Claims Reserving under the Double Chain Ladder Model

Description

This package provides functions for statistical modelling and forecasting in claims reserving in non-life insurance under the Double Chain Ladder framework by Martinez-Miranda, Nielsen and Verrall (2012). Using specific functions, the user will be able generate plots to visualize and gain intuition about the data (run-off triangles), break down classical chain ladder under the DCL model, visualize the underlying delay function and the inflation, introduce expert knowledge about the severity inflation, the zero-claims etc. Besides a validation exercise can be performed through a back-test on the data.

Details

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

Maintainer: Maria Dolores Martinez-Miranda <mmiranda@ugr.es>

References

Martinez-Miranda M.D., Nielsen B, Nielsen J.P and Verrall, R. (2011) Cash flow simulation for a model of outstanding liabilities based on claim amounts and claim numbers. Astin Bulletin, 41/1, 107-129.

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 American Actuarial Journal, 17(2), 101-113.

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 more at http://www.cassknowledge.com/research/article/double-chain-ladder-cass-knowledge

Examples

data(NtriangleDCL)
data(XtriangleDCL)

# Classical chain ladder parameters
my.clm.par<-clm(XtriangleDCL)
Plot.clm.par(my.clm.par)

# Estimation of the DCL parameters (break-down of the chain ladder parameters)
my.dcl.par<-dcl.estimation(XtriangleDCL,NtriangleDCL)
Plot.dcl.par(my.dcl.par)

# DCL Predictions by diagonals (future calendar years)
# Splitting the chain ladder reserve into RBNR and IBNR claims (ignoring the tail)
preds.dcl.diag<-dcl.predict(my.dcl.par,Model=0,Tail=FALSE,num.dec=0)

# Full cashflow considering the tail (only the variance process)
# Below only B=200 simulations for faster calculations in the example
boot1<-dcl.boot(dcl.par=my.dcl.par,Ntriangle=NtriangleDCL,boot.type=1,B=200)
Plot.cashflow(boot1)

Switch to a higher level of aggregation

Description

From the input run-off triangle (weekly, monthly, quarterly, etc.) the function creates another triangle on a higher level of aggregation.

Usage

Aggregate(triangle, freq = 4)

Arguments

.

Details

If the input triangle does not consist of complete periods (for example a quarterly triangle with only three quarters in the last year), then the last (lower level) periods will been removed to get full aggregated periods.

Value

A run-off triangle in the specified higher level of aggregation.

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

See Also

get.incremental,get.cumulative,Plot.triangle

Examples

## A dummy example: a run-off triangle with 5*4=20 quarters
m<-20
my.square<-matrix(1,m,m)
# Now my.triangle is a quarterly triangle (the upper left triangle from my.square)
my.triangle<-my.square
my.triangle[row(my.square)+col(my.square)>(m+1)]<-NA
my.yearly.triangle<-Aggregate(my.triangle)
list(original=my.triangle,yearly=my.yearly.triangle)

Incurred data (BDCL example)

Description

Real motor data from a major insurer. It is a yearly run-off (incremental) triangle consisting of the incurred data during 19 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen and Verrall (2013).

Usage

data(ItriangleBDCL)

Format

Matrix with dimension 19 by 19: 19 undewriting years and 19 development years.

References

Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2013) Double Chain Ladder and Bornhuetter-Fergusson. North American Actuarial Journal, 17(2), 101-113.

Examples

data(ItriangleBDCL)
data(XtriangleBDCL)
m<-nrow(XtriangleBDCL)

clm.I<-clm(ItriangleBDCL)
alpha.I<-clm.I$alpha
# The total paid for each accident year in the past
Ri.X<-rowSums(XtriangleBDCL,na.rm=TRUE)
# Incurred outstanding numbers
Ri.CL.incurred<-alpha.I-Ri.X
Total.CL.incurred<- sum(Ri.CL.incurred,na.rm=TRUE)

## Compare with CL on paid data
clm.X<-clm(XtriangleBDCL)
Xhat<-as.matrix(clm.X$triangle.hat)
Ri.CL.paid<-rowSums(Xhat)-rowSums(XtriangleBDCL,na.rm=TRUE)
Total.CL.paid<- sum(Ri.CL.paid,na.rm=TRUE)

