Statistical comparison of international size-based equity index using a mixture distribution

Investors and financial analysts spend an inordinate amount of time, resources and effort in an attempt to perfect the science of maximising the level of financial returns. To this end, the field of distribution modelling and analysis of firm size effect is important as an investment analysis and appraisal tool. Numerous studies have been conducted to determine which distribution best fits stock returns (Mandelbrot, 1963; Fama, 1965 and Akgiray and Booth, 1988). Analysis and review of earlier research has revealed that researchers claim that the returns follow a normal distribution. However, the findings have not been without their own limitations in terms of the empirical results in that many also say that the research done does not account for the fat tails and skewness of the data. Some research studies dealing with the anomaly of firm size effect have led to the conclusion that smaller firms tend to command higher returns relative to their larger counterparts with a similar risk profile (Banz, 1981). Recently, Janse van Rensburg et al. (2009a) conducted a study in which both non- normality of stock returns and firm size effect were addressed simultaneously. They used a scale mixture of two normal distributions to compare the stock returns of large capitalisation and small capitalisation shares portfolios. The study concluded that in periods of high volatility, the small capitalisation portfolio is far more risky than the large capitalisation portfolio. In periods of low volatility they are equally risky. Janse van Rensburg et al. (2009a) identified a number of limitations to the study. These included data problems, survivorship bias, exclusion of dividends, and the use of standard statistical tests in the presence of non-normality. They concluded that it was difficult to generalise findings because of the use of only two (limited) portfolios. In the extension of the research, Janse van Rensburg (2009b) concluded that a scale mixture of two normal distributions provided a more superior fit than any other mixture. The scope of this research is an extension of the work by Janse van Rensburg et al. (2009a) and Janse van Rensburg (2009b), with a view to addressing several of the limitations and findings of the earlier studies. The Janse van rensburg (2009b) study was based on data from the Johannesburg Stock Exchange (JSE); this study seeks to compare their research by looking at the New York Stock Exchange (NYSE) to determine if similar results occur in developed markets. For analysis purposes, this study used the statistical software package R (R Development Core Team 2008) and its package mixtools (Young, Benaglia, Chauveau, Elmore, Hettmansperg, Hunter, Thomas, Xuan 2008). Some computation was also done using Microsoft Excel. This dissertation is arranged as follows: Chapter 2 is a literature review of some of the baseline studies and research that supports the conclusion that earlier research finding had serious limitations. Chapter 3 describes the data used in the study and gives a breakdown of portfolio formation and the methodology used in the study. Chapter 4 provides the statistical background of the methods used in this study. Chapter 5 presents the statistical analysis and distribution fitting of the data. Finally, Chapter 6 gives conclusions drawn from the results obtained in the analysis of data as well as recommendations for future work