t - Institut des Actuaires

Théorie du risque et corrélations
Stéphane Loisel
ISFA, Univ. Lyon 1
[email protected]
Plan de l’exposé
• Solvabilité II: à qui la faute?
• Motivations
• Résultats de maths (sieste)
Solvabilité II (S2):
ce qu’en pensent les universitaires
• Une mesure de risque liée à Bâle II et aux agences de
notation: la VaR à 99,5%
• Mathématiquement et statistiquement impossible à
quantifier
• Une intention louable au début du processus
• Qui a contribué à diffuser la culture du risque dans le secteur
de l’assurance
• MAIS une formule standard bien trop complexe
• Calendrier intenable de manière à ne pas avoir trop de
modèles internes à auditer en 1ère vague
• De nouvelles incitations à abandonner certaines pratiques de
bon sens, notamment en gestion actif-passif, en stratégie
d’investissement, en réassurance, …
• Nécessité de garder ses pratiques saines de gestion des
risques, tout en les adaptant partiellement en fonction de S2.
2
99,5
%
3
Gestion des risques
1
Scénarios
2
3
VaR à 90%, TVaR à 80%
Modèle interne (partiel ou non) adapté au pilotage
et à la gestion des risques
(stratégies de réassurance et d’investissement, risk limits, …)
VaR à 99,5%
Modèle interne pour Solvabilité II: si une grippe (type 1918)
se produisait, quel serait le choc sur les actifs?
Risques NBC, risques terroristes, rachats, taux, contrepartie:
VaR à 99,5%
Corrélations? Corrélations des extrêmes?
Solution proposée (trop tard!)
• Ne pas parler de VaR à 99,5% mais réfléchir
aux scénarios adaptés
• Prise en compte de la réass
• Prise en compte du risque endogène
• Prise en compte des autres crises de
corrélation
• Construire les scénarios: le + dur!
• Que faire des chiffres issus des scénarios?
Risque endogène, crises de corrélation
• Risque endogène et cercles vicieux
• Crises de corrélation
• Incertitude de modèle, hétérogénéité et
corrélation
• Mesures de corrélation: 1 piège révélateur
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)
• Relevant when individual entities are similar in terms of
forecasts and likely reactions to events
• Relevant when outcomes are sensitive to concerted actions
Le risque endogène est
implémentable assez facilement
• Travaux de Rama Cont
• Travaux avec X. Milhaud sur les crises de
corrélation et risque de rachat
When diversification fails:
correlation, contagion and endogenous risk
Rama CONT
Joint work with: Lakshithe Wagalath (Univ. Pierre & Marie Curie, Paris-VI)
& Laurent PICHARD (Columbia University, New York)
The economic origin of correlations in returns
Two different origins:
Correlation in fundamentals: common factors in returns (usual
explanation)→ correlation in ”fundamentals”, should not vary
strongly in time
Correlation from trading: generated by systematic supply/demand
generated by specific (often rule-based) trading strategies →
depends on market liquidity
8
Price impact from trading
Portfolio allocation of a fund 𝜙𝑎 (𝑡) = (𝜙𝑎1 (𝑡), ..., 𝜙𝑎𝑛 (𝑡))
Variation in portfolio from trading: Δ𝜙𝑎 (𝑡) = 𝜙𝑎 (𝑡) − 𝜙𝑎 (𝑡 − ℎ)
Aggregate excess demand of institutional investors/ fund managers:
∑
Δ𝜙(𝑡) = (Δ𝜙𝑖 (𝑡), 𝑖 = 1..𝑛) =
Δ𝜙𝑎 (𝑡)
(3)
𝑎∈𝔸
We model the evolution of the price as the sum of a random
component reflecting “fundamental” volatility + a price impact
term due to institutional trading:
price impact function
Δ𝑆𝑡𝑖
𝑆𝑡𝑖
=
𝜇𝑖 ℎ
Expected return
Δ𝜙𝑖 (𝑡)
) +
𝜎𝑖 𝜖𝑖 (𝑡)
(4)
𝜆𝑖
random component
+ 𝑓(
market depth
9
0.5
0.4
0.3
λ=10
λ=100
λ=2
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−10
−5
0
5
Figure 1: Price impact function
10
10
Example 1: fixed-mix strategies
A large proportion of mutual funds follow fixed-mix strategies.
Fund manager has a (medium term) target allocation in terms of
proportions allocated to each asset class 𝑖.
To maintain the target allocation, the fund manager needs to
rebalance periodically his/her portfolio as prices moves.
