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# The results in Table reveal

The results in Table 2 reveal that the cardinality-constrained portfolios outperform the benchmark index in all cases. For instance, under daily re-balancing frequency the minimum variance portfolio with 3, 5 and 10 assets deliver statistically lower portfolio risk (1.209, 1.120, and 0.956, respectively) in comparison to the Ibovespa index (1.436) and also in comparison to the equally weighted portfolio (1.298). This result also holds when the optimal portfolios are re-balanced on a weekly and monthly basis. The most important message from this result is that it SNDX275 cost is possible to build a portfolio less riskier that the market portfolio with very few assets, therefore corroborating the evidence in Jacob (1974) and Johnson and Shannon (1974).
To further illustrate the results, we plot in Fig. 1 the cumulative returns under daily re-balancing of the cardinality-constrained minimum variance portfolios with 3 (blue line), 5 (green line), and 10 (red line) assets along with the cumulative returns of the benchmark index and of the equally weighted portfolio. The figure shows a stellar difference in cumulative returns across the out-of-sample period: while the minimum variance portfolio achieves a cumulative portfolio return of approximately 45% throughout the period, the benchmark index and the equally weighted portfolio deliver cumulative returns of −25% and −15% approximately.

Concluding remarks
The implementation of the mean-variance portfolio policy of Markowitz (1952) for small investors can be problematic, as efficient portfolios sometimes contain too many securities. One alternative to overcome this difficulty is the cardinality-constrained version of the mean-variance portfolio optimization problem, in which the optimal portfolio is restricted to a limited number of assets. Despite its practical relevance, this approach has a fundamental difficulty as the resulting optimization problem becomes non-differential as discontinuous due to the inclusion of a cardinality constraint.
The empirical evidence of this paper provides favorable evidence to the cardinality-constrained portfolios. We find that optimal portfolios containing only 3 assets outperform the market portfolio in terms of lower risk and higher Sharpe ratios, and that this result is robust to the choice of portfolio re-balancing frequency. Overall, our results corroborate the evidence in Jacob (1974) and Johnson and Shannon (1974) as we find that it is possible to obtain better risk-adjusted performance with fewer securities in the portfolio by using an alternative allocation scheme.

Introduction
Banking spread in Brazil has been for long at high levels and, although with a downward trend in recent years, it still remains at a relatively higher level when compared to the values observed in the rest of the world. A commonly used explