Regularly Varying Asymptotics for Tail Risk

Regularly Varying Asymptotics for Tail Risk Haijun Li Department of Mathematics Washington State University Humboldt Univ-Berlin Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 1 / 26

Risk Factors and Risk Measures Fix a probability space (Ω, F, P) generated by a stochastic system (e.g., financial portfolio). Let L denote a convex cone of random risk factors defined on (Ω, F, P); e.g., X L is interpreted as a (aggregated) loss in a financial portfolio at the end of a given period. Note that X, where loss X L, represents the net worth of a financial position. Assume that loss X L is continuous and heavy-tailed. A (univariate) risk measure ϱ : L R is a functional satisfying some operational properties. Multivariate (or set-valued) risk measures have also been studied in the literature. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 2 / 26

Risk Factors and Risk Measures Fix a probability space (Ω, F, P) generated by a stochastic system (e.g., financial portfolio). Let L denote a convex cone of random risk factors defined on (Ω, F, P); e.g., X L is interpreted as a (aggregated) loss in a financial portfolio at the end of a given period. Note that X, where loss X L, represents the net worth of a financial position. Assume that loss X L is continuous and heavy-tailed. A (univariate) risk measure ϱ : L R is a functional satisfying some operational properties. Multivariate (or set-valued) risk measures have also been studied in the literature. We are interested in measuring and analyzing risk associated with extreme events. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 2 / 26

Tail Conditional Expectation Two popular risk measures: Value-at-Risk (VaR) at confidence level p, defined by VaR p (X) := inf{x R : P(X > x) 1 p}. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 3 / 26

Tail Conditional Expectation Two popular risk measures: Value-at-Risk (VaR) at confidence level p, defined by VaR p (X) := inf{x R : P(X > x) 1 p}. Tail Conditional Expectation at confidence level p (TCE, Expected Shortfall, Tail VaR, Conditional VaR), defined by TCE p (X) := E(X X > VaR p (X)). TCE is also known as Average VaR: TCE p (X) = 1 1 p 1 p VaR λ(x)dλ. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 3 / 26

Tail Conditional Expectation Two popular risk measures: Value-at-Risk (VaR) at confidence level p, defined by VaR p (X) := inf{x R : P(X > x) 1 p}. Tail Conditional Expectation at confidence level p (TCE, Expected Shortfall, Tail VaR, Conditional VaR), defined by TCE p (X) := E(X X > VaR p (X)). TCE is also known as Average VaR: TCE p (X) = 1 1 p 1 p VaR λ(x)dλ. lim p 0 TCE p (X) = E(X). lim p 1 TCE p (X) =. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 3 / 26

Tail Conditional Expectation Two popular risk measures: Value-at-Risk (VaR) at confidence level p, defined by VaR p (X) := inf{x R : P(X > x) 1 p}. Tail Conditional Expectation at confidence level p (TCE, Expected Shortfall, Tail VaR, Conditional VaR), defined by TCE p (X) := E(X X > VaR p (X)). TCE is also known as Average VaR: TCE p (X) = 1 1 p 1 p VaR λ(x)dλ. lim p 0 TCE p (X) = E(X). lim p 1 TCE p (X) =. Non-negativity Assumption Since we are interested in analyzing tail risk as p 1, we assume w.l.o.g. that loss variables X 0 a.s.. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 3 / 26

VaR p (X) = TCE p (X) = 0 0 g VaR (P(X > x))dx, g VaR (x) := I {x>1 p} g TCE (P(X > x))dx, g TCE (x) := min{x/(1 p), 1}. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 4 / 26

VaR p (X) = TCE p (X) = 0 0 g VaR (P(X > x))dx, g VaR (x) := I {x>1 p} g TCE (P(X > x))dx, g TCE (x) := min{x/(1 p), 1}. Distortion Risk (Denneberg, 1989, 1994) For a given nondecreasing function g : [0, 1] [0, 1] such that g(0) = 0 and g(1) = 1, a distortion risk measure H g (X) of loss X 0 with df F( ) is defined as the following Choquet integral: H g (X) = 0 g(p(x > x))dx = E Pg,F (X), where P g,f denotes the probability measure induced by the distribution function ḡ(f( )), where ḡ(u) = 1 g(1 u). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 4 / 26

