Robustness and Sensitivity Analysis of Risk Measurement Procedures

Quantitative Finance, Vol. 10, No. 6, June–July 2010, 593–606

Robustness and sensitivity analysis of risk measurement procedures

RAMA CONT*yz, ROMAIN DEGUESTy and GIACOMO SCANDOLOx

yIEOR Department, Columbia University, New York, NY, USA zLaboratoire de Probabilite´s et Mode` les Ale´atoires, CNRS, Universite´de Paris VI, Paris, France xDipartimento di Matematica per le Decisioni, Universita` di Firenze, Firenze, Italia

(Received 18 December 2008; in final form 1 February 2010)

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.

Keywords: Risk management; Risk measurement; Coherent risk measures; Law invariant risk measures; Value-at-Risk; Expected shortfall

1. Introduction portfolio loss distribution, an implicit starting point is

Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 the knowledge of the loss distribution. In applications, One of the main purposes of quantitative modeling in however, this probability distribution is unknown and finance is to quantify the risk of financial portfolios. In should be estimated from (historical) data as part of the connection with the widespread use of Value-at-Risk and risk measurement procedure. Thus, in practice, measuring related risk measurement methodologies and the Basel the risk of a financial portfolio involves two steps: committee guidelines for risk-based requirements for estimating the loss distribution of the portfolio from regulatory capital, methodologies for measuring of the available observations and computing a risk measure that risk of financial portfolios have been the focus of recent summarizes the risk of this loss distribution. While these attention and have generated a considerable theoretical two steps have been considered and studied separately, literature (Artzner et al. 1999, Acerbi 2002, 2007, they are intertwined in applications and an important Fo¨llmer and Schied 2002, 2004, Frittelli and Rosazza criterion in the choice of a risk measure is the availability Gianin 2002). In this theoretical approach to risk of a method for accurately estimating it. Estimation or measurement, a risk measure is represented as a map mis-specification errors in the portfolio loss distribution assigning a number (a measure of risk) to each random can have a considerable impact on risk measures, and it is payoff. The focus of this literature has been on the important to examine the sensitivity of risk measures to properties of such maps and requirements for the risk these errors (Gourieroux et al. 2000, Gourieroux and measurement procedure to be coherent, in a static or Liu 2006). dynamic setting. Since most risk measures such as Value-at-Risk or Expected Shortfall are defined as functionals of the 1.1. A motivating example Consider the following example, based on a data set of *Corresponding author. Email: [email protected] 1000 loss scenarios for a derivatives portfolio

Quantitative Finance ISSN 1469–7688 print/ISSN 1469–7696 online 2010 Taylor & Francis http://www.informaworld.com DOI: 10.1080/14697681003685597 594 R. Cont et al.

12 7 Sensitivity of historical VaR Sensitivity of historical ES Sensitivity of historical ES Sensitivity of Gaussian ES 10 6 Sensitivity of Laplace ES

5 8 4 6 3 4 2 Percentage change Percentage change 2 1

0 0

−1 –2 −1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 –2 –1.5 –1 −0.5 Loss size x 107 Loss size x 107

Figure 1. Empirical sensitivity (in percentage) of the historical Figure 2. Empirical sensitivity (in percentage) of the ES at 99% VaR at 99% and historical ES at 99%. estimated with diverse methods.

incorporating hundreds of different risk factors.y measure and its estimation method used for computing it, The historical Value-at-Risk (VaR) i.e. the quantile of we define the notion of risk measurement procedure as a the empirical loss distribution, and the Expected Shortfall two-step procedure that associates with a payoff X and a b (Acerbi 2002) of the empirical loss distribution, computed data set x ¼ (x1, ..., xn) of size n a risk estimate ðxÞ at the 99% level are, respectively, 8.887 M$ and 9.291 M$. based on the data set x. This estimator of the ‘theoretical’ To examine the sensitivity of these estimators to the risk measure (X ) is said to be robust if small variations addition of a single observation in the data set, we in the loss distribution—resulting either from estimation compute the (relative) change (in %) in the estimators or mis-specification errors—result in small variations in when a new observation is added to the data set. Figure 1 the estimator. displays this measure of sensitivity as a function of the size of the additional observation. While the levels of the 1.2. Contribution of the present work two risk measures are not very different, they display quite different sensitivities to a change in the data set, the In the present work, we propose a rigorous approach for Expected Shortfall being much more sensitive to large examining how estimation issues can affect the computa- observations while VaR has a bounded sensitivity. While tion of risk measures, with a particular focus on robustness Expected Shortfall has the advantage of being a coherent and sensitivity analysis of risk measurement procedures, risk measure (Artzner et al. 1999, Acerbi 2002), it appears using tools from robust statistics (Huber 1981, Hampel et to lack robustness with respect to small changes in the al. 1986). In contrast to the considerable literature on risk Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 data set. measures (Artzner et al. 1999, Kusuoka 2001, Acerbi 2002, Another point, which has been left out of most studies 2007, Fo¨llmer and Schied 2002, Rockafellar and Uryasev on risk measures (with the notable exception of 2002, Tasche 2002), which does not discuss estimation Gourieroux and Liu 2006), is the impact of the estimation issues, we argue that the choice of the estimation method method on these sensitivity properties. A risk measure and the risk measure should be considered jointly using the such as Expected Shortfall (ES) can be estimated in notion of risk estimator. different ways: either directly from the empirical loss We introduce a qualitative notion of ‘robustness’ for a distribution (‘historical ES’) or by first estimating a risk measurement procedure and a way of quantifying it parametric model (Gaussian, Laplace, etc.) from the via sensitivity functions. Using these tools we show that observed sample and computing the Expected Shortfall there is a conflict between coherence (more precisely, using the estimated distribution. the subadditivity) of a risk measure and the robustness, Figure 2 shows the sensitivity of the Expected Shortfall in the statistical sense, of its commonly used estimators. for the same portfolio as above, but estimated using three This consideration goes against the traditional arguments different methods. We observe that different estimators for the use of coherent risk measures and therefore merits for the same risk measure exhibit very different sensitiv- discussion. We complement this abstract result by ities to an additional observation (or outlier). computing measures of sensitivity, which allow us to These examples motivate the need for assessing the quantify the robustness of various risk measures with sensitivity and robustness properties of risk measures in respect to the data set used to compute them. In conjunction with the estimation method being used to particular, we show that the same ‘risk measure’ may compute them. In order to study the interplay of a risk exhibit quite different sensitivities depending on the

yData courtesy of Socie´te´Ge´ne´rale Risk Management unit. Robustness and sensitivity analysis of risk measurement procedures 595

