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![Arthur Charpentier, UQ`aM Economics Seminar 2019
Risk Transfert without Segmentation
Insured Insurer
Loss E[S] S − E[S]
Average Loss E[S] 0
Variance 0 Var[S]
All the risk - Var[S] - is kept by the insurance company.
Remark: all those interpretation are discussed in Denuit & Charpentier (2004).
@freakonometrics freakonometrics freakonometrics.hypotheses.org 3](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-3-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
Risk Transfert with Segmentation and Perfect Information
Assume that information Ω is observable,
Insured Insurer
Loss E[S|Ω] S − E[S|Ω]
Average Loss E[S] 0
Variance Var E[S|Ω] Var S − E[S|Ω]
Observe that Var S − E[S|Ω] = E Var[S|Ω] , so that
Var[S] = E Var[S|Ω]
→ insurer
+ Var E[S|Ω]
→ insured
.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 4](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-4-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
Risk Transfert with Segmentation and Imperfect Information
Assume that X ⊂ Ω is observable
Insured Insurer
Loss E[S|X] S − E[S|X]
Average Loss E[S] 0
Variance Var E[S|X] E Var[S|X]
Now
E Var[S|X] = E E Var[S|Ω] X + E Var E[S|Ω] X
= E Var[S|Ω]
perfect pricing
+ E Var E[S|Ω] X
misfit
.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 5](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-5-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
Actuarial Pricing Model
Premium is E S|X = x = E
N
i=1
Yi X = x = E [N|X = x] · E [Y |X = x]
Statistical and modeling issues to approximate based on some training datasets,
with claims frequency {ni, xi} and individual losses {yi, xi}.
Use GLM to approximate E [N|X = x] and E [Y |X = x]
Recall that E E S|X = E S
How can we claim that model π(x) = E S|X = x is “good” ?
@freakonometrics freakonometrics freakonometrics.hypotheses.org 6](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-6-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
How can we visualize the goodness of a model ?
In the context of classification, use ROC curve
sensitivity = TNR =
TN
TN + FP
= P[m(X) > s
Y =1
|Y = 1] vs. specificity = TPR =
TP
TP + FN
Specificity
Sensitivity
1.0 0.8 0.6 0.4 0.2 0.0
0.00.20.40.60.81.0
q
q
0.285 (0.860, 0.927)
0.500 (0.907, 0.860)
logit
bagging
random forest
boosting
Specificity
Sensitivity
1.0 0.8 0.6 0.4 0.2 0.0
0.00.20.40.60.81.0
logit
bagging
random forest
boosting
Source : Charpentier, Flachaire & Ly (2018)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 7](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-7-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
From Econometric to ‘Machine Learning’ Techniques
(see Charpentier & Denuit (2005) or Kaas et al. (2008)) or any other statistical
model
πj(x) where πj ∈ argmin
m∈Fj :Xj →R
n
i=1
(si, m(xi))
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 Charpentier et al. (2017) for
a description and a discussion).
With d competitors, each insured i has to choose among d premiums,
πi = π1(xi), · · · , πd(xi) ∈ Rd
+.
@freakonometrics freakonometrics freakonometrics.hypotheses.org 26](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-26-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
Machine Learning & Credit
Before discussing the use of those models in insurance, note that the same issues
exist in credit, see Hardt, Price & Srebro (2017).
“the shift from traditional to machine learning lending models may have
important distributional effects for consumers [... ] machine learning would offer
lower rates to racial groups who already were at an advantage under the
traditional model, but it would also benefit disadvantaged groups by enabling them
to obtain a mortgage in the first place ” Fuster, Goldsmith-Pinkham, Ramadorai &
Walther (2017)
@freakonometrics freakonometrics freakonometrics.hypotheses.org 27](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-27-320.jpg)
![Arthur Charpentier, UQ`aM Economics Seminar 2019
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 informa-
tion (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 40](https://image.slidesharecdn.com/slidesuqameco2019-190311150414/85/Econ-Seminar-Uqam-40-320.jpg)
Uploaded byArthur Charpentier
Econ. Seminar Uqam
Arthur Charpentier presented at the UQ'aM Economics Seminar on insurance segmentation and mutualization. He discussed how insurance allows for risk sharing by pooling together individuals with different risk levels. Insurers can differentiate prices through segmentation based on observable risk factors, but this leads to a "spiral of segmentation" where each group is charged their own risk-based price. Charpentier presented models to transfer risk with and without segmentation using perfect and imperfect information. He proposed using Lorenz curves and pseudo-Lorenz curves to visualize the performance of actuarial pricing models and their ability to correctly order risks. Machine learning techniques can also be applied to approximating insurance premiums based on risk characteristics.
