Robust Methods In Portfolio Theory

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MASTER THESISLucia PetrusovaRobust Methods in Portfolio TheoryDepartment of Probability and Mathematical StatisticsSupervisor of the master thesis:Study programme:Study branch:RNDr. Martin Branda, Ph.D.MathematicsProbability, Mathematical Statisticsand EconometricsPrague 2016

I declare that I carried out this master thesis independently, and only with thecited sources, literature and other professional sources.I understand that my work relates to the rights and obligations under the ActNo. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that theCharles University has the right to conclude a license agreement on the use ofthis work as a school work pursuant to Section 60 subsection 1 of the CopyrightAct.In . date .signature of the authori

Title: Robust Methods in Portfolio TheoryAuthor: Lucia PetrusovaDepartment: Department of Probability and Mathematical StatisticsSupervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathematical StatisticsAbstract: This thesis is concerned with the robust methods in portfolio theory.Different risk measures used in portfolio management are introduced and the corresponding robust portfolio optimization problems are formulated. The analyticalsolutions of the robust portfolio optimization problem with the lower partial moments (LPM), value-at-risk (VaR) or conditional value-at-risk (CVaR), as a riskmeasure, are presented. The application of the worst-case conditional value-atrisk (WCVaR) to robust portfolio management is proposed. This thesis considersWCVaR in the situation where only partial information on the underlying probability distribution is available. The minimization of WCVaR under mixturedistribution uncertainty, box uncertainty, and ellipsoidal uncertainty are investigated. Several numerical examples based on real market data are presented toillustrate the proposed approaches and advantage of the robust formulation overthe corresponding nominal approach.Keywords: robust methods, portfolio selection, risk measures, conditional valueat-risk.ii

I would like to thank my supervisor RNDr. Martin Branda, Ph.D. for his ideasand guidance throughout the preparation of this thesis. I am also grateful to myfamily and friends for their patience and support.iii

ContentsIntroduction21 Risk Measures1.1 Lower Partial Moments . . . . . . . . . . . . .1.1.1 Upper Bounds for the Univariate Cases1.2 Value-at-risk . . . . . . . . . . . . . . . . . . .1.3 Conditional Value-at-risk . . . . . . . . . . . .1.4 Worst-case VaR and Worst-case CVaR . . . .2 Minimization of Risk Measures2.1 Minimization of CVaR . . . . . . . . . . . . . . . . . .2.2 Minimization of Worst-case CVaR . . . . . . . . . . . .2.2.1 Mixture Distribution . . . . . . . . . . . . . . .2.2.2 Discrete Distribution . . . . . . . . . . . . . . .2.2.3 Box Uncertainty in Discrete Distribution . . . .2.2.4 Ellipsoidal Uncertainty in Discrete Distribution.455788.101011121616183 Robust Portfolio Selection3.1 Portfolio Selection with CVaR . . . . . . . . . . . . . . .3.2 Robust Portfolio Selection Using LPM . . . . . . . . . .3.2.1 Explicit Solution of the Robust Portfolio Problem3.3 Robust Portfolio Selection Using WCVaR . . . . . . . . .3.3.1 Mixture Distribution Uncertainty . . . . . . . . .3.3.2 Box Uncertainty in Discrete Distribution . . . . .3.3.3 Ellipsoidal Uncertainty in Discrete Distribution .20212122252626274 Numerical Application4.1 CVaR Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2 WCVaR under Box Uncertainty . . . . . . . . . . . . . . . . . . .4.3 WCVaR under Mixture Distribution . . . . . . . . . . . . . . . .29293234Conclusion38Bibliography39List of Figures40List of Tables41Attachments421

