This document discusses modeling and estimating extreme risks and quantiles in non-life insurance. It introduces the generalized extreme value distribution and Pickands–Balkema–de Haan theorem, which state that maximums of iid random variables converge to one of three extreme value distributions. It also discusses estimators for the shape parameter of these distributions, such as the Hill estimator, and using the generalized Pareto distribution above a threshold to estimate value-at-risk quantiles. Examples are given applying these methods to Danish fire insurance loss data.
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Basic concepts and how to measure price volatility
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Talk at the modcov19 CNRS workshop, en France, to present our article COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability
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1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
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Poonawalla Fincorp and IndusInd Bank Introduce New Co-Branded Credit Cardnickysharmasucks
The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
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The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
Could this growth lead to a “Roaring Twenties”? As quickly as the U.S. economy contracted, experiencing a 9.1% drop in economic output relative to the business cycle in Q2 2020, the largest in recorded history, it has rebounded beyond expectations. This surprising growth seems to be fueled by the U.S. government’s aggressive fiscal and monetary policies, and an increase in consumer spending as mobility restrictions are lifted. Unemployment rates between June 2020 and June 2021 decreased by 5.2%, while the demand for labor is increasing, coupled with increasing wages to incentivize Americans to rejoin the labor force. Schools and businesses are expected to fully reopen soon. In parallel, vaccination rates across the country and the world continue to rise, with full vaccination rates of 50% and 14.8% respectively.
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“In order for the U.S. economy to continue growing, whether there is another wave or not, the U.S. needs to focus on diversifying supply chains, supporting business investment, and maintaining consumer spending,” says Grace Feeley, a research analyst at M Capital Group.
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The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
Which Crypto to Buy Today for Short-Term in May-June 2024.pdf
Slides ensae-2016-8
1. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Actuariat de l’Assurance Non-Vie # 8
A. Charpentier (UQAM & Université de Rennes 1)
ENSAE ParisTech, Octobre / Décembre 2016.
http://freakonometrics.hypotheses.org
@freakonometrics 1
2. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Les grands risques
On avait vu dans la section #7 l’importance de l’écrêtement des grands risques.
On va revenir ici sur la modélisation de ces coûts importants.
Références: Beirlant et al. (2004), Embrechts et al. (2013) ou Reiss & Thomas
(2007)
@freakonometrics 2
3. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Le théorème central limite des extrêmes
Théorème Supposons qu’il existe des constantes de normalisation an ∈ R et
bn > 0, et une loi non-dégénérée H telles que
b−1
n {Xn:n − an}
L
→ H.
Alors H est du même type qu’une trois lois suivantes (données par leur fonction
de répartition),
1. Loi de Fréchet, Φξ(x) = exp −x−ξ
I(x > 0), ξ > 0,
2. Loi de Weibull, Ψξ(x) = exp −x−ξ
si x ≤ 0, et 1 sinon, ξ > 0,
3. Loi de Gumbel, Λ(x) = exp (− exp(−x)).
On notera que
X ∼ Φξ ⇐⇒ log Xξ
∼ Λ ⇐⇒ −1/X ∼ Ψξ.
@freakonometrics 3
4. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Les trois lois limites
Ces trois lois sont en fait les trois cas particulier de la distribution GEV -
Generalized Extreme Value (représentation de Jenkinson-von Mises)
H (x) =
exp − [1 − ξ (x − µ) /σ]
1/ξ
si ξ = 0
exp (− exp [− (x − µ) /σ]) si ξ = 0
dès lors que µ + ξx/σ > 0.
Remarque S’il existe des constantes de normalisation an ∈ R et bn > 0, et une
loi non-dégénérée GEV(ξ) telles que
b−1
n {Xn:n − an}
L
→ GEV(ξ).
on dira que FX appartient au max-domain d’attraction (MDA) de GEV(ξ).
@freakonometrics 4
5. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Les grands risques
On peut définir des risques sous-exponentiels, si Xi est une suite de variables i.i.d.
lim
x→∞
P [X1 + ... + Xn > x]
P [max {X1, ..., Xn} > x]
= 1, n ≥ 1.
Définition Si FX est une fonction de répartition continue d’espérance E [X], on
définit l’indice de grands risques par
DFX
(p) =
1
E [X]
1
1−p
F−1
X (t) dt pour p ∈ [0, 1] .
La version empirique est alors Tn (p) la proportion des [np] plus gros sinistres par
rapport à la somme totale, i.e.
Dn (p) =
X1:n + X2:n + ... + X[np]:n
X1 + ... + Xn
où
1
n
< p ≤ 1.
