Estimation de mesures de risques à partir des L p -quantiles

1/ 42 Estimation de mesures de risques à partir des L p -quantiles extrêmes Stéphane GIRARD (Inria Grenoble Rhône-Alpes) collaboration avec Abdelaati DAOUIA (Toulouse School of Economics), & Gilles STUPFLER (University of Nottingham) Mai 2017

2/ 42 Outline Risk measures Tail risk estimation based on extreme L p quantiles Estimation Intermediate levels Extreme levels Additional remarks Illustration on simulations Illustration on real data Conclusion

3/ 42 A simple way to assess the (environmental) risk is to compute a measure linked to the value X of the phenomenon of interest (rainfall height, wind speed, river flow, etc.): quantiles (= Value at Risk = return level), expectiles, tail conditional moments, spectral risk measures, distortion risk measures, etc. Here, we focus on the first two measures: quantiles and expectiles. We shall see how to estimate a simple extension of such measures.

4/ 42 Quantiles Let X be a random variable with cumulative distribution function F. Definition (Quantile) The quantile associated with X is the function q defined on (0, 1) by q(τ) = inf{t R F (t) τ}. In other words, the quantile q(τ) of level τ is the smallest real value exceeded by X with probability less than 1 τ. For the sake of simplicity, we assume X 0 and F is continuous and strictly increasing. Then, the quantile q(τ) is the unique real value such that F (q(τ)) = τ.

5/ 42 Estimation of extreme quantiles 1) Intermediate level. Let {X 1,..., X n } be a n sample from F. The empirical estimator of q(τ) is given by q n (τ) := X nτ,n where X 1,n X n,n are the order statistics. The asymptotic properties of this estimator are well-known when τ is fixed. Here, we focus on the asymptotic behaviour of this estimator when τ = τ n 1 as n. In such a case, q(τ) = q(τ n ) is an extreme quantile. In the situation where, additionally, n(1 τ n ), τ n is referred to as an intermediate level.

It can be shown that A is necessarily regularly varying with index ρ. The larger ρ is, the smaller the approximation error A is. 6/ 42 In the following, we assume that X has a right heavy tail. Assumption (First order condition C 1 (γ)) The survival function F := 1 F is regularly varying at + with index 1/γ < 0: F (tx) x > 0, lim t + F (t) = x 1/γ. The next condition controls the rate of convergence in C 1 (γ). Assumption (Second order condition C 2 (γ, ρ, A)) There exist γ > 0, ρ < 0 and a function A tending to zero at + with asymptotically constant sign such that: x > 0, [ ] 1 F (tx) lim t A(1/F (t)) F (t) x 1/γ = x 1/γ x ρ/γ 1. γρ

7/ 42 Under the second order condition, estimator q n (τ n ) is asymptotically Gaussian with (relative) rate of convergence n(1 τ n ): Theorem (Intermediate extreme quantiles, Theorem 2.4.1, de Haan & Ferreira, 2006) Suppose C 2 (γ, ρ, A) holds. If τ n 1 with n(1 τ n ) and n(1 τn )A(1/(1 τ n )) λ R, then ( ) qn (τ n ) n(1 τn ) q(τ n ) 1 d N (0, γ 2 ) as n.

2) Arbitrary level. Extreme quantiles of arbitrary large order τ n i.e. such that n(1 τ n) c < can be estimated thanks to Weissman s approximation deduced from condition C 1 (γ): ( ) 1 γ q(τ n) τn 1 τ n q(τ n ). One chooses τ n such that n(1 τ n ) as well as an estimator γ n of γ (Hill estimator for instance) to compute Weissman estimator (1978): ( 1 ) γn q n W (τ n τ τn n ) = 1 τ n q n (τ n ). q W n (τ n τ n ) inherits its asymptotic distribution from γ n with a slightly reduced rate of convergence. 8/ 42

9/ 42 Estimators of γ Hill estimator (1975), which is the MLE for Pareto distributions, γ τn = n(1 τ 1 n) log (X n i+1,n ) log ( ) X n(1 τ n ) n n(1 τn),n i=1 with τ n 1, n(1 τ n ) and a bias condition. Unbiased versions: Peng (1998), Fraga Alves et al. (2003), Caeiro & Gomes (2005),... Using a finite number of order statistics: Pickands (1975), Maximum Likelihood estimator: Smith (1987), Probability Weighted Moments estimator: Hosking & Wallis (1987), and much more!