# the predictions by rows
data.frame(underw.year=c(1:m,"Total"),CLM.paid=c(Ri.CL.paid,Total.CL.paid),
           CLM.incurred=round(c(Ri.CL.incurred,Total.CL.incurred),4))

# now the predictions by diagonals
inflat.factor<-Ri.CL.incurred/Ri.CL.paid
inflat.factor[Ri.CL.paid==0]<-1
# the lower triangle from incurred chain ladder is defined as:
Ihat<-Xhat
for (i in 1:m) Ihat[i,]<-Xhat[i,]*inflat.factor[i]
# now the sums by diagonals
Diag.CL.paid<-sapply(split(Xhat, row(Xhat)+col(Xhat)), sum, na.rm=TRUE)
Dclm.paid<-c(Diag.CL.paid[-(1:m)])
Total.CL.paid<- sum(Dclm.paid,na.rm=TRUE)
Dclm.paid<-c(Dclm.paid,Total.CL.paid)

Diag.CL.inc<-sapply(split(Ihat, row(Ihat)+col(Ihat)), sum, na.rm=TRUE)
Dclm.inc<-c(Diag.CL.inc[-(1:m)])
Total.CL.inc<- sum(Dclm.inc,na.rm=TRUE)
Dclm.inc<-c(Dclm.inc,Total.CL.inc)

# A table with the chain ladder predictions (paid and incurred data)
data.frame(Future.years=c(1:(m-1),'Tot.'),
    clm.paid=round(Dclm.paid),clm.incurred=round(Dclm.inc))

Number of non-zero payments (adding prior knowledge example)

Description

It is a yearly run-off (incremental) triangle consisting of the number of non-zero payments during 14 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen, Verrall and Wuthrich (2013).

Usage

data(NpaidPrior)

Format

Matrix with dimension 14 by 14: 14 undewriting years and 14 development years.

Source

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.

Examples

data(NpaidPrior)
Plot.triangle(NpaidPrior)

Number of reported claims (BDCL example)

Description

Real motor data from a major insurer. It is a yearly run-off (incremental) triangle consisting of the number of reported claims during 19 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen and Verrall (2013).

Usage

data(NtriangleBDCL)

Format

Matrix with dimension 19 by 19: 19 undewriting years and 19 development years.

References

Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2013) Double Chain Ladder and Bornhuetter-Fergusson. North American Actuarial Journal, 17(2), 101-113.

Examples

data(NtriangleBDCL)

Plot.triangle(NtriangleBDCL, Histogram=TRUE)
Plot.triangle(NtriangleBDCL)

# Classical chain ladder method
clm(NtriangleBDCL)

Number of reported claims (DCL example)

Description

Real motor data from a major insurer. It is a yearly run-off (incremental) triangle consisting of the number of reported claims during 10 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen and Verrall (2012).

Usage

data(NtriangleDCL)

Format

Matrix with dimension 10 by 10: 10 undewriting years and 10 development years.

Source

Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2012) Double Chain Ladder. Astin Bulletin, 42/1, 59-76.

Examples

data(NtriangleDCL)

Plot.triangle(NtriangleDCL, Histogram=TRUE)
Plot.triangle(NtriangleDCL)

# Classical chain ladder method
clm(NtriangleDCL)

Number of reported claims (adding prior knowledge example)

Description

It is a yearly run-off (incremental) triangle consisting of the number of reported claims during 14 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen, Verrall and Wuthrich (2013).

Usage

data(NtrianglePrior)

Format

Matrix with dimension 14 by 14: 14 undewriting years and 14 development years.

Source

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.

Examples

data(NtrianglePrior)

Plot.triangle(NtrianglePrior, Histogram=TRUE)
Plot.triangle(NtrianglePrior)

# Classical chain ladder method
clm(NtrianglePrior)

Plotting the full cashflow (bootstrap distribution)

Description

Provide histograms and boxplots of the RBNS, IBNR and total(=RBNS+IBNR) cashflows. The boxplots corresponds to the distribution of the outstanding liabities in the future calendar periods. The histograms show the distribution of the reserve (overall total).

Usage

Plot.cashflow( cashflow )

Arguments

Details

The cashflow should be derived by specifying the parameter summ.by="diag" in the function dcl.boot or dcl.boot.prior.

Value

No returned value.

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.

