Nonparametric estimation of extreme risks from heavy-tailed distributions

Nonparametric estimation of extreme risks from heavy-tailed distributions Laurent GARDES joint work with Jonathan EL METHNI & Stéphane GIRARD December 2013

1 Introduction to risk measures 2 Introduction to Extreme Value Analysis 3 Estimation of risk measures 4 Extrapolation 5 Application 2 / 41

Plan 1 Introduction to risk measures 2 Introduction to Extreme Value Analysis 3 Estimation of risk measures 4 Extrapolation 5 Application 3 / 41

The Value-At-Risk Let Y R be a random loss variable. For example, Y can represent the loss of a portfolio during some fixed interval time. A risk measure is a real-valued function R(.) quantifying the risk associated to a given loss variable. The most known risk measure is the Value-at-Risk of level α (0, 1) is the α-quantile defined by R(Y ) = VaR α(y ) := F (α) = inf{t, F (t) α}, where F (.) is the generalized inverse of the survival function of Y. 4 / 41

Illustration of the Value-At-Risk 5 / 41

Drawback of the Value-At-Risk Let Y 1 and Y 2 be two loss variables with survival functions F 1(.) and F 2(.) represented below : Full line : F 1(.), dashed line : F 2(.). For α 0.22, VaR α(y 1) = VaR α(y 2). Clearly, the loss Y 1 is more dangerous than Y 2. 6 / 41

Other risk measures To overcome this drawback, other risk measures have been proposed : The Conditional Tail Expectation of level α (0, 1) defined by CTE α(y ) := E(Y Y > VaR α(y )). The Conditional-Value-at-Risk of level α (0, 1) introduced by Rockafellar et Uryasev [2000] defined by with 0 λ 1. CVaR λ,α (Y ) := λvar α(y ) + (1 λ)cte α(y ), The Conditional Tail Variance of level α (0, 1) introduced by Valdez [2005] is defined by CTV α(y ) := E{(Y CTE α(y )) 2 Y > VaR α(y )}. 7 / 41

First contribution : definition of 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(y ) := E(Y a Y > VaR α(y )), 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(y ) = CTM 1,α(Y ), CVaR λ,α (Y ) = λvar α(y ) + (1 λ)ctm 1,α(Y ), CTV α(y ) = CTM 2,α(Y ) CTM 2 1,α(Y ). = All the previous risk measures depend on the VaR and the CTM. 8 / 41

Second contribution : Risk measure for extreme losses in a regression case Generalization of risk measures : 1 For extreme losses : the fixed level α (0, 1) is replaced by a sequence α n 0. n 2 Regression case : a covariate X R p is recorded simultaneously with Y. Denoting by F (. x) the conditional survival distribution function of Y given X = x, the Regression Value-at Risk is defined by : RVaR αn (Y x) := F (α n x) = inf{t, F (t x) α n}, and the Regression Conditional Tail Moment of order a is defined by : RCTM a,αn (Y x) := E(Y a Y > RVaR αn (Y x), X = x), where a > 0 is such that the moment of order a of Y exists. 9 / 41

Second contribution : Risk measure for extreme losses in a regression case This yields the following risk measures : RCTE αn (Y x) = RCTM 1,αn (Y x), RCVaR λ,αn (Y x) = λrvar αn (Y x) + (1 λ)rctm 1,αn (Y x), RCTV αn (Y x) = RCTM 2,αn (Y x) RCTM 2 1,α n (Y x). All the risk measures depend on the RVaR and the RCTM. To deal with extreme losses (i.e. when α n 0), we need to use Extreme Value Analysis results. 10 / 41

Plan 1 Introduction to risk measures 2 Introduction to Extreme Value Analysis 3 Estimation of risk measures 4 Extrapolation 5 Application 11 / 41

Extreme Value Analysis Let Z 1,..., Z n be n independent copies of a random value Z R with survival function F (.). We denote by Z 1,n... Z n,n the ordered statistics. The main goal of Extreme Value Analysis is to estimate extreme quantiles defined by : F (α n) = VaR αn (Z), for α n 0. The simplest way to estimate quantile consists in using the empirical survival function ˆ F n(z) := 1 n I{Z i > z}. n i=1 Clearly, this is not a good idea since when α n < 1/n, ˆ F n (α n) = Z n,n. 12 / 41

