Contributions to quantitative risk management in insurance Stphane - - PowerPoint PPT Presentation
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Contributions to quantitative risk management in insurance Stphane Loisel ISFA, Universit Lyon 1 Habilitation diriger les recherches Stphane
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes
Contributions to quantitative risk management in insurance
Stéphane Loisel
ISFA, Université Lyon 1
Habilitation à diriger les recherches
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 1 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Classical model
A recent correlation crisis in Kruger Park
Correlation between short-term mortality indicators can suddenly increase. Here correlation and the marginal risks increase at the same time. Another example: a drive with your mother-in-law.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 4 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Classical model
Classical assumptions and our problem
Classical assumptions: R(t) = u + ct − N(t)
i=1 Xi
claim amounts (Xi)i≥1: sequence of i.i.d. r.v.’s, with finite mean, The (Xi)i≥1 are independent from (N(t))t≥0 (e.g. renewal process). Problem: Derive asymptotics of finite-time ruin probabilities for large risks ψ(u, t) = P(∃τ ∈ [0, t] , R(τ) < 0|R(0) = u), in more general models with dependence between claim amounts and possible correlation crises.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 5 / 37
Claims: dependent Pareto
Ruin probability is Ψ(u) = λ/c 1 · βα Γ(α)θα−1e−βθdθ + ∞
λ/c
λ θc e−θue
λ c u
· βα Γ(α)θα−1e−βθ
dθ
Claims: dependent Pareto
Ψ(u) = 1 − Γ(α, βθ0) Γ(α) + β Γ(α)(βθ0)α−1e−βθ0 (u + β)−1
Γ(α − 1, (β + u)θ0) ((β + u)θ0)α−2e−(β+u)θ0
One can see that the probability of ruin decays to a constant lim
u→∞ Ψ(u) = 1 − Γ(α, βλ c )
Γ(α) > 0 as fast as u−1!! Compared to the independent case...
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Classical model
Some types of correlation (not developed in this talk)
Classical assumptions: R(t) = u + ct − N(t)
i=1 Xi
claim amounts (Xi)i≥1: sequence of i.i.d. r.v.’s, with finite mean, The (Xi)i≥1 are independent from (N(t))t≥0 (e.g. renewal process). Some models with embedded correlations: c is not constant over time and is adjusted to the observed previous claims (with Bühlmann linear credibility premium principle): impact of using credibility theory on the ruin probability (joint work with J. Trufin, 2009) dependence between the claim arrival process and the claim sizes (earthquake risk, flooding and drought risk, ...). Works of Boudreault et al. (2006), Albrecher et al. (2007), joint work with R. Biard, C. Lefèvre and H. Nagaraja (2010). dependence between claim arrivals and the intensity process: shot-noise processes, cycles influenced by large claims (joint work with
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 6 / 37
Introduction Simple structure of dependence Dependence on the history of the process Our problems
Flooding-type risk process
Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 8 / 30
Introduction Simple structure of dependence Dependence on the history of the process Our problems
Rewording flooding risk with simple ideas
Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 9 / 30
Introduction Simple structure of dependence Dependence on the history of the process Our problems
Earthquake-type risk process
Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 10 / 30
Introduction Simple structure of dependence Dependence on the history of the process Our problems
Rewording earthquake risk with simple ideas
Stéphane Loisel (ISFA, Lyon) A class of path-dependent claim amounts Istanbul 11 / 30
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts
Impact of dependence between claim amounts
In practice, claim amounts are influenced by common factors. This leads to stochastic correlation models. This correlation may change during time, due to endogenous risk or external shocks (joint works with Wayne Fisher and Shaun Wang (2007) and with Pierre Arnal and Romain Durand (2010)), parameter uncertainty (see Meyers (1999)), ... What happens
◮ if dependence between claim amounts is governed by a Markovian
environment process?
◮ if claim amounts of different lines of business suddenly become more
dependent (in a common shock model)?
◮ if those correlation crises are triggered by some large claims?
