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  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Gestion des risques bancaires et nanciersrisques extrêmeset risques corrélésArthur CharpentierEdF, formation continuearthur.charpentier@univ-rennes1.fr1
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.-3 -1 1 3X-13Y(Xi,Yi)0.2 0.5 0.8U (rank of X)0.40.9V(rankofY)Density of the copulaIsodensity curves of the density(Ui,Vi)2
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Agenda• General introductionModelling correlated risks• A short introduction to copulas• Quantifying dependence• Statistical inference• Agregation properties3
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Agenda• General introductionModelling correlated risks• A short introduction to copulas• Quantifying dependence• Statistical inference• Agregation properties4
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Some references on large and correlated risksRank, J. (2006). Copulas: From Theory to Application in Finance. Risk Book ,Nelsen, R. (1999,2006). An introduction to copulas. Springer Verlag ,Cherubini, U., Luciano, E. & Vecchiato, W. (2004). Copula Methods inFinance. Wiley,Beirlant, J., Goegebeur, Y., Segers, J. & Teugels, J. (2004). Statistics ofExtremes: Theory and Applications. Wiley,McNeil, A. Frey, R., & Embrechts, P. (2005). Quantitative RiskManagement: Concepts, Techniques, and Tools. Princeton University Press,5
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Agenda• General introductionModelling correlated risks• A short introduction to copulas• Quantifying dependence• Statistical inference• Agregation properties6
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas, an introduction (in dimension 2)Denition 1. A copula C is a joint distribution function on [0, 1]2, with uniformmargins on [0, 1].Set C(u, v) = P(U ≤ u, V ≤ v), where (U, V ) is a random pair with uniformmargins.C is a distribution function on [0, 1]2, and thus C(0, v) = C(u, 0) = 0, C(1, 1) = 1.Furthermore C is increasing: since P is a positive measure, for all u1 ≤ u2 andv1 ≤ v2,P(u1 < U ≤ u2, v1 < V ≤ v2) ≥ 0,thusC(u2, v2) − C(u1, v2)−C(u2, v1) + C(u1, v1) ≥ 0. 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Copula, positive area7
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.C has uniform margins, and thusC(u, 1) = P(U ≤ u, V ≤ 1) = P(U ≤ u) = u on [0, 1].Proposition 2. C is a copula if and only if C(0, v) = C(u, 0) = 0, C(u, 1) = uand C(1, v) = v for all u, v, with the following 2-increasingness propertyC(u2, v2) − C(u1, v2) − C(u2, v1) + C(u1, v1) ≥ 0,for any u1 ≤ u2 and v1 ≤ v2.8
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Borders of the copula function!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4!0.20.00.20.40.60.81.01.21.4!0.20.00.20.40.60.81.01.2Figure 1: Value of the copula on the border of the unit square.9
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.XYZFonction de répartition à marges uniformesFigure 2: Graphical representation of a copula.10
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.If C is twice dierentiable, one can dene its density asc(u, v) =∂2C(u, v)∂u∂v.11
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.x xzDensité d’une loi à marges uniformesFigure 3: Density of a copula.12
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Fonction de répartition à marges uniformes Densité d’une loi à marges uniformesFonction de répartition à marges uniformes Densité d’une loi à marges uniformesFigure 4: Distribution functions and densities.13
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Fonction de répartition à marges uniformes Densité d’une loi à marges uniformesFonction de répartition à marges uniformes Densité d’une loi à marges uniformesFigure 5: Distribution functions and densities.14
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Sklars theoremTheorem 3. (Sklar) Let C be a copula, and FX and FY two marginaldistributions, then F(x, y) = C(FX(x), FY (y)) is a bivariate distributionfunction, with F ∈ F(FX, FY ).Conversely, if F ∈ F(FX, FY ), there exists C such thatF(x, y) = C(FX(x), FY (y)). Further, if FX and FY are continuous, then C isunique, and given byC(u, v) = F(F−1X (u), F−1Y (v)) for all (u, v) ∈ [0, 1] × [0, 1]We will then dene the copula of F, or the copula of (X, Y ).In that case, the copula of (X, Y ) is the distribution function of (FX(X), FY (Y )).Proposition 4. If (X, Y ) has copula C, the copula of (g(X), h(Y )) is also C forany increasing functions g and h.15
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas, an introduction (in dimension d ≥ 2)Denition 5. A copula C is a joint distribution function on [0, 1]d, withuniform margins on [0, 1].Let U = (U1, ..., Ud) denote a random pair with uniform margins.C is a distribution function on [0, 1]d, and thus C(u) = 0 if ui = 0 for somei ∈ {1, . . . , d}, and C(1) = 1.Furthermore C satises some increasing property since P is a positive measure(for all 0 ≤ u ≤ v ≤ 1, P(u < U ≤ v) ≥ 0), thuszsign(z)C(z) ≥ 0,where the sum is taken over all vertices of [u × v], and where sign(z) is +1 ifzi = ui for an even number of i (and −1 otherwise, see Figure 6). And nally Chas uniform margins, and thusC(1, . . . , 1, ui, 1, . . . , 1) = ui on [0, 1].16
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Increasing functions in dimension 3Figure 6: The notion of 3-increasing functions.17
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Theorem 6. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginaldistributions, then F(x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, withF ∈ F(F1, . . . , Fd).Conversely, if F ∈ F(F1, . . . , Fd), there exists C such thatF(x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fis are continuous, then C isunique, and given byC(u) = F(F−11 (u1), . . . , F−1d (ud)) for all (ui) ∈ [0, 1]We will then dene the copula of F, or the copula of X.In that case, the copula of (X = (X1, . . . , Xd) is the distribution function ofU = (F1(X1), . . . , Fd(Yd)).Again, if C is dierentiable, one can dene its density,c(u1, . . . , ud) =∂dC(u1, . . . , ud)∂u1 . . . ∂ud.18
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas in high dimension, a dicult problemIt is usually dicult to represent dependence in dimension d > 2, and it isusually studied by pairs.In dimension d = 2, one can dene the following Fréchet class F(FX, FY , FZ)dened by its marginal distributions. But it can also be interested to studyF(FXY , FXZ, FY Z) dened by it paired distributions.One of the problem that arises is the compatibility of marginals: one has toverify thatCXY (x, y) = CX|Z(x|z)CY |Z(y|z)dz,for instance.19
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.00.00.20.40.60.81.00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Composante 1p0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Composante 2p0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Composante 3pFigure 7: Scatterplot in dimension 3 including projections.20
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas and ranksThe copula of X = (X1, . . . , Xd) is the distribution function ofU = (F1(X1), . . . , Fd(Yd)).In practice, since marginal distributions are unknown, the idea is to substituteempirical distribution function,Fi(xi) =#{observations Xi,js lower than xi}#{observations }=1nnj=11(Xi,j ≤ xi).Note thatFi(Xi,j0) =#{observations Xi,js lower than Xi,j0}#{observations }=1nnj=11(Xi,j ≤ Xi,j0) =Ri,j0n,where Ri,j0 denotes the rank of Xi,j0 within {Xi,1, ..., Xi,n}.On a statistical point of view, studying the copula means studying ranks.21
