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1. 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. 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. 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 claims’life dynamics, from the occurrence until closure, in life insurance in nonlife insurance 3
4. 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. 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. 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. 7. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lexis diagram in insurance and ﬁnally, 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. 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. 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. 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,j−1 , • 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,j−1 , • Ni,j , total number of claims occurred year i, seen after j years, 10
11. 11. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Hachemeister (1975), Kremer (1985) and ﬁnally 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(µi,j ) where µi,j = exp[αi + βj ]. It is then extremely simple to calibrate the model. 11
12. 12. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Assume that Yi,j ∼ P(µi,j ) where µi,j = exp[αi + βj ]. the occurrence factor αi the development factor βj 12
13. 13. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Assume that Yi,j ∼ P(µi,j ) where µi,j = exp[αi + βj ]. It is then extremely simple to calibrate the model, Yi,j = exp[αi + βj ] Yi,j = exp[αi + βj ] on past observations on the future 13
14. 14. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Assume that Yi,j ∼ P(µi,j ) where µi,j = exp[αi + βj ]. q q q q q q q q q q q q q q q q q q 2 4 6 8 10 2 4 6 8 10 14
15. 15. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance The log-Poisson regression model Consider here another triangle with monthly health claims, q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 20 25 30 35 2 4 6 8 10 12 Month of occurence Delay 15
16. 16. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Additional remarks Since we consider a Poisson model, then Var(Y |α, β) = E(Y |α, β). Note that overdispersed Poisson distributions are usually considered, i.e. Var(Y |α, β) = φ · E(Y |α, β), with φ > 1. Further, the idea of using only two factors can be found in de Vylder (1978), i.e. Yi,j = ri · cj . But other factor based models have been considered e.g. Taylor (1977), Yi,j = di+j · cj where di+j denotes a calendar factor, interpreted as an inﬂation eﬀect. 16
17. 17. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Quantifying uncertainty in a stochastic model The goal in claims reserving is to quantify E [R − R]2 Fi+j where R= Yi,j is the amount of reserves. i,j,i+j>t Classically, bootstrap techniques are considered, i.e. generate pseudo-triangles, Yi,j = Yi,j + Yi,j · εi ,j then ﬁt a log-Poisson model Yi,j ∼ P(µi,j ) where µi,j = exp[αi + βj ]. Then generate P(µi,j ) for future payments, i.e. i + j > t. 17
18. 18. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Bootstrap and GLM log-Poisson in triangles 6 q 5 4 q q 3 q 2 q 1 q q 0 0 2 4 6 8 10 18
19. 19. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Bootstrap and GLM log-Poisson in triangles 8 6 q 4 q q 2 q q q 0 0 2 4 6 8 10 19
20. 20. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Bootstrap and GLM log-Poisson in triangles 8 q 6 4 q 2 q q q 0 0 2 4 6 8 10 20
21. 21. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Bootstrap and GLM log-Poisson in triangles 8 6 q 4 2 q q q 0 0 2 4 6 8 10 21
22. 22. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Bootstrap and GLM log-Poisson in triangles 10 15 20 25 30 Total amount of reserves ('000 000\$) (log-Poisson + bootstrap versus lognormal distribution + Mack) 22
