![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Introduction the risk measures, and risk perception
S. Clam [...] once said: I dene a coward as someone who will not bet when
you oer him two-to-one odds and let him choose his side .
With the centuries old St. Petersburg paradox in my mind, I pedantically
corrected him: You mean will not make a suciently small bet (so that the
change in the marginal utility of money will not contaminate his choice). .
Recalling this conversation, a few years ago I oered some lunch colleagues to
bet each $200 to $100 that the side of a coin they specied would not appear at
the rst tom. One distinguished scholar - who lays no claim to advanced
mathematical skills - gave the following answer:
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Risk measures: the expected utility approach
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
) .
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Risk measures: Yarri's dual approach
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 ≥ α) Rg(X) = V aR(X, α), and if g(x) = min{x/α, 1}
Rg(X) = TV aR(X, α) (also called expected shortfall),
Rg(X) = EP(X|X V aR(X, α)).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Yarri's dual approach: capacities and Choquet's integral
Here Rg(X) = g(P(X x))dx = g(FX(x))dx with g : [0, 1] → [0, 1]
increasing. Thus, g ◦ FX is a decreasing function taking values in [0, 1] on [0, ∞):
g ◦ FX is a survival function.
Can Rg(X) be seen as an expected value of X with a change of measure ?
Yes if there exists a probability measure Q such that g ◦ FX(x) = Q(X x). If it
is possible to dene such a measure Q, generally Q is not a probability measure.
In fact, Q satises
• Q(∅) = 0 (since FX(∞) = 0 and g(0) = 0),
• Q(Ω) = 1 (since FX(0) = 1 and g(1) = 1),
• Q(A) ≤ Q(B) if A ⊂ B (since FX(·) is decreasing and g(·) is increasing).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Axiomatic approach for risk measures
There are three way to describe risk measure: characterizing natural properties
that should satisfy
• the risk measure R(·), e.g. R(·) is subadditive (R(X + Y ) ≤ R(X) + R(Y )),
• induced stochastic ordering , i.e. X Y (Y is more risky than X) if and
only if R(X) ≤ R(Y ) [Economics],
• induced set of acceptable risks A, i.e. X ∈ A (X is is acceptable) if and
only if R(X) ≤ 0 [Financial Mathematics].
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
More dicult to quantify than to compare
Denition 7. satises the continuity axiom (or Archimedean axiom) if for
any FX, FY and FZ such that FX FY FZ, then for all α, β ∈ (0, 1)
αFX + [1 − α]FZ FY βFX + [1 − β]FZ.
Proposition 8. If satises the continuity and associativity axioms,
X Y ⇐⇒ R(X) ≤ R(Y )
where
R(X) = EP(X) = xdFX(x) =
∞
0
P(X x)dx.
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The expected utility approach
Denition 9. satises the independence axiom if for any distribution function
FX, FY and FZ such that FX FY , then for all λ ∈ [0, 1]
λFX + [1 − λ]FZ λFX + [1 − λ]FZ.
or equivalently
(λX) ⊕ ([1 − λ]Z) (λY ) ⊕ ([1 − λ]Z),
where ⊕ denotes a mixture.
Hence, ordering are not modied when mixing risks with a third one. Recall that
(λX) ⊕ ([1 − λ]Z) = (λX) + ([1 − λ]Z).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
The expected utility approach
The insurance premium is then obtained by the null utility principle: π(X)
satises
EP(u(π(X) − X)) = 0.
Example 12. With an exponential utility, u(x) = [1 − e−αx
]/α, alors
π(X) =
1
α
log EP(eαX
) .
Note that the exponential utility does not exist for heavy tailed risks.
Example 13. With a quadratic utility, u(x) = x − x2
/2s where x s, then
π(X) ∼ EP(X) +
κ
2
V arP(X).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Changing probabilities ?
Le prix de l'option aujourd'hui s'écrit
α + βS0 =
1
1 + r
1 + r − d
u − d
Cu +
u − (1 + r)
u − d
Cd ,
qui peut s'écrire
π =
1
1 + r
(qCu + (1 − q)Cd) , où q =
1 + r − d
u − d
.
Notons que q ∈ [0, 1], c'est à dire que le prix de l'option est l'espérance
mathématique, sous une probabilité Q appelée probabilité risque neutre du payo
à échéance: π = EQ (payo). Notons que Q n'a rien n'a voir avec la probabilité
dite historique P qu'a le sous-jacent de monter ou de descendre: le prix d'un
payo aléatoire X ne s'écrit pas EP (X).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
La distortion de probabilités, Yaari (1987)
Example 16. Un exemple de relation d'ordre est la dominance stochastique à
l'ordre 1. X 1 Y si et seulement si une des conditions suivantes (équivalentes)
sont satisfaites,
• E(g(X)) ≤ E(g(Y )) pour g croissante,
• pour tout x ∈ R, P(X ≤ x) ≥ P(Y ≤ x),
• pour tout x ∈ R, P(X x) ≤ P(Y x),
• pour tout x ∈]0, 1[, V aR(X, α) ≤ V aR(Y, α).
Cette relation d'ordre est notée V aR dans Denuit Charpentier (2004).
Example 17. Un exemple de relation d'ordre est la dominance stochastique à
l'ordre 2. X 2 Y si et seulement si une des conditions suivantes (équivalentes)
sont satisfaites,
• E(g(X)) ≤ E(g(Y )) pour g croissante et convexe,
• E((X − t)+) ≤ E((Y − t)+) pour t ∈ R,
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
• pour tout x ∈]0, 1[,
α
0
V aR(X, p)dp ≥
α
0
V aR(Y, p)dp,
• pour tout x ∈]0, 1[,
∞
α
V aR(X, p)dp ≤
∞
α
V aR(Y, p)dp,
• pour tout x ∈ [0, 1[, TV aR(X, α) ≤ TV aR(Y, α).
Cette relation d'ordre est notée T V aR dans Denuit Charpentier (2004).
Denition 18. est une relation vériant l'axiome d'indépendance comonotone
si X Y implique X + Z Y + Z pour tout Z tel que les couples (X, Z) et
(Y, Z) soient comonotones.
Remark 19. X et Z sont comonotones s'il n'existe pas ω, ω tels que
X(ω) X(ω ) et Y (ω) Y (ω ).
Denition 20. est une relation vériant l'axiome de cohérence si pour des
variables X et Y constantes (P(X = x) = P(Y = y) = 1), FX FY implique
x ≤ y.
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Proposition 21. Si est une relation d'ordre vériant les axiomes de
continuité, d'indépendance comonotone, monotonie, et est compatible avec la
dominance stochastique à l'ordre 1, alors il existe une unique fonction de
distortion croissante g : [0, 1] → [0, 1] telle que
X Y ⇐⇒ Rg(X) ≤ Rg(Y )
où
Rg(X) = xdg ◦ P = g(P(X x))dx =
1
0
F−1
X (1 − p)dg(p).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Convex risk measures
A risk measure is said to be convex (from Artzner, Delbaen, Eber Heath
(1999)) if
• R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ),
• R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ,
• R(·) is convex, i.e. R(λX + (1 − λ)Y ) ≤ λR(X) + (1 − λ)R(Y ), for any
λ ∈ [0, 1].
Hence, if a convex measure satises the homogeneity condition, it is coherent.
Remark A natural way to dene a convex measure satisfying the small size
coherent condition is adding a coherent measure a liquidity charge,
Rconvex(X) = Rcoherent(X) + Cliquidity(X).
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on Shapley value
One idea is to use the Shapley value.
Let Si denote all the subsets of {1, 2, ..., n} that contain i.
The Shapley-value of risk i is
ki =
S∈Si
(|s| − 1)!)(n − |s|)!
n!
· [Γ(S) − Γ(S {i})],
where | · | denote the size of the subset (cardinality).
It is simply a weighted average of all possible congurations.
Note that here k1 + ... + kn = k.
This capital allocation has been used by Swiss Re.
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation based on Aumann-Shapley value
An alternative is to use the Aumann-Shapley value.
Here the idea is to consider sensitivity of marginal cost. Dene the following cost
function, C : [0, 1]n
→ R,
C(ω) = R(ωt
X) = R
n
i=1
ωiXi .
Note that C(1) = R(X) = k.
Dene Ci(ω) =
∂C(ω)
∂ωi
, and dene the Aumann-Shapley value as
ki =
1
0
Ci(u1)du.
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Capital allocation and decision loss function
Capital allocation can also be seen as the solution of distance minimization
problem.
Consider some distance d(ki, Xi|R, k, X). Then
ki = argminκ(κ, Xi|R, k, X).
Example 35. Consider the following expected value distance,
d(ki, Xi) = E([Xi − ki]2
).
Then ki =
k
n
+ E(Xi) −
E(X)
n
.
Example 36. Consider the following tail VaR distance,
d(ki, Xi|α, k, X) = E([Xi − ki]2
|X F−1
X (α)).
Then ki =
k
n
+ E(Xi|X F−1
X (α)) −
E(X|X F−1
X (α))
n
.
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Example 37. Consider the following stop loss distance,
d(ki, Xi) = E([Xi − ki]+).
Then ki = F−1
Xi
(FX+ (k)).
Example 38. Consider the following distance on risk measures,
d(ki, Xi) = R(Xi) −
ki
k
(R(X1) + ... + R(Xn))
2
.
Then ki =
R(Xi)
R(X1) + ... + R(Xn)
· k.
104](https://image.slidesharecdn.com/slides-edf-1-130617053457-phpapp01/85/Slides-edf-1-104-320.jpg)
![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
And nally a similar idea is to write V aR(X, α) = TV aR(X, a(X)) for some a(·)
with values in [0, 1], and to consider capital allocation based on TVaR (see
Overbeck (2000)).
Numerical example based on a simulated portfolio
Consider three lines of business, with rather dierent behaviors,
• an exponential distribution X1 ∼ E(0.03), E(X1) ∼ 33,
• a log-normal distribution X2 ∼ LN(3, 1), E(X2) ∼ 33,
• a Pareto distribution X3 ∼ P(2, 30), E(X3) ∼ 33,
Assume further that the three risks are independent.
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using a parametric models
Actually, is the Gaussian model does not t very well, it is still possible to use
Gaussian approximation
If the variance is nite, (X − E(X))/σ might be closer to the Gaussian
distribution, and thus, consider the so-called Cornish-Fisher approximation (see
[?] or [?]), i.e.
Q(X, α) ∼ E(X) + zα V (X), (1)
where
zα = Φ−1
(α)+
ζ1
6
[Φ−1
(α)2
−1]+
ζ2
24
[Φ−1
(α)3
−3Φ−1
(α)]−
ζ2
1
36
[2Φ−1
(α)3
−5Φ−1
(α)],
(2)
where ζ1istheskewnessofX, andζ2
is the excess kurtosis, i.e. i.e.
ζ1 =
E([X − E(X)]3
)
E([X − E(X)]2)3/2
and ζ1 =
E([X − E(X)]4
)
E([X − E(X)]2)2
− 3. (3)
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![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Using a parametric models
Denition 45. Given a n sample {X1, · · · , Xn}, the Cornish-Fisher estimation
of the α-quantile is
qn(α) = µ + zασ, where µ =
1
n
n
i=1
Xi and σ =
1
n − 1
n
i=1
(Xi − µ)
2
,
and
zα = Φ−1
(α)+
ζ1
6
[Φ−1
(α)2
−1]+
ζ2
24
[Φ−1
(α)3
−3Φ−1
(α)]−
ζ2
1
36
[2Φ−1
(α)3
−5Φ−1
(α)],
where ζ1 is the natural estimator for the skewness of X, and ζ2
is the natural
estimator of the excess kurtosis, i.e. ζ1 =
n(n − 1)
n − 2
√
n
n
i=1(Xi − µ)3
(
n
i=1(Xi − µ)2)
3/2
and
ζ2 = n−1
(n−2)(n−3) (n + 1)ζ2 + 6 whereζ2 =
n n
i=1(Xi−µ)4
( n
i=1(Xi−µ)2
)
2 − 3.
121](https://image.slidesharecdn.com/slides-edf-1-130617053457-phpapp01/85/Slides-edf-1-121-320.jpg)
![Arthur CHARPENTIER - Gestion des risques bancaires et financiers.
Smoothing nonparametric estimators
Two techniques have been considered to smooth estimation of quantiles, either
implicit, or explicit.
• consider a linear combinaison of order statistics,
The classical empirical quantile estimate is simply
Qn (p) = F−1
n
i
n
= X(i) = X([np]) where [·] denotes the integer part. (5)
The estimator is simple to obtain, but depends only on one observation. A
natural extention will be to use - at least - two observations, if np is not an
integer. The weighted empirical quantile estimate is then dened as
Qn (p) = (1 − γ) X([np]) + γX([np]+1) where γ = np − [np] .
124](https://image.slidesharecdn.com/slides-edf-1-130617053457-phpapp01/85/Slides-edf-1-124-320.jpg)
Uploaded byArthur Charpentier
Slides edf-1
The document outlines two sessions on risk management in banking and finance. Session 9 will cover risk measures, regulatory aspects, and basic principles, including defining risk measures, academic vs accounting standards, desirable properties, and estimating risks from samples. Session 10 will cover correlations, copulas, modeling dependencies between risks, diversification effects, comparing risks under dependence vs independence, and analyzing individual risk contributions. Examples of applications to finance, environmental risks, and credit risk are also provided.
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- 1. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Gestion des risques bancaires et nanciers mesures de risques et cadre règlementaire Arthur Charpentier EdF, formation continue arthur.charpentier@univ-rennes1.fr 1
- 2. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Overview of the two sessions 1. Notions de base en probabilité 2. Processus stochastiques en temps discret 3. Marchés nanciers en temps discret 4. Intégration stochastique en temps continu 5. Formule d'Itô et formule de Feynman-Kac 6. Changement de probabilité 7. Introduction aux processus stochastiques discontinus 8. Introduction aux modèles de taux d'intérêt 9. Mesures de risques, aspect règlementaires et principes de base 10. Corrélations, copules 11. Modèles statiques 12. Modèles dynamiques 2
- 3. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Overview of the two sessions Consider a set of risks, denoted by a random vector X = (X1, . . . , Xd) The interest is an agregation function of those risks g(X), where g : Rd → R, and we wish to measure the risk of this quantity R(g(X)), for some risk measure R. • Séance 9 : Mesures de risques, aspect règlementaires et principes de base what is a risk measure? what are academic or pragmatic risks measures, and what about accounting standards? what are desirable properties of R? how to estimate R(Z) given a sample {Z1, . . . , Zn}? • Séance 10 : Corrélations, copules 3
- 4. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Overview of the two sessions Consider a set of risks, denoted by a random vector X = (X1, . . . , Xd) The interest is an agregation function of those risks g(X), where g : Rd → R, and we wish to measure the risk of this quantity R(g(X)), for some risk measure R. • Séance 9 : Mesures de risques, aspect règlementaires et principes de base • Séance 10 : Corrélations, copules how to model X? what about diversication eects? what is the correlation of risks in X? can we compare R(g(X)) and R(g(X⊥ )) (i.e. under independence)? what is the contribution of Xi in the overall risk? 4
- 5. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Some possible motivations... in nance Consider a set of stock prices at time T denoted X = (X1, . . . , Xd) and Y = (Y1, . . . , Yd) the ratio of the price at time T divided by the price at time 0, and and let g(X) denote the payo at time T of some nancial derivative, • e.g. spread derivatives, g(x1, x2) = (x1 − x2 − K)+ based on the spread between two assets, or more generally any extreme spread options, dual spread options, correlation options or ratio spread options, • e.g. buttery derivatives, g(x) = (a x − K)+, i.e. call option on a portfolio of d assets, • e.g. min-max derivatives or rainbow, g(x) = (min{x} − K)+, g(x) = (max{x} − K)+, i.e. call option on the minimum or maximum of d assets, • e.g. Atlas derivatives, g(x) = ( i=i+ i=i− Yi − K)+, where the sum is considered skipping the i− lowest and the d − i+ largest returns, or Himalaya derivatives, g(x) = ( i=d i=i+ Yi − K)+, 5
- 6. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Some possible motivations... in environmental risks 6
- 7. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Some possible motivations... in credit risk Applications with a high number of risks can also be considered, in credit risk for instance. Let X = (X1, ..., Xd) denote the vector of indicator variables, indicating if the i-th contract defaulted during a given period of time. If a credit derivative is based on the occurrence of k defaults among d companies, and thus, the pricing is related to the distribution of the number of defaults, N, dened as N = X1 + ... + Xd. Under the assumption of possible contagious risks, the distribution of N should integrate dependencies. CreditMetrics in 1995 suggested a Gaussian model for credit changes, based on a probit approach, Xi = 1(X∗ i ui), where X∗ i ∼ N(0, 1). 7
- 8. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. !6 !4 !2 0 2 4 6 0.00.10.20.30.40.50.6 Value of the company Probit model in dimension 1 DEFAULT !4 !2 0 2 4 !4!2024 Value of company (1) Valueofcompany(2) (1) DEFAULTS (2)DEFAULTS Probit model in dimension 2 Figure 1: Modeling defaults based on a probit approach. 8
- 9. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Some possible motivations... in risk management Consider a set a risks X = (X1, ..., Xd) (returns in a portfolio, losses per line of business, positions of nancial desks) A classical risk measure (as in Markowitz (1959)) is the standard deviation, σ(Xi) = V ar(Xi) = E((Xi − E(X)) 2 ). The risk of the portfolio S = X1 + . . . + Xd is σ(S) = σ(X1)2 + . . . + σ(Xd)2 + 2 ij r(Xi, Xj)σ(Xi)σ(Xj). Risks are now measured using Value-at-Risk (i.e. quantiles), and there is no relationship between q(X1 + . . . + Xd) and q(X1) + . . . + q(Xd). 9
- 10. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. qqq q q q qq qq qq q q q qq q q q q q q q q q q q q q q q qqq qq qq qqq q q q q q q q qq q qq q q qq q qq q q q q q qqq q q q q q qq q q qqq 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 qqq q qq q q q q qq q q q q q qqqq qq q q q qq q q q qq q q q q qq qq q q q q q q q qq q qq q qq q q qq q q q q q qq q q q qqq q q q q qq q q q q q q qq q q q qqq qqq q q q qq qq q q q q q q q q qq qq q q qq qq q q q q q qq q qq q q q q q q q q qqq q q q qq q q q q q qq q q q q qqqq q q q qqq q qqq q q q q q q q q q q q q q q q q q q q qqq q q q q q qq q qqq q q q q q q q qqq q q q q q q q q q q q q q qq q qqq qq q q q qqq qq q q q qq q qqq q q q qq q qqqq q q q qqq qqqq qq q q q q qqq q q q q q q q qq q qq q q qq q q q q q q qq q q q q q q q q qq 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 q q q q qq q q qq q q q q qq q q q qq q q q q q q q q q q qq q q q q q q q q q qqq q q q q qq qqq q q q q q q q q q qq qqqq q q q q q q 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q q q q q q q q q q q qq q q q qq q qq q q qq q q qqqq q qq q q q qqqqqq q qq qq q q q qqq qq q q q q qq q q qq q qq q q q q q q q q qqq q q qq q q qqq qqqq q q q q q q q q q q q q qq q q qq q qq q qq q q qqqqq q qq qq qq q q q q q q qqqqqqqqq q q qqqqq q q qqqq qq q q qq q qqqq q q q q q q qq q q qqqq qqq q q qqqqq q q q qq q qqqqqq q qqqq q q q qqq q q qqq q q q q q q q qq q qq q q q qq qqq q qq qqqqqqq q qqq qq qq qq q q qqqq qq qq q qq q q qqq qqqqqq q q qq q q q qqq q q q q q qq qqq qqqqqq q q q q q q qq qqqqq q qq q q qq q q q q q q q qqq qq qqqqq qqqq q q q q qq qq qqq q qqq q q q qqq q q q qqqq q q q q q q q qq qq q qqqqq q q q q q q qq qqqq q q q q q q qq qqqq q q qq qqq qqq q q qq q q q q qqqqqq q qq qqqqqqqq qq q q q q q q q q q q qq q q q q q q q q q q q q q q q qq qq qqqqqqq qqq q q q qq q q q qq q q q qq q qq q q q q q q q qq q q q q q qq q q q q q q q qq q qq q q q q q q qq q q q q q q q qq q qq q q q q q qqq q qq q q qqq q qqq q q qqq q q qq q qqqqqq qq q qq qq qq q q q q q q qq qq qqq q q qq q q q qqq q q q q q q qq q qq q qq qq qqqq q q q q qq qq q q q q qq q q qq qq q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q qq q q q q qq q q q qq qq q q q qqq q q q q q q q q q q q q q qqq q qq qqq q q qq q q q qq q qqq qq q q q q q q q qq q q q q q q q q q q q qqq q q q qqq q q q q q q q q q q qq q qqqq q qq q q q q qqq q q qq q q q qq q qq qq qq q q q qq qqq qq qq q q qq qqqq q q q q q q qq qqqqq q qq qq qq qq q q q q q q q qq q q q q q q q q q qqq q qq q q q q q q q qq q qqqq q qqq q q qq qq q q q q q qq q qq qq qqq q q q q qq qqq q qq qq qq q q q q q q qq qqq q q q q q q q qqq q qq qq q qq q qq q q q qq qq qqq q qq q q qqq qq q q q q q q q q q q q q qqq qq q q q q q q q q q qq q q q q q q qq q qqq q q q q q q q qq q qq q q q q q q q q q q q q qq qqq q q q q q q q q q q q q q q qq q q q q qq qq qq q qq q qq q qq q q qqq q q q qqqq qq qqq q q q q q q q q q q q qq q q qq q q q q q qq q qq qq q q qqq qq qqq q qq qqqqq q qqq q q q q qq qq q q q qq q q q q q q qqq q q q q q q q qqq q q q q qq q qq q q q qqq q qq q q q q q q qqq q q q q q qq q q qqqq q qqq q qqqq q q q qq q qq q qqq q q q q q q q qq qqqq q q q q q q q qqq q q q q qq q q q qq q q q qq q qq q q q qq q q qq q q qq qq q qq qq q q q q q q q qq qqq q q q q q qq q q q q q q qqqq q qqq q q q qq q q q qqq q qq q q qqqqq q qqq qqq q q qq qq qqqq q q q q q q q qq q q q q qq q qqqq q q q q qq q qq q qq qqqqqq qq qq q q q qqq q q q q qq qqqq q q q qqqqq q q q q qq qq q q qqq q q qqq q qq q q qq q q qq q q q q qq qqq q qq q qqq q q qqqqqq q q qqqqq qqq q qq q qq qq qqqq q q q q qqq qq qqq qq q q q q q qq q qq q q q q q q qq q q qq q q q q q q q q q q q q q q qq qqq q q q q q q q qqq qq q q q qq q q q qq q qqqqq qqq q q q q q q qq q qq q q q q q qqq qq q qq q q qqq q q qqqqq q q q qqqq qq q q qqq q qqq qqq q q q q q qq qq qq q q q qq q q q qqq q q q q q qq qqq qqqqq q qqqq qq q q q qq qq q qqq q qq q q q q q q qq q q q q q q q qqq qq qqq q qqq q q q q qq qqq qq qqq q q q q q qq q q qqq qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq q qqq q q q q q qqq q q q qqq q qqq qq q qq q qq qqqqq q q qq qq q q qqq q q qq q qq q qq q q q q q q q q qq q q q q q q q qqq q q q q qq q q qq qq q q qq q q q q q q qq q qq q q q q qq q q qqqq q qq q q q q qqqq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qqq q qq q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q qqq q q q qq qqq q q q q q qq q qqq q q q q qq q q q q q q q q q q q q q q q q q q q q q q qqqqq q q q q q q q q q q qqqq qqq q q q q q qq qq q q qqq q qqq qq qq q qq q qqqq qq q q q q qq q q q q q qq q q q q q q q q q q q q q q qq q qq q q q qq qqq q q qqqq q q qq q q q q qq q q q qq q q qq qqq q q q qq qq q q q q q qq qq q qq q q q q qqq q q q qq q q q qq q q q qq q q q q q q qqq qq q q q qq qq q q q q qq qq qq q q q q q q qq q q q qq qqq q qq qq q q qq q q q q q q q qqq q q q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q qq q q qq q q q q q q q q q qq q qq q q q qq q q qq qqq q q q q q qq qq q q q q q qq q qq q q q q q qqq q q q qq qq q q q qq qq q q qq q q q q q q q q q qqqq q q q q q q q qqq qq q q q q qq q q qq q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q qqq q q q q qq q q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q q q qq q q q q q q qq qq q q q q q qq qq q q q q qqq q q qq q q q q q qq qq qq qq qq q qq q q qqq q q q q qq qq qq q q q qq q qq q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q qqq q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q qq q qqq q q q qq q q q q q qq q q q q q q q q q q q q q qqq q q q qq qq q q q qqqq 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 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 q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q qq q q q q q q q q q q qq q q q qq q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq qqq q q q q q q qq q qq q q q q q qq q q q q q q q q q q qq qq q q q qq q qq qq q q q q q q q qq q q qq q q qq q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq 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 q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q qq q q q q q q q q q q q q q q q q qq q q qq 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 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 q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq 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 q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q qqq q q q q q q q q q q q q q q q q q qq q q q q q q q q q qqq q q qq 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 qq q q q q q q q q q q q q q q q q q q q q q qq 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 q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq qq q q q q q q q q qqq q q q q q q q q q q q q q qq q q q q q qq q q q qq q q q q q q q q q q q q qq q q q q q q q qq q qqq q q q q q q qq q q q q q q q q q q q qqq q q q q q q q q qqq q q qq q q qq q qq q q q q q q q q qq 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 q q q qqq 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 q q q q q q q q q q q q qq 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 q q qq q q qq q q q q q q q q q qq qq q q q q q q q qq q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q qq 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 qq q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q qq q q q qq q q q qq q q q q q qq q q q q q q q q q q qq q q q q q q q q q q qq qq 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 q q qq q q q qq q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q qq 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 qq q q q q q q q q q q q qq q qqq q q q q qq qq 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 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 q q qq qqq q qq qq q qq qq q qqqqq q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q qq q q q q q q qq q q q q q q qq q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q qq q qq q q q q q q q q q q qqqq qqq qq q qq q q q q q q q q qq q q qqq q q qq q q q qqq q qq q q qqq q q q qq q q q q qq qq q q q q q q q qq q q q q q q q qq q q q q q qq q q qq q q qq q q q qq q q q q qqq q q q q qq q q q q q qq q q q q q q q q q q q qq q qq q q q q qq q q q qqq q q q q q q q q qq q q qqq q qqqq qq q q q q q q q q q q qq qq q q q q q q qqq q q q q q qq q q q q q q q q qq qq q q q q q qq q q q qq q q q q qq q qqq qq q qq q q q q qqq q q q q q q q q qq q qq q q qqq q q q qq q q q q qqqq q qq q q q q qq qqqqqq q q qq q q q qq q qq q q q q q q q q q q q q q qq qqq q q q q q q q q q qq qq q q q q q q qq qq q q qq q q q qq q q q q q q q qqq qq q q q q qq q q q q q qqq q qq q q qq q q q q q qq q q q qq q q q q q q q q qqqq q q q q qq q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q qq qqq q q q q q q qq q q q qq q qq q qq q q qq q q qqq q q qq q q qq q q q q q q q q q q qq q q qq q q q q q q q q qq q qq q qqq qq q qq q q q q q q q q q q q qq q q q q q qq qq qq qq q q q qq q qq q q q q q q q q qqqq qq qq q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q qq q qq qqq q qq q q q q q q qqq q qq q qq q q q qqqqq q q qq q q qq q q q qqqqqq q q q q q q q q q qq q q q q q q qq q qqqq q q q q q q q q q q The Nasdaq index, 1971−2007 1970 1980 1990 2000 −0.10−0.050.000.050.10 010002000300040005000 Dailylogreturn LeveloftheNasdaqindex Figure 3: Measuring risks of nancial assets, Nasdaq (1971-2007). 11
- 12. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • Operational risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 12
- 13. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 13
- 14. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Some references on risk management Föllmer, H. Schied, A. (2004). Stochastic nance. An introduction in discrete time. Gruyter Studies in Mathematics, Jorian, P. (2007). Value-at-Risk, McNeil, A. Frey, R., Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Roncalli, T. (2004). La gestion des risques nanciers. Economica. Basel Committee on Banking Supervision. International convergence of capital measurement and capital standards. http://www.bis.org/publ/bcbs128.pdf 14
- 15. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Introduction to risk management: crisis a statistical model is a probability distribution constructed to enable inferences to be drawn or decisions made from data (Jorion (2007)). 1974 Herstatt Bank, 620 million USD (⇒ Real Time Gross Settlement Systems) 1994 Metallgesellschaft, 1340 million USD (oil futures) 1994 Orange County, 1810 million USD (derivaties) 1994 Procter Gamble, 102 million USD (derivaties) 1995 Barings, 1330 million USD (fraudulent manipulations) 1997 Natwest, 127 million USD (fraudulent manipulations) 1998 LTCM (Long Term Capital Management), 2000 million USD (liquidity crisis) 15
- 16. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Distinguishing among risks • market risk • interest rate risk • currency risk • volatility risk • credit risk • default risk • downgrading risk • operational risk • disaster risk • fraud risk • technologic risk • litigation risk • liquidity risk • model risk 16
- 17. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. History of risk measures The evolution of (analytical) Risk Management Tools (from Jorion (2007)) 1938 bond duration 1952 Markowitz mean-variance framework 1963 Sharpe's single beta model 1973 Black Scholes option pricing formula 1983 RAROC, Risk Adjusted Return 1992 Stress testing 1993 Value-at-Risk (VaR) 1994 RiskMetrics 1997 CreditMetrics 1998 integration of credit and market risk 1999 coherent risk measures 17
- 18. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 18
- 19. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Market risks Classical models for stock prices, • dynamic models (Bachelier (1900), Black Scholes (1973)), Brownian geometric dSt = µStdt drift + √ V StdWt random part , where (Wt)t≥0 is a standard brownian motion, • more advanced dynamic models (Heston (1993)) have stochastic volatility dSt = µStdt + √ VtdWS t dVt = κ(θ − Vt)dt + ξ √ VtdWV t , where (WS t )t≥0 and (WV t )t≥0 are two standard brownian motions (possibly correlated). 19
- 20. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 0.0 0.2 0.4 0.6 0.8 1.0 50100150200 Stock price over 1 year, large volatility Time 0.0 0.2 0.4 0.6 0.8 1.0 50100150200 Stock price over 1 year, large volatility Time Figure 4: Random generation of a stock price, dSt = µStdt + σStdWt. 20
- 21. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Market risks Note that continuous GARCH processes can also be considered dSt = µStdt + √ VtdWS t dVt = κ(θ − Vt)dt + ξVtdWV t , 21
- 22. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Market risks • the capital asset pricing model (Markowitz (1970) or the Sharpe index are based on the mean-variance framework, 0 5 #0 #5 !#.0!0.50.00.5#.0#.5%.0%.5 ca)t!type sp/)ance 0 #0 # !#.0!0.0.00.#.0#.%.0%. ca)t!t+pe sp/)ance Figure 5: Capital asset pricing model, the mean-variance framework. 22
- 23. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. How to quantify market risks : volatility All the information about uncertainty is summarized by the volatiliy - or variance - parameter. Note that this is one of the drawback of the use of the Gaussian distribution. 23
- 24. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. A (very) short word on diversication Naturally, in higher dimension (when dealing with multiple stocks), Gaussian vectors are considered X = X1 X2 . . . Xd ∼ N µ1 µ2 . . . µd , σ2 1 ρ1,2σ1σ2 · · · ρ1,dσ1σd ρ2,1σ2σ1 σ2 2 · · · ρ2,dσ2σd . . . . . . . . . ρd,1σdσ1 ρd,2σdσ2 · · · σ2 d All the information about marginal risks is in the variances (σ2 i ) while all the information on the dependence is in the correlation coecients (ρi,j). 24
- 25. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. On the Gaussian distribution The Gaussian distribution is very important for many reasons, • it is a stable distribution, i.e. it appears as a limiting distribution in the central limit theorem: for i.i.d. Xi's with nite variance, √ n X − E(X) √ V X L → N(0, 1). • it is an elliptic distribution, i.e. X = µ + AX0 where A A = Σ, and where X0 has a spheric distribution, i.e. f(x0) is a function of x0x0 (spherical level curves), 25
- 26. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. −3 −2 −1 0 1 2 3 −3−2−10123 Level curves of a spherical distribution −3 −2 −1 0 1 2 3 −3−2−10123 Level curves of a elliptical distribution Figure 6: The Gaussian distribution. 26
- 27. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. On the Gaussian distribution As a consequence, if X ∼ N(µ, Σ), and if X = X1 X2 ∼ N µ1 µ2 , Σ11 Σ12 Σ21 Σ22 • Xi ∼ N(µi, Σi), for all i = 1, · · · , d, • α X = α1X1 + · · · + αdXd ∼ N(α µ, α Σα), • X1|X2 = x2 ∼ N(µ1 + Σ12Σ−1 2,2(x2 − µ2), Σ1,1 − Σ12Σ−1 2,2Σ21) 27
- 28. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.00 0.05 0.10 0.15 0.20 Density of the Gaussian distribution −3 −2 −1 0 1 2 3 −3−2−10123 Level curves of a elliptical distribution Figure 7: The Gaussian distribution. 28
- 29. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 29
- 30. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Credit risk Credit risk is the risk that an obligor does not full his payment obligations, either wholly or in part. For single obligors, one should model • risk arrival (given a time horizon, it there a default ?) • risk timing (when does the default occur ?) • risk magnitude (how high is the loss ?) and on a portfolio level • risk correlation (will there be any contagion ?) Here, loss distributions have low probabilities but high losses (high downside risk, strongly skewed). Hence, for credit risk, volatiliy has no meaning. Main problem: no much historical data. 30
- 31. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Classical credit risk models • credit scoring, logit/probit models to model default occurence • models for rating transition based on Markov chains • models for bond prices, intensity based 31
- 32. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The binomial model for credit default Consider defaults in a portfolio, until a xed time-horizon T, with no interest rate. The exposures are of identical size L, identical recovery rate c. Assume also that the portfolio is homogeneous, i.e. each obligor defaults with a probability p before time-horizon T. Assume that defauts happen independenty of each other. Let X denote the number of default in the portfolio, so that the loss is X(1 − c)L. If defaults are independent, P(X = k) = d k pk (1 − p)d−k = d! k!(d − k)! pk (1 − p)d−k = (k; d, p) and P(X ≤ k) = k i=0 d k pk (1 − p)d−k = B(k; d, p) 32
- 33. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 0 5 10 15 20 25 30 0.000.050.100.150.20 Distribution of number of defaults, n=30, p=20% Number of defaults Probability 0 5 10 15 20 25 30 0.000.050.100.150.200.250.300.35 Distribution of number of defaults, n=30, p= 5% Number of defaults Probability 0 5 10 15 20 25 30 0.20.40.60.81.0 Distribution of number of defaults, n=30, p=20% Number of defaults Cumulativeprobability 0 5 10 15 20 25 30 0.20.40.60.81.0 Distribution of number of defaults, n=30, p= 5% Number of defaults Cumulativeprobability Figure 8: Distribution of the number of defaults, independent case. 33
- 34. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Extending the binomial model for credit default A simplied rm's value model can be considered, where the default of each obligor k is triggered by the change of the value Vk(t) of the assets of its rm. Assume that Vk(T) is normally distributed, and standardized, i.e. Vk(T) ∼ N(0, 1). Assume more generally that V (T) = (V1(T), · · · , Vd(T)) ∼ N(0, Σ), i.e. the asset values of dierent obligors might be correleted with each other. Obligor k defaults if its rm's value falls below a barrier Vk(T) ≤ Bk. This can be seen as a multivariate probit model. 34
- 35. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. A one-factor model for credit default Assume that the values of the assets of the obligors are driven by one common factor Y , and an idiosyncratic standard normal noise component εk, so that Vk(T) = √ ρY + 1 − ρεk, k = 1, · · · , d, where Y and the εk's are independent and N(0, 1) distributed. This is an exchangeable model: conditional on the realisation of the systematic factor Y , the rm's values and the defaults rae independent. Assuming that Bk = B and that the exposure is Lk = 1, then P(X = k) = P(X = k|Y = y)φ(y)dy, and conditional on Y = y, the probability to have k defaults is P(X = k|Y = y) = d k p(y)k (1 − p(y))d−k 35
- 36. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. A one-factor model for credit default where p(y) is the probability that Vi(T) falls below B, i.e. p(y) = P(Vi(T) ≤ B|Y = y) = P εi ≤ B − √ Y √ 1 − ρ ≤ B Y = y = Φ K − √ ρy √ 1 − ρ . Substituting yields P(X = k) = d k Φ K − √ ρy √ 1 − ρ k 1 − Φ K − √ ρy √ 1 − ρ d−k φ(y)dy 36
- 37. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 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 q q q q q 0 5 10 15 20 25 30 0.000.050.100.15 Distribution of number of defaults, n=30, p=20%, r=0.1 Number of defaults Probability 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 q q q q q 0 5 10 15 20 25 30 0.000.050.100.15 Distribution of number of defaults, n=30, p=20%, r=0.2 Number of defaults Probability 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 q q q q q 0 5 10 15 20 25 30 0.000.050.100.15 Distribution of number of defaults, n=30, p=20%, r=0.3 Number of defaults Probability 0 5 10 15 20 25 30 0.00.20.40.60.81.0 Distribution of number of defaults, n=30, p=20%, r=0.1 Number of defaults Cumulativeprobability 0 5 10 15 20 25 30 0.00.20.40.60.81.0 Distribution of number of defaults, n=30, p=20%, r=0.2 Number of defaults Cumulativeprobability 0 5 10 15 20 25 30 0.20.40.60.81.0 Distribution of number of defaults, n=30, p=20%, r=0.3 Number of defaults Cumulativeprobability Figure 9: Distribution of the number of defaults, nonindependent case. 37
- 38. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using variance for credit risk ? Is variance relevant to measure risk ? No since it is not a downside risk. 38
- 39. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 0.0 0.2 0.4 0.6 0.8 020406080100 Evolution of the variance of X as a function of rho, p=20% Evolution of parameter rho Variance Figure 10: Evolution of the variance of X as a function of ρ. 39
- 40. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using variance for credit risk ? One has to nd a coecient which measures properly downside risk. An idea is to use a quantile Denition 1. The α-quantile of a distribution FX is qX(α), such that qX(α) = F−1 X (α) where F−1 X (u) = inf{x, FX(x) ≥ u}, where u ∈ (0, 1). Then F−1 X (·) is increasing on (0, 1), continuous from the left, with limits from the right, and further F−1 X ◦ FX(x) ≤ x for any x and FX ◦ F−1 X (u) ≥ u for any p. Remark Almost equivalently, it is possible to dene F−1 X (u) = sup{x, FX(x) ≤ u}, where u ∈ (0, 1). 40
- 41. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 0.0 0.2 0.4 0.6 0.8 20406080100 Evolution of the variance of X as a function of rho, p= 5% Evolution of parameter rho Quantile 90% quantile 95% quantile 99% quantile 0.0 0.2 0.4 0.6 0.8 406080100 Evolution of the variance of X as a function of rho, p=20% Evolution of parameter rho Quantile 90% quantile 95% quantile 99% quantile Figure 11: Evolution of the quantile of X as a function of ρ. 41
- 42. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 42
- 43. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Value-at-Risk The expression is quite recent and its origin is uncertain: in the 80's, some papers introduced dollars-at-risk, capital-at-risk, income-at-risk, earning-at-risk and nally value-at-risk Denomination has been stabilized after the publication of RiskMetrics Technical Document in 1994, by JPMorgan. Note that the work accomplished by JPMorgan was more a pulic relation campaign than an advanced technical study: VaR is more a practice than a theory. VaR summarizes the worst loss ever on a target horizon that will not be exceeded with a given level of condence, i.e. formaly it is a quantile of the projected distibution of gains and losses over the target horizon Till Guldimann (1992) created the term value-at-risk while head of global research at JP Morgan in the late 80's. It appeared in the G30 report (group of thirty) in July 1993. 43
- 44. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. A technical denition for the Value-at-Risk Denition 2. Let X denote the loss distribution, then V aR(X, α) = qX(α), for all α ∈ (0, 1). 44
- 45. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The Basel II accord (2004) June 2004, the Basel Committee nalized the Basel Accords, based on three pillars • minimum regulatory requierements, i.e. some risk-based capital requirements: set capital charges against credit risk (internal rating based), market risk (internal model approach) and operational risk. the goal is to keep constant the level of capital in the global banking syste: 8% of risk weighted assets, • supervisorv review, i.e. expanded role for bank regulartors, to ensure that banks operate above the minimum regulatory capital ratios, that banks have appropriate processes for assessing their risks, and appropriate corrective actions • market discipline, i.e. set of disclosure recommendations, encouraging to publish informations about exposures, risk proles, capital cushion... From the rst pillar, there should be a credit risk charge (CRC), a market risk charge (MRC) and an operationnal risk charge (ORC), and the bank's total 45
