Can robust optimization offer improved portfolio performance?: An empirical study of Indian market
Shashank Oberoi, Mohammed Bilal Girach, Siddhartha P. Chakrabarty
aa r X i v : . [ q -f i n . P M ] A ug C AN ROBUST OPTIMIZATION OFFER IMPROVED PORTFOLIO PERFORMANCE ?: A
NEMPIRICAL STUDY OF I NDIAN MARKET S HASHANK O BEROI ∗ M OHAMMED B ILAL G IRACH † S IDDHARTHA
P. C
HAKRABARTY ‡ Abstract
The emergence of robust optimization has been driven primarily by the necessity to address the demeritsof the Markowitz model. There has been a noteworthy debate regarding consideration of robust approachesas superior or at par with the Markowitz model, in terms of portfolio performance. In order to address thisskepticism, we perform empirical analysis of three robust optimization models, namely the ones based on box,ellipsoidal and separable uncertainty sets. We conclude that robust approaches can be considered as a viablealternative to the Markowitz model, not only in simulated data but also in a real market setup, involving theIndian indices of S&P BSE 30 and S&P BSE 100. Finally, we offer qualitative and quantitative justificationregarding the practical usefulness of robust optimization approaches from the point of view of number of stocks,sample size and types of data.
Keywords: Robust portfolio optimization; Worst case scenario; Uncertainty sets; S&P BSE 30; S&P BSE100
NTRODUCTION
The risk associated with individual assets can be reduced through investment in a diversified portfolio com-prising of several assets. For optimal allocation of weights in a diversified portfolio, one of the well establishedmethods is the classical mean-variance portfolio optimization introduced by Markowitz [9, 10]. Despite beingconsidered as the basic theoretical framework in the field of portfolio optimization, the Markowitz model is notwidely accepted among investment practitioners. One of the most major limitations of the mean-variance model isthe sensitivity of the optimal portfolios to the errors in the estimation of return and risk parameters. These param-eters are estimated using sample mean and sample covariance matrix, which are maximum likelihood estimates(MLEs) (calculated using historical data) under the assumption that the asset returns are normally distributed.According to DeMiguel and Nogales [4], since the efficiency of MLEs is extremely sensitive to deviations ofthe distribution of asset returns from the assumed normal distribution, it results in the optimal portfolios beingvulnerable to the errors in estimation of input parameters. While referring to the Markowitz model as “estimation-error maximizers”, Michaud [11] argues that it often overweights those assets having higher estimated expectedreturn, lower estimated variance of returns and negative correlation between their returns (and vice versa). Bestand Grauer [1] study the sensitivity of weights of optimal portfolios with respect to changes in estimated expectedreturns of individual assets. Upon imposition of no short selling constraint, they observe that small changes inestimated expected return of individual assets can result in the assignment of zero weights to almost half the as-sets comprising the portfolio (which is counterintuitive), leading to a large adjustment in portfolio weights. In anempirical study, Broadie [2] reports evidence of overestimation of expected returns of optimal portfolios obtainedusing the Markowitz model by observing that the estimated efficient frontier lies above the actual efficient frontier. ∗ Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India, e-mail: [email protected] † Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India, e-mail: [email protected] ‡ Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India, e-mail: [email protected], Phone: +91-361-2582606,Fax: +91-361-2582649
OBUST P ORTFOLIO O PTIMIZATION A PPROACHES
The determination of the structure of uncertainty sets, so as to obtain computationally tractable solutions,is a key step in robust optimization. In the real world, even the distribution of asset returns has an uncertaintyassociated with it. In order to address this issue, a frequently used technique is to find an estimate of the uncertainparameter and define a geometric bound around it. Empirically, historical data is used to compute estimates ofthese uncertain parameters. For a given optimization problem, determining the geometry of the uncertainty setis a difficult task. For the purpose of this work, we will use three types of uncertainty sets, namely, box andellipsoidal (for expected returns) [5, 7] and separable (for both expected returns and covariance matrix of returns)[8]. Accordingly, we first introduce the notations to be used in this work.1. N : Number of assets.2. x : Weight vector for a portfolio.3. a = (cid:0) a , a , . . . , a N (cid:1) : Vector of N uncertain parameters.4. ˆa = (cid:0) ˆ a , ˆ a , . . . , ˆ a N (cid:1) : Estimate for a . 