introduction to generalized linear models prepared by louise francis francis analytics and actuarial...
TRANSCRIPT
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Introduction to Generalized Linear Models
Prepared byLouise Francis Francis Analytics and Actuarial Data Mining, Inc.October 3, 2004
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ObjectivesGentle introduction to Linear Models and Generalized Linear ModelsIllustrate some simple applicationsShow examples in commonly available softwareWhich model(s) to use?Practical issues
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A Brief Introduction to Regression
One of most common statistical methods fits a line to dataModel: Y = a+bx + error Error assumed to be Normal
Chart1
199119911991199119915410.36894801813715.4316490031
199219921992199219924868.22401916743985.4514762824
199319931993199319934392.97032555464255.4713035617
199419941994199419944190.79939668184525.4911308411
199519951995199519953892.14956055424795.5109581204
199619961996199619963493.68593076815065.5307853997
199719971997199719974529.44911890785335.5506126791
199819981998199819984977.45413790195605.5704399584
199919991999199919995452.77807391365875.5902672377
200020002000200020005726.90586661316145.6100945171
200120012001200120016686.81635002886415.6299217964
200220022002200220026885.12381307026685.6497490757
200320032003200320038855.43242364836955.6695763551
Severity
Fitted Y
Year
Severity
Workers Comp Sevirity Trend
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.7
197110.9
197211.1
19739.6
197415.7
197514.2
197620.4
197716.1
197810.1
197916.5
198015.2
198115.5
198213.2
19838
19848.2
198510
19865
198710.9
198815
198912.9
199012.9
199110.9
19928
19937.3
19941.8
19953.1
19961.9
19973.6
19984
19993
20007
20019.4
20029.7
Sheet1
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeveritySUMMARY OUTPUT
Accident Year
PriortRegression Statistics
(1)(2)(3)
Multiple R0.7219364186
YearSeverityFitted YXR Square0.5211921926
19913944772913017,291.00$5,410$3,7151Adjusted R Square0.4776642101
19923251066783416,678.00$4,868$3,9852Standard Error1052.729195738
19933049669422816,942.00$4,393$4,2553Observations13
19942778566303316,630.00$4,191$4,5254
19952612867134416,713.00$3,892$4,7965ANOVA
19962351667315716,731.00$3,494$5,0666dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367Regression113269748.696560913269748.696560911.97372730570.0053302696
19983204764321151.0016,438.43$4,977$5,6068Residual1112190626.35514981108238.75955907
19993467863471831.0026,359.69$5,453$5,8769Total1225460375.0517107
20003677063573391.016,420.57$5,727$6,14610
20013949157906161.025,905.80$6,687$6,41611CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612Intercept3445.4118217237619.37154364585.56275446790.00016948892082.18355612554808.6400873222082.18355612554808.640087322
200342232414711671.154,769.05$8,855$6,95613X Variable 1270.019827279378.0334796973.46030740050.005330269698.2692096133441.770444945498.2692096133441.7704449454
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Sheet1
1694.937299015
882.772542885
137.4990219929
-334.6917341593
-903.3613975662
-1571.8448546316
-806.1014937713
-628.1163020565
-422.8121933241
-418.704227904
271.1864282324
199.4740639945
1899.7628472933
X Variable 1
Residuals
X Variable 1 Residual Plot
Sheet2
3715.43164900313715.4316490031
3985.45147628243985.4514762824
4255.47130356174255.4713035617
4525.49113084114525.4911308411
4795.51095812044795.5109581204
5065.53078539975065.5307853997
5335.55061267915335.5506126791
5605.57043995845605.5704399584
5875.59026723775875.5902672377
6145.61009451716145.6100945171
6415.62992179646415.6299217964
6685.64974907576685.6497490757
6955.66957635516955.6695763551
Y
Predicted Y
Year Index
Y
Workers Compensation Claim Severity
Sheet3
3493.6859307681
3892.1495605542
4190.7993966818
4392.9703255546
4529.4491189078
4868.2240191674
4977.4541379019
5410.3689480181
5452.7780739136
5726.9058666131
6686.8163500288
6885.1238130702
8855.4324236483
Sample Percentile
Y
Normal Probability Plot
Severity
Fitted Y
Year
Severity
Workers Comp Sevirity Trend
-
A Brief Introduction to Regression
Fits line that minimizes squared deviation between actual and fitted values
-
Simple Formula for Fitting Line
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.7
197110.9
197211.1
19739.6
197415.7
197514.2
197620.4
197716.1
197810.1
197916.5
198015.2
198115.5
198213.2
19838
19848.2
198510
19865
198710.9
198815
198912.9
199012.9
199110.9
19928
19937.3
19941.8
19953.1
19961.9
19973.6
19984
19993
20007
20019.4
20029.7
Sheet1
WC
SUMMARY OUTPUT
Regression Statistics
(1)(2)(3)(4)(5)(6)(7)(8)
AccidentSeverityMultiple R0.7219364186
YearYFitted YXR Square0.5211921926
19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.4776642101
19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.729195738
19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13
19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259
19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA
19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.696560913269748.696560911.97372730570.0053302696
19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.35514981108238.75955907
19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.0517107
