Robustness and sensitivity analysis of risk measurement ...rama/papers/robustrisk.pdf · Robustness...

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Robustness and sensitivity analysis of risk measurement procedures * Rama CONT 1 , Romain DEGUEST 1,2 and Giacomo SCANDOLO 3 1) IEOR Dept, Columbia University, New York. 2) CMAP, Ecole Polytechnique, France. 3) Dipartimento di Matematica per le Decisioni, Universit` a di Firenze, Italia. Financial Engineering Report No. 2007-06 Center for Financial Engineering, Columbia University. Abstract Measuring the risk of a financial portfolio involves two steps: esti- mating the loss distribution of the portfolio from available observations and computing a “risk measure” which summarizes the risk of the port- folio. We define the notion of “risk measurement procedure”, which includes both of these steps and study the robustness of risk measure- ment procedures and their sensitivity to a change in the data set. After introducing a rigorous definition of ’robustness’ of a risk measurement procedure, we illustrate the presence of a conflict between subaddi- tivity and robustness of risk measurement procedures. We propose a measure of sensitivity for risk measurement procedures and compute the sensitivity function of various examples of risk estimators used in financial risk management, showing that the same risk measure may exhibit quite different sensitivities depending on the estimation pro- cedure used. Our results illustrate in particular that using historical Value at Risk leads to a more robust procedure for risk measurement than recently proposed alternatives like CVaR. We also propose other risk measurement procedures which possess the robustness property. * We thank Hans F¨ ollmer, Peter Bank, Paul Embrechts, Gerhard Stahl and seminar participants at QMF 2006 (Sydney), Humboldt University (Berlin), Torino, Lecce, IN- FORMS Applied Probability Days (Eindhoven), Cornell ORIE seminar and the Harvard Statistics Seminar for helpful comments. This project has benefited from partial fund- ing by the European Research Network “Advanced Mathematical Methods for Finance” (AMAMEF). 1

Transcript of Robustness and sensitivity analysis of risk measurement ...rama/papers/robustrisk.pdf · Robustness...

Page 1: Robustness and sensitivity analysis of risk measurement ...rama/papers/robustrisk.pdf · Robustness and sensitivity analysis of risk measurement procedures∗ Rama CONT 1, Romain

Robustness and sensitivity analysis of

risk measurement procedures∗

Rama CONT1, Romain DEGUEST 1,2 and Giacomo SCANDOLO3

1) IEOR Dept, Columbia University, New York.2) CMAP, Ecole Polytechnique, France.

3) Dipartimento di Matematica per le Decisioni,Universita di Firenze, Italia.

Financial Engineering Report No. 2007-06Center for Financial Engineering, Columbia University.

AbstractMeasuring the risk of a financial portfolio involves two steps: esti-

mating the loss distribution of the portfolio from available observationsand computing a “risk measure” which summarizes the risk of the port-folio. We define the notion of “risk measurement procedure”, whichincludes both of these steps and study the robustness of risk measure-ment procedures and their sensitivity to a change in the data set. Afterintroducing a rigorous definition of ’robustness’ of a risk measurementprocedure, we illustrate the presence of a conflict between subaddi-tivity and robustness of risk measurement procedures. We propose ameasure of sensitivity for risk measurement procedures and computethe sensitivity function of various examples of risk estimators used infinancial risk management, showing that the same risk measure mayexhibit quite different sensitivities depending on the estimation pro-cedure used. Our results illustrate in particular that using historicalValue at Risk leads to a more robust procedure for risk measurementthan recently proposed alternatives like CVaR. We also propose otherrisk measurement procedures which possess the robustness property.

∗We thank Hans Follmer, Peter Bank, Paul Embrechts, Gerhard Stahl and seminarparticipants at QMF 2006 (Sydney), Humboldt University (Berlin), Torino, Lecce, IN-FORMS Applied Probability Days (Eindhoven), Cornell ORIE seminar and the HarvardStatistics Seminar for helpful comments. This project has benefited from partial fund-ing by the European Research Network “Advanced Mathematical Methods for Finance”(AMAMEF).

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Keywords: robust statistics, coherent risk measures, law invariant riskmeasures, Value at Risk, expected shortfall.

Contents

1 Introduction 3

2 Risk measures 52.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Coherent risk measures . . . . . . . . . . . . . . . . . . . . . . 52.3 Distribution-based risk measures . . . . . . . . . . . . . . . . 6

3 Risk measurement procedures 73.1 Estimation of risk measures . . . . . . . . . . . . . . . . . . . 73.2 Historical risk estimators . . . . . . . . . . . . . . . . . . . . . 93.3 Parametric risk estimators . . . . . . . . . . . . . . . . . . . . 103.4 Effective risk measures . . . . . . . . . . . . . . . . . . . . . . 12

4 Qualitative robustness of risk estimators 144.1 C-robustness of a risk estimator . . . . . . . . . . . . . . . . . 144.2 Qualitative robustness of historical risk estimators . . . . . . 15

4.2.1 Historical VaR α . . . . . . . . . . . . . . . . . . . . . 174.2.2 Historical CVaR and spectral risk measures . . . . . . 174.2.3 A robust tail risk measure . . . . . . . . . . . . . . . . 18

4.3 Qualitative robustness of parametric risk estimators . . . . . 19

5 Sensitivity analysis of risk measurement procedures 225.1 Historical VaR . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Historical estimators of spectral risk measures and CVaR . . 245.3 Parametric estimators: Gaussian model . . . . . . . . . . . . 255.4 Parametric estimators: double-exponential model . . . . . . . 26

6 An example 27

7 Discussion 297.1 Summary of main results . . . . . . . . . . . . . . . . . . . . 297.2 Re-examining subadditivity . . . . . . . . . . . . . . . . . . . 30

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1 Introduction

One of the main purposes of mathematical modeling in finance is to quan-tify the risk of financial portfolios. In connection with the widespreaduse of Value at Risk and related risk measurement methodologies and theBasel committee guidelines for risk-based requirements for regulatory capi-tal, methodologies for measuring of the risk of financial portfolios have beenthe focus of recent attention and have generated a considerable theoreticalliterature [1, 2, 3, 9, 8, 10]. In this theoretical approach to risk measurement,a risk measure is represented as a map assigning a number (a measure ofrisk) to each random payoff. The focus of this literature has been on theproperties of such maps and requirements for the risk measurement proce-dure to be coherent, in a static or dynamic setting.

An implicit starting point of these developments is that the underlyingprobability measure describing market events is known. This is crucial sincemost risk measures such as Value-at-Risk or Expected Shortfall are definedas functionals of the distribution of the considered payoff. In applications,however, this probability distribution is unknown and should be estimatedfrom (historical) data as part of the risk measurement procedure. Thus,in practice, measuring the risk of a financial portfolio involves two steps:estimating the loss distribution of the portfolio from available observationsand computing a risk measure of this distribution which summarizes the riskof the portfolio. While these two steps have been considered and studiedseparately, they are intertwined in applications and an important criterion inthe choice of a risk measure is the availability of good estimation procedures.In order to study the interplay of a risk measure and its estimation methodused for computing it, we define the notion of risk measurement procedure, asa two–step procedure which maps a couple (Dn, X), where X is a payoff andDn a data set on which the estimation is based, to a risk estimate ρ(X) forX. This quantity is supposed to estimate the “abstract” risk measure ρ(X)estimated from the data set Dn. This distinction between an ”abstract”risk measure and its estimation procedure allows us to study the interplaybetween the specification of risk measures and the choice of the estimator.