2. µ : Vector for expected return.6. ˆ µ : Estimate for µ .7. Σ : Covariance matrix for asset returns.8. Σ µ : Covariance matrix for errors in estimation.9. λ : Risk aversion.10. U µ, Σ : General uncertainty set with µ and Σ as uncertain parameters.11. : Unity vector of length N .The classical Markowitz model formulation with no short selling constraint is given by the following problem(hereafter referred to as Mark ): max x (cid:8) µ ⊤ x − λ x ⊤ Σ x (cid:9) such that x ⊤ = 1 and x ≥ . (2.1)Robust portfolio optimization involves enhancing the robustness of the portfolio obtained using the Markowitzmodel, by optimizing the portfolio performance in worst-case scenarios. Most of the robust models deal withoptimizing a given objective function with a predefined “uncertainty set” for obtaining computationally tractablesolutions. For any general uncertainty set U µ, Σ , the worst case classical Markowitz model formulation [6, 7] withno short selling constraint is given as: max x (cid:26) min ( µ , Σ ) ∈ U µ, Σ µ ⊤ x − λ x ⊤ Σ x (cid:27) such that x ⊤ = 1 and x ≥ , (2.2)2.1 R OBUST P ORTFOLIO O PTIMIZATION WITH B OX U NCERTAINTY S ET A general polytopic [5] uncertainty set which resembles a box, is defined as, U δ ( ˆa ) = { a : | a i − ˆ a i | ≤ δ i , i = 1 , , , . . . , N } , (2.3)where δ i represents the value which determines the confidence interval region for asset i . As we intend to modelthe uncertainty in expected returns ( µ ) using box uncertainty sets, we use, U δ ( ˆ µ ) = { µ : | µ i − ˆ µ i | ≤ δ i , i = 1 , , , . . . , N } . (2.4)Accordingly, using (2.4), the max-min robust formulation (2.2) reduces to the following maximization problem(hereafter referred to as Box ): max x (cid:8) ˆ µ ⊤ x − λ x ⊤ Σ x − δ ⊤ | x | (cid:9) such that x ⊤ = 1 and x ≥ . (2.5)While dealing with box uncertainty set, we assume that the returns follow normal distribution. Therefore, wedefine δ i for − α )% confidence level as, δ i = σ i z α n − , where z α represents the inverse of standard normal distribution, σ i is the standard deviation of returns of asset i and n is the number of observations of returns for asset i .3.2 R OBUST P ORTFOLIO O PTIMIZATION WITH E LLIPSOIDAL U NCERTAINTY S ET In order to capture more information from the data, the consideration of the second moment gives rise toanother class of uncertainty sets, namely, ellipsoidal uncertainty sets. The ellipsoidal uncertainty set for expectedreturn ( µ ) is expressed as: U δ ( ˆ µ ) = (cid:8) µ : ( µ − ˆ µ ) ⊤ Σ − µ ( µ − ˆ µ ) ≤ δ (cid:9) . (2.6)Therefore, the max-min robust formulation (2.2) in conjunction with (2.6) results in the following maximizationproblem (hereafter referred to as Ellip ): max x (cid:26) ˆ µ ⊤ x − λ x ⊤ Σ x − δ q x ⊤ Σ µ x (cid:27) such that x ⊤ = 1 and x ≥ . (2.7)If the uncertainty set follows ellipsoid model, the condidence level is set using a chi-square ( χ ) distribution withthe number of assets being the degrees of freedom (df). Accordingly, for − α )% confidence level, δ isdefined as [3, 13]: δ = χ N ( α ) (2.8)where χ N ( α ) is the inverse of a chi square distribution with N degrees of freedom.2.3 R OBUST P ORTFOLIO O PTIMIZATION WITH S EPARABLE U NCERTAINTY S ET The above two robust approaches model only the expected returns using uncertainty sets. Hence, in order toalso encapsulate the uncertainly in the covariances, the box uncertainty set for the covariance matrix of returns isdefined akin to that for expected returns. The lower bound Σ ij and the upper bound Σ ij can be specified for eachentry Σ ij , resulting in the following constructed box uncertainty set for the covariance matrix [14]: U Σ = { Σ : Σ ≤ Σ ≤ Σ , Σ (cid:23) } . (2.9)In the above equation, the condition Σ (cid:23) implies that Σ is a symmetric positive semidefinite matrix. T¨ut¨unc¨u andKoenig [14] define the uncertainty set for expected returns as, U µ = { µ : µ ≤ µ ≤ µ } , (2.10)where µ and µ represent lower and upper bounds on expected return vector µ respectively. Consequently, themax-min robust formulation (2.2) transforms to the following maximization problem (hereafter referred to as Sep ): max x (cid:8) µ ⊤ x − λ x ⊤ Σ x (cid:9) such that x ⊤ = 1 and x ≥ . (2.11)The above approach involves the use of “separable” uncertainty sets [8], which implies that the uncertainty setsfor expected returns and covariance matrix are defined independent of each other.3 C OMPUTATIONAL R ESULTS