For a long-only portfolio this leads to buy if price decreases, sell if
price increases, until strategic allocation is revised.
11
Simulation example
3 asset classes
100 funds
Input noise covariance (“fundamental”
⎛
0.2 0
⎜
Σ=⎜
⎝ 0 0.1
0
0
13
covariance)
⎞
0
⎟
0 ⎟
⎠
0.3
240
220
200
180
160
140
120
100
80
0
1
2
3
4
5
Figure 2: A joint evolution scenario for 3 asset classes.
14
Covariance of fundamentals:
⎛
0.2
⎜
Σ=⎜
⎝ 0
0
Realized correlations:
⎛
1
⎜
ˆ
𝐶=⎜
⎝ 20%
21%
0
0
⎞
⎟
0.1 0 ⎟
⎠
0 0.3
20% 21%
⎞
⎟
1
52% ⎟
⎠
52%
1
Simulation of a 5- year scenario for 3 asset classes: realized vs
fundamental correlations: 𝜆 = 0.5.
Zero fundamental correlation, significant realized correlation: effect
entirely due to price impact of trading.
15
Proposition 1 (Continuous-time limit). Consider the above model
with an “aggregate” fund with allocation 𝑋 = (𝑥𝑖 , 𝑖 = 1..𝑛).
As the time step ℎ → 0, the Markov chain (𝑆 ℎ , Φℎ ) converges
weakly to a diffusion limit
(𝑆 ℎ , Φℎ ) ⇒ (𝑆, Φ)
where
)
𝑥𝑖 𝑉𝑡 (( ′
𝑑𝑆𝑖 (𝑡)
= 𝜇𝑖 𝑑𝑡 +
𝑋 𝜇 − 𝜇𝑖 𝑑𝑡
𝑆𝑖 (𝑡)
𝜆𝑖 𝑆𝑖 (𝑡)
)
𝑥𝑖 𝑉𝑡 ( ′
( 𝑋 𝜇 − 𝜇𝑖 ](𝐴𝑑𝑊 )𝑖 + 𝑋 ′ 𝐴𝑑𝑊𝑡
+[1 −
𝜆𝑖 𝑆𝑖 (𝑡)
𝑛
∑
𝑉𝑡
𝑋𝑖
with 𝑑𝑉𝑡 = 𝑉𝑡
𝑑𝑆𝑖 (𝑡)
𝜙𝑖 (𝑡) = 𝑥𝑖
𝑆𝑖 (𝑡)
𝑆
(𝑡)
𝑖=1 𝑖
where 𝐴 = Σ1/2 is a positive square root of the (fundamental)
covariance matrix and 𝑉𝑡 is the value of the aggregate fund
portfolio.
20
Rachats: mélange de GLM et
corrélations (X. Milhaud)
Rachats: mélange de GLM et
corrélations (X. Milhaud)
Mesures de corrélation:
1 piège révélateur
1421
1680
1736
2639
2674
4046
5306
6524
9499
82250
1064
1253
693
2639
1883
5929
2072
4109
5166
59850
Corrélation entre sinistres (journaliers) de 2 branches (4500 points)
Avec la dernière ligne : corrél. égale à 99%
Sans la dernière ligne : corrél. égale à 79%
Tau de Kendall plus robuste mais corrél. beaucoup plus faible ( 10% )
Prise en compte de corrélation des extrêmes…
Some explicit formulas for ruin
probabilities in models with
dependence among risks
Stéphane Loisel
ISFA, Université Lyon 1
Seminario CMM, Abril 2011
Con H. Albrecher y C. Constantinescu
http://dx.doi.org/10.1016/j.insmatheco.2010.11.007
Mixing → Pareto distribution
Given a r.v.
X ∼ Exp(Θ)
distributed, with Θ ∼ Γ(α, β) distributed with density
fΘ (θ) =
β α α−1 −βθ
θ e ,
Γ(α)
θ > 0,
the resulting mixed survival function of X is given by
−α
Z ∞
x
−θx
1 − FX (x) =
e fΘ (θ)dθ = 1 +
, x ≥0
β
0
Mixing → Weibull distribution
If we mix an exponential with a stable distribution of order
1/2, we obtain a Weibull distribution with fixed shape
parameter p = 1/2. Namely, if X ∼ Exp(Θ), with
α
2
e −α /4θ ,
fΘ (θ) = √
3
2 πθ
then the resulting mixed distribution for X is
1 − FX (x) = exp{−αx 1/2 },
x ≥ 0.