Once we have acknowledged the uncertainty in the risk estimates result-ing either from estimation or mis-specification errors in the loss distribution,it is natural to examine the sensitivity of the results with respect to theseerrors. Of particular concern is the robustness of the risk measurement:the estimator for the portfolio’s risk is said to be robust if small variationsin the loss distribution –resulting either from estimation or mis-specification

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errors– should result in small variations in the estimator.We propose a rigorous approach for examining these issues, using tools

from robust statistics. We introduce a qualitative notion of ’robustness’ for arisk measurement procedure and a way of quantifying it via sensitivity func-tions. Using these tools we show that there is a conflict between coherence(more precisely, the sub-additivity) of a risk measure and the robustness, inthe statistical sense, of its commonly used estimators. This considerationgoes against the traditional arguments for the use of coherent risk measuresin risk measurement and deserves discussion. We complement this abstractresult by computing measures of sensitivity, which allow to quantify therobustness of various risk measures with respect to the data set used tocompute them. In particular, we show that the same “risk measure” mayexhibit quite different sensitivities depending on the estimation procedureused. These properties are studied in detail for some well known examplesof risk measures: Value at Risk, Expected Shortfall/ CVaR [1, 16, 17] andthe class of spectral risk measures introduced by Acerbi [2]. Our resultsillustrate in particular that using historical Value at Risk instead of alter-native risk measures, suggested in the recent theoretical literature, leads toa more robust procedure for risk measurement.

The article is structured as follows. Section 2 recalls some basic notionson distribution-based risk measures. In Section 3 we establish the distinctionbetween an abstract risk measure and a risk measurement procedure. Wethen show that a risk measurement procedure applied to a data set canbe viewed as the application of an effective risk measure to the empiricaldistribution obtained from this data. We give examples of effective riskmeasures associated to various risk measurement procedures.

Section 4 defines the notion of robustness for a risk measurement pro-cedure and examines whether this property holds for commonly used riskmeasurement procedures. We show in particular that there exists a conflictbetween the sub-additivity of a risk measure and robustness of its estimationprocedure.

In section 5 we define the notion of sensitivity function for a risk mea-sure and we compute sensitivity functions for some commonly used riskmeasurement procedures. The behavior of these sensitivity functions al-low us to compare different risk measurement procedures in terms of theirsensitivity to a change in the underlying data set.

We discuss in section 7 some implications of our findings for the choiceof risk measures and the design of risk measurement procedures in finance.

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2 Risk measures

2.1 Notations

We introduce here some basic notations to be used throughout the paper.We shall denote D = D(R) the (convex) set of cumulative distribution func-tions (cdf) on R. The weak topology on D is defined by the Levy distance:

dL(F,G) , infε > 0 : F (x− ε)− ε ≤ G(x) ≤ F (x+ ε) + ε ∀x ∈ R,

or by the equivalent Prohorov distance [12]. The upper and lower quantilesof F ∈ D of order α ∈ (0, 1) are defined, respectively, by:

q+α (F ) , infx ∈ R : F (x) > α and q−α (F ) , infx ∈ R : F (x) ≥ α.

Plainly q+α (F ) ≥ q−α (F ), but, for any fixed F , equality holds for almost allα ∈ (0, 1). Moreover, the maps α 7→ q+α (F ), q−α (F ) are increasing, thusmeasurable; therefore, integrals involving these maps are well defined.For p ≥ 1 let Dp be the set of distributions having finite p-th moment andby Dp− the set of distributions whose left tail has finite p-th moment. Wedenote µ(F ) the mean of F ∈ D1 and σ2(F ) the variance of F ∈ D2. Forany n ≥ 1 and any x = (x1, . . . , xn) ∈ Rn, let

Fx(x) ,1n

n∑i=1

Ix≥xi

the associated empirical cdf; Demp will denote the set of all empirical cdf.If X is a random variable defined on some probability space (Ω,F ,P), thenFX ∈ D denotes the distribution of X under P, i.e. FX(x) = P(X ≤ x).

2.2 Coherent risk measures

Let (Ω,F ,P) be a probability space representing market scenarios and let L0

denote the space of random variables defined on it. Each random variableX ∈ L0 represents the ”Profit and Loss” (P&L) deriving from the holdingof a portfolio over a specified horizon; in our framework, negative valuesfor X correspond to losses, as in [3] and subsequent works, but unlike mostliterature in actuarial risks. Consider a convex cone V ⊂ L0 containingall the constant r.v. Generally speaking, a risk measure on V is a mapρ : V → R assigning to each P&L X ∈ V its degree of riskiness. Artzneret al [3] formulated a set of requirements, known as coherency axioms thatmonetary measures of risk should verify:

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Definition 1 (Coherent risk measure). A risk measure ρ : V → R is coherentif it is (X,Y ∈ V ):

1. Monotone (decreasing). ρ(X) ≤ ρ(Y ) provided X ≥ Y

2. Cash (or translation) invariant. ρ(X + c) = ρ(X)− c for any c ∈ R.

3. Subadditive. ρ(X + Y ) ≤ ρ(X) + ρ(Y )

4. Positive homogeneous. ρ(λX) = λρ(X) for any λ ∈ R+:

2.3 Distribution-based risk measures

Most risk measures used in practice are distribution-based, i.e. they dependonly on the distribution of the portfolio gain/loss.1

Definition 2 (Distribution-based risk measure). A risk measure ρ : V → Ris said to be distribution-based if ρ(X) = ρ(Y ) whenever FX = FY .

Putting ρ(FX) , ρ(X), we can represent a distribution-based risk mea-sure ρ on V as a map (also denoted ρ) defined on the set of loss distributionsDρ = FX ∈ D : X ∈ V .

A relevant class of distribution-based risk measures which includes allexamples used in applications is given by

ρm(F ) = −∫ 1

0q+u (F )m(du), (1)

where m is a probability measure on [0, 1]. We will denote Dm the set ofdistributions for which the integral above is finite. Notice that if the supportof m does not contain 0 nor 1, then Dm = D. For any choice of the weightm, ρm is monotone, translation invariant and positive homogeneous. Thesubadditivity of such risk measures can be easily characterized [2, 9, 15]:

Proposition 1. A risk measure ρm as in (1) is sub-additive (hence coherent)on Dm if and only if m has a decreasing density: m(du) = φ(u)du where φis a positive decreasing function.

Three particular cases deserve attention:1In the literature such risk measures are often called “law-invariant”, an awkward name

which conveys in fact the opposite of what it should.

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Value at Risk It corresponds to the choice m = δα for a fixed α ∈ (0, 1)(usually α ≤ 5%) and therefore is defined as:

VaR α(F ) , −q+α (F ). (2)

VaR α is not subadditive, and its domain of definition is all D.

Expected shortfall (CVaR) corresponds to choosing m as the uniformdistribution over (0, α), where α ∈ (0, 1) is fixed (again, usually α =1% or 5%):

CVaR α(F ) ,1α

∫ α

0VaR u(F ) du. (3)

In this case, Dm = D1−, the set of distributions having integrable left

tail. Contrarily to VaR, CVaR is a coherent risk measure [1, 2, 9].