Some useful references for sums of dependent risks: among others, Barbe et al. (2006), Albrecher et al. (2006), Kortschak and Albrecher (2008), Demarta (2002). Papers of Vladimir Kaishev and coauthors.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 7 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
Systemic risk: securitization
Does securitization really atomize risk?
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 8 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
Systemic risk: securitization
Risk is just transferred and recombined, but does not disappear. If a large risk becomes reality, all counterparts may be affected and/or downgraded. Reinsurer’s default can lead to sudden increase in frequency and correlation
that case ? Work in progress with C. Blanchet, D. Dorobantu and S. Louhichi.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 9 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
How can independent risks become suddenly strongly correlated?
Endogenous uncertainty: Uncertainty is generated/modified by response of individual entities to events Feedback loop: outcomes → forecasts → decisions → outcomes → revised forecasts → revised decisions → . . . (Millennium Bridge) Statistical relationships are endogenous to the model, and may undergo structural shifts (Goodhart’s Law: Any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes) Relevant when individual entities are similar in terms of forecasts and likely reactions to events Relevant when outcomes are sensitive to concerted actions
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 10 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
How can independent risks become suddenly strongly correlated?
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 11 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
How can independent risks become suddenly strongly correlated?
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 12 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
How can independent risks become suddenly strongly correlated?
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 13 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Systemic risk
Analogy with LTCM
This analogy is drawn from Danielsson and Shin (2003). Similar potential feedback loops for surrender risk in life insurance.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 14 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Our framework in Biard et al. (2008)
Focus on some heavy-tailed distributions
Definition (Regular variation class (R))
F belongs to R−α, α ≥ 0 if and only if lim
x→∞
F(xy) F(x) = y −α for any y > 0. Note that if 0 < α < 1, any X with c.d.f. F has infinite mean.
Definition (Multivariate Regular variation class (MR))
A random vector X = (X1, ..., Xn) belongs to MR−α, α > 0 if and only if there exists a θ ∈ Sn−1, where Sn−1 is the unit sphere with respect to a norm |·|; such that P (|X| > tu, X/ |X| ∈ ·) P (|X| > u)
v
→ t−αPSn−1 (θ ∈ ·) , where
v
→ denotes vague convergence on Sn−1.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 15 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Static dependence models
A basic situation
∀n ≥ 1, Xn = InW0 + (1 − In)Wn, where (Wn)n≥0 : i.i.d. sequence, common cdf FW ∈ R−α, α ≥ 0, (In)n≥1 is a sequence of i.i.d Bernoulli random variables, with P(I1 = 1) = p ∈ [0, 1], The Wn, n ≥ 0 are independent from the Ik, k ≥ 1. Denote the aggregate claim amount (with Sp
t = 0, if Nt = 0) by
Sp
t = N(t)
Xk. First, we calculate P(Sp
t > x) for large x, then we approximate ψp(u, t) by
P(Sp
t > u) and, finally, we compare ψp(u, t) and ψq(u, t) for 0 ≤ p < q ≤ 1.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 16 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Static dependence models
Important properties and consequences
Max-sum property: if F1 and F2 belong to R−α, α > 0 then F1 ∗ F2(x) ∼ F1(x) + F2(x) for large x, Convolution closure: if F1 and F2 belong to R−α, α > 0 then so does F1 ∗ F2. Since FW belongs to R−α, for any k ≥ 1, 1 ≤ j ≤ k − 1 and any pairwise distinct n1, ..., nk−j ≥ 1, for large x, P
(k − j)FW (x) + FW x j
k − j + FW
j
FW (x) ∼ (k − j + jα)FW (x) (1)
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 17 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Static dependence models
P(Sp(t) > x) ∼
∞
e−λt (λt)k k!
