On the Conditional Value at Risk (CoVaR) from the copula perspective

Transcription

1 On the Conditional Value at Risk (CoVaR) from the copula perspective Piotr Jaworski Institute of Mathematics, Warsaw University, Poland 1

2 Overview 1. Basics about VaR, CoVaR and copulas. 2. Standard CoVaR. 3. Modified CoVaR. 4. Estimation. 2

3 Value at Risk (VaR) We recall that Value-at-Risk, at a given significance level α (0, 1), of a random variable X modelling a position, is defined as follows: V ar α (X) = inf{v R : P(X + v 0) α}. The above can be expressed in terms of quantiles. Namely Value-at- Risk at a level α is equal to minus upper α quantile V ar α (X) = Q + α (X). 3

4 Conditional Value at Risk (CoVaR) Let X and Y be random variables modelling returns of market indices. CoVaR is defined as VaR of Y conditioned by X. In more details: CoV ar(y X) = V ar β (Y X E), where E, the Borel subset of the real line, is modelling some adverse event concerning X. 4

5 Standard CoVaR A standard Conditional-VaR at a level (α, β) is defined as VaR at level β of Y under the condition that X = V ar α (X). CoV ar α,β (Y X) = V ar β (Y X = V ar α (X)). The above can be expressed in terms of quantiles. Namely CoV ar α,β (Y X) = Q + β (Y X = Q+ α (X)). 5

6 Modified CoVaR A modified Conditional-VaR at a level (α, β) is defined as VaR at level β of Y under the condition that X V ar α (X). CoV ar α,β (Y X) = V ar β (Y X V ar α (X)). The above can be expressed in terms of quantiles. Namely CoV ar α,β (Y X) = Q + β (Y X Q+ α (X)). 6

7 The axiomatic definition of copula, n = 2 Definition 1 The function C : [0, 1] 2 [0, 1] is called a copula if the following three properties hold: (c1) u 1, u 2 [0, 1] C(u 1, 0) = 0, C(0, u 2 ) = 0; (c2) u 1, u 2 [0, 1] C(u 1, 1) = u 1, C(1, u 2 ) = u 2 ; (c3) u 1, u 2, v 1, v 2 [0, 1], u 1 v 1, u 2 v 2 C(v 1, v 2 ) C(u 1, v 2 ) C(v 1, u 2 ) + C(u 1, u 2 ) 0. 7

8 Inclusion-exclusion principle 1 v 2 + u u 1 v 1 1 8

9 The probabilistic definition of copulas Theorem 1 For a function C : [0, 1] 2 [0, 1] the following conditions are equivalent : 1. C is a copula. 2. There exist random variables U, V, which are uniformly distributed on [0, 1], such that C is a restriction to the unit square [0, 1] 2 of their joint distribution function. Random variables U and V are called the representers of copula C. 9

10 Sklar Theorem Theorem 2 Let F be a 2-dimensional distribution function and F 1, F 2 its marginal distribution functions, then there is a copula C such that for each x = (x 1, x 2 ) R 2 F (x 1, x 2 ) = C(F 1 (x 1 ), F 2 (x 2 )). Furthermore, the copula C is uniquely determined when the boundary distribution functions F i are continuous. Conversely, if C is a 2-dimensional copula and F 1, F 2 are univariate distribution functions then the function F is a 2-dimensional distribution function and F 1, F 2 are its boundary distribution functions. 10

11 The invariance of copulas The copulas are true measures of interdependence between random phenomena. Namely they do not depend on the scale in which these phenomena are quantified. Proposition 1 Let C be a copula of a random variable X = (X 1, X 2 ). If the functions f 1, f 2 are defined and strictly increasing on the supports of X 1, X 2, then C is also a copula of the random variable Y = (f 1 (X 1 ), f 2 (X 2 )). 11

12 Copulas with nontrivial tail expansions Definition 2 We say that a copula C has a tail expansion at the vertex (0, 0) of the unit square if the limit C(tx, ty) lim t 0 + t exists for all nonnegative x, y. The function L : [0, ] 2 C(tx, ty) [0, ), L(x, y) = lim, t 0 t is called the tail dependence function or the leading term of the tail expansion. L(1, 1) is called the (lower) tail coefficient. L is homogeneous of degree 1 and concave. 12

13 Conditional probability by copulas Theorem 3 Let C(u, v) be a copula of random variables X and Y having continuous distribution functions F X and F Y, then P(Y y X = x) = lim η y D uc(f X (x), F Y (η)), where D u denotes the partial left-sided upper Dini derivative with respect to first variable u C(u, v) C(u h, v) D u C(u, v) = lim sup, h 0 + h is a version of conditional probability. 13

