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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance: Risk Pooling and Price Segmentation Using Information in a Big Data Context
A. Charpentier (Universit de Rennes 1)
GT Big Data, Institut des ActuairesOctobre 2017
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Brief Introduction
A. Charpentier (Universit de Rennes 1)
Professor Economics Department, Universit de Rennes 1Director Data Science for Actuaries Program, Institute of Actuaries(previously Actuarial Sciences, UQM & ENSAE Paristechactuary in Hong Kong, IT & Stats FFA)
PhD in Statistics (KU Leuven), Fellow of the Institute of ActuariesMSc in Financial Mathematics (Paris Dauphine) & ENSAEResearch Chair :ACTINFO (valorisation et nouveaux usages actuariels de linformation)Editor of the freakonometrics.hypotheses.orgs blogEditor of Computational Actuarial Science, CRCAuthor of Mathmatiques de lAssurance Non-Vie (2 vol.), Economica
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http://freakonometrics.hypotheses.orghttps://www.crcpress.com/Computational-Actuarial-Science-with-R/Charpentier/p/book/9781138033788https://twitter.com/freakonometricshttps://freakonometrics.github.io/https://freakonometrics.hypotheses.org/
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Pricing in a Nutshell
Insurance is the contribution of the many to the misfortune of the few
Finance: risk neutral valuation = EQ[S1F0] = EQ0[S1], where S1 = N1
i=1Yi
Insurance: risk sharing (pooling) = EP[S1]
or, with segmentation / price differentiation () = EP[S1 = ] for some
(unobservable?) risk factor
imperfect information given some (observable) risk variables X = (X1, , Xk)(x) = EP
[S1X = x] = EPX [S1|x]
Insurance pricing is not only data driven, it is also essentially model driven (seePricing Game)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Pricing in a Nutshell
Premium is = EPX[S1]
It is datadriven (or portfolio driven) since PX is based on the portfolio.
click to visualize the construction
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Pricing in a Nutshell
Premium is E[S1|X = x
]= E
[Ni=1
Yi
X = x]
= E [N |X = x] E [Yi|X = x]
Statistical and modeling issues to approximate based on some training datasets,with claims frequency {ni,xi} and individual losses {yixi}
depends on the model used to approximate E [N |X = x] and E [Yi|X = x]
depends on the choice of meta-parameters
depends on variable selection / feature engineering
Try to avoid overfit
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Risk Sharing in Insurance
Important formula E[S]
= E[E[S|X
]and its empirical version
1n
ni=1
Si 1n
ni=1
(Xi) (as n, from the law of large number)
interpreted as on average what we pay (losses) is the sum of what we earn(premiums).
This is an ex-post statement, where premiums were calculated ex-ante.
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Risk Transfert without Segmentation
Insured Insurer
Loss E[S] S E[S]Average Loss E[S] 0Variance 0 Var[S]
All the risk - Var[S] - is kept by the insurance company.
Remark: all those interpretation are discussed in Denuit & Charpentier (2004).
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance, Risk Pooling and Solidarity
La Nation proclame la solidarit et lgalit de tous les Franais devant lescharges qui rsultent des calamits nationales (alina 12, prambule de laConstitution du 27 octobre 1946)
31 zones TRI (Territoires Risques dInondation) on the left, and flooded areas.
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http://www.rhone-mediterranee.eaufrance.fr/docs/dir-inondations/cartes/Carte_TRI_retenu_RhoM.pdfhttps://twitter.com/freakonometricshttps://freakonometrics.github.io/https://freakonometrics.hypotheses.org/
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance, Risk Pooling and SolidarityHere is a map with a risk score -{1, 2, , 6} scale
One can look at Lorenz curve
South Other Total
% portfolio 11% 89% 100%% claims 51% 49% 100%
Premium 463 55 100
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Risk Transfert with Segmentation and Perfect Information
Assume that information is observable,
Insured Insurer
Loss E[S|] S E[S|]Average Loss E[S] 0Variance Var
[E[S|]
]Var[S E[S|]
]Observe that Var
[S E[S|]
]= E
[Var[S|]
], so that
Var[S] = E[Var[S|]
] insurer
+Var[E[S|]
] insured
.
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Risk Transfert with Segmentation and Imperfect Information
Assume that X is observable
Insured Insurer
Loss E[S|X] S E[S|X]Average Loss E[S] 0Variance Var
[E[S|X]
]E[Var[S|X]
]Now
E[Var[S|X]
]= E
[E[Var[S|]
X]]+ E[Var[E[S|]X]]= E
[Var[S|]
]
pooling
+E{Var[E[S|]
X]} solidarity
.
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Risk Transfert with Segmentation and Imperfect Information
With imperfect information, we have the popular risk decomposition
Var[S] = E[Var[S|X]
]+ Var
[E[S|X]
]= E
[Var[S|]
]
pooling
+E[Var[E[S|]
X]] solidarity
insurer
+Var[E[S|X]
] insured
.
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
More and more price differentiation ?
