Aggregation-robustness and model uncertainty of regulatory risk measures
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DOI: 10.1007/s00780-015-0273-z
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- Embrechts, P., Wang, B. & Wang, R. Finance Stoch (2015) 19: 763. doi:10.1007/s00780-015-0273-z
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Abstract
Research related to aggregation, robustness and model uncertainty of regulatory risk measures, for instance, value-at-risk (VaR) and expected shortfall (ES), is of fundamental importance within quantitative risk management. In risk aggregation, marginal risks and their dependence structure are often modelled separately, leading to uncertainty arising at the level of a joint model. In this paper, we introduce a notion of qualitative robustness for risk measures, concerning the sensitivity of a risk measure to the uncertainty of dependence in risk aggregation. It turns out that coherent risk measures, such as ES, are more robust than VaR according to the new notion of robustness. We also give approximations and inequalities for aggregation and diversification of VaR under dependence uncertainty, and derive an asymptotic equivalence for worst-case VaR and ES under general conditions. We obtain that for a portfolio of a large number of risks, VaR generally has a larger uncertainty spread compared to ES. The results warn that unjustified diversification arguments for VaR used in risk management need to be taken with much care, and they potentially support the use of ES in risk aggregation. This in particular reflects on the discussions in the recent consultative documents by the Basel Committee on Banking Supervision.
Keywords
Value-at-risk Expected shortfall Dependence uncertainty Risk aggregation Aggregation-robustness Inhomogeneous portfolio Basel IIIMathematics Subject Classification
62G35 60E15 62P05JEL Classification
C101 Introduction
Risk measurement, with its crucial importance for financial institutions such as banks, insurance companies and investment funds, has drawn a lot of attention in both academia and industry over the past several decades. Although a financial risk, often modelled by a probability distribution, cannot be characterized by a single number, sometimes one needs to assign a number to a risk position. The determination of regulatory capital is one such example, the ranking of risks another. For such purposes, quantitative tools that map risks to numbers were introduced, and they are called risk measures.
Over the past three decades, value-at-risk (VaR) became the benchmark (Jorion [22]). Expected shortfall (ES), an alternative to VaR which is coherent (Artzner et al. [3]), is arguably the second most popular risk measure in use. In two recent consultative documents BCBS [4, 5], the Basel Committee on Banking Supervision proposed to take a move from VaR to ES for the measurement of market risk in banking. Under Solvency 2 and the Swiss Solvency Test, the same discussion takes place within insurance regulation; see, for instance, Sandström [34] and SCOR [35]. As a consequence, there have been extensive debates on issues related to diversification, aggregation, economic interpretation, optimization, extreme behaviour, robustness and backtesting of VaR and ES. We omit a detailed analysis here and refer to Embrechts et al. [15], Emmer et al. [16] and the references therein.
Here are some of the issues raised. VaR is not coherent, but it is elicitable (see Gneiting [18, Theorem 9]; that paper also contains some earlier references), easy to backtest and more robust with respect to statistical uncertainty, as argued in Gneiting [18] and Cont et al. [10]. ES is coherent, but not elicitable and difficult to backtest. There have been extensive discussions on the problematic diversification and aggregation issues of VaR due to its lack of subadditivity; see, for example, Embrechts et al. [14]. Daníelsson et al. [11] argue that the violation of subadditivity for VaR is rare in practice. VaR, being a quantile, does not address the crucial “what if” question, i.e. what are the consequences if a particular rare event (measured by VaR) occurs. Whereas this was clear since its introduction within the financial industry around 1994, it took some serious financial crises to bring this issue fully onto the regulatory agenda.
The importance of robustness properties of risk measures has only fairly recently become a focal point of regulatory attention. By now, numerous academic as well as applied papers address the topic. Conflicting views typically result from different notions of robustness; Embrechts et al. [15] contains a brief discussion and some references. In the present paper, the measurement of aggregated risk positions under uncertainty with respect to the dependence structure of the underlying risk factors will be discussed. We show that ES enjoys a new property of aggregation-robustness which VaR generally does not have.