In this section, we analyze the performance of the robust portfolio optimization approaches discussed inSection 2 vis-`a-vis the Markowitz model, using the historical data from the market, as well as simulated data.For the purpose of this analysis, we consider two scenarios in terms of number of stocks N , being and ,with the goal of observing the effect of increase in the number of stocks on the performance of robust portfoliooptimization approaches. These numbers were chosen since they represent the number of stocks in S&P BSE 30and S&P BSE 100 indices, respectively.For the first scenario ( N = 31 ), we make use of the daily log-returns, based on the adjusted daily closing pricesof the stocks comprising the S&P BSE 30, obtained from Yahoo Finance [17]. Accordingly, we consider theperiod of our analysis to be from 18th December, 2017 to 30th September, 2018 (both inclusive) which had atotal of active trading days i.e., daily log-returns. Corresponding to this historical data from S&P BSE30, we generate two sets of simulated data for all the assets, by sampling returns using a multivariate normal4istribution whose mean and covariance matrix are set to those obtained for the historical S&P BSE 30 data. Thefirst set of sample returns comprises of the number of samples to be the same as in the historical data, namely , in this case. On the other hand, the second set comprises of a larger number of samples, namely, .The two sets of simulated sample returns of different sizes were used to facilitate the study of the impact of thenumber of samples in simulated data on the performance of the robust portfolio optimization approaches. Wemake a comparative study of robust portfolio optimization approaches, in case of the historical S&P BSE 30 data,as well as the two sets of simulated data, in order to analyze whether the worst case robust portfolio optimizationapproaches are useful in a real market setup. For the second scenario ( N = 98 ), we use the daily log-returns,based on the adjusted daily closing prices of the stocks comprising the S&P BSE 100, obtained from YahooFinance [17], for the period 18th December, 2016 to 30th September, 2018 (both inclusive) with trading days i.e., daily log-returns. The two sets of simulated data were generated in a manner akin to the scenario of N = 31 assets. Similar kind of comparative study is performed for the second scenario.For Box and Ellip model, we construct uncertainty sets in expected return with − α )% confidence levelby considering α = 0 . . Separable uncertainty set in Sep model is constructed as a − α )% confidenceinterval for both µ and Σ using non-parametric Boostrap Algorithm with same α as in other robust models andassuming β , i.e the number of simulations, equal to .The performance analysis for these robust portfolio models vis-`a-vis the Mark model is performed using the Sharpe Ratio of the constructed portfolios, with λ representing the risk-aversion in the ideal range i.e., λ ∈ [2 , [5]. Further, since the T-bill rate in India from 2016 to 2018 was observed to oscillating around [16], so wehave assumed the annualized riskfree rate to be equal to . We now present the computational results observedin case of the two scenarios, as discussed above.3.1 P ERFORMANCE WITH N = 31 ASSETS
We begin with the analysis for N = 31 assets, in the case of the simulated data with samples andpresent the results in Figure 1 and Table 1. From Figure 1 we observe that the efficient frontiers for the Ellipand the Sep models lie below the efficient frontier for the Mark model, which supports the argument made in[2] regarding over-estimation of the efficient frontier for the Mark model. Further, the observed overlap of theefficient frontiers for the Mark and the Box models suggest that the utilizing box uncertainty sets for robustportfolio optimization does not prove to be of much use in this case. Also, from Figure 1, we observe that theMark model starts outperforming the Sep model in terms of the Sharpe Ratio after the risk-aversion λ crosses .The above observations are supported quantitatively by the results tabulated in Table 1 as well, since the averageSharpe Ratio for portfolios constructed in the ideal range of risk-aversion λ ∈ [2 , is the same in case of boththe Mark and the Box models. Also, we infer from Table 1, that the Sep model performs at par with the Markmodel by taking the average Sharpe Ratio into consideration, with the Ellip model performing the best among allthe models.The analysis with the number of simulated samples being the same as the number of log-returns in the caseof S&P BSE 30 data is