mages' blog

We released version 0.2.2 of ChainLadder a few weeks ago. This version adds back the functionality to estimate the index parameter for the compound Poisson model in glmReserve using the cplm package by Wayne Zhang.

Ok, what does this all mean? I will run through a couple of examples and look behind the scene of glmReserve. However, the clue is in the title, glmReserve is a function that uses a generalised linear model to estimate future claims, assuming claims follow a Tweedie distribution. I should actually talk about a family of distributions that is known as Tweedie, named by Bent Jørgensen after Maurice Tweedie. Joe Rickert published a nice post about the Tweedie distribution last year.

Like most other functions in ChainLadder, glmReserve purpose is to predict future insurance claims based on historical data.

The data at hand is often presented in form of a claims triangle, such as the following example data set from the ChainLadder package:

library(ChainLadder)
cum2incr(UKMotor) 
##       dev
## origin    1    2    3    4    5   6   7
##   2007 3511 3215 2266 1712 1059 587 340
##   2008 4001 3702 2278 1180  956 629  NA
##   2009 4355 3932 1946 1522 1238  NA  NA
##   2010 4295 3455 2023 1320   NA  NA  NA
##   2011 4150 3747 2320   NA   NA  NA  NA
##   2012 5102 4548   NA   NA   NA  NA  NA
##   2013 6283   NA   NA   NA   NA  NA  NA

The rows present different origin periods in which accidents occurred and the columns along each row show the incremental reported claims over the years (the data itself is stored in a cumulative form).

Suppose all claims will be reported within 7 years, then I'd like to know how much money should be set aside for the origin years 2008 to 2013 for claims that have incurred but not been reported (IBNR) yet. Or, to put it differently, I have to predict the NA fields in the bottom right hand triangle.

First, let's reformat the data as it would be stored in a database, that is in a long format of incremental claims over the years (I add the years also as factors, which I will use later):

claims <- as.data.frame(cum2incr(UKMotor)) # convert into long format
library(data.table)
claims <- data.table(claims)
claims <- claims[ , ':='(cal=origin+dev-1, # calendar period
                         originf=factor(origin),
                         devf=factor(dev))]
claims <- claims[order(dev), cum.value:=cumsum(value), by=origin]
setnames(claims, "value", "inc.value")
head(claims)
##    origin dev inc.value  cal originf devf cum.value
## 1:   2007   1      3511 2007    2007    1      3511
## 2:   2008   1      4001 2008    2008    1      4001
## 3:   2009   1      4355 2009    2009    1      4355
## 4:   2010   1      4295 2010    2010    1      4295
## 5:   2011   1      4150 2011    2011    1      4150
## 6:   2012   1      5102 2012    2012    1      5102

Let's visualise the data:

library(lattice)
xyplot(cum.value/1000 + log(inc.value) ~ dev , groups=origin, data=claims, 
       t="b", par.settings = simpleTheme(pch = 16),
       auto.key = list(space="right",
                       title="Origin\nperiod", cex.title=1,
                       points=FALSE, lines=TRUE, type="b"),
       xlab="Development period", ylab="Amount",
       main="Claims development by origin period",
       scales="free")

The left plot of the chart above shows the cumulative claims payment over time, while the right plot shows the log-transformed incremental claims development for each origin/accident year.

One of the oldest methods to predict future claims development is called chain-ladder, which can be regarded as a weighted linear regression through the origin of cumulative claims over the development periods. Multiplying those development factors to the latest available cumulative position allows me to predict future claims in an iterative fashion.

It is well know in actuarial science that a Poisson GLM produces the same forecasts as the chain-ladder model.

Let's check:

# Poisson model
mdl.pois <- glm(inc.value ~ originf + devf, data=na.omit(claims), 
                family=poisson(link = "log"))
# predict claims
claims <- claims[, ':='(
  pred.inc.value=predict(mdl.pois, 
                         .SD[, list(originf, devf, inc.value)],
                         type="response")), by=list(originf, devf)]
# sum of future payments
claims[cal>max(origin)][, sum(pred.inc.value)]
## [1] 28655.77
# Chain-ladder forecast
summary(MackChainLadder(UKMotor, est.sigma = "Mack"))$Totals[4,]
## [1] 28655.77

Ok, this worked. However, both of these models make actually fairly strong assumptions. The Poisson model by its very nature will only produce whole numbers, and although payments could be regarded as whole numbers, say in pence or cents, it does feel a little odd to me. Similarly, modelling the year on year developments via a weighted linear regression through the origin, as in the case of the chain-ladder model, sounds not intuitive either.

