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  1. 1. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Actuarial Pricing Game A. Charpentier (UQAM & Université de Rennes 1) with R. Élie & J. Jakubowicz Paris, 100% Actuaires, November 2015. http://freakonometrics.hypotheses.org @freakonometrics 1
  2. 2. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Pricing Game, Charpentier, Denuit & Élie (2015) @freakonometrics 2
  3. 3. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Pricing Game, Charpentier, Denuit & Élie (2015) @freakonometrics 3
  4. 4. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Pricing Game, with a Toy Dataset @freakonometrics 4
  5. 5. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium q qq qqqqq q q q qqq q qq q q qq q q q q qq q q q q q q qq q q q q q qqq qq q q q q qqq qq q qq q q q q q q q q q q q q q q q qq qq q q q qq qqq q q q qq qq q q q qq q q q q q qqqq qqqq q qq q q qq qqq qq qq q q q q qqq q q q qq q q q qq qq q q qq q qqq qq q q q q q q q q qq q qq q q q q qqq q q qq qqq qq qqq q qq q qq q q q qqqqq q q q q q q q q qq qq q q q q q qq q qqq q q q q q qq q q q q q q q qqq q q q q qq q q q q q q 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 qq qq q q q qqq q q q q q q q q q q q q qq qq q q q q q q q q qq q qq 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 q q q q q q q qq q q q q q q q q q qq q q q q q qq 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 q q q q qq q q q q q q qq q q q q qqq 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 qq q q q qq q q q qq q q q q q q qq qqq qqq q q q qqq 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 qq q q q q q q q q q q q q q q q q q q q qqqq 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 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 qqq qq q q qq qq q q q q q q q q q q q qq q q q q q q qq qq 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 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 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 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 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 qq 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 q q q q q q q q q q q q q q qq q q qq qq 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 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 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 qqqq 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 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 qq q q q q q q q q q q q qq q q q q q q q q q qq 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 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 qq q qq 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 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 qq 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 qq 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 qq q q q q q q q q q q qq q q q qq q q q q q q qq q qq qq qq 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 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 qqqq q q q q qqq qq q qq q qq qqqq q q q q q q q q qqq q q q q q q q q q q q q q q q qqq q qq q q q q qq q q qq q q qqq qqq q q q q q q q q qqq qqq qq q q qq qq qq q q q q q qqq qq q qq q q q q qqq q q q q q qq qq q q q q q q q q qq qq qq q q q q q q qq q q q q qq q q qq qqq q q q q qq q q q q qq q qq q q q qq qqqq q q q q q q q q qqq qq q q q q q qq qqqq q q q q q qq q q q qq q q qqqq qqq q q q q q q q q q q q q qq q q qq q q q q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Gender = 'Male' @freakonometrics 5
  6. 6. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium q q q qq 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 qq 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 qqq q q q qq q q q q q q q q q q q q q qq q q q q q q qq 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 qq 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 qq q q q q q qq qq q q q qqq 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 qq 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 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 qq qq 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 q q q q q q q q q qq q q q q q q q q q q qqq 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 q q q q q qqq 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 qq q q q qq 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 qq q q q q q q q q q q q q qq q q q q q q qq q q q qq q q q q q q q qq q q qq q q q qqq 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 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 q q q qq q q q q q q q q q q q q q qq 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 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 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 q q 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 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 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 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 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 qq q q q q q q q qq q q qq 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 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 qq q q q q q q q qqq q q q q q q q qq 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 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 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 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 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 qq q q q q q q q q q q qq q q q q qq q q q q q qq qq 