980996.26865671646.904016956900

19811054.42176870756.960747809101

AccidentMonths of Development19821307.42049469967.17581138702

Year12243648607200000000000019832105.83941605847.652469439103

19741,3811,3361,4621,6601,4067724061919857231334000019841892.19330855027.545491915904

19751,2891,7271,7301,9131,3106493581677330964221119852530.12048192777.836022201905

19761,6051,9771,9471,7091,00654026816679483218141010719863534.42622950828.170306253806

19772,1012,1592,0501,73598858233213966382721218319874798.8338192428.476128213107

19781,9551,9431,8171,3848304601939356311597219885011.42857142868.519476297408

19792,2592,0251,5481,2737523401506836241813419894665.88785046738.448033416809

19802,8151,9911,5581,107540228885528148619907436.42611683858.9141456518010

19812,4081,9731,6059544802281155227151119916343.23432343238.755144063011

19822,3881,8351,2808193541636744211019926169.81132075478.7274235363012

19832,6411,7651,08266333513462341819936790.08746355688.8232191017013

19842,4171,65489667728490421519948549.87212276229.0536716054014

19851,9241,202941610268985519958884.52655889159.0920664537015

19861,8101,59195664820294199611425.89118198879.3436372155016

19872,2731,7921,059626242199716079.64601769919.6853095287017

19882,4031,9661,166693198032548.087550165310

19892,4712,0091,142198140658.310199282311

19902,6422,007198242748.360338964112

19912,366198354848.609541550813

198468668.834375044714

Outstanding Losses198574758.919280356915

198672868.893759321716

AccidentMonths of Development198790009.104979856317

Year122436486072000000000000198890639.111933615318

1974$5,275$8,867$12,476$11,919$8,966$5,367$3,281$1,524$667$348$123$82$18$40$0$0$0$0198984559.042519377319

19756,61711,30613,77314,38610,5934,2342,1101,051436353931011055331990114929.3494091935110

19767,65811,06413,65513,3527,5924,0641,8951,003683384216102935750331991109289.2990901638111

19778,73514,31814,89712,9787,7414,3552,1329104983231769910132141992123739.4232325045112

19788,72215,07015,25711,1895,9593,4731,531942547286177616771993125779.439640178113

19799,34916,47014,32010,5746,5612,8641,328784424212146113381994148459.6054185888114

198011,14516,35114,63611,2735,1592,5881,29057340513481541995176059.7759293059115

198110,93315,01214,7289,0675,1072,4561,400584269120931996173659.7621851627116

198213,32316,21812,6766,2903,3551,407613398192111

198313,89916,95812,4147,7004,1121,637576426331

198414,27215,80610,1568,0053,604791379159

198513,90115,38412,5397,9113,8091,404827

198615,95222,79916,0168,9642,9291,321

198722,77224,14618,3978,3763,373

198825,21626,94717,9508,610

198924,98130,57419,621

199030,38934,128

199128,194

Incurred Losses


Year1224364860728496108120132144156168180192204216

1974$5,542$10,842$17,063$19,294$19,627$20,599$21,169$20,065$19,604$19,478$19,312$19,291$19,252$19,274$19,246$19,246$19,246$19,246

1975$6,927$14,115$19,459$23,772$25,477$24,888$24,127$23,580$23,208$23,174$23,135$23,161$23,137$23,132$23,132$23,130$23,162

1976$8,028$13,808$20,936$26,639$27,365$27,952$27,069$26,822$26,732$26,564$26,484$26,466$26,464$26,436$26,447$26,430

1977$9,312$18,195$24,509$29,940$31,505$31,067$30,525$30,566$30,337$30,267$30,173$30,098$30,100$30,081$30,063

1978$9,231$19,588$27,324$32,407$33,153$33,090$32,385$32,182$32,145$32,175$32,179$32,008$32,032$31,993

1979$9,979$22,233$30,692$34,679$35,652$35,395$35,206$34,969$34,714$34,632$34,625$34,611$34,562

