mages' blog

I continue with the growth curve model for loss reserving from last week's post. Today, following the ideas of James Guszcza [2] I will add an hierarchical component to the model, by treating the ultimate loss cost of an accident year as a random effect. Initially, I will use the nlme R package, just as James did in his paper, and then move on to Stan/RStan [6], which will allow me to estimate the full distribution of future claims payments.

Last week's model assumed that cumulative claims payment could be described by a growth curve. I used the Weibull curve and will do so here again, but others should be considered as well, e.g. the log-logistic cumulative distribution function for long tail business, see [1].
The growth curve describes the proportion of claims paid up to a given development period compared to the ultimate claims cost at the end of time, hence often called development pattern. Cumulative distribution functions are often considered, as they increase monotonously from 0 to 100%. Multiplying the development pattern with the expected ultimate loss cost gives me then the expected cumulative paid to date value.

However, what I'd like to do is the opposite, I know the cumulative claims position to date and wish to estimate the ultimate claims cost instead. If the claims process is fairly stable over the years and say, once a claim has been notified the payment process is quite similar from year to year and claim to claim, then a growth curve model is not unreasonable. Yet, the number and the size of the yearly claims will be random, e.g. if a windstorm, fire, etc occurs or not. Hence, a random effect for the ultimate loss cost across accident years sounds very convincing to me.

Here is James' model as described in [2]:
\[
\begin{align}
CL_{AY, dev} & \sim \mathcal{N}(\mu_{AY, dev}, \sigma^2_{dev}) \\
\mu_{AY,dev} & = Ult_{AY} \cdot G(dev|\omega, \theta)\\
\sigma_{dev} & = \sigma \sqrt{\mu_{dev}}\\
Ult_{AY} & \sim \mathcal{N}(\mu_{ult}, \sigma^2_{ult})\\
G(dev|\omega, \theta) & = 1 - \exp\left(-\left(\frac{dev}{\theta}\right)^\omega\right)
\end{align}
\]The cumulative losses \(CL_{AY, dev}\) for a given accident year \(AY\) and development period \(dev\) follow a Normal distribution with parameters \(\mu_{AY, dev}\) and \(\sigma_{dev}\).

The mean itself is modelled as the product of an accident year specific ultimate loss cost \(Ult_{AY}\) and a development period specific parametric growth curve \(G(dev | \omega, \theta)\). The variance is believed to increase in proportion with the mean. Finally, the ultimate loss cost is modelled with a Normal distribution as well.

Assuming a Gaussian distribution of losses doesn't sound quite intuitive to me, as loss are often skewed to the right, but I shall continue with this assumption here to make a comparison with [2] possible.

Using the example data set given in the paper I can reproduce the result in R with nlme:

The fit looks pretty good, with only 5 parameters. See James' paper for a more detailed discussion.

Let's move this model into Stan. Here is my attempt, which builds on last week's pooled model. With the generated quantities code block I go beyond the scope of the original paper, as I try to estimate the full posterior predictive distribution as well.
The 'trick' is the line mu[i] <- ult[origin[i]] * weibull_cdf(dev[i], omega, theta); where I have an accident year (here labelled origin) specific ultimate loss.

The notation ult[origin[i]] illustrates the hierarchical nature in Stan's language nicely.

Let's run the model:
The estimated parameters look very similar to the nlme output above.

Let's take a look at the parameter traceplot and the densities of the estimated ultimate loss costs by origin year.
This looks all not too bad. The trace plots don't show any particular patterns, apart from \(\sigma_{ult}\), which shows a little skewness.

The generated quantities code block in Stan allows me to get also the predictive distribution beyond the current data range. Here I forecast claims up to development year 12 and plot the predictions, including the 95% credibility interval of the posterior predictive distribution with the observations.

The model seems to work rather well, even with the Gaussian distribution assumptions. Yet, it has still only 5 parameters. Note, this model doesn't need an additional artificial tail factor either.

Conclusions

The Bayesian approach sounds to me a lot more natural than many classical techniques around the chain-ladder methods. Thanks to Stan, I can get the full posterior distributions on both, the parameters and predictive distribution. I find communicating credibility intervals much easier than talking about the parameter, process and mean squared error.

James Guszcza contributed to a follow-up paper with Y. Zhank and V. Dukic [3] that extends the model described in [2]. It deals with skewness in loss data sets and the autoregressive nature of the errors in a cumulative time series.

Frank Schmid offers a more complex Bayesian analysis of claims reserving in [4], while Jake Morris highlights the similarities between a compartmental model used in drug research and loss reserving [5].

