A new approach to assessing model risk in high dimensions

Journal of Banking & Finance 58 (2015) 166–178

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Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf

A new approach to assessing model risk in high dimensions q Carole Bernard a, Steven Vanduffel b,⇑ a b

Grenoble Ecole de Management, Department of Finance, 12 rue Pierre Sémard, 38003 Grenoble cedex 01, France Department of Economics and Political Science at Vrije Universiteit Brussel (VUB), Belgium

a r t i c l e

i n f o

Article history: Received 9 February 2014 Accepted 31 March 2015 Available online 29 April 2015 JEL classification: G32 G28 C35 C52 C60 Keywords: Model risk VaR Rearrangement Algorithm Tail dependence Outlier detection Minimum variance portfolio Credit risk management

a b s t r a c t A central problem for regulators and risk managers concerns the risk assessment of an aggregate portfolio defined as the sum of d individual dependent risks X i . This problem is mainly a numerical issue once the joint distribution of ðX 1 ; X 2 ; . . . ; X d Þ is fully specified. Unfortunately, while the marginal distributions of the risks X i are often known, their interaction (dependence) is usually either unknown or only partially known, implying that any risk assessment of the portfolio is subject to model uncertainty. Previous academic research has focused on the maximum and minimum possible values of a given risk measure of the portfolio when only the marginal distributions are known. This approach leads to wide bounds, as all information on the dependence is ignored. In this paper, we integrate, in a natural way, available information on the multivariate dependence. We make use of the Rearrangement Algorithm (RA) of Embrechts et al. (2013) to provide bounds for the risk measure at hand. We observe that incorporating the information of a well-fitted multivariate model may, or may not, lead to much tighter bounds, a feature that also depends on the risk measure used. In particular, the risk of underestimating the Value-at-Risk at a very high confidence level (as used in Basel II) is typically significant, even if one knows the multivariate distribution almost completely. Our results make it possible to determine which risk measures can benefit from adding dependence information (i.e., leading to narrower bounds when used to assess portfolio risk) and, hence, to identify those situations for which it would be meaningful to develop accurate multivariate models. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The risk assessment of high dimensional portfolios X is a core issue in the regulation of financial institutions and in quantitative risk management. In this regard, one usually attempts to measure the risk of the aggregate portfolio (defined as the sum of individual risks X i ) using a risk measure, such as the standard deviation or the Value-at-Risk (VaR). It is clear that solving this problem is mainly a numerical issue once the joint distribution of X :¼ ðX 1 ; X 2 ; . . . ; X d Þ is completely specified. Unfortunately, estimating a multivariate

q C. Bernard gratefully acknowledges support from the NSERC, from the project on Systemic Risk funded by the GRI in financial services and the Louis Bachelier Institute, and from the Humboldt Research Foundation, as well as the hospitality of the Chair of Mathematical Statistics at the Technische Universität München. S. Vanduffel acknowledges the financial support of the BNP Paribas Fortis Chair in Banking and of the Stewardship of Finance Chair. We thank Kris Boudt, Giovanni Puccetti and Ludger Rüschendorf for very helpful discussions. This paper received the PRMIA Annual Frontiers in Risk Management Award at the 2014 ERM symposium in Chicago. ⇑ Corresponding author. E-mail addresses: [email protected] (C. Bernard), steven. [email protected] (S. Vanduffel).

http://dx.doi.org/10.1016/j.jbankfin.2015.03.007 0378-4266/Ó 2015 Elsevier B.V. All rights reserved.

distribution is a difficult task, and thus the assessment of portfolio risk is prone to model misspecification (model risk). At present, there is no generally accepted framework for quantifying model risk. A natural way to do so consists in finding the minimum and maximum possible values of a chosen risk measure evaluated in a family of candidate models. For example, Cont (2006) found bounds on prices of contingent claims, incorporating model risk on the choice of the risk neutral measure used for pricing. In the same spirit, Kerkhof et al. (2010) assessed model risk in the context of management of market risk by computing the worst-case VaR across a range of models chosen based on econometric estimates involving past and present data. A related approach can be found in Alexander and Sarabia (2012), where the authors compare VaR estimates of the model actually used with those of a benchmark model (i.e., the regulatory model) and use the observed deviations to estimate a capital charge supplement to cover for model risk. More recently, Embrechts et al. (2013) proposed the Rearrangement Algorithm (RA) to find (approximate) bounds on the VaR of high dimensional portfolios, assuming that marginal distributions of the individual risks are known (or prone to negligible model risk) and that the dependence structure (also called the copula) among the risks is not specified. This assumption is

C. Bernard, S. Vanduffel / Journal of Banking & Finance 58 (2015) 166–178

natural, as fitting the marginal distribution of a single risk X i ði ¼ 1; 2; . . . ; dÞ can often be performed in a relatively accurate manner, whereas fitting a multivariate model for X is challenging, even when the number of observations is large. The bounds derived by Embrechts et al. (2013) are wide, as they neglect all information on the interaction among the individual risks. In this paper, we propose to integrate in a natural way the information from a fitted multivariate model. Standard approaches to portfolio modeling use a multivariate Gaussian distribution or a multivariate Student’s t distribution; however, there is ample evidence that these models are not always adequate. Specifically, while the multivariate Gaussian distribution can be suitable as a fit to a dataset ‘‘as a whole,’’ this approach is usually a poor choice if one wants to use it to obtain accurate estimates of the probability of simultaneous extreme (‘‘tail’’) events, or if one wants to estimate the VaR of the aggregate portfolio P S ¼ di¼1 X i at a given high confidence interval; see e.g., McNeil et al. (2010). There is recent literature dealing with the development of flexible multivariate models that allow a better fit to the data. However, no model is perfect, and while such developments are needed for an accurate assessment of portfolio risk, they are only useful to regulators and risk managers if they are able to significantly reduce the model risk inherent in risk assessments. In this paper, we develop a framework that allows for practical quantification of model risk. Our results make it possible to identify risk measures for which the additional information of a well-fitted multivariate model reduces the model risk significantly, making these measures meaningful candidates for use by risk managers and regulators. In particular, we observe from numerical experiments that the portfolio VaR at a very high confidence level (as used in the current Basel regulation) might be prone to such a high level of model risk that, even if one knows the multivariate distribution nearly perfectly, its range of possible values remains wide. In fact, one may then not even be able to reduce the model risk as computed in Embrechts et al. (2013), where no information on the dependence among the risks is used. The idea pursued in our approach is intuitive and corresponds to real-world situations. Let us assume that we have N observations for X, i.e. our dataset consists of N vectors of dimension d; fxi gi¼1;...;N where xi ¼ ðxi1 ; . . . ; xid Þ. We also assume that a multivariate model has already been fitted to this dataset. This fitted distribution is a candidate joint distribution of X (benchmark model). However, we are aware that the model is subject to misspecification, and we split Rd into two disjoint subsets: F will be referred to as the ‘‘fixed’’ or ‘‘trusted ’’ area and U as the ‘‘unfixed’’ or ‘‘untrusted’’ area. Specifically, U reflects the area in which the fitted model is not considered appropriate. Note that

Rd ¼ F

[

U;

F

\

U ¼ £:

If one has perfect trust in the model, then all realizations of X reside in the ‘‘trusted’’ part (U ¼ £) and there is no model risk. By contrast, F ¼ £ when there is no trust in the fit of the dependence, which corresponds to the case studied by Embrechts et al. (2013). A closely related problem has already been studied for two-dimensional portfolios (d ¼ 2) when some information on the dependence (copula) is available; see e.g., Tankov (2011) and Bernard et al. (2012). Tankov (2011) uses extreme dependence scenarios to find model-free bounds for the prices of some bivariate derivatives, whereas Bernard et al. (2014) use such scenarios to determine optimal investment strategies for investors with state-dependent constraints. While both applications show that finding bounds on copulas in the bivariate case can be of interest, portfolio risk management involves more than two risks. Unfortunately, finding bounds on copulas in the general d-dimensional case in the presence of constraints is not only more difficult