Extreme Value Analysis We thus need some information beyond the maximum observation. This information is given by the following result. Fisher-Tippett theorem If there exist two sequences (a n) > 0 and (b n) and a non-degenerate distribution function H(.) such that ( ) Zn,n b n P z H(z), as n goes to infinity, a n then { H(y) = where z + = max(z, 0). [ exp { 1 + γ z µ σ exp { exp ( z µ σ } 1/γ )} + ] if γ 0, if γ = 0, The distribution H(.) is called the Extreme Value Distribution (EVD). The shape parameter γ R is called the extreme value index. It controls the heaviness of the tail. 13 / 41

Extreme Value Analysis If γ > 0, F (z) decreases at a polynomial rate to zero as z goes to infinity. More precisely, F (z) = z 1/γ L(z), where L(.) is a slowly varying function satisfying for all t > 0 : L(tz) lim z L(z) = 1. Such distributions are known as heavy-tail distributions (Pareto, student, log-gamma, etc...) If γ < 0, F (z) decreases at a polynomial rate to zero as z goes to the right endpoint y F < (Uniform, Beta, etc...) If γ = 0, F (z) decreases at an exponential rate to zero as z goes to the right endpoint (Normal, exponential, gamma, etc...) 14 / 41

Plan 1 Introduction to risk measures 2 Introduction to Extreme Value Analysis 3 Estimation of risk measures 4 Extrapolation 5 Application 15 / 41

Recall... Recall that we are interested in the quantities : and RVaR αn (Y x) := F (α n x) = inf{t, F (t x) α n}, RCTM a,αn (Y x) := E(Y a Y > RVaR αn (Y x), X = x) = 1 E (Y a I{Y > RVaR αn (Y x)} X = x), α n where a > 0 is such that the moment of order a of Y exists. 16 / 41

Conditional moment We introduce the conditional moment of order a 0 of Y given X = x 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 αn (Y x) = ϕ 0 (α n x), RCTM a,αn (Y x) = 1 ϕ a(ϕ 0 (α n x) x). α n Goal : estimate ϕ a(. x) and ϕ a (. x). 17 / 41

Inference Estimator of ϕ a(. x) : We propose to use a classical kernel estimator given by n ( ) / n ( x Xi ϕ a,n(y x) = K Yi a x Xi I{Y i > y} K i=1 h n 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) : i=1 h n ). Since ˆϕ a,n(. x) is a non-increasing function, an estimator of ϕ a defined for α (0, 1) by (α x) can be ˆϕ a,n(α x) = inf{t, ˆϕ a,n(t x) < α}. 18 / 41

Inference The Regression Value-At-Risk of order α n is thus estimated by : RVaR αn (Y x) = ϕ 0 (α n x), and the Regression Conditional Tail Moment of order a is estimated by : RCTM a,αn (Y x) = 1 α n ϕ a,n ( ϕ 0 (α n x) x) We thus can now estimate all the above mentioned risk measures. To do... Find the asymptotic joint distribution of {( ) } RCTM aj,α n (Y x), j = 1,..., J, RVaR αn (Y x), where 0 a 1 < a 2 <... < a J. 19 / 41

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 (y x) = y 1/γ(x) L(y x), where for x fixed, L(. x) is a slowly varying function. In this context, γ(.) is a positive function of the covariate x and is referred to as the conditional tail index. Condition (F.1) also implies that for a [0, 1/γ(x)) and all α (0, 1), RCTM a,α(y x) exists and, RCTM a,α(y x) = α aγ(x) l a(α 1 x), where for x fixed, l a(. x) is a slowly-varying function 20 / 41

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 ( y ) ε a(u x) l a(y x) = c a(x) exp du, u where c a(.) is a positive function and ε a(y x) 0 as y. 1 (F.3) ε a(. x) is continuous and ultimately non-increasing for all a [0, 1/γ(x)). 21 / 41

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 { } ϕ a(z x) ω n(y, ξ) = sup ϕ a(z x ) 1, z [(1 ξ)y, (1 + ξ)y] and d(x, x ) h. 22 / 41

Main result 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 nh p nα n as n, ξ > 0 such that nh p nα n {h n ω n(rvar(α n x), ξ)} 0, ) RCTM aj,α n (Y x) nhp α n ( RCTM aj,α n (Y x) 1 j {1,...,J} is asymptotically Gaussian, centered, with covariance matrix K 2 2γ 2 (x)σ(x)/g(x) where 23 / 41 Σ(x) = a i a j (2 (a i +a j )γ(x)) (1 (a i +a j )γ(x)), ( ) RVaR αn (Y x) RVaR α(y x) 1 a 1. a J a 1 a J 1.

