Tag Archives: insurance

Variance decomposition and price segmentation in Insurance

Today, I was giving a talk at the Economics department, and I got a very interesting question about some tables I keep showing to explain why insurance companies like segmentation. The tables illustrate three different case. Here, S stands for the individual (random) loss.

  • the first one is the case where the premium asked is the same for all the insured – i.e. the pure premium \mathbb{E}[S]

As explain, the loss is here on an individual basis, so, per policy, the insurer faces the (random) loss S-\mathbb{E}[S], which is, on average, null. That’s the second line. For the last line, I keep saying that we look at the overall loss of the insurer, but that’s not correct. Here, with a factor n, we would have the variance of the total loss for the insurance company. We just removed the n factor in the table

  • then, we have perfectly observable heterogeneity : insured have a risk factor \Omega, obervable, and in that case, the ‘pure’ premium is \mathbb{E}[S|\Omega]

That’s what we have below. Here again, on average, the insured should have a null profit. And the total variance (which was \text{Var}[S] in our previous example) is now splitted in two parts (that’s basically Pythagoras theorem).

The interpreration is the following

And then, I usually mention the third and last case, more realistic

  • the risk factor \Omega is not obervable, but segmentation is still possible using some proxy of the risk factor, obtained using some covariates, and the ‘pure’ premium is \mathbb{E}[S|\boldsymbol{X}]

And here also, there is a nice interpretation, because of the variance decomposition : there is one part that we observed previously, with some ‘perfect pricing’ and an additional part (that is positive) that is related to the fact that the covariates are just a proxy of the risk factor….

The term on the left is then a lower bound, obtained if actually, using our covariate, available for the pricing, we can get the risk factor.

That was my story, but the fact that n (the portfolio size) was not mentioned in the tables was a bit confusing… So I decided to create some graphs to illustrate those three cases

  • same premium for everyone

Consider some simple simulations. On the graph on the left, we have on the x-axis the risk factor, and on the y-axis, the loss (going roughly from 0 to 20). The pure premium is the average of those losses. Here, it’s 10. That’s the plain red line (on the left). In the middle, the y-axis is the insured profit/loss per policy. Someone with a loss close to 0 means a gain of 10, someone with a loss close to 20 means a loss of 10. On average, there is no profit (that’s the plain line). And then, on the right, we have the distribution of the profit/loss (per contract). Again, on average it’s 0, with some variance,

  • premium based on covariates

Consider here is simple covariate x : assume here that’s we’ve been able to create a binary variable, that can distinguish the low risks and the high risks. Here, there are two levels for the premium. The low premium is close to 6, and the high one is close to 14. That’s again the graph on the left

Then we have the profit/loss per policy for the insured, in the middle. Here, when the loss was close to 0, the gain is smaller : it is 6 (while it was 10 before). When it was close to 10, previously, it meant a 0 profit, but now it’s either a loss of 4, or a gain of 4. The profit/loss distribution is now on the right. There is less dispersion, and less variance. That the decrease of variance we’ve discussed before. To summarize, segmentation does reduce the variability of the result for the insurance company. That’s what we observe on the right.

  • premium based on the risk factor

Assume now that \Omega is observable. And that we use it for our pricing. The premium is now continuous, and it is the red line, on the left. The profit/loss (in the middle) is the difference between the loss, and its expected value (conditional on the risk factor). And on the right, we have the distribution.

As expected, there is much less variability on the profit/loss distribution of the insurance company in that case. And actually, that’s a lower bound for the variance of result of the insurance company… I hope that the graph clarify what’s going on here…

Networks to reinvent insurance?

The theory of networks, or graphs, was born in 1735, following the work of Leonard Euler, who tried to find a walk – starting from a given point – that would bring us back to that point by passing once and only once through each of the seven bridges in the city of Königsberg. These networks can be compared to metro networks, consisting of stations (nodes), linked between two by rails, or not, or more generally a road network, which can give rise to congestion studies, for example. But today, networks are mainly social, connecting people through friendships, professional, family, or monetary ties. Network analysis makes it possible to create relatively homogeneous communities, accepting to share a risk, recreating a mutualisation.

Network and credit

In genealogy, we will have hierarchical networks, a child being linked to his parents, who are themselves linked to their parents. In sociology, social networks make it possible to analyze the links between individuals (or organizations) within a group. Friendships can be studied in a schoolyard (a link that could be an invitation to a birthday party) or e-mail exchanges in a company (the Enron e-mail database has been widely used, with over 180,000 messages exchanged between 36,000 employees). Figure 1 shows two networks of 20 individuals (A, B, …, T).

