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Solvency II newspeak one year uncertainty for IBNR the boostrap - - PowerPoint PPT Presentation

Feb 15, 2024 126 likes •835 views

Arthur CHARPENTIER, IBNR and one-year uncertainty Solvency II newspeak one year uncertainty for IBNR the boostrap approach Arthur Charpentier 1 & Laurent Devineau 2 1 Universit Ecole Polytechnique, 2 Milliman e Rennes 1 - CREM

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Solvency II’ newspeak ‘one year uncertainty for IBNR’ the boostrap approach

Arthur Charpentier1 & Laurent Devineau2

1 Universit´

e Rennes 1 - CREM & ´ Ecole Polytechnique, 2 Milliman

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

Lyon Summer School, July 2010 1

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Agenda of the talk

  • Solvency II : CP 71 and the one year horizon
  • Solvency II : new way of looking at the ‘uncertainty’
  • From MSE to MSEP (MSE of prediction)
  • From MSEP to MSEPC (conditional MSEP)
  • CDR, claims development result
  • From Mack (1993) to Merz & W¨

uthrich (2009)

  • Updating Poisson-ODP bootstrap technique
  • ne year

ultimate China ladder Merz & W¨ uthrich (2008) Mack (1993) GLM+boostrap x Hacheleister & Stanard (1975) England & Verrall (1999) 2

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Arthur CHARPENTIER, IBNR and one-year uncertainty

‘one year horizon for the reserve risk’

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Arthur CHARPENTIER, IBNR and one-year uncertainty

‘one year horizon for the reserve risk’

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Arthur CHARPENTIER, IBNR and one-year uncertainty

‘one year horizon for the reserve risk’

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Arthur CHARPENTIER, IBNR and one-year uncertainty

‘one year horizon for the reserve risk’

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Arthur CHARPENTIER, IBNR and one-year uncertainty

‘one year horizon for the reserve risk’

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Arthur CHARPENTIER, IBNR and one-year uncertainty

‘one year horizon for the reserve risk’

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder Ci,j+1 = λj · Ci,j and Mack
  • Factor models Yi,j ∼ L(Ai, Bj) and GLM’s (ODP-bootstrap)
  • and the Bayesian approach

9

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder Ci,j+1 = λj · Ci,j and Mack
  • Factor models Yi,j = ϕ(Ai, Bj) and GLM’s (ODP-bootstrap)
  • expert opinion and the Bayesian approach

10

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder Ci,j+1 = λj · Ci,j and Mack
  • Factor models Yi,j = ϕ(Ai, Bj) and GLM’s (ODP-bootstrap)
  • expert opinion and the Bayesian approach

11

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder Ci,j+1 = λj · Ci,j and Mack
  • Factor models Yi,j = ϕ(Ai, Bj) and GLM’s (ODP-bootstrap)
  • expert opinion and the Bayesian approach

12

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder Ci,j+1 = λj · Ci,j and Mack
  • Factor models Yi,j = ϕ(Ai, Bj) and GLM’s (ODP-bootstrap)
  • expert opinion and the Bayesian approach

13

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder Ci,j+1 = λj · Ci,j and Mack
  • Factor models and GLM’s (ODP-bootstrap), E(Yi,j|F) = ϕ(Ai, Bj)
  • expert opinion and the Bayesian approach

14

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Standard models in IBNR models

  • Chain Ladder and Mack (1993) E(Ci,j+1|F) = λj · Ci,j
  • Factor models and GLM’s (ODP-bootstrap), E(Yi,j|F) = ϕ(Ai, Bj)
  • expert opinion and the Bayesian approach

15

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Notations for triangle type data

  • Xi,j denotes incremental payments, with delay j, for claims occurred year i,
  • Ci,j denotes cumulated payments, with delay j, for claims occurred year i,

Ci,j = Xi,0 + Xi,1 + · · · + Xi,j,

1 2 3 4 5 3209 1163 39 17 7 21 1 3367 1292 37 24 10 2 3871 1474 53 22 3 4239 1678 103 4 4929 1865 5 5217 et et 1 2 3 4 5 3209 4372 4411 4428 4435 4456 1 3367 4659 4696 4720 4730 2 3871 5345 5398 5420 3 4239 5917 6020 4 4929 6794 5 5217

  • Ft denotes information available at time t, available at time t, based on the

first k years, only Ft = {(Ci,j), 0 ≤ i + j ≤ t} = {(Xi,j), 0 ≤ i + j ≤ t} 16

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Notations for triangle type data

