Effects of Size and Allocation Method on Stock Portfolio Performance: A Simulation Study

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1 2011 3rd International Conference on Information and Financial Engineering IPEDR vol.12 (2011) (2011) IACSIT Press, Singaore Effects of Size and Allocation Method on Stock Portfolio Performance: A Simulation Study eat-yung g and im-leng Goh Faculty of Economics & Administration, University of Malaya, uala Lumur, Malaysia Abstract. This aer examines the erformance of ortfolios of stocks listed in the Malaysian exchange through a simulation study. The effects of different ortfolio sizes and fund allocation methods on return er unit of risk, or risk reward, were analyzed. Risk rewards increase with the inclusion of a larger number of stocks in a ortfolio but at a decreasing rate. The results show that a ortfolio size of 11 stocks is generally sufficient to generate reasonable risk rewards. The results, confirmed by holdout validation, also suggest that the conditional otimal and minimized variance allocation methods yield high risk reward, while the equal weight method has the oorest erformance. eywords: otimization, return, risk reward, simulation. 1. Introduction The number of stocks to be included and the method to allocate funds among the selected stocks are two imortant criteria in forming a stock ortfolio. Many of the studies conducted to find otimal ortfolio size did not reach a consensus, and some even suggested that large ortfolios with 30 stocks or more may not be well diversified [1, 2]. Another dimension of roblem to ortfolio formation is that the unconstrained ortfolio otimization as imlied in the Markowitz s mean-variance aroach introduces difficulty in arriving at an otimal solution that is ractical [3]. Constrained otimization methods are recommended to avoid comlex unrealistic solutions. Taking these difficulties into consideration, this study exlores for strategies for creating ortfolios consisting of stocks listed in the Malaysian stock market. Using a simulation study, we attemt to determine the referred ortfolio size and method of fund allocation. Many studies [2, 4] comared the risk erformance of ortfolio in the context of the modern ortfolio theory where risk (tyically the variance) is minimized for a given level of exected return. We roose the use of risk reward in our analysis of ortfolio erformance because evaluation of ortfolio based on variance has a weakness in its imlicit assumtion of a constant mean return. Further, both mean returns and variance were shown to decline [5] as ortfolio size increases. 2. Data and Samle Period The samle of this study is the comonent stocks of the uala Lumur Comosite Index (LCI). A ortfolio created from the comonent stocks of LCI has the advantages of liquidity, rotection and diversification (see [5]). The rice data were obtained from the Perfect Analysis database, while data on dividends and caital changes were extracted from the Perfect Analysis database, LSETRACER.COM and Bursa Malaysia Comany Announcements. The average one-month fixed deosit interest rate was used as the risk-free rate, and stock returns were adjusted for caital changes including rights issue, bonus issue, caital reayment, share slit and reverse slit. Monthly data from anuary 2000 through May 2006 (estimation eriod) were used to determine the weight for each stock in the ortfolio. The data from une 2006 through May 2007 were used for holdout validation (holdout eriod). A total of 83 comonent stocks were selected and the remaining 17 stocks were excluded because of incomleteness due to new listing or susension over a long eriod. 538

2 3. Performance Measure and Allocation Methods The ortfolio return is comuted as R w R P 1 1 T Pi Pi 1 + D where R 100%, 1, 2,..., T i 1 P, w is the roortion of investment on stock in the ortfolio, P i is the end-of-month rice of stock in month i, D is dividend, is the number of stocks in the ortfolio and T is the total number of observations. The ortfolio risk is measured by standard deviation given by T S ( Ri R )( Ri R ) /( T 1), 1, 2,...,, R i and i 1 i 1 S P w w S 1 1 where R i are the monthly returns of stock and stock in month i resectively. The risk reward for measuring ortfolio erformance is defined the average return er unit of risk given by R / S. Conditional otimal, equal weight, minimized variance and minimized standard error allocation methods were considered. The conditional otimal allocation method maximizes ( R R f ) / S, where R f is the risk free rate, subject to: w 1 (1) 1 w (2) if 3 then w 0.7 (3) if 5 then w 0.6 (4) if 7 then w minimum (0.5, 5/ ) (5) Constraint (2) is set to avoid small weights so that no stock in the ortfolio is under reresented. Constraints (3), (4) and (5) are to eliminate excessively big weights being assigned to one single stock. Constraint (5) is to kee the weights no greater than five times of the equal weight, but subject to a maximum of 50 ercent. The equal weight allocation method assigns a weight of 1 w. The weight for the minimized variance 2 2 allocation method is given by w (1 S ) /( (1 S )). The minimized standard error allocation method uses 1 the weight w (1 SE ) /( (1 SE )) where SE is the standard error of the caital asset ricing model 1 estimated by regressing the returns of individual stock on the market returns comuted from LCI. 4. Simulation and Results A simulation was erformed to examine how ortfolio size and allocation method affect risk reward. Stocks were selected randomly and ortfolios with size 3, 5, 7, 9, 11, 13, 15, 18, 20, 25, 30, 35, and 50 were formed. The simulation assumes that investors will make a monthly adjustment on the allocation of their ortfolios following any rice changes so that the initial roortion of fund allocation in the ortfolio is maintained. The simulation rocess was reeated for relication sizes of 10, 30, 50, 100, 200, and 500. The weight for each stock in the ortfolio was derived according to the four allocation methods using data for the estimation eriod. The risk rewards were comuted for each ortfolio. The same roortions of allocation to each stock in the ortfolio were used in the holdout validation to examine the robustness of the results. The mean risk rewards for ortfolio of different sizes were found to be significantly different and the results are consistent for all allocation methods and for both estimation and holdout samles (results not reorted, see [5] for details). Figures 1 and 2 clearly show that ortfolio risk reward is ositively correlated with the ortfolio size. Thus, diversification of investment by holding a ortfolio with a larger size can achieve better erformance, although it is also clear that the marginal contribution of a larger size increases at a decreasing rate. The equal weight allocation method roduces the lowest risk reward, and more aarently so when the number of relications increases. The results of the holdout analysis are consistent 539