167

but also less useful for risk management applications. The reason is that when d > 2, in most cases, the worst copula (under constraints) of a vector X does not give rise to the highest possible P value of the risk measure at hand of S ¼ di¼1 X i , because the marginal distributions also have an impact; see e.g., Bernard et al. (2014) for illustrations of this feature. Hence, in this paper we study bounds for risk measures of the aggregate risk S by using information on the multivariate joint distribution of its components X i (which embeds information on the dependence) rather than using copula information. Some previous papers have dealt explicitly with the presence of (partial) information on the dependence structure: Embrechts and Puccetti (2010) and Embrechts et al. (2013) consider the situation in which some of the bivariate distributions are known; Denuit et al. (1999) study VaR bounds assuming that the joint distribution of the risks is bounded by some distribution; and Bernard et al. (2015) compute VaR bounds when the variance of the sum is known. However, the setup in these papers is often difficult to reconcile with the information that is available in practice; or, it does not make use of all available dependence information. Furthermore, while the bounds that are proposed in these papers might be sharp (attainable), they do not always make it possible to strengthen the unconstrained1 bounds in a significant way and are often difficult to compute numerically, especially for higher dimensions with inhomogeneous risks. The paper is organized as follows. We lay out our setting in Section 2. Sections 3 and 4 are devoted to the development of a practical method for deriving bounds on risk measures. This method relies on a (discretized) matrix representation of the portfolio X and builds on the Rearrangement Algorithm that was recently developed by Puccetti and Rüschendorf (2012) and further studied by Embrechts et al. (2013). We illustrate the results using various examples. In Section 5 we provide two applications. First, we find the minimum variance portfolio under model uncertainty, and next we assess the model risk of a credit portfolio. The numerical results show that the proposed bounds, which take into account dependence information, typically outperform the (unconstrained) ones already available in the literature and thus allow for more realistic assessment of model risk. However, model risk remains a significant concern, especially when using a risk measure that focuses on ‘‘tail type’’ events, such as the VaR computed at very high confidence level. 2. Setting Let X :¼ ðX 1 ; X 2 ; . . . ; X d Þ be some random vector of interest having finite mean and defined on an atomless probability space. Let F  Rd and U ¼ Rd n F . We assume that we know. (i) the marginal distribution F i of X i on R for i ¼ 1; 2; . . . ; d, (ii) the distribution of X j fX 2 F g, (iii) the probability pF :¼ PðX 2 F Þ and pU :¼ PðX 2 UÞ ¼ 1  pF . Without loss of generality, we can assume that F ¼ £ if and only if pF ¼ 0. As the joint distribution of X is only fully specified on the subarea F of Rd , risk measures (e.g., the VaR) of the aggreP gate risk di¼1 X i cannot be computed precisely (unless pF ¼ 1). In fact, there are many vectors X that satisfy the properties (i), (ii) and (iii) but have a different risk measure of their sum. In order to derive the maximum (minimum) possible value, it is convenient to consider a mixture representation. Specifically, consider the indicator variable I corresponding to the event ‘‘X 2 F ’’ 1 Here, ‘‘unconstrained bounds’’ refers to bounds that are obtained when only the marginal distributions are fixed, but no dependence information is used.

168

C. Bernard, S. Vanduffel / Journal of Banking & Finance 58 (2015) 166–178

ð1Þ

I :¼ 1X2F ;

so that one can express the probabilities that a random vector takes values in F , respectively in U, as

pF ¼ PðI ¼ 1Þ and pU ¼ PðI ¼ 0Þ:

ð2Þ

It is then clear that for a given random vector X satisfying properties (i), (ii) and (iii), there exists a multivariate vector ðZ 1 ; Z 2 ; . . . Z d Þ that we can take independent of I, such that

! d d d X X X X i ¼d I X i þ ð1  IÞ Z i ; i¼1

i¼1

ð3Þ

i¼1

where ‘‘¼d ’’ denotes the equality in distribution. Here, for i ¼ 1; 2; . . . ; d; Z i is distributed with

F Zi ðzÞ ¼ F X i jX2U ðzÞ:

i ¼ 1; 2; . . . ; d

ð5Þ

for some uniformly distributed random variable U independent of I. Note that all Z ci are indeed increasing in each other and are distributed with F Zi ði ¼ 1; 2; . . . ; dÞ. 2.1. Model risk We aim at finding the extreme possible values of the risk meaP sure at hand for the sum di¼1 X i . We use the gap between the minimum and the maximum value of the risk measure to assess model risk. Formally, we use in this paper the following definition of model risk, which is in the same spirit as in Barrieu and Scandolo (2015). Definition 2.1 (Model risk). Let X be a random vector satisfying (i), (ii) and (iii) and assume that a (law-invariant) risk measure .ðÞ is P used to assess the risk of di¼1 X i . Define

(

 F

. :¼ inf .

d X Yi i¼1

!)

;

(

þ F

. :¼ sup .

d X Yi

!)

;

ð6Þ

i¼1

where the infimum and the supremum are taken over all other (joint distributions of) random vectors ðY 1 ; Y 2 ; . . . ; Y d Þ that comply  with X for the properties (i), (ii) and (iii). We refer to .þ F and .F as the best-possible bounds. They may, or may not, be sharp  (attainable). Assume that .þ F > 0 and .F > 0. The model risk that one underestimates the risk by computing a direct estimate of P .ð X i Þ in some chosen benchmark model (i.e., when some multivariate distribution for X has been fully specified) is then defined as

P

d i¼1 X i

.þF  .

 ð7Þ

.þF and, similarly, the model risk for overestimation is given as

.

P d

i¼1 X i

.

 F



 .F

:

ing their properties. Recent literature on model risk estimation has dealt mainly with the situation in which there is full uncertainty on the dependence among the risks X i ði ¼ 1; 2; . . . ; dÞ, i.e., when F ¼ £, as discussed in the introduction; see Embrechts et al. (2014) for a recent account in the context of Basel III regulation. In this paper, we consider the case in which information on the dependence translates into joint distributions that are ‘‘partially’’  known, and we analyze to which extent the values .þ F and .F obtained in the case of full dependence uncertainty can be reduced. 2.2. Choice of the trusted area

ð4Þ

Since the marginal distributions F i of the X i are known, the marginal distributions F Zi of the Z i are given as well. However, the joint distribution of ðZ 1 ; Z 2 ; . . . Z d Þ remains unspecified. In other words, all portfolios X that satisfy the properties (i), (ii) and (iii) only differ in the choice of dependence among the Z i , showing that the assessment of model uncertainty is intimately connected with the analysis of extreme dependence among the Z i . A special role in the analysis of model risk is played by the comonotonic dependence, i.e, when all Z i are increasing in each other. For this particular dependence we denote Z i by Z ci . Formally,

Z ci ¼ F 1 Z i ðUÞ;

The remainder of the paper aims at obtaining the maximum and P  .þF and .F of . di¼1 X i and at examin-

minimum possible values

ð8Þ

Our approach requires specification of an area F on which we ‘‘know’’ or ‘‘trust’’ the model. The basic idea to construct F is that it should contain all possible realizations that ‘‘agree’’ with a certain multivariate (benchmark) distribution for X. Hence, a natural criterion by which to determine the trusted part of the multivariate distribution consists in selecting a multivariate density f (coming for instance from a multivariate Student’s t model or a Pair-Copula Construction model (Aas et al., 2009) that fits well with the available data and, next, to define the trusted area F as the set of possible realizations that are ‘‘likely enough’’ under the model. Hence, a natural way to select F is by selecting the subset of observations xi :¼ ðxi1 ; . . . ; xid Þ such that

  F :¼ xi 2 Rd jf ðxi Þ P e ;

ð9Þ

where e > 0 is some cutoff level that indicates the point beyond which we cannot trust the model. Its choice will be discussed further below. The main difficulty is how to choose f. Of course, one can resort to traditional maximum likelihood estimation (MLE) techniques and select a suitable density f from a family of candidate densities. However, the specification of f can be seriously affected by observations that are a result of recording errors or of exceptional phenomena such as extreme stress, or that appear as realizations of another (undetected) distribution (Tukey, 1962; Rousseeuw and Leroy, 2005). Hence, traditional MLE only works well when there are no outliers, which is rarely the case. In statistics, the outlyingness of an observation is typically measured using the concept of (squared) Mahalonobis distance (Healy, 1968). Hence, in this case, the set of realizations xi that agree with a model for X is given as an ellipsoid,   F :¼ xi 2 Rd j ½xi1  l1 ; ...; xid  ld R1 ½xi1  l1 ;. ..;xid  ld t 6 c ; ð10Þ where c is the appropriate cutoff level and l; R are the mean vector and covariance matrix, respectively. In practice, these parameters have to be estimated from data. As already indicated, however, their ^ respectively, should not be easily affected by ^ and R, estimates l outliers in the data (which at this point are still unknown). To remedy this problem, statistical techniques have been developed, i.e., the so-called robust (or resistant) methods. In this regard, it appears from the literature that the minimum covariance estimator (MCD) as proposed in Rousseeuw and Van Zomeren (1990) is a suitable ^ that are used in crite^ and R robust approach2 for estimating the l rion (10); see also Hadi (1992). A fast algorithm is presented in Rousseeuw and Driessen (1999). The analysis carried out so far already suggests that criterion (10) appears to be a viable approach to selecting F . Moreover, criterion (10) can also be seen as a special case of criterion (9) in the 2 When the number of observations N is not significantly larger (or even smaller) than d, one first needs to apply dimension reduction techniques (see e.g., Filzmoser et al., 2008).