 

Figure 1: Random networks, 20 nodes (Watts-Strogatz and Barbasi)

In a Facebook or Linkedin type vision, we will say that E and F are linked, in the sense of “friends”, if there is a segment linking points E and F. A network can be directed, for example if we study the exchange of messages (E wrote to F), or money loans (E lent money to F). If historically only adjacency was studied (existence or not of links), we can now add weights, for example the amount of a financial loan. Babutsidze (2012) thus studies the positions of French and German banks in interbank lending within the European zone (the nodes are then the banks). The study of networks within village communities in developing countries has led to a better understanding of informal finance mechanisms. Banerjee et al (2013) study the dissemination of information in a network, and more particularly microfinance loans.

While networks are useful for better organizing microcredit, CNN noted in 2015 that Facebook allowed credit organizations to use a borrower’s social network to determine whether or not it represents a good credit risk. In particular, if friends’ credit scores were too low, a person could be denied credit. This situation is dangerous because of the particular properties of networks, and more particularly the paradox of friends.

From the very small world to the paradox of friends

In 1929, Frigyes Karinthy hypothesized that any person on earth could be connected to any other person by a succession of individual relationships involving at most 6 links. “We should select anyone from the world’s 1.5 billion people, anyone, anywhere. It seems that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the chosen individuals using nothing other than the network of personal acquaintances. This theory of six handshakes originated in a new literary novel. It will be necessary to wait for the work of Michael Gurevich in the 1960s, then Stanley Milgram ten years later, to see the first attempts to quantify these relationships appear, under the name “Small World Problem”.

While Leskovec & Horvitz (2008) confirmed this order of magnitude, by analyzing several billion messages exchanged using the Windows Live Messenger platform, more recently, Baghat et al (2016) estimated that any two people on Facebook were connected by an average of three and a half people. On the random network on the left, a person has, on average, 2 friends, while a random friend has, on average, 2.25 friends. On the right-hand network, the gap is even greater, because if there too a person has, on average, 2 friends, a random friend will have on average more than 4 friends.

 

Figure 2: Random networks, 500 nodes (Watts-Strogatz and Barbasi)

This paradox, observed in 1991 by sociologist Scott Feld, is very easily demonstrated. Heuristically, we can see a link with the probabilistic property \frac{\mathbb{E}[X^2]}{\mathbb{E}[X]}=\mathbb{E}[X]+\frac{\text{Var}[X]}{\mathbb{E}[X]}>\mathbb{E}[X] where the term on the left is the number of friends of my friends, divided by my number of friends. The difference is all the greater the greater the dispersion of the number of friends. If the left-hand network is very dense, the right-hand network on the other hand has a power law property: the distribution of the number of friends follows a power law (or Zipf law, or Pareto’s law). Figure 3 shows the distribution of the number of friends on a network, in a double logarithmic scale: linearity indicates a distribution according to power. This type of distribution can be found in a very large number of networks, particularly Facebook, as shown by Wohlgemuth & Matache (2014).

 

Figure 3: Distribution of the number of friends on simulated random networks (Watts-Strogatz and Barbasi in red)

The classic interpretation is that some people are central in the network, with a very large number of connections. This property is well known in marketing (we will then speak of a “peer effect“) but it also has impacts in risk management or public health. Chrisakis & Fowler (2010) have shown that influenza epidemics can be detected almost two weeks in advance, by monitoring the infection in a social network. In particular, the analysis of the health of central people in a network is “an ideal way to predict outbreaks, but detailed information doesn’t exist for most groups, and to produce it would be time-consuming and costly”. To return to the example of the credit score, if it is found to be correlated to the number of friends, the friends paradox makes it dangerous to use the friends’ score as an indicator of an individual’s risk!

The importance of homophilia

Another important feature of networks is the notion of homophilia, introduced in 2001 in sociology by two important articles, corresponding to the tendency to be connected to one’s peers. McPherson et al (2001) assumed that similarity generates connection, and therefore people’s personal networks are homogeneous across many socio-demographic, behavioural and intrapersonal characteristics. Moody (2001) studied friendships in elementary school playgrounds in the United States, with a focus on interracial friendships. Easley & Kleinberg (2010) thus presents a number of consequences of homophilia, ranging from the creation of tables at business meals to the granting of credit in the United States. The measurement of homophilia is the same as asking, taking into account pre-existing groups (according to gender, age, socio-professional category, etc.) how the links are distributed, between groups, or within groups.

 

Figure 4: Low homophilia (left) and high homophilia (right)

In an insurance context, an actuary seeks to create tariff classes, groups that are homogeneous in terms of risks, according to explanatory variables (the so-called tariff variables). People who live in the same place, drive the same types of vehicles, and have the same characteristics, are likely to be in the same class. But if homophilia exists in a population, a tariff group could perhaps be observed on a network of friends. Why not then consider creating groups within a network?

