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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
modeling analogiesin nonlife and life insurance
Arthur Charpentier1
1 Universite Rennes 1 - CREM & Ecole Polytechnique
http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/
Reserving seminar, SCOR, May 2010
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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
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Lexis diagram in insurance
Lexis diagrams have been designed to visualize dynamics of life among severalindividuals, but can be used also to follow claims’life dynamics, from theoccurrence until closure,
in life insurance in nonlife insurance
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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 untilclosure,
in life insurance in nonlife insurance
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Lexis diagram in insurance
individual lives or claims can also be followed looking at diagonals, occurrenceuntil closure,occurrence until closure,occurrence until closure,occurrence untilclosure,
in life insurance in nonlife insurance
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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 followobservations per year of birth, or year of occurrence occurrence untilclosure,occurrence until closure,occurrence until closure,occurrence until closure,
in life insurance in nonlife insurance
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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 thetransposed triangle occurrence until closure,occurrence until closure,occurrenceuntil closure,occurrence until closure,
in life insurance in nonlife insurance
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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 timecalendar time
in life insurance in nonlife insurance
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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 timecalendar time calendar time calendar time calendar time calendar time calendartime calendar timer time calendar time
in life insurance in nonlife insurance
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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,
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
The log-Poisson regression model
Hachemeister (1975), Kremer (1985) and finally Mack (1991) suggested alog-Poisson regression on incremental payments, with two factors, the year ofoccurence 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.
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
The log-Poisson regression model
Assume thatYi,j ∼ P(µi,j) where µi,j = exp[αi + βj ].
the occurrence factor αi the development factor βj
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
The log-Poisson regression model
Assume thatYi,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
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
The log-Poisson regression model
Assume thatYi,j ∼ P(µi,j) where µi,j = exp[αi + βj ].
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
The log-Poisson regression model
Consider here another triangle with monthly health claims,
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Month of occurence
5 10 15 20 25 30 35
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Delay
2 4 6 8 10 12
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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 thatoverdispersed 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, interpretedas an inflation effect.
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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 =∑
i,j,i+j>t
Yi,j is the amount of reserves.
Classically, bootstrap techniques are considered, i.e. generate pseudo-triangles,
Y ?i,j = Yi,j +√Yi,j · εi?,j?
then fit a log-Poisson model Y ?i,j ∼ P(µ?i,j) where µ?i,j = exp[α?i + β?j ].
Then generate P(µ?i,j) for future payments, i.e. i+ j > t.
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Bootstrap and GLM log-Poisson in triangles
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Bootstrap and GLM log-Poisson in triangles
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Bootstrap and GLM log-Poisson in triangles
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Bootstrap and GLM log-Poisson in triangles
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Bootstrap and GLM log-Poisson in triangles
Total amount of reserves ('000 000$)
10 15 20 25 30
(log-Poisson + bootstrap versus lognormal distribution + Mack)
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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
m(j, t) =# deaths during calendar year t aged x last birthday
average population during calendar year t aged j last birthday
logm(j, t) = αj + βjγt
– Lee & Carter (1992), logm(x, t) = β(1)x + β
(2)x κ
(2)t ,
– Renshaw & Haberman (2006), logm(x, t) = β(1)x + β
(2)x κ
(2)t + β
(3)x γ
(3)t−x,
– Currie (2006), logm(x, t) = β(1)x + κ
(2)t + γ
(3)t−x,
– Cairns, Blake & Dowd (2006),logitq(x, t) = logit(1− e−m(x,t)) = κ
(1)t + (x− α)κ(2)
t ,– Cairns et al. (2007),
logitq(x, t) = logit(1− e−m(x,t)) = κ(1)t + (x− α)κ(2)
t + γ(3)t−x.
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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 beconsidered. Then
Dj,t ∼ P(Ej,t · µj,t) where µj,t = exp[αj + βjγt]
Brillinger (1986) and Brouhns, Denuit and Vermunt (2002)
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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 beconsidered. Then
Dj,t ∼ P(Ej,t · µj,t) where µj,t = exp[αj + βjγt]
the age factors (αj , βj) the time factor t
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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) andβ = (β0, β1, · · · , β110).