IntroductionOn the financial markets, investors constantly face to a trade-off between adjusting potential returns for higher risk. Recently, there is a number of ways thatrisk can be defined and measured.In 1952, Harry Markowitz introduced modern porfolio theory [13], or meanvariance analysis. Due to symmetrical nature of the variance, which is the reasonwhy the variance does not differentiate the gain from the loss, even Markowitzhimself later proposed using the semivariance instead. To be better convenient fordifferent risk profiles of the investors, Bawa [2] and Fishburn [8] introduced a classof downside risk measures known as the lower partial moments, or LPM. Value-atrisk, or VaR, is a popular measure of risk in financial risk management. However,VaR has been critized in recent years in several aspects. VaR is not subadditivein general distribution case and thus it is not a coherent risk measure in the senseof Artzner [1]. A very serious shortcoming of VaR is that it is just a percentileof a loss distribution, so it does not show the nature of extreme losses exceedingit. These troubles motivated the search for a better measure of risk than VaR forpractical applications. Conditional value-at-risk, or CVaR, roughly defined as themean of the tail distribution exceeding VaR, is a measure of risk with significantadvantages over VaR. It is able to quantify dangers beyond VaR and it is coherent.Rockafellar and Uryasev [15] introduced a fundamental minimization formula forCVaR and showed that CVaR can be calculated by minimizing a more tractableauxiliary function without predetermining the corresponding VaR. Moreover, VaRcan be calculated as a by-product. The CVaR minimization formula usuallyresults in convex programs, and even linear programs. Therefore, CVaR attractedmuch attention in recent years and is applied to financial optimization and riskmanagement.As Black and Litterman [3] noticed, in the classical mean variance model, theportfolio selection is very sensitive to the mean and the covariance matrix. Theyshowed that even a small change in the mean can produce a large change in theportfolio position. Thus, the associated risk grows due to the uncertainty of theunderlying probability distribution. The relevant keywords in this context arerobustness and robust portfolio selection.Chen, He and Zhang [6] pointed that the assumptions on the distributionare arguably always subjective. Therefore, estimation on the moments of assetreturns using the historical data may be considered more objective measurement.Using the knowledge of the mean and the covariance, they introduced (see [6])analytical solution of the robust portfolio selection based on LPM, VaR or CVaR,as a risk measure.Lobo and Boyd [11], Costa and Paiva [5], Goldfarb and Iyengar [9], and Lu[12] studied the robust portfolio in the mean variance framework. Instead of theprecise information on the mean and the covariance matrix of asset returns, theyintroduced some types of uncertainties, such as polytopic, box and ellipsoidaluncertainty.This thesis is outlined as follows: In the first chapter, we introduce risk measures that are often applied to robust portfolio management. In the second chapter, we formulate the corresponding minimization problems for the proposed mea-2

sures of risk and make further investigation on some special cases of underlyingprobability distribution. We are particularly interested in the problem of minimizing the worst-case CVaR, or WCVaR, associated with mixture distributionuncertainty, box uncertainty, and ellipsoidal uncertainty in the distributions. Inthe third chapter, we present the application of WCVaR, introduced by Zhu andFukushima [17], to robust portfolio optimization. In the last chapter, we discuss the results of numerical applications on portfolio selection performed via themethods proposed in this thesis. Finally, we conclude the results and outline thefuture directions.3

