Slides nantes

Arthur CHARPENTIER - Pricing insurance linked securities: interplay between finance and insurance.
Pricing insurance linked securities :
interplay between ﬁnance and insurance
Arthur Charpentier
http ://perso.univ-rennes1.fr/arthur.charpentier/
Atelier Finance & Risque
Universit´e de Nantes, Avril 2008
1

survey of literature
• Fundamental asset pricing theorem, in finance, Cox & Ross (JFE, 1976),
Harrison & Kreps (JET, 1979), Harrison & Pliska (SPA, 1981, 1983).
Recent general survey
– Dana & Jeanblanc-Picqué (1998). Marchés financiers en temps continu :
valorisation et équilibre. Économica.
– Duffie (2001). Dynamic Asset Pricing Theory. Princeton University Press.
– Bingham & Kiesel (2004). Risk neutral valuation. Springer Verlag
• Premium calculation, in insurance.
– Bühlmann (1970) Mathematical Methods in Risk Theory. Springer Verlag.
– Goovaerts, de Vylder & Haezendonck (1984). Premium Calculation in
Insurance. Springer Verlag.
– Denuit & Charpentier (2004). Mathématiques de l’assurance non-vie, tome
1. Économica.
2

survey of literature
• Price of uncertain quantities, in economics of uncertainty, von Neumann
& Morgenstern (1944), Yaari (E, 1987). Recent general survey
– Quiggin (1993). Generalized expected utility theory : the rank-dependent
model. Kluwer Academic Publishers.
– Gollier (2001). The Economics of Risk and Time. MIT Press.
• Bentoglio & Betbeze (2005). L’Etat et l’assurance des risques nouveaux.
La Documentation Fran¸caise.
3

Agenda
A short introduction to insurance risks
• Catastrophe and (very) large risks
• Mortality risks, from short term pandemic to long term risk
Insurance linked securities
• Insurance linked securities
• Catastrophe or mortality bonds
Financial versus insurance pricing
• Insurance : from pure premium to other techniques
• Finance : from complete to incomplete markets
Pricing Insurance linked
• Distorted premium
• Indiﬀerence utility
4

Agenda
5

from mass risk to large risks
insurance is “the contribution of the many to the misfortune of the few”.
1. judicially, an insurance contract can be valid only if claim occurrence satisfy
some randomness property,
2. the “game rule” (using the expression from Berliner (Prentice-Hall, 1982),
i.e. legal framework) should remain stable in time,
3. the possible maximum loss should not be huge, with respect to the insurer’s
solvency,
4. the average cost should be identiﬁable and quantiﬁable,
5. risks could be pooled so that the law of large numbers can be used
(independent and identically distributed, i.e. the portfolio should be
homogeneous),
6. there should be no moral hazard, and no adverse selection,
7. there must exist an insurance market, in the sense that demand and supply
should meet, and a price (equilibrium price) should arise.
6

risk premium and regulatory capital (points 4 and 5)
Within an homogeneous portfolios (Xi identically distributed), sufficiently large
(n → ∞),
X1 + ... + Xn
n
→ E(X). If the variance is finite, we can also derive a
confidence interval (solvency requirement), i.e. if the Xi’s are independent,
n
i=1
Xi ∈


nE(X) ± 1.96
√
nVar(X)
risk based capital need


 with probability 95%.
High variance, small portfolio, or nonindependence implies more volatility, and
therefore more capital requirement.
7

independent risks, large portfolio (e.g. car insurance)
q
q
independent risks, 10,000 insured
q
q
Fig. 1 – A portfolio of n = 10, 000 insured, p = 1/10.
8

q
q
independent risks, 10,000 insured, p=1/10
900 950 1000 1050 1100 1150 1200
0.0000.0020.0040.0060.0080.0100.012
casindépendant,p=1/10,n=10,000
distribution de la charge totale, N((np,, np((1 −− p))))
RUIN
(1% SCENARIO)
RISK−BASED CAPITAL
NEED +7% PREMIUM
969
9

q
q
independent risks, 10,000 insured, p=1/10
900 950 1000 1050 1100 1150 1200
0.0000.0020.0040.0060.0080.0100.012
casindépendant,p=1/10,n=10,000
RUIN
(1% SCENARIO)
NEED +7% PREMIUM
986
10