There is another aspect to highlight with the Poisson model; its variance is equal to the mean. Yet, in real data I often observe that the variance increases in proportion to the mean. Well, this can be remedied by using an over-dispersed quasi-Poisson model.

I think a more natural approach would be to assume a compound distribution that models the frequency and severity of claims separately, e.g. Poisson frequency and Gamma severity.

Here the Tweedie distributions comes into play.

Tweedie distributions are a subset of what are called Exponential Dispersion Models. EDMs are two parameter distributions from the linear exponential family that also have a dispersion parameter \(\Phi\).

Furthermore, the variance is a power function of the mean, i.e. \(\mbox{Var}(X)=\Phi \, E[X]^p\).

The canonical link function for a Tweedie distribution in a GLM is the power link \(\mu^q\) with \(q=1-p\). Note, \(q=0\) is interpreted as \(\log(\mu)\).

Thus, let \(\mu_i = E(y_i)\) be the expectation of the ith response. Then I have the following model.
\[
y \sim \mbox{Tweedie}(q, p)\\
E(y) = \mu^q = Xb \\
\mbox{Var}(y) = \Phi \mu^p
\]The variance power \(p\) characterises the distribution of the responses \(y\). The following are some special cases:

• Normal distribution, p = 0
• Poisson distribution, p = 1
• Compound Poisson-Gamma distribution, 1 < p < 2
• Gamma distribution, p = 2
• Inverse-Gaussian, p = 3
• Stable, with support on the positive real numbers, p > 2

Finally, I get back to the glmReserve function, which Wayne Zhang, the other author of the cplm package, contributed to the ChainLadder package.

With glmReserve I can model a claims triangle using the Tweedie distribution family.

In my first example I set use the parameters \(p=1, q=0\), which should return the results of the Poisson model.

(m1 <- glmReserve(UKMotor, var.power = 1, link.power = 0))
##       Latest Dev.To.Date Ultimate  IBNR       S.E         CV
## 2008   12746   0.9732000    13097   351  125.8106 0.35843464
## 2009   12993   0.9260210    14031  1038  205.0826 0.19757473
## 2010   11093   0.8443446    13138  2045  278.8519 0.13635790
## 2011   10217   0.7360951    13880  3663  386.7919 0.10559429
## 2012    9650   0.5739948    16812  7162  605.2741 0.08451188
## 2013    6283   0.3038201    20680 14397 1158.1250 0.08044210
## total  62982   0.6872913    91638 28656 1708.1963 0.05961042

Perfect, I get the same results, plus further information about the model.

Setting the argument var.power=NULL will estimate \(p\) in the interval \((1,2)\) using the cplm package.

(m2 <- glmReserve(UKMotor, var.power=NULL))
##       Latest Dev.To.Date Ultimate  IBNR       S.E         CV
## 2008   12746   0.9732000    13097   351  110.0539 0.31354394
## 2009   12993   0.9260870    14030  1037  176.9361 0.17062307
## 2010   11093   0.8444089    13137  2044  238.5318 0.11669851
## 2011   10217   0.7360951    13880  3663  335.6824 0.09164138
## 2012    9650   0.5739948    16812  7162  543.6472 0.07590718
## 2013    6283   0.3038201    20680 14397 1098.7988 0.07632138
## total  62982   0.6873063    91636 28654 1622.4616 0.05662252
m2$model
## 
## Call:
## cpglm(formula = value ~ factor(origin) + factor(dev), link = link.power, 
##     data = ldaFit, offset = offset)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -6.7901  -1.6969   0.0346   1.6087   8.4465  
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         8.25763    0.04954 166.680  < 2e-16 ***
## factor(origin)2008  0.03098    0.05874   0.527 0.605588    
## factor(origin)2009  0.09999    0.05886   1.699 0.110018    
## factor(origin)2010  0.03413    0.06172   0.553 0.588369    
## factor(origin)2011  0.08933    0.06365   1.403 0.180876    
## factor(origin)2012  0.28091    0.06564   4.279 0.000659 ***
## factor(origin)2013  0.48797    0.07702   6.336 1.34e-05 ***
## factor(dev)2       -0.11740    0.04264  -2.753 0.014790 *  
## factor(dev)3       -0.62829    0.05446 -11.538 7.38e-09 ***
## factor(dev)4       -1.03168    0.06957 -14.830 2.28e-10 ***
## factor(dev)5       -1.31346    0.08857 -14.829 2.28e-10 ***
## factor(dev)6       -1.86307    0.13826 -13.475 8.73e-10 ***
## factor(dev)7       -2.42868    0.25468  -9.536 9.30e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Estimated dispersion parameter: 10.788
## Estimated index parameter: 1.01 
## 
## Residual deviance: 299.27  on 15  degrees of freedom
## AIC:  389.18 
## 
## Number of Fisher Scoring iterations:  4

From the model I note that the dispersion parameter \(\phi\) was estimated as 10.788 and the index parameter \(p\) as 1.01.