C. Bernard, S. Vanduffel / Journal of Banking & Finance 58 (2015) 166–178

sense that the equivalence between the two criteria occurs by assuming a multivariate elliptical density with robust location vec^ (for the appropriate choices of e and ^ and robust covariance R tor l c). When data do not make it possible to propose an elliptical model as the benchmark (in particular, when data are skewed) one can apply ‘‘symmetrizing’’ transformations (e.g., a logarithmic or a Box-Cox transformation) or adjust the measure of outlyingness by using a robust measure for skewness as well (see Hubert and Van der Veeken, 2008). However, in high dimensions, skewness is not really an issue because data become more and more concentrated in an outside shell and thus exhibit symmetry. In other words, location and covariance are the dominant factors in the detection of outliers; see Hall et al. (2005) and Hubert and Van der Veeken (2008). In summary, in the remainder of the paper, we stick to criterion (10) when choosing the trusted area F . As for the choice of the cutoff level c in (10), several possibilities have been described in the literature. Rousseeuw and Van Zomeren (1990) use a cutoff value for distinguishing outliers from non-outliers, which corresponds to considering for F the ellipsoid that covers a fixed percentage of the total probability under the model (e.g., 97.5%). This approach implies that some data are considered as outliers even when they are not. An interesting variant on this approach was considered in Lee and Mykland (2008). These authors reject observations when their distance is not in the usual range of maximum distances (its distribution can be derived based on the available data). Thus, in the same spirit, one may choose c equal to the maximum observed (empirical) distance or use this value as an upper bound for any other possible choice (see also Titterington (1978) for a related idea). Approaches that select c such that the empirical distribution of observed distances matches closely with its theoretical counterpart have been proposed by Garrett (1989) and Filzmoser (2005). For our practical applications in Section 5, we rely on the work of Garrett (1989), i.e., we plot the quantiles of both distributions against each other (Q–Q plot) and remove the extreme cases.

169

distributions F k ðk ¼ 1; 2; . . . ; dÞ and do not allow for conclusions regarding the dependence. It is then important to observe that rearranging the values xik ði ¼ 1; 2; . . . ; N; k ¼ 1; 2; . . . ; dÞ within the kth column does not affect the empirical marginal distribution of X k but changes the dependence (‘‘interaction between elements of different columns’’). Let us denote by ‘F the number of elements in F N and by ‘U the number of elements in U N , such that

N ¼ ‘F þ ‘U : Without loss of generality, it is convenient to modify the matrix M by changing the order of the rows so that the ‘‘trusted area’’ corresponds to the ‘F first rows and the untrusted area corresponds to the last ones. By doing so, we have only reallocated the states i ¼ 1; 2; . . . ; N, without impact on the adequacy of M to describe the distributional (law-invariant) properties of X. Similarly, as per definition of the subset U N , we are allowed to rearrange the values within the columns of U N (and thus within the corresponding parts of M), as this operation generates a new matrix that is considered to be as trustworthy as the initial one, i.e., we still comply with the basic assumptions (i), (ii) and (iii) (note, indeed, that we do not know the dependence between the X i , conditionally on X 2 U). Without loss of generality, we can thus always assume that the matrix U N depicts a comonotonic dependence (in each column, the values are sorted in decreasing order; that is, such that xm1 k P xm2 k P . . . P xm‘U k for all k ¼ 1; 2; . . . ; d). Finally, for F N (and thus also for the corresponding part of M) we can assume that the ‘F observations (xij 1 ; xij 2 . . . xij d ) appear in such a way that for the sums of the components, i.e., sj :¼ xij 1 þ xij 2 þ    þ xij d (j ¼ 1; 2; . . . ; ‘F ), it holds that s1 P s2 P. . .P s‘F . From now on, without any loss of generality, the observations are reported in the following matrix M

2.3. Matrix representation of the portfolio X In this paper we employ the RA of Embrechts et al. (2013) to approximate risk bounds for the portfolio X. In this regard, it is convenient to represent the portfolio X through a matrix. Hence, let us consider N d-dimensional vectors xi ði ¼ 1; 2; . . . ; NÞ that can be seen as realizations (sample values) stemming from a fitted multivariate distribution of X. Each observation xi occurs with probability N1 naturally (possibly involving repetitions). Denote by M ¼ ðxij Þ the corresponding N  d matrix. We assume that the matrix M contains sufficient data so that it can effectively be seen as a representation of the random vector of interest X. Define SN by P SN ðiÞ ¼ dk¼1 xik for ði ¼ 1; 2; . . . ; NÞ. In other words, SN can be seen as a random variable that takes the value SN ðiÞ in ‘‘state’’ i for i ¼ 1; 2; . . . ; N. In general, it might be difficult to find sharp bounds P for risk measures of S ¼ i X i . The purpose of what follows is to deal with this problem using the ‘‘sampled’’ counterpart SN of S, rather than S itself. As before, we suppose that the joint distribution of X is not completely specified. In the context of the matrix representation M for the vector X, we assume that the matrix M is effectively split into two parts. There is a subset F N of trusted observations xi , and a subset U N containing the rest of the observations. In the sequel, the set F N will be referred to as the ‘‘fixed’’ or ‘‘trusted’’ part and U N as the ‘‘untrusted’’ part. In the case in which one has perfect trust in all observations, the ‘‘untrusted’’ part contains no elements (U N ¼ £) and SN can be used to assess the risk of S. By contrast, if one has no trust in the observed dependence, then F N ¼ £. In this case, the N observations xi are useful only in modeling marginal

ð11Þ

where the gray area reflects F N and the white area reflects U N . The corresponding vectors SFN and SUN consist of sums of the components for each observation in the trusted (respectively untrusted) part:

ð12Þ

While s1 P s2 P    P s‘F are trusted, the sums ~si change when the choice of dependence in U N is varied. In fact, the set fi1 ; . . . ; i‘F g can be seen as the collection of states (scenarios) in which the corresponding observations are trusted, whereas the set fm1 ; . . . ; m‘U g provides the states in which there is doubt with respect to the dependence structure.

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Finally, with some abuse of notation (completing by zeros so that SFN and SUN take N values), one also has the following representation of SN :

SN ¼ ISFN þ ð1  IÞSUN ;

ð13Þ

where I ¼ 1 if xi 2 F N ði ¼ 1; 2; . . . ; NÞ. The representation (13) can be seen as the sampled counterpart of (3). In particular, note that SUN is a comonotonic sum and corresponds to the sampled version Pd c of i¼1 Z i . In this paper, we aim to find worst case dependences allowing for a robust risk assessment of the portfolio sum S (using SN ). This amounts to rearranging the outcomes in the columns of the untrusted part U N (i.e., changing the comonotonic dependence there) such that the risk measure at hand for SN is maximized or minimized.

3. Bounds for risk measures that are consistent with convex order A risk measure .ðÞ summarizes the information about some risk into a single number. It is natural to require that this operation agrees with the risk averse behavior of decision makers in the sense that if all risk averse decision makers (those who maximize expected utility of wealth with a concave utility function) agree that X is less risky than Y, then .ðXÞ 6.ðYÞ should hold. This order is also called convex order. Formally, X is smaller than Y in convex order if and only if Eðv ðXÞÞ 6 Eðv ðYÞÞ for all convex functions v(such that the expectations exist). Examples of risk measures that are consistent with convex order include the standard deviation and the concave distortion risk measures (e.g., Tail Value-at-Risk (TVaR)).3 However, VaR, which is undoubtedly the most used risk measure, is not consistent with the preferences of a risk-averse agent. In Section 4, we will study VaR bounds for portfolios. Let us recall the definitions of VaR and TVaR. For p 2 ð0; 1Þ, we denote by VaRp ðSÞ the VaR of S at level p,

VaRp ðSÞ ¼ F 1 S ðpÞ ¼ inf fs 2 R j F S ðsÞ P pg:

ð14Þ

By convention, inff£g ¼ 1 and inffRg ¼ 1, so that VaRp ðSÞ is properly defined by (14) for all p 2 ½0; 1. Furthermore, TVaRp ðSÞ denotes the Tail Value-at-Risk (TVaR) at level p

TVaRp ðSÞ ¼

1 1p

Z

1

VaRu ðSÞdu;

p 2 ð0; 1Þ:

p

Observe that p ! TVaRp is continuous. We define TVaR1 ðSÞ ¼ limp%1 TVaRp ðSÞ. TVaRp is a weighted average of all upper VaRs from probability level p onwards. Similarly, we can define the left Tail Value-at-Risk (LTVaR) at level p as the average Rp of the VaRs below p, i.e. LTVaRp ðSÞ ¼ 1p 0 VaRu ðSÞdu and LTVaR0 ðSÞ ¼ LTVaRp&0 ðSÞ. 3.1. Convex order bounds In what follows, we aim at deriving the best-possible bounds .þ F and . F when .ðÞ is consistent with convex order. Recall that the   þ range .F ; .F is fully driven by the unknown joint behavior of the random vector ðZ 1 ; Z 2 ; . . . ; Z d Þ in which the Z i have known marginal distribution F Zi ðzÞ ¼ F X i jX2U ðzÞ (see (4)). The following propo sition provides an expression for .þ F and a lower bound for .F . 3 TVaR is closely related to conditional tail expectation (CTE) or expected shortfall (ES). They are equivalent notions for continuous variables. See for instance Kaas et al. (2008) for details.