1980$12,223$24,417$32,154$37,364$36,966$36,471$36,110$36,055$36,012$36,071$36,038$36,016

1981$12,579$24,390$32,762$35,719$36,360$35,832$35,687$35,569$35,391$35,281$35,265

1982$15,077$27,474$33,300$34,147$34,715$34,738$34,674$34,625$34,509$34,489

1983$15,896$27,586$33,429$36,714$37,900$37,966$38,022$37,997$38,012

1984$16,436$27,344$31,705$37,172$38,044$37,319$37,329$37,258

1985$15,823$26,323$33,896$36,399$36,791$36,734$36,886

1986$17,914$35,852$43,885$47,524$47,390$47,309

1987$25,101$42,232$56,496$60,329$61,402

1988$28,559$51,753$70,004$74,813

1989$28,828$64,745$78,853

1990$36,479$67,520

1991$33,645

Cum Mean Settlement Size


Year122436486072000000000000

1974$996$3,254$5,346$6,766$7,998$8,739$8,944$8,931$8,962$8,985$8,979$8,972$8,975$8,975$8,972$8,972$8,972$8,972

1975$1,054$4,065$6,228$7,854$9,188$9,949$9,855$9,825$9,816$9,790$9,851$9,850$9,871$9,871$9,871$9,871$9,8801.05837091391.2493821099

1976$1,307$4,274$7,576$9,443$9,916$10,058$10,054$10,129$10,097$10,108$10,119$10,140$10,135$10,134$10,141$10,1411.23994072721.0514179501

1977$2,106$5,484$8,173$10,049$10,355$10,496$10,559$10,679$10,623$10,630$10,622$10,619$10,619$10,633$10,6331.61068258041.2830019258

1978$1,892$6,866$9,827$11,665$12,266$11,966$11,808$11,696$11,742$11,785$11,809$11,784$11,778$11,7730.89854586921.2521142142

1979$2,530$7,475$10,355$11,473$11,508$11,552$11,562$11,545$11,534$11,554$11,547$11,546$11,5391.33713636471.0886139833

1980$3,534$7,286$10,227$11,266$11,575$11,517$11,511$11,637$11,625$11,679$11,678$11,6761.39693989070.9748018736

1981$4,799$9,000$11,215$11,793$12,039$12,208$12,241$12,340$12,306$12,298$12,2981.35774055181.235184726

1982$5,011$9,063$10,730$11,573$11,785$11,761$11,798$11,794$11,789$11,7931.04430133661.0069779925

1983$4,666$8,455$11,415$12,373$12,593$12,734$12,877$12,867$12,8820.9310494570.9329401408

1984$7,436$11,492$13,665$14,200$14,314$14,142$14,092$14,0741.59378586781.3591911991

1985$6,343$10,928$13,560$13,696$13,495$13,662$13,7790.85299500380.9509260024

1986$6,170$12,373$14,622$15,277$15,470$15,5470.97266016141.1321770144

1987$6,790$12,577$15,981$16,379$16,3051.10053405371.0165430186

1988$8,550$14,845$16,890$17,5561.259169661.1803115284

1989$8,885$17,605$18,2761.03914145511.1859103615

1990$11,426$17,3651.28604389960.9863498763

1991$16,0801.4072990685

0

Cum CWP Rate0


Year1224364860728496108120132144156168180192204216

19740.1400.2130.2560.2760.3290.4250.4820.5000.5070.5110.5130.5140.5140.5140.5140.5140.5140.514

19750.1320.2090.2330.2680.3510.4440.4760.4870.4930.4950.4960.4960.4970.4970.4970.4970.497

19760.1210.1730.2250.2950.4060.4770.5000.5090.5150.5170.5180.5190.5190.5190.5190.519