20003677063573391.016,420.57$5,727$6,14610$39131,174.079
20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.4118217237619.37154364585.56275446790.00016948892082.18355612554808.6400873222082.18355612554808.640087322
200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.019827279378.0334796973.46030740050.005330269698.2692096133441.770444945498.2692096133441.7704449454
Sum/Average$5,336749,143.61182.00
B270.02
3,445.41
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Sheet1
1694.937299015
882.772542885
137.4990219929
-334.6917341593
-903.3613975662
-1571.8448546316
-806.1014937713
-628.1163020565
-422.8121933241
-418.704227904
271.1864282324
199.4740639945
1899.7628472933
X Variable 1
Residuals
X Variable 1 Residual Plot
Sheet2
3493.6859307681
3892.1495605542
4190.7993966818
4392.9703255546
4529.4491189078
4868.2240191674
4977.4541379019
5410.3689480181
5452.7780739136
5726.9058666131
6686.8163500288
6885.1238130702
8855.4324236483
Sample Percentile
Y
Normal Probability Plot
Sheet3
199119911991199119915410.36894801813715.4316490031
199219921992199219924868.22401916743985.4514762824
199319931993199319934392.97032555464255.4713035617
199419941994199419944190.79939668184525.4911308411
199519951995199519953892.14956055424795.5109581204
199619961996199619963493.68593076815065.5307853997
199719971997199719974529.44911890785335.5506126791
199819981998199819984977.45413790195605.5704399584
199919991999199919995452.77807391365875.5902672377
200020002000200020005726.90586661316145.6100945171
200120012001200120016686.81635002886415.6299217964
200220022002200220026885.12381307026685.6497490757
200320032003200320038855.43242364836955.6695763551
Y
Fitted Y
Year
Severity
Workers Comp Sevirity Trend
MBD001271B2.unknown
MBD0012AE08.unknown
MBD0012EFC8.unknown
MBD00129396.unknown
MBD00126489.unknown
-
Excel Does RegressionInstall Data Analysis Tool Pak (Add In) that comes with ExcelClick Tools, Data Analysis, Regression
-
Goodness of Fit StatisticsR2: (SS Regression/SS Total)percentage of variance explainedF statistic: (MS Regression/MS Resid)significance of regressionT statistics: Uses SE of coefficient to determine if it is significant significance of coefficientsIt is customary to drop variable if coefficient not significant
Note SS = Sum squared of errors
-
Output of Excel Regression Procedure
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.7
197110.9
197211.1
19739.6
197415.7
197514.2
197620.4
197716.1
197810.1
197916.5
198015.2
198115.5
198213.2
19838
19848.2
198510
19865
198710.9
198815
198912.9
199012.9
199110.9
19928
19937.3
19941.8
19953.1
19961.9
19973.6
19984
19993
20007
20019.4
20029.7
Sheet1
WC
SUMMARY OUTPUT
tRegression Statistics
(1)(2)(3)(4)(5)(6)(7)(8)
AccidentSeverityMultiple R0.72
YearYFitted YXR Square0.52
19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.48
19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73
19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00
19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259
19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA
19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01
19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76
19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05
20003677063573391.016,420.57$5,727$6,14610$39131,174.079
20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64
200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77
Sum/Average$5,336749,143.61182.00
B270.02
3,445.41
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Sheet1
X Variable 1
Residuals
X Variable 1 Residual Plot
Sheet2
Sample Percentile
Y
Normal Probability Plot
Sheet3
Y
Fitted Y
Year
Severity
Workers Comp Sevirity Trend
MBD001271B2.unknown
MBD0012AE08.unknown
MBD0012EFC8.unknown
MBD00129396.unknown
MBD00126489.unknown
-
Assumptions of RegressionErrors independent of value of XErrors independent of value of YErrors independent of prior errorsErrors are from normal distributionWe can test these assumptions
-
Other Diagnostics: Residual PlotPoints should scatter randomly around zeroIf not, a straight line probably is not be appropriate
Chart1
1694.937299015
882.772542885
137.4990219929
-334.6917341593
-903.3613975662
-1571.8448546316
-806.1014937713
-628.1163020565
-422.8121933241
-418.704227904
271.1864282324
199.4740639945
1899.7628472933
X Variable 1
Residuals
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.7
197110.9
197211.1
19739.6
197415.7
197514.2
197620.4
197716.1
197810.1
197916.5
198015.2
198115.5
198213.2
19838
19848.2
198510
19865
198710.9
198815
198912.9
199012.9
199110.9
19928
19937.3
19941.8
19953.1
19961.9
19973.6
19984
19993
20007
20019.4
20029.7
Sheet1
WC
SUMMARY OUTPUT
tRegression Statistics
(1)(2)(3)(4)(5)(6)(7)(8)
AccidentSeverityMultiple R0.72
YearYFitted YXR Square0.52
19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.48
19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73
19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00
19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259
19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA
19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01
19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76
19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05