More Related Content
Econ. Seminar Uqam
- 1. Arthur Charpentier, UQ`aMEconomics Seminar 2019 ‘Segmentation’ & ‘Mutualisation’ Arthur Charpentier (Universit´e du Qu´ebec `a Montr´eal) UQ`aM, Economics Seminar, March 2019, Montr´eal @freakonometrics freakonometrics freakonometrics.hypotheses.org 1
- 2. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance, “segmentation” & “mutualisation” “Insurance is the contribution of the many to the misfortune of the few ” Insurance: risk sharing (pooling) π = EP S1 segmentation / price differentiation π(ω) = EP S1 Ω = ω for some (unobservable) risk factor Ω imperfect information given some (observable) risk variables x π(x) = EP S1 X = x = EPx S1 why a “spirale de la segmentation” ? @freakonometrics freakonometrics freakonometrics.hypotheses.org 2
- 3. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Risk Transfert without Segmentation Insured Insurer Loss E[S] S − E[S] Average Loss E[S] 0 Variance 0 Var[S] All the risk - Var[S] - is kept by the insurance company. Remark: all those interpretation are discussed in Denuit & Charpentier (2004). @freakonometrics freakonometrics freakonometrics.hypotheses.org 3
- 4. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Risk Transfert with Segmentation and Perfect Information Assume that information Ω is observable, Insured Insurer Loss E[S|Ω] S − E[S|Ω] Average Loss E[S] 0 Variance Var E[S|Ω] Var S − E[S|Ω] Observe that Var S − E[S|Ω] = E Var[S|Ω] , so that Var[S] = E Var[S|Ω] → insurer + Var E[S|Ω] → insured . @freakonometrics freakonometrics freakonometrics.hypotheses.org 4
- 5. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Risk Transfert with Segmentation and Imperfect Information Assume that X ⊂ Ω is observable Insured Insurer Loss E[S|X] S − E[S|X] Average Loss E[S] 0 Variance Var E[S|X] E Var[S|X] Now E Var[S|X] = E E Var[S|Ω] X + E Var E[S|Ω] X = E Var[S|Ω] perfect pricing + E Var E[S|Ω] X misfit . @freakonometrics freakonometrics freakonometrics.hypotheses.org 5
- 6. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Actuarial Pricing Model Premium is E S|X = x = E N i=1 Yi X = x = E [N|X = x] · E [Y |X = x] Statistical and modeling issues to approximate based on some training datasets, with claims frequency {ni, xi} and individual losses {yi, xi}. Use GLM to approximate E [N|X = x] and E [Y |X = x] Recall that E E S|X = E S How can we claim that model π(x) = E S|X = x is “good” ? @freakonometrics freakonometrics freakonometrics.hypotheses.org 6
- 7. Arthur Charpentier, UQ`aMEconomics Seminar 2019 How can we visualize the goodness of a model ? In the context of classification, use ROC curve sensitivity = TNR = TN TN + FP = P[m(X) > s Y =1 |Y = 1] vs. specificity = TPR = TP TP + FN Specificity Sensitivity 1.0 0.8 0.6 0.4 0.2 0.0 0.00.20.40.60.81.0 q q 0.285 (0.860, 0.927) 0.500 (0.907, 0.860) logit bagging random forest boosting Specificity Sensitivity 1.0 0.8 0.6 0.4 0.2 0.0 0.00.20.40.60.81.0 logit bagging random forest boosting Source : Charpentier, Flachaire & Ly (2018) @freakonometrics freakonometrics freakonometrics.hypotheses.org 7
- 8. Arthur Charpentier, UQ`aMEconomics Seminar 2019 How can we visualize the goodness of a model ? Source : https://www.progressive.com/jobs/analyst-program/ @freakonometrics freakonometrics freakonometrics.hypotheses.org 8
- 9. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Constructing the (pseudo)-Lorenz curve Sort the n risks according to the model π(x1) ≥ π(x2) ≥ · · · ≥ π(xn) @freakonometrics freakonometrics freakonometrics.hypotheses.org 9
- 10. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Constructing the (pseudo)-Lorenz curve On the x-axis, xi = i/n, on the y-axis, yi = i j=1 yj/ n j=1 yj @freakonometrics freakonometrics freakonometrics.hypotheses.org 10
- 11. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Constructing the (pseudo)-Lorenz curve Connect points (xi, yi) @freakonometrics freakonometrics freakonometrics.hypotheses.org 11
- 12. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Constructing the (pseudo)-Lorenz curve see Frees, Meyers & Cummins (2014). @freakonometrics freakonometrics freakonometrics.hypotheses.org 12
- 13. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Practice of (pseudo)-Lorenz curves What if π and π are not perfectly correctly correlated...? @freakonometrics freakonometrics freakonometrics.hypotheses.org 13
- 14. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Practice of (pseudo)-Lorenz curves What if m and m are not perfectly correctly correlated...? @freakonometrics freakonometrics freakonometrics.hypotheses.org 14