independent risks, small portfolio (e.g. ﬁre insurance)
q
q
independent risks, 400 insured
q
q
Fig. 4 – A portfolio of n = 400 insured, p = 1/10.
11

q
q
independent risks, 400 insured, p=1/10
30 40 50 60 70
0.000.010.020.030.040.050.06
casindépendant,p=1/10,n=400
RUIN
(1% SCENARIO)
NEED +35% PREMIUM
39
12

q
q
independent risks, 400 insured, p=1/10
30 40 50 60 70
0.000.010.020.030.040.050.06
casindépendant,p=1/10,n=400
RUIN
(1% SCENARIO)
NEED +35% PREMIUM
48
13

nonindependent risks, large portfolio (e.g. earthquake)
q
q
independent risks, 10,000 insured
q
q
Fig. 7 – A portfolio of n = 10, 000 insured, p = 1/10, nonindependent.
14

q
q
non−independent risks, 10,000 insured, p=1/10
1000 1500 2000 2500
0.0000.0020.0040.0060.0080.0100.012
distribution de la charge totale
nonindependantcase,p=1/10,n=10,000
RUIN
(1% SCENARIO)
NEED +105% PREMIUM
897
15

q
q
non−independent risks, 10,000 insured, p=1/10
1000 1500 2000 2500
0.0000.0020.0040.0060.0080.0100.012
distribution de la charge totale
nonindependantcase,p=1/10,n=10,000
RUIN
(1% SCENARIO)
NEED +105% PREMIUM
2013
16

some stylized facts about natural disasters
“climatic risk in numerous branches of industry is more important than the risk
of interest rates or foreign exchange risk” (AXA 2004, quoted in Ceres (2004)).
Fig. 10 – Major natural catastrophes (from Munich Re (2006).)
17

Some stylized facts : natural catastrophes
Includes hurricanes, tornados, winterstorms, earthquakes, tsunamis, hail,
drought, ﬂoods...
Date Loss event Region Overall losses Insured losses Fatalities
25.8.2005 Hurricane Katrina USA 125,000 61,000 1,322
23.8.1992 Hurricane Andrew USA 26,500 17,000 62
17.1.1994 Earthquake Northridge USA 44,000 15,300 61
21.9.2004 Hurricane Ivan USA, Caribbean 23,000 13,000 125
19.10.2005 Hurricane Wilma Mexico, USA 20,000 12,400 42
20.9.2005 Hurricane Rita USA 16,000 12,000 10
11.8.2004 Hurricane Charley USA, Caribbean 18,000 8,000 36
26.9.1991 Typhoon Mireille Japan 10,000 7,000 62
9.9.2004 Hurricane Frances USA, Caribbean 12,000 6,000 39
26.12.1999 Winter storm Lothar Europe 11,500 5,900 110
Tab. 1 – The 10 most expensive natural catastrophes, 1950-2005 (from Munich
Re (2006)).
18

basics on extreme value theory
When modeling large claims (industrial ﬁre, business interruption,...) : extreme
value theory framework is necessary.
The Pareto distribution appears naturally when modeling observations over a
given threshold,
F(x) = P(X ≤ x) = 1 −
x
x0
b
, where x0 = exp(−a/b)
Then equivalently log(1 − F(x)) ∼ a + b log x, i.e. for all i = 1, ..., n,
log(1 − Fn(Xi)) ∼ a + b · log Xi.
Remark : if −b ≥ 1, then EP(X) = ∞, the pure premium is inﬁnite.
The estimation of b is a crucial issue (see Zajdenweber (JRI, 1998) or from
Charpentier (BFA, 2005).)
19

goodness of ﬁt of the Pareto distribution
! " # $ % &!
!'!#!(!"!&!
!o#!lo# &areto +lot, -urri0ane losses
)oga-it01 o3 t0e loss a1ount
)oga-it01o39u1ulate:;-o<a<ilites
=>'?@ slo;e> ! &A"'B
=>"'?@ slo;e> !!A%$#
0 20 40 60 80 1000.51.01.52.0
Hill estimator of the tail index
Percentage of bservations exceeding the threshold
Tailindex,with95%confidenceinterval
Fig. 11 – Pareto modeling of hurricanes losses (Pielke & Landsea (WF, 1998)).
20