  • Xi,j denotes incremental payments, with delay j, for claims occurred year i,
  • Ci,j denotes cumulated payments, with delay j, for claims occurred year i,

Ci,j = Xi,0 + Xi,1 + · · · + Xi,j,

1 2 3 4 5 3209 1163 39 17 7 21 1 3367 1292 37 24 10 2 3871 1474 53 22 3 4239 1678 103 4 4929 1865 5 5217 et et 1 2 3 4 5 3209 4372 4411 4428 4435 4456 1 3367 4659 4696 4720 4730 2 3871 5345 5398 5420 3 4239 5917 6020 4 4929 6794 5 5217

  • Ftk denotes partial information available at time t, based on the first k years,
  • nly

Ft

k = {(Ci,j), 0 ≤ i + j ≤ t, i ≤ k} = {(Xi,j), 0 ≤ i + j ≤ t, i ≤ k}

17

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Chain Ladder estimation

1 2 3 4 5 3209 4372 4411 4428 4435 4456 1 3367 4659 4696 4720 4730 2 3871 5345 5398 5420 3 4239 5917 6020 4 4929 6794 5 5217 et et 1 2 3 4 5 3209 4372 4411 4428 4435 4456 1 3367 4659 4696 4720 4730 4752.4 2 3871 5345 5398 5420 5430.1 5455.8 3 4239 5917 6020 6046.15 6057.4 6086.1 4 4929 6794 6871.7 6901.5 6914.3 6947.1 5 5217 7204.3 7286.7 7318.3 7331.9 7366.7

with the following link ratios

1 2 3 4 n λj 1,38093 1,01143 1,00434 1,00186 1,00474 1,0000

One the triangle has been completed, we obtain the amount of reserves, with respectively 22, 36, 66, 153 and 2150 per accident year, i.e. the total is 2427. 18

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Arthur CHARPENTIER, IBNR and one-year uncertainty

How to quantify uncertainty in Solvecny II

In Solvency II, uncertainty is quantified as a dispersion measure (variance or quantile) of changes in prediction, with one year of additional information. The best estimate at time t is Rt = E(C∞|Ft) while it become, at time t + 1

  • Rt+1 = E(C∞|Ft+1).

The goal is to estimate E

  • [E(C∞|Ft+1) − E(C∞|Ft)]2|Ft
  • 19
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Quantifying uncertainty in odds/tails games

In statistics, the mean squared error is a standard measure to quantify the uncertainty of an estimator, i.e. mse( θ) = E

  • θ − θ

2 In order to formalize the prediction process in claims reserving consider the following simpler case. Let {x1, · · · , xn} denote an i.i.d. B(p) sample. We want to predict Sh = Xn+1 + · · · + Xn+h. Let n Sh = ψ(Xn+1, · · · , Xn+h) = h · pn denote the natural predictor for Sh, at time n. Since Sh is a random variable (θ was a constant) define mse(n Sh) = E

  • n

Sh − E(Sh) 2 20

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Arthur CHARPENTIER, IBNR and one-year uncertainty

and msep(n Sh) = E

  • n

Sh − Sh 2 Note that msep(n Sh) = E

  • n

Sh − E(Sh) 2 + E

  • [E(Sh) − Sh]2

= mse(n Sh) + Var(Sh) where the first term is a process error and the second term a estimation error. It is also possible to calculate the information given the information available at time n, i.e. a conditional msep, msepcn(n Sh) = E

  • n

Sh − Sh 2 |Fn

  • denoted E(msepcn(n

Sh)) = msep(n Sh). 21

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What are we looking for ?

In Solvency II requirements, CDRn+1 = [n Sh] − [xn+1 + n+1 Sh−1] This defines a martingale since E(CDRn+1|Fn) = 0 and what is required is to estimate msepcn(CDRn+1) i.e. find msepcn(CDRn+1). 22

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What are we looking for ?

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What are we looking for ?

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What are we looking for ?