20003677063573391.016,420.57$5,727$6,14610$39131,174.079
20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64
200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77
Sum/Average$5,336749,143.61182.00
B270.02
3,445.41
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Sheet1
X Variable 1
Residuals
Sheet2
Sample Percentile
Y
Normal Probability Plot
Sheet3
Y
Fitted Y
Year
Severity
Workers Comp Sevirity Trend
MBD001271B2.unknown
MBD0012AE08.unknown
MBD0012EFC8.unknown
MBD00129396.unknown
MBD00126489.unknown
-
Other Diagnostics: Normal PlotPlot should be a straight lineOtherwise residuals not from normal distribution
Chart2
3493.6859307681
3892.1495605542
4190.7993966818
4392.9703255546
4529.4491189078
4868.2240191674
4977.4541379019
5410.3689480181
5452.7780739136
5726.9058666131
6686.8163500288
6885.1238130702
8855.4324236483
Sample Percentile
Y
Normal Probability Plot
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.7
197110.9
197211.1
19739.6
197415.7
197514.2
197620.4
197716.1
197810.1
197916.5
198015.2
198115.5
198213.2
19838
19848.2
198510
19865
198710.9
198815
198912.9
199012.9
199110.9
19928
19937.3
19941.8
19953.1
19961.9
19973.6
19984
19993
20007
20019.4
20029.7
Sheet1
WC
SUMMARY OUTPUT
tRegression Statistics
(1)(2)(3)(4)(5)(6)(7)(8)
AccidentSeverityMultiple R0.72
YearYFitted YXR Square0.52
19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.48
19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73
19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00
19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259
19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA
19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01
19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76
19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05
20003677063573391.016,420.57$5,727$6,14610$39131,174.079
20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64
200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77
Sum/Average$5,336749,143.61182.00
B270.02
3,445.41
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Sheet1
X Variable 1
Residuals
Sheet2
Sample Percentile
Y
Normal Probability Plot
Sheet3
Y
Fitted Y
Year
Severity
Workers Comp Sevirity Trend
MBD001271B2.unknown
MBD0012AE08.unknown
MBD0012EFC8.unknown
MBD00129396.unknown
MBD00126489.unknown
-
Test for autocorrelated errorsAutocorrelation often present in time series dataDurban Watson statistic:If residuals uncorrelated, this is near 2
-
Durban Watson StatisticIndicates autocorrelation present
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.71.1571
197110.91.1091.109
197211.11.1111.232099
19739.61.0961.350380504
197415.71.1571.5623902431
197514.21.1421.7842496577
197620.41.2042.1482365878
197716.11.1612.4941026785
197810.11.1012.746007049
197916.51.1653.1990982121
198015.21.1523.6853611403
198115.51.1554.256592117
198213.21.1324.8184622765
198381.085.2039392586
19848.21.0825.6306622778
1985101.16.1937285056
198651.056.5034149309
198710.91.1097.2122871583
1988151.158.2941302321
198912.91.1299.364073032
199012.91.12910.5720384531
199110.91.10911.7243906445
199281.0812.6623418961
19937.31.07313.5866928545
19941.81.01813.8312533259
19953.11.03114.260022179
19961.91.01914.5309626004
19973.61.03615.054077254
199841.0415.6562403442
199931.0316.1259275545
200071.0717.2547424833
20019.41.09418.8766882768
20029.71.09720.7077270396
2003151.1523.8138860955
Line Fit
WC
SUMMARY OUTPUT
tRegression Statistics
(1)(2)(3)(4)(5)(6)(7)(8)
AccidentSeverityMultiple R0.72
YearYFitted YXR Square0.52
19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.48
19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73
19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00
19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259
19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA
19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01
19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76
19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05
20003677063573391.016,420.57$5,727$6,14610$39131,174.079
20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64
200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77
Sum/Average$5,336749,143.61182.00
B270.02
3,445.41
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Line Fit
0
0
0
0
0
0
0
0
0
0
0
0
0
X Variable 1
Residuals
Expo Fit
0
0
0
0
0
0
0
0
0
0
0
0
0
Sample Percentile
Y
Normal Probability Plot
Econo Model
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
Y
Fitted Y
Year
Severity
Workers Comp Severity Trend
Econo Model (2)
WC
SUMMARY OUTPUT
Regression Statistics
(1)(2)(3)(4)(5)(5)(6)(7)(8)Multiple R0.6992898719
AccidentSeverityln(Severity)R Square0.489006325
YearYYFitted SevXAdjusted R Square0.4425523545
YearSeverityFittedStandard Error0.1904278372
19913944772913017,291.00$5,4108.60$3,9321$75-6(448.91)361991$5,410$3,932Observations13
19923251066783416,678.00$4,8688.49$4,1162$(467)-52,336.63251992$4,868$4,116
19933049669422816,942.00$4,3938.39$4,3093$(943)-43,770.32161993$4,393$4,309ANOVA
19942778566303316,630.00$4,1918.34$4,5114$(1,145)-33,434.2591994$4,191$4,511dfSSMSFSignificance F
19952612867134416,713.00$3,8928.27$4,7225$(1,443)-22,886.8041995$3,892$4,722Regression10.38172667260.381726672610.52668523720.0078114247