@freakonometrics 5
6. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
02468101214
Index
X
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0102030405060
Index
X
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0500010000150002000025000
Index
X
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.50.60.70.80.91.0
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.50.60.70.80.91.0
0e+00 2e+04 4e+04 6e+04 8e+04 1e+05
0.50.60.70.80.91.0@freakonometrics 6
7. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Proposition(théorème de Pickands-Balkema-de Haan) soit ξ ∈ R, les deux
résultats suivants sont équivalents,
1. F ∈ MDA (Hξ), i.e. il existe des suites (an) et (bn) telles que
lim
n→∞
P (Xn:n ≤ anx + bn) = Hξ (x) , x ∈ R.
2. Il existe une fonction positive a (·) telle que pour 1 + ξx > 0,
lim
u→∞
F (u + xa (u))
F (u)
= lim
u→∞
P
X − u
a (u)
> x X > u
=
(1 + ξx)
−1/ξ
if ξ = 0,
exp (−x) if ξ = 0.
Aussi, F ∈ MDA (Gξ) si et seulement si il existe σ (·) > 0 telle que
lim
u→xF
sup
0<x<xF
P (X − u ≤ x|X > u) − Hξ,σ(u) (≤ x) = 0
@freakonometrics 7
8. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Loi de type Pareto
La loi de X est à variation régulière d’indice α ∈ (0, +∞) si pour tout x
lim
t→∞
P[X > tx]
P[X > t]
= lim
t→∞
F(tx)
F(t)
= x−α
ou encore
P[X > x] = x−α
L(x) où L est à variation lente, i.e. lim
t→∞
L(tx)
L(t)
= 1.
@freakonometrics 8
9. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Estimateur de Hill et Pareto Plot
Si u est suffisement grand
P(X ≤ x|X > u) ≈ 1 − 1 + ξ
x + u
σ
−1/ξ
,
pour un σ > 0, soit
log 1 −
F(x)
1 − F(u)
≈ −
1
ξ
log 1 + ξ
x + u
σ
≈ −
1
ξ
log x + constant.
Un estimateur naturel de ξ est alors la pente de la droite passant au mieux par le
nuage des (log Xi, log(1 − F(Xi))).
ξHill
n,k =
1
k
k
i=1
log Xn−i:n − log Xn−k+1:n
@freakonometrics 9
10. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Quelques estimateurs classiques de ξ
ξP ickands
n,k =
1
log 2
log
Xn−k:n − Xn−2k:n
Xn−2k:n − Xn−4k:n
ξHill
n,k =
1
k
k
i=1
log Xn−i:n − log Xn−k+1:n
ξDEdH
n,k = ξ
H(1)
n,k + 1 −
1
2
1 −
ξ
H(1)
n,k
2
ξ
H(2)
n,k
−1
,
où
ξ
H(r)
n,k =
1
k
k−1
i=1
[log Xn−i:n − log Xn−k:n]
r
, r = 1, 2, ..
@freakonometrics 10
11. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Lois à queue fine
• loi exponentielle, f (x) = β exp (−βx), pour β > 0,
• loi Gamma, f (x) = βα
Γ (α)
−1
xα−1
exp (−βx) pour α, β > 0,
• loi de Weibull, f (x) = cβxβ−1
exp −cxβ
pour c > 0 et β ≥ 1,
Lois à queue épaisse
• loi de Weibull, f (x) = cβxβ−1
exp −cxβ
pour c > 0 et β < 1,
• loi lognormale, f (x) = 1√
2πσx
exp −(log x−µ)2
2σ2 pour σ > 0
• loi de Pareto, f (x) = α
θ+x
θ
θ+x
α
pour α, θ > 0
• loi de Burr, f (x) = (θ + xγ
)
−(α+1)
αγθα
xγ−1
pour α, γ, θ > 0.
@freakonometrics 11
12. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
La fonction de moyenne en excès
Fonction de répartition Fonction en excédent
Loi lognormale
x
0
1
√
2πσt
exp −
(log t − µ)
2
2σ2
∼
σ2
u
log u − µ
Exponentielle 1 − exp (−λx)
1
λ
Pareto 1 − θ
θ+x
α θ + x
α − 1
Gamma
x
0
β (βt)
α−1 exp (−βt)
Γ (α)
dt ∼
1
β
Weibull 1 − exp −λxβ
∼
u1−β
λβ
@freakonometrics 12
13. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
0 5 10 15 20
0246810
Fonction de moyenne en excès loi log normale
Seuil u
Moyenneenexcédent
0 5 10 15 20
0246810
Fonction de moyenne en excès loi exponentielle
Seuil u
Moyenneenexcédent
@freakonometrics 13
14. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
0 5 10 15 20
0246810
Fonction de moyenne en excès loi Pareto
Seuil u
Moyenneenexcédent
0 5 10 15 20
0246810
Fonction de moyenne en excès loi Weibull
Seuil u
Moyenneenexcédent
@freakonometrics 14
15. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Estimation de quantiles extrêmes
Le quantile d’ordre p ∈]0, 1[ - correspondant à la Value-at-Risk - associé à X, de
fonction de répartition FX, se définie par
V aR(X, p) = xp = F−1
(p) = sup{x ∈ R, F(x) ≥ p}.
