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Slides pricing-game-3

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Présentation GT Big Data, Institut des Actuaires, Octobre 2017

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Slides pricing-game-3

  1. 1. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 Insurance: Risk Pooling and Price Segmentation — Using Information in a ‘Big Data’ Context — A. Charpentier (Université de Rennes 1) GT Big Data, Institut des Actuaires Octobre 2017 @freakonometrics freakonometrics freakonometrics.hypotheses.org 1
  2. 2. 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 @freakonometrics freakonometrics freakonometrics.hypotheses.org 2
  3. 3. 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 π = EQ S1 F0 = EQ0 S1 , where S1 = N1 i=1 Yi 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) @freakonometrics freakonometrics freakonometrics.hypotheses.org 3
  4. 4. 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 @freakonometrics freakonometrics freakonometrics.hypotheses.org 4
  5. 5. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 Insurance Pricing in a Nutshell Premium is π≈E S1|X = x = E N 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 {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 @freakonometrics freakonometrics freakonometrics.hypotheses.org 5
  6. 6. 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 1 n n i=1 Si ∼ 1 n n i=1 π(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. @freakonometrics freakonometrics freakonometrics.hypotheses.org 6
  7. 7. 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 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 7
  8. 8. 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 freakonometrics freakonometrics.hypotheses.org 8
  9. 9. 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 @freakonometrics freakonometrics freakonometrics.hypotheses.org 9
  10. 10. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 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 10
  11. 11. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 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] pooling + E Var E[S|Ω] X solidarity . @freakonometrics freakonometrics freakonometrics.hypotheses.org 11
  12. 12. 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|Ω] pooling + E Var E[S|Ω] X solidarity →insurer + Var E[S|X] → insured . @freakonometrics freakonometrics freakonometrics.hypotheses.org 12
  13. 13. 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 Observe that E π(X) = x∈X π(x) · P[x] = x∈X1 π(x) · P[x] + x∈X2 π(x) · P[x] • Insured with x ∈ X1 : choose Ins1 • Insured with x ∈ X2: choose Ins2 Ins1: x∈X1 π1(x) · P[x] = E[S|X ∈ X1] Ins2: x∈X2 π2(x) · P[x] = E[S|X ∈ X2] @freakonometrics freakonometrics freakonometrics.hypotheses.org 13 20 30 40 50 60 70 80 90 0.000.050.100.150.20 Age of the Driver ClaimsAnnualFrequency
  14. 14. 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 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 C., Denuit & Élie (2015) @freakonometrics freakonometrics freakonometrics.hypotheses.org 14
  15. 15. 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 82.3 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 15
  16. 16. 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% @freakonometrics freakonometrics freakonometrics.hypotheses.org 16
  17. 17. 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 @freakonometrics freakonometrics freakonometrics.hypotheses.org 17
  18. 18. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 Model Comparison and Lorenz curves Source: Progressive Insurance @freakonometrics freakonometrics freakonometrics.hypotheses.org 18
  19. 19. 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 n and Li = i j=1 yj n j=1 yj We have observed losses yi and premiums π(xi). Con- sider an ordered sample by the model, see Frees, Meyers & Cummins (2014), π(x1) ≥ π(x2) ≥ · · · ≥ π(xn), then plot {Fi, Li} with Fi = i n and Li = i j=1 yj n j=1 yj 0 20 40 60 80 100 020406080100 Proportion (%) Income(%) poorest ← → richest 0 20 40 60 80 100 020406080100 Proportion (%) Losses(%) more risky ← → less risky @freakonometrics freakonometrics freakonometrics.hypotheses.org 19
  20. 20. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 Model Comparison for Life Insurance Models 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] click to visualize the construction @freakonometrics freakonometrics freakonometrics.hypotheses.org 20 Population sorted by predicted risk (%) CumulativeLosses(%) Most Risky Less Risky 'Perfect' Pricing 2 classes Pricing Average Pricing qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 0 20 40 60 80 100 020406080100 100 90 80 70 60 50 40 30 Age
  21. 21. 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) 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 (see C. & 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)) @freakonometrics freakonometrics freakonometrics.hypotheses.org 21
  22. 22. 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 = π1(xi), · · · , πd(xi) ∈ Rd +. @freakonometrics freakonometrics freakonometrics.hypotheses.org 22
  23. 23. 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  24. 24. 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 Ratio99%vs1%quantile 020406080100 @freakonometrics freakonometrics freakonometrics.hypotheses.org 24
  25. 25. Arthur Charpentier, Insurance: Risk Pooling and Price Segmentation - 2017 Insurance Ratemaking Before Competition @freakonometrics freakonometrics freakonometrics.hypotheses.org 25 qq 0 20 40 60 80 100 020406080100 Insured (%) CumulatedPremium(%) ← lowest premium highest premium → Insurer 1 qq 0 20 40 60 80 100 020406080100 Insured (%) CumulatedClaims(%) ← highest premium lowest premium → Insurer 1

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