14 Standard CoVaR by copulas In the following we assume that random variables X and Y have continuous distribution functions F X and F Y. We select the version of conditional probability from Theorem 3. This leads to the following definition of CoVaR. CoV ar α,β (Y X) = sup{y : F Y (y) = v } = Q + v (Y ) = V ar v (Y ), where v = inf{v : D u C(α, v) > β}. When C is continuously differentiable we get C u (α, F Y ( CoV ar α,β (Y X))) = 0. 14

15 Examples: product copula Π Π(u, v) = uv, Π(u, v) u = v, v = β, CoV ar α,β (Y X) = V ar β (Y ). 15

16 Examples: FGM copulas C F GM (u, v; θ) = uv(1 + θ(1 u)(1 v)), θ [ 1, 1], C F GM (u, v; θ) u = v + θv(1 v)(1 2u), v = 2β 1 + θ(1 2α) +, = (1 + θ(1 2α)) 2 4βθ(1 2α), lim v = 1 + θ + 2β 1 + 2θ(1 2β) + θ 2. 16

17 Examples: comonotonic copula M M(u, v) = min(u, v), D u M(u, v) = { 0 for v < x, 1 for v x, v = α, CoV ar α,β (Y X) = V ar α (Y ). 17

18 Examples: Marshall-Olkin copulas C MO (u, v) = { u 1 a v for u a v b, uv 1 b for u a < v b, a, b (0, 1). v = D u C MO (u, v) = { (1 a)u a v for u a > v b, v 1 b for u a v b, βα a 1 a for β (0, (1 a)α (1 b)a/b ), α a/b for β [(1 a)α (1 b)a/b, α (1 b)a/b ], β 1/(1 b) for β (α (1 b)a/b, 1), lim v = β 1 b. 1 18

19 Examples: Gaussian copula, r ( 1, 1) ( C Ga (u, v; r) = Φ N(0,R) Φ 1 (u), Φ 1 N(0,1) N(0,1) (v)), R = C Ga (u, v; r) u = Φ N(0,1) Φ 1 N(0,1) (v) rφ 1 1 r 2 ( 1 r r 1 N(0,1) (u), ( ) v = Φ N(0,1) 1 r 2 Φ 1 (β) + rφ 1 N(0,1) N(0,1) (α), lim v = 0 r > 0, β r = 0, 1 r < 0, ), lim v α = Sgn(r). 19

20 Examples: t-student copulas C t (u, v; ν, r) u ( C t (u, v; ν, r) = Φ t(ν,r) Φ 1 (u), Φ 1 t(ν,1) t(ν,1) (v)), R = ( 1 r r 1 = Φ t(ν+1,1) ), r ( 1, 1), ν >= 1, ( Φ 1 t(ν,1) (v) rφ 1 t(ν,1) (u)) 1 + 1/ν 1 r Φ 1 t(ν,1) (u)2 /ν, v = Φ t(ν,1) 1 r Φ 1 t(ν,1) (α)2 /ν Φ 1 (β) + rφ 1 t(ν+1,1) 1 + 1/ν t(ν,1) (α), 20

21 Examples: t-student copulas cont. lim v = 0 β < β, 1/2 β = β, 1 β > β, lim v α = { w β < β, w β > β, β = Φ t(ν+1,1) r 1 + ν 1 r 2, w = 1 r ν Φ 1 t(ν+1,1) (β) r ν. 21

22 Examples: Clayton copulas, θ > 0 C Cl (u, v; θ) = ( u θ + v θ 1 ) 1θ, C Cl (u, v; θ) u = ( u θ + v θ 1 ) 1+θ θ u θ 1, v = α (β θ 1+θ 1 + α θ ) 1 θ, lim v = 0, lim v α = (β 1+θ θ ) 1 θ 1. 22

23 Examples: Survival Clayton copulas, θ > 0 Ĉ Cl (u, v; θ) = ( (1 u) θ + (1 v) θ 1 ) 1θ + u + v 1, ĈCl(u, v; θ) u = ( (1 u) θ + (1 v) θ 1 ) 1+θ θ (1 u) θ 1 + 1, v = 1 (1 α) ((1 β) θ 1+θ 1 + (1 α) θ ) 1 θ, lim v = 1 (1 β) 1+θ, 1 23

24 Tail dependence Theorem 4 Let the copula C have a continuously differentiable, nonzero tail dependence function L lim t 0 C(tu, tv) If furthermore L(1,v) u t = L(u, v) and lim t 0 C(tu, tv) u = L(u, v). u is strictly increasing, then for β < L(1, ) u lim v = 0, lim v α = v (0, ), where L(1, v ) u = β. 24