Notons que les assureurs parle plutôt de période de retour. Si T est la première
année où on observe un phénomène qui se produit annuellement avec probabilité
p. Alors
P[T = k] = [1 − p]k−1
p de telle sorte que E[T] =
1
p
Parmi les autres mesures pertinantes (et intéressantes en réassurance), on
retiendra la TV aR - ou ES - correspondant à la valeur moyenne sachant que la
V aR a été dépassée,
TV aR(X, p) = E(X|X > V aR(X, p)).
@freakonometrics 15
16. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Utilisation de l’approximation GPD pour estimer V aR(X, p)
La première méthode (Smith (1987)) respose sur la modélisation de la queue de la
distribution, au delà d’un seuil u. Notons
Nu =
n
i=1
I(Xi > u),
le nombre de dépassement de u dans un échantillon X1, ..., Xn. Pour x > u,
F(x) = P(X > x) = P(X > u)P(X > x|X > u) = F(u)P(X > x|X > u),
où
P(X > x|X > u) = F(x)u(x − u)
avec Fu(t) = P(X − u ≤ t|X > u) ∼ Hξ,β(t), pour des valeurs ξ et β (Hξ,β
désignant la loi GPD).
@freakonometrics 16
17. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Utilisation de l’approximation GPD pour estimer V aR(X, p)
Aussi, un estimateur naturel de F(x) repose sur l’utilisation d’un estimateur
empirique de F(u), et de l’approximation GPD de Fu(x), i.e.
F(x) = 1 −
Nu
n
1 + ξ
x − u
β
−1/ξ
pour tout x > u, et u suffisement grand.
Aussi, un estimateur nature de la V aR(X, p) est Qp défini par
Qp = u +
β
ξ
n
Nu
(1 − p)
−ξ
− 1
.
Notons qu’un intervalle de confiance asymptotique peut être obtenu en
considérant la méthode de vraisemblance profilée (profile likelihood).
@freakonometrics 17
18. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
1 > library(evir)
2 > data(danish)
3 > plot(danish ,type="h")
on peut estimer ξ avec l’estimateur de Hill ξHill
n,k ,
1 > hill(danish ,option="xi")
0 500 1000 1500 2000
050100150200250
Index
danish
15 196 400 604 808 1035 1284 1533 1782 2031
0.30.40.50.60.70.80.9
29.00 4.71 3.02 2.23 1.85 1.61 1.40 1.24 1.07
Order Statistics
xi(CI,p=0.95)
Threshold
@freakonometrics 18
19. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
ou directement un quantile Qp pour une probabilité
élevée, e.g. = 99.5%
1 > hill(danish ,option="quantile",p=.995)
Sinon, on peut utiliser l’approximation GPD. On effet,
le Pareto plot semble confirmer une loi de Pareto
1 > n=length(danish)
2 > Xs=sort(danish ,decreasing=TRUE)
3 > plot(log(Xs),log ((1:n)/(n+1)))
15 196 400 604 808 1035 1284 1533 1782 2031
404550556065
29.00 4.71 3.02 2.23 1.85 1.61 1.40 1.24 1.07
Order Statistics
Quantile,p=0.995
Threshold
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
0 1 2 3 4 5
−6−4−20
@freakonometrics 19
20. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
L’estimation de la loi GPD donne
1 > u=5
2 > nu=sum(Xs >u)
3 > p=.995
4 > gpd(danish ,5)
5 $par.ests
6 xi beta
7 0.6320499 3.8074817
8
9 $par.ses
10 xi beta
11 0.1117143 0.4637269
15 196 400 604 808 1035 1284 1533 1782 2031
404550556065
29.00 4.71 3.02 2.23 1.85 1.61 1.40 1.24 1.07
Order Statistics
Quantile,p=0.995
Threshold
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
0 1 2 3 4 5
−6−4−20
@freakonometrics 20
21. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
La densité de la loi de Pareto, en fonction de ξ et σ est
gξ,σ(x) =
1
σ
1 +
ξx
σ
− 1
ξ −1
que l’on peut écrire en fonction de ξ et Qp, si on considère une loi de Pareto au
delà d’un seuil u,
gξ,Qp
(x) =