estimation procedure used. These properties are studied in 2. Estimation of risk measures detail for some well-known examples of risk measures: Value-at-Risk, Expected Shortfall/CVaR (Acerbi 2002, Let (, F, P) be a given probability space representing Rockafellar and Uryasev 2002, Tasche 2002) and the class market outcomes and L0 be the space of all random of spectral risk measures introduced by Acerbi (2007). variables. We denote by D the (convex) set of cumulative Our results illustrate, in particular, that historical distribution functions (cdf ) on R. The distribution of a Value-at-Risk, while failing to be sub-additive, leads to random variable X is denoted FX 2D, and we write X F a more robust procedure than alternatives such as if FX ¼ F. The Le´vy distance (Huber 1981) between two Expected shortfall. cdfs F, G 2D is Statistical estimation and sensitivity analysis of risk measures have also been studied by Gourieroux et al. d ðF, GÞ¼4 inff" 4 0 : (2000) and Gourieroux and Liu (2006). In particular, F ðx "Þ" GðxÞF ðx þ "Þþ", 8x 2 Rg: Gourieroux and Liu (2006) consider non-parametric estimators of distortion risk measures (which includes The upper and lower quantiles of F 2Dof order 2 (0, 1) the class studied in this paper) and focus on the are defined, respectively, by asymptotic distribution of these estimators. By contrast, þ 4 R we study their robustness and sensitivity using tools from q ðF Þ¼ inffx 2 : F ðxÞ 4 gq ðF Þ robust statistics. ¼4 fx 2 R F ðxÞg: Methods of robust statistics are known to be relevant in inf : quantitative finance. Czellar et al. (2007) discuss robust Abusing notation, we denote q ðX Þ¼q ðFXÞ. For p 1 methods for estimation of interest-rate models. we denote by Dp the set of distributions having a finite pth Dell’Aquila and Embrechts (2006) discuss robust estima- moment, i.e. tion in the context of extreme value theory. Heyde et al. Z (2007), which appeared simultaneously with the first jxjpdF ðxÞ 5 1, version of this paper, discuss some ideas similar to those R discussed here, but in a finite data set (i.e. Dp non-asymptotic) framework. We show that, using appro- and by the set of distributions whose left tail has a finite p moment. We denote (F ) the mean of F 2D1 and priate definitions of consistency and robustness, the 2 2 discussion can be extended to a large-sample/asymptotic (F ) the variance of F 2D . For any n 1 and any Rn framework which is the usual setting for discussion of x ¼ (x1, ..., xn) 2 , estimators. Our asymptotic framework allows us to 1 Xn establish a clear link, absent in Heyde et al. (2007), F empðxÞ¼4 I x n fxxig between properties of risk estimators and those of risk i¼1 measures. denotes the empirical distribution of the data set x; Demp will denote the set of all empirical distributions. 1.3. Outline Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 Section 2 recalls some basic notions on distribution-based risk measures and establishes the distinction between a 2.1. Risk measures risk measure and a risk measurement procedure. We show The ‘Profit and Loss’ (P&L) or payoff of a portfolio over a that a risk measurement procedure applied to a data set specified horizon may be represented as a random can be viewed as the application of an effective risk variable X 2 L L0(, F, P), where negative values for measure to the empirical distribution obtained from this X correspond to losses. The set L of such payoffs is data and give examples of effective risk measures assumed to be a convex cone containing all constants. associated with various risk estimators. A risk measure on L is a map : L ! R assigning to each Section 3 defines the notion of robustness for a risk P&L X 2 L a number representing its degree of riskiness. measurement procedure and examines whether this prop- Artzner et al. (1999) advocated the use of coherent risk erty holds for commonly used risk measurement proce- measures, defined as follows. dures. We show, in particular, that there exists a conflict between the subadditivity of a risk measure and the Definition 2.1 (coherent risk measure (Artzner et al. robustness of its estimation procedure. 1999)): A risk measure : L ! R is coherent if it is In section 4 we define the notion of sensitivity function (1) monotone (decreasing): (X ) (Y ) provided for a risk measure and compute sensitivity functions for X Y; some commonly used risk measurement procedures. (2) cash-additive (additive with respect to cash reserves): In particular, we show that, while historical VaR has a (X þ c) ¼ (X ) c for any c 2 R; bounded sensitivity to a change in the underlying data (3) positive homogeneous: (X ) ¼ (X ) for any 0; set, the sensitivity of Expected Shortfall estimators is (4) sub-additive: (X þ Y ) (X ) þ (Y ). unbounded. We discuss in section 5 some implications of our findings for the design of risk measurement proce- The vast majority of risk measures used in finance dures in finance. are statistical, or distribution-based risk measures, 596 R. Cont et al.

i.e. they depend on X only through its distribution FX: As a consequence, we recover the well-known facts that ES is a coherent risk measure, while VaR is not. FX ¼ FY ) ðX Þ¼ðY Þ: In this case, can be represented as a map on the set of 2.2. Estimation of risk measures probability distributions, which we still denote by . Once a (distribution-based) risk measure has been Therefore, by setting chosen, in practice one has first to estimate the P&L 4 distribution of the portfolio from available data and then ðF Þ¼ðX Þ, X apply the risk measure to this distribution. This can be we can view as a map defined on (a subset of ) the set of viewed as a two-step procedure. probability distributions D. (1) Estimation of the loss distribution FX: one can use We focus on the following class of distribution-based either an empirical distribution obtained from a risk measures, introduced by Acerbi (2002) and Kusuoka historical or simulated sample or a parametric (2001), which contains all examples used in the literature: form whose parameters are estimated from avail- Z 1 able data. This step can be formalized as a function ðX Þ¼ qðX ÞmðduÞ, ð1Þ n m u from X¼[n1R , the collection of all possible data 0 b sets, to D;ifx 2X is a data set, we denote Fx the where m is a probability measure on (0, 1). Let Dm be the corresponding estimate of FX. set of distributions of r.v. for which the above risk (2) Application of the risk measure to the estimated b measure is finite. m can then be viewed as a map P&L distribution Fx, which yields an estimator R 4 b m : Dm } . Notice that if the support of m does not b ðxÞ¼ðFxÞ for (X ). contain 0 or 1, then Dm ¼D. Three cases deserve particular attention. We call the combination of these two steps a risk measurement procedure. . Value at Risk (VaR). This is the risk measure that is frequently used in practice and corre- Definition 2.3 (risk measurement procedure): A risk sponds to the choice m ¼ for a fixed 2 (0, 1) measurement procedure (RMP) is a couple (M, ), where R (usually 10%), that is : D ! is a risk measure and M : X!D an estimator for the loss distribution. 4 ð2Þ VaRðF Þ¼ q ðF Þ: The outcome of this procedure is a risk estimator b R Its domain of definition is all D. : X! defined as . Expected shortfall (ES). This corresponds to 4 b x °b ðxÞ¼ðFxÞ, choosing m as the uniform distribution over that estimates (X ) given the data x (see diagram). (0, ), where 2 (0, 1) is fixed: Z 1 ES ðF Þ¼4 VaR ðF Þdu: ð3Þ u

Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 0 1 In this case, Dm ¼D, the set of distributions having an integrable left tail. . Spectral risk measures (Acerbi 2002, 2007). This R class of risk measures generalizes ES and = ◦ corresponds to choosing m(du) ¼ (u)du, where : [0, 1] ! [0, þ1) is a density on [0, 1] and u } (u) is decreasing. Therefore, 2.2.1. Historical risk estimators. The historical estimator bh Z associated with a risk measure is the estimator 1 4 obtained by applying to the empirical P&L distribution ðF Þ¼ VaRuðF ÞðuÞdu: ð4Þ b emp 0 (sample cdf ) Fx ¼ Fx : q qþ" bh emp If 2 L (0, 1) (but not in L ) and 0 ðxÞ¼ðFx Þ: p 1 1 around 1, then D Dm, where p þ q ¼ 1. For a risk measure m, as in (1), Xn For any choice of the weight m, defined in (1) is m b hðxÞ¼ ðF empÞ¼ w x , x 2 Rn, monotone, additive with respect to cash and positive m m x n,i ðiÞ i¼1 homogeneous. The subadditivity of such risk measures has been characterized as follows (Kusuoka 2001, where x(k) is the kth least element of the set {xi}in, and the weights are equal to Fo¨llmer and Schied 2002, Acerbi 2007). 4 i 1 i Proposition 2.2 (Kusuoka 2001, Fo¨llmer and Schied 2002, wn,i ¼ m , for i ¼ 1, ..., n 1, n n Acerbi 2007): The risk measure m defined in (1) is sub-additive (hence coherent) on D if and only if it is a n 1 m wn,n ¼ m ,1 : spectral risk measure. n Robustness and sensitivity analysis of risk measurement procedures 597

Historical estimators are L-estimators in the sense of Let be a positively homogeneous risk measure, Huber (1981). then we have

Example 2.4: Historical VaR is given by ðF ð j ÞÞ ¼ ðF Þ: d h Therefore, if the scale parameter is estimated by maxi- VaR ðxÞ¼xðbncþ1Þ, ð5Þ mum likelihood, the associated risk estimator of is then R where bac denotes the integer part of a 2 . given by mle Example 2.5: The historical expected shortfall ES is b ðxÞ¼cb ðxÞ: given by ! Example 2.7 (MLRE for a Gaussian family): The MLE 1 Xbnc ESchðxÞ¼ x þ x ðn bncÞ : ð6Þ of the scale parameter in the Gaussian scale family is n ðiÞ ðbncþ1Þ i¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn bðxÞ¼ x2: ð10Þ Example 2.6: The historical estimator of the spectral n i i¼1 risk measure associated with is given by Z Xn i=n The resulting risk estimators are given by b ðxÞ¼cbðxÞ, bh ðxÞ¼ wn,ixðiÞ, where wn,i ¼ ðuÞdu: ð7Þ where, depending on the risk measure considered, c is i¼1 ði1Þ=n given by

c ¼ VaRðF Þ¼z, 2 2.2.2. Maximum likelihood estimators. In the para- expfz=2g c ¼ ESðF Þ¼ pffiffiffiffiffiffi , metric approach to loss distribution modeling, a para- 2p Z metric model is assumed for FX and parameters are 1 estimated from data using, for instance, maximum c ¼ ðF Þ¼ zuðuÞdu, 0 likelihood. We call the risk estimator obtained the ‘maximum likelihood risk estimator’ (MLRE). We discuss where z is the -quantile of a standard normal these estimators for scale families of distributions, which distribution. include as a special case (although a multidimensional Example 2.8 (ML risk estimators for Laplace one) the common variance–covariance method for VaR distributions): The MLE of the scale parameter in the estimation. Let F be a centered distribution. The scale Laplace scale family is family generated by the reference distribution F is Xn defined by b 1 ðxÞ¼ jxij: ð11Þ x n D ¼4 fF ð j Þ : 4 0g, where F ðx j Þ¼4 F : i¼1 F Note that this scale estimator is not the standard deviation If F 2Dp ( p 1), then D Dp and it is common to F but the Mean Absolute Deviation (MAD). The resulting choose F with location 0 and scale 1, so that F (|) has 2 risk estimator is Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 location 0 and scale . In line with common practice in short-term risk management we assume that the risk b ðxÞ¼cbðxÞ, ð12Þ factor changes have location equal to zero. Two examples of scale families of distributions that we will study are: where c takes the following values, depending on the risk measure considered (we assume 0.5): . the Gaussian family where F has density 2 c ¼ VaRðF Þ¼lnð2Þ, 1ffiffiffiffiffiffi x f ðxÞ¼p exp , c ¼ ESðF Þ¼1 lnð2Þ, 2p 2 Z Z 1=2 1 . the Laplace or double exponential family where c ¼ ðF Þ¼ lnð2uÞðuÞdu þ lnð2 2uÞðuÞdu: F has density 0 1=2 1 f ðxÞ¼ expðÞjxj : 2 2.3. Effective risk measures The Maximum Likelihood Estimator (MLE) b ¼ b mleðxÞ of is defined by In all of the above examples we observe that the risk estimator b ðxÞ, computed from a data set x ¼ (x1, ..., xn), Xn can be expressed in terms of the empirical distribution b ¼ arg:max ln f ðx j Þ, ð8Þ 40 i Femp; in other words, there exists a risk measure i¼1 x eff such that, for any data set x ¼ (x1, ..., xn), the risk and solves the following nonlinear equation: estimator b ðxÞ is equal to the new risk measure eff Xn 0 b applied to the empirical distribution f ðxi= Þ b xi ¼n: ð9Þ 4 f ðxi=b Þ emp b i¼1 eff ðFx Þ¼ ðxÞ: ð13Þ 598 R. Cont et al.

bh We will call eff the effective risk measure associated with with coincides with . The same remains true even if the risk estimator b. In other words, while is the risk the density is not decreasing, so that is not a spectral measure we are interested in computing, the effective risk risk measure. measure is the risk measure that the procedure defined eff Example 2.12 (Gaussian ML risk estimator): Consider in definition 2.3 actually computes. the risk estimator introduced in example 2.7. The So far, the effective risk measure is defined for all eff associated effective risk measure is defined on D2 and empirical distributions by (13). Consider now a risk b given by estimator that is consistent with the risk measure at sZffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D F , that is 2 eff ðF Þ¼cðF Þ, where ðF Þ¼ x dF ðxÞ: n!1 R b ðX1, ..., XnÞ!ðF Þ a.s., Example 2.13 (Laplace ML risk estimator): Consider for any i.i.d. sequence Xi F. Consistency of a risk estimator for a class of distributions of interest is a the risk estimators introduced in example 2.8. The 1 minimal requirement to ask for. If b is consistent with the associated effective risk measure is defined on D and risk measure for F 2D D , we can extend to D given by eff eff eff Z as follows: for any sequence (xi)i1 such that eff ðF Þ¼cðF Þ, where ðF Þ¼ jxjdF ðxÞ: Xn R 1 d I ! F ðÞ 2 D , n fxig eff i¼1 Notice that, in both of these examples, the effective risk measure eff is different from the original risk measure . we define

4 effðF Þ¼ lim b ðx1, ..., xnÞ: ð14Þ n!1 3. Qualitative robustness Consistency guarantees that eff (F ) is independent of the chosen sequence. We now define the notion of qualitative robustness of a Definition 2.9 (effective risk measure): Let b : X!R be risk estimator and use it to examine the robustness of the a consistent risk estimator of a risk measure for a class various risk estimators considered above. Deff of distributions. There is a unique risk measure : D } R such that eff eff 3.1. C-Robustness of a risk estimator . ðF empÞ¼4 b ð Þ for any data set eff x x Fix a set CD representing the set of ‘plausible’ loss x ¼ (x1, ..., xn) 2X; 4 distributions, containing all the empirical distributions: . eff ðF Þ¼ limn!1 b ðx1, ..., xnÞ for any Demp C. In most examples the class C is specified via an sequence (xi)i1 such that integrability condition (e.g., existence of moments) in Xn 1 d which case it automatically contains all empirical

Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 I ! F ðÞ 2 D : n fxig eff distributions. Consider an interior element F 2C, i.e. for i¼1 any 40, there exists G 2C, with G 6¼ F, such that d(G, F ) . We call a (risk) estimator C-robust at F if Equation (13), defining the effective risk measure, the law of the estimator is continuous with respect to a allows us in most examples to characterize eff explicitly. change in F (remaining in C ) uniformly in the size n of the As shown in the examples below, eff may be quite data set. different from and lacks many of the properties was b initially chosen for. Definition 3.1: A risk estimator is C-robust at F if, for any "40, there exist 40 and n0 1 such that, for all Example 2.10 (historical VaR): The empirical quantile G 2C, d h VaR is a consistent estimator of VaR for any F 2Dsuch b b þ d h d ðG, F Þ ¼) d ðLnð, GÞÞ, Lnð, F ÞÞ ", 8n n0, that q ðF Þ¼q ðF Þ. Otherwise, VaRðX1, ..., XnÞ may not have a limit as n !1. Therefore, the effective risk where d is the Le´vy distance. d h measure associated with VaR is VaR restricted to the set When C¼D, i.e. when any perturbation of the distribution is allowed, the previous definition corresponds to þ Deff ¼fF 2D: q ðF Þ¼q ðF Þg: the notion of qualitative robustness (also called asymptotic robustness) as outlined by Huber (1981). This case is Example 2.11 (historical estimator of ES and spectral risk not generally interesting in econometric or financial measures): A general result on L-estimators by Van applications since requiring robustness against all pertur- Zwet (1980) implies that the historical estimator of any bations of the model F is quite restrictive and excludes spectral risk measure (in particular of the ES) is even estimators such as the sample mean. consistent with at any F where the risk measure is Obviously, the larger the set of perturbations C, the defined. Therefore, the effective risk measure associated harder it is for a risk estimator to be C-robust. In the Robustness and sensitivity analysis of risk measurement procedures 599

remainder of this section we will assess whether the risk Proposition 3.5: The historical estimator of VaR is estimators previously introduced are C-robust w.r.t. a C-robust at any F 2C, where suitable set of perturbations C. 4 þ C ¼ fF 2D: q ðF Þ¼q ðF Þg: In other words, if the quantile of the (true) loss distribution is uniquely determined, then the empirical 3.2. Qualitative robustness of historical risk estimators quantile is a robust estimator. The following generalization of a result of Hampel et al. (1986) is crucial for the analysis of robustness of historical risk estimators. 3.2.2. Historical estimator of ES and spectral risk Proposition 3.2: Let be a risk measure and F 2CD. measures. Let defined in (4) be in terms of a density If b h, the historical estimator of , is consistent with at q p in L (0, 1), so that D ¼D ( p and q are conjugate.) every G 2C, then the following are equivalent: However, here we do not assume that is decreasing, so (1) the restriction of to C is continuous (w.r.t. the Le´vy that need not be a spectral risk measure, although it is distance) at F; still in the form (1). bh (2) is C-robust at F. Proposition 3.6: For any F 2Dp, the historical estimator p A proof of proposition 3.2 is given in appendix A. of is D -robust at F if and only if, for some "40, From this proposition, we obtain the following corollary ð Þ¼ 8 2ð Þ[ð Þ ð Þ that provides a sufficient condition on the risk measure to u 0, u 0, " 1 ",1 , 15 ensure that the corresponding historical/empirical estima- i.e. vanishes in the neighborhood of 0 and 1. tor is robust. bh Proof: We have seen in section 2.3 that is consistent h Corollary 3.3: If is continuous in C, then b is C-robust with at any F 2D. If (15) holds for some ", then the at any F 2C. support of m (recall that m(du) ¼ (u)du) does not contain 0or1.AsA is empty, theorem 3.4 yields continuity of Proof: Fix G 2C and let (X ) be an i.i.d. sequence m n n1 at any distribution in D . Hence, we have D -robustness distributed as G. Then, by the Glivenko–Cantelli of b at F thanks to corollary 3.3. Theorem we have, for almost all !, On the contrary, if (15) does not hold for any ", then 0 emp n!1 d ðFXð!Þ, GÞ! 0, X ¼ðX1, ..., XnÞ: or 1 (or both) are in the support of m and therefore is By continuity of at G it holds that, again for not continuous at any distribution in D, in particular at b almost all !, F. Therefore, by proposition 3.2 we conclude that is not b emp D-robust at F. œ ðXð!ÞÞ ¼ ðFXð!ÞÞ!ðGÞ, An immediate, but important consequence is the and therefore b is consistent with at G. A simple following. application of proposition 3.2 concludes. œ

Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 Corollary 3.7: The historical estimator of any spectral Our analysis will use the following important result risk measure defined on Dp is not Dp-robust at any adapted from Huber (1981, theorem 3.1). For a measure F 2Dp. In particular, the historical estimator of ES is not m on [0, 1] let D1-robust at any F 2D1. A ¼4 f 2½0, 1 : mðfgÞ 4 0g m Proof: It is sufficient to observe that, for a spectral risk be the set of values where m puts a positive mass. measure, the density is decreasing and therefore it We remark that Am is empty for a continuous m (as in the cannot vanish around 0, otherwise it would vanish on the definition of spectral risk measures). entire interval [0, 1]. œ Theorem 3.4: Let m be a risk measure of the form (1). Proposition 3.6 illustrates a conflict between subaddi- If the support of m does not contain 0 or 1, then m is tivity and robustness: as soon as we require a þ continuous at any F 2D such that q ðF Þ¼q ðF Þ for any (distribution-based) risk measure m to be coherent, its 2Am. Otherwise, m is not continuous at any F 2D. historical estimator fails to be robust (at least when all In other words, a risk measure of the form (1) can be possible perturbations are considered). continuous at some F if and only if its computation does not involve any extreme quantile (close to 0 or 1.) In this case, continuity is ensured provided F is contin- 3.2.3. A robust family of risk estimators. We have just uous at all points where m has a point mass. seen that ES has a non-robust historical estimator. However, we can remove this drawback by slightly modifying its definition. Consider 0515251 and define the risk measure 3.2.1. Historical VaR. In this case, Am ¼ {}, so Z 1 2 combining corollary 3.3 and theorem 3.4 we have the VaR ðF Þdu: u following. 2 1 1 600 R. Cont et al.

