Motor Claims Reserve
Date : 02/05/2022
- Introduction
- Part One : The data set.
- Part two : mackchainladder
- Part Three : data distribution
- Part Four : reserve using GLM model
- Part Four : Paid-incurred chain model
- Part Four : One year claims development
- Part Five : glmreserve
- Part Five : Clark’s methods
Introduction
The purpose of this project is to predict the best claims reservation (incurred claims+IBNR) noting that the incurred claims means the claims incurred and have been reported to the insurance company,while the IBNR is the claims incurred but not reported (pure IBNR , or incurred but not enough reported IBNER due to claims developments)
We will be using - autoPaid - data set in triangle format/type which can be found on chainladder package,noting that we will be using chainladder package from R analysis software especially for achieving our project.
Part One : The data set.
1. the data set in triangle format :
## dev
## origin 1 2 3 4 5 6 7 8 9 10
## 1 101125 209921 266618 305107 327850 340669 348430 351193 353353 353584
## 2 102541 203213 260677 303182 328932 340948 347333 349813 350523 NA
## 3 114932 227704 298120 345542 367760 377999 383611 385224 NA NA
## 4 114452 227761 301072 340669 359979 369248 373325 NA NA NA
## 5 115597 243611 315215 354490 372376 382738 NA NA NA NA
## 6 127760 259416 326975 365780 386725 NA NA NA NA NA
## 7 135616 262294 327086 367357 NA NA NA NA NA NA
## 8 127177 244249 317972 NA NA NA NA NA NA NA
## 9 128631 246803 NA NA NA NA NA NA NA NA
## 10 126288 NA NA NA NA NA NA NA NA NAthe above data set show the claims incurred development by each development year for each origin year.
in instance the claims incurred for the 1st year were 101125 , then it became 209921 in the 2nd development year due to claims developments which include IBNR and IBNER , and the same apply for all origin and dev years.
note that the 10nth development year for the 1st origin year has the ultimate reserve which include incurred and IBNR
- on graph:
2. the development years cumulative claims each origin year:
## dev
## origin 1 2 3 4 5 6 7 8 9
## 1 101125 311046 577664 882771 1210621 1551290 1899720 2250913 2604266
## 2 102541 305754 566431 869613 1198545 1539493 1886826 2236639 2587162
## 3 114932 342636 640756 986298 1354058 1732057 2115668 2500892 NA
## 4 114452 342213 643285 983954 1343933 1713181 2086506 NA NA
## 5 115597 359208 674423 1028913 1401289 1784027 NA NA NA
## 6 127760 387176 714151 1079931 1466656 NA NA NA NA
## 7 135616 397910 724996 1092353 NA NA NA NA NA
## 8 127177 371426 689398 NA NA NA NA NA NA
## 9 128631 375434 NA NA NA NA NA NA NA
## 10 126288 NA NA NA NA NA NA NA NA
## dev
## origin 10
## 1 2957850
## 2 NA
## 3 NA
## 4 NA
## 5 NA
## 6 NA
## 7 NA
## 8 NA
## 9 NA
## 10 NA- on graph:
3. the difference between the development years claims for each origin year:
