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Last data update: 2014.03.03

R: Parameter estimation - DCL model reproducing the incurred...

idcl.estimation

R Documentation

Parameter estimation - DCL model reproducing the incurred reserve.

Description

Estimate the parameters in the Double Chain Ladder model model: delay parameters, severity mean and variance. The inflation parameter is corrected using the incurred data to provide the incurred cashflow.

Usage

idcl.estimation( Xtriangle , Ntriangle , Itriangle , adj = 1 , 
    Tables = TRUE , num.dec = 4 , n.cal = NA , 
    Fj.X = NA , Fj.N = NA , Fj.I = NA)

Arguments

`Xtriangle`	The paid run-off triangle: incremental aggregated payments. It should be a matrix with incremental aggregated payments located in the upper triangle and the lower triangle consisting in missing or zero values.
`Ntriangle`	The counts data triangle: incremental number of reported claims. It should be a matrix with the observed counts located in the upper triangle and the lower triangle consisting in missing or zero values. It should has the same dimension as `Xtriangle` (both in the same aggregation level (quarters, years,etc.))
`Itriangle`	The incurred triangle. It should be a matrix with incurred data located in the upper triangle. It is an incremental run-off triangle with the same dimension as `Xtriangle` (both in the same aggregation level (quarters, years,etc.))
`adj`	Method to adjust the estimated delay parameters for the distributional model. It should be 1 (default value) or 2. See more in details below.
`Tables`	Logical. If TRUE (default) it is showed a table with the estimated parameters.
`num.dec`	Number of decimal places used to report numbers in the tables (if Tables=TRUE).
`n.cal`	Integer specifying the number of most recent calendars which will be used to calculate the development factors. By default `n.cal=NA` and all the observed calendars are used (classical chain ladder).
`Fj.X`	Optional vector with lentgth m-1 (m being the dimension of the triangles) with the development factors to calculate the chain ladder estimates from `Xtriangle`. See more details in `clm`.
`Fj.N`	Optional vector with lentgth m-1 with the development factors to calculate the chain ladder estimates from `Ntriangle`.
`Fj.I`	Optional vector with lentgth m-1 with the development factors to calculate the chain ladder estimates from `Itriangle`.

Details

Two model are estimated in the double chain ladder framework as with the dcl.estimation function. In this case the DCL inflation parameter estimated by dcl.estimation from Ntriangle and Xtriangle is adjusted so that the derived predicted reserve is equal to the incurred reserve. Use this estimation method if you want the RBNS/IBNR split the incurred reserve and the incurred full cashflow.

Value

`pi.delay`	General delay parameters
`mu`	Mean severity factor
`inflat`	Underwriting severity inflation (BDCL inflation)
`inflat.DCL`	Underwriting severity inflation (DCL inflation)
`pj`	Delay probabilities (under a Multinomial assumption)
`mu.adj`	Adjusted mean factor corresponding to the `pj` parameters
`sigma2`	Variance severity factor
`phi`	Overdispersion parameter used to derive the estimate `sigma2`
`Ey`	Severity mean for each underwriting period
`Vy`	Severity variance for each underwriting period
`adj`	Type of adjusted used to derive the `pj` probabilities
`alpha.N`	Underwriting chain ladder parameter in the (OD)-Poisson model. Counts triangle (Ntriangle)
`beta.N`	Underwriting chain ladder parameter in the (OD)-Poisson model. Counts triangle (Ntriangle)
`Nhat`	The chain ladder preditions (counts triangle). It is a matrix having the chain ladder predictions in the future (lower triangle) and the fitted values in the past (upper triangle).
`alpha.X`	Underwriting chain ladder parameter in the (OD)-Poisson model. Paid triangle (Xtriangle)
`beta.X`	Underwriting chain ladder parameter in the (OD)-Poisson model. Paid triangle (Xtriangle)
`Xhat`	The chain ladder preditions (paid triangle). It is a matrix having the chain ladder predictions in the future (lower triangle) and the fitted values in the past (upper triangle).
`alpha.I`	Underwriting chain ladder parameter in the (OD)-Poisson model. Incurred triangle (Itriangle)
`beta.I`	Underwriting chain ladder parameter in the (OD)-Poisson model. Incurred triangle (Itriangle)
`CL.I.i`	Outstanding incurred numbers (row sums of the lower predicted triangle) from classical chain ladder on the incurred triangle.

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

References

Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2012) Double Chain Ladder. Astin Bulletin, 42/1, 59-76. Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2013) Double Chain Ladder and Bornhuetter-Ferguson. North Americal Actuarial Journal, 17(2), 101-113.