- 46. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. capital must exceed the total-risk charge (TRC) Capital TRC = CRC + MRC + ORC. Why using VaR as a risk measure ? Markowitz (1952) claimed that standard deviation should be an intuitive and appropriate risk measure (leading to the mean-variance trade-o). The same year, Roy (1952) claimed that the optimal bundle of assets (investment) for investors who employ the safety rst principle is the portfolio that minimizes the probability of disaster. Roy A. D. (1952), Safety rst and the holding of assets, Econometrica, 20, 431-449. Markowitz H. M. (1952), Portfolio selection, Journal of Finance, 7, 77-91. 46
- 47. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 47
- 48. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Introduction the risk measures, and risk perception S. Clam [...] once said: I dene a coward as someone who will not bet when you oer him two-to-one odds and let him choose his side . With the centuries old St. Petersburg paradox in my mind, I pedantically corrected him: You mean will not make a suciently small bet (so that the change in the marginal utility of money will not contaminate his choice). . Recalling this conversation, a few years ago I oered some lunch colleagues to bet each $200 to $100 that the side of a coin they specied would not appear at the rst tom. One distinguished scholar - who lays no claim to advanced mathematical skills - gave the following answer: 48
- 49. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Introduction the risk measures, and risk perception I won't bet because I would fell the $100 loss more than the $200 gain. But I'll take you on if you promise to let me make 100 such bets . What was behind this interesting answer ? He, and many others, have given something liko tho following explanation. One toss is not enough to make it reasonably sure that the law of averages will turn out in my favor. But in a hundred tosses of a coin, the law of large numbers will make a dam good bet. I am, so to speak, virtually sure to come out ahead in such a sequence, and that is why I accept the sequence while rejecting the single toss. . One can check that P(gain 0) = P(at least 34 odds) ∼ 99.91%. However, with one toss, the maximal loss is $100 but it becomes $10,000 with 100 tosses. 49
- 50. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Notations Let X be a real valued random variable, interpreted as a (net) loss. Denition 3. A risk measure is a function R : X → R, interpreted as the capital necessary. Example 4. R(X) = sup{X(ω), ω ∈ Ω}, R(X) = sup{EQ(X), Q ∈ Q} where Q is a set of probabilities (called scenarios), R(X) = F−1 X (α) where α ∈ (0, 1) ...etc. 50
- 51. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Risk measures and price of a risk 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 = EP(X), based on the règle des parties. For Quételet, the expected value was, in the context of insurance, the price that guarantees a nancial 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 (1953)). 51
- 52. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. What is probability P ? my dwelling is insured for $ 250,000. My additional premium for earthquake insurance is $ 768 (per year). My earthquake deductible is $ 43,750... The more I look to this, the more it seems that my chances of having a covered loss are about zero. I'm paying $ 768 for this ? (Business Insurance, 2001). • Estimated annualized proability in Seatle 1/250 = 0.4%, • Actuarial probability 768/(250, 000 − 43, 750) ∼ 0.37% The probability for an actuary is 0.37% (closed to the actual estimated probability), but it is much smaller for anyone else. 52
- 53. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Saint Pétersbourg's paradox Problem proposed by Bernoulli (1713), Une pièce de monnaie est lancée jusqu'à ce que pile apparaisse. Le joueur A reçoit alors de la banque B la somme de 2n francs, ou n est le nombre total de lancers. Quelle mise doit disposer A avant le premier jet pour que la partie soit équitable ? It is a paradox since the expected value is innite ∞ i=1 P( stop after n draw) · 2n = ∞ i=1 1 2n · 2n = ∞ i=1 1 = ∞. 53
- 54. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Saint Pétersbourg's paradox Many answers have been investigated • the bank does not have innite liabilities, and thus, the player can play only a nite time (Buffon (1777), Poisson (1837), Borel (1949)), • the player has a moral utility of money (Cramer(1728), Bernoulli (1738), von Neumann Morgenstern (1956)) where a concave utility function is considered, • the player bets using subjective probabilities, were rare events are assumed to be impossible (D'Alembert (1754), Menger (1934), Yaari (1987)) 54
- 55. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Risk measures: the expected utility approach 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 ) . 55
- 56. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Risk measures: Yarri's dual approach 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 ≥ α) Rg(X) = V aR(X, α), and if g(x) = min{x/α, 1} Rg(X) = TV aR(X, α) (also called expected shortfall), Rg(X) = EP(X|X V aR(X, α)). 56
- 57. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Yarri's dual approach: capacities and Choquet's integral Here Rg(X) = g(P(X x))dx = g(FX(x))dx with g : [0, 1] → [0, 1] increasing. Thus, g ◦ FX is a decreasing function taking values in [0, 1] on [0, ∞): g ◦ FX is a survival function. Can Rg(X) be seen as an expected value of X with a change of measure ? Yes if there exists a probability measure Q such that g ◦ FX(x) = Q(X x). If it is possible to dene such a measure Q, generally Q is not a probability measure. In fact, Q satises • Q(∅) = 0 (since FX(∞) = 0 and g(0) = 0), • Q(Ω) = 1 (since FX(0) = 1 and g(1) = 1), • Q(A) ≤ Q(B) if A ⊂ B (since FX(·) is decreasing and g(·) is increasing). 57
- 58. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Yarri's dual approach: capacities and Choquet's integral Such a measure Q satises only Q(A) ≤ Q(B) if A ⊂ B: Q is a capacity. With this notation, Rg(X) = xdg ◦ P = g(P(X x))dx = Q(X x)dx, but since Q is not a probability measure, Rg(·) is not an expected value: it is the so-called Choquet's integral. 58
- 59. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Distortion of values versus distortion of probabilities 0 1 2 3 4 5 6 0.00.20.40.60.81.0 Calcul de l’esperance mathématique Figure 12: Expected value xdFX(x) = P(X x)dx. 59
- 60. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Distortion of values versus distortion of probabilities 0 1 2 3 4 5 6 0.00.20.40.60.81.0 Calcul de l’esperance d’utilité Figure 13: Expected utility u(x)dFX(x). 60
- 61. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Distortion of values versus distortion of probabilities 0 1 2 3 4 5 6 0.00.20.40.60.81.0 Calcul de l’intégrale de Choquet Figure 14: Distorted probabilities g(P(X x))dx. 61
- 62. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Axiomatic approach for risk measures There are three way to describe risk measure: characterizing natural properties that should satisfy • the risk measure R(·), e.g. R(·) is subadditive (R(X + Y ) ≤ R(X) + R(Y )), • induced stochastic ordering , i.e. X Y (Y is more risky than X) if and only if R(X) ≤ R(Y ) [Economics], • induced set of acceptable risks A, i.e. X ∈ A (X is is acceptable) if and only if R(X) ≤ 0 [Financial Mathematics]. 62
- 63. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Ordering and comparing risks Assume that risks are positive random variables. The higher R(X), the risker X is. Y will be said to be more risky than X will be denoted X Y . In Pascal's approach FX(x) = P(X ≤ x) X Y ⇐⇒ R(X) ≤ R(Y ) where R(X) = EP(X) = xdFX(x). 63
- 64. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. More dicult to quantify than to compare Denition 5. is an ordering relationship if it is reexive (FX FX), transitive (if FX FY and FY FZ then FX FZ) and antisymmetric (if FX FY and FY FX then FX = FY ). Note that the ordering on the set of distribution functions will be extended to the set of positive random variables (with X ∼ Y if FX = FY , i.e. X L = Y ). Denition 6. satises the additivity axiom if for any risks X, Y and Z such that X Y , then X + Z Y + Z. It denotes the invariance of perception in case of a common variation. It might also be called the linearity axiom. 64
- 65. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. More dicult to quantify than to compare Denition 7. satises the continuity axiom (or Archimedean axiom) if for any FX, FY and FZ such that FX FY FZ, then for all α, β ∈ (0, 1) αFX + [1 − α]FZ FY βFX + [1 − β]FZ. Proposition 8. If satises the continuity and associativity axioms, X Y ⇐⇒ R(X) ≤ R(Y ) where R(X) = EP(X) = xdFX(x) = ∞ 0 P(X x)dx. 65
- 66. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The expected utility approach In order to answer Saint Petersbourg' paradox, one solution, proposed by Bernoulli was to introduce a moral value of money, i.e. a nonlinear perception of gains: he suggests to consider log(1 + X) instead of X. The price of the game was then EP(log(1 + X)). Analogously, Cramèr suggested to consider √ X, so that the price was EP( √ X). Hence, the idea was to consider a utility function of gains, u(·), which can change for all players. Several mathematicians, for example Laplace, discussed the Bernoulli principle in the following century, and its relevance to insurance systems seems to have been generally recognized. In 1832, Barrois presented a fairly complete theory of re insurance based on Laplace's work on the Bernoulli principle. (Borch (1974)). 66
- 67. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The expected utility approach Denition 9. satises the independence axiom if for any distribution function FX, FY and FZ such that FX FY , then for all λ ∈ [0, 1] λFX + [1 − λ]FZ λFX + [1 − λ]FZ. or equivalently (λX) ⊕ ([1 − λ]Z) (λY ) ⊕ ([1 − λ]Z), where ⊕ denotes a mixture. Hence, ordering are not modied when mixing risks with a third one. Recall that (λX) ⊕ ([1 − λ]Z) = (λX) + ([1 − λ]Z). 67