19962351667315716,731.00$3,4948.16$4,9436$(1,842)-11,841.8611996$3,494$4,943Residual110.39889037280.0362627612
19972853162997416,299.00$4,5298.42$5,1757$(806)00.001997$4,529$5,175Total120.7806170454
19983204764321151.0016,438.43$4,9778.51$5,4178$(358)1(358.10)11998$4,977$5,417
19993467863471831.0026,359.69$5,4538.60$5,6719$1172234.4541999$5,453$5,671CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20003677063573391.016,420.57$5,7278.65$5,93710$39131,174.0792000$5,727$5,937Intercept8.23034398770.112037914373.460346314907.98375007628.47693789937.98375007628.4769378993
20013949157906161.025,905.80$6,6878.81$6,21511$1,35145,405.06162001$6,687$6,215X Variable 10.04579737220.01411545043.24448535780.00781142470.01472945960.07686528480.01472945960.0768652848
20023882953715211.055,639.55$6,8858.84$6,50712$1,55057,747.87252002$6,885$6,507
200342232414711671.154,769.05$8,8559.09$6,81113$3,520621,119.29362003$8,855$6,811
Sum/Average$5,3368.58$5,175749,143.61182.00
B270.02RESIDUAL OUTPUTPROBABILITY OUTPUT
3,445.41
ObservationPredicted YResidualsStandard ResidualsPercentileY
18.27614135990.3199312071.75477100673.84615384628.1587125985
28.32193873210.16854573970.924446164611.53846153858.2667168703
38.36773610430.02002478470.109832710419.23076923088.3406467815
48.4135334765-0.072886695-0.39977175226.92307692318.387760889
58.4593308487-0.1926139784-1.056456567734.61538461548.4183556038
68.5051282209-0.3464156224-1.900033748242.30769230778.4904844718
78.5509255931-0.1325699893-0.7271249834508.512673822
88.5967229653-0.0840491433-0.460995978357.69230769238.5960725669
98.6425203375-0.0386398414-0.211933290265.38461538468.6038804961
108.6883177097-0.0353870342-0.194092168173.07692307698.6529306755
118.73411508190.07377807610.404660833280.76923076928.8078931579
128.77991245410.05720593990.313765342788.46153846158.837118394
138.82570982630.26307655661.442932430296.15384615389.0887863829
Econo Model (2)
0.0200247847
-0.072886695
-0.1926139784
-0.3464156224
-0.1325699893
-0.0840491433
-0.0386398414
-0.0353870342
0.0737780761
0.0572059399
0.2630765566
12
13
X Variable 1
Residuals
Econo Model Test
19.2307692308
26.9230769231
34.6153846154
42.3076923077
50
57.6923076923
65.3846153846
73.0769230769
80.7692307692
88.4615384615
96.1538461538
Sample Percentile
Y
Normal Probability Plot
Sheet3
199119911991199119915410.36894801818.3677361043
199219921992199219924868.22401916748.4135334765
199319931993199319934392.97032555468.4593308487
199419941994199419944190.79939668188.5051282209
199519951995199519953892.14956055428.5509255931
199619961996199619963493.68593076818.5967229653
199719971997199719974529.44911890788.6425203375
199819981998199819984977.45413790198.6883177097
199919991999199919995452.77807391368.7341150819
200020002000200020005726.90586661318.7799124541
200120012001200120016686.81635002888.8257098263
200220022002200220026885.12381307020
200320032003200320038855.43242364830
Y
Fitted Sev
Year
Severity
Workers Comp Severity Trend
0.319931207
0.1685457397
0.0200247847
-0.072886695
-0.1926139784
-0.3464156224
-0.1325699893
-0.0840491433
-0.0386398414
-0.0353870342
0.0737780761
0.0572059399
0.2630765566
X Variable 1
Residuals
X Variable 1 Residual Plot
8.59607256698.2761413599
8.49048447188.3219387321
8.3877608898.3677361043
8.34064678158.4135334765
8.26671687038.4593308487
8.15871259858.5051282209
8.41835560388.5509255931
8.5126738228.5967229653
8.60388049618.6425203375
8.65293067558.6883177097
8.80789315798.7341150819
8.8371183948.7799124541
9.08878638298.8257098263
Y
Predicted Y
X Variable 1
Y
X Variable 1 Line Fit Plot
8.1587125985
8.2667168703
8.3406467815
8.387760889
8.4183556038
8.4904844718
8.512673822
8.5960725669
8.6038804961
8.6529306755
8.8078931579
8.837118394
9.0887863829
Sample Percentile
Y
Normal Probability Plot
0
0
0
0
0
0
0
0
0
0
0
0
0
Residuals
Predicted Y
Residual
Plot of Residuals
00
00
00
00
00
00
00
00
00
00
00
00
00
Severity
Fitted
Year
Severity
Exponential Trend
(1)(2)(3)(4)(5)(6)
WCHealth
YearSeverityIns IndexCPIFittedStd ResidualSUMMARY OUTPUT
19915,41011.7136.24,158.62.0
19924,86812.7140.34,496.60.6Regression Statistics
19934,39313.6144.54,818.6(0.7)Multiple R0.8988759252
19944,19113.8148.34,718.3(0.8)R Square0.8079779288
19953,89214.3152.44,716.4(1.3)Adjusted R Square0.7695735146
19963,49414.5156.94,581.0(1.7)Standard Error699.2105512592
19974,52915.1160.64,670.2(0.2)Observations13
19984,97715.7163.04,900.00.1
19995,45316.1166.64,960.60.8ANOVA
20005,72717.3172.25,320.60.6dfSSMSFSignificance F
20016,68718.9177.16,062.61.0Regression2.0020,571,421.1010285710.550894121.03867341820.0002610693
20026,88520.7179.97,084.8(0.3)Residual10.004,888,953.95488895.394992245
20038,85523.8184.08,873.9(0.0)Total12.0025,460,375.05
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
Intercept5,903.563,623.591.60.1-2170.313977.4-2170.313977.4
X Variable 1667.63172.153.90.0284.11051.2284.11051.2
Ln(severity)Ln(HI)Ln(CPI)X Variable 2-70.2837.96-1.90.1-154.914.3-154.914.3
8.59607256692.46167134244.9141243937
8.49048447182.53863238354.9437829871
8.3877608892.60909084724.9732795076
8.34064678152.62693076534.9989000379RESIDUAL OUTPUTPROBABILITY OUTPUT
8.26671687032.65745997035.0261805054
8.15871259852.67628172465.0552899346ObservationPredicted YResidualsStandard ResidualsPercentileY