- 15. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Practice of (pseudo)-Lorenz curves What if m and m are not perfectly correctly correlated...? @freakonometrics freakonometrics freakonometrics.hypotheses.org 15
- 16. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Practice of (pseudo)-Lorenz curves What if m and m are not perfectly correctly correlated...? @freakonometrics freakonometrics freakonometrics.hypotheses.org 16
- 17. Arthur Charpentier, UQ`aMEconomics Seminar 2019 What is the “average” model ? What is this “average pricing” ? @freakonometrics freakonometrics freakonometrics.hypotheses.org 17
- 18. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Can it be worst than the “average” model ? Is it a lower bond ? Is it possible to be below that curve ? @freakonometrics freakonometrics freakonometrics.hypotheses.org 18
- 19. Arthur Charpentier, UQ`aMEconomics Seminar 2019 What is in the upper corner ? What is the upper bond ? Ex-post pricing... @freakonometrics freakonometrics freakonometrics.hypotheses.org 19
- 20. Arthur Charpentier, UQ`aMEconomics Seminar 2019 How to understand this (pseudo)-Lorenz curve ? Is there a continuity between mutualization and hyper-segmentation ? @freakonometrics freakonometrics freakonometrics.hypotheses.org 20
- 21. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance, Risk Pooling and Solidarity Consider flood risk, in France One can look at the “Lorenz curve” @freakonometrics freakonometrics freakonometrics.hypotheses.org 21
- 22. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Price Differentiation, a Toy Example Claims frequency N ∈ {0, 1} (average cost = 1,000) X1 Young Experienced Senior Total X2 Town 12% (500) 9% (2,000) 9% (500) 9.5% (3,000) Outside 8% (500) 6.67% (1,000) 4% (500) 6.33% (2,000) Total 10% (1,000) 8.22% (3,000) 6.5% (1,000) 8.23% (5,000) from Charpentier, Denuit & ´Elie (2015) @freakonometrics freakonometrics freakonometrics.hypotheses.org 22
- 23. Arthur Charpentier, UQ`aMEconomics Seminar 2019 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 82.3 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 @freakonometrics freakonometrics freakonometrics.hypotheses.org 23
- 24. Arthur Charpentier, UQ`aMEconomics Seminar 2019 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% @freakonometrics freakonometrics freakonometrics.hypotheses.org 24
- 25. Arthur Charpentier, UQ`aMEconomics Seminar 2019 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) πj(x) = E N1 X = x · E Y X = x = exp(αT x) Poisson P(λx) · exp(β T x) Gamma G(µX ,ϕ) that can be extended to GAMs, πj(x) = exp d k=1 sk(xk) Poisson P(λx) · exp d k=1 tk(xk) Gamma G(µX ,ϕ) or some Tweedie model on St (compound Poisson, see Tweedie (1984)) conditional on X @freakonometrics freakonometrics freakonometrics.hypotheses.org 25
- 26. Arthur Charpentier, UQ`aMEconomics Seminar 2019 From Econometric to ‘Machine Learning’ Techniques (see Charpentier & Denuit (2005) or Kaas et al. (2008)) or any other statistical model πj(x) where πj ∈ argmin m∈Fj :Xj →R n i=1 (si, m(xi)) 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 Charpentier et al. (2017) for a description and a discussion). With d competitors, each insured i has to choose among d premiums, πi = π1(xi), · · · , πd(xi) ∈ Rd +. @freakonometrics freakonometrics freakonometrics.hypotheses.org 26
- 27. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Machine Learning & Credit Before discussing the use of those models in insurance, note that the same issues exist in credit, see Hardt, Price & Srebro (2017). “the shift from traditional to machine learning lending models may have important distributional effects for consumers [... ] machine learning would offer lower rates to racial groups who already were at an advantage under the traditional model, but it would also benefit disadvantaged groups by enabling them to obtain a mortgage in the first place ” Fuster, Goldsmith-Pinkham, Ramadorai & Walther (2017) @freakonometrics freakonometrics freakonometrics.hypotheses.org 27
- 28. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Field experiment: the actuarial pricing games Actuarial pricing is data based, and model based To understand how model influence pricing we ran some actuarial pricing games @freakonometrics freakonometrics freakonometrics.hypotheses.org 28