Spectral risk measures [15, 1, 2] This class of risk measures generalizesCVaR and corresponds to choosing m(du) = φ(u)du, where φ is adecreasing probability density on (0, 1). Therefore:

ρφ(F ) ,∫ 1

0VaR u(F )φ(u) du. (4)

In view of Proposition 1, spectral risk measures are exactly the riskmeasures in (1) which are coherent. If φ ∈ Lq(0, 1) (but not in Lq+ε)and φ ≡ 0 around 1, then Dm = Dp−, where

1p

+1q

= 1

3 Risk measurement procedures

3.1 Estimation of risk measures

Once a risk measure ρ has been chosen, in practice one has to estimatethe loss distribution of the portfolio from available data, then compute therisk measure using this distribution. Assume that the portfolio loss can bewritten as X = h(Z), where Z ∈ RK is a random vector describing riskfactors. The estimation of a risk measure for the portfolio then involves thefollowing steps:

1. Estimating the distribution of the risk factors from observations z1, . . . , zNof the vector Z. This estimation may be using nonparametric or para-metric methods, leading to an estimated distribution FZ.

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2. Computing the loss distribution of the portfolio from the distributionFZ of the risk factors. In simple cases (e.g. linear portfolios) it is pos-sible to compute this distribution analytically but in most cases (e.g.portfolios with options) a Monte Carlo simulation may be needed atthis stage. The result of this second step is an estimated loss distribu-tion FX .

3. Applying the risk measure to the estimated loss distribution FX , inorder to obtain an estimate ρ(FX).

For ease of presentation, we shall consider the setting where the data areassumed to come from an IID sample of portfolio losses (time series or MonteCarlo simulation), that is X = Z. In this setting the procedure consists in:

1. Estimating the loss portfolio distribution from a data set. This es-timation step can be represented as a map M : X → Dρ, whereX = ∪n≥1Rn represents data sets.

2. Computing the risk associated to the loss portfolio through a riskmeasure ρ : Dρ → R.

X R-

@

@@@R

M ρ

ρ = ρ M

This naturally leads to the notion of risk measurement procedure, whichintegrates these two steps:

Definition 3 (Risk measurement procedure). A risk measurement procedure(RMP) is a couple (M,ρ), where ρ : Dρ → R is a risk measure and M : X →Dρ an estimator for the loss distribution.

From the operational viewpoint, what really matters is the compositionρ = ρ M : X → R which gives us a recipe to directly compute an estimatefor the risk measure out of the data-set (see diagram). We will call the mapρ the risk estimator for ρ associated to the estimation method M .

Given a risk measure ρ, various estimation procedures can lead to dif-ferent risk measurement procedures. We now describe some important ex-amples.

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3.2 Historical risk estimators

Given a data set x ∈ X = ∪n≥1Rn, the historical risk estimator ρh associatedto a risk measure ρ is the estimator obtained by applying the risk measureρ to the empirical loss distribution (sample cdf) Fx:

ρh(x) = ρ(Fx).

For a risk measure ρm, defined in (1) we can easily compute:

ρhm(x) = ρm(Fx) = −n∑i=1

wn,i x(i),

where x(k) is the k-th least element of the set xii≤n, wn,i , m(i−1n , in

]for

i = 1, . . . , n− 1, and wn,n = m(n−1n , 1

). It follows that:

Example 3.1. The historical estimator of VaR α, α ∈ (0, 1) is:

VaRh

α (x) = −x(bnαc+1), (5)

where bac denotes the integer part of a ∈ R.

Example 3.2. The historical estimator of CVaR α, α ∈ (0, 1) is:

CVaRh

α(x) = − 1nα

bnαc∑i=1

x(i) + x(bnαc+1)(nα− bnαc)

(6)

Example 3.3. The historical estimator of the spectral risk measure ρφassociated to φ : [0, 1] → [0,+∞) is given by:

ρhφ(x) = −n∑i=1

wn,i x(i), (7)

where, for any n ≥ 1 and i = 1, . . . , n

wn,i =∫ i/n

(i−1)/nφ(u) du.

Historical risk estimators thus belong to the class of L-estimators in thesense of Huber [12].

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3.3 Parametric risk estimators

In the parametric approach, we choose a parametric family DΘ = (Fθ)θ∈Θ ⊂D of loss distributions, where Θ ⊂ RK (K ≥ 1) is the set of (real) parametersθ = (θ1, . . . , θK) and we assume that the loss distribution lies in this family.For a risk measure ρ : Dρ → R (with DΘ ⊆ Dρ), we can map the parameterof the model to the risk estimate by

r(θ) = ρ(Fθ), θ ∈ Θ

Then, an estimator of θ is chosen, i.e. a map θ : X → Θ assigning to eachdata set x ∈ X a value θ(x) ∈ Θ of the parameter(s). The risk estimatorassociated to the parametric family (Fθ)θ∈Θ is then defined as

ρ(x) = ρ(Fθ(x)

) = r(θ(x)), x ∈ X .

The main example we will study is the maximum likelihood estimator. Moregenerally one can consider the class of M-estimators, obtained by solving aminimization problem of the form:

θ(x) = arg.maxθ∈Θ

n∑i=1

Ψ(xi,θ), x = (x1, . . . , xn) ∈ X , (8)

for some function Ψ = Ψ(x,θ).If Fθ admits a strictly positive density fθ and we take Ψ(x,θ) = log fθ(x)in (8), then we obtain the Maximum Likelihood Estimator (MLE) of θ,denoted by θMLE .

We will focus in what follows on Maximum Likelihood Estimators forlocation-scale families. This includes, for instance, the variance-covarianceapproach to estimate Value at Risk.

Let F? ∈ D be a fixed distribution; the location-scale family associatedwith F? is defined as:

DF? , Fµ,s : µ ∈ R, s > 0, where Fµ,s(x) , F?

(x− µ

s

).

Observe that if F? is the distribution of some X, then Fµ,s is the distributionof sX+µ. It easily turns out that Fµ,s has finite mean (variance) if and onlyif F? does. If F? ∈ D2, we choose to be centered with unit scale parameter:F? = F0,1. Fµ,s has a density if and only if F? has a density and in this casewe have:

fµ,s(x) =1sf?

(x− µ

s

).

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A relevant subfamily is obtained by fixing µ = µ0 (in practice, often µ0 = 0)thus obtaining the scale family DF?,µ0 , Fµ0,s : s > 0. Two importantlocation-scale (or scale) families of (symmetric) distributions that we willstudy are:

• the Gaussian family where

f?(x) =1√2π

exp(−x

2

2

).

• the Laplace or double exponential family where

f?(x) =12

exp (−|x|) .

For a location-scale family derived from a density f?, the MLE µMLE andsMLE of µ and s are defined by the equations:

∑ni=1

f ′?

(xi−µMLE

sMLE

)f?

(xi−µMLE

sMLE

) = 0

∑ni=1

[(xi−µMLEsMLE

) f ′?

(xi−µMLE

sMLE

)f?

(xi−µMLE

sMLE

) + 1

]= 0

Let ρ be a translation invariant and homogeneous risk measure, for instanceany risk measure ρm of type (1) such as VaR , CVaR or any spectral riskmeasure. For the location-scale family DF? , we have

ρ(Fµ,s) = −µ+ c s, ∀Fµ,s ∈ DF? ,

where c = ρ(F?) does not depend on µ and s. As a consequence the ML riskestimator of ρ under the family DF? is

ρMLE(x) = −µMLE(x) + c sMLE(x).

