Robust Optimization Approaches for Portfolio Selection: A Computational and Comparative Analysis
TTECHNICAL UNIVERSITY OF CRETESCHOOL OF PRODUCTION ENGINEERING AND MANAGEMENT
Robust Optimization Approaches for PortfolioSelection: A Computational and ComparativeAnalysis byAntonios GeorgantasA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE M.Sc. DEGREE INOPERATIONS RESEARCHJune 2018THESIS COMMITTEEProfessor Michalis Doumpos (DPEM),
Thesis Supervisor
Professor Constantin Zopounids (DPEM)Professor Fotios Pasiouras (DPEM) a r X i v : . [ q -f i n . P M ] O c t ΟΛΥΤΕΧΝΕΙΟ ΚΡΗΤΗΣΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΑΡΑΓΩΓΗΣ ΚΑΙ ΔΙΟΙΚΗΣΗΣ
Μεθοδολογίες Ευσταθούς Βελτιστοποίησης ΕπενδυτικώνΧαρτοφυλακίων: Μια Υπολογιστική Συγκριτική Ανάλυση
Αντώνιος ΓεωργαντάςΔΙΑΤΡΙΒΗ ΓΙΑ ΤΗ ΜΕΡΙΚΗ ΕΚΠΛΗΡΩΣΗΤΩΝ ΑΠΑΙΤΗΣΕΩΝ ΓΙΑ ΤΟ ΜΔΕΣΤΗΝ ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΕΡΕΥΝΑΙΟΥΝΙΟΣ 2018ΕΞΕΤΑΣΤΙΚΗ ΕΠΙΤΡΟΠΗΚαθηγητής Μιχάλης Δούμπος (ΜΠΔ), ΕπιβλέπωνΚαθηγητής Κωνσταντίνος Ζοπουνίδης (ΜΠΔ)Καθηγητής Φώτιος Πασιούρας (ΜΠΔ)
Abstract
The field of portfolio selection is an active research topic, which combines ele-ments and methodologies from various fields, such as optimization, decision anal-ysis, risk management, data science, forecasting, etc. The modeling and treatmentof deep uncertainties for the future asset returns is a major issue for the success ofanalytical portfolio selection models. Recently, robust optimization (RO) modelshave attracted a lot of interest in this area. RO provides a computationally tractableframework for portfolio optimization based on relatively general assumptions onthe probability distributions of the uncertain risk parameters. Thus, RO extendsthe framework of traditional linear and non-linear models (e.g., the well-knownmean-variance model), incorporating uncertainty through a formal and analyticalapproach into the modeling process. Robust counterparts of existing models canbe considered as worst-case re-formulations as far as deviations of the uncertainparameters from their nominal values are concerned. Although several RO modelshave been proposed in the literature focusing on various risk measures and differ-ent types of uncertainty sets about asset returns, analytical empirical assessmentsof their performance have not been performed in a comprehensive manner. Theobjective of this study is to fill in this gap in the literature. More specifically, weconsider different types of RO models based on popular risk measures and con-duct an extensive comparative analysis of their performance using data from theUS market during the period 2005-2016. For the analysis, three different robustversions of the mean-variance model are considered, together with two other ro-bust models for conditional value-at-risk and the omega ratio. The robust versionsare compared against standard (non-robust) models through various portfolio per-formance metrics, focusing on out-of-sample results. The analysis is based on arolling window approach.
Acknowledgments
By completing the present dissertation, I would like to thank Professor Doumposfor giving me the opportunity to work under his guidance in the field of FinancialEngineering, as also for the patience and perseverance he has shown throughoutthis period. Professor Doumpos provided me with everything he deemed impor-tant in order to grasp effectively the notions affiliated with the Thesis, with respectto what we were opting to accomplish. The interaction was more than satisfyingand this acted as the foundation for many of the methodologies I explain in themain body of the Thesis. Furthermore, I am grateful to my family and the friendsclose to me for their support during this period.
Contents
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Conic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Robust Linear Optimization . . . . . . . . . . . . . . . . . . . . 122.3 Selection of Uncertainty Set . . . . . . . . . . . . . . . . . . . . 122.3.1 Ellipsoidal Uncertainty . . . . . . . . . . . . . . . . . . 132.3.2 Polyhedral Uncertainty . . . . . . . . . . . . . . . . . . 142.3.3 Cardinality Constrained Uncertainty . . . . . . . . . . . 152.3.4 Norm Uncertainty . . . . . . . . . . . . . . . . . . . . . 162.3.5 Perturbation Vectors . . . . . . . . . . . . . . . . . . . . 172.4 Computational Issues . . . . . . . . . . . . . . . . . . . . . . . . 18 . . . . . . . . . . . . . . . . . . . . . . . 193.1 Classical Portfolio Optimization . . . . . . . . . . . . . . . . . . 193.1.1 Mean Variance Optimization . . . . . . . . . . . . . . . 213.1.2 Multi-Objective Optimization . . . . . . . . . . . . . . . 233.1.3 Omega ratio Optimization . . . . . . . . . . . . . . . . . 253.1.4 CVaR Optimization . . . . . . . . . . . . . . . . . . . . 273.2 Robust Portfolio Optimization . . . . . . . . . . . . . . . . . . . 283.2.1 Mean Variance with Box Uncertainty . . . . . . . . . . . 283.2.2 Mean-Variance with Ellipsoidal Uncertainty . . . . . . . 293.2.3 Robust Multi-Objective Optimization . . . . . . . . . . . 31.2.4 Worst-case Omega ratio . . . . . . . . . . . . . . . . . . 353.2.5 Worst-case CVaR . . . . . . . . . . . . . . . . . . . . . 37 . . . . . . . . . . . . . . . . . . . . . . . . 424.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1.1 Rolling Windows Approach . . . . . . . . . . . . . . . . 424.2 Portfolio Performance Indicators . . . . . . . . . . . . . . . . . . 454.3 Composition of Portfolios . . . . . . . . . . . . . . . . . . . . . 464.4 Portfolio Performance Results . . . . . . . . . . . . . . . . . . . 494.4.1 Comparison of non-robust models . . . . . . . . . . . . . 494.4.2 Comparison of robust models . . . . . . . . . . . . . . . 504.4.3 Robust versus non-robust models . . . . . . . . . . . . . 524.5 Validation of the Uncertainty Sets . . . . . . . . . . . . . . . . . 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 1Introduction
An area of science which constantly admits novel research is financial engineer-ing. Financial engineering is a multi-disciplinary field involving financial theory,the tools of mathematics and the practice of programming. One popular fieldof research in this area concerns the portfolio selection and management. Theseminal approach of Markowitz during 1950s focusing on the development ofa mean-variance model through quadratic mathematical modeling tools, was thecornerstone in this area and altered the philosophy in the financial domain. Al-though the mean-variance model is extensively employed by practitioners, it hasseveral impractical aspects. One discrete element is that Markowitz’s frameworkconsiders the first two moments of the distribution and therefore implies that theunderlying asset returns are normally distributed [13]. In addition to unrealisticportfolio weights, one of the major discrepancies with the mean-variance modelis the high sensitivity of the parameter estimations to small changes in the inputs.This doesn’t seem to be that appealing to the broader community and a differentscheme which accounts for less restrictive and more realistic assumptions needsto be formulated. An idea which could alleviate the impact of highly concentratedundiversified asset allocations lies within the scope of robust optimization models.Robust models include methods to improve the accuracy of inputs and to applyrobust optimization frameworks to portfolio optimization [13]. Worst-case opti-mization incorporates uncertainty directly into the optimization process. Alongwith the development of various robust models, much effort has been devotedto test the performance of these robust portfolios. Although many researchershave conducted out-of-sample performance tests to contrast the classical mean-variance model and robust models, there has not been a dominating conclusionto the performance of such approaches [23]. Our aim in this Thesis is to providea thorough investigation between the efficiency of classical portfolio approachesValue at Risk, Conditional Value at Risk etc) and their robust counterparts.
To the best of our knowledge, no work to date has attacked this specific problemheads on. Thus our main contribution lies in providing an extensive framework forevaluating robust optimization techniques under different architectures. We wereparticularly interested in investigating the efficiency of the models considered,in periods, which could behave “out-of-the-box”, by incorporating a high levelof turmoil, such as year 2008 where the global financial crisis took place. Thisaddition resulted in an extra degree of complexity, which we should account forin the Thesis, rendering the corresponding results more realistic. At the sametime, we compared the performance of the employed robust framework to thenon-robust variants of each respective model with respect to certain performanceindicators. By doing so, we opt for a concrete understanding in terms of theachieved superiority of robust models as opposed with their non-robust “enemies”.Overall we can deduce that the performance of the robust models outperforms thenon-robust models, for most of the metrics used.
This thesis is organized into the following chapters: Chapter 2 provides a briefoverview of robust optimization framework. Chapter 3 then describes the portfolioselection problem, models employed for the computational analysis along withthe solution methods( exact and approximate). Chapter 4 presents our simulationexperiments that we undertake to verify the robustness of the findings; and, finally,Chapter 5 concludes and outlines future work.
