aa r X i v : . [ ec on . GN ] F e b THE VAR AT RISK

ALFRED GALICHON † Abstract.

I show that the structure of the ﬁrm is not neutral with respect to regulatorycapital budgeted under rules which are based on the Value-at-Risk. Indeed, when a holdingcompany has the liberty to divide its risk into as many subsidiaries as needed, and when thesubsidiaries are subject to capital requirements according to the Value-at-Risk budgetingrule, then there is an optimal way to divide risk which is such that the total amount ofcapital to be budgeted by the shareholder is zero. This result may lead to regulatoryarbitrage by some ﬁrms.

Keywords : value-at-risk.

MSC 2000 subject classiﬁcation : 91B06, 91B30, 90C08 Introduction

The single most used measure of ﬁnancial risk is undoubtedly the Value-at-Risk (VaR).The VaR at level 95% is deﬁned as the minimal amount of capital which is required to coverthe losses in 95% of cases. In statistical terms, the value-at-risk is the quantile of level α ofthe losses, namely V aR α ( X ) = inf { x : Pr ( X ≤ x ) > α } . Date : First version is June 3, 2008. The present version is of January 14, 2010. † † (note that unlike the most widely adopted convention in the literature, we chose to countpositively an eﬀective loss).The widespread popularity of the Value-at-Risk is due to its adoption as a the “1st pillar”in the Basle II agreements. Despite its widespread use and simplicity, the Value-at-Riskis highly criticized among academics. The literature on risk measures classically deﬁnesa set of axioms which satisfactory risk measures should satisfy (see [1]). Among these, subadditivity : a risk measure ρ should satisfy ρ ( X + Y ) ≤ ρ ( X ) + ρ ( Y ). While this axiomis widely accepted in the academic community, it is perhaps ironical that VaR fails to satisfythis subadditivity axiom as it has been widely documented (we come back to that below).The subadditivity axiom is generally motivated by a loose invocation of risk aversion, orpreference for diversiﬁcation. In this note I propose a quite diﬀerent argument to motivatethe importance of the subadditivity axiom. Assuming the value-at-risk is used to budgetregulatory capital requirements, and assuming that the managers have an incentive tominimize these capital requirements, I show that the lack of the subadditivity propertyinduces the possibility for the management to optimally divide their risk in order to minimizetheir budgeted capital: the structure of the ﬁrm is not neutral to the aggregated capitalrequirement. More precisely, I show that for any level of the value-at-risk there is a divisionof the risk which sets the aggregated capital requirement to zero.2. The problem

Consider a trading ﬂoor which is organized into N various desks. For each trading desk i = 1 , ..., N , call X i the random variable of the contingent loss of trading ﬂoor i . The totalrandom loss of the trading ﬂoor is X = P Ni =1 X i , we suppose that this amount is bounded: X ∈ [0 , M ] almost surely.We suppose that each desk budgets a regulatory capital equal to its VaR at level α ∈ (0 , V aR α ( X i ). Consequently the total amount of regulatory capital which the managementneeds to budget is P Ni =1 V aR α ( X i ). It is therefore easy to formulate the manager’s problem:inf N X i =1 V aR α ( X i ) s.t. N X i =1 X i = X a.s., and V aR α ( X i ) ≥ . (2.1) AR AT RISK 3

As the management has full control over the structure of the ﬁrm, it results that it has thechoice over N and over the random variables ( X , ..., X N ) which satisﬁes the constraints.Note that the economical risk which the trading ﬂoor bears is X = P Ni =1 X i , and thusthe regulatory capital to be budgeted should be V aR α ( X ) >

0. However, I shall show that,under mild assumptions, there is a structure of the ﬁrm such that the capital budgetedunder the rule above is zero. The assumption needed is the following:

Assumption.

The distribution of X is absolutely continuous with respect to the Lebesguemeasure. As we shall see in Point 6. in the discussion below this assumption can in fact be removed.

The optimal structure.