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Having an estimator of the uncertainty

Let us continue with our repeated tails/heads game. Let pn = [x1 + · · · + xn]/n, so that Var( pn) = p(1 − p) n thus mse(n Sh) = mse(h · pn) = h2 · mse( pn) = h2 n p(1 − p),

  • r

msep(n Sh) = nhp(1 − p) + h2 n p(1 − p) = nh + h2 n p(1 − p) i.e. msep(n Sh) = h(n + h) n p(1 − p). 26

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Having an estimator of the uncertainty

Thus, this quantity can be estimated as

  • msep(n

Sh) = h(n + h) n

  • pn(1 −

pn). while the mse estimator was

  • mse(n

Sh) = h2 n pn(1 − pn) Looking at the msepc at time n, we have msepcn(n Sh) = Var(S|Fn) + mse(n Sh|Fn) 27

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Having an estimator of the uncertainty

Looking at the msepc at time n, we have msepcn(n Sh) = Var(S|Fn) + mse(n Sh|Fn) where Var(S|Fn) = Var(Xn+1 + · · · + Xn+h|x1, · · · , xn) = Var(Xn+1 + · · · + Xn+h) = hp(1 − p) and mse(n Sh|Fn) =

  • E(Sh|Fn) − n

Sh 2 which can be written msepcn(n Sh) = hp(1 − p) + h2 (p − pn)2 28

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Having an estimator of the uncertainty

This quantity can be estimated as

  • msepcn(n

Sh) = h pn(1 − pn) + 0. i.e. we keep only the variance process term. Mack (1993) suggested to use partial information to estimate the second term. Define D = {Xi, i ≤ n} and Bk = {Xi, i ≤ n, i ≤ k} with k ≤ n. Define

  • msepc

k n(n

Sh) = h pn(1 − pn) + h2 ( pn − pk)2 29

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The one year horizon uncertainty

In Solvency II, insurance companies are requiered to estimate the msepc, at time n, of the difference between Xn+1 + n+1 S(h−1) and n S(h). Those two quantities estimate the same things, at different dates,

  • n

S(h) is a predictor for Sh at time n

  • Xn+1 + n+1

S(h−1) is a predictor for Sh at time n + 1, If we admit that we are looking for the following quantity (as in Merz & W¨ uthrich (2008)) msepcn = E

  • [Xn+1 + (h − 1) ·

pn+1 − h · pn]2 |Fn

  • i.e.

msepcn = E n + h n + 1 Xn+1 + n − h n + 1 pn 2 |Fn

  • msepcn = (n + h)2

(n + 1)2 p + (n + h)(n − h) (n + 1)2 p · pn + (n − h)2 (n + 1)2 p2

n

30

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Updating an estimator, an econometric introduction

If βn =

  • X′

nXn

−1 X′

nY n denotes the OLS estimate, if a new observation

becomes available (yn+1, xn+1), then

  • βn+1 =

βn +

  • X′

nXn

−1 xn+1 1 + x′

n+1

  • X′

nXn

−1 xn+1

  • yn+1 − x′

n+1

βn

  • r
  • βn+1 =

βn +

  • X′

n+1Xn+1

−1 xn+1

  • yn+1 − x′

n+1

βn

  • The CDR for a new observation X = x is then

CDRn = x′([ βn+1 − βn] i.e. CDRn = x′ X′

n+1Xn+1

−1 xn+1

  • yn+1 − x′

n+1

β

  • 31
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Mack’s ultimate uncertainty

As shown in Mack (1993),

  • msep(

Ri) = C2

i,∞ n−1

  • j=n−i+1
  • σ2

j

  • λ2

j

  • 1
  • Ci,j

+ 1

  • Sj
  • where Sj is the sum of cumulated payments on accident years before year n − j,

Sj =

n−j

  • i=1

Ci,j. Finally, it is possible also to derive an estimator for the aggregate msep (all accident years)

  • msep(

R) =

  • msep(

Ri) + 2 C2

i,∞ n

  • k=i+1
  • Ck,n

n−1

  • j=n−i+1
  • σ2

j

  • λ2

jSj

32

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Mack’s ultimate uncertainty

> library(ChainLadder) > source("http://perso.univ-rennes1.fr/arthur.charpentier/bases.R") > MackChainLadder(PAID) MackChainLadder(Triangle = PAID) Latest Dev.To.Date Ultimate IBNR Mack.S.E CV(IBNR) 1 4,456 1.000 4,456 0.0 0.000 NaN 2 4,730 0.995 4,752 22.4 0.639 0.0285 3 5,420 0.993 5,456 35.8 2.503 0.0699 4 6,020 0.989 6,086 66.1 5.046 0.0764 5 6,794 0.978 6,947 153.1 31.332 0.2047 6 5,217 0.708 7,367 2,149.7 68.449 0.0318 Totals Latest: 32,637.00 Ultimate: 35,063.99 IBNR: 2,426.99 Mack S.E.: 79.30 CV(IBNR): 0.03

i.e. msepc6( R) = 79.30. 33

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Merz & W¨ uthrich’s one year uncertainty