Finally, Glenn Meyers published a monograph on Stochastic Loss Reserving Using Bayesian MCMC Models earlier this year [7] that is worth taking a look at.

References

[1] David R. Clark. LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach. Casualty Actuarial Society, 2003. CAS Fall Forum.

[2] James Guszcza. Hierarchical Growth Curve Models for Loss Reserving, 2008, CAS Fall Forum, pp. 146–173.

[3] Y. Zhang, V. Dukic, and James Guszcza. A Bayesian non-linear model for forecasting insurance loss payments. 2012. Journal of the Royal Statistical Society: Series A (Statistics in Society), 175: 637–656. doi: 10.1111/j.1467-985X.2011.01002.x

[4] Frank A. Schmid. Robust Loss Development Using MCMC. Available at SSRN. See also http://lossdev.r-forge.r-project.org/

[5] Jake Morris. Compartmental reserving in R. 2015. R in Insurance Conference.

[6] Stan Development Team. Stan: A C++ Library for Probability and Sampling, Version 2.8.0. 2015. http://mc-stan.org/.

[7] Glenn Meyers. Stochastic Loss Reserving Using Bayesian MCMC Models. Issue 1 of CAS Monograph Series. 2015.

Session Info

R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.1 (El Capitan)

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_0.2.3 rstan_2.8.0   ggplot2_1.0.1  Rcpp_0.12.1      
[5] lattice_0.20-33  

loaded via a namespace (and not attached):
 [1] nloptr_1.0.4       plyr_1.8.3         tools_3.2.2       
 [4] digest_0.6.8       lme4_1.1-10        statmod_1.4.21    
 [7] gtable_0.1.2       nlme_3.1-122       mgcv_1.8-8        
[10] Matrix_1.2-2       parallel_3.2.2     biglm_0.9-1       
[13] SparseM_1.7        proto_0.3-10       coda_0.18-1       
[16] gridExtra_2.0.0    stringr_1.0.0      MatrixModels_0.4-1
[19] lmtest_0.9-34      stats4_3.2.2       grid_3.2.2        
[22] nnet_7.3-11        tweedie_2.2.1      inline_0.3.14     
[25] cplm_0.7-4         minqa_1.2.4        actuar_1.1-10     
[28] reshape2_1.4.1     car_2.1-0          magrittr_1.5      
[31] scales_0.3.0       codetools_0.2-14   MASS_7.3-44       
[34] splines_3.2.2      rsconnect_0.3.79   systemfit_1.1-18  
[37] pbkrtest_0.4-2     colorspace_1.2-6   quantreg_5.19     
[40] labeling_0.3       sandwich_2.3-4     stringi_1.0-1     
[43] munsell_0.4.2      zoo_1.7-12

Last week I posted a biological example of fitting a non-linear growth curve with Stan/RStan. Today, I want to apply a similar approach to insurance data using ideas by David Clark [1] and James Guszcza [2].

Instead of predicting the growth of dugongs (sea cows), I would like to predict the growth of cumulative insurance loss payments over time, originated from different origin years. Loss payments of younger accident years are just like a new generation of dugongs, they will be small in size initially, grow as they get older, until the losses are fully settled.

Here is my example data set:

Following the articles cited above I will assume that the growth can be explained by a Weibull curve, with two parameters \(\theta\) (scale) and \(\omega\) (shape):
\[
G(dev|\omega, \theta) = 1 - \exp\left(-\left(\frac{dev}{\theta}\right)^\omega\right)
\]Inspired by the classical over-dispersed Poisson GLM in actuarial science, Guszcza [2] assumes a power variance structure for the process error as well. For the purpose of this post I will assume that claims cost follow a Normal distribution, to make a comparison with a the classical least square regression easier. With a prior estimate of the ultimate claims cost the cumulative loss can be modelled as:
\[
\begin{align}
CL_{AY, dev} & \sim \mathcal{N}(\mu_{dev}, \sigma^2_{dev}) \\
\mu_{dev} & = Ult \cdot G(dev|\omega, \theta)\\
\sigma_{dev} & = \sigma \sqrt{\mu_{dev}}
\end{align}
\]Perhaps, the above formula suggests a hierarchical modelling approach, with different ultimate loss costs by accident year. Indeed, this is the focus of [2] and I will endeavour to reproduce the results with Stan in my next post, but today I will stick to a pooled model that assumes a constant ultimate loss across all accident years, i.e. \(Ult_{AY} = Ult \;\forall\; AY\).

