Efficient and robust portfolio optimization in the multivariate Generalized Hyperbolic framework

Transcription

1 Quantitative Finance, Vol., No., October 2, Efficient and robust ortfolio otimization in the multivariate Generalized Hyerbolic framework MRTIN HELLMICHy and STEFN KSSERGER*z yfrankfurt School of Finance and Management, Sonnemannstraße 9, 634 Frankfurt, Germany zinstitute of Mathematical Finance, Ulm University, Helmholtzstraße 8, 8969 Ulm, Germany (Received 4 October 27; in final form 8 May 29) In this aer, we aly the multivariate Generalized Hyerbolic (MGH) distribution to ortfolio modeling, using Conditional Value at Risk (CVaR) as a risk measure. Exloiting the fact that ortfolios whose constituents follow an mgh distribution are univariate GH distributed, we rove some results relating to measurement and decomosition of ortfolio risk, and show how to efficiently tackle ortfolio otimization. Moreover, we develo a robust ortfolio otimization aroach in the mgh framework, using Worst Case Conditional Value at Risk (WCVaR) as risk measure. Keywords: Portfolio otimization; Robust otimization; sset allocation; Risk management; Multivariate Generalized Hyerbolic distribution; Conditional Value at Risk; Worst Case Conditional Value at Risk JEL Classification: C6; C52; C6; C63; G32. Introduction Modern ortfolio otimization is a far cry from the classical mean variance aroach ioneered by Markowitz (952). The dearture from the time-honored Markowitz framework has been surred by two intimately related insights. First, the use of a Gaussian distribution to describe the returns of financial assets will inevitably lead to what can at best be called a rough aroximation to reality. Second, variance can be an inadequate risk measure if a more flexible, non-gaussian return distribution is adoted. It has become a generally acceted fact, suorted by numerous emirical studies, that emirical asset return distributions are non-normal. In fact, they are almost always found to exhibit skewness (i.e. asymmetry) and excess kurtosis, which renders the normal (Gaussian) distribution an inadequate model (Prause 999, Raible 2, Schoutens 23, Cont and Tankov 24). Thus, realistic modelling calls for alternative robability distributions. In recent years, several viable alternatives to the Gaussian distribution, caable of caturing commonly observed emirical features, have been roosed *Corresonding author. s.kassberger@fs.de for use in financial modelling. For examle, Madan and Seneta (99) suggest the Variance Gamma distribution, Eberlein and Keller (995) and ingham and Kiesel (2) advocate the use of the Hyerbolic distribution, arndorff-nielsen (997) rooses the Normal Inverse Gaussian distribution, Eberlein (2) alies the Generalized Hyerbolic distribution, and as and Haff (26) find that the Generalized Hyerbolic Skew Student s t distribution matches emirical data very well. While these studies document the suerior caabilities of the Generalized Hyerbolic class and its subclasses when it comes to realistically describing univariate financial data, recent emirical studies conducted in a multivariate setting make a convincing case for the multivariate Generalized Hyerbolic (mgh) distribution and its subclasses as a model for multivariate financial data as well. For instance, McNeil et al. (25) calibrate the mgh model and its subclasses to both multivariate stock- and multivariate exchange-rate returns. In a likelihood-ratio test against the general mgh model, the Gaussian model is always rejected. as et al. (25) and Kassberger and Kiesel (26) emloy the multivariate NIG (Normal Inverse Gaussian) distribution successfully for risk management uroses. The latter study demonstrates that the NIG distribution rovides a much better Quantitative Finance ISSN rint/issn online ß 2 Taylor & Francis htt:// DOI:.8/

2 54 M. Hellmich and S. Kassberger fit to the emirical distribution of hedge fund returns than the normal distribution. The Gaussian distribution is found to seriously understate the robability of tail events, while the heavier tails of the mgh class seem to describe actual tail behavior well. Tail-related risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR) are shown to be severely misleading when calculated on the basis of the Gaussian distribution. This roblem is found to carry over into the ortfolio context. ll the aforementioned distributions have two imortant features in common. First, they can all be considered as marginal distributions of (multivariate) Lévy rocesses (i.e. rocesses with indeendent and stationary, but not necessarily Gaussian increments). Second, they all belong to the class of (multivariate) Generalized Hyerbolic distributions, which encomasses the Gaussian distribution as a limiting case. Therefore, the mgh class offers a natural generalization of the multivariate Gaussian class. The dearture from the normal distribution and the adotion of more realistic distributions, however, call for adequate risk measures and comutational tools. Portfolio otimization using non-gaussian distributions should not be erformed in a mean variance framework, because in the non-gaussian case it is inaroriate to describe the riskiness of a financial asset solely by the variance of its returns (thereby ignoring higher moments). In recent years, CVaR, also known as Exected Shortfall or Tail-VaR, has been embraced by academics and ractitioners alike as a tractable and theoretically well-founded alternative to classical risk measures such as VaR or variance. In addition to being based on realistic distributional assumtions and an informative risk measure, an alternative ortfolio otimization aroach should be comutationally feasible even for roblems involving a large number of assets in order to be alicable to real-world situations. Moreover, it should be amenable to a robust formulation of the ortfolio otimization roblem. Robust formulations are based on the insight that otimal ortfolios can be remarkably sensitive to only slight variations in the inut arameters, which are often fraught with estimation error. The combined effect of the uncertainties in the arameters can render the result of a ortfolio otimization rocedure highly unreliable. To counteract this henomenon, robust aroaches rely on uncertainty sets that contain the true arameters for a secific