n
Nu
(1 − p)
−ξ
− 1
ξ[Qp − u]
1 +
n
Nu
(1 − p)
−ξ
− 1
[Qp − u]
· x
− 1
ξ −1
1 > gq=function(x,xi ,q){
2 + ( (n/nu*(1-p) )^(-xi) -1)/(xi*(q-u))*(1+((n/nu*(1-p))^(-xi) -1)/(q-u)
*x)^(-1/xi -1)}
@freakonometrics 21
22. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
On cherche le maximum de vraisemblance
(ξ, Qp) = argmax
Nu−1
i=0
log[gξ,Qp
(xn−i:n)]
log[L(ξ,Qp)]
1 > loglik=function(param){
2 + xi=param [2];q=param [1]
3 + lg=function(i) log(gq(Xs[i],xi ,q))
4 + return(-sum(Vectorize(lg)(1: nu)))}
50 100 150 200
0.20.40.60.81.0
Q
XI
−920
−930
−950
−975
−975
−1000
−1000
−1500
@freakonometrics 22
23. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
ou mieux, le maximum de la vraisemblance profilée (seul Qp nous intéresse),
Qp = argmax max
ξ
{log[L(ξ, Qp)]}
1 > PL=function(Q){
2 + profilelikelihood =function(xi){
3 + loglik(c(Q,xi))}
4 + return(optim(par =.6,fn= profilelikelihood )$
value)} 30 40 50 60 70 80 90 100
−960−950−940−930−920
99.5% quantile
ProfileLikelihood
1 > (OPT=optimize(f=PL ,interval=c(10 ,100)))
2 $minimum
3 [1] 51.42182
4
5 $objective
6 [1] 918.2469
@freakonometrics 23
24. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Mise en oeuvre pratique, quantile bi-centennaire
On peut utiliser le test de rapport de vraisemblance pour obtenir un intervalle de
confiance, puisque
2 log[Lp(Qp)] − log[Lp(Qp)]
L
→ χ2
(1), lorsque n → ∞.
1 > XQ=seq (30 ,100 , length =101)
2 > L=Vectorize(PL)(XQ)
3 > up=OPT$objective
4 > I=which(-L>=-up -qchisq(p=.95 ,df=1)/2)
5 > range(XQ[I])
6 [1] 45.4 60.1
On peut utiliser le code suivant
1 > gpd.q(tailplot(gpd(Xs ,u)) ,.995)
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
5 10 20 50 100 200
2e−041e−035e−032e−021e−01
x (on log scale)
1−F(x)(onlogscale)
9995
@freakonometrics 24
25. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Choix du seuil u
Supposons que l’on choisisse un seuil u tel que u = Xn−k:n, alors
xp,k = Xn−k:n +
βk
ξk
n
k
(1 − p)
−ξk
− 1 ,
pour k > n(1 − p). On notera que si k = n(1 − p), xp,k coïncide avec l’estimateur
empirique du quantile, i.e. X[np+1]:n. Notons de plus que
MSE(xp) = V ar(xp) + E(xp − V aR(X, p))2
.
@freakonometrics 25
26. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Estimation de la période de retour
Pour rappel, le niveau Pm période de retour associée à une période de m années
(niveau qui sera excédé, en moyenne, toutes les m observations) est la solution de
P(X > u) 1 + ξ
Pm − u
σ
1−ξ
=
1
m
,
si X|X > u peut être modélisé par une loi de Pareto généralisée, soit
Pm = u +
σ
ξ
[mP(X > u)]ξ
− 1 ,
sous l’hypothèse où ξ = 0, et
Pm = u + σ log(mP(X > u)),
si ξ = 0.
@freakonometrics 26
27. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Utilisation de l’estimateur de Hill
Rappelons que l’estimateur de Hill basé sur la statistique d’ordre est
ξHill
n,k =
1
k
k
i=1
log Xn−i:n − log Xn−k+1:n,
qui est particulièrement intéressante si ξ > 0.
Rappelons que F(x) = P(X > x) = x−1/ξ
L(x), et donc, pour x ≥ Xn−k:n et k
suffisement faible,
F(x)
F(Xn−k:n)
=
L(x)
L(Xn−k:n)
x
Xn−k:n
−1/ξ
.
Si l’on suppose que le ration des fonctions à variation lente peut être supposé
comme négligeable, alors
F(x) ∼ F(Xn−k:n)
x
Xn−k:n
−1/ξ
@freakonometrics 27
28. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Aussi, un estimateur naturel de la fonction de répartition est
F(x) = 1 −
k
n
F(Xn−k:n)
x
Xn−k:n
−1/ξHill
n,k
, pour x ≥ Xn−k:n.