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.boot, dcl.boot.prior, dcl.estimation

Examples

# Results described in the data application by Martinez-Miranda, Nielsen and Verrall (2012)
data(NtriangleDCL)
data(XtriangleDCL)

# Estimation of the DCL parameters
est<-dcl.estimation(XtriangleDCL,NtriangleDCL)
# Full cashflow considering the tail (only variance process)
# Below only B=200 simulations for a fast example
boot1<-dcl.boot(dcl.par=est,Ntriangle=NtriangleDCL,boot.type=1,B=200)
Plot.cashflow(boot1)

Plotting the estimated chain ladder parameters

Description

Show a plot with the estimates of the chain ladder parameters and the development factors

Usage

Plot.clm.par( clm.par )

Arguments

Value

No returned value

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.

See Also

clm, dcl.estimation, Plot.triangle

Examples

data(NtriangleDCL)
my.clm.par<-clm(NtriangleDCL)

Plot.clm.par(my.clm.par)

Plotting the estimated parameters in the DCL model

Description

Show a two by two plot with the estimated parameters in the Double Chain Ladder model

Usage

Plot.dcl.par( dcl.par , type.inflat = 'DCL' )

Arguments

Value

No returned value

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

dcl.estimation, bdcl.estimation, idcl.estimation

Examples

data(NtriangleDCL)
data(XtriangleDCL)

# Estimation of the DCL parameters described in Martinez-Miranda, Nielsen and Verrall (2012)
my.dcl.par<-dcl.estimation(XtriangleDCL,NtriangleDCL)
Plot.dcl.par(my.dcl.par)

Plotting an incremental run-off triangle

Description

Graphical representaion of an incremental run-off triangle.

Usage

Plot.triangle( triangle , Histogram = FALSE , tit='' )

Arguments

Details

A histogram representation of the histogram is consistent with the run-off triangles of counts such as the number of reported claims, number of payments, etc. See Martinez-Miranda, Nielsen, Sperlich and Verrall (2013) for a further explanation.

Value

No returned value

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., Sperlich, S., Verrall, R. (2013). Continuous Chain Ladder: Reformulating and generalizing a classical insurance problem. Experts Systems with Applications, 40(14), 5588-5603.

See Also

get.incremental, clm

Examples

## Plotting a counts triangle (number of reported claims)
data(NtriangleDCL)
Plot.triangle(NtriangleDCL, Histogram=TRUE,tit=expression(paste('Counts: ',N[ij])))
# Classical CL predictions
clm.N<-clm(NtriangleDCL)
Nhat<-clm.N$triangle.hat
# Compare the original histogram with the CL projections
Plot.triangle(Nhat, Histogram=TRUE,tit='CL Predictions')

## Plotting a paid triangle (number of reported claims)
data(XtriangleDCL)
Plot.triangle(XtriangleDCL)

Paid data (BDCL example)

Description

Real motor data from a major insurer. It is a run-off (incremental) triangle consisting of the aggregated payments during 19 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen and Verrall (2013).

Usage

data(XtriangleBDCL)

Format

Matrix with dimension 19 by 19: 19 undewriting years and 19 development years.

References

Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2013) Double Chain Ladder and Bornhuetter-Fergusson. North American Actuarial Journal, 17(2), 101-113.

Examples

data(XtriangleBDCL)

Plot.triangle(XtriangleBDCL)
# Classical chain ladder method
clm(XtriangleBDCL)

Paid data (DCL example)

Description

Real motor data from a major insurer. It is a yearly run-off (incremental) triangle consisting of the aggregated payments during 10 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen and Verrall (2012).

Usage

data(XtriangleDCL)

Format

Matrix with dimension 10 by 10: 10 undewriting years and 10 development years.

Source

Martinez-Miranda, M.D., Nielsen, J.P. and Verrall, R. (2012) Double Chain Ladder. Astin Bulletin, 42/1, 59-76.

Examples

data(XtriangleDCL)
Plot.triangle(XtriangleDCL)
# Classical chain ladder method
clm(XtriangleDCL)

Paid data (adding prior knowledge example)

Description

It is a run-off (incremental) triangle consisting of the aggregated payments during 19 years. These data were used in the empirical illustration provided by Martinez-Miranda, Nielsen, Verrall and Wuthrich (2013).

Usage

data(XtrianglePrior)

Format

Matrix with dimension 14 by 14: 14 undewriting years and 14 development years.