Not surprisingly the estimated reserves are similar to the Poisson model, but with a smaller predicted standard error.

Intuitively the modelling approach makes a lot more sense, but I end up with one parameter for each origin and development period, hence there is a danger of over-parametrisation.

Looking at the plots above again I note that many origin periods have a very similar development. Perhaps a hierarchical model would be more appropriate?

For more details on glmReserve see the help file and package vignette.

Session Info

R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.5 (Yosemite)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] data.table_1.9.6  ChainLadder_0.2.2

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.1        nloptr_1.0.4       plyr_1.8.3        
 [4] tools_3.2.2        digest_0.6.8       lme4_1.1-9        
 [7] statmod_1.4.21     gtable_0.1.2       nlme_3.1-121      
[10] lattice_0.20-33    mgcv_1.8-7         Matrix_1.2-2      
[13] parallel_3.2.2     biglm_0.9-1        SparseM_1.7       
[16] proto_0.3-10       coda_0.17-1        stringr_1.0.0     
[19] MatrixModels_0.4-1 stats4_3.2.2       lmtest_0.9-34     
[22] grid_3.2.2         nnet_7.3-10        tweedie_2.2.1     
[25] cplm_0.7-4         minqa_1.2.4        ggplot2_1.0.1     
[28] reshape2_1.4.1     car_2.1-0          actuar_1.1-10     
[31] magrittr_1.5       scales_0.3.0       MASS_7.3-44       
[34] splines_3.2.2      systemfit_1.1-18   pbkrtest_0.4-2    
[37] colorspace_1.2-6   quantreg_5.19      sandwich_2.3-4    
[40] stringi_0.5-5      munsell_0.4.2      chron_2.3-47      
[43] zoo_1.7-12        

Over the weekend we released version 0.2.1 of the ChainLadder package for claims reserving on CRAN.

New Features

• New function PaidIncurredChain by Fabio Concina, based on the 2010 Merz & Wüthrich paper Paid-incurred chain claims reserving method
• Functions plot.MackChainLadder and plot.BootChainLadder gained new argument which, allowing users to specify which sub-plot to display. Thanks to Christophe Dutang for this suggestion.

Output of plot(MackChainLadder(MW2014, est.sigma="Mack"), which=3:6)

Changes

• Updated NAMESPACE file to comply with new R CMD checks in R-3.3.0
• Removed package dependencies on grDevices and Hmisc
• Expanded package vignette with new paragraph on importing spreadsheet data, a new section "Paid-Incurred Chain Model" and an added example for a full claims development picture in the "One Year Claims Development Result" section, see also [1] .

Binary versions of the package will appear on the various CRAN mirrors over the next couple of days. Alternatively you can install ChainLadder directly from GitHub using the following R commands:

install.packages(c(“systemfit”, “actuar", "statmod", "tweedie", "devtools"))
library(devtools)
install_github("mages/ChainLadder")
library(ChainLadder)

Completely new to ChainLadder? Start with the package vignette.

References

[1] Claims run-off uncertainty: the full picture. (with M. Merz) SSRN Manuscript, ID 2524352, 2014.

ChainLadder is an R package that provides statistical methods and models for claims reserving in general insurance.

With version 0.2.0 we added new functions to estimate the claims development result (CDR) as required under Solvency II. Special thanks to Alessandro Carrato, Giuseppe Crupi and Mario Wüthrich who have contributed code and documentation.

New Features

• New generic function CDR to estimate the one year claims development result. S3 methods for the Mack and bootstrap model have been added already:
  
  • CDR.MackChainLadder to estimate the one year claims development result of the Mack model without tail factor, based papers by Merz & Wüthrich (2008, 2014)
  • CDR.BootChainLadder to estimate the one year claims development result of the bootstrap model.
• New function tweedieReserve to estimate reserves in a GLM framework, including the one year claims development result.
• Package vignette has a new chapter on One Year Claims Development Result
• New example data MW2008 and MW2014 form the Merz & Wüthrich (2008, 2014) papers

Changes

• Source code development moved from Google Code to GitHub
• as.data.frame.triangle now gives warning message when dev. period is a character.
• Alessandro Carrato, Giuseppe Crupi and Mario Wüthrich have been added as authors, thanks to their major contribution to code and documentation.
• Christophe Dutang, Arnaud Lacoume and Arthur Charpentier have been added as contributors, thanks to their feedback, guidance and code contribution.