confidence level, instead of oint estimates of the arameters, thereby taking arameter uncertainty into account. For a survey of robust otimization, the interested reader is referred to ertsimas et al. (28). modern ortfolio otimization aroach is thus characterized by the following desirable features: allowance for realistic return distributions, use of a realistic risk measure, comutational tractability, and admissibility of a tractable robust formulation. We contribute to the literature by roosing a ortfolio otimization aroach that incororates all of these features. Our aroach is based on the mgh distribution, relies on CVaR, leads to a convex otimization roblem, and allows for a robust formulation that can be solved just as efficiently as the original roblem. The remainder of this aer is structured as follows. Section 2 introduces CVaR as an alternative risk measure and gives an overview of several standard forms of the ortfolio otimization roblem. In section 3, the multivariate Generalized Hyerbolic class of distributions is introduced. In addition, results relating to the deteration of the CVaR for mgh ortfolios are established, and a decomosition formula for the CVaR of a ortfolio is resented and roved. These results, while interesting in their own right for risk management uroses, form the foundation for an efficient formulation of the ortfolio otimization roblem in the mgh framework, which is the subject of section 4. Furthermore, section 4 introduces a robust formulation of the ortfolio otimization roblem, which relies on Worst Case Conditional Value at Risk (WCVaR) as a risk measure. It is shown that the robust ortfolio otimization roblem can be solved as efficiently as the original roblem. Section 5 is devoted to a numerical study in which the methodologies develoed in the aer are alied to emirical data. Section 6 sums u the main insights and concludes. 2. Risk measures, erformance measures, and ortfolio otimization 2.. Coherent measures of risk Since a non-gaussian distribution cannot be characterized solely in terms of its means and its covariance matrix, a deviation from the multivariate Gaussian aradigm of ortfolio otimization has to be suorted by the adotion of alternative risk measures, such as Value at Risk or CVaR. In their seal aer, rtzner et al. (999) secify a number of desirable roerties a risk measure should have and introduce the notion of a coherent risk measure (see also Malevergne and Sornette (26)). In the following definition, L and L 2 can be interreted as random losses. risk measure that mas a random loss to a real number is said to be coherent if it satisfies the following axioms.. () Translation invariance: (L þ l ) ¼ (L ) þ l, for all random losses L and all l 2 R.. (2) Subadditivity: (L þ L 2 ) (L ) þ (L 2 ), for all random losses L, L 2.. (3) Positive homogeneity: (L ) ¼ (L ), for all random losses L and all 4.. (4) Monotonicity: (L ) (L 2 ), for all random losses L, L 2 with L L 2 almost surely. It is worth noting that subadditivity and ositive homogeneity imly convexity, whereas the converse generally does not hold. Value at Risk (VaR) has become an industry standard for measuring financial risks. VaR has derived much of its oularity from the fact that it gives a handy and easy-to-understand reresentation of otential losses.

3 Efficient and robust ortfolio otimization 55 If X is the random return associated with an asset, then L ¼ X is the relative loss, and the VaR at level 2 (, ), denoted by VaR (L), is defined as VaR (L) X inf{l 2 R : P(L4l) } ¼ inf{l 2 R : F L (l ) }. Hence, VaR (L) is the smallest relative loss level whose robability of being exceeded is at most. For continuous, strictly increasing loss distribution functions (which we will assume throughout the aer), VaR can be more simly exressed as the -quantile of the loss distribution function F L : VaR ðlþ ¼FL ðþ: Of course, VaR can also be defined in terms of absolute losses. However, as we are going to model returns rather than rices, the above definition is more aroriate for our uroses. VaR suffers from several shortcogs that become articularly evident when it is to be used as a risk measure in the ortfolio context. s rtzner et al. (999) oint out, VaR can lack subadditivity when alied to non-ellitical distributions, which amounts to ignoring the benefits of ortfolio diversification. Moreover, VaR is generally a non-convex function of ortfolio weights. s nonconvexity normally leads to multile local extrema, it renders ortfolio otimization a comutationally exensive roblem. Recently, there has been increasing interest in CVaR as a closely related alternative to the VaR aroach. CVaR does not suffer from any of the above-mentioned shortcogs; in articular, it is a coherent risk measure (see, e.g., cerbi and Tasche 22) and thus has several desirable roerties that VaR lacks, such as subadditivity and convexity. The CVaR at level 2 (, ) is defined as the exectation of the relative loss conditional on the relative loss being at least VaR (L): CVaR ðlþ¼ 4 E½L j L VaR ðlþš: straightforward consequence of this definition is the relation CVaR (L) VaR (L). CVaR is more informative than VaR, as CVaR (L) takes the loss distribution beyond the oint VaR (L) into account and thus also measures the severity of losses that exceed VaR (L). VaR, in contrast, ignores losses beyond VaR (L) and thus discards information imlicit in the loss distribution. CVaR is well suited as a risk measure in the context of ortfolio otimization, for reasons that will be elaborated on in what follows Portfolio otimization using CVaR Portfolio otimization roblems aear in various guises. The following result, which is roved by Krokhmal et al. (22), establishes a link among three of the most common formulations. Let : X } R be a convex risk measure, and let R : X } R be a concave reward function, both defined on the convex set XR d. Let x 2X be a vector of ortfolio weights, i.e. assume that P d i¼ x i ¼. Then the following three otimization roblems lead to the same efficient frontiers when varying the arameters,, and!, resectively: x2x subject to, x2x subject to ðxþ RðxÞ, ðxþ, RðxÞ, ðpþ ðp2þ max RðxÞ, x2x ðp3þ subject to ðxþ!: In other words, a ortfolio that is efficient for one of these three roblem formulations will also be efficient for the other two. In our subsequent considerations, we will identify R(x) with the exected ortfolio return, which is a linear (and thus concave) function of ortfolio weights, and (x) with the ortfolio CVaR, which is convex in the ortfolio weights. While the above formulations involving the imization of a linear functional of risk and reward are very common in the literature, other formulations of the ortfolio otimization roblem entail the maximization of a reward risk ratio. For instance, the use of Return-on-Risk-Caital (RORC for short), defined as R(x)/(x), is motivated by Fischer and Roehrl (25). Rachev et al. (27) rovide an overview of various other reward risk ratios. 