1. Risk MeasuresMeasures of risk have an important role in optimization under uncertainty, especially in finance and insurance industry. In this chapter, we introduce the mostpopular risk measures used in risk management, and discuss their fundamentalproperties.In the return-risk trade-off analysis, the risk is explicitly quantified by a riskmeasure that maps the loss to a real number. In general, loss can be expressed asa function Z f (x, y) of a decision vector x X Rn representing portfolio,where X expresses decision constraints, and a random vector y Y Rmrepresenting the future values, e.g., interest rates, random rates of return. Wheny has a known probability distribution, a random variable Z has its distributiondependent on the choice of x. Therefore, if we want to choose x within terms ofany optimization problem, then we should take into account not just expectations,but also the “riskness” of x.There is a number of ways that risk can be defined. The important questionis, how a suitable risk measure looks like, and what a risk measure might or notmight have. In 1999, Artzner et al. [1] presented their essential work on coherentrisk measure. They presented the following set of consistency rules for a riskmeasure ρ mapping a random loss Z to a real number:(i) Subadditivity: For all random losses Z and Y , ρ(Z Y ) ρ(Z) ρ(Y );(ii) Positive homogenity: For positive constant λ, ρ(λZ) λρ(X);(iii) Monotonicity: If Z Y a.s. for each outcome, then ρ(Z) ρ(Y );(iv) Translation invariance: For any constant c, ρ(Z c) ρ(X) c.A risk measure that satisfies the above axioms is called a coherent risk measure.Let random loss Z be defined on some probability space (Ω, F , P ). In thesituation that a probability measure P is ambiguous and characterized as a certainset P, then we generally define the worst-case risk measure ρW related to ρ asfollows:ρW (Z) sup ρ(Z).(1.1)P PProposition 1 ([17]). If ρ associated with crisp probability measure P is a coherent risk measure, then the corresponding ρW associated with ambiguous probabilitymeasure P remains a coherent risk measure.Coherent risk measure, in the sense introduced by Artzner [1], is supposed tobe a “good” measure of risk because it has four desirable properties. In this thesis,different risk measures are discussed and they are not necessarily coherent. We areparticularly interested in CVaR, that is a well known coherent risk measure, andthe worst-case CVaR. By Proposition 1, the worst-case CVaR is also a coherentrisk measure.Throughout the thesis, we also present the results on a class of downsiderisk measures known as the lower partial moments, introduced by Bawa [2] andFishburn [8].4

1.1Lower Partial MomentsThe lower partial moments are defined as followsLPMm (r) E [(r X)m ],(1.2)where (t) max{t, 0}, X is the asset return (X Z), r is the return ona benchmark index, and m is a parameter, that can take any nonnegative value.Specifically, if m 0, then LPM0 is clearly the probability of the asset return fallingbelow the benchmark index; if m 1, then LPM1 is the expected shortfall of the investment, fallingbelow the benchmark index; if m 2, then LPM2 is almost an analog of the semivariance, but thedeviation references to the benchmark instead of the mean.As we can see, LPM is specified by r and m. The return r is often set to therisk-free rate, or simply to zero. By choosing the degree m an investor can specifythe measure of his/her risk attitude. Intuitively, large values of m will penalizelarge deviations more than low values.We denoteP {p E p [ξ] µ, Covp [ξ] Γ 0},where P is the set of probability distributions with mean µ Rn and covariancen, which is positive semidefinite. We denote R (µ, Γ) to reprematrix Γ S sent the fact that the random vector R belongs to the set whose elements havemean µ and covariance matrix Γ.1.1.1Upper Bounds for the Univariate CasesIn this section, we discuss the moment upper bounds, using the information aboutthe mean and the covariance of the underlying distribution. These bounds leadto robust portfolio optimization models, as we will see later.As we remarked, LPM0 (r) measures the probability that a random return fallsbelow the target r. Its upper bound, which is presented in the following lemma,is exactly set by Chebyshev-Cantelli inequality [4].Lemma 2 ([6]). It holds thatsup P{X r}sup LPM0 (r) X (µ,σ 2 )X (µ,σ 2 )( 1,1 (r µ)2 /σ 2if r µ,1,if r µ.LPM1 (r) is the expected shortfall of X below the benchmark r. As we willsee later, this measure of risk is highly related to CVaR. In this case, Jensen’sinequality is used to derive its upper bound.5