19770.0920.1630.2330.2960.3940.4340.4570.4710.4760.4780.4790.4790.4790.4790.479

19780.1060.1680.2650.3550.4230.4690.4940.5050.5080.5100.5110.5110.5110.512

19790.0700.1490.2740.3390.4000.4450.4620.4670.4680.4690.4700.4710.471

19800.0670.1810.2570.3290.3860.4120.4230.4260.4280.4300.4300.431

19810.0770.1750.2470.3340.3790.3980.4070.4120.4140.4150.415

19820.0790.2120.3070.3690.4050.4300.4380.4400.4410.442

19830.0880.1950.2680.3290.3730.3960.4030.4050.405

19840.0690.1780.2610.3200.3700.3960.4020.404

19850.0810.2070.2960.3640.4230.4460.451

19860.0890.2090.3370.4180.4710.484

19870.0810.2380.3520.4400.489

19880.0880.2490.4080.465

19890.0870.2620.391

19900.1090.263

19910.084

Cum Mean Settlement Size Age To Age Factors


Year122436486072000000000000

19743.2661.6431.2661.1821.0931.0230.9991.0031.0030.9990.9991.0001.0001.0001.0001.0001.000$514

19753.8551.5321.2611.1701.0830.9910.9970.9990.9971.0061.0001.0021.0001.0001.0001.001

19763.2691.7731.2461.0501.0141.0001.0080.9971.0011.0011.0020.9991.0001.0011.000

19772.6041.4901.2291.0301.0141.0061.0110.9951.0010.9991.0001.0001.0011.000

19783.6291.4311.1871.0520.9760.9870.9911.0041.0041.0020.9980.9991.000

19792.9541.3851.1081.0031.0041.0010.9980.9991.0020.9991.0000.999

19802.0621.4041.1021.0270.9950.9991.0110.9991.0051.0001.000

19811.8751.2461.0521.0211.0141.0031.0080.9970.9991.000

19821.8081.1841.0791.0180.9981.0031.0001.0001.000

19831.8121.3501.0841.0181.0111.0110.9991.001

19841.5451.1891.0391.0080.9880.9970.999

19851.7231.2411.0100.9851.0121.009

19862.0051.1821.0451.0131.005

19871.8521.2711.0250.995

19881.7361.1381.039

19891.9821.038

19901.520

All yr Avg2.3231.3441.1181.0411.0161.0021.0020.9991.0011.0011.0001.0001.0001.0001.0001.0001.000

5 Yr Avg1.8191.1741.0321.0041.0031.0041.0030.9991.0021.0001.0001.0001.000

Sel1.8191.1741.0321.0041.0031.0041.0030.9991.0021.0001.0001.0001.0001.0001.0001.0001.000

ATU2.2391.2311.0491.0171.0131.0101.0051.0021.0031.0011.0011.0011.0011.0011.0001.0001.000

Ultimate$36,009$21,377$19,168$17,847$16,512$15,699$13,852$14,102$12,918$11,803$12,306$11,685$11,547$11,779$10,638$10,146$9,880

OS$19,929$21,377$19,168$17,847$16,512$15,699$13,852$14,102$12,918$11,803$12,306$11,685$11,547$11,779$10,638$10,146$9,880

Total241,189

(1)(2)(3)(4)(5)(6)(7)

(5)*(6)(6) - '(5)

Table 4: Development Factors for Paid LossesSelectedAge toCumulativeEstimatedEstimated

Age to AgeUltimateAccidentPaidUltimateOutstanding

AccidentMonths of DevelopmentAgeFactorsFactorsYearLossesLossesLosses

Year1224364860728496108120132144156168180192204216127.24124.9571991$5,451136,042130,591136,042.10130,591.10

19747.3972.3231.6081.4461.4291.1741.0371.0211.0101.0031.0011.0011.0001.0011.0001.0001.000242.0053.4471990$33,392115,09381,701115,093.0081,701.00

19759.0612.0241.6511.5861.3881.0661.0231.0111.0021.0101.0011.0031.0001.0001.0001.001361.3411.7191989$59,232101,81542,583101,815.2142,583.21

19767.4162.6531.8251.4881.2081.0541.0261.0091.0051.0031.0041.0001.0001.0011.000481.1551.2821988$66,20384,84218,63984,841.7318,638.73

