基于扭曲风险度量的鲁棒投资策略.pdf

应用概率统计第 40 卷第 1 期2024 年 2 月Chinese Journal of Applied Probability and StatisticsFeb.,2024,Vol.40,No.1,pp.122-138doi:10.3969/j.issn.1001-4268.2024.01.008Worst-Case Distortion Risk Measure with Application toRobust Portfolio SelectionYAN Xuechen(Department of Statistics and Finance,School of Management,University of Science andTechnology of China,Hefei,230026,China)LI Lu(Postdoctoral Programme of Bank of Communications,Shanghai,200120,China)WANG Yashi(School of Information Management for Law,China University of Political Science and Law,Beijing,100027,China)Abstract:Portfolio selection depends heavily on the underlying distribution of loss.When thedistribution information of loss can only be observed through a limited sample of data,robustnessof the portfolio selection model is of crucial importance.Assuming that the underlying distributionof loss has a known mean and variance and lies within a ball centred on the reference distributionwith the Wasserstein distance as the radius,this paper proposes a robust portfolio strategy modelbased on the distortion risk measure and translates it into a simpler equivalent form.Furthermore,simulation and empirical study are used to demonstrate the validity of the model.Keywords:distortion risk measure;portfolio strategy;robust model;Wasserstein distance2020 Mathematics Subject Classification:91G10;60A10Citation:YAN X C,LI L,WANG Y S.Worst-case distortion risk measure with application torobust portfolio selectionJ.Chinese J Appl Probab Statist,2024,40(1):122138.1IntroductionThe core of modern portfolio theory includes balancing return and risk,determiningeffective portfolio strategy and allocating capital to multiple available assets.In general,the mean value is used to describe the return,and risk measures are used for risk man-agement to map the distribution of loss to the real line to quantify the risk.Markowitz1laid the foundation of modern portfolio theory,who proposed to use variance as a measureThe project was supported by Qian Duansheng Distinguished Scholar Support Program of ChinaUniversity of Political Science and Law(Grant No.DSJCXZ180403).Co-first author.Corresponding author,E-mail:.Received January 24,2022.Revised March 8,2022.No.1YAN X.C.,et al.:WDRM with Application to Robust Portfolio Selection123of risk,and introduced the mean-variance model,which is a representative approach inthe return-risk trade-offanalysis framework.After that,a new measure of downside risk:Value-at-Risk(VaR),has become popular in financial risk management(see,e.g.,2).The Basel Committee has recommended it as a standard for bank supervision.However,VaR has two obvious defects.One is that it lacks subadditivity and therefore it is nota coherent risk measure as described in 3.The other one is that VaR is a quantile ofthe loss distribution,so it could not capture the tail risk.Based on those observations,lots of researchers have proposed using Conditional Value-at-Risk(CVaR,also known asExpected Shortfall,ES),which is defined as the mean of the tail distribution beyond VaR(see,e.g.,4,5).As a measure of risk,CVaR has some desired properties as a risk measuresuch as subadditivity and convexity.It is well-known that both VaR and CVaR belong tothe class of distortion risk measures(see,e.g.,6,7).The class of distortion risk measuresstems from Yaaris8dual theory of choice under risk that in the case in which the decisionmaker is not risk neutral,the risk attitude can be captured by some form of expectationwith the probability P re-weighted,or distorted,according to some distortion functions.It has been widely applied in quantitative risk management of finance,insurance and etc.(see,e.g.,6,912).When the distortion function is concave,the corresponding risk mea-sure is coherent and is called the spectral risk measure described in13.Various distortionrisk measures,especially the spectral risk measure,have been widely used in portfolio s-election(see,e.g.,11,12).As CVaR is a typical example of spectral risk measure,thereis also a large amount of literature applying it as a risk metric to study portfolio strategy(see,e.g.,1416).The form of classical portfolio selection optimization problem based on risk measurescan be formulated as follows:minRn(TX)s.t.T1=1,EP(TX),where =(1,2,n)Tis the decision vector;X=(X1,X2,Xn)Tis a randomvector representing the loss of n assets.The subscript P represents that the expectationis calculated under the constraint that the underlying probability measure ofX is P,and represents the minimum expected return acceptable to the investor.In addition,is arisk measure.Classical model assumes that the underlying distribution of portfolio loss isknown,usually empirically.In this case,the portfolio strategy is based solely on historicaldata.However,due to the ambiguity of the underlying distribution,the lack of historicaldata and the presence of measurement error,in real financial markets we can only obtainpartial information about the underlying distribution of