Consider 1 = E[S1]and 2(x) = E
[S1|X = x
]Observe that E
[(X)
]=xX
(x) P[x]
=
xX1
(x) P[x] +
xX2
(x) P[x]
Insured with x X1 : choose Ins1 Insured with x X2: choose Ins2
Ins1:
xX1
1(x) P[x] 6= E[S|X X1]
Ins2:
xX2
2(x) P[x] = E[S|X X2]
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Price Differentiation, a Toy Example
Claims frequency Y (average cost = 1,000)
X1
Young Experienced Senior Total
X2Town
12%(500)
9%(2,000)
9%(500)
9.5%(3,000)
Outside8%(500)
6.67%(1,000)
4%(500)
6.33%(2,000)
Total10%
(1,000)8.22%(3,000)
6.5%(1,000)
8.23%(5,000)
from C., Denuit & lie (2015)
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https://f.hypotheses.org/wp-content/blogs.dir/253/files/2015/10/Risques-Charpentier-Denuit-Elie.pdfhttps://twitter.com/freakonometricshttps://freakonometrics.github.io/https://freakonometrics.hypotheses.org/
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Price Differentiation, a Toy Example
Y-T(500)
Y-O(500)
E-T(2,000)
E-O(1,000)
S-T(500)
S-O(500)
none 82.3 82.3 82.3 82.3 82.3 82.3X1 X2 120 80 90 66.7 90 40
market 82.3 80 82.3 66.7 82.3 40
none 82.3 82.3 82.3 82.3 82.3 82.3X1 100 100 82.2 82.2 65 65X2 95 63.3 95 63.3 95 63.3
X1 X2 120 80 90 66.7 90 40
market 82.3 63.3 82.2 63.3 65 40
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Price Differentiation, a Toy Example
premium losses loss 99.5% Marketratio quantile Share
none 247 285 115.4% (8.9%) 66.1%X1 X2 126.67 126.67 100.0% (10.4%) 33.9%
market 373.67 411.67 110.2% (5.1%)
none 41.17 60 145.7% (34.6%) 189% 11.6%X1 196.94 225 114.2% (11.8%) 140% 55.8%X2 95 106.67 112.3% (15.1%) 134% 26.9%
X1 X2 20 20 100.0% (41.9%) 160% 5.7%
market 353.10 411.67 116.6% (5.3%) 130%
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Model Comparison (and Inequalities)
Use of statistical techniques to get price differentiationsee discriminant analysis, Fisher (1936)
In human social affairs, discrimination is treatment orconsideration of, or making a distinction in favor of oragainst, a person based on the group, class, or categoryto which the person is perceived to belong rather thanon individual attributes (wikipedia)
For legal perspective, see Canadian Human Rights Act
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http://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1936.tb02137.x/abstract;jsessionid=DBF4533A7EFB45D5CF8EDA619AC8EE3F.f02t02https://en.wikipedia.org/wiki/Discriminationhttps://en.wikipedia.org/wiki/Canadian_Human_Rights_Acthttps://twitter.com/freakonometricshttps://freakonometrics.github.io/https://freakonometrics.hypotheses.org/
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Model Comparison and Lorenz curves
Source: Progressive Insurance
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Model Comparison and Lorenz curves
Consider an ordered sample {y1, , yn} of incomes,with y1 y2 yn, then Lorenz curve is
{Fi, Li} with Fi =i
nand Li =
ij=1 yjnj=1 yj
We have observed losses yi and premiums (xi). Con-sider an ordered sample by the model, see Frees, Meyers& Cummins (2014), (x1) (x2) (xn), thenplot
{Fi, Li} with Fi =i
nand Li =
ij=1 yjnj=1 yj
0 20 40 60 80 100
020
4060
8010
0
Proportion (%)
Inco
me
(%)
poorest richest
0 20 40 60 80 100
020
4060
8010
0
Proportion (%)
Loss
es (
%)
more risky less risky
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http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2438129http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2438129https://twitter.com/freakonometricshttps://freakonometrics.github.io/https://freakonometrics.hypotheses.org/
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Model Comparison for Life Insurance Models
Consider the case of a death insurance contract, thatpays 1 if the insured deceased within the year.(x) = E
[Tx t+ 1|Tx > t
] No price discrimination = E[(X)] Perfect discrimination (x) Imperfect discrimination = E[(X)|X < s] and + = E[(X)|X > s]
click to visualize the construction
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
From Econometric to Machine Learning Techniques
In a competitive market, insurers can use different sets of variables and differentmodels, e.g. GLMs, Nt|X P(X t) and Y |X G(X , )
j(x) = E[N1X = x] E[Y X = x] = exp(Tx)
Poisson P(x)
exp(Tx)
Gamma G(X ,)
that can be extended to GAMs,
j(x) = exp(
dk=1
sk(xk))
Poisson P(x)
exp(
dk=1
tk(xk))
Gamma G(X ,)
or some Tweedie model on St (compound Poisson, see Tweedie (1984)) conditionalon X (see C. & Denuit (2005) or Kaas et al. (2008)) or any other statistical model
j(x) where j argminmFj :XjR
{ni=1
`(si,m(xi))}
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https://www.ams.org/mathscinet-getitem?mr=786162https://www.economica.fr/livre-mathematiques-de-l-assurance-non-vie-tome-ii-tarification-et-provisionnement-denuit-michel-charpentier-arthur,fr,4,9782717848601.cfmhttp://www.springer.com/us/book/9783540709923https://twitter.com/freakonometricshttps://freakonometrics.github.io/https://freakonometrics.hypotheses.org/
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
From Econometric to Machine Learning Techniques
For some loss function ` : R2 R+ (usually an L2 based loss, `(s, y) = (s y)2
since argmin{E[`(S,m)], m R} is E(S), interpreted as the pure premium).
For instance, consider regression trees, forests, neural networks, or boostingbased techniques to approximate (x), and various techniques for variableselection, such as LASSO (see Hastie et al. (2009) or C., Flachaire & Ly (2017) fora description and a discussion).
With d competitors, each insured i has to choose among d premiums,i =
(1(xi), , d(xi)
) Rd+.
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance and Risk Segmentation: Pricing Game
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance and Risk Segmentation: Pricing Game
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
Rat
io 9
9% v
s 1%
qua
ntile
020
4060
8010
0
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition Gas Type Diesel
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050
100
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miu
m (
Die
sel)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition Gas Type Regular
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miu
m (
Reg
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Gas
)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition Paris Region
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition Car Weight
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ght