This paper provides a plausible explanation for why the optimum number of stocks in a portfolio is elusive, and suggests a way to determine this optimal number. Diversification is dependent on the number of stocks in a portfolio and the correlation structure. Adding stocks to a portfolio increases the level of diversification, and consequently leads to risk reduction. However the risk reduction effect dissipates after a certain number of stocks, beyond which additional stocks do not contribute to risk reduction. To explain this phenomenon, this paper investigates the relationship between portfolio diversification and concentration using a genetic algorithm.
To quantify diversification, we use the Portfolio Diversification Index (PDI). In the case of concentration, we introduce a new quantification method. Concentration is quantified as complexity of the correlation matrix. The proposed method quantifies the level of dependency (or redundancy) between stocks in a portfolio. By contrasting the two methods it is shown that the optimal number of stocks that optimizes diversification depends on both number of stocks and average correlation. Our result shows that, for a given universe, there is a set of Pareto optimal portfolios each containing a different number of stocks that simultaneously maximizes diversification and minimizes concentration. The preferred portfolio among the Pareto set will depend on the preference of the investor. Our result also suggests that an ideal condition for the optimal number of stocks is when the variance reduction as a result of adding a stock is off-set by the the variance contribution of complexity.