Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Competitive insurance markets in a context of big data and machine learning A. Charpentier (Université de Rennes 1)
ESSEC, Paris November 2017
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Brief Introduction A. Charpentier (Université de Rennes 1) Professor Economics Department, Université de Rennes 1 Director Data Science for Actuaries Program, Institute of Actuaries (previously Actuarial Sciences, UQàM & ENSAE Paristech actuary in Hong Kong, IT & Stats FFA) PhD in Statistics (KU Leuven), Fellow of the Institute of Actuaries MSc in Financial Mathematics (Paris Dauphine) & ENSAE Research Chair : ACTINFO (valorisation et nouveaux usages actuariels de l’information) Editor of the freakonometrics.hypotheses.org’s blog Editor of Computational Actuarial Science, CRC Author of Mathématiques de l’Assurance Non-Vie (2 vol.), Economica
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance vs. Credit
click to visualize the construction
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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 N1 X Finance: risk neutral valuation π = EQ S1 F0 = EQ0 S1 , where S1 = Yi i=1
Insurance: risk sharing (pooling) π = EP S1 or, with segmentation / price differentiation π(ω) = EP S1 Ω = ω for some (unobservable?) risk factor Ω imperfect information given some (observable) risk variables X = (X1 , · · · , Xk ) π(x) = EP S1 X = x = EPX S1 |x Insurance pricing is not only data driven, it is also essentially model driven (see Pricing Game)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Pricing in a Nutshell Premium is π = 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 π≈E S1 |X = x = E
N X i=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 {yi xi } • 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 n
n
1X 1X Si ∼ π(X i ) (as n → ∞, from the law of large number) n i=1 n i=1 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]
0
0
Var[S]
Variance
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 solidarité et l’égalité de tous les Français devant les charges qui résultent des calamités nationales” (alinéa 12, préambule de la Constitution du 27 octobre 1946)
31 zones TRI (Territoires à Risques d’Inondation) on the left, and flooded areas. @freakonometrics
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance, Risk Pooling and Solidarity Here 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 Ω is observable,
Loss
Insured
Insurer
E[S|Ω]
S − E[S|Ω]
Average Loss
E[S] 0 h i h i Variance Var E[S|Ω] Var S − E[S|Ω] h i h i Observe that Var S − E[S|Ω] = E Var[S|Ω] , so that h i h i Var[S] = E Var[S|Ω] + Var E[S|Ω] . | {z } | {z } → insurer
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→ insured
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Risk Transfert with Segmentation and Imperfect Information Assume that X ⊂ Ω is observable
Loss Average Loss Variance
Insured
Insurer
E[S|X]
S − E[S|X]
E[S] h i Var E[S|X]
0 h i E Var[S|X]
Now h i E Var[S|X]
= =
ii ii h h h h E E Var[S|Ω] X + E Var E[S|Ω] X io n h h i E Var[S|Ω] + E Var E[S|Ω] X . | {z } | {z } pooling
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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 h i h i Var[S] = E Var[S|X] + Var E[S|X] ii h i h h = E Var[S|Ω] + E Var E[S|Ω] X | {z } | {z } pooling
|
solidarity
{z
→insurer
}
h i + Var E[S|X] . | {z } → insured
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
More and more price differentiation ? Consider π1 = E S1 and π2 (x) = E S1 |X = x X Observe that E π(X) = π(x) · P[x]
• Insured with x ∈ X1 : choose Ins1 • Insured with x ∈ X2 : choose Ins2 Ins1:
X
π1 (x) · P[x] 6= E[S|X ∈ X1 ]
x∈X1
Ins2:
X
0.15 0.10
x∈X2
0.05
x∈X1
π(x) · P[x]
0.00
π(x) · P[x] +
Claims Annual Frequency
=
X
0.20
x∈X
X
20
π2 (x) · P[x] = E[S|X ∈ X2 ]
