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Math. Program., Ser. ADOI 10.1007/s10107-009-0271-z A computational study on robust portfolio selection
based on a joint ellipsoidal uncertainty set

Zhaosong Lu
Received: 11 December 2006 / Accepted: 11 January 2009 Springer and Mathematical Programming Society 2009 The "separable" uncertainty sets have been widely used in robust portfolio selection models [e.g., see Erdo˘gan et al. (Robust portfolio management. manuscript,Department of Industrial Engineering and Operations Research, Columbia University,New York, 2004), Goldfarb and Iyengar (Math Oper Res 28:1–38, 2003), Tütüncüand Koenig (Ann Oper Res 132:157–187, 2004)]. For these uncertainty sets, eachtype of uncertain parameters (e.g., mean and covariance) has its own uncertainty set.
As addressed in Lu (A new cone programming approach for robust portfolio selec-tion, technical report, Department of Mathematics, Simon Fraser University, Burnaby,2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set, manu-script, Department of Mathematics, Simon Fraser University, Burnaby, 2008), these"separable" uncertainty sets typically share two common properties: (i) their actualconfidence level, namely, the probability of uncertain parameters falling within theuncertainty set is unknown, and it can be much higher than the desired one; and(ii) they are fully or partially box-type. The associated consequences are that theresulting robust portfolios can be too conservative, and moreover, they are usuallyhighly non-diversified as observed in the computational experiments conducted inthis paper and Tütüncü and Koenig (Ann Oper Res 132:157–187, 2004). To combatthese drawbacks, the author of this paper introduced a "joint" ellipsoidal uncertaintyset (Lu in A new cone programming approach for robust portfolio selection, technicalreport, Department of Mathematics, Simon Fraser University, Burnaby, 2006; Robustportfolio selection based on a joint ellipsoidal uncertainty set, manuscript, Departmentof Mathematics, Simon Fraser University, Burnaby, 2008) and showed that it can be Z. Lu was supported in part by SFU President's Research Grant and NSERC Discovery Grant.
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canadae-mail: zhaosong@sfu.ca constructed as a confidence region associated with a statistical procedure applied toestimate the model parameters. For this uncertainty set, we showed in Lu (A new coneprogramming approach for robust portfolio selection, technical report, Departmentof Mathematics, Simon Fraser University, Burnaby, 2006; Robust portfolio selectionbased on a joint ellipsoidal uncertainty set, manuscript, Department of Mathematics,Simon Fraser University, Burnaby, 2008) that the corresponding robust maximumrisk-adjusted return (RMRAR) model can be reformulated and solved as a cone pro-gramming problem. In this paper, we conduct computational experiments to comparethe performance of the robust portfolios determined by the RMRAR models with our"joint" uncertainty set (Lu in A new cone programming approach for robust portfo-lio selection, technical report, Department of Mathematics, Simon Fraser University,Burnaby, 2006; Robust portfolio selection based on a joint ellipsoidal uncertainty set,manuscript, Department of Mathematics, Simon Fraser University, Burnaby, 2008)and Goldfarb and Iyengar's "separable" uncertainty set proposed in the seminal paper(Goldfarb and Iyengar in Math Oper Res 28:1–38, 2003). Our computational resultsdemonstrate that our robust portfolio outperforms Goldfarb and Iyengar's in terms ofwealth growth rate and transaction cost, and moreover, ours is fairly diversified, butGoldfarb and Iyengar's is surprisingly highly non-diversified.
Robust portfolio selection · Ellipsoidal uncertainty set · Mathematics Subject Classification (2000)
91B28 · 90C20 · 90C22 It is well known that the optimal portfolios determined by the classical mean-variance model are often sensitive to perturbations in the problem parameters(e.g., see Recently, robust portfolio selection models have been proposed to alle-viate such a sensitivity ]. The "separable" uncertainty sets have been widelyused in the models (e.g., see For these uncertainty sets, each type of uncer-tain parameters (e.g., mean and covariance) has its own uncertainty set. As addressedin ], these "separable" uncertainty sets typically share two common properties:(i) their actual confidence level, namely, the probability of uncertain parameters fall-ing within the uncertainty set is unknown, and it can be much higher than the desiredone; and (ii) they are fully or partially box-type. The associated consequences are thatthe resulting robust portfolios can be too conservative, and moreover, they are usuallyhighly non-diversified as observed in the computational experiments conducted in thispaper and ].
