R in Insurance Conference, London, 14 July 2013

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Not only verbs but also believes can be conjugated

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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:

0.75*(0*0.7 + 1*0.2 + 2*0.1) + 0.25*(0*0.5 + 1*0.3 + 2*0.2) = 0.475

Now a customer of two years asks for his renewal. Suppose he had no claims in the first year and one claim last year, how many claims should I predict for next year? Or in other words, how much credibility should I give him?

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

Why models need a certain culture to flourish

About half a year ago Ian Branagan, Chief Risk Officer of Renaissance Re - a Bermudian reinsurance company with a focus on property catastrophe insurance, gave a talk about the usage of models in risk management and how they evolved over the last twenty years. Ian's presentation, titled with the famous quote of George E.P. Box: "All models are wrong, but some are useful", was part of the lunch time lecture series of talks at Lloyd's, organised by the Insurance Institute of London.

I re-discovered the talk online over the weekend and found it most enlightening again.

So, what makes models useful? And here I mean models that estimate extreme outcomes / percentiles. Three factors are critical, according to Ian, to embed models successfully in risk management and decision making processes.

1. Need - A clear defined need for the model.
2. Capabilities - The skills and resources to build and maintain the model.
3. Culture - An organisational culture that embraces, understands and challenges the model.

The need, if not driven internally, is often imposed by external requirements, such as regulation, e.g. banks and insurers have to use models to estimate the risk of insolvency in many countries. Building capabilities can largely be achieved by investing in people, technology and data. However, the last factor culture, so Ian, is often the most challenging one. Changing business processes, particularly in decision making at senior level requires people to change.

Where in the past senior management may have relied on advisors' expert judgement to guide them in their decision makings, they have to use models in a similar way now as well. I suppose, in the same way as it takes time and effort to build effective relationships with people, it is true for models as well. And equally, decisions should never rely purely on either other people's opinion or indeed model output. As Ian put it, outsourcing all modelling/thinking, and with that the decision making to vendors of models, such as catastrophe modelling companies or rating agencies, who both aim to provide probabilities for extreme events (catastrophes and companies failures) may be sufficient to tick a risk management box, but can ultimately put the company at risk, if model assumptions and limitations are not well understood.

Perhaps we are at the dawn of another enlightenment? Recall Kant's first sentence of his essay What is enlightenment?: "Enlightenment is man's emergence from his self-incurred immaturity." Indeed, it doesn't matter if we use experts' opinions or the output of models, relying blindly on them is dangerous and foolish. Don't stop thinking for yourself. Be critical! Remember, all models are wrong, but some are useful.

ChainLadder 0.1.6 released with chain-ladder factor models

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

R in Insurance: Presentations are online

The programme and the presentation files of the first R in Insurance conference have been published on GitHub.

Front slides of the conference presentations

Additionally to the slides many presenters have made their R code available as well:

• Alexander McNeil shared the examples of the CreditRisk+ model he presented.
• Lola Miranda made a Windows version of the double chain-ladder package DCL available via the Cass knowledge web site.
• Alessandro Carrato's 1-year re-reserving code is hosted on the ChainLadder project web site.
• Giorgio Spedicato's life contingencies package is on CRAN already.
  
• Simon Brickman and Adam Rich's HTML presentation and underlying R code for Automated Reporting is included in the GitHub repository.
  
• Stefan Eppert pointed out that KatRisk published an illustrative catastrophe model in R.
• Hugh Shanahan's code to integrate R with Azure for high-throughput analysis is on GitHub.

Thanks again to all the presenters who have helped to make the event a success.

Hopefully, we see you again next year!

Quick review: R in Insurance Conference

Yesterday the first R in Insurance conference took place at Cass Business School in London.

I think the event went really well, but as a member of the organising committee my view is probably skewed. Still, we had a variety of talks, a full house, a great conference dinner and to top it all, the Tower Bridge opened while we had our drinks at the end of the evening.

I will post a more complete review in the future with links to the files of the presentations and R code, once we had a chance to collate all the information.

Many thanks again to all who helped to make this event a success, particularly Andreas Tsanakas at Cass and to our sponsors Mango Solutions and CYBAEA.

Lecture room

Breakout area

Conference dinner at Cantina del Ponte

Tower Bridge

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.

R in Insurance: Programme and Abstracts published

I am delighted to announce that the programme and abstracts for the first R in Insurance conference at Cass Business School in London, 15 July 2013, have been published.