longevity and mortality risks
Year
Age
0 20 40 60 80 100
5e−055e−045e−035e−02
Age
1899
1948
1997
1900 1920 1940 1960 1980 2000
5e−042e−035e−032e−025e−02
Age
60 years old
40 years old
20 years old
Fig. 12 – Mortality rate surface (function of age and year).
21

longevity and mortality risks
Main problem= forecasting future mortality rates, i.e.
m(x, t) =
# deaths during calendar year t aged x last birthday
average population during calendar year t aged x last birthday
cf. LifeMetrics (JPMorgan),
– Lee & Carter (JASA, 1992), log m(x, t) = β
(1)
x + β
(2)
x κ
(2)
t ,
– Renshaw & Haberman (IME, 2006), log m(x, t) = β
(1)
x + β
(2)
x κ
(2)
t + β
(3)
x γ
(3)
t−x,
– Currie (2006), log m(x, t) = β
(1)
x + κ
(2)
t + γ
(3)
t−x,
1950 1960 1970 1980 1990 2000
−15−10−50510
Lee Carter model, total France (fit701)
Year (y)
Kappa(LeeCarter)
55 60 65 70 75 80 85
−4.5−4.0−3.5−3.0−2.5−2.0
Age (x)
Beta1(LeeCarter)
55 60 65 70 75 80 85
0.0300.0320.0340.036
Age (x)
Beta2(LeeCarter)
22

Agenda
23

insurance linked securities, from insurance to ﬁnance
Finn & Lane (1995) “there are no right price of insurance, there is simply the
transacted market price which is high enough to bring forth sellers and low
enough to induce buyers”.
traditional indemni industry parametr ilw
reinsurance securitiza loss securitiza derivatives
securitization
traditional indemnity indust parametric ilw
reinsurance securitization loss securitization derivatives
securitiza
24

(re)insurance principle
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
0.0 0.2 0.4 0.6 0.8 1.0
02468101214
Reinsurance Excess−of−Loss Indemnity
Time (exposure period)
Insurancelosses
q
q
INSURANCE COMPANY
REINSURANCE COMPANY
PREMIUM (2)
Fig. 13 – (re)insurance excess-of-loss principle.
25

(re)insurance principle
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q q
q q
0.2 0.4 0.6 0.8
02468101214
Reinsurance Excess−of−Loss Indemnity
Insurancelosses
q
q
INSURANCE COMPANY
REINSURANCE COMPANY
PREMIUM (2)INDEMNITY (5)
Fig. 14 – (re)insurance excess-of-loss principle.
26

catastrophe or mortality bonds
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
0.0 0.2 0.4 0.6 0.8 1.0
02468101214
Parametric cat bond
Insurancelosses
q
q
INSURANCE COMPANY
SPECIAL PURPOSE VEHICLE
PREMIUM (2)
FINANCIAL INVESTORS
CAT BOND
NOMINAL (100)
Fig. 15 – catastrophe or mortality bonds principle.
27

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
0.0 0.2 0.4 0.6 0.8 1.0
02468101214
Parametric cat bond
Insurancelosses
q
q
INSURANCE COMPANY
PREMIUM (2)
FINANCIAL INVESTORS
CAT BOND
NOMINAL (100)
NOMINAL + RISKY
COUPON (100+5+2)
28

q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q q
q q
0.2 0.4 0.6 0.8
02468101214
Parametric cat bond
Insurancelosses
q
q
INSURANCE COMPANY
PREMIUM (2)
FINANCIAL INVESTORS
CAT BOND
NOMINAL (100)
NOMINAL + RISKY
COUPON (100+5−9)
INDEMNITY (9)
29

comparing (re)insurance and ﬁnancial markets
From AIG (1997),
q
q
q
q
q
q
q
q
q
q
q
World Equity market, $15,000 billion
World governement bond
market, $9,000 billion
US non−life insurance
and reinsurance capital
$200 billion
Hurricane Andrew
$20 billion
Hurricane in Florida
$100 billion (potential)
=⇒ capital markets have signiﬁcantly greater capacity then (re)insurance markets
30