In that case, there exists a sequence of real numbers x = 0

Discussion

1. This example shows that it is possible to tear down the risk into small pieces which areindetectable by the Value-at-Risk. Quite obviously, it is far too stylized to aim at modelling this is a slight abuse of terminology as a digital option would more correctly characterize x i { X ∈ ( x i , x i +1 ) } , but both expressions are close when x i and x i +1 are close. ALFRED GALICHON † any of the real-life events that took place on the credit market in 2007-2008. Instead itsambition is to providing an alternative illustration of a major ﬂaw in the Value-at-Risk –namely the fact that it is insensitive to the seriousness of the events in the tail. It takesinto account the probability that the threshold point V aR α be reached, not how serious thelosses are beyond that point.2. Since the seminar work of Modigliani-Miller, a vast literature in corporate ﬁnance hasdiscussed whether the structure of the capital of the ﬁrm does impact or not the value ofthe ﬁrm for the stakeholders. In Modigliani-Miller’s result the value of the ﬁrm dependslinearly on the risky capital claims, thus yielding to the celebrated structural neutralityresult. When one introduces nonlinear tax on risky capital (such as the Value-at-Risk inthe present setting), the result is reversed, and one has to conclude to the non-neutrality ofthe capital structure.3. An obvious consequence of this paper’s point is that the VaR is not additive. Note thatthere are cases where VaR is additive, notably when all the X i ’s are comonotonic . Comono-tonicity indeeds turns out to be the regulator’s best case which justiﬁes the comonotonicadditivity axiom put forward a large literature: see [4].4. The requirement that a measure of risk should be immune to regulatory arbitrage hasled [2] to deﬁne the axiom of strong coherence , which is a natural requirement so that thestructure of the ﬁrm be neutral to risk measurement. In that paper, strong coherence isshown to be equivalent to the classical risk measures axioms in [1].5. The assumption that the distribution of the risk X be absolutely continuous was nec-essary for the argument presented above to work, and there are connections to be exploredwith the theory of portfolio diversiﬁcation under thick-tailedness, see [3].6. However when the distribution of the risk has atoms of mass greater than α , it is stillpossible decrease the regulatory capital to zero by randomizing , i.e. randomly assigningthe risk to one out of several subsidiaries which shall then take on the total loss. The ideais the following: consider N ≥ /α and incorporate a number N of subsidiaries indexedby i ∈ { , ..., N } . When the parent company faces loss X , an index i is drawn randomly,uniformly in { , ..., N } , and the parent company turns to subsidiary i to cover all oﬀ the AR AT RISK 5 company’s loss X . The parent company has no residual risk: in it in any case insured byone of its subsidiaries. All the subsidiaries have a probability 1 /N which is less than α ofbeing called, so each of their V aR α is zero.7. When there is an upper bound to the number N of subsidiaries that can be created,the problem of the minimal capital cost of risk X given by (2.1) is still open.4. Possible extension

It would be interesting to consider the above problem with overhead costs, namely whendividing the ﬁrm into several desks is costly. The problem would then becomeinf N X i =1 V aR α ( X i ) + α ( N ) s.t. N X i =1 X i = X a.s., and V aR α ( X i ) ≥ . where α ( N ) is increasing with N and can be interepreted as an a priori penalization of theﬁrm’s complexity by the regulator. The problem of determining an optimal α ( N ) from thepoint of view of the regulator in a properly deﬁned setting is of interest. † ´Ecole polytechnique, Department of Economics. E-mail: [email protected] References [1] Artzner P., and F. Delbaen and J.-M. Eber and D. Heath, “Coherent measures of risk,”

MathematicalFinance , 9, pp. 203–228, 1999.[2] Ekeland, I., Galichon, A., and Henry, M., “Comonotonic measures of multivariate risks,” to appear in

Mathematical Finance .[3] Ibragimov, R., “Portfolio diversiﬁcation and value at risk under thick-tailedness,” Harvard Institute ofEconomic Research Discussion Paper

Advances in Mathematical Economics

3, pp.83–95, 2001.[5] R¨uschendorf, L., “Law invariant convex risk measures for portfolio vectors,”