Example 3.4 (ML risk estimators for a Gaussian family).Consider the Gaussian scale-location family, which depends on the param-eters θ = (µ, σ) ∈ Θ = R × R+. The ML estimators θ = (µ, σ) are the”natural” ones:

µ(x) =1n

n∑i=1

xi, s(x) = σ(x) =

√√√√ 1n

n∑i=1

(xi − µ(x))2. (9)

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The resulting usual Gaussian ML risk estimators are:

ρ(x) = −µ(x) + c σ(x), (10)

where c takes the following values, depending on the risk measure considered:

c = VaR α(F?) = −q+α (F?) is (minus) the α-quantile of the N(0, 1) distribution

c = CVaR α(F?) =1α

∫ α

0VaR u(F?) du =

exp−VaR α(F?)2/2α√

c = ρφ(F?) =∫ 1

0VaR u(F?)φ(u) du.

Example 3.5 (ML risk estimators for the Laplace family).Consider now the Laplace with parameters θ = (µ, λ) ∈ Θ = R× R+. The

ML estimators θ = (µ, λ) are

µ(x) =1n

n∑i=1

xi, s(x) = λ(x) =1n

n∑i=1

|xi − µ(x)|. (11)

The resulting Laplace ML risk estimators are therefore:

ρ(x) = −µ(x) + c λ(x), (12)

where c takes the following values, depending on the risk measure considered:

c = VaR α(F?) = − log(2α) for α ∈ (0, 1/2])

c = CVaR α(F?) = − 1α

∫ α

0log(2u) du = − log(2α− 1) for α ∈ (0, 1/2])

c = ρφ(F?) = −∫ 1/2

0log(2u)φ(u) du+

∫ 1

1/2log(2− 2u)φ(u) du.

3.4 Effective risk measures

Let ρ : Dρ → R be a risk measure and ρ : X → R a risk estimator. For asample size n ≥ 1 and any F ∈ D, let Ln(ρ, F ) ∈ D be the law of the riskestimator

Ln(ρ, F ) = Law(ρ(X1, . . . , Xn)), for X1, . . . , Xn ∼ F iid.

A sensible requirement for a risk estimator is consistency:

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Definition 4 (Consistency).A risk estimator ρ is (weakly) consistent with ρ at F ∈ Dρ if

Ln(ρ, F ) → δρ(F ) as n→∞,

where δx denotes the point mass concentrated at x ∈ R.

Given a risk measure ρ that is continuous at some F ∈ Dρ, as a simpleconsequence of the Glivenko-Cantelli Theorem, we can see that the historicalrisk estimator ρh is consistent with ρ at F . For instance, we know thatρ = VaR α is continuous ar F ∈ D if and only if q−α (F ) = q+α (F ). It followsthat the historical estimator of VaR α is consistent with VaR α at any suchF . On the other hand, nothing can be said a priori about consistency atthose F where ρ is not continuous.

Consider now a consistent risk estimator ρ(x1, .., xn) computed froman IID sample X1, ...Xn. A key observation is that, since the estimatorshould be invariant under permutation of the data, its properties shouldonly depend on the sample distribution Fx. We can therefore define a mapρeff : Demp → R such that

ρ(x1, .., xn) = ρeff(Fx) (13)

The map ρeff associates a number to each (empirical) distribution and canthus be seen as a risk measure itself. The two-step risk estimation procedurecan thus be represented as a one-step procedure in which the effective riskmeasure ρ is applied to the empirical (i.e. historical) loss distribution Fx.The above definition relation defines the effective risk measure ρeff(F ) forempirical distributions F ∈ Demp. This definition can be extended to alarger set of loss distributions using consistency:

Definition 5 (Effective risk measure). The effective risk measure associatedto a risk measurement procedure with risk estimator ρ : X → R is

ρ(x1, .., xn) = ρeff(Fx) (14)

If ρ is a consistent risk estimator, the effective risk measure admits a uniqueconsistent extension to

Deff = F ∈ D : Ln(ρ, F ) → δc for some c ∈ R (15)by ρeff(F ) = limn→∞ρ(X1, .., Xn) (16)

where (Xi)i≥1 is any IID sequence with law F .

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We note that by construction the risk estimator ρ is consistent with ρeff

at all F ∈ Deff.For a historical risk estimator ρh which derives from ρ, the set of distri-

butions for which ρ is consistent with ρ is exactly Deff so for all F ∈ Deff,ρeff(F ) = ρ(F ) and the effective risk measure coincides with the initial riskmeasure ρ.

But for parametric risk estimators, the effective risk measure associatedcan be very different from the risk measure initially considered. In thefollowing we give some examples of effective risk measures associated tocommon parametric estimators.

Example 3.6 (Gaussian loss distribution). Consider the risk estimatorsintroduced in Example 3.4. The associated effective risk measure is definedon D2 and given by

ρeff(F ) = −µ(F ) + c σ(F ), (17)

where µ(F ) =∫xF (dx) and σ(F ) =

√∫(x−

∫xF (dx))2F (dx).

Example 3.7 (Laplace loss distribution).Consider the risk estimators introduced in Example 3.5. The associatedeffective risk measure is defined on D1 and given by

ρeff(F ) = −µ(F ) + c λ(F ), (18)

where µ(F ) =∫xF (dx) and λ(F ) =

∫|x−

∫xF (dx)|F (dx).

4 Qualitative robustness of risk estimators

If we take into account the errors in the estimation of the loss distributionof the portfolio, an important question is the impact of these errors on therisk measure of the portfolio. A risk estimator is said to be robust if a smallvariation in the loss distribution of the portfolio results in a small change inthe distribution of the risk estimator.

4.1 C-robustness of a risk estimator

In order to make this definition precise, one needs to quantify the notionof “small change” in the loss distribution and in the law of the estimator.We use here the Prohorov distance dP (see paragraph 2.1) to quantify thecloseness of probability measures. We will denote Bδ(F ) the ball centered

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in F with radius δ. Fix a set C ⊆ D of plausible loss distributions andF ∈ C. We assume F is not an isolated point of C, i.e. for any δ > 0, thereexists G 6= F such that G ∈ C and dP (G,F ) < δ. The intuitive notion ofrobustness can now be made precise using the following definition.

Definition 6 (C-robustness of a risk estimator).A risk estimator ρ is C-robust at F if for any ε > 0 there exist δ > 0 andn0 ≥ 1 such that:

G ∈ C, dP (F,G) ≤ δ =⇒ dP (Ln(ρ, F ),Ln(ρ, G)) ≤ ε, ∀n ≥ n0.

Alternatively, the Levy metric can be used instead of the Prohorov met-ric. When C = D, i.e. when any perturbation of the loss distribution isallowed, the previous definition corresponds to the notion of qualitative ro-bustness as proposed by Huber [12]. However, this case is not generallyinteresting in econometric or financial applications since requiring robust-ness against all perturbations of the model F is quite restrictive.

In our case, the notion of C-robustness depends on the class C of admis-sible distributions: for example the sample mean, which is well known notto be robust in the sense of Huber [12], is C-robust at any F ∈ D1 if C is aset of distributions with support included in ]F−1(α), F−1(1− α)[ for some0 < α < 1.

Note that this notion of C–robustness is local in the following sense. Ifρ is C-robust at F and C′ is another set such that C′ ∩ Bδ(F ) = C ∩ Bδ(F )for some δ > 0, then ρ is also C′-robust.

