RM-CVaR: Regularized Multiple β -CVaR Portfolio
RRM-CVaR: Regularized Multiple β -CVaR Portfolio Kei Nakagawa , Shuhei Noma , Masaya Abe Innovation Lab, Nomura Asset Management Co.,Ltd. ,Japan { kei.nak.0315, shuhei.mi.st, masaya.abe.428 } @gmail.com Abstract
The problem of finding the optimal portfolio for in-vestors is called the portfolio optimization problemthat mainly uses the expectation and variability ofreturn (i.e., mean and variance). Although the vari-ance would be the most fundamental risk measureto be minimized, it has several drawbacks. Con-ditional Value-at-Risk (CVaR) is a relatively newrisk measure that addresses some of the shortcom-ings of the well-known variance-related risk mea-sures, and because of its computational efficiencieshas gained popularity. CVaR is defined as the ex-pected value of the loss that occurs beyond a cer-tain probability level ( β ). However, portfolio op-timization problems that use CVaR as a risk mea-sure are formulated with a single β and may out-put significantly different portfolios depending onhow the β is selected. We confirm even smallchanges in β can result in huge changes in thewhole portfolio structure. In order to improve thisproblem, we propose RM-CVaR: Regularized Mul-tiple β -CVaR Portfolio. We perform experimentson well-known benchmarks to evaluate the pro-posed portfolio. Compared with various portfolios,RM-CVaR demonstrates a superior performance ofhaving both higher risk-adjusted returns and lowermaximum drawdown. The problem of finding the optimal portfolio for investors iscalled the portfolio optimization problem that mainly usesthe expectation and variability of return (i.e., mean andvariance[Markowitz, 1952]). Although the variance wouldbe the most fundamental risk measure to be minimized, it hasseveral drawbacks. Controlling the variance does not onlylead to low deviation from the expected return on the down-side, but also on the upside. Hence, quantile based risk mea-sures have been suggested such as Value-at-Risk (VaR) thatmanage and control risk in terms of percentiles of the loss dis-tribution. Instead of regarding both upside and downside ofthe expected return, VaR considers only the downside of theexpected return as risk and represents the predicted maximumloss with a specified confidence level (e.g., 95%). VaR is incorporated into several regulatory requirements, like BaselAccord II, and hence plays a particularly important role in riskanalysis. However, VaR, if studied in the framework of coher-ent risk measures [Artzner et al. , 1999], lacks subadditivity,and therefore convexity, in the case of general loss distribu-tions (although it may be subadditive for special classes ofthem, e.g. for normal distributions). This drawback entailsboth inconsistencies with the well-accepted principle of di-versification (diversification reduces risk). For example, VaRof two different investment portfolios may be greater than thesum of the individual VaRs. Also, VaR is nonconvex and non-smooth and has multiple local minimum, while we seek theglobal minimum [McNeil et al. , 2005]. Besides, both vari-ance and VaR ignores the magnitude of extreme or rare lossesby their definition.Conditional Value-at-Risk (CVaR) that addresses theseshortcomings of the variance and VaR is a relatively new riskmeasure and has gained popularity. CVaR is defined as theexpected value of the loss that occurs beyond a certain proba-bility level ( β ). [Pflug, 2000] proved that CVaR is a coherentrisk measure having subadditivity and convexity. Addition-ally, [Rockafellar et al. , 2000] show that the minimization ofCVaR results in a tractable optimization problem. For exam-ple, when the loss is defined as the minus return and a finitenumber of historical observations of returns are used in es-timating CVaR, its minimization can be written as a linearprogram and solved efficiently.However, portfolio optimization problems that use CVaRas a risk measure are formulated with a single β and may out-put significantly different portfolios depending on how the β is selected. We evaluate how the portfolio changes as the β level changes with well-known benchmarks. This is similar tothe ”error maximization” that [Michaud, 1989] points out inthe case of the mean-variance portfolio. [Ardia et al. , 2017;Nakagawa et al. , 2018] empirically showed minimum vari-ance portfolio weights are highly sensitive to the inputs. Weconfirm even small changes in β can result in huge changesin the whole portfolio structure.On the other hand, many papers either ignore transac-tion costs or only subtract ad hoc transaction costs after-ward [Shen et al. , 2014]. When buying and selling assetson the markets, the investors incur in payment of commis-sions and other costs, globally defined transaction costs, thatare charged by the brokers or the financial institutions playing a r X i v : . [ q -f i n . P M ] M a y he