10/ 42 Drawbacks of quantiles The quantile of level τ does not provide information on the extremes of X over q(τ). For instance, two distributions may have the same quantile of level 99% but different tail indices γ. Similarly, the estimator q n (τ n ) does not use the most extreme values of the sample. Loss of information. Our goal: Adapt the definition of quantiles to take into account the whole distribution tail.

11/ 42 Quantiles from an optimization point of view From Koenker & Bassett (1978), q(τ) = arg min E( τ 1l {X q} X q τ 1l {X 0} X ). q R The initial motivation was to estimate quantiles in a regression framework, thanks to a minimization problem. The second term E( τ 1l {X 0} X ) does not play any role in the minimization, but it ensures that the cost function exists even when E X =. In particular, the median is the best L 1 predictor of X : q(1/2) = arg min E X q E X. q R

12/ 42 Expectiles Let us assume that E X <. Definition (Expectiles, Newey & Powell, 1987) The expectile associated with X is the function ξ defined on (0, 1) by ξ(τ) = arg min E( τ 1l {X q} (X q) 2 τ 1l {X 0} X 2 ). q R To define the expectile, the check function η τ (x, 1) = τ 1l {x 0} x introduced by Koenker & Bassett (1978) is replaced in the optimization problem by the function η τ (x, 2) = τ 1l {x 0} x 2.

13/ 42 Comparison of cost functions Red: expectiles η τ (., 2), blue: quantiles η τ (., 1) with τ = 1/3.

14/ 42 The new cost function is continuously differentiable, the associated first order condition is: (1 τ)e( X ξ(τ) 1l {X ξ(τ)} ) = τe( X ξ(τ) 1l {X >ξ(τ)} ). In particular, ξ(1/2) = E(X ), and more generally: τ = E( X ξ(τ) 1l {X ξ(τ)}). E( X ξ(τ) ) An expectile is thus defined in terms of mean distance with respect to X, and not only in terms of frequency. Besides, the computation of an empirical expectile takes into account the whole tail information.

15/ 42 Estimation of extreme expectiles 1) Intermediate level. The empirical estimator of ξ(τ) is ξ n (τ) = arg min q R 1 n n η τ (X i q, 2). Under the first order condition, estimator ξ n (τ n ) is asymptotically Gaussian with (relative) rate of convergence n(1 τ n ): Theorem (Intermediate extreme expectiles, Daouia, Girard & Stupfler, 2016) Assume C 1 (γ) holds with 0 < γ < 1/2. If τ n 1 such as n(1 τ n ), then ( ) ( ξn (τ n ) n(1 τn ) ξ(τ n ) 1 d N 0, γ 2 2γ ). 1 2γ i=1

16/ 42 2) Arbitrary level. Proposition (Bellini et al. 2014 and Daouia, Girard & Stupfler, 2016) Assume C 1 (γ) holds with 0 < γ < 1. Then, ξ(τ) q(τ) (γ 1 1) γ as τ 1. Weissman s approximation thus still holds for expectiles: ( ) 1 γ ξ(τ n) τn 1 τ n ξ(τ n ). A Weissman type estimator can then be derived: ( 1 ) γn ξ n W (τ n τ τn n ) = ξn 1 τ n (τ n ) where n(1 τ n ) and n(1 τ n) c <.

17/ 42 Drawbacks of expectiles Existence of expectiles requires E X < which amounts to supposing γ < 1. In practice, to obtain reasonable estimations, even at the intermediate level, one needs γ < 1/2. Similar problems occur when dealing with Expected Shortfall (or Conditional Tail Expectation). Restricts the potential fields of applications.

18/ 42 L p quantiles: Basic idea Quantile of level τ: solution of the minimization problem q(τ) = arg min E( τ 1l {X q} X q τ 1l {X 0} X ). q R Expectile of level τ: when E X <, solution of the minimization problem ξ(τ) = arg min E( τ 1l {X q} X q 2 τ 1l {X 0} X 2 ). q R Definition (L p quantile, Chen 1996) Assume E X p 1 <. The L p -quantile associated with X is the function q(, p) defined on (0, 1) as q(τ, p) = arg min E( τ 1l {X q} X q p τ 1l {X 0} X p ). q R

19/ 42 Advantages and drawbacks When p > 1, the L p quantile q(τ, p) exists, is unique and verifies τ = E( X q(τ, p) p 1 1l {X q(τ,p)} ) E( X q(τ, p) p 1. ) It can thus be interpreted in terms of (pseudo-)distance to X in the space L p 1. The condition for the existence of L p quantiles is E X p 1 <. When 1 < p < 2, it is a weaker condition than the existence condition of expectiles. When p 2, the L p quantiles do not define a coherent risk measure (Bellini et al., 2014) since they are not subadditive.

