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Tweedie Stochastic Reserving Model

tweedieReserve.Rd

This function implements loss reserving models within the generalized linear model framework in order to generate the full predictive distribution for loss reserves. Besides, it generates also the one year risk view useful to derive the reserve risk capital in a Solvency II framework. Finally, it allows the user to validate the model error while changing different model parameters, as the regression structure and diagnostics on the Tweedie p parameter.

Usage

tweedieReserve(triangle, var.power = 1, 
                    link.power = 0, design.type = c(1, 1, 0), 
                    rereserving = FALSE, cum = TRUE, exposure = FALSE, 
                    bootstrap = 1, boot.adj = 0, nsim = 1000, 
                    proc.err = TRUE, p.optim = FALSE,
                    p.check = c(0, seq(1.1, 2.1, by = 0.1), 3),
                    progressBar = TRUE, ...)

Arguments

triangle

An object of class triangle.

var.power

The index (p) of the power variance function \(V(\mu)=\mu^p\). Default to p = 1, which is the over-dispersed Poisson model. If NULL, it will be assumed to be in (1, 2) and estimated using the cplm package. See tweedie.

The index of power link function. The default link.power = 0 produces a log link. See tweedie.

design.type

It's a 3 dimension array that specifies the design matrix underlying the GLM. The dimensions represent respectively: origin period, development and calendar period. Accepted values are: 0 (not modelled), 1 (modelled as factor) and 2 (modelled as variable). Default to c(1,1,0), which is the common specification in actuarial literature (origin and development period as factors, calendar period not modelled). If a parameter for the calendar period is specified, a linear regression on the log CY parameter is fitted to estimate future values, thus is recommended to validate them running a plot of the gamma values (see output gamma_y) .

rereserving

Boolean, if TRUE the one year risk view loss reserve distribution is derived. Default to FALSE. Note, the runtime can materially increase if set to TRUE.

cum

Boolean, indicating whether the input triangle is cumulative or incremental along the development period. If TRUE, then triangle is assumed to be on the cumulative scale, and it will be converted to incremental losses internally before a GLM is fitted.

exposure

Boolean, if TRUE the exposure defined in the triangle object is specified as offset in the GLM model specification. Default to FALSE.

bootstrap

Integer, it specifies the type of bootstrap for parameter error. Accepted values are: 0 (disabled), 1 (parametric), 2 (semi-parametric). Default to 1.

boot.adj

Integer, it specified the methodology when using semi-parametric bootstrapping. Accepted values are: 0 (cycles until all the values of the pseudo-triangle are >= 0), 1 (overwrite negative values to 0.01). Default to 0. Note, runtime can materially increase when set to 0, as it could struggle to find pseudo-triangles >= 0)

nsim

Integer, number of simulations to derive the loss reserve distribution. Default to 1000. Note, high num of simulations could materially increase runtime, in particular if a re-reserving algorithm is used as well.

proc.err

Boolean, if TRUE a process error (coherent with the specified model) is added to the forecasted distribution. Default to TRUE.

p.optim

Boolean, if TRUE the model estimates the MLE for the Tweedie's p parameter. Default to FALSE. Recommended to use to validate the Tweedie's p parameter.

p.check

If p.optim=TRUE, a vector of p values for consideration. The values must all be larger than one (if the response variable has exact zeros, the values must all be between one and two). Default to c(0,seq(1.1,2.1,by=0.1),3). As fitting the Tweedie p-value isn't a straightforward process, please refer to tweedie.profile, p.vec argument.

progressBar

Boolean, if TRUE a progress bar will be shown in the console to give an indication of bootstrap progress.

...

Arguments to be passed onto the function glm or cpglm such as contrasts or control. It is important that offset and weight should not be specified. Otherwise, an error will be reported and the program will quit.

Value

The output is an object of class "glm" that has the following components:

call

the matched call.

summary

A data frame containing the predicted loss reserve statistics. The following items are displayed:

  • Latest: Latest paid

  • Det.Reserve: Deterministic reserve, i.e. the MLE GLM estimate of the Reserve

  • Ultimate: Ultimate cost, defined as Latest+Det.Reserve

  • Dev.To.Date: Development to date, defined as Latest/Ultimate

The following items are available if bootstrap>0

  • Expected.Reserve: The expected reserve, defined as the average of the reserve simulations. Should be roughly as Det.Reserve.

  • Prediction.Error: The prediction error of the reserve, defined as sqrt of the simulations. Please note that if proc.err=FALSE, this field contains only the parameter error given by the bootstrap.

  • CoV: Coefficient of Variation, defined as Prediction. Error/Expected.Reserve.

  • Expected Ultimate: The expected ultimate, defined as Expected.Reserve+Latest.

The following items are availbale if bootstrap>0 & reserving=TRUE

  • Expected.Reserve_1yr: The reserve derived as sum of next year payment and the expected value of the re-reserve at the end of the year. It should be similar to both Expected.Reserve and Det.Reserve. If it isn't, it's recommended to change regression structure and parameters.

  • Prediction.Error_1yr: The prediction error of the prospective Claims Development Result (CDR), as defined by Wüthrich (CDR=R(0)-X-R(1)).

