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DCL (version 0.1.2)

dcl.estimation: 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.

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).

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.

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.

See Also

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

Examples

Run this code
# NOT RUN {
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)
# }

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