- 68. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Example 10. If X, Y are two Bernoulli variables B(2/3) and B(1/3) respectively , independent, X = 0 p = 1/3 1 p = 2/3 and Y = 0 p = 2/3 1 p = 1/3 X ⊕ Y = X p = 1/2 Y p = 1/2 = 0 p = 1/3 × 1/2 1 p = 2/3 × 1/2 0 p = 2/3 × 1/2 1 p = 1/3 × 1/2 = 0 p = 1/2 1 p = 1/2 X + Y = 0 + 0 p = 1/3 × 2/3 0 + 1 p = 1/3 × 1/3 1 + 0 p = 2/3 × 2/3 1 + 1 p = 2/3 × 1/3 = 0 p = 2/9 1 p = 5/9 2 p = 2/9 68
- 69. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Proposition 11. If satises the continuity and independence axioms, there exists a function u with values in R, continuous, strictly increasing, unique up to an ane transformation, such that X Y ⇐⇒ Ru(X) ≤ Ru(Y ) where Ru(X) = EP(u(X)) = u(x)dFX(x). Proof. von Neumann Morgenstern (1944) or Fishburn (1970). The continuity of u comes from the continuity assumption of the ordering. If u is concave, the risk taker is said to be risk adverse since (Jensen's inequality) EP(u(X)) ≤ u(EP(X)). 69
- 70. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The expected utility approach The insurance premium is then obtained by the null utility principle: π(X) satises EP(u(π(X) − X)) = 0. Example 12. With an exponential utility, u(x) = [1 − e−αx ]/α, alors π(X) = 1 α log EP(eαX ) . Note that the exponential utility does not exist for heavy tailed risks. Example 13. With a quadratic utility, u(x) = x − x2 /2s where x s, then π(X) ∼ EP(X) + κ 2 V arP(X). 70
- 71. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Risk aversion and nance One can introduce, Arrow-Pratt coecient of absolute risk aversion, RA(x) = − u (x) u (x) , and the coecient of relative risk aversion, RR(x) = − xu (x) u (x) CARA (Constant Absolute Risk Aversion) means that RA(·) is constant, i.e. u(x) = − 1 α exp(−αx). CRRA (Constant Relative Risk Aversion) means that RA(·) is constant, i.e. u(x) = − x1−α 1 − α , for α 0, including the limiting case u(x) = log(x) (when α → 1. • modeling portfolios with Gaussian returns and CARA utility Assume that X ∼ N(µ, σ2 ) and u(x) = − 1 α exp(−αx), for some α 0. By solving u(EP(X) − π) = EP(u(X)), using the expression of the Laplace transform of the Gaussian distribution, EP(u(X)) = − 1 α EP(exp(−αX)) = − 1 α exp −αµ + α2 2 σ2 , 71
- 72. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. and u(EP(X) − π) = − 1 α exp(−α(µ − π)), thus, one gets that π = α 2 σ2 . • modeling portfolios with lognormal risks and CRRA utility Assume that log X ∼ N(µ, σ2 ) and u(x) = − x1−α 1 − α , for some α 0. By solving u(EP(X) − π) = EP(u(X)), one gets that π = ασ2 2 × EP(X). • quadratic utility principle If u is quadratic u(x) = x − x2 /2s, then EP(u(X)) = EP(X) 1 − 1 2s EP(X) − 1 2s V arP(X), hence only the mean and the variance matter. 72
- 73. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Changing probabilities ? Question: what is probability P ? Il y a donc une aversion particulière pour l'incertitude liée à l'ignorance. On préfère avoir un modèle probabiliste que pas de modèle du tout, on préfère évaluer raisonnablement ses chances de succès, fussent-elles minces, que de n'en avoir aucune idée. (Ekeland (1991)). Idée de Ramsey (1931), formalisée par Savage (1972): les individus ne raisonne pas sous P, la probabilité réelle (inconnue), mais sous une probabilité subjective Q. Problème: dicile d'estimer une probabilité d'évènement rare. Travaux de Selvige (1975): importance des évènements rares aux conséquences importantes. Approche psychologique du risque: besoin de comparer à des évènements rares quantiables (taux de mortalité infantile, quinte ush au pocker...). 73
- 74. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Changing probabilities ? Denition 14. est une relation vériant l'axiome de monotonie si P(X + ε ≤ Y ) = 1 implique X Y , pour tout ε 0. Proposition 15. Si est une relation d'ordre vériant les axiomes de continuité, d'additivité et de monotonie, alors il existe une probabilité Q telle que X Y ⇐⇒ RQ(X) ≤ RQ(Y ) où RQ(X) = EQ(X) = ∞ 0 Q(X x)dx. 74
- 75. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using subjective probabilities Considérons un call européen, dont le payo actualisé est e−rT (ST − K)+. Le prix n'est pas EP e−rT (ST − K)+ . Le prix d'un call européen proposant de toucher (ST − K)+ à maturité. La valorisation de l'option, à la date d'aujourd'hui est basée sur la notion de portefeuille de réplication: deux portefeuilles orant le même payo à une date T ont nécessairement le même prix aujourd'hui (sinon il serait possible de constituer une opportunité d'arbitrage). 75
- 76. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Changing probabilities ? Considérons le modèle de Cox, Ross Rubinstein (1979), avec un actif sans risque valant 1 aujourd'hui, et 1 + r dans un an, et un actif risqué valant S0 aujourd'hui, et, dans un an S1, valant soit Su, soit Sd, avec d 1 + r u, suivant l'état de la nature. Considérons un call européen donnant le droit d'acheter le sous-jacent à maturité (dans un an) à la valeur K. Le payo dans un an est alors (S1 − K)+. Construisons un portefeuille α + βS0 permettant de répliquer la valeur de l'option dans un an: • si le marché monte, le portefeuille vaudra α (1 + r) + βSu, et l'option Cu = (Su − K)+ • si le marché baisse, le portefeuille vaudra α (1 + r) + βSd, et l'option Cd = (Sd − K)+ 76
- 77. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Changing probabilities ? Dans un marché avec absence d'opportunité d'arbitrage, si ces deux produits ont la même valeur dans un an, c'est donc qu'ils ont le même prix aujourd'hui. Le portefeuille qui permet de répliquer le payo de l'option est obtenu en résolvant α (1 + r) + βSu = Cu α (1 + r) + βSd = Cd c'est à dire que α = Cu − Cd S0u − S0d et β = 1 1 + r Cu − S0u Cu − Cd S0u − S0d . Notons au passage qu'il est ainsi toujours possible de constituer un unique portefeuille de réplication. 77
- 78. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Changing probabilities ? Le prix de l'option aujourd'hui s'écrit α + βS0 = 1 1 + r 1 + r − d u − d Cu + u − (1 + r) u − d Cd , qui peut s'écrire π = 1 1 + r (qCu + (1 − q)Cd) , où q = 1 + r − d u − d . Notons que q ∈ [0, 1], c'est à dire que le prix de l'option est l'espérance mathématique, sous une probabilité Q appelée probabilité risque neutre du payo à échéance: π = EQ (payo). Notons que Q n'a rien n'a voir avec la probabilité dite historique P qu'a le sous-jacent de monter ou de descendre: le prix d'un payo aléatoire X ne s'écrit pas EP (X). 78
- 79. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Les paradoxes de Allais et Ellsberg Allais (1953), l'eet de certitude ou l'eet de sécurité Choisir entre les deux lotteries suivantes Loterie A 100% de chance de recevoir 1 million , Loterie B 10% de chance de recevoir 5 millions 89% de chance de recevoir 1 million 1% de chance de ne rien recevoir, puis entre Loterie C 11% de chance de recevoir 1 million 89% de chance de ne rien recevoir, Loterie D 10% de chance de recevoir 5 millions 90% de chance de ne rien recevoir, 79
- 80. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Préférer A à B, et D à C (observé empiriquement) viole l'hypothèse d'indépendance (et même le principe de la chose sûre). Ellsberg (1961), l'eet d'ambiguité Choisir entre les deux lotteries suivantes Loterie A win 1000 si la boule tirée est rouge Loterie B win 1000 si la boule tirée est bleue, puis entre Loterie C win 1000 si la boule tirée n'est pas rouge Loterie D win 1000 si la boule tirée n'est pas bleue, Préférer A à B, et C à D (observé empiriquement) viole l'hypothèse d'indépendance (et même le principe de la chose sûre). 80
- 81. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. La distortion de probabilités, Yaari (1987) Example 16. Un exemple de relation d'ordre est la dominance stochastique à l'ordre 1. X 1 Y si et seulement si une des conditions suivantes (équivalentes) sont satisfaites, • E(g(X)) ≤ E(g(Y )) pour g croissante, • pour tout x ∈ R, P(X ≤ x) ≥ P(Y ≤ x), • pour tout x ∈ R, P(X x) ≤ P(Y x), • pour tout x ∈]0, 1[, V aR(X, α) ≤ V aR(Y, α). Cette relation d'ordre est notée V aR dans Denuit Charpentier (2004). Example 17. Un exemple de relation d'ordre est la dominance stochastique à l'ordre 2. X 2 Y si et seulement si une des conditions suivantes (équivalentes) sont satisfaites, • E(g(X)) ≤ E(g(Y )) pour g croissante et convexe, • E((X − t)+) ≤ E((Y − t)+) pour t ∈ R, 81
- 82. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. • pour tout x ∈]0, 1[, α 0 V aR(X, p)dp ≥ α 0 V aR(Y, p)dp, • pour tout x ∈]0, 1[, ∞ α V aR(X, p)dp ≤ ∞ α V aR(Y, p)dp, • pour tout x ∈ [0, 1[, TV aR(X, α) ≤ TV aR(Y, α). Cette relation d'ordre est notée T V aR dans Denuit Charpentier (2004). Denition 18. est une relation vériant l'axiome d'indépendance comonotone si X Y implique X + Z Y + Z pour tout Z tel que les couples (X, Z) et (Y, Z) soient comonotones. Remark 19. X et Z sont comonotones s'il n'existe pas ω, ω tels que X(ω) X(ω ) et Y (ω) Y (ω ). Denition 20. est une relation vériant l'axiome de cohérence si pour des variables X et Y constantes (P(X = x) = P(Y = y) = 1), FX FY implique x ≤ y. 82
- 83. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Proposition 21. Si est une relation d'ordre vériant les axiomes de continuité, d'indépendance comonotone, monotonie, et est compatible avec la dominance stochastique à l'ordre 1, alors il existe une unique fonction de distortion croissante g : [0, 1] → [0, 1] telle que X Y ⇐⇒ Rg(X) ≤ Rg(Y ) où Rg(X) = xdg ◦ P = g(P(X x))dx = 1 0 F−1 X (1 − p)dg(p). 83
- 84. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 84
- 85. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Coherent risk measures and the axiomatic approach A risk measure is said to be coherent (from Artzner, Delbaen, Eber Heath (1999)) if • R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ), • R(·) is positively homogeneous, i.e. for any λ ≤ 0, R(λX) = λR(X), • R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ, • R(·) is subadditive, i.e. R(X + Y ) ≤ R(X) + R(Y ). subadditivity can be interpreted as diversication does not increase risk. Example: the Expected-Shortfall is coherent, the Value-at-Risk is not. 85
- 86. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Convex risk measures A risk measure is said to be convex (from Artzner, Delbaen, Eber Heath (1999)) if • R(·) is monotonic, i.e. X ≤ Y implies R(X) ≤ R(Y ), • R(·) is invariant by translation, i.e. for any κ, R(X + κ) = R(X) + κ, • R(·) is convex, i.e. R(λX + (1 − λ)Y ) ≤ λR(X) + (1 − λ)R(Y ), for any λ ∈ [0, 1]. Hence, if a convex measure satises the homogeneity condition, it is coherent. Remark A natural way to dene a convex measure satisfying the small size coherent condition is adding a coherent measure a liquidity charge, Rconvex(X) = Rcoherent(X) + Cliquidity(X). 86