The mathematical property of (non-)subadditivity of a risk measure becomes relevant upon analysing the aggregate position of a portfolio. As is often the case in practice, the dependence structure among individual risks in a portfolio is difficult to obtain from a statistical point of view, while the marginal distributions of the individual risks (assets) may typically be easier to model; see, for instance, Embrechts et al. [14] and Bernard et al. [7]. Modelling a high-dimensional dependence structure is well known to be data-costly, and dimension reduction techniques such as vine copulas, hierarchical structures and very specific parametric models often have to be implemented. Whereas such simplifying techniques in general create computational and modelling ease, they typically involve considerable model uncertainty. This leads to a notion of dependence uncertainty (DU) in risk aggregation, a concept of main interest for this paper.
From a mathematical or statistical point of view, it is clearly better to look at robustness properties of a model at the level of the joint distribution of the risk factors. The main reason for separating the two (marginals, dependence) is because of processes in practice, where indeed the two are often modelled separately. This is particularly true in a stress testing environment.
Hence for this paper, we introduce the notion of aggregation-robustness to study properties of risk measures for aggregation in the presence of dependence uncertainty. The new notion is based on the classic notion of robustness for statistical functionals in, e.g. Huber and Ronchetti [21]. However, as opposed to the conclusions in Cont et al. [10], we show that when model uncertainty lies solely at the level of the dependence structure, coherent distortion risk measures (such as ES) are continuous with respect to weak convergence of the underlying distributions, whereas VaR in general is not. This result supports the use of ES for risk aggregation, especially when statistical information on marginal distributions is reliable.
Under DU, the attainable values of a risk measure lie in an interval. This interval can be seen as a measurement of model uncertainty for that risk measure. When a risk measure is used to quantify the riskiness of an aggregate position of a portfolio, the ratio between the risk measure of the aggregate risk and the sum of the risk measures of the marginal risks is called a diversification ratio. The diversification ratio measures how good the risks in a portfolio hedge (compensate for) each other. With only models for marginal distributions available, the diversification ratio also takes values in a DU-interval.
To study the DU-interval of VaR and ES and their diversification ratios, one needs to calculate the worst-case and best-case values of VaR and ES under dependence uncertainty. Due to the subadditivity of ES, the worst-case value of ES is the sum of the ES of the marginal risks. However, the other three quantities (best- and worst-case VaR, best-case ES) are in general unknown. Partial results exist. The worst-case value of VaR for \(n=2\) was given in Makarov [27] based on early results in multivariate probability theory. Embrechts and Puccetti [13] gave a dual bound for the worst-case VaR for \(n\geqslant 3\) in the homogeneous model, i.e. when all marginal risks have the same distribution. Partial solutions for the worst-case and best-case values of VaR are to be found in Wang et al. [40], Puccetti and Rüschendorf [30] and Bernard et al. [7], based on the notion of complete mixability (CM) introduced in Wang and Wang [37]. A fast algorithm to numerically calculate the worst-case and best-case values of VaR under general conditions was introduced in Embrechts et al. [14]; this is the so-called rearrangement algorithm (RA). For the best-case ES, some partial analytical results can be found in Bernard et al. [7] and Cheung and Lo [9], and a numerical procedure was proposed by Puccetti [29].
In most of the existing analytical results, it is assumed that the marginal distributions are identical (homogeneous case), with some extra conditions on the shape of the underlying risk factor densities (assumed to exist). In this paper, we relax the assumptions on the marginal distributions. Instead of explicit values for the worst-case and best-case VaR, we obtain approximations. The new results obtained can be used within a discussion on capital requirement; they, moreover, yield a DU-interval for VaR and its diversification ratio.
Further understanding of the worst-case VaR can be obtained through the asymptotic behaviour as the number of risks in the portfolio grows to infinity, i.e. for a large portfolio regime. In the homogeneous case, Puccetti and Rüschendorf [31] obtained an asymptotic equivalence between the worst-case VaR and the worst-case ES under dependence uncertainty, and this under a strong condition on the identical marginal distributions. The required condition was later weakened by Puccetti et al. [32] (based on further results on complete mixability) and Wang [39] (based on a duality result obtained in Rüschendorf [33]). It was finally removed by Wang and Wang [38] (based on the notion of extreme negative dependence). When the marginal distributions are not identical, Puccetti et al. [32] also obtained the asymptotic equivalence under the assumption that only finitely many different choices of the marginal distributions can appear; this mathematically allows a reduction to the case of identical marginal distributions. In this paper, we give a unifying result on this asymptotic equivalence, by allowing the marginal distributions to be arbitrary. Only weak uniformity conditions on the moments of the marginal distributions are required for our results to hold. These conditions are easily justified in practice and are necessary for the most general equivalence to hold. The new results lead to the asymptotic DU-spread of VaR and ES, and show that VaR in general yields a larger DU-spread compared to ES.
