# Insurance: Risk Pooling and Price Segmentation

Talk this afternoon at the seminar of Telecom ParisTech

Insurance is usually defined as “the contribution of the many to the misfortune of the few”. This idea of pooling risks together using the law of large number legitimates the use of  the expected value as actuarial “fair” premium. In the context of heterogeneous risks, nevertheless, it is possible to legitimate price segmentation based on observable characteristics. But nowadays, intensive segmentation can be observed, with a much wider range of offered premium, on a given portfolio. In this talk, we will briefly get back on statistical approaches of insurance pricing (classical econometric tools vs machine learning). We will then get back on recent experiments (so-called “actuarial pricing game”) organized since 2015, where real actuaries are playing in competitive (artificial) market, that mimic real insurance market. We will get back on conclusions obtained on two editions, the first one, and the most recent one, where a dynamic version of the game was launched.

By the way, there will be soon a fourth version of the “Actuarial Pricing Game”… some information soon, on this blog…

# Insurance: Risk Pooling or Price Segmentation, in Paris

I will be tomorrow at the ESSEC-Risk seminar, to give a talk on insurance pricing. Slides are now online.

# Insurance: Risk Pooling or Price Segmentation, in London

This week, I will be in London

# R in Insurance, in Paris

The 5th conference on R in Insurance will be organized on Thursday 8 June 2017 at ENSAE , Paris. I will attend the conference and the program is really nice (I was in the scientific committee – with Christophe Dutang, Markus Gesmann, Giorgio Alfredo Spedicato and Andreas Tsanakas – and I have to admit that was received many interesting submissions). Furthermore, the gala dinner will take place at the restaurant of Musée d’Orsay. I really can’t miss it…

# R in Insurance, 2017

Following the successfull conferences in London (2013, 2014, 2016) and in Amsterdam (2015), the next edition will take place in Paris. The R in insurance 2017 will take place in ENSAE on June 8.

This one-day conference will focus again on applications in insurance and actuarial science that use R, the lingua franca for statistical computation. The intended audience of the conference includes both academics and practitioners who are active or interested in the applications of R in insurance. The two invited speakers are Katrien Antonio (KU Leuven) and Julie Seguela (Covea). It will be a nice event !

A few month ago, I did mention a graph, of some so-called Lorenz curves to compare regression models, see e.g. Progressive’s slides (thanks Guillaume for the reference)

The idea is simple. Consider some model for the pure premium (in insurance, it is the quantity that we like to model), i.e. the conditional expected valeur

$m(\boldsymbol{x})=\mathbb{E}[Y\vert \boldsymbol{X}=\boldsymbol{x}]$

On some dataset, we have our predictions,$\widehat{y}_i=m(\boldsymbol{x}_i)$ as well as observed quantities, $y_i$. The curve are obtained simply :

• sort the observations so that

$\widehat{y}_1\geq \widehat{y}_2 \geq \cdots \geq \widehat{y}_i \geq \cdots\geq \widehat{y}_n$

• based on that ordering (from high risks to low risks, based on our predictions), we plot Lorenz curve

$\left(\frac{i}{n},\frac{\displaystyle\sum_{k=1}^i y_k}{\displaystyle\sum_{k=1}^n y_k}\right)$

# Pricing Game

In November, with Romuald Elie and Jérémie Jakubowicz, we will organize a session during the 100% Actuaires day, in Paris, based on a “pricing game“. We provide two datasets, (motor insurance, third party claims), with 2  years of experience, and 100,000 policies. Each ‘team’ has to submit premium proposal for 36,000 potential insured for the third year (third party, material + bodily injury).

We will work as a ‘price aggergator’ with all the teams, with simple matching rules (the cheapest is chosen, or more complex rules, based on random selection among cheap insurers). The complete description is available on line.

R codes to read the datasets are

> training <- read.csv2(
+ "http://freakonometrics.free.fr/training.csv")
> dim(training)
[1] 100021     20
+ "http://freakonometrics.free.fr/pricing.csv")
> dim(pricing)
[1] 36311    15

Everyone is invited to play! The more, the merrier….

# Computational Actuarial Science, with R

The book Computational Actuarial Science, with R is officially out. In the introduction of the book, and on the website of CRC, it is mentioned that the datasets can be found “in an R package on CRAN“, which is unfortunately incorrect. Some datasets are too large, so the package can not be uploaded on CRAN. Hopefully, Christophe host the package on his website.