30
40
50
60
70
80
Age of the Driver
x∈X2
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90
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Price Differentiation, a Toy Example Claims frequency Y (average cost = 1,000) X1
X2
Young
Experienced
Senior
Total
12%
9%
9%
9.5%
(500)
(2,000)
(500)
(3,000)
8%
6.67%
4%
6.33%
(500)
(1,000)
(500)
(2,000)
10%
8.22%
6.5%
8.23%
(1,000)
(3,000)
(1,000)
(5,000)
Town
Outside
Total from C., Denuit & Élie (2015)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Price Differentiation, a Toy Example
Y-T
Y-O
E-T
E-O
S-T
S-O
(500)
(500)
(2,000)
(1,000)
(500)
(500)
none
82.3
82.3
82.3
82.3
82.3
82.3
X1 × 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
82.3
X1
100
100
82.2
82.2
65
65
X2
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%
Market
ratio
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 differentiation see discriminant analysis, Fisher (1936) “In human social affairs, discrimination is treatment or consideration of, or making a distinction in favor of or against, a person based on the group, class, or category to which the person is perceived to belong rather than on individual attributes” (wikipedia) For legal perspective, see Canadian Human Rights Act
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Model Comparison and Lorenz curves
Source: Progressive Insurance @freakonometrics
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
100
Model Comparison and Lorenz curves
poorest ←
60 40 20
Consider an ordered sample {y1 , · · · , yn } of incomes, with y1 ≤ y2 ≤ · · · ≤ yn , then Lorenz curve is
Income (%)
80
→ richest
Pi
0
i j=1 yj {Fi , Li } with Fi = and Li = Pn n j=1 yj
0
20
40
60
80
100
80 60 40 20
i j=1 yj {Fi , Li } with Fi = and Li = Pn n j=1 yj
more risky ←
0
Pi
Losses (%)
We have observed losses yi and premiums π b(xi ). Consider an ordered sample by the model, see Frees, Meyers & Cummins (2014), π b(x1 ) ≥ π b(x2 ) ≥ · · · ≥ π b(xn ), then plot
100
Proportion (%)
0
20
→ less risky 40
60
80
100
Proportion (%)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Model Comparison for Life Insurance Models Age
60 40 20
Cumulative Losses (%)
80
100
100
0
Consider the case of a death insurance contract, that pays 1 if the insured deceased within the year. π(x) = E Tx ≤ t + 1|Tx > t — No price discrimination π = E[π(X)] — Perfect discrimination π(x) — Imperfect discrimination π− = E[π(X)|X < s] and π+ = E[π(X)|X > s]
90
80
70
60
50
40
30
●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ●● ●● ●● ●● ● ● ●● ●● ●● ●● ● ● ●●
'Perfect' Pricing 2 classes Pricing Average Pricing
Most Risky
Less Risky
click to visualize the construction 0
20
40
60
80
100
Population sorted by predicted risk (%)
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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 different models, e.g. GLMs, Nt |X ∼ P(λX · t) and Y |X ∼ G(µX , ϕ) b T x) b b b T x) · exp(β π bj (x) = E N1 X = x · E Y X = x = exp(α | {z } | {z } Poisson P(λx ) Gamma G(µX ,ϕ)
that can be extended to GAMs, π bj (x) = exp
d X
! sbk (xk ) · exp
{z
Poisson P(λx )
! b tk (xk )
k=1
k=1
|
d X
} |
{z
Gamma G(µX ,ϕ)
}
or some Tweedie model on St (compound Poisson, see Tweedie (1984)) conditional on X (see C. & Denuit (2005) or Kaas et al. (2008)) or any other statistical model ( n ) X π bj (x) where π bj ∈ argmin `(si , m(xi )) m∈Fj :Xj →R
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i=1
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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 boosting based techniques to approximate π(x), and various techniques for variable selection, such as LASSO (see Hastie et al. (2009) or C., Flachaire & Ly (2017) for a description and a discussion). With d competitors, each insured i has to choose among d premiums, πi = π b1 (xi ), · · · , π bd (xi ) ∈ Rd+ .