The rest of the paper is organized as follows. In Sect. 2, we introduce the notion of aggregation-robustness and show that ES is aggregation-robust but VaR is not. In Sect. 3, we give new bounds on the diversification ratios under dependence uncertainty, and establish an asymptotic equivalence between VaR and ES under a worst-case scenario. The dependence uncertainty spreads of VaR and of ES are derived and compared in Sect. 4. In Sect. 5, numerical examples are presented to illustrate our results. Section 6 draws some conclusions. All proofs are put in the Appendix.
Throughout the paper, we let\((\varOmega, \mathcal{A}, \mathbb {P})\)be an atomless probability space and \(L^{0}:=L^{0}(\varOmega, \mathcal{A}, \mathbb {P})\) the set of all (equivalence classes of) real-valued random variables on that probability space. Elements of \(L^{0}\) are often referred to as risks. Their distribution functions are simply referred to as distributions. We write \(X\sim F\) to denote \(F(x)=\mathbb {P}[X\leqslant x]\), \(x\in \mathbb {R}\). We also denote the generalized inverse function of \(F\) by \(F^{-1}\), that is, \(F^{-1}(p)=\inf\{t\in \mathbb {R}:F(t)\geqslant p\}\) for \(p\in(0,1]\), and \(F^{-1}(0)=\inf\{t\in \mathbb {R}:F(t)> 0\}\).
2 Robustness of VaR and ES for risk aggregation
2.1 Robustness of risk measures
The robustness of a statistical functional or an estimation procedure describes the sensitivity to underlying model deviations and/or data changes. Different definitions and interpretations of robustness exist in the literature; see, for example, Huber and Ronchetti [21] from a purely statistical perspective, Hansen and Sargent [20] in the context of economic decision making, and Ben-Tal et al. [6] within optimization. In statistics, robustness mainly concerns the so-called distributional (or Hampel–Huber) robustness: the statistical consequences when the shape of the actual underlying distribution deviates slightly from the assumed model.
A risk measure \(\rho\) is a function which maps a risk in a set \(\mathcal{X}\) to a number, \(\rho: \mathcal{X}\rightarrow(-\infty ,+\infty]\), where \(\mathcal{X}\subset L^{0}\) typically contains \(L^{\infty}\) and is closed under addition and positive scalar multiplication. A risk measure is law-invariant if it only depends on the distribution of the risk. We omit the general introduction of risk measures, and refer the interested reader to Föllmer and Schied [17, Chap. 4]. Since law-invariant risk measures are a specific type of statistical functionals, their robustness properties are already extensively studied in the statistical literature; see, e.g. Huber and Ronchetti [21, Chap. 3].
In the following, we introduce a new, in our opinion practically relevant notion of robustness for risk aggregation, which turns out to favour ES over VaR.
2.2 Aggregation-robustness
In this section, we show that VaR is more sensitive to model uncertainty at the level of dependence than ES. For single risks \(X_{i}\), \(i=1,\dots,n\), the aggregate risk \(S\) is simply defined as \(S=X_{1}+\cdots+X_{n}\). Often in practice, a joint model of \(X_{1},\dots,X_{n}\) is chosen in two stages: \(n\) marginal distributions \(F_{1},\dots,F_{n}\) and a dependence structure (often through a copula \(C\)). Whereas the modelling of marginal distributions is fairly standard, the dependence structure can be really difficult to model, statistically estimate and test. Considerable model uncertainty, which is often different in nature from the model uncertainty of marginal distributions, arises from modelling the dependence structure. In the following, we study sensitivity with respect to uncertainty in the dependence structure; for the purpose of this paper, we assume the marginal distributions\(F_{1},\dots,F_{n}\)are given.