> install.packages("CASdatasets", repos = "http://dutangc.free.fr/pub/RRepos/")

or

> install.packages("CASdatasets", repos = "http://dutangc.free.fr/pub/RRepos/",
type = "source")

Here are the files :

## Insurance datasets

A collection of datasets, originally for the book ‘Computational Actuarial Science with R’ edited by Arthur Charpentier (CAS with R). Now, the package contains a large variety of actuarial datasets.

 Version: 0.9-8 Published: 2014-05-21 Author: Christophe Dutang Maintainer: Christophe Dutang License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] NeedsCompilation: no

 Reference manual: CASdatasets.pdf Package source: CASdatasets_0.9-8.tar.gz Package installation: go to this page Windows binaries: r-release: CASdatasets_0.9-8.zip OS X Snow Leopard binaries: r-release: CASdatasets_0.9-8.tgz OS X Mavericks binaries: r-release: CASdatasets_0.9-8.tgz Old sources: CASdatasets archive

(more to come…)

# London, Bayes and the Lloyd’s

Monday, we really had a great conference in London.

It was a great pleasure since I did learn a lot of things. And also a great honor to be the last speaker. Tuesday morning, I wanted to go to Thomas Bayes’grave, which is the the graveyard next to the CASS Business School. I had a good a apriori about where the grave should be,

but to be honest, it was not possible to get close enough to be able to read the name on it (even if I now know that it is the large one in the right lower corner of the picture)

Actually, on the internet, you can find some picture where the stone is clean, so you can learn that the grave is the “cotton” one – at least, you can easily read that name.

It was actualy more simple to see William Blake’s grave, as well as Daniel Defoe’s.

Then, with Leo, we’ve been to the Lloyd’s to see some friends, as well as Richard Rogers’s building.

At the 11th floor, you have a lot of rooms for meetings, as well as old paintings, to tell a bit more about the history of the company,

The building is just amazing. Unfortunately, to get in, there is a dress code. A sort of strict one actually. Leo is working for RBC, so he casually wears a suit. But I don’t. I mean, I did have a shirt, but as someone mentioned, “there is no collar !” (I don’t want to put my friend into trouble for helping me getting in).

So, after going throught the basement, we’ve been able to reach the elevator, and go on top.

The building is not exactly located where Edward Lloyd got his coffee shop (even after moving at the end of 1691 on Lombard street), but the Lloyd’s is still a legend for anyone interested in the history of insurance, and more generally, the history of risk modeling (and management).

# London, July 2014

I will be in London in July 2014 for the second R in Insurance conference. I had good feedback from the first one, but I could not get there, last July. I will upload more information once I have some (check also on Markus’s blog)

# R for actuarial science

As mentioned in the Appendix of Modern Actuarial Risk Theory, “R (and S) is the ‘lingua franca’ of data analysis and statistical computing, used in academia, climate research, computer science, bioinformatics, pharmaceutical industry, customer analytics, data mining, finance and by some insurers. Apart from being stable, fast, always up-to-date and very versatile, the chief advantage of R is that it is available to everyone free of charge. It has extensive and powerful graphics abilities, and is developing rapidly, being the statistical tool of choice in many academic environments.

R is based on the S statistical programming language developed by Joe Chambers at Bell labs in the 80’s. To be more specific, R is an open-source implementation of the S language, developed by Robert Gentlemn and Ross Ihaka. It is a vector based language, which makes it extremely interesting for actuarial computations. For instance, consider some Life Tables,

> TD[39:52,]       > TV[39:52,]
Age    Lx         Age    Lx
39  38 95237          38 97753
40  39 94997          39 97648
41  40 94746          40 97534
42  41 94476          41 97413
43  42 94182          42 97282
44  43 93868          43 97138
45  44 93515          44 96981
46  45 93133          45 96810
47  46 92727          46 96622
48  47 92295          47 96424
49  48 91833          48 96218
50  49 91332          49 95995
51  50 90778          50 95752
52  51 90171          51 95488

Those (French) Life Tables can be found here

> TD <- read.table(
+ "https://perso.univ-rennes1.fr/arthur.charpentier/TV8890.csv",sep=";",header=TRUE)

From those vectors, it is possible to construct the matrix of death probabilities, $\boldsymbol{P}=[\text{ }_{k}p_x]$, using for instance