Classical ruin model
R(t) = u + ct −
N(t)
X
Xk ,
k=1
• u ≥ 0: initial surplus
• (N(t))t≥0 claim counting process (Poisson or renewal
process)
• Xk iid random variables: claim amounts, E {X } < ∞
• c > 0: constant premium intensity
• ct > E {N(t)}E {X }: net profit condition
Ruin
Time of ruin
Tu = inf (R(t) < 0 | R(0) = u)
t≥0
Probability of ruin
ψ(u) = P (Tu < ∞)
Classical result (Cramer, 1930) for compound Poisson model
with X ∼ Exp(θ) claim amounts
ψ(u) = min
nλ
o
λ
e −(θ− c )u , 1 ,
θc
u ≥ 0.
Mixing idea
Denote by
ψθ (u) = P (Tu < ∞ | Θ = θ)
Then, the ruin probability is given by
Z ∞
ψθ (u)dFΘ (θ).
ψ(u) =
0
Claims: dependent Pareto
• Compound Poisson risk model ( τ ∼ Exp(λ))
• Claims X ∼ Exp(Θ), where Θ ∼ Γ(α, β)
Ruin probability is
Z λ/c
β α α−1 −βθ
Ψ(u) =
1·
θ e dθ
Γ(α)
Z0 ∞
λ −θu λ u β α α−1 −βθ
+
e ec ·
θ e
dθ
θc {z } Γ(α)
λ/c |
|
{z
}
Ψθ (u)
fΘ (θ), Θ∼Γ(α,β)
Claims: dependent Pareto
Ψ(u) = 1 −
+
Γ(α, βθ0 )
Γ(α)
Γ(α − 1, (β + u)θ0 )
β
(βθ0 )α−1 e −βθ0 (u + β)−1
| {z } ((β + u)θ0 )α−2 e −(β+u)θ0
Γ(α)
{z
}
→u→∞ 0 |
→u→∞ 1
One can see that the probability of ruin decays to a constant
lim Ψ(u) = 1 −
u→∞
Γ(α, βλ
)
c
>0
Γ(α)
as fast as u −1 !! Compared to the independent case...
Claims: dependent Pareto
At u = 0,
Γ(α, βλ
) βλ Γ(α − 1, βλ
)
c
c
Ψ(0) = 1 −
+
,
Γ(α)
c
Γ(α)
where
Z
Γ(α, x) =
∞
w α−1 e −w dw
x
is the incomplete Gamma function.
Mixing dependence structure
For X ∼ Exp(Θ), with Θ ∼ FΘ , consider the classical
compound Poisson risk model with exponential claim sizes
that fulfill, for each n,
P (X1 > x1 , . . . , Xn > xn | Θ = θ) =
n
Y
e −θxk .
k=1
That is, given Θ = θ, the Xk (k ≥ 1) are conditionally
independent and distributed as Exp(θ).
(1)
Net profit condition
Since for θ ≤ θ0 = λ/c the net profit condition is violated and
consequently ψθ (u) = 1 for all u ≥ 0, this can be rewritten as
Z ∞
Z ∞
ψ(u) =
ψθ (u)dFΘ (θ) = FΘ (θ0 ) +
ψθ (u)dFΘ (θ).
0
θ0
An immediate consequence is that in this dependence model
lim ψ(u) = FΘ (θ0 ),
u→∞
which is positive whenever the random variable Θ has
probability mass at or below θ0 = λ/c (probability of net profit
condition not being fulfilled).
Claims: dependent Weibull
For the classical Cramér Lundberg risk model with claims
X ∼ Exp(Θ),
where Θ has a stable distribution of order 1/2,
α
2
e −α /4θ ,
fΘ (θ) = √
2 πθ3
the probability of ruin is
Z
θ0
α
2
1· √
e −α /4θ dθ
3
Z0 ∞ 2 πθ
λ −θu λ u
α
2
e ec · √
e −α /4θ dθ
+
3
θc {z } 2 πθ
θ0 |
|
{z
}
Ψ(u) =
Ψθ (u)
fΘ (θ)
Here the net profit condition holds for any θ > θ0 = λc .
Claims :dependent Weibull
Ψ(u)
=
+
−
Erfc
α
√
2 θ0
√
p
√
√
1
α
θ0 u
−α u √
√
π
1
+
α
u
+
(1
+
α
u)Erf
−
uθ
e
{e
0
πα2
2 θ0
√ √
p
√
√
α
2α −θ u− α2
√ + uθ0
e α u π 1 − α u + (−1 + α u)Erf
− √ e 0 4θ0 }
2 θ0
θ0
θ0 √
Since
Erf(x) = 1 −
Γ
1
2
2, x
√
π
equivalent to
Γ
1 2
,x
2
=
√
π(1 − Erf(x)),
one can write the result in terms of incomplete Gamma functions.