Definition 2.1
(Aggregation-robustness)
We say that a law-invariant risk measure \(\rho: \mathcal {X}\rightarrow (-\infty,+\infty]\) is aggregation-robust if \(\rho\) is continuous with respect to weak convergence in each admissible class \(\mathfrak{S}_{n}\) compatible with \(\rho\).
The robustness character of Definition 2.1 in intuitively clear. If the joint distributions of \((X_{1},\dots,X_{n})\) and \((Y_{1},\dots ,Y_{n})\) are close according to the Lévy metric, then the distributions of \(X_{1}+\cdots+X_{n}\) and \(Y_{1}+\cdots+Y_{n}\) are also close according to the Lévy metric. As a consequence, \(\rho\) is insensitive to small perturbations of the joint distribution of the underlying risk factors, keeping the marginal distributions of the individual risks fixed. It is clear that Hampel’s robustness, as discussed above, without the restriction of risks being in a common admissible class, implies aggregation-robustness. When the dependence structure is modelled by copulas, our definition of robustness implies that a risk measure is insensitive to the copula of the individual risks when the marginal distributions are assumed to be known. The fact that in Definition 2.1 we look at risks in \(\mathfrak{S}_{n}\) reflects our interest in aggregation and diversification. One could, of course, look at other functional-robustness definitions beyond aggregation (summation).
Example 2.2
The non-aggregation-robustness of \(\mathrm {VaR}_{p}\) essentially comes from the fact that it is not continuous with respect to weak convergence (Hampel’s robustness). Suppose that \(\mathrm {VaR}_{p}\) is not continuous at some distribution, say \(F_{0}\). One may find \(F_{n}, n\in \mathbb {N}\), which converge to \(F_{0}\) weakly, but \(F_{n}^{-1}(p),n\in \mathbb {N}\), do not converge to \(F_{0}^{-1}(p)\); if in addition, such \(F_{n},n\in \mathbb {N}\), and \(F_{0}\) lie in the same admissible class, then \(\mathrm {VaR}_{p}\) is not aggregation-robust. That leads to the construction in Example 2.2.
In the above example, the joint distribution \(C_{t}\) with a small \(t>0\) can be seen as the joint distribution \(C_{0}\) influenced by a small perturbation. It is moreover worth noting that in Example 2.2, the marginal distributions of \(X_{t}\) and \(Y_{t}\) are continuous with positive densities. Hence, even if the true marginal distributions are known to have positive densities, VaR can still be discontinuous in aggregation. When one considers absolutely continuous models for a single risk, one has Hampel’s robustness for \(\mathrm {VaR}_{p}\); however, considering absolutely continuous marginal models is not sufficient for the aggregation-robustness of \(\mathrm {VaR}_{p}\). On the other hand, we shall see that ES is aggregation-robust, although it is well known to be non-robust in Hampel’s sense (see Cont et al. [10]) since it is discontinuous at any distribution with respect to the weak topology.
Theorem 2.3
All coherent distortion risk measures on\(\mathcal {X}_{0}\)with a continuous distortion function are aggregation-robust.
As a coherent distortion risk measure has a convex distortion function, assuming continuity only excludes a jump of the distortion function at 1. Theorem 2.3 says that when the model uncertainty lies at the level of dependence but not at the level of the marginal distributions, coherent distortion risk measures such as ES are continuous with respect to weak convergence.
Our result can be interpreted as follows. For a distribution \(F\) and a random variable \(X \sim F\), even adding constraints on marginal distributions of the aggregation model of \(X\), \(F\mapsto \mathrm {VaR}_{p}(X)\) is still not continuous (with respect to weak convergence), whereas \(F\mapsto \mathrm {ES}_{p}(X)\) is continuous with these constraints. It should not be interpreted as an argument against the classic continuity results of VaR, noting that VaR is continuous at most commonly used distributions in financial risk management.