4.2 Qualitative robustness of historical risk estimators

In this paragraph we focus on historical estimators ρh(x) = ρ(Fx), andstudy which among them are C-robust, in the sense of Definition 6, whereC is a suitable set of plausible loss distributions. The following proposi-tion characterizes robustness among historical risk measures; a related, butslightly different result is due to Hampel [11].

Proposition 2. Let ρ be a risk measure, C ⊆ D and F ∈ C. If ρh isconsistent with ρ at every G ∈ C, the following are equivalent:

1. the restriction of ρ to C is weakly continuous at F ;

2. ρh is C-robust at F .

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Proof. First, observe that for any fixed ε > 0 and G ∈ C, as ρh is consistentwith ρ at F and G, there exists n∗ ≥ 1 such that

dP (Ln(ρh, F ), δρ(F )) + dP (Ln(ρh, G), δρ(G)) <2ε3, ∀n ≥ n∗. (19)

”1. ⇒ 2”. Assume that ρ|C is continuous at F and fix ε > 0. Then thereexists δ > 0 such that if dP (F,G) < δ, then dP (δρ(F ), δρ(G)) = |ρ(F ) −ρ(G)| < ε/3. Thus C-robustness readily follows from (19) and the triangularinequality

dP (Ln(ρh, F ),Ln(ρh, G)) ≤≤ dP (Ln(ρh, F ), δρ(F )) + dP (δρ(F ), δρ(G))+ ≤ dP (Ln(ρh, G), δρ(G)).

”2. ⇒ 1”. Conversely, assume that ρh is C-robust at F and fix ε > 0. Thenthere exists δ > 0 and n ≥ 1 such that

dP (F,G) < δ, G ∈ C ⇒ dP (Ln(ρh, F ),Ln(ρh, G)) < ε/3.

As a consequence, from (19) and the triangular inequality

|ρ(F )− ρ(G)| = dP (δρ(F ), δρ(G)) ≤≤ dP (δρ(F ),Ln(ρh, F )) + dP (Ln(ρh, F ),Ln(ρh, G)) + dP (Ln(ρh, G), δρ(G)),

it follows that ρ|C is continuous at F .

In view of the fact that a weakly continuous risk measure ρ has a con-sistent historical risk estimator we have:

Corollary 1. If ρ is weakly continuous at any G ∈ C, then ρh is C-robustat any F ∈ C.

Our analysis will use the following important result [12, Theorem 3.1]:

Theorem 1. Let ρm be a risk measure of the form (1), where m is a probabil-ity measure on (0, 1). Let α be the largest α ≥ 0 such that supp(m) ⊆ [α, 1−α] and αnn≥1 the (countable) set of point masses of m (i.e. m(αn) > 0).Then:

1. if α > 0 then ρm is weakly continuous at any F ∈ Dρ such thatq+F (αn) = q−F (αn) for any n

2. if α = 0 then ρm is not continuous at any F ∈ Dρ

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4.2.1 Historical VaR α

Using the previous two results, we now show that the historical VaR isC-robust, where C is the set of all distributions continuous at α:

Proposition 3 (Historical risk estimator of VaR α).For α ∈ (0, 1), let

Cα , F ∈ D : q+F (α) = q−F (α).

Then the historical VaR α is Cα-robust at any F ∈ Cα.

Proof. By using Theorem 1, we know that VaR α is weakly continuous atany F ∈ Cα. Therefore, by applying Corollary 1 we obtain that the historicalVaR α is Cα-robust at any F ∈ Cα.

This Proposition confirms the intuition that if the quantile of a dis-tribution is uniquely determined, then the empirical quantile is a robustestimator.

4.2.2 Historical CVaR and spectral risk measures

Let us now give a characterization of robustness when m(du) = φ(u) du forsome density φ on (0, 1). This case will cover spectral risk measures and inparticular expected shortfall/CVaR . For p ∈ [1,∞), we denote Dp the setof distributions with finite p-th moment.

Proposition 4. Let φ be a density in Lq(0, 1), with q ∈ (1,∞], 1/p+1/q =1. Let ρφ be the risk measure defined on Dp by

ρφ(F ) =∫ 1

0VaR u(F )φ(u) du.

Pick F ∈ Dp, such that no discontinuity of φ coincides with a discontinuityof qF , the quantile function of F . Then the historical estimator of ρφ isDp-robust at F if and only

supp(φ) ⊆ [α, 1− α]

for some α > 0.

Proof. Consider F ∈ Dp, such that no discontinuity of φ coincides with adiscontinuity of qF . By contraposition, we assume that α = 0. Then byusing Theorem 1 we know that ρφ is not continuous at F . As ρφ is defined

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on Dp, it is equivalent to saying that the restriction of ρφ to Dp is not con-tinuous at F . A result of Van Zwet [18] implies that the historical estimatorof ρφ is consistent with ρφ at any G ∈ Dp. Thus by applying Proposition 2we conclude that the historical estimator of ρφ is not Dp-robust at F .Now we assume that α > 0, then Theorem 1 shows that ρφ is weakly con-tinuous at F , which is equivalent to saying that the restriction of ρφ to Dpis weakly continuous at F . Therefore, by using Proposition 2, we see thatthe historical estimator of ρφ is Dp-robust at F .

Corollary 2. Proposition 4 implies that:

1. The historical CVaR α is not robust on D1 = F ∈ P,∫ +∞−∞ |x| dF <

∞. Indeed, in this case φ(u) = 1[0,α]

α and supp(φ) = [0, α].

2. More generally, historical estimators of spectral risk measures, i.e.with a decreasing function φ in Lq(0, 1), are not Dp-robust at anyF ∈ Dp for 1/p+ 1/q = 1.

This proposition stresses a conflict between subadditivity and robust-ness, which means that requiring coherence to a risk measure contrasts withthe robustness of its historical estimator.

More generally, a historical estimator of any coherent risk measure de-fined as the supremum of a finite number of coherent spectral risk measures:

ρφ(F ) = supi=1,..,K

∫ 1

0VaR α(F ) φi(α) dα (20)

with decreasing functions φi ≥ 0 in Lq(0, 1) for i = 1, ..,K, is notDp-robust at any F ∈ Dp.

Kusuoka [15] has shown that any coherent distribution-based risk mea-sure can be represented in the form above where the supremum is takenover a possibly infinite family of decreasing weight functions φ; so the abovestatement has a fairly general flavor.

4.2.3 A robust tail risk measure

From Proposition 1, the sub-additivity of ρφ implies that φ is decreasingwhile the robustness of the historical estimator ρh implies that supp(φ) ⊂[α, 1−α] for some α > 0. These two requirements are clearly not compatible,which reveals a conflict between robustness and subadditivity.

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Although historical estimators of CVaR α are not robust, one can easilymodify them to obtain a robust weighted VaR, defined for F ∈ D1 by

WV aRα1,α2(F ) =1

α2 − α1

∫ α2

α1

dαVaRα(F ) (21)

Corollary 3. Let us consider 0 < α1 < α2 < 1 and a distribution F inD1 such that q+F (α1) = q−F (α1) and q+F (α2) = q−F (α2). Then the historicalestimator of WV aRα1,α2 is D1-robust at F .

Due to the conflict between subadditivity and robustness noted above,WV aRα1,α2 is not (asymptotically) sub-additive, but for small α1 it is infact empirically undistinguishable from CVaR α2 . This point will be furtherdiscussed in Section 7.