To combat the aforementioned drawbacks, we considered the following factor model for asset returns in which was first studied in ]. Suppose that a dis-crete-time market has n traded assets. The vector of asset returns over a single marketperiod is denoted by r ∈ n. The returns on the assets in different market periodsare assumed to be independent. The single period return r is assumed to be a randomvector given by A computational study on robust portfolio selection r = µ + V T f + , where µ ∈ n is the vector of mean returns, f N (0, F) ∈ m denotes the returnsof the m factors driving the market, V ∈ m×n denotes the factor loading matrix of then assets, and  N (0, D) ∈ n is the vector of residual returns. Further, it is assumedthat D is a positive semidefinite diagonal matrix, and the residual return vector  isindependent of the factor return vector f . Suppose the market data consists of assetreturns {rt : t = 1, . . , p} and factor returns { f t : t = 1, . . , p} for p trading peri-ods. Let B = ( f 1, f 2, . . , f p) ∈ m×p denote the matrix of factor returns, and let e ∈ p denote an all-one vector. Further, let A = (eBT ), yi = (r1, r2, . . , r )T , ¯x (AT A)−1 AT yi, s2 = y i A ¯ xi 2/( p m −1) for i = 1, . . , n. In we proposed a "joint" ellipsoidal uncertainty set of (µ, V ) with ω-confidence level in the form of Sµ,v Sµ,v(ω) = ( ˜µ, ˜ xi − ¯xi )T (AT A)( ˜xi − ¯xi ) V ) ∈ n × m×n : ≤ (m + 1)˜c(ω) for some ˜c(ω), where ˜xi = ( ˜ µi, ˜V1i, ˜V2i, . . , ˜Vmi)T for i = 1, . . , n. We showed that it can be constructed as an ω-confidence region associated with a statistical procedureapplied to estimate the model parameters (µ, V ). For the details, see Based on the "joint" ellipsoidal uncertainty set Sµ,v, we studied the following robust maximum risk-adjusted return (RMRAR) problem in ]: φ (µ,V )Sµ,v where θ ≥ 0 represents a risk-aversion parameter, and E[] = φT µ, Var[] = φT (V T F V + D)φ,  = {φ : eT φ = 1, φ ≥ 0}. We showed that the RMRAR problem can be reformulated and solved as a coneprogramming problem (see Theorems 4.4 and 4.5 of ]).
In this paper, we conduct computational experiments to compare the performance of the robust portfolios determined by the RMRAR models with our "joint" uncer-tainty set and Goldfarb and Iyengar's "separable" uncertainty set proposed in theseminal paper Our computational results demonstrate that our robust portfoliooutperforms Goldfarb and Iyengar's in terms of wealth growth rate and transactioncost, and moreover, ours is fairly diversified, but Goldfarb and Iyengar's is highlynon-diversified.
2 Computational results
In this section, we present computational experiments on the RMRAR models. Weconduct two types of computational tests. The first type of tests are based on simulated data, and the second type of tests use real market data. The main objective ofthese computational tests is to compare the performance of the RMRAR models withour "joint" uncertainty set and Goldfarb and Iyengar's "separable" uncertainty setdescribed in (3) and (4) of ]. All computations are performed using SeDuMi V1.1R2Throughout this section, the symbols "LROB" and "GIROB" are used to label therobust portfolios determined by the RMRAR models with our "joint" and Goldfarband Iyengar's "separable" uncertainty sets, respectively. The following terminologywill also be used in this section.
Definition 1 The diversification number of a portfolio is defined as the number of its
components that are above 1%.
2.1 Computational results for simulated data In this subsection, we conduct computational tests for simulated data. The data isgenerated in the same manner as described in Section 7 of ]. Indeed, we fix the num-ber of assets n = 50 and the number of factors m = 5. A symmetric positive definitefactor covariance matrix F is randomly generated, and it is assumed to be certain. Thenominal factor loading matrix V is also randomly generated. The covariance matrix D of the residual returns  is assumed to be certain and set to D = 0.1 diag(V T F V ), that is, the linear model explains 90% of the asset variance. The nominal asset returnsµ ∈ n are chosen independently according to a uniform distribution on [0.5, 1.5%].
