dcl.predict.prior {DCL}

Pointwise predictions (RBNS/IBNR split) adding prior knowledge

Description

Pointwise predictions by calendar years and rows of the outstanding liabilities. The predictions are splitted between RBNS and IBNR claims.

Usage

dcl.predict.prior( Ntriangle , Xtriangle , inflat.i , inflat.j , Qi , 
  Model = 2, adj = 2, Tail = FALSE, Tables = TRUE, 
  summ.by = "diag", num.dec = 2 )

Arguments

Ntriangle

Optional. 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 the Xtriangle (both in the same aggregation level (quarters, years,etc.)) used to derive dcl.par

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.

inflat.i

Optional. A vector with dimension m (the dimension of the input triangles) specifying the severity inflation in the underwriting direction. If not specified it will be estimated using dcl.estimation.

inflat.j

Optional. A vector with dimension m specifying the severity inflation in the development direction. If not specified it will be assumed to be 1 and then the severity mean not depending on the development period.

Qi

Optional. A vector with dimension m specifying the probability of zero-claims for each underwriting period. If not specified then it will be assumed no zero-payments.

Model

Possible values are 0, 1 or 2 (default). See dcl.estimation for more in details.

adj

Method to adjust the estimated delay parameters for the distributional model. It should be 1 (default value) or 2. See dcl.estimation for more in details.

Tail

Logical. If TRUE (default) the tail is provided.

Tables

Logical. If TRUE (default) it is showed a table with the predicted outstanding liabilities in the future calendar periods (summ.by="diag") or by underwriting period (summ.by="row").

summ.by

A character value such as "diag", "row" or "cell".

num.dec

Number of decimal places used to report numbers in the tables. Used only if Tables=TRUE

Details

The predictions are calculated under the first moment assumptions in the DCL model (see M1-M3) in Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2012). In this case the severity mean is specified as

inflat.i * (1-Qi) * inflat.j * mu

where inflat.i, Qi, inflat.j and mu are prior information specified by the user. With this specification, the prediction formula consists of the expectation (conditional expectation -if Ntriangle is given and Model=0) of the future (RBNS/IBNR) aggregated payments. See formulas (8)-(9) in the paper.

If the prior information is not provided the function will return the DCL predictions as dcl.predict. The information about Qi, inflat.j can be extracted through DCL using extract.prior.

Value

Xrbns

A matrix with dimension m by 2m-1 (m being the dimension of the input triangles in DCL) having the outstanding RBNS numbers as the entries.

Drbns

A vector with dimension 2m-1 with elements being the outstanding liabilities for RBNS claims in the future calendar periods (sums by diagonals). The last value is the RBNS reserve (overall sum).

Rrbns

A vector with dimension m with elements being the outstanding liabilities for RBNS claims at each underwriting period (sums by rows). The last value is the RBNS reserve (overall sum).

Xibnr

A matrix with dimension m by 2m-1 (m being the dimension of the input triangles in DCL) having the outstanding IBNR numbers as the entries.

Dibnr

A vector with dimension 2m-1 with elements being the outstanding liabilities for IBNR claims in the future calendar periods (sums by diagonals). The last value is the IBNR reserve (overall sum).

Ribnr

A vector with dimension m with elements being the outstanding liabilities for IBNR claims at each underwriting period (sums by rows). The last value is the RBNS reserve (overall sum).

Xtotal

A matrix with dimension m by 2m-1 (m being the dimension of the input triangles in DCL) having the outstanding total (=RBNS+IBNR) numbers as the entries.

Dtotal

A vector with dimension 2m-1 with elements being the outstanding liabilities for all claims in the future calendar periods (sums by diagonals). The last value is the total (=RBNS+IBNR) reserve (overall sum).

Rtotal

A vector with dimension m with elements being the outstanding liabilities for all claims at each underwriting period (sums by rows). The last value is the total (=RBNS+IBNR) reserve (overall sum).

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 Wuthrich, M.V. (2013) Double Chain Ladder, Claims Development Inflation and Zero Claims. Scandinavian Actuarial Journal. In press.

See Also

dcl.estimation, bdcl.estimation, idcl.estimation, dcl.predict,extract.prior

Examples

## Data application by in Martinez-Miranda, Nielsen, Verrall and Wuthrich (2013)
data(NtrianglePrior)
data(NpaidPrior)
data(XtrianglePrior)

Ntriangle<-NtrianglePrior
Xtriangle<-XtrianglePrior
Npaid<-NpaidPrior

## Extract information about zero-claims and severity dev. inflation
my.priors<-extract.prior(Xtriangle,Npaid,Ntriangle)
my.inflat.j<-my.priors$inflat.j
my.Qi<-my.priors$Qi

# Reproducing the poinwise predicions (tables 3,4,5) in the paper
# Note: in the paper we did not use Ntriangle in the predictions 
# when modelling the predictions are slightly different

## Prior A: only using development year inflation
m<-nrow(Ntriangle)
preds.prior.A.gen<-dcl.predict.prior(Ntriangle,Xtriangle,
          inflat.j=my.inflat.j,Qi=rep(0,m),Model=0,adj=1,
          Tail=FALSE,Tables=TRUE,summ.by="diag",num.dec=2)

preds.prior.A.mod<-dcl.predict.prior(Ntriangle,Xtriangle,
          inflat.j=my.inflat.j,Qi=rep(0,m),Model=2,adj=2,
          Tail=FALSE,Tables=TRUE,summ.by="diag",num.dec=2)

## Prior B: only using zero claims inflation
preds.prior.B.gen<-dcl.predict.prior(Ntriangle,Xtriangle,
          inflat.j=rep(1,m),Qi=my.Qi,Model=0,adj=1,
          Tail=FALSE,Tables=TRUE,summ.by="diag",num.dec=2)

preds.prior.B.mod<-dcl.predict.prior(Ntriangle,Xtriangle,
          inflat.j=rep(1,m),Qi=my.Qi,Model=2,adj=2,
          Tail=FALSE,Tables=TRUE,summ.by="diag",num.dec=2)

## Prior C: only using development inflation and zero claims inflation
preds.prior.C.gen<-dcl.predict.prior(Ntriangle,Xtriangle,
          inflat.j=my.inflat.j,Qi=my.Qi,Model=0,adj=1,
          Tail=FALSE,Tables=TRUE,summ.by="diag",num.dec=2)

preds.prior.C.mod<-dcl.predict.prior(Ntriangle,Xtriangle,
          inflat.j=my.inflat.j,Qi=my.Qi,Model=2,adj=2,
          Tail=FALSE,Tables=TRUE,summ.by="diag",num.dec=2)