role of intermediary. Transaction costs represent the mostimportant feature to account for when selecting a real portfo-lio, since they diminish net returns and reduce the amount ofcapital available for future investments [Mansini et al. , 2015].In order to improve these problems, in this paper, we pro-pose RM-CVaR: Regularized Multiple β -CVaR portfolio tobridge the gap between risk minimization and cost reduction.To control transaction cost, we impose L et al. , 2009; Shen et al. , 2014]. Weprove that the RM-CVaR Portfolio optimization problem iswritten as a linear programming problem like the single β -CVaR portfolio. We also perform experiments to evaluate theproposed portfolio. Compared with various portfolios, theRM-CVaR portfolio demonstrates a superior performance ofhaving both higher risk-adjusted returns and lower maximumdrawdown. Because of these regularization and sparsity-inducing prop-erties, there has been substantial recent interest in L et al. , 2009; Fan et al. , 2012; Shen et al. ,2014]. They show superior portfolio performances when var-ious types of norm regularities are combined. Analogously,[Gotoh and Takeda, 2011] consider L L et al. , 2014]. Both estimatemodels that would achieve good out-of-sample performance.[Shen et al. , 2015; Shen and Wang, 2016] propose toemploy the bandit learning framework to attack portfolioproblems. [Shen et al. , 2015] presented a bandit algo-rithm for conducting online portfolio choices by effectuallyexploiting correlations among multiple arms. [Shen andWang, 2016] proposed an online algorithm that leveragesThompson sampling into the sequential decision-making pro-cess for portfolio blending. Also, [Shen and Wang, 2017;Shen et al. , 2019] apply a subset resampling algorithm intothe mean-variance portfolio and the Kelly growth optimalportfolio to obtain promising results. Through resamplingsubsets of the original large datasets, [Shen and Wang, 2017;Shen et al. , 2019] constructed the associated subset portfolioswith more accurately estimated parameters without requiringadditional data. However, these studies do not take transac-tion costs or turnover into account.Interactions from portfolio optimization to machine learn-ing include [Gotoh and Takeda, 2005; Takeda and Sugiyama,2008]. [Gotoh and Takeda, 2005] first have pointed out thecommon mathematical structure employed both in the classof machine learning methods known as ν -support vector ma-chines ( ν -SVMs) and in the CVaR minimization. On the D e n s i t y VaR Loss
CVaR
Figure 1: Illustration of VaR and CVaR. other hand, [Takeda and Sugiyama, 2008] were the first topoint out that the model of [Gotoh and Takeda, 2005] isequivalent to the machine learning methods called E ν -SVC.Applying our method to machine learning algorithms is amajor future task. In this section, we define VaR and CVaR, and then formulatea portfolio optimization problem using them.Let r i be the return of stock i (1 ≤ i ≤ n ) and w i be theportfolio weight for stock i . Here, r i is a random variable andfollows the continuous probability density function p ( r ) .We denote r = ( r , ..., r n ) T and w = ( w , ..., w n ) T . Let L ( w, r ) be loss function e.g. L ( w, r ) = − w T r . The proba-bility that the loss function is less than α is Φ( w, α ) = (cid:90) L ( w,r ) ≤ α p ( r ) dr (1)When w is fixed, Φ( w, α ) is non-decreasing as a functionof α and is continuous from the right, but is generally notcontinuous from the left. For simplicity, assume Φ( w, α ) iscontinuous function with respect to α . Then, VaR and CVaRare defined as follows (Figure 1). Definition 3.1.
V aR ( w | β ) := α β ( w ) = min( α : Φ( w, α ) > β ) (2) Definition 3.2.
CV aR ( w | β ) := φ ( w | β ) (3) = (1 − β ) − (cid:90) L ( w,r ) ≥ α β ( w ) L ( w, r ) p ( r ) dr It is difficult to directly optimize the above CVaR becausethe integration interval depends on VaR. Therefore, to calcu-late φ β ( w ) , we also define F ( w, α | β ) as Definition 3.3. F ( w, α | β ) := α +(1 − β ) − (cid:90) R n [ L ( w, r ) − α ] + p ( r ) dr (4) where [ t ] + := max( t, . Then, the following relationship holds between φ ( w | β ) and F ( w, α | β ) . emma 3.1. For an arbitrarily fixed w , F ( w, α | β ) is convexand continuously differentiable as a function of α . φ ( w | β ) isgiven by minimizing F ( w, α | β ) with respect to α . min α F ( w, α | β ) = φ ( w | β ) (5) In this formula, the set consisting of the values of α for whichthe minimum is attained, namely A β = arg min α F ( w, α | β ) (6) is a nonempty closed bounded interval.Proof. Proof are given in [Rockafellar et al. , 2000].CVaR is defined by the value of VaR, but it is possibleto obtain the CVaR without obtaining VaR according to thisLemma. If X is a constraint that the portfolio must satisfy,the following Lemma holds for the formulation of a portfoliooptimization problem using CVaR as the risk measure. Lemma 3.2.