3. eyond Gaussian mean variance otimization: Using the mgh distribution for ortfolio modelling 3.. Modelling multivariate returns with the mgh distribution s already ointed out, there is comelling emirical evidence that returns of financial assets are not Gaussian. s a consequence, a more realistic model is called for. ecause of its great generality and relatively high numerical tractability, the mgh distribution is an ideal candidate The mgh distribution as a normal mean variance mixture. random variable W 2 R þ is said to have a Generalized Inverse Gaussian (GIG) distribution with arameters,, and, denoted by W GIG(,, ), if its density is given by f GIG ð y;,, Þ 8 ð Þ =2 < ffiffiffiffiffiffi y ex y þ y, y 4, ¼ 2K ð Þ 2 :, y, ðþ where, for x4, K (x) is the modified essel function of the third kind with index : K ðxþ ¼ Z y ex xð y þ y Þ dy: 2 2

4 56 M. Hellmich and S. Kassberger The arameters in () are assumed to satisfy 4 and if5; 4 and 4 if ¼ ; and and 4 if 4. The exected value of Y can be exressed as ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi = K þ ð Þ EðW Þ¼ ffiffiffiffiffiffi : ð2þ K ð Þ The class of mgh distributions can now be introduced as the class of normal mean variance mixtures with a GIG-distributed mixing variable. random variable X ¼ (X,..., X d ) is said to follow a d-dimensional mgh distribution with arameters,,,,, and, denoted by X GH d (,,,,, ), if X ¼ d þ W þ ffiffiffiffiffi W Z, where, 2 R d are deteristic, Z N k (, I k ) follows a k-dimensional normal distribution, W GIG(,, ) is a ositive, scalar random variable indeendent of Z, 2 R dk denotes a d k matrix, and ¼. We find that X W ¼ w N d ( þ w, w), i.e. that the conditional distribution of X given W is normal, which exlains the name normal mean variance mixture. The mixing variable W can be thought of as a stochastic volatility factor. From the above definition, it follows directly that E(X ) ¼ þ E(W ) and Cov(X ) ¼ E(W ) þ Var(W ). It is interesting to note that the absence of correlation of the comonents of X imlies indeendence if and only if W is almost surely constant, i.e. if X is multivariate normal. For ¼, the class of normal variance mixture distributions is obtained. These distributions fall into the class of ellitical distributions, which will be formally introduced later. For non-singular, it can be shown that the following reresentation for the density f GHd of a d-dimensional GH d (,,,,, ) distributed random variable holds: f GHd ð y;,,,,, Þ ( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) K d=2 ð ð þðy Þ ð y ÞÞð þ ÞÞ ¼ c exðð y Þ Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð þðy Þ ð y ÞÞð þ ÞÞ, ð3þ with c a normalizing constant, ð Þ =2 ð þ Þ d=2 c ¼ ð2þ d=2 jj =2 ffiffiffiffiffiffi, ð4þ K ð Þ where jj denotes the deterant. Observe that, for every a4, the distributions GH d (,,,,, ) and GH d (, /a, a,, a, a) coincide, since, for all y 2 R, f GHd ð y;,,,,, Þ ¼f GHd ð y;, =a, a,, a, aþ: ð5þ This non-uniqueness gives rise to an identifiability roblem when trying to calibrate the arameters. However, this roblem can be addressed in several ways, for examle by requiring the deterant of to assume a re-secified value, or by fixing the value of either or Subclasses of the mgh class. The mgh class of distributions is very general and accommodates many subclasses that have become oular in financial modelling. The urose of this section is to rovide a brief survey of some of these subclasses. Comare also McNeil et al. (25), who rovide a discussion of the tail behavior of these classes. Hyerbolic distributions. For ¼ 2 ðd þ Þ, one arrives at the d-dimensional Hyerbolic distribution. However, the univariate marginals of a d-dimensional Hyerbolic distribution with d 2 are not univariate Hyerbolic distributions. See Eberlein and Keller (995) for an alication of univariate Hyerbolic distributions to financial modelling. Normal Inverse Gaussian (NIG) distributions. For ¼ 2, one obtains the class of NIG distributions, which has become widely alied to financial data (see, e.g., as et al. (25) and Kassberger and Kiesel (26)). The tails of NIG distributions are slightly heavier than those of the Hyerbolic class. Variance Gamma (VG) distributions. For 4 and ¼, one obtains a limiting case which is known as the Variance Gamma class. See Madan and Seneta (99) for an alication of univariate VG distributions to equity return modelling. Skew Student s t distributions. For ¼ 2 and ¼, another limiting case is obtained, which is often called the Skew Student s t distribution. The interesting asect of this distribution is that, in contrast to the aforementioned ones, it is able to account for heavy-tailedness. See as and Haff (26) for an alication in a univariate setting. Ellitically symmetric mgh (symgh) distributions. For ¼, one obtains the subclass of ellitically symmetric mgh distributions, which are henceforth called symmetric mgh or symgh distributions. Comared with the general mgh distribution, the density of a symgh distribution simlifies considerably: f symghd ð y;,,,, Þ =2 d=2 ð Þ ¼ ð2þ d=2 jj =2 ffiffiffiffiffiffi K ð Þ K ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d=2ð ð þðy Þ ð y ÞÞ Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð6þ ð ð þðy Þ d=2 ð y ÞÞ Þ The symgh class belongs to the class of ellitical distributions Distributional roerties, CVaR, and ortfolio risk decomosition of mgh ortfolios In this section, we take advantage of the analytical tractability of the mgh class to state the distribution of a ortfolio whose constituents follow an mgh distribution. This result will be used to derive analytical exressions for the ortfolio s CVaR and for the risk contribution of a single asset to overall ortfolio risk.