En considérant l’inverse de cette fonction, on en déduit l’estimation naturelle de
la V aR suivante,
xHill
p = Xn−k:n
n
k
(1 − p)
−ξHill
n,k
,
pour des quantiles xp tels que p > 1 − k/n. Notons que cet estimateur peut
également s’écrire
xHill
p,k = Xn−k:n + Xn−k:n
n
k
(1 − p)
−ξHill
n,k
− 1 ,
qui peut ainsi être comparé à l’estimateur obtenu par maximum de
vraisemblance sur le modèle GPD,
xp,k = Xn−k:n +
βk
ξk
n
k
(1 − p)
−ξk
− 1 ,
@freakonometrics 28
29. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Dans quels cas peut-on utiliser cette méthode ?
Rappelons que l’estimateur de Hill n’est pertinent que si ξ > 0 (et donc pas si
ξ = 0). Dans ce dernier cas, des mauvaises conclusions peuvent être déduites.
De plus les propriétés de cet estimateur à distance finie (en particulier pour des
petits échantillons) sont relativement décevantes. Néanmoins, les propriétés de
l’estimateur de Hill ont été beaucoup plus étudiées dans la littérature que celui
du maximum de vraisemblance.
@freakonometrics 29
30. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
La tarification en réassurance
Le principe de base est le calcul de la prime pure, correspondant à l’espérance
mathématique des paiements, avec franchise.
Pour un traité stop-loss, de portée infinie, le montant de l’indemnité versée par le
réassureur est (X − d)+, et la prime pure est donc
E[(X − d)+] = E[(X − d)+|X ≤ d] · P[X ≤ d] + E[(X − d)+|X > d] · P[X > d],
ce qui se simplifie simplement en
πd = E[(X − d)+] = E[X − d|X > d] · P[X > d].
La quantité E[X − d|X > d] correspond à la fonction de moyenne en excès, notée
e(d),
e(x) =
1
P[X > x]
∞
x
P[X > t]dt.
@freakonometrics 30
31. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
La méthode du burning-cost
On utilise ici tout simple un estimateur empirique (non paramétrique) de la
prime pure,
πd =
1
n
n
i=1
(Xi − d)+
Par très robuste, impossible de valoriser une tranche de niveau élevé.
@freakonometrics 31
32. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
L’approche paramétrique et la loi de Pareto
Rappelons que les sinistres suivent ici une loi de Pareto, i.e. la densité s’écrit
f(x) = α · θα
· x−α−1
, x ≥ θ,
et la fonction de répartition
F(x) = P(X ≤ x) = 1 −
θ
x
α
, x ≥ θ.
(θ est parfois noté OP la point d’observation (observation point)).
Remarque L’estimateur du maximum de vraisemblance de α vérifie
1
α
=
1
n
n
i=1
log
Xi
θ
,
si θ est connu (sinon, on considère θ = min{X1, ..., Xn}).
On décompose alors le calcul de la prime pure de façon usuelle
espérance = fréquence × coût moyen
@freakonometrics 32
33. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Coût à charge de la tranche, expected excess loss
Pour les sinistres dont le montant est compris entre DE (la franchise) et EP (le
maximum), le montant de la perte en excès s’écrit min{CO, max{X − DE, 0}},
et donc
EL =
EP
DE
(x − DE)f(x)dx +
∞
EP
COf(x)dx =
DE
1 − α
(RL1−α
− 1) si α = 1
(et EL = DE · log(RL) si α = 1), où pour rappel, EL = (CO + DE)/CO désigne
la taille relative de la tranche.
La fréquence
On notera également que la fréquence associée à un niveau x ≥ θ est
FQ(x) = FQ(θ) ·
θ
x
α
@freakonometrics 33
34. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
La prime pure
Enfin, la prime est alors donnée par
π = FQ(DE) · EL = FQ(θ) ·
θ
DE
α
·
DE
1 − α
(RL1−α
− 1) si α = 1.
Le rate-on-line est alors donnée, pour α = 1,
ROL =
π
CO
=
FQ(θ) · θα
· (RL1−α
− 1)
DEα · (RL − 1) · (1 − α)
Peut-on toujours utiliser la loi de Pareto ?
Cette approche a été proposée, étudiée et implémantée depuis longtemps (Swiss
Re (2001), Huyghues-Beaufond (1991)).
Cette loi peut effectivement être souvent utile, elle est simple d’utilisation, et
facile à ajuster (un seul paramètre si on suppose connu θ).
@freakonometrics 34
35. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
La tarification par exposition
Il s’agit ici de calculer la prime pour des traités non-proportionels.
Soit M la somme assurée. On notera xi = Xi/M les montants de sinistres (i.i.d.)
et d = D/M où D désigne la priorité du traité en excédent de sinistre. La prime
pure, du point de vue de l’assureur est
E [min {X, D}] E [N] = ME [min {x, d}] E [N]
et la prime pure dite de base sera
E [X] E [N] = ME [min {x, 1}] E [N] .