P  d Proposition 3.1 (Bounds on . i¼1 ðX i Þ ). Let X be a random vector that satisfies properties (i), (ii) and (iii), and let I and ðZ c1 ; Z c2 ; . . . ; Z cd Þ be defined as in (5). Assume that the risk measure .ðÞ is consistent with convex order. Then,

 F

. P A :¼ .

d d X X I X i þ ð1  IÞ EðZ ci Þ i¼1

¼

I

! and

.þF ¼ B :

i¼1

! d d X X X i þ ð1  IÞ Z ci : i¼1

i¼1

Proof. It is well-known that for any vector ðY 1 ; Y 2 ; . . . ; Y d Þ and any convex function v ðxÞ, it holds that

E

v

d X Yi

!! 6E

i¼1

v

d X F 1 Y i ðUÞ

!! ð15Þ

;

i¼1

where U is a uniformly distributed random variable on ð0; 1Þ; see Meilijson and Nádas (1979). A simple conditioning argument and taking into account Jenssen’s inequality then shows that for all convex functions v ðxÞ,

E

v

d X 

IX i þ ð1 

IÞEðZ ci Þ



!!

6E

v

i¼1

d X Xi

!!

i¼1

6E

v

d X 

IX i þ ð1 

IÞZ ci



!! ;

i¼1 þ Hence, we have that . F P A and .F 6 B. Note that the multivariate vector

 IX 1 þ ð1  IÞZ c1 ; IX 2 þ ð1  IÞZ c2 ; . . . ; IX d þ ð1  IÞZ cd satisfies conditions (i), (ii) and (iii) so that

. ¼ B. þ F

ð16Þ h

The stated lower and upper bounds A, resp. B, in Proposition 3.1 P are intuitive. When computing the risk of the portfolio sum di¼1 X i , we consider the events fX 2 F g and fX 2 Ug separately. The distriP bution of di¼1 X i is known on the event fX 2 F g, but unknown on P the event fX 2 U g. On U, one then substitutes di¼1 X i either by the Pd c constant i¼1 EðZ i Þ (to compute the lower bound A), or by the P comonotonic sum di¼1 Z ci (to compute the upper bound B). In particular, when U ¼ £, the upper bound is equal to the lower bound and there is no model risk. The bounds A and B are useful in deriving approximations for þ the best-possible bounds . F resp. .F . These approximations are explained in the next section. 3.2. Approximating the best-possible bounds From the proof of Proposition 3.1, we observe that the upper bound Bð¼ .þ F Þ is sharp. Hence, in order to maximize the risk measure of SN (e.g., standard deviation) one just needs to employ a comonotonic scenario on U N . However, we have already initialized a comonotonic structure (without loss of generality), and the corresponding values of the sums are exactly the values ~si ði ¼ 1; 2; . . . ; ‘U Þ reported for SUN in (12). The upper bound .þ F can then be computed using the values si ði ¼ 1; 2; . . . ; ‘F Þ and ~si ði ¼ 1; 2; . . . ; ‘U Þ in the appropriate way. By contrast, the stated lower bound A may not be sharp because it is not straightforward to find a vector ðZ 1 ; Z 2 ; . . . ; Z d Þ with given P P marginal distributions as in (4) such that I di¼1 X i þ ð1  IÞ di¼1 Z i Pd Pd c and I i¼1 X i þ ð1  IÞ i¼1 EðZ i Þ have the same distribution. In order to obtain this situation we need that the vector ðZ 1 ; Z 2 ; . . . ; Z d Þ

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P P P satisfies di¼1 Z i ¼ di¼1 EðZ i Þ, i.e., the sum di¼1 Z i has a flat quantile function on ð0; 1Þ. The literature refers to this situation as ‘‘joint mixability’’ (for a homogeneous portfolio this concept is known as ‘‘complete mixability’’ ), a concept that can essentially be traced back to a paper of Gaffke and Rüschendorf (1981) and has been extensively studied in a series of papers including Wang and Wang (2011) and Puccetti et al. (2012, 2013). In these papers it is shown, among other results, that in several theoretical cases of interest one can construct a dependence among the risks that lead to mixability. Furthermore, in many practical cases one can construct a dependence such that the quantile function of the sum becomes approximately flat; see e.g., Embrechts et al. (2013) and Bernard et al. (2015) for illustrations. In this regard, we note that a large class of distributions exhibits asymptotic mixability implying that in high-dimensional problems the lower bound A that is stated in Proposition 3.1 is expected to be approximately sharp; see e.g., Puccetti et al. (2013) and Puccetti and Rüschendorf (2013). Hence, to achieve the minimum bound . F in Proposition 3.1, the values of SUN must be as close as possible to each other; ideally, SUN must be constant. To do so, we apply the Rearrangement Algorithm (RA) of Embrechts et al. (2013) to the matrix U N (untrusted part). Denote by ~sm i the corresponding values of the sums of SUN after applying the RA. The lower bound can now be computed using the values si ði ¼ 1; 2; . . . ; ‘F Þ and ~sm i ði ¼ 1; 2; . . . ; ‘U Þ in the appropriate way. We illustrate the procedure for approximating the best-possible bounds with a simple generic example, where the risk measure at hand is the variance. Example 3.2 (Variance). Using the notation from above, the upper bound on the variance is computed as

! ‘F ‘U X 1 X 2 2 ðsi  sÞ þ ð~si  sÞ ; N i¼1 i¼1

ð17Þ

where the average sum s is given by

!

s ¼

‘F ‘U N X d X 1X 1 X ~si : xij ¼ si þ N i¼1 j¼1 N i¼1 i¼1

ð18Þ

We then compute the minimum variance as follows:

! ‘F ‘U X 1 X 2 2 m ðsi  sÞ þ ð~si  sÞ ; N i¼1 i¼1

ð19Þ

where s is computed as in (18). We further illustrate Proposition 3.1 and its consequences for deriving (sharp) approximations for the best-possible bounds, i.e, the approach explained in Section 3.2 with two examples of risk measures that are consistent with convex order. In the first example, we use the standard deviation (Example 3.3) and in the second example we use TVaR (Example 3.6). 3.3. Examples We consider a (discretized version) of a random vector X with standard Student’s t distributed marginals that follows a multivariate standard Student’s t distribution on a trusted area F . Specifically, the density of X on F is given by

Cððd þ mÞ=2Þ xT R1 x f X ðxÞ ¼ pffiffiffiffiffiffi 1 þ d=2 m ðmpÞ Cðm=2Þ jRj

Table 1 P20 In the first column we report the standard deviation of i¼1 X i when there is no dependence uncertainty. Approximations of the lower and upper bounds of the P20 þ standard deviation of i¼1 X i are reported as pairs ð. F ; .F Þ for various probability levels pF and correlation coefficients q. All digits reported in the table are significant.a F ¼ C pF ;q;m;d

U¼£ pF ¼ 1

pF ¼ 0:98

pF ¼ 0:8

pF ¼ 0:2

U ¼ Rd pF ¼ 0

q¼0 q ¼ 0:1 q ¼ 0:5

5.0 8.5 16.2

(4.8, 7.9) (8.2, 10.3) (15.5, 16.8)

(3.9, 14.7) (6.6, 15.7) (12.5, 18.9)

(1.4, 21.5) (2.4, 21.6) (4.6, 21.9)

(0.0, 22.4) (0.0, 22.4) (0.0, 22.4)

a We assessed the standard error on each estimate in Table 1 by repeating the sampling from X thirty times and we found that the maximum standard deviation (over all cases in the table) was 0.007, confirming that the first digit of each number in the table can be considered exact.

and Nadarajah (2004). In the remainder of the paper, we assume that m > 2 in which case the covariance matrix R exists and is given m R. Consistent with the analysis carried out in Section 2.2, by R ¼ m2 the subset F is taken as an ellipsoid,

n o F ¼ CpF ;q;v ;d :¼ x 2 Rd j xR1 xt 6 cðpF Þ ;

ð20Þ

where cðpF Þ is the appropriate cutoff value corresponding to PðX 2 F Þ ¼ pF . To determine cðpF Þ, we can use the fact that the scaled squared Mahalanobis distance 1d XR1 Xt follows a F-distribution with parameters d and m (i.e., 1d XR1 Xt  Fðd; mÞ).4 In the examples, we consider a portfolio of d ¼ 20 risks and a multivariate Student’s t distribution with m ¼ 10 degrees of freedom. We obtain the matrix M by sampling N ¼ 3; 000; 000 realizations of the multivariate Student’s t distribution and we allocate each realization to either the untrusted subset U N or to the trusted subset F N . Armed with this information, we next apply the algorithm described in Section 3.2 for approximating the  best-possible bounds .þ F and .F . Example 3.3 (Standard deviation – Multivariate Student’s t distribution). In Table 1, we provide the upper and lower bounds for the standard deviation of the portfolio for various probability levels pF and correlation levels q. The first column (pF ¼ 1) provides results for cases in which there is no uncertainty on the multivariate distribution; as such, it provides a benchmark for assessing model risk (see Definition 2.1). The last column (pF ¼ 0) provides bounds for cases in which there is full uncertainty on the dependence; it corresponds to the situation that is traditionally studied in the literature. One observes from Table 1 that adding some partial information on the dependence (i.e., when pF > 0) significantly reduces the gap between the lower and upper bounds and confirms that information on dependence is important when assessing the risk of a portfolio using the standard deviation. For instance, when pF ¼ 0:8 and q ¼ 0, the unconstrained upper bound for the standard deviation shrinks by approximately 52% (from 22.4 to 14.7). However, the impact of model risk on the standard deviation can be substantial even when the joint distribution X is almost perfectly known, i.e., when pF is close to 1. Consider, for instance, pF ¼ 0:98 and q ¼ 0. We find that using a multivariate Student’s t assumption (as the benchmark) might underestimate the standard deviation by (7.9–5)/7.9 = 36.7% and overestimate it by (5– 4.8)/4.8 = 4.2% (using the measures of model risk in (6)).