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 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 q q q q qqq 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 q q q q q q q q q q q q q q q q q q q q q q q 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 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 qq 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 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 qq q q q q q q q q q q qq q q q q q q q qq q qq q q q q q q qq 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 qq qq q q q q qq q q q q q q q qqq 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 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 qq 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 qq qq 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 qq 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 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 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 qq q qq 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 qq q q q q q q q q qqq q 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 qq q q q q q q qq q q q q q q q q q q qq q qq q q q q q q qq q q q q q q qq q q q q q qqq q q q q q qqq qq 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 q q q q q q q q q q q q qq 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 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 qq 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 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 qq 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqq qq qq q q q qq q q qq q q q q q qqq q q q qq q qq qqq q qq q q q q q q q qq q q q q q q q q q q q qq q qqqq q q q q q qq q qq qqqq qqqqq qq q q qqqqqq qq q qq q qqq q q q q qq q q q q q q q qq q q q q q q q qqq q q qq qq q qq q qq q q qqq 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 qq q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q qq q q qq q q q q qq q q qq q q qq q q q q qqqq qq q q q qqq q q q q qq qq qq q q q q q q q q q q q qqq q qqq q q qqqq qqqq q q q q q q qq q q q q q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q qqq q qqq qq qqqqqq q q qqq qq q qqqqqq qq qqqqqq qqq qqq q q qqq q q qq qqq q qqqq qq qq qq q qqqq qqqq q qq q q q qqqqq qqqq qqq q q q q qq qqqqq qq qq qqqqq q q qq q qq q q q qq q q q q qqq qqq qq q qqqq q qqq q q qq qq q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Gender = 'Female' @freakonometrics 6
  7. 7. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium qqqqqqqqqqqqq A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Age in [17,25] @freakonometrics 7
  8. 8. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium 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 qq q q q q q q q q q q q qq q qq qq 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 qq 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 q q q q q q q q q q q q q q 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 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 qq 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 qq q q q q q q q q q q q q q q qq q q q q qq 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 qq 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 qq 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 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 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 qq 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 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 qq 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 qqq 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 qq q q q q q q q q q q q q q q q qq q qqqq q q q q q q q q q q q q q q 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 qq q qq q q qq 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 q qq qq q q q qq q q qq 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 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 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 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 qq q qq 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 qq q q q q q q q q q q q qq 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 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 qq 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 qq qq q q q q q q q q q q q q q q 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 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 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 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 qq 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 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 qq q q q q qqq 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 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 qq 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 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 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 qqq 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 qq q q q q q q q q q q q q q q q q q qq 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 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 qq q q qq q q qq q q q q q q q q q q q qq q q q q q q q q qq q q q qq q qq qq q q q q 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 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 qq q 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 qq q q q q q q q qq q q q q q q q q qqq 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 qq q q q q q q q qq qq q q qq qq q qq qqq q q q qq 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 