This is simply the average of VaR levels across a range of The following result exhibits conditions on the function loss probabilities. As under which the MLE of the scale parameter is weakly 1 continuous on D . ðuÞ¼ I15u52 2 1 Theorem 3.10 (weak continuity of the scale MLE): Let vanishes around 0 and 1, proposition 3.6 shows that the DF be the scale family associated with the distribution F 2D historical (i.e. using the empirical distribution directly) and assume that (F ) ¼ 0, (F ) ¼ 1 and F admits a risk estimator of this risk measure is D1-robust. Of course, density f. Suppose now that , defined as in (16), is even, the corresponding risk measure is not coherent since is increasing on Rþ, and takes values of both signs. Then, the not decreasing. Note that, for 151/n, where n is the following two assertions are equivalent: sample size, this risk estimator is indistinguishable from mle Rþ the historical Expected Shortfall! Yet, unlike Expected . : D } , defined as in (17), is weakly Shortfall estimators, it has good robustness properties as continuous at F 2D ; R . is bounded and (, F ) X (x/)F (dx) ¼ 0 n !1. One can also consider a discrete version of the mle above risk measure: has a unique solution ¼ (F ) for all F 2D . 1 Xk A proof is given in appendix A. Using the above result, VaRuj ðF Þ,05 u1 5 5 uk 5 1, we can now study the qualitative robustness of parametric k j¼1 risk estimators for Gaussian or Laplace scale families. which enjoys similar robustness properties. Proposition 3.11 (non-robustness of Gaussian and 3.3. Qualitative robustness of the maximum likelihood Laplace MLRE): Gaussian (respectively Laplace) risk estimator MLRE of cash-additive and homogeneous risk measures are not D2-robust (respectively D1-robust) at any F in D2 We now discuss the qualitative robustness for MLRE in a (respectively in D1). scale family of a risk measure m defined as in (1). First, we generalize the definition of the scale Maximum Proof: We detail the proof for the Gaussian scale family. bmle 4 mle emp The same arguments hold for the Laplace scale family. Likelihood Estimator ðxÞ¼ ðFx Þ given in equations (8) and (9) for distributions that do not belong Let us consider a Gaussian MLRE of a translation to Demp. invariant and homogeneous risk measure, denoted by b bmle Let DF be the scale family associated with the distri- ðxÞ¼c ðxÞ. First of all we notice that the function g bution F 2D and assume that (F ) ¼ 0, (F ) ¼ 1 and F associated with the MLE of the scale parameter of a admits a density f. Define the function distribution belonging to the Gaussian scale family is þ 0 even, and increasing on R . Moreover, it takes values of 4 @ 1 x f ðxÞ ðxÞ¼ ln f ¼1 x , x 2 R: both signs. Secondly, we recall that the effective risk @ f ðxÞ ¼1 measure associated with the Gaussian ML risk estimator ð16Þ mle 2 is eff (F ) ¼ c (F ) for all F 2Deff ¼D g ¼D . Therefore, The ML estimator mle(G ) of the scale parameter (G ) as g is unbounded, by using theorem 3.10, we know that 2 for G 2D corresponds to the unique that solves eff is not continuous at any F 2D . As the

Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 F Z b b c h 4 x Gaussian MLRE considered verifies ðxÞ¼eff ðxÞ, ð, GÞ¼ GðdxÞ¼0, for G 2D : ð17Þ 2 F and is consistent with eff at all F 2D by construction, we R can apply proposition 3.2 to conclude that, for F 2D2, b is By defining D ¼ {G 2D: j (x)jG(dx)51}, we can 2 mle not D -robust at F. œ extend the definition of (G ) to all G 2D . Note that mle when G 2D= F, (G ) may exist if G 2D but does not correspond to the ML estimator of the scale parameter of G. Moreover, from definition (17), we notice that if we 4. Sensitivity analysis compute the ML estimator of the scale parameter for a emp bmle distribution Gx 2Demp we recover the MLE ðxÞ In order to quantify the degree of robustness of a risk introduced in equations (8) and (9). In the examples estimator, we now introduce the concept of the sensitivity below, we have computed the function for the Gaussian function. and Laplace scale families. Definition 4.1 (sensitivity function of a risk g Example 3.8 (Gaussian scale family): The function estimator): The sensitivity function of a risk estimator for the Gaussian scale family is at F 2Deff is the function defined by gð Þ¼ þ 2 ð Þ x 1 x , 18 4 eff ð"z þð1 "ÞF Þeff ðF Þ 2 SðzÞ¼Sðz; F Þ¼ lim , and we immediately conclude that D g ¼D . "!0þ " Example 3.9 (Laplace scale family): The function l for for any z 2 R such that the limit exists. the Laplace scale family is equal to S(z; F ) measures the sensitivity of the risk estimator to the addition of a new data point in a large sample. lð Þ¼ þj j ð19Þ x 1 x , It corresponds to the directional derivative of the effective 1 and we obtain D l ¼D . risk measure eff at F in the direction z 2D. In the Robustness and sensitivity analysis of risk measurement procedures 601

language of robust statistics, S(z; F ) is the influence The case q()5z is handled in a very similar way. Finally, function (Costa and Deshayes 1977, Deniau et al. 1977, if z ¼ q(), then, again by (21), we have q(F") ¼ z for any Huber 1981, Hampel et al. 1986) of the effective risk " 2 [0, 1). Hence, measure and is related to the asymptotic variance of eff d the historical estimator of (Huber 1981, Hampel SðzÞ¼ q ðF Þj ¼ 0, d" " "¼0 et al. 1986). and we conclude. œ Remark 1: If D is convex and contains all empirical distributions, then "z þ (1 ")F 2D for any " 2 [0, 1], This example shows that the historical VaR has a z 2 R and F 2D. These conditions hold for all the risk bounded sensitivity to a change in the data set, which measures we are considering. means that this risk estimator is not very sensitive to a small change in the data set.

4.1. Historical VaR 4.2. Historical estimators of Expected Shortfall and We have seen above that the effective risk measure d h spectral risk measures associated with VaR is the restriction of VaR to We now consider a spectral risk measure defined by a D ¼C ¼f 2D þð Þ¼ ð Þg eff F : q F q F : weight function as in (4). The sensitivity function of the historical VaR has the Proposition 4.3: Consider a distribution F with a density following simple explicit form. f 40. Assume the following. Proposition 4.2: If F 2D admits a strictly positive (1) density f, then the sensitivity function at F of the historical Z 1 VaR is u 8 ðuÞdu 5 1: 1 0 f ðquðF ÞÞ > , if z 5 qðF Þ, < f ðqðF ÞÞ 0, if z ¼ q ðF Þ, (2) The risk measure is only sensitive to the left tail SðzÞ¼> ð20Þ : (losses) , if z 4 qðF Þ: f ðqðF ÞÞ 9u0 2ð0, 1Þ, 8u 4 u0, ðuÞ¼0: Proof: First we observe that the map u } q(u) X q (F )is u (3) The density f is increasing in the left tail: 9x 50, the inverse of F and so it is differentiable at any u 2 (0, 1) 0 f increasing on (1, x ). and we have 0 The sensitivity function at F 2D of the historical 1 1 0 estimator of is then given by q ðuÞ¼ 0 ¼ : F ðqðuÞÞ f ðquðF ÞÞ Z Z F ðzÞ u 1 1 u Fix z 2 R and set, for " 2 [0, 1), F" ¼ "z þ (1 ")F, so that SðzÞ¼ ðuÞdu þ ðuÞdu: Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 0 f ðquðF ÞÞ F ðzÞ f ðquðF ÞÞ F F0. For "40, the distribution F" is differentiable at 0 any x 6¼ z, with F" ðxÞ¼ð1 "Þ f ðxÞ 4 0, and has a jump We note that the assumptions in the proposition are (of size ") at the point x ¼ z. Hence, for any u 2 (0, 1) and verified for Expected Shortfall and for all commonly used þ 4 " 2 [0, 1), F" 2Cu, i.e. qu ðF"Þ¼qu ðF"Þ¼qu ðF"Þ.Inparametric distributions in finance: Gaussian, Student-t, particular, double-exponential, Pareto (with exponent 41), etc. 8 ÀÁ < qÀÁ1" , for 5 ð1 "ÞF ðzÞ, Proof: Using the notation introduced in the proof of q ðF Þ¼ " ð21Þ " : q 1" , for ð1 "ÞF ðzÞþ", proposition 4.2 we have z, otherwise. Z 1 VaR ðF ÞVaR ðF Þ Assume now that z4q(), i.e. F(z)4; from (21) SðzÞ¼ lim u " u ðuÞdu: þ it follows that "!0 0 " qðF"Þ¼q , for " 5 1 : We will now show that the order of the integral and 1 " F ðzÞ the limit ! 0 can be interchanged using a domi- As a consequence, nated convergence argument. First note that 1 lim"!0þ " (VaRu(F")) VaRu(F )) exists, and is finite VaRðF"ÞVaRðF0Þ d for all u 2 (0, 1). Define u ¼ inf{F(x0), u0} and consider SðzÞ¼ lim ¼ qðF"Þj"¼0 "!0þ " d" 40. By the mean value theorem, for any u u(1 "): 0 d 1 ¼ q ¼ 2 8 , 9 2ð0, Þ, d" 1 " "¼0 f ðqð=ð1 "ÞÞÞ ð1 "Þ "¼0 0 u ¼ : VaRuðF"Þ¼VaRuðF0Þ" 2 : f ðqðF ÞÞ ð1 Þ f ðqðu=ð1 ÞÞÞ 602 R. Cont et al.