insurance linked securities, from insurance to finance
Mortality/mortality risk :
• deviation of the trend (e.g. consequence of obesity)
• one off deviation (e.g. avian flu, Spanish influenza)
Natural catastrophe risk :
• wind related (e.g. hurricanes in Florida or winter storms in Europe)
• flood related (e.g. Paris)
• soil related (e.g. earthquakes in California or Japan, or volcanic eruptions)
Man-made disasters :
• terrorism (e.g. 9/11)
• institutional investors : pension funds, (re)insurance ?
• financial markets ?
• governments ?
31

Agenda
32

the pure premium as a technical benchmark
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century
proposed to evaluate the “produit scalaire des probabilités et des gains”,
< p, x >=
n
i=1
pixi =
n
i=1
P(X = xi) · xi = EP(X),
based on the “règle des parties”.
For Quételet, the expected value was, in the context of insurance, the price that
guarantees a financial equilibrium.
From this idea, we consider in insurance the pure premium as EP(X). As in
Cournot (1843), “l’espérance mathématique est donc le juste prix des chances”
(or the “fair price” mentioned in Feller (AS, 1953)).
Problem : Saint Peterburg’s paradox, i.e. infinite mean risks (cf. natural
catastrophes)
33

the pure premium as a technical benchmark
For a positive random variable X, recall that EP(X) =
∞
0
P(X > x)dx.
q
q
q
q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Expected value
Loss value, X
Probabilitylevel,P
Fig. 18 – Expected value EP(X) = xdFX(x) = P(X > x)dx.
34

from pure premium to expected utility principle
Ru(X) = u(x)dP = P(u(X) > x))dx
where u : [0, ∞) → [0, ∞) is a utility function.
Example with an exponential utility, u(x) = [1 − e−αx
]/α,
Ru(X) =
1
α
log EP(eαX
) ,
i.e. the entropic risk measure.
See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern
(PUP, 1994), Rochet (E, 1994)... etc.
35

Distortion of values versus distortion of probabilities
q
q
q
q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Expected utility (power utility function)
Loss value, X
Probabilitylevel,P
Fig. 19 – Expected utility u(x)dFX(x).
36

q
q
q
q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Expected utility (power utility function)
Loss value, X
Probabilitylevel,P
Fig. 20 – Expected utility u(x)dFX(x).
37

from pure premium to distorted premiums (Wang)
Rg(X) = xdg ◦ P = g(P(X > x))dx
where g : [0, 1] → [0, 1] is a distorted function.
Example
• if g(x) = I(X ≥ 1 − α) Rg(X) = V aR(X, α),
• if g(x) = min{x/(1 − α), 1} Rg(X) = TV aR(X, α) (also called expected
shortfall), Rg(X) = EP(X|X > V aR(X, α)).
See D’Alembert (1754), Schmeidler (PAMS, 1986, E, 1989), Yaari (E, 1987),
Denneberg (KAP, 1994)... etc.
Remark : Rg(X) will be denoted Eg ◦ P. But it is not an expected value since
Q = g ◦ P is not a probability measure.
38

q
q
q
q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Distorted premium beta distortion function)
Loss value, X
Probabilitylevel,P
Fig. 21 – Distorted probabilities g(P(X > x))dx.
39

q
q
q
q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Distorted premium beta distortion function)
Loss value, X
Probabilitylevel,P
Fig. 22 – Distorted probabilities g(P(X > x))dx.
40

some particular cases a classical premiums
The exponential premium or entropy measure : obtained when the agent
as an exponential utility function, i.e.
π such that U(ω − π) = EP(U(ω − S)), U(x) = − exp(−αx),
i.e. π =
1
α
log EP(eαX
).
Esscher’s transform (see Esscher (SAJ, 1936), B¨uhlmann (AB, 1980)),
π = EQ(X) =
EP(X · eαX
)
EP(eαX)
,
for some α > 0, i.e.
dQ
dP
=
eαX
EP(eαX)
.
Wang’s premium (see Wang (JRI, 2000)), extending the Sharp ratio concept
E(X) =
∞
0
F(x)dx and π =
∞
0
Φ(Φ−1
(F(x)) + λ)dx
41

pricing options in complete markets : the binomial case
In complete and arbitrage free markets, the price of an option is derived using
the portfolio replication principle : two assets with the same payoff (in all
possible state in the world) have necessarily the same price.
Consider a one-period world,
risk free asset 1 → (1+r), and risky asset S0 → S1 =