19776.7192.4791.7651.4011.1241.0631.0441.0061.0041.0021.0001.0001.0021.000601.0611.1101987$58,02964,4086,37964,408.116,379.11

19788.8762.6711.7581.2821.0891.0421.0131.0111.0091.0040.9981.0011.001721.0221.0461986$45,98848,1162,12848,115.912,127.91

19799.1482.8411.4721.2071.1181.0411.0091.0031.0041.0021.0011.001841.0101.0231985$36,05936,90084136,899.96840.96

19807.4822.1721.4891.2191.0651.0281.0191.0041.0091.0011.000961.0031.0131984$37,09937,57647737,576.33477.33

19815.6971.9231.4781.1731.0681.0271.0201.0041.0011.0001081.0051.0101983$37,68138,04436338,043.56362.56

19826.4171.8321.3511.1261.0631.0221.0051.0031.0021201.0021.0051982$34,37834,53515734,535.02157.02

19835.3221.9771.3811.1651.0751.0311.0031.0031321.0011.0031981$35,17235,27710535,276.93104.93

19845.3321.8681.3541.1811.0611.0121.0041441.0011.0021980$35,96236,0508836,049.8887.88

19855.6911.9521.3341.1581.0711.0211561.0011.0021979$34,52434,5775334,577.3253.32

19866.6532.1351.3841.1531.0341681.0001.0011978$31,98632,0193332,018.5832.58

19877.7662.1071.3641.1171801.0001.0011977$30,04930,0702130,069.7920.79

19887.4202.0981.2721921.0011.0011976$26,39726,4151826,415.2618.26

19898.8831.7332041.0001.0001975$23,15923,159023,159.000.00

19905.483

All yr Avg7.1042.1741.4991.2641.1381.0481.0181.0071.0051.0031.0011.0011.0011.0001.0001.0011.000

5 Yr Avg7.2412.0051.3411.1551.0611.0221.0101.0031.0051.0021.0011.0011.001

Selected7.2412.0051.3411.1551.0611.0221.0101.0031.0051.0021.0011.0011.0011.0001.0001.0011.000

Age to Ult24.9573.4471.7191.2821.1101.0461.0231.0131.0101.0051.0031.0021.0021.0011.0011.0011.000

Ultimate$136,042$115,093$101,815$84,842$64,408$48,116$36,900$37,576$38,044$34,535$35,277$36,050$34,577$32,019$30,070$26,415$23,159

OS$130,591$81,701$42,583$18,639$6,379$2,128$841$477$363$157$105$88$53$33$21$180.0

Total284,177

Incurred Age-to-Age Factors


Year1224364860728496108120132144156168180192204216

19741.9561.5741.1311.0171.0501.0280.9480.9770.9940.9910.9990.9981.0010.9991.0001.0001.000

19752.0381.3791.2221.0720.9770.9690.9770.9840.9990.9981.0010.9991.0001.0001.0001.001

19761.7201.5161.2721.0271.0210.9680.9910.9970.9940.9970.9991.0000.9991.0000.999

19771.9541.3471.2221.0520.9860.9831.0010.9930.9980.9970.9981.0000.9990.999

19782.1221.3951.1861.0230.9980.9790.9940.9991.0011.0000.9951.0010.999

19792.2281.3801.1301.0280.9930.9950.9930.9930.9981.0001.0000.999

19801.9981.3171.1620.9890.9870.9900.9980.9991.0020.9990.999

19811.9391.3431.0901.0180.9850.9960.9970.9950.9971.000

19821.8221.2121.0251.0171.0010.9980.9990.9970.999

19831.7351.2121.0981.0321.0021.0010.9991.000

19841.6641.1591.1721.0230.9811.0000.998

19851.6641.2881.0741.0110.9981.004

19862.0011.2241.0830.9970.998

19871.6821.3381.0681.018

19881.8121.3531.069

19892.2461.218

19901.851

All yr Avg1.9081.3281.1341.0230.9980.9930.9910.9930.9980.9980.9990.9991.0001.0001.0001.0011.000

