Data Mining: Neural Network Applications by Louise Francis CAS Annual Meeting, Nov 11, 2002 Francis...
-
Upload
marshall-aron-underwood -
Category
Documents
-
view
222 -
download
0
Transcript of 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
Regression Statistics
Multiple R0.9710307205
R Square0.9429006601
Adjusted R Square0.9393319514
Standard Error0.2029924091
Observations18
ANOVA
dfSSMSFSignificance F
Regression110.887155609910.8871556099264.21339705260
Residual160.65929469020.0412059181
Total1711.5464503001
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
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
Regression Statistics
Multiple R0.9526579653
R Square0.9075571988
Adjusted R Square0.9017795237
Standard Error0.2523772415
Observations35
ANOVA
dfSSMSFSignificance F
Regression220.010192466610.0050962333157.07999968160
Residual322.03821670560.063694272
Total3422.0484091722
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
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)
Predicted 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)
Predicted Log(severity)
1989 - Year
Log(severity)
Line Fit Plot
SUMMARY OUTPUT
Regression Statistics
Multiple R0.9631293004
R Square0.9276180493
Adjusted R Square0.9230941774
Standard Error0.2130170023
Observations35
ANOVA
dfSSMSFSignificance F
Regression218.60875949369.3043797468205.04958277710
Residual321.45203978410.0453762433
Total3420.0607992777
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
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
AccidentMonths of Development
Year12-2424-3636-4848-60
19807.3972.3231.6081.446
19817.3972.3231.6081.446
19827.3972.3231.608
19837.3972.323
19847.397
Cumulative Paid Losses
AccidentMonths of Development
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
Cumulative Paid Losses
AccidentMonths of Development
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