Using insurance networks

In this spirit, in 2010, Friendsurance was launched in Germany and has more than 100,000 insured in 2018. In France, a short collaborative insurance experiment was launched in 2015, with Inspeer, offering to share damage insurance deductibles (in car or home insurance) with friends. These types of collaborative insurance, sometimes called peer-to-peer insurance, are based on the formation of small groups by a broker. A portion of the insurance premiums paid is paid to a group fund, the other portion to a third party insurance company. Minor damage suffered by the policyholder is first covered by this group fund. For claims exceeding the deductible, the usual insurer is used. A group can be formed by the insured, forming a social network a bit like Facebook. In this model, the only requirement is that all group members must have the same type of insurance (e. g. liability insurance with legal expenses insurance).

As Schiller (2013) noted, this type of mechanism has many virtues, the first being to reduce costs and the risk of fraud. There is no tendency to cheat on the cost of a claim when the risk is borne by family members or friends. The anonymity of mutuality that exists in the law of large numbers is disappearing. But aren’t we reinventing version 2.0. of the tontine associations, with the strong return of risk sharing within close-knit communities?

References

Joshua Angrist. The perils of peer effects. Labor Economics, 30, 98-108, 2014

Zakaria Babutsidze. Positions of French and German Banks in European interbank lending network. OFCE, Mars 2012.

Abhijit Banerjee, Arun Chandrasekhar, Esther Duflo & Matthew Jackson. Diffusion of Microfinance. Science, 341,

Smriti Bhagat, Moira Burke, Carlos Diuk, Ismail Onur Filiz & Sergey Edunov. Three and a half degrees of separation. Facebook Research, 2016.

Ananya Bhattacharya. Facebook patent: Your friends could help you get a loan – or not. 4 août 2015, CNN,

Nicholas Christakis & James Fowler Social Network Sensors for Early Detection of Contagious Outbreaks. PLoS ONE. 5 (9): e12948. arXiv:1004.4792 2015

David Easley & Jon Kleinberg. Networks, Crowds, and Markets. Cambridge University Press. 2010.

Scott Feld. Why your friends have more friends than you do, American Journal of Sociology, 96 (6): 1464–1477, 1991.

Matthew Jackson. Social and Economic Networks. Princeton University Press, 2010.

Jure Leskovec & Eric Horvitz. Planetary-Scale Views on a Large Instant-Messaging Network. Microsoft Research, 2008.

Miller McPherson, Lynn Smith-Lovin & James Cook. Birds of a Feather: Homophily in Social Networks. Annual Review of Sociology. 27: 415–444, 2001.

James Moody. Race, School Integration, and Friendship Segregation in America. American Journal of Sociology, 107 (3): 679-716, 2001.

Wesley Perkins, Michael Haines & Richard Rice. Misperceiving the college drinking norm and related problems: a nationwide study of exposure to prevention information, perceived norms and student alcohol misuse. Journal of Studies on Alcohol 66 (4) : 470-478, 2005.

Ben Schiller. A Social Network For Insurance That Cuts Costs And Reduces Fraud. Fast Company, October 2013,

Brad Walker. How Peer-to-Peer Companies Are Transforming the Insurance Sector. The Street, Avril 2016,

Jason Wohlgemuth & Mihaela Matache. Small-World Properties of Facebook Group

Networks. Complex Systems, 23 (2014).

[i] Complete data can be downloaded from https://snap.stanford.edu/data/email-Enron.html

[ii] https://www.friendsurance.com/ and https://www.inspeer.me/ respectively

Insurance, Actuarial Science, Data and Models

Our research chaire ACTINFO, with our colleagues from Lyon, at the DAMI chaire,  PREVENT’HORIZON chaire & ACTUARIAT DURABLE chaire, will organize a 2 day conference in Paris, on Insurance, Actuarial Science, Data & Models, in ten days.

We invited Katrien ANTONIO (KU Leuven), Alexandre BOUMEZOUED (Milliman Paris), Alfred GALICHON (New-York University), Pierre-Yves GEOFFARD (Paris School of Economics), Meglena JELEVA (University of Paris Nanterre), Julie JOSSE (Ecole Polytechnique), Florence JUSOT (Paris Dauphine University), Michael LUDKOWSKI (University of California Santa Barbara), François PANNEQUIN (CREST and ENS Paris-Saclay), Florian PELGRIN (Edhec Business School), Dylan POSSAMAI (Columbia University) and Julien TRUFIN (ULB Brussels). More information (including the program) is online.

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…

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 !

Additional thoughts about ‘Lorenz curves’ to compare models

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

On some dataset, we have our predictions, as well as observed quantities, . The curve are obtained simply :

  • sort the observations so that

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

Continue reading Additional thoughts about ‘Lorenz curves’ to compare models

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
> pricing <- read.csv2(
+ "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 <christophe.dutang at ensimag.fr>
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation: no

Downloads:

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

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).