Remark 2.4
Cont et al. [10] also introduced the notion of \(\mathcal{C}\)-robustness, where \(\mathcal{C}\) is a set of distributions. A risk measure \(\rho\) is \(\mathcal{C}\)-robust if \(\rho\) is continuous in \(\mathcal{C}\) with respect to the Lévy distance; see Cont et al. [10, Proposition 2]. Using this notion, \(\mathrm {VaR}_{p}\) is \(\mathcal{C}_{p}\)-robust, where \(\mathcal{C}_{p}\) is the set of distributions \(F\) for which \(F^{-1}\) is continuous at \(p\). If we denote by \(\mathfrak{D}(\mathfrak{S}_{n})\) the set of all possible distributions for an admissible class \(\mathfrak{S}_{n}\), then \(\rho\) is aggregation-robust if and only if \(\rho\) is \(\mathfrak{D}(\mathfrak{S}_{n})\)-robust for all possible choices of \(n\in \mathbb {N}\) and \(\mathfrak{D}(\mathfrak{S}_{n})\), in which \(\mathfrak{S}_{n}\) is compatible with \(\rho\).
In the case \(\mathcal {X}=L^{\infty}\), we obtain that a continuous distortion function is a necessary and sufficient condition for the aggregation-robustness of distortion risk measures.
Theorem 2.5
A distortion risk measure on\(\mathcal {X}=L^{\infty}\)is aggregation-robust if and only if its distortion function\(h\)is continuous on\([0,1]\).
Finally, we remark that it would be of much interest to characterize aggregation-robust statistical functionals (risk measures) other than the class of distortion risk measures. Such a characterization is beyond the scope of this paper, and we leave it for future work.
3 Bounds on VaR aggregation
In Sect. 2, we mainly looked at the sensitivity properties of risk measures on aggregated risks under small changes of the underlying dependence assumptions. In this section, for VaR, we concentrate on deviations (possibly) far away from some true underlying, though unknown, dependence structure. Such results can be used to analyse extreme scenarios for risk aggregation and may be helpful in order to determine conservative capital requirements under model (i.e. dependence) uncertainty; for a real life example on this, see Aas and Puccetti [1].
3.1 Aggregation and diversification under dependence uncertainty
(Conservative capital requirement) \(\overline{\varDelta }^{p}_{n}\mathrm {VaR}_{p}^{+}(S_{n})\) can be used as the most conservative capital requirement in the case of given (or estimated) marginal distributions \(F_{1},\dots,F_{n}\) of the individual risks.
(Measurement of model uncertainty) If \(\overline{\varDelta }^{p}_{n}\) is small, then the model uncertainty is small, and the risk measure \(\mathrm {VaR}\) is considered as less problematic in risk aggregation; capital requirement principles based on \(\mathrm {VaR}_{p}^{+}\) become more plausible. If \(\overline{\varDelta }^{p}_{n}\) is large, then the model uncertainty is severe, and arguments of diversification benefit need to be taken with care.
3.2 Bounds on VaR aggregation for a finite number of risks
The following theorem contains our main result regarding approximations of \(\overline {\mathrm {VaR}}_{p}(S_{n})\) and \(\underline {\mathrm {VaR}}_{p}(S_{n})\).
Theorem 3.1
Note that in the case when all marginal distributions are bounded, \(\overline {\mathrm {VaR}}_{p}(S_{n})\) and \(\overline {\mathrm {ES}}_{p}(S_{n})\) differ by at most a constant which does not depend on \(n\). Theorem 3.1 can also be formulated for the worst diversification ratio of VaR.
Corollary 3.2
In the homogeneous case \(F:=F_{1}=F_{2}=\cdots{}\), the left- and right-hand sides of (3.8) both converge to \(\frac{\mathrm {ES}_{p}(X)}{\mathrm {VaR}_{p}(X)}\) as \(n\rightarrow\infty\), where \(X\sim F\), assuming \(\mathrm {VaR}_{p}(X) \ne0\). In the following, we study the limit, as \(n\) goes to infinity, of the worst- and best-case VaR and its diversification ratio under general marginal assumptions.