## dev
## origin 1 2 3 4 5 6 7 8 9 10
## 1 101125 108796 56697 38489 22743 12819 7761 2763 2160 231
## 2 102541 100672 57464 42505 25750 12016 6385 2480 710 NA
## 3 114932 112772 70416 47422 22218 10239 5612 1613 NA NA
## 4 114452 113309 73311 39597 19310 9269 4077 NA NA NA
## 5 115597 128014 71604 39275 17886 10362 NA NA NA NA
## 6 127760 131656 67559 38805 20945 NA NA NA NA NA
## 7 135616 126678 64792 40271 NA NA NA NA NA NA
## 8 127177 117072 73723 NA NA NA NA NA NA NA
## 9 128631 118172 NA NA NA NA NA NA NA NA
## 10 126288 NA NA NA NA NA NA NA NA NA4. the percent of changing in the development years claims each origin year.
## dev
## origin 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10
## 1 2.076 1.270 1.144 1.075 1.039 1.023 1.008 1.006 1.001
## 2 1.982 1.283 1.163 1.085 1.037 1.019 1.007 1.002 NA
## 3 1.981 1.309 1.159 1.064 1.028 1.015 1.004 NA NA
## 4 1.990 1.322 1.132 1.057 1.026 1.011 NA NA NA
## 5 2.107 1.294 1.125 1.050 1.028 NA NA NA NA
## 6 2.030 1.260 1.119 1.057 NA NA NA NA NA
## 7 1.934 1.247 1.123 NA NA NA NA NA NA
## 8 1.921 1.302 NA NA NA NA NA NA NA
## 9 1.919 NA NA NA NA NA NA NA NA
## smpl 1.993 1.286 1.138 1.065 1.031 1.017 1.006 1.004 1.001
## vwtd 1.990 1.285 1.137 1.064 1.031 1.017 1.006 1.004 1.001we can see that for the 1st origin year, the differnce percent between the 1st &2nd development years is 2.076 and between the 2nd & 3rd development years is 1.270 and so on, this means, that the iccured claims increased by 2.076 on the 2nd development year & in creased by 1.270 on the 3rd development year
Part two : mackchainladder
Applying mackchainladder
now will apply the mackchainladderfuntion model to forecasts future claims developments based on a historical claims development triangle and estimates the standard error around those.
## MackChainLadder(Triangle = autoPaid, est.sigma = "Mack")
##
## Latest Dev.To.Date Ultimate IBNR Mack.S.E CV(IBNR)
## 1 353,584 1.000 353,584 0 0 NaN
## 2 350,523 0.999 350,752 229 998 4.3544
## 3 385,224 0.995 387,054 1,830 1,713 0.9360
## 4 373,325 0.989 377,481 4,156 1,885 0.4537
## 5 382,738 0.973 393,454 10,716 2,872 0.2680
## 6 386,725 0.943 409,932 23,207 3,847 0.1658
## 7 367,357 0.887 414,305 46,948 6,405 0.1364
## 8 317,972 0.780 407,609 89,637 9,177 0.1024
## 9 246,803 0.607 406,593 159,790 12,532 0.0784
## 10 126,288 0.305 414,021 287,733 19,085 0.0663
##
## Totals
## Latest: 3,290,539.00
## Dev: 0.84
## Ultimate: 3,914,785.82
## IBNR: 624,246.82
## Mack.S.E 30,358.21
## CV(IBNR): 0.05we can see that:
1. the Ultimate for the 1st origin year is 353,584 & for the 2nd year origin 350,752 and so on.
2. the IBNR for the 2nd year is 229 & for the 3rd year 1830 and so on.
3. the total of latest claims is 3,290,539.00 & the total ultimate claims is 3,914,785.82 with .87 Dev to date in total.
4. the total IBNR is 624,246.82 with mack standard error of 0.05 in total (30,358.21 in numbers)
- on graph :
The standard error in triangles :
## dev
## origin 1 2 3 4 5 6 7 8
## 1 0 0.00 0.000 0.000 0.000 0.000 0.000 0.000
## 2 0 0.00 0.000 0.000 0.000 0.000 0.000 0.000
## 3 0 0.00 0.000 0.000 0.000 0.000 0.000 0.000
## 4 0 0.00 0.000 0.000 0.000 0.000 0.000 846.437
## 5 0 0.00 0.000 0.000 0.000 0.000 2094.555 2279.905
## 6 0 0.00 0.000 0.000 0.000 2401.343 3251.425 3391.420
## 7 0 0.00 0.000 0.000 4812.878 5519.778 6013.675 6118.106
## 8 0 0.00 0.000 5881.012 7866.651 8457.300 8860.678 8961.120
## 9 0 0.00 6668.704 9588.736 11258.223 11852.202 12237.891 12347.534
## 10 0 8667.76 13017.925 15941.930 17630.574 18339.307 18769.975 18910.504
## dev
## origin 9 10
## 1 0.000 0.0000
## 2 0.000 997.8177
## 3 1332.813 1712.9000
## 4 1561.920 1885.4599
## 5 2656.495 2872.4144
## 6 3676.537 3846.5568
## 7 6299.784 6404.7907
## 8 9103.105 9177.4357
## 9 12474.455 12532.4060
## 10 19039.129 19085.1603we can see that the 2nd origin year will have almost 998 S.E at the 10th Dev. year, & the 3rd origin year will have almost 1333 S.E at the 9th Dev. year ,and almost 1713 at the 10th Dev. yearm ans so on.
The full triangle :
## dev
## origin 1 2 3 4 5 6 7 8
## 1 101125 209921.0 266618.0 305107.0 327850.0 340669.0 348430.0 351193.0
## 2 102541 203213.0 260677.0 303182.0 328932.0 340948.0 347333.0 349813.0
## 3 114932 227704.0 298120.0 345542.0 367760.0 377999.0 383611.0 385224.0
## 4 114452 227761.0 301072.0 340669.0 359979.0 369248.0 373325.0 375696.3
## 5 115597 243611.0 315215.0 354490.0 372376.0 382738.0 389122.5 391594.1
## 6 127760 259416.0 326975.0 365780.0 386725.0 398766.6 405418.4 407993.6
## 7 135616 262294.0 327086.0 367357.0 390850.8 403020.9 409743.7 412346.3
## 8 127177 244249.0 317972.0 361419.5 384533.7 396507.0 403121.2 405681.7
## 9 128631 246803.0 317179.7 360518.9 383575.5 395519.0 402116.7 404670.8
## 10 126288 251311.7 322974.1 367105.1 390582.8 402744.5 409462.7 412063.6
## dev
## origin 9 10
## 1 353353.0 353584.0
## 2 350523.0 350752.1
## 3 386801.2 387054.0
## 4 377234.4 377481.1
## 5 393197.4 393454.4
## 6 409663.9 409931.8
## 7 414034.5 414305.2
## 8 407342.6 407608.9
## 9 406327.6 406593.2
## 10 413750.6 414021.1Part Three : data distribution
Data distribution using bootchainladder function :
## BootChainLadder(Triangle = autoPaid, R = 999, process.distr = c("gamma",
## "od.pois"))
##
## Latest Mean Ultimate Mean IBNR IBNR.S.E IBNR 75% IBNR 95%
## 1 353,584 353,584 0 0 0 0
## 2 350,523 350,764 241 514 395 1,280
## 3 385,224 387,088 1,864 1,175 2,577 3,966
## 4 373,325 377,511 4,186 1,750 5,283 7,274
## 5 382,738 393,540 10,802 2,616 12,421 15,281
## 6 386,725 410,009 23,284 3,876 25,772 29,947
## 7 367,357 414,657 47,300 5,622 50,986 56,020
## 8 317,972 407,930 89,958 8,057 95,419 103,850
## 9 246,803 407,026 160,223 11,603 168,089 179,969
## 10 126,288 414,814 288,526 22,281 304,326 325,857
##
## Totals
## Latest: 3,290,539
## Mean Ultimate: 3,916,925
## Mean IBNR: 626,386
## IBNR.S.E 30,292
## Total IBNR 75%: 646,509
## Total IBNR 95%: 675,553we can see that:
1. the mean ultimate incurred claims is : 3,915,580
2. the mean IBNR is : 625,041 with standard error of : 30,442
3. the IBNR at the 75% quantile is:645,833 & and at 95% is :675,558
4. this means that the IBNR is almost normally distributed .
- on graph:
Part Four : reserve using GLM model
the generalized linear model for loss reserving
- some insurance companies use the generalized linear model for loss reserving .
## Latest Dev.To.Date Ultimate IBNR S.E CV
## 2 350523 0.9993471 350752 229 464.3474 2.02771788
## 3 385224 0.9952720 387054 1830 1187.4583 0.64888432
## 4 373325 0.9889902 377481 4156 1700.7849 0.40923602
## 5 382738 0.9727643 393454 10716 2650.8531 0.24737338
## 6 386725 0.9433882 409932 23207 3867.6316 0.16665797
## 7 367357 0.8866825 414305 46948 5552.4815 0.11826875
## 8 317972 0.7800907 407609 89637 8024.2919 0.08951986
## 9 246803 0.6070026 406593 159790 11982.6054 0.07498971
## 10 126288 0.3050280 414021 287733 22428.0768 0.07794753
## total 2936955 0.8247089 3561202 624247 30832.5263 0.04939155extract the underlying GLM model :
##
## Call:
## glm(formula = value ~ factor(origin) + factor(dev), family = fam,
## data = ldaFit, offset = offset)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.588524 0.040888 283.421 < 2e-16 ***