5 Efficient and robust ortfolio otimization Distribution of ortfolio returns and CVaR. From the arametrization of the mgh class, it can easily be inferred that it is closed under linear transformations. More recisely, let X GH d (,,,,, ) and Y ¼ X þ b, where 2 R kd and b 2 R k. Then Y ¼ X þ b ¼ þ b þ W þ ffiffiffiffiffi W Z GH k ð,,, þ b,, Þ: Thus, linear transformations of mgh random variables leave the distribution of the GIG mixing variable unchanged. In articular, it follows that every comonent X i of X is governed by a univariate GH distribution: X i GH (,,, i, i, ii ). Furthermore, for x ¼ (x,..., x d ) 2 R d, x X ¼ Xd i¼ x i X i GH ð,,, x, x, x xþ: If, in addition, the x i are required to sum to unity (i.e. that x ¼, where ¼ (,...,) 2 R d ), and thus can be interreted as the weights of the individual assets in a ortfolio, we can conclude that if the returns of the constituents of a ortfolio follow an mgh distribution, then the return of the ortfolio is univariate GH distributed. Now, we derive the univariate GH density of ortfolio returns, which will turn out to be considerably simler than its multivariate counterart. Using (5), we can reresent a univariate GH density of the form f GH (y;,,,,, ) as f GH (y;,, /,, /, ). This shows that, in the univariate case, without loss of generality, the disersion arameter 2 R þ can be assumed to be, and the density thus simlifies (comare (3) and (4)) f GH ð y;,,,, Þ ð Þ =2 ð þ 2 Þ =2 ¼ ð2þ =2 ffiffiffiffiffiffi K ð Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K =2 ð þðy Þ 2 Þð þ 2 Þ exðð y ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi =2 : ð þðy Þ 2 Þð þ 2 Þ Now assume that the returns of d assets X ¼ (X,..., X d ) are distributed according to GH d (,,,,, ). Then the return x X of a ortfolio with asset weights x with x ¼ follows a GH (,,, x, x, x x) distribution. Denoting by f GH (y) the density of the ortfolio return x X evaluated at y, CVaR can be comuted as follows: CVaR ð x X Þ¼E ½ x X j x X VaR ð x X ÞŠ ¼ E ½x X j x X VaR ðx X ÞŠ ¼ Z GH ð Þ y f GH ð yþ dy: ð7þ The quantile-function GH ðþ of the ortfolio return distribution can be calculated using standard numerical root-finding methods. Having the ortfolio distribution available in closed form is of great advantage, as it allows fast, exact, analytical comutation of risk figures or of moments of ortfolio returns without having to fall back on tyically time-consug Monte Carlo simulations. This key feature makes it ossible to set u efficient ortfolio otimization algorithms, as will be demonstrated later Decomosition of ortfolio risk. When investigating the risk rofile of a ortfolio, not only are its aggregated risk characteristics of interest, but so too are the contributions of the individual constituents to its overall risk. The CVaR framework rovides a very intuitive decomosition of overall risk into its individual building blocks. Such a decomosition was resented by Panjer (2) for multivariate normal distributions, and generalized to ellitical distributions by Landsman and Valdez (23). Here, we rove a decomosition formula for the mgh distribution. Let the ortfolio returns be X GH d (,,,,, ), and let x 2 R d denote the asset weights (with x ¼ ). y additivity of conditional exectation, CVaR ð x XÞ¼E½ x X j x X VaR ð x XÞŠ ¼ Xd i¼ E½ x i X i j x X VaR ð x XÞŠ, which can be interreted as follows. The ortfolio CVaR is the sum of the risk contributions of the individual assets in case a shortfall event occurs, i.e. in case the relative ortfolio loss exceeds VaR ( x X ). It is imortant to note, however, that in general the ortfolio CVaR is different from the sum of the CVaR values for the individual assets. The following roosition shows how to comute the individual CVaR contribution of a osition in a secific asset. Proosition 3.: Let X GH d (,,,,, ), and let x 2 R d with x ¼. Then, the CVaR contribution of the osition in asset i is E½ x i X i j x X VaR ð x XÞŠ ¼ Z Z GH ð Þ y f GH2 ð y, y 2 Þ dy 2 dy, where GH is the quantile function of a GH (,,, x, x, x x) distribution, and f GH2 is the density function of a x i i x 2 i GH 2,,, x, ii x i Pj x! j ij x i Pj x j ij x, x x i! i x distribution. Proof: See aendix. œ

6 58 M. Hellmich and S. Kassberger 4. CVaR-based ortfolio otimization in the mgh framework 4.. The general case First, we study a ortfolio otimization roblem of the class (P2), which is reresentative of the class of roblems (P) through (P3). We choose (P2) because of its similarity to the classical Markowitz roblem, which involves imizing risk (as measured by ortfolio variance) under a imum constraint for the exected return. Consider the ortfolio otimization roblem (P2 ): x CVaR ð x XÞ, subject to x 2X¼fx 2 R d þ : x, x ¼ g, ðp2 Þ where X GH d (,,,,, ), ¼ (,..., d ) 2 R d, and i ¼ E(X i ) is the exected return of asset i. Hence, the objective is to imize CVaR under the condition that the exected ortfolio return is at least. Since both the objective function and the constraints in (P2 ) are convex, this roblem falls into the category of convex otimization roblems, which makes the sohisticated machinery of convex otimization available. In articular, convex otimization roblems do not suffer from the existence of local ima which are not at the same time global ima. However, directly evaluating the objective function via formula (7) might be undesirable from a numerical oint of view, since it entails use of a numerical root-finding rocedure. This roblem can be circumvented by alying the insights of Rockafellar and Uryasev (2, 22), who introduce an auxiliary function