8.41835560382.71164886845.078605420514158.58748209141251.78146592671.96115160083.84615384623493.6859307681
8.5126738222.75086958155.093750200824496.6316319442371.59238722330.582169511911.53846153853892.1495605542
8.60388049612.78042838385.115595729734818.5675545838-425.5972290292-0.666778275419.23076923084190.7993966818
8.65293067552.84808703235.14865659244718.2828539551-527.4834572734-0.826402255326.92307692314392.9703255546
8.80789315792.93792773635.176432178654716.3820496521-824.2324890979-1.291315544634.61538461544529.4491189078
8.8371183943.03050691765.19240114164580.9972951344-1087.3113643663-1.703478187642.30769230774868.2240191674
9.08878638293.17026885995.214663981674670.1975931414-140.7484742336-0.220509013504977.4541379019
84900.026380199277.42775770270.121305175957.69230769235410.3689480181
94960.5852423141492.19283159950.771112838765.38461538465452.7780739136
105320.631376125406.27449048810.636505563573.07692307695726.9058666131
116062.6180685223624.19828150650.977924256280.76923076926686.8163500288
127084.7668306752-199.643017605-0.312778415688.46153846156885.1238130702
138873.8836064896-18.4511828413-0.028907255696.15384615388855.4324236483
X Variable 1
Residuals
X Variable 1 Residual Plot
X Variable 2
Residuals
X Variable 2 Residual Plot
Y
Predicted Y
X Variable 1
Y
X Variable 1 Line Fit Plot
Y
Predicted Y
X Variable 2
Y
X Variable 2 Line Fit Plot
Sample Percentile
Y
Normal Probability Plot
Std Residual
Fitteed Severity
Standardized Residual
Fitted vs Standardized Residual
(1)(2)(3)(4)(1)(2)(3)(4)(5)(2)
WCHealthLag
YearSeverityIns IndexCPIResidualResidual(e(t)-e(t-1))^2e(t)^2FittedStd ResidualSUMMARY OUTPUT
19915,41011.7136.21,251.81,566,956.84,158.62.0
19924,86812.7140.3371.61,251.8774,732.8138,080.94,496.60.6Regression Statistics
19934,39313.6144.5(425.6)371.6635,511.3181,133.04,818.6(0.7)Multiple R0.8988759252
19944,19113.8148.3(527.5)(425.6)10,380.8278,238.84,718.3(0.8)R Square0.8079779288
19953,89214.3152.4(824.2)(527.5)88,060.0679,359.24,716.4(1.3)Adjusted R Square0.7695735146
19963,49414.5156.9(1,087.3)(824.2)69,210.51,182,246.04,581.0(1.7)Standard Error699.2105512592
19974,52915.1160.6(140.7)(1,087.3)895,981.319,810.14,670.2(0.2)Observations13
19984,97715.7163.077.4(140.7)47,600.95,995.14,900.00.1
19995,45316.1166.6492.277.4172,030.1242,253.84,960.60.8ANOVA
20005,72717.3172.2406.3492.27,382.0165,059.05,320.60.6dfSSMSFSignificance F
20016,68718.9177.1624.2406.347,490.8389,623.56,062.61.0Regression2.0020,571,421.1010285710.550894121.03867341820.0002610693
20026,88520.7179.9(199.6)624.2678,714.539,857.37,084.8(0.3)Residual10.004,888,953.95488895.394992245
20038,85523.8184.0(18.5)(199.6)32,830.5340.48,873.9(0.0)Total12.0025,460,375.05
Sum3,459,925.34,888,953.9
DW =0.7077025815CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
Intercept5,903.563,623.591.60.1-2170.313977.4-2170.313977.4
X Variable 1667.63172.153.90.0284.11051.2284.11051.2
Ln(severity)Ln(HI)Ln(CPI)X Variable 2-70.2837.96-1.90.1-154.914.3-154.914.3
8.59607256692.46167134244.9141243937
8.49048447182.53863238354.9437829871
8.3877608892.60909084724.9732795076
8.34064678152.62693076534.9989000379RESIDUAL OUTPUTPROBABILITY OUTPUT
8.26671687032.65745997035.0261805054
8.15871259852.67628172465.0552899346ObservationPredicted YResidualsStandard ResidualsPercentileY
8.41835560382.71164886845.078605420514158.58748209141251.78146592671.96115160083.84615384623493.6859307681
8.5126738222.75086958155.093750200824496.6316319442371.59238722330.582169511911.53846153853892.1495605542
8.60388049612.78042838385.115595729734818.5675545838-425.5972290292-0.666778275419.23076923084190.7993966818
8.65293067552.84808703235.14865659244718.2828539551-527.4834572734-0.826402255326.92307692314392.9703255546
8.80789315792.93792773635.176432178654716.3820496521-824.2324890979-1.291315544634.61538461544529.4491189078
8.8371183943.03050691765.19240114164580.9972951344-1087.3113643663-1.703478187642.30769230774868.2240191674
9.08878638293.17026885995.214663981674670.1975931414-140.7484742336-0.220509013504977.4541379019
84900.026380199277.42775770270.121305175957.69230769235410.3689480181
94960.5852423141492.19283159950.771112838765.38461538465452.7780739136
105320.631376125406.27449048810.636505563573.07692307695726.9058666131
116062.6180685223624.19828150650.977924256280.76923076926686.8163500288
127084.7668306752-199.643017605-0.312778415688.46153846156885.1238130702
138873.8836064896-18.4511828413-0.028907255696.15384615388855.4324236483
X Variable 1
Residuals
X Variable 1 Residual Plot
X Variable 2
Residuals
X Variable 2 Residual Plot
Y
Predicted Y
X Variable 1
Y
X Variable 1 Line Fit Plot
Y
Predicted Y
X Variable 2
Y
X Variable 2 Line Fit Plot
Sample Percentile
Y
Normal Probability Plot
Std Residual
Fitteed Severity
Standardized Residual
Fitted vs Standardized Residual
(1)(2)(3)(4)(5)(6)
WCHealthLaw
YearSeverityIns IndexCPIReformFittedStd ResidualSUMMARY OUTPUT
19915,41011.7136.26.94,158.62.0
19924,86812.7140.37.84,496.60.6Regression Statistics
19934,39313.6144.574,818.6(0.7)Multiple R0.8988759252
19944,19113.8148.36.14,718.3(0.8)R Square0.8079779288
19953,89214.3152.45.64,716.4(1.3)Adjusted R Square0.7695735146
19963,49414.5156.95.34,581.0(1.7)Standard Error699.2105512592