3.3 Asymptotic equivalence and limit of the worst diversification ratio
Theorem 3.3
Theorem 3.3 establishes the asymptotic equivalence of the worst-case ES and the worst-case VaR for risk aggregation for general, possibly inhomogeneous portfolios. As mentioned in Sect. 3.1, homogeneous or almost-homogeneous cases for which (3.13) holds were previously obtained in the literature. While existing methods of proof were mainly based on the theory of complete mixability, an extension using the same techniques to arbitrarily many different marginal distributions was not possible.
The following corollary is obtained from Theorem 3.3 by symmetry:
Corollary 3.4
Remark 3.5
4 Uncertainty spread of VaR and ES
Theorem 4.1
- (i)Suppose that the distributions\(F_{i},i\in \mathbb {N}\), satisfy (3.9), (3.15) and (4.1). Then$$\begin{aligned} \liminf_{n\rightarrow\infty}\frac{\overline{\mathrm {VaR}}_{q}(S_{n})-\underline {\mathrm {VaR}}_{q}(S_{n})}{\overline{\mathrm {ES}}_{p}(S_{n})-\underline {\mathrm {ES}}_{p}(S_{n})} =&\liminf_{n\rightarrow\infty}\frac{\overline {\mathrm {ES}}_{q}(S_{n})-\underline {\mathrm {LES}}_{q}(S_{n})}{\overline {\mathrm {ES}}_{p}(S_{n})-\underline {\mathrm {ES}}_{p}(S_{n})} \\ \geqslant & \liminf_{n\rightarrow\infty}\frac{\overline {\mathrm {ES}}_{q}(S_{n})-\mu_{n}}{\overline {\mathrm {ES}}_{p}(S_{n})-\mu_{n}}\geqslant 1. \end{aligned}$$(4.2)
- (ii)Suppose that the distributions\(F_{i},i\in \mathbb {N}\), are identical and equal to a non-degenerate distribution\(F\), and\(\mathbb {E}[|X|^{k}]<\infty\)for some\(k>1\), where\(X\sim F\). Then$$ \liminf_{n\rightarrow\infty }\frac {\overline{\mathrm {VaR}}_{q}(S_{n})-\underline{\mathrm {VaR}}_{q}(S_{n})}{\overline{\mathrm {ES}}_{p}(S_{n})-\underline {\mathrm {ES}}_{p}(S_{n})}\geqslant \frac{\mathrm {ES}_{q}(X)-\mathrm {LES}_{q}(X)}{\mathrm {ES}_{p}(X)-\mathbb {E}[X]}\geqslant 1. $$(4.3)
Theorem 4.1 suggests that VaR is overall more sensitive to dependence uncertainty for large \(n\), compared to ES. Numerical evidence of the comparison of the DU-spreads for VaR and ES at the same level can be found in Sect. 5, even for small values of \(n\). Note that although the DU-spread of ES is smaller than that of VaR asymptotically, both risk measures have a rather large uncertainty spread in general, suggesting that dependence uncertainty in risk aggregation must be taken with care, no matter whether ES or VaR is chosen as the underlying risk measure; see Aas and Puccetti [1] for values in the context of a real life example.
Remark 4.2
In the homogeneous case, for any continuous distribution \(F\), the limit of the DU-spread ratio in (4.3) is strictly greater than 1 since \(\mathrm {LES}_{q}(X)<\mathbb {E}[X]\) and \(\mathrm {ES}_{q}(X)>\mathrm {ES}_{p}(X)\). In the case \(q=p\), we note that for light-tailed risks \(X\), \(\mathrm {LES}_{p}(X)\) is slightly smaller than \(\mathbb {E}[X]\); for heavy-tailed risks \(X\), \(\mathrm {LES}_{p}(X)\) can be significantly smaller than \(\mathbb {E}[X]\), leading to a much larger DU-spread of VaR. From Theorem 4.1, we can also see that, approximately, the \(\mathrm {VaR}_{q}\) interval under DU is \([\sum _{i=1}^{n}\mathrm {LES}_{q}(X_{i}),\ \sum_{i=1}^{n}\mathrm {ES}_{q}(X_{i})]\) and the \(\mathrm {ES}_{p}\) interval under DU is given by \([\sum_{i=1}^{n} \mathbb {E}[X_{i}],\ \sum_{i=1}^{n}\mathrm {ES}_{p}(X_{i})]\).