Minimizing the CVaR over all w ∈ X is equiv-alent to minimizing F ( w, α | β ) over all ( w, α ) ∈ X × R , inthe sense that min w ∈ X φ ( w | β ) = min ( w,α ) ∈ X × R min α F ( w, α | β ) . (7) Furthermore, if f ( w, r ) is convex with respect to w , then F ( w, α | β ) is convex with respect to ( w, α ) , and φ ( w | β ) isconvex with respect to w . If X is a convex set, the minimiza-tion problem of φ ( w | β ) on w ∈ X can be formulated as aconvex programming problem.Proof. Proof are given in [Rockafellar et al. , 2000].We approximate the function F ( w, α | β ) by sampling arandom variable r from the density function p ( r ) . When r [1] , r [2] , ..., r [ q ] are obtained by sampling or simple histori-cal data, the function F ( w, α | β ) is approximated as follows. F ( w, α | β ) = α + ( q (1 − β )) − q (cid:88) k =1 [ − w T r k − α ] + (8)Finally, we formulate the portfolio optimization problemwith CVaR as a linear programming problem as bellow. min w,α,u ,...,u q α + ( q (1 − β )) − q (cid:88) k =1 u k (9) s.t.u k ≥ − w T r [ k ] − α ( k = 1 , ..., q ) (10) u k ≥ k = 1 , ..., q ) (11) T w = 1 (12) w j ≥ j = 1 , ..., n ) (13)Here, T w = 1 indicates the sum of all the portfolioweights always equals one, and 1 (left side) denotes a columnvector with ones. Beside, w j ≥ indicates that investors takea long position of the j -th asset, β -CVaRPortfolio In this section, we propose a model that takes into accountthe multiple CVaR values. The formulation is to minimizethe margin between multiple β levels of CVaR.Let C β k be the value of CVaR obtained by solving Eq. (9)-(13). Then, minimizing C considering C β k is a main problemof this research. Problem 1. min ( w,C ) ∈ X × R C (14) s.t.φ ( w | β k ) ≤ C + C β k ( k = 1 , . . . , K ) (15)Let F ( w, α | β ) be the function likewise Lemma 3.1 φ ( w | β ) = min α F ( w, α | β ) (16)Using Eq. (16), Problem 1 can be written as follows. Problem 2. min ( w,C ) ∈ X × R C (17) s.t. min α k F ( w, α k | β k ) ≤ C + C β k ( k = 1 , . . . , K ) (18)Let α = ( α , · · · , α K ) T and we consider the followingProblem 3. Problem 3. min ( w,C,α ) ∈ X × R × R m C (19) s.t.F ( w, α k | β k ) ≤ C + C β k ( k = 1 , . . . , K ) (20)Here, the following Lemma holds between Problem 2 and3. Lemma 4.1. (1)If ( w ∗ , C ∗ ) is the optimal value for Eq.(2), ( w ∗ , C ∗ , α ∗ ) is the optimal value of Eq. (3). (2)If ( w ∗∗ , C ∗∗ , α ∗∗ ) is the optimal value for Eq. (3), ( w ∗∗ , C ∗∗ ) is the optimal value for Eq. (2).Proof. Assume that ( w ∗ , C ∗ ) is the optimal value for Prob-lem 2. Because ( w ∗ , C ∗ ) is a feasible solution of Prob-lem 2, min α i F ( w ∗ , α k | β k ) ≤ C ∗ + C β k holds. Define α ∗ = ( α , . . . , α K ) T as α ∗ k := argmin α k F ( w ∗ , α k | β k ) . Then, ( w ∗ , C ∗ , α ∗ ) is a feasible solution of Problem 3since F ( w ∗ , α ∗ k | β k ) ≤ C ∗ + C β k holds. If ( w ∗ , C ∗ , α ∗ ) is not the optimal solution of Problem 3, there exists afeasible solution ( ˆ w, ˆ C, ˆ α ) satisfying ˆ C < C ∗ . Then, min α k F ( ˆ w, ˆ α k | β k ) ≤ ˆ C + C β k ( k = 1 , ..., K ) holds. There-fore ( ˆ w, ˆ C ) is a feasible solution of Problem 2, which con-tradicts that C ∗ is the optimal solution of Problem 2. As-sume that ( w ∗∗ , C ∗∗ , α ∗∗ ) is the optimal value for Problem3. Then, because ( w ∗∗ , C ∗∗ , α ∗∗ ) is a feasible solution