!ðdþmÞ=2 :

Here, m is the number of degrees of freedom and jRj is the determinant of the correlation matrix R satisfying Ri;j ¼ q ð1 < q < 1Þ for all pairs ðX i ; X j Þ with i – j (homogeneous portfolio); see also Kotz

4 The F-distribution with parameters ðd; mÞ is the distribution of the ratio of two scaled chi-squared v2 ðdÞ and v2 ðmÞ variables. In the case of a multivariate normal distribution, we would find a v2 ðdÞ distribution for the squared Mahalanobis distance XR1 Xt .

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Remark 3.4. The (sharp) approximations for . F can be compared with the non-sharp bound A that can be evaluated using Monte-Carlo simulations in a straightforward way. In Example 3.3, it turns out that they correspond closely to each other implying that the approximations for the best-possible bounds are accurate. It also provides evidence that the RA-based method that we propose to approximate the bounds can be expected to perform well in general. Remark 3.5. Given that the untrusted area depends on the level pF , the size of matrix on which we apply the RA (to obtain the approximation for . F ) is decreasing in pF . Using the same laptop, and the RA implemented in Matlab, with a sample of size 3,000,000 for each of the 20 variables, we need about 90 s when pF ¼ 0:98, 400 s when pF ¼ 0:8, 18 min when pF ¼ 0:2 and 30 min when pF ¼ 0. On the other hand, the computation of the non-sharp bound A using Monte-Carlo simulations requires the same time in all cases of the table, i.e., less than 1 min per value in Table 1. In the following example, we assume that the risk measure used to assess aggregate risk is TVaR. Example 3.6 (TVaR – Multivariate Student’s t distribution). Table 2 provides the bounds on TVaR for various confidence levels p, probability levels pF and correlation levels q. The results in Table 2 confirm that adding dependence information makes it possible to narrow the distance between the upper and lower bounds. For example, when p ¼ 95% and q ¼ 0:1 then for pF ¼ 0 we find that the distance between the upper and lower bound for TVaR is equal to 48.2. When pF ¼ 0:8, this distance is reduced to 25.8, i.e., the gap is reduced by 46.5%. We also observe that for TVaR the degree of model risk strongly depends on the interplay between the probability level p used to assess the TVaR and the degree of uncertainty on the dependence as measured by pF . When p is large (e.g., p ¼ 99:5%), even a small proportion of model uncertainty (e.g., pF ¼ 0:98) may have a tremendous effect on the model risk of underestimation. We can explain this observation as follows. The TVaR is essentially measuring the average of all upper VaRs, and its level is thus driven mainly by scenarios in which one or more outcomes of the risks involved are high. These scenarios, however, are typically not considered trustworthy for depicting the (tail) dependence (as they tend to exhibit a Mahalanobis distance that is too large), and enforcing comonotonicity has a clear negative impact on the level of the TVaR. In fact, for a given level of p the model risk of underestimation increases sharply with an increase in the level of uncertainty pU ð¼ 1  pF Þ and approaches its (unconstrained) maximum already for small to moderate values of pU . This effect is further emphasized when the level of p increases. In other words, the TVaR is highly vulnerable to underestimation, especially when it is assessed at high probability levels. By contrast, the risk of overestimating appears to be less pronounced. It is only when the level of uncertainty pU is nearly equal to one that the minimum value for TVaR approaches its (unconstrained) minimum. Finally, note that similarly as in the case of the variance, the (non-sharp) lower bound A can be assessed using Monte-Carlo simulations. We found that doing so yields values for A that are close to the values reported in Table 2. In other words, in this example the bound A can be seen as sharp (after rounding up the first digit to the nearest number) confirming once more that the RA of Embrechts et al. (2013) is superior in yielding excellent approximations for risk bounds. We end this section with a remark.

Remark 3.7 (Bounds on copulas). Proposition 3.1 is stated in the general case of a d-dimensional problem. In particular, we can exhibit the dependence structure such that the upper bound is sharp. For the lower bound, given that the lower Fréchet bound exists for d ¼ 2, it is straightforward to construct a sharp bound using the antimonotonic dependence. In fact, for all risk measures . that are consistent with convex order,

.ðIðX 1 þ X 2 Þ þ ð1  IÞðA1 þ A2 ÞÞ 6 .ðX 1 þ X 2 Þ 6 .ðIðX 1 þ X 2 Þ  þ 1  IÞðZ c1 þ Z c2 Þ ;

1 where A1 ¼ F 1 X 1 jðX 1 ;X 2 Þ2U ðUÞ and A2 ¼ F X 2 ðX 1 ;X 2 Þ2U ð1  UÞ for a uniform

ð0; 1Þ random variable U that is independent of I. Hence, for d ¼ 2 the lower bound in Proposition 3.1 is sharp if and only if A1 þ A2 is constant, which is usually not the case. However, we will show further in this paper that in high dimensions the lower bound can be expected to be nearly sharp. Our approach is connected to the work of Tankov (2011), who provides (for the case in which d ¼ 2) the pointwise minimum and maximum dependence structure (copula) among all copulas that are fixed on some compact subset F of ½0; 1:2 Generally, the bounds obtained from Tankov (2011) are not copulas but quasi-copulas, and thus they lead to non-sharp bounds for .ðX 1 þ X 2 Þ. Our setting does not require assumptions regarding the compactness of F and generates sharp upper bounds and approximate sharp lower ones (and sharp for d ¼ 2). Importantly, the results in Tankov (2011) and extensions of Bernard et al. (2012) are restricted to d ¼ 2, whereas our approach holds for general d, making it suitable for application in portfolio risk management.

4. Bounds for Value-at-Risk VaR is a widely used risk measure in financial services. Unfortunately, it is not consistent with convex order5 so that the bounds of the previous section do not apply. 4.1. VaR bounds The following proposition provides bounds on VaR to assess model risk on VaR estimation. The proof is provided in Appendix A.1. P Proposition 4.1 (VaR Bounds for di¼1 X i ). Let X be a random vector that satisfies properties (i), (ii) and (iii), and let I; ðZ c1 ; Z c2 ; . . . ; Z cd Þ and U be defined as in (1) and (5). Define the variables Li and Hi as

  Li ¼ LTVaRU Z ci and Hi ¼ TVaRU Z ci : Furthermore, define

mp :¼ VaRp

! d d X X I X i þ ð1  IÞ Li ; i¼1

¼ VaRp I

Mp :

i¼1

! d d X X X i þ ð1  IÞ Hi : i¼1

i¼1

The best-possible bounds satisfy

.F and .þF for the VaR of the aggregate risk

.F P mp and .þF 6 Mp :

ð21Þ

5 When X is smaller than Y in the sense of convex order, then X and Y must have the same mean (since also X will be convex smaller than Y). The distribution functions will then cross at least once (or coincide) so that VaRs cannot be ordered.

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Table 2 P In the first column we report TVaR95% and TVaR99:5% of 20 i¼1 X i in the absence of uncertainty. Bounds are then given for various confidence levels p, correlation coefficients q and probabilities pF . All digits reported are significant.a F ¼ C pF ;q;m;d

a

U ¼ £ pF ¼ 1

pF ¼ 0:98

pF ¼ 0:8

pF ¼ 0:2

U ¼ Rd p F ¼ 0

p = 95%

q¼0 q ¼ 0:1 q ¼ 0:5

10.8 18.3 34.9

(10.3, 15.9) (17.5, 22.2) (33.4, 36.4)

(8.6, 40.3) (14.6, 40.4) (27.8, 43.1)

(4.0, 48.1) (6.9, 48.1) (13.0, 48.1)

(0.0, 48.2) (0.0, 48.2) (0.0, 48.2)

p = 99.5%

q¼0 q ¼ 0:1 q ¼ 0:5

16.9 28.8 54.8

(15.5, 56.4) (26.3, 56.6) (50.1, 63.8)

(12.4, 75.1) (21.1, 75.0) (40.3, 75.1)

(7.3, 75.7) (12.4, 75.7) (23.5, 75.7)

(0.0, 75.7) (0.0, 75.7) (0.0, 75.7)

The maximum estimated standard error on the estimate was 0.019, so the first digit of each number can be considered exact.