Chapter 2Robust Optimization
The field of portfolio selection remains today a particularly active area of re-search, which combines elements and methodologies from various fields, suchas optimization, decision analysis under uncertainty, financial risk management,data science, forecasting etc. Common ground for all these methodologies usedin this specific field is the necessity of modeling and handling the uncertaintyin asset returns. To this end, dynamic and stochastic programming models areoften employed. However, an issue that arises with such approaches is that it isoften computationally cumbersome to obtain detailed information about the prob-ability distributions of the uncertainties in the model. This is a common reason,which explains why dynamic and stochastic programming methodologies havenot become extensively adopted in various applications. Robust optimization is arecently developed technique, which accounts for the same type of problems asstochastic programming. However, it typically makes relatively general assump-tions on the probability distributions of the uncertain parameters in order to workwith problem formulations that are more computationally tractable [13].Robust optimization extends the framework of traditional linear and non-linearmodels (e.g., the mean-variance model), by incorporating instantaneously the un-certainty as a parameter of the problem. Robust logic deals with making optimiza-tion models robust with respect to constraint violations by solving robust counter-parts of these problems for appropriately defined uncertainty sets for the uncertainparameters. These robust counterparts are in fact worst-case formulations of theoriginal problem as far as deviations of the parameters from their nominal valuesare concerned; however, typically the worst case scenarios are defined in smartways that do not lead to overly conservative formulations [13]. An element inrobust optimization is that one makes the problem well defined, by assuming thatthe uncertain parameters vary in a particular set defined by one’s knowledge aboutheir probability distributions and then takes a worst case (max-min) approach. Inthe optimization literature, the term ”robust optimization” has been used to de-scribe several different concepts; however, robust optimization refers to an area ofoptimization whose roots are in the robust control engineering literature.The way to compute the worst case is also open to debate: should it use afinite number of scenarios, such as historical data, or continuous, convex un-certainty sets, such as polyhedra or ellipsoids? The answers to these questionswill determine the formulation and the type of the robust counterpart [16]. Aspinpointed by Goldfarb and Iyengar [18], a wide range of robust optimizationproblems corresponding to the natural class of uncertainty sets (defined by the es-timation procedures) can be cast as second-order cone programs (SOCPs) and canbe solved very efficiently using interior-point algorithms [26], [29], [34]. In fact,both the worst case and practical computational effort required to solve an SOCPis comparable to that for solving a convex quadratic program of similar size andstructure; i.e., in practice, the computational effort required to solve these robustportfolio selection problems is comparable to that required to solve the classicalMarkowitz mean-variance portfolio selection problems [18].Our focus in this Thesis is to provide a more recent approach to optimizationunder uncertainty, in which the uncertainty model is not stochastic, but ratherdeterministic and set-based [6].Given an objective f ( x ) to optimize subject to constraints f i ( x , u i ) ≤ withuncertain parameters, { u i } , the general Robust Optimization formulation is: min f ( x )s . t . f i ( x , u i ) ≤ , ∀ u i ∈ U i , i = 1 , . . . , m (2.1)Here x ∈ R N is a vector of decision variables, f , f i : R N → R are functions,and the uncertain parameters u i ∈ R k are assumed to take arbitrary values in the uncertainty sets U i ⊆ R k . The goal of (2.1) is to compute minimum cost solu-tions x ∗ among all those solutions which are feasible for all realizations of thedisturbances u i within U i . Thus, if some of the U i , are continuous sets, (2.1) asstated, has an infinite number of constraints. Intuitively, this problem offers somemeasure of feasibility protection for optimization problems containing parametershich are not known exactly [6].Ben-Tal and Nemirovski( [3], [4], [5]) stated that for suitably defined uncertaintysets U , the robust counterparts of linear programs, quadratic programs, and gen-eral convex programs are themselves tractable optimization problems [18]. A large set of optimization models, some of which, would be described to a greaterextent in the following segments of the Thesis, can be treated as conic optimiza-tion problems. A conic optimization (CO) problem (called also conic program) isof the form min c (cid:62) x + d s . t . A x = b , x ∈ K (2.2)where x ∈ R N is the decision vector, K ⊂ R M represents a closed pointed con-vex cone with a nonempty interior and the constraint is a given affine mappingfrom R n to R m . Virtually, any convex program can be represented as a conicoptimization problem by specifying K appropriately.According to Ben-Tal, El Ghaoui and Nemirovski [2], a wide variety of convexprograms are covered by just three types of cones. For the sake of the modelspresented here, we will narrow down the possible varieties to just one, that being:• Direct products of Lorentz (or Second-order, or Ice Cream) cones L k = { x ∈ R k : x k ≥ (cid:113)(cid:80) k − j =1 x j } . These cones give rise to Conic QuadraticOptimization (called also Second Order Conic Optimization). The mathe-matical form of a CQO problem is min c (cid:62) x s . t . (cid:107) A i x − b i (cid:107) ≤ ψ (cid:62) i x − d i , ≤ i ≤ m (2.3) A set K is a cone if for all x ∈ K it follows that α x ∈ K for all α ≥ . A convex cone is acone with the property that x + y ∈ K for all x , y ∈ K [13]. eeping in mind the above formulation, we can detect that researchers internation-ally adopt this form, since it is general enough to encompass linear programs, con-vex quadratic programs, and constrained convex quadratic programs. At the sametime, the problems in this class share many of the properties of linear programs,making the corresponding optimization algorithms used for solving these prob-lems very efficient and highly scalable. Many robust portfolio allocation problemscan be formulated as Second Order Cone Programming (SOCP) problems [13]. A large class of optimization problems can often be cast as robust linear optimiza-tion problems by taking the robust counterpart of a linear optimization problem.So, without loss of generality we get: min c (cid:62) x s . t . A x ≤ b , ∀ a ∈ U , . . . , a m ∈ U m , (2.4)where a i represents the i -th row of the uncertain matrix A and takes values in theuncertainty set U i ⊆ R N . Then, a (cid:62) i x ≤ b i , ∀ a i ∈ U i if and only if max { a i ∈U i } a (cid:62) i x ≤ b i , ∀ i . This is the subproblem which must be solved. Ben-Tal and Nemirovski [4]show that the robust LP is essentially always tractable for most practical uncer-tainty sets of interest. We can’t always expect to get a corresponding linear pro-gram, by doing this formulation. A relatively simple way to model uncertainty is to generate scenarios for the possi-ble values of the uncertain parameters using, for example, future asset returns. Asexplained in Section 2.2, scenario optimization can be incorporated in the robustoptimization framework by specifying an uncertainty set that is a collection of sce-narios for the uncertain parameters. The robust formulation of the original prob-lem would then contain a set of constraints one of each scenario in the uncertaintyset and the optimization would make sure that the original constraint is satisfiedor the worst-case scenario in the set [13]. Uncertainty sets are usually extendedto richer sets ranging from polytopes to more advanced conic-representable setsderived from statistical procedures. For instance,one can frequently obtain con-fidence levels for the uncertain parameters [13]. A central feature that RobustOptimization (RO) tries to tackle is the probability guarantees on feasibility underparticular distributional assumptions for the disturbance vectors [6]. Specifically,what does robust feasibility imply about the probability of feasibility, i.e. what isthe smallest (cid:15) we can find such that x ∈ X ( U ) ⇒ P ( f i ( x , u i ) > ≤ (cid:15), under (ideally mild) assumptions on a distribution for u i ? Such implications maybe used as guidance for selection of a parameter representing the size of the un-certainty set.At this point, we must give the definition introduced by Bertsimas, Brown,and Caramanis [6], regarding classes of functions f i , coupled with the types ofuncertainty sets U i , that yield tractable robust counterparts. So, the robust feasibleset could be X ( U ) = { x | f i ( x , u i ) ≤ , ∀ u i ∈ U i , i = 1 , . . . , m } . Ellipsoidal uncertainty sets allow for including second moment information aboutthe distributions of uncertain parameters and have been used extensively. Bertsi-mas, Brown, and Caramanis [6], mention that controlling the size of these ellip-soidal sets, as in the theorem below, has the interpretation of a budget of uncer-tainty that the decision-maker selects in order to easily trade-off robustness andperformance.
THEOREM 2.1 (Ben-Tal and Nemirovski [4]). Let U be ellipsoidal i.e, U = U (Π , Q ) = { Π( u ) |(cid:107) Q u (cid:107) ≤ ρ } , where u → Π( u ) is an affine embedding of R L into R m × N and Q ∈ R M × L . Thenroblem (2.2) is equivalent to an (SOCP). Explicitly, if we have the uncertainoptimization min c (cid:62) x s . t . a i x ≤ b i , ∀ a i ∈ U i , ∀ i = 1 . . . , m, (2.5) where the uncertainty set is given as U = { ( a , . . . , a m ) : a i = a i + ∆ i u i , i = 1 , . . . , m, (cid:107) u (cid:107) ≤ ρ } ( a i denotes the nominal value), then the robust counterpart is min c (cid:62) x s . t . a i x ≤ b i − ρ (cid:107) ∆ i x (cid:107) , ∀ i = 1 . . . , m. (2.6)The intuition is the following: for the case of ellipsoidal uncertainty, the subprob-lem max { a i ∈U i } a (cid:62) i x ≤ b i , ∀ i is an optimization over a quadratic constraint. Thedual, therefore, involves quadratic functions, which leads to the resulting SOCP. Polyhedral Uncertainty can be viewed as a special case of ellipsoidal uncertainty [4].When U is polyhedral, the subproblem becomes linear, and the robust counterpartis equivalent to a linear optimization problem. We consider the following prob-lem: min c (cid:62) x s . t . max D i a i ≤ d i a (cid:62) i x ≤ b i , i = 1 . . . , m. (2.7)The dual of the subproblem ( x is not a variable of optimization in the inner max)becomes max a (cid:62) i x s . t . D i a i ≤ d i = ⇒ min p (cid:62) i d i s . t . p (cid:62) i D i = x , p i ≥ (2.8)nd therefore the robust linear optimization now becomes min c (cid:62) x s . t . p (cid:62) i d i ≤ b i , i = 1 , . . . , m, p (cid:62) i D i = x i = 1 , . . . , m, p i ≥ i = 1 , . . . , m. (2.9)Thus the size of such problems grows polynomially in the size of the nominalproblem and the dimensions of the uncertainty set. A cardinality constraint can be defined as the number of parameters of the prob-lem, which are allowed to deviate from their nominal values. In general, givenan uncertainty matrix, A = (cid:16) a ij (cid:17) we assume that each component a ij lies in [ a ij − ˆ a ij , a ij + ˆ a ij ] . Following the principle proposed by Bertsimas et al. [6], weallow at most Γ i coefficients of row i to deviate. The positive number Γ i controlsthe trade-off between the optimality of the solution and its robustness to parameterperturbation. Given values Γ , . . . , Γ m , the problem is transformed in the robustsense as min c (cid:62) x s . t . (cid:88) j a ij x j + max S i ⊆ J i : | S i | =Γ i (cid:88) j ∈ S i ˆ a ij (cid:37) j ≤ b i , ≤ i ≤ m, − (cid:37) j ≤ x j ≤ (cid:37) j , ≤ j ≤ n, l ≤ x ≤ κ , (cid:37) ≥ . (2.10)Because of the set selection in the inner maximization, this problem is nonconvex.We can though, take the dual of the inner maximization problem and acquire anquivalent linear formulation, which is tractable nonetheless. This is what we get: max c (cid:62) x s . t . (cid:88) j a ij x j + υ i Γ i + (cid:88) j n ij ≤ b i , ∀ i,υ i + n ij ≥ ˆ a ij (cid:37) j , ∀ i, j, − (cid:37) j ≤ x j ≤ (cid:37) j , ∀ j, l ≤ x ≤ κ , n ≥ , (cid:37) ≥ . (2.11) According to Bertsimas, Pachamanova, and Sim [7], robust linear optimizationproblems with uncertainty sets described by more general norms can be cast asconvex problems with constraints related to the dual norm. Subsequently, we usethe notation vec( A ) to denote the vector formed by concatenating all of the rowsof matrix A [6]. THEOREM 2.2 (Bertsimas, Pachamanova, and Sim [6]). With the uncer-tainty set U = (cid:8) A | (cid:107) M ( vec ( A ) − vec ( ¯ A )) (cid:107) ≤ ∆ (cid:9) , where M is an invertible matrix, ¯ A is any constant matrix, and (cid:107) · (cid:107) is any norm,problem (2.4) is equivalent to min c (cid:62) x s . t . ¯ A (cid:62) i x + ∆ (cid:107) ( M (cid:62) ) − x i (cid:107) ∗ ≤ b i , i = 1 , . . . , m (2.12)where x i ∈ R ( m · n ) × is a vector that contains x ∈ R n in entries ( i − · n + 1 through i · n and everywhere else and (cid:107) · (cid:107) ∗ is the corresponding dual norm of (cid:107) · (cid:107) [6]. Considering the above, Theorem 2.2 leads to an equivalent problem ashe norm-based model, with corresponding dual norm constraints. In particular,the l and l ∞ norms yield linear optimization problems, and the l norm results inan SOCP [6]. In various applications, a pretty small (and unavoidable in reality) perturbationof the data may render the nominal optimal solution infeasible [2]. Moreover, astraightforward adjustment of the optimal solution to the actual data may have anegative impact on the quality of the solution.For the sake of completeness, we give the definition introduced by Ben-Tal etal. [2].