5 Yr Avg1.9191.2841.0931.0160.9961.0000.9980.9970.9990.9990.9981.0001.000

Sel1.9191.2841.0931.0160.9961.0000.9980.9970.9990.9990.9981.0001.0001.0001.0001.0011.000

ATU2.7011.4081.0961.0030.9870.9910.9910.9930.9960.9960.9970.9991.0001.0001.0001.0011.000

Ultimate$90,868$95,049$86,450$75,031$60,594$46,873$36,546$36,980$37,852$34,368$35,173$35,991$34,550$31,994$30,076$26,448$23,162

OS$85,417$61,657$27,218$8,828$2,565$885$487$(119)$171$(10)$(784)$1,493$2,585$1,945$3,679$3,321$3,916

Total203,254

Reported Age-to-Age Factors


Year122436486072000000000000

19741.4931.1741.1781.0281.0121.0111.0011.0021.0011.0001.0001.0001.0001.0001.0001.0001.000

19751.4881.1861.1401.0351.0121.0051.0021.0011.0011.0001.0011.0001.0001.0001.0001.000

19761.5771.1561.1151.0311.0141.0041.0011.0011.0001.0001.0001.0001.0001.0001.000

19771.4571.1631.1271.0211.0071.0041.0011.0011.0001.0011.0001.0001.0001.000

19781.5391.1861.1061.0231.0061.0021.0011.0011.0011.0001.0001.0001.000

19791.4631.1131.0741.0171.0031.0021.0011.0011.0001.0001.0001.000

19801.3321.0901.0561.0141.0021.0011.0001.0001.0001.0001.000

19811.3471.0911.0411.0121.0031.0011.0011.0001.0001.000

19821.3271.0711.0421.0071.0031.0011.0001.0011.000

19831.3241.0671.0391.0091.0011.0011.0001.000

19841.3351.0721.0601.0141.0031.0011.000

19851.2961.1021.0741.0101.0041.001

19861.4171.1211.0681.0091.002

19871.4201.1191.0651.011

19881.5141.1271.074

19891.4801.119

19901.496

All yr Avg1.4301.1221.0841.0171.0051.0031.0011.0011.0001.0001.0001.0001.0001.0001.0001.0001.000

5 Yr Avg1.4651.1171.0681.0111.0031.0011.0001.0001.0001.0001.0001.0001.000

Sel1.4651.1171.0681.0111.0031.0011.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.000

ATU1.7771.2131.0851.0161.0051.0031.0021.0011.0011.0011.0001.0001.0001.0001.0001.0001.000

Ultimate$7,186$8,868$8,993$8,233$7,320$6,127$5,812$6,540$7,220$6,606$6,897$7,156$6,349$5,307$5,901$5,016$4,717

OS$3,142$1,554$706$128$38$16$10$9$6$4$2$2$1$1$1$10.0

Total5,619

CWP Age-to-Age Factors


Year122436486072000000000000

19742.2651.4141.2701.2231.3081.1471.0381.0181.0081.0041.0021.0011.0001.0011.0001.0001.000

19752.3501.3211.3091.3561.2811.0761.0261.0121.0051.0031.0011.0011.0001.0001.0001.000

19762.2691.4971.4641.4171.1911.0541.0181.0121.0041.0021.0021.0011.0001.0001.000

19772.5801.6631.4351.3601.1091.0571.0331.0121.0031.0021.0001.0001.0001.000

19782.4461.8661.4811.2191.1161.0561.0221.0071.0061.0011.0001.0011.001

19793.0962.0511.3291.2031.1141.0401.0111.0041.0021.0021.0011.001

19803.6301.5471.3521.1871.0711.0281.0081.0051.0051.0011.000

19813.0381.5431.4051.1491.0531.0251.0121.0071.0021.000

19823.5491.5481.2521.1061.0651.0191.0051.0031.001

19832.9371.4651.2741.1441.0631.0191.0041.002

19843.4501.5711.3021.1711.0741.0151.005

19853.3041.5731.3211.1751.0581.012

19863.3181.8071.3241.1391.029

19874.1921.6581.3311.122

19884.2741.8441.224

19894.4831.670

19903.608

All yr Avg3.2231.6271.3381.2121.1181.0461.0171.0081.0041.0021.0011.0011.0001.0001.0001.0001.000

