Showing posts with label Actuarial. Show all posts
Showing posts with label Actuarial. Show all posts
Predicting claims with a Bayesian network
Here is a little Bayesian Network to predict the claims for two different types of drivers over the next year, see also example 16.15 in [1].
Let's assume there are good and bad drivers. The probabilities that a good driver will have 0, 1 or 2 claims in any given year are set to 70%, 20% and 10%, while for bad drivers the probabilities are 50%, 30% and 20% respectively.
Further I assume that 75% of all drivers are good drivers and only 25% would be classified as bad drivers. Therefore the average number of claims per policyholder across the whole customer base would be:
To answer the above question I present the data here as a Bayesian Network using the
Next, I set the client's evidence (0 claims in year one and 1 claim in year two) and propagate these back through my network to estimate the probabilities that the customer is either a good (73.68%) or a bad (26.32%) driver. Knowing that a good driver has on overage 0.4 claims a year and a bad driver 0.7 claims I predict the number of claims for my customer with the given claims history as 0.4789.
Alternatively I could have added a third node for year 3 and queried the network for the probabilities of 0, 1 or 2 claims given that the customer had zero claims in year 1 and one claim in year 2. The sum product of the number of claims and probabilities gives me again an expected claims number of 0.4789.
[2] Søren Højsgaard (2012). Graphical Independence Networks with the gRain Package for R. Journal of Statistical Software, 46(10), 1-26. URL http://www.jstatsoft.org/v46/i10/
Let's assume there are good and bad drivers. The probabilities that a good driver will have 0, 1 or 2 claims in any given year are set to 70%, 20% and 10%, while for bad drivers the probabilities are 50%, 30% and 20% respectively.
Further I assume that 75% of all drivers are good drivers and only 25% would be classified as bad drivers. Therefore the average number of claims per policyholder across the whole customer base would be:
0.75*(0*0.7 + 1*0.2 + 2*0.1) + 0.25*(0*0.5 + 1*0.3 + 2*0.2) = 0.475
To answer the above question I present the data here as a Bayesian Network using the
gRain package [2]. I start with the contingency probability tables for the driver type and the conditional probabilities for 0, 1 and 2 claims in year 1 and 2. As I assume independence between the years I set the same probabilities. I can now review my model as a mosaic plot (above) and as a graph (below) as well.
Next, I set the client's evidence (0 claims in year one and 1 claim in year two) and propagate these back through my network to estimate the probabilities that the customer is either a good (73.68%) or a bad (26.32%) driver. Knowing that a good driver has on overage 0.4 claims a year and a bad driver 0.7 claims I predict the number of claims for my customer with the given claims history as 0.4789.
Alternatively I could have added a third node for year 3 and queried the network for the probabilities of 0, 1 or 2 claims given that the customer had zero claims in year 1 and one claim in year 2. The sum product of the number of claims and probabilities gives me again an expected claims number of 0.4789.
References
[1] Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2004), Loss Models: From Data to Decisions, Wiley Series in Proability and Statistics.[2] Søren Højsgaard (2012). Graphical Independence Networks with the gRain Package for R. Journal of Statistical Software, 46(10), 1-26. URL http://www.jstatsoft.org/v46/i10/
Session Info
R version 3.0.2 (2013-09-25)
Platform: x86_64-apple-darwin10.8.0 (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] grid stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] Rgraphviz_2.6.0 gRain_1.2-2 gRbase_1.6-12 graph_1.40.0
loaded via a namespace (and not attached):
[1] BiocGenerics_0.8.0 igraph_0.6.6 lattice_0.20-24 Matrix_1.1-0
[5] parallel_3.0.2 RBGL_1.38.0 stats4_3.0.2 tools_3.0.2
Claims Inflation - a known unknown
Over the last year I worked with two colleagues of mine on the subject of inflation and claims inflation in particular. I didn't expect it to be such a challenging topic, but we ended up with more questions than answers. The key question and biggest challenge is to define what inflation, or indeed claims inflation actually is and how to measure it. We published a summary of our thoughts and findings in this month's issue of The Actuary.
Last year's discussion about the differences between the retail price index (RPI) and consumer price index (CPI) in the UK only exemplified the challenge. The economist Tim Harford illustrated the differences between the RPI and CPI with a simple example of price changes for a shirt and blouse in his Radio 4 programme More or Less. The radio podcast is still available from the BBC. Start listening after about 18 minutes into the show.
Last year's discussion about the differences between the retail price index (RPI) and consumer price index (CPI) in the UK only exemplified the challenge. The economist Tim Harford illustrated the differences between the RPI and CPI with a simple example of price changes for a shirt and blouse in his Radio 4 programme More or Less. The radio podcast is still available from the BBC. Start listening after about 18 minutes into the show.
ChainLadder 0.1.5-6 released on CRAN
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.
Since the last CRAN release Dan Murphy added new features to the
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.
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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.
Submit a talk for the first R in Insurance conference
The registration for the first R in Insurance is open and there is still time to submit a talk / lightning talk.
The conference will take place at Cass Business School in London on Monday, 15 July 2013. This is the Monday following the useR! 2013 conference in Spain. Thus, if you come from overseas to Spain, why not stop in London on your way back?
All further information and registration details are available on the Cass Business School conference site.
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All further information and registration details are available on the Cass Business School conference site.
Reserving based on log-incremental payments in R, part III
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.
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.
Read more »# 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$logvalue.ind.inf <- with(dat, log(value/volume.index*earning.index))Indeed, the interaction plot shows the various origin years now to be much more closely grouped. Only the single point of the last origin period stands out now. Christofides tests several models with different numbers of origin levels, but I am happy with the minimal model using only one parameter for the origin period, namely the intercept:with(dat, interaction.plot(dev, origin, logvalue.ind.inf))
points(1+dat$dev, dat$logvalue.ind.inf, pch=16, cex=0.8)
Reserving based on log-incremental payments in R, part II
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:
Before I plot the data, I will transform the triangle into a data frame and add extra columns:
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:Read more »
# 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))
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)))
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)
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:Read more »
Reserving based on log-incremental payments in R, part I
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:
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.
Read more »
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)))
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.
Read more »
Claims reserving in R: ChainLadder 0.1.5-4 released
Last week we released version 0.1.5-4 of the
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
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.
Read more »
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.
Read more »
Using R in Insurance, Presentation at GIRO 2012
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.
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.
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