## factor(origin)2 -0.008041 0.051794 -0.155 0.877489
## factor(origin)3 0.090443 0.050637 1.786 0.082506 .
## factor(origin)4 0.065399 0.051048 1.281 0.208340
## factor(origin)5 0.106844 0.050796 2.103 0.042484 *
## factor(origin)6 0.147870 0.050751 2.914 0.006105 **
## factor(origin)7 0.158482 0.051518 3.076 0.003990 **
## factor(origin)8 0.142187 0.053672 2.649 0.011908 *
## factor(origin)9 0.139692 0.058005 2.408 0.021275 *
## factor(origin)10 0.157796 0.073551 2.145 0.038740 *
## factor(dev)2 -0.010061 0.029810 -0.338 0.737684
## factor(dev)3 -0.566599 0.036779 -15.406 < 2e-16 ***
## factor(dev)4 -1.051402 0.046362 -22.678 < 2e-16 ***
## factor(dev)5 -1.682513 0.064812 -25.960 < 2e-16 ***
## factor(dev)6 -2.340274 0.095993 -24.380 < 2e-16 ***
## factor(dev)7 -2.933742 0.143040 -20.510 < 2e-16 ***
## factor(dev)8 -3.882731 0.263858 -14.715 < 2e-16 ***
## factor(dev)9 -4.315592 0.406793 -10.609 1.25e-12 ***
## factor(dev)10 -6.146107 1.430109 -4.298 0.000125 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Tweedie family taken to be 472.0577)
##
## Null deviance: 2379116 on 54 degrees of freedom
## Residual deviance: 16899 on 36 degrees of freedom
## AIC: NA
##
## Number of Fisher Scoring iterations: 4Part Four : Paid-incurred chain model
this model uses both : the claims payments and incurred losses(the number of incurred losses) to get a unified ultimate loss prediction.
will use the same paid claims data used earlier, and the incurred claims data set we will use is the -auto Incurred- which can be found on chainladder package.
lets have a look at both data sets:
auto Paid data set:
## dev
## origin 1 2 3 4 5 6 7 8 9 10
## 1 101125 209921 266618 305107 327850 340669 348430 351193 353353 353584
## 2 102541 203213 260677 303182 328932 340948 347333 349813 350523 NA
## 3 114932 227704 298120 345542 367760 377999 383611 385224 NA NA
## 4 114452 227761 301072 340669 359979 369248 373325 NA NA NA
## 5 115597 243611 315215 354490 372376 382738 NA NA NA NA
## 6 127760 259416 326975 365780 386725 NA NA NA NA NA
## 7 135616 262294 327086 367357 NA NA NA NA NA NA
## 8 127177 244249 317972 NA NA NA NA NA NA NA
## 9 128631 246803 NA NA NA NA NA NA NA NA
## 10 126288 NA NA NA NA NA NA NA NA NAauto Incurred data set:(no. the incurred losses)
## dev
## origin 1 2 3 4 5 6 7 8 9 10
## 1 325423 336426 346061 347726 350995 353598 354797 355025 354986 355363
## 2 323627 339267 344507 349295 351038 351583 352050 352231 352193 NA
## 3 358410 386330 385684 384699 387678 387954 388540 389436 NA NA
## 4 405319 396641 391833 384819 380914 380163 379706 NA NA NA
## 5 434065 429311 422181 409322 394154 392802 NA NA NA NA
## 6 417178 422307 413486 406711 406503 NA NA NA NA NA
## 7 398929 398787 398020 400540 NA NA NA NA NA NA
## 8 378754 361097 369328 NA NA NA NA NA NA NA
## 9 351081 335507 NA NA NA NA NA NA NA NA
## 10 329236 NA NA NA NA NA NA NA NA NAthe model outputs:
## $Ult.Loss.Origin
## [,1]
## [1,] 352566.7
## [2,] 389696.9
## [3,] 380137.3
## [4,] 394357.5
## [5,] 410371.3
## [6,] 411260.9
## [7,] 394319.5
## [8,] 386660.9
## [9,] 369855.4
##
## $Ult.Loss
## [1] 3489226
##
## $Res.Origin
## [,1]
## [1,] 2043.682
## [2,] 4472.911
## [3,] 6812.299
## [4,] 11619.455
## [5,] 23646.289
## [6,] 43903.948
## [7,] 76347.542
## [8,] 139857.949
## [9,] 243567.371
##
## $Res.Tot
## [1] 552271.4
##
## $s.e.
## [1] 21612.81we can see that
1. the ultimate loss will be 3489226
2. the total reserve will be 552271.4
3.the total S.E will be 21612.81
Part Four : One year claims development
model simple explanation
this model means a short-term view and assessments of the one-year changes of the claims predictions when one updates the available information at the end of each accounting year