presented in Figure 2 and Table 2. The efficient frontiers for the Sep and Ellip modellie below that of the Mark model. We observe results similar to the case when simulated samples wereconsidered, upon comparison of the Mark model and the Box model. However, we observe a slight inconsistencyin the performance of Box model as evident from the plot of the Sharpe Ratio in Figure 2. We also infer thatthe Ellip model and the Sep model outperform the Mark model in terms of the Sharpe Ratio in the ideal range ofrisk-aversion λ ∈ [2 , . It is difficult to compare the performance of the Ellip model with that of the Sep modelin this case, since the average Sharpe Ratio for both of them is almost the same (Table 2).For the historical market data involving the stocks comprising S&P BSE 30, we observe from Figure 3, thatthe efficient frontiers for the Mark model and the Box model almost overlap with each other. Further, the efficientfrontier for the Sep model lies below that of the Mark model with further widening of the gap between the plots,in case of the Ellip model. However, the performance of the Box model, in terms of the Sharpe ratio is quiteinconsistent as evident from Figure 3. We also observe that the Sep model outperforms the Mark model in theideal range of risk-aversion λ ∈ [2 , upon taking the Sharpe Ratio into consideration as the performance measure.5his is not true in case of the Ellip Model, as evident from the Sharpe Ratio plot in Figure 3. Even from Table3, we observe that average Sharpe Ratio for the Ellip model is only slightly greater than that for the Mark model.We also note that the Sep model outperforms all the other three models. A common observation that could be inferred from three cases considered in the scenario involving less num-ber of assets ( N = 31 ) is that the Sep and the Ellip models perform superior or equivalent in comparison to theMark model in the ideal range of risk-aversion. ERFORMANCE WITH N = 98 ASSETS
We now analyze the scenario involving N = 98 assets. On applying robust model along with the Markmodel on the simulated data having samples, we observe results similar to the corresponding case for theprevious scenario when we compared the Box model with the Mark model. This is evident from the coincidingplots of the efficient frontier and the plots for the Sharpe Ratio for both the models in Figure 4. However, incontrast to the scenario of N = 31 assets, we observe that not only does the Ellip model but also the Sep modeloutperforms the Mark model when considering the portfolios constructed in the ideal range of risk-aversion λ ∈ [2 , . Additionally, from Table 4, we can infer that the Ellip model exhibits superior performance in comparisonto the Sep model, in terms of greater average value of the Sharpe Ratio.In Figure 5 and Table 5 we present the results of the study for the simulated data with the number of samplesbeing the same as that of log-returns of S&P BSE 100 data. The comparative results observed for the Box modeland the Mark model are similar to the previous case of simulated samples. In the ideal range of risk aversion λ ∈ [2 , , one observes that the efficient frontier for both the Ellip as well as the Sep model lie below the efficientfrontier for the Mark model and both the models perform better than the Mark model in terms of the SharpeRatio. Additionally, from the Sharpe ratio plot in Figure 5, any comparative inference of the Sep model and theEllip model is difficult, since each outperforms the other in a different sub-interval of the risk-aversion range. Thesimilar values of the average Sharpe Ratio in Table 5 supports the claim of almost equivalent performance of thesetwo models in this case.Finally, the results for the historical market data, involving stocks comprising S&P BSE 100 are presented inFigure 6 and Table 6. While the efficient frontier plot leads to observations similar to the previous case, however,there is a slight inconsistency in the performance of the Box model as can be seen from the plot of the SharpeRatio in Figure 6. The robust portfolios constructed using the Sep and the Ellip models outperform the onesconstructed using the Mark model in the ideal range of risk-aversion λ ∈ [2 , . Additionally, the performance ofthe Ellip model is marginally better than the Sep model as evident from the Sharpe Ratio plot, an inference that issupported by the marginal difference in average Sharpe Ratio of both (Table 6). We draw a common inference from the three cases considered in the scenario involving greater number ofassets, i.e., the Sep and the Ellip model outperform the Mark model in the ideal range of risk aversion.