Therefore, we have As a consequence, ðF Þ ðF Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 VaRu " VaRu 0 d 2 c z ð0Þ¼c ðF"Þ ¼ 1 : " d" "¼0 2 u œ ð1 Þ2 f ðqðu=ð1 ÞÞÞ u 2 , 5 " "0, ð1 "0Þ f ðqðu=ð1 ÞÞÞ 4.4. ML risk estimators for Laplace distributions u 1 2 2 L ðÞ, f, q increasing, ð1 "0Þ fqðÞðuÞ We have seen that the effective risk measure of the Laplace so that we can apply dominated convergence MLRE of VaR, ES, or any spectral risk measure is Z 1 1 VaR ðF ÞVaR ðF Þ eff ðF Þ¼cðF Þ, F 2D, SðzÞ¼ lim u " u ðuÞdu "!0þ " Z0 where c ¼ (G ), G is theR distribution with density 1 jxj d g(x) ¼ e /2, and (F ) ¼ Rjxj dF(x). ¼ q ðF Þ ðuÞdu d" u " 0Z "¼0 Z Proposition 4.6: Let be a positively homogeneous risk F ðzÞ u 1 1 u measure. The sensitivity function at F 2D1 of its Laplace ¼ ðuÞdu þ ðuÞdu, 0 f ðquðF ÞÞ F ðzÞ f ðquðF ÞÞ MLRE is thanks to proposition 4.2. œ SðzÞ¼cðjzjðF ÞÞ: Since the effective risk measure associated with histor- Proof: As usual, we have, for z 2 R, S(z) ¼ 0(0), where ical ES R is ES itself, defined on (") ¼ c(F"), F" ¼ (1 ")F þ "z and c is defined above. 1 D ¼fF 2D: x F ðdxÞ 5 1g, an immediate conse- We have quence of the previous proposition is the following. ð"Þ¼cð1 "ÞðF Þþc"jzj, 1 Corollary 4.4: The sensitivity function at F 2D for and we conclude that historical ES is 0 ð0Þ¼cjzjcðF Þ: z þ 1 q ðF ÞES ðF Þ, if z q ðF Þ, SðzÞ¼ œ qðF ÞESðF Þ, if z qðF Þ. This proposition shows that the sensitivity of the This result shows that the sensitivity of historical ES Laplace MLRE at any F 2D1 is not bounded, but linear is linear in z, and thus unbounded. This means that this in z. Nonetheless, the sensitivity of the Gaussian MLRE risk measurement procedure is less robust than the is quadratic at any F 2D2, which indicates a greater historical VaR. sensitivity to outliers in the data set.

4.3. ML risk estimators for Gaussian distributions Table 1. Behavior of sensitivity functions for some risk

Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 estimators. We have seen that the effective risk measure associated with Gaussian maximum likelihood estimators of Risk estimator Dependence in z of S(z) VaR, ES, or any spectral risk measure is Historical VaR Bounded ð Þ¼ ð Þ 2D ¼D2 Gaussian ML for VaR Quadratic eff F c F , F eff , Laplace ML for VaR Linear Historical Expected Shortfall Linear where c ¼ (Z ), Z N(0, 1), is a constant depending only Gaussian ML for Expected Shortfall Quadratic on the risk measure (we are not interested in its explicit Laplace ML for Expected Shortfall Linear value here). 2 Proposition 4.5: The sensitivity function at F 2D of the 4.5. Finite-sample effects Gaussian ML risk estimator of a positively homogeneous risk measure is The sensitivity functions computed above are valid for (asymptotically) large samples. In order to assess the c z 2 SðzÞ¼ 1 : finite-sample relevance accuracy of risk estimator sensitiv- 2 ities, we compare them with the finite-sample sensitivity R Proof: Let, for simplicity, ¼ (F ). Fix z 2 and set, as b ðX1, ..., XN, zÞb ðX1, ..., XNÞ 2 SNðz; F Þ¼ : usual, F" ¼ (1 ")F þ "z (" 2 [0, 1)); observe that F" 2D 1=ðN þ 1Þ for any ". If we set (") X c(F"), with c ¼ (N(0, 1)), then we have S(z) ¼ 0(0). It is immediate to compute Figure 3 compares the empirical sensitivities of historical, Z Z Gaussian, and Laplace VaR and historical, Gaussian, and 2 2 2 2 2 ðF"Þ¼ x F"ðdxÞ¼ð1 "Þ x F ðdxÞþ"z "z Laplace ES with their theoretical (large sample) counter- R R parts. We have used the same set of N ¼ 1000 loss 2 2 2 2 2 2 2 2 2 ¼ð1 "Þð Þþ"z " z ¼ þ "½z " : scenarios as in section 1.1. Robustness and sensitivity analysis of risk measurement procedures 603

x 107 x 107 6 12 Empirical Sensitivity of historical VaR Empirical Sensitivity of historical CVaR Theorical Sensitivity of historical VaR Theorical Sensitivity of historical CVaR 5 10 = 1% = 1% α α 4 8

3 6

2 4

1 2

0 0 Sensitivity of historial VaR for Sensitivity of historial CVaR for −1 −2 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Perturbation x 106 Perturbation x 106 x 107 x 107 2.5 2.5 Empirical Sensitivity of gaussian VaR Empirical Sensitivity of gaussian CVaR Theorical Sensitivity of gaussian VaR Theorical Sensitivity of gaussian CVaR = 1%

= 1% 2 2 α α

1.5 1.5

1 1

0.5 0.5

0 0 Sensitivity of gaussian VaR for Sensitivity of gaussian CVaR for −0.5 −0.5 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Perturbation x 106 Perturbation x 106 x 106 x 107 20 2.5 Empirical Sensitivity of laplace VaR Empirical Sensitivity of laplace CVaR Theorical Sensitivity of laplace VaR Theorical Sensitivity of laplace CVaR 2 = 1% = 1%

15 α α 1.5

10 1

5 0.5

0 Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 0 −0.5 Sensitivity of laplace VaR for Sensitivity of laplace CVaR for −5 −1 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Perturbation x 106 Perturbation x 106

Figure 3. Empirical versus theoretical sensitivity functions of risk estimators for ¼ 1% at a 1 day horizon. Historical VaR (upper left), historical ES (upper right), Gaussian VaR (left), Gaussian ES (right), Laplace VaR (lower left), Laplace ES (lower right).