In the following, we give a result for finite \(n\), in the case of bounded risks. A proof can be directly obtained from Theorem 3.1.
Corollary 4.3
Note that in Corollary 4.3, since \(\mathrm {ES}_{q}(X_{i})\geqslant \mathrm {ES}_{p}(X_{i})\) and \(\mathbb {E}[X_{i}]\geqslant \mathrm {LES}_{q}(X_{i})\), the left-hand side of (4.4) is the summation of \(n\) nonnegative terms, while the right-hand side of (4.4) is a constant. Hence (4.4) holds for \(n\) sufficiently large as long as the summation of the left-hand side of (4.4) diverges as \(n\rightarrow\infty\).
We remark that it remains theoretically unclear under what conditions the DU-spread of \(\mathrm {VaR}_{q}\) is larger than (or equal to) that of \(\mathrm {ES}_{p}\) for finite \(n\) and \(q\geqslant p\). In all our numerical examples (see Sect. 5 below), \(\mathrm {VaR}_{q}\) always has a larger DU-spread than \(\mathrm {ES}_{p}\).
5 Numerical examples
- (A)
(Mixed portfolio) \(S_{n}=X_{1}+\cdots+X_{n}\), where \(X_{i}\sim\mathrm{Pareto}(2+0.1i)\), \({i=1,\ldots,5}\), and \(X_{i}\sim\mathrm{Exp}(i-5)\), \({i=6,\ldots,10}\), and \(X_{i}\sim\mathrm{lognormal} (0,(0.1(i-10))^{2})\), \(i=11,\ldots,20\).
- (B)
(Light-tailed portfolio) \(S_{n}=Y_{1}+\cdots+Y_{n}\), where \(Y_{i}\sim\mathrm{Exp}(i), {i=1,\ldots,5}\); \(Y_{i}\sim\mathrm{Weibull}(i-5,1/2)\), \({i=6,\ldots,10}\); \(Y_{i}\stackrel{d}{=}Y_{i-10}, i=11,\ldots,20\).
- (C)
(Pareto portfolio) \(S_{n}=Z_{1}+\cdots+Z_{n}\), where \(Z_{i}\sim \mathrm {Pareto}(1.5)\), \(i=1,\dots,20\).
Bounds obtained with RA (\(\varDelta x=10^{-6}\)), model (A), mixed portfolio
n = 5 | n = 10 | n = 20 | |||||||
---|---|---|---|---|---|---|---|---|---|
best | worst | spread | best | worst | spread | best | worst | spread | |
\(\mathrm{ES}_{0.975 }(S_{n})\) | 22.48 | 44.88 | 22.40 | 22.52 | 55.59 | 33.07 | 29.15 | 102.35 | 73.20 |
\(\mathrm{VaR}_{0.975 }(S_{n})\) | 9.79 | 41.46 | 31.67 | 10.04 | 52.67 | 42.63 | 21.44 | 100.65 | 79.21 |
\(\mathrm{VaR}_{0.9875}(S_{n})\) | 12.06 | 56.21 | 44.16 | 12.06 | 69.03 | 56.98 | 22.12 | 126.63 | 104.51 |
\(\mathrm{VaR}_{0.99 }(S_{n})\) | 12.96 | 62.01 | 49.05 | 12.96 | 75.34 | 62.38 | 22.29 | 136.30 | 114.01 |
\(\frac{\overline {\mathrm {ES}}_{0.975}(S_{n})}{\overline {\mathrm {VaR}}_{0.975}(S_{n})}\) | 1.08 | 1.06 | 1.02 |
Bounds obtained with RA (\(\varDelta x=10^{-6}\)), model (B), light-tailed portfolio
n = 5 | n = 10 | n = 20 | |||||||
---|---|---|---|---|---|---|---|---|---|
best | worst | spread | best | worst | spread | best | worst | spread | |
\(\mathrm{ES}_{0.975 }(S_{n})\) | 4.72 | 10.71 | 5.99 | 24.55 | 63.19 | 38.63 | 31.33 | 126.38 | 95.04 |
\(\mathrm{VaR}_{0.975 }(S_{n})\) | 3.69 | 10.57 | 6.88 | 13.61 | 61.41 | 47.81 | 13.61 | 125.73 | 112.13 |
\(\mathrm{VaR}_{0.9875}(S_{n})\) | 4.38 | 12.15 | 7.77 | 19.20 | 78.75 | 59.55 | 19.20 | 160.75 | 141.55 |
\(\mathrm{VaR}_{0.99 }(S_{n})\) | 4.61 | 12.66 | 8.05 | 21.21 | 84.80 | 63.59 | 21.21 | 172.96 | 151.75 |
\(\frac{\overline {\mathrm {ES}}_{0.975}(S_{n})}{\overline {\mathrm {VaR}}_{0.975}(S_{n})}\) | 1.01 | 1.03 | 1.01 |