Source

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.

Examples

data(XtrianglePrior)

Plot.triangle(XtrianglePrior)
# Classical chain ladder method
clm(XtrianglePrior)

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

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

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.

See Also

get.incremental, Plot.dcl.par, dcl.predict, dcl.estimation, idcl.estimation, clm

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)

Classical Chain Ladder Method

Description

Provide the classical chain ladder output consisting of the development (forward) factors and the predictions in the full square. Besides it provides the estimated parameters under the (over-dispersion) Poisson model for the double chain ladder estimation.

Usage

clm( triangle , n.cal = NA , Fj = NA )

Arguments

Details

By default Fj=NA and then classical chain ladder with the common calculation of the development factors (or using the most recent calendars -if 0<n.cal<m is provided), is performed. By specifying a valid vector with the development factors (it should has lenght equal to m-1), the user is allowed to use his own values in the algorithm. If valid values are specified for both n.cal and Fj, the first one (n.cal) will be ignored and the given development factors will be used in the calculations.

Value

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.

Verrall, R. (1991) Chain ladder and Maximum Likelihood. Journal of the Institute of Actuaries 118, 489-499.

See Also

get.incremental, Plot.clm.par, Plot.triangle

Examples

data(NtriangleDCL)
clm.N<-clm(NtriangleDCL)
# The alpha's
clm.N$alpha
# The beta's
clm.N$beta
# The development factors
clm.N$Fj
# Plotting the parameters and the dev. factors
Plot.clm.par(clm.N)
# The predictions
Nhat<-clm.N$triangle.hat
Plot.triangle(Nhat,Histogram=TRUE)

## Trying variations from classical chain ladder
# Try CLM only using the more recent 2 calendars in the development
# factors calculation
clm(NtriangleDCL,n.cal=2)

# Try CLM providing a vector with given development factors
my.Fj<-c(1.4,1.1,1.0,1.1,1.1,1.0,1.0,1.0,1.1)
clm(NtriangleDCL,Fj=my.Fj)

Bootstrap distribution: the full cashflow

Description

Provide the distribution of the IBNR, RBNS and total (RBNS+IBRN) reserves by calendar years and rows using bootstrapping.

Usage

dcl.boot( dcl.par , sigma2 , Ntriangle , boot.type = 2 , B = 999 , 
  Tail = TRUE , summ.by = "diag" , Tables = TRUE , num.dec = 2 , n.cal = NA)

Arguments

Details

If the calculated severity variance using the function dcl.estimation is not a valid estimate then it is recommended to provide directly this value through the argument sigma2. It can be calculated using the function var applied to a pilot sample of individual payments observed in the first underwriting period.

Value

Note

If boot.type=2 the function will take some time to perform the calculations. It increases with the dimension of the triangles and the specified number of simulations B.

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.

See Also

Plot.cashflow, dcl.boot.prior

Examples

# Results described in the data application by Martinez-Miranda, Nielsen and Verrall (2012)
data(NtriangleDCL)
data(XtriangleDCL)

# Estimation of the DCL parameters
est<-dcl.estimation(XtriangleDCL,NtriangleDCL)

# Full cashflow considering the tail (only the variance process)
# Below only B=200 simulations to be faster in the example
boot1<-dcl.boot(dcl.par=est,Ntriangle=NtriangleDCL,boot.type=1,B=200)
Plot.cashflow(boot1)

# Full cashflow with tail and taking into account the parameters uncertainty
# and B=999 simulations. Do not run it unless you can wait about one minute
# boot2<-dcl.boot(dcl.par=est,Ntriangle=NtriangleDCL,boot.type=2)
# Plot.cashflow(boot2)

Bootstrap distribution (the full cashflow) adding prior knowledge

Description

Provide the distribution of the IBNR, RBNS and total (RBNS+IBRN) reserves by calendar years and rows using bootstrapping.

Usage

dcl.boot.prior( Xtriangle , Ntriangle , sigma2 , mu , inflat.i , inflat.j , Qi , 
    Model = 2 , adj = 1 , boot.type = 2, B = 999 ,  
    Tail = TRUE , summ.by = "diag", Tables = TRUE, num.dec = 2 , n.cal = NA ,
    Fj.X = NA , Fj.N = NA )

Arguments

Details

If proper values are provided for the arguments sigma2, mu, inflat.i, inflat.j and Qi then, they will be considered fixed as prior knowledge. Otherwise, if not specified, inflat.j will be assumed to be a vector of ones, Qi a vector of zeros, and the rest will be estimated using dcl.estimation.