qq qq q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qqqq q q q q q q q qqq q q q qq qq q q q q qq q q q q q q q q q qq q q qq q qq qq q q q q q q q q q q q q q q qqq 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 qq q q qq q q q q q qq q q q q qqqq q q qq q q q q qqq 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 qq qqq q q q q q q qq q qq q q q q q qq q qqq q qq q qq q q q q q q qq 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 q q q q qq q q q q q qq q q q q q q q q q qq q q q qq q q qqq 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 qq q q qq q q q q q q q q q q qq 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 q q q q q q q q q q q q q qq q q q q q qqq q 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q qq qq q q q q q q q q q q q q q q qq q qq q q q q q q qq 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 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 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 q q q q qq 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 qq qq 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 qq qqqqq q q 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 qq qq 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 qq q q q q q q qq q q qq q q q qq q q q q q q q q q qq qq 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 qq q q q q q q q q q q q q q qq q q q qq 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 qq q q q q q q q q q q q q qq q q q qqq 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 qqq q q q q q q q q q q q q q q q qq 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 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 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 qq q q q q q q q q qq q q q qq q q q q q q q qq q q q q qq q q q q qq q q q q q qq q q q q q q qq 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 q q q q q q q q q q q q q q q q q q 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 qq 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 q q q q q q q q q q q q q q q q q q q q qqq q q qqq 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 qqq 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 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 qq 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 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 qq q q q q q q q q q q q q q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Age in [30,50] @freakonometrics 8
  9. 9. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium q q q q q q q q q q q q q q q q q q q q qqq q q qq q qqq q q q q q q q qq q q q q q q q q q q q q qq 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 qq q q qqq q q qq q qqq q q q q q q q qq q q q q q q q q q q q q qq q q qq q q q q q q q q qq q qq q q q qq 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 qq 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 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 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 qq q q q q qq 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 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 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 qq 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 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 qq 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 qq q q q q q qq q q q q q q q q q q qq q q q q q q q qq q q q qq q q qq qq q qqq qq q q qq qqq q q qqqqqq qq q q q qq qqq 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 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 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 qqq 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 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 q q q q q q q q q q q qq qq q qq q q q q q q q q q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Age in [70,100] @freakonometrics 9
  10. 10. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium q q qq q qq q q q qq q q q q q qq q q q q qq q q q qqq qq q q q qq q qq q q q qqq q qq q q q qq q qq q q q q qq q q 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 qq qq q qqqq q q q qq qq qqq 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 qq q q q q q q q qq q qq q qq q q q q qq q q q qq q qqq q q q qqq q qq qqq 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 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 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 qq q q q q q q q q q qq q q q q q q q q q q q qq qq q q q qq q qq q q q q q q q qq q q q q qq q q q q q qq q q q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Category = 'Small' @freakonometrics 10
  11. 11. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium q qq q q q q qq q q q q q q q qq q qq q qq q q q q qq q q qq qq 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 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 qq 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 qq q q q q q q qq q q q q q q q q q qq qq q q q qqqq 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 qq q q q q qq q q q q q qq q qq q q q q qq q q q q qq q q qq q qqq q q q q q q qq qq qq q q q q q q q q q q qqq qqq 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 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 qq q q qq q qq qq q q q q q q qq q q q q q q q qqqq 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 qq 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 qq q qq q q q qq 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 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 