5 Yr Avg3.9751.7101.3001.1501.0581.0181.0071.0041.0031.0011.0011.0011.000

Sel3.9751.7101.3001.1501.0581.0181.0071.0041.0031.0011.0011.0011.0001.0001.0001.0001.000

ATU11.1452.8041.6391.2611.0961.0361.0181.0111.0071.0041.0021.0021.0011.0001.0001.0001.000

Ultimate$3,778$5,392$5,313$4,754$3,901$3,065$2,664$2,665$2,945$2,926$2,867$3,085$2,994$2,718$2,827$2,604$2,344

OS$3,439$3,469$2,072$983$342$107$47$29$20$11$7$5$2$1$1$10.0

Total10,535

CWP Rate Age-to-Age Factors


Year122436486072000000000000

19741.5171.2041.0791.1891.2931.1351.0371.0161.0071.0031.0021.0011.0001.0011.0001.0001.000

19751.5791.1141.1481.3101.2661.0711.0251.0111.0041.0031.0001.0011.0001.0001.0001.000

19761.4381.2951.3141.3751.1751.0501.0171.0111.0041.0021.0011.0011.0001.0001.000

19771.7711.4301.2741.3311.1011.0521.0311.0111.0031.0021.0001.0001.0001.000

19781.5891.5731.3401.1911.1091.0541.0211.0061.0051.0011.0001.0011.001

19792.1171.8421.2371.1831.1111.0391.0101.0031.0021.0021.0011.001

19802.7241.4201.2801.1711.0691.0271.0081.0041.0041.0011.000

19812.2551.4141.3501.1351.0501.0231.0111.0061.0011.000

19822.6741.4451.2021.0981.0621.0181.0051.0021.001

19832.2181.3721.2261.1341.0621.0181.0041.001

19842.5851.4651.2281.1561.0711.0141.005

19852.5491.4281.2291.1631.0541.011

19862.3421.6111.2401.1281.027

19872.9521.4821.2491.110

19882.8221.6371.139

19893.0291.492

19902.412

All yr Avg2.2691.4521.2361.1911.1121.0431.0161.0071.0031.0021.0011.0011.0001.0001.0001.0001.000

5 Yr Avg2.7111.5301.2171.1381.0551.0171.0071.0041.0031.0011.0011.0011.000

Sel2.7111.5301.2171.1381.0551.0171.0071.0041.0031.0011.0011.0011.0001.0001.0001.0001.000

ATU6.2682.3121.5111.2411.0901.0331.0161.0101.0061.0031.0021.0011.0011.0001.0001.0001.000

Ultimate0.5250.6080.5910.5770.5330.5000.4580.4070.4080.4430.4160.4310.4720.5120.4790.5190.497