23. 23. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Lee & Carter’s approach of mortality Dynamic models for mortality became popular following the publication of Lee & Carter (1992)’s models. The idea is that if # deaths during calendar year t aged x last birthday m(j, t) = average population during calendar year t aged j last birthday log m(j, t) = αj + βj γt (1) (2) (2) – Lee & Carter (1992), log m(x, t) = βx + βx κt , (1) (2) (2) (3) (3) – Renshaw & Haberman (2006), log m(x, t) = βx + βx κt + βx γt−x , (1) (2) (3) – Currie (2006), log m(x, t) = βx + κt + γt−x , – Cairns, Blake & Dowd (2006), (1) (2) logitq(x, t) = logit(1 − e−m(x,t) ) = κt + (x − α)κt , – Cairns et al. (2007), (1) (2) (3) logitq(x, t) = logit(1 − e−m(x,t) ) = κt + (x − α)κt + γt−x . 23
24. 24. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance A stochastic model for mortality Assume here that E(D|α, β, γ) = Var(D|α, β, γ), thus a Poisson model can be considered. Then Dj,t ∼ P(Ej,t · µj,t ) where µj,t = exp[αj + βj γt ] Brillinger (1986) and Brouhns, Denuit and Vermunt (2002) 24
25. 25. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance A stochastic model for mortality Assume here that E(D|α, β, γ) = Var(D|α, β, γ), thus a Poisson model can be considered. Then Dj,t ∼ P(Ej,t · µj,t ) where µj,t = exp[αj + βj γt ] the age factors (αj , βj ) the time factor t 25
26. 26. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance A stochastic model for mortality Two sets of parameters depend on the age, α = (α0 , α1 , · · · , α110 ) and β = (β0 , β1 , · · · , β110 ). qq qq q q qq q qq q q qq 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 q qq q q q q q q q q qq q q q q q q q qq q q qq q qq q q q q qq q q qq q q qq q qq q q q q qq q qq q q q qq qq q q qq qq q qq q q q qqqq qqq qq qq q qq q q q qq qqq q qqq qqqq q q q qqqq qqq q q q q qq q q qq qq q q qq q q qq q q qq qq qq qq q 0 20 40 60 80 0 20 40 60 80 26
27. 27. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance A stochastic model for mortality and one set of parameters depends on the time, γ = (γ1899 , γ1900 , · · · , γ2005 ). q q q q q qq q qq q q qqqqqq q q q qq qq q qqqqq q qq q qqqq qq qqqqq q q q q q qq qq qqq q qqq qq q qqqqqq qqqq qqq qq qqq qq qq q qq qq q qq qqq qq q qq qq qq 1900 1920 1940 1960 1980 2000 27
28. 28. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Errors and predictions exp[αj + βj γt ] exp[αj + βj γt ] ˜ on past observations on the future 28
29. 29. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Forecasting γ Based on γ = (γ1899 , · · · , γ2005 ), we need to forecast γ = (γ2006 , · · · , γ2050 ). qqq q qqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q q q q q qq qq q q qqqqqqqqqq qqqqqqq qq q q q q q qq qqqqq q qqqqqq qqqqqqqq qqqqq qqqqq qqqq qq qqqq qqqq qqq qqq qq qqq 1900 1950 2000 2050 29
30. 30. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Forecasting γ Classically integrated ARIMA processes are considered, qqq q qqqqqqqqqqqqqqqq q qqqqqqqqqqqqqq q q q q q qq qq q q qqqqqqqqqq qqqqqqq qq q q q q q qq qqqqq q qqqqqq qqqqqqqq qqqqq qqqqq qqqq qq qqqq qqqq qqq qqq qq qqq 1900 1950 2000 2050 30
31. 31. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Understanding errors in stochastic models Dj,t − Dj,t Pearson’s residuals, εj,t = , as a function of age j Dj,t q q q q q q q q q q q qqq q q q q q q q q q q q q qqqqq qq qqq qqq q qqq q qqq qqq q 1990 q q q q q qqq qq q q qqq q q qq qqqqqqqq qqq q qqq qqqq q q q q q qqqq qq q qqqqq q q qq qqqqqqqqqqqq qqqqqqqqqqqqqq qqqqqqq q qqqqqqqqq q qqqqq q q q q 1980 qqqqqq qq qqq qq q qqqq qqqqqqqqqqqq q q qqqqqqq qqqqqqqqqqqqqqqqqq q qqqqqqq q q qqqqqqqqq qq q qqqqqq q qqqqqqqqqq q qq q qq qqqqqqqqqqqqqqqqqqqq q q qq qqqqqqqqqqqqqqqqqqqqqqq q qqq qq qqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqq q q qq qqq qqqqqq q q qq q qqqqqqqqqqqqqqq q q qqq qqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqq q q qq qqq qqqqqqqqqqq q q q qq q qq qqqqqqqq qqq q qqqq q qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqq qq q qqq q qqq q qq q q q qq q 1970 qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qq q qqq qqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq q q q qq q qqqqqq qq q q q qq q qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q