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
2000
4000
6000
8000
10000
Insurance and Risk Segmentation: Pricing Game
0
● ● ● ● ● ● ● ● ● ● ●
p1
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p2
● ●
●
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p3
p4
@freakonometrics
●
●
p5
● ● ● ● ● ●
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● ● ● ● ● ●
● ● ● ● ● ● ●
p6
p7
p8
p9
freakonometrics
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p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
60 40 20 0
Ratio 99% vs 1% quantile
80
100
Insurance and Risk Segmentation: Pricing Game
p1
p2
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p3
p4
p5
p6
p7
freakonometrics
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
100
highest premium →
60 40
Cumulated Claims (%)
60 40 20
20
← highest premium
Insurer 1 ●●
0
0
0
Cumulated Premium (%)
Insurer 1
80
← lowest premium
80
100
Insurance Ratemaking Before Competition
20
40
60
80
100
0
Insured (%)
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lowest premium →
●●
20
40
60
80
100
Insured (%)
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Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition
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200
400
600
800
1000
●
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●
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
0
● ● ● ● ● ● ● ●
p1
p2
@freakonometrics
p3
p4
p5
p6
freakonometrics
p7
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
freakonometrics.hypotheses.org
27
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition Gas Type Diesel ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
150 100
●
50
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
Premium (Diesel)
200
250
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
p1
p2
@freakonometrics
p3
p4
p5
p6
freakonometrics
p7
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
freakonometrics.hypotheses.org
28
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
150
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
100
●
50
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
Premium (Regular Gas)
200
250
Insurance Ratemaking Before Competition Gas Type Regular
p1
p2
@freakonometrics
p3
p4
p5
p6
freakonometrics
p7
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
freakonometrics.hypotheses.org
29
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Before Competition Paris Region ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
150
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
100
●
50
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
Premium (Paris Region)
200
250
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
p1
p2
@freakonometrics
p3
p4
p5
p6
freakonometrics
p7
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
freakonometrics.hypotheses.org
30
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
150
200
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
100
●
50
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
Premium (Car Weight < 11,000)
250
Insurance Ratemaking Before Competition Car Weight
p1
p2
@freakonometrics
p3
p4
p5
p6
freakonometrics
p7
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
freakonometrics.hypotheses.org
31
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
150
200
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ●
100
●
50
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
0
Premium (Car Value < 15,000)
250
Insurance Ratemaking Before Competition Car Value
p1
p2
@freakonometrics
p3
p4
p5
p6
freakonometrics
p7
p8
p9
p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23
freakonometrics.hypotheses.org
32
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
800
1000
1000 200 0 0
200
800
1000
0
02
04
06
08
10
nsu e 13
@freakonometr cs
600
800
1000
08
10
10 06
nsu e 17
02 00 00
02
04
06
08
nsu e 17
freakonometr cs
400
08
10 08 06
nsu e 21
02 00 00
200
nsu e 10
04
08 06 04 02 00
nsu e 17
600
nsu e 17
10
nsu e 13
400
04
600
400
nsu e 17
200 400
600
800
1000 800 200
0
400 0
0
600
nsu e 21
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nsu e 13
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freakonometr cs.hypotheses.org
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nsu e 10
34
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Competition We need a Decision Rule to select premium chosen by insured i cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc
@freakonometrics
Ins1
Ins2
Ins3
Ins4
Ins5
Ins6
787.93
706.97
1032.62
907.64
822.58
603.83
170.04
197.81
285.99
212.71
177.87
265.13
473.15