Initially, the appearance of the variables Hi and Li may seem odd. However, note that the variables Z ci , which played crucial roles in deriving the bounds in Proposition 3.1 can also be expressed as    Z ci ¼ VaRU Z ci , and here we merely use TVaRU Z ci and LTVaRU Z ci instead. Thus, Proposition 4.1 has a similar form to that of Proposition 3.1. In the case of no uncertainty (i.e., U ¼ £) there is no model risk, as I ¼ 1. When there is full uncertainty, i.e., U ¼ Rd , then I ¼ 0, and we recover the bounds on the VaR of the portfolio given in Theorem 2.1 of Bernard et al. (2015). Despite the similarities in their form, the VaR bounds mp and M p that are provided in Proposition 4.1 are more difficult to compute than the convex order bounds A and B of Proposition 3.1. The basic reason for this observation is that it is easy to simulate possible realizations for ðZ c1 ; Z c2 ; . . . ; Z cd Þ, as these bounds follow in a straightforward way from the simulated ones for X under the benchmark model. By contrast, the realizations for ðL1 ; L2 ; . . . ; Ld Þ and ðH1 ; H2 ; . . . ; Hd Þ do not follow immediately, which leads to nested simulations when computing mp and M p . Indeed, the simulation of a single realization for Li (or for Hi ) requires, for each simulated value u of the uniformly distributed variable U, a large number of draws from the variable Z i in order to estimate LTVaR at the level U ¼ u. Furthermore, the inequalities in (21) are typically strict so that the bounds mp and M p are not sharp in general; see also the discussion in Section 3.2. Nevertheless, they play a crucial role in deriving (sharp) þ approximations for the best-possible bounds . F resp. .F .

4.2. Approximations The method that we propose for obtaining sharp approximations of the VaR bounds is rooted in the RA of Embrechts et al. (2013) and in a formula for the VaR of a mixture that we apply to the bounds M p and mp . To provide the intuition, let us first consider a mixture S ¼ IXþ ð1  IÞY, where I is a Bernoulli distributed random variable with parameter pF and where the components X and Y are independent of I. Assuming for now that the distribution

functions of X and Y are continuous and strictly increasing with unbounded support, we have that

VaRp ðSÞ ¼ VaRa ðXÞ ¼ VaRb ðYÞ

ð22Þ

for some 0 < a < 1 and 0 < b < 1 that satisfy pF a þ ð1  pF Þb ¼ p. Next, we observe from Proposition 4.1 that for all p 2 ð0; 1Þ and all vectors ðZ 1 ; Z 2 ; . . . ; Z d Þ that are consistent with (4),

VaRp

! ! d d d d X X X X I X i þ ð1  IÞ Z i 6 M p ¼ VaRp I X i þ ð1  IÞ Hi : i¼1

i¼1

i¼1

i¼1

ð23Þ Using the formula (22) for the VaR of a mixture to the right-hand side yields that the bound Mp satisfies d X Hi

Mp ¼ VaRb

! ¼ TVaRb

i¼1

d X Z ci

!

i¼1

for the appropriate choice of 0 < b < 1. Applying formula (22) to the left-hand side of the inequality (23) then leads to the conclusion that this inequality would turn into an equality if there exists a vector ðZ 1 ; Z 2 ; . . . ; Z d Þ such that

VaRb

d X

! Zi

¼ TVaRb

i¼1

! d X c Zi :

ð24Þ

i¼1

In other words, Mp is a sharp bound for VaRp

P d

i¼1 X i



if there is a

vector ðZ 1 ; Z 2 ; . . . ; Z d Þ that yields a flat quantile function of its sum Pd i¼1 Z i on ½b ; 1. It thus follows that the best approximation for P  d the upper bound on VaRp i¼1 X i is likely to occur when the quanP tile (VaR) function of the di¼1 Z i can be made (approximately) flat on ½b ; 1. Similar reasoning shows that in order to reach the stated lower bound as closely as possible, one should make the quantile function of the portfolio sum as flat as possible on the interval ½0; b . As before, we operate in the theoretical context of mixability and in many cases of interest one can construct a dependence among risks that leads to quantile functions that are approximately flat; see also the discussion in Section 3.2.

Table 3 P20 VaR95% ; VaR99:5% and VaR99:95% of i¼1 X i are reported in the absence of uncertainty (i.e., U ¼ £). Approximate VaR Bounds are then given for various confidence levels p, correlation coefficients q and probabilities pf. All digits reported are significant. F ¼ C pF ;q;m;d

U ¼ £ pF ¼ 1

pF ¼ 0:98

pF ¼ 0:8

pF ¼ 0:2

U ¼ Rd p F ¼ 0

p = 95%

q¼0 q ¼ 0:1 q ¼ 0:5

8.1 13.8 26.3

(7.9, 9.0) (13.4, 15.1) (25.4, 27.8)

(6.6, 40.3) (11.3, 40.4) (21.4, 40.8)

(2.2, 48.1) (3.6, 48.1) (7.0, 48.0)

(2.5, 48.2) (2.5, 48.2) (2.6, 48.2)

p = 99.5%

q¼0 q ¼ 0:1 q ¼ 0:5

14.2 24.2 45.9

(13.4, 56.6) (22.8, 56.5) (43.4, 58.5)

(11.0, 75.2) (18.7, 75.1) (35.7, 75.0)

(6.2, 75.7) (10.5, 75.7) (19.9, 75.5)

(0.4, 75.7) (0.4, 75.7) (0.4, 75.7)

p = 99.95%

q¼0 q ¼ 0:1 q ¼ 0:5

20.7 34.8 66.1

(18.2, 103.3) (30.7, 102.4) (58.4, 103.0)

(14.2, 106) (24.2, 106) (46.2, 106)

(8.6, 106) (14.5, 106) (27.7, 106)

(0.1, 106) (0.1, 106) (0.1, 106)

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To compute the (approximate) maximum VaR, we present an algorithm that can be applied directly to the matrix M of the observed data. Recall that the first ‘F rows of the matrix M correspond to F N , whereas ‘U denotes the number of rows of U N (N ¼ ‘F þ ‘U ). In the algorithm, we also make use of SFN and SUN , which we treat as random variables. Here, SFN plays the role of T P and SUN plays the role of di¼1 Z i (see also (13)). Without loss of generality, assume that SFN takes values s1 P s2 P . . . P s‘F . We compute the VaR at probability level p, so that, for ease of exposition, k :¼ Nð1  pÞ, where we assume that k is an integer. The basic idea of the algorithm is then to first compute the value b (or equivalently, the bound Mp ) and next to use the RA of Embrechts et al. (2013) to change the dependence among the risks (on U N ) such that the quantile function of SUN becomes flat on ½b ; 1. Note that in order to compute b we essentially use formula (22) for the VaR of a mixture, but in order to account for all possible distributions a more general version of this formula is needed. It is provided in Appendix A.2. Algorithm for computing the maximum VaR for 0 6 pF < 1 and k :¼ Nð1  pÞ 2 N (1) Compute a where (

(

a :¼ inf a 2 ð0; 1Þ j 9b 2 ð0; 1Þ

pF a þ ð1  pF Þb ¼ p

)

VaRa ðSFN Þ P TVaRppF a ðSUN Þ

is ‘‘mostly’’ known (i.e., pF is high). In this regard, the precise degree of model error highly depends on the level of the probability p that is used to assess VaRp . Let us consider the benchmark model with q ¼ 0 (the risks are uncorrelated and standard Student’s t distributed) and pF ¼ 1 (no uncertainty). We find that P  P  20 20 and, similarly, VaR99:5% VaR95% i¼1 X i ¼ 8:1 i¼1 X i ¼ P  20 14:2; VaR99:95% i¼1 X i ¼ 20:7. When pF ¼ 0:98, then pU ¼ 0:02, and the benchmark model might overestimate the VaR95% by (8.1– 7.9)/7.9 = 2.5% or underestimate it by (9–8.1)/9 = 10%. However, when using the VaR99:5% , the degree of underestimation may rise to (56.6–14.2)/56.6 = 75%, whereas the degree of overestimation is equal only to (14.2–13.4)/13.4 = 6.0%. Hence, the risk of underestimation is sharply increasing in the probability level that is used to assess VaR. Finally, note that when very high probability levels are used in VaR calculations (p ¼ 99:95%; see the last three rows in Table 3), the constrained upper bound is very close to the unconstrained upper bound, even when there is almost no uncertainty on the dependence (pF ¼ 0:98 ). The upper bound computed by Embrechts et al. (2013) is thus nearly the best possible bound, even though it seems that the multivariate model is known at a very high confidence level as F nearly contains all Rd . This implies that any effort to fit a multivariate model accurately will not reduce the model risk on the assessment of (upper) Value-at-Risk at very high confidence levels.

