Nonparametric estimation of tail risk measures from heavy-tailed distributions

Transcription

1 Nonparametric estimation of tail risk measures from heavy-tailed distributions Jonthan El Methni, Laurent Gardes & Stéphane Girard 1 Tail risk measures Let Y R be a real random loss variable. The Value-at-Risk of level α (0, 1) denoted by VaR(α) is the α-quantile of the survival function F (x) = P (Y > x) VaR(α) := F 1 (α) = inf{t, F (t) α} Fonction de survie α VaR(α) Consider Y 1 and Y 2 two random loss variables with associated survival functions F 1 and F 2. 1

2 Fonction de survie α VaR(α) Random variables with light tail probabilities and with heavy tail probabilities may have the same VaR(α). This is one of the main criticisms against the VaR, Embrechts et al. [1997]. The Conditional Tail Expectation of level α (0, 1) denoted by CTE(α) is defined by CTE(α) := E(Y Y > VaR(α)). Fonction de survie α VaR(α) CTE(α) The CTE takes into account the whole information contained in the upper part of the tail distribution. 2

3 Let Y R be a real random loss variable. The Value-at-Risk of level α (0, 1) is the α-quantile defined by VaR(α) := F 1 (α), where F 1 is the (generalized) inverse of the survival function of Y. The Conditional Tail Expectation of level α (0, 1) is defined by CTE(α) := E(Y Y > VaR(α)). The Conditional-Value-at-Risk of level α (0, 1) introduced by Rockafellar et Uryasev [2000] is defined by with 0 λ 1. CVaR λ (α) := λvar(α) + (1 λ)cte(α), The Conditional Tail Variance of level α (0, 1) introduced by Valdez [2005] is defined by CTV(α) := E((Y CTE(α)) 2 Y > VaR(α)). The first goal of this work is to unify the definitions of the previous risk measures. To this end, the Conditional Tail Moment of level α (0, 1) is introduced: CTM a (α) := E(Y a Y > VaR(α)), where a 0 is such that the moment of order a of Y exists. All the previous risk measures of level α can be rewritten as CTE(α) = CTM 1 (α), CVaR(α) = λvar(α) + (1 λ)ctm 1 (α), CTV(α) = CTM 2 (α) CTM 2 1(α). All the risk measures depend on the VaR and the CTM a. Our second aim is to estimate these risk measures in case of extreme losses and in the case where a covariate X R p is recorded simultaneously with Y. The fixed level α (0, 1) is replaced by a sequence α n 0. n Denoting by F (. x) the conditional survival distribution function of Y given X = x, the Regression Value-at Risk is defined by: RVaR(α n x) := F 1 (α n x) = inf{t, F (t x) α n }, 3

4 and the Regression Conditional Tail Moment of order a is defined by: RCTM a (α n x) := E(Y a Y > RVaR(α n x), X = x), where a > 0 is such that the moment of order a of Y exists.this yields the following risk measures: RCTE(α n x) = RCTM 1 (α n x), RCVaR λ (α n x) = λrvar(α n x) + (1 λ)rctm 1 (α n x), RCTV n (α n x) = RCTM 2 (α n x) RCTM 2 1(α n x). All the risk measures depend on the RVaR and the RCTM a. The conditional moment of order a 0 of Y given X = x is defined by ϕ a (y x) = E (Y a I{Y > y} X = x), where I{.} is the indicator function. Since ϕ 0 (y x) = F (y x), it follows RVaR(α n x) = ϕ 1 0 (α n x), RCTM a (α n x) = 1 α n ϕ a (ϕ 1 0 (α n x) x). Goal: estimate ϕ a (. x) and ϕ 1 a (. x). 2 Estimators and asymptotic results Estimator of ϕ a (. x) : given by n ( ) / n x Xi ϕ a,n (y x) = K Yi a I{Y i > y} K i=1 We propose to use a classical kernel estimator h n i=1 ( x Xi h n is a sequence called the window-width such that h n 0 as n, K is a bounded density on R p with support included in the unit ball of R p. Estimator of ϕ 1 a (. x) : Since ˆϕ a,n (. x) is a non-increasing function, an estimator of ϕ 1 a (α x) can be defined for α (0, 1) by ˆϕ 1 a,n(α x) = inf{t, ˆϕ a,n (t x) < α}. h n ). 4