ISCUSSION
In this concluding section we analyze the different kinds of scenarios in the context of trends of the SharpeRatio. Recall that, we have considered the “adjusted closing prices” data of S&P BSE 30 and S&P BSE 100 toillustrate our analysis. Further, we have also generated simulated samples using the true mean and covariancematrix of log-returns obtained from the aforesaid actual market data of “adjusted closing prices”. Since thenumber of instances in market data for the assets comprising the two indices, was very less, we simulated twosets of samples, one where the number of simulated samples matches the number of instances of real marketdata available, say ζ ( < and another where the number of simulated samples is large (a constant, which inour case was taken to be ), irrespective of the number of stocks. The motivation behind this setup was tounderstand if the market data we obtained (which was limited) is able to capture the trends and results in betterportfolio performance. 6.1 F ROM THE S TANDPOINT OF N UMBER OF S TOCKS
We begin with a description of the results summarized in Table 7, wherein for a particular row and a particularcolumn, we presented the maximum possible Sharpe Ratio that was obtained for that particular scenario. Forexample, in case of the tabular entry for the case when N = 98 where we simulated ζ samples using true meanvector and the true covariance matrix of S&P BSE 100, we refer to Table 5 (which explains the simulationcorresponding to S&P BSE 100 with ζ simulated samples) and take the maximum of its last row i.e. , maximumof average Sharpe ratios that was attained using the available robust and Mark models.Larger the number of stocks, better is the performance of the portfolios constructed using robust optimization.This claim can be supported both qualitatively and quantitatively. Qualitatively, the number of stocks in a portfoliorepresents its diversification. According to Modern Portfolio Theory (MPT), investors get the benefit of betterperformance from diversifying their portfolios since it reduces the risk of relying on only one (small number) asset(assets) to generate returns. Based on the analysis by Value Research Online [15] one observes that on an averagebasis, the large-cap funds hold around shares while the mid-cap funds hold around − assets for balancedfunds, in which around − of the assets are held in equity. This is because of great stability of returns incase of companies with large market capitalization, whereas this is not the case with mid-cap companies. Hencediversification requirements drives greater percentages in equities in case of mid-cap funds. From Table 7, we canprovide quantitative justification by observing that the Sharpe Ratio was more for portfolios with larger numberof stocks as compared to portfolios with smaller number of stocks. However, we observe opposite behavior forthe market data which can be attributed to the following two reasons:1. The insufficient availability of market data, when it comes to larger number of stocks.2. The error in the estimation of return and covariance matrix accumulating as the number of stocks increases,impacting the performance of the model [11].4.2 F ROM THE S TANDPOINT OF N UMBER OF S IMULATED S AMPLES
We now focus on the performance of the portfolio when different number of samples were simulated andtabulate the results in Table 8 in the same way as was done in the preceding discussion. Here several interestingperformance trends can be noticed. We observe that in the case of smaller number of stocks, the performance whenthe number of simulated samples is ζ ( < is better than the case when a large ( ) simulated samples weregenerated. On the other hand, the exactly opposite trend can be observed when higher number of stocks are takeninto consideration. This observation can be explained as follows: In the case of real market, the number of datainstances being available is relatively low. So, when larger number of samples were generated, we observe higherSharpe Ratio as compared to ζ number of simulations. However, the reason behind such a pattern of oppositebehavior, when smaller number of stocks are considered, is not obvious.4.3 F ROM THE S TANDPOINT OF THE K IND OF D ATA