Definition 2 . An uncertain Linear Optimization problem is a collection min x c (cid:62) x + d s . t . A x ≤ b , c , d, A , b ∈ U (LOu)of LO problems (instances) min (cid:8) c (cid:62) x + d : A x ≤ b (cid:9) of common structure (i.e.with common numbers m of constraints and n of variables) with the data varyingin a given uncertainty set U ⊂ R ( m +1) × ( n +1) [2].Perturbation vectors are directly associated with the rules of robustness, sincethey incorporate the uncertainty in a parameterized manner, allowing for somedegree of immunization against deviations of the nominal value of the estimatedparameter.Mathematically, Ben-Tal et al. [2] deal with the uncertainty in an affine fash-ion, by perturbation vector Ξ varying in a given perturbation set J : U = (cid:34) c (cid:62) d A b (cid:35) = (cid:34) c (cid:62) d A b (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) nominal data D + L (cid:88) l =1 Ξ l (cid:34) c (cid:62) l d l A l b l (cid:35)(cid:124) (cid:123)(cid:122) (cid:125) basic shifts D l : Ξ ∈ J ⊂ R L (2.13)e must note, that the basic shifts D l within the specified uncertainty set U denotethe volatility of the given perturbation vector Ξ , within the linear optimizationproblem. By making this addition to the uncertainty set, we aim to account forunforeseen discrepancies in the performance of the model considered, making theresults we get more robust.Ben-Tal, El Ghaoui and Nemirovski [2] explain that when speaking about per-turbation sets with simple geometry (parallelotopes, ellipsoids) , we can normalizethese sets to be standard. Perturbation vectors are designed for the uncertain data,in a style to generate a reliable solution, which is immunized against uncertainty.For instance, a parallelotope is by definition an affine image of a unit box { ξ ∈ R k : − ≤ ξ j ≤ , j = 1 , . . . , k , which allows the opportunity to tacklethese problems by using the unit box instead of general parallelotope [2]. Several primal-dual interior-point methods have been developed in the last fewyears for SOCPs. For instance, Lobo et al. [26] shows that the number of iterationsrequired to solve a SOCP grows at most as the square root of the problem size,while their practical numerical experiments indicate that the typical number ofiterations ranges between 5 and 50 - more or less independent of the problemsize [26]. A feature we want to address in the Thesis is the efficiency of thealgorithms emulated for the classical portfolio optimization in comparison withtheir robust variants. Although robust optimization poses a new trend in portfoliooptimization, we want to investigate the complexity of its architecture as opposedto their classic ”enemies”.9
Chapter 3Portfolio Selection Analysis
As mentioned in Chapter 2 the field of portfolio optimization has received muchattention among operations researchers. More advanced techniques allow formore complex representations of portfolio procedures under certain conditions athand. In the present work, we focus on portfolio models, which among others con-sist of mean-variance framework (MV), which was firstly introduced by the workof Markowitz [27]. In its simplest form, mean-variance analysis provides a frame-work to construct and select portfolios, based on the expected performance of theinvestments and the risk appetite of the investor. Markowitz reasoned that in-vestors should decide on the basis of a trade-off between risk and expected return.He suggested that risk should be measured by the variance of returns-the averagesquared deviation around the expected return [13] , [27]. Moreover, Markowitzargued that for any given level of expected return, a rational investor would choosethe portfolio with minimum variance from the set of all possible portfolios. Theset of all possible portfolios that can be constructed is called the feasible set . Min-imum variance portfolios are called mean-variance efficient portfolios . The setof all mean-variance efficient portfolios, for different desired levels of expectedreturn, is called the efficient frontier . In the following Figure 3.1 we provide agraphical depiction of the efficient frontier of risky assets. The feasible set isbounded by the black bold curve.
Harry Markowitz was the first to model the trade-off between risk and return inportfolio selection as an optimization problem [27]. However, more than 50 yearsafter Markowitz’s seminal work, it appears that full risk-return optimization at theportfolio level is only done at the more quantitative firms, where processes forigure 3.1: Feasible and Markowitz Efficient Portfoliosautomated forecast generation and risk control are already in place [13]. Froma practical point of view, it is important to make the portfolio selection processrobust to different sources of risk-including estimation risk and model risk. Laterin the Thesis, we show applications of variations of Markowitz’s mean-varianceportfolio optimization formulation and show explicitly how to make the problemrobust with respect to errors in expected return and covariances estimates. Ro-bustness can be incorporated in the construction of modern portfolios as well, soas to amend their performance in terms of expected returns and the minimizationof the respective covariance of returns.Previously in Section 2.3 we gave a brief introduction to uncertainty sets. Forthe sake of our computational procedure, we incorporate uncertainty about theaccuracy of estimates directly in the portfolio optimization process. .1.1 Mean Variance Optimization
In this Section, we give a more detailed explanation of mean-variance models withthe underlying mathematical manipulations required to make the correspondingoptimization models tractable. First of all, we assume that an investor has tochoose a portfolio comprised of N risky assets. Each asset is associated with arespective weight w i , which represents the percentage of the i -th asset held in theportfolio and, N (cid:88) i =1 w i = 1 We suppose the assets’ returns R = ( R , R , . . . , R N ) (cid:62) have expected returns µ = ( µ , µ , . . . , µ N ) (cid:62) and an N × N covariance matrix given by Σ = σ σ . . . σ N σ σ . . . σ N ... ... . . . ... σ N σ N . . . σ NN where σ ij denotes the covariance between asset i and asset j such that σ ii = σ i , σ ij = ρ ij σ i σ j and ρ ij is the correlation between asset i and j . Under theseassumptions, the return of a portfolio with weights w = ( w , w , . . . , w N ) (cid:62) ∈ W ⊆ R N is a random variable R p = w (cid:62) R with expected return and variancegiven by µ p = w (cid:62) µ σ p = w (cid:62) Σw By picking the portfolio’s weights, an investor chooses among the available mean-variance pairs. To calculate the weights for one possible pair, we choose a targetmean-return, µ . Then, the investor’s problem is a constrained minimization prob-em: min w (cid:62) Σw s . t . µ = w (cid:62) µ , N (cid:88) i =1 w i = 1 (3.1)This version of the classical mean-variance optimization problem is known as the risk minimization formulation .However, the mean-variance optimization problem can be expressed in variousequivalent forms, in the sense that they all lead to the same efficient frontier asthey trade expected portfolio return versus risk in a similar way [13]. Hence, wedo present the Risk Aversion Formulation .This alternative formulation explicitly models the trade-off between risk andreturn in the objective function using a risk aversion coefficient λ . We denote thefollowing mathematical formulation as (Mv): max w (cid:62) µ − λ w (cid:62) Σw s . t . N (cid:88) i =1 w i = 1 (Mv)When λ is small, the penalty from the contribution of the portfolio risk is alsosmall, leading to more risky portfolios. If we gradually increase λ from zeroand for each instance solve the optimization problem, we end up calculating eachportfolio along the efficient frontier. It is a common practice to calibrate λ suchthat a particular portfolio has the desired risk profile. The calibration is oftenperformed via backtests with historical data.The estimation error in forecasts may significantly influence the resulting opti-mized portfolio weights. As explained by Black and Litterman [8], small changesin the expected returns, in particular, may have a substantial impact. Indeed, ifestimation errors in expected returns are large, they will influence the optimalallocation.As will be explained later on, mean-variance formulation can be cast withdifferent robust formulations leading to different optimization problems. Morepecifically, Mean Variance with Box Uncertainty formulation in 3.2.1,
MeanVariance with Ellipsoidal Uncertainty formulation in 3.2.2 and
Robust Multi-Objective Optimization formulation in 3.2.3 are considered variations of classicalmean-variance procedure, under different uncertainty set assumptions.
