Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

TOP DOI 10.1007/s11750-013-0303-y ORIGINAL PAPER Robust ortfolio choice with CVaR and VaR under distribution and mean return ambiguity A. Burak Paç Mustafa Ç. Pınar Received: 8 July 013 / Acceted: 16 October 013 Ó Sociedad de Estadística e Investigación Oerativa 013 Abstract We consider the roblem of otimal ortfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short ositions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust ortfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the otimal ortfolio rules when the uncertainty in the return vector is modeled via an ellisoidal uncertainty set. In the resence of a riskless asset, the robust CVaR and VaR measures, couled with a minimum mean return constraint, yield simle, mean-variance efficient otimal ortfolio rules. In a market without the riskless asset, we obtain a closed-form ortfolio rule that generalizes earlier results, without a minimum mean return restriction. Keywords Robust ortfolio choice Ellisoidal uncertainty Conditional Value-at-Risk Value-at-Risk Distributional robustness Mathematics Subject Classification (010) 91G10 91B30 90C90 1 Introduction The Value-at-Risk (VaR) is widely used in the financial industry as a downside risk measure. Since VaR does not take into account the magnitude of otential losses, the Conditional Value-at-Risk (CVaR), defined as the mean losses in excess of VaR, was roosed as a remedy and results usually in convex (linear) ortfolio A. B. Paç M. Ç. Pınar (&) Dearment of Industrial Engineering, Bilkent University, 06800 Ankara, Turkey e-mail: mustafa@bilkent.edu.tr

A. B. Paç, M. Ç. Pınar otimization roblems (Rockafellar and Uryasev 000, 00). The urose of the resent work is to give an exlicit solution to the otimal ortfolio choice roblem by minimizing the CVaR and VaR measures under distribution and mean return ambiguity when short ositions are allowed. Distribution ambiguity is understood in the sense that no knowledge of the return distribution for risky assets is assumed while the mean and variance/covariance are assumed to be known. The otimal ortfolio choice roblem using the aforementioned risk measures under distribution ambiguity and allowing short ositions was studied by Chen et al. (011) in a recent aer in the case of n risky assets, extending the work of Zhu and Fukushima (009) where the authors treat robust ortfolio choice under distribution ambiguity. Chen et al. assumed that the mean return vector l and variance covariance matrix C of risky assets are known, and comute ortfolios that are robust in the sense that they minimize the worst-case CVaR risk measure over all distributions with fixed first and second moment information. They obtained closed-form robust otimal ortfolio rules. The reader is referred to El Ghaoui et al. (003), Natarajan et al. (010) and Poescu (005) for other related studies on ortfolio otimization with distributional robustness, and to Tong et al. (010) for a comutational study of scenario-based CVaR in ortfolio otimization. A recent reference work on ortfolio otimization (using the mean-variance aroach as well as semi-variance and utility functions) in both single and multi-eriod frameworks is by Steinbach (001). In articular, Natarajan et al. (010) study exected utility models in ortfolio otimization under distribution ambiguity using a iecewise-linear concave utility function. They obtain bounds on the worst-case exected utility, and comute otimal ortfolios by solving conic rograms. They also relate their bounds to convex risk measures by defining a worst-case Otimized-Certainty- Equivalent (OCE) risk measure. It is well known that one of the two risk measures used in the resent aer, namely CVaR, can be obtained using the OCE aroach for a class of utility functions; see Ben-Tal and Teboulle (007). Thus the results of the resent aer comlement the revious work of Natarajan et al. (010) by roviding closed-form otimal ortfolio rules for worst-case CVaR (and worst-case VaR) under both distribution and mean return ambiguity. In the resent aer, we first extend, in Sect., the results of Chen et al. to the case where a riskless asset is also included in the asset universe, a case which is usually an integral art of otimal ortfolio choice theory. The inclusion of the riskless asset in the asset universe leads to extreme ositions in the ortfolio, which imlies that the robust CVaR and VaR measures as given in the resent aer have to be utilized with a minimum mean return constraint in the resence of a riskless asset to yield closedform otimal ortfolio rules. The distribution robust ortfolios of Chen et al. (011) are criticized in Delage and Ye (010) for their sensitivity to uncertainties or estimation errors in the mean return data, a case that we refer to as Mean Return Ambiguity; see also Best and Grauer (1991a, b) and Black and Litterman (199) for studies regarding sensitivity of otimal ortfolios to estimation errors. To (artially) address this issue, we analyze in Sect. 3, the roblem when the mean return is subject to ellisoidal uncertainty (Ben-Tal and Nemirovski 1999, 1998; Garlai et al. 007; Goldfarb and Iyengar 003; Ling and Xu 01) in addition to distribution ambiguity, and derive a closed-form ortfolio rule. The ellisoidal

