Nonparametric estimation of extreme risks from heavy-tailed - PowerPoint PPT Presentation

Nonparametric estimation of extreme risks from heavy-tailed distributions St´ ephane GIRARD joint work with Jonathan EL METHNI & Laurent GARDES September 2013

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Extreme risk measures 1 Estimators and asymptotic results 2 Extrapolation 3 Application 4 2 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Some risk measures Let Y ∈ R be a random loss variable. The Value-at-Risk of level α ∈ (0 , 1) is the α -quantile defined by ← ( α ) = inf { t , F ( t ) ≤ α } , VaR ( α ) := F ← ( . ) is the generalized inverse of the survival function of Y . where F 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 CVaR λ ( α ) := λ VaR ( α ) + (1 − λ ) CTE ( α ) , with 0 ≤ λ ≤ 1. 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 )) . 3 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application A new risk measure : the Conditional Tail Moment 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 ) , λ VaR ( α ) + (1 − λ ) CTM 1 ( α ) , CVaR ( α ) = CTM 2 ( α ) − CTM 2 CTV ( α ) = 1 ( α ) . = ⇒ All the risk measures depend on the VaR and the CTM a . 4 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Extreme losses and regression case Our second aim is to estimate these risk measures in case of extreme losses and to 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 n →∞ 0. → 1 Denoting by F ( . | x ) the conditional survival distribution function of Y 2 given X = x , the Regression Value-at Risk is defined by : ← ( α n | x ) = inf { t , F ( t | x ) ≤ α n } , RVaR ( α n | x ) := F 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. 5 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Extreme regression risk measures This yields the following risk measures : RCTE ( α n | x ) RCTM 1 ( α n | x ) , = RCVaR λ ( α n | x ) = λ RVaR ( α n | x ) + (1 − λ ) RCTM 1 ( α n | x ) , RCTM 2 ( α n | x ) − RCTM 2 RCTV n ( α n | x ) = 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 ϕ ← RVaR ( α n | x ) = 0 ( α n | x ) , 1 α n ϕ a ( ϕ ← RCTM a ( α n | x ) = 0 ( α n | x ) | x ) . Goal : estimate ϕ a ( . | x ) and ϕ ← a ( . | x ). 6 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Inference Estimator of ϕ a ( . | x ) : We propose to use a classical kernel estimator given by , „ x − X i « „ x − X i « n n X X Y a ϕ a , n ( y | x ) = b i I { Y i > y } . K K h n h n i =1 i =1 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 ← a ( . | x ) : ϕ a , n ( . | x ) is a non-increasing function, an estimator of ϕ ← Since ˆ a ( α | x ) can be defined for α ∈ (0 , 1) by ϕ ← ˆ a , n ( α | x ) = inf { t , ˆ ϕ a , n ( t | x ) < α } . 7 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Heavy-tail 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 ( λ y | x ) = λ − 1 /γ ( x ) . lim y →∞ F ( y | 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 a γ ( 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 ( λ y | x ) lim ℓ a ( y | x ) = 1 . y →∞ 8 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Heavy-tail assumptions (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 „Z y « ε a ( u | x ) ℓ a ( y | x ) = c a ( x ) exp , du u 1 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 )). 9 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Regularity assumptions 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 ˛ ˛ ff ˛ ˛ ϕ a ( z | x ) ˛ ˛ ˛ , z ∈ [(1 − ξ ) y , (1 + ξ ) y ] and d ( x , x ′ ) ≤ h ω n ( y , ξ ) = sup ϕ a ( z | x ′ ) − 1 . ˛ 10 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Main result Theorem 1 : Suppose (F.1) , (F.2) and (L) hold. Let 0 ≤ a 1 < a 2 < · · · < a J , x ∈ R p such that g ( x ) > 0 and 0 < γ ( x ) < 1 / (2 a J ), α n → 0 and nh p α n → ∞ as n → ∞ , ξ > 0 such that √ nh p α n ( h ∨ ω n ( RVaR ( α n | x ) , ξ )) → 0, Then, 8 !9 � ! < = √ RCTM a j , n ( α n | x ) � RVaR n ( α n | x ) nh p α n RCTM a j ( α n | x ) − 1 RVaR ( α n | x ) − 1 , : ; j ∈{ 1 ,..., J } is asymptotically Gaussian, centered, with covariance matrix � K � 2 2 γ 2 ( x )Σ( x ) / g ( x ) where 0 1 a 1 B C . a i a j (2 − ( a i + a j ) γ ( x )) B . C . Σ( x ) = (1 − ( a i + a j ) γ ( x )) A . B C @ a J a 1 · · · a J 1 11 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Conditions on the sequences α n and h n nh p n α 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 . √ nh p α n ( h ∨ ω n ( RVaR ( α n | x ) , ξ )) → 0 : The biais induced by the smoothing is negligible compared to the standard-deviation. 12 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application Consequences Suppose the assumptions of Theorem 1 hold. Then, if 0 < γ ( x ) < 1 / 2, ! „ « √ � 0 , 2(1 − γ ( x )) γ 2 ( x ) � K � 2 RCTE n ( α n | x ) d 2 nh p α n RCTE ( α n | x ) − 1 − → N 1 − 2 γ ( x ) g ( x ) ! „ « 0 , γ 2 ( x )( λ 2 + 2 − 2 λ − 2 γ ( x )) √ � � K � 2 RCVaR λ, n ( α n | x ) d 2 nh p α n RCVaR λ ( α n | x ) − 1 − → N 1 − 2 γ ( 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, ! „ « √ � � K � 2 RCTV n ( α n | x ) d 2 nh p α n RCTV ( α n | x ) − 1 − → N 0 , V γ ( x ) , g ( x ) where V γ ( x ) = 8(1 − γ ( x ))(1 − 2 γ ( x ))(1 + 2 γ ( x ) + 3 γ 2 ( x )) . (1 − 3 γ ( x ))(1 − 4 γ ( x )) 13 / 26

Outline Extreme risk measures Estimators and asymptotic results Extrapolation Application A Weissman type estimator In Theorem 1, the condition nh p α 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. Definition 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 „ α n « a ˆ γ n ( x ) W � a , n ( β n | x ) = � RCTM a , n ( α n | x ) RCTM β n | {z } extrapolation 14 / 26