La courbe d’exposition G est le rapport entre la prime de l’assureur et la prime
de base, G : [0, 1] → [0, 1]
G (d) =
E [min {x, d}]
E [min {x, 1}]
pour d ∈ [0, 1] .
@freakonometrics 35
36. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
La courbe d’exposition
La courbe d’exposition est définie simplement par
G(d) =
E[min{X, d}]
E([X]
=
1
E([X]
d
0
(1 − F(t))dt.
On notera que cette fonction G vérifie G(0) = 0, G(1) = 1, qu’elle est croissante
et concave. En effet,
∂G(d)
∂d
=
1 − F(d)
f(d)
≥ 0,
et
∂2
G(d)
∂d2
=
−f(d)
f(d)
≤ 0,
La concavité reflète la part des grands risques: plus la courbe est proche de la
diagonale, plus la part des grands risques dans la charge totale tend à être
négligeable.
@freakonometrics 36
37. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Example Pour une loi de Pareto P(θ, α),
G(d) =
1 − θα−1
(θ + d)1−α
1 − θα−1(θ + 1)1−α
.
Example Bernegger (1997) a suggéré une modélisation de G à l’aide de la
fonction suivante à deux paramètres,
Ga,b (x) =
log (a + bx
) − log (a + 1)
log (a + b) − log (a + 1)
,
correspond au modèle dit MBBEFD
(Maxwell-Boltzman-Bose-Einstein-Fermi-Dirac), issu de la littérature de physique
statistique. La distribution X sous-jacente est alors
FX (x) =
1 − (a+1)bt
a+bt pour 0 ≤ t < 1
1 pour t = 1
@freakonometrics 37
38. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple des Ouragans
On utilise la base Normalized Hurricane Damages in the United States pour la
période 1900-2005, à partir de Pielke et al. (2008).
1 > library(gdata)
2 > db=read.xls(
3 + "http:// sciencepolicy .colorado.edu/ publications /special/public_data
_may_2007. xls",sheet =1)
4 > tail(db)
5 Year Hurricane. Description State Category Base.Economic.Damage
6 202 2005 Cindy LA 1 320 ,000 ,000
7 203 2005 Dennis FL 3 2 ,230 ,000 ,000
8 204 2005 Katrina LA ,MS 3 81 ,000 ,000 ,000
9 205 2005 Ophelia NC 1 1 ,600 ,000 ,000
10 206 2005 Rita TX 3 10 ,000 ,000 ,000
11 207 2005 Wilma FL 3 20 ,600 ,000 ,000
@freakonometrics 38
40. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple des Ouragans
et on obtient alors des chiffres
1 > base=db[ ,1:4]
2 > base$Base.Economic.Damage=Vectorize( stupidcomma )(db$Base.Economic.
Damage)
3 > base$Normalized.PL05=Vectorize( stupidcomma )(db$Normalized.PL05)
4 > base$Normalized.CL05=Vectorize( stupidcomma )(db$Normalized.CL05)
et on obtient la base suivante
1 > tail(base)
2 Year Hurricane. Description State Category Base.Economic.Damage
3 202 2005 Cindy LA 1 3.20e+08
4 203 2005 Dennis FL 3 2.23e+09
5 204 2005 Katrina LA ,MS 3 8.10e+10
6 205 2005 Ophelia NC 1 1.60e+09
7 206 2005 Rita TX 3 1.00e+10
8 207 2005 Wilma FL 3 2.06e+10
@freakonometrics 40
41. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple des Ouragans
La base historique de coûts d’ouragans est
1 > plot(base$Normalized .PL05/1e9 ,type="h",ylim=
c(0 ,155))
0 50 100 150 200
050100150
Index
L’analyse se fait (comme en assurance non-vie classique) en deux temps : on va
modéliser la fréquence annuelle, et les coûts individuels. La base des fréquence
s’obtient avec
1 > TB <- table(base$Year)
2 > years <- as.numeric(names(TB))
3 > counts <- as.numeric(TB)
4 > years0 =(1900:2005) [which(! (1900:2005)%in%years)]
5 > db <- data.frame(years=c(years ,years0),
@freakonometrics 41
42. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
6 + counts=c(counts ,rep(0, length(years0))))
7 > db [88:93 ,]
8 years counts
9 88 2003 3
10 89 2004 6
11 90 2005 6
12 91 1902 0
13 92 1905 0
14 93 1907 0
On a, en moyenne, deux ouragans par an,
1 > mean(db$counts)begin{lstlisting }[ escapechar=
,style=mystyle,firstnumber=1][1]
1.95283
On peut aussi tenter un modèle de régression, linéaire
1 > reg0 <- glm(counts~years ,data=db ,family=poisson(link="identity"),
2 + start=lm(counts~years ,data=db)$ coefficients )
@freakonometrics 42
43. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
ou exponentiel
1 > reg1 <- glm(counts~years ,data=db ,family=poisson(link="log"))
1 > (predictions=cbind(constant=mean(db$counts),linear= cpred0 [126] ,
exponential=cpred1 [126]))
2 constant linear exponential
3 126 1.95283 3.573999 4.379822
@freakonometrics 43
44. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Pour modéliser les coûts individuels des ouragans
1 > library(evir)
2 > hill(base$Normalized .PL05)
15 30 45 60 75 90 107 126 145 164 183 202
02468
1.65e+10 3.57e+09 9.81e+08 2.46e+08 7.75e+07 4.29e+06
Order Statistics
xi(CI,p=0.95)
Threshold
Considérons une compagnie qui a 5% de part de marché, et qui veut valoriser un
traité de réassurance, de franchise 2 milliards, avec une limite de 4 milliards.
