SPSS analysis on modern portfolio theory-optimal portfolio strategies in today’s capital market

Abstract
This paper provides information on specific ideas embedded in single index model/construction of optimal portfolios compared to the classic Markowitz model. Important arguments are presented regarding the validity of these two models. The researcher utilises SPSS analysis to demonstrate important research findings. This type of analysis is conducted to explore the presence of any significant statistical difference between the variance of the single index model and the Markowitz model. The paper also includes implications for investors.
Introduction

In the contemporary environment involving business investments, selecting appropriate investments is a relevant task of most organisations. Rational investors try to minimise risks as well as maximise returns on their investments (Better, 2006). The ultimate goal is to reach a level identified as optimal portfolios. The focus in this process is on initiating the portfolio selection models, which are essential for optimising the work of investors. Research shows that the Markowitz model is the most suitable model for conducting stock selection, as this is facilitated through the use of a full covariance matrix (Bergh and Rensburg, 2008).
The importance of this study reflects in the application of different models so as to develop adequate portfolios in organisations. It is essential to compare certain models because investors may be provided with sufficient knowledge about how they can best construct their portfolios. In this context, the precise variance of the portfolio selection model is important, as it reflects portfolio risk (Bergh and Rensburg, 2008). Information on the parameters of different models is significant to make the most appropriate decisions regarding portfolio creation. Markowitz is a pioneer in the research on portfolio analysis, as his works have contributed to enhancing investors’ perspectives on the available options regarding specific models of constructing optimal portfolios (Fernandez and Gomez, 2007).
Research Methodology
The research question presented in this study referred to the exploration of ideas embedded in single index model/construction of optimal portfolios and comparing them with the classic Markowitz model. The focus was on the construction of optimal portfolios, as the researcher was concerned with the evaluation of constructed portfolios with specific market parameters (Better, 2006). Moreover, the researcher paid attention to the stock market price index, including stocks of organisations distributed in three major sectors: services, financial, and industrial (Fernandez and Gomez, 2007). The behaviour of this index was explored through the implementation of SPSS analysis. The data covered a period of seven years, starting on January 1, 2000 and ending on December 31, 2006. It was essential to evaluate the effectiveness parameters of the single index model/construction of optimal portfolios and the Markowitz model. The criteria for the selection of companies included that all organisations shared the same fiscal year (ending each year on December 31) as well as they have not demonstrated any change in position.
Results and Data Analysis
The research methodology utilised in the study is based on the model of single index/optimal portfolios and the Markowitz model. The exploration of the relationship between these two models required the selection of 35 equally weighted optimal portfolios, as two sizes of portfolio were outlined. An approximate number of 10 optimal portfolios represented the first size, which further generated 12 portfolios. In addition, the researcher considered the option of simulating of optimal portfolios represented at second sizes (Bergh and Rensburg, 2008). The criterion of queuing randomise portfolio selection has been used to generate approximate 23 portfolios from the second size category. The researcher selected five and 10 stocks to analyse the data. The portfolio size split allowed the researcher to explore how the portfolio size could be used to affect the relationship between the single index model/optimal portfolios and the Markowitz model (Fernandez and Gomez, 2007). Results of testing the data are provided in the table below:
Optimal portfolio numberVariance of Single Index ModelVariance of the Markowitz ModelOptimal portfolio numberVariance of the Single Index ModelVariance of the Markowitz Model
100.00370.003950.00210.0023
100.00140.001750.00280.0038
100.00210.002850.00420.0051
100.00200.002150.00250.0030
100.00310.003550.00260.0024
100.00190.001950.00330.0038
100.00880.008650.00670.0071
100.00280.003750.00370.0053
100.00250.002450.00380.0043
100.00220.002350.00210.0020
100.00190.002050.00630.0061
100.00230.002650.02120.0202
Table 1: Variance of Five and 10 Optimal Portfolios
Based on the results provided in the table, it can be concluded that the variance between the single index model/construction of optimal portfolios and the Markowitz model is similar. For instance, values of 0.0020 and 0.0019 for the variance of the two models are similar. This means that the results do not show substantial statistical differences between the two models. The tables below contain a descriptive summary of the results presented in the previous table:
MeasureSingle Index ModelMarkowitz Model
Mean0.00440.0047
Minimal0.00210.0020
Maximum0.02120.0202
Standard Deviation0.00370.0035
Table 2: Descriptive Summary of 10 Optimal Portfolios
The results in Table 2 were derived from testing the performance of 10 optimal portfolios. It has been indicated that the mean for the single index model of 10 portfolios is 0.0044, while the mean for the Markowitz model is 0.0047, implying an insignificant statistical difference. The minimal value of the single index model is reported at 0.0021, while the minimal value of the Markowitz model is 0.0020. The difference is insignificant. The maximum value of the single index model is 0.0212, while the same value of the Markowitz model is 0.0202. Based on these values, it can be argued that there is a slight difference existing between the two models. The standard deviation of the single index model is 0.0037, while the standard deviation of the Markowitz model is 0.0035, which also reflects an insignificant statistical difference.
