Simplify the input and portfolio-building measures, hazardous optimal Markowitz model the framework for a single index
THE IRAQI MAGAZINE FOR ADMINISTRATIVE SCIENCES,
Volume 8, Issue 31, Pages 87-121
AbstractPortfolio selection has been one of the most important research field in contemporary finance. In this context Markowitz's model (1952) has been considered as a pioneering solution for this problem. A lot of models and extensions have been proposed to improve the performance of portfolio which have led to a great number of researches and studies with the aim of formulating risk and returns of economic factors and understanding diversification in investment strategies. Markowitz's portfolio selection model present two main difficulties for being applied. First one, data required. If we could accurate expectations about future returns for each asset and the correlation of returns between each pair of assets then, the Markowitz's model under certain conditions and supposed known the investors' utility function, would produce the optimal portfolio. The obtaining of accurate forecast of input data needed for this model is a difficult task, particularly in the case of covariance matrix between securities. The estimation can get very complex as the portfolio size becomes large. For instance, if the number of stocks in a portfolio is (3000), as it is in NYSE, we need to estimate approximately (4.5) millions correlation coefficients!. The large number of inputs can be computationally impractical due to the large number of estimates that have to be made. Second one, there is a computational difficulty associated to the resolution of large-scale quadratic programming problems with a dense covariance matrix. To simplify analysis, the single-index model assumes that there is only one economic factor that causes the systematic risk affecting all stock returns and this factor can be represented by the rate of return on a market index. According to this model, the return and risk of any stock can be decomposed into two parts, one due to firm-specific factors and the other due to general factors that affect the all market. With this model, only the betas of the individual securities and the market variance need to be estimated to calculate covariance(Markowitz's model problem). Hence, the index model greatly reduces the quality and quantity of inputs and procedures that would have to be made to constructing optimal portfolio. But the model's beta suffers from imprecision in its estimation. So, developed techniques to modify it and make it more accurate.
Thus, this research will aim to introduce the single-index model as a potential solution to simplify the inputs and procedures of the optimal portfolio calculation. To achieve this objective, we will discuss the assumptions and concepts underlying the single-index model. We also will outline the estimation procedures of the model and highlight issues about model's beta estimation and the most important techniques to solve it, and test all that in the Iraq stock exchange for the period (January 2007- March 2011).
The research found a number of conclusions and the most important of these is that the index portfolio is an efficient portfolio only if all alpha values are zero. This makes intuitive sense. Unless security analysis reveals that a security has a nonzero alpha, including it in the active portfolio would make the portfolio less attractive. In addition to the security's systematic risk, which is compensated for by the market risk premium (through beta), the security would add its specific risk to portfolio variance. With a zero alpha, however, the latter is not compensated by an addition to the nonmarket risk premium. Hence, if all securities have zero alphas, the optimal weight in the active portfolio will be zero, and the weight in the index portfolio will be(1.0). However, when security analysis uncovers securities with nonmarket risk premiums(nonzero alphas), the index portfolio is no longer efficient.
The research found a number of recommendations and the most important of these is necessity to adopt the most widely used model in the simplification of building the optimal portfolio, which is a single-index model and follow the best techniques to estimate and modify its parameters, especially beta.
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