Lemma 3 ([6]). It holds thatsup LPM1 (r) X (µ,σ 2 )sup E [(r X) ] X (µ,σ 2 )r µ pσ 2 (r µ)2.2A tight upper bound on LPM2 (r) can be also established by Jensen’s inequality.Lemma 4 ([6]). It holds thatsup LPM2 (r) X (µ,σ 2 )sup E [(r X)2 ] [(r µ) ]2 σ 2 .X (µ,σ 2 )Proof. Firstly, it holdsσ 2 var X E X 2 (E X)2 E X 2 µ2 ,and thusE X 2 µ2 σ 2 .Further, for any X (µ, σ 2 ), by Jensen’s inequality we haveE [(r X)2 ] E [(r X)2 ] E [(r X)2 ]E [(r X)2 ] E [(r X) ]2E [r2 2rX X 2 ] ([E r E X] )2r2 2rX E X 2 [(r µ) ]2r2 2rX (µ2 σ 2 ) [(r µ) ]2(r2 2rX µ2 ) σ 2 [(r µ) ]2(r µ)2 σ 2 [(r µ) ]2σ 2 [(r µ) ]2 .To show the tightness of the bound, consider a sequence of distributions(σµ n 1,with probability n 1,nXn µ σ n 1, with probability 1/n.We determine mean and variance of XnE Xn n 1σ1µ µ σ n 1nnn 1 1 1nµ µ σ n 1 µ σ n 1nn 1nµ µ σ n 1 µ σ n 1n1nµnµ6

var Xn E Xn2 E (Xn )2 2 2n 1σ1 µ µ σ n 1 µ2 nnn 1 2 2µσσ1 2n 12µ µ 2µσ n 1 (n 1)σ 2 nnn 1 n 12 µ 1 2 1 (n 1)µ2 2µσ n 1 σ 2 µ 2µσ n 1 σ 2 n σ 2nn µ2 1(nµ2 µ2 2µσ n 1 σ 2 µ2 2µσ n 1 σ 2 n σ 2 µ2 n 1 nµ2 nσ 2 µ2n σ2(1.3)It means that Xn (µ, σ 2 ), andE [(r Xn )2 ] σ 2 [(r µ) ]2 , as n .This indicates that the upper bound is indeed tight, which completes the proof.Remark. If m 2, thensup E [(r X)m ] , see [6].X (µ,σ 2 )1.2Value-at-riskValue-at-risk is one of the most popular risk measures. However, it is unstable andnumerical application is difficult when losses does not follow normal distribution.Beyond the treshold amount indicated by this measure, there is no handle on theextent of the losses that might occure. It is incapable of distinguishing betweensituations when losses are only a little bit worse and those which are enormous.Moreover, VaR is not a coherent risk measure in the sense of Artzner [1]. Despiteof all shortcomings, VaR is frequently used in risk management.In everything that follows, we suppose that random vector y is governed bya probability measure P on Y (a Borel measure) that is independent of x. Foreach x, we denote by Ψ(x, ·) on R the resulting distribution function for the lossf (x, y), i.e.,Ψ(x, ζ) P{y f (x, y) ζ}.(1.4)The previous function (1.4) actually represents the probability that f (x, y) doesnot exceed a treshold ζ.Given a confidence level α (0, 1) (usually greater than 0.9), the value-at-riskis defined as follows.Definition 1. The VaRα of the loss associated with a decision x is the valueζα (x) min{ζ ψ(x, ζ) α}.7(1.5)