Conditions on the sequences α n and h n nh 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 αn (Y x), + ) of R p R. nh p n α n {h n ω n(rvar αn (Y x), ξ)} 0 : The biais induced by the smoothing is negligible compared to the standard-deviation. 24 / 41

Consequences Suppose the assumptions of Theorem 1 hold. Then, if 0 < γ(x) < 1/2, ( ) ( RCTE αn (Y x) nhp α n RCTE αn (Y x) 1 d N 0, 2(1 ) γ(x))γ2 (x) K 2 2 1 2γ(x) g(x) ( ) ( RCVaR λ,αn (Y x) nhp α n RCVaR λ,αn (Y x) 1 d N 0, γ2 (x)(λ 2 + 2 2λ 2γ(x)) 1 2γ(x) The RCTV αn (Y x) estimator involves the computation of a second order moment, it requires the stronger condition 0 < γ(x) < 1/4, ( ) ( ) RCTV αn (Y x) nhp α n RCTV αn (Y x) 1 d K 2 2 N 0, V γ(x), g(x) ) K 2 2 g(x) where V γ(x) = 8(1 γ(x))(1 2γ(x))(1 + 2γ(x) + 3γ2 (x)). (1 3γ(x))(1 4γ(x)) 25 / 41

Plan 1 Introduction to risk measures 2 Introduction to Extreme Value Analysis 3 Estimation of risk measures 4 Extrapolation 5 Application 26 / 41

A Weissman type estimator In Theorem 1, the condition nh p nα n provides a lower bound on the level of the risk measure to estimate. In consequence, α n must be an order of an extreme quantile within the sample. 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). 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 ( ) aˆγn(x) αn RCTM W a,β n (Y x) = RCTM a,αn (Y x) β n }{{} extrapolation 27 / 41

Idea Recall that RCTM a,αn (Y x) = α aγ(x) n l a(α 1 x). n Thus, RCTM a,βn (Y x) RCTM a,αn (Y x) = ( αn β n ) aγ(x) l a(βn 1 x) l a(αn 1 x). Since l a(. x) is a slowly varying function, RCTM a,βn (Y x) RCTM a,αn (Y x) ( αn β n ) aγ(x). Replacing RCTM a,αn (Y x) and γ(x) by their estimators lead to the Weissmann estimator. 28 / 41

Asymptotic result 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 ( ) nh p n α n(ˆγ n(x) γ(x)) d N 0, v 2 (x), 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 nh p n α n RCTM W a,β n (Y x) ( log(α n/β n) RCTM a,βn (Y 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. 29 / 41

Estimation of the conditional tail index Without covariate : Hill [1975] Let Z 1,n Z n,n be the order statistics associated with i.i.d. heavy-tailed random variables Z 1,..., Z n and (α n) n 1 (0, 1). The Hill estimator is given by ˆγ (H) n, nα = 1 nα n n log Z n j+1,n log Z n nαn,n. nα n j=1 With a covariate : Z n j+1,n RVaR τj α n (Y x) A kernel version of the Hill estimator is given by / J J ˆγ n,αn (x) = (log RVaR τj α n (Y x) log RVaR τ1 α n (Y x)) log(τ 1/τ j ), j=1 j=1 where J 1 and (τ j ) j 1 is a decreasing sequence of weights. 30 / 41

Extrapolation The asymptotic normality of ˆγ n,αn (x) has been established by Daouia et al. [2011]. Suppose the assumptions of Theorem 1 hold together with (F.3), ( ) nh p n α n(ˆγ n,αn (x) γ(x)) d N 0, K 2 2γ 2 (x)v J /g(x), with V J = J j=1 ( ) /( 2(J j) + 1 J J 2 log(1/τ j ) τ j j=1 ) 2 In particular, if β n 0 and β n/α n 0, we obtain nh p n α n RCTE W β n (Y x) ( ) RCTE βn (Y x) 1 d N 0, K 2 2γ 2 (x)v J /g(x). 31 / 41

Plan 1 Introduction to risk measures 2 Introduction to Extreme Value Analysis 3 Estimation of risk measures 4 Extrapolation 5 Application 32 / 41