F ðx, Þ¼ 4 þ Z ½ x y Š þ ð yþ dy, y2r d where is a real number, x y denotes the loss associated with the vector of ortfolio weights x 2 R d and the return vector y 2 R d, and : R d } R þ is the d-dimensional robability density function of the asset returns. Rockafellar and Uryasev demonstrate that F (x, ) is convex with resect to (x, ), and that, given x, CVaR can be calculated by imizing F (x, ) with resect to : CVaR ð x X Þ¼ F ðx, Þ: ð8þ 2R If we ostulate that the asset returns follow a d-dimensional mgh distribution with density function f GHd (y;,,,,, ), then F (x, ) can be considerably simlified; in articular, d-dimensional integration can be avoided. Denoting by f GH (z;,,,,, ) the (univariate) density of the ortfolio return x X, we obtain F ðx, Þ ¼ þ Z ð x y Þ y2r d : x y f GHd ð y;,,,,, Þdy ¼ þ Z ð z Þ f GH ðz;,,, x, x, x xþ dz: ð9þ Hence, the original roblem can be recast as the convex rogram ðx,þ F ðx, Þ, subject to x 2X¼fx 2 R d þ : x, x ¼ g, 2 R, ðp2 Þ which, in contrast to the original roblem (P2 ), does not require numerical root-finding. For roblems involving the maximization of RORC, i.e. roblems of the form x max x CVaR ð x XÞ, subject to x 2X¼fx2 R d þ : x ¼ g, ðp4þ the objective function will in general be non-convex. For this tye of roblem, we can take advantage of the fact that a solution to (P4) will lie on the efficient frontier induced by the resective roblem (P2 ); evidently, ortfolios that are inefficient in the sense of (P2 ) cannot be solutions to (P4). Thus, in order to find a solution to (P4), one can calculate the efficient frontier of (P2 ) or, more recisely, efficient ortfolios for several values of, to aroximate the efficient frontier and then simly evaluate the objective function of (P4) for these ortfolios. Thus, we are able to reduce the non-convex otimization roblem (P4) to a fixed number (deending on the accuracy required) of convex otimization roblems that are efficiently solvable. This is highly advantageous, since for non-convex ortfolio otimization roblems, one tyically has to fall back on heuristic otimization rocedures. Fischer and Roehrl (25), for instance, advocate using swarm-intelligence methods for RORC otimization, which are comutationally exensive and cannot guarantee that a global otimum is attained The ellitical case In this section, we recall the notion of ellitical (also called ellitically symmetric or ellitically contoured) distributions, and demonstrate how to exloit their structural roerties in the context of ortfolio otimization. Ellitical distributions have been alied to financial modelling since the seal aer by Owen and Rabinovitch (983), and they remain oular to this day (see, e.g., ingham and Kiesel 22, Landsman and Valdez 23, Hamada and Valdez 28). We confine our considerations to absolutely continuous multivariate distributions with mean vector and ositive definite disersion matrix, as these are the ones that are relevant from a ractical standoint. distribution of this tye is said to be ellitical if its density f : R d } R has the form f ðxþ ¼ gððx Þ ðx ÞÞ jj =2,

7 Efficient and robust ortfolio otimization 59 where g : R } R is a scalar function termed the density generator. Thus, the density of an ellitical distribution is a function of the quadratic form (x ) (x ), and its level sets are ellitically symmetric in R d, which exlains the name. Insecting formula (6), one recognizes the symmetric mgh distribution to be a member of the ellitical family. More generally, normal variance mixtures (and thus, in articular, the multivariate normal distribution) can be shown to be ellitical. Ellitical distributions have several nice roerties, which facilitate their alication to ractical roblems. For examle, linear combinations of the comonents of ellitical random vectors remain ellitical and have the same characteristic generator. In articular, the univariate marginal distributions of ellitical distributions are also ellitical and inherit the generator of the arent distribution. The essential roerty for ortfolio-otimization uroses, however, was formulated by Embrechts et al. (22): Suose that X follows a d-dimensional ellitical distribution. Let i ¼ E(X i ), and let be a ositive homogeneous, translation-invariant risk measure. Define the subset of ortfolios having exected return as Then X¼ 4 fx 2 R d þ : x ¼, x ¼ g: arg x2x ð x X Þ¼arg x2x Varðx X Þ: Thus, for an ellitical ortfolio distribution, instead of solving a ortfolio otimization roblem with a ositive homogeneous, translation-invariant risk measure as objective function under the condition that a given exected return is attained, one can solve the corresonding roblem with the variance as objective function. Hence, the otimization roblem reduces to a simle Markowitz-tye otimization. Furthermore, it is imortant to note that the otimal allocation will be indeendent of the risk measure used. Of course, the values of the objective functions will differ in general, but the set of solutions will not. The above insight alies to risk measures such as VaR and CVaR, since both are ositive homogeneous and translation invariant. Now let X be an ellitical mgh distributed random variable, i.e. X GH d (,,,,,), and let x 2 R d (with x ¼ ) be the ortfolio comosition. Then x X GH (,,, x, x, x), and thus x X is also ellitical. Moreover, E(x X) ¼ x and Var(x X ) ¼ E(W)x x, where W GIG(,, ). Therefore, a ortfolio otimization roblem of the above tye can be recast as the quadratic rogram x x x, subject to x 2X¼fx2R d þ : x ¼, x ¼ g: ðqpþ The advantage this formulation offers over the general (non-ellitical) mgh case is twofold. First, the simlicity and additional structure of the objective function can be exloited to solve quadratic rograms more efficiently than general convex rograms using dedicated quadratic otimization algorithms. Second, once an otimal solution x of the above roblem has been found, it has a universal character: Not only does it solve the Markowitz-tye variance imization roblem above, but it also imizes ortfolio VaR and ortfolio CVaR for all levels. Thus, if the aim is to imize CVaR for different levels, one needs to solve the otimization roblem only once. This is in contrast to the situation in the general (non-ellitical) case, where otimization with resect to different risk measures or different levels tyically leads to different otimal allocations. Once the quadratic rogram has been solved, one may wish to calculate the CVaR corresonding to the otimal solution x ot, which can be done using formula (7) Robust ortfolio otimization using Worst Case CVaR The alicability of a ortfolio otimization aroach to real-world roblems not only hinges on its numerical tractability but also crucially deends on its robustness: Small changes in inut data should have only a or imact on otimization results. This insight has surred interest in robust ortfolio otimization aroaches (see, e.g., El Ghaoui et al. 23, Goldfarb and Iyengar 23, Halldo rsson and Tu tu ncu 23, or Zhu and Fukushima 29). The central idea of robust ortfolio otimization is to use uncertainty sets for the unknown arameters instead of only oint estimates, and to comute ortfolios whose worst-case erformance (meaning the erformance under the least favorable arameters in the uncertainty set) is otimal. In this section, we develo a robust otimization aroach in the mgh framework, using Worst Case Conditional Value at Risk (WCVaR) as a risk measure. The resulting robust otimization roblem will be demonstrated to be as tractable as its classical (non-robust) CVaR-based counterart exaed above. Let P be a class of multivariate asset return distributions, let X P be a random vector of asset returns with distribution P 2P, and let x 2X be a vector of ortfolio weights. The WCVaR of a ortfolio with weights x at level 2 (, ) is defined as WCVaR P ðxþ¼4 su CVaR ð x X P Þ: P2P s demonstrated by Zhu and Fukushima (29), WCVaR inherits subadditivity, ositive homogeneity, monotonicity, and translation invariance from CVaR and therefore, just as CVaR, is a coherent risk measure. Moreover, the authors show that WCVaR is convex in x. The robust counterart of the classical ortfolio otimization roblem involves imizing WCVaR on a non-emty, comact, convex set of ortfolio weights XR d þ : x2x WCVaRP ðxþ: ðrþ The solution x ot of (R) will then be the allocation with otimal worst-case roerties.

8 5 M. Hellmich and S. Kassberger Robust otimization within the mgh class. Our objective in this subsection is to state a robust otimization roblem in the mgh class, making use of its secific characteristics. First, a arametric family P of distributions needs to be secified. The arameter sace can be chosen in several ways; see ertsimas et al. (28) for an overview of different aroaches. We consider searable uncertainty sets, which have been used extensively in the literature on robust ortfolio otimization (e.g. by Halldo rsson and Tu tu ncu 23, Tu tu ncu and Koenig 24, and Kim and oyd 27). ssume that P X {GH d (,,,,, ):(,, ) 2M} is a family of mgh distributions with,, fixed. (,, ) is assumed to be an element of a searable olyhedral uncertainty set M X I I I with I ¼ 4 f 2 R d : L U g, I ¼ 4 f 2 R d : L U g, I ¼ 4 f 2 R dd : L U, os. definiteg comact intervals. ll inequalities in the set definitions are to be understood comonent-wise. lthough the arameters governing the mixing distribution are ket fixed, our aroach is flexible enough to incororate uncertainty not only in the means and covariances of the return distribution but also in skewness and kurtosis, since, which influences the latter two moments, is incororated into the uncertainty set. Since M is comact, the suremum is attained: WCVaR P ðxþ ¼su CVaR ð x X P Þ P2P ¼ max CVaR ð x X P Þ: ðþ P2P For the auxiliary function F (x, ) (comare (9)), we introduce the more exlicit notation F ðx, ;,,,,, Þ ¼ 4 Z ðz þ Þ f GH ðz;,,, x, x, x xþ dz: For all P ¼ GH d (,,,,, ) 2P and x 2X, we find by equation (8) that CVaR ð x X P Þ¼ F ðx, ;,,,,, Þ: 2R Combining this with () leads to WCVaR P ðxþ ¼ max F ðx, ;,,,,, Þ: ð,,þ2m 2R Consequently, the robust otimization roblem (R) reads as follows when stated in exlicit form: x2x max F ðx, ;,,,,, Þ: ð,,þ2m 2R ðr Þ Towards an efficient formulation of the robust roblem. In the following, we demonstrate how (R ) can be substantially simlified by exloiting the secific structure of the mgh class. To this end, we collect some essential roerties of F in the following lemma. Lemma 4.: Let XR d þ be a convex set. (a) F (x, ;,,,,, ) is comonent-wise monotonically decreasing in and and comonent-wise monotonically increasing in. In articular, for any (x, ) 2XR, max F ðx, ;,,,,, Þ ð,,þ2m ¼ F ðx, ;,,, L, L, U Þ: (b) F (x, ;,,,,, ) is convex in (x, ) on XR. Proof: See aendix. œ Lemma 4. is key to the roof of roosition 4.2. Proosition 4.2: The following relations hold: (a) WCVaR P ðxþ ¼ ðx, ;,,, L, L, U Þ, 2R with F (x, ;,,, L, L, U ) convex in on R; (b) x2x WCVaRP ðxþ ¼ ðx, ;,,, L, L, U Þ, ðx,þ2xr ðr Þ with F (x, ;,,, L, L, U ) convex in (x, ) on XR. Proof: See aendix C. œ The above roosition demonstrates that the WCVaR for any ortfolio x 2X is attained for the arameters