OS0.4420.3450.2000.1120.0440.0160.0070.0040.0020.0010.0010.0010.0000.0000.0000.0000.000

Total1.176

Revised and Filled in

AccidentEarned

YearExposures

197411,000

197511,000

197611,000

197712,000

AccidentEarned197812,000

YearExposures197912,000

198011,000198012,000

198111,000198112,000

198211,000198211,000

198312,000198311,000

198412,000198411,000

198512,000198511,000

198612,000198612,000

198712,000198713,000

198811,000198814,000

198911,000198914,000

199011,000199014,000

199111,000199113,000

199212,000

199313,000

199414,000

199514,000

199614,000

199713,000

Cape Cod Model

Cumulative Paid Losses


Year1224364860

1980$267$1,975$4,587$7,375$10,661

1981$267$1,975$4,587$7,375

1982$267$1,975$4,587

1983$267$1,975

1984$267

Incremental Paid Losses


Year1224364860

1980$267$1,708$2,612$2,788$3,286

1981$267$1,708$2,612$2,788

1982$267$1,708$2,612

1983$267$1,708

1984$267

Incremental Paid Losses


Year12-2424-3636-4848-60

19806.3971.5291.0671.179

19816.3971.5291.0671.179

19826.3971.5291.067

19836.3971.529

19846.397

Cumulative Paid Losses Age to Age Factors


Year12-2424-3636-4848-60

19807.3972.3231.6081.446

19817.3972.3231.6081.446

19827.3972.3231.608

19837.3972.323

19847.397



Year1224364860

1980$267$1,975$4,587$7,375$10,661

1981$294$2,173$5,046$8,113

1982$323$2,390$5,550

1983$355$2,629

1984$391



Year1224364860

1980$267$1,975$4,587$7,375$10,661

1981$294$2,173$5,046$8,113

1982$243$1,795$4,170

1983$255$1,885

1984$294

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

Accident Year

Average Paid

Average Paid Losses at Dev Age One

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

&A

Page &P

Actual

Linear

Exponential

Actual vs Fitted Line

An Example of a Nonlinear Function

_1094548634.bin

Conventional Statistics: Regression

One of the most common methods of fitting a function is linear regressionModels a relationship between two variables by fitting a straight line through pointsMinimizes a squared deviation between an observed and fitted value

Neural NetworksAlso minimizes squared deviation between fitted and actual values

Can be viewed as a non-parametric, non-linear regression

The Feedforward Neural Network

The Activation FunctionThe sigmoid logistic function

The Logistic Function

Simple Example: One Hidden Node

LINK Excel.Sheet.8 "C:\\Business\\Diagram JT.xls" "Simple Neural Net!R3C2:R16C11" \p

_1065514631.unknown

Function if Network has One Hidden Node

EMBED Equation.3

EMBED Equation.3

_1065515299.unknown

_1066636152.unknown

_1066636185.unknown

_1066636088.unknown

_1065515017.unknown

Development Example: Incremental Payments Used for Fitting

_1094548634.bin

_1094555220.bin

Two Methods for Fitting Development CurveNeural NetworksSimpler model using only development age for predictionMore complex model using development age and accident yearGLM modelExample uses Poisson regressionLike OLS regression, but does not require normalityFits some nonlinear relationshipsSee England and Verrall, PCAS 2001

The Chain Ladder Model

Cumulative paid:

Age to age factor:

Estimate of age to age factor using mean:

_1097848377.unknown

_1097848445.unknown

_1097848361.unknown

Common Approach: The Deterministic Chain Ladder

Estimate of paid at 24 months:

Estimate of Ultimate Paid:

_1097849326.unknown

_1097849345.unknown

_1097848377.unknown

GLM ModelA Stochastic Chain Ladder Model

Poisson Model:

Data often normalized by dividing by an exposure base

_1097849778.unknown

Hidden Nodes for Paid Chain Ladder Example

_1094548634.bin

_1094823369.bin

NN Chain Ladder Model with 3 Nodes

_1094895475.bin

_1098422721.bin

_1094548634.bin

Universal Function ApproximatorThe feedforward neural network with one hidden layer is a universal function approximator

Theoretically, with a sufficient number of nodes in the hidden layer, any continuous nonlinear function can be approximated

Neural Network Curve with Dev Age and Accident Year

_1095057577.bin

_1097930226.bin

_1097935972.bin

_1094548634.bin

GLM Poisson Regression Curve

_1095063284.bin

How Many Hidden Nodes for Neural Network?Too few nodes: Dont fit the curve very well

Too many nodes: Over parameterizationMay fit noise as well as pattern

How Do We Determine the Number of Hidden Nodes?Use methods that assess goodness of fit Hold out part of the sampleResamplingBootstrappingJacknifingAlgebraic formulaUses gradient and Hessian matrices