Based on some martingale properties, one can prove that E(CDRi(n + 1)|Fn) = 0 (neither boni nor mali can be expected). Further, it can be proved that (CDRi(n + 1))n’s are non correlated, and thus msepcn(CDRi(n + 1)) = Var(CDRi(n + 1)|Fn) = E(CDRi(n + 1)2|Fn) Merz & W¨ uthrich (2008) proved that the one year horizon error can be estimated with a formula similar to Mack (1993)

  • msepcn(CDRi(n + 1)) =

C2

i,∞

  • Γi,n +

∆i,n

  • where
  • ∆i,n =
  • σ2

n−i+1

  • λ2

n−i+1Sn+1 n−i+1

+

n−1

  • j=n−i+2
  • Cn−j+1,j

Sn+1

j

2

  • σ2

j

  • λ2

jSn j

34

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and

  • Γi,n =
  • 1 +
  • σ2

n−i+1

  • λ2

n−i+1Ci,n−i+1

  • n−1
  • j=n−i+2
  • 1 +
  • σ2

j

  • λ2

j[Sn+1 j

]2 Cn−j+1,j

  • − 1

Merz & W¨ uthrich (2008) mentioned that this term can be approximated as

  • Γi,n ≈
  • σ2

n−i+1

  • λ2

n−i+1Ci,n−i+1

+

n−1

  • j=n−i+2
  • Cn−j+1,j

Sn+1

j

2

  • σ2

j

  • λ2

jCn−j+1,j

using a simple development of (1 + ui) ≈ 1 + ui, but which is valid only if ui is extremely small, i.e.

  • σ2

j

  • λ2

j

<< Cn−j+1,j 35

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Implementing Merz& W¨ uthrich’s formula

> source("http://perso.univ-rennes1.fr/arthur.charpentier/merz-wuthrich-triangle.R") > MSEP_Mack_MW(PAID,0) MSEP Mack MSEP observable approche MSEP observable exacte 1 0.0000000 0.000000 0.000000 2 0.6393379 1.424131 1.315292 3 2.5025153 2.543508 2.543508 4 5.0459004 4.476698 4.476698 5 31.3319292 30.915407 30.915407 6 68.4489667 60.832875 60.832898 7 79.2954414 72.574735 72.572700

36

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Implementing Merz& W¨ uthrich’s formula

Could Merz & W¨ uthrich’s formula end up with more uncertainty than Mack’s

> Triangle = read.table("http://perso.univ-rennes1.fr/arthur.charpentier/ + GAV-triangle.csv",sep=";")/1000000 > MSEP_Mack_MW(Triangle,0) MSEP Mack MSEP observable approche MSEP observable exacte 1 0.00000000 0.0000000 0.0000000 2 0.01245974 0.1296922 0.1526059 3 0.20943114 0.2141365 0.2144196 4 0.25800338 0.1980723 0.1987730 5 3.05529740 3.0484895 3.0655251 6 58.42939329 57.0561173 67.3757940 7 58.66964613 57.3015524 67.5861066

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GLM log-Poisson in triangles

Recall that we while to estimate E([R − R]2) =

  • E(R) − E(

R) 2 + Var(R − R) ≈ Var(R) + Var( R) Classically, consider a log-Poisson model, were incremental payments satisfy Xi,j ∼ P(µi,j) where µi,j = exp[ηi,j] = exp[γ + αi + βj] Using the delta method, we get that asymptotically Var( Xi,j) = Var( µi,j) ≈

  • ∂µi,j

∂ηi,j

  • 2

Var( ηi,j) where, since we consider a log link, ∂µi,j ∂ηi,j = µi,j 38

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i.e., with an ODP distribution (i.e. Var(Xi,j = ϕE(Xi,j), E

  • [Xi,j −

Xi,j]2 ≈ ϕ · µi,j + µ2

i,j ·

Var(ηi,j) and Cov(Xi,j, Xk,l) ≈ µi,j · µk,l · Cov ( ηi,j, ηk,l) Thus, since the overall amount of reserves satisfies E

  • [R −

R]2 ≈

  • i+j−1>n
  • ϕ ·

µi,j + µ′ Var( η) µ.