Robust ortfolio choice with CVaR and VaR uncertainty is regulated by a arameter that can be interreted as a measure of confidence in the mean return estimate. In the resence of the riskless asset, a robust otimal ortfolio rule under distribution and mean return ambiguity is obtained if the quantile arameter of CVaR or VaR measures is above a threshold deending on the otimal Share ratio of the market and the confidence regulating arameter, or no such otimal rule exists (the roblem is infeasible). The key to obtain otimal ortfolio rules in the resence of a riskless under distribution and mean return ambiguity asset is again to include a minimum mean return constraint to trace the efficient robust CVaR (or robust VaR) frontier; Steinbach (001). The incremental imact of adding robustness against mean return ambiguity in addition to distribution ambiguity is to alter the otimal Share ratio of the market viewed by the investor. The investor views a smaller otimal Share ratio decremented by the arameter reflecting the confidence of the investor in the mean return estimate. In the case the riskless asset is not included in the ortfolio roblem, in Sect. 4, we derive in closed form the otimal ortfolio choice robust against distribution and ellisoidal mean return ambiguity without using a minimum mean return constraint, which generalizes a result of Chen et al. (011) stated in the case of distribution ambiguity only, i.e., full confidence in the mean return estimate. Minimizing robust CVaR and VaR in the resence of a riskless asset under distribution ambiguity We work in a financial market with a riskless asset with return rate R in addition to n risky assets. We investigate robust solutions minimizing CVaR and VaR measures. The unit initial wealth is allocated into the riskless and risky assets so as to allow short ositions, thus we can define the loss function as a function of the vector x R n of allocations to n risky assets and the random vector n of return rates for these assets: f ðx; nþ ¼ ðx T n þ Rð1 x T eþþ; ð1þ where e denotes an n-vector of all ones. For the calculation of CVaR and VaR, we use the method in Rockafellar and Uryasev (000), minimizing over c the auxiliary function: F h ðx; cþ ¼c þ 1 E½ðf ðx; nþ cþ þš; ðþ i.e., the h-cvar is calculated as follows: CVaR h ðxþ ¼min F hðx; cþ; ð3þ cr where h is the threshold robability level or the quantile arameter, which is generally taken to be in the interval [0.95,1). The convex set consisting of c values that minimize F h contains the h-var, VaR h ðxþ; which is the minimum value in the set.