Value

Note

If boot.type=2 the function will take some time to perform the calculations. It increases with the dimension of the triangles and the specified number of simulations B.

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

References

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

Plot.cashflow, dcl.boot

Examples

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

## Extract information about zero-claims and severity dev. inflation
my.priors<-extract.prior(XtrianglePrior,NpaidPrior,NtrianglePrior,Plots=FALSE)
my.inflat.j<-my.priors$inflat.j
my.Qi<-my.priors$Qi

## Bootstrap cashflow incorporating prior knowledge about
##      severity inflation and zero claims
# Only variance process
# Below only B=200 simulations for a fast example
dist.priorC.I<-dcl.boot.prior(NtrianglePrior,XtrianglePrior,
  inflat.j=my.inflat.j,Qi=my.Qi,adj=2,Tail=FALSE,boot.type=1,B=200)
Plot.cashflow(dist.priorC.I)

## Try to compare with DCL with no prior knowledge: 
# Only variance process
# dist.dcl.I<-dcl.boot.prior(NtrianglePrior,XtrianglePrior,adj=2,
#    Tail=FALSE,boot.type=1)
# Plot.cashflow(dist.dcl.I)

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

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

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.

See Also

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

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)

Pointwise predictions (RBNS/IBNR split)

Description

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

Usage

dcl.predict( dcl.par , Ntriangle , Model = 2 , Tail = TRUE , 
  Tables = TRUE , summ.by="diag", num.dec = 2 )                     

Arguments

Details

If Model=0 or Model=1 then the predictions are calculated using the DCL model parameters in assumptions M1-M3 (general delay parameters, see Martinez-Miranda, Nielsen and Verrall 2012). If Model=2 the adjusted delay probabilities (distributional model D1-D4) are considered. By choosing Model=0 the predictions are calculated ignoring the observed counts in Ntriangle (also if the Ntriangle is not specified). It should be specified to reproduce get the IBNR/RBNS split of classical paid chain ladder.

Choose summ.by="diag" to calculate the predicted outstanding liabilities in the future calendar periods (diagonal sums), summ.by="row" for sums by underwriting periods (row sums); or summ.by="cell" to get only the the individual cell predictions.

Value

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.

See Also

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

Examples

## Data application by in Martinez-Miranda, Nielsen and Verrall (2012)

data(NtriangleDCL)
data(XtriangleDCL)

# Estimation of the DCL parameters described 
est<-dcl.estimation(XtriangleDCL,NtriangleDCL)

# with general delay parameters and ignoring Ntriangle to reproduce exactly chain ladder
pred1<-dcl.predict(dcl.par=est,Model=1,Tail=FALSE)

# with Modeled parameters (distributional Model) and ignoring Ntriangle
pred2<-dcl.predict(dcl.par=est,Model=2,Tail=FALSE)

# with Modeled parameters (distributional Model) using observed Ntriangle
pred3<-dcl.predict(dcl.par=est,Ntriangle=NtriangleDCL,Model=2,Tail=FALSE)

# providing the Tail, with Modeled parameters (distributional Model)
pred4<-dcl.predict(dcl.par=est,Ntriangle=NtriangleDCL,Model=2,Tail=TRUE)

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

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

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)

Extracting information about zero-claims and severity inflation

Description

A way of extracting information about zero-claims and severity development inflation through the DCL method applied to two counts triangles: number of payments and number of reported claims.

Usage

extract.prior(Xtriangle, Npaid, Ntriangle, Plots = TRUE , n.cal = NA ,
  Fj.X = NA , Fj.N = NA , Fj.Npaid = NA )

Arguments

Details

The function implements the strategy proposed in the paper by Martinez-Miranda, Nielsen, Verrall and Wuthrich (2013) to extract information for additional triangles (see "Section 5: An example showing how other data can be used to provide prior information in practice"). The derived severity inflation inflat.j does not extend to the tail. If you want provide the tail, by using dcl.predict.prior, the vector should be extended to have dimension 2m-1, otherwise the tail will be not provided (as was done in the cited paper).