20/ 42 Estimation of extreme L p quantiles 1) Intermediate levels. Introducing one has η τ (x, p) = τ 1l {x 0} x p, q(τ, p) = arg min E(η τ (X q, p) η τ (X, p)). q R The empirical estimator of q(τ, p) is obtained by minimizing the empirical counterpart of the previous criterion: q n (τ, p) = arg min q R 1 n n η τ (X i q, p). According to Geyer (1996), since the empirical criterion is convex, the asymptotic behaviour of the minimizer directly depends on the asymptotic behaviour of the criterion itself. i=1

21/ 42 Theorem (Intermediate extreme L p quantiles, Daouia, Girard & Stupfler, 2016) Let p > 1. Assume C 2 (γ, ρ, A) holds with 0 < γ < [2(p 1)] 1. If τ n 1 such that n(1 τ n ) and n(1 τ n )A(1/(1 τ n )) λ R then, ( ) qn (τ n, p) n(1 τn ) q(τ n, p) 1 d N ( 0, γ 2 V (γ, p) ) where V (γ, p) = Γ(2p 1)Γ(γ 1 2p + 2) Γ(p)Γ(γ 1 and Γ(x) is the Gamma p + 1) function.

22/ 42 Behaviour of variances γ (0, 1/2) V (γ, p) for some values of p [1, 2].

23/ 42 2) Arbitrary levels. In order to remove the condition on τ n, one has to show that Weissman s approximation is still valid for L p quantiles. Proposition (Daouia, Girard & Stupfler, 2016) Let p > 1. Assume C 1 (γ) with γ < 1/(p 1). Then, q(τ, p) lim = C(γ, p), τ 1 q(τ, 1) [ ] γ γ where C(γ, p) = B(p, γ 1 and B(x, y) is the Beta function. p + 1) Extreme L p quantiles are asymptotically proportional to usual extreme quantiles, for all p > 1. For p = 2, one has C(γ, 2) = (γ 1 1) γ, which coincides with the previous result on expectiles.

24/ 42 Behaviour of constants γ (0, 1/2) C(γ, p) for some values of p [1, 2].

25/ 42 Weissman s approximation is thus still valid for L p quantiles: ( ) 1 γ q(τ n, τn p) 1 τ n q(τ n, p). A Weissman type estimator can be derived for L p quantiles: ( 1 ) γn q n W (τ n τ τn n, p) = 1 τ n q n (τ n, p) where n(1 τ n ) and n(1 τ n) c <. Establishing the asymptotic behaviour of this estimator requires to investigate the error term in Weissman s approximation.

26/ 42 Theorem (Arbitrary extreme L p quantiles, Daouia, Girard & Stupfler, 2016) Suppose C 2 (γ, ρ, A) holds with γ < [2(p 1)] 1. Let τ n, τ n 1 such that n(1 τ n ), n(1 τ n) c < and ( ( )) 1 1 n(1 τn ) max q(τ n, 1), 1 τ n, A = O(1) 1 τ n d n(1 τn )( γ n γ) N. Then, ( n(1 τn ) q W n (τ ) n τ n, p) d log([1 τ n ]/[1 τ n]) q(τ n, 1 N. p)

27/ 42 3) Exploiting links between quantiles and L p quantiles First, remark that from the property q(τ, p) C(γ, p)q(τ, 1), one can build other estimators of extreme L p quantiles: at intermediate levels: q n (τ n, p) := C(ˆγ n, p) q n (τ n, 1) where ˆγ n is an estimator of the tail index and q n (τ n, 1) = X nτn,n. at arbitrary levels: q W n (τ n τ n, p) := C(ˆγ n, p) q W n (τ n τ n, 1) where q W n (τ n τ n, 1) is Weissman s estimator. Asymptotic normality results have been established under the condition γ < (p 1) 1 instead of γ < [2(p 1)] 1 for the previous theorem.

28/ 42 Second, recall that the L p quantile q(τ n, p) exists is unique and verifies τ n = E( X q(τ n, p) p 1 1l {X q(τn,p)}) E( X q(τ n, p) p 1. ) It is thus possible to find levels α n and τ n such that q(τ n, p) = q(α n, 1) by imposing τ n = E( X q(α n, 1) p 1 1l {X q(αn,1)}) E( X q(α n, 1) p 1. ) Asymptotically, as n, one can show that 1 τ n 1 α n 1 γ B (p, 1γ p + 1 ). Starting from the two equations in red, one can estimate extreme quantiles from extreme L p quantiles.