A. B. Paç, M. Ç. Pınar The worst-case CVaR, when n may assume a distribution from the set D ¼fjE ½nŠ ¼l; Cov ½nŠ ¼C 0g [i.e., the set of all distributions with (known and/or trusted) mean l and covariance C], is defined as follows: RCVaR h ðxþ ¼max CVaR hðxþ ð4þ D ¼ max min F hðx; cþ: ð5þ D cr For convenience, in the rest of the aer, we relace occurrences of the su oerator with max since in all cases we consider the su is attained. We assume l is not a multile of e, as usual. Let us define the excess mean return ~l ¼ l Re; and the highest attainable Share ratio in the market ¼ ~l T C 1 ~l; see Best (010). The following Theorem 1 gives an exlicit solution of the ortfolio choice roblem of minimizing worst-case CVaR under a minimum mean return constraint min RCVaR h ðxþ; ð6þ xr n :ðl ReÞ T x d R where d R þ is a minimum target mean return arameter. We assume the minimum mean target return is larger than the riskless return, i.e., d [ R. Theorem 1 For h þ1 ; the roblem Eq. (6) admits the otimal ortfolio rule x ¼ d R C 1 ~l: For h\ þ1 ; the roblem Eq. (6) is unbounded. Proof Since the set of distributions D is convex and the function c þ 1 1 h E½ðf ðx; nþ cþ þš is convex in c for every n; we can interchange suremum and minimum; see Theorem.4 of Shairo (011). More recisely, one considers first the roblem min cr max D F h ðx; cþ and one finds a unique otimal solution as we shall do below. Then, using convexity of D and the convexity of the function c þ 1 1 h E½ðf ðx; nþ cþ þš in c one invokes Theorem.4 of Shairo (011) that allows interchange of min and max. We carry out this sequence of oerations below: min max F hðx; cþ ¼min c þ 1 cr D cr max D E½ð xt n R þ Rx T e cþ þ Š ¼ min cr c þ 1 max n ðl;cþ E½ð R c xt ðn ReÞÞ þ Š ¼ min cr c þ 1 max g ðm;r Þ E½ð R c gþ þ Š ð7þ ð8þ 1 While our roof is similar to the roof in Chen et al. (011) in essence, their roof is faulty because their argument for exchanging max and min relies on a result of Zhu and Fukushima (009) which is valid for discrete distributions. Our setting here, like that of Chen et al. (011) is not confined to discrete distributions. ence, a different justification is needed for exchanging max and min.

Robust ortfolio choice with CVaR and VaR ¼ min c þ 1 R c m þ cr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r þð R c mþ : ð9þ Maximum oerators in Eqs. (7) and (8) are equivalent, since by a slight modification of Lemma.4 in Chen et al. (011) we can state the equivalence of following sets of univariate random distributions: D 1 ¼fx T ðn ReÞ : n ðl; CÞg D ¼fg : g ðm; r Þg; where m ¼ x T ðl ReÞ; and r ¼ x T Cx: Equality Eq. (9) follows by Lemma. of Chen et al. (011). Then RCVaR h ðxþ is the minimization of the following function over c: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h x ðcþ ¼c þ 1 R c m þ r þðr þ c þ mþ : c * x minimizing h x ðcþ can be found by equating the first derivative to zero: h 0 1 1 þðr þ c þ mþðr þðr þ c þ mþ Þ 1 xðcþ ¼1 þ ence, we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 1 ¼ðR þ c þ mþ= r þðr þ c þ mþ ; (recall that h [ 0:5 so that h 1 [ 0) or, equivalently, or, whence we obtain ðh 1Þ ðr þðr þ c þ mþ Þ¼ðR þ c þ mþ ; ðh 1Þ r ¼ðR þ c þ mþ ð1 ðh 1Þ Þ ðh 1Þr R þ c þ m ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4h 4h c ðh 1Þr x ¼ R m þ ffiffi ffiffiffiffiffiffiffiffiffiffiffi ; h since h x ðcþ is a convex function as is verified immediately using the second derivative test: h 00 x ðcþ ¼ 1 ðþ ðr þðr þ c þ mþ Þ 3 r 0: The minimum value of h x ðcþ being known, we can calculate the min max value: :