Examples

The examples below use the triangle of the 2008 Merz & Wüthrich paper and illustrate how the one year claims development result can be estimated using the new CDR function for output of MackChainLadder and BootChainLadder. Also the tweedieReserve function is demonstrated, which can estimate the one year CDR as well, by setting the argument rereserving to TRUE.

For further details see package vignette and the help pages of the respective functions.

References

Michael Merz and Mario V. Wüthrich. Modelling the claims development result for solvency purposes. CAS E-Forum, Fall:542–568, 2008

Michael Merz and Mario V. Wüthrich. Claims run-off uncertainty: the full picture. SSRN Manuscript, 2524352, 2014.

Session Info

R version 3.1.3 (2015-03-09)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.2 (Yosemite)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats graphics grDevices utils datasets methods base     

other attached packages:
[1] ChainLadder_0.2.0 statmod_1.4.20 systemfit_1.1-14 lmtest_0.9-33    
[5] zoo_1.7-12        car_2.0-25     Matrix_1.1-5     

loaded via a namespace (and not attached):
 [1] acepack_1.3-3.3     actuar_1.1-8        cluster_2.0.1      
 [4] colorspace_1.2-6    digest_0.6.8        foreign_0.8-63     
 [7] Formula_1.2-0       ggplot2_1.0.0       grid_3.1.3         
[10] gtable_0.1.2        Hmisc_3.15-0        lattice_0.20-30    
[13] latticeExtra_0.6-26 lme4_1.1-7          MASS_7.3-39        
[16] mgcv_1.8-5          minqa_1.2.4         munsell_0.4.2      
[19] nlme_3.1-120        nloptr_1.0.4        nnet_7.3-9         
[22] parallel_3.1.3      pbkrtest_0.4-2      plyr_1.8.1         
[25] proto_0.3-10        quantreg_5.11       RColorBrewer_1.1-2 
[28] Rcpp_0.11.5         reshape2_1.4.1      rpart_4.1-9        
[31] sandwich_2.3-2      scales_0.2.4        SparseM_1.6        
[34] splines_3.1.3       stringr_0.6.2       survival_2.38-1    
[37] tools_3.1.3         tweedie_2.2.1

Taking the first step is often the hardest: getting data from Excel into R.

Suppose you would like to use the ChainLadder package to forecast future claims payments for a run-off triangle that you have stored in Excel.


How do you get the triangle into R and execute a reserving function, such as MackChainLadder?

Well, there are many ways to do this and the ChainLadder package vignette, as well as the R manual on Data Import/Export has all of the details, but here is a quick and dirty solution using a CSV-file.

Open a new Excel workbook and copy your triangle into cell A1, with the first column being the accident or origin period and the first row describing the development period or age. You find an example CSV-file on GitHub.

Make sure that your triangle has no formatting, such a commas to separate thousands, as Excel will save those cell as characters. Now open R and go through the following commands:

# The first command will open a window and 
# ask you to select your CSV-file
myCSVfile <- file.choose()
# Read file into R
dat <- read.csv(file=myCSVfile) 
# use read.csv2 if semicolons are used as a separator
# likely to be the case if you are in continental Europe

dat # to see your data

     AY   X1    X2    X3    X4    X5    X6    X7    X8    X9   X10
1  1981 5012  8269 10907 11805 13539 16181 18009 18608 18662 18834
2  1982  106  4285  5396 10666 13782 15599 15496 16169 16704    NA
3  1983 3410  8992 13873 16141 18735 22214 22863 23466    NA    NA
4  1984 5655 11555 15766 21266 23425 26083 27067    NA    NA    NA
5  1985 1092  9565 15836 22169 25955 26180    NA    NA    NA    NA
6  1986 1513  6445 11702 12935 15852    NA    NA    NA    NA    NA
7  1987  557  4020 10946 12314    NA    NA    NA    NA    NA    NA
8  1988 1351  6947 13112    NA    NA    NA    NA    NA    NA    NA
9  1989 3133  5395    NA    NA    NA    NA    NA    NA    NA    NA
10 1990 2063    NA    NA    NA    NA    NA    NA    NA    NA    NA

Ok, the data is in R, but now you have to convert it into a triangle. A triangle is basically a matrix with extra attributes. To do this execute the following steps.