Definition 2. The VaR α (“upper” VaRα ) of the loss associated with a decisionx is the valueζα (x) min{ζ ψ(x, ζ) α}.(1.6)Remark. The minimum in (1.5) is achieved because ψ(x, ζ) is nondecreasing andright-continuous in ζ. In the situation when ψ(x, ζ) is strictly increasing andcontinuous, ζα (x) is equal to the unique ζ satisfying ψ(x, ζ) α. In all othercases, this equation can have no solution or a whole range of solutions.1.3Conditional Value-at-riskA coherent risk measure that quantifies the losses that might occure in the tailis conditional value-at-risk. This risk assessment technique is derived by takinga weighted average between the value-at-risk and losses exceeding the value-atrisk. Therefore, the term is also known as “Mean Excess Loss” or “Tail VaR”.We make a technical assumption that f (x, y) is continuous in x and measurable in y. We also assume that E ( f (x, y) ) for each x X .Definition 3. The CVaRα of the loss associated with a decision x is the valueφα (x) mean of the α tail distribution of f (x, y),where the distribution in question is the one with distribution function ψα (x, ·)defined by(0,for ζ ζα (x)ψα (x, ζ) (1.7)[ψ(x, ζ) α]/[1 α], for ζ ζα (x).Definition 4. The CVaR α (“upper” CVaRα ) of the loss associated with a decisionx is the valueψα E {f (x, y) f (x, y) ζα (x)},(1.8)whereas the CVaR α (“lower” CVaRα ) of the loss associated with a decision x isthe valueψα E {f (x, y) f (x, y) ζα (x)}.(1.9)We note that the conditional expectation in (1.9) is well defined becauseP{f (x, y) f (x, y) ζα (x)} 1 α 0, but (1.8) only makes sense as longas P{f (x, y) f (x, y ζα (x)} 0, which is not assured merely through theassumption that α (0, 1). For more details see [16].1.4Worst-case VaR and Worst-case CVaRInstead of assuming the precise knowledge of the distribution of the random vectory, we assume that the density function is only known to belong to a certain setP of distributions, i.e., p(·) P.According to general definition of the worst-case risk measure (1.1), we definethe worst-case VaR and the worst-case CVaR as follows.Definition 5. The WVaRα of the loss associated with a decision x is the valueWVaRα (x) sup VaRα (x).p(·) P8

Definition 6. The WCVaRα of the loss associated with a decision x is the valueWCVaRα (x) sup CVaRα (x).p(·) P9

2. Minimization of RiskMeasuresIn this chapter, we discuss the minimization problem of CVaR and WCVaR.For WCVaR we make further investigation on some special cases of P, formulate the corresponding minimization problems, that can be efficiently solved, anddiscuss their computational aspects. We are particularly interested in mixturedistribution, box uncertainty in discrete distribution and ellipsoidal uncertaintyin discrete distribution, which are the most often used uncertainty structures inrobust optimization.2.1Minimization of CVaRSuppose that y follows a continuous distribution. As Rockafellar and Uryasevdemonstrate [15], the calculation of CVaR can be achieved by minimizing of thefollowing auxiliary function with respect to the variable ζ R:Z1[f (x, y) ζ] p(y)dy,(2.1)Fα (x, ζ) ζ 1 α y Rmwhere [t] max{t, 0}, and p(·) denotes a density function of y. Thus, we havethe fundamental minimimization formula specified in the following theorem. Formore details and the proof of this theorem see [15].Theorem 5 (Fundamental minimization formula [15]). As a function of ζ R,Fα (x, ζ) is finite and convex (hence continuous), withφα (x) min Fα (x, ζ)ζ Rand moreoverζα (x) lower endpoint of arg min Fα (x, ζ),ζ Rζα (x) upper endpoint of arg min Fα (x, ζ),ζ Rwhere the argmin refers to the set of ζ for which the minimum is attained andin this case has to be a nonempty, closed, bounded interval (perhaps reducing toa single point). In particular, one always hasζα (x) arg min Fα (x, ζ),φα (x) Fα (x, ζα (x)).ζ RAs noticed in [15], Theorem 5 shows the difference between CVaR and VaR,and present the fundamental reason why CVaR is much better behaved thanVaR when dependence on a choice of x must be handled. The reason is the fact,that the optimal value in a problem of minimization, in this case φα (x), is moreagreeable as a function of parameters than is the optimal solution set, which ishere the argmin interval with ζα (x) as its lower endpoint.10