A. B. Paç, M. Ç. Pınar min max F hðx; cþ ¼min h xðcþ cr D cr ¼ h x ðc x Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðh 1Þr ¼ R m þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffi þ 1 ðh 1Þr ffiffi ffiffiffiffiffiffi þ r þ ðh 1Þ h 1 h 4hð1 hþ r h ¼ R m þ ð h 1Þr ffiffiffi ffiffiffiffiffiffiffiffiffiffiffi þ 1 r h þ 1 þ 1 h ffiffi ffiffiffiffiffiffiffiffiffiffiffi h ðh 1Þr ¼ R m þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffi þ 1 rðþ h ffiffi ffiffiffiffiffiffiffiffiffiffiffi h hr ¼ R m þ ffiffiffi ffiffiffiffiffiffiffiffiffiffiffi h ffiffiffi h ¼ R m þ ffiffiffiffiffiffiffiffiffiffiffi r ffiffiffi ¼ R ðl ReÞ T h ffiffiffiffiffiffiffiffiffiffi x þ ffiffiffiffiffiffiffiffiffiffiffi x T Cx: ð10þ ence, by Theorem.4 of (Shairo 011) we have ffiffi RCVaR h ðxþ ¼ R ðl ReÞ T h ffiffiffiffiffiffiffiffiffiffi x þ ffiffiffiffiffiffiffiffiffiffiffi x T Cx: Minimizing the above exression for worst-case CVaR under the minimum mean return constraint, the robust otimal ortfolio selection can be found.using a nonnegative multilier k, the Lagrange function is ffiffiffi Lðx; kþ ¼ R ðl ReÞ T h ffiffiffiffiffiffiffiffiffiffi x þ ffiffiffiffiffiffiffiffiffiffiffi x T Cx þ kðd R ðl ReÞ T xþ: The first-order conditions are: l þ Re þ jcx ffiffiffiffiffiffiffiffiffiffi kðl ReÞ ¼0; x T Cx where we defined j ffiffi ffiffiffiffiffiffi h for convenience. We make the suosition that x = 0 1 h and define r ffiffiffiffiffiffiffiffiffiffi x T Cx: We get the candidate solution rðk þ 1Þ x c ¼ C 1 ~l: j Now, using the identity x T c Cx c ¼ r ; we obtain the equation It is a simle exercise to show that in the absence of the minimum mean return constraint the ortfolio osition in the ith risky asset tends to 1 deending on the sign of the ith comonent of ~C 1 ~l:

Robust ortfolio choice with CVaR and VaR which imlies that k ¼ j ffiffiffi ðk þ 1Þ ¼ j ; 1: Under the condition j ffiffiffi 1 we ensure k C 0. Now, utilizing the constraint which we assumed would be tight, the resulting equation yields (after substituting for x c and k) r ¼ d ffiffiffiffi R ; which is ositive under the assumtion d [ R ( is ositive by ositive definiteness of C and the assumtion that l is not a multile of e). Now, substitute the above exressions for r and k into x c, and the desired exression is obtained after evident simlification. If jffiffiffi \1 then the roblem Eq. (6) is unbounded by the convex duality theorem since it is always feasible. h The VaR measure can also be calculated using the auxiliary function Eq. (): VaR h ðxþ ¼arg min F hðx; cþ: cr The worst-case VaR is now calculated as follows: RVaR h ðxþ ¼max VaR hðxþ D ¼ max arg min F hðx; cþ D cr ¼ arg min max F hðx; cþ cr D ¼ arg min h xðcþ cr ¼ c x : Again, a change of the order of oerators makes this calculation ossible by Theorem.4 of Shairo (011). With a similar aroach to that followed for CVaR, we ose the roblem of minimizing the robust VaR: min R xr xt l þ Rx T ðh 1Þ e þ n ffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi x T Cx h subject to ðl ReÞ T x d R: We obtain a relica of the revious result in this case. ence the roof, which is identical, is omitted. Theorem For h 1 þ 1 ffiffiffi ffiffiffiffiffiffiffi ; the roblem of minimizing robust VaR under þ1 distribution ambiguity and a minimum mean return restriction admits the otimal ortfolio rule x ¼ d R C 1 ~l:

For h\ 1 þ 1 ffiffiffi ffiffiffiffiffiffiffi ; the roblem is unbounded. þ1 In both results stated above, the otimal ortfolios are mean-variance efficient. We lot the critical thresholds for h given in Theorems 1 and in Fig. 1 below. Both thresholds tend to one as goes to infinity and the threshold curve for robust VaR dominates that of robust CVaR. ence, robust VaR leads to aggressive ortfolio behavior in a larger interval for h than robust CVaR. In other words, one has to choose a larger confidence level h for robust VaR as comared to robust CVaR to make an otimal ortfolio choice. Thus we can affirm that robust VaR is more conservative than robust CVaR. Based on the above results, it is straightforward to derive the equations of the robust CVaR efficient frontier and the robust VaR efficient frontier. Both robust frontiers are the straight lines governed by the equation ffiffiffiffi j d ¼ j ffiffiffiffi R þ j ffiffiffiffi f ; ð11þ where for f = RCVaR and j ¼ ffiffi ffiffiffiffiffiffi h we obtain the efficient frontier for robust 1 h CVaR; and for f = RVaR and j ¼ ðh 1Þ ffiffiffiffiffiffi we get the robust VaR efficient frontier. ffiffi h 1 h A. B. Paç, M. Ç. Pınar Fig. 1 The critical thresholds for robust CVaR and robust VaR. The uer curve is the threshold value curve for robust VaR