financial assets from the empirical124Chinese Journal of Applied Probability and StatisticsVol.40distribution.Risk measures are very sensitive to changes in the underlying distribution.Hence classical model can lead to adverse decisions.This has driven the development of robust portfolio strategy model(see,e.g.,17,18).One of the key points in building this model is the construction of distribution ambiguitysets,which typically uses the first and second order moments of the underlying distributionor utilises the Wasserstein distance(see,e.g.,19,20).The first method is relativelysimple to solve,but does not reflect the distance between two distributions.Wassersteindistance in the second method is a popular way for calculating the distance between twodistributions,and it is a major breakthrough in that it can calculate the distance betweendiscrete and continuous distributions(see,e.g.,21).However,the disadvantage of thismethod is that it is difficult to simplify.So in this article,we combine these two methodsto construct ambiguity sets.There is also a large amount of literature on robust portfolio management.Lobo andBoyd22provided a worst-case mean-variance analysis with respect to the second-ordermoment ambiguity.Based on Wasserstein distance,Blanchet et al.23studied robustmean-variance portfolio selection and simplified this problem to an empirical varianceminimization problem with an additional regularization term.However,using variance asa measure of risk is considered undesirable,because it will be punished equally regardless ofthe downside risk or upside potential.Li12made an analysis of the worst-case distortionrisk measure when the first two moments of the underlying distribution were known,andnaturally extended his results to the application of portfolio selection.On the basis of thiswork,Bernard et al.24added the restriction of Wasserstein distance to the distributionambiguity sets and figured out the upper and lower bounds of distortion risk measures.In this paper,we consider the problem of robust portfolio selection based on theworst-case distortion risk measure and translate it into a simpler equivalent form.Slightlydifferent from the traditional method,we regard the portfolio as a whole and get someenlightening results for high-dimensional problems.The analysis reveals that the robustmodel is more sensitive to changes in the expected return than the classical model,whichallows robust models to better cater for people with different risk preferences.We alsofind that the more convex the risk measure model is,that is,the more robust the modelitself is,the weight allocation changes more slowly with the ambiguity of distribution.Moreover,in the period of financial crisis,the ambiguity of distribution increases and therobust investment strategy outperforms other strategies significantly.The rest of the paper is organized as follows:Section 2 reviews the relevant definitions.Section 3 converts our robust optimization model into a simpler equivalent form.SectionNo.1YAN X.C.,et al.:WDRM with Application to Robust Portfolio Selection1254 and Section 5 verify the validity of the model through simulation and empirical research,respectively.The results are summarized in Section 6.2Minimization of Worst-Case Distortion with Wasserstein DistancesLet(,A,P)be the atomless probability space that we work on throughout the paper,and L2=L2(,A,P)the set of all square integrable random variables on(,A,P).Wedenote by M2be the corresponding space of all distribution functions of the randomvariables in L2.For a distribution F,the(left-continuous)inverse of F is F1(u)=infy R|F(y)u,u (0,1,and F1(0)=infy R|F(y)0.Let U U(0,1)represent a standard uniform random variable.2.1Distortion Risk MeasuresFor a given random variable X that represents loss,we define its cumulative distri-bution function FX.Then a distortion risk measure is defined via the Choquet integralg(X)=01 g1 F(x)dx+0g1 F(x)dx,whenever at least one of the two integrals is finite.The function g:0,1 0,1 iscalled the distortion function which is a non-decreasing function satisfying g(0)=0 andg(1)=1.When g is concave,the distortion risk measure is coherent(see,e.g.,3,25).Our paper focuses on the coherent distortion risk measure,that is,g is concave.When gis concave,one can rewrite the distorted expectation g(X)asg(X)=10F1(u)h(u)dx,(1)where h(u)=g(x)|x=1u,0 u 0 and10h(u)du=1,in addition,g denotes the left derivative of g(see,e.g.,26).In this case,we call ga spectralrisk measure.Since in this paper,we confine to spectral risk measures,we stick to theexpression(1)and denote it by hwhich is a little abuse of notation,that is,h(X)=10F1(u)h(u)dx.Assumption 1We assume thath 0,10h(u)du=1,and that10h(u)2du +.For a function h satisfying Assumption 1,we can verify that for any X F M2,h(X).To see it,note that by H olders inequality,we haveh(X)=10F1(u)h(u)dx 610F1(u)2dx10h(u)2dx1/2126Chinese Journal of Applied Probability and StatisticsVol.40=E(X2)10h(u)2dx1/2 0,and 0 is the feasible region of portfolio.3Main ResultsIn order to make our study more reasonable,we need to propose an additional con-straint.Assumption 3We assume thatM,P(,)contains at least two elements,thatis (P)2+(P)2,otherwise problem(3)is trivial.To solve the problem(3),we need to solve the inner maximization problemh(Y)=maxGM,P(,)Gh(Y).