  • Emergence.Pattern: It's the emergence pattern defined as Prediction.Error_1yr/Prediction.Error.

Triangle

The input triangle.

FullTriangle

The completed triangle, where empty cells in the original triangle are filled with model predictions.

model

The fitted GLM, a class of glm or cpglm. It is most convenient to work with this component when model fit information is wanted.

scale

The dispersion parameter phi

bias

The model bias, defined as bias<-sqrt(n/d.f)

GLMReserve

Deterministic reserve, i.e. the MLE GLM estimate of the Reserve

gamma_y

When the calendar year is used, it displays the observed and fitted calendar year (usually called "gamma"") factors.

res.diag

It's a data frame for residual diagnostics. It contains:

  • unscaled: The GLM Pearson residuals.

  • unscaled.biasadj: The GLM Person residuals adjusted by the bias, i.e. unscaled.biasadj=unscaled*bias.

  • scaled: The GLM Person scaled residuals, i.e. scaled=unscaled/sqrt(phi).

  • scaled, biasadj: The GLM Person scaled residuals adjusted by the bias, i.e. scaled.biasadj=scaled*bias.

  • dev: Development year.

  • origin: Origin year.

  • cy: Calendar year.

[If boostrap>1]

distr.res_ult

The full distribution "Ultimate View"

[If rereserve=TRUE]

distr.res_1yr

The full distribution "1yr View"

References

Gigante, Sigalotti. Model risk in claims reserving with generalized linear models. Giornale dell'Istituto Italiano degli Attuari, Volume LXVIII. 55-87. 2005

England, Verrall. Stochastic claims reserving in general insurance. B.A.J. 8, III. 443-544. 2002

England, Verrall. Predictive distributions of outstanding liabilities in general insurance. A.A.S. 1, II. 221-270. 2006

Peters, Shevchenko, Wüthrich, Model uncertainty in claims reserving within Tweedie's compound poisson models. Astin Bulletin 39(1). 1-33. 2009

Renshaw, Verrall. A stochastic model underlying the chain-ladder technique. B.A.J. 4, IV. 903-923. 1998

Author

Alessandro Carrato MSc FIA OA alessandro.carrato@gmail.com

Note

This function was born initially as a fork of the glmReserve by Wayne Zhang. I would like to thank him for his work that permitted me to speed up my coding.

Warning

Note that the runtime can materially increase for certain parameter setting. See above for more details.

See also

See also summary.tweedieReserve.

Examples

if (FALSE) { # \dontrun{
## Verrall's ODP Model is a Tweedie with p=1, log link and 
## origin/development periods as factors, thus c(1,1,0)
res1 <- tweedieReserve(MW2008, var.power=1, link.power=0, 
                           design.type=c(1,1,0), rereserving=TRUE,
                           progressBar=TRUE)

## To get directly ultimate view and respective one year view 
## at selected percentiles
summary(res1) 

#To get other interesting statistics
res1$summary

## In order to validate the Tweedie parameter 'p', it is interesting to 
## review its loglikelihood profile. Please note that, given the nature 
## of our data, it is expected that we may have some fitting issues for 
## given 'p' parameters, thus any results/errors should be considered 
## only indicatively. Considering different regression structures is anyway 
## recommended. Different 'p' values can be defined via the p.check array 
## as input of the function. 
## See help(tweedie.profile), p.vec parameter, for further information.
## Note: The parameters rereserving and bootstrap can be set to 0 to speed up 
## the process, as they aren't needed. 

## Runs a 'p' loglikelihood profile on the parameters 
## p=c(0,1.1,1.2,1.3,1.4,1.5,2,3)
res2 <- tweedieReserve(MW2008, p.optim=TRUE, 
                       p.check=c(0,1.1,1.2,1.3,1.4,1.5,2,3), 
                       design.type=c(1,1,0), 
                        rereserving=FALSE, bootstrap=0, 
                        progressBar=FALSE)

## As it is possible to see in this example, the MLE of p (or xi) results 
## between 0 and 1, which is not possible as Tweedie models aren't 
## defined for 0 < p < 1, thus the Error message. 
## But, despite this, we can conclude that overall the value p=1 could be 
## reasonable for this dataset, as anyway it seems to be near the MLE. 

## In order to consider an inflation parameter across the origin period, 
## it may be interesting to change the regression structure to c(0,1,1) 
## to get the same estimates of the Arithmetic Separation Method, as 
## referred in Gigante/Sigalotti. 
res3 <- tweedieReserve(MW2008, var.power=1, link.power=0, 
                           design.type=c(0,1,1), rereserving=TRUE,
                           progressBar=TRUE)
res3

## An assessment on future fitted calendar year values (usually defined 
## as "gamma") is recommended
plot(res3$gamma_y)

## Model residuals can be plotted using the res.diag output
plot(scaled.biasadj ~ dev, data=res3$res.diag) # Development year
plot(scaled.biasadj ~ cy, data=res3$res.diag) # Calendar year
plot(scaled.biasadj ~ origin, data=res3$res.diag) # Origin year
} # }