25 Modified CoVaR by copulas Let C(u, v) be a copula of random variables X and Y having continuous distribution functions F X and F Y, then P(Y y X Q + α (X)) = P(Y y X Q+ α (X)) α = C(α, F Y (y)). α Therefore mcov ar α,β (Y X) = V ar w (Y ), where w is the solution of the equation C(α, w ) = αβ. 25

26 Examples: product copula Π Π(u, v) = uv, αw = αβ, w = β, mcov ar α,β (Y X) = V ar β (Y ). 26

27 Examples: comonotonic copula M M(u, v) = min(u, v), min(α, w ) = αβ, w = αβ, CoV ar α,β (Y X) = V ar αβ (Y ). 27

28 Examples: FGM copulas C F GM (u, v; θ) = uv(1 + θ(1 u)(1 v)), θ [ 1, 1], w = 2β 1 θ(1 α) +, = (1 θ(1 α)) 2 + 4βθ(1 α), lim w = 1 θ + 2β 1 + 2θ(2β 1) + θ 2. 28

29 Examples: Gaussian copula, r ( 1, 1) ( C Ga (u, v; r) = Φ N(0,R) Φ 1 (u), Φ 1 N(0,1) N(0,1) (v)), R = ( 1 r r 1 ), lim w = lim w α = 0 r > 0, β r = 0, 1 r < 0, r > 0, 0 r = 0, r < 0, 29

30 Examples: Archimedean copulas We recall that the n-variate copula C is called Archimedean if there exist generators ψ and ϕ such that C(x, y) = ψ(ϕ(x) + ϕ(y)). The generators are convex nonincreasing functions such that ψ : [0, ] [0, 1], ϕ : [0, 1] [0, ], ψ(0) = 1, ϕ(1) = 0 and t [0, 1] ψ(ϕ(t)) = t. We get w = ψ(ϕ(αβ) ϕ(α)). 30

31 Examples: Archimedean copulas cont. If ϕ is nonstrict i.e. ϕ(0) < + then lim w = 1. If ϕ is strict, i.e. ϕ(0) =, and regularly varying at 0, d > 0 x > 0 lim t 0 + ϕ(tx) ϕ(t) = x d, then lim w = 0, w lim α = β (1 β d ) 1/d. 31

32 Examples: DJM copulas Let f : [0, + ] [0, 1] be a surjective, monotonic function and g its right inverse (f(g(y)) = y). If furthermore f is concave and nondecreasing or convex and nonincreasing then the function C f : [0, 1] 2 [0, 1], C f (x, y) = is a copula. { 0 for x = 0, xf ( ) g(y) x for x > 0 We get w = f(αg(β)), lim w = f(0), w lim α = f (0 + )g(β). 32

33 Tail dependence Theorem 5 Let the copula C have a nonzero tail dependence function L C(tu, tv) lim = L(u, v). t 0 t Then for β < L(1, ) = L(0+,1) u lim w lim w = 0, α = w (0, ), where L(1, w ) = β. 33

34 Estimation Let (x t, y t ) n t=1 be a given sample. To each pair we associate a pair of ranks (rk x t, rk y t ). We determine a size of the tail fixing a constant γ (0, 0.5). We select the indices of the elements from the γ-tail Γ = {t : rk x t + rk y t γn}. We estimate L as a distribution function of a singular measure on [0, ) 2 which is uniformly distributed on half-lines starting from the origin and crossing the points (rk x t, rk y t ) : t Γ. l 1 (v) = L(1, v) = 1 γn t Γ rk x t + rk y t rk y t min ( v, rk y t rk x t ). 34

35 Estimation cont. Since L is homogeneous we get the following estimate of its derivative with respect to first variable: l 2 (v) = 1 L(1, v) = 1 γn t Γ rk x t + rk y t rk x t 1l v>rkyt /rkx t. For sufficiently small α and β < l 1 (γn) = l 2 (γn) we put: mcov ar α,β (Y X) V arŵ (Y ), CoV arα, β(y X) V ar ˆv (Y ), ŵ = α max(β, l1 1 (β)), ˆv = α l2 (β). 35

36 Bibliography 1. Adrian, T., Brunnermeier, M.K.: CoVaR, Working paper, Federal Reserve Bank of New York, Gigardi G., Ergün T.A.: Systemic risk measurement: Multivariate GARCH estimation of CoVar, Journal of Banking & Finance 37, (2013) Jaworski P.: On uniform tail expansions of bivariate copulas, Applicationes Mathematicae 31.4 (2004) Jaworski P.: On uniform tail expansions of multivariate copulas and wide convergence of measures, Applicationes Mathematicae 33.2 (2006) Jaworski P.: Tail Behaviour of Copulas, In: P. Jaworski, F. Durante, W. Härdle and T. Rychlik, editors, Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw, September 2009, Lecture Notes in Statistics - Proceedings 198. Springer 2010, pp