- 29. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Before Competition q q q q q q qq q q q qqq q q q q q q q q q q q qq q q q q q q q q qq q q qq qqq q q q q qq q q q qq qq qq q q q q q q q q qq qq q q q qq q q qqqqq q q q q q qq q q q q q q q q qq q q qq q q q q qq q q q q q qq q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q qq q q q qq q q qq qqq q q q q q q q q q qq qq q q qq q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q qq q q qq q q q q q q q q q q q q q q qqqqq q q q q q q q qq qq q q q q q q q q q q q q q qq q q q q q qq q qqq q q q q q q qqq qq q q q qq q q qq q qq q q q q q qqq q q q q q q q q q q q q q q q qq qqq q qq q q q q q qq q q q qq q q q q q q qq q q qqq q q qq q q q q q q q q q q q q q q q q q qq qq q q qq q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q qq q q qq q q q q qqq q qq q qq q q qq qqqq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q qqq q qq q q q q q q q q q q q q q q qq q q q q q qqq q q q q qq q q qq q q q q q q q q q qqq q q q q qq q q q q q q q q qq qqq q qq q q q q q qq q q q q q qq q q q q q qq q q q q q q q q q q q q qq qq q q q q qq q q q q qq q q q q qq q q q qqq q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q qq q qq qqq qqq q qq q qq q q q q q qq q q q q q q q q q q qq q qq q q q qqq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q qq qqq q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq qq q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q qqq q qqqq q q q qqq q q q q q q q q qq q q qqq q q q qq q q q q qq q q q qq q qq q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q qq q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qqq q q qq q q q q q q q q q qq q q qq q q q q qqq q q q q q q q qq q q q q q qq q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q qqq q q q qq qqqq q qq q q q q q q q q q q q q q q q qqq qq qq q q qq q qq q q q q qqq q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q q q qqqq q q q q q qqq q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q qq q q q q q q qq q qq q q q q q q q qq q q q q qq q qq q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q qq q qq q qq q q q qq q q q q q q qq q q q q q q q q q q q q qqq q q q q q qq q q q q q q q q q q q q q qqq q q q q q qqq q q q q q q q q qq q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq q qq q q q q q qq q 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Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Before Competition Gas Type Regular qqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqq q qqqqqqq q qqqqqqqqqqqq q qqqqq q qq q q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q q q qqq q 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Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Before Competition Car Weight qqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqq q qqqqqqq q qqqqqqqqqqqq q qqqqq q qq q q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q q q qqq q qqqqqq q 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- 34. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Before Competition Car Value qqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqq q qqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqq q qqqqqqq q qqqqqqqqqqqq q qqqqq q qq q q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqq q qqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q q q qqq q qqqqqq q 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q q q q q q qq q qq q q q q q q q q qqq p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 p13 p14 p15 p16 p17 p18 p19 p20 p21 p22 p23 050100150200250 @freakonometrics freakonometrics freakonometrics.hypotheses.org 34
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Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Competition : High Correlation ? q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q 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q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q 0 200 400 600 800 1000 02004006008001000 nsu e 13 nsue17 0 0 0 2 0 4 0 6 0 8 1 0 000204060810 nsu e 13 nsue17 0 200 400 600 800 1000 02004006008001000 nsu e 17 nsue21 0 0 0 2 0 4 0 6 0 8 1 0 000204060810 nsu e 17 nsue21 0 200 400 600 800 1000 02004006008001000 nsu e 10 nsue17 0 0 0 2 0 4 0 6 0 8 1 0 000204060810 nsu e 10 nsue17 @freakonometr cs freakonometr cs freakonometr cs.hypotheses.org 35
- 36. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Competition : High Correlation ? q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q qq q q q q q q q q q q q q q qq q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q 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q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q qq q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 200 400 600 800 1000 02004006008001000 nsu e 3 nsue4 0 0 0 2 0 4 0 6 0 8 1 0 000204060810 nsu e 3 nsue4 0 200 400 600 800 1000 02004006008001000 nsu e 3 nsue13 0 0 0 2 0 4 0 6 0 8 1 0 000204060810 nsu e 3 nsue13 0 200 400 600 800 1000 02004006008001000 nsu e 10 nsue13 0 0 0 2 0 4 0 6 0 8 1 0 000204060810 nsu e 10 nsue13 @freakonometr cs freakonometr cs freakonometr cs.hypotheses.org 36