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.05.56.06.57.07.58.08.59.0Scatterplot of (X,Y)X (raw data)Y(rawdata)5 10 15 205101520Scatterplot of the ranks of (X,Y)Ranks of the Xi’sRanksoftheYi’s0.2 0.4 0.6 0.8 1.00.20.40.60.81.0Scatterplot of the ranks of (X,Y), divided by nRanks of the Xi’s/nRanksoftheYi’s/n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Scatterplot o+ ,-,/0, t1e copula!t3pe tran+orm o+ ,6,70Ui=Ranks of the Xi’s/n+1Vi=RanksoftheYi’s/n+1Figure 8: Copulas, ranks and parametric inference, from (Xi, Yi) to (Ui, Vi).22
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Some very classical copulas• The independent copula C(u, v) = uv = C⊥(u, v).The copula is standardly denoted Π, P or C⊥, and an independent version of(X, Y ) will be denoted (X⊥, Y ⊥). It is a random vector such that X⊥ L= X andY ⊥ L= Y , with copula C⊥.In higher dimension, C⊥(u1, . . . , ud) = u1 × . . . × ud is the independent copula.• The comonotonic copula C(u, v) = min{u, v} = C+(u, v).The copula is standardly denoted M, or C+, and an comonotone version of(X, Y ) will be denoted (X+, Y +). It is a random vector such that X+ L= X andY + L= Y , with copula C+.(X, Y ) has copula C+if and only if there exists a strictly increasing function hsuch that Y = h(X), or equivalently (X, Y )L= (F−1X (U), F−1Y (U)) where U isU([0, 1]).23
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Note that for any u, vP(U ≤ u, V ≤ v) = P({U ∈ [0, u]} ∩ {V ∈ [0, v]})≤ min{P(U ∈ [0, u]), P(V ∈ [0, v])}thus, C(u, v) ≤ min{u, v} = C+(u, v). Thus, C+is an upper bound for the set ofcopulas.In higher dimension, C+(u1, . . . , ud) = min{u1, . . . , ud} is the comonotoniccopula.• The contercomotonic copula C(u, v) = max{u + v − 1, 0} = C−(u, v).The copula is standardly denoted W, or C−, and an contercomontone version of(X, Y ) will be denoted (X−, Y −). It is a random vector such that X− L= X andY − L= Y , with copula C−.(X, Y ) has copula C−if and only if there exists a strictly decreasing function hsuch that Y = h(X), or equivalently (X, Y )L= (F−1X (1 − U), F−1Y (U)) where U isU([0, 1]).24
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Note that for any u, v,P(U ≤ u, V ≤ v) = P({U ∈ [0, u]} ∩ {V ∈ [0, v]})= P(U ∈ [0, u]) + P(V ∈ [0, v]) − P({U ∈ [0, u]} ∪ {V ∈ [0, v]})thus, C(u, v) ≥ u + v − 1 since P({U ∈ [0, u]} ∪ {V ∈ [0, v]}) ≤ 1, and sinceC(u, v) ≥ 0, C(u, v) ≥ max{u + v − 1, 0} = C−(u, v). Thus, C−is a lower boundfor the set of copulas.In higher dimension, C−(u1, . . . , ud) = max{u1 + . . . + ud − (d − 1), 0} is not acopula: if (X, Y ) and (X, Z) are countercomonotonic, (Y, Z) is necessarilycomonotonic - it is not possible to have all component highly negativelycorrelated.Anyway, it is still the best pointwise lower bound.25
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.20.40.60.8u_10.20.40.60.8u_200.20.40.60.81Frechetlowerbound0.20.40.60.8u_10.20.40.60.8u_200.20.40.60.81Independencecopula0.20.40.60.8u_10.20.40.60.8u_200.20.40.60.81FrechetupperboundFréchet Lower Bound0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Independent copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Fréchet Upper Bound0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Scatterplot, Lower Fréchet!Hoeffding bound0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Scatterplot, Indepedent copula random generation0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Scatterplot, Upper Fréchet!Hoeffding boundFigure 9: Contercomontonce, independent, and comonotone copulas.26
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Pitfalls on independence and comonotonicityThe following proposition is false,Uncorrect Proposition 7. If X and Y are independent, if Y and Z areindependent, then X and Z are independent.If(X, Y, Z) = (1, 1, 1) with probability 1/4,(1, 2, 1) with probability 1/4,(3, 2, 3) with probability 1/4,(3, 1, 3) with probability 1/4,then X and Y are independent, and Y and Z are independent, but X = Z.27
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 1 2 3 401234X and Y independentComponent XComponentY0 1 2 3 401234Y and Z independentComponent YComponentZ0 1 2 3 401234X and Z comonotonicComponent XComponentZFigure 10: Mixing independence and comonotonicity.28
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Pitfalls on independence and comonotonicityThe following proposition is false,Uncorrect Proposition 8. If X and Y are comonotonic, if Y and Z arecomonotonic, then X and Z are comonotonic.If(X, Y, Z) = (1, 1, 1) with probability 1/4,(1, 2, 3) with probability 1/4,(3, 2, 1) with probability 1/4,(3, 3, 3) with probability 1/4,then X and Y are comonotonic, and Y and Z are comonotonic, but X and Z areindependent.29
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 1 2 3 401234X and Y comonotonicComponent XComponentY0 1 2 3 401234Y and Z comonotonicComponent YComponentZ0 1 2 3 401234X and Z independentComponent XComponentZFigure 11: Mixing independence and comonotonicity.30
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Pitfalls on independence and comonotonicityThe following proposition is false,Uncorrect Proposition 9. If X and Y are comonotonic, if Y and Z areindependent, then X and Z are independent.If(X, Y, Z) = (1, 1, 3) with probability 1/4,(2, 1, 1) with probability 1/4,(2, 3, 3) with probability 1/4,(3, 3, 1) with probability 1/4,then X and Y are comonotonic, and Y and Z are independent, but X and Z areanticomonotonic.31
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.If(X, Y, Z) = (1, 1, 1) with probability 1/4,(2, 1, 3) with probability 1/4,(2, 3, 1) with probability 1/4,(3, 3, 3) with probability 1/4,then X and Y are comonotonic, and Y and Z are independent, but X and Z arecomonotonic.32
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 1 2 3 401234X and Y comonotonicComponent XComponentY0 1 2 3 401234Y and Z independentComponent YComponentZ0 1 2 3 401234X and Z comonotonicComponent XComponentZFigure 12: Mixing independence and comonotonicity.33
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Elliptical (Gaussian and t) copulasThe idea is to extend the multivariate probit model, Y = (Y1, . . . , Yd) withmarginal B(pi) distributions, modeled as Yi = 1(Xi ≤ ui), where X ∼ N(I, Σ).• The Gaussian copula, with parameter α ∈ (−1, 1),C(u, v) =12π√1 − α2Φ−1(u)−∞Φ−1(v)−∞exp−(x2− 2αxy + y2)2(1 − α2)dxdy.Analogously the t-copula is the distribution of (T(X), T(Y )) where T is the t-cdf,and where (X, Y ) has a joint t-distribution.• The Student t-copula with parameter α ∈ (−1, 1) and ν ≥ 2,C(u, v) =12π√1 − α2t−1ν (u)−∞t−1ν (v)−∞1 +x2− 2αxy + y22(1 − α2)−((ν+2)/2)dxdy.34
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Archimedean copulasDenition of Archimedean copulas• Archimedian copulas C(u, v) = φ−1(φ(u) + φ(v)), where φ is decreasingconvex (0, 1), with φ(0) = ∞ and φ(1) = 0.Example 10. If φ(t) = [− log t]α, then C is Gumbels copula, and ifφ(t) = t−α− 1, C is Claytons. Note that C⊥is obtained when φ(t) = − log t.How Archimedean copulas were introduced ?1. The frailty approach (Oakes (1989)).Assume that X and Y are conditionally independent, given the value of anheterogeneous component Θ. Assume further thatP(X ≤ x|Θ = θ) = (GX(x))θand P(Y ≤ y|Θ = θ) = (GY (y))θfor some baseline distribution functions GX and GY .ThenF(x, y) = P(X ≤ x, Y ≤ y) = E(P(X ≤ x, Y ≤ y|Θ = θ))35
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.thus, since X and Y are conditionally independent,F(x, y) = E(P(X ≤ x|Θ = θ) × P(Y ≤ y|Θ = θ))and thereforeF(x, y) = E (GX(x))Θ× (GY (y))Θ= ψ(− log GX(x) − log GY (y))where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ). SinceFX(x) = ψ(− log GX(x)) and FY (y) = ψ(− log GY (y))and thus, the joint distribution of (X, Y ) satisesF(x, y) = ψ(ψ−1(FX(x)) + ψ−1(FY (y))).Example 11. If Θ is Gamma distributed, the associated copula is Claytons. IfΘ has a stable distribution, the associated copula is Gumbels.36