we will use the functions CDR : -one year claim development result -
## IBNR CDR(1)S.E. CDR(2)S.E. CDR(3)S.E. CDR(4)S.E. CDR(5)S.E.
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 2 229.1499 997.8177 0.0000 0.0000 0.0000 0.0000
## 3 1830.0181 1442.0056 924.4707 0.0000 0.0000 0.0000
## 4 4156.0534 1105.4590 1268.4483 850.8574 0.0000 0.0000
## 5 10716.4040 2253.6958 964.0382 1237.4955 843.1222 0.0000
## 6 23206.7589 2637.3737 2183.9375 916.5884 1231.9929 844.2364
## 7 46948.1842 5206.7444 2529.0008 2135.1706 884.2348 1215.0949
## 8 89636.9105 6755.6483 5049.2556 2443.4813 2078.2032 851.3035
## 9 159790.2318 8687.7932 6639.8133 4979.0587 2403.1289 2050.3506
## 10 287733.1080 14401.9739 8669.9570 6639.9295 4985.4753 2401.8493
## Total 624246.8188 22752.3275 14773.0832 10307.9026 6884.0988 4133.2281
## CDR(6)S.E. CDR(7)S.E. CDR(8)S.E. CDR(9)S.E. CDR(10)S.E. Mack.S.E.
## 1 0.0000 0.0000 0.0000 0.0000 0 0.0000
## 2 0.0000 0.0000 0.0000 0.0000 0 997.8177
## 3 0.0000 0.0000 0.0000 0.0000 0 1712.9000
## 4 0.0000 0.0000 0.0000 0.0000 0 1885.4599
## 5 0.0000 0.0000 0.0000 0.0000 0 2872.4144
## 6 0.0000 0.0000 0.0000 0.0000 0 3846.5568
## 7 835.4823 0.0000 0.0000 0.0000 0 6404.7907
## 8 1186.7369 817.9051 0.0000 0.0000 0 9177.4357
## 9 833.6339 1172.9711 809.6488 0.0000 0 12532.4060
## 10 2052.7089 830.7797 1175.6989 812.2527 0 19085.1603
## Total 2948.3560 1853.6696 1480.8285 812.2527 0 30358.2137Part Five : glmreserve
many issuance uses for loss reserving:
why? when over-dispersed Poisson model is used, it reproduces the estimates from chain-ladder;
it provides a more coherent modelling framework than the Mack method;
all the relevant established statistical theory can be directly applied to perform hypothesis testing and diagnostic checking;
## Latest Dev.To.Date Ultimate IBNR S.E CV
## 2 350523 0.9993471 350752 229 464.3474 2.02771788
## 3 385224 0.9952720 387054 1830 1187.4583 0.64888432
## 4 373325 0.9889902 377481 4156 1700.7849 0.40923602
## 5 382738 0.9727643 393454 10716 2650.8531 0.24737338
## 6 386725 0.9433882 409932 23207 3867.6316 0.16665797
## 7 367357 0.8866825 414305 46948 5552.4815 0.11826875
## 8 317972 0.7800907 407609 89637 8024.2919 0.08951986
## 9 246803 0.6070026 406593 159790 11982.6054 0.07498971
## 10 126288 0.3050280 414021 287733 22428.0768 0.07794753
## total 2936955 0.8247089 3561202 624247 30832.5263 0.04939155to extract the underlying GLM model:
##
## Call:
## glm(formula = value ~ factor(origin) + factor(dev), family = fam,
## data = ldaFit, offset = offset)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.588524 0.040888 283.421 < 2e-16 ***
## factor(origin)2 -0.008041 0.051794 -0.155 0.877489
## factor(origin)3 0.090443 0.050637 1.786 0.082506 .
## factor(origin)4 0.065399 0.051048 1.281 0.208340
## factor(origin)5 0.106844 0.050796 2.103 0.042484 *
## factor(origin)6 0.147870 0.050751 2.914 0.006105 **
## factor(origin)7 0.158482 0.051518 3.076 0.003990 **
## factor(origin)8 0.142187 0.053672 2.649 0.011908 *
## factor(origin)9 0.139692 0.058005 2.408 0.021275 *
## factor(origin)10 0.157796 0.073551 2.145 0.038740 *
## factor(dev)2 -0.010061 0.029810 -0.338 0.737684
## factor(dev)3 -0.566599 0.036779 -15.406 < 2e-16 ***
## factor(dev)4 -1.051402 0.046362 -22.678 < 2e-16 ***
## factor(dev)5 -1.682513 0.064812 -25.960 < 2e-16 ***
## factor(dev)6 -2.340274 0.095993 -24.380 < 2e-16 ***
## factor(dev)7 -2.933742 0.143040 -20.510 < 2e-16 ***
## factor(dev)8 -3.882731 0.263858 -14.715 < 2e-16 ***
## factor(dev)9 -4.315592 0.406793 -10.609 1.25e-12 ***
## factor(dev)10 -6.146107 1.430109 -4.298 0.000125 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Tweedie family taken to be 472.0577)
##
## Null deviance: 2379116 on 54 degrees of freedom
## Residual deviance: 16899 on 36 degrees of freedom
## AIC: NA
##
## Number of Fisher Scoring iterations: 4Part Five : Clark’s methods
Clark’s methods work on incremental losses. His likelihood function is based on the assumption that incremental losses follow an over-dispersed Poisson (ODP) process.