19974,52915.1160.654,670.2(0.2)Observations13
19984,97715.7163.04.54,900.00.1
19995,45316.1166.64.34,960.60.8ANOVA
20005,72717.3172.245,320.60.6dfSSMSFSignificance F
20016,68718.9177.14.56,062.61.0Regression2.0020,571,421.1010285710.550894121.03867341820.0002610693
20026,88520.7179.95.87,084.8(0.3)Residual10.004,888,953.95488895.394992245
20038,85523.8184.06.38,873.9(0.0)Total12.0025,460,375.05
CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
Median days away from workIntercept5,903.563,623.591.60.1-2170.313977.4-2170.313977.4
X Variable 1667.63172.153.90.0284.11051.2284.11051.2
YearAnnualLn(HI)Ln(CPI)X Variable 2-70.2837.96-1.90.1-154.914.3-154.914.3
199262.46167134244.9141243937
199362.53863238354.9437829871
199452.60909084724.9732795076
199552.62693076534.9989000379RESIDUAL OUTPUTPROBABILITY OUTPUT
199652.65745997035.0261805054
199752.67628172465.0552899346ObservationPredicted YResidualsStandard ResidualsPercentileY
199852.71164886845.078605420514158.58748209141251.78146592671.96115160083.84615384623493.6859307681
199962.75086958155.093750200824496.6316319442371.59238722330.582169511911.53846153853892.1495605542
200062.78042838385.115595729734818.5675545838-425.5972290292-0.666778275419.23076923084190.7993966818
200162.84808703235.14865659244718.2828539551-527.4834572734-0.826402255326.92307692314392.9703255546
8.80789315792.93792773635.176432178654716.3820496521-824.2324890979-1.291315544634.61538461544529.4491189078
8.8371183943.03050691765.19240114164580.9972951344-1087.3113643663-1.703478187642.30769230774868.2240191674
9.08878638293.17026885995.214663981674670.1975931414-140.7484742336-0.220509013504977.4541379019
84900.026380199277.42775770270.121305175957.69230769235410.3689480181
94960.5852423141492.19283159950.771112838765.38461538465452.7780739136
105320.631376125406.27449048810.636505563573.07692307695726.9058666131
116062.6180685223624.19828150650.977924256280.76923076926686.8163500288
127084.7668306752-199.643017605-0.312778415688.46153846156885.1238130702
138873.8836064896-18.4511828413-0.028907255696.15384615388855.4324236483
1251.7814659267
371.5923872233
-425.5972290292
-527.4834572734
-824.2324890979
-1087.3113643663
-140.7484742336
77.4277577027
492.1928315995
406.2744904881
624.1982815065
-199.643017605
-18.4511828413
X Variable 1
Residuals
X Variable 1 Residual Plot
1251.7814659267
371.5923872233
-425.5972290292
-527.4834572734
-824.2324890979
-1087.3113643663
-140.7484742336
77.4277577027
492.1928315995
406.2744904881
624.1982815065
-199.643017605
-18.4511828413
X Variable 2
Residuals
X Variable 2 Residual Plot
5410.36894801814158.5874820914
4868.22401916744496.6316319442
4392.97032555464818.5675545838
4190.79939668184718.2828539551
3892.14956055424716.3820496521
3493.68593076814580.9972951344
4529.44911890784670.1975931414
4977.45413790194900.0263801992
5452.77807391364960.5852423141
5726.90586661315320.631376125
6686.81635002886062.6180685223
6885.12381307027084.7668306752
8855.43242364838873.8836064896
Y
Predicted Y
X Variable 1
Y
X Variable 1 Line Fit Plot
5410.36894801814158.5874820914
4868.22401916744496.6316319442
4392.97032555464818.5675545838
4190.79939668184718.2828539551
3892.14956055424716.3820496521
3493.68593076814580.9972951344
4529.44911890784670.1975931414
4977.45413790194900.0263801992
5452.77807391364960.5852423141
5726.90586661315320.631376125
6686.81635002886062.6180685223
6885.12381307027084.7668306752
8855.43242364838873.8836064896
Y
Predicted Y
X Variable 2
Y
X Variable 2 Line Fit Plot
3493.6859307681
3892.1495605542
4190.7993966818
4392.9703255546
4529.4491189078
4868.2240191674
4977.4541379019
5410.3689480181
5452.7780739136
5726.9058666131
6686.8163500288
6885.1238130702
8855.4324236483
Sample Percentile
Y
Normal Probability Plot
1.9611516008
0.5821695119
-0.6667782754
-0.8264022553
-1.2913155446
-1.7034781876
-0.220509013
0.1213051759
0.7711128387
0.6365055635
0.9779242562
-0.3127784156
-0.0289072556
Std Residual
Fitteed Severity
Standardized Residual
Fitted vs Standardized Residual
MBD001F6C34.unknown
MBD001F6C36.unknown
MBD001F6C37.unknown
MBD001F6C35.unknown
MBD00129396.unknown
MBD0012EFC8.unknown
MBD001F6C33.unknown
MBD0012AE08.unknown
MBD00126489.unknown
MBD001271B2.unknown
-
Non-Linear RelationshipsThe model fit was of the form:Severity = a + b*YearA more common trend model is:SeverityYear=SeverityYear0*(1+t)(Year-Year0)T is the trend rateThis is an exponential trend modelCannot fit it with a line
-
Transformation of VariablesSeverityYear=SeverityYear0*(1+t)(Year-Year0)Log both sidesln(SevYear)=ln(SevYear0)+(Year-Year0)*ln(1+t) Y = a + x * bA line can be fit to transformed variables where dependent variable is log(Y)
-
Exponential Trend Cont.R2 declines and Residuals indicate poor fit
Chart1
0.319931207
0.1685457397
0.0200247847
-0.072886695
-0.1926139784
-0.3464156224
-0.1325699893
-0.0840491433
-0.0386398414
-0.0353870342
0.0737780761
0.0572059399
0.2630765566
Residuals
Predicted Y
Residual
Plot of Residuals
wORKERS cOMP
Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase
Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year
PriorPriorPriorPrior0.00.00.0PriorPrior
19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991
199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992