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Consider two risks, X and Y , such thatX|Θ = θG ∼ E(θG) and Y |Θ = θG ∼ E(θG) are independent,X|Θ = θB ∼ E(θB) and Y |Θ = θB ∼ E(θB) are independent,(unobservable good (G) and bad (B) risks).The following gures start from 2 classes of risks, then 3, and then a continuousrisk factor θ ∈ (0, ∞).37
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 5 10 1505101520Conditional independence, two classes!3 !2 !1 0 1 2 3!3!2!10123Conditional independence, two classesFigure 13: Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).38
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 5 10 15 20 25 30010203040Conditional independence, three classes!3 !2 !1 0 1 2 3!3!2!10123Conditional independence, three classesFigure 14: Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).39
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 20 40 60 80 100020406080100Conditional independence, continuous risk factor!3 !2 !1 0 1 2 3!3!2!10123Conditional independence, continuous risk factorFigure 15: Continuous classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).40
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.2. The survival approach: assume that there is a convex survival function S,with S(0) = 1, such thatP(X > x, Y > y) = S(x + y),then the joint survival copula of (X, Y ) isS(S−1(u) + S−1(v)).Example 12. If S is the Pareto survival distribution, the associated copula isClaytons. If S is the Weibull survival distribution, the associated copula isGumbels.41
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.3. The use of Kendalls distribution function K(t) = P(C(U, V ) ≤ t) where(U, V ) is a random pair with distribution function C.Then, for Archimedean copulas,K(t) = t −φ (t)φ(t)= t − λ(t),which can be inverted easily inφ(t) = φ(t0) exp1t01λ(t)dt ,for some 0 < t0 < 1 and 0 ≤ u ≤ 1.42
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Some more examples of Archimedean copulasψ(t) range θ(1) 1θ(t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)(2) (1 − t)θ [1, ∞)(3) log1−θ(1−t)t[−1, 1) Ali-Mikhail-Haq(4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986)(5) − log e−θt−1e−θ−1(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987)(6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993)(7) − log{θt + (1 − θ)} (0, 1](8)1−t1+(θ−1)t[1, ∞)(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)(10) log(2t−θ − 1) (0, 1](11) log(2 − tθ) (0, 1/2](12) ( 1t− 1)θ [1, ∞)(13) (1 − log t)θ − 1 (0, ∞)(14) (t−1/θ − 1)θ [1, ∞)(15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994)(16) ( θt+ 1)(1 − t) [0, ∞)43
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Some characterizations of Archimedean copula• Frank copula is the only Archimedean such that (U, V )L= (1 − U, 1 − V )(stability by symmetry),• Clayton copula is the only Archimedean such that (U, V ) has the samecopula as (U, V ) given (U ≤ u, V ≤ v) (stability by truncature),• Gumbel copula is the only Archimedean such that (U, V ) has the samecopula as (max{U1, ..., Un}, max{V1, ..., Vn}) for all n ≥ 1 (max-stability),44
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Extreme value copulas• Extreme value copulasC(u, v) = exp (log u + log v) Alog ulog u + log v,where A is a dependence function, convex on [0, 1] with A(0) = A(1) = 1, etmax{1 − ω, ω} ≤ A (ω) ≤ 1 for all ω ∈ [0, 1] .An alternative denition is the following: C is an extreme value copula if for allz > 0,C(u1, . . . , ud) = C(u1/z1 , . . . , u1/zd )z.Those copula are then called max-stable: dene the maximum componentwise ofa sample X1, . . . , Xn, i.e. Mi = max{Xi,1, . . . , Xi,n}.45
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.The joint distribution of M isP(M ≤ x) = C(F1(x1, . . . , Fd(xd))n,where C is the copula of the Xis. Since P(Mi ≤ xi) = Fi(xi)n, it can be writtenP(M ≤ x) = C(P(M1 ≤ x1)1/n, . . . , P(Md ≤ xd)1/n)n.Thus, C(u1/n1 , . . . , u1/nd )nis the copula of the n maximum componentwise from asample with copula C.Example 13. : If A is constant (1 on [0, 1]), then X and Y are independent,and if A(ω) = max {ω, 1 − ω}, X and Y are comonotonic. Gumbels copula isobtained ifA(ω) = ((1 − ω)α+ ωα+ 1)(1/α),for all 0 ≤ ω ≤ 1 and α ≥ 1.46
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.50.60.70.80.91.0Pickands dependence function AFigure 16: Shape of Gumbels dependence function A(ω).47
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.How to construct much more copulas ?Using geometric transformationsFrom a given copula C, cdf of random pair (U, V ), dene• the copula of (U, 1 − V ),C(U,1−V )(u, v) = u − C(u, 1 − v)• the copula of (1 − U, V ),C(1−U,V )(u, v) = v − C(1 − u, v)• the copula of (1 − U, 1 − V ), the rotated or survival copula,C(1−U,1−V )(u, v) = C∗(u, v) = u + v − 1 + C(1 − u, 1 − v)Note that if P(X ≤ x, Y ≤ y) = C(P(X ≤ x), P(Y ≤ y)), thenP(X > x, Y > y) = C∗(P(X > x), P(Y > y)).48
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.40.80.0 0.2 0.4 0.6 0.8 1.00.00.40.8Figure 17: Using geometric transformation to generate new copulas.49
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.40.80.0 0.2 0.4 0.6 0.8 1.00.00.40.8Figure 18: Using geometric transformation to generate new copulas.50
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.40.80.0 0.2 0.4 0.6 0.8 1.00.00.40.8Figure 19: Using geometric transformation to generate new copulas.51
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.40.80.0 0.2 0.4 0.6 0.8 1.00.00.40.8Figure 20: Using geometric transformation to generate new copulas.52
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Using mixture of copulasLemma 14. The set of copulas is convex, i.e. if {Cθ, θ ∈ Ω} is a collection ofcopulas,C(u, v) =RCθ(u, v)dΠ(θ)is a copula, where Π is a distribution on ΩThus C = αC1 + (1 − α)C2 denes a copula for all α ∈ [0, 1].Example 15. Fréchet (1951) suggested a mixture of the lower and the upperbound,C(u, v) = αC−(u, v) + (1 − α)C+(u, v), for some α ∈ [0, 1].Example 16. Mardia (1970) suggested a mixture of the lower, the upperbound, and the independent copulaC(u, v) =α22C−(u, v) + (1 − α2)C⊥(u, v) +α22C+(u, v), α ∈ [0, 1].53
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Using distortion functionsDenition 17. A distortion function is a function h : [0, 1] → [0, 1] strictlyincreasing such that h(0) = 0 and h(1) = 1.The set of distortion function will be denoted H.Note that h ∈ H if and only if h−1∈ H. Given a copula C, deneCh(u, v) = h−1(C(h(u), h(v))).If h is convex, then Ch is a copula, called distorted copula.Example 18. if h(x) = x1/n, the distorted copula isCh(u, v) = Cn(u1/n, v1/n), for all n ∈ N, (u, v) ∈ [0, 1]2.if the survival copula of the (Xi, Yi)s is C, then the survival copula of(Xn:n, Yn:n) = (max{X1, ..., Xn}, max{Y1, ..., Yn}) is Ch.54
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example 19. if C(u, v) = uv = C⊥(u, v) (the independent copula), andφ(·) = log h(·), thenCh(u, v) = h−1(h(u)h(v)) = φ−1(φ(u) + φ(v)).Example 20. if h(x) = [1 − e−αx]/[1 − e−α] (an exponential distortion), and ifC = C⊥, thenCh(u, v) = −1αlog 1 +(e−αu− 1)(e−αv− 1)e−α − 1,which is Frank copula.55
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Distorted Frank copula, h(x) = x Distorted Frank copula, h(x) = x(1 2)Distorted Frank copula, h(x) = x(1 3)Distorted Frank copula, h(x) = x(1 4)Figure 21: Distorted copula, from Frank copula.56
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Monte Carlo and copulasGeneration of independent variables can be done using a Random function.Denition 21. Function Random should satisfy the following properties (i) forall 0 ≤ a ≤ b ≤ 1,P (Random ∈ ]a, b]) = b − a.(ii) successive calls of function Random should generate independent draws, i.e.0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1P (Random1 ∈ ]a, b] , Random2 ∈ ]c, d]) = (b − a) (d − c) ,or more generally, dene k-uniformity for all 0 ≤ ai ≤ bi ≤ 1, i = 1, ..., k,P (Random1 ∈ ]a1, b1] , ..., Randomk ∈ ]ak, bk]) =ki=1(bi − ai) .Thus, one can generate easily random vectors U = (U1, ..., Ud) with independentcomponent.57