- 87. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Sets of acceptable risks Denition 22. Given a risk measure R, a risk X is acceptable if X ∈ AR where AR = {Y ∈ X such that R(Y ) ≤ 0}. Conversely, Theorem 23. Given a set of acceptable risks A, the associated risk measure is the smallest capital amont m such that X − m is acceptable, i.e. RA(X) = inf{m ∈ R such that X − m ∈ A}. Then RAR (·) = R(·) and ARA = A. Proposition 24. If R is a convex risk measure, then AR is convex. Conversely, if A is convex, then RA is a convex risk measure. Proposition 25. If R is a positively homogeneous risk measure, then AR is a positive cone. Conversely, if A is a positive cone, then RA is a positively homogeneous risk measure. Example 26. If R(X) = sup{X(ω), ω ∈ Ω}, then AR = {Y, Y ≤ 0}. If R(X) = F−1 X (α) where α ∈ (0, 1), then AR = {Y, P(Y ≤ 0) ≥ α}. 87
- 88. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Characterizations of coherent risk measures Proposition 27. If R is a coherent risk measure, then there exists a set of probability measures Q such that R(X) = sup Q∈Q {EQ(X)}. 88
- 89. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Value-at-Risk: going further Proposition 28. The Value-at-Risk is (generally) not a coherent risk measure. If X, Y ∼ B(92.5%), independent, then V ar(X, 90%) + V ar(Y, 90%) = 0 + 0 ≤ V ar(X + Y, 90%) = 1. q 0.80 0.85 0.90 0.95 1.00 0.00.51.01.52.0 Proposition 29. The Value-at-Risk is a coherent risk measure for elliptical risks. Proposition 30. For all X, note that V aR(X, α) = inf{R(X) such that R is coherent and V aR(X, α) ≤ R(X)}. 89
- 90. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Tail Value-at-Risk (or Expected Shortfall) Dene TV aR(X, α) = E(X|X V aR(X, α)) = 1 1 − α 1 α FX−1(u)du. In some sense, the TailVaR is the average of worst cases, while V aR was the best worst case. Proposition 31. Tail Value-at-Risk is a coherent risk measure. 90
- 91. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 91
- 92. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Some simple example of capital allocation Let X1, ..., Xn denote a set of risks, and dene the associated portfolio X = X1 + ... + Xn. Consider a risk measure R. The objective is to distribute (or allocate) risk capital k = R(X) of the portfolio to its components, i.e. compute k1, ..., kn such that k1 + ... + kn = k. A rst approach would be to consider a proportional capital allocation, i.e. ki = R(Xi) R(X1) + . . . + R(Xn) · R(X). 92
- 93. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation based on marginal capital cost Idea from cooperative game theory. Let S denote a subset of {1, 2, ..., n} (a subportfolio). Dene the associated cost function, Γ(S) = R j∈S Xj , satisfying Γ({1, ..., n}) = R(X) = k. The marginal cost of risk i is Mi = Γ({1, ..., n}) − Γ({1, ..., n} {i}) = R j Xj − R j=i Xj , Problem is that M1 + ... + Mn = k. 93
- 94. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation based on Shapley value One idea is to use the Shapley value. Let Si denote all the subsets of {1, 2, ..., n} that contain i. The Shapley-value of risk i is ki = S∈Si (|s| − 1)!)(n − |s|)! n! · [Γ(S) − Γ(S {i})], where | · | denote the size of the subset (cardinality). It is simply a weighted average of all possible congurations. Note that here k1 + ... + kn = k. This capital allocation has been used by Swiss Re. 94
- 95. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation based on Shapley value Note that if n = 2, R(X1 + X2) = 1 2 (R(X1 + X2) − R(X2) + R(X1)) allocation for X1 + 1 2 (R(X1 + X2) − R(X1) + R(X2)) allocation for X2 . 95
- 96. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation based on Aumann-Shapley value An alternative is to use the Aumann-Shapley value. Here the idea is to consider sensitivity of marginal cost. Dene the following cost function, C : [0, 1]n → R, C(ω) = R(ωt X) = R n i=1 ωiXi . Note that C(1) = R(X) = k. Dene Ci(ω) = ∂C(ω) ∂ωi , and dene the Aumann-Shapley value as ki = 1 0 Ci(u1)du. 96
- 97. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Aumann-Shapley value and capital allocation Example 32. Consider the variance risk measure, R(X) = E(X) + θvar(X), and assume that X ∼ N(µ, Σ), then ki = E(Xi) + θcov(Xi, X) . Example 33. Consider the standard deviation risk measure, R(X) = E(X) + θ var(X), and assume that X ∼ N(µ, Σ), then ki = E(Xi) + θ cov(Xi, X) var(X) . 97
- 98. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Aumann-Shapley value and capital allocation Assume that R is (positively) homogeneous , then so is C, i.e. C(λω) = λC(ω), and thus Ci(λω)Ci(ω), i.e. the Aumann-Shapley value reduces to marginal costs, ki = 1 0 Ci(u1)du = 1 0 Ci(1) = Ci(1). Note that this allocation can also be derived directly using Euler's theorem for homogeneous functions, C(ω) = ω1 ∂C(ω) ∂ω1 + ... + ωn ∂C(ω) ∂ωn . 98
- 99. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation using distortion risk measures If R is a distortion risk measure, R(X) = ∞ 0 g(1 − Fx)dx. Writing R as R(X) = EQX (X), where dQX/dP = g(1 − FX(X)) then k = R(X) = EQX (X) and ki = EQX (Xi). 99
- 100. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation and dependence Since E(XY ) = E(X)E(Y ) + cov(X, Y ), we can write for distortion risk measures ki = EQX (Xi) = ki = E(Xi · g(1 − FX(x))) = E(Xi) + cov(Xi, g(1 − FX(X))), since E(g(1 − FX(X))) = 1. Hence, the capital is maximum if Xi and X are comonotonic, i.e. ki = E(Xi · g(FX(X))) ≤ E(X+ i · g(FX(X+ ))) = R(X+ i ) = R(Xi). The allocated capital is always lower than the stand-alone risk, i.e. the allocation belongs to the core (in game theory terms). Further, one can obtain ki 0 even if R(Xi) 0. 100
- 101. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Aumann-Shapley value for TVaR TVaR is a distortion risk measure, R(X) = E(X|X F−1 X (α)) and thus, ki = E(Xi|X F−1 X (α)). Aumann-Shapley value for VaR VaR is a positively homogenous, and marginal costs works, R(X) = F−1 X (α) and thus, ki = E(Xi|X = F−1 X (α)). 101
- 102. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation based on expected utility Consider for instance exponential risk measures, R(X) = 1 θ log E(eθX ). The Aumann-Shapley allocation is then ki = 1 0 E(XieθuX ) E(eθuX) du = E Xi · 1 0 eθuX E(eθuX) du here again, change of measure representation. Remark 34. If the Xi's are independent, then ki = R(Xi). 102
- 103. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Capital allocation and decision loss function Capital allocation can also be seen as the solution of distance minimization problem. Consider some distance d(ki, Xi|R, k, X). Then ki = argminκ(κ, Xi|R, k, X). Example 35. Consider the following expected value distance, d(ki, Xi) = E([Xi − ki]2 ). Then ki = k n + E(Xi) − E(X) n . Example 36. Consider the following tail VaR distance, d(ki, Xi|α, k, X) = E([Xi − ki]2 |X F−1 X (α)). Then ki = k n + E(Xi|X F−1 X (α)) − E(X|X F−1 X (α)) n . 103
- 104. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Example 37. Consider the following stop loss distance, d(ki, Xi) = E([Xi − ki]+). Then ki = F−1 Xi (FX+ (k)). Example 38. Consider the following distance on risk measures, d(ki, Xi) = R(Xi) − ki k (R(X1) + ... + R(Xn)) 2 . Then ki = R(Xi) R(X1) + ... + R(Xn) · k. 104
- 105. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Axiomatic approach for capital allocation Based on previous results, a standard assumption is to assume that capital allocated • depends on Xi and X, • but not on the decomposition of the rest X − Xi = j=i Xj. Denition 39. A capital allocation is a function Λ : R × R → R of two arguments (x, y) where x is risk Xi and y is the portfolio X. Denition 40. A capital allocation Λ is called capital allocation with respect to risk measure R if Λ(X, X) = R(X). The capital allocated to X is then the risk capital R(X) of X. 105
- 106. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The three axioms for capital allocation The rst axiom is linear aggregation: the risk capital of the portfolio equals the sum of the risk capital of risks. Hence, Λ(X1, X) + Λ(X2, X) + . . . + Λ(Xn, X) = Λ(X, X) = R(X). Denition 41. A capital allocation Λ satises the linear aggregation axiom if Λ(x, x + y) + Λ(y, x + y) = Λ(x + y, x + y). 106
- 107. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The three axioms for capital allocation The second axiom is diversication: the risk capital for a component should not exceed the overall risk capital of the risk considered as a stand-alone portfolio. Hence, Λ(Xi, X) ≤ Λ(Xi, Xi) for all i = 1, ..., n. Denition 42. A capital allocation Λ saties the diversication axiom if Λ(x, y) ≤ Λ(x, x). 107
- 108. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. The three axioms for capital allocation The third axiom is continuity: small changes to the portfolio have a limited eect on the risk capital. Hence, Λ(Xi, X + εXi) ∼ Λ(Xi, X) if εissmall. Denition 43. A capital allocation Λ saties the continuity axiom if lim ε→0 Λ(x, y + εy) = Λ(x, y). 108
- 109. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Questions about capital allocation axioms The existence issue: given a risk measure R, is it possible to derive capital allocations satisfying those axioms ? The completeness issue: if the allocation exists, is it unique ? If not, is it possible to add more axioms ? The explicit formulation issue: given a risk measure R, is it possible to specify capital allocation? 109
- 110. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Existence of capital allocation Recall that a risk measure is • (positively) homogeneous if R(λX) = λR(X), λ ≥ 0, • subadditive if R(X + Y ) ≤ R(X) + R(Y ). If R is a positively homogeneous and subadditive risk measure such that the following functional derivative lim ε→0 R(X + εZ) − R(X) ε exists for every Z, then Λ(x, y = lim ε→0 R(y + εx) − R(y) ε . 110
- 111. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Unicity of capital allocation based on R Based on the three axioms, the risk capital allocation Λ is uniquely determined as the derivative of the risk measure R at X in direction of risk Xi. 111