In (2.1) we assume that y follows a continuous distribution. However, toconsider a discrete distribution make sense even for a continuous distribution inCVaR formulation, because we usually use summation to approximate the integralin (2.1). Let S denote the number of sample points. In the following, we assumethat the sample space of the random vector y is given by {y(1) , y(2) , . . . , y(S) },SPπk 1, πk 0, k 1, . . . , S. Further, denotewhere P{y(k) } πk andk 1π (π1 , π2 , . . . , πS )T and defineSX1Gα (x, ζ, π) ζ πk [f (x, y(k) ) ζ] .(1 α) k 1(2.2)For given x and π, Rockafellar and Uryasev [16] defined the corresponding CVaRasCVaRα (x, π) min Gα (x, ζ, π).(2.3)ζ RWe can formulate minimizing of CVaR as the following minimization programwith decision variables (x, u, ζ) Rn RS R:1(π)T u1 αs.t. x X ,uk f (x, y(k) ) ζ, k 1, . . . , S,uk 0, k 1, . . . , S,minζ (2.4)(2.5)(2.6)(2.7)where u (u1 , . . . , uS ) is an auxiliary vector utilized to deal with the computationof [·] in the original objective function.If the function f (x, y) is linear with respect to x and the set X is a convexpolyhedron, then the problem (2.4) - (2.7) can be solved by a linear programmingmethod.2.2Minimization of Worst-case CVaRIn this section, we present the results formulated by Zhu and Fukushima [17].First of all, we quote the following lemma (minimax theorem), which will be usedto formulate the minimization problems in a tractable way.Lemma 6 ([17]). Suppose that X and Y are nonempty compact convex sets inRn and Rm , respectively, and the function φ(x, y) is convex in x for any given y,and concave in y for any given x. Then, we havemin max φ(x, y) max min φ(x, y)x X y Yy Y x XOne can find the details and the proof of Lemma 6 for example in [7].11

2.2.1Mixture DistributionIn this section, we assume that about the distribution of y we only know it belongs to a set of distributions that consists of all mixtures of some predeterminedlikelihood distributions, i.e.,( l)lXXp(·) PM λi pi (·) :λi 1, λi 0, i 1, . . . , l ,(2.8)i 1i 1iwhere p (·) denotes the ith likelihood distribution, and l denotes the number ofthe likelihood distributions. Denote()lXΛ λ (λ1 , . . . , λl ) :λi 1, λi 0, i 1, . . . , l .(2.9)i 1DefineFαi (x, ζ)1 ζ 1 αZ[f (x, y) ζ] pi (y)dy, i 1, . . . , l.(2.10)y RmUsing Lemma 6, we get the following theorem.Theorem 7 ([17]). For each x, W CV aRα (x) with respect to PM can be computedasW CV aRα (x) min max Fαi (x, ζ),(2.11)ζ R i Lwhere L {1, 2, . . . , l}.Proof. For given x X , we define the following functionHα (x, ζ, λ) ζ lX11 α"Z[f (x, y) ζ] y RmlX#λi pi (y) dyi 1λi Fαi (x, ζ)(2.12)i 1where λ Λ, and the set Λ is specified as (2.9). The function Hα (x, ζ, λ) isconvex in ζ and concave in λ [16]. Moreover, min Hα (x, ζ, λ) is a continuousζ Rfunction with respect to λ. By the definition of WCVaRα (x), and the fact thatΛ is a compact set, we can writeWCVaRα (x) max min Hα (x, ζ, λ)λ Λ ζ R max minλ Λ ζ RlXλi Fαi (x, ζ)(2.13)i 1As Rockafellar and Uryasev proved [16], for fixed x and each i, the optimalsolution set of min Fαi (x, ζ) is a nonempty, closed, and bounded interval. Thus,ζ Rwe denote12