3 with those of the within-samle analysis. While the conditional otimal method yields the highest risk reward, its sueriority does not hold for the holdout eriod. The second best minimized variance method in the estimation samle turned to yield the highest risk reward for the ortfolios constructed for the holdout samle. Fig. 1: 95% Confidence interval for risk rewards of ortfolios by number of stocks Ex-ost analysis Fig. 2: 95% Confidence interval for risk rewards of ortfolios by number of stocks Ex-ante analysis The grouing of ortfolios according to homogenous risk reward using the Games-Howell rocedure suggests an otimal ortfolio size of between 5 to 11 stocks, deending on the method of allocation used (see Table 1). The analysis shows that a ortfolio weighted by the conditional otimal method requires a minimum of 11 stocks in order to achieve a considerable high and accetable level of risk reward. For the other three allocation methods, at least 5 stocks are required to form a ortfolio with reasonable erformance. For the holdout samle, the lowest otimal ortfolio size among the four methods is 5 stocks for the conditional otimal allocation method, followed by 9 stocks for the equal weight method. The minimized 540

4 variance and minimized standard error allocation methods require at least 20 and 11 stocks, resectively. Overall, the otimal size is higher than that suggested from the within-samle analysis. The results in Table 2 show that a ortfolio s risk reward is related to the method of allocation. The Games-Howell comarison rocedure indicates that the mean difference of risk reward between each allocation method is highly significant, and this is true of both the samle eriods. For the estimation eriod, the conditional otimal method is the best, followed by the minimized variance allocation method. In contrast, the equal weight method rovides the lowest risk reward. When extended to the holdout samle eriod, the minimized variance method generates the highest value of risk reward. Table 1: Grous of ortfolio with homogeneous risk rewards using Games-Howell rocedure Grou Conditional otimal Equal weight Minimized variance Minimized standard error Estimation eriod 1 3 3, , 7, 9, 11, 13, 15 5, 7, 9 5, 7, 9, 11, 13, 15, , 9, 11, 13, 15, 18, 20 7, 9, 11, 13 7, 9, 11, 13, 15, 18, , 11, 13, 15, 18, 20, 25 7, 11, 13, 15 7, 11, 13, 15, 18, 20, 25, 30, , 13 11, 18, 20, 25, 30, 35 11, 13, 15, 18, 20 7, 11, 18, 20, 25, 30, 35, , 15 30, 35, 50 15, 18, 20, , 20 18, 20, 25, 30, , 30, 35, , Holdout eriod , 7, 9 5, 7 5 5, 7 3 7, 9, 11 7, 9 7, 9 7, , 13, 15 9, 11 11, 13, 15 9, , 15, 18 11, 13, 15 18, 20, 25 11, 13, , 18, 20 13, 15, 18 20, 25, 30 18, 20, , 20, 25, 18, 20, 30 25, 30, 35 20, 25, 30, , 50 20, 25, 30, 35 30, 35, 50 35, , 35, 50 Table 2: difference (i j) of risk rewards between allocation methods using Games-Howell rocedure Allocation method (i) Equal weight Allocation method (j) Minimized variance Minimized standard error Estimation eriod Conditional otimal Equal weight Minimized variance Holdout eriod Conditional otimal Equal weight Minimized variance

5 5. Conclusion This aer shows that both ortfolio size and allocation method affect the erformance of a ortfolio. Increasing ortfolio size yields higher return er unit of risk, albeit at a decreasing rate. While the simulation suggests an otimal ortfolio size that ranges from 5 to 20 stocks, most of the allocation methods yield reasonably high risk reward with a ortfolio size of about 11 stocks. The allocation method of minimized variance generates a high risk reward, while the equal weight method has the oorest erformance. 6. References [1] D.L. Domian, D.A. Louton, M.D. Racine. Diversification in ortfolios of individual stocks: 100 stocks are not enough. Finan. Rev. 2007, 42(4): [2] M. Statman. How many stocks make a diversified ortfolio.. Finan. Quant. Anal. 1987, 22(3): [3] T.. Chang,. Meade,.E. Beasley, and Y.M. Sharaiha. Heuristics for cardinality constrained ortfolio otimisation. Comut. Oer. Res. 2000, 27(13): [4] W. H. Wagner and S.C. Lau. The effect of diversification on risk. Financ. Anal , 27(6): [5].Y. g. The effect of size, selection strategy and allocation method on stock ortfolio erformance. Research Paer, Faculty of Economics & Administration, University of Malaya