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.The idea to generate correlated vectors U = (U1, ..., Ud), the idea is to use rstP(U1 ≤ u1, . . . , Ud ≤ ud) = P(Ud ≤ ud|U1 ≤ u1, . . . , Ud−1 ≤ ud−1)×P(Ud−1 ≤ ud−1|U1 ≤ u1, . . . , Ud−2 ≤ ud−2)× . . .×P(U3 ≤ u3|U1 ≤ u1, U2 ≤ u2)×P(U2 ≤ u2|U1 ≤ u1) × P(U1 ≤ u1).58
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Starting from the end, P(U1 ≤ u1) = u1 since U1 is uniform, whileP(U2 ≤ u2|U1 = u1)= P(U2 ≤ u2, U3 ≤ 1, . . . Ud ≤ 1|U1 = u1)= limh→0P(U2 ≤ u2, U3 ≤ 1, . . . Ud ≤ 1|U1 ∈ [u1, u1 + h])= limh→0P(u1 ≤ U1 ≤ u1 + h, U2 ≤ u2, U3 ≤ 1, . . . Ud ≤ 1)P(U1 ∈ [u1, u1 + h])= limh→0P(U1 ≤ u1 + h, U2 ≤ u2, U3 ≤ 1, . . . Ud ≤ 1) − P(U1 ≤ u1, U2 ≤ u2, U3 ≤ 1, . . .P(U1 ∈ [u1, u1 + h])= limh→0C(u1 + h, u2, 1, . . . , 1) − C(u1, u2, 1, . . . , 1)h=∂C∂u1C(u1, u2, 1, . . . , 1).and more generally,P(Uk ≤ uk|U1 = u1, . . . , Uk−1 = uk−1) =∂k−1∂u1 . . . ∂uk−1C(u1, . . . , uk, 1, . . . , 1).59
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Thus, U = (U1, .., Un) with copula C could be simulated using the followingalgorithm,• simulate U1 uniformly on [0, 1],u1 ← Random1,• simulate U2 from the conditional distribution ∂1C(·|u1),u2 ← [∂1C(·|u1)]−1(Random2),• simulate Uk from the conditional distribution ∂1,...,k−1C(·|u1, ..., uk−1),uk ← [∂1,...,k−1C(·|u1, ..., uk−1)]−1(Randomk),...etc, where the Randomis are independent calls of a Random function.This is the underlying idea when using Cholesky decomposition.60
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example: for Claytons copula, C(u, v) = (u−α+ v−α− 1)−1/α, (U, V ) has jointdistribution C if and only if U is uniform on on [0, 1] and V |U = u hasconditional distributionP(V ≤ v|U = u) = ∂2C(v|u) = (1 + uα[v−α− 1])−1−1/α.The algorithm to generate Claytons copula is the• simulate U1 uniformly on [0, 1],u1 ← Random1,• simulate U2 from the conditional distribution ∂2C(·|u),u2 ← [∂1C(·|u1)]−1(Random2),i.e.u2 ← [(Random2)−α/(1+α− 1]u−α1 + 1−1/α.61
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0Distribution of v given u=0.30.0 0.2 0.4 0.6 0.8 1.00.00.51.01.5Distribution of v given u=0.50.0 0.2 0.4 0.6 0.8 1.00.00.40.8Generation of Clayton’s copula0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.5Distribution of v given u=0.8Figure 22: Simulation of Claytons copula.62
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.!i#tribution +e -q/0.0 0.2 0.4 0.6 0.8 1.00100300500!i#tribution +e 9q/0.0 0.2 0.4 0.6 0.8 1.002004000.0 0.2 0.4 0.6 0.8 1.00.00.40.8-ni:orm mar=in#!4 !2 0 2 4!4!2024 Stan+ar+ ?au##ian mar=in#Figure 23: Simulation of the independent copula.63
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.!is$ri&u$i(n +e -./010 012 014 014 015 6100200400!is$ri&u$i(n +e 7./010 012 014 014 015 6100200400010 012 014 014 015 610010014015-ni8(r9 9argins!2 0 2 4!2024 <$an+ar+ =aussian 9arginsFigure 24: Simulation of the comontone copula.64
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Distribution de -q/0.0 0.2 0.4 0.6 0.8 1.00200400Distribution de Vq/0.0 0.2 0.4 0.6 0.8 1.002004000.0 0.2 0.4 0.6 0.8 1.00.00.40.8-niform margins!2 0 2 4!4!202 <tandard =aussian marginsFigure 25: Simulation of the contercomonotone copula.65
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.!istri&tion de -./010 012 014 014 015 1100200400!istri&tion de 7./010 012 014 014 015 1100100300900010 012 014 014 015 110010014015-ni:orm m<r=ins!4 !2 0 2 4!4!2024 St<nd<rd ?<ssi<n m<r=insFigure 26: Simulation of the Gaussian copula.66
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.D"#$%"&u$"on +e U.y010 012 014 014 015 1100200400D"#$%"&u$"on +e V.y010 012 014 014 015 1100200400010 012 014 014 015 110010014015Un"fo%m ma%;"n#!2 0 2 4!2024 S$an+a%+ =au##"an ma%;"n#Figure 27: Simulation of Claytons copula.67
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Distribution de Uqy0.0 0.2 0.4 0.6 0.8 1.00200400Distribution de Vqy0.0 0.2 0.4 0.6 0.8 1.002004000.0 0.2 0.4 0.6 0.8 1.00.00.40.8Uniform margins!4 !2 0 2 4!4!2024 Standard Gaussian marginsFigure 28: Simulation of Claytons survival copula.68
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Distribution de Uqy0.0 0.2 0.4 0.6 0.8 1.00200400Distribution de Vqy0.0 0.2 0.4 0.6 0.8 1.002004000.0 0.2 0.4 0.6 0.8 1.00.00.40.8Uniform margins!4 !2 0 2!4!2024 Standard Gaussian marginsFigure 29: Simulation of a copula mixture.69
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas in nance: options on multiple assetsRemark 22. Recall that Breeden & Litzenberger (1978) proved that the riskneutral probability can be obtrained from option prices: consider the price of a callC(T, K) = e−rTEQ((ST − K)+). Since (ST − K)+ =∞K1(ST > x)dx, one getsC(T, K) = e−rT∞KQ(ST > x)dx,henceQ(ST ≤ x) = −e−rT ∂C∂K(T, x), or Q(ST ≤ x) = −erT ∂P∂K(T, x)where P denotes the price of a put option.Consider an option on 2 assets, with payo h(S1T , S2T ). The price at time 0 ise−rTEQ(h(S1T , S2T )).70
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas in nance: call on maximumHere the payo is h(S1T , S2T ) = (max{S1T , S2T } − K)+. The price is thenC(T, K) = e−rTEQ((max{S1T , S2T } − K)+)= e−rTEQ∞K1 − 1(max{S1T , S2T } ≤ x)dx= e−rT∞K1 − Q(max{S1T , S2T } ≤ x)Q(S1T ≤x,S2T ≤x)dx,hence, if (S1T , S2T ) has copula C (under Q), thenC(T, K) = e−rT∞K1 − C erT ∂P1∂K(T, x), erT ∂P2∂K(T, x) dx.71
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas in nance: call on spreadsHere the payo is h(S1T , S2T ) = ([S1T − S2T ] − K)+. The price is thenC(T, K) = e−rTEQ((S1T − S2T − K)+) = e−rTEQ∞−∞1(S2T + K ≤ x ≤ S1T )dx= e−rT∞−∞Q(K + S2T ≤ x) − Q(S2T + K ≤ x, S1T ≤ x} ≤ x)Q(S1T ≤x,S2T ≤x+K)dx,hence, if (S1T , S2T ) has copula C (under Q), thenC(T, K) = e−rT∞−∞erT ∂P2∂K(T, x−K)−C erT ∂P1∂K(T, x), erT ∂P2∂K(T, x − K) dx.72
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Copulas in nance: bonds on option pricesUsing Tchens inequality, it is possible to derive bounds for options when thepayo is supermodular.73
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Agenda• General introductionModelling correlated risks• A short introduction to copulas• Quantifying dependence• Statistical inference• Agregation properties74
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Natural properties for dependence measuresDenition 23. κ is measure of concordance if and only if κ satises1. κ is dened for every pair (X, Y ) of continuous random variables,2. −1 ≤ κ (X, Y ) ≤ +1, κ (X, X) = +1 and κ (X, −X) = −1,3. κ (X, Y ) = κ (Y, X),4. if X and Y are independent, then κ (X, Y ) = 0,5. κ (−X, Y ) = κ (X, −Y ) = −κ (X, Y ),6. if (X1, Y1) P QD (X2, Y2), then κ (X1, Y1) ≤ κ (X2, Y2),7. if (X1, Y1) , (X2, Y2) , ... is a sequence of continuous random vectors thatconverge to a pair (X, Y ) then κ (Xn, Yn) → κ (X, Y ) as n → ∞.75
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.As pointed out in Scarsini (1984), most of the axioms are self-evident.If κ is measure of concordance, then, if f and g are both strictly increasing, thenκ(f(X), g(Y )) = κ(X, Y ). Further, κ(X, Y ) = 1 if Y = f(X) with f almostsurely strictly increasing, and analogously κ(X, Y ) = −1 if Y = f(X) with falmost surely strictly decreasing (see Scarsini (1984)).76
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Association measures: Kendalls τ and Spearmans ρRank correlations can be considered, i.e. Spearmans ρ dened asρ(X, Y ) = corr(FX(X), FY (Y )) = 121010C(u, v)dudv − 3and Kendalls τ dened asτ(X, Y ) = 41010C(u, v)dC(u, v) − 1.Historical version of those coecientsSpearmans rho was introduced in Spearman (1904) asρ(X, Y ) = 3[P((X1 − X2)(Y1 − Y3) > 0) − P((X1 − X2)(Y1 − Y3) < 0)],where (X1, Y1), (X2, Y2) and (X3, Y3) denote three independent versions of(X, Y ) (see Nelsen (1999)).77