Operational Properties H g (X) = E Pg,F (X) = the expected value of X under the distorted probability measure P g,f that may describe a possible scenario. H g (X) satisfies the following as a risk measure ϱ : L R. 1 (monotonicity or ordering preference) For X, Y L with X Y almost surely, ϱ(x) ϱ(y). 2 (positive homogeneity) For all X L and every λ > 0, ϱ(λx) = λϱ(x). 3 (translation invariance) For all X L and every l R, ϱ(x + l) = ϱ(x) + l. 4 (subadditivity) If g is concave, then ϱ(x + Y) ϱ(x) + ϱ(y). 5 (superadditivity) If g is convex, then ϱ(x + Y) ϱ(x) + ϱ(y). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 5 / 26

Operational Properties H g (X) = E Pg,F (X) = the expected value of X under the distorted probability measure P g,f that may describe a possible scenario. H g (X) satisfies the following as a risk measure ϱ : L R. 1 (monotonicity or ordering preference) For X, Y L with X Y almost surely, ϱ(x) ϱ(y). 2 (positive homogeneity) For all X L and every λ > 0, ϱ(λx) = λϱ(x). 3 (translation invariance) For all X L and every l R, ϱ(x + l) = ϱ(x) + l. 4 (subadditivity) If g is concave, then ϱ(x + Y) ϱ(x) + ϱ(y). 5 (superadditivity) If g is convex, then ϱ(x + Y) ϱ(x) + ϱ(y). Any risk measure ϱ : L R satisfying (1)-(4) is called a coherent risk measure (Artzner, Delbaen, Eber, & Heath, 1999). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 5 / 26

Dual Representation (Delbaen, 2000) Under some regularity conditions (Fatou property,...), a coherent risk measure ϱ(x) arises as the supremum of expected values of loss X under various scenarios: ϱ(x) = sup Q S E Q (X) where S is a (closed) convex set of probability measures on physical states, that are absolutely continuous with respect to the underlying measure P. Q P (absolute continuity): P(A) = 0 implies that Q(A) = 0, A F. Q P implies that Q(A) = A f (ω)p(dω), where f = dq/dp is known as the Radon-Nikodym derivative. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 6 / 26

Axiom-Duality Approach The use of this axiom-duality approach from functional analysis can be traced back to Lloyd Shapley (1953) in deriving an optimal solution of a cooperative game. Early work on distortion risk measures and their representation can be found in Quiggin (1982) and Yaari (1987) The connection of distortion risk to coherent risk was first established in Schmeidler (1986, 1989). Early work on ordering preferences and their dual integral representation can be found in Huber (1981) and Gilboa and Schmeidler (1989). Distortion risk measures were introduced in the actuarial literature in Denneberg (1994) and Wang (1995, 1996). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 7 / 26

Worst Conditional Expectation Consider the following class of scenario probability measures: S p := {P( A) : P(A) > 1 p, A F} = {Q : dq/dp L 1/(1 p)}, 0 < p < 1. The corresponding coherent risk is called the worst conditional expectation and given by, WCE p (X) = sup{e(x A) : P(A) > 1 p, A F}. Since 1 p is usually small, S p contains all the scenario probability measures conditioning on events with occurring probability at least 1 p, including rare events. WCE p (X) is the worst expected loss which could be incurred from various random events with occurring probability at least 1 p. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 8 / 26

In general, VaR p (X) WCE p (X) TCE p (X). If X is continuous, WCE p (X) = TCE p (X). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 9 / 26

In general, VaR p (X) WCE p (X) TCE p (X). If X is continuous, WCE p (X) = TCE p (X). VaR p ( ) is a non-coherent, distortion measure. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 9 / 26