29/ 42 Letting τ n(p, α n ) := 1 (1 α n ) 1 γ n B ( p, 1 γ ) p + 1, n the extreme quantile q(α n, 1) can be estimated by q n ( τ n(p, α n ) τ n, p) where q n ( τ n, p) is an estimator of the extreme L p quantile q(, p) i.e. q n ( τ n, p) = q W n ( τ n, p) or q n ( τ n, p) = q W n ( τ n, p). Asymptotic normality results have been established for both estimators q W n ( τ n(p, α n ) τ n, p) and q W n ( τ n(p, α n ) τ n, p).

30/ 42 Illustration on simulations We focus on L p quantiles for p (1, 2) (alternative risk measure to expectiles with a weaker existence condition). The accuracy is assessed by computing the relative mean-squared error (MSE) on 3000 replications of samples of size n = 200 from Pareto F (x) = 1 x 1/γ, x > 1 and Fréchet distributions: F (x) = e x 1/γ, x > 0. 1) Intermediate level Which L p quantiles can be estimated accurately with q n (τ n, p)? We consider a L p quantile of level τ n = 0.9. In the following, γ {0.1, 0.15,..., 0.45} and p {1, 1.05,..., 2}.

Relative MSE - Pareto distribution - Intermediate level Pareto distribution (tau=0.90) 0.03 RMSE (in log scale) 0.02 variable gam=.45 gam=.40 gam=.35 gam=.30 gam=.25 gam=.20 gam=.15 gam=.10 0.01 1.00 1.25 1.50 1.75 2.00 p Horizontally: p, Vertically: relative MSE (in log scale) for different values of γ (small values: bottom curves, large values: top curves). 31/ 42

Relative MSE - Fréchet distribution - Intermediate level Fréchet distribution (tau=0.90) 0.03 RMSE (in log scale) 0.02 variable gam=.45 gam=.40 gam=.35 gam=.30 gam=.25 gam=.20 gam=.15 gam=.10 0.01 1.00 1.25 1.50 1.75 2.00 p Horizontally: p, Vertically: relative MSE (in log scale) for different values of γ (small values: bottom curves, large values: top curves). 32/ 42

33/ 42 First conclusions: The estimation accuracy is getting lower when γ increases. When γ 0.2, the estimation of expectiles (p = 2) is more difficult than the estimation of quantiles (p = 1). The value of p minimizing the relative MSE depends on the tail index γ. However, p [1.2, 1.4] seems to be a good compromise. 2) Extreme level Comparison of: q W n (τ n τ n, p) (based on empirical criterion + extrapolation), q W n (τ n τ n, p) (based on the extreme L 1 quantile). We consider a L p quantile of level τ n = 1 1/n. In the following, τ n = 1 k/n where k {2,..., n 1}, γ n is Hill s estimator, γ {0.1, 0.45} and p {1.2, 1.5, 1.8}.

34/ 42 Relative MSE - Fréchet distribution - Extreme level Fréchet : p=1.2, gamma=0.1 Fréchet : p=1.2, gamma=0.45 0.25 0.4 0.20 0.3 RMSE (in log scale) 0.15 0.10 variable hat tilde RMSE (in log scale) variable hat tilde 0.2 0.05 0.00 0.1 0 50 100 150 200 k 0 50 100 150 200 k Horizontally: k, Vertically: relative MSE (in log scale) of q W n (τ n τ n, p = 1.2) and q W n (τ n τ n, p = 1.2) as a function of k {2,..., n 1} (left: γ = 0.1, right: γ = 0.45).

35/ 42 Relative MSE - Fréchet distribution - Extreme level Fréchet : p=1.5, gamma=0.1 Fréchet : p=1.5, gamma=0.45 0.25 0.4 0.20 0.3 RMSE (in log scale) 0.15 0.10 variable hat tilde RMSE (in log scale) variable hat tilde 0.2 0.05 0.00 0.1 0 50 100 150 200 k 0 50 100 150 200 k Horizontally: k, Vertically: relative MSE (in log scale) of q W n (τ n τ n, p = 1.5) and q W n (τ n τ n, p = 1.5) as a function of k {2,..., n 1} (left: γ = 0.1, right: γ = 0.45).