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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).
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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
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Forecasting γ
Based on γ = (γ1899, · · · , γ2005), we need to forecast γ = (γ2006, · · · , γ2050).
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Forecasting γ
Classically integrated ARIMA processes are considered,
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Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Understanding errors in stochastic models
Pearson’s residuals, εj,t =Dj,t − Dj,t√
Dj,t
, as a function of age j
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31
Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Understanding errors in stochastic models
Pearson’s residuals, εj,t =Dj,t − Dj,t√
Dj,t
, as function of time t
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32
Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Understanding outliers
Outliers can simply be understood in a univariate context. To extand it in higherdimension, Tukey (1975) defined the
depth(y) = minu,u6=0
{1n
n∑i=1
1(Xi ∈ Hy,u)
}
where Hy,u = {x ∈ Rd such that u′x ≤ u′y} and for α > 0.5, defined the depthset 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
Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Understanding outliers
−2 −1 0 1
−1.
5−
1.0
−0.
50.
00.
51.
0
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51.
0
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where the blue set is the empirical estimation for Dα, α = 0.5.
34
Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Understanding outliers
It is possible to extend it to define (past) functional outliers,
0 20 40 60 80
−8
−6
−4
−2
Age
Log
Mor
talit
y R
ate
−10 −5 0 5 10 15
−1
01
23
4PC score 1
PC
sco
re 2
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19141915
1916
1917
1918
1919
1940
1942
1943
1944
(here male log-mortality rates in France from 1899 to 2005).
35
Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
Understanding outliers when generating scenarios
Based on the log-Poisson Lee & Carter model, it is possible to generate scenarios,
0 20 40 60 80
−16
−14
−12
−10
−8
−6
−4
−2
Age
Log
Mor
talit
y R
ate
−10 0 10
−0.
10−
0.05
0.00
0.05
0.10
0.15
PC score 1
PC
sco
re 2
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2058
2089
=⇒ this stochastic model does not generate extremal scenarios
36
Arthur CHARPENTIER, modeling analogies in life and nonlife insurance
ReferencesHachemeister, C.A. & Stanard, J. N. (1975). IBNR claims count estimation with static lag functions. XIIth Astin
Colloquium, Portugal.
Kremer, E. (1982). IBNR-claims and the two-way model of ANOVA. Scandinavian Actuarial Journal, pp. 4755.
Mack, T. (1991). Which stochastic model is underlying the chain ladder method ? Insurance : mathematics & economics,15(2/3), 133-138.
De Vylder, F. (1978). Estimation of IBNR claims by least squares. Mitteilungen der Vereinigung Schweizerischer
Versicherungsmathematiker, pp. 249254.
Taylor, G. (1977). Separation and other effects from the distribution of non-life insurance claim delays. Astin Bulletin,9, 217-230.
Lee, R. D., & Carter, L. (1992), Modeling and forecasting U.S. mortality, Journal of the American Statistical Association,87, 659671.
Renshaw, A.E. & Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for mortality reductionfactors. Insurance : Mathematics & Economics, 38, 556-570.
Currie, I.D. (2006). Smoothing and forecasting mortality rates with p-splines
Cairns , A., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., & Khalaf-Allah, M. (2008), Mortality DensityForecasts : An Analysis of Six Stochastic Mortality Models. Pensions Institute Discussion Paper PI-0801.
Brillinger, D. R. (1986). The natural variability of vital rates and associated statistics. Biometrics, 42, 693734.
Brouhns, N., Denuit, M. & Vermunt, J.K. (2002). A Poisson log-bilinear regression approach to the construction ofprojected life-tables. Insurance : Mathematics and Economics, 31, 373-393.
Tukey, J.W. (1975). Mathematics and the picturing of data. Proceedings of the International Congress of Mathematicians, 2,523-531.
Ruts, I. & Rousseeuw, P.J. (1996). Computing depth contours of bivariate point clouds, Computational Statistics & Data
Analysis, 23. 153168.
Rousseeuw, P.J., Ruts, I., & Tukey, J.W. (1999). The Bagplot : A Bivariate Boxplot, Annals of Statistics, 53, 382387.
37