Showing posts with label cplm. Show all posts
Showing posts with label cplm. Show all posts
ChainLadder 0.2.2 is out with improved glmReserve function
We released version 0.2.2 of
Ok, what does this all mean? I will run through a couple of examples and look behind the scene of
Like most other functions in ChainLadder,
The data at hand is often presented in form of a claims triangle, such as the following example data set from the ChainLadder package:
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
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):
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:
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:
Finally, I get back to the
With
In my first example I set use the parameters \(p=1, q=0\), which should return the results of the Poisson model.
Setting the argument
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
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 NASuppose 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
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")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] 28655.77There 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 0.05961042Setting 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
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 R in the insurance industry
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:
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!
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.
 |
| Using R in Insurance |
For the historians under the readers, here is my five year old poster from GIRO in Vienna 2006.
 |
| Poster session at useR! 2011 in Warwick, UK |
Here are a few more insurance related packages:
- 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
Other useful documents:
- 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.
Help! There is a special interest group for R in insurance:
