The Optimal Number of Securities in a Risky Asset Portfolio When There Are Fixed Costs of Transacting: Theory and Some Empirical Results

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The normative theory of portfolio selection has, since Markowitz, proceeded for the most part on the assumption that there are no costs of transacting in securities markets. Exceptions to this generalization are the work of Pogue who proposes a quadratic programming solution to the portfolio selection problem with variable transactions costs, and, in a multiperiod context, the ad hoc portfolio revision models of Smith, and the more rigorous, though computationally burdensome dynamic programming models of Chen et al. All of these models focus exclusively on the variable costs of transacting. Mao deals implicitly with the fixed costs of purchasing securities and the limited diversification which these will imply. However, his model both lacks an explicit optimization criterion for determining the number of securities to include in the portfolio and assumes a homogeneous security universe; this latter assumption is relaxed when he later considers the problem of which securities should be included in the portfolio. Clearly an ideal solution to the problem must consider simultaneously both how many and which securities to include. More recently, Jacob has developed some simplified models for selecting optimal portfolios, given a constraint on the number of securities which may be included in the portfolio. While her models are superior to Mao's in taking account of the residual risk of the securities as well as their systematic risk, they do not contain any explicit procedure for determining the optimal number of securities to include in the portfolio.