Interarrival times: dependent Pareto
If one considers the classical Cramér Lundberg risk model with
exponential claims, where the parameter Λ of the exponential
inter-arrival time is a r v.
Λ ∼ Γ(α, β)
one obtains Pareto inter-arrival times and the probability of
ruin can be calculated explicitly as well. Similarly as before, for
λ0 = θc, one can write
Z
λ0
Ψ(u) =
0
λ −θu λ u β α α−1 −βλ
e ec
λ e
dλ
|θc {z } |Γ(α) {z
}
Ψλ (u)
Z
∞
α
1·
+
λ0
fΛ (λ), Λ∼Γ(α,β)
β
λα−1 e −βλ dλ
Γ(α)
Interarrival times: dependent Pareto
u
Γ(α + 1) − Γ(α + 1, cβθ − uθ)
Ψ(u) = e −uθ β α (β − )−α−1
c
cθΓ(α)
Γ(α, βcθ)
+
Γ(α)
Thus, the probability of ruin decays again as fast as u −1 to
lim Ψ(u) =
u→∞
Γ(α, βcθ)
Γ(α)
the probability of net profit condition not being fulfilled.
At u = 0
Ψ(0) =
Γ(α + 1) − Γ(α + 1, cβθ) Γ(α, βcθ)
+
βcθΓ(α)
Γ(α)
Proposition
The dependence model characterized by
P (X1 > x1 , . . . , Xn > xn | Θ = θ) =
n
Y
e −θxk
k=1
can equivalently be described by having marginal claim sizes
X1 , X2 , . . . that are completely monotone, with a dependence
structure due to anArchimedean survival copula with
−1
generator ϕ = FeΘ
for each subset (Xj , . . . , Xjn ) (for
1
j1 , . . . , jn pairwise different), where FeΘ denotes the
Laplace-Stieltjes transform of FΘ .
Archimedean generator
−1
ϕ(t) = FeΘ
(t)
As the inverse of a Laplace-Stieltjes transform of a cdf,
• ϕ : [0, 1] → [0, ∞] is a continuous strictly decreasing
function
• ϕ(0) = ∞ and ϕ(1) = 0
• ϕ−1 is completely monotone
Thus, the Archimedean copula is well-defined for all n (see e.g.
Nelsen, 2006).
Message: X is completely monotone.
Dubey 1977
For N(t) ∼ Poisson(Λ)
Z
t
λ̂(s)ds −
R(t) = c
0
N(t)
X
Yj ,
t≥0
j=0
Estimates for λ
• λ̂(t) = E {Λ | N(t)} → exact form for the ruin probability
• λ̂(t) =
• λ̂(t) =
N(t)
→ moments of ruin
t
a+N(t)
→ approximation
b+t
of the probability of ruin,
considering λ̂ to be the “credibility”estimate.
Note: This model was theoretical basis for the Bonus-Malus system
for the Swiss obligatory car insurance.
Thank you for your attention!
[email protected]
Proof
The joint distribution of the tail of X1 , . . . , Xn is
Z
P (X1 > x1 , . . . , Xn > xn )
∞
P (X1 > x1 , . . . , Xn > xn | Θ = θ) dFΘ (θ)
=
0
Z
=
∞
e −θ(x1 +···+xn ) dFΘ (θ)
0
= FeΘ (x1 + · · · + xn ).
At the same time, the representation with survival copula Ĉ is
given by
P (X1 > x1 , . . . , Xn > xn ) = Ĉ (F X (x1 ), . . . , F X (xn )),
where F X (xi ) = 1 − FX (xi ) is the tail of the marginal claim
size Xi (note that the Xi ’s are all identically distributed).
If the survival copula is Archimedean with generator ϕ, then it
admits the representation
Ĉ (F X (x1 ), . . . , F (xn )) = ϕ−1 (ϕ(F X (x1 )) + · · · + ϕ(F X (xn ))),
which due to
Z
F X (xi ) =
∞
e −θxi dFΘ (θ) = FeΘ (xi ),
i = 1, . . . , n,
(2)
0
exactly matches with (2) when the generator is chosen to be
−1
ϕ(t) = FeΘ
(t).