1pF

F a and let b ¼ pp 2 ½0; 1. 1p

5. Applications

F

(2) Apply the RA of Embrechts et al. (2013) to the first bð1  b Þ‘U c rows of the untrusted part U N of the matrix M, where bc denotes the floor. (3) By abuse of notation, denote the rearranged sums in the untrusted part as SUN . This is the dependence that potentially achieves the maximum VaR by making TVaRb ðSUN Þ as close as possible to VaRb ðSUN Þ. To compute this maximum possible VaR, calculate all (row) sums for U N and F N and sort them from maximum to minimum value, ~s1 P ~s2 P    P ~sk P    P ~sN . Then, the VaR is ~sk .

The above algorithm is a quick way to derive potentially attainable bounds for the VaR of the aggregate risk. It requires running the Rearrangement Algorithm only once. However, as the RA will rarely generate a perfectly constant sum on the area where it is applied, and it is possible that a better bound might be obtained by applying step 3 to the first bð1  bÞ‘U c rows of U N for some other bð0 < b < 1Þ. The algorithm for computing the minimum VaR is similar and omitted. 4.3. Example The following example illustrates the algorithm for computing the maximum VaR with a multivariate Student’s t distribution as benchmark. Example 4.2 (VaR – Multivariate Student’s t distribution). The VaR bounds reported in Table 3 are obtained using a matrix of 20 risks obtained from 3,000,000 samples. We make the following observations. First, adding dependence information typically makes it possible to significantly reduce the gap between the upper and lower VaR bounds, compared to the case in which no dependence information is used (pF ¼ 0). However, model risk remains present even when the dependence

In this section, we present two applications. Our first application concerns portfolio optimization in the presence of uncertainty on dependence. The second application deals with the risk measurement of a credit risk portfolio under incomplete information. Specifically, we assess model risk on the VaR estimation of a portfolio of defaultable exposures. In both applications, randomness is driven by stocks that are listed in the S&P 500 index. Let X 1 ; X 2 ; . . . X 500 denote the weekly log-returns of these stocks. Using Bloomberg, we built a history of the observed weekly stock log-returns from January 1st, 1999 to July 25th, 2014, i.e., we have 812 observations in total. We only considered stocks for which the return history was complete (no interruption). We then selected the first fifty in alphabetical order. Hence, we consider (after renumbering) the return vector X :¼ ðX 1 ; X 2 , . . ., X 50 Þ of these stocks. As a benchmark model for X we consider a multivariate Student’s t distribution. First, using the fast MCD (Minimum Covariance Determinant) approach developed by Rousseeuw and Van Zomeren (1990), we compute robust estimators for the location l and covariance matrix R and use maximum likelihood estimation to estimate the number of degrees of ^ ¼ 4Þ: In order to assess the performance of the profreedom mðm posed benchmark model in fitting the data we follow the approach of Garrett (1989) and Filzmoser (2005). Hence, we compute the scaled squared Mahalanobis distances for each of the 812 observations and order them to obtain the empirical quantiles. Under the multivariate Student’s t distribution the scaled squared Mahalanobis distance has a Fð50; 4Þ distribution. We compute its theoretical quantiles and compare these with the empirical quantiles using a Q–Q plot in Fig. 1. The Q–Q plot shows that the benchmark model does not provide a good fit of the observations that exhibit a high Mahalanobis distance. Specifically, the plot supports the choice of the benchmark model on a subarea F ,

n o ^ 1 ½x  l ^ R ^ t 6 cðpF Þ ; F ¼ x :¼ ðx1 ;x2 ;. .. ;x50 Þ 2 R50 j ½x  l

ð25Þ

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Fig. 1. Q-Q plot for the scaled squared Mahalanobis distances.

where cðpF Þ is the cut-off value that we take equal to 10 (corresponding to a probability pF ¼ 0:98). In the applications below, we also consider other values of pF , which makes it possible to gain more insight into the interplay between the degree of dependence uncertainty and optimal portfolios (application 1), as well as into VaR assessment of a credit risk portfolio (application 2). 5.1. Minimum variance portfolio under uncertainty We consider an investor who is composing an investment portfolio using the subset of 50 stocks of the S&P 500 index. We assume that the proportions that are invested in the different assets are kept constant during the investment horizon (constant mix portfolio). We denote them by pi ; i ¼ 1; 2; . . . ; 50, and the weekly log-return of the investment portfolio can then be computed as

Sp ¼ p1 X 1 þ p2 X 2 þ    þ p50 X 50 ; where p :¼ ðp1 ; p2 ; . . . ; p50 Þ. Markowitz (1952) provided a quantitative approach to determine the optimal weights by optimizing the trade-off between mean (expected return) and variance (risk). Critical inputs are the covariances among the assets and their expected return parameters. Since the estimation risk inherent in forecasting expected log-returns is well known, there has always been an interest in optimal portfolios that rely only on covariances (i.e., volatilities and correlations). Let us also remark that the recent global financial crisis has raised the awareness of risk. These observations explain why there is a growing capital inflow6 in ‘‘low risk’’ investment solutions, such as the equally weighted portfolio, the minimum variance portfolio or, more generally, the so-called smart beta portfolio. In the literature on low risk portfolios, the covariance matrix is estimated using a robust method and the optimal weights are derived using the estimated covariances. Here, we realize that covariances are also subject to uncertainty, and we aim to study portfolios that provide the best protection in the presence of this uncertainty. Formally, we consider the problem

min max var ðSp Þ s:t: pT 1 ¼ 1 and p

Sp 2A

pi P 0 ði ¼ 1; 2; . . . ; 50Þ;

ð26Þ

6 Flood (2013) states that the total amount held in low-risk strategies grew to 142 billion USD at the end of March 2013, up from just 58 billion USD at the end of 2010.

where n o P A ¼ Sp :¼ 50 i¼1 pi X i j X satisfies properties ðiÞ; ðiiÞ and ðiiiÞ ; see also Section 2. Note that the requirement pi P 0 ði ¼ 1; 2; . . . ; 50Þ is consistent with the economic theory that efficient portfolios must have non-negative weights. For a given portfolio p, we determine its worst-case variance (among all possible dependence structures for XÞ. Next, we find the optimal portfolio as the one that yields the lowest worst-case variance (among all possible portfolios). This idea is consistent with the worst-case expected utility approach of Gilboa and Schmeidler (1989) for making choices under ambiguity. As the weights pi are positive, it is clear that the worst-case variance of Sp is always attained for the same covariance matrix. This worst-case covariance matrix is essentially obtained by taking X as a multivariate Student’s t distributed vector (with covariance b on F , but with comonotonic dependence on U. Both F matrix R) and U are directly affected by the choice of pF and the corresponding cut-off value cðpF Þ. It is then straightforward to obtain the worst-case covariance matrix by simulation, and we denote it by ^ wc . Our basic problem (26) is then equivalent to R

b wc pt s:t: pT 1 ¼ 1 and min p R p

pi P 0 ði ¼ 1; 2; . . . ; 50Þ:

ð27Þ

Problem (27) can be solved using standard optimization tools. We let pF vary and report for the optimal portfolio the number of weights that are significantly different from zero as well as the well-known Herfindhal concentration index, which is computed as the sum of the squares of the proportions invested. The first column in Table 4 (pF ¼ 1) provides the classical minimum variance portfolio. From the table we observe that concentration is increasing in the degree of uncertainty (i.e., decreasing in the probability pF ). This finding is intuitive, as decreasing pF leads to worst-case variances that are increasingly affected by scenarios that depict perfect dependence among the assets. Hence, diversification pays off less, and one tends to overweight low volatility assets. In the special case when pF ¼ 0, the only information is obtained from the marginal distributions and the optimal portfolio is obtained by investing 100% in the asset that has the smallest volatility. In this regard, we note that it is common to observe a spike in the correlation among assets in turbulent

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Table 4 We report as a function of the level of pF the number of non-zero weights in the optimal portfolio as well as its Herfindahl index and annualized volatility.