# Load the ChainLadder package
library(ChainLadder)
# Ignore first column which holds accident year information 
tri <- dat[,-1]
# Convert to matrix
tri <- as.matrix(tri)
# Add dimension names
dimnames(tri) <- list(origin=dat[,1], dev=1:ncol(tri))
# Convert into a triangle class
tri <- as.triangle(tri)
tri
      dev
origin    1     2     3     4     5     6     7     8     9    10
  1981 5012  8269 10907 11805 13539 16181 18009 18608 18662 18834
  1982  106  4285  5396 10666 13782 15599 15496 16169 16704    NA
  1983 3410  8992 13873 16141 18735 22214 22863 23466    NA    NA
  1984 5655 11555 15766 21266 23425 26083 27067    NA    NA    NA
  1985 1092  9565 15836 22169 25955 26180    NA    NA    NA    NA
  1986 1513  6445 11702 12935 15852    NA    NA    NA    NA    NA
  1987  557  4020 10946 12314    NA    NA    NA    NA    NA    NA
  1988 1351  6947 13112    NA    NA    NA    NA    NA    NA    NA
  1989 3133  5395    NA    NA    NA    NA    NA    NA    NA    NA
  1990 2063    NA    NA    NA    NA    NA    NA    NA    NA    NA

With those preparations done you can execute the MackChainLadder function:

M <- MackChainLadder(tri, est.sigma = "Mack")
M
     Latest Dev.To.Date Ultimate   IBNR Mack.S.E CV(IBNR)
1981 18,834       1.000   18,834      0        0      NaN
1982 16,704       0.991   16,858    154      206    1.339
1983 23,466       0.974   24,083    617      623    1.010
1984 27,067       0.943   28,703  1,636      747    0.457
1985 26,180       0.905   28,927  2,747    1,469    0.535
1986 15,852       0.813   19,501  3,649    2,002    0.549
1987 12,314       0.694   17,749  5,435    2,209    0.406
1988 13,112       0.546   24,019 10,907    5,358    0.491
1989  5,395       0.336   16,045 10,650    6,333    0.595
1990  2,063       0.112   18,402 16,339   24,566    1.503

               Totals
Latest:    160,987.00
Dev:             0.76
Ultimate:  213,122.23
IBNR:       52,135.23
Mack S.E.:  26,909.01
CV(IBNR):        0.52

To copy the full triangle back into Excel you can use the clipboard:
write.table(M$FullTriangle, file="clipboard", sep="\t").
Go back to Excel and hit <Ctrl> + V on your keyboard to paste the data into R.

For more details see the package vignette and Dan's post on pasting triangles from Excel into R via the clipboard.

If you are after a thorough overview of R in insurance take a look at the book Computational Actuarial Science with R.

Finally, join the Special Interest Group on using R in actuarial science and insurance to share your questions and answers.

Session Info

R version 3.1.2 (2014-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods  
[7] base     

other attached packages:
[1] ChainLadder_0.1.9 systemfit_1.1-14  lmtest_0.9-33    
[4] zoo_1.7-11        car_2.0-21        Matrix_1.1-4     

loaded via a namespace (and not attached):
 [1] acepack_1.3-3.3     actuar_1.1-6        cluster_1.15.3     
 [4] foreign_0.8-61      Formula_1.1-2       grid_3.1.2         
 [7] Hmisc_3.14-5        lattice_0.20-29     latticeExtra_0.6-26
[10] MASS_7.3-35         nnet_7.3-8          plyr_1.8.1         
[13] RColorBrewer_1.0-5  Rcpp_0.11.3         reshape2_1.4       
[16] rpart_4.1-8         sandwich_2.3-2      splines_3.1.2      
[19] statmod_1.4.20      stringr_0.6.2       survival_2.37-7    
[22] tools_3.1.2         tweedie_2.2.1 

Over the weekend we released version 0.1.8 of the ChainLadder package for claims reserving on CRAN.

What is claims reserving?

The insurance industry, unlike other industries, does not sell products as such but promises. An insurance policy is a promise by the insurer to the policyholder to pay for future claims for an upfront received premium.

As a result insurers don't know the upfront cost for their service, but rely on historical data analysis and judgement to predict a sustainable price for their offering. In General Insurance (or Non-Life Insurance, e.g. motor, property and casualty insurance) most policies run for a period of 12 months. However, the claims payment process can take years or even decades. Therefore often not even the delivery date of their product is known to insurers. The money set aside for those future claims payments are called reserves.

Version 0.1.6 of the ChainLadder package has been released and is already available from CRAN.

The new version adds the function CLFMdelta. CLFMdelta finds consistent weighting parameters delta for a vector of selected age-to-age chain-ladder factors for a given run-off triangle.

The added functionality was implemented by Dan Murphy, who is the co-author of the paper A Family of Chain-Ladder Factor Models for Selected Link Ratios by Bardis, Majidi, Murphy. You find a more detailed explanation with R code examples on Dan's blog and see also his slides from the CAS spring meeting.