Daily rainfalls in the Cévennes-Vivarais region The Cévennes-Vivarais region raingauge stations Daily rainfalls (in mm) measured at N = 523 stations from 1958 to 2000 33 / 41

Objective The loss variable Y is the daily rainfall (in mm). The random value Y is observed with a covariate X = (latitude,longitude,altitude,etc...). Goal : for all x, estimate the following risk measures : RVaR β (Y x) with β = 1/(365.25 T ) T -year return level in hydrology. RCTE β (Y x) with β = 1/(365.25 T ) average rainfall over the T -year return level in hydrology. We consider the case T = 100 years (extrapolation needed). 34 / 41

A cross validation procedure to choose h n and α n : Step 1 Double loop on H = {h i ; i = 1,..., M} and on A = {α j ; j = 1,..., R}. Loop on all raingauge stations {x t; t = 1,..., N}. 35 / 41

A cross validation procedure to choose h n and α n : Step 1 Double loop on H = {h i ; i = 1,..., M} and on A = {α j ; j = 1,..., R}. Loop on all raingauge stations {x t; t = 1,..., N}. Consider one raingauge station x t 35 / 41

A cross validation procedure to choose h n and α n : Step 1 Double loop on H = {h i ; i = 1,..., M} and on A = {α j ; j = 1,..., R}. Loop on all raingauge stations {x t; t = 1,..., N}. Remove all other raingauge stations 35 / 41

A cross validation procedure to choose h n and α n : Step 1 Double loop on H = {h i ; i = 1,..., M} and on A = {α j ; j = 1,..., R}. Loop on all raingauge stations {x t; t = 1,..., N}. Remove all other raingauge stations Estimate γ > 0 using the classical Hill estimator. It only depends on α j. 35 / 41

A cross validation procedure to choose h n and α n : Step 1 Double loop on H = {h i ; i = 1,..., M} and on A = {α j ; j = 1,..., R}. Loop on all raingauge stations {x t; t = 1,..., N}. Remove all other raingauge stations Estimate γ > 0 using the classical Hill estimator. It only depends on α j. = We obtain ˆγ (H) n,t,α j 35 / 41

A cross validation procedure to choose h n and α n : Step 2 Remove the station x t 36 / 41

A cross validation procedure to choose h n and α n : Step 2 Work in B(x t, h i ) \ {x t} 36 / 41

A cross validation procedure to choose h n and α n : Step 2 Work in B(x t, h i ) \ {x t} Estimate γ(x) > 0 using the kernel version of the Hill estimator. It depends on α j and on h i. 36 / 41

A cross validation procedure to choose h n and α n : Step 2 Work in B(x t, h i ) \ {x t} Estimate γ(x) > 0 using the kernel version of the Hill estimator. It depends on α j and on h i. = We obtain ˆγ n,hi,α j (x t) 36 / 41

A cross validation procedure to choose h n and α n : Step 2 Work in B(x t, h i ) \ {x t} Estimate γ(x) > 0 using the kernel version of the Hill estimator. It depends on α j and on h i. = We obtain ˆγ n,hi,α j (x t) (h emp, α emp) = arg min median{(ˆγ (H) n,t,α j ˆγ n,hi,α j (x t)) 2, t {1,..., N}}. (h i,α j ) H A 36 / 41

Computation of RVaR W and RCTE W 523 Stations Regular grid : 200 200 Two dimensional covariate X =(latitude, longitude). Bi-quadratic kernel : K(x) (1 x 2 ) 2 I { x 1}. Harmonic sequence of weights : (τ j ) j {1,...,9} = 1/j. Results of the procedure (h emp, α emp) = (24, 1/(3 365.25)). 37 / 41

Estimated conditional tail index 38 / 41

RVaR W β (Y x) : 100-year return level 39 / 41

RCTE W β (Y x) above the 100-year return level 40 / 41

References Daouia, A., Gardes, L., Girard, S. and Lekina, A. (2011). Kernel estimators of extreme level curves, Test, 20, 311 333. Gardes, L. and Girard, S. (2008). A moving window approach for nonparametric estimation of the conditional tail index, Journal of Multivariate Analysis, 99, 2368 2388. Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution, The Annals of Statistics, 3, 1163 1174. Rockafellar, R.T. and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21 42. Valdez, E.A. (2005). Tail conditional variance for elliptically contoured distributions, Belgian Actuarial Bulletin, 5, 26 36. Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations, Journal of the American Statistical Association, 73, 812 815. 41 / 41