dcl.estimation: Parameter estimation - Double Chain Ladder model
In DCL: Claims Reserving under the Double Chain Ladder Model

dcl.estimation

R Documentation

Parameter estimation - Double Chain Ladder model

Description

Compute the estimated parameters in the model (delay parameters, severity underwriting inflation, severity mean and variance) using the Double Chain Ladder method.

Usage

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

Details

Two models are estimated in the double chain ladder framework (Martinez-Miranda, Nielsen and Verrall 2012).

The basic DCL model only makes assumption on the first moments (see assumptions M1-M3 in Section 2 of the paper). From the two input triangles (Ntriangle,Xtriangle) the parameters involved in this model are estimated: pi.delay (delay parameters that could be negative values and/or sum up above 1, by solving the linear system (7) in Section 3), mu (mean of the individual payments in the first underwriting period, from expression (9)), inflat (the underwriting severity mean inflation, from expression (8)), alpha.N and beta.N (the chain ladder parameters in the (OD)Poisson model for Ntriangle from expressions (10)-(12)). Using the estimated parameters in this simpler model the predicted outstanding numbers (calculated from equations (14) and (15)) are exactly the classical chain ladder predictions (see Theorem 1 in pp. 67).

The second model is a distributional model (assumptions D1-D4 in Section 5) which allows to provide the full cash-flow. In this model the parameters are adjusted to match with the assumptions: pj are delay probabilities resulting from adjusting the general parameters pi.delay (defined in expressions (21)-(22)), mu.adj is the corresponding adjusted mean factor and sigma2 is the variance factor (in expression (24)). The function dcl.estimation suggest two different adjustments of the general pi.delay, the user should choose the adjustment which does not modify substantially the IBNR/RBNS split in the basic model (M1-M3), see Martinez-Miranda, Nielsen, Verrall and W|thrich (2013) for a discussion.

Value

`pi.delay`	General delay parameters
`mu`	Mean severity factor
`inflat`	Underwriting severity 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).

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., Verrall, R. and W|thrich, M.V. (2013) Double Chain Ladder, Claims Development Inflation and Zero Claims. Scandinavian Actuarial Journal.

Examples

data(NtriangleDCL)
data(XtriangleDCL)
# Estimation of the DCL parameters described in Martinez-Miranda, Nielsen and Verrall (2012)
est1<-dcl.estimation(XtriangleDCL,NtriangleDCL)
Plot.dcl.par(est1)
# Compare two possible adjustmets to get distributional parameters
# est1 with adj=1
pj.1<-est1$pj
pi.delay<-est1$pi.delay
est2<-dcl.estimation(XtriangleDCL,NtriangleDCL,adj=2,Tables=FALSE)
pj.2<-est2$pj
data.frame(pi.delay=pi.delay,pj.adj.1=pj.1,pj.adj.2=pj.2)

DCL documentation built on May 5, 2022, 5:06 p.m.