4.3 Qualitative robustness of parametric risk estimators

We now discuss robustness properties of parametric risk estimators of trans-lation invariant and homogeneous risk measures (for instance any risk mea-sure ρm of type (1)). Pick a distribution F? ∈ D and fix the location param-eter to µ0. We consider in this section scale parameters that are computedvia the Maximum Likelihood (ML). For F ∈ DF?,µ0 we denote by s(F ) thescale parameter of F . Define

ψ?(x) = −1− xf ′?(x)f?(x)

(22)

Then the scale parameter is a solution of

λ(s, F ) ,∫ψ?

(x− µ0

s

)F (dx) = 0 for F ∈ DF?,µ0 (23)

By defining Dψ? = F ∈ D|∫ψ?(x)F (dx) <∞, we can define s(F ) for any

F ∈ Dψ? . Note that if F /∈ DF?,µ0 , s(F ) does not correspond to the “scaleparameter” of F . Moreover if we compute the scale parameter of Fx ∈ Demp

we recover the classical ML estimator s(x) presented in section 3.

Example 4.1 (Gaussian family). The function ψg? for the Gaussian familycan be easily computed

ψg?(x) = −1 + x2, (24)

and we get Dψg?

= D2 and recover the well-known MLE of the Gaussianvariance

s(x) =

√√√√ 1n

n∑i=1

(xi − µ0)2. (25)

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Example 4.2 (Laplace family). The scale parameter of the Laplace familycorresponds to λ. The function ψl? is the following

ψl?(x) = −1 + |x|, (26)

and we get Dψl?

= D1 and obtain the following ML estimator of s

s(x) =1n

n∑i=1

|xi − µ0|. (27)

The following result exhibits conditions on the function ψ? under whichthe maximum likelihood estimator of the scale parameter is weakly contin-uous on Dψ? :

Theorem 2 (Weak continuity of the MLE). Let us consider Dψ?, a scalefamily associated to ψ? and s : Dψ? → R+ the scale function defined onDψ?. Suppose now that ψ? is even, monotone and increasing on R+ andtakes values of both signs. Then, these two assertions are equivalent

• s is weakly continuous at F ∈ Dψ?

• ψ? is bounded and λ(s, F ) ,∫ψ?

(x−µ0

s

)F (dx) = 0 has a unique

solution s = s(F ) for F ∈ Dψ?.

Proof. Consider Dψ? a scale family associated to ψ?. For simplicity, weconsider here that µ0 = 0. We can assume that F ∈ Dψ? is the distributionof the random variable X representing the profits and losses of a portfolio.We will show that the continuity problem of the scale function s : Dψ? → R+

of portfolios X can be reduced to the continuity (on a properly definedspace) issue of the location function of portfolios Y = log(X2). The changeof parameter here is made to use results of [12] concerning weak continuityof location parameters.

By denotingG the distribution of Y , we getG(y) = P (Y < y) = P (X2 <ey) = F (ey/2)− F (−ey/2), with density g(y) = G′(y) = ey/2 f(ey/2). More-over, if we denote τ the location function defined on the location family ofdistributions Dϕ? , G :

∫ϕ?(y)G(dy) <∞, where ϕ?(y− τ) = −g′?(y−τ)

g?(y−τ) ,then it is defined by the solution of the following implicit relation∫

ϕ?(y − τ)G(dy) = 0 (28)

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for G ∈ Dϕ? . Let us now replace g? by its value w.r.t f?

ϕ?(y − τ) = −g′?(y − τ)g?(y − τ)

= −12e

(y−τ)/2f?(e(y−τ)/2) + 12ey−τf ′?(e

(y−τ)/2)e(y−τ)/2f?(e(y−τ)/2)

= −12

[1 + e(y−τ)/2

f ′?(e(y−τ)/2)

f?(e(y−τ)/2)

]

= −12

[1 +

x

s

f ′?(xs )

f?(xs )

]=

12ψ?

(xs

). (29)

Moreover, we can also rewrite G(dy) as

G(dy) = g′(y)dy = xf(x)d(log(x2)

)= 2f(x)dx = 2F (dx),

which ends to show the equivalence between the function s(F ) for F ∈ Dψ?

and the function τ(G) for G ∈ Dϕ? . We have shown that a scale functioncharacterized by the function ψ?, can also be interpreted as a location func-tion characterized by the function ϕ?. Therefore, the weak continuity ofthe scale function will depend on the behavior of ϕ?. From Equation (29),we see that for all x ∈ R, 2ϕ?(x) = ψ?[ex/2]. Therefore, as the associatedfunction ψ? to the function s(.) is assumed to be even, and monotone in-creasing on R+, it implies that ϕ? is monotone increasing on R. Moreover,as ψ? takes values of both signs it is also true for ϕ?. To conclude, we ap-ply [12, Theorem 2.6] which states that a location function associated toϕ? is weakly continuous at G if and only if ϕ? is bounded and the locationfunction computed at G is unique.

Theorem 2 enables us to study the robustness of parametric risk es-timators for Gaussian or Laplace scale families i.e. with a fixed locationparameter µ = µ0:

Corollary 4 (Parametric risk estimators for Gaussian and Laplace scalefamilies). Gaussian (resp. Laplace) risk estimators of translation invariantand homogeneous risk measures are not D2-robust (resp. D1-robust) at anyF in D2 (resp. in D2).

Proof. We detail the proof for the Gaussian scale family. The same argumentcan be used for the Laplace scale family as well. Let us consider a Gaus-sian risk estimator of a translation invariant and homogeneous risk measure,

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denoted ρ(x) = −µ0 + c σ(x). First of all we notice that the function ψg?associated to the ML estimation of the scale parameter of a distributionbelonging to the Gaussian scale family is even, and increasing on R+. More-over it takes values of both signs. Secondly, we recall that the effective riskmeasure associated to the Gaussian risk estimator is ρeff(F ) = −µ0 + c σ(F )for all F ∈ Deff = Dψg

?= D2. Therefore, as ψg? is unbounded, by using The-

orem 2, we know that ρeff is not continuous at any F ∈ D2. As the Gaussianrisk estimator considered ρ verifies ρ(x) = ρeff

h(x), and is consistent withρeff at all F ∈ D2 by construction, we can apply Proposition 2 to concludethat, for F ∈ D2, ρ is not D2-robust at F .

5 Sensitivity analysis of risk measurement proce-dures

The discussion above leads to distinguish robust from non-robust risk mea-surement procedures. This qualitative analysis can be further refined througha quantitative sensitivity analysis that we now discuss. Intuitively, the em-pirical sensitivity of a risk estimator, computed from an IID sample of sizeN with law F , to the addition of a new observation can be measured by

SN (x, F ) =ρ(X1, ..., XN , XN+1)− ρ(X1, ..., XN )

1N+1

=ρeff( 1

N+1

∑Ni=1 1u≥Xi + 1u≥x)− ρeff( 1

N

∑Ni=1 1u≥Xi)

1N+1

We will call this quantity the empirical sensitivity function of the risk esti-mator ρ. This motivates the following definition in the large sample case:

Definition 7 (Sensitivity function of a risk measurement procedure). Thesensitivity function of a risk measurement procedure (M,ρ) at F ∈ Deff isthe real-valued map S defined by

S(z;F ) , limε→0+

ρeff(εδz + (1− ε)F )− ρeff(F )ε

for any z ∈ R such that the limit exists.