In general, VaR p (X) WCE p (X) TCE p (X). If X is continuous, WCE p (X) = TCE p (X). VaR p ( ) is a non-coherent, distortion measure. Deeper Properties (Delbaen, 2000; Kusuoka, 2001) Suppose that (Ω, F, P) is atomless, X L is continuous, and ϱ is coherent and law-invariant with the Fatou property. 1 ϱ(x) VaR p (X) implies that ϱ(x) TCE p (X) for 0 < p < 1. 2 ϱ(x) = sup TCE λ (X) µ(dλ), µ M [0,1] for a convex set M of probability measures on [0, 1]. 3 If ϱ is comonotone, then ϱ(x) = [0,1] TCE λ(x) m(dλ) for some probability measure m. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 9 / 26

In general, VaR p (X) WCE p (X) TCE p (X). If X is continuous, WCE p (X) = TCE p (X). VaR p ( ) is a non-coherent, distortion measure. Deeper Properties (Delbaen, 2000; Kusuoka, 2001) Suppose that (Ω, F, P) is atomless, X L is continuous, and ϱ is coherent and law-invariant with the Fatou property. 1 ϱ(x) VaR p (X) implies that ϱ(x) TCE p (X) for 0 < p < 1. 2 ϱ(x) = sup TCE λ (X) µ(dλ), µ M [0,1] for a convex set M of probability measures on [0, 1]. 3 If ϱ is comonotone, then ϱ(x) = [0,1] TCE λ(x) m(dλ) for some probability measure m. TCEs (distortion, coherent, comonotone) are building blocks. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 9 / 26

(Cai and Li, 2005) For any light-tailed loss variable X (i.e., its distribution tails decay exponentially), lim p 1 TCE p(x) VaR p(x) 1. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 10 / 26

(Cai and Li, 2005) For any light-tailed loss variable X (i.e., its distribution tails decay exponentially), lim p 1 TCE p(x) VaR p(x) 1. For a heavy-tailed loss variable X, as p 1, TCEp(X) VaR p(x)? Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 10 / 26

(Cai and Li, 2005) For any light-tailed loss variable X (i.e., its distribution tails decay exponentially), TCE lim p(x) p 1 VaR 1. p(x) For a heavy-tailed loss variable X, as p 1, TCEp(X) VaR? p(x) It boils down to analyzing tail behaviors of P(X tb), as var.jpg (JPEG Image, 468x394 pixels) http://www.ima.umn.edu/2009-2010/mm8.2-11.10/var.jpg t, for various upper critical region B. Figure: α = 1 p Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 10 / 26

Regular variation A loss variable X 0 with df F has a regularly varying right tail at with tail index α > 0, denoted as X RV α, if for all s > 0, P(X > ts) P(X > t) s α, as t. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 11 / 26

Regular variation A loss variable X 0 with df F has a regularly varying right tail at with tail index α > 0, denoted as X RV α, if for all s > 0, P(X > ts) P(X > t) s α, as t. In fact, X RV α implies that the above limit converges uniformly on intervals of the form s (b, ), b > 0. X RV α if and only if its survival function is given by F(s) := P(X > s) = s α L(s), s > 0, where L > 0 is slowly varying: lim t L(ct)/L(t) = 1, for every c > 0. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 11 / 26

Regular variation A loss variable X 0 with df F has a regularly varying right tail at with tail index α > 0, denoted as X RV α, if for all s > 0, P(X > ts) P(X > t) s α, as t. In fact, X RV α implies that the above limit converges uniformly on intervals of the form s (b, ), b > 0. X RV α if and only if its survival function is given by F(s) := P(X > s) = s α L(s), s > 0, where L > 0 is slowly varying: lim t L(ct)/L(t) = 1, for every c > 0. F( ) behaves asymptotically like a power function or homogeneous function (Bingham, Goldie, & Teugels, 1987). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 11 / 26

Asymptotic Relation Between TCE and VaR Karamata s Theorem If X RV α with α > 1, then t P(X > s)ds t P(X > t), for sufficiently large t. α 1 Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 12 / 26

Asymptotic Relation Between TCE and VaR Karamata s Theorem If X RV α with α > 1, then t P(X > s)ds If X RV α with α > 1, then t P(X > t), for sufficiently large t. α 1 TCE p (X) lim p 1 VaR p (X) = α α 1. The limit holds even if X is not non-negative. The rate of convergence of TCEp(X) VaR to α p(x) α 1 can be assessed by the second order regular variation (Hua, 2012). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 12 / 26