To prepare my analysis I read in the data as a long data frame instead of the matrix structure above. Additionally, I compose another data frame that I will use later to predict payments two years beyond the first 10 years. Furthermore, to align the output with [2] I relabelled the development periods from years to months, so that year 1 becomes month 6, year 2 becomes month 18, etc. The accident years run from 1991 to 2000, while the variable origin maps those years from 1 to 10.

To get a reference point for Stan I start with a non-linear least square regression:

The output above doesn't look unreasonable, apart from the accident years 1991, 1992 and 1998. The output of gnls gives me an opportunity to provide my prior distributions with good starting points. I will use an inverse Gamma as a prior for \(\sigma\), constrained Normals for the parameters \(\theta, \omega\) and \(Ult\) as well. If you have better ideas, then please get in touch.

The Stan code below is very similar to last week. Again, I am interested here in the posterior distributions, hence I add a block to generate quantities from those. Note, Stan comes with a build-in function for the cumulative Weibull distribution weibull_cdf.

I store the Stan code in a separate text file, which I read into R to compile the model. The compilation takes a little time. The sampling itself is done in a few seconds.

Let's take a look at the output:

The parameters are a little different to the output of gnls, but well within the standard error. From the plots I notice that the samples for the ultimate loss as well as for \(\theta\) are a little skewed to the right. Well, assuming cumulative losses to follow a Normal distribution was a bit a of stretch to start with. Still, the samples seem to converge.

Finally, I can plot the 90% credible intervals of the posterior distributions.

The 90% prediction credible interval captures most of the data and although this model might not be suitable for reserving individual accident years, it could provide an initial starting point for further investigations. Additionally, thanks to the Bayesian model I have access to the full distribution, not just point estimators and standard errors.

My next post will continue with this data set and the ideas of James Guszcza by allowing the ultimate loss cost to vary by accident year, treating it as a random effect. Here is a teaser of what the output will look like:

References

[1] David R. Clark. LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach. Casualty Actuarial Society, 2003. CAS Fall Forum.

[2] Guszcza, J. Hierarchical Growth Curve Models for Loss Reserving, 2008, CAS Fall Forum,
pp. 146–173.

Session Info

R version 3.2.2 (2015-08-14)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.1 (El Capitan)

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] rstan_2.8.0    ggplot2_1.0.1     Rcpp_0.12.1      
[4] nlme_3.1-122   lattice_0.20-33   ChainLadder_0.2.3

loaded via a namespace (and not attached):
 [1] nloptr_1.0.4       plyr_1.8.3         tools_3.2.2       
 [4] digest_0.6.8       lme4_1.1-10        statmod_1.4.21    
 [7] gtable_0.1.2       mgcv_1.8-8         Matrix_1.2-2      
[10] parallel_3.2.2     biglm_0.9-1        SparseM_1.7       
[13] proto_0.3-10       gridExtra_2.0.0    coda_0.18-1       
[16] stringr_1.0.0      MatrixModels_0.4-1 stats4_3.2.2      
[19] lmtest_0.9-34      grid_3.2.2         nnet_7.3-11       
[22] inline_0.3.14      tweedie_2.2.1      cplm_0.7-4        
[25] minqa_1.2.4        reshape2_1.4.1     car_2.1-0         
[28] actuar_1.1-10      magrittr_1.5       codetools_0.2-14  
[31] scales_0.3.0       MASS_7.3-44        splines_3.2.2     
[34] systemfit_1.1-18   rsconnect_0.3.79   pbkrtest_0.4-2    
[37] colorspace_1.2-6   labeling_0.3       quantreg_5.19     
[40] sandwich_2.3-4     stringi_1.0-1      munsell_0.4.2     
[43] zoo_1.7-12

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

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.

There can never be too many examples for transforming data with R. So, here is another example of reshaping a data.frame into a matrix.

Here I have a data frame that shows incremental claim payments over time for different loss occurrence (origin) years.


The format of the data frame above is how this kind of data is usually stored in a data base. However, I would like to see the payments of the different origin years in rows of a matrix.

The first idea might be to use the reshape function, but that would return a data.frame. Yet, it is actually much easier with the matrix function itself. Most of the code below is about formatting the dimension names of the matrix. Note that I use the with function to save me a bit of typing.


An elegant alternative to matrix provides the acast function of the reshape2 package. It has a nice formula argument and allows me not only to specify the aggregation function, but also to add the margin totals.

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.

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")))