ofProblem 3, F ( w ∗∗ , α ∗∗ i | β i ) ≤ C ∗∗ + C β i ( i = 1 , ..., m ) holds. ( w ∗∗ , C ∗∗ ) is a feasible solution for Problem 2 since min α i F ( w ∗∗ , α i | β i ) ≥ F ( w ∗∗ , α ∗∗ i | β i ) ≤ C ∗∗ + C β i ( i = , ..., m ) holds. if ( w ∗∗ , C ∗∗ ) is not the optimal solutionof Problem 2, there exists a feasible solution ( ˆ w, ˆ C ) satis-fying ˆ C < C ∗∗ . Define ˆ α = (ˆ α , ..., ˆ α m ) T as ˆ α i :=arg min α i F β i ( ˆ w, α i ) . Then, ( ˆ w, ˆ C, ˆ α ) is a feasible solution ofProblem 3, which contradicts that C ∗∗ is the optimal solutionof Problem 3.According to Lemma 4.1, Problem 1 and 3 are a equivalentproblem. When r [1] , . . . , r [ Q ] are obtained by sampling, thefunction F ( w, α | β ) is approximated as follows. F ( w, α | β ) (cid:39) α + 1 Q (1 − β ) Q (cid:88) q =1 [ − w (cid:62) r [ q ] − α ] + (21)Finally, we derive the Regularized Multiple β -CVaR Port-folio, where the objective is to minimize multiple CVaR val-ues and control the portfolio turnover. The changes of theturnover during each rebalancing period are directly relatedto transaction costs, market impacts and taxes. Controlingthe portfolio turnover is realized through imposing the L -regularization term as (cid:107) w − w − (cid:107) = n (cid:88) i =1 | w i − w − i | (22)where w − i denotes the portfolio weight before rebalancing.From the above, the Regularized Multiple β -CVaR Portfo-lio optimization problem can be formulated as follows: Problem 4. min ( w,C,α ) ∈ X × R × R K C + λ (cid:107) w − w − (cid:107) (23) s . t . ˜ F ( w, α k | β k ) ≤ C + C β k ( k = 1 , . . . , K ) (24)We can easily proof Problem is linear programming prob-lem similar to the usual CVaR minimization problem. Theorem 4.1.
The Regularized Multiple β CVaR Portfoliooptimization problem is equivalent to the following linearprogramming problem. min
C,w,α,t,u C + n (cid:88) i =1 u i subject to u i ≥ λ (cid:0) w i − w − i (cid:1) u i ≥ − λ (cid:0) w i − w − i (cid:1) t qk ≥ t qk ≥ − w (cid:62) r [ q ] − α k α k + 1 Q (1 − β k ) Q (cid:88) q =1 t qk ≤ C + C β k T w = 1 w i ≥ i = 1 , ..., n ) Proof.
Using a standard approach in optimization, we replaceeach absolute value term λ (cid:107) w − w − (cid:107) with sof tmax . Thenobjective and constraints are all linear. In this section, we will report the results of our empirical stud-ies with well-known benchmarks. First, we evaluate how theportfolio changes as the β level changes. Depending on how β is chosen, a completely different portfolio may be con-structed. Next, we compare the out-of-sample performanceamong several portfolio strategy including our proposed. In the experiments, we use well-known academic benchmarkscalled Fama and French (FF) datasets [Fama and French,1992] to ensure the reproducibility of the experiment. ThisFF dataset is public and is readily available to anyone. TheFF datasets have been recognized as standard datasets andheavily adopted in finance research because of its extensivecoverage to asset classes and very long historical data series.We use FF25 dataset and FF48 dataset. For example, FF25dataset includes 25 portfolios formed on the basis of size andbook-to-market ratio and FF48 dataset contains monthly re-turns of 48 portfolios representing different industrial sectors.We use both datasets as monthly data from January 1989 toDecember 2018.