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 qq q q q q qq q q q q q q q q q qq qq 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 qq q q q q q q q qqq q q q q q q qq 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 qq q q q qq q q q q q q q q q q q qq qq qq q q q q q q q q q q qq qqq qqq q q q qq 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 qq 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 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 qqq 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 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 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 qq qq 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 qq q q q q q q q q q q q q qq q q q q q q qq 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 q q q q q q q q q qq q q qq 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 qq q q q q q q q q qqq 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 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 qq q q q q q q q qq q q qqq q qq q qq 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 qq q q q q q qq 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 qqq q qqq q q q qq q q q q q q qqqq q q qq qqq q qq qq q q q q qq qq q q q q qqqq qq q qq qq q qq q qqq q q q qq q q qqqqq q q qq q qq q qqq q qq qq q qqq q q qq q qqq q q q qqq q q q q q q q qqq q q qqqqq qq q qq q q q qq q q qqq 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 qq q q q q q q q q q qq qq q qq q q q qqq q q qqq q q q q q q qqq q q qq qq qq q q qq q q q qq qq qq 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 q q q qq q q q q qq q q q q qq qq qqq q q q qq q q qq qq q q qq 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 qq 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 qq qq q q q q q q qq q q q q q q q q q qq qq qqq q q q qqqqqq A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Category = 'Medium' @freakonometrics 11
  12. 12. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium qq q q q q q q q qq 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 qqqq 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 q q q q q q q q q 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 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 q q q q q q q q q q q qq q 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 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 qq q qqq 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 qqq q q q q q q q q q q q q q q q qq q qq q qq q q q q q q q q q qq qq qq q qqq 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 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 qq q q q q q q q qq 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 q q q q qqq 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 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 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 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 qqq 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 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 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 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 qq qq qq 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 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 qq 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 q q q q q q q q q q q q qqq qqq q q q qq q q q q q qqq qq q qq qq q q q q qqq q qqqqqq qq q q q q qqq q qq q qq q q q q qqq q q qqqqqq 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 q q q q q q q q q q q q q q 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 qq q q q qq q q q q q qq 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 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 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 qqq 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 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 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 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 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 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q q q q q q q qqq q qq qq q qqqqqq q qqq q q q qqqqqq q qq q qq q q q q qq qqq qq qq q q q q q q qqq q q q q qq qq q qq q q q qqq q qq q qq q qqqq qq qqqqq q q q qqqq qq qq qq qqq q q q q q qq q qqqq qq q q qqq q qqqqqq qq q q q q q qqq q qq q q qqq q qq q q qq q qq qq q q q q q q q qq q q q q q q q q qq q qq qq q q q q q q qq q qqqqq q q qq q q q qqqq qqq q qqqqq q q q q q q q qq q qq q q q qqqq q q qq q q q q qq qqq q qqq q q q q q q q q q q q q q qq qqq qq q qq 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 qq qq q q q q q q q q q q q q q q q q qq q q q qq q qq qq q q q q q q qqq q q q q q q q qq qq q q q q q q q q q q q q q q q q qqq 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 qq 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 q q q qq q q q q q q q q q q q q q q qq q q q qq q qqq q q q q q q q q q q qq qq q q q qq q q q qq q q q q qq q q qq 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 qq q q q q qq q q q q qq 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 qq q q q q qq q q q q q qq q q qq 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 qqq q q q q qq q q q q q q qqq q qq 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 qq qq q q q q q q q q q q q q q qq qq qq q qq q q q qq q q q qq q q qq q q q q q q q qq qq q q q q q q q q q qq q q q q q qq qq q q qq q q q q qq q q q q q q q q q q qq 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 q q q q q q q q q q q q q q q qq q q q qq 