36/ 42 Relative MSE - Fréchet distribution - Extreme level Fréchet : p=1.8, gamma=0.1 Fréchet : p=1.8, gamma=0.45 0.25 0.4 0.20 RMSE (in log scale) 0.15 0.10 variable hat tilde RMSE (in log scale) 0.3 variable hat tilde 0.2 0.05 0.00 0.1 0 50 100 150 200 k 0 50 100 150 200 k Horizontally: k, Vertically: relative MSE (in log scale) of q W n (τ n τ n, p = 1.8) and q W n (τ n τ n, p = 1.8) as a function of k {2,..., n 1} (left: γ = 0.1, right: γ = 0.45).

Illustration on real data S&P500 index from Jan, 4th, 1994 to Sep, 30th, 2016 (5727 trading days). 0.10 0.05 Logreturns 0.00-0.05-0.10 1995 2000 2005 2010 2015 Time To reduce the potential serial dependence, we use lower frequency data by choosing weekly (Wednesday to Wednesday) returns in the same sample period (Cai et al., 2015). This results in a sample {X 1,..., X 1176 } of size 1176. 37/ 42

38/ 42 Computation of prediction errors For t = 1,..., 656 Starting from {X t,..., X t+n 1 } a training sample with n = 520, Our goal is to estimate q(1/n, 1) which can be viewed as the weekly loss return for a once-per-decade financial crisis. Three estimators are computed: q W n (1/n τ n, p = 1) (Weissman estimator for L 1 quantiles), q W n ( τ n(p, 1/n) τ n, p) and q W n ( τ n(p, 1/n) τ n, p) (based on estimators for L p quantiles). The associated prediction errors are computed with respect to X t+n.

Prediction errors - Weekly loss returns 0.00030 0.00030 0.00025 variable 0.00025 variable p=2 p=2 p=1.9 p=1.9 p=1.8 p=1.8 Realized Loss 0.00020 p=1.7 p=1.6 p=1.5 p=1.4 Realized Loss 0.00020 p=1.7 p=1.6 p=1.5 p=1.4 p=1.3 p=1.3 p=1.2 p=1.2 p=1.1 p=1.1 quantile quantile 0.00015 0.00015 0.00010 0.00010 0 40 80 120 Sample fraction k 0 40 80 120 Sample fraction k Horizontally: k, Vertically: Prediction error for q W n ( τ n(p, 1/n) τ n, p) (left), q W n ( τ n(p, 1/n) τ n, p) (right) and q W n (1/n τ n, p = 1) (magenta) as a function of k. 39/ 42

40/ 42 Conclusion The extreme behaviour of L p quantile has been established. Classical quantiles as well as expectiles are particular cases of L p quantiles. In contrast to quantiles, L p quantiles take into account the whole tail of the distribution. The condition for existence of L p quantiles is weaker than for expectiles. It is possible to extrapolate to arbitrarily large levels. The theory has been extended to a mixing dependence framework and to real losses (non necessarily positive).

41/ 42 Some references Bellini, F., Klar, B., Müller, A., Gianina, E.R. (2014). Generalized quantiles as risk measures, Insurance: Mathematics and Economics 54: 41 48. Cai, J., Einmahl, J., de Haan, L. and Zhou, C. (2015). Estimation of the marginal expected shortfall: the mean when a related variable is extreme, JRSS B, 77: 417 442. Chen, Z. (1996). Conditional L p quantiles and their application to testing of symmetry in non-parametric regression, Statistics and Probability Letters 29: 107 115. Daouia, A., Girard, S., Stupfler, G. (2016). Estimation of tail risk based on extreme expectiles, https://hal.archives-ouvertes.fr/hal-01142130 El Methni, J., Gardes, L., Girard, S. (2014). Nonparametric estimation of extreme risks from conditional heavy-tailed distributions, Scandinavian Journal of Statistics 41: 988 1012. Girard, S., Stupfler, G. (2015). Extreme geometric quantiles in a multivariate regular variation framework, Extremes 18(4): 629 663.

42/ 42 Girard, S., Stupfler, G. (2017). Intriguing properties of extreme geometric quantiles, REVSTAT: Statistical Journal, 15: 107 139. Geyer, C.J. (1996). On the asymptotics of convex stochastic optimization, unpublished manuscript. Koenker, R., Bassett, G.S. (1978). Regression quantiles, Econometrica 46: 33 50. Newey, W.K., Powell, J.L. (1987). Asymmetric least squares estimation and testing, Econometrica 55: 819 847. Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations, Journal of the American Statistical Association 73: 812 815.