Robust ortfolio choice with CVaR and VaR 3 Robust CVaR in the resence of riskless asset under distribution and mean return ambiguity We consider now the roblem of choosing a ortfolio x R n that minimizes the function RRCVaR h ðxþ ¼ max CVaR h ðxþ ð1þ D; lu l or equivalently RRCVaR h ðxþ ¼max lu l max D min cr F hðx; cþ ð13þ where we define the ellisoidal uncertainty set U l ¼fljkC 1= ðl l nom Þk ffiffi g for the mean return denoted l; where l nom denotes a nominal mean return vector which can be considered as the available estimate of mean return. The arameter acts as a measure of confidence in the mean return estimate. We consider now the roblem of choosing a ortfolio x R n that minimizes the function RRCVaR h ðxþ ¼max max min F hðx; cþ ð14þ lu l D cr subject to ðl ReÞ T x d R: As in the revious section we define D ¼fjE ½nŠ ¼l; Cov ½nŠ ¼C 0g: Considering the inner max min roblem over D and c R; resectively, as in the roof of Theorem 1 of revious section while we kee l fixed, we arrive at the inner roblem for the objective function (recall Eq. 10) that we can transform immediately: ffiffiffi h ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ¼ max R ðl ReÞ T h x þ ffiffiffiffiffiffiffiffiffiffiffi lu l ¼ R ðl Þ T x þ ffiffi ffiffi ffiffiffiffiffiffiffiffiffiffi x T h Cx þ max RCVaR h ðxþ ¼max R ðl ReÞ T x þ lu l lu l ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi x T Cx ffiffiffiffiffiffiffiffiffiffi x T Cx ffiffiffiffiffiffiffiffiffiffi x T Cx; where the last equality follows using a well-known transformation result in robust otimization (see e.g., Ben-Tal and Nemirovski 1999), and l ¼ l nom Re: Using the same transformation on the minimum mean return constraint as well, we obtain the second-order conic roblem min xr n R xt l þ ffiffi þ ffiffi h ffiffiffiffiffiffiffiffiffiffiffi! ffiffiffiffiffiffiffiffiffiffi x T Cx subject to where d [ R. ðl Þ T x ffiffi ffiffiffiffiffiffiffiffiffiffi x T Cx d R;

A. B. Paç, M. Ç. Pınar Theorem 3 ffiffiffi 1. Under the Slater constraint qualification, if h [ ð ffiffiffi ð roblem Eq. (14) admits the otimal ortfolio rule x d R ¼ ffiffiffiffi ffiffi ffiffiffiffi C 1 ~l: ð Þ. If ¼ ; the roblem is unbounded. 3. If [ ; the roblem is infeasible. ffi Þ ffi Þ þ1 ; and \ then the Proof Under the Slater constraint qualification the Karush Kuhn Tucker otimality conditions are both necessary and sufficient. Using a non-negative multilier k we have the Lagrange function ffiffi! Lðx; kþ ¼ R x T l ffiffi h ffiffiffiffiffiffiffiffiffiffi þ þ ffiffiffiffiffiffiffiffiffiffiffi x T Cx þ kðd R ðl Þ T x þ ffiffi ffiffiffiffiffiffiffiffiffiffi x T CxÞ: Going through the usual stes as in the roof of Theorem 1 under the suosition ffiffiffiffiffiffiffiffiffiffi that r (defined as x T Rx) is a finite non-zero ositive number, we have the candidate solution: rðk þ 1Þ x c ¼ j þ ffiffi C 1 l : ðk þ 1Þ From the identity x T c Cx c ¼ r we obtain the quadratic equation in k: ð Þk þð ffiffi ffiffi jþk þ j j ¼ 0 with the two roots j þ ffiffiffiffi ffiffi ffiffiffiffi ffiffi þ j þ ffiffiffiffi ffiffi ; ffiffiffiffi ffiffi : þ The left root cannot be ositive, so it is discarded. The right root is ositive if [ and j ffiffiffiffi ffiffi : Now, returning to the conic constraint which is assumed to be tight we obtain an equation in r after substituting for k in the right root, and solving for r using straightforward algebraic simlification we get r ¼ ffiffiffiffi d R ffiffi ; which is ositive rovided [ : Now, the exression for x * is obtained after substitution for r and k into x c and evident simlification. Part is immediate from the result of Part 1. For art 3, if ; our hyothesis of a finite ositive non-zero r is false, in which case the only ossible value for r is zero, achieved at the zero risky ortfolio which is infeasible.h