Considérons une loi de Pareto, au delà de u = 500 millions,
1 > threshold =.5
2 > (gpd.PL <- gpd(base$Normalized .PL05/1e9/20, threshold)$par.ests)
3 xi beta
4 0.4424669 0.6705315
Pour rappel 1 ouragan sur 8 atteint ce seuil u,
@freakonometrics 44
45. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
1 > mean(base$Normalized .CL05/1e9/20 >.5)
2 [1] 0.1256039
On peu alors valoriser un contrat de réassurance To compute it we can use
1 > E <- function(yinf ,ysup ,xi ,beta){
2 + as.numeric(integrate(function(x) (x-yinf)*dgpd(x,xi ,mu=threshold ,
beta), lower=yinf ,upper=ysup)$value +(1- pgpd(ysup ,xi ,mu=threshold ,
beta))*(ysup -yinf)) }
Si on espère être touché par 2 ouragans, une année donnée
1 > predictions [1]
2 [1] 1.95283
que chaque ouragan a 12.5% de chancess de coûter plus de 500 million
1 > mean(base$Normalized .PL05/1e9/20 >.5)
2 [1] 0.1256039
et que si un ouragan dépasse 500 million, le coût moyen en excès (en millions) est
@freakonometrics 45
46. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
1 > E(2,6,gpd.PL[1],gpd.PL [2])*1e3
2 [1] 330.9865
de telle sorte que la prime pure d’un contrat de réassurance serait
1 > predictions [1]*mean(base$Normalized.PL05/1e9/20 >.5)*
2 + E(2,6,gpd.PL[1], gpd.PL [2])*1e3
3 [1] 81.18538
en millions d’euros, pour une couverture de 4 milliards, en excès de 2.
@freakonometrics 46
47. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple de la Perte d’Exploitation
Considérons les données suivantes, de perte d’exploitation,
1 > library(gdata)
2 > db=read.xls("http://perso.univ -rennes1.fr/arthur. charpentier /SIN_
1985_2000 -PE.xls", sheet =1)
Comme pour tout contrat d’assurance, on distingue
• le nombre moyen de sinistres
• le coût moyen des sinistres
La encore (pour l’instant), on n’utilise pas de covariables.
@freakonometrics 47
48. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Fréquence des sinistres en Perte d’Exploitation
1 > date=db$DSUR
2 > D=as.Date(as.character(date),format="%Y%m%d"
)
3 > vD=seq(min(D),max(D),by=1)
4 > sD=table(D)
5 > d1=as.Date(names(sD))
6 > d2=vD[-which(vD%in%d1)]
7 > vecteur.date=c(d1 ,d2)
8 > vecteur.cpte=c(as.numeric(sD),rep(0, length(
d2)))
9 > base=data.frame(date=vecteur.date ,cpte=
vecteur.cpte)
1985 1990 1995 2000
05101520
Utilisons une régression de Poisson pour modéliser la fréquence journalière de
sinistres.
1 > regdate=glm(cpte~date ,data=base ,family=poisson(link="log"))
@freakonometrics 48
49. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
2 > nd2016=data.frame(date=seq(as.Date(as.character (20160101) ,format="
%Y%m%d"),
3 + as.Date(as.character (20161231) ,format="%Y%m
%d"),by=1))
4 > pred2016 =predict(regdate ,newdata=nd2016 ,type="response")
5 > sum(pred2016)
6 [1] 163.4696
Il y a environ 165 sinistres, par an.
@freakonometrics 49
50. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
1985 1990 1995 2000
050100150
Date
Coûts('000000)
Sur 16 ans, le 32ème plus gros sinistre (2 par an) était de l’ordre de 15 million.