L, L, and U. Furthermore, it shows that the original robust otimization roblem can be massively simlified and cast as a convex rogram (R ), which is as efficiently solvable as the corresonding classical otimization roblem Introducing a imum return constraint. This subsection exaes robust ortfolio otimization using WCVaR under a imum constraint for the worst-case exected ortfolio return. Let P, X P, and X be defined as above. The worst-case exected return for a ortfolio x 2X is X d P2P Eðx X P Þ¼ x i ð i þ i EðW ÞÞ ð,þ2i I i¼ ¼ x L þ x L EðW Þ, with W GIG(,, ) and E(W ) as in formula (2). arently, the worst-case exected ortfolio return is assumed for the arameter vectors L and L, which also aear in the arameter set for which the WCVaR is attained. If we use the notation WC X L þ L, E(W ) for the worst-case exected return vector, then the robust otimization roblem under a imum return constraint for the worst-case exected return can be stated as the convex rogram ðx,þ F ðx,;,,, L, L, U Þ, subject to x 2X¼fx2R d þ : x ¼, WC x g, 2 R: ðr2þ

9 Efficient and robust ortfolio otimization 5 5. Numerical results In this section, we resent a numerical examle, based on emirical data, in which the theory develoed above is alied. We consider four indices: two stock rice indices, namely the Dow Jones Eurostoxx 5 and the S&P 5, a bond index, namely the ioxx Euro, and a commodity index, namely the S&P GSCI. We calibrate an mgh model to 2 weekly returns of these indices observed between Setember 24 and Setember 28. Using the EM algorithm (see, e.g., McNeil et al. 25) for calibration, we obtained the following arameter estimates for the joint return distribution, where the order of the elements in the following vectors and matrices corresonds to the order in the above enumeration: ¼ :725, ¼ 2:74, ¼ 6:766, 2:465 :4 6:67 ¼ :265 C, ¼ 5:33 :79 C 8:75 6:563 3:822 2:72 :22 :77 2:72 3:96 :23 :9 ¼ 4 :22 :23 :86 :5 C :77 :9 :5 :6 Let X follow an mgh distribution with the above arameters. Then :45 :34 EðX Þ¼ :474 C 2:87 4:72 2:829 :84 :622 2:829 3:5 :95 :84 CovðX Þ¼ :84 :95 :87 :6 C :622 :84 :6 :5 :777 :234 :86 :777 :275 :44 CorrðX :234 :275 :4 C :86 :44 :4 :335 :738 :77 SkewðX :3 C :653 :632 C :22 :634 arently, the GSCI commodity index features the highest exected return (.287 ercentage oints er week), while the ioxx bond index has the lowest exected return. The volatility of the ioxx is by far the lowest, the stock rice indices exhibit a similar level of volatility, and the GSCI can be seen to be substantially more volatile than the other indices. The stock rice indices are strongly ositively correlated, but slightly negatively correlated to both the bond and the commodity indices, whereas the latter two are almost uncorrelated. ll indices exhibit only moderate negative skewness and excess kurtosis when observed on a weekly basis. However, as Konikov and Madan (22) oint out, the skewness of the marginal distribution of a Le vy rocess decreases as the recirocal of the square root of time, whereas its excess kurtosis decreases as the recirocal of time. earing in d that the mgh distribution is infinitely divisible and therefore can be regarded as the marginal distribution of a multivariate Le vy rocess, we can conclude that for daily returns, both negative skewness and (esecially) excess kurtosis would be considerably ffiffiffi more ronounced (scale the weekly figures by 5 and 5, resectively). ased on these arameters and ¼.95, we erform a mean-cvar otimization under imum return constraints, i.e. we solve (P2 ). Figure resents the mean-cvar efficient frontier (to left grah), the comositions of the efficient ortfolios (to right grah), and the CVaR contributions of the individual assets in the efficient ortfolios (bottom grah). The weekly CVaR ranges from.79 ercentage oints for the imum-cvar ortfolio (which at the same time has the imum exected return among all efficient ortfolios) to 7. ercentage oints for the ortfolio with maximum exected return. The imum-cvar ortfolio is made u mainly of a osition in the ioxx, while encomassing only small ositions in the S&P 5 and the GSCI. The maximum-return ortfolio consists solely of a osition in the GSCI, the index with maximum exected return. The grah at the bottom of figure dislays the CVaR contributions of the individual assets given the ortfolio comositions shown in the to right grah. The uer boundary of the colored area corresonds exactly to the efficient frontier shown in the to left grah, the only difference being that, in the to left grah, exected return corresonds to the vertical axis and CVaR to the horizontal axis. n interesting henomenon can be witnessed when relating ortfolio comositions to CVaR contributions: lthough the weights of the individual assets change linearly when moving towards higher returns, their risk contributions do not. This effect becomes articularly evident for ortfolios that consist only of the Eurostoxx 5 and the GSCI: linear decrease in the weight of the Eurostoxx 5 induces a suerlinear decrease in its risk contribution. This observation can be exlained by the relatively higher diversification benefits and thus the relatively lower risk contributions associated with smaller ositions. Now we tackle the corresonding robust ortfolio otimization roblem (R ). First of all, the uncertainty set needs to be secified. In the sirit of the aroach taken by Kim and oyd (27), we assume that the uncertainty sets arise from the base-case arameters