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Similarly Kendalls tau was not dened using copulae, but as the probability ofconcordance, minus the probability of discordance, i.e.τ(X, Y ) = 3[P((X1 − X2)(Y1 − Y2) > 0) − P((X1 − X2)(Y1 − Y2) < 0)],where (X1, Y1) and (X2, Y2) denote two independent versions of (X, Y ) (seeNelsen (1999)).Equivalently, τ(X, Y ) = 1 −4Qn(n2 − 1)where Q is the number of inversionsbetween the rankings of X and Y (number of discordance).78
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0!0.50.00.51.01.5 Concordant pairsXY!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0!0.50.00.51.01.5Discordant pairsXYFigure 30: Concordance versus discordance.79
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.The case of the Gaussian random vectorIf (X, Y ) is a Gaussian random vector with correlation r, then (Kruskal (1958))ρ(X, Y ) =6πarcsinr2and τ(X, Y ) =2πarcsin (r) .80
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Link between Kendalls tau and Spearmans rhoNote that Kendalls tau and Spearmans are linked: it is impossible to have atthe same time τ ≥ 0.4 and ρ = 0.Hence ρ and τ satisfy3τ − 12≤ ρ ≤1 + 2τ − τ22if τ ≥ 0τ2+ 2τ − 12≤ ρ ≤1 + 3τ2if τ ≤ 0.which yield the area given below.81
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.-1.0 -0.5 0.0 0.5 1.0Tau de Kendall-1.0-0.50.00.51.0RhodeSpearmanFigure 31: Admissible region of ρ and τ.82
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.From Kendalltau to copula parametersKendalls τ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Gaussian θ 0.00 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1.00Gumbel θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞Plackett θ 1.00 1.57 2.48 4.00 6.60 11.4 21.1 44.1 115 530 +∞Clayton θ 0.00 0.22 0.50 0.86 1.33 2.00 3.00 4.67 8.00 18.0 +∞Frank θ 0.00 0.91 1.86 2.92 4.16 5.74 7.93 11.4 18.2 20.9 +∞Joe θ 1.00 1.19 1.44 1.77 2.21 2.86 3.83 4.56 8.77 14.4 +∞Galambos θ 0.00 0.34 0.51 0.70 0.95 1.28 1.79 2.62 4.29 9.30 +∞Morgenstein θ 0.00 0.45 0.90 - - - - - - - -83
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.From Spearmans rho to copula parametersSpearmans ρ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Gaussian θ 0.00 0.10 0.21 0.31 0.42 0.52 0.62 0.72 0.81 0.91 1.00Gumbel θ 1.00 1.07 1.16 1.26 1.38 1.54 1.75 2.07 2.58 3.73 +∞A.M.H. θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞Plackett θ 1.00 1.35 1.84 2.52 3.54 5.12 7.76 12.7 24.2 66.1 +∞Clayton θ 0.00 0.14 0.31 0.51 0.76 1.06 1.51 2.14 3.19 5.56 +∞Frank θ 0.00 0.60 1.22 1.88 2.61 3.45 4.47 5.82 7.90 12.2 +∞Joe θ 1.00 1.12 1.27 1.46 1.69 1.99 2.39 3.00 4.03 6.37 +∞Galambos θ 0.00 0.28 0.40 0.51 0.65 0.81 1.03 1.34 1.86 3.01 +∞Morgenstein θ 0.00 0.30 0.60 0.90 - - - - - - -84
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Alternative expressions of those coecientsNote that those coecients can also be expressed as followsρ(X, Y ) =[0,1]×[0,1]C(u, v) − C⊥(u, v)dudv[0,1]×[0,1]C+(u, v) − C⊥(u, v)dudv(1)(the normalized average distance between C and C⊥), for instance.A dependence measure in higher dimension ?From equations 1 and ??, it is possible to obtain a natural mutlidimensionalextention (see Wolf (1980), Joe (1990) or Nelsen (1996)),ρ(X) =[0,1]d C(u) − C⊥(u)du[0,1]×[0,1]C+(u) − C⊥(u)du=d + 12d − (d + 1)2d[0,1]dC(u)du − 1(2)and similarlyτ(X) ==12d−1 − 1)2d[0,1]dC(u)dCu − 1 (3)85
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Note that a lower bound for τ is then −1/(2d−1− 1), while it is(2d− (d + 1)!)/(d!(2d− (d + 1))).In dimension 3, Kendalls τ is the average of the three 2-dimensional Kendalls τ,τ(X, Y, Z) =13(τ(X, Y ) + τ(X, Z) + τ(Y, Z)).86
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Tail concentration functionsVenter (2002) suggest to use several Tail Concentration FunctionsDenition 24. For lower tails, deneL(z) = P(U < z, V < z)/z = C(z, z)/z = Pr(U < z|V < z) = Pr(V < z|U < z),and for upper tails,R(z) = P(U > z, V > z)/(1 − z) = Pr(U > z|V > z).Joe (1990) uses the term upper tail dependence parameter forR = R(1) = limz→1 R(z), and lower tail dependence parameter forL = L(0) = limz→0 L(z).87
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Functional correlation measuresConsider also Kendalls tau, dened as −1 + 41010C(u, v)dC(u, v).Denition 25. The cumulative tau can be dened asJ(z) = −1 + 4z0z0C(u, v)dC(u, v)/C(z, z)2.88
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Gaussian copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0L and R concentration functionsL function (lower tails) R function (upper tails)GAUSSIANqqFigure 32: L and R cumulative curves.89
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Gumbel copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0L and R concentration functionsL function (lower tails) R function (upper tails)GUMBELqqFigure 33: L and R cumulative curves.90
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Clayton copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0L and R concentration functionsL function (lower tails) R function (upper tails)CLAYTONqqFigure 34: L and R cumulative curves.91
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Student t copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0L and R concentration functionsL function (lower tails) R function (upper tails)STUDENT (df=5)qqFigure 35: L and R cumulative curves.92
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Student t copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0L and R concentration functionsL function (lower tails) R function (upper tails)STUDENT (df=3)qqFigure 36: L and R cumulative curves.93
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Dependence in independenceColes, Heffernan & Tawn (1999) propose another function,χ(z) =2 log(1 − z)log C(z, z)− 1Then set η = (1 + limz→1 χ(z))/294
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Gaussian copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Chi dependence functionslower tails upper tailsGAUSSIANqqFigure 37: χ functions.95
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Gumbel copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Chi dependence functionslower tails upper tailsGUMBELqqFigure 38: χ functions.96
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Clayton copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Chi dependence functionslower tails upper tailsCLAYTONqqFigure 39: χ functions.97
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Student t copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0qqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Chi dependence functionslower tails upper tailsSTUDENT (df=3)qqFigure 40: χ functions.98
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.(Strong) tail dependence measureJoe (1993) dened, in the bivariate case a tail dependence measure.Denition 26. Let (X, Y ) denote a random pair, the upper and lower taildependence parameters are dened, if the limit exist, asλL = limu→0P X ≤ F−1X (u) |Y ≤ F−1Y (u) ,andλU = limu→1P X > F−1X (u) |Y > F−1Y (u) .As mentioned in Fougères (2004), this coecient can be obtained dierently:setθ(x) =log P(max{X, Y } ≤ x)log P(X ≤ x).ThenλU = 2 − limx→∞θ(x),99
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.since when x → ∞2 −log P(max{X, Y } ≤ x)log P(X ≤ x)∼P(X > x, Y > x)1 − P(X > x)= P(Y > x|X > x).Note that these coecient can be expressed only through the copula,Proposition 27. Let (X, Y ) denote a random pair with copula C, the upper andlower tail dependence parameters are dened, if the limit exist, asλL = limu→0C(u, u)uand λU = limu→1C∗(u, u)1 − u.Does λ = 0 implies that extremal events are independent ?Example 28. If (X, Y ) has a Gaussian copula with parameter θ < 1, then λ = 0.Hence, visually, dependence is weaker than any Gumbels copula (even with θ israther small), but are extremal events independent ?100