1 > quantile(db$COUTSIN ,1 -32/2400)/1e6
2 98.66667%
3 15.34579
@freakonometrics 50
51. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple de la Perte d’Exploitation
Considérons un contrat ave une franchise de 15 millions, pour une converture
totale de 35 millions. Pour la compagnie de réassurance, le coût espéré est
E(g(X)) avec
g(x) = min{35, max{x − 15, 0}}
La fonction d’indemnité est
1 > indemn=function(x) pmin ((x -15)*(x >15) ,50-15)
Sur 16 ans, l’indemnité moyenne aura été de
1 > mean(indemn(db$COUTSIN/1e6))
2 [1] 0.1624292
@freakonometrics 51
52. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple de la Perte d’Exploitation
Par sinistre, la compagnie de réassurance pait, en moyenne, 162,430 euros. Donc
avec 160 sinistres par an, la prime pure - burning cost - est de l’ordre de 26
millions.
1 > mean(indemn(db$COUTSIN/1e6))*160
2 [1] 25.98867
On peut aussi tenter un modèle paramétrique, de type Pareto. Les trois
paramètres sont
• le seuil µ (considéré comme fixe)
• le paramètre d’échelle (scale) σ (noté aussi β)
• the tail index ξ
@freakonometrics 52
53. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple de la Perte d’Exploitation
Prenons un seuil u plus faible que la franchise, e.g. 12 million.
1 > gpd.PL <- gpd(db$COUTSIN ,12 e6)$par.ests
2 > gpd.PL
3 xi beta
4 7.004147e-01 4.400115e+06
et on pose
1 > E <- function(yinf ,ysup ,xi ,beta ,threshold){
2 + as.numeric(integrate(function(x) (x-yinf)*dgpd(x,xi ,mu=threshold ,
beta),lower=yinf ,upper=ysup)$value +(1- pgpd(ysup ,xi ,mu=threshold ,
beta))*(ysup -yinf))
3 + }
@freakonometrics 53
54. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple de la Perte d’Exploitation
Sachant qu’un sinistre dépasse 12 million, le remboursement moyen est de l’ordre
de 6 millions
1 > E(15e6 ,50e6 ,gpd.PL[1], gpd.PL [2] ,12e6)
2 [1] 6058125
La probabilité qu’un sinistre dépasse u est
1 > mean(db$COUTSIN >12 e6)
2 [1] 0.02639296
Aussi, avec 160 sinistres par an,
1 > p
2 [1] 159.4757
et 2.6% des sinistres qui dépassent 12 millions,
1 > mean(db$COUTSIN >12 e6)
2 [1] 0.02639296
@freakonometrics 54
55. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Réassurance, Exemple de la Perte d’Exploitation
Finallement, la prime pure est
1 > p*mean(db$COUTSIN >12 e6)*E(15e6 ,50e6 ,gpd.PL[1], gpd.PL[2] ,12 e6)
2 [1] 25498867
qui est proche de la valeur empirique.
On peut regarder la robustesse au choix de u,
1 > esp=function(threshold =12e6 ,p=sum(pred2010)){
2 + (gpd.PL <- gpd(db$COUTSIN ,threshold)$par.
ests)
3 + return(p*mean(db$COUTSIN >threshold)*E(15e6
,50e6 ,gpd.PL[1], gpd.PL[2], threshold))}
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
2 4 6 8 10 12 14
2628303234
Threshold
Prixdelacouverture('000000)
@freakonometrics 55
56. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Retour aux données RC
On peut utiliser un modèle avec comme loi pour les coûts une loi de Pareto
1 > ?gamlss.family
2 > ?PARETO2
La loi de Pareto a ici pour densité
f(y|µ, σ) =
1
σ
µ
1
σ [y + µ]− 1
σ −1
et sa moyenne est alors
E[Y ] =
µσ
1 − σ
avec ici
E[Y |X = x] =
exT
α
exT
β
1 − exTβ
@freakonometrics 56
57. Arthur CHARPENTIER - Actuariat de l’Assurance Non-Vie, # 8
Retour aux données RC
1 > seuil =1500
2 > regpareto=gamlss(cout~ageconducteur , sigma.formula= ~ageconducteur
,data=base_RC[base_RC$cout >seuil ,], family=PARETO2(mu.link = "log",
sigma.link = "log"))
3 -------------------------------------------------------------------
4 Mu link function: log
5 Mu Coefficients :
6 Estimate Std. Error t value Pr(>|t|)
7 (Intercept) 9.166591 0.212249 43.188 <2e-16 ***
8 ageconducteur 0.007024 0.004420 1.589 0.113
9
10 -------------------------------------------------------------------
11 Sigma link function: log
12 Sigma Coefficients :
13 Estimate Std. Error t value Pr(>|t|)
14 (Intercept) -0.793548 0.166166 -4.776 2.58e -06 ***
15 ageconducteur -0.009316 0.003513 -2.652 0.00835 **
@freakonometrics 57