Su = S0u( increase, d > 1)
Sd = S0d( decrease, u < 1)
The price C0 of a contingent asset, at time 0, with payoff either Cu or Cd at time
1 is the same as any asset with the same payoff. Let us consider a replicating
portfolio, i.e. 


α (1 + r) + βSu = Cu = max {S0u − K, 0}
α (1 + r) + βSd = Cd = max {S0d − K, 0}
42

pricing options in complete markets : the binomial case
The only solution of the system is
β =
Cu − Cd
S0u − S0d
and α =
1
1 + r
Cu − S0u
Cu − Cd
S0u − S0d
.
C0 is the price at time 0 of that portfolio.
C0 = α + βS0 =
1
1 + r
(πCu + (1 − π) Cd) where π =
1 + r − d
u − d
(∈ [0, 1]).
Hence C0 = EQ
C1
1 + r
where Q is the probability measure (π, 1 − π), called risk
neutral probability measure.
43

ﬁnancial versus actuarial pricing, a numerical example
risk-free asset risky asset contingent claim
1 →



1.05
1.05
100 →



110
70
??? →



150
10
probability 75%
probability 25%



Actuarial pricing : pure premium EP(X) =
3
4
× 150 +
1
4
× 10 = 115 (since
p = 75%).
Financial pricing :
1
1 + r
EQ(X) = 126.19 (since π = 87.5%).
The payoﬀ can be replicated as follows,



−223.81 · 1.05 + 3.5 · 110 = 150
−223.81 · 1.05 + 3.5 · 70 = 10
and thus −223.81 · 1 + 3.5 · 100 = 126.19.
44

ﬁnancial versus actuarial pricing, a numerical example
0.00 0.01 0.02 0.03 0.04 0.05 0.06
115120125130135140145
Comparing binomial risks, from insurance to finance
Alpha or lambda coefficients
Prices
ACTUARIAL PURE PREMIUM
FINANCIAL PRICE
(UNDER RISK NEUTRAL MEASURE)
WANG DISTORTED PREMIUM
ESSCHER
TRANSFORM
EXPONENTIAL
UTILITY
q
Fig. 23 – Exponential utility, Esscher transform, Wang’s transform...etc.
45

risk neutral measure or deflators
The idea of deflators is to consider state-space securities
contingent claim 1 contingent claim 2
??? →



1
0
??? →



0
1
probability 75%
probability 25%



Then it is possible to replicate those contingent claims



−1.667 · 1.05 + 0.025 · 110 = 1
−1.667 · 1.05 + 0.025 · 70 = 0



2.619 · 1.05 + −0.02 · 110 = 0
2.619 · 1.05 + −0.02 · 70 = 1
The market prices of the two assets are then 0.8333 and 0.119. Those prices can
then be used to price any contingent claim.
E.g. the final price should be 150 × 0.8333 + 10 × 0.119 = 126.19.
46

pricing options in incomplete markets
The valuation of option is based on the idea of replicating portfolios. What if we
cannot buy the underlying asset.
risk-free asset risky asset contingent claim
1 →



1.05
1.05
100 →



110
70
??? →



150
10
probability 75%
probability 25%



Actuarial pricing : pure premium EP(X) =
3
4
× 150 +
1
4
× 10 = 115.
Financial pricing : ...
The payoﬀ cannot be replicated =⇒ incomplete market
Remark : this model can be extended in continuous time, with continuous
prices (driven by Browian diﬀusion).
47

pricing options in incomplete markets
• the market is not complete, and catastrophe (or mortality risk) cannot be
replicated,
• the guarantees are not actively traded, and thus, it is diﬃcult to assume
no-arbitrage
• underlying diﬀusions are not driven by a geometric Brownian motion process
48