[α i , ᾱi ] arg min Fαi (x, ζ), i 1, . . . , l.ζ RSuppose that g1 (t) and g2 (t) are two convex functions defined on R. Letthe nonempty, closed, and bounded intervals [t 1 , t̄ 1 ] and [t 2 , t̄ 2 ] denote their setsof minima. It holds that for any β1 0 and β2 0 such that β1 β2 0,β1 g1 (t) β2 g2 (t) is also a convex function, and its set of minima must lie in thenonempty, closed, and bounded interval [min{t 1 , t 2 }, max{t̄ 1 , t̄ 2 }]. By this fact,we getarg min Hα (x, ζ, λ) A , λ Λ,ζ Rwhere A is the nonempty, closed, and bounded interval given byA [min α i , max ᾱi ],i Li Lwhich impliesmin Hα (x, ζ, λ) min Hα (x, ζ, λ).ζ Rζ AThus, by Lemma 6max min Hα (x, ζ, λ) max min Hα (x, ζ, λ)λ Λ ζ Rλ Λ ζ A min max Hα (x, ζ, λ).ζ A λ Λ(2.14)Obviously, it holdsmin max Hα (x, ζ, λ) inf max Hα (x, ζ, λ)ζ A λ Λζ R λ Λ(2.15)By (2.14), (2.15), and the well-known result on the min-max inequalityinf max Hα (x, ζ, λ) max min Hα (x, ζ, λ),ζ R λ Λλ Λ ζ Rwe getmax min Hα (x, ζ, λ) min max Hα (x, ζ, λ).λ Λ ζ Rζ R λ ΛTherefore, we can writeWCVaRα (x) min max Hα (x, ζ, λ)ζ R λ Λ min maxζ R λ ΛlXλi Fαi (x, ζ).(2.16)i 1Now we only need to verify that the righ-hand sides of (2.11) and (2.16)are equivalent. The right-hand side of (2.16) can be written as the followingoptimization problem()lXminθ:λi Fαi (x, ζ) θ, λ Λ .(2.17)(ζ,θ) R Ri 113

By the specification of the set Λ (2.9), it is clear that any feasible solution of(2.17) satisfiesFαi (x, ζ) θ, i 1, . . . , l.(2.18)On the other hand, if (2.18) holds, then for any λ Λ, we havelXλi Fαi (x, ζ) lXi 1λi θ θ.i 1Thus, we can see that the problem (2.17) is equivalent to minθ : Fαi (x, ζ) θ, i 1, . . . , l ,(ζ,θ) R Rwhich is in fact the righ-hand side of (2.11) written as an optimization problem.This completes the proof.DenoteFαL (x, ζ) max Fαi (x, ζ).i L(2.19)By Theorem 7, we get the following corollary.Corollary 8 ([17]). Minimizing W CV aRα (x) over X can be achieved by minimizing FαL (x, ζ) over X R, i.e.,min W CV aRα (x) x Xmin(x,ζ) X RFαL (x, ζ).(2.20)More specifically, if (x , ζ ) attains the right-hand side minimum, then x attains the left-hand side minimum, and ζ attains the minimum of FαL (x , ζ),and vice versa.As Rockafellar and Uryasev demonstrate [16], the function Fα (x, ζ) definedby 2.1 is convex in (x, ζ) if the function f (x, y) is convex in x. It holds thatthe function g(t) max{g1 (t), g2 (t)} is convex whenever both g1 (t) and g2 (t) areconvex. Thus, we can see that if f (x, y) is convex in x, then FαL (x, ζ) is convexin (x, ζ). Moreover, if f (x, y) is convex in x and X is a convex set, then theproblem of WCVaR minimization is a convex program.Using Theorem 7 and Corollary (8), we get that the WCVaR minimizationproblem is equivalent to min(x,ζ,θ) X R R1θ:ζ 1 αZ i [f (x, y) ζ] p (y)dy θ, i 1, . . . , l ,y Rm(2.21)which is more tractable problem in comparison with the original one. The calculation of the integral is probably the most difficult part in this computation.Monte Carlo simulation is one of the most popular and efficient approximationmethods used to deal with this complexity. Rockefellar and Uryasev [16] use thismethod to approximate F̃α (x, ζ) as follows14