# Weights > 0.03% Annual volatility Herfindhal index

pF ¼ 1

pF ¼ 0:9

pF ¼ 0:7

pF ¼ 0:5

pF ¼ 0:3

pF ¼ 0:1

pF ¼ 0

25 10.8 0.064

24 16.5 0.064

23 18.7 0.065

22 19.9 0.073

17 20.6 0.084

11 20.9 0.13

1 21.1 1

Table 5 All VaR bounds are obtained based on 3,000,000 samples of the joint distribution for X. The RA-based algorithm, as discussed in Section 4.2, is then used to derive the VaR bounds. All digits reported are significant. F ¼ C pF ;q;m;d p = 95% p = 99.5% p = 99.95%

pi empirical pi empirical pi empirical

U ¼ £ pF ¼ 1

pF ¼ 0:98

pF ¼ 0:8

pF ¼ 0:2

U ¼ Rd p F ¼ 0

0 4 16

(0, 0) (2, 11) (5, 50)

(0, 1) (1, 15) (1, 50)

(0, 1) (1, 16) (1, 50)

(0, 1) (1, 16) (1, 50)

Table 6 All VaR bounds are obtained based on 3,000,000 samples of the joint distribution for X. The RA-based algorithm, as discussed in Section 4.2, is then used to derive the VaR bounds. All digits reported are significant. F ¼ C pF ;q;m;d

U ¼ £ pF ¼ 1

pF ¼ 0:98

pF ¼ 0:8

pF ¼ 0:5

U ¼ Rd p F ¼ 0

p = 95%

pi = 0.25% pi = 0.5% pi = 0.75%

0 1 2

(0, 0) (1, 2) (1, 2)

(0, 2) (0, 4) (0, 7)

(0, 2) (0, 5) (0, 7)

(0, 2) (0, 5) (0, 7)

p = 99.5%

pi = 0.25% pi = 0.5% pi = 0.75%

7 11 14

(4, 13) (6, 18) (10, 23)

(0, 25) (1, 49) (2, 50)

(0, 25) (1, 50) (1, 50)

(0, 25) (0, 50) (1, 50)

p = 99.95%

pi = 0.25% pi = 0.5% pi = 0.75%

21 26 28

(11, 50) (18, 50) (22, 50)

(1, 50) (2, 50) (3, 50)

(1, 50) (1, 50) (1, 50)

(1, 50) (1, 50) (1, 50)

market conditions. In our setting, this can be modeled by considering low values of pF . Diversification is rendered inefficient, and the best protection is obtained through concentrated portfolios. Of course, the protection is limited, casting doubt on the use of minimum variance portfolios as a vehicle to limit downside risk. 5.2. Credit portfolio risk Financial institutions are inherently exposed to credit risk. The benchmark7 approach to assessing the risk of credit portfolios relies on ‘‘Merton’s model of the firm.’’ The baseline of this approach is simple: a default is an event in which the asset value drops below a threshold value (a liability that is due). Formally, after normalization, default of the ith risk ði ¼ 1; 2; . . . ; 50Þ occurs when fNi < ci g where N i is the normalized asset log-return of the ith risky asset and ci is the threshold value such that the individual default probability of the ith institution is given by pi ¼ PðN i < ci Þ. The total loss S can then be written as

S :¼

50 X

v i 1X
i

i¼1

in which v i is the exposure on the ith firm. In what follows, we consider the portfolio loss that arises from the activity of a bank in selling credit default swap contracts that are written on the fifty companies described above. We assume that the portfolio is homogeneous in its exposures, so that we can assume v i ¼ 1 for all i. Furthermore, in line with industry practice, we use weekly (equity) log-returns X i as a suitable proxy for describing the dynamics of the corresponding asset log-returns Ni ; see Mashal et al. (2003). Ratings for each of the fifty selected companies are obtained from Datastream. These ratings are AAA, AA+, AA, AA, A+, A, A, 7 Note that the standard approaches in the Basel III and Solvency II regulation also build on this idea.

BBB+, BBB, BBB, BB+, BB, BB, B+, B, B, CCC, and they correspond to an average one-year global corporate default rate that we use as an estimate of pi for each company i. In our sample, we find that the average default probability in the portfolio is equal to 0.18%. After inferring from the pi the corresponding threshold values ci in a straightforward way (the marginal distributions of the X i are Student’s t distributed), we then apply the approach8 outlined in Section 4.2 for deriving VaR bounds. The results are reported in Table 5. We observe from Table 5 that unless one has near-perfect knowledge of the multivariate distribution, i.e., when pF is close to one, the VaR bounds are not very sensitive to the choice of the trusted area and match closely the unconstrained bounds. In other words, the information on the ‘‘middle’’ of the distribution does not significantly reduce uncertainty in credit risk assessment. This observation is not surprising, as the scenarios that are included in U typically contain one or more asset log-returns that are either high or low. Hence, the set of untrusted scenarios can be used (by rearranging) to describe a risky situation (in which most defaults essentially occur together) or a non-risky one. Finally, we also tested the impact of the marginal default probabilities on the results. Specifically, we report in Table 6 the VaR bounds for the values pi ¼ 0:25%; pi ¼ 0:5% and pi ¼ 0:75% ði ¼ 1; 2; . . . ; 50Þ. The results confirm those shown in Table 5.

6. Conclusion Recent turbulent events, such as the subprime crisis, have increased the pressure on regulators and financial institutions to P Note that S writes as a sum S ¼ 50 i¼1 f i ðX i Þ in which f i ðxÞ ¼ Ix
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carefully reconsider risk models and to understand the extent to which the outcomes of risk assessments based on these models are robust with respect to changes in the underlying assumptions. The measurement of model risk may be considered to be reasonably under control when only the marginal distributions are assumed to be known (unconstrained bounds); see Embrechts et al. (2013). However, these bounds are wide, as they ignore the (partial) information on dependence that might be available. In this paper, we integrate in a natural way information on the multivariate structure. We propose bounds that are easy to compute but not sharp in general, and we show that the RA of Embrechts et al. (2013) can be applied conditionally for approximating sharp bounds. Our approach may lead to bounds that are significantly tighter than the (unconstrained) ones available in the literature, accounting for the available information coming from a multivariate fitted model and allowing for a more realistic assessment of model risk. However, model risk remains a significant concern, and we recommend caution regarding regulation based on Value-at-Risk at a very high confidence level since such an assessment is unable to benefit from careful risk management attempts to fit a multivariate model. Finally, note that we assume that the marginal distributions are fixed and known. To capture the possible uncertainty of the marginal distributions, one might consider amplifying their tails. For example, a distortion (e.g., the Wang transform) could be applied.

VaRp

d X Xi

!

d d X X Z ci 6 VaRp I X i þ ð1  IÞTVaRU

i¼1

i¼1

¼ VaRp

! d X d I X i þ ð1  IÞ m Hi ;

!!

i¼1

i¼1

i¼1

where the last equality follows from the fact that TVaR is additive Pd c i¼1 Z i . The proof for the lower bound is

for the comonotonic sum similar and omitted. h A.2. VaR of a mixture

The following lemma provides an expression for the VaR of a mixture. The proof of this lemma is omitted as it is lengthy and it can be found in Bernard and Vanduffel (2014). Lemma A.1 (Computing VaR). Consider a sum S ¼ IXþ ð1  IÞY, where I is a Bernoulli distributed random variable with parameter pF and where the components X and Y are independent of I. Define a 2 ½0; 1 by



a :¼ inf a 2 ð0; 1Þ j 9b 2 ð0; 1Þ and let b ¼

ppF a 1pF



pF a þ ð1  pF Þb ¼ p



VaRa ðXÞ P VaRb ðYÞ

2 ½0; 1. Then, for p 2 ð0; 1Þ,

  VaRp ðSÞ ¼ max VaRa ðXÞ; VaRb ðYÞ :



ð29Þ

Appendix A. Proofs References A.1. Proof of Proposition 4.1 For a given random vector X satisfying properties (i), (ii) and (iii), there exists a vector ðY 1 ; Y 2 ; . . . Y d Þ with marginals Y i that have the same distribution as Z ci such that

! d d d X X X X i ¼d I X i þ ð1  IÞ Y i : i¼1

i¼1

i¼1

As per definition of the VaR, it follows for all p 2 ð0; 1Þ, !  d X ðxÞ þ ð1  p ÞF Pd VaRp X i ¼ inf x 2 R j pF F Pd F X jI¼1 i¼1 i

i¼1

Y i¼1 i

ðxÞ P p ;

ð28Þ where pF :¼ PðI ¼ 1Þ. Note that for all p 2 ð0; 1Þ, 1

F Pd

Y i¼1 i

ðpÞ 6 TVaRp

d X Yi

!

6 TVaRp

i¼1

! d X c Zi ; i¼1

where the second inequality follows from the fact that the Z ci are comonotonic (while having the same distribution as the Y i ). Thus,

Pd F 1

Y i¼1 i

ðUÞ 6 R :¼ TVaRU

d X Z ci

! a:s:

i¼1

which can be also written in terms of their cdf. Therefore, for all x 2 R; F Pd ðxÞ P F R ðxÞ. Thus, Y i¼1 i

VaRp

!  d X X i 6 inf x 2 R j pF F Pd i¼1

X jI¼1 i¼1 i

ðxÞ þ ð1  p ÞF R ðxÞ P p : F

We observe that the right-hand side of the above equation is by definition the VaR of a sum of mutually exclusive variables, it follows that

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