>  Lx <- TD$Lx > m <- length(Lx) > p <- matrix(0,m,m); d <- p > for(i in 1:(m-1)){ + p[1:(m-i),i] <- Lx[1+(i+1):m]/Lx[i+1] + d[1:(m-i),i] <- (Lx[(1+i):(m)]-Lx[(1+i):(m)+1])/Lx[i+1]} > diag(d[(m-1):1,]) <- 0 > diag(p[(m-1):1,]) <- 0 > q <- 1-p One can compute easily, e.g., the (curtate) expectation of life defined as $e_x =\mathbb{E}(K_x)=\sum_{k=1}^\infty k\cdot \text{ }_{k|1}q_x = \sum_{k=1}^\infty \text{ }_{k}p_x$ and one can compute the vector of life expectancy, at various ages $\boldsymbol{e}=[e_x]$, as > life.exp = function(x){sum(p[1:nrow(p),x])} > e = Vectorize(life.exp)(1:m) An actually, any kind of actuarial quantity can be derived from those matrices. The expected present value (or actuarial value) of a temporary life annuity-due is, for instance, $\ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1} \nu^k \cdot {}_{k}p_x =\frac{1-A_{x:\overline{n}|}}{1-\nu}$ The code to compute those functions is here > for(j in 1:(m-1)){ adots[,j]<-cumsum(1/(1+i)^(0:(m-1))*c(1,p[1:(m-1),j])) } or consider the expected present value of a term insurance $A^1_{x:\overline{n}|} =\sum_{k=0}^{n-1} \nu^{k+1} \cdot \text{ }_{k|}q_x$ with the following code > for(j in 1:(m-1)){ A[,j]<-cumsum(1/(1+i)^(1:m)*d[,j]) } Some more details can be found in the first part of the notes of the crash courses of last summer, in Meielisalp. Vector – or matrices – are extremely convenient to work with, when dealing with life contingencies. It is also possible to model prospective mortality. Here, the mortality is not only function of the age $x$, but also time $t$, > t(DTF)[1:10,1:10] 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 0 64039 61635 56421 53321 52573 54947 50720 53734 47255 46997 1 12119 11293 10293 10616 10251 10514 9340 10262 10104 9517 2 6983 6091 5853 5734 5673 5494 5028 5232 4477 4094 3 4329 3953 3748 3654 3382 3283 3294 3262 2912 2721 4 3220 3063 2936 2710 2500 2360 2381 2505 2213 2078 5 2284 2149 2172 2020 1932 1770 1788 1782 1789 1751 6 1834 1836 1761 1651 1664 1433 1448 1517 1428 1328 7 1475 1534 1493 1420 1353 1228 1259 1250 1204 1108 8 1353 1358 1255 1229 1251 1169 1132 1134 1083 961 9 1175 1225 1154 1008 1089 981 1027 1025 957 885 Thus, we now have a force of mortality matrix $\boldsymbol{\mu}=[\mu_{x,t}]$, or surface It is also possible to use R packages to estimate a Lee-Carter model of the mortality rate, $\log \mu _{x,t} =\alpha _{x} +\beta _{x} \cdot \kappa_{t} +\varepsilon _{x,t}$ > library(demography) > MUH =matrix(DEATH$Male/EXPOSURE$Male,nL,nC) > POPH=matrix(EXPOSURE$Male,nL,nC)
> BASEH <- demogdata(data=MUH, pop=POPH, ages=AGE, years=YEAR, type="mortality",
+ label="France", name="Hommes", lambda=1)
> RES=residuals(LCH,"pearson")

One can easily study residuals, for instance as a function of the age,

or a function of the year,

Some more details can be found in the second part of the notes of the crash courses of last summer, in Meielisalp.

R is also interesting because of its huge number of libraries, that can be used for predictive modeling. One can easily use smoothing functions in regression, or regression trees,

> TREE = tree((nbr>0)~ageconducteur,data=sinistres,split="gini",mincut = 1)
> age = data.frame(ageconducteur=18:90)
> y1 = predict(TREE,age)
> reg = glm((nbr>0)~bs(ageconducteur),data=sinistres,family="binomial")
> y = predict(reg,age,type="response")

Some practitioners might be scared because the legend claims that R is not as good as SAS to handle large databases. Actually, a lot of functions can be used to import datasets. The most convenient one is probably