- LDF : Loss Development factor/Format Clark’s method/ Losses develop according to a theoretical growth curve.
## Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
## 1 353,584 1.100 388,960 35,376 9,724 27.5
## 2 350,523 1.113 390,304 39,781 10,430 26.2
## 3 385,224 1.131 435,606 50,382 12,089 24.0
## 4 373,325 1.154 430,747 57,422 13,075 22.8
## 5 382,738 1.186 453,878 71,140 14,937 21.0
## 6 386,725 1.233 476,961 90,236 17,324 19.2
## 7 367,357 1.310 481,318 113,961 20,008 17.6
## 8 317,972 1.454 462,403 144,431 23,179 16.0
## 9 246,803 1.810 446,807 200,004 28,987 14.5
## 10 126,288 3.814 481,723 355,435 51,988 14.6
## Total 3,290,539 4,448,705 1,158,166 113,917 9.8## Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
## 1 353,584 1.053 372,464 18,880 6,332 33.5
## 2 350,523 1.066 373,751 23,228 7,115 30.6
## 3 385,224 1.083 417,132 31,908 8,538 26.8
## 4 373,325 1.105 412,479 39,154 9,630 24.6
## 5 382,738 1.136 434,629 51,891 11,397 22.0
## 6 386,725 1.181 456,733 70,008 13,687 19.6
## 7 367,357 1.255 460,906 93,549 16,410 17.5
## 8 317,972 1.393 442,793 124,821 19,809 15.9
## 9 246,803 1.734 427,858 181,055 25,920 14.3
## 10 126,288 3.653 461,294 335,006 49,227 14.7
## Total 3,290,539 4,260,040 969,501 85,817 8.9- on graph:
but; using The Weibull growth curve tends to be faster developing than the log-logistic
## Origin CurrentValue Ldf UltimateValue FutureValue StdError CV%
## 1 353,584 1.007 355,911 2,327 1,388 59.6
## 2 350,523 1.011 354,275 3,752 1,788 47.7
## 3 385,224 1.018 391,986 6,762 2,460 36.4
## 4 373,325 1.029 384,185 10,860 3,170 29.2
## 5 382,738 1.049 401,468 18,730 4,262 22.8
## 6 386,725 1.084 419,291 32,566 5,761 17.7
## 7 367,357 1.150 422,595 55,238 7,685 13.9
## 8 317,972 1.287 409,308 91,336 10,231 11.2
## 9 246,803 1.637 403,968 157,165 14,716 9.4
## 10 126,288 3.396 428,903 302,615 28,967 9.6
## Total 3,290,539 3,971,890 681,351 42,396 6.2- on grapgh:
- LDF Clark’s Cap Cod method/expected ultimate losses in each origin year are the product of earned premium that year and a theoretical loss ratio.
## Origin CurrentValue Premium ELR FutureGrowthFactor FutureValue
## 1 353,584 1,000,000 0.392 0.0059 2,311
## 2 350,523 1,000,000 0.392 0.0097 3,784
## 3 385,224 1,000,000 0.392 0.0159 6,231
## 4 373,325 1,000,000 0.392 0.0263 10,323
## 5 382,738 1,000,000 0.392 0.0440 17,226
## 6 386,725 1,000,000 0.392 0.0740 28,995
## 7 367,357 1,000,000 0.392 0.1259 49,333
## 8 317,972 1,000,000 0.392 0.2173 85,149
## 9 246,803 1,000,000 0.392 0.3829 150,083
## 10 126,288 1,000,000 0.392 0.7020 275,129
## Total 3,290,539 10,000,000 628,564
## UltimateValue StdError CV%
## 355,895 1,470 63.6
## 354,307 1,907 50.4
## 391,455 2,482 39.8
## 383,648 3,238 31.4
## 399,964 4,226 24.5
## 415,720 5,509 19.0
## 416,690 7,158 14.5
## 403,121 9,277 10.9
## 396,886 12,081 8.0
## 401,417 16,129 5.9
## 3,919,103 32,262 5.1- on grapgh:
