基于扭曲风险度量的鲁棒投资策略.pdf
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1、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 Managemen
2、t,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 h
3、eavily 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 wit
4、hin 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 th
5、e 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 Stati
6、st,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-agemen
7、t 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(
8、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-offana
9、lysis 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 i
10、s 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
11、 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
12、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
13、 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
14、 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 measuresc
15、an 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 represent
16、s 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 under
17、lying 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
18、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,
19、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 w
20、ay 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 m
21、ethodsto 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
22、 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
23、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 distributionambiguit
24、y 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
25、 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
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- 基于 扭曲 风险 度量 投资 策略
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