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 qq 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 q q q q q q q q q q qqq q q qq q q q q q q q qq q q q q q q qq q q qqq 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 qq q q q q q q qq q q qq 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 qq q q q q qq q q q q qqq q qq qq q q q q qq q q q q q qq qq qq q q q q q q qq q q qq qqq q q q qq q q q q q q q qqq 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 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 qq 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 qq q q q q q q qq 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 qq q q q qq q q q q q q q qq qq q q qq q qq q q q q q qq 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 qq q q q q q q qqq q q q q q qq 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 qq q q q q q q q q q qqqq 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 qq 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 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 qq 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 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Category = 'Large' @freakonometrics 12
  13. 13. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Premiums, Brief Summary A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Premium 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 qq qq q q q q q q q q q q q q q q q q q qq qq q qq q q qq q q q qq qq qq q q q q q q q q q q q q q q q q qq qq q q q q qq q q q q q q q q qq q q q q q qq 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 qq q q q q q q q q q qq qq qqq q qq q q qq q q q qq q q q q q qqq q q q q q q q qq q q q q qqq q q q q q qq 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 qq q q q q q qq 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 qq qq q q q q q q q q q q q q q q q q q qq qq q qq q q qq q q q qq qq qq q q q q q q q q q q q q q qq q q qq qq q q q q qq q q q q q q q q qq q q q q q qq 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 qq q q q q q qq q q q q q q q q q qq qq qqq q qq q q qqq q q qq q q q q qqqq q q q q q q q qq qq q q qqq q q q q q qq 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 qq q q q q q q q q q q q q qq 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 qq 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 qqq 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 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 qq q 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 qq 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 q q q q q q q q q q q q q q 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 qqq 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 q q q q qq q q q qq q q q qq q q q q q q q q qq qq q qqq qq q q q q q q q q q qq q q q q q q q qqq 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 qq q qq q q q qq 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 qq q qq q q qq q q q q qq q q qq q q q q q q qq q qq qq 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 qq 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 q q q q q q qq q q q q q qq q qq q q q q q q q qq q q q q q q qq q q qqq q q qqq q q q q q q q qq q q q q q q qq q q q q q q qq q q qq qq 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 qq q q q q q q q qq 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 q q q q q q qq q q q q q q q q qqq 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 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 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 qq q 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 qq q q q qq qqq 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 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 qq q q q q qq q qq q q q q q qqq q q q q q q qq q q q q q q q q q q q qq qq q q q q qq 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 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 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 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 qqq qq q q q q qq 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 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 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 qq 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 qq q q q qq qq q q q q q q q q qq 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 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 qqq q q q q q q q q q q q q q qq q qq q qq 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 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 qq q q q qq 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 qq qq q q qq q q qq q q q qqq q qq 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 qq q qq 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 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 qq q q q q q q q q q q q q q qq q q qq q q q q qq q q q q q q q q q q qq q q q q q qq 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 q q q q q q q q q q 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 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 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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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 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 q q q q q q q q q q q q q q q q q 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 qq q q q q q q qq q q q q q q q q q q q q q qqqq qqq qqqq q qqq q qqqqqqqqqqqqq q qqqqqqqq qqq qqqqqqqqqqqqqqqqqqqq q qqq q qqqqqq q qq q qq q q q q q q qqq