10 52 M. Hellmich and S. Kassberger Exected return (weekly, in ercentage oints) Mean-CVaR efficient frontier CVaR (weekly, in ercentage oints) Cumulative ortfolio weights Comosition of efficient ortfolio s DJ Eurostoxx 5 S&P 5 ioxx Euro S&P GSCI Exected return (weekly, in ercentage oints) Portfolio CVaR (weekly, in ercentage oints) CVaR contributions of individual assets DJ Eurostoxx 5 S&P 5 ioxx Euro S&P GSCI Exected return (weekly, in ercentage oints) Figure. Efficient frontier, ortfolio comosition, and CVaR contributions in the base case. resented above, with a shift of the latter either u or down by %: I ¼f 2 R 4 : L U g¼f 2 R 4 : j i i j: i g, I ¼f 2 R 4 : L U g¼f 2 R 4 : j i i j: i g, I ¼f 2 R 44 : L U, os. definiteg ¼f ¼ð ij Þ2R 44 : j ij ij j: ij, os. definiteg: y roosition 4.2, the worst-case scenario is fully detered by the interval bounds L, L, and U, which can be calculated to be L ¼ :9, L ¼ :, 4:24 2:983 :82 :694 2:983 3:45 :83 :89 U ¼ :82 :83 :24 :4 C :694 :89 :4 2:7 where the ositive definiteness of U is easily verified. Thus, the worst-case returns will have lower means, more ronounced negative skewness (recall that all comonents of are negative in our examle, and thus L 5), and higher variances and covariances, leading to lower exected returns and higher risk of efficient ortfolios. The deteriorated risk return rofile becomes evident when comaring the robust efficient frontier (i.e. the worst-case otimal ortfolios) dislayed in the to left grah of figure 2 with the efficient frontier in the base case. Comaring the comositions of base-case and worstcase efficient ortfolios (to right grahs of figures and 2, resectively), one recognizes that the weight of the ioxx osition (the least risky investment) has increased throughout the full sectrum of exected returns, while the weight of the Eurostoxx 5 is essentially zero throughout. Looking more closely at the CVaR contributions, we note that even for ortfolios where the noal weight of the GSCI is significantly smaller than that of the ioxx, the GSCI s risk contribution can be the higher of the two, on account of its far higher volatility. s in the baseline scenario, we witness linearly decreasing ortfolio weights giving rise to suerlinearly decreasing CVaR contributions.

11 Efficient and robust ortfolio otimization 53 Exected return (weekly, in ercentage oints) Mean-CVaR efficient frontier CVaR (weekly, in ercentage oints) Cumulative ortfolio weights Comosition of efficient ortfolios DJ Eurostoxx 5 S&P 5 ioxx Euro S&P GSCI Exected return (weekly, in ercentage oints) Portfolio CVaR (weekly, in ercentage oints) CVaR contributions of indivdual assets DJ Eurostoxx 5 S&P 5 ioxx Euro S&P GSCI Exected return (weekly, in ercentage oints) Figure 2. Efficient frontier, ortfolio comosition, and CVaR contributions for robust ortfolios. Exected return (weekly, in ercentage oints) Mean-CVaR efficient frontiers Classical efficient ortfolios in base case Classical efficient ortfolios in worst case Robust efficient ortfolios in worst case Robust efficient ortfolios in base case CVaR (weekly, in ercentage oints) Figure 3. Classical and robust efficient frontiers in the base case and worst case. Finally, we comare the erformance of both classical and robust ortfolios in the baseline and worst-case scenarios (see figure 3). The blue and red curves reresent the efficient frontiers of figures and 2, resectively. The black curve reflects the erformance of robust efficient ortfolios under the arameters of the base case, while the green curve reresents the erformance of classical efficient ortfolios in the worst case. s evidenced by the green curve, classical efficient ortfolios erform quite badly should the worst case obtain, and some even lead to negative exected returns. Moreover, because of the lack of monotonicity in the green curve, some allocations are severely inefficient, leading to lower exected returns while at the same time being riskier than other feasible ortfolios. In contrast, robust ortfolios (red curve) erform substantially better than the classical ones in the worst case while being only slightly worse in the base case. Furthermore, excet for very low-risk ortfolios, robust ortfolios can be seen to exhibit a monotonic relation of risk and return in the base case. It is worth noting that, since the classical and robust imum- CVaR ortfolios are not identical, both the blue and black curves and the red and green curves start from slightly different (though visually almost indistinguishable) oints, resectively. Overall, one notes that robust ortfolios erform reasonably well in both scenarios, while classical ortfolios exhibit ronounced sensitivity to the scenario that actually obtains and thus might lead to severely inefficient allocations.

12 54 M. Hellmich and S. Kassberger 6. Conclusion In this aer, we develo a tractable yet flexible aroach to ortfolio risk management and ortfolio otimization based on the mgh distribution. s the normal distribution is a limiting case of the mgh class, the aroach resented in this aer can be considered a natural generalization of the Markowitz aroach. Exloiting the fact that ortfolios whose constituents follow an mgh distribution are univariate GH distributed, we rovide analytical formulas for ortfolio CVaR and the contributions of individual assets thereto. Then we demonstrate how to efficiently comute otimal ortfolios in the mgh framework. Using WCVaR as a risk measure, we formulate a robust otimization aroach within the mgh framework, which is shown to lead to otimization roblems that can be solved as efficiently as their classical counterarts. 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