Asymptotic Relation Between TCE and VaR Karamata s Theorem If X RV α with α > 1, then t P(X > s)ds If X RV α with α > 1, then t P(X > t), for sufficiently large t. α 1 TCE p (X) lim p 1 VaR p (X) = α α 1. The limit holds even if X is not non-negative. The rate of convergence of TCEp(X) VaR to α p(x) α 1 can be assessed by the second order regular variation (Hua, 2012). If 0 < α 1, then E(X) =. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 12 / 26

Tail Distortion Definition (Zhu & Li, 2012) For a given nondecreasing function g : [0, 1] [0, 1] such that g(0) = 0 and g(1) = 1, the tail distortion risk measure H g ( ) of loss X 0 is defined as follows: H g (X X > VaR p (X)) = 0 g(p(x > x X > VaR p (X)))dx. The tail distortion risk of a continuous loss X is a distortion risk measure, but can be viewed as the expected value of X under a change of the underlying measure that is deformed only on the tail of the loss distribution. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 13 / 26

Tail Distortion Definition (Zhu & Li, 2012) For a given nondecreasing function g : [0, 1] [0, 1] such that g(0) = 0 and g(1) = 1, the tail distortion risk measure H g ( ) of loss X 0 is defined as follows: H g (X X > VaR p (X)) = 0 g(p(x > x X > VaR p (X)))dx. The tail distortion risk of a continuous loss X is a distortion risk measure, but can be viewed as the expected value of X under a change of the underlying measure that is deformed only on the tail of the loss distribution. If g = 1, then H g (X X > VaR p (X)) = TCE p (X). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 13 / 26

Asymptotics of Tail Distortion Theorem (Zhu & Li, 2012) If X RV α, and g( ) is any distortion function with 1 g(w α+δ )dw < for some 0 < δ < α, then as p 1, H g (X X > VaR p (X)) ( 1 + 1 ) g(w α )dw VaR p (X). If g = 1 and α > 1, then H g (X X > VaR p (X)) α α 1 VaR p(x). Example: Let g(x) = x k, 0 x 1, k 0. This function is known as proportional hazards distortion (Wang, 1995, 1996). If 0 < α 1, set k > 1/α, and use the tail distortion risk measure H g (X X > VaR p (X)) αk αk 1 VaR p(x), as p 1. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 14 / 26

Multivariate Risks Consider X = (X 1,..., X d ) from a multi-asset portfolio at the end of a given period, where X i = loss of the position on the i-th market. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 15 / 26

Multivariate Risks Consider X = (X 1,..., X d ) from a multi-asset portfolio at the end of a given period, where X i = loss of the position on the i-th market. Motivation: Investors are sometimes not able to aggregate their multivariate portfolios on various security markets because of liquidity problems and/or transaction costs... A risk measure R(X) for loss vector X corresponds to an upper closed subset of R d consisting of all the deterministic portfolios x such that the modified positions x X is acceptable to regulator/supervisor. That is, any deterministic portfolio in set R(X) cancels the risk imposed by losses X on various markets. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 15 / 26

Coherent Risks (Jouini, Meddeb & Touzi, 2004) A multivariate risk measure R(X) with 0 R(0) R d is called coherent if (Monotonicity) For any X and Y, X Y component-wise implies that R(X) R(Y), (Subadditivity) For any X and Y, R(X + Y) R(X) + R(Y), (Positive Homogeneity) For any X and positive s, R(sX) = sr(x), (Translation Invariance) For any X and any deterministic vector l, R(X + l) = R(X) + l. Here we use the component-wise ordering to simplify the discussion. In general, a partial ordering induced by a closed convex cone K R(0) R d + can be used to account for some frictions on the financial market such as transaction costs, liquidity problems, irreversible transfers, etc... Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 16 / 26

Scenario-Based Representation R(X) = {x R d : E Q (x X) 0, Q S}, where S is a closed convex set of probability measures that are absolutely continuous with respect to the probability measure P. If S = the set of probability measures conditioning on tail events, R(X) is the worst conditional expectation: WCE p (X) := {x R d : E(x X B) 0, B F, P(B) 1 p}. If X is continuous, then WCE p (X) equals the vector-valued tail conditional expectation: TCE p (X) = (E(X X A) + R d +), 0 < p < 1, A Q p(x) where Q p (X) = {A R d : A is upper, P(X A) 1 p}. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 17 / 26