199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993
199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994
199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995
199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)
19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)
19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)
19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)
20004,534,0352002200220000.020000.02000(4,534,035)
246195205056278
$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)
Age to Age FactorsAge to Age Factors
Paid Losses$39,739Incurred(75,792,474)
Ratio IBNR to CASE(0.00)
Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)
PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)
19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012
19921.0221.02119921.0171.013
19931.0311.03019931.0171.019
19941.0451.035Claims19941.0281.022
19951.0731.054Alt19951.0421.039
19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047
19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076
19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134
19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450
20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000
Bests DevRemainder
Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012
Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07
3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070
1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111
2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700
20001,386,125(4,534,035)(3,271.01)
4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006
WC
Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity
Policy Year
Prior
19913944772913017,291.00$5,410.37
19923251066783416,678.00$4,868.22
19933049669422816,942.00$4,392.974375.3228120516
19942778566303316,630.00$4,190.804170.0435239382
19952612867134416,713.00$3,892.153866.8047950274
19962351667315716,731.00$3,493.693464.3488509134
19972853162997416,299.00$4,529.454476.8554840734
19983204764321151.0016,438.43$4,977.454894.9137009317
19993467863471831.0026,359.69$5,452.785310.5666156202
20003677063573391.016,420.57$5,726.91365491.3381123058
20013949157906161.025,905.80$6,686.82246164.6893537309
20023882953715211.055,639.55$6,885.12246678.1688919716
200342232414711671.154,769.05$8,855.4312
197015.71.1571
197110.91.1091.109
197211.11.1111.232099
19739.61.0961.350380504
197415.71.1571.5623902431
197514.21.1421.7842496577
197620.41.2042.1482365878
197716.11.1612.4941026785
197810.11.1012.746007049
197916.51.1653.1990982121
198015.21.1523.6853611403
198115.51.1554.256592117
198213.21.1324.8184622765
198381.085.2039392586
19848.21.0825.6306622778
1985101.16.1937285056
198651.056.5034149309
198710.91.1097.2122871583
1988151.158.2941302321
198912.91.1299.364073032
199012.91.12910.5720384531
199110.91.10911.7243906445
199281.0812.6623418961
19937.31.07313.5866928545
19941.81.01813.8312533259
19953.11.03114.260022179
19961.91.01914.5309626004
19973.61.03615.054077254
199841.0415.6562403442
199931.0316.1259275545
200071.0717.2547424833
20019.41.09418.8766882768
20029.71.09720.7077270396
2003151.1523.8138860955
Line Fit
WC
SUMMARY OUTPUT
tRegression Statistics
(1)(2)(3)(4)(5)(6)(7)(8)
AccidentSeverityMultiple R0.72
YearYFitted YXR Square0.52
19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.48
19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73
19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00
19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259
19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA
19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F
19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01
19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76
19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05
20003677063573391.016,420.57$5,727$6,14610$39131,174.079
20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64
200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77
Sum/Average$5,336749,143.61182.00
B270.02
3,445.41
RESIDUAL OUTPUTPROBABILITY OUTPUT
ObservationPredicted YResidualsPercentileY
13715.43164900311694.9372990153.84615384623493.6859307681
23985.4514762824882.77254288511.53846153853892.1495605542
34255.4713035617137.499021992919.23076923084190.7993966818
44525.4911308411-334.691734159326.92307692314392.9703255546
54795.5109581204-903.361397566234.61538461544529.4491189078
65065.5307853997-1571.844854631642.30769230774868.2240191674
75335.5506126791-806.1014937713504977.4541379019
85605.5704399584-628.116302056557.69230769235410.3689480181
95875.5902672377-422.812193324165.38461538465452.7780739136
106145.6100945171-418.70422790473.07692307695726.9058666131
116415.6299217964271.186428232480.76923076926686.8163500288
126685.6497490757199.474063994588.46153846156885.1238130702
136955.66957635511899.762847293396.15384615388855.4324236483
Line Fit
0
0
0
0
0
0
0
0
0
0
0
0
0
X Variable 1
Residuals
Expo Fit
0
0
0
0
0
0
0
0
0
0
0
0
0
Sample Percentile
Y
Normal Probability Plot
Sheet2
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
Y
Fitted Y
Year
Severity
Workers Comp Severity Trend
Sheet3
WC
SUMMARY OUTPUT
Regression Statistics
(1)(2)(3)(4)(5)(5)(6)(7)(8)Multiple R0.6992898719
AccidentSeverityln(Severity)R Square0.489006325
