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R: Parameter estimation - DCL model using the BDCL method

bdcl.estimation

R Documentation

Parameter estimation - DCL model using the BDCL method

Description

Estimate the parameters in the Double Chain Ladder model (delay parameters, severity mean and variance) using the Double Chain Ladder method with a Bornhuetter-Ferguson adjustment. The Bornhuetter-Ferguson tecnhique is applied to stabilise the underwriting inflation parameters using incurred data

Usage

bdcl.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 inflation parameter (inflat) is estimated from the incurred triangle (see BF adjustment in the description of the BDCL method in Martinez-Miranda, Nielsen and Verrall 2013). The predicted reserve using these estimates is different from the incurred reserve. If you want to reproduce exactly the incurred reserve (by splitting it into its RBNS and IBNR components) then use the function idcl.estimation.

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)

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.

Examples

# Reproducing the data analysis in the paper by Martinez-Miranda, Nielsen and Verrall (2013) 
data(NtriangleBDCL)
data(XtriangleBDCL)
data(ItriangleBDCL)

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/bdcl.estimation.Rd_%03d_medium.png", width=480, height=480)
> ### Name: bdcl.estimation
> ### Title: Parameter estimation - DCL model using the BDCL method
> ### Aliases: bdcl.estimation
> ### Keywords: models
> 
> ### ** Examples
> 
> # Reproducing the data analysis in the paper by Martinez-Miranda, Nielsen and Verrall (2013) 
> data(NtriangleBDCL)
> data(XtriangleBDCL)
> data(ItriangleBDCL)
> 
> 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 
>