> baseCOUT = read.table("http://freakonometrics.free.fr/baseCOUT.csv",
>  tail(baseCOUT,4)
numeropol  debut_pol    fin_pol freq_paiement langue  type_prof alimentation type_territoire
6512     87291 2002-10-16 2003-01-22       mensuel      A Professeur   Vegetarien          Urbain
6513     87301 2002-10-01 2003-09-30       mensuel      A Technicien   Vegetarien          Urbain
6514     87417 2002-10-24 2003-10-21       mensuel      F Technicien   Vegetalien     Semi-urbain
6515     88128 2003-01-17 2004-01-16       mensuel      F     Avocat   Vegetarien     Semi-urbain
utilisation presence_alarme marque_voiture sexe exposition age duree_permis age_vehicule i   coutsin
6512 Travail-occasionnel             oui           FORD    M  0.2684932  47           29           28 1 1274.5901
6513              Loisir             oui          HONDA    M  0.9972603  44           24           25 1  278.0745
6514 Travail-occasionnel             non     VOLKSWAGEN    F  0.9917808  23            3           11 1  403.1242
6515              Loisir             non           FIAT    F  0.9972603  23            4           11 1  230.9565

But if the dataset is too large, it is also possible to specify which variables might be interesting, using

> mycols = rep("NULL", 18)
> mycols[c(1,4,5,12,13,14,18)] <- NA
numeropol freq_paiement langue sexe exposition age    coutsin
1         6        annuel      A    M  0.9945205  42   279.5839
2        27       mensuel      F    M  0.2438356  51   814.1677
3        27       mensuel      F    M  1.0000000  53   136.8634
4        76       mensuel      F    F  1.0000000  42   608.7267

It is also possible (before running a code on the entire dataset) to import only the first lines of the dataset.

> baseCOUTsubCR = read.table("http://freakonometrics.free.fr/baseCOUT.csv",
> tail(baseCOUTsubCR,4)
numeropol freq_paiement langue sexe exposition age   coutsin
97       1193       mensuel      F    F  0.9972603  55  265.0621
98       1204       mensuel      F    F  0.9972603  38 9547.7267
99       1231       mensuel      F    M  1.0000000  40  442.7267
100      1245        annuel      F    F  0.6767123  48  179.1925

It is also possible to import a zipped file. The file itself has a smaller size, and it can usually be imported faster.

> import.zip = function(file){
+ temp = tempfile()
> system.time(import.zip("http://freakonometrics.free.fr/baseFREQ.csv.zip"))
trying URL 'http://freakonometrics.free.fr/baseFREQ.csv.zip'
Content type 'application/zip' length 692655 bytes (676 Kb)
opened URL
==================================================
user  system elapsed
0.762       0.029       4.578
user  system elapsed
0.591       0.072       9.277

Finally, note that it is possible to import any kind of dataset, not only a text file. Even a Microsoft Excel folder. On a Windows computer, one can use SQL queries

> sheet = "c:\\Documents and Settings\\user\\excelsheet.xls"
> connection = odbcConnectExcel(sheet)
> query = paste("SELECT * FROM",spreadsheet$TABLE_NAME[1],sep=" ") > result = sqlQuery(connection,query) Then, once the dataset is imported, several functions can be used, > cost = aggregate(coutsin~ AgeSex,mean, data=baseCOUT) > frequency = merge(aggregate(nbsin~ AgeSex,sum, data=baseFREQ), + aggregate(exposition~ AgeSex,sum, data=baseFREQ)) > frequency$freq = frequency$nbsin/frequency$exposition
> base.freq.cost = merge(frequency, cost)

Finally, R is interesting for its graphical interface. “If you can picture it in your head, chances are good that you can make it work in R. R makes it easy to read data, generate lines and points, and place them where you want them. Its very flexible and super quick. When youve only got two or three hours until deadline, R can be brilliant” as said Amanda Cox, a graphics editor at the New York Times. “R is particularly valuable in deadline situations when data is scant and time is precious.”.
Several cases were considered on the blog http ://chartsnthings.tumblr.com/…. First, we start with a simple graph, here State Government control in the US

Then try to find a nice visual representation, e.g.