q qqqqqqq q q qqq q q qqq q qq q qqqqqqqqqq q q qqqqq q qqqqqqqqqq q qqqqqq q qqq q qqqqqqqqq q qqqqqqqqqqqqqqqqq q q q qq q qqqqqq q qqqqqqqqq q qq qq qqqqqqqq q q qq q qqqqq q qqqqqqqqqq qq qqqqqq q qq q qqqqqqq q qqqq q qqqqqqqqq qqq qqq q qqqqq q qqq q q q qqqqqqqqqqqqqqq q qq q qqq q qqq q qq q qqq q qqqqq q qqq q qqqqqqqqqq q qqqqqqqqqqqqqqq q qqq q qqqqqqqqqqqq q q q q q q q qq q q qq qq qq qqqqqq q q q qqqqqqqqqqq q q qqqqq qq qqqqqq q qqqqq q qqqqqqq q qqqqqqq q qqqqqqqqqqqqqqqqqqqq q qqqq qq q qqqqq q q q qqq qq qqqqqqqqqqqq q qqqqqqqqq q qqq q qq q qq q qqqqqqqq q qq qq q qqqqqqqqqqqqq q qqqqqqq q qqqqq q qqqqqqq q qqqqq q q q qqqqqqqqqqq q qq q q q q qqqq q qqqqqq q q q qqqq q qqqqqqqqqq q qqq qqq qqqqqqqqq q qq q qqq q q q q qq qqqqqq q qqqqq q q q qqq q qq q qq q qqqqq q qq q q qq qqqq q qqq q qqqq q qqqq q qqq q q q qqq q q q qq q qq q qqq q qqqq q qqqqqqqqqqqqq q qqqqqqqqqq q qq q qqqqqqq q qqqqq q qqqqqqq q qqqqq q q qq qqqqqq q q q qqqq q qqqqqqqqq q qqq q qqqqqqqqq q q q q qqqqqqqqqq q qqqqqq q qq q qqqqqqqqqqqqqqq q qqqqq q q q qq q qqqq q qqqqqqqqqqqq q qqqqqqqqqq q qqqqqqqq q qqqq q q qq qqqq q q q qqqqqqqqq q qqqqqqq q qqqqqqqqq qq qq q q q q qqqq q q qqq q q qq qq qqqqqq qq q q q q qqqq q qqqqq q qqqqqqqqqqq q qqqq q q q qqqqqqqqqq q q q qqqqqqqq q qqqqqq qq qqqqq q qqq q qqqqqqqqqqqqqqqq q q q q q q q qqq qq qqq q q q qqqqqqqqqqqqqqqqq q q q qqqqq q q q qqqqqqqq q qqqqq q qqqqqqqqqq qq qqqqqq q qq q qqqqq q qqqq q qqqqqq q qq q qqqqqqqqqqqq qq q q qqqqqq q qqqqqqqqqqq q q q qqqqqqqq q qqqq q qqqqq q q q q qqqqqqqqq qq qqqqqqq q qqq q qqqqq q q q qq q qqqqqqq q q q q q qqq qqq qqqq q qq q q q qqqqq q q q qqqqqq q qqqqqqqqq q qqqq q qqqqqqqq q qqqqqqqqqqqqqq q qqq q qqqq qqq q q q qqqqqqqqq q qqqqqqqqqqqqqqqq q qqq q qqq qq q q qqqqqqqqqq q q q q q q q q q qq q qqq q qqqqqqqqqqq q qqqqqq qq qqqqqqqqqqqq q qqq q q qq q qqq q qqqq q qq q qqqqqqqq q qq q q q qqqqqq q qqqqqqqqqqqqqqqq q qqqqqqqq q qq q q q qqqqq q qqq q q qqqqqq q q q q q qqqqq q qqq q qqqqqqqqqq q qqq q qqqqqqq q qqqqq q qqqqqqqqqqqqq q qq q qqqqqqqqqqq q qqqqqq q qqq qq qqqqqqq qq qq q qqqqqq q qqq qq qqqqqqqq q qqqqqqqqq q qqq q qqqqqqq q qqqqq q q q qqqqqq q qqqqqqqqqqqq q qqqqqqq q qq q qqqqqqqqq q qqqqqqqqqq q q q qq q qqqqqqq q qq q qq qq qqqqqq q qqqqqq q qqq q qqqq q qqq q qq q qq q qqq q qqqqqq q qqqqq q q q qq qq qqq qq qqq q qqqqqqqq q qqq q qqq qqq q q qq q q q q q q q qqq q qqqqqqqqqq q qqqqq qq qqqqqqqqqqqqq q qqqqqqq q qqqqq q qq q qqqqqqqq q qqq qq qqqqqq q qqq q qqqqqqqqqqqqqqqq q qqq q qqqqqqq qq qqqqqqq 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 qq q qq 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 qq q q q q q q q q q q q q q q q q q q q qq 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 qq q qqq q q q q q q q qq q q q q q q q q qqq 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 qq q q q q q q qq q qq qq 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 qq q q q q q q q q q q q q q q qqq qq 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 qq 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 q q q q q q q q q q q q q q q q q q q qq q q q qq qqq q q q q q qq qq 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 qq 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 qq 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 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 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 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 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 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 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 qqqq q q q q q q q q q q q 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 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 qq q q q q qqqq q q qq q qqqq q q q 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 q q q qq q q q q q q q qq q q q q qq qq q q q qqq 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 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0100200300400500600700 Bonus = −50 @freakonometrics 13
  14. 14. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Comparing Models Source: unkown, so far... @freakonometrics 14
  15. 15. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Comparing Models 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 15
  16. 16. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 The Models, Brief Summary @freakonometrics 16
  17. 17. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition Decision Rule: the insured selects the cheapeast premium, cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc A B C D E F 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 17
  18. 18. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition Decision Rule: the insured selects randomly from the three cheapeast premium cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc cccccccccccc ccccccccccccc A B C D E F 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 18
  19. 19. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition Decision Rule: the insured were assigned randomly to some insurance company for year n − 1. For year n, they stay with their company if the premium is one of the three cheapeast premium, if not, random choice among the four A B C D E F 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 19
  20. 20. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Financial Results, Market Share, Method 2 q q qq q q q q q q q q q q qq A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 100020003000400050006000 NumberofContracts @freakonometrics 20
