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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

[email protected]

http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/

Reserving seminar, SCOR, May 2010

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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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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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but usually we do not work on continuous time individual observations(individuals or claims) : we summarized information per year occurrence untilclosure,


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individual lives or claims can also be followed looking at diagonals, occurrenceuntil closure,occurrence until closure,occurrence until closure,occurrence untilclosure,


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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,


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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,


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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


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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


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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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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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Assume thatYi,j ∼ P(µi,j) where µi,j = exp[αi + βj ].

the occurrence factor αi the development factor βj

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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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Consider here another triangle with monthly health claims,

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Month of occurence

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Delay

2 4 6 8 10 12

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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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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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Bootstrap and GLM log-Poisson in triangles

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Total amount of reserves ('000 000$)

10 15 20 25 30

(log-Poisson + bootstrap versus lognormal distribution + Mack)

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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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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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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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Two sets of parameters depend on the age, α = (α0, α1, · · · , α110) andβ = (β0, β1, · · · , β110).

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and one set of parameters depends on the time, γ = (γ1899, γ1900, · · · , γ2005).

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Errors and predictions

exp[αj + βj γt] exp[αj + βj γt]

on past observations on the future

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Forecasting γ

Based on γ = (γ1899, · · · , γ2005), we need to forecast γ = (γ2006, · · · , γ2050).

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Forecasting γ

Classically integrated ARIMA processes are considered,

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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


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


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



−2 −1 0 1

−1.

5−

1.0

−0.

50.

00.

51.

0

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where the blue set is the empirical estimation for Dα, α = 0.5.

34



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


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


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

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