YearYYFitted SevXAdjusted R Square0.4425523545
YearSeverityFittedStandard Error0.1904278372
19913944772913017,291.00$5,4108.60$3,9321$75-6(448.91)361991$5,410$3,932Observations13
19923251066783416,678.00$4,8688.49$4,1162$(467)-52,336.63251992$4,868$4,116
19933049669422816,942.00$4,3938.39$4,3093$(943)-43,770.32161993$4,393$4,309ANOVA
19942778566303316,630.00$4,1918.34$4,5114$(1,145)-33,434.2591994$4,191$4,511dfSSMSFSignificance F
19952612867134416,713.00$3,8928.27$4,7225$(1,443)-22,886.8041995$3,892$4,722Regression10.38172667260.381726672610.52668523720.0078114247
19962351667315716,731.00$3,4948.16$4,9436$(1,842)-11,841.8611996$3,494$4,943Residual110.39889037280.0362627612
19972853162997416,299.00$4,5298.42$5,1757$(806)00.001997$4,529$5,175Total120.7806170454
19983204764321151.0016,438.43$4,9778.51$5,4178$(358)1(358.10)11998$4,977$5,417
19993467863471831.0026,359.69$5,4538.60$5,6719$1172234.4541999$5,453$5,671CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%
20003677063573391.016,420.57$5,7278.65$5,93710$39131,174.0792000$5,727$5,937Intercept8.23034398770.112037914373.460346314907.98375007628.47693789937.98375007628.4769378993
20013949157906161.025,905.80$6,6878.81$6,21511$1,35145,405.06162001$6,687$6,215X Variable 10.04579737220.01411545043.24448535780.00781142470.01472945960.07686528480.01472945960.0768652848
20023882953715211.055,639.55$6,8858.84$6,50712$1,55057,747.87252002$6,885$6,507
200342232414711671.154,769.05$8,8559.09$6,81113$3,520621,119.29362003$8,855$6,811
Sum/Average$5,3368.58$5,175749,143.61182.00
B270.02RESIDUAL OUTPUTPROBABILITY OUTPUT
3,445.41
ObservationPredicted YResidualsStandard ResidualsPercentileY
18.27614135990.3199312071.75477100673.84615384628.1587125985
28.32193873210.16854573970.924446164611.53846153858.2667168703
38.36773610430.02002478470.109832710419.23076923088.3406467815
48.4135334765-0.072886695-0.39977175226.92307692318.387760889
58.4593308487-0.1926139784-1.056456567734.61538461548.4183556038
68.5051282209-0.3464156224-1.900033748242.30769230778.4904844718
78.5509255931-0.1325699893-0.7271249834508.512673822
88.5967229653-0.0840491433-0.460995978357.69230769238.5960725669
98.6425203375-0.0386398414-0.211933290265.38461538468.6038804961
108.6883177097-0.0353870342-0.194092168173.07692307698.6529306755
118.73411508190.07377807610.404660833280.76923076928.8078931579
128.77991245410.05720593990.313765342788.46153846158.837118394
138.82570982630.26307655661.442932430296.15384615389.0887863829
Sheet3
0.0200247847
-0.072886695
-0.1926139784
-0.3464156224
-0.1325699893
-0.0840491433
-0.0386398414
-0.0353870342
0.0737780761
0.0572059399
0.2630765566
12
13
X Variable 1
Residuals
19.2307692308
26.9230769231
34.6153846154
42.3076923077
50
57.6923076923
65.3846153846
73.0769230769
80.7692307692
88.4615384615
96.1538461538
Sample Percentile
Y
Normal Probability Plot
199119911991199119915410.36894801818.3677361043
199219921992199219924868.22401916748.4135334765
199319931993199319934392.97032555468.4593308487
199419941994199419944190.79939668188.5051282209
199519951995199519953892.14956055428.5509255931
199619961996199619963493.68593076818.5967229653
199719971997199719974529.44911890788.6425203375
199819981998199819984977.45413790198.6883177097
199919991999199919995452.77807391368.7341150819
200020002000200020005726.90586661318.7799124541
200120012001200120016686.81635002888.8257098263
200220022002200220026885.12381307020
200320032003200320038855.43242364830
Y
Fitted Sev
Year
Severity
Workers Comp Severity Trend
0.319931207
0.1685457397
0.0200247847
-0.072886695
-0.1926139784
-0.3464156224
-0.1325699893
-0.0840491433
-0.0386398414
-0.0353870342
0.0737780761
0.0572059399
0.2630765566
X Variable 1
Residuals
X Variable 1 Residual Plot
8.59607256698.2761413599
8.49048447188.3219387321
8.3877608898.3677361043
8.34064678158.4135334765
8.26671687038.4593308487
8.15871259858.5051282209
8.41835560388.5509255931
8.5126738228.5967229653
8.60388049618.6425203375
8.65293067558.6883177097
8.80789315798.7341150819
8.8371183948.7799124541
9.08878638298.8257098263
Y
Predicted Y
X Variable 1
Y
X Variable 1 Line Fit Plot
8.1587125985
8.2667168703
8.3406467815
8.387760889
8.4183556038
8.4904844718
8.512673822
8.5960725669
8.6038804961
8.6529306755
8.8078931579
8.837118394
9.0887863829
Sample Percentile
Y
Normal Probability Plot
Residuals
Predicted Y
Residual
Plot of Residuals
Severity
Fitted
Year
Severity
Exponential Trend
WCHealthSUMMARY OUTPUT
YearSeverityIns IndexCPIFittedResidualSUMMARY OUTPUT
19915,41011.7243906445136.24158.58748209141251.7814659267Regression Statistics
19924,86812.6623418961140.34496.6316319442371.5923872233Regression StatisticsMultiple R0.8229062628
19934,39313.5866928545144.54818.5675545838-425.5972290292Multiple R0.8988759252R Square0.6771747174
19944,19113.8312533259148.254718.2828539551-527.4834572734R Square0.8079779288Adjusted R Square0.6126096608
19953,89214.260022179152.354716.3820496521-824.2324890979Adjusted R Square0.7695735146Standard Error0.1587459978
19963,49414.5309626004156.854580.9972951344-1087.3113643663Standard Error699.2105512592Observations13
19974,52915.054077254160.554670.1975931414-140.7484742336Observations13
19984,97715.65624034421634900.026380199277.4277577027ANOVA
19995,45316