447.58
343.64
410.76
414.23
425.23
337.98
336.20
468.45
339.33
383.55
672.91
freakonometrics
freakonometrics.hypotheses.org
35
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Competition Basic ‘rational rule’ πi = min{b π1 (xi ), · · · , π bd (xi )} = π b1:d (xi ) cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc
@freakonometrics
Ins1
Ins2
Ins3
Ins4
Ins5
Ins6
787.93
706.97
1032.62
907.64
822.58
603.83
170.04
197.81
285.99
212.71
177.87
265.13
473.15
447.58
343.64
410.76
414.23
425.23
337.98
336.20
468.45
339.33
383.55
672.91
freakonometrics
freakonometrics.hypotheses.org
36
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Insurance Ratemaking Competition A more realistic rule πi ∈ {b π1:d (xi ), π b2:d (xi ), π b3:d (xi )} cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc
@freakonometrics
Ins1
Ins2
Ins3
Ins4
Ins5
Ins6
787.93
706.97
1032.62
907.64
822.58
603.83
170.04
197.81
285.99
212.71
177.87
265.13
473.15
447.58
343.64
410.76
414.23
425.23
337.98
336.20
468.45
339.33
383.55
672.91
freakonometrics
freakonometrics.hypotheses.org
37
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
A Game with Rules... but no Goal Two datasets : a training one, and a pricing one (without the losses in the later) Step 1 : provide premiums to all contracts in the pricing dataset Step 2 : allocate insured among players Season 1 13 players Season 2 14 players Step 3 [season 2] : provide additional information (premiums of competitors) Season 3 23 players (3 markets, 8+8+7) Step 3-6 [season 3] : dynamics, 4 years
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
38
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
@freakonometrics
freakonometrics
10
150 50
100
Loss Ratio (in %)
8 6 4
Market Share (in %)
0
2 0
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
A1 A2 A3 A4 A5 A6 A7 A8 A9
A11
A13
150 50
100
Loss Ratio (in %)
6 4
0
Market Share (in %)
8
10
200
A11
2
Insurer 11 Use of two XGBoost models (bodily injury and material), with correction for negative premiums Actuary working for a private insurance company
A1 A2 A3 A4 A5 A6 A7 A8 A9
0
Insurer 4 GLM for frequency and standard cost (large claimes were removed, above 15k), Interaction Age and Gender Actuary working for a mutuelle company
200
Pricing Game in 2015
freakonometrics.hypotheses.org
39
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Pricing Game in 2017 Insurer 6 (market 3) Team of two actuaries (degrees in Engineering and Physics), in Vancouver, Canada. Used GLMs (Tweedie), no territorial classification, no use of information about other competitors
99% vs 1% 95% vs 5%
2
5
10
20
6
1
Ratio of Quantiles of Premiums
50
100
“Segments with high market share and low loss ratios were also given some premium increase”
1
2
3
4
Year
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
40
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Pricing Game in 2017 Insurer 7 (market 1)
40
220
30 20
Market Share (%)
180 160
7
5
95% vs 5%
140
7
Loss Ratio (%)
20 10
99% vs 1%
100
10
120
7
1
2
3
4
Year
@freakonometrics
0
80
2 1
Ratio of Quantiles of Premiums
50
200
100
Actuary in France, used random forest for variable selection, and GLMs
1
2
3
Year (market 1)
freakonometrics
freakonometrics.hypotheses.org
4
1
2
3
4
Year (market 1)
41
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Pricing Game in 2017 Insurer 15 (market 2)
40
220
30 10
15
1
2
3
4
Year
@freakonometrics
0
80
2
100
120
15
20
Market Share (%)
180 160 140
10 5
95% vs 5%
Loss Ratio (%)
20
15
99% vs 1%
1
Ratio of Quantiles of Premiums
50
200
100
Actuary,working as a consultant, Margin Method with iterations, MS Access & MS Excel
1
2
3
Year (market 2)
freakonometrics
freakonometrics.hypotheses.org
4
1
2
3
4
Year (market 2)
42
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Pricing Game in 2017 Insurer 21 (market 1) Actuary, working as a consultant, used GLMs, with variable selection using LARS and LASSO
40
220
30 20
Market Share (%)
180 160
21
10
21
1
2
3
4
Year
@freakonometrics
0
80
2
100
120
5
140
Loss Ratio (%)
95% vs 5%
10
20
21
99% vs 1%
1
Ratio of Quantiles of Premiums
50
200
100
Iterative learning algorithm (codes available on github)
1
2
3
Year (market 1)
freakonometrics
freakonometrics.hypotheses.org
4
1
2
3
4
Year (market 1)
43
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Pricing Game in 2017 Insurer 4 (market 2)
40
220
30
4
2
3
4
Year
@freakonometrics
0
4 1
80
2
100
10
120
5
20
Market Share (%)
180 160
4 140
Loss Ratio (%)
95% vs 5%
10
20
99% vs 1%
1
Ratio of Quantiles of Premiums
50
200
100
Actuary, working as a consultat,used XGBOOST, used GLMs for year 3.