The conference committee received strong abstracts from academia and the industry, covering:

• Pricing
• Reserving
• Data mining
• Capital modelling
• Automate reporting
• Catastrophe modelling
• High-performance computing
• Software development management

Register by the end of May to get the early bird booking fee.

We gratefully acknowledge the sponsorship of Mango Solutions and CYBAEA, without whom the event wouldn't be possible.

Programme and Abstracts

Register by the end of May to get the early bird booking fee.

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.

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.

Registration for 'R in Insurance' conference has opened

The registration for the first conference on R in Insurance on Monday 15 July 2013 at Cass Business School in London has opened.

The intended audience of the conference includes both academics and practitioners who are active or interested in the applications of R in insurance.

The 2013 R in Insurance conference builds upon the success of the R in Finance and R/Rmetrics events. We expect invited keynote lectures by:

• Professor Alexander McNeil, Department of Actuarial Science & Statistics
  Heriot-Watt University: Implementing CreditRisk+ in R with the Faster Fourier Transform
• Trevor Maynard, Head of Exposure Management and Reinsurance, Lloyd's: There is an R in Lloyd's

We invite you to submit a one-page abstract for consideration. Both academic and practitioner proposals related to R are encouraged.

The submission deadline for abstracts is 28 March 2013. Please email your abstract (in txt or pdf format) to rinsuranceconference at gmail dot com.

The conference committee consists of:

• Jens Perch Nielsen
• Andreas Tsanakas
• Christophe Dutang
• Markus Gesmann

Details about the registration and abstract submission are given on the new dedicated R in Insurance page at Cass Business School.

The organisers, Andreas Tsanakas and Markus Gesmann, gratefully acknowledge the sponsorship of Mango Solutions.

New Data Scientist role at Lloyd's

Lloyd's of London is looking for a Data Scientist as part of the Analysis team. See Lloyd's career web site for more details.

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.

# 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:

Read more »

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:

# 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: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:

## 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.
Read more »

R in Insurance Conference, London, 15 July 2013

The first conference on R in Insurance will be held on Monday 15 July 2013 at Cass Business School in London, UK.

The intended audience of the conference includes both academics and practitioners who are active or interested in the applications of R in insurance.

This one-day conference will focus on applications in insurance and actuarial science that use R, the lingua franca for statistical computation. Topics covered may include actuarial statistics, capital modelling, pricing, reserving, reinsurance and extreme events, portfolio allocation, advanced risk tools, high-performance computing, econometrics and more. All topics will be discussed within the context of using R as a primary tool for insurance risk management, analysis and modelling.

The intended audience of the conference includes both academics and practitioners who are active or interested in the applications of R in insurance.

The 2013 R in Insurance conference builds upon the success of the R in Finance and R/Rmetrics events. We expect invited keynote lectures by:

• Professor Alexander McNeil, Department of Actuarial Science & Statistics
  Heriot-Watt University
• Trevor Maynard, Head of Exposure Management and Reinsurance, Lloyd's

We invite you to submit a one-page abstract for consideration. Both academic and practitioner proposals related to R are encouraged.

Details about the registration and abstract submission are given on the R in Insurance event web site of Cass Business School.

You can contact us via rinsuranceconference at gmail dot com.

The organisers, Andreas Tsanakas and Markus Gesmann, gratefully acknowledge the sponsorship of Mango Solutions.

Data.table rocks! Data manipulation the fast way in R

I really should make it a habit of using data.table. The speed and simplicity of this R package are astonishing.

Here is a simple example: I have a data frame showing incremental claims development by line of business and origin year. Now I would like add a column with the cumulative claims position for each line of business and each origin year along the development years.

It's one line with data.table! Here it is:

myData[order(dev), cvalue:=cumsum(value), by=list(origin, lob)]

It is even easy to read! Notice also that I don't have to copy the data. The operator ':=' works by reference and is one of the reasons why data.table is so fast.

And it is getting even better. Suppose you want to get the latest claims development position for each line of business and origin year. Again, it is only one line: Read more »

Claims reserving in R: ChainLadder 0.1.5-4 released

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.
Read more »

From guts to data driven decision making

Source: Wikipedia, License: CC0

There is a wonderful cartoon by Loriot, a German humorist (1923 - 2011), about a couple sitting at a breakfast table, arguing about how to boil a four-and-a-half minute egg. The answer appears simple, but husband and wife argue about how to measure the time using experience, feelings and expert judgment (wife) or a clock (husband).

The whole sketch is hilarious and is often regarded as a fine observation of miss-communication.

Yet, I think it really points out two different approaches in decision making: You can trust your guts or use data/measurements to support your decision.
Read more »