Multivariate Regular Variation Consider random loss vectors on R d + = [0, ] d. The extreme value analysis of TCE TCE p (X) as p 1 boils down to analyzing tail behaviors of P(X tb) and E(X X tb) as t for various upper set B. A random vector X is said to have a multivariate regularly varying distribution F (MRV, Resnick, 2007) if there exists a Radon measure µ, called the intensity measure, on R d +\{0} such that lim t P(X tb) P( X > t) = µ(b), for any relatively compact set B R d +\{0} with µ( B) = 0, where denote a norm on R d. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 18 / 26

Homogeneity in the Tails For simplicity, we assume the margins are tail equivalent: P(X i > t) 1, as t. P(X 1 > t) Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 19 / 26

Homogeneity in the Tails For simplicity, we assume the margins are tail equivalent: P(X i > t) 1, as t. P(X 1 > t) Note that µ(sb) = s α µ(b) for any s > 0 and any subset B that is bounded away from the origin. We assume that tail index α > 1. Any margin of F is regularly varying with tail index α. In fact, for any s > 0 lim t P(X i > ts) = µ((s, ] Rd 1 + ) = s α µ((1, ] R d 1 + ) P( X > t) Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 19 / 26

Homogeneity in the Tails For simplicity, we assume the margins are tail equivalent: P(X i > t) 1, as t. P(X 1 > t) Note that µ(sb) = s α µ(b) for any s > 0 and any subset B that is bounded away from the origin. We assume that tail index α > 1. Any margin of F is regularly varying with tail index α. In fact, for any s > 0 lim t P(X i > ts) = µ((s, ] Rd 1 + ) = s α µ((1, ] R d 1 + ) P( X > t) Example: Multivariate t distribution, multivariate Pareto distributions, certain elliptical distributions,... Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 19 / 26

Tail Estimates Using MRV MRV Rewrite: For any subset B bounded away from the origin, µ(b) P(X tb) µ((1, ] R d 1 + ) P(X i > t), as t. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 20 / 26

Tail Estimates Using MRV MRV Rewrite: For any subset B bounded away from the origin, µ(b) P(X tb) µ((1, ] R d 1 + ) P(X i > t), as t. That is, multivariate heavy-tails can be approximated proportionally by univariate heavy-tails where the proportionality constant µ(b)/µ((1, ] R d 1 + ) encodes the extremal dependence information of multivariate extremes. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 20 / 26

Tail Estimates Using MRV MRV Rewrite: For any subset B bounded away from the origin, µ(b) P(X tb) µ((1, ] R d 1 + ) P(X i > t), as t. That is, multivariate heavy-tails can be approximated proportionally by univariate heavy-tails where the proportionality constant µ(b)/µ((1, ] R d 1 + ) encodes the extremal dependence information of multivariate extremes. In particular, the tail of the -aggregation of X is given by P( X > t) µ({x : x > 1}) µ((1, ] R d 1 + ) P(X i > t), as t. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 20 / 26

Tail Estimates of VaR of Sums Once again, assume the margins are tail equivalent. Take the l 1 -norm, x 1 := d i=1 x i. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 21 / 26

Tail Estimates of VaR of Sums Once again, assume the margins are tail equivalent. Take the l 1 -norm, x 1 := d i=1 x i. VaR for the tail risk of portfolio aggregations can be approximated proportionally by marginal VaRs: ( d ) µ({x : d VaR p X i i=1 x i > 1}) i=1 µ((1, ] R d 1 VaR p (X i ), as p 1. + ) Aggregated dependent losses under Archimedean copula structure: Wüthrich (2003), and Alink, Löwe & Wüthrich (2004). Under general copula structure: Albrecher, Asmussen & Kortschak (2006); Barbe, Fougères & Genest (2006); Alink, Löwe & Wüthrich (2007). Under general MRV: Kortschak & Albrecher (2009); Li & Wu (2012). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 21 / 26