MeasureSingle Index ModelMarkowitz Model
Mean0.00280.0031
Minimal0.00140.0017
Maximum0.00880.0086
Standard Deviation0.00200.0019
Table 3: Descriptive Summary of 5 Optimal Portfolios
Table 3 provides the results for five optimal portfolios. These results are similar to the ones reported previously (10 optimal portfolios). The mean for the single index model of 5 optimal portfolios is 0.0028, while the mean for the Markowitz model is 0.0031, implying an insignificant statistical difference. There are insignificant differences between the two models regarding other values, such as minimal and maximum value as well as standard deviation.
Furthermore, the researcher performed an ANOVA analysis of 10 optimal portfolios, which are presented in the table below. It has been indicated that the effective score for the single index model and the Markowitz model is almost the same. Yet, an insignificant difference was reported between the two means and standard deviations for both models.
ANOVA AnalysisSum of squaresDfConditionMeanStandard DeviationStandard Error MeanFSig.
Between Groups.00011.000.003125.0018704.0005399.089.768
Within Groups.000222.000.002892.0019589.0005655
Total.00023
Table 4: ANOVA Analysis for the Variance between the Single Index Model and the Markowitz Model of 10 Portfolios
From the conducted analysis, it can be also concluded that the F-test presents an insignificant statistical value, implying that the researcher rejected the hypothesis of a significant difference existing between portfolio selections with regards to risk in both models used in the study (Fernandez and Gomez, 2007). Hence, the hypothesis of a significant difference between the variance of the single index model and the Markowitz model was rejected (Lediot and Wolf, 2003). In the table below, the researcher provided the results of an ANOVA analysis conducted on five optimal portfolios:
ANOVA AnalysisSum of SquaresDfConditionMeanStandard DeviationStandard Error MeanFSig.
Between Groups.00011.000.004852.0036535.0007618.096.758
Within Groups.001442.000.004509.0038595.0008048
Total.00145
Table 5: ANOVA Analysis for the Variance between the Single Index Model and the Markowitz Model of 5 Portfolios
The results from Table 5 show that the variance between the single index model and the Markowitz model of five optimal portfolios is almost the same. Regardless of the stock number in the selected optimal portfolios, there is no significant statistical difference between the single index model and the Markowitz model.
The main finding based on the reported data is that the single index model/construction of optimal portfolios is similar to the Markowitz model with regards to the formation of specific portfolios (Bergh and Rensburg, 2008). As indicated in this study, the precise number of stocks in the constructed optimal portfolios does not impact the final result of comparing the two analysed models. The fact that these models are not significantly different from each other can prompt investors to use the most practical approach in constructing optimal portfolios (Haugen, 2001). Placing an emphasis on efficient frontiers is an important part of investors’ work, as they are focused on generating the most efficient portfolios at the lowest risk. As a result, optimally selected portfolios would be able to generate positive returns for organisations. This applies to both the single index model and the Markowitz model (Fernandez and Gomez, 2007).
Conclusion and Implications of Research Findings
The results obtained in the present study are important for various parties. Such results may be of concern to policy makers, investors as well as financial market participants. In addition, the findings generated in the study are similar to findings reported by other researchers in the field (Bergh and Rensburg, 2008). It cannot be claimed that either of the approaches has certain advantages over the other one. Even if the number of stocks is altered, this does not reflect in any changes of the results provided by the researcher in this study. Yet, the major limitation of the study is associated with the use of monthly data. It can be argued that the use of daily data would be a more viable option to ensure accuracy, objectivity as well as adherence to strict professional standards in terms of investment (Better, 2006).
In conclusion, the similarity of the single index model and the Markowitz model encourage researchers to use both models equally because of their potential to generate optimal portfolios. Moreover, the lack of significant statistical differences between the variance of the single index model and the Markowitz model can serve as an adequate basis for investors to demonstrate greater flexibility in the process of making portfolio selection decisions (Haugen, 2001). The results obtained in the study were used to reject the hypotheses that were initially presented. As previously mentioned, the conducted F-test additionally indicates that the single index model and the Markowitz model are almost similar in scope and impact (Fernandez and Gomez, 2007).
Investors should consider that portfolio selection models play an important role in determining the exact amount of risk taking while constructing optimal portfolios. Hence, investors are expected to thoroughly explore those models while they select their portfolios (Garlappi et al., 2007). Both individual and institutional investors can find the results generated in this study useful to facilitate their professional practice. A possible application of the research findings should be considered in the process of embracing new investment policies in the flexible organisational context (Bergh and Rensburg, 2008). Future research may extensively focus on the development of new portfolio selection models that may further expand the capacity of organisations to improve their performance on investment risk taking indicators.
References
Bergh, G. and Rensburg, V. (2008). ‘Hedge Funds and Higher Moment Portfolio Performance Appraisals: A General Approach’. Omega, vol. 37, pp. 50-62.
Better, M. (2006). ‘Selecting Project Portfolios by Optimizing Simulations’. The Engineering Economist, vol. 51, pp. 81-97.
Fernandez, A. and Gomez, S. (2007). ‘Portfolio Selection Using Neutral Networks’. Computers & Operations Research, vol. 34, pp. 1177-1191.
Garlappi, L., Uppal, R., and Wang, T. (2007). ‘Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach’. The Review of Financial Studies, vol. 20, pp. 41-81.
Haugen, R. (2001). Modern Investment Theory. New Jersey: Prentice Hall.
Lediot, O. and Wolf, M. (2003). ‘Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection’. Journal of Finance, vol. 10, pp. 603-621.