The asymptotic and empirical sensitivities coincide for This is useful since theoretical sensitivity functions are all risk estimators except for historical risk measurement analytically computable, whereas empirical sensitivities procedures. For the historical ES, the theoretical sensi- require perturbating the data sets and recomputing the tivity is very close to the empirical one. Nonetheless, we risk measures. note that the empirical sensitivity of the historical VaR can be equal to 0 because it is strongly dependent on the integer part of N, where N is the number of scenarios and the quantile level. This dependency disappears 5. Discussion asymptotically for large samples. The excellent agreement shown in these examples 5.1. Summary of main results illustrates that the expressions derived above for theoretical sensitivity functions are useful for evaluating the Let us now summarize the contributions and main sensitivity of risk estimators for realistic sample sizes. conclusions of this study. 604 R. Cont et al.

First, we have argued that when the estimation step is measurement procedures over non-robust procedures for explicitly taken into account in a risk measurement the sole reason of saving subadditivity. procedure, issues like robustness and sensitivity to the data set are important and need to be accounted for with at least the same attention as the coherence properties set 5.3. Beyond distribution-based risk measures forth by Artzner et al. (1999). Indeed, we do think that it While the ‘axiomatic’ approach to risk measurement in is crucial for regulators and end-users to understand the principle embodies a much wider class of risk measures robustness and sensitivity properties of the risk estimators than distribution-based (or ‘law-invariant’) risk measures, they use or design to assess the capital requirement, or research has almost exclusively focused on this rather manage their portfolio. Indeed, an unstable/non-robust restrictive class of risk measures. Our result, that coher- risk estimator, be it related to a coherent measure of risk, ence and robustness cannot coexist within this class, can is of little use in practice. also be seen as an argument for going beyond Second, we have shown that the choice of the estima- distribution-based risk measures. This also makes sense tion method matters when discussing the robustness of in the context of the ongoing discussion on systemic risk: risk measurement procedures: our examples show that evaluating exposure to systemic risk requires considering different estimation methods coupled with the same risk the joint distribution of a portfolio’s losses with other, measure lead to very different properties in terms of external, risk factors, not just the marginal distribution robustness and sensitivity. of its losses. In fact, risk measures that are not Historical VaR is a qualitatively robust estimation distribution-based are routinely used in practice: the procedure, whereas the proposed examples of coherent Standard Portfolio Analysis of Risk (SPAN) margin (distribution-based) risk measures do not pass the test of system, and cited as the original motivation by Artzner qualitative robustness and show high sensitivity to ‘out- et al. (1999), is a well-known example of such a method liers’. This explains perhaps why many practitioners have used by many clearinghouses and exchanges. been reluctant to adopt ‘coherent’ risk measures. Also, We hope to have convinced the reader that there is most parametric estimation procedures for VaR and ES more to risk measurement than the choice of a ‘risk lead to non-robust estimators. On the other hand, measure’: statistical robustness, and not only ‘coherence’, weighted averages of historical VaR such as Z should be a concern for regulators and end-users when 1 2 choosing or designing risk measurement procedures. VaR ðF Þdu, u The design of robust risk estimation procedures requires 2 1 1 the explicit inclusion of the statistical estimation step with 1424140, have robust empirical estimators. in the analysis of the risk measurement procedure. We hope this work will stimulate further discussion on these important issues. 5.2. Re-examining subadditivity The conflict we have noted between robustness of a risk measurement procedure and the subadditivity of the risk Acknowledgements Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 measure shows that one cannot achieve robust estimation in this framework while preserving subadditivity. While a We thank Hans Fo¨llmer, Peter Bank, Paul Embrechts, strict adherence to the coherence axioms of Artzner et al. Gerhard Stahl and seminar participants at QMF 2006 (1999) would lead to choosing subadditivity over robust- (Sydney), Humboldt University (Berlin), Torino, Lecce, ness, several recent studies (Danielsson et al. 2005, Heyde INFORMS Applied Probability Days (Eindhoven), the et al. 2007, Ibragimov and Walden 2007, Dhaene et al. Cornell ORIE Seminar, the Harvard Statistics Seminar 2008) have provided reasons for not doing so. and EDHEC for helpful comments. This project has Danielsson et al. (2005) explore the potential for benefited from partial funding by the European Research violations of VaR subadditivity and report that, for Network ‘Advanced Mathematical Methods for Finance’ most practical applications, VaR is sub-additive. They (AMAMEF). conclude that, in practical situations, there is no reason to choose a more complicated risk measure than VaR, solely for reasons of subadditivity. Arguing in a different direction, Ibragimov and Walden (2007) show that, for References very ‘heavy-tailed’ risks defined in a very general sense, Acerbi, C., Spectral measures of risk: a coherent representation diversification does not necessarily decrease tail risk but of subjective risk aversion. J. Bank. Finance, 2002, 26, actually can increase it, in which case requiring sub- 1505–1518. additivity would in fact be unnatural. Finally, Finally, Acerbi, C., Coherent measures of risk in everyday market Heyde et al. (2007) argue against subadditivity from an practice. Quant. Finance, 2007, 7, 359–364. axiomatic viewpoint and propose to replace it by a weaker Artzner, P., Delbaen, F., Eber, J. and Heath, D., Coherent measures of risk. Math. Finance, 1999, 9, 203–228. property of co-monotonic subadditivity. All these objec- Costa, V. and Deshayes, J., Comparaison des RLM estimateurs. tions to the subadditivity axiom deserve serious consid- Theorie de la robustesse et estimation d’un parametre. eration and further support the choice of robust risk Asterisque, 1977, 43–44. Robustness and sensitivity analysis of risk measurement procedures 605

