idcl.estimation {DCL}

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.

See Also

Plot.dcl.par, dcl.predict, dcl.estimation, bdcl.estimation

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)