Asymptotics of Vector-Valued TCEs Let X be a non-negative loss vector that has an MRV df with intensity measure µ and tail index α > 1. Define A j (w) := {(x 1,..., x d ) R d : x j > w}, 1 j d. Tail Estimates of TCE (Joe & Li, 2010) Let B be an upper set bounded away from 0, then E(X j X tb) lim = t t 0 µ(a j (w) B) dw µ(b) Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 22 / 26

Asymptotics of Vector-Valued TCEs Let X be a non-negative loss vector that has an MRV df with intensity measure µ and tail index α > 1. Define: S d 1 + is the unit sphere and Bd 1 (0) is the unit open ball (w.r.t. the norm ). Q := {B R d : B is upper, B S d 1 +, B [Bd 1 (0)]c }. u j (B; µ) := 0 µ(a j (w) B) µ(b) dw. Tail Estimates of TCE (Joe & Li, 2010) As p 1, TCE p (X) ( VaR 1 (1 p)/µ(b) ( X ) (u1 (B; µ),..., u d (B; µ)) + R d). B Q Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 23 / 26

Elliptical Distributions Σ is a d d positive-semidefinite matrix. A is a d m matrix with AA = Σ. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 24 / 26

Elliptical Distributions Σ is a d d positive-semidefinite matrix. A is a d m matrix with AA = Σ. U = (U 1,..., U m ) is uniformly distributed on the unit sphere in R m. R > 0 is a random variable independent of U. Consider the stochastic representation X = (X 1,..., X d ) = RA(U 1,..., U m ), (T 1,..., T d ) := A(U 1,..., U m ). Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 24 / 26

Elliptical Distributions Σ is a d d positive-semidefinite matrix. A is a d m matrix with AA = Σ. U = (U 1,..., U m ) is uniformly distributed on the unit sphere in R m. R > 0 is a random variable independent of U. Consider the stochastic representation X = (X 1,..., X d ) = RA(U 1,..., U m ), (T 1,..., T d ) := A(U 1,..., U m ). If R has a regularly varying survival function with tail index α, then X is multivariate regularly varying. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 24 / 26

Y + := max{y, 0} for any random variable Y. Estimates of TCE for Elliptical Distributions Let S := d i=1 X i. If R has a regularly varying survival function with tail index α > 1, then for 1 i d, as p 1, E ( X i S > VaR p (S) ) α E α 1 ( T i ( d i=1 T i ) α 1 + E( d j=1 T j) α + ) VaR p ( S ), Landsman & Valdez (2003) obtained the exact formula of E ( X i S > VaR p (S) ) for X with joint elliptical densities. Zhu & Li (2012) showed that the difference between our tail estimate and the Landsman-Valdez formula is in order of O(10 3 ) (t distribution with ν = 2.1, n = 100, 000, p = 0.95 and various choices of covariances). In contrast, our tail estimates do not require existence of joint densities. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 25 / 26

Remarks Two ingredients: duality and homogeneity in the tails. These allow us to estimate the ratios of tail probabilities and their integrals for analyzing tail risk. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 26 / 26

Remarks Two ingredients: duality and homogeneity in the tails. These allow us to estimate the ratios of tail probabilities and their integrals for analyzing tail risk. Many structural properties (positive dependence, monotonicity, subadditivity, or stochastic order relation,...) emerge only from extremes. Heavy tail asymptotics of risk measures are often more tractable and the effects of emergent structural properties on tail risk can then be analyzed. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 26 / 26

Remarks Two ingredients: duality and homogeneity in the tails. These allow us to estimate the ratios of tail probabilities and their integrals for analyzing tail risk. Many structural properties (positive dependence, monotonicity, subadditivity, or stochastic order relation,...) emerge only from extremes. Heavy tail asymptotics of risk measures are often more tractable and the effects of emergent structural properties on tail risk can then be analyzed. For example, the tail estimate of VaR under Archimedean copula dependence was applied in Embrechts, Nešlehová & Wüthrich (2009) to study the sub- and superadditivity properties of VaR when the confidence level p is close to 1. Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin 26 / 26