k
k j
FW (x). (2) can be rewritten as P[Sp(t) > x] ∼ {λt + E [(Z p(t))α − Z p(t)]} FW (x), (3) where Z p(t) denotes a binomial random variable Bin[N(t), p]. The mixed binomial law MBin[N(t), p] is stochastically increasing in p (see, e.g., Lefèvre and Utev (1996)). Since the function f (x) = xα − x, x ∈ {0, 1, . . .}, is decreasing (resp. increasing) when α < 1 (resp. α > 1), we get (for u large enough), for α < 1 (infinite mean case), 0 ≤ p < q ≤ 1 ⇒ ψp(u, t) > ψq(u, t), and for α > 1 (finite mean case), 0 ≤ p < q ≤ 1 ⇒ ψp(u, t) < ψq(u, t).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 18 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts influenced by an environment process
Correlation crises
Dependence between claim amounts, claim size distribution and intensity are modulated by a Markovian environment process. Markov environment process (J(t))t≥0 with J ≥ 2 states 1, . . . , J
◮ with initial distribution π0 ◮ and transition rate matrix Q.
Claim amounts: for 1 ≤ i ≤ n, sequence of i.d. random vectors (X i
m)m≥1 s.t.
∀n ≥ 1, X i
n = I i nW i 0 + (1 − I i n)W i n,
◮ where the (W i
n)n≥0 are i.i.d. r.v.’s with cdf F i W ∈ R−αi ,
◮ the (I i
n)n≥1 are i.i.d. Bernoulli r.v.’s with parameter pi ∈ [0, 1],
◮ the W i
n, n ≥ 0 are independent from the I i k, k ≥ 1
◮ and the W i
n, n ≥ 0 and I i k, k ≥ 1 are independent from a Poisson
process Ni(t) with parameter λi. Define the J independent processes (1 ≤ i ≤ J) as Y i(t) = cit −
Ni (t)
X i
mi
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 19 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts influenced by an environment process
Correlation crisis
Let Tp be the instant of the pth jump of the process Jt, and define (R(t))t≥0 by R(t) = u +
(Y i(Tp) − Y i(Tp−1))1{JTp−1=i,Tp≤t} +
(Y i(t) − Y i(Tp−1))1{JTp−1=i,Tp−1≤t<Tp}. A typical modulated risk process with two states (red and blue).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 20 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts influenced by an environment process
Correlation crises: case where α1 < αi for all i ≥ 2
Theorem
As u → +∞, we have for any t > 0 ψ(u, t) ∼ J
π0(i)E
i +
i
]α1 ¯ F 1(u), where W 1
i (t) is the time spent by the environment process in state 1 during
[0, t] given that J(0) = i, M⊥
i
follows a mixed Poisson distribution with random parameter λ1(1 − p1)W 1
i (t),
and Mcom
i
follows a mixed Poisson distribution with random parameter λ1p1W 1
i (t).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 21 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts influenced by an environment process
Correlation crises: case where α1 < αi for all i ≥ 2
Theorem
If besides α1 ∈ N, we have E
i
i (t)
i (1, t),
and E
i
]α1 = E
α1
S(α1, k)(λ1p1)k W 1
i (t)
k =
α1
S(α1, k)(λ1p1)kD1
i (k, t),
where S(α1, k) is the (α1, k) Stirling number of the second kind, and where for m ≥ 1, D1
i (m, t) = E
i (t)
m | J(0) = i
is the ith component of vector D1(m, t) defined by D1(0, t) = 1 and for m ≥ 1, D1(m, t) = r t eQ(t−u)A11D1(m − 1, u)du where A11 is J × J with coeff. δi1δj1.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 22 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Dependence between claim amounts influenced by an environment process
Pure correlation crises: case where αi = α ∈ N \ {0} for all 1 ≤ i ≤ 3
Assume that in state 1, the W 1
n , n ≥ 1 are i.i.d.,
in state 2, there is a light correlation: the W 2
n , n ≥ 1 have Gaussian copulas,
and in state 3, the W 3
n , n ≥ 1 are given by the basic dependence model with
parameter p3. Then,
Theorem
We have for any t > 0, as u → +∞, ψ(u, t) ∼ 3
π0(i)[λ1D1
i (1, t) + λ2D2 i (1, t)
+λ3 1 − p3 D3
i (1, t) + α
S(α3, k)(λ3p3)kD3
i (k, t)]
F(u). Similar results may be obtained with classical copulas, and dependence between different state processes. See Biard, Lefèvre and L. (2008) for further details.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 23 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes
1
Ruin and correlation
2
Optimal reserve allocation Optimal allocation problem Heavy-tailed case Light-tailed case
3
Sensitivity and robustness
4
New themes
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 24 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Optimal allocation problem
The allocation problem is the minimization of the risk measure IT (u1, u2) = E
T (u1)
T (u2)
for u1 ≥ 0 and u2 ≥ 0 under the constraint u1 + u2 = u for large u where I i
T (ui) = T
i = 1, 2.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 25 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Heavy-tailed case
Heavy-tailed case and T = ∞
No environment process. FW1 ∈ R−α1 and FW2 ∈ R−α2 with α1 < α2.