- 37. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Competition We need a Decision Rule to select premium chosen by insured i cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc 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 freakonometrics.hypotheses.org 37
- 38. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Competition Basic ‘rational rule’ πi = min{π1(xi), · · · , πd(xi)} = π1:d(xi) cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc 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 freakonometrics.hypotheses.org 38
- 39. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Insurance Ratemaking Competition A more realistic rule πi ∈ {π1:d(xi), π2:d(xi), π3:d(xi)} cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc 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 freakonometrics.hypotheses.org 39
- 40. Arthur Charpentier, UQ`aMEconomics Seminar 2019 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 informa- tion (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 40
- 41. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Pricing Game in 2015 Insurer 4 GLM for frequency and standard cost (large claimes were removed, above 15k), Interaction Age and Gender Actuary working for a mutuelle com- pany Insurer 11 Use of two XGBoost models (bodily injury and material), with correction for negative premiums Actuary working for a private insur- ance company A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13 MarketShare(in%) 0246810 A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13 LossRatio(in%) 050100150200 A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13 MarketShare(in%) 0246810 A1 A2 A3 A4 A5 A6 A7 A8 A9 A11 A13 LossRatio(in%) 050100150200 @freakonometrics freakonometrics freakonometrics.hypotheses.org 41
- 42. Arthur Charpentier, UQ`aMEconomics Seminar 2019 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 “Segments with high market share and low loss ratios were also given some premium increase” Year RatioofQuantilesofPremiums 125102050100 1 2 3 4 699% vs 1% 95% vs 5% @freakonometrics freakonometrics freakonometrics.hypotheses.org 42
- 43. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Pricing Game in 2017 Insurer 7 (market 1) Actuary in France, used random forest for variable selection, and GLMs Year RatioofQuantilesofPremiums 125102050100 1 2 3 4 799% vs 1% 95% vs 5% Year (market 1) LossRatio(%) 80100120140160180200220 1 2 3 4 7 Year (market 1) MarketShare(%) 010203040 1 2 3 4 7 @freakonometrics freakonometrics freakonometrics.hypotheses.org 43
- 44. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Pricing Game in 2017 Insurer 15 (market 2) Actuary,working as a consultant, Margin Method with iterations, MS Access & MS Excel Year RatioofQuantilesofPremiums 125102050100 1 2 3 4 1599% vs 1% 95% vs 5% Year (market 2) LossRatio(%) 80100120140160180200220 1 2 3 4 15 Year (market 2) MarketShare(%) 010203040 1 2 3 4 15 @freakonometrics freakonometrics freakonometrics.hypotheses.org 44
- 45. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Pricing Game in 2017 Insurer 21 (market 1) Actuary, working as a consultant, used GLMs, with variable selection using LARS and LASSO Iterative learning algorithm (codes available on github) Year RatioofQuantilesofPremiums 125102050100 1 2 3 4 2199% vs 1% 95% vs 5% Year (market 1) LossRatio(%) 80100120140160180200220 1 2 3 4 21 Year (market 1) MarketShare(%) 010203040 1 2 3 4 21 @freakonometrics freakonometrics freakonometrics.hypotheses.org 45
- 46. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Pricing Game in 2017 Insurer 4 (market 2) Actuary, working as a consultant,used XGBOOST, used GLMs for year 3. Year RatioofQuantilesofPremiums 125102050100 1 2 3 4 4 99% vs 1% 95% vs 5% Year (market 2) LossRatio(%) 80100120140160180200220 1 2 3 4 4 Year (market 2) MarketShare(%) 010203040 1 2 3 4 4 @freakonometrics freakonometrics freakonometrics.hypotheses.org 46
- 47. Arthur Charpentier, UQ`aMEconomics Seminar 2019 Pricing Game in 2017 Insurer 8 (market 3) Mathematician, working on Solvency II software in Austria Generalized Additive Models with spatial variable @freakonometrics freakonometrics freakonometrics.hypotheses.org 47
- 48. Arthur Charpentier, UQ`aMEconomics Seminar 2019 What’s Next ? • hard to derive theoretical properties of model competition equilibrium • more on-going field studies... but hard to get players... • 2018 game, more (too) complicated design... • 2019 game(s)... to be continued... @freakonometrics freakonometrics freakonometrics.hypotheses.org 48