Value

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

References

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.predict.prior, dcl.estimation

Examples

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

extract.prior(XtrianglePrior,NpaidPrior,NtrianglePrior)

Cumulative triangle

Description

Switch from an incremental to a cumulative triangle

Usage

get.cumulative( triangle )

Arguments

Value

The cumulative triangle

Note

The methods in this the DCL package works normally on incremental triangles

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

See Also

get.incremental

Examples

data(NtriangleDCL)
get.cumulative(NtriangleDCL)

Incremental triangle

Description

Switch from an cumulative to an incremental triangle

Usage

get.incremental( triangle )

Arguments

Value

The incremental triangle

Note

The methods in this the DCL package works normally on incremental triangles

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

See Also

get.cumulative

Examples

data(NtriangleDCL)
Ntriangle.cum<-get.cumulative(NtriangleDCL)
get.incremental(Ntriangle.cum)

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

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

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)

Back-test: testing against the experience

Description

A back-test to validate incurred reserve (IDCL) against paid reserve (DCL) or the paid with a Bornhuetter-Fergusson adjustment (BDCL). The validation strategy consists of: (1) Cut ncut=1,2,. diagonals from the observed paid triangle. (2) Apply the three methods (DCL, BDCL and IDCL), and (3) compare forecasts and actual values.

Usage

validating.incurred( ncut = 0 , Xtriangle , Ntriangle , Itriangle , 
  Model = 0 , Plot.box = TRUE , Tables = TRUE , num.dec = 4 , 
  n.cal = NA , Fj.X = NA , Fj.N = NA , Fj.I = NA)

Arguments

Details

If ncut=0 the test is not computed but a plot showing the difference among the three methods is shown. It is recommended to start with this step to have some insight about the problem. Note that the first part in the IDCL inflation is usually very volatile since no many outstanding liabitities arise from the first underwriting periods.

The predicion errors provided through the value pe.vector are calculated as follow:

For individual cells: pe.cells = sum(ce.dif^2) / sum(ce.obs^2)

with ce.dif being the vector with the differences between the predicted cells and the actual cells (ce.obs).

For diagonals: pe.diags = sum(ca.dif^2) / sum(ca.obs^2)

with ca.dif being the vector with the differences between the predicted calendars and the actual calendars (ca.obs).

For the total reserve: pe.tot = sum(tot.dif^2) / sum(tot.obs^2)

with tot.dif the absolute difference between the predicted reserve and the actual reserve (tot.obs).

Value

Note

To validate classical chain ladder on paid data against classical chain ladder on incurred data it should be used Model=0 (see dcl.predict) for more details.

Author(s)

M.D. Martinez-Miranda, J.P. Nielsen and R. Verrall

References

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.

Martinez-Miranda M.D., Nielsen, J.P., Sperlich, S., Verrall, R. (2013). Continuous Chain Ladder: Reformulating and generalizing a classical insurance problem. Experts Systems with Applications, 40(14), 5588-5603.

See Also

dcl.estimation, bdcl.estimation, idcl.estimation, ,dcl.predict

Examples

data(NtriangleBDCL)
data(XtriangleBDCL)
data(ItriangleBDCL)

Ntriangle<-NtriangleBDCL
Xtriangle<-XtriangleBDCL
Itriangle<-ItriangleBDCL
## First compare the three methods to be validated (three different inflations)
validating.incurred(ncut=0,Xtriangle,Ntriangle,Itriangle)

## Now perform a backtest cutting up to four calendars backward
test.res<-matrix(NA,4,10)
par(mfrow=c(2,2),cex.axis=0.9,cex.main=1)
par(mar=c(1.5,1.5,1.5,1.5),oma=c(1,0.5,0.5,0.2),mgp=c(3,0.5,0)) 
for (i in 1:4)
{
  res<-validating.incurred(ncut=i,Xtriangle,Ntriangle,Itriangle,Tables=FALSE)
  test.res[i,]<-as.numeric(res$pe.vector)
}
test.res<-as.data.frame(test.res)
names(test.res)<-c("num.cut","pe.point.DCL","pe.point.BDCL","pe.point.IDCL",
"pe.calendar.DCL","pe.calendar.BDCL","pe.calendar.IDCL",
"pe.total.DCL","pe.total.BDCL","pe.total.IDCL")
print(test.res)