Robust ortfolio choice with CVaR and VaR The roblem of minimizing the robust VaR under distribution and mean return ambiguity in the resence of a minimum target mean return constraint is osed as min R xr xt l ffiffi ðh 1Þ þ þ n ffiffi ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi x T Cx h subject to ðl Þ T x ffiffi ffiffiffiffiffiffiffiffiffiffi x T Cx d R: Again, we obtain a result similar to the revious theorem in this case. The roof is a verbatim reetition of the roof of the revious theorem, hence omitted. Theorem 4 1. Under the Slater constraint qualification, if h [ 1 ffiffiffi þ ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; and \ 1þð Þ then the roblem of minimizing the robust VaR under distribution and mean return ambiguity in the resence of a minimum target mean return constraint admits the otimal ortfolio rule x d R ¼ ffiffiffiffi ffiffi ffiffiffiffi C 1 ~l: ð Þ. If ¼ ; the roblem is unbounded. 3. If [ ; the roblem is infeasible. Notice that we obtain identical and mean-variance efficient ortfolio rules for both CVaR and VaR under distribution and mean return ambiguity. Furthermore, the otimal ortfolio rules reduce to those of Theorems 1 and, resectively, for ¼ 0; the case of distribution ambiguity only. The robust CVaR and VaR efficient frontiers are straight lines given by the equation: ffiffiffiffi ffiffi j d ¼ j ffiffiffiffi ffiffi R þ þ j ffiffiffiffi ffiffi f ; þ where for f = RRCVaR and j ¼ ffiffi ffiffiffiffiffiffi h we obtain the efficient frontier for robust 1 h CVaR; and for f = RRVaR and j ¼ ðh 1Þ ffiffi ffiffiffiffiffiffi we get the robust VaR efficient h 1 h frontier. Comaring the above results and efficient frontier to the results of the revious section and to the efficient frontier Eq. (11) for the case of ambiguity distribution only, we notice that the effect of introducing mean return ambiguity in addition to ffiffiffiffi ffiffiffiffi ffiffi distribution ambiguity has the effect of relacing by : More recisely, the mean return ambiguity decreases the otimal Share ratio of the market viewed by the investor. The investor can form an otimal ortfolio in the risky assets as long as his/her confidence in the mean return vector is not too low, i.e., his does not exceed the otimal Share ratio of the market.

A. B. Paç, M. Ç. Pınar The efficient frontier line for the case of robust ortfolios in the face of both distribution and mean return ambiguity is less stee than the efficient robust ortfolios for distribution ambiguity only. This simle fact can be verified by direct comutation: ffiffiffiffi ffiffiffiffi ffiffi j ffiffiffiffi ffiffi þ j ffiffiffiffi j ffiffi ¼ ðj ffiffiffiffi ffiffiffiffi ffiffi \0: Þðj þ Þ The d-intercet for the former is also smaller than for the latter as can be easily seen. We rovide a numerical illustration in Fig. with ¼ 0:47;¼ 0:4; R ¼ 1:01 and h = 0.95. The efficient ortfolios robust to distribution ambiguity are reresented by the steeer line. It is clear that the incremental effect of mean return ambiguity and robustness is to render the investor more risk averse and more cautious. 4 Without the riskless asset We consider now the roblems of the revious section as treated in Chen et al. (011), i.e., without the riskless asset and without the minimum mean return constraint since we do not need this restriction to obtain an otimal ortfolio rule. In that case we are dealing with the loss function: Fig. The efficient frontier lines for robust CVaR and robust VaR for ¼ 0:47;¼ 0:4; R ¼ 1:01 and h = 0.95. The steeer line corresonds to distribution ambiguity case while the oint line corresonds to distribution and mean return ambiguity case