S(z, F ) measures the sensitivity of the risk estimator based on a largesample to the addition of a new data point. S(z;F ) is nothing but thedirectional derivative of the effective risk measure ρeff at F in the directionδz ∈ D. In the language of robust statistics, it is the influence function of the

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risk estimator ρ and is related to the asymptotic variance of the historicalestimator of ρ [12, 11].

Remark 1. If Dρ is convex and contains all empirical distributions, thenεδz + (1− ε)F ∈ Dρ for any ε ∈ [0, 1], z ∈ R and F ∈ Dρ. These conditionshold for all the risk measures we are considering.

5.1 Historical VaR

We have seen before that the effective risk measure associated to VaRh

α isthe restriction of VaR α to

Deff = Cα = F ∈ D : q−α (F ) = q+α (F ).

We want to compute the sensitivity function of the historical VaR α, at leastin the plain case when F ∈ D admits a strictly positive density f ; it isimmediate to see that in such a case F ∈ Cu for any u ∈ (0, 1).

Proposition 5. If F ∈ D admits a strictly positive density f , then thesensitivity function at F of the historical VaR α is

S(z) =

1− α

f(qα(F ))if z < qα(F )

0 if z = qα(F )

− α

f(qα(F ))if z > qα(F )

(30)

Proof. First we observe that the map u 7→ q(u) , qu(F ) is the inverse of Fand so it is differentiable at any u ∈ (0, 1) and we have:

q′(u) =1

F ′(q(u))=

1f(qu(F ))

.

Fix z ∈ R and set, for ε ∈ [0, 1), Fε = εδz + (1 − ε)F , so that F ≡ F0.For ε > 0, the distribution Fε is differentiable at any x 6= z, with F ′

ε(x) =(1 − ε)f(x) > 0, and has a jump (of size ε) at the point x = z. Hence, forany u ∈ (0, 1) and ε ∈ [0, 1), Fε ∈ Cu, i.e. q−u (Fε) = q+u (Fε) , qu(Fε). Inparticular we can easily compute:

qα(Fε) =

q( α

1−ε) for α < (1− ε)F (z)

q(α−ε1−ε ) for α ≥ (1− ε)F (z) + ε

z otherwise

(31)

Assume now that z > q(α), i.e. F (z) > α; from (31) it follows that

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qα(Fε) = q( α1−ε), for ε < 1− α

F (z) .

As a consequence

S(z) = limε→0+

VaR α(Fε)−VaR α(F0)ε

= − d

dεqα(Fε)|ε=0

= − d

dεq

1− ε

)∣∣∣∣ε=0

= −

[1

f(q( α1−ε))

α

(1− ε)2

]ε=0

= − α

f(qα(F ))

The case q(α) < z is handled in a very similar way. Finally, if z = q(α),then, again by (31) we have qα(Fε) = z for any ε ∈ [0, 1). Hence, in thiscase

S(z) = − d

dεqα(Fε)|ε=0 = 0,

and we conclude.

This example shows that the historical VaR α has a bounded sensitiv-ity to a change in the data set, which means that this risk measurementprocedure is not very sensitive to a small change in the data set.

5.2 Historical estimators of spectral risk measures and CVaR

Consider a distribution F having positive density f > 0. Assume that:∫ 1

0

φ(u)f(qu(F ))

du <∞.

Proposition 6. The sensitivity function at F ∈ Dφ of the historical esti-mation procedure of ρφ is

S(z) = −∫ 1

0

u

f(qu(F ))φ(u) du+

∫ 1

F (z)

1f(qu(F ))

φ(u) du

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Proof. Using the notations of the previous proof we have:

S(z) = limε→0+

∫ 1

0

VaR u(Fε)−VaR u(F )ε

φ(u) du

=∫ 1

0limε→0+

VaR u(Fε)−VaR u(F )ε

φ(u) du

=∫ 1

0−

[d

dεqu(Fε)

]ε=0

φ(u) du

=∫ F (z)

0

−uf(qu(F ))

φ(u) du+∫ 1

F (z)

1− u

f(qu(F ))φ(u) du,

thanks to Proposition 5. We stress that changing the integral with the limitin the second equality above is legitimate. Indeed, limε→0+ ε−1(VaR u(Fε))−VaR u(F )) exists, is finite for all u ∈ (0, 1), and for ε small∣∣∣∣VaR u(Fε)−VaR u(F )

ε

∣∣∣∣ < 1f(qu(F ))

∈ L1(φ),

so that we can apply dominated convergence.

Since the effective risk measure associated to historical CVaR α is CVaR α

itself, defined on D− = F ∈ D :∫x−F (dx) < ∞, an immediate conse-

quence of the previous proposition is the following

Corollary 5. The sensitivity function at F ∈ D− for historical CVaR α is

S(z) =

− zα

+1− α

αqα(F )− CVaR α(F ) if z ≤ qα(F )

−qα(F )− CVaR α(F ) if z ≥ qα(F )

This result shows that the sensitivity of historical CVaR α is linear in z,and thus unbounded. It means that this risk measurement procedure is lessrobust than the historical VaR α.

5.3 Parametric estimators: Gaussian model

We have seen that the effective risk measure of the Gaussian estimators ofVaR , CVaR , or any spectral risk measure is

ρeff(F ) = −µ(F ) + cσ(F ), F ∈ Deff = D2,

where c = ρ(N(0, 1)) is a constant depending only on the risk measure ρ(we are not interested in its explicit value here).

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Proposition 7. The sensitivity function at F ∈ D2 of the Gaussian esti-mator associated to a translation invariant and homogeneous risk measureρ is

S(z) = µ− z +σ c

2

[(z − µ

σ

)2

− 1

]where c = ρ(N(0, 1)). For the Gaussian scale family (µ = µ0) we have:

S(z) =σ c

2

[(z − µ0

σ

)2

− 1

].

Proof. Let, for simplicity, µ = µ(F ) and σ = σ(F ). Fix z ∈ R and set, asusual, Fε = (1− ε)F + εδz (ε ∈ [0, 1)); observe that Fε ∈ D2 for any ε. If weset ψ(ε) , −µ(Fε)+cσ(Fε), with c = ρ(N(0, 1)), then we have S(z) = ψ′(0).It is immediate to compute µ(Fε) = (1− ε)µ+ εz and

σ2(Fε) =∫

Rx2 Fε(dx)− µ(Fε)2

= (1− ε)∫

Rx2 F (dx) + εz2 − [(1− ε)µ+ εz]2

= (1− ε)(σ2 + µ2) + εz2 − (1− ε)2µ2 − ε2z2 − 2ε(1− ε)µz= σ2 + ε[(z − µ)2 − σ2]− ε2(µ2 + σ2 − 2µz)

As a consequence

ψ′(0) =d

[−(1− ε)µ− εz + c

√σ2(Fε)

]ε=0

= µ− z +σ c

2

[(z − µ

σ

)2

− 1

]as we desired. Similarly, starting from ψ(ε) = −µ0 + cσ(Fε), we obtain thesecond statement.

5.4 Parametric estimators: double-exponential model

Consider now a translation invariant and homogeneous risk measure ρ andits MLE estimator under the Laplace family. We have seen that the effectiverisk measure associated to this risk estimator is

ρeff(F ) = −µ(F ) + c λ(F ), F ∈ Deff = D1,

where c = ρ(G), and G is the distribution with density g(x) = e−|x|/2.