A major factor we aim to incorporate in the Thesis is the approach established byFliege and Werner [15], where the principle is to start with the multiobjective for-mulation of the mean-variance portfolio problem. In terms of the robustificationprocedure, a detailed outline is provided in Section 3.2.3.Formally, a multiobjective optimization problem formulation (MOP) is de-fined as: min { f ( w ) , f ( w ) , . . . , f k ( w ) } , w ∈ W (3.2)The feasible set W ⊆ R N is implicitly determined by a set of equality and in-equality constraints. The vector function f : R N → R k is composed by k scalarobjective functions f i : R N → R ( i = 1 , . . . , k ; k ≥ . In multiobjective opti-mization, the sets R N and R k are known as decision variable space and objectivefunction space, respectively. The image of W under the function f is a subset ofthe objective function space denoted by Z = f ( W ) and referred to as the feasibleset in the objective function space. In multiobjective optimization problems, thereis no canonical order R k and thus, we need weaker definitions to compare vectorsin R k [19]. Most of the solution concepts in MCDM come from the old idea ofPareto-efficiency. Any solution is deemed efficient if it is impossible to move toanother solution which would improve at least one criterion and make no criterionworse. This can be understood from the following figure.igure 3.2: Illustration of the feasible set of solutionsConsequently, the feasible set of solutions lies within the Pareto front and arethe solutions which will be taken into consideration to designate the optimal pairof solutions from the two objective functions in terms of the quantity we wantto optimize at each case. The Pareto front is only composed of non-dominatedvectors.As established by Fliege and Werner [15], multicriteria optimization is theideal setting to analyze portfolio optimization in the sense of Markowitz. From afinancial perspective, we can simply set k = 2 , let say f ( w ) = s ( w ) = w (cid:62) Σw be the risk function for some covariance matrix Σ and let f ( w ) = − m ( w ) = − µ (cid:62) w be the return function for some vector of expected returns µ ∈ R N .Fliege and Werner [15] prove that the generation of almost all efficient port-folios, based on the above assumption can be attained by solving all problems ofthe form min λ s ( w ) − λ m ( w ) = λ w (cid:62) Σw − λ µ (cid:62) w s . t . w ∈ W (3.3), for all λ , λ ≥ with λ + λ = 1 . .1.3 Omega ratio Optimization The Omega ratio is a recent performance measure proposed to counterattack theknown shortcomings of the Sharpe ratio. The Sharpe ratio is the first attemptto quantify the trade-off between risk and reward in investment under uncer-tainty [20]. However, its underlying assumptions have been widely criticized [25].Omega ratio entails the partitioning of the returns into losses and gains in excess ofa predetermined threshold and it can be defined as the probability-weighted gainsby the probability-weighted losses. In theory, the Omega ratio can be used for anydistribution of asset returns. Nevertheless, it assumes accurate knowledge of thedistribution. The existence of uncertain input parameters can lead to many solu-tions for the Omega ratio maximization problem: one solution for each possiblerealization of the uncertain input. We tackle this problem in the robust sense, byassuming that the realization of the input parameters will be within an uncertaintyset. This worst-case approach based on the assumption of only partial information(Ben-Tal and Nemirovski [3], T ¨ut¨unc¨u and Koenig [35]) provides an immuniza-tion against the worst-case scenario for all possible realizations of the uncertaininput. We establish the worst-case Omega ratio maximization in Section 3.2.4under a mixture distribution with uncertain mixing probabilities distributions.Let y i denote the random return of asset i and the i -th element of the vector y ∈ R m . Similarly, w is the vector of weights, whose components add up to1. The random return of a portfolio of assets is given by w (cid:62) y . With f ( y i ) and F ( y i ) , we denote the probability density and cumulative distribution functions,respectively.According to Keating and Shadwick [22], Omega ratio is defined as Ω ( y i ) = (cid:90) + ∞ τ [1 − F ( y i )] dy i (cid:90) τ −∞ [ F ( y i )] dy i (3.4)The above can be simplified to Ω ( y i ) = E ( y i ) − τE [ τ − y i ] + + 1 (3.5)here τ denotes a threshold that partitions the returns to desirable (gain) and un-desirable (loss). Omega ratio can be distinguished into two categories; continuousand discrete. In the Thesis, we will deal with the discrete case, where the discreteprobability distribution is characterized by a mass function: (cid:88) u P r ( y = u ) = 1 Considering a discrete probability distribution the Omega ratio for a portfolio isdefined as Ω ( w ) = w (cid:62) ( Y (cid:62) π ) − τ π (cid:62) [ τ − ( Y w )] + + 1 (3.6)where Y ∈ R S × N is the matrix that contains the S sample returns for the N assetsand π is the vector with the probabilities for each sample return.As illustrated in Kapsos, Christofides and Rustem [20], the scalar form of theOmega ratio maximization problem employing the linear-fractional programmingmethod under the discrete distribution becomes max x , v , ζ x (cid:62) ( Y (cid:62) π ) − τ ζ s . t . π (cid:62) v = 1 , x (cid:62) ( Y (cid:62) π ) ≥ τ ζ, v ≥ τ ζ − Y x , v ≥ , N (cid:88) i =1 x i = ζ,ζ x ≤ x ≤ ζ x ,ζ ≥ (OR)Note that the asset weights w have changed to the variables x . .1.4 CVaR Optimization Since the middle of the 1990s, Value-at-Risk (VaR; RiskMetrics [28]), a newmeasure of downside risk grew in population in financial risk management [37].However, several shortcomings were identified in terms of its performance andreliability. Following the conceptual logic behind the mean-variance formulation,here we do present a measure of more complex architecture than VaR, entitledConditional Value at Risk (CVaR). Defined as the mean of the tail distributionexceeding VaR, CVaR has attracted much attention, since it possesses some bet-ter properties than VaR. Let f ( w , y ) denote the loss associated with the decisionvector w ∈ W ⊆ R N and the random vector y ∈ R m . Initially, we assume that y follows a continuous distribution and its corresponding density function is p ( · ) .Motivated by the theoretical limitations of VaR, Rockafellar and Urysaev [31]propose an alternative risk measure, CVaR that is defined as the conditional ex-pectation of the loss of the portfolio or equal to VaR , that isCVaR β ( w ) = 11 − β (cid:90) f ( w , y ) ≥ VaR β ( w ) f ( w , y ) p ( y ) dy (3.7)Rockafellar and Urysaev [31] prove that CVaR is sub-additive and can be castfor a given confidence level β into the following convex optimization problem:CVaR β ( w ) = min α ∈ R F β ( w , α ) , where F β ( w , α ) is expressed as F β ( w , α ) = α + 11 − β (cid:90) y ∈ R m [ f ( w , y ) − α ] + p ( y ) d y (CVaR)where [ · ] + is defined as [ t ] + = max { , t } for any t ∈ R [11].Alternatively, CVaR optimization problem can be formulated in a linear program-ming fashion overriding the integral estimation in (CVaR), as pinpointed by Rock-afellar and Urysaev [31]. With more delicate assumptions on portfolio returns, CVaR is also called Extreme Value The-ory VaR, Mean-Excess Loss, Mean Shortfall, Tail VaR, Expected Shortfall, or Conditional TailExpectation (Embrechts et al. [10]; Artzner et al. [1]) in ξ + ϑ − T − T (cid:88) i =1 ν i s . t . ν i ≥ , i = 1 . . . , T,ν i ≥ f ( w , y i ) − ξ i = 1 . . . , T (3.8)where ξ is equivalent to α in the initial (CVaR) form, ϑ is the same as − β andT denotes the number of the different scenarios.Assuming that f ( w , y ) is linear in w , (3.8) is linear and can be solved very effi-ciently by standard linear programming techniques. Having provided a brief review of the traditional portfolio models in Section 3.1our next target is to provide the robust variants of each of the models describedabove. More specifically, we want to get a more concrete picture of how to con-struct a portfolio so that the risk is as small as possible with respect to the worst-case scenario of the uncertain parameters. In Chapter 2 a brief introduction wasgiven in terms of tractable reformulations of the uncertainty. Here, we surveysome recent advances in portfolio selection with parameter uncertainty.
A reasonable way to incorporate uncertainty caused by estimation discrepancies isto require that the investor be protected if the estimated return ˆ µ i for each asset isaround the true expected return µ i . The error from the estimation can be assumedto be not larger than some small number δ i ≥ . The simplest possible choice forthe uncertainty set for µ is the “box” [13]. U δ ( ˆ µ ) = { µ | µ i − ˆ µ i | ≤ δ i , i = 1 , . . . , N } A rational treatment for δ (cid:48) i s could be the incorporation of some confidence levelaround the estimated expected return. In our case, we consider that the individualreturn of the risky assets is normally distributed, meaning that µ i − ˆ µ i σ i / √ T i follows atandard normal distribution, and a % confidence level for µ i can be obtainedby setting δ i = 1 . σ i / √ T i , where T i is the sample size used in the estimationand σ i is the standard deviation of asset i . Subsequently, we do get the robustformulation of the mean-variance problem under the assumption on µ i , which wedenote with (MvBU). max w (cid:62) ˆ µ − δ (cid:62) | w | − λ w (cid:62) Σw s . t . N (cid:88) i =1 w i = 1 (MvBU)As explained in [13], if the weight of asset i in the portfolio is negative, the worst-case expected return for asset i is µ i + δ i (we lose the largest amount possible).If the weight of asset i in the portfolio is positive, then the worst-case expectedreturn for asset i is µ i − δ i (we gain the smallest amount possible). In this robustversion of the mean-variance formulation, assets whose mean return estimatesare less accurate (have a larger estimation error δ i ) are penalized in the objectivefunction and would prospectively lead to having smaller weights in the optimalportfolio allocation. Even though more general uncertainty sets lead to more complicated optimizationproblems, the intuition behind the uncertainty remains the same. We introduce analternative form of the uncertainty which is incorporated in the expected returnsvector µ , the ellipsoidal uncertainty . U δ ( ˆ µ ) = (cid:110) µ | ( µ − ˆ µ ) (cid:62) Σ − µ ( µ − ˆ µ ) ≤ δ (cid:111) where ˆ µ is the vector of mean estimated returns, µ is the vector of mean true re-turns from all the stocks considered respectively and Σ µ represents the covariancematrix of the errors in the estimation of the expected (average) returns.The adoption of this specific uncertainty set envisages the idea that the investorwould like to be protected in instances in which the total scaled deviation of therealized average returns from the estimated returns is within δ . This uncertaintyet cannot be interpreted as individual confidence levels around each point esti-mate. However, its representation resembles