Slides by Dan Murphy

Last week we released version 0.1.5-6 of the ChainLadder package on CRAN. The ChainLadder package provides statistical models, which are typically used for the estimation of outstanding claims reserves in general insurance. The package vignette gives an overview of the package functionality.

Output of plot(MackChainLadder(GenIns))

Since the last CRAN release Dan Murphy added new features to the MackChainLadder function and we fixed a bug in BootChainLadder. Here are he details:

New Features

• The list output of the MackChainLadder function now includes the parameter risk and process risk breakdowns of the total risk estimate for the sum of projected losses across all origin years by development age.
  
• The Mack Method's recursive parameter risk calculation now enables Mack's original two-term formula (the default) and optionally the three-term formula found in Murphy's 1994 paper and in the 2006 paper by Buchwalder, Bühlmann, Merz, and Wüthrich.
  
• A few more Mack Method examples.
  

Bug Fixes

• The phi-scaling factor in BootChainLadder was incorrect. Instead of calculating the number of data items in the upper left triangle as n*(n+1)/2, n*(n-1)/2 was used. Thanks to Thomas Girodot for reporting this bug.
  

Please get in touch if you would like to collaborate or find any issues or bugs.

Join me at the first R in Insurance conference at Cass Business School in London, 15 July 2013.

This is the third post about Christofides' paper on Regression models based on log-incremental payments [1]. The first post covered the fundamentals of Christofides' reserving model in sections A - F, the second focused on a more realistic example and model reduction of sections G - K. Today's post will wrap up the paper with sections L - M and discuss data normalisation and claims inflation.

I will use the same triangle of incremental claims data as introduced in my previous post. The final model had three parameters for origin periods and two parameters for development periods. It is possible to reduce the model further as Christofides illustrates in section L onwards by using an inflation index to bring all claims payments to current value and a claims volume adjustment or weight for each origin period to normalise the triangle.

In his example Christofides uses claims volume adjustments for the origin years and an earning or inflation index for the different payment calendar years. The claims volume adjustments aims to normalise the triangle for similar exposures across origin periods, while the earnings index, which measures largely wages and other forms of compensations, is used as a first proxy for claims inflation. Note that the earnings index shows significant year on year changes from 5% to 9%. Barnett and Zehnwirth [2] would probably recommend to add further parameters for the calendar year effects to the model.

# Page D5.36
ClaimsVolume <- data.frame(origin=0:6, 
  volume.index=c(1.43, 1.45, 1.52, 1.35, 1.29, 1.47, 1.91))
# Page D5.36
EarningIndex <- data.frame(cal=0:6, 
  earning.index=c(1.55, 1.41, 1.3, 1.23, 1.13, 1.05, 1))
# Year on year changes
round((1-EarningIndex$earning.index[-1]/EarningIndex$earning.index[-7]),2)
# [1] 0.09 0.08 0.05 0.08 0.07 0.05

dat <- merge(merge(dat, ClaimsVolume), EarningIndex)

# Normalise data for volume and earnings
dat$logvalue.ind.inf <- with(dat, log(value/volume.index*earning.index))

with(dat, interaction.plot(dev, origin, logvalue.ind.inf))
points(1+dat$dev, dat$logvalue.ind.inf, pch=16, cex=0.8)

Indeed, the interaction plot shows the various origin years now to be much more closely grouped. Only the single point of the last origin period stands out now. Christofides tests several models with different numbers of origin levels, but I am happy with the minimal model using only one parameter for the origin period, namely the intercept:

Following on from last week's post I will continue to go through the paper Regression models based on log-incremental payments by Stavros Christofides [1]. In the previous post I introduced the model from the first 15 pages up to section F. Today I will progress with sections G to K which illustrate the model with a more realistic incremental claims payments triangle from a UK Motor Non-Comprehensive account:

# Page D5.17
tri <- t(matrix(
  c(3511, 3215, 2266, 1712, 1059, 587, 340,
    4001, 3702, 2278, 1180,  956, 629,  NA,
    4355, 3932, 1946, 1522, 1238,  NA,  NA,
    4295, 3455, 2023, 1320,   NA,  NA,  NA,
    4150, 3747, 2320,   NA,   NA,  NA,  NA,
    5102, 4548,   NA,   NA,   NA,  NA,  NA,
    6283,   NA,   NA,   NA,   NA,  NA,  NA), nc=7))

The rows show origin period data, e.g. accident years, underwriting years or years of account and the columns present the development periods or lags. The triangle appears to be fairly well behaved. The last two years in rows 6 and 7 appear to be slightly higher than rows 2 to 5 and the values in row 1 are lower in comparison to the later years. The last payment of £1,238 in the third row stands out a bit as well.