- 112. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Examples of capital allocation: standard deviation For a risk measure based on standard deviation, R(X) = E(X) + λ · var(X), the derivative can be derived simply as Λ(X, Y ) = E(X) + λ · cov(X, Y ) var(X) . 112
- 113. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Examples of capital allocation: expected shortfall Here also an explicit expression can be derived. If R(X) = 1 1 − α 1 α VaR(X, u)du = E(X · I(X VaR(X, α))) 1 − α (continuous random variable), then Λ(Xi, X) = 1 1 − α xi · I(X VaR(X, α))dFXi (xi). If P(X = V aR(X, α)) 0, set β = P(X ≤ V aR(X, α)) − α P(X = V aR(X, α)) , then Λ(Xi, X) = 1 1 − α xi · I(X VaR(X, α))dFXi (xi) + β xi · I(X = VaR(X, α))dFXi 113
- 114. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Examples of capital allocation: Value-at-Risk The VaR is not subadditive, and thus there does not exist a linear, diversifying capital allocation with respect to VaR. Nevertheless, the directional derivative lim ε→0 V aR(X + εXi, α) − V aR(X, α) ε might exist for certain portfolios X. This was suggested by Hallerbach (1999), and works well - under continuity assumptions (Tasche (1999) or Gouriéroux, Laurent Scaillet (2000)). Alternative is to write V aR(X, α) = E(X) + λ(X) · var(X), and to use a covariance allocation: if R(X)) = E(X) + λ · var(X), then Λ(X, Y ) = E(X) + λ · cov(X, Y ) var(X) . This technique is very popular in credit risk applications (see Kalkbrener et al. (2004)). 114
- 115. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. And nally a similar idea is to write V aR(X, α) = TV aR(X, a(X)) for some a(·) with values in [0, 1], and to consider capital allocation based on TVaR (see Overbeck (2000)). Numerical example based on a simulated portfolio Consider three lines of business, with rather dierent behaviors, • an exponential distribution X1 ∼ E(0.03), E(X1) ∼ 33, • a log-normal distribution X2 ∼ LN(3, 1), E(X2) ∼ 33, • a Pareto distribution X3 ∼ P(2, 30), E(X3) ∼ 33, Assume further that the three risks are independent. 115
- 116. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 0 50 100 150 200 0.0000.0100.0200.030 Densities of the three risks 0 50 100 150 200 0.00.20.40.60.81.0 Cumulative distribution function of the three risks Figure 15: The three lines of business. 116
- 117. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. risk 1 risk 2 risk 3 Allocation of the pure premium risk 1risk 2 risk 3 Allocation based on 90%!TVaR risk 1 risk 2 risk 3 Allocation based on 95%!TVaR risk 1 risk 2 risk 3 Allocation based on stop!loss distance risk 1 risk 2 risk 3 Relative (proportional) capital allocation ! VaR 90% risk 1 risk 2 risk 3 Relative (proportional) capital allocation ! VaR 99% risk 1 risk 2 risk 3 Relative (proportional) capital allocation ! Tail VaR 90% risk 1 risk 2 risk 3 Relative (proportional) capital allocation ! Tail VaR 99% Figure 16: Comparing capital allocations for the three independent lines of business 117
- 118. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Agenda • General introduction Financial risks • Market risks • Credit risk • From variance to Value-at-Risk Risk measures and capital allocation • Risk measures: an axiomatic introduction • Risk measures: convexity and coherence • Capital allocation: an axiomatic introduction Risk measures and statistical inference 118
- 119. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using a parametric models A natural idea (that can be found in classical nancial models) is to assume Gaussian distributions: if X ∼ N(µ, σ), then the α-quantile is simply q(α) = µ + Φ−1 (α)σ, where Φ−1 (α) is obtained in statistical tables (or any statistical software), e.g. u = −1.64 if α = 90%, or u = −1.96 if α = 95%. Denition 44. Given a n sample {X1, · · · , Xn}, the (Gaussian) parametric estimation of the α-quantile is qn(α) = µ + Φ−1 (α)σ, where µ = 1 n n i=1 Xi and σ = 1 n − 1 n i=1 (Xi − µ) 2 119
- 120. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using a parametric models Actually, is the Gaussian model does not t very well, it is still possible to use Gaussian approximation If the variance is nite, (X − E(X))/σ might be closer to the Gaussian distribution, and thus, consider the so-called Cornish-Fisher approximation (see [?] or [?]), i.e. Q(X, α) ∼ E(X) + zα V (X), (1) where zα = Φ−1 (α)+ ζ1 6 [Φ−1 (α)2 −1]+ ζ2 24 [Φ−1 (α)3 −3Φ−1 (α)]− ζ2 1 36 [2Φ−1 (α)3 −5Φ−1 (α)], (2) where ζ1istheskewnessofX, andζ2 is the excess kurtosis, i.e. i.e. ζ1 = E([X − E(X)]3 ) E([X − E(X)]2)3/2 and ζ1 = E([X − E(X)]4 ) E([X − E(X)]2)2 − 3. (3) 120
- 121. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using a parametric models Denition 45. Given a n sample {X1, · · · , Xn}, the Cornish-Fisher estimation of the α-quantile is qn(α) = µ + zασ, where µ = 1 n n i=1 Xi and σ = 1 n − 1 n i=1 (Xi − µ) 2 , and zα = Φ−1 (α)+ ζ1 6 [Φ−1 (α)2 −1]+ ζ2 24 [Φ−1 (α)3 −3Φ−1 (α)]− ζ2 1 36 [2Φ−1 (α)3 −5Φ−1 (α)], where ζ1 is the natural estimator for the skewness of X, and ζ2 is the natural estimator of the excess kurtosis, i.e. ζ1 = n(n − 1) n − 2 √ n n i=1(Xi − µ)3 ( n i=1(Xi − µ)2) 3/2 and ζ2 = n−1 (n−2)(n−3) (n + 1)ζ2 + 6 whereζ2 = n n i=1(Xi−µ)4 ( n i=1(Xi−µ)2 ) 2 − 3. 121
- 122. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using a parametric models More generally, if FX ∈ F = {Fθ, θ ∈ Θ} (assumed to be continuous), qX(α) = F−1 θ (α), and thus, a natural estimator is qX(α) = F−1 θ (α), (4) where θ is an estimator of θ (maximum likelihood, moments estimator...). 122
- 123. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using a nonparametric estimator For continuous distribution q(α) = F−1 X (α), thus, a natural idea would be to consider q(α) = F−1 X (α), for some nonparametric estimation of FX. Denition 46. The empirical cumulative distribution function FX, based on sample {X1, . . . , Xn} is Fn(x) = 1 n n i=1 1(Xi ≤ x). Denition 47. The kernel based cumulative distribution function, based on sample {X1, . . . , Xn} is Fn(x) = +∞ −∞ t − x h dFn(t) = 1 nh n i=1 x −∞ k Xi − t h dt = 1 n n i=1 K Xi − x h where K(x) = x −∞ k(t)dt, k being a kernel and h the bandwidth. 123
- 124. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Smoothing nonparametric estimators Two techniques have been considered to smooth estimation of quantiles, either implicit, or explicit. • consider a linear combinaison of order statistics, The classical empirical quantile estimate is simply Qn (p) = F−1 n i n = X(i) = X([np]) where [·] denotes the integer part. (5) The estimator is simple to obtain, but depends only on one observation. A natural extention will be to use - at least - two observations, if np is not an integer. The weighted empirical quantile estimate is then dened as Qn (p) = (1 − γ) X([np]) + γX([np]+1) where γ = np − [np] . 124
- 125. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. 0.0 0.2 0.4 0.6 0.8 1.0 2468 The quantile function in R probability level quantilelevel q q q q q q q q q q type=1 type=3 type=5 type=7 0.0 0.2 0.4 0.6 0.8 1.0 234567 The quantile function in R probability level quantilelevel q q q q qq q qq qqq q q qq qqqqq qqq qqqqqqqq qqqqqq q qqq q qqq q q q q q q q q qq q qq qqq q q qq qqqqq qqq qqqqqqqq qqqqqq q qqq q qqq q q q q type=1 type=3 type=5 type=7 Figure 17: Several quantile estimators in R. 125
- 126. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Smoothing nonparametric estimators In order to increase eciency, L-statistics can be considered i.e. Qn (p) = n i=1 Wi,n,pXi:n = n i=1 Wi,n,pF−1 n i n = 1 0 F−1 n (t) k (p, h, t) dt (6) where Fn is the empirical distribution function of FX, where k is a kernel and h a bandwidth. This expression can be written equivalently Qn (p) = n i=1 i n (i−1) n k t − p h dt X(i) = n i=1 i n − p h − i−1 n − p h X(i) (7) where again (x) = x −∞ k (t) dt. The idea is to give more weight to order statistics X(i) such that i is closed to pn. 126
- 127. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. E.g. the so-called Harrell-Davis estimator is dened as Qn(p) = n i=1 i n (i−1) n Γ(n + 1) Γ((n + 1)p)Γ((n + 1)q) y(n+1)p−1 (1 − y)(n+1)q−1 Xi:n, • nd a smooth estimator for FX, and then nd (numerically) the inverse, The α-quantile is dened as the solution of FX ◦ qX(α) = α. If Fn denotes a continuous estimate of F, then a natural estimate for qX(α) is qn(α) such that Fn ◦ qn(α) = α, obtained using e.g. Gauss-Newton algorithm. 127
- 128. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Parametrics or nonparametrics? 0 1 2 3 4 5 0.00.20.40.60.8 Density, theoritical versus empirical Theoritical lognormal Fitted lognormal Fitted gamma −4 −2 0 2 4 0.00.10.20.3 Density, theoritical versus empirical Theoritical Student Fitted lStudent Fitted Gaussian Figure 18: Estimation of Value-at-Risk, model error. 128
- 129. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using extreme value results, a semi-parametric approach Quantile estimation based on extreme value results can be seen as semi parametric. Given a n-sample {Y1, . . . , Yn}, let Y1:n ≤ Y2:n ≤ . . .≤ Yn:n denotes the associated order statistics. One possible method is to use Pickands-Balkema-de Haan theorem. The idea is to assume that for u large enough, Y − u given Y u has a Generalized Pareto distribution with parameters ξ and β, which can be estimated using standard maximum likelihood techniques. And therefore, if u = Yn−k:n for k large enough, and if ξ0, denote by βk and ξk maximum likelihood estimators of the Genralized Pareto distribution of sample {Yn−k+1:n − Yn−k:n, ..., Yn:n − Yn−k:n}, Q(Y, α) = Yn−k:n + βk ξk n k (1 − α) −ξk − 1 (8) 129
- 130. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using extreme value results, a semi-parametric approach An alternative is to use Hill's estimator if ξ 0, Q(Y, α) = Yn−k:n n k (1 − α) −ξk , (9) where ξk = 1 k k i=1 log Yn+1−i:n − log Yn−k:n (if ξ 0). 130
- 131. Arthur CHARPENTIER -Gestion des risques bancaires et financiers. Using extreme value results, a semi-parametric approach 10 20 30 40 50 70 0.0020.0050.0200.0500.2000.500 x (on log scale) 1!F(x)(onlogscale) 99959995 10 20 30 40 50 60 70 0.010.020.050.100.20 x (on log scale)1!F(x)(onlogscale) 99959995 Figure 19: Estimating V aR and TV aR using EVT. 131