a joint confidence region used, forexample, in Wald tests [12]. For the emulated model we approximate δ with theinverse χ distribution. Similarly to the first specification of uncertainty (box),here again, we want to detect the ”worst” estimates of the expected returns andhow would this impact the allocation of the portfolio. Mathematically, this can beexpressed as max w min µ ∈ (cid:110) µ | ( µ − ˆ µ ) (cid:62) Σ − µ ( µ − ˆ µ ) ≤ δ (cid:111) w (cid:62) ˆ µ − λ w (cid:62) Σw s . t . N (cid:88) i =1 w i = 1 (MvEU)We denote this problem as (MvEU), which stands for the Mean Variance withEllipsoidal Uncertainty formulation and is not in a form that can be input into astandard optimization solver. We need to solve the ”inner” problem first whileholding the vector of weight w fixed and compute the worst expected portfolioreturn over the set of possible values for µ .The robust problem that occurs after some algebra manipulations is the fol-lowing: max w w (cid:62) µ − λ w (cid:62) Σw − δ (cid:113) w (cid:62) Σ µ w s . t . N (cid:88) i =1 w i = 1 (3.9)Just as in the previous problem, we interpret the term δ (cid:112) w (cid:62) Σ µ w as the penaltyfor estimation risk, where δ reflects the degree of the investor’s aversion to esti-mation risk. It is not immediately obvious how one can estimate Σ µ . Accordingto Fabozzi, Kolm, Pachamanova and Focardi [12] critics of this approach haveargued that the realized returns typically have large stochastic components thatbelittle the expected returns, and hence estimating Σ µ accurately from historicaldata is very hard, if not impossible [24]. Several approximate methods for esti-mating Σ µ have been found to work well in practice [33]. In our case, assumingthat returns in a given sample of size T come from a normal distribution, we con-ider Σ µ = (1 /T ) · Σ [12], where Σ is the covariance matrix of asset returns asdepicted in Section 3.1.1.At this phase, we provide the Cholesky Decomposition of the covariance ma-trix Σ µ for making the optimization problem above solver-friendly and thus con-verting it into a Second Order Cone Programming (SOCP) problem. We introducea new variable z to replace the quantity δ (cid:112) w (cid:62) Σ µ w and another one variable q .As occurs from this transformation of the problem at hand, Σ µ = C (cid:62) C . Af-ter these reformulations, the Cholesky Decomposition for this problem can beexpressed as max w (cid:62) µ − λ w (cid:62) Σw − δ z s . t . Cw − q = 0 , N (cid:88) i =1 q i − z ≤ , N (cid:88) i =1 w i = 1 , w ≥ , q , z ∈ R (3.10) The general setting we employ for this class of problems is the following convexparametric optimization problemefmin w ∈ W f ( w , u ) s.t. g ( w , u ) ≤ In the above formulation, it is assumed that W subsumes all certain constraints,whereas all uncertain constraints explicitly depending on u are handled by theinequality g ( w , u ) ≤ [15]. The operator efmin in a given non-empty and convexset M ⊂ R N is searching for efficient points at each order relation according tothe dimensions considered in the application at hand, as explained by Fliege andicente [14] .Equivalently, the multi-objective robust counterpart for the multi-objectiveproblem with uncertainties (as formulated above) is defined asefmin w ∈ W f RCU ( w ) s.t. g RCU ( w ) ≤ with f RCU ( w ) := max u ∈ U f ( w , u ) ... max u ∈ U f k ( w , u ) g RCU ( w ) := max u ∈ U g ( w , u ) ... max u ∈ U g m ( w , u ) i.e. each component of the objective function and the constraints is replaced by itsrobust counterpart. We realize that for k = 1 the definition of the multiobjectiverobust counterpart coincides with the definition of the usual robust counterpartand hence provides a proper generalization of this concept to multiobjective op-timization. As pinpointed by Fliege [15] the robustification of the k -dimensionalobjective is now the very same as the robustification of the m -dimensional con-straint.Here we do consider the formulation (3.3) from the perspective of uncertainty.There have been many studies focusing on the uncertainty for the expected returnsalone, others on covariance alone by specifying confidence levels with lower andupper bounds for individual elements. In this work, we will consider an uncer-tainty set, which will include expected returns and covariance in a unified way.No estimated parameter acts independently and the task is to examine their effi-ciency when these two estimated parameters deviate from their real value. Forsimplicity of the exposition that follows, we choose an ellipsoid around a nominalpoint ( ˆ µ , ˆ Σ ) of size ε . U ε ( ˆ µ , ˆ Σ ) = { ( µ , Σ ) ∈ R N × S N + : (cid:107) µ − ˆ µ (cid:107) + c (cid:107) Σ − ˆ Σ (cid:107) ≤ ε } (3.11)here S N + denotes the cones of positive semidefinite matrices.Our main motivation for the consideration of multiobjective problems under datauncertainty stems from mean-variance portfolio optimization. In this present work,what will be of our primary importance, is the reformulation of a multiobjectivemodel into its robust variant, which will incorporate the uncertainty as stated inequation (3.11). The model we will consider is adopted from Fliege and Werner[15].Taking into account the uncertainty in expected returns and in the covariancematrix, we use the previously introduced joint uncertainty set as shown in (3.11)and derive the following robust multiobjective mean-variance formulation: f = min w (cid:62) Σw + εc (cid:107) w (cid:107) f = min − µ (cid:62) w + ε (cid:107) w (cid:107) s . t . N (cid:88) i =1 w i = 1 , w ≥ (3.12)Based on the specific choice of the uncertainty set (3.11) the robustified versions(i.e. the robust counterparts) of s ( w ) = w (cid:62) Σw and − m ( w ) = − µ (cid:62) w can beanalytically obtained as : s RC ( w ) = max( µ , Σ ) ∈U ε (cid:18) ˆ µ , ˆ Σ (cid:19) w (cid:62) Σw = w (cid:62) (cid:16) ˆ Σ + εc I (cid:17) w = w (cid:62) ˆ Σw + εc (cid:107) w (cid:107) − m RC ( w ) = max( µ , Σ ) ∈U ε (cid:18) ˆ µ , ˆ Σ (cid:19) − µ (cid:62) w = − ˆ µ (cid:62) w + ε (cid:107) w (cid:107) (3.13) hebyshev Scalar Transformation We employ the Chebyshev scalarizing function proposed by Steuer and Choo [32]in order to render (3.12) in a structure that could be efficiently solved by a standardoptimization software. For the robust counterpart problem in (3.13) we have twoobjective functions that we aim to maximize. So, respectively η , η ≥ are givenweights such that η + η = 1 . The outline of the scalarization technique appliedin (3.13) is the following : min α s . t . α ≥ η (cid:110) w (cid:62) (cid:16) Σ + εc I (cid:17) w − f ∗ (cid:111) ,α ≥ η (cid:110) f ∗ − ( µ (cid:62) w − ε (cid:107) w (cid:107) ) (cid:111) , N (cid:88) i =1 w i = 1 , w , α ≥ (3.14)where f ∗ and f ∗ are the optimal values of the problems below: f ∗ = min w (cid:62) (cid:16) Σ + εc I (cid:17) w s . t . N (cid:88) i =1 w i = 1 , w ≥ f ∗ = max µ (cid:62) w − ε (cid:107) w (cid:107) s . t . N (cid:88) i =1 w i = 1 , w ≥ (3.15)In model (3.14) η and η represent the the weights of the two objective functions,accordingly. This is a convex non-linear problem and can’t be fed to the softwarein this form. By adding a pseudo-parameter ω , we treat this problem into a moreomprehensive mathematical form, which we denote as RMu : min α s . t . (cid:110) w (cid:62) (cid:16) Σ + εc I (cid:17) w − η α ≤ f ∗ (cid:111) , (cid:110) µ (cid:62) w + 1 η α − δω = f ∗ (cid:111) , (cid:107) w (cid:107) − ω ≤ , N (cid:88) i =1 w i = 1 , w , α, ω ≥ (RMu) As explained in Section 3.1.3 the optimization of Omega ratio requires exactknowledge of the probability distribution of asset returns y . Since partial knowl-edge of estimation errors can lead to overoptimistic solutions, we introduce theworst-case Omega ratio.The worst-case Omega ratio (WO) for a fixed w ∈ W with the assumption ofthe discrete analog of a set of probability distributions is defined as [20]WO ( w ) ≡ inf π ∈ Π w (cid:62) ( Y (cid:62) π ) − τ π (cid:62) [ τ − ( Y w )] + (3.16)where the density function is only known to belong to a set Π of distributions.We do consider the mixture distribution uncertainty, where it is known thatthe underlying distribution is a mixture distribution with known continuous mix-ture components but unknown mixture weights. We employ the efficient frontierapproach. Mixture distribution is defined as a convex combination of probabilitydensity functions, known as mixture components. The weights associated withthe mixture components are called mixture weights. First of all, we assume thatthe distribution of y is characterized by the mixture of a set of prespecified distri-butions with unknown mixture weights. So, ≡ (cid:110) λ = ( λ , . . . , λ l ) : l (cid:88) i =1 λ i = 1 , λ ≥ , i = 1 , . . . , l (cid:111) . Let the distribution of y being characterized by a mixture of a set of distributionswith unknown mixing parameters such that p ( y ) ∈ P = (cid:110) l (cid:88) i =1 λ i p i ( y ) : λ ∈ Λ (cid:111) where λ i is the unknown mixture weight of the probability distribution p i ( y ) .Employing a robust counterpart approach and using the efficient frontier method [21]the optimization program becomes: max w ∈ W , θ ∈ R θ s . t . γ ( w (cid:62) E p ( y ) i − τ ) − (1 − γ ) E p i ([ τ − w (cid:62) y ] + ) ≥ θ, ∀ i = 1 , . . . , l (3.17)In order to obtain the portfolio with the maximum worst-case Omega ratio, theabove problem needs to be solved for different values of γ . An algorithm for per-forming this task is presented below.Set γ = 0 , wcor = −∞ , w ∗ = 0 while γ ≤ do Solve (3.17) and get w candidate Set minOR = min { Omega ratio for each distribution } if minOR > wcor then wcor = minOR, w ∗ = w candidate end γ = γ + step endreturn w ∗ , wcor Algorithm 1:
Designation of maximum worst-case Omega ratio [20]Different uncertainty sets may lead to significantly different decisions. The trade-off between robustness and performance must be taken into account. Under mix-ture distribution uncertainty, the modeler has to determine the mixture compo-ents. In this current work, the results obtained are contingent on the analysis ofhistorical data using different subsets.We do present the worst-case Omega ratio application under mixture distribu-tion, which will be applied to real data as shown more explicitly in Chapter 4. Weare employing the discrete analog of (3.17), as described in Kapsos et al. [21], i.e. max θ s . t . γ ( w (cid:62) µ i − τ ) − (1 − γ ) 1 S i (cid:62) u i ≥ θ, ∀ i = 1 , . . . , l, u i ≥ τ − Y i w , ∀ i = 1 , . . . , l, u i ≥ , ∀ i = 1 , . . . , l, ≤ w ≤ , N (cid:88) i =1 w i = 1 (WCOR)where µ i is the vector with the expected returns for the i -th mixture component, S i is the number of samples from the i -th mixture component, u i an auxiliaryvariable introduced to linearize the max function in (WCOR) and Y i the S i × N matrix that contains the sample returns from the i -th distribution for the N assets. In this Section, we assume that the density function of the portfolio return p ( · ) isonly known to belong to a certain set P of distributions, i.e., p ( · ) ∈ P . Zhu andFukushima [37] define the worst-case CVaR (WCVaR) for fixed w ∈ W withrespect to P as: WCVaR β ( w ) = sup p ( · ) ∈P CVaR β ( w ) where the computation of CVaR was explained in closer detail in Section 3.1.4. ixture Distribution As explained previously for the Worst-case Robust Omega ratio in Section (3.2.4),here we do assume that the density function of y is only known to belong to a set ofdistributions which consists of all the mixture distributions of some predeterminedlikelihood distributions, i.e., p ( · ) ∈ P ∆ = (cid:110) l (cid:88) i =1 λ i p i ( · ) : l (cid:88) i =1 λ i = 1 , λ i ≥ , i = 1 , . . . , l (cid:111) (3.18)where p i ( · ) signifies the i -th distribution scenario, and i denotes the number ofpossible scenarios.With respect to the uncertainty set we have defined, we realize that F iβ ( w , α ) = α + 11 − β (cid:90) y ∈ R m [ f ( w , y ) − α ] + p i ( y ) d y . i = 1 , . . . , l (3.19)We reformulate the original problem to a more tractable one. It can be seen thatthe WCVaR minimization is equivalent to min w , α, θ θ s . t . α + 11 − β (cid:90) y ∈ R m [ f ( w , y ) − α ] + p i ( y ) d y ≤ θ, i = 1 , . . . , l (3.20)An approximation method can be used to tackle the difficulty of the computa-tion of the integral of a multivariate and nonsmooth function in (3.20). Zhu etal. [37] mention that Monte Carlo simulation is one of the most effective methodsfor high-dimensional integral calculation. Rockafellar and Urysaev [31] use thismethod to approximate F β ( w , α ) as ˜ F β ( w , α ) = α + 1 S (1 − β ) S (cid:88) k =1 [ f ( w , y [ k ] ) − α ] + (3.21)here y [ k ] is the k -th sample generated by simple random sampling with respectto y according to its density function p ( · ) , and S denotes the number of samples.Replacing the integral in (3.20) with (3.21) yields min w , α, θ θ s . t . α + 1 S i (1 − β ) S i (cid:88) k =1 [ f ( w , y i [ k ] ) − α ] + ≤ θ, i = 1 , . . . , l (3.22)where y ik denotes the k -th sample with respect to the i -th distribution scenario p i ( · ) and S i denotes the number of corresponding samples. The approximation ofproblem (3.20) could as well be formulated as min w , α, θ θ s . t . α + 11 − β S i (cid:88) k =1 π ik [ f ( w , y i [ k ] ) − α ] + ≤ θ, i = 1 , . . . , l (3.23)where π ik denotes the probability according to the k -th sample with respect to the i -th likelihood distribution p ( · ) i . If π ik is equal to S i for all k , then (3.23) reducesto (3.22). We denote π i = ( π i , . . . , π iS i ) (cid:62) .By introducing some new variables the optimization problem (3.23) can be rewrit-ten as the following minimization problem with variables ( w , u , α, θ ) ∈ R N × R m × R × R . min θ s . t . α + 1(1 − β ) ( π i ) (cid:62) u i ≤ θ, i = 1 , . . . , l, u ik ≥ f ( w , y i [ k ] ) − α, k = 1 , . . . , S i , i = 1 . . . , l, u ik ≥ , k = 1 , . . . , S i , i = 1 . . . , l, w ∈ W (3.24)s defined in Section 3.1.3 the random vector y = ( y , y , . . . , y N ) (cid:62) ∈ R N repre-sents the uncertain returns of the N risky assets. Adjusting the above formulationto incorporate some additional elements, we consider the loss function to be de-fined as f ( w , y ) = − w (cid:62) y Portfolio optimization tries to locate an optimal trade-off between the risk and thereturn according to the investor’s preference, whereas the robust portfolio selec-tion is performed through the worst-case analysis of risk and return [37]. Thus,the robust portfolio selection problem using WCVaR as a risk measure can berepresented as min w ∈ W WCVaR ( w ) We complete the formulation of robust portfolio selection model, by specifyingthe constraint set W . We suppose that the investor has an initial wealth w . Thusthe portfolio selection satisfies e (cid:62) w = w In the case of mixture distribution uncertainty given by (3.18) let ˜ y i denote theexpected value of y with respect to the likelihood distribution p ( · ) . In terms ofthe worst-case minimum expected return φ required by the investor the followingcondition must hold w (cid:62) ˜ y i ≥ φ, i = 1 , . . . , l (3.25)The robust portfolio selection problem, under the mixture distribution probabil-ity, is formulated as the following linear program with variables ( w , u , α, θ ) ∈ R N × R m × R × R . in θ s . t . α + 1(1 − β ) ( π i ) (cid:62) u i ≤ θ, i = 1 , . . . , l, u ik ≥ − w (cid:62) y ik − α, k = 1 , . . . , S i , i = 1 . . . , l, u ik ≥ , k = 1 , . . . , S i , i = 1 . . . , l, w (cid:62) ˜ y i ≥ φ, i = 1 , . . . , l, e (cid:62) w = w , w ∈ W (WCVaR)2 Chapter 4Experiments and Results
Having provided a solid explanation of the models we have selected to incorpo-rate into the simulation analysis, the next step would be to evaluate their in-sampleand out-of-sample performance, across one empirical dataset of daily returns, us-ing certain performance criteria. To assess the magnitude of the potential gainsthat can actually be realized by an investor, it is necessary to analyze the out-of-sample performance of the strategies from the optimizing models. Afterwards,the performance of the non-robust portfolio models would be contradicted withtheir robust variants. The non-robust models considered along with their robustvariants are shown in Table 4.1.
We perform simulations based on historical data publicly available from Yahoo fi-nance, acquired from time series for the index S & P 500 spanning the period fromJanuary 1, 2005, to December 31, 2016. S & P 500 is the most common equity in-dex and is often used as a benchmark for the developed equities [20]. This datasetcomprises approximately 500 stocks from the New York market. For this analysis,we use 20 portfolios to optimize. This intriguing period would be a challengingtest-bed for the framework considered, since it incorporates the year, where theglobal financial crisis took place in 2008 [30], leading to a highly unpredictablefactor in terms of the performance of the models considered.
Inspired by the procedure proposed by Gilli and Schumann [17],we conduct rolling-window backtests with a historical window of length H, and an out-of-sampleable 4.1: List of asset-allocation models
Model Classification Abbreviation
Non-Robust Models
Mean Variance (Mv)Omega ratio (OR)CVaR (CVaR)
Robust Models
Robust Mean Variance (MvBU)Robust Ellipsoidal (MvEU)Robust Multi-objective (RMu)Worst-case Omega (WCOR)Worst-case CVaR (WCVaR) holding period of length F. We set H to business days, F to business days.Thus we optimize at point in time t on data from t − H to t − , the resultingportfolio is held until t = t + F . At this point, a new optimal portfolio is com-puted, using data from t − H until t − , and the existing portfolio is rebalanced.This new portfolio is then held until t = t + F , and so on. This is illustrated inFigure 4.1 for the first two periods. With our dataset, we have exactly 44 invest-ment periods. From each run, we estimate the parameters needed to implementa particular strategy. These estimated parameters are then used to determine theportfolio weights for the assets. We optimize the first time in January 2005 ( t ) ,the last date being 31 December 2016 ( t ) . The outcome of this rolling-windowapproach is a series of t n − H monthly out-of-sample returns, where t n denotesthe length of the rolling window approach.• Estimation period: One-year rolling window• Test period: One quarter rolling windowTherefore, from the historical window in every period, the simulation incor-porates four quarters of each year (in-sample) and one quarter (out-of-sample),where the quarters do not necessarily originate from the beginning of each year;each run could start from the middle of each year, but the spanning period willlways be one year. Based on the outputs acquired from the in-sample period, thesimulation uses these data to evaluate the efficiency of the portfolios created in theout-of-sample period. Our goal is to study the performance of the aforementionedmodels based on the data acquired from the international portfolio market of NewYork. period 1t - H t H t + F Fperiod 2 t rebalance t + Ft - H Figure 4.1: Illustration of rolling windows optimizationAt this point, we make a clarification about the procedure adopted exclusivelyfor the Worst-case Omega ratio (WCOR) and Worst-case CVaR (WCVaR). For thenominal Omega ratio (OR) and the nominal CVaR (CVaR), the one year empiricaldistribution is used as input, whereas for the Worst-case Omega ratio (WCOR)and Worst-case CVaR (WCVaR) four additional empirical distributions are pro-vided by partitioning the one year period into four quarters of a year sub-periods.Note that for l = 1 the corresponding formulations in Sections 3.2.4 and 3.2.5are equivalent to their non-robust variants. In terms of (RMu), the parameter ε de-picted in (3.11), was computed as the 95 % percentile of the bootstrap procedureby performing 1000 statistical computations, assuming that the respective samplerequired for a specific period (approximately 60 days), is a subset of the historicalreturns of the preceding year. All in all, we consider the same amount of historicaldata, as pointed out earlier on in this Section. .2 Portfolio Performance Indicators Taking into account the time series of daily out-of-sample returns generated byeach of the optimized models for the dataset S & P 500, we compute the followingquantities.In order to assess the performance of the above simulation framework, foreach of the optimization models used, we are going to use some metrics. First ofall, we compute the mean return of model k , which is defined as the product of theaverage returns of the stocks in the portfolios ¯ r k with the weights of the portfoliosobtained from the simulation w k . Hence, ˆ µ k = ¯ r k w k (4.1)In addition, we calculate the standard deviation of portfolio returns of model k ,which is defined as: ˆ σ k = (cid:113) w (cid:62) k Σ k w k (4.2)where Σ k denotes the covariance matrix of stock returns.Additionally, we compute the Sharpe ratio of model k , which stands for thesample mean of excess returns (over the risk-free rate) ˆ µ k divided by their samplestandard deviation, ˆ σ k : SR k = ˆ µ k ˆ σ k (4.3)We also include the Sortino ratio of model k , which is defined as the ratio ofmean returns of model k with the standard deviation of negative returns for thismodel. Thus, SoR k = ˆ µ k S [ max (0 , − Rw k )] (4.4)where R signifies the return data and S [] denotes the calculation of the standarddeviation of model k .Another instance we want to capture is the Omega ratio of model k . For thisetric, we employ the quantity (3.5) in Section 3.1.3, fed with the weights whichwe acquire from the simulation horizon of model k .In a similar manner to Omega ratio, we compute the Conditional Value at Risk (CVaR)for each model k . For this metric we employ the quantity (CVaR) in Section 3.1.4,adjusted with the respective weights we obtain from the simulation data for eachmodel k .Those computations are being performed for both the in-sample data and theout-of-sample data of the simulation procedure explained earlier on. Except for the above risk-return performance measures, the composition of theportfolios is also analyzed. The