In our empirical studies, the tested portfolio models have thefollowing meanings: • “1/N” stands for equally-weighted (1/N) portfolio[DeMiguel et al. , 2007]. • “MV” stands for minimum-variance portfolio. We usethe latest 10 years (120 months) to calculate covariancematrix. • “DRP” stands for the doubly regularized miminum-variance portfolio [Shen et al. , 2014]. We use the latest10 years (120 months) to calculate covariance matrix,and set combinations of two coefficients for regulariza-tion terms to λ = { . , . , . , . } and λ = { . , . , . , . } . • “EGO” stands for the Kelly growth optimal portfoliowith ensemble learning [Shen et al. , 2019]. We set n (number of resamples) = 50, n (size of each resample)= 5 τ , τ (number of periods of return data) = 120, n (number of resampled subsets) = 50, n (size of eachsubset) = n . , n is number of assets. • “CVaR” stands for minimum CVaR portfolio with β . Weimplemnet 5 patterns of β = { } , and use the latest 10 years (120 months) to calu-late each model. • “ACVaR” stands for the average portfolio calculated bythe average of minimum CVaR portfolio of different β = { } at each time point. • “RM-CVaR” stands for our proposed portfolio. We set K = 5 ( k = 1 , ..., K ) as 5 patterns of β k = { } to calculate C β k and set Q (numberof sampling periods of return data) as {
10 years (120months), 7 years (84 months) } . For the coefficient ofthe regularization term, we implement 4 patterns of λ { . , . , . , . } . We also implement λ =0 to compare with the best RM-CVaR. The RM-CVaRportfolio presented in Algorithm 1 is straightforward toimplement.We use the first-half period, from January 1989 to De-cember 2003, as the in-sample period to decide the hyper-parameters of each model. After that, we use the second half-period, from January 2004 to December 2018 as the out-of-sample periods. Each portfolio is updated by sliding one-month-ahead and carrying out a monthly forecast. Algorithm 1
RM-CVaR Portfolio
Input: K probability levels β k ∈ (0 ,
1) ( k = 1 , . . . , K ) ,a number of sampling periods Q ∈ Z + ,a coeffient of the regularization term λ ∈ R + anda return matrix Y ∈ R n × ( T + Q ) Output: a weight matrix W ∈ R n × ( T +1) for t = 1 , . . . , T + 1 do R ← Y [ t : Q + t − Solve the linear programming introduced inTheorem 4.1 Contain the solution w ∗ to W [ t ] end for return W In the first experiment, we define the weight difference oftwo minimum CVaR portfolios which have β i and β j are asbellow. Diff = 1 T T (cid:88) t =1 || w β i t − w β j t || (25)We set β i and β j as { . , . } , { . , . } , { . , . } ,and { . , . } . A large Diff indicates that the two portfo-lios are different. This measure is similar to turnover definedbelow.Next, we compare the out-of-sample performance of theportfolios. In evaluating the portfolio strategy, we use thefollowing measures that are widely used in the field of finance[Brandt, 2010].The portfolio return at time t is defined as R t = n (cid:88) i =1 r it w it − (26)where r it is the return of i asset at time t , w it − is the weightof i asset in the portfolio at time t − , and n is the numberof asset. We evaluate the portfolio strategy by its annualizedreturn (AR), risk as the standard deviation of return (RISK),risk/return (R/R) as return divided by risk as for the portfoliostrategy. R/R is a risk-adjusted return measure for a portfolio Table 1: The weight difference of two minimum CVaR portfolios inthe out-of-sample period. β β β β AR = T (cid:89) t =1 (1 + R t ) /T − (27) RISK = (cid:114) T − × ( R t − µ ) (28) R / R = AR / RISK (29)Here, let µ = (1 /T ) (cid:80) Tt =1 R t be the average return of theportfolio.We also evaluate maximum drawdown (MaxDD), which isyet another widely used risk measures [Magdon-Ismail andAtiya, 2004; Shen and Wang, 2017], for the portfolio strat-egy: Namely, MaxDD is defined as the largest drop from anextremum: MaxDD = min k ∈ [1 ,T ] (cid:18) , W k max j ∈ [1 ,k ] W j − (cid:19) (30) W k = k (cid:89) l =1 (1 + R l ) . (31)where W k be the cumulative return of the portfolio until time k . The turnover (TO) indicates the volumes of rebalancing.Since a high turnover inevitably generates high explicit andimplicit trading costs thus reducing the portfolio return, it hasbeen recognized as an important performance metric. Theone-way annualized turnover is calculated as an average ab-solute value of the rebalancing trades over all the trading pe-riods: TO = 122( T − T − (cid:88) t =1 || w t − w + t − || (32)where T − indicates the total number of the rebalancingperiods and w + t − is the re-normalized portfolio weight vectorbefore rebalance. w + t − = w t − ⊗ r t w t − (cid:62) r t (33)where r t is the return vector of the assets at time t , w t − isthe weight vector at time t − and the operator ⊗ denotes theHadamard product. Table 1 shows the weight difference of two minimum CVaRportfolios in the out-of-sample period. Only 1% difference inthe β level changes the portfolio weight on average by 48% in EWMV
DRP
EGOCVaR
ACVaR
RM-CVaR
Figure 2: The cumulative return in the out-of-sample period for FF25datasets. -50%0%50% EW MV DRPEGO
CVaR
ACVaRRM-CVaR
Figure 3: The cumulative return in the out-of-sample period for FF48datasets.Table 2: The performance of each portfolio in out-of-sample period for FF25 dataset (upper panel) and FF48 dataset (lower panel).FF25 EW MV DRP EGO ACVaR CVaR RM-CVaR95 96 97 98 99 λ = 0 Best λ AR [%] 8.27 8.45 8.48 8.58 8.48 8.46 8.36 8.35 8.73 8.42
R/R 0.46 0.55 0.54 0.46 0.54 0.52 0.54 0.54 0.56 0.53 0.56
MaxDD [%] -57.63 -58.14 -61.21 -61.76 -59.44 -57.75 -56.75 -60.81 -59.54 -62.21 -54.14 -52.81
TO [%] 16.95 31.10 λ = 0 Best λ AR [%] 8.14 8.99 9.09 11.11 11.83 11.68 10.79 11.54 11.96 12.92 15.75
RISK [%] 19.27
MaxDD [%] -59.81 -50.84 -50.25 -57.39 -47.22 -46.98 -45.21 -45.36 -48.23 -50.38 -35.29 -34.93
TO [%] 36.73 27.48
FF25 and 39% in FF48. We confirm CVaR portfolio weightsare highly sensitive to the β levels.Table 2 reports the overall performance measures of RM-CVaR, our proposed portfolio, and the compared 10 portfo-lios introduced in Section 5.2. Among the comparisons ofvarious portfolios, where the best performance is highlightedin bold. In both datasets, the proposed RM-CVaR with λ achieves the highest R/R and the lowest MaxDD. Not sur-prisingly, RM-CVaR differs from ACVaR, which is the sim-ple average of five CVaR portfolios. RM-CVaR also exceedsindividual β levels of CVaR by R/R and MaxDD. In FF25datasets, RM-CVaR without λ outperforms all the comparedportfolios in terms of AR but has the worst TO. Introduc-ing the regularization term λ , the TO is considerably sup-pressed, and RISK, R/R and maxDD are also the best. InFF48 datasets, In FF48, RM-CVaR with λ has the best AR,R/R, and MaxDD, but the TO is very high. This is becausethe λ selected in this experiment does not sufficiently sup-press TO.Furthermore, in order to compare the trend and dynamicsof the each portfolio return, Figure 2 and 3 show the cumu-lative return over the out-of-sample periods for the FF25 andFF48 datasets. Although there is not much difference in theFF25 dataset, RM-CVaR apparently outperforms the otherswith the visible margins in the FF48 datasets in most of the time periods. We can confirm that RM-CVaR avoides a largedrawdown. Our study makes the following contributions: • We propose RM-CVaR: Regularized Multiple β -CVaRPortfolio and prove that the optimization problem iswritten as a linear programming. • We demonstrate that the CVaR portfolio dramaticallychanges depending on the β level. • RM-CVaR is a superior performance of having bothhigher risk-adjusted returns and lower maximum draw-down.Our future work includes incorporating the subsamplingmethod such as [Shen and Wang, 2017; Shen et al. , 2019]. eferences [Ardia et al. , 2017] David Ardia, Guido Bolliger, KrisBoudt, and Jean-Philippe Gagnon-Fleury. The impact ofcovariance misspecification in risk-based portfolios.
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