Theorem
If we denote u1 = (1 − β(u))u and u2 = β(u)u with β(u) ∈ (0, 1) we have for large u β(u) ∼
′
2
D
′
1
FW2(u) FW1(u) 1/(α2−2) , where for i = 1, 2, D
′
i = (αi − 3)−1 1
ci 1 1 − ψi(0) λi + λ c − (λi + λ)µi 1 (αi − 1)(αi − 2)(αi − 3). The finite-time case can also be treated (see Biard, Macci, L. and Veraverbeke (2010) for details).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 26 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Light-tailed case
Light-tailed case and T = ∞
No environment process. We assume that the Cramer-Lundberg exponent exists for each process and is equal to Ri, i = 1, 2 with R1 < R2.
Theorem
For large u, the solution is given by u1 = R2 R1 + R2 u + 1 R1 + R2 log
′
2
M
′
1
u2 = u − u1 + o(1), where M
′
i = −Ri
1 ci 1 1 − ψi(0) 1 − (λi + λ)µi R2
i ((λi + λ) ˆ
FWi
′
(Ri) − 1) i = 1, 2. Other allocation problems could be considered (see Cénac et al. (2010)).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 27 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes
1
Ruin and correlation
2
Optimal reserve allocation
3
Sensitivity and robustness Robustness analysis and ERSM Sensitivity analysis of finite-time ruin probabilities
4
New themes
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 28 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Robustness analysis and ERSM
Motivation
500 1000 1500 2000 2500 3000 3500 4000 0,025 0,03 0,035 0,04 0,045 0,05 Q95For initial reserve u ≥ 0 (see L., Mazza and Rullière (2008, 2009)), we show the convergence (in distribution) of the renormalized difference √ N
ψ(u, t) − ¯ ψN(u, t)
ψ(u, t) and its equivalent obtained when claims are drawn from the empirical distribution
asymptotic variance can be obtained from easy-to-simulate quantities, thanks to so-called partly shifted processes.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 29 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Robustness analysis and ERSM
OM (or partly shifted) risk processes
Offset-modified or partly shifted risk process (link with Rama Cont’s vertical derivative)
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 30 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Robustness analysis and ERSM
Reliable ruin probability
Set φY (u, t) = P
s ≥ 0 ∀s < t | RY 0 = u
where RY
s = Rs − 1{U<s}Y , s ≥ 0, and U is uniform on [0, t]. U, Y and
(Rt)t≥0 are mutually independent. Then the variance of the ruin probability is Var
As Takács’s lemma remains valid for partly shifted processes, those computations can be done quite quickly.
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 31 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Robustness analysis and ERSM
Estimation Risk Solvency Margin (ERSM)
10 10.5 11 11.5 12 12.5 13 13.5 14 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 ERSMIf uη and uη,ε are respectively defined as the initial capital required to ensure that ψ(uη, t) ≤ η and ψN,reliable
1−ε
(uη,ε, t) ≤ η, the Estimation Risk Solvency Capital ERSMη,1−ε can be defined as the additional capital needed to take estimation risk into account : ERSMη,1−ε = uη,ε − uη. Other estimators may also be used (see Susan Pitts’s recent work).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 32 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Sensitivity analysis of finite-time ruin probabilities
Sensitivity analysis
Because C 2 condition is not satisfied, Malliavin calculus on the Poisson space (see Privault and Wei, 2004) cannot be used. We had to use integration by parts to compute the sensitivity of the finite-time ruin probability w.r.t. initial reserve u ≥ 0 (joint work with Nicolas Privault (2009)).