Robust ortfolio choice with CVaR and VaR Fig. 3 The behavior of robust CVaR as a function of in the case without the riskless asset. The uer curve is for h = 0.99 and the lower curve is for h = 0.95 and the auxiliary function f ðx; nþ ¼ x T n ð15þ F h ðx; cþ ¼c þ 1 E½ð xt n cþ þ Š: ð16þ In Chen et al. (011) Theorem.9 the authors solve the roblem min max CVaR hðxþ xr n :e T x¼1 D in closed form. We shall now attack the roblem under the assumtion that the mean returns are subject to errors that we confine to the ellisoid: U l ¼fljkC 1= ðl l nom Þk ffiffi g for the mean return denoted l; where l nom denotes a nominal mean return vector as in the revious section. That is, we are interested in solving min max max CVaR hðxþ: xr n :e T x¼1 lu l D Using artially the roof of Theorem.9 in Chen et al. (011), and Theorem.4 of Shairo (011) as in the revious section, we have for arbitrary l max CVaR hðxþ ¼ D and the maximum is attained at rffiffiffiffiffiffiffiffiffiffiffi h ffiffiffiffiffiffiffiffiffiffi x T Cx x T l;

A. B. Paç, M. Ç. Pınar h 1 ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x T Cx x T l hðþ which haens to be equal to the robust VaR under distribution ambiguity. Therefore, we obtain rffiffiffiffiffiffiffiffiffiffiffi max max CVaR h ffiffiffiffiffiffiffiffiffiffi hðxþ ¼ x lu l D T Cx x T l nom þ ffiffi ffiffiffiffiffiffiffiffiffiffi x T Cx: Now, we are ready to rocess the roblem min xr n :e T x¼1 rffiffiffiffiffiffiffiffiffiffiffi h ffiffiffiffiffiffiffiffiffiffi x T Cx x T l nom þ ffiffi ffiffiffiffiffiffiffiffiffiffi x T Cx: From the first-order conditions we obtain x ¼ r j þ ffiffi C 1 ðl nom þ keþ where r ðx T CxÞ 1= ; j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h=ðþ and k is the Lagrange multilier. Using the equation r ¼ x T Cx we obtain the quadratic equation: Ck þ Bk þ A ðj þ ffiffi Þ ¼ 0 where A ¼ðl nom Þ T C 1 l nom ; C ¼ e T C 1 e; B ¼ e T C 1 l nom : We solve for k under the condition Cðj þ ffiffi Þ [ AC B (note that AC- B [ 0 by Cauchy Schwarz inequality): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ B þ B AC þ Cðj þ ffiffi Þ : C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We discard the root B B ACþCðjþ Þ C because it leads to a negative value for r; see the exression for r below. In this case, the dual roblem is infeasible and the rimal is unbounded since it cannot be infeasible. The condition Cðj þ ffiffi Þ [ AC B is equivalent to [ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A B =C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; h=ðþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [we do not allow ¼ð A B =C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h=ðþþ since it results in a r that grows without bound, hence the roblem is unbounded]. Using the above exression for k in the equation e T x = 1 we solve for r to get j þ ffiffi r ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B AC þ Cðj þ ffiffi : Þ Substituting k and r to the exression for x obtained from the first-order conditions we obtain the solution