Examples

data(NtriangleBDCL)
data(XtriangleBDCL)
data(ItriangleBDCL)

my.idcl.par<-idcl.estimation(XtriangleBDCL,NtriangleBDCL,ItriangleBDCL)
# Parameters 
Plot.dcl.par(my.idcl.par,type.inflat='IDCL')
# IDCL Predictions by diagonals (future calendar years)
preds.idcl.diag<-dcl.predict(my.idcl.par,NtriangleBDCL,num.dec=0)

# Comparing with the BDCL method  
my.bdcl.par<-bdcl.estimation(XtriangleBDCL,NtriangleBDCL,ItriangleBDCL)
# Parameters shown in Table 1
Plot.dcl.par(my.bdcl.par,type.inflat='BDCL')
# BDCL Predictions by diagonals (future calendar years)
preds.bdcl.diag<-dcl.predict(my.bdcl.par,NtriangleBDCL,num.dec=0)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(DCL)
Loading required package: lattice
Loading required package: latticeExtra
Loading required package: RColorBrewer
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/DCL/idcl.estimation.Rd_%03d_medium.png", width=480, height=480)
> ### Name: idcl.estimation
> ### Title: Parameter estimation - DCL model reproducing the incurred
> ###   reserve.
> ### Aliases: idcl.estimation
> ### Keywords: models
> 
> ### ** Examples
> 
> data(NtriangleBDCL)
> data(XtriangleBDCL)
> data(ItriangleBDCL)
> 
> my.idcl.par<-idcl.estimation(XtriangleBDCL,NtriangleBDCL,ItriangleBDCL)
       delay.par   delay.prob inflat.DCL inflat.IDCL severity.mean severity.var
1   5.922242e-02 0.0592224162   1.000000   1.0000000     2579.0642    286808926
2   3.097740e-01 0.3097739983   1.117293   1.1172929     2881.5702    358036053
3   2.031802e-01 0.2031802197   1.494734   1.4947337     3855.0143    640796811
4   1.996404e-01 0.1996404449   1.746091   1.7460907     4503.2802    874432486
5   1.388353e-01 0.1388353306   2.107455   2.4540246     6329.0871   1727231272
6   4.403213e-02 0.0440321332   2.093575   0.8239007     2124.8928    194689440
7   2.267573e-02 0.0226757333   2.249536   0.1435635      370.2594      5911265
8   9.489989e-03 0.0094899895   2.125004   0.7926218     2044.2225    180187507
9   1.757430e-03 0.0017574297   1.902800   0.2847205      734.3124     23250382
10  2.879106e-03 0.0028791056   2.019675   0.7969141     2055.2927    182144352
11  2.016829e-04 0.0002016829   2.070358   0.6567019     1693.6764    123688473
12  2.589893e-03 0.0025898926   2.266601  -0.5239147    -1351.2097     78725218
13  1.886956e-03 0.0018869556   2.315662   2.0509174     5289.4477   1206393545
14  3.185318e-03 0.0031853177   2.474680   1.9798681     5106.2070   1124255898
15 -1.668436e-04 0.0006493504   2.382877   1.8410459     4748.1757    972124543
16  1.252765e-03 0.0000000000   2.839129   1.2605700     3251.0910    455749929
17 -4.231763e-04 0.0000000000   3.181535   1.7695988     4563.9089    898136368
18  4.147243e-05 0.0000000000   4.174702   2.1597728     5570.1929   1337854289
19 -4.044053e-05 0.0000000000   6.750140   2.6702731     6886.8059   2045050482
  mean.factor mean.factor.adj variance.factor
1    2579.002        2579.064       286808926
> # Parameters 
> Plot.dcl.par(my.idcl.par,type.inflat='IDCL')
> # IDCL Predictions by diagonals (future calendar years)
> preds.idcl.diag<-dcl.predict(my.idcl.par,NtriangleBDCL,num.dec=0)
   Future.years     rbns     ibnr    total       clm
1             1 28718323   561964 29280288  61090913
2             2 19828074  3005177 22833251  48061355
3             3 14264538  2295308 16559846  36266482
4             4  7717656  2252489  9970145  22989797
5             5  2899658  1674712  4574370  10439464
6             6  1396182   720796  2116978   4913941
7             7   703796   399955  1103752   2380121
8             8   387307   208731   596039   1174087
9             9   346716    93249   439964    848056
10           10   225582    73417   298999    599856
11           11   253657    32739   286396    593718
12           12   192835    42556   235391    495823
13           13   143149    31833   174982    397095
14           14    24845    40456    65301    135553
15           15        0    15837    15837    109485
16           16        0     5468     5468         0
17           17        0     2937     2937         0
18           18        0     1627     1627         0
19           19        0      822      822        NA
20           20        0      446      446        NA
21           21        0      237      237        NA
22           22        0      122      122        NA
23           23        0       82       82        NA
24           24        0       42       42        NA
25           25        0       34       34        NA
26           26        0       20       20        NA