5 Assumptions (F.1) The conditional survival distribution function of Y given X = x is assumed to be heavy-tailed i.e. for all λ > 0, F (λy x) lim y F (y x) = λ 1/γ(x). In this context, γ(.) is a positive function of the covariate x and is referred to as the conditional tail index since it tunes the tail heaviness of the conditional distribution of Y given X = x. Condition (F.1) also implies that for a [0, 1/γ(x)), RCTM a (. x) exists, and for all y > 0, RCTM a (1/y x) = y aγ(x) l a (y x), where for x fixed, l a (. x) is a slowly-varying function i.e. for all λ > 0, l a (λy x) lim y l a (y x) = 1. (F.2) l a (. x) is normalized for all a [0, 1/γ(x)). In such a case, the Karamata representation of the slowly-varying function can be written as ( y ) ε a (u x) l a (y x) = c a (x) exp du, 1 u where c a (.) is a positive function and ε a (y x) 0 as y. (F.3) ε a (. x) is continuous and ultimately non-increasing for all a [0, 1/γ(x)). A Lipschitz condition on the probability density function g of X is also required: (L) There exists a constant c g > 0 such that g(x) g(x ) c g d(x, x ). where d(x, x ) is the Euclidean distance between x and x. Finally, for y > 0 and ξ > 0, the largest oscillation of the conditional moment of order a [0, 1/γ(x)) is defined by { } ϕ a (z x) ω(x, y, a, ξ, h n ) = sup ϕ a (z x ) 1, z [(1 ξ)y, (1 + ξ)y], x B(x, h n ), where B(x, h n ) denotes the ball centred at x with radius h n. 5

6 Theorem 1 Suppose (F.1), (F.2) and (L) hold. Let Then, 0 a 1 < a 2 < < a J, x R p such that g(x) > 0 and 0 < γ(x) < 1/(2a J ), α n 0 and p nα n as n, ξ > 0 such that p nα n (h n max a ω(x, RVaR(α n x), a, ξ, h n )) 0, ) p RCTM aj,n(α n x) n α n ( RCTM aj (α n x) 1 j {1,...,J}, ( ) RVaR n (α n x) RVaR(α n x) 1 is asymptotically Gaussian, centered, with covariance matrix K 2 2 γ2 (x)σ(x)/g(x) where Σ(x) = a i a j (2 (a i +a j )γ(x)) (1 (a i +a j )γ(x)) a 1 a J a 1 a J 1.. Conditions on the sequences α n and h n. p nα n : Necessary and sufficient condition for the almost sure presence of at least one point in the region B(x, h n ) [RVaR(α n x), + ) of R p R. p n α n (h n max a ω(x, RVaR(α n x), a, ξ, h n )) 0: The biais induced by the smoothing is negligible compared to the standard-deviation. 6

7 Corollary 1 Suppose the assumptions of Theorem 1 hold. Then, if 0 < γ(x) < 1/2, ( ) ( p RCTE n (α n x) n α n RCTE(α n x) 1 d N 0, 2(1 γ(x))γ2 (x) K 2 ) 2 1 2γ(x) g(x) ( ) p ( RCVaR λ,n (α n x) n α n RCVaR λ (α n x) 1 d N 0, γ2 (x)(λ λ 2γ(x)) K 2 ) 2 1 2γ(x) g(x) The RCTV(α n x) estimator involves the computation of a second order moment, it requires the stronger condition 0 < γ(x) < 1/4, ( ) ( p RCTV n (α n x) n α n RCTV(α n x) 1 d K 2 ) 2 N 0, V γ(x), g(x) where V γ(x) = 8(1 γ(x))(1 2γ(x))(1 + 2γ(x) + 3γ2 (x)). (1 3γ(x))(1 4γ(x)) 3 Illustration on a simulated dataset A sample {(X i, Y i ), i = 1,..., n} of size n = 1000 is generated. The covariate X is uniform on [0, 1]. The conditional distribution of Y X = x is chosen in the Hall class : F (y x) = y 1/γ(x) a(1 + by ρ/γ(x) ) }{{} l(y x) with a = 1/2, b = 1, ρ = 1 and conditional tail index function x [0, 1] γ(x) = 1 ( ) ( ( sin(πx) ( 2 exp 64 x 1 ) )) 2. 2 We have chosen a bi-quadratic kernel K(u) (1 u 2 ) 2 I { u 1} with smoothing parameter h n = 0.1. Our goal if to estimate RCTE(α n x) and RVaR(α n x) with α n =

8 Theoretical RCTE(α n x) and RVaR(α n x) with a logarithmic scale. Theoretical and estimated RVaR(α n x) with a logarithmic scale. 8

9 Theoretical and estimated RCTE(α n x) with a logarithmic scale. Theoretical and estimated RCTE(α n x) with a logarithmic scale. 9

10 Theoretical and estimated RCTE(α n x) with a logarithmic scale. 4 Extrapolation: A Weissman type estimator In Theorem 1, the condition p nα n provides a lower bound on the level of the risk measure to estimate. This restriction is a consequence of the use of a kernel estimator which cannot extrapolate beyond the maximum observation in the ball B(x, h n ). In consequence, α n must be an order of an extreme quantile within the sample. Let us consider (α n ) n 1 and (β n ) n 1 two positive sequences such that α n 0, β n 0 and 0 < β n < α n. A kernel adaptation of Weissman s estimator [1978] is given by ( ) aˆγn(x) αn RCTM W a,n(β n x) = RCTM a,n (α n x) β n }{{} extrapolation Theorem 2 Suppose the assumptions of Theorem 1 hold together with (F.3). Let ˆγ n (x) be an estimator of the conditional tail index such that p n α n (ˆγ n (x) γ(x)) d N ( 0, v 2 (x) ), 10

11 with v(x) > 0. If, moreover (β n ) n 1 is a positive sequence such that β n 0 and β n /α n 0 as n, then p n α n RCTM W a,n(β n x) log(α n /β n ) RCTM a (β n x) 1 d N ( 0, (av(x)) 2). The condition β n /α n 0 allows us to extrapolate and choose a level β n arbitrarily small. Estimation of the conditional tail index Without covariate : Hill [1975]. Let (k n ) n 1 be a sequence of integers such that k n {1... n}. The Hill estimator is given by ˆγ n,αn = 1 k n 1 log Z n i+1,n log Z n kn+1,n, k n 1 i=1 where Z 1,n Z n,n are the order statistics associated with i.i.d. random variables Z 1,..., Z n and α n = k n /n. With a covariate : A kernel version of the Hill estimator is given by / J J ˆγ n,αn (x) = (log RVaR n (τ j α n x) log RVaR n (τ 1 α n x)) log(τ 1 /τ j ), j=1 where J 1 and (τ j ) j 1 is a decreasing sequence of weights. j=1 The asymptotic normality of ˆγ n,αn (x) has been established by Daouia et al. [2011]. As a consequence, replacing RVaR n by RVaR W n and RCTM a,n by RCTM W a,n provides (asymptotically Gaussian) estimators for all the risk measures considered here, and for arbitrarily small levels. In particular, since RCTE(α n x) = RCTM 1 (α n x), we obtain ( RCTE W )ˆγn(x) αn n (β n x) = RCTE n (α n x). β n 11

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