Finally, we generate a sequence of asset and factor return vectors r and f according tothe normal distributions N (µ, V T F V + D) and N (0, F) for an investment period oflength p = 90, respectively. We also randomly generate the asset returns, denoted by R ∈ n×p, for next period of length p according to N (µ, V T F V + D). In addition, given a desired confidence level ω > 0, our "joint" uncertainty set Sµ,v is built as inSection 3 of and Goldfarb and Iyengar's "separable" uncertainty set Sm × Svis built as in Section 2 of ] with ˜ ω = ω1/n. As discussed in Section 2 of ], Sm × Sv has at least ω-confidence level, but its actual confidence level is unknown.
Let φr be a robust portfolio computed from the data of the current period. Suppose that φr is held constant for the investment over next period. The wealth growth rateof φr over next period is defined as (1≤kp(e + Rk))T φr − 1, where e ∈ n denotes the all-one vector and Rk ∈ n denotes the kth column of Rfor k = 1, . . , p.
We next report the performance of the RMRAR models with our "joint" uncertainty set and Goldfarb and Iyengar's "separable" uncertainty set described in (3) and(4) of ] as the risk aversion parameter θ ranges from 0 to 10. The computationalresults averaged over 10 randomly generated instances are shown in Fig. that consistsof three groups of plots for ω = 0.05, 0.50, 0.95, respectively. In each of these threegroups, the left plot is about the diversification number of the robust portfolios, and theright plot is about the wealth growth rate of the robust portfolios over next period. Wefirst observe that our robust portfolio is fairly diversified, but Goldfarb and Iyengar's A computational study on robust portfolio selection wealth growth rate 2.85 wealth growth rate diversification number diversification number wealth growth rate diversification number Fig. 1 Performance of portfolios for ω = 0.05, 0.50 and 0.95
is highly non-diversified. Indeed, for ω = 0.05, 0.50, 0.95, the diversification numberof our robust portfolio is around 26, and that of Goldfarb and Iyengar's is around oneor two. One possible interpretation of this phenomenon is that our uncertainty set Sµ,vis ellipsoidal, but Goldfarb and Iyengar's uncertainty set Sm ×Sv is partially box-type.
It seems that the ellipsoidal uncertainty structure tends to produce more diversifiedrobust portfolio than does the fully or partially box-type one. In addition, we observethat for ω = 0.05 or ω = 0.50 with a relatively small θ, the wealth growth rate ofour robust portfolio is lower than that of Goldfarb and Iyengar's. But for ω = 0.95or ω = 0.50 with a relatively large θ, our wealth growth rate is higher than Goldfarband Iyengar's. This phenomenon is actually not surprising. Indeed, we know that Sµ,vhas confidence ω while Sm × Sv has at least ω confidence, but its actual confidencelevel can be much higher than ω. Hence for a small ω, the associated model with Sm × Sv can be more robust than that with Sµ,v. However, for a relatively large ω,the uncertainty set Sm × Sv can be over confident, and its corresponding robust modelcan be conservative. This phenomenon becomes more prominent as the risk aversionparameter θ gets larger.
2.2 Computational results for real market data In this subsection, we perform experiments on real market data for the RMRAR mod-els with our "joint" and Goldfarb and Iyengar's "separable" uncertainty sets. The Table 1 Assets
Aerospace and Defense Boeing Corp.
Verizon Communications United Technologies Honeywell Intl.
Semiconductors and Other Electronic Components Computer Software Texas Instruments Computers and Office Equipment Intl. Business Machines Johnson & Johnson Abbott Laboratories Bristol-Myers Squibb Network and Other Communications Equipment Lucent Technologies Lyondell Chemical Electronics and Electrical Equipment Utilities (Gas & Electric) Dominion Resources Rockwell Automation Public Service Enterprise Group universe of assets that are chosen for investment are those ranked at the top of eachof 10 industry categories by Fortune 500 in 2006. In total there are n = 47 assets inthis set (see Table The set of factors are 10 major market indices (see Table Thedata sequence consists of daily asset returns from 25July 2002 through 10 May 2006.
It shall be mentioned that the data used in this experiment was collected on 11 May2006. The most recent data available at that time was the one on 10 May 2006.
