Conditional VAR and Expected Shortfall: A New Functional Approach

Frédéric Ferraty, Alejandro Quintela-Del-Río
2014 Econometric Reviews  
We estimate two well-known risk measures, the Value-at-risk and the expected shortfall, conditionally to a functional variable (i.e., a random variable valued in some semi(pseudo)-metric space). We use nonparametric kernel estimation for constructing estimators of these quantities, under general dependence conditions. Theoretical properties are stated whereas practical aspects are illustrated on simulated data: nonlinear functional and GARCH(1,1) models. Some ideas on bandwidth selection using
more » ... th selection using bootstrap are introduced. Finally, an empirical example is given through data of the S&P 500 time series. Corresponding author: Alejandro Quintela-del-Río. A major concern for regulators and owners of financial institutions is the risk analysis. The Value-at-Risk (VaR) (Embrechts et al., 1997) is one of the most common risk measures used in finances. It measures down-side risk and is determined for a given probability level p. In a typical situation, measuring losses, the VaR is the lowest value which exceeds this level (that is, the quantile of the loss distributions). The expected shortfall (ES) (Acerbi, 2002) is the expected loss given the loss exceeds the VaR threshold. From the Basel Accords (1996, 2006), the VaR (and more recently the ES) forms the essential basis of the determination of market risk capital. Many banks compute VaR for managing the financial hazard of their portfolios (Gilli and Këllezi, 2006) . Several methods have been developed to calculate the VaR. See, for example, the paper of Bao et al. (2006) for an exhaustive description of up to 16 methods to estimate the VaR, with applications to different financial time series. Usually, one collects additional information I (i.e., past observed returns, economical exogenous covariates, etc.) and to take into account such relevant information I, the conditional VaR (CVaR) of the variable Y t (being Y t the risk or loss variable which can be the negative logarithm of returns at time t) is defined, for a fixed level p, as the value I]. Most studies estimate CVaR through quantile estimation (see for instance Gaglianone et al., 2009 and references therein). When the conditional information is a finitedimensional predictor, Scaillet (2004 and 2005) proposed to estimate CES and CVaR nonparametrically by using kernel estimators. Other works based on nonparametric estimators are those of Chen (2007), Chen and Tang (2005) or Cai and Wang (2008). A related approximation is that of Cosma et al. (2007), where they introduce a new approach on shape preserving estimation of cumulative distribution functions and probability density functions, using the wavelet methodology for multivariate dependent data (these estimators preserve shape constraints such as monotonicity, positivity and integration to one, and allow 2 for low spatial regularity of the underlying functions). The paper also discusses CVaR estimation for financial time series data. Other noteworthy works are those of Fermanian and Scaillet (2005) , discussing nonparametric estimation of the VaR and the expected shortfall in a credit environment where netting agreements are present; or Linton and Xiao (2011) and Zhu and Galbraith (2011), estimating the expected shortfall in the heavy tail case. Comparable studies of nonparametric functional techniques applied to quantile estimation (but in an environmental setting) can be seen in Quintela and Francisco (2011). In this paper, we consider the estimation of the CES and CVaR when one has at hand a covariate X valued in some semi-metric 1 space (F, d(·, ·)). Such a random variable is called functional covariate. This amounts equivalently to considering an infinite-dimensional covariate instead of a finite one as in Cai and Wang (2008) . This situation may occur when a continuous process of returns {Z t } t∈[0,(n+1)τ ) is cut into n + 1 pieces X i = {X i (δ) = Z (i−1)τ +δ ; 0 ≤ δ < τ } (i = 1, . . . , n + 1). This mechanism which consists in building the process {X i } i=1,...,n of functional variables (also called functional process) from the continuous process {Z t } t∈(0,nτ ] is quite standard now (for more details, see the monograph of Bosq, 2000, and precursor references therein). Moreover, for i = 1, . . . , n, the building of a X i+1 -measurable real random variable (r.r.v.) Y i allows us to consider a new statistical sample of n pairs (X 1 , Y 1 ), . . . , (X n , Y n ) identically distributed but not necessarily independent. Various choices for the Y i 's are possible: a future return (i.e., Y i = X i+1 (δ) with δ ∈ [0, τ )), or the supremum return over a future range (i.e., Y i = sup δ∈[0,τ ) X i+1 (δ)), or any other interesting scalar quantity computed from X i+1 . It makes sense now to study features of the distribution of the r.r.v. Y conditionally to the functional variable X . This statistical issue has been thoroughly investigated in the functional data literature, and the reader can find numerous works dealing with conditional cumulative distribution, conditional density, conditional mean, conditional mode or conditional quantiles (for general overviews, see the monograph of Ferraty 1 A semi-metric (also called pseudo-metric) d(·, ·) is a metric allowing d(χ1, χ2) = 0 for some χ1 = χ2 3 and Vieu, 2006, and the collective book edited by Ferraty and Romain, 2011; for more recent advances, see for instance Lemdani et al., 2009 , or Ferraty et al., 2010 . In such a conditional functional situation, the CVaR is the quantity ν p (χ) such that 1 − F (ν p (χ)|χ) = p where F (·|χ) is the cumulative distribution function (c.d.f.) conditionally to X = χ and the CES is given by µ p (χ) = E(Y |Y > ν p (χ), X = χ) =
doi:10.1080/07474938.2013.807107 fatcat:vxwn3excavebpca7z6baqpdzoa