Hold Out Part of SampleFit model on 1/2 to 2/3 of data

Test fit of model on remaining data

Need a large sample

Cross-ValidationHold out 1/n (say 1/10) of dataFit model to remaining dataTest on portion of sample held outDo this n (say 10) times and average the resultsUsed for moderate sample sizesJacknifing similar to cross-validation

BootstrappingCreate many samples by drawing samples, with replacement, from the original dataFit the model to each of the samplesMeasure overall goodness of fit and create distribution of resultsUsed for small and moderate sample sizes

Jackknife of 95% CI for 2 and 5 Nodes

_1097929787.bin

_1097929937.bin

Another Complexity of Data: Interactions

_1095057577.bin

_1097931533.bin

_1097931659.bin

_1097931712.bin

_1095147464.bin

_1094548634.bin

Technical Predictors of Stock PriceA Complex Multivariate Example

Stock Prediction: Which Indicator is Best?

Moving AveragesMeasures of VolatilitySeasonal IndicatorsThe January effectOscillators

The DataS&P 500 Index since 1930Open High Low Close

Moving AveragesA very commonly used technical indicator1 week MA of returns2 week MA of returns1 month MA of returnsThese are trend following indicatorsA more complicated time series smoother based on running medians called T4253H

Volatility MeasuresFinance literature suggests volatility of market changes over timeMore turbulent market -> higher volatilityMeasuresStandard deviation of returnsRange of returnsMoving averages of above

Seasonal Effects

OscillatorsMay indicate that market is overbought or oversoldMay indicate that a trend is nearing completionSome oscillatorsMoving average differencesStochastic

Stochastic and Relative Strength IndexStochastic based on observation that as prices increase closing prices tend to be closer to upper end of range%K = (C L5)/(H5 L5)C is closing prince, L5 is 5 day low, H5 is 5 day high%D = 3 day moving average of %KRS =(Average x days up closes)/(Avg of x days down Closes)RSI = 100 100/(1+RS)

Measuring Variable Importance

Look at weights to hidden layer

Compute sensitivities: a measure of how much the predicted values error increases when the variables are excluded from the model one at a time

Neural Network ResultVariable ImportanceSmoothed return%K (from stochastic)Smoothed %K2 Week %D 1 Week range of returnsSmoothed standard deviationMonthR2 was .13 or 13% of variance explained

Understanding Relationships Between Predictor and Target Variables: A Tree Example

_1097958741.bin

_1098420950.bin

_1098421725.bin

_1066579412.bin

What are the Relationships between the Variables?

_1066579412.bin

_1097958741.bin

Visualization Method for Understanding Neural Network Functions

Method was published by Plate et al. in Neural Computation, 2000Based on Generalized Additive ModelsDetailed description by Francis in Neural Networks Demystified, 2001Hastie, Tibshirini and Friedman present a similar method

VisualizationMethod is essentially a way to approximate the derivatives of the neural network model

_1097921315.unknown

_1097921387.unknown

Neural Network Result for Smoothed Return

_1066128142.bin

_1097959346.bin

Neural Network Result for Oscillator

_1066144470.bin

_1066144608.bin

_1066132652.bin

Neural Network Result for Smoothed Return and Oscillator

_1066129375.bin

_1097959792.bin

Neural Network Result for Standard Deviation

_1066131935.bin

_1066145135.bin

_1066131417.bin

ConclusionsNeural Networks are a lot like conventional statisticsThey address some problems of conventional statistics: nonlinear relationships, correlated variables and interactionsDespite black block aspect, we now can interpret themFind further information, including many references, at www.casact.org/aboutcas/mdiprize.htmNeural Networks for Statistical Modeling, M. SmithReference for Kohonen Networks:Visual Explorations in Finance, Guido Debok and Teuvo Kohone (Eds), Springer, 1998

Data Mining: Neural Network Applications by Louise Francis CAS Annual Meeting, Nov 11, 2002 Francis...

Documents

Transcript of Data Mining: Neural Network Applications by Louise Francis CAS Annual Meeting, Nov 11, 2002 Francis...