# Application of the Special Constrained Multiparametric Linear Program to Portfolio Selection Decisions

## Article excerpt

INTRODUCTION

Individual investors who are faced with portfolio selection decisions may choose from diversification techniques which range from the most simple and naive forms to more highly complex schemes requiring quadratic programming. Regardless of the technique employed, it is widely recognized that the foremost goal of most investors is a common one: to achieve the maximum rate of return at the level of risk considered appropriate for the individual investor's portfolio.

While the risk to total return tradeoff is important, investors do exhibit individual references for other investment features as well. This is evident when one considers the many types of investments available from which to choose. Some investments offer maximum growth potential in capital appreciation; some periodic income distributions; some liquidity; and still others some combination of all the above.

Given the myriad of investment opportunities available to the individual investor, the need exists for a "tool" which will permit the investor to combine the different types of investments in such a way as to minimize investment risk and maximize investment returns while satisfying individual preferences. The purpose of this research is to demonstrate how the Special Constrained Multiparametric Linear Program (SCMLP) model can facilitate this complex decision process in such a manner as no other method can.

The next section of this research explores some of the widely accepted theoretical and practical means that have been devised to make portfolio selection decisions. The third section presents the SCMLP and related theory. The fourth section demonstrates how the SCMLP may be employed in the portfolio selection decision process. Finally, the last section provides a summary and the conclusions of this research.

THEORETICAL BACKGROUND: PORTFOLIO SELECTION TECHNIQUES

The classical model for portfolio analysis was developed by Markowitz (1952). The theoretical substance of the Markowitz model is brilliant and is rightfully accepted in finance literature. However, its practical usage is limited by the following factors:

(1) The quadratic programming solution procedure requires a considerable amount of computer time and space.

(2) The complexity of the model makes it difficult to explain to an individual user.

(3) The Markowitz model focuses only on risk and total returns and does not consider other individual preferences.

Various attempts have been made to overcome the practical problems posed by the Markowitz model. For instance, Sharpe's single index model greatly simplifies the solution procedure and the time and cost of obtaining a solution. However, this approach may not lead to the optimal mean--variance portfolio and does not take into consideration other preferences of individual investors (Sharpe, 1963).

A variation of Sharpe's single index model, developed by Elton, Gruber, and Padberg (1976), uses the risk measure from Sharpe's model (BETA) in a returns to risk ratio which is then used to determine the proportion of the portfolio to be invested in each asset. In fact, Burgess and Bey (1988) demonstrate that the Elton, Gruber, and Padberg procedure is effective in estimating the Markowitz efficient portfolio and can be an effective screening procedure for large number of securities. Again, however, regardless of the simplification and effectiveness of the Elton, Gruber, and Padberg approach, individual references for investment factors other than risk and total returns are not taken into consideration in the portfolio selection process.

The literature is also replete with articles on linear programming approaches to portfolio selection and optimization which do take into consideration the individual preferences of investors (e.g., Lee & Lerro, 1973; Beazer, 1976; Mitchell & Rosebery, 1977; Courtney, 1979; Avery; Hodges & Schaefer, 1977). …

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