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Marges uniformesCopuledeGumbel!2 0 2 4!2024Marges gaussiennesFigure 41: Simulations of Gumbels copula θ = 1.2.101
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Marges uniformesCopuleGaussienne!2 0 2 4!2024Marges gaussiennesFigure 42: Simulations of the Gaussian copula (θ = 0.95).102
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example 29. Consider the case of Archimedean copulas, thenλU = 2 − limx→01 − φ−1(2x)1 − φ−1(x)and λL = limx→0φ−1(2φ(x))x= limx→∞φ−1(2x)φ−1(x).Further, properties can be derived for distorted generators, φα,β(·) = φ(·α)β,upper and lower tails coecients are respectivelyλU and λ1/αL for φα,1(·) = φ(·α)and2 − (2 − λU )1/βand λ1/βL for φ1,β(·) = φ(·)β(Weak) tail dependence measureLedford & Tawn (1996) propose the following model to study tail dependence.Consider a random vector with identically distributed marginals, XL= Y .• under independence, P(X > t, Y > t) = P(X > t) × P(Y > t) = P(X > t)2,• under comonotonicity, P(X > t, Y > t) = P(X > t) = P(X > t)1,103
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Assume that P(X > t, Y > t) ∼ P(X > t)1/ηas t → ∞, where η ∈ (0, 1] will becalled coecient of tail dependenceMore precisely,• η = 1, perfect positive dependence (tail comontonicity),• 1/2 < η < 1, more dependent than independence, but asymptoticallyindependent,• η = 1/2, tail independence• 0 < η < 1/2 less dependent than independence.104
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.One can then dene upper tail coecient ηU and lower tail coecient ηL.Example 30. : If (X, Y ) has Gumbel copula,P(X ≤ x, Y ≤ y) = exp(−(x−α+ y−α)1/α), α ≥ 0then ηU = 1. Further, ηL = 1/2α.Example 31. : If (X, Y ) has a Clayton copula, then ηU = 1/2 while ηL = 1.Example 32. : If (X, Y ) has a Gaussian copula, then ηU = ηL = (1 + r)/2.105
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Agenda• General introductionModelling correlated risks• A short introduction to copulas• Quantifying dependence• Statistical inference• Agregation properties106
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Estimation of copulasSince C(u, v) = F(F−1X (u), F−1Y (v)), copula has be estimated only afterestimating marginal distribution.Margins CopulaParametric Fα and Fβ CθNonparametric FX and FY C(Fully) parametric estimation of copulasStep 1: t the 2 univariate marginal cdfs FX and FY with the help of theobservations {x1, x2, . . . , xn} and {y1, y2, . . . , yn} respectively; let α and β be thecorresponding MLEs of α and β.Step 2: estimate θ with the parameters α = α and β = β xed at the estimatedvalues from Step 1; i.e. on pseudo-observations (Ui, Vi)s, whereUi = Fα(Xi) and Vi = Fβ(Yi),107
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.let the result be θ.Step 3: using α, β and θ as starting values, determine the global MLEs α, β andθ of the parameters α, β and θ.Parametric estimation of copulasAn alternative is to use nonparametric estimation of margins, FX and FY .Step 1: estimate θ based on pseudo-observations (Ui, Vi)s, whereUi = FX(Xi) and Vi = FY (Yi),let the result be θ.Nonparametric estimation of copulasGiven an estimation of marginal distributions (parametric or nonparametric), the108
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.idea is to consider the empirical copula C, dened asC(u, v) =#{i such that Ui ≤ u and Vi ≤ v}#{i}=1nni=11(FX(Xi) ≤ u) × 1(FY (Yi) ≤ v).109
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example Loss-ALAE: consider the following dataset, were the Xis are lossamount (paid to the insured) and the Yis are allocated expenses. Denote by Riand Si the respective ranks of Xi and Yi. Set Ui = Ri/n = ˆFX(Xi) andVi = Si/n = ˆFY (Yi).Figure 43 shows the log-log scatterplot (log Xi, log Yi), and the associate copulabased scatterplot (Ui, Vi).Figure 44 is simply an histogram of the (Ui, Vi), which is a nonparametricestimation of the copula density.Note that the histogram suggests strong dependence in upper tails (theinteresting part in an insurance/reinsurance context).110
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.1 2 3 4 5 612345 Log!log scatterplot, Loss!ALAElog(LOSS)log(ALAE)0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Copula type scatterplot, Loss!ALAEProbability level LOSSProbabilitylevelALAEFigure 43: Loss-ALAE, scatterplots (log-log and copula type).111
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Figure 44: Loss-ALAE, histogram of copula type transformation.112
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.The basic idea to get an estimator of the density at some point x is to count howmany observation are in the neighborhood of x (e.g. in [x − h, x + h) for someh > 0).Therefore, consider the moving histogram or naive estimator as suggested byRosenblatt (1956),f(x) =12nhni=1I(Xi ∈ [x − h, x + h)).Note that this can be easily extended using other denitions of the neighborhoodof x,f(x) =1nhni=1Kx − Xih,where K is a kernel function (e.g. K(ω) = I(|ω| ≤ 1)/2).113
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.2 0.4 0.6 0.80.20.40.60.8012345Estimation of Frank copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Figure 45: Theoretical density of Frank copula.114
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.2 0.4 0.6 0.80.20.40.60.8012345Estimation of Frank copula0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Figure 46: Estimated density of Frank copula, using standard Gaussian (indepen-dent) kernels, h = h∗.115
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Problem of nonparametric estimation with kernel: bias on the borders.Let K denote a symmetric kernel with support [−1, 1]. Note thatE(f(0, h) =12f(0) + O(h)116
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2 Kernel based estimation of the uniform density on [0,1]Density0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2Kernel based estimation of the uniform density on [0,1]DensityFigure 47: Density estimation of an uniform density on [0, 1].117
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Several techniques have been introduce to get a better estimation on the border,• boundary kernel (Müller (1991))• mirror image modication (Deheuvels & Hominal (1989), Schuster(1985))• transformed kernel (Devroye & Györfi (1981), Wand, Marron &Ruppert (1991))• Beta kernel (Brown & Chen (1999), Chen (1999, 2000)),see Charpentier, Fermanian & Scaillet (2006) for a survey withapplication on copulas.118
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Consider the kernel estimator of the density of the(Xi, Yi) = (G−1(Ui), G−1(Vi))s, where G is a strictly increasing distributionfunction R → [0, 1], with a dierentiable density.119
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Since density f of (X, Y ) is continuous, twice dierentiable, and bounded above,for all (x, y) ∈ R2, considerf(x, y) =1nh2ni=1Kx − XihKy − Yih.Sincef(x, y) = g(x)g(y)c[G(x), G(y)]. (4)can be inverted inc(u, v) =f(G−1(u), G−1(v))g(G−1(u))g(G−1(v)), (u, v) ∈ [0, 1] × [0, 1], (5)one gets, substituting f in (5)c(u, v) =1nh · g(G−1(u)) · g(G−1(v))ni=1KG−1(u) − G−1(Ui)h,G−1(v) − G−1(Vi)h,(6)120
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.2 0.4 0.6 0.80.20.40.60.8012345Estimation of Frank copula0.2 0.4 0.6 0.80.20.40.60.8Figure 48: Estimated density of Frank copula, using a Gaussian kernel, after aGaussian normalization.121
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.The Beta-kernel based estimator of the copula density at point (u, v), is obtainedusing product beta kernels, which yieldsc(u, v) =1nni=1K Xi,ub+ 1,1 − ub+ 1 · K Yi,vb+ 1,1 − vb+ 1 ,where K(·, α, β) denotes the density of the Beta distribution with parameters αand β.122