  21. 21. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Financial Results, Market Share, Method 3 q qq q q q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 2000300040005000 NumberofContracts @freakonometrics 21
  22. 22. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Financial Results, Loss Ratio, Method 2 Market Loss Ratio ∼ 154%. q q q q q q q qq q q q q q A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 100150200250 LossRatio @freakonometrics 22
  23. 23. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies q −2 −1 0 1 2 3 −1.0−0.50.00.51.01.5 MCA factor map Dim 1 (11.02%) Dim2(7.25%) Female Male A B C DE F Large Medium Small Employed Housewife Retired Self−employed Unemployed (17,25] (25,35] (35,50] (50,70] (70,101] A1 A10 A11 A12 A13 A2 A3 A4 A5 A6A7 A8 A9 @freakonometrics 23
  24. 24. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −5.5 −4.0 −2.0 0.0 2.0 4.0 5.4 Pearson residuals: p−value = < 2.22e−16 A Gender MaleFemale A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 24
  25. 25. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −11 −4 −2 0 2 4 13 Pearson residuals: p−value = < 2.22e−16 A Category SmallMediumLarge A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 25
  26. 26. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −19 −4 −2 0 2 4 15 Pearson residuals: p−value = < 2.22e−16 A Occupation UnemployedSelf−employedRetiredHousewifeEmployed A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 26
  27. 27. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −20 −4 0 4 48 Pearson residuals: p−value = < 2.22e−16 A AgeF (70,101](50,70](35,50](25,35](17,25] A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 27
  28. 28. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −26 −4 −2 0 2 4 31 Pearson residuals: p−value = < 2.22e−16 A DensityF (200,300](125,200](70,125](40,70](10,40] A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 28
  29. 29. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −25 −4 −2 0 2 4 29 Pearson residuals: p−value = < 2.22e−16 A BonusF (100,200](30,100](0,30](−10,0](−30,−10](−49,−30][−50,−49] A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 29
  30. 30. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Insurers and Insurance Companies −7.5 −4.0 −2.0 0.0 2.0 4.0 8.7 Pearson residuals: p−value = < 2.22e−16 A PoldurF (10,16](5,10](1,5][0,1] A1 A10 A11 A12 A13 A2 A3 A4 A5 A6 A7 A8 A9 @freakonometrics 30
  31. 31. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition, A2 No segmentation, unique price 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 31
  32. 32. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition, A1 Three models (GLM) frequency bodily injury - fre- quency material damage - loss material Age in 4 categories, 30−, 30 − 45, 45 − 60 and 60+, an interaction with occupation. Manual Smoothing of parameters Done with SAS and Excel. Sent by an actuary working for a mutuelle 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 32
  33. 33. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition, A8-A9 GLM for frequency and standard cost (large claimes were removed, above 15k) Interaction Age and Gender All variables but density and group2. Done with a software developped by Actuaris. Sent by an actuary working for a mutuelle 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 33
  34. 34. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition, A11 Use of all variables, but Subgroup2, Use of two XGBoost models (bodily injury and material), Correction for negative premiums Use of Python Sent by an actuary working for a private insurance com- pany (sudent at the Data Science for Actuaries pro- gram) 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 34
  35. 35. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Market Competition, A12 Use of two xgboost models (bodily injury and material), Correction for negative premiums Use of R Sent by an actuary working for a private insurance com- pany, in Europe, but not in France. 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 35
  36. 36. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Distorting the Market: the Price Aggregator as a Market Player Partnership between A4 and the price aggregator: if A4 is either the 4th or 5th , it returns the 3rd price (−ε) A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 Partdemarché(%) 0510152025 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 Partdemarché(%) 0510152025 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 0510152025 Happens for 33% of the prices. Market share 8.8% up to 22.5% @freakonometrics 36
  37. 37. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Distorting the Market: the Price Aggregator as a Market Player Increase of loss ratio of A4 from 128% to 143% A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 Partdemarché(%) 0.00.51.01.52.02.5 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A13 A14 Partdemarché(%) 0.00.51.01.52.02.5 Much higher volatility on other companies... @freakonometrics 37
  38. 38. Arthur CHARPENTIER, Romuald Élie & Jérémie Jakubowicz, 2015 Take-Home Message • different but consistent models, from different actuaries, different backgrounds, different softwares, different languages, different models • hard to predict model’s behavior in a competitive market from standard tools (lift curves) • the choice of the insurance company has no big impact on market results, but has a big impact on how to use information on competitors @freakonometrics 38

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