Czellar, V., Karolyi, G.A. and Ronchetti, E., Indirect robust Then, by using that jC is continuous at F, there exists estimation of the short-term interest rate process. J. Empir. 4 0 such that, for each G 2Cthat satisfies d ðF, G Þ2, Finance, 2007, 14, 546–563. Danielsson, J., Jorgensen, B., Samorodnitsky, G., Sarma, M. we have d((F ), (G )) ¼j(F ) (G )j5"/2. As Demp C, we obtain that and de Vries, C., Subadditivity re-examined: the case for Value-at-Risk. Preprint, London School of Economics, 2005. " Pðd ðF, G empÞ2ÞP jðF ÞðG empÞj : Dell’Aquila, R. and Embrechts, P., Extremes and robustness: x x 2 a contradiction? Financial Mkts Portfol. Mgmt, 2006, 20, P emp 103–118. Therefore, it suffices to show that ðd ðF, Gx Þ2Þ Deniau, C., Oppenheim, G. and Viano, C., Courbe d’influence 1 ð"=2Þ. et sensibilite. Theorie de la robustesse et estimation d’un Now, using that Glivenko–Cantelli convergence is parametre. Asterisque, 1977, 43–44. uniform in G, we have for each "40 and 40 the Dhaene, J., Laeven, R., Vanduffel, S., Darkiewicz, G. and n G 2C Goovaerts, M., Can a coherent risk measure be too existence of 0 1 such that, for all , subadditive? J. Risk Insurance, 2008, 75, 365–386. " Pðd ðG, G empÞÞ1 , 8n n : Fo¨llmer, H. and Schied, A., Convex measures of risk and x 2 0 trading constraints. Finance Stochast., 2002, 6, 429–447. Fo¨llmer, H. and Schied, A., Stochastic Finance: An Introduction Therefore, the C-robustness follows from the triangular in Discrete Time, 2004 (Walter de Gruyter: Berlin). emp emp inequality d ðF, Gx Þd ðF, G Þþd ðG, Gx Þ and by Frittelli, M. and Rosazza Gianin, E., Putting order in risk taking . measures. J. Bank. Finance, 2002, 26, 1473–1486. Gourieroux, C., Laurent, J. and Scaillet, O., Sensitivity analysis ‘2. ) 1’. Conversely, assume that b h is C-robust at F and of values at risk. J. Empir. Finance, 2000, 7, 225–245. fix "40. Then there exists 40 and n 1 such that, for Gourieroux, C. and Liu, W., Sensitivity analysis of distortion 0 risk measures. Working Paper, 2006. all G 2C, Hampel, F., Ronchetti, E., Rousseeuw, P. and Stahel, W., " d ðF, G Þ 5 ) d ðL ðb h, F Þ, L ðb h, G ÞÞ 5 , 8n n : Robust Statistics: The Approach Based on Influence Functions, n n 3 0 1986 (Wiley: New York). Heyde, C., Kou, S. and Peng, X., What is a good risk measure: As a consequence, from the triangular inequality bridging the gaps between data, coherent risk measures, and insurance risk measures. Preprint, Columbia University, jðFÞðGÞj ¼ dððFÞ,ðGÞÞ 2007. bh bh bh Huber, P., Robust Statistics, 1981 (Wiley: New York). dððFÞ,Lnð ,FÞÞ þ dðLnð ,FÞ,Lnð ,GÞÞ Ibragimov, R. and Walden, J., The limits of diversification when h þ dðLnðb ,GÞ,ðGÞÞ, losses may be large. J. Bank. Finance, 2007, 31, 2551–2569. Kusuoka, S., On law invariant coherent risk measures. and the consistence of b h with at F and any G 2C, Adv. Math. Econ. , 2001, 3, 83–95. j Rockafellar, R. and Uryasev, S., Conditional Value-at-Risk for it follows that C is continuous at F. general distributions. J. Bank. Finance, 2002, 26, 1443–1471. Tasche, D., Expected shortfall and beyond. In Statistical Data Analysis Based on the L1-Norm and Related Methods, edited A.2. Proof of theorem 3.10 by Y. Dodge, pp. 109–123, 2002 (Birkha¨user: Boston, Basel, Berlin). We will show that the continuity problem of the ML scale Van Zwet, W., A strong law for linear functions of order mle þ function : D ! R of portfolios X can be reduced to Downloaded By: [Cont, Rama] At: 09:32 9 June 2010 statistics. Ann. Probab., 1980, 8, 986–990. the continuity (on a properly defined space) issue of the ML location function of portfolios Y ¼ ln(X2). The change of variable here is made to use the results of Appendix A: Proofs Huber (1981) concerning the weak continuity of location parameters. The distribution F can be seen as the A.1. Proof of proposition 3.2 distribution of a portfolio X0 with (X0) ¼ 0 and 2 (X0) ¼ 1. Then, by setting Y0 ¼ lnðX0Þ, and denoting ‘1. ) 2’. Assume that jC is continuous at F and fix "40. Since for all G 2C by G the distribution of Y0 we have ð Þ¼ ð Þ¼ ð 2 yÞ¼ ð y=2Þ ð y=2Þ h h G y P Y0 y P X0 e F e F e , d ðLnðb , F Þ, Lnðb , G ÞÞ ð Þ¼ 0ð Þ¼ y=2 ð y=2Þ h h g y G y e f e : d ðLnðb , FÞ, ðF ÞÞþd ððF Þ, Lnðb , G ÞÞ, Moreover, by introducing the function h Z and b is consistent with at F, it suffices to prove that 0 g ð yÞ 4 there exists 40 and n 1 such that, for all G 2C, ’ð yÞ¼ , D’ ¼ G : ’ð yÞGðdyÞ 5 1 , 0 gð yÞ " d ðG, F Þ ¼) d ð , L ðb h, G ÞÞ , 8n n : we can define, as in Huber (1981), the ML location ðF Þ n 0 mle 2 function (H ) for any distribution H 2Du as the solution of the following implicit relation Note that Strassen’s Theorem (Huber 1981, theorem 3.7) Z gives us the following sufficient condition to obtain the ’ð y ÞHðdyÞ¼0: ðA1Þ desired result: " " " Now we consider the distribution F 2D of the random P jðFÞðGempÞj 1 ¼)dð ,L ðb h,GÞÞ : X x 2 2 ðFÞ n 2 variable X representing the P&L of a portfolio and 606 R. Cont et al.

assumeR that FX has density fX and that the solution to Noticing that mle (x/)FX(dx) ¼ 0 has a unique solution ¼ (FX). 2 2 FYðdyÞ¼fYðyÞdy ¼ xfXðxÞdðlnðx ÞÞ ¼ 2fXðxÞdx ¼ 2FXðdxÞ, Denoting by FY the distribution of Y ¼ ln(X ), it is easy to check that F 2D since, for y ¼ ln(x2), we have we immediately obtain from equations (17), (A1) and Y u mle mle (A2) that (FX) ¼ (FY) when Y ¼ 2 ln(X ). We have g0ð yÞ therefore shown that a scale function characterized by the ’ð yÞ¼ gð yÞ function can also be interpreted as a location function characterized by the function u. From equation (A2), we 1 ð yÞ=2 ð yÞ=2 1 y 0 ð yÞ=2 R x/2 2 e f ðe Þþ2 e f ðe Þ see that, for all x 2 ,2u(x) ¼ [e ]. Therefore, as is ¼ þ e ð yÞ=2f ðe ð yÞ=2Þ assumed to be even and increasing on R , it implies that u is increasing on R. Moreover, as takes values of both 1 f 0ðe ð yÞ=2Þ signs it is also true for u. To conclude, we apply ¼ 1 þ e ð yÞ=2 2 f ðe ð yÞ=2Þ theorem 2.6 of Huber (1981), which states that a location function associated with u is weakly continuous at G if 1 f 0ðxÞ 1 ¼ 1 þ x ¼ ðxÞ: ðA2Þ and only if u is bounded and the location function 2 f ðxÞ 2 computed at G is unique. Downloaded By: [Cont, Rama] At: 09:32 9 June 2010