(4)According to the Theorem 1 of 24,we can get the following theorem.Theorem 4If we denote byc0=corr(P1(U),h(U).Then,the following state-ments hold:(i)If(P)2+(P)2 (P)2+(P)2+2P(1c0),thenthe solution to problem(4)is unique and the worst case quantile function ofGis givenbyh(u)=+h(u)+P1(u)ab,0 u 0denotes the unique solution todW(P1,h)=.In this case,the corresponding worst-case valueh(Y)=h(h)=+std(h(U)corr(h(U),h(U)+P1(U).(6)(ii)If(P)2+(P)2+2P(1c0)6,then the solution to problem(4)ish(Y)=h(h0)=+std(h(U).(7)From this theorem,we can naturally get the following two cases.(i)If(P)2+(P)2 (P)2+(P)2+2P(1c0),thenthe problem(3)is equivalent tominW+std(h(U)corr(h(U),h(U)+P1(U).(8)(ii)If(P)2+(P)2+2P(1c0)6,then the problem(3)is equivalenttominW+std(h(U).(9)Combining the above two cases,we conclude the result.In formula(8)and(9),is the unique positive solution to dW(P1,h)=.Inorder to make it more convenient to use computer for calculation,we can simplify itfurther.Proposition 5If(P)2+(P)2 (P)2+(P)2+2P(1 c0),we denote byk=1 (P)2(P)2/(2P),a=(1k2)2P,b=2(1k2)c0Pandc=c202k22,then=(b+b2 4ac)/(2a).ProofDenote by k=corr(P1(U),h(U).By(5),we havek=Cov(P1(U),h(U)+P1(U)P1(U)h(U)+P1(U)=Cov(P1(U),h(U)+2PP2+22P+2Cov(h(U),P1(U)=c0+P2+22P+2,c0P,No.1YAN X.C.,et al.:WDRM with Application to Robust Portfolio Selection129where c0=corr(P1(U),h(U).The above equation can be rewritten as(1 k2)2P2+2(1 k2)c0P+c202 k22=0.(10)On the other hand,noting that h M,P(,)and by computingdW(P1,h)2=EP1(U)h(U)2=2P+2P+2+2 2Cov(P1(U),h(U)+P=(P)2+(P)2+2P1 corr(P1(U),h(U),we havek=1 (P)2(P)22P.(11)By Theorem 4,when(P)2+(P)2 (P)2+(P)2+2P(1 c0),we have that is the positive solution to(10)where k is given by(11).That is,a2+b+c=0.Therefore,we have =(b+b2 4ac)/(2a),which completes the proof.?By Proposition 5,we get the explicit solution of.Substituting this solution into(8),we can obtain the optimisation expression associated with alone.Example 6When we considerCVaR,that is,the distortion function are defined by(2),then we can get that:(i)If(P)2+(P)2 (P)2+(P)2,namely 0,we use the robust optimization model given by(8)and(9).When =0,the M,P(P,P)is singleton,containing only one distribution P,then the modelbecomes classical optimal model.Assuming the initial asset value is 100,and using 1267 daily return vectors(02/01/2003 to 31/12/2007)as the initial window through which we obtain the first weight vectorunder different models.Then we use this weight vector for the next days investment andtake advantage of the return vector of this day to get a new asset value.We add the newdata to the initial window and remove an old data.By rolling forward,we can obtain thechanges in asset values for different investment strategies from 2008 to 2011.These resultsare displayed in Figures 5 and 6.2008200920102011Year30405060708090100110120130140Cumulative wealthWT(0.8,=1*10-5)RobustWT(0.8,=1*10-6)RobustWT(0.8)Classical1/n PortfolioMax ReturnFigure 5Cumulative asset value of various investment strategies under WT0.8distortion risk measureIn terms of the general trend in value changes,it is clear that there had been marketdeclines from 2008 to 2009 as a result of the so-called“sub-prime mortgage financial crisis”.After this period of time,the values of the portfolio gradually increases.As can be seenfrom both two figures,during the financial crisis,the robust investment strategy clearlyoutperformed the other strategies.This is intuitive because in the period of financialcrisis,the financial market is highly volatile,leading to an increase in the ambiguity of136Chinese Journal of Applied Probability and StatisticsVol.402008200920102011Year30405060708090100110120130140Cumulative wealthCVaR(0.9,=1*10-4)RobustCVaR(0.9,=1*10-5)RobustCVaR(0.9)Classical1/n PortfolioMax ReturnFigure 6Cumulative asset value of various investment strategies under CVaR0.9distortion risk measurethe underlying distribution of stock market.In such circumstance,classical investmentstrategy and max return investment strategy that rely exclusively on historical data donot take this variation into account and will lead to unfavourable decisions.The returnsachieved by the 1/n investment strategy are entirely determined by the market and aretherefore more influenced by general market trends.In addition,during the economic upward period,the returns obtained by the robustmodel are more stable compared to the other models.In some periods,the 1/n modelyields higher returns than the robust model,but the asset values obtained through the1/n model also fluctuate significantly,which is undesirable to risk-averse investors.6Concluding RemarksWe propose a robust portfolio selection model based on the distortion risk measureand transl