Finally, we discuss about the performance of the portfolio from the standpoint of kind of data that we haveused in this work. Accordingly, the relevant results are tabulated in Table 9, from where the behavior is observedto be fairly consistent. For both the cases, the performance in case of the simulated data is better than in case of thereal market data. This is clear from the fact that the real market data is difficult to model as it hardly follows anydistribution, whereas the simulated is generated from multivariate normal distribution with mean and covariancesas the true values obtained from the data. 5 C
ONCLUSION
Robust optimization is an emerging area of portfolio optimization. Various questions have been raised onthe advantages of robust methods over the Markowitz model. Through computational analysis of various robustoptimization approaches followed by a discussion from different standpoints, we try to address this skepticism. Weobserve that robust optimization with ellipsoidal uncertainty set performs superior or equivalent as compared to7he Markowitz model, in the case of simulated data, similar to the results reported by Santos [12]. In addition, weobserve favorable results in the case of market data as well. Better performance of the robust formulation havingseparable uncertainty set in comparison to the Markowitz model is in line with the previous study on the samerobust model by T¨ut¨unc¨u and Koenig [14]. Empirical results presented in this work advocate enhanced practicaluse of the robust models involving ellipsoidal uncertainty set and separable uncertainty set and accordingly, thesemodels can be regarded as possible alternatives to the classical mean-variance analysis in a practical setup.R
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Risk Aversion S ha r pe R a t i o Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Figure 1: Efficient Frontier plot and Sharpe Ratio plot for different portfolio optimization models in case ofSimulated Data with 1000 samples (31 assets)
Standard Deviation -3 R e t u r n -3 Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Risk Aversion S ha r pe R a t i o Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Figure 2: Efficient Frontier plot and Sharpe Ratio plot for different portfolio optimization models in case ofSimulated Data with same number of samples as market data (31 assets)9 .005 0.01 0.015
Standard Deviation R e t u r n -3 Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Risk Aversion S ha r pe R a t i o Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Figure 3: Efficient Frontier plot and Sharpe Ratio plot for different portfolio optimization models in case ofMarket Data (31 assets)
Standard Deviation -3 R e t u r n -3 Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Risk Aversion S ha r pe R a t i o Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Figure 4: Efficient Frontier plot and Sharpe Ratio plot for different portfolio optimization models in case ofSimulated Data with 1000 samples (98 assets)
Standard Deviation R e t u r n -3 Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Risk Aversion S ha r pe R a t i o Vanilla MarkowitzWith Box uncertainty
With Ellipsoid uncertaintyWith Separable uncertainty
Figure 5: Efficient Frontier plot and Sharpe Ratio plot for different portfolio optimization models in case ofSimulated Data with same number of samples as market data (98 assets)10
Standard Deviation -3 R e t u r n -3 Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Risk Aversion S ha r pe R a t i o Vanilla MarkowitzWith Box uncertaintyWith Ellipsoid uncertaintyWith Separable uncertainty
Figure 6: Efficient Frontier plot and Sharpe ratio plot for different portfolio optimization models in case of MarketData (98 assets) λ SR
Mark SR Box SR Ellip SR Sep λ SR
Mark SR Box SR Ellip SR Sep λ SR
Mark SR Box SR Ellip SR Sep SR Mark SR Box SR Ellip SR Sep λ SR
Mark SR Box SR Ellip SR Sep λ SR
Mark SR Box SR Ellip SR Sep ζ ( < ζ ( <1000)