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Beta (independent) bivariate kernel , x=0.0, y=0.0 Beta (independent) bivariate kernel , x=0.2, y=0.0 Beta (independent) bivariate kernel , x=0.5, y=0.0Beta (independent) bivariate kernel , x=0.0, y=0.2 Beta (independent) bivariate kernel , x=0.2, y=0.2 Beta (independent) bivariate kernel , x=0.5, y=0.2Beta (independent) bivariate kernel , x=0.0, y=0.5 Beta (independent) bivariate kernel , x=0.2, y=0.5 Beta (independent) bivariate kernel , x=0.5, y=0.5Figure 49: Shape of bivariate Beta kernels K(·, x/b + 1, (1 − x)/b + 1) × K(·, y/b +1, (1 − y)/b + 1) for b = 0.2.123
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Estimation of the copula density (Beta kernel, b=0.1) Estimation of the copula density (Beta kernel, b=0.1)0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Figure 50: Estimated density of Frank copula, Beta kernels, b = 0.1124
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.2 0.4 0.6 0.80.20.40.60.80.00.51.01.52.02.53.0Estimation of the copula density (Beta kernel, b=0.05) Estimation of the copula density (Beta kernel, b=0.05)0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Figure 51: Estimated density of Frank copula, Beta kernels, b = 0.05125
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.001234Standard Gaussian kernel estimator, n=100Estimation of the density on the diagonalDensityoftheestimator0.0 0.2 0.4 0.6 0.8 1.001234Standard Gaussian kernel estimator, n=1000Estimation of the density on the diagonalDensityoftheestimator0.0 0.2 0.4 0.6 0.8 1.001234Standard Gaussian kernel estimator, n=10000Estimation of the density on the diagonalDensityoftheestimatorFigure 52: Density estimation on the diagonal, standard kernel.126
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.001234Transformed kernel estimator (Gaussian), n=100Estimation of the density on the diagonalDensityoftheestimator0.0 0.2 0.4 0.6 0.8 1.001234Transformed kernel estimator (Gaussian), n=1000Estimation of the density on the diagonalDensityoftheestimator0.0 0.2 0.4 0.6 0.8 1.001234Transformed kernel estimator (Gaussian), n=10000Estimation of the density on the diagonalDensityoftheestimatorFigure 53: Density estimation on the diagonal, transformed kernel.127
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.001234Beta kernel estimator, b=0.05, n=100Estimation of the density on the diagonalDensityoftheestimator0.0 0.2 0.4 0.6 0.8 1.001234Beta kernel estimator, b=0.02, n=1000Estimation of the density on the diagonalDensityoftheestimator0.0 0.2 0.4 0.6 0.8 1.001234Beta kernel estimator, b=0.005, n=10000Estimation of the density on the diagonalDensityoftheestimatorFigure 54: Density estimation on the diagonal, Beta kernel.128
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Tail dependence and statistical inferenceConsider an i.i.d. sample (X1, Y1) , ...., (Xn, Yn).Consider unit Pareto transformation of margins: setT =11 − FX (X)∧11 − FY (Y ).Observe that the survival distribution function of T, FT , is regularly varyingwith parameter η. But because FX and FY are unknown, dene the pseudoobservations Tis asTi =11 − FX,n (Xi)∧11 − FY,n (Yi)=n + 1n + 1 − Ri∧n + 1n + 1 − Si,where Ri and Si denote the ranks of the Xis and Yis. Hill estimator can thenbe used, based on the k + 1 largest values of the Tis,ηHill =1kki=1logTn−i+1:nTn−k:n.129
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Estimation of η (Hills estimator)Proposition 33. Assume that (X, Y ) has upper tail dependence, with tail indexη, with additional regularity conditions, then√k (ηHill − η) is asymptoticallynormally distributed, with mean 0 and varianceσ2= η2(1 − l) 1 − 2∂c (1, 1)∂x∂c (1, 1)∂y.Remark 34. From this Proposition, a test for asymptotic dependence (i.e.η = 1) can de dened: asymptotic dependence is accepted if1 − ηHillσ (η = 1)≤ Φ−1(95%)130
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Estimation of η (Pengs estimator)SetSn (k) =ni=1I (Xi > Xn−k:n, Yi > Yn−k:n)andηPeng =1log 2logSn (k)Sn ( k/2 )−1.Proposition 35. Assume that (X, Y ) has upper tail dependence, with tail indexη, and the same technical assumption as before, then√k (ηPeng − η) isasymptotically normally distributed, with mean 0.131
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Estimation of λ (Huangs estimator)It is also possible to estimate λU : Huang-estimator of is based on the denitionof the upper tail index.λHuang =nkP (Ui, Vi) ∈ 1 −kn× 1 −kn=1kni=1I (Ri > n − k, Si > n − k)=1kni=1I (Xi > Xn−k:n, Yi > Yn−k:n) .132
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 100 200 300 4000.50.70.91.1Eta (Hill estimate), Gaussian copula, tau=0.30 100 200 300 4000.50.70.91.1Eta (Peng estimate), Gaussian copula, tau=0.30 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), Gaussian copula, tau=0.30 100 200 300 4000.50.70.91.1Eta (Hill estimate), Clayton copula, tau=0.30 100 200 300 4000.50.70.91.1Eta (Peng estimate), Clayton copula, tau=0.30 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), Clayton copula, tau=0.3Figure 55: Estimation of η and λ for Gaussian and Clayton copulas, with Kendallstau equal to 0.3133
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 100 200 300 4000.50.70.91.1Eta (Hill estimate), survival Clayton copula, tau=0.30 100 200 300 4000.50.70.91.1Eta (Peng estimate), survival Clayton copula, tau=0.30 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), survival Clayton copula, tau=0.30 100 200 300 4000.50.70.91.1Eta (Hill estimate), Gumbel copula, tau=0.30 100 200 300 4000.50.70.91.1Eta (Peng estimate), Gumbel copula, tau=0.30 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), Gumbel copula, tau=0.3Figure 56: Estimation of η and λ for survival Clayton and Gumbel copulas, withKendalls tau equal to 0.3134
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 100 200 300 4000.50.70.91.1Eta (Hill estimate), Gaussian copula, tau=0.70 100 200 300 4000.50.70.91.1Eta (Peng estimate), Gaussian copula, tau=0.70 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), Gaussian copula, tau=0.70 100 200 300 4000.50.70.91.1Eta (Hill estimate), Clayton copula, tau=0.70 100 200 300 4000.50.70.91.1Eta (Peng estimate), Clayton copula, tau=0.70 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), Clayton copula, tau=0.7Figure 57: Estimation of η and λ for Gaussian and Clayton copulas, with Kendallstau equal to 0.7135
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 100 200 300 4000.50.70.91.1Eta (Hill estimate), survival Clayton copula, tau=0.70 100 200 300 4000.50.70.91.1Eta (Peng estimate), survival Clayton copula, tau=0.70 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), survival Clayton copula, tau=0.70 100 200 300 4000.50.70.91.1Eta (Hill estimate), Gumbel copula, tau=0.70 100 200 300 4000.50.70.91.1Eta (Peng estimate), Gumbel copula, tau=0.70 100 200 300 4000.00.20.40.60.81.0Lambda (Huang estimate), Gumbel copula, tau=0.7Figure 58: Estimation of η and λ for survival Clayton and Gumbel copulas, withKendalls tau equal to 0.7136
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Agenda• General introductionModelling correlated risks• A short introduction to copulas• Quantifying dependence• Statistical inference• Agregation properties137
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Risk measures and diversicationAny copula C is bounded by Fréchet-Hoeding bounds,maxdi=1ui − (d − 1), 0 ≤ C(u1, . . . , ud) ≤ min{u1, . . . , ud},and thus, any distribution F on F(F1, . . . , Fd) is boundedmaxdi=1Fi(xi) − (d − 1), 0 ≤ F(x1, . . . , xd) ≤ min{F1(x1), . . . , Ff (xd)}.Does this means the comonotonicity is always the worst-case scenario ?Given a random pair (X, Y ), let (X−, Y −) and (X+, Y +) denotecontercomonotonic and comonotonic versions of (X, Y ), do we haveR(φ(X−, Y −))?≤ R(φ(X,Y ))?≤ R(φ(X+, Y +)).138