characteristics of a portfolio’s composition relateto management issues, such as the monitoring and rebalancing of the portfolio,as well as its management (transaction) costs. To this end, first, we consider thenumber of assets in the portfolios constructed by each model.To get a sense of the amount of trading required to implement each portfoliostrategy, we compute the portfolio turnover, defined as the sum of the absolutevalue of the trades across the N variable assets [9]. Hence,Turnover = N (cid:88) j =1 ( | ˆ w k,j,t +1 − ˆ w k,j,t | ) (4.5)in which ˆ w k,j,t is the portfolio weight of asset j at time t under each respectiveoptimized model k . The turnover quantity defined above can be interpreted as theaverage percentage of wealth traded in each period. Turnover refers to the sum ofabsolute differences in portfolio weights compared to the previous quarter.An additional feature we want to capture is the average diversification indexfrom all the periods for every single run. As explained by Kim et al. [23] one ofthe shortcomings of the mean-variance model is its tendency to put much weighton only a few assets. Having this in mind, if robust portfolios consist of more as-sets, the higher correlation with fundamental factors that we observe could be dueto diversification. Blume and Freund [36] introduced a portfolio diversificationeasure: the deviation of a portfolio from the market portfolio. Since the weightof each security in the market portfolio would be very small, they proceeded to anapproximation scheme with the sum of the squares of the proportions invested ineach stock.Diversification = N (cid:88) j =1 ( w k,j,t − w m ) = N (cid:88) j =1 (cid:18) w k,j,t − N m (cid:19) ≈ N (cid:88) j =1 w k,j,t (4.6)where N is the number of stocks in the portfolio N m is the number of stocks in themarket portfolio, and w m is the weight given to a security in the market portfolio.In terms of the diversification index, we can see in Table 4.2 that among thethree extensions of the mean-variance framework in the robust sense, compris-ing (MvBU), (MvEU) and (RMu), only (MvBU) performs worse than the (Mv).Moreover, (OR) and (CVaR) attain a lower level of diversification compared totheir robust variants (WCOR), (WCVaR) respectively. It is rather difficult to con-clude that robust portfolios are more diversified than mean-variance portfolios.This makes the task of determining, whether the robust models are systematicallysuperior to the non-robust models used in portfolio optimization more challeng-ing. Table 4.2: Descriptive statistics for the composition of the portfolios (Mv) (OR) (CVaR) (MvBU) (MvEU) (RMu) (WCOR) (WCVaR) Assets in portfolio . . . . . . . . Diversification index . . . . . . . . Turnover . . . . . . . . Based on Table 4.2 , we detect that for the three robust models (MvBU),(MvEU)and (RMu) there is a significant increase between the number of stocks in theportfolios (which is not desirable), compared to the non-robust model (Mv). Thisisn’t the case, however with the remaining two robust models (WCOR),(WCVaR).More specifically, the portfolios developed with the (WCOR) model have a slightlylower number of assets compared to the portfolios developed with (OR). On theother hand, for this metric explicitly, we note that (CVaR) performs slightly betterompared to the robust counterpart (WCVaR).The turnover ratio is illustrated in Figure 4.2 for the non-robust models alongwith their robust counterparts across the simulation horizon. Comparing the port-folio turnover for the different models, we see that the turnover for the robustvariant of the sample-based mean-variance portfolio, equipped with the box un-certainty (MvBU) is greater than the rest of the models. A general finding suggeststhat with the exception of (MvBU), each other robust model achieves systemati-cally a lower turnover than its non-robust counterpart. Moreover, it is interestingto note a peak value for the robust mean-variance model somewhere around the th period of the simulation, which coincides with the year 2008, when the col-lapse of the U.S. housing market triggered the financial crisis, leading to dramaticplunge of major stock markets [30]. This is an indicator, that the simulation pro-cedure can accurately replicate the incident of the financial crisis carried out in2008. Figure 4.2: Turnover ratio for the considered strategies .4 Portfolio Performance Results In this Section, we report the results of the computational experiments with the ro-bust portfolio selection framework proposed in the Thesis. The objective of thesecomputational experiments was to contrast the performance of the non-robustportfolio selection strategies with that of the robust portfolio selection strategies.The purpose of these experiments was to focus on the benefit accrued from ro-bustness; All the computations were performed in MATLAB (R2016b) using theGurobi solver.There are certain features we want to capture through this experimental proce-dure. Opting for a more concrete interpretation of the results obtained from all thecorresponding simulation periods, the results shown hereafter were averaged overeach run for all efficient portfolios derived through each model (20 portfolios foreach model). Furthermore, we give the in-sample and out-of-sample performanceof all the strategies considered in the simulation in an average form spanning allthe horizon. To assess the magnitude of the potential gains that can actually berealized by an investor, it is necessary to analyze the out-of-sample performanceof the strategies from the optimization models [9].
Taking a look at the mean return accumulated from each model, we can realizethat for the non-robust models, there is a difference between the values obtainedfor the in-sample data and for the out-of-sample data.Overall, the (OR) model does perform in a superior manner compared to (Mv)and (CVaR) for each performance indicator we have incorporated in Section 4.2,in terms of the out-of-sample data (except the mean return, where these modelsperform equally). Regarding the Conditional Value at Risk metric, three differentconfidence levels are employed. We start with a 90% confidence level in thefirst case and then we augment it to 95% in the second case reaching 99% inthe third case. We notice, that even though the (CVaR) model should possessa higher value for each one of the different confidence levels imposed, for theConditional Value at Risk metric, this model does attain nevertheless the worstalue among (Mv) and (OR) for a 99% confidence level, in terms of the out-of-sample data, that being 0.0302 compared to 0.247 and 0.0285, for (OR) and(Mv), respectively. Furthermore, we can detect that the results we get for thestandard deviation and for the Conditional Value at Risk (along with 3 confidencelevels) are quite consistent with respect to the in-sample data compared to theout-of-sample data, for these non-robust models, with almost indistinguishablediscrepancies. The detailed results for the non-robust models are shown in Table4.3. Table 4.3: Performance metrics for the non-robust models (Mv) (OR) (CVaR)
Mean return (in-sample) . . . Mean return (out-of-sample) . . . Standard deviation (in-sample) . . . Standard deviation (out-of-sample) . . . Sharpe ratio (in-sample) . . . Sharpe ratio (out-of-sample) . . . Sortino ratio (in-sample) . . . Sortino ratio (out-of-sample) . . . Omega ratio (in-sample) . . . Omega ratio (out-of-sample) . . .
90% CVaR (in-sample) (%) . . .
90% CVaR (out-of-sample) (%) . . .
95% CVaR (in-sample) (%) . . .
95% CVaR (out-of-sample) (%) . . .
99% CVaR (in-sample) (%) . . .
99% CVaR (out-of-sample) (%) . . . In terms of the robust variants for the computation of mean-return, all modelsperform better in-sample rather than out-of-sample. Among the in-sample Sharperatios for the robust models at hand, (MvBU) attains the highest value (0.4027) forthe in-sample data, implying that this model can perform in a satisfying mannerwithin the bounds of the uncertainty (”box”) imposed. We see, that the Condi-tional Value at Risk metric we get explicitly from the optimization of (WCVaR)doesn’t attain the best value among the robust models for none of the confidenceevels considered, as far as the in-sample data are concerned. An instance, whichcomes as a surprise is, that for the out-of-sample data, (WCVaR) model does ac-quire the second worst value in terms of Conditional Value at Risk metric amongthe robust models for each of the confidence levels mentioned. This behaviourcould be attributed to the fact, that the value of (WCVaR) is actually the averageof the 20 portfolios considered, where each one corresponds to a certain risk andreturn, depending on its location in the efficient frontier curve. Indeed, one port-folio does minimize the risk, but since we are interested in the average form of therisk, it is possible to get an inferior value for the Conditional Value at Risk metricfor model (WCVaR) in comparison with the other robust models. Additionally,another factor which could cause this behaviour lies in the fact of the step im-posed for each robust model, to construct the efficient frontier. With the exceptionof (WCOR), which represents a single portfolio, every other robust model used analternative step to designate the efficient frontier. In a respective manner, (WCOR)does acquire the worst value as far as the Omega ratio metric is concerned for theout-of-sample data, among the rest of the robust models. The fact, in this case, isthat as was mentioned before, (WCOR) poses a sole portfolio, so no average formwas taken and nevertheless, it attained a lesser value (1.2351) in comparison withthe other robust models, which used an average form to account for the Omegaratio metric.Another finding which is suggested by the results shown in Table 4.4 , is that(RMu) performs in quite close proximity, not only for the in-sample data but alsofor the out-of-sample data, as well. The in-sample values for (MvBU) with respect to the Sortino ratio and Omega ratio, were notleft intentionally blank. This model had an extremely low standard deviation of negative returns(nearly 0) and so the computation in (4.4) was infeasible. For the Omega ratio, the daily returnsfor the in-sample data were always positive and the computation of (3.5) measures the ratio ofgains to losses. Considering that there are no losses for (MvBU) we couldn’t get a value for thismetric either. able 4.4: Performance metrics for the robust models (MvBU) (MvEU) (RMu) (WCOR) (WCVaR)
Mean return (in-sample) . . . . . Mean return (out-of-sample) . . . . . Standard deviation (in-sample) . . . . . Standard deviation (out-of-sample) . . . . . Sharpe ratio (in-sample) . . . . . Sharpe ratio (out-of-sample) . . . . . Sortino ratio (in-sample) − . . . . Sortino ratio (out-of-sample) . . . . . Omega ratio (in-sample) − . . . . Omega ratio (out-of-sample) . . . . .
90% CVaR (in-sample) (%) .
083 0 . . . .
90% CVaR (out-of-sample) (%) . . . . .
95% CVaR (in-sample) (%) . . . . .
95% CVaR (out-of-sample) (%) . . . . .
99% CVaR (in-sample) (%) . . . . .