> an <- 6; ligne = rep(1:an, each=an); colonne = rep(1:an, an) > passe = (ligne + colonne - 1)<=an; np = sum(passe) > futur = (ligne + colonne - 1)> an; nf = sum(passe) > INC=PAID > INC[,2:6]=PAID[,2:6]-PAID[,1:5] > Y = as.vector(INC) > lig = as.factor(ligne); col = as.factor(colonne) > CL <- glm(Y~lig+col, family=quasipoisson) > Y2=Y; Y2[is.na(Y)]=.001

39

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> CL2 <- glm(Y2~lig+col, family=quasipoisson) > YP = predict(CL) > p = 2*6-1; > phi.P = sum(residuals(CL,"pearson")^2)/(np-p) > Sig = vcov(CL) > X = model.matrix(CL2) > Cov.eta = X%*%Sig%*%t(X) > mu.hat = exp(predict(CL,newdata=data.frame(lig,col)))*futur > pe2 = phi.P * sum(mu.hat) + t(mu.hat) %*% Cov.eta %*% mu.hat > cat("Total reserve =", sum(mu.hat), "prediction error =", sqrt(pe2),"\n") Total reserve = 2426.985 prediction error = 131.7726

i.e. E( R − R) = 131.77. 40

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Bootstrap and unccertainty

Bootstrap is now a standard nonparametric technique used to quantify uncertainty. In the linear model, Y (x) = E(Y |X = x) = x′ β while Y (x) = E(Y |X = x) + ε, and the uncertainty is related to Var( Y (x)) = Var(x′ β) = x′Var( β)x Var(Y (x)) = Var(x′ β + ε) ≈ Var( Y (x)) + σ2 To derive confidence interval or quantiles of Y (x) or Y (x) we need further assuming, like a distribution for residuals ε Instead of giving an analytic formula, monte carlo simulations can be used. The idea is to generate samples {(X⋆

i , Y ⋆ i ), i = 1, ..., n} or {(X⋆ i ,

Y (X⋆

i ) + ε⋆ i ), i = 1, ..., n}

41

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Bootstrap and unccertainty

  • 2

4 6 8 10 5 10

  • 42
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Bootstrap and unccertainty

  • 2

4 6 8 10 5 10

  • 2

4 6 8 10 5 10

43

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Bootstrap and unccertainty

  • 2

4 6 8 10 5 10

  • 44
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Bootstrap and unccertainty

  • 2

4 6 8 10 5 10

  • 2

4 6 8 10 5 10

  • 45
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Bootstrap and unccertainty

  • 2

4 6 8 10 5 10

  • 2

4 6 8 10 5 10

46

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Bootstrap and unccertainty

  • 2

4 6 8 10 5 10

  • 2

4 6 8 10 5 10

  • 47
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Bootstraping errors ?

Parametric generation : if Z has distribution F(·), then F −1(Random) is randomdly distributed according to F(·). Nonparametric generation : we do not know F(·), it is still possible to estimate it

  • Fn(z) = 1

n

n

  • i=1

1(Xi ≤ x) Then

  • F −1

n (u) = Xi:n where i

n ≤ u < i + 1 n where Xi:n denotes the order statistics, X1:n ≤ X2:n ≤ · · · ≤ Xn−1:n ≤ Xn:n Thus,

  • F −1

n (Random) = Xi with probability 1

n for all i. 48

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Parametric versus nonparametric random generation

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0

49

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Parametric versus nonparametric random generation

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 50
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Parametric versus nonparametric random generation

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0

  • 51
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Bootstrap and ultimate uncertainty

From triangle of incremental payments, (Yi,j) assume that Yi,j ∼ P( Yi,j) where Yi,j = exp( Li + Cj)

  • 1. Estimate parameters

Li and Cj, define Pearson’s (pseudo) residuals

  • εi,j = Yi,j −

Yi,j

  • Yi,j
  • 2. Generate pseudo triangles on the past, {i + j ≤ t}

Y ⋆

i,j =

Yi,j + ε⋆

i,j

  • Yi,j
  • 3. (re)Estimate parameters

L⋆

i and

C⋆

j , and derive expected payments for the

future, Y ⋆

i,j.

  • R =
  • i+j>t
  • Y ⋆

i,j

52

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Arthur CHARPENTIER, IBNR and one-year uncertainty

is the best estimate.