A complete description of our experimental procedure is as follows. The entire data sequence is divided into investment periods of length p = 90 days. For eachinvestment period t, the factor covariance matrix F is computed based on the factorreturns of the previous p trading days, and the variance di of the residual return isset to di = s2, where s2 is given in Sect. In addition, given a desired confidence level ω > 0, our "joint" uncertainty set Sµ,v is built as in Section 3 of ,and A computational study on robust portfolio selection Table 2 Factors
Dow Jones Composite 65 Stock Average Dow Jones Industrials Dow Jones Utilities Dow Jones Transportation S&P 500 Composite Dow Jones Wilshire 5000 Composite Goldfarb and Iyengar's "separable" uncertainty set Sm × Sv is built as in Sect. 2 ofwith ˜ ω = ω1/n. The robust portfolios are then obtained by solving the RMRAR models with these uncertainty sets, and they are held constant for the investment ateach period t.
Since a block of data of length p = 90 is required to construct uncertainty sets or estimate the parameters, the first investment period indexed by t = 1 starts from (p + 1)th day. The time period 25 July 2002–10 May 2006 contains 11 periods oflength p = 90, and hence in all there are 10 investment periods. Given a sequence ofportfolios {φt }10 , the corresponding overall wealth growth rate is defined as 1≤t≤10 t pk(t+1) p(e + rk ) and the average diversification number is defined as 10, where I (φt ) denotes the diversification number of the portfolio φt .
We now report the performance of the RMRAR models with our "joint" uncer- tainty set Sµ,v, and Goldfarb and Iyengar's "separable" uncertainty set Sm × Svas the risk aversion parameter θ ranges from 0 to 104. The computational resultsfor the confidence level ω = 0.05, 0.50, 0.95 are shown in Fig. that consistsof three group of plots. In each of these groups, the left plot is about the averagediversification number of robust portfolios, and the second plot is about the overallwealth growth rate over next 10 periods of the investment using robust portfolios.
We observe that our robust portfolio is fairly diversified, but Goldfarb and Iyen-gar's is highly non-diversified. Also, the overall wealth growth rate of the investmentbased on our robust portfolio is higher than that using Goldfarb and Iyengar's robustportfolio.
The realization cost is another natural concern for any investment strategy. We next compare the cost of implementing the above investment strategies. For a sequence of portfolios {φt }10 , its average transaction cost is defined as φt φt−1 (see also the discussion in ]). In Fig. we report the average transaction costs of theinvestments using the robust portfolios for the confidence levels ω = 0.05, 0.50, 0.95,respectively. We observe that the investment based on our robust portfolio incurs loweraverage transaction cost than that using Goldfarb and Iyengar's robust portfolio.
overall wealth growth rate overall wealth growth rate average diversification number average diversification number overall wealth growth rate average diversification number Fig. 2 Performance of portfolios for ω = 0.05, 0.50 and 0.95
average transaction cost average transaction cost average transaction cost Fig. 3 Average cost of portfolios for ω = 0.05, 0.50 and 0.95
3 Concluding remarks
In this paper, we conducted computational experiments to compare the performanceof the RMRAR model with our "joint" uncertainty set and Goldfarb and Iyengar's"separable" uncertainty set. We observed that the RMRAR model with our uncer-tainty set is usually less conservative than that based on Goldfarb and Iyengar's. Inparticular, our robust portfolio outperforms Goldfarb and Iyengar's in terms of wealth A computational study on robust portfolio selection growth rate and transaction cost. In addition, our robust portfolio is fairly diversified,but Goldfarb and Iyengar's is highly non-diversified. Though we only considered theRMRAR model in this paper, we expect that the similar phenomenon can also beobserved in other robust portfolio selection models, e.g., robust maximum Sharperatio and robust value-at-risk models (see ]).
The author is grateful to Antje Berndt for stimulating discussions on selecting suitable factors and providing me the real market data. The author is also indebted to Garud Iyengarfor making his code available to me. We also gratefully acknowledge comments from Dimitris Bertsimas,Victor DeMiguel, Darinka Dentcheva, Andrzej Ruszczy´nski and Reha Tütüncü at the 2006 INFORMSAnnual Meeting in Pittsburgh, USA.
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Source: http://www.math.sfu.ca/~zhaosong/ResearchPapers/robport-MPA.pdf

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