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Tchens theorem and bounding some pure premiumsIf φ : R2→ R is supermodular, i.e.φ(x2, y2) − φ(x1, y2) − φ(x2, y1) + φ(x1, y1) ≥ 0,for any x1 ≤ x2 and y1 ≤ y2, then if (X, Y ) ∈ F(FX, FY ),E φ(X−, Y −) ≤ E (φ(X, Y )) ≤ E φ(X+, Y +) ,as proved in Tchen (1981).Example 36. the stop loss premium for the sum of two risks E((X + Y − d)+) issupermodular.139
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example 37. For the n-year joint-life annuity,axy:n =nk=1vkP(Tx > k and Ty > k) =nk=1vkkpxy.Thena−xy:n ≤ axy:n ≤ a+xy:n ,wherea−xy:n =nk=1vkmax{kpx + kpy − 1, 0}( lower Fréchet bound ),a+xy:n =nk=1vkmin{kpx, kpy}( upper Fréchet bound ).140
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example 38. For the n-year last-survivor annuity,axy:n =nk=1vkP(Tx > k or Ty > k) =nk=1vkkpxy,where kpxy = P(Tx > k or Ty > k) = kpx + kpy − kpxy.Thena−xy:n ≤ axy:n ≤ a+xy:n ,wherea−xy:n =nk=1vk(1 − min{kqx, kqy}) ( upper Fréchet bound ),a+xy:n =nk=1vk(1 − max{kqx + kqy − 1, 0}) ( lower Fréchet bound ).141
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Example 39. For the widows pension annuity,ax|y = ay − axy =∞k=1vkkpy −∞k=1vkkpxy.Thena−x|y ≤ ax|y ≤ a+x|y,wherea−x|y = ay − axy =∞k=1vkkpy −∞k=1vkmin{kpx, kpy}.( upper Fréchet bound ),a+x|y = ay −axy =∞k=1vkkpy −∞k=1vkmax{kpx +kpy −1, 0}.( lower Fréchet bound ).142
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Value-at-Risk fails to be subadditiveOne of the axiom of coherence is subadditivity, i.e. R(X + Y ) ≤ R(X) + R(Y ).But Value-at-Risk is not coherent: one can haveVaR(X + Y, α)≥VaR(X, α) + VaR(Y, α) e.g. (see Embrechts (2007))• X and Y are independent but very skew (see e.g. credit risk, with defaultprobabilities < α),• X and Y are independent but very heavy-tailed (see e.g. the case of innitevariance i.e. X ∈ Lpwith p < 1),• X and Y are N(0, 1) with (special) tail dependence.143
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Makarovs theorem and bounding Value-at-RiskIn the case where R denotes the Value-at-Risk (i.e. quantile function of the P&Ldistribution),R−≤ R(X−+ Y −)≤R(X + Y )≤R(X++ Y +) ≤ R+,where e.g. R+can exceed the comonotonic case. Recall thatR(X + Y ) = VaRq[X + Y ] = F−1X+Y (q) = inf{x ∈ R|FX+Y (x) ≥ q}.If X ∼ E(α) and Y ∼ E(β),P(X > x) = exp(−x/α), P(Y > y) = exp(−y/β) and x ∈ R+.The inequalitiesexp(−x/ max{α, α}) ≤ Pr[X + Y > x] ≤ exp(−(x − ξ)+/(α + β))hold for all x ∈ R+, whatever the dependence between X and Y , whereξ = (α + β) log(α + β) − α log α − β log β.144
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Therefore, the inequalities− max{α, β} log(1 − q) ≤ VaRq[X + Y ] ≤ ξ − (α + β) log(1 − q)hold for all q ∈ (0, 1).Recall that in the independence case independence X + Y ∼ G(2, 1) and underperfect positive dependence X + Y ∼ E(2).145
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.0!4!2024Bornes de la VaR d’un portefeuilleSomme de 2 risques GaussiensFigure 59: Value-at-Risk for 2 Gaussian risks N(0, 1).146
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.90 0.92 0.94 0.96 0.98 1.000123456Bornes de la VaR d’un portefeuilleSomme de 2 risques GaussiensFigure 60: Value-at-Risk for 2 Gaussian risks N(0, 1).147
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.2 0.4 0.6 0.8 1.005101520Bornes de la VaR d’un portefeuilleSomme de 2 risques GammaFigure 61: Value-at-Risk for 2 Gamma risks G(3, 1).148
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.90 0.92 0.94 0.96 0.98 1.0005101520Bornes de la VaR d’un portefeuilleSomme de 2 risques GammaFigure 62: Value-at-Risk for 2 Gamma risks G(3, 1).149
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Bounding Value-at-Risk for a sum of 2 risksIn a general (and theoretical) context, Schweizer & Sklar (1981), studied thedistribution of ψ(X, Y ) for some R2→ R function ψ, where (X, Y ) ∈ F(FX, FY ),using the concept of supremal and inmal convolutions,Fsup (FX, FY ) (z) = sup {C (FX (x) , FY (y)) , ψ (x, y) = z} (7)Finf (FX, FY ) (z) = inf {C (FX (x) , FY (y)) , ψ (x, y) = z} (8)Williamson (1991) and Embrechts, Hoing & Juri (2002) proposednumerical algorithm to calculate those bounds.The idea is to observe that bounds for the distribution of the sum of X and Y isgiven by the distribution of Smin and Smax, whereP(Smax < s) = supx∈Rmax{P(X < x) + P(Y < s − x) − 1, 0}andP(Smin ≤ s) = infx∈Rmin{P(X ≤ x) + P(Y ≤ s − x), 1}.150
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Note that those bounds can also be written as follows,Proposition 40. Let (X, Y ) ∈ F(FX, FY ) then for all s ∈ R,τC− (FX, FY )(s) ≤ P(X + Y ≤ s) ≤ ρC− (FX, FY )(s),whereτC(FX, FY )(s) = supx,y∈R{C(FX(x), FY (y)), x + y = s}and, if ˜C(u, v) = u + v − C(u, v),ρC(FX, FY )(s) = infx,y∈R{ ˜C(FX(x), FY (y)), x + y = s}.151
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.5 1.0 1.5 2.00.00.20.40.60.8 Density of the sum of two uniform random variablesDensityIndependenceClayton, tau=0.5Clayton, tau=0.2Gumbel, tau=0.5Gumbel, tau=0.2Comonotonicity0"#0 0"#$ 0"%0 0"%$ 1"001"21"(1")1"#2"0Quantile of the sum of two uniform random variables*+,-.-/0/1230454067.81/0439:.074!.1!;/<=>?8@4A48@48B4C0.21,8D31.7E0"$C0.21,8D31.7E0"2F7G-40D31.7E0"$F7G-40D31.7E0"2C,G,8,1,8/B/12Figure 63: Sum to two U([0, 1]) random variables.152
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.7 Density of the sum of three uniform random variablesDensityIndependenceClayton, tau=0.5Clayton, tau=0.2Gumbel, tau=0.5Gumbel, tau=0.2Comonotonicity0"#0 0"#$ 0"%0 0"%$ 1"001"$2"02"$3"0Quantile of the sum of three uniform random variables)ro,-,i/it1 /e4e/56-nti/e89-/6e!-t!:is<=>ndependenceB/-1ton, t-6D0"$B/-1ton, t-6D0"2E6m,e/, t-6D0"$E6m,e/, t-6D0"2Bomonotonicit1Figure 64: Sum to three U([0, 1]) random variables.153
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.0 1 2 3 4 50.00.10.20.30.40.5 Density of the sum of five uniform random variablesDensityIndependenceClayton, tau=0.5Clayton, tau=0.2Gumbel, tau=0.5Gumbel, tau=0.2Comonotonicity!"#! !"#$ !"%! !"%$ &"!!"!"$4"!4"$$"!Quantile of the sum of five uniform random variables)robabilit1 l343l5uantil3(9alu3!at!:is<)>nd3p3nd3nc3Bla1tonC tauD!"$Bla1tonC tauD!"2Fumb3lC tauD!"$Fumb3lC tauD!"2Bomonotonicit1Figure 65: Sum to ve U([0, 1]) random variables.154
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Bounding Value-at-Risk for a sum of d ≥ 3 risksThe standard bounds (derived in dimension 2) can be extended, and still holds inarbitrary dimensions,VaR(X1 + ... + Xd, α) ≤ G−1(α)whereG(s) = sup(x1,...,xd−1) C−(F1(x1) + ... + Fd−1(xd−1) + Fd(s − x1 − ... − xd−1). Butwhen d ≥ 3, it fails to be sharp.Embrechts & Puccetti (2006) suggested to use the dual optmizationproblem, and thusVaR(X1 + ... + Xd, α) ≤ H−1(α)where H(s) = 1 − d infr∈[0,s/d)1s − rds−(d−1)rr[1 − F(x)]dx, if X1, ..., Xd haveidentical marginal distributions (denotes F).155
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Numerical implicationsConsider 3 risks, and try to get the VaR of the sum of log-normal risks,α independence comonotonicity dual standardbound bound0.90 7.54 8.85 14.44 15.380.95 9.71 12.73 19.50 20.630.99 16.06 25.16 35.31 37.030.999 29.78 53.99 69.98 73.81Table 1: Range for VaR(X1 + X2 + X3) for a LN(−0.2, 1) portfolio.156
  • Arthur CHARPENTIER - Gestion des risques bancaires et financiers.Numerical implicationsIn the context of operational risks, Moscadelli (2004) considered 8 businesslinesα comonotonic dual standard(Basel II) bound bound99% 2.8924 ×1041.4778 ×1052.6950 ×10599.5% 6.7034 ×1043.3922 ×1056.1114 ×10599.9% 4.8347 ×1052.3807 ×1064.1685 ×10699.99% 8.7476 ×1064.0740 ×1076.7936 ×107Table 2: Range for VaR(X1 + ... + X8).Remark 41. In banks, Basel II suggests to use the comonotonic case as a worstcase scenario.157