  • 4. Generate a scenario for future payments, Y ⋆

i,j e.g. from a Poisson distribution

P( Y ⋆

i,j)

R =

  • i+j>t

Y ⋆

i,j

One needs to repeat steps 2-4 several times to derive a distribution for R. 53

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Bootstrap and GLM log-Poisson in triangles

  • 2

4 6 8 10 2000 4000 6000 8000

  • 54
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Arthur CHARPENTIER, IBNR and one-year uncertainty

Bootstrap and GLM log-Poisson in triangles

  • 2

4 6 8 10 2000 4000 6000 8000

  • 55
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Bootstrap and GLM log-Poisson in triangles

  • 2

4 6 8 10 2000 4000 6000 8000

  • 56
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Bootstrap and GLM log-Poisson in triangles

  • 2

4 6 8 10 2000 4000 6000 8000

  • 57
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Bootstrap and GLM log-Poisson in triangles

If we repeat it 50,000 times, we obtain the following distribution for the mse.

50 100 150 200 0.000 0.002 0.004 0.006 0.008 0.010 0.012 msep of overall reserves Density

MACK GLM

58

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Bootstrap and one year uncertainty

  • 2. Generate pseudo triangles on the past and next year {i + j ≤ t + 1}

Y ⋆

i,j =

Yi,j + ε⋆

i,j

  • Yi,j
  • 3. Estimate parameters

L⋆

i and

C⋆

j , on the past, {i + j ≤ t}, and derive expected

payments for the future, Y ⋆

i,j.

  • Rt =
  • i+j>t
  • Yi,j
  • 4. Estimate parameters

L⋆

i and

C⋆

j , on the past and next year, {i + j ≤ t + 1},

and derive expected payments for the future, Y ⋆

i,j.

  • Rt+1 =
  • i+j>t
  • Yi,j
  • 5. Calculate CDR as CDR=

Rt+1 − Rt. 59

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Ultimate versus one year uncertainty

ultimate (R − E(R)) versus one year uncertainty,

−200 −100 100 200 0.000 0.002 0.004 0.006 0.008 0.010

60

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Bootstrap and GLM log-Poisson in triangles

  • 2

4 6 8 10 2000 4000 6000 8000

  • 61
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Bootstrap and GLM log-Poisson in triangles

  • 2

4 6 8 10 2000 4000 6000 8000

  • 62
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Arthur CHARPENTIER, IBNR and one-year uncertainty

Why a Poisson model for IBNR ?

Hachemeister & Stanard (1975), Kremer (1985) and Mack(1991) proved that with a log-Poisson regression model on incremental payments, the sum of predicted payments corresponds to the Chain Ladder estimator. Recall that Yi,j ∼ P(Li + Cj), i.e.

  • we consider two factors, line Li and column Cj
  • we assume that E(Yi,j|F) = exp[Li + Cj] (since the link function is log)
  • we assume further that Var(Yi,j|F) = exp[Li + Cj] = E(Yi,j (since we consider

a Poisson regression) 63

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Why a Poisson model for IBNR ?

Adding additional factors is complex (too many parameters, and need to forecast a calendar factor, if any). Changing the link function is not usual, and having a multiplicative model yield to natural interpretations, Why not changing the distribution (i.e. the variance function) ? = ⇒ consider Tweedie models. 64

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Tweedie models

Assume here that the variance function is V ar(Y ) = ϕE(Y )p for some p ∈ [0, 1]. p = 1 is obtained with a Poisson model, p = 2 with a Gamma model. If p ∈ (1, 2), we obtain a compound Poisson distribution. 65

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Best estimate and Tweedie parameter

1.0 1.2 1.4 1.6 1.8 2.0 −1200 −1000 −800 −600 −400 −200 Tweedie power parameter log likelihood

66

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Best estimate and Tweedie parameter

1.0 1.2 1.4 1.6 1.8 2.0 −200 −190 −180 −170 −160 −150 −140 Tweedie power parameter log likelihood

67

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Best estimate and Tweedie parameter

1.0 1.2 1.4 1.6 1.8 2.0 2430 2435 2440 Tweedie power parameter Amount of reserves

68

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Arthur CHARPENTIER, IBNR and one-year uncertainty

Best estimate and Tweedie parameter

Best estimate amount of reserve with Tweedie power p, with the 95% quantile and the 99.5% quantile

1.0 1.2 1.4 1.6 1.8 2.0 2400 2450 2500 2550 2600 2650 2700 Tweedie power Reserves

69