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 33 / 37
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes
1
Ruin and correlation
2
Optimal reserve allocation
3
Sensitivity and robustness
4
New themes Longevity risk Other perspectives
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 34 / 37
Introduction Characteristics of longevity risk Modeling longevity risk Longevity risk and new regulations Transferring longevity risk Modeling issues for pricing Basis Risk Detection of drift changes
FIGURE: Migrations and life expectation
4/ 57 Stéphane Loisel, ISFA- U. Lyon 1 Longevity Risk
Examining Structural Shifts in Mortality Using the Lee-Carter Method Lawrence R. Carter, Alexia Prskawetz
Introduction Characteristics of longevity risk Modeling longevity risk Detection of drift changes Longevity and mortality risks: a natural hedge? Transferring longevity risk Modeling issues for pricing
PURE LONGEVITY RISK
change of the average trend short-term oscillations around the average trend (risk of over-reactions) Heterogeneity and basis risk : the evolution of the policyholders mortality is usually different from that of the national population (selection effects). Financial Risk Long term interest rate risk Counterparty risk
6/ 40 Stéphane Loisel, ISFA- U. Lyon 1 Longevity Risk
Ruin and correlation Optimal reserve allocation Sensitivity and robustness New themes Other perspectives
Other perspectives
s-convex extrema for t-monotone distributions in ruin theory (univariate and multivariate issues) about variable annuities Climate change risk and actuarial science (MIRACCLE project) Enterprise Risk Management Solvency II; accelerating nested simulations Value of insurance portfolios
Stéphane Loisel (ISFA, Lyon) QRM in insurance November 2010 36 / 37
St´ ephane Loisel
ISFA, Universit´ e Lyon 1
Montr´ eal, Nov. 2011 With H. Albrecher and C. Constantinescu
http://dx.doi.org/10.1016/j.insmatheco.2010.11.007
1
− Ruine économique = situation où la valeur de marché l’actif est inférieure à la fair value des passifs − L’horizon d’une année impose de pouvoir disposer de la distribution des fonds propres économiques dans un an − Le niveau 99,5% permet de garantir la solvabilité de la compagnie. La probabilité de l’événement « ruine économique » est dans ce cas inférieure à 0,5%.
2
première période
valeur » de la distribution de FP1
2
t=0 t=1
Temps
Scenario 1 Scenario i Scenario N
Scenario 1 Scenario j Scenario M
…
Scenario 1 Scenario j Scenario M (issu du scenario 1) (issu du scenario N)
Simulations Primaires « Monde Réel » Simulations secondaires « market consistent » issues des simulations primaires
A1
1 FP1 1
VEP1
1
Bilan en t=1 - Scenario 1
A1
N
FP1
N
VEP1
N
… …
t=T
…
A0 FP0 VEP0
Bilan en t=0
Bilan en t=1 - Scenario N
3
1 2 3 4
1 2 3 4
Stock ZC
1 2 3 4
1 2 3 4 <0,5% 0,5% - 5% 5% - 10% >10%
Approche « SdS exhaustif »
Action
Approche « Accélérateur SdS »
Région à explorer
(Région où les scénarios sont les plus adverses en terme de solvabilité) Chaque simulation primaire est caractérisée par un couple de facteurs de risque Action x ZC ZC
4
Rang FP1 1
2
3
4
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1 2 3 4
1 2 3 4
5
Rang FP1 1
2
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1 2 3 4
1 2 3 4
6
Rang FP1 1
2
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1 2 3 4
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May 3, 2012
8
May 3, 2012
Types of guarantees offered in Variable Annuities Risk analysis and Solvency II pitfalls Confronting points of view of insu Guarantees and surrender risk
GMWB
Stéphane Loisel (ISFA, Lyon) Variable annuities
11 / 27
9
May 3, 2012
10
May 3, 2012
11
May 3, 2012
12
May 3, 2012
13
May 3, 2012
14
May 3, 2012