And finally, you can just print it in your favorite newspaper,

And you can get any kind of graphs,

Graphs are important. “Its not just about producing graphics for publication. Its about playing around and making a bunch of graphics that help you explore your data. This kind of graphical analysis is a really useful way to help you understand what you’re dealing with, because if you cant see it, you cant really understand it. But when you start graphing it out, you can really see what you’ve got” as said Peter Aldhous, San Francisco bureau chief of New Scientist magazine. Even for actuaries. “The commercial insurance underwriting process was rigorous but also quite subjective and based on intuition. R enables us to communicate our analytic results in appealing and innovative ways to non-technical audiences through rapid development lifecycles. R helps us show our clients how they can improve their processes and effectiveness by enabling our consultants to conduct analyses efficiently”, as explained by John Lucker, team of advanced analytics professionals at Deloitte Consulting Principal, in http://blog.revolutionanalytics.com/r-is-hot/. See also Andrew Gelman’s view, on graphs, http://www.stat.columbia.edu/…

So yes, actuaries might be interested to use R for actuarial communication, as mentioned in http ://www.londonr.org/…

The Actuarial Toolkit (see http ://www.actuaries.org.uk/…) stresses the interest of R, “The power of the language R lies with its functions for statistical modelling, data analysis and graphics ; its ability to read and write data from various data sources; as well as the opportunity to embed R in excel or other languages like VBA. In the way SAS is good for data manipulations, R is superior for modelling and graphical output“.

From 2011, Asia Capital Reinsurance Group (ACR) uses R to Solve Big Data Challenges (see http ://www.reuters.com/…). And Lloyd’s uses motion charts created with R to provide analysis to investors (as discussed on http ://blog.revolutionanalytics.com/…)

A lot of information can be found on http ://jeffreybreen.wordpress.com/…

Markus Gesmann mentioned on his blog a lot of interesting graphs used for actuarial reporting, http ://lamages.blogspot.ca/…

Further, R is free. Which can be compared with SAS, $6,000 per PC, or$28,000 per processor on a server (as mentioned on http ://en.wikipedia.org/…)

It is also becoming more and more popular, as a programming language. As mentioned on this month Transparent Language Popularity (see http ://lang-index.sourceforge.net/), R is ranked 12. Far away after C or Java, but before Matlab (22) or SAS (27). On StackOverFlow (see http ://stackoverflow.com/) is also far being C++ (399,232 occurrences) or Java (348,418), but with 21,818 occurrences, it appears before Matlab (14,580) and SAS (899). As mentioned on http ://r4stats.com/articles/popularity/ R is becoming more and more popular, on listserv discussion traffic

It is clearly the most popular software in data analysis, as mentioned by the Rexer Analytics survey, in 2009

What about actuaries ? In a survey (see http ://palisade.com/…), R was not extremely popular.

If we consider only statistical softwares, SAS is still far ahead, among UK and CAS actuaries

But, as mentioned by Mike King, Quantitative Analyst, Bank of America, “I cant think of any programming language that has such an incredible community of users. If you have a question, you can get it answered quickly by leaders in the field. That means very little downtime.” This was also mentioned by Glenn Meyers, in the Actuarial Review “The most powerful reason for using R is the community” (in http ://nytimes.com/…). For instance, http ://r-bloggers.com/ has contributions from more than 425 R users.

As said by Bo Cowgill, from Google “The best thing about R is that it was developed by statisticians. The worst thing about R is that it was developed by statisticians.

# Natural Catastrophe Insurance: How Should the Government Intervene?

An updated version of the joint paper with Benoit Le Maux is online on http://papers.ssrn.com/.

The present paper develops a new theoretical framework for analyzing the decision to provide or buy insurance against the risk of natural catastrophes. In contrast with conventional models of insurance, the insurer has a non-zero probability of insolvency that depends on the distribution of the risks, the premium rate, and the amount of capital in the company. Among several results, we show that risk-averse policyholders will accept to pay higher rates for a government-provided insurance with unlimited guarantee. However, depending on the correlation between and within the regional risks, a government program can be more attractive to high-correlation than to low correlation areas, which may lead to inefficiencies if the insurance ratings are not appropriately chosen.

# Course on risk, insurance, and uncertainty

The course on risk and insurance in Luminy, starts at 10.30 on Friday (here) instead of Thursday (I switched with Patrice Bertail). The slides can be found here,

Then, it will be time to leave Marseille,

# Lecture notes on risk and insurance

I just finished some lectures notes on risk and insurance. The notes, that can be downloaded [pdf], are in French, and will be used at the JES (Journées d’Etudes Statistiques), organised at the CIRM (mentioned here). Previous notes on risk measures [pdf] and copulas [pdf]. Again, all comments are welcome…