Robust ortfolio choice with CVaR and VaR x 1 ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðj þ ffiffi C 1 l nom þ 6 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ D C Cðj þ ffiffi þ 1 7 4 5C 1 e; Þ C D where we have defined D ¼ AC B for simlicity. The ortfolio roblem for robust VaR under distribution and ellisoidal mean return ambiguity is resolved similarly. In fact, the only change is in the definition of j. Therefore, we have roved the following result. Theorem 5 1. The distribution and mean return ambiguity robust CVaR ortfolio choice, i.e., the solution to roblem min max max CVaR hðxþ xr n :e T x¼1 lu l D is given by 3 x 1 ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðj þ ffiffi C 1 l nom þ 6 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ D C Cðj þ ffiffi þ 1 7 4 5C 1 e; ð17þ Þ C D where D ¼ AC B ; A ¼ðl nom Þ T C 1 l nom ; C ¼ e T C 1 e; B ¼ e T C 1 l nom ; and j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h=ðþ rovided that [ ð A B =C jþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : If ð A B =C jþ the roblem is unbounded.. The distribution and mean return ambiguity robust VaR ortfolio choice, i.e., the solution to roblem min max h 1 ffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x T Cx x T l xr n :e T x¼1 lu l hðþ 3 is given by 3 x 1 ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðj þ ffiffi C 1 l nom þ 6 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ D C Cðj þ ffiffi þ 1 7 4 5C 1 e: Þ C D ð18þ where D ¼ AC B ; A ¼ðl nom Þ T C 1 l nom ; C ¼ e T C 1 e; B ¼ e T C 1 l nom ; and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ¼ h 1 ffiffiffiffiffiffiffiffiffiffi rovided that [ ð A B =C jþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : If ð A B =C jþ the hð1 hþ roblem is unbounded. Notice that the otimal ortfolios are mean-variance efficient. Using the data in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h Chen et al. (011) we lot the otimal CVaR function 1 h x T Cx x T l nom þ ffiffiffiffiffiffiffiffiffiffiffiffi x T Cx evaluated at x * as a function of in Fig. 3 for two different values of h. As the confidence arameter increases (i.e., confidence in the mean return estimate diminishes), the otimal robust CVaR increases, which imlies an increase in risk (Fig. 3). A similar behavior occurs for robust VaR otimal value function.

A. B. Paç, M. Ç. Pınar Interestingly the otimal robust CVaR and VaR increase for constant when the quantile arameter h increases (Fig. 3). When the uncertainty set U l is reduced to a single oint l nom = l, i.e., for ¼ 0 we obtain recisely the result of Chen et al. (011), namely, Theorem.9: Corollary 1 1. The distribution ambiguity robust CVaR ortfolio choice, i.e., the solution to roblem min max CVaR hðxþ xr n :e T x¼1 D is given by x 1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 B l þ Cj D C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 C 1 e: ð19þ Cj D C where D ¼ AC B ; A ¼ l T C 1 l; C ¼ e T C 1 e; B ¼ e T C 1 l; and j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h=ðþ rovided that h/(1 - h) [ A - B /C. If h/(1 - h) B A - B /C the roblem is unbounded.. The distribution ambiguity robust VaR ortfolio choice, i.e., the solution to roblem h 1 ffiffiffiffiffiffiffiffiffiffi min ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x T Cx x T l xr n :e T x¼1 hðþ is given by x 1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 B l þ Cj D C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 C 1 e: ð0þ Cj D C where D ¼ AC B ; A ¼ l T C 1 l; C ¼ e T C 1 e; B ¼ e T C 1 l; and j ¼ h 1 ffiffiffiffiffiffiffiffiffiffi hð1 hþ h 1 rovided that ffiffiffiffiffiffiffiffiffiffi [ A B h 1 =C: If ffiffiffiffiffiffiffiffiffiffi A B =C the roblem is hð1 hþ hð1 hþ unbounded. Acknowledgments anonymous referee. The revised version of the aer benefited greatly from the comments of an References Ben-Tal A, Nemirovski A (1999) Robust solutions to uncertain linear rogramming roblems. Oer Res Lett 5(1):1 13 Ben-Tal A, Nemirovski A (1998) Robust convex otimization. Math Oer Res 3(4):769 805 Ben-Tal A, Teboulle M (007) An old new concet of convex risk measures: the otimized certainty equivalent. Math Finance 17:449 476 Best MJ, Grauer R (1991) Sensitivity analysis for mean-variance ortfolio roblems. Manag Sci 37(8):980 989 Best MJ, Grauer RR (1991) On the sensitivity of mean-variance-efficient ortfolios to changes in asset means: some analytical and comutational results. Rev Financial Stud 4():315 34

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