Before I plot the data, I will transform the triangle into a data frame and add extra columns:

m <- dim(tri)[1]; n <- dim(tri)[2]
dat <- data.frame(
  origin=rep(0:(m-1), n),
  dev=rep(0:(n-1), each=m),
  value=as.vector(tri))

## Add dimensions as factors
dat <- with(dat, data.frame(origin, dev, cal=origin+dev, 
                            value, logvalue=log(value),
                            originf=factor(origin),
                            devf=as.factor(dev),
                            calf=as.factor(origin+dev)))

I am particularly interested in the decay of claims payments in the development year direction for each origin year on the original and log-scale. The interaction.plot of the stats package does an excellent job for this:

op <- par(mfrow=c(2,1), mar=c(4,4,2,2))
with(dat, interaction.plot(x.factor=dev, trace.factor=origin, 
          response=value))
points(dat$devf, dat$value, pch=16, cex=0.5)
with(dat, interaction.plot(x.factor=dev, trace.factor=origin, 
          response=logvalue))
points(dat$devf, dat$logvalue, pch=16, cex=0.5)
par(op)

Indeed the origin years 1 to 4 (rows 2 to 5) look quite similar and the decay of claims in development year direction appears to be linear on a log-scale from development year 1 onwards.

Based on those observations Christofides suggests two models; the first one will have a unique level for each origin year and a unique level for the zero development period. The parameters for development periods 1 to 6 are assumed to follow a linear relationship with the same slope \(s\):
\begin{align}
\ln(P_{ij}) & = Y_{ij} = a_i + d_j + \epsilon_{ij}
&\mbox{for } i,\,j \mbox{ from } 0 \mbox{ to } 6\\
\mbox{where } d_0 &= d,\quad d_j = s \cdot j
&\mbox{for } j > 0
\end{align}and \(\epsilon_{ij} \sim N(0, \sigma^2)\). The second model will be a reduced version of the above with only two levels for the origin years 5 and 6. Hence, I add four more columns to my data frame:

A recent post on the PirateGrunt blog on claims reserving inspired me to look into the paper Regression models based on log-incremental payments by Stavros Christofides [1], published as part of the Claims Reserving Manual (Version 2) of the Institute of Actuaries.

The paper is available together with a spread sheet model, illustrating the calculations. It is very much based on ideas by Barnett and Zehnwirth, see [2] for a reference. However, doing statistical analysis in a spread sheet programme is often cumbersome. I will go through the first 15 pages of Christofides' paper today and illustrate how the model can be implemented in R.

Let's start with the example data of an incremental claims triangle:

## Page D5.4
tri <- t(matrix(
  c(11073,  6427, 1839, 766,
    14799,  9357, 2344,  NA,
    15636, 10523,   NA,  NA,
    16913,    NA,   NA,  NA), 
  nc=4, dimnames=list(origin=0:3, dev=0:3)))

The above triangle shows incremental claims payments for four origin (accident) years over time (development years). It is the aim to predict the bottom right triangle of future claims payments, assuming no further claims after four development years.

Christofides model assumes the following structure for the incremental paid claims \(P_{ij}\):
\begin{align}
\ln(P_{ij}) & = Y_{ij} = a_i + b_j + \epsilon_{ij}
\end{align}where i and j go from 0 to 3, \(b_0=0\) and \(\epsilon_{ij} \sim N(0, \sigma^2)\). Unlike the basic chain-ladder method, this is a stochastic model that allows me to test my assumptions and calculate various statistics, e.g. standards errors of my predictions.

Last week we released version 0.1.5-4 of the ChainLadder package on CRAN. The R package provides methods which are typically used in insurance claims reserving. If you are new to R or insurance check out my recent talk on Using R in Insurance.

The chain-ladder method which is a popular method in the insurance industry to forecast future claims payments gave the package its name. However, the ChainLadder package has many other reserving methods and models implemented as well, such as the bootstrap model demonstrated below. It is a great starting point to learn more about stochastic reserving.

Since we published version 0.1.5-2 in March 2012 additional functionality has been added to the package, see the change log, but in particular the vignette has come a long way.

Many thanks to my co-authors Dan Murphy and Wayne Zhang.

Every year the UK’s general insurance actuarial community organises a big conference, which they call GIRO, short for General Insurance Research Organising committee.

This year's conference is in Brussels from 18 - 21 September 2012. Despite the fact that Brussels is actually in Belgium the UK actuaries will travel all the way to enjoy good beer and great talks.

On Wednesday morning I will run a session on Using R in insurance. It would be great to see some of you there.


I prepared the slides with R, RStudio, knitr, pandoc and slidy again. My title page shows a word cloud about the GIRO conference. It uses the wordcloud package and was inspired by Ian Fellows' post on FellStats.