99% CVaR (out-of-sample) (%) . . . . . We notice that for all the models (robust and non-robust) with the exceptionof (RMu), there is a smaller standard deviation for the in-sample data, rather thanfor the out-of-sample data.From the results presented in Tables 4.3 and 4.4, it is evident that all modelsperform considerably better for the in-sample tests compared to the out-of-sampleones, in terms of the Sharpe ratio. Although this behaviour verifies the well-knownweaknesses of using classical sample-based estimates of the moments of asset re-turns to implement Markowitz’s mean-variance portfolios, the difference betweenthe in-sample Sharpe ratio (0.0772) and out-of-sample Sharpe ratio (0.0704) for(RMu) is only marginal. This could be attributed to the joint uncertainty set es-tablished to construct the robust multi-objective model (RMu), rendering it moreresilient to uncertainty shortcomings.Additionally, non-robust models perform much better in terms of the Sortinoratio for the in-sample data in contrast with the out-of-sample data. This applies,as well to the robust models.We shift our attention to the Omega ratio. As was expected for the in-sampleata (WCOR) attains the highest value among the robust models (1.9592), aswell as (OR) does among the non-robust models (2.082). Moreover, it is evidentthat the non-robust model (OR) performs slightly better than the robust model(WCOR) in terms of in-sample data. On the contrary, taking a closer look in theout-of-sample data, it is not a trivial task to pinpoint why (WCOR) performs inthe worst style, attaining a value (1.2351) among the rest of the robust models,considering that the Omega ratio we should get for (WCOR) stems from the opti-mization process of (WCOR) and should theoretically give us the highest value.As reported in Table 4.3 the values for the out-of-sample Conditional Valueat Risk metric show a steady rise as the confidence increases, not only for thenon-robust models but also for the robust models as well.Aiming to present a more concrete realization of the behaviour of the robustmodels in comparison with their non-robust variants, we consider the followingTable. Here, we state in which metrics, do the non-robust models acquire bettervalues than their robust equivalents. We realize, that robust optimization tech-niques would not always yield better performance. (RO) tries to designate the beststrategy, using historical data, however, the prediction cannot always be accurateand errors might arise through this procedure.Table 4.5: Efficiency of Robust Extensions of the models
Mv OR CVaRMvBU - MvEU - RMu
Standard deviationSortino ratio90% CVaR95% CVaR99% CVaR
WCOR
Mean returnSharpe ratioSortino90,95,99% CVaR
WCVaR -eeping in mind these explanations, we can rationally assume, that the ma-jority of the results we obtain from the robust optimization framework are overallsuperior from those we get from classical optimization techniques, without thismeaning that robust optimization is the only efficient way of handling problemsof this architecture. In some instances, as the ones examined in the Thesis, thereare metrics for which non-robust models perform better than robust models forthe out-of-sample data.
During the process of the evaluation procedure, certain assumptions were made,regarding the range and the architecture of the uncertainty sets employed to reach atractable solution. In this Section, we present the results regarding the uncertaintysets employed for each model and tests and whether the out-of-sample data arein accordance with the uncertainty sets formulated based on historical data (in-sample).Model (MvBU) assumes that the unknown future ( out-of-sample ) mean return µ fi of stock i will be such that | µ fi − ˆ µ i | ≤ δ i , where δ i was introduced previouslyin Section 3.2.1 and ˆ µ i is the mean return of the stock according to the in-sample data. Our aim hereafter is to investigate whether the true mean return of the stockscalculated from the out-of-sample data does indeed satisfy this assumption. Ouraim is to measure the frequency, where this condition is verified, within the con-sidered bounds.Model (MvEU) assumes that ( µ − ˆ µ ) T Σ − µ ( µ − ˆ µ ) ≤ δ , where δ was in-troduced as well in Section 3.2.2 with ˆ µ being the vector of the mean stock re-turns from in-sample data and µ the vector of out-of-sample mean returns of thestocks and Σ µ defined using the in-sample data. As in the previous model, we testwhether this assumption holds or not for each of the tests performed during theexamined time period.Model (RMu) assumes that (cid:107) µ − ˆ µ (cid:107) + c (cid:107) Σ − ˆ Σ (cid:107) ≤ ε where parameter ε sig-nifies the boundary under which lie the 950 sorted random values of the bootstrapprocedure with respect to the distribution followed by (cid:107) µ − ˆ µ (cid:107) + c (cid:107) Σ − ˆ Σ (cid:107) forthe in-sample data, mentioned in Section 4.1.1 and setting c = 1 as explained inection 3.2.3.Model (WCOR) assumes that with multiple estimates for the omega ratio { Ω , Ω , Ω , Ω } , the best portfolio is the one that maximizes the worst of theomegas. As was formulated in the Thesis, Worst-case Omega ratio was solvedwith 4 mixtures , each corresponding to 4 quarters prior to the current quarterT, to obtain an optimal robust portfolio. We calculate the corresponding omegasof the optimal portfolio by Ω , . . . , Ω for each of the past 4 quarters and alsocompute the omega ratio Ω RT of the optimal robust portfolio for the out-of-samplequarter T, according to (3.5). Moreover, we solve non-robust (OR) and calculateits Omega ratio Ω T for the out-of-sample quarter T. Then,• The robustification can be considered as “fully successful” if Ω RT ≥ min { Ω , Ω , Ω , Ω } and Ω T < min { Ω , Ω , Ω , Ω } • The robustification can be considered as “partially successful” if Ω RT > Ω T • The robustification can be considered as “totally unsuccessful” if Ω RT < min { Ω , Ω , Ω , Ω } and Ω T ≥ min { Ω , Ω , Ω , Ω } • The robustification can be considered as “partially unsuccessful” if Ω RT < Ω T A similar approach is employed to compare the (WCVaR) model to its nominal(non-robust) counterpart (CVaR). In particular, let CVaR , . . . , CVaR denote thelast years quarterly CVaRs used to derive the worst-case CVaR portfolio with thethe (WCVaR) model. The out-of-sample CVaR for the corresponding portfoliois CVaR RT , whereas the out-of-sample CVaR of the portfolio derived from thenominal (CVaR) model is denoted by CVaR T . Then:• The robustification can be considered as “fully successful” if CVaR RT ≤ max { CVaR , CVaR , CVaR , CVaR } andCVaR T > max { CVaR , CVaR , CVaR , CVaR } • The robustification can be considered as “partially successful” if CVaR RT < CVaR T able 4.6: Performance Checks (MvBU) (MvEU) (RMu) (WCOR) (WCVaR) Simulation Outputs . . . − . − . • The robustification can be considered as “totally unsuccessful” if CVaR RT > max { CVaR , CVaR , CVaR , CVaR } andCVaR T ≤ max { CVaR , CVaR , CVaR , CVaR } • The robustification can be considered as “partially unsuccessful” if CVaR RT > CVaR T For the two latter evaluation checks; (WCOR) and (WCVaR), we make an ad-justment, which will account for the level of impact we assign to each of thestatements. So, the adjustment lies within the concept of the weighted average,where we give a weight equal to 1 if the statement for each single run of the sim-ulation is “totally successful”, 0.5 if the statement is “partially successful”, -0.5 ifthe statement is “partially unsuccessful” and -1 if the statement is “totally unsuc-cessful”. Considering that each of the statements is denoted as C ( i ) , with i = 1 representing the first statement and i = 4 , the last statement, we can express math-ematically the reward function for each one of the 44 periods of the simulation,each period representing j . Hence,Gain j = 1 × C (1) + 0 . × C (2) − × C (3) − . × C (4) (4.7)All in all, having estimated how frequent a specific condition holds, namely for(MvBU),(MvEU),(RMu) and the level of impact that (WCOR) and (WCVaR)pose according to (4.7), we subsequently take the average of these quantities fromall the assets participating in the portfolios at each run, for each of the 44 periodsconsidered. We present the outcomes in Table 4.6.The next step of the process is to interpret the effectiveness of these validationchecks, by counting the number of time periods for which each condition holds,either in terms of the bounded uncertainty for models (MvBU),(MvEU),(RMu) oreither in terms of the gain function employed for models (WCOR) and (WCVaR).iewing Table 4.6, we realize that (MvBU) verifies the uncertainty condition im-posed in Section 3.2.1 for nearly the 96 % of the simulation periods, an elementwhich implies that this model does have an exceptional behaviour out-of-sample.As far as (MvEU) model is concerned, we realize that it performs within thegiven bounds for almost the 66 % of the simulation runs.(RMu) performs within the given threshold ε for nearly the 71 % of the simu-lation periods.Models (WCOR) and (WCVaR) attain nearly the same score, which is a neg-ative one. This implies, that there is an inclination towards the satisfaction of theterm “partial unsuccessfulness”, meaning that these models didn’t perform in thedesired manner for the out-of-sample data.8 Chapter 5Conclusions
In summary, although robust models decrease the sensitivity in parameter estima-tion errors, it is not a trivial task to measure how successfully the proposed modelsachieve their goals under practical settings. The verdict from this comparison be-tween robust and non-robust models is, that there seems to be an ameliorationin the results we get, without that being universal. Robust optimization modelscannot always cope with the uncertainty in a convincing manner, that being theirmajor limitation. However, as depicted from the results shown above, they doindeed present a satisfying performance in terms of the different uncertainty ar-chitectures imposed on the robust models. In general, the out-of-sample resultswith respect to the robust models are superior to those we get for the non-robustmodels. (MvEU) seems to perform in a superior manner out-of-sample for nearlyall the metrics considered, in terms of the robust models. (RMu) presents the mostconsistent behaviour among the robust models, since the values attained from themetrics considered, for the in-sample and the out-of-sample data, present infinites-imal deviations. Some models perform better than others judging by different met-rics, but the whole picture is that the results are promising and pose the need forthe investigation of more advanced techniques in the field of robust optimization.At a subsequent stage of the evaluation, the validation of the uncertainty sets wasexamined in Section 4.5, to check whether the robust models do indeed performduring the simulation runs, within their respective bounds. Based on the resultsacquired, we can deduce that (MvBU) presents an excellent performance, con-sidering that it verifies the box uncertainty mentioned in Section 3.2.1 for nearlythe 96 % of the simulation periods. Models (MvEU) and (RMu) perform in asatisfying manner, verifying their specific uncertainty assumptions for the 66 % and 69 % of the simulation periods accordingly. We considered these specificmodels, so as to capture the effects of robust optimization framework under dif-erent mathematical representations of the models, each one tackling a differentobjective. A next step of this procedure, could be to incorporate even more flexiblemodels of data-driven uncertainty and not just of predetermined architecture, as inthe proposed methodology perceived in the Thesis. Another instance that could beexamined, is the impact of robust optimization for portfolio selection on industrieswith different investing policies.0 References [1] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk.
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