Top Rated Pune Call Girls Viman Nagar ⟟ 6297143586 ⟟ Call Me For Genuine Sex...
Slides scor-reserves
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 claims’life 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 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. 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,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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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. 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. 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
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. 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