1
2
3
Year (market 2)
freakonometrics
freakonometrics.hypotheses.org
4
1
2
3
4
Year (market 2)
44
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Pricing Game in 2017 Insurer 8 (market 3) Mathematician, working on Solvency II sofware in Austria Generalized Additive Models with spatial variable
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
45
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Cluster, Segmentation and (Social) Networks Social networks could be used to get additional information about insured people...
Why not using social networks to create (more) solidarity ? @freakonometrics
freakonometrics
freakonometrics.hypotheses.org
46
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Cluster, Segmentation and (Social) Networks Homophily is the tendency of individuals to associate and bond with similar others, “birds of a feather flock together”
from Moody (2001) Race, School Integration and Friendship Segregation in America
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
47
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Cluster, Segmentation and (Social) Networks
●
Nymeria Sand
●
Tyene Sand
● ●
So far, risk classes are based on covariates X, correlated (causal effect?) with claims occurence (or severity).
Obara Sand
●
●
●
Dorna Willem Swyft Lannister
●
Ermesande Hayford
Princess of Dorne
●
Mors Martell brother of Doran
●
●
●
Gilliane Glover
●
Jonnel Stark
Elia Martell Aegon Rhaenys Targaryen son Targaryen of daughter Rhaegar of Rhaegar
Maron Martell Hizdahr zo Loraq Drogo
Brandon Stark Burner
Daenerys Targaryen
●
Brandon Stark son of Cregan
Rodwell Stark
Rhaella Targaryen Aerys II Targaryen
Viserys Targaryen
●
Alys Karstark
Beron Stark
●
●
Willam Stark
Jaehaerys II Shaera Targaryen Targaryen
Myriame Manderly
Artos Stark
Eddard Stark
Robert Baratheon
Jenny of Oldstones
Shireen Baratheon
Tyrion Lannister
Ella Lannister Damon Lannister son of Jason
●
Alys Stackspear
Damion Lannister
●
●
Cerissa Brax Damon Lannister
Tybolt Lannister
●
Teora Kyndall
Stafford Lannister
Daven Lannister Myranda Lefford Cerenna Myrielle Lannister Lannister
●
●
●
Willas Tyrell Alerie Hightower Mace Tyrell
Olenna Redwyne
Hobber Redwyne Paxter Redwyne
●
Lanna Lannister Antario Jast
Alysanne Farman
●
Rohanne Gerold Webber Lannister
Jason Lannister
Joffrey Tommen Baratheon Baratheon
●
Leonette Fossoway
Garlan Tyrell
Medwick Tyrell Luthor Elyn Tyrell Norridge son of Moryn Olene Tyrell
●
Loras Tyrell Moryn Tyrell
Unknown father Unknown Tyrell mother Tyrell
Luthor Tyrell
Theodore Tyrell
Luthor Tyrell son of Theodore Lia Serry
● ●
Elinor Tyrell
●
●
Leo Blackbar
●
Mina Tyrell
Rodrik Greyjoy
Horas Redwyne Lady of Robin Greyjoy House Piper
See InsPeer experience.
