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  • 1. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance modeling analogies in nonlife and life insurance Arthur Charpentier1 1 Universit Rennes 1 - CREM & Ecole Polytechnique e arthur.charpentier@univ-rennes1.fr http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/ Reserving seminar, SCOR, May 2010 1
  • 2. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Agenda Lexis diagam in life and nonlife insurance From Chain Ladder to the log Poisson model From Lee & Carter to the log Poisson model Generating scenarios and outliers detection 2
  • 3. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance Lexis diagrams have been designed to visualize dynamics of life among several individuals, but can be used also to follow claimslife dynamics, from the occurrence until closure, in life insurance in nonlife insurance 3
  • 4. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance but usually we do not work on continuous time individual observations (individuals or claims) : we summarized information per year occurrence until closure, in life insurance in nonlife insurance 4
  • 5. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance individual lives or claims can also be followed looking at diagonals, occurrence until closure,occurrence until closure,occurrence until closure,occurrence until closure, in life insurance in nonlife insurance 5
  • 6. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance and usually, in nonlife insurance, instead of looking at (calendar) time, we follow observations per year of birth, or year of occurrence occurrence until closure,occurrence until closure,occurrence until closure,occurrence until closure, in life insurance in nonlife insurance 6
  • 7. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance and nally, recall that in standard models in nonlife insurance, we look at the transposed triangle occurrence until closure,occurrence until closure,occurrence until closure,occurrence until closure, in life insurance in nonlife insurance 7
  • 8. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance note that whatever the way we look at triangles, there are still three dimensions, year of occurrence or birth, age or development and calendar time,calendar time calendar time in life insurance in nonlife insurance 8
  • 9. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance and in both cases, we want to answer a prediction question...calendar time calendar time calendar time calendar time calendar time calendar time calendar time calendar timer time calendar time in life insurance in nonlife insurance 9
  • 10. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance What can be modeled in those triangles ? In life insurance, Li,j , number of survivors born year i, still alive at age j Di,j , number of deaths of individuals born year i, at age j, Di,j = Li,j Li,j1 , Ei,j , exposure, i.e. i, still alive at age j (if we cannot work on cohorts, exposure is needed). In life insurance, Ci,j , total claims payments for claims occurred year i, seen after j years, Yi,j , incremental payments for claims occurred year i, Yi,j = Ci,j Ci,j1 , Ni,j , total number of claims occurred year i, seen after j years, 10
  • 11. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Hachemeister (1975), Kremer (1985) and nally Mack (1991) suggested a log-Poisson regression on incremental payments, with two factors, the year of occurence and the year of development Yi,j P(i,j ) where i,j = exp[i + j ]. It is then extremely simple to calibrate the model. 11
  • 12. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Assume that Yi,j P(i,j ) where i,j = exp[i + j ]. the occurrence factor i the development factor j 12
  • 13. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Assume that Yi,j P(i,j ) where i,j = exp[i + j ]. It is then extremely simple to calibrate the model, Yi,j = exp[i + j ] Yi,j = exp[i + j ] on past observations on the future 13
  • 14. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Assume that Yi,j P(i,j ) where i,j = exp[i + j ]. q q q q q q q q q q q q q q q