SX1F̃α (x, ζ) ζ [f (x, y(k) ) ζ] ,S(1 α) k 1(2.22)where y(k) denotes the kth sample point that is generated by simple randomsampling according to density function p(·) of y, and S denotes the number ofsample points. When the number of sample points used in approximation is largeenough, then the approximation accuracy (or convergence) is guaranteed by thelaw of large numbers.Remark. We note that the function Gα (x, ζ, π) (2.2) is equal to the functionF̃α (x, ζ) (2.22), if πk is equal to 1/S.In the following, we replace the integral in (2.21) with the summation used in(2.22) Si X1[f (x, y(k) )i ζ] θ, i 1, . . . , l ,minθ:ζ i (x,ζ,θ) X R R S (1 α) k 1(2.23)denotes the kth sample point with respect to the ith likelihood distriwherebution p (·), and S i denotes the number of sample points.In general, the problem (2.21) can be formulated using the approximation asyi(k)i min(x,ζ,θ) X R R iθ:ζ 1(1 α)SXπki [f (x, y(k) )i ζ] θ, i 1, . . . , l , k 1(2.24)wheredenotes the probability with respect to the ith likelihood distributionpi (·) according to the kth sample point.Remark. Obviously, if πki is equal to 1/S i for all k, then (2.24) can be reducingto problem (2.23).This general form (2.24) of the optimization problem can be reformulated asthe following problem with decision variables (x, u, ζ, θ) Rn Rm R R:πkimin θs.t. x X ,1ζ (π i )T ui θ, i 1, . . . , l,1 αiuik f (x, y(k)) ζ, k 1, . . . , S i , i 1, . . . , l,uik 0, k 1, . . . , S i ,i 1, . . . , l.(2.25)(2.26)(2.27)(2.28)(2.29)where π i (π1i , . . . , πSi i ), and u (u1 ; u2 ; . . . ; ul ) Rm is an auxiliary vector,lPwhere m S i.i 1If the function f (x, y) is linear with respect to x and the set X is a convexpolyhedron, then the problem (2.25) - (2.29) is a linear program.Remark. Especially if l 1, then the problem (2.25) - (2.29) is exactly that ofRockafellar and Uryasev [16] with πki 1/S 1 .15

2.2.2Discrete DistributionIn this section, we assume that y follows a discrete distribution. We are particularly interested in minimizing of WCVaR under box uncertainty and ellipsoidaluncertainty. These two types of uncertainty sets are easy to be specified andthe optimization problem can be formulated in a tractable way. In the previouschapter, we defined WCVaRα (x) for the general distribution case. In the case ofdiscrete distribution, we denote P as Pπ , that we may identify as a subset ofRS . By the formula (2.3), WCVaR for fixed x X with respect to Pπ is thendefined asWCVaRα (x) sup CVaRα (x, π),π Pπand it is equivalent toWCVaRα (x) sup min Gα (x, ζ, π).π Pπ ζ RUsing Lemma 6, we get the following theorem. The proof of this theorem isgiven for example in [17].Theorem 9 ([17]). Suppose that Pπ is a compact convex set. Then, for each x,we haveW CV aRα (x) min max Gα (x, ζ, π).ζ R π PπBy Theorem 9, if Pπ is a compact convex set, the minimization problemof WCVaRα (x) over X can be also formulated as the following problem withdecision variables (x, u, ζ, θ) Rn RS R R:mins.t.θx(2.30)(2.31) X,1π T u θ,1 α f (x, y(k) ) ζ, k 1, . . . , S, 0, k 1, . . . , S.max ζ π Pπukuk(2.32)(2.33)(2.34)Problem (2.30) - (2.34) includes the max operation in the constraints andthus it is not suitable for numerical application. If f (x, y) is linear in x and Xis a convex polyhedron, then under box uncertainty in distribution, this problem can be formulated as a linear program, and under ellipsoidal uncertainty indistribution, as a second-order cone program [17].2.2.3Box Uncertainty in Discrete DistributionSuppose that π belongs to a box, i.e.,π PπB {π : π π 0 η, eT η 0, η η η̄},16(2.35)

where π 0 denotes nominal distribution (the most likely distribution), e denotesthe vector of ones, and η and η̄ are given constant vectors. We can see that theconstraint eT η 0 ensures π to be a probability distribution, and the nonnegativity constraint π 0 is included in th

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