q q qq qq qqq qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qq q qqqqqq q q qq q qqqq q qqq qqqqqqqq qqqqq qqqqqqqqqqq qqq qqqq qq qqq qqqq qqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qq q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq q qqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qq qq q q qqqqq qq qqqqqq q qqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqq qq q q q qqqq q qq q q qqqq qqqq qq qqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqq qq q q q q q q qqqqqq q q qqqq q qqq qqqqqqqqqqqqqqqqqq qq q qq q qqq qq qqqq q qq qqqqq qqqq qqq qqqq q qqqqqqqqq q qqq q qq qqqqqqq qq qqqqqqqqqqqqqqq qqqqqqqqqqqqq qq q qqq qqqqqqqqqqq q qq qqqqqqqq qqq q q q qq q qqqqqqqqqqqqqqqq qqqqqqqqq q q qqqqq qqqq q q qqqqqqqqqqqqqqqqqq 1960 q q q qqqq qqqqqqqqqq q q qqqqq qqq q q q qqq qq qqq qqqqq q qq qqq q q q qq 1950 q q q 1940 q q q q 1930 q 1920 1910 q q 0 20 40 60 80 100 31
32. 32. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Understanding errors in stochastic models Dj,t − Dj,t Pearson’s residuals, εj,t = , as function of time t Dj,t q q q q q q q q q q q q q qq q q q q q q q q q qq qqq qqq qqq q q q q q q q q qq q 90 q qq q q q q q q q qq q q q q q q q qq q q q qqqq q qq q q qq qqqqqqqq q qqqq q q q q q q qq qqqqqq q q qq q q q q q q q q q q qqqqq qq qq qq q qq qq q 80 q q q qq q qq q q q q q q qqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqq q q q qq qqqqqqqqqqqqqqqqqq q qqqqqqq q q q q q q qqq qqqqq qq qqqqqqqqqqq qqqqqqqqqqqq qq qqqq qqqqqqqqqqqqqqqq q q q qq q q q qq q q q qqqqqqq q q q q q q q q qqq qq q q qq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqq qq qq qq qq qq q q q q qq qqqqqqqqqq qqqqq qqqqqqq q qqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qqqqqqqqqq qq qq qq q qq q q q q q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqq q qqq q q q qqq qqq q qq qqqqqqqqqqqqqq qq q q qq q q q qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqq qqqqqq qqq q q qqqqq qqqqqqqq qqqqq q qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qq q qq q qqq q q q q q q q q q q q qqqqqqqqqq q qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqqqq qqqqq q qq qq qqqqqqq qqqqqqqqqq qqq q qqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q q q qq q 70 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqqqqq q qqqqqq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qq q q q q q q q q q q q qqq qqqqqqq q q q q q qqqqqqqq qqqqqqq qqqqqqqqqqqqqqqqqqqqq qqq q qqqqqqqqqq qqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q qq qqq qqqqq q qqqq q qq q qqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qqq qq q qq qq q qqqqq q qq q qqqqqqqqqq q q qq qqqqqqq qqqq qq q qq qqqqqq q q q q q qqqqqqqq qqqqqqqq qq q q qqqqq qqqqqqqqqqqqqqqqqq qqqqqq q qqqq q q qqq qq q qqq qqqq q q qq q qqqq q qqqq qqqqqqqqqq q q qqq q q qqqqqqqqqqqqqqqqqq q qq qqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqq q qqqqqq qqqq qqqqqqqqqq q q q qqq q qqq q q q q q q q qqq q q q q q q qq q q qqq qqqq qqq q qq q 60 qq qqq qqq qq q qqq q q q q qqq q qq qq qqq q qq q q q q q qq qqqqq qqq q q q q q q q q q 50 qq q q 40 qq q q 30 q 20 10 q q 1900 1920 1940 1960 1980 2000 32
33. 33. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Understanding outliers Outliers can simply be understood in a univariate context. To extand it in higher dimension, Tukey (1975) deﬁned the n 1 depth(y) = min 1(X i ∈ Hy,u ) u,u=0 n i=1 where Hy,u = {x ∈ Rd such that u x ≤ u y} and for α > 0.5, deﬁned the depth set as Dα = {y ∈ R ∈ Rd such that depth(y) ≥ 1 − α}. The empirical version is called the bagplot function (see e.g. Rousseeuw & Ruts (1999)). 33
34. 34. Arthur CHARPENTIER, modeling analogies in life and nonlife insurance Understanding outliers q q q q q q 1.0 1.0 q q q q 0.5 0.5 q q q q q q 0.0 0.0 q q q q −0.5 −0.5 q q q q q q q q q q q q −1.0 −1.0 q q q q −1.5 −1.5 q q q q −2 −1 0 1 −2 −1 0 1 where the blue set is the empirical estimation for Dα , α = 0.5. 34
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