Impact of WTC 9/11 on stock prices (Munich Re and SCOR)
2001 2002
30354045505560
250300350
MunichRestockprice
Fig. 24 – Catastrophe event and stock prices (Munich Re and SCOR).
49

Agenda
50

pricing insurance linked securities using distorted premium
Lane (AB, 2000) proposed to ﬁt an econometric model on yield spreads (ER,
excess return), as a function of the CEL (conditional expected loss) and the PFL
(probability of ﬁrst loss), as a Cobb-Douglas function,
ER = 0.55 · PFL0.495
· CEL0.574
.
Wang (GPRI, 2004) Assume that X has a normal distribution (where µ and σ
are functions of CEL and PFL), and with λ = 0.45...
51

Property Catastrophe Risk Linked Securities, 2001
0
2
4
6
8
10
12
14
16 Yield spread (%)
Mosaic2A
Mosaic2B
HalyardRe
DomesticRe
ConcentricRe
JunoRe
ResidentialRe
Kelvin1stevent
Kelvin2ndevent
GoldEagleA
GoldEagleB
NamazuRe
AtlasReA
AtlasReB
AtlasReC
SeismicLtd
Lane model
Wang model
Empirical
Fig. 25 – Cat bonds yield spreads, empirical versus models.
52

using expected utility principle to price securities
Consider
– a risk-free bond,
– a risky traded asset, with price Xt,
– a non traded insurance index, with value St
We want to price the option with payoff payoff (ST − K)+.
In complete market (the index is traded), we price a replicating portfolio.
In incomplete market, we create a portfolio δ = (α, β) of tradable assets which
should not be too far from the true payoff. See the super-replication price or risk
minimization principle idea.
If X is a random payoff, the classical Expected Utility based premium is obtain
by solving
u(ω, X) = U(ω − π) = EP(U(ω − X)).
53

using expected utility principle to price securities
Consider an investor selling an option with payoﬀ X at time T,
– either he keeps the option, uδ (ω, 0) = supδ∈A EP U(ω + (δ · S)T ) ,
– either he sells the option, uδ (ω + π, X) = supδ∈A EP U(ω + (δ · S)T − X) .
The price obtained by indiﬀerence utility is the minimum price such that the two
quantities are equal, i.e.
π(ω, X) = inf {π ∈ R such that uδ (ω + π, X) − uδ (ω, 0) 0} .
This price is the minimal amount such that it becomes interesting for the seller
to sell the option : under this threshold, the seller has a higher utility keeping the
option, and not selling it.
54

conclusion
• classical finance, π = EP[(S − d)+]
• classical insurance, π = e−rT
EQ[(ST − K)+]
• large risks, expected utility approach U(ω − π) = EP(U(ω − [S − d]+))
• large risks, dual (Yaari) approach ω − π = Eg◦P(ω − S)
• incomplete market, dual (Yaari) approach, i.e. Esscher
• incomplete market, expected utility approach, i.e. indifference utility
=⇒ a general framework for pricing of both insurance and financial risks.
E.g. capital requirements of Basle II and Solvency II.
55

perspectives
• diversiﬁcation issues
Consider two risks S1 and S2, π1+2 = EP[S1 + S2] = EP[S1] + EP[S2] = π1 + π2
What about other premiums ? E.g. Eg◦P[S1 + S2]
?
= Eg◦P[S1] + Eg◦P[S2]
For some non subadditive risk measure (e.g. VaR) π1+2>π1 + π2.
• econometric and heterogeneity issues
In the case of pure premium, EP(S) = EP
N
i=1
Yi = EP(N)
frequency
× EP(Y )
average cost
With heterogeneity πi = EP(S|X = xi) = EP(N|X = xi) × EP(Y |X = xi)
What about other premiums ?
56

Slides nantes

Recommended

Recommended

More Related Content

What's hot

What's hot (12)

Viewers also liked

Viewers also liked (11)

Similar to Slides nantes

Similar to Slides nantes (20)

More from Arthur Charpentier

More from Arthur Charpentier (20)

Recently uploaded

Recently uploaded (20)

Slides nantes