Bounds obtained with RA (\(\varDelta x=10^{-6}\)), model (C), Pareto portfolio
n = 5 | n = 10 | n = 20 | |||||||
---|---|---|---|---|---|---|---|---|---|
best | worst | spread | best | worst | spread | best | worst | spread | |
\(\mathrm{ES}_{0.975 }(S_{n})\) | 103.8 | 172.6 | 68.8 | 166.2 | 345.1 | 178.9 | 266.2 | 690.3 | 424.1 |
\(\mathrm{VaR}_{0.975 }(S_{n})\) | 15.7 | 130.6 | 114.9 | 21.8 | 291.3 | 269.5 | 43.5 | 620.8 | 577.3 |
\(\mathrm{VaR}_{0.9875}(S_{n})\) | 22.6 | 207.3 | 184.7 | 27.6 | 462.4 | 434.8 | 46.7 | 985.5 | 938.8 |
\(\mathrm{VaR}_{0.99 }(S_{n})\) | 25.5 | 240.5 | 215.0 | 30.5 | 536.5 | 506.0 | 47.5 | 1143.6 | 1096.0 |
\(\frac{\overline {\mathrm {ES}}_{0.975}(S_{n})}{\overline {\mathrm {VaR}}_{0.975}(S_{n})}\) | 1.32 | 1.19 | 1.11 |
- (i)
The worst-case VaR at level 0.975 and the worst-case ES at level 0.975 are very close, even for small values of \(n\), in all models considered (cf. Theorem 3.3, (3.13)).
- (ii)
The ratio between the worst-case VaR at level 0.975 and the worst-case ES at level 0.975 goes to 1 as \(n\) grows large. In the heavy-tailed model (C), the convergence is relatively slow (cf. Theorem 3.3, (3.14)).
- (iii)
The DU-spreads of \(\mathrm {VaR}_{0.99}\), \(\mathrm {VaR}_{0.985}\) and \(\mathrm {VaR}_{0.975}\) are larger than those of \(\mathrm {ES}_{0.975}\) in all considered models (cf. Theorem 4.1).
- (iv)
In the heavy-tailed model (C), the DU-spreads of VaR are significantly larger than those of ES (cf. Remark 4.2).
6 Conclusion
In this paper, we have considered the risk measures VaR and ES under dependence uncertainty. We have introduced the notion of aggregation-robustness and have shown that all coherent distortion risk measures, including ES, are aggregation-robust, but VaR is not. We have also derived bounds for the worst- and best-case VaR in aggregation and its diversification ratio under dependence uncertainty. An asymptotic equivalence between VaR and ES for inhomogeneous portfolios under the weakest so far known conditions on the marginal distributions has been established. It has been shown that when the number of risks in aggregation is large, VaR generally exhibits a larger uncertainty spread compared to ES at the same or a lower confidence level. Numerical examples have been provided to support our theoretical results. The main results in this paper suggest that ES is less sensitive with respect to dependence uncertainty in aggregation, and it typically has a smaller uncertainty spread compared to VaR.
Acknowledgements
The authors would like to thank two referees, an Associate Editor and the Editor for helpful comments which have substantially improved the paper, and Edgars Jakobsons (ETH Zurich) for his kind help on some numerical examples in this paper. Paul Embrechts thanks the Oxford-Man Institute for its hospitality during his visit as 2014 Oxford-Man Chair. Ruodu Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Forschungsinstitut für Mathematik (FIM) at ETH Zurich during his visits in 2013 and 2014.