●
Shiera Crakehall
●
Jeyne Westerling
Joanna Lannister
Renly Baratheon
Stannis Baratheon Selyse Florent
●
Lucion Lannister
●
Tysha
Ormund Baratheon
●
●
Elys Waynwood
Talisa Stark
Robb Stark
Rodrik Stark
Margaery Tyrell
●
●
●
Alys Arryn
●
Myrcella Baratheon
Steffon Cassana Baratheon Baratheon
Jasper Arryn
●
Rowena Arryn
Catelyn Stark
Cersei Lannister
●
●
●
Jaime Lannister
Melantha Blackwood
Rhaelle Targaryen
Jon Arryn
●
Aegon Betha V TargaryenBlackwood
Duncan Targaryen
●
Jeyne Royce
Robert Arryn
Marla Prester
●
Maekar I Targaryen
Tywin Lannister
●
Lyarra Stark Rickard Stark
Lyanna Glover
●
Petyr Baelish
Lysa Arryn
Sansa Stark Bran Stark
●
●
Dyanna Dayne
Edmure Tully
●
Melesa Crakehall
Ramsay Bolton
Marna Locke
Lysara Karstark
A lot of cofounding variables (age, profession, location, etc.)
●
Jeyne Tytos Marbrand Lannister
Benjen Stark
Lorra Royce
Tywin Frey
Lyanna Stark
●
Edwyle Stark
Cleos Frey
●
Minisa Whent Hoster Tully
Jon Snow
Arya Flint
Rodrik Stark son of Beron
●
Jeyne Darry
Tygett Lannister
Brandon Stark
Lynara Stark
●
Robyn Ryswell
Willem Frey
Roslin Frey Walder Emmon Frey Frey son of Emmon Tion Frey Lyonel Frey Genna Roose Lannister Bolton
Arya Stark Rickon Stark
Rhaegar Targaryen
●
●
●
Darlessa Marbrand
Gerion Lannister Rickon Stark son of Benjen Arra Norrey
Walder Frey
●
Jyanna Frey
Bethany Rosby
●
Kevan Lannister
Tyrek Lannister
Olyvar Willamen Frey Perwyn Frey Benfrey Frey Frey
Perra Royce
Walda Frey daughter of Merrett
Lancel Lannister
Sarella Sand
●
●
Amarei Crakehall
Frey
Dorea Sand
●
Cregan Stark
Why not consider clusters in (social) networks, too?
●
●
Obella Sand
Loreza Ellaria Sand Sand Oberyn Martell Elia Sand
Mellario Doran of Norvos Martell
Walder Marissa Frey Frey son of Merrett Merrett Mariya Frey Darry Amerei
Pate of the Blue Fork
●
Harlon GreyjoyLady of Donel House Greyjoy Stonetree Quenton Greyjoy
●
Victarion Greyjoy Urrigon Greyjoy Lady of Quellon House Greyjoy Sunderly
●
Balon Greyjoy
Asha (Yara) Greyjoy
Erik Ironmaker
Alannys Harlaw Maron Greyjoy Theon Greyjoy
Euron Aeron Greyjoy Greyjoy
via shiring.github.io @freakonometrics
freakonometrics
freakonometrics.hypotheses.org
48
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
(Social) Networks and Credit Used already on credit (see cnn or or digitaltrends) E.g Lenddo or Lendup It does mean that homophily can be seen as a substitute to standard credit ‘explanatory’ variales...
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
49
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Information and Networks But other kinds of networks can be used, e.g. (genealogical) trees
See Ewen Gallic’s ongoing work (actinfo chair). @freakonometrics
freakonometrics
freakonometrics.hypotheses.org
50
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Privacy Issues See General Data Protection Regulation (EU 2016/679) : what about aggregation ? Consider a population {1, · · · , n} and a partition {I1 , · · · , Ik } (e.g. geographical 1 X areas Z), with respective sizes {n1 , · · · , nk }. Set Y j = Yi . nj i∈Ij
For continous covariates, set X k,j
1 X = Xk,i , nk i∈Ij
For categorical variables, consider the associate composition variable 1 X X k,j = (X k,1,j , · · · , X k,dk ,j ) where X k,`,j = 1(Xk,i = `). nk i∈Ij
See e.g. C. & Pigeon (2016) on micro-macro models and Enora Belz’s ongoing work.
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
51
Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017
Privacy Issues See Verbelen, Antonio & Claeskens (2016) and Antonio & C. (2017) on GPS data
@freakonometrics
freakonometrics
freakonometrics.hypotheses.org
52