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Risk measurement procedure Dependence in z of S(z)Historical VaR boundedGaussian VaR quadraticLaplace VaR linear

Historical CVaR linearGaussian CVaR quadraticLaplace CVaR linear

Table 1: Behavior of sensitivity functions for some important examples ofrisk measurement procedures.

Proposition 8. Let ρ be a translation invariant and homogeneous risk mea-sure. The sensitivity function at F ∈ D1 of its parametric estimator basedon the Laplace scale family with location parameter µ = µ0 is

S(z) = λ c

[|z − µ0|

λ− 1

]where λ = λ(F ).

Proof. As usual, we have, for z ∈ R, S(z) = ψ′(0), where ψ(ε) = −µ0 +c λ(Fε), Fε = (1 − ε)F + εδz and c is defined above. For simplicity, let usdenote λ = λ(F ). We have

ψ(ε) = c (1− ε)λ+ c ε |z − µ0|,

henceψ′(0) = c|z − µ0| − c λ.

This proposition shows that the sensitivity of the Laplace risk estimatorat any F ∈ D1 is not bounded but linear in z. Nonetheless, the sensitivityof the Gaussian risk estimator is quadratic at any F ∈ D2, which indicatesa higher sensitivity to outliers in the data set.

6 An example

We consider in this section an application of the above concepts to a measureof counterparty risk for a derivatives portfolio. The data for the examplestudied here has been provided to us by Societe Generale Risk Management

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unit: it consists of 1000 risk scenarios for the bank portfolio, simulated usingthe internal model of the bank and incorporating hundreds of different riskfactors (market risk, interest rate risk, counterparty default). Each scenariois simulated over 30 years and the objective is to quantify the counterpartyrisk exposure of the bank’s derivatives portfolio via an appropriate tail riskmeasure of the loss distribution at various horizons (e.g. 1 day, 1 month, 1year and 10 years). In our study, we have considered the 1 day horizon andplotted the portfolio histogram in figure 1.

Figure 1: Histogram of portfolio gain at a 1 day horizon.

In table 2, we have computed for different quantile levels α = 1%, 0.4%the following risk estimators: historical, gaussian, and laplace VaR andhistorical, gaussian, and laplace CVaR .

Then, in order to analyze the accuracy of our theoretical risk estima-tor sensitivities, we have computed the empirical sensitivities of historical,gaussian, and laplace VaR and historical, gaussian, and laplace CVaR andplotted them with the theoretical one. The result are shown in figures 2 and3, where the x-axis represents the value of the portfolio, and the y-axis thesensitivity to the corresponding value of the portfolio.

One can notice that the theoretical and empirical sensitivities coincidefor all risk estimators except for historical risk measurement procedures. Forthe historical CVaR , the theoretical sensitivity is very close to the empirical

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Risk measurement procedure α = 1% α = 0.4%Historical VaR 8.887 9.193

Historical CVaR 9.291 9.606Gaussian VaR 8.876 9.351

Gaussian CVaR 9.370 9.802Laplace VaR 9.970 11.021

Laplace CVaR 11.117 12.167

Table 2: Risk estimators for α = 1%, 0.4% at a 1 day horizon computed forseveral risk measurement procedures (in million euro).

one. Nonetheless, we notice that the empirical sensitivity of the historicalVaR can be equal to 0 because it is strongly dependent on the integer partof N α, where N is the number of scenarios and α the order of the quantile.This dependency disappears asymptotically.

While the empirical sensitivities, which require perturbating the datasets and recomputing the risk measures, are quite costly to compute, thesensitivity functions are analytically and thus easily computable. The excel-lent agreement for realistic sample sizes shown in these examples implies thatour theoretical sensitivity functions are useful for evaluating the sensitivityof risk estimators in such practical settings.

7 Discussion

7.1 Summary of main results

Let us now summarize the contributions and main conclusions of this study.First, we have argued that when the estimation step is explicitly taken

into account in a risk measurement procedure, issues like robustness andsensitivity to the data set are important and need to be accounted for withat least the same attention as the coherence properties set forth by Artzneret al [3]. Indeed, an unstable/non-robust risk estimator, be it related to acoherent measure of risk, is of little use in practice.

Second, we have shown that the choice of the estimation method mat-ters when discussing the robustness of risk measurement procedures: ourexamples show that different estimation methods coupled with the samerisk measure lead to very different properties in terms of robustness andsensitivity.

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Nonparametric (“historical”) VaR is a qualitatively robust estimationprocedure, whereas the proposed examples of coherent (distribution-based)risk measures do not pass the test of qualitative robustness. Most paramet-ric estimation procedures for VaR and CVaR lead to nonrobust estimators.These observations plead against the use of CVaR . On the other hand his-torical VaR and more generally weighted averages of historical VaR of thetype

1α2 − α1

∫ α2

α1

VaRu(F ) du

lead to robust empirical estimators.

7.2 Re-examining subadditivity

The conflict we have noted between robustness of a risk measurement proce-dure and the subadditivity of the risk measure shows that one cannot achieverobust estimation in this framework while preserving subadditivity. Whilea strict adherence to the coherence axioms of Artzner et al [3] would pushus to choose subadditivity over robustness, several recent studies [7, 13, 14]have provided reasons for not doing so.

Danielsson et al. [7] explore the potential for violations of VaR subad-ditivity and report that for most practical applications VaR is sub-additive.They conclude that in practical situations there is no reason to choose amore complicated risk measure than VaR, solely for reasons of subadditiv-ity. Arguing in a different direction, Ibragimov & Walden [13] show thatfor very “heavy-tailed” risks defined in a very general sense, diversificationdoes not necessarily decrease tail risk but actually can increase it, in whichcase requiring sub-additivity would in fact be unnatural. Finally, Kou etal [14] argue against subadditivity from an axiomatic viewpoint and pro-pose to replace it by a weaker property of co-monotonic subadditivity. Allthese objections to the sub-additivity axiom deserve serious considerationand further support the choice of robust risk measurement procedures overnon-robust ones for the sole reason of saving sub-additivity.

Without taking an axiomatic stance on these issues, we hope to haveconvinced the reader that there is more to risk measurement than the choiceof a “risk measure”. We think that the property of robustness - and notonly the coherence - should be a concern for regulators and end-users whendesigning risk measurement procedures. What our study illustrates is thatthe design of robust risk estimation procedures requires the inclusion of thestatistical estimation step in the risk measurement procedure. We hope thiswork will stimulate further discussion on these important issues.

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[14] Kou S., Heyde C.C. and Peng X.H. (2007) What Is a Good Risk Mea-sure: Bridging the Gaps between Data, Coherent Risk Measures, andInsurance Risk Measures, Financial Engineering Report 2007-09, Cen-ter for Financial Engineering, Columbia University.

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Figure 2: Sensitivity of risk estimators for α = 1% at a 1 day hori-zon. Historical VaR (upper left), Historical CVaR (upper right), Gaus-sian VaR (left), Gaussian CVaR (right), Laplace VaR (lower left), LaplaceCVaR (lower right).

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Figure 3: Sensitivity of risk estimators for α = 0.4% at a 1 day hori-zon. Historical VaR (upper left), Historical CVaR (upper right), Gaus-sian VaR (left), Gaussian CVaR (right), Laplace VaR (lower left), LaplaceCVaR (lower right).

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