The last slide shows the output of sessionInfo(). I am sure it will become helpful one day, when I have to remind myself how I actually created the slides and which packages and versions I used.

Today we published version 0.1.5-1 of the ChainLadder package for R. It provides methods which are typically used in insurance claims reserving to forecast future claims payments.

Claims development and chain-ladder forecast of the RAA data set using the Mack method

The package started out of presentations given at the Stochastic Reserving Seminar at the Institute of Actuaries in 2007, 2008 and 2010, followed by talks at CAS meetings in 2008 and 2010.

Initially the package came with implementations of the Mack-, Munich- and Bootstrap Chain-Ladder methods. Since version 0.1.3-3 it also provides general multivariate chain ladder models by Wayne Zhang. Version 0.1.4-0 introduced new functions on loss development factor fitting and Cape Cod by Daniel Murphy following a paper by David Clark. Version 0.1.5-0 has added loss reserving models within the generalized linear model framework following a paper by England P. and Verrall R. (1999) implemented by Wayne Zhang.

For more details see the project web site: http://code.google.com/p/chainladder/ and an early blog entry about R in the insurance industry.

Changes in version 0.1.5-1:

• Internal changes to plot.MackChainLadder to pass new checks introduced by R 2.14.0.
  
• Commented out unnecessary creation of 'io' matrix in ClarkCapeCod function. Allows for analysis of very large matrices for CapeCod without running out of RAM. 'io' matrix is an integral part of ClarkLDF, and so remains in that function.
  
• plot.clark method
  
  • Removed "conclusion" stated in QQplot of clark methods.
    
  • Restore 'par' settings upon exit
    
  • Slight change to the title
    
• Reduced the minimum 'theta' boundary for weibull growth function
  
• Added warnings to as.triangle if origin or dev. period are not numeric
  

Here is a little example using the googleVis package to display the RAA claims development triangle:

library(ChainLadder)
library(googleVis)
data(RAA) # example data set of the ChainLadder package
class(RAA) <- "matrix" # change the class from triangle to matrix
df <- as.data.frame(t(RAA)) # coerce triangle into a data.frame
names(df) <- 1981 : 1990
df$dev <-  1:10
plot(gvisLineChart(df, "dev", options=list(gvis.editor="Edit me!", hAxis.title="dev. period")))

Let's talk about R in the insurance industry today.  David Smith's blog entry reminded me about our poster at the R user conference in Warwick in August 2011:

Using R in Insurance

We presented examples on how R can be used in the insurance industry. We had a lot of fun presenting our poster. By accident we had printed the poster with quite a bit of access white space to the right. So we asked everyone who came along to sign it and by the end of the evening we had over 100 signatures!

Poster session at useR! 2011 in Warwick, UK

Yesterday Wayne Zhang, with whom I collaborate on the ChainLadder package, released the first version of his new cplm package on CRAN.  The name cplm is short for compound Poisson linear models. The cplm package is for fitting Tweedie compound Poisson linear models using the Monte Carlo EM algorithm. The form of the models that are handled in the package are generalized linear models, mixed-effect models and Bayesian models. For non-Bayesian models, maximum likelihood estimations are obtained for all parameters in the model, especially for the index parameter. Estimation for the Bayesian model is performed by Markov Chain Monte Carlo simulations. These models find their application in actuarial science, see also his paper.  

• ChainLadder - Reserving methods in R. The package provides Mack-, Munich-, Bootstrap, and Multivariate-chain-ladder methods, as well as the LDF Curve Fitting methods of Dave Clark and GLM-based reserving models.
• cplm - Monte Carlo EM algorithms and Bayesian methods for fitting Tweedie compound Poisson linear models.
• lossDev - A Bayesian time series loss development model. Features include skewed-t distribution with time-varying scale parameter, Reversible Jump MCMC for determining the functional form of the consumption path, and a structural break in this path.
• actuar: Loss distributions modelling, risk theory (including ruin theory), simulation of compound hierarchical models and credibility theory.
• fitdistrplus: Help to fit of a parametric distribution to non-censored or censored data
• favir: Formatted Actuarial Vignettes in R. FAViR lowers the learning curve of the R environment. It is a series of peer-reviewed Sweave papers that use a consistent style.
• mondate: R packackge to keep track of dates in terms of months
• lifecontingencies - Package to perform actuarial evaluation of life contingencies

• Introduction to R for Actuaries by Nigel de Silva
• An Interactive Introduction To R by Michael Driscoll and Dan Murphy
• If you have trouble with your IT department to get R on your machine this document might help you to put some good arguments forward. The report "An Actuarial Toolkit" was presented at the GIRO convention 2006 in Vienna.