27           27        0       15       15        NA
28           28        0        3        3        NA
29           29        0        0        0        NA
30           30        0        0        0        NA
31           31        0        0        0        NA
32           32        0        0        0        NA
33           33        0        0        0        NA
34           34        0        0        0        NA
35           35        0        0        0        NA
36           36        0        0        0        NA
37         Tot. 77102319 11461074 88563393 190495745
> 
> # Comparing with the BDCL method  
> my.bdcl.par<-bdcl.estimation(XtriangleBDCL,NtriangleBDCL,ItriangleBDCL)
   delay.par delay.prob inflat.DCL inflat.BDCL severity.mean severity.var
1     0.0592     0.0592     1.0000      1.0000      2579.064    350497302
2     0.3098     0.3098     1.1173      1.1173      2881.570    437541022
3     0.2032     0.2032     1.4947      1.4955      3856.956    783880192
4     0.1996     0.1996     1.7461      1.7445      4499.233   1066687565
5     0.1388     0.1388     2.1075      2.1078      5436.207   1557228621
6     0.0440     0.0440     2.0936      2.0914      5393.831   1533045263
7     0.0227     0.0227     2.2495      2.2396      5776.131   1758062863
8     0.0095     0.0095     2.1250      2.1158      5456.839   1569071251
9     0.0018     0.0018     1.9028      1.8878      4868.678   1249057800
10    0.0029     0.0029     2.0197      2.0067      5175.412   1411400536
11    0.0002     0.0002     2.0704      2.0504      5288.050   1473504248
12    0.0026     0.0026     2.2666      2.2135      5708.847   1717343276
13    0.0019     0.0019     2.3157      2.3068      5949.332   1865076855
14    0.0032     0.0032     2.4747      2.4427      6299.903   2091356383
15   -0.0002     0.0006     2.3829      2.3109      5959.973   1871754667
16    0.0013     0.0000     2.8391      2.3875      6157.427   1997831581
17   -0.0004     0.0000     3.1815      2.4944      6433.119   2180738246
18    0.0000     0.0000     4.1747      2.7498      7091.924   2650260077
19    0.0000     0.0000     6.7501      2.8539      7360.359   2854686119
  mean.factor mean.factor.adj variance.factor
1    2579.002        2579.064       350497302
> # Parameters shown in Table 1
> Plot.dcl.par(my.bdcl.par,type.inflat='BDCL')
> # BDCL Predictions by diagonals (future calendar years)
> preds.bdcl.diag<-dcl.predict(my.bdcl.par,NtriangleBDCL,num.dec=0)
   Future.years     rbns     ibnr     total       clm
1             1 37812985   615136  38428120  61090913
2             2 25878325  3293679  29172004  48061355
3             3 17804231  2536746  20340978  36266482
4             4  9485413  2494820  11980232  22989797
5             5  3698865  1866861   5565726  10439464
6             6  1839293   820892   2660185   4913941
7             7   904735   461593   1366327   2380121
8             8   512417   246100    758516   1174087
9             9   457254   113302    570555    848056
10           10   328835    87417    416252    599856
11           11   336960    40177    377137    593718
12           12   242186    49345    291531    495823
13           13   163171    36768    199939    397095
14           14    27580    45697     73278    135553
15           15        0    18383     18383    109485
16           16        0     6630      6630         0
17           17        0     3569      3569         0
18           18        0     1970      1970         0
19           19        0      997       997        NA
20           20        0      547       547        NA
21           21        0      283       283        NA
22           22        0      151       151        NA
23           23        0      100       100        NA
24           24        0       56        56        NA
25           25        0       41        41        NA
26           26        0       24        24        NA
27           27        0       16        16        NA
28           28        0        3         3        NA
29           29        0        0         0        NA
30           30        0        0         0        NA
31           31        0        0         0        NA
32           32        0        0         0        NA
33           33        0        0         0        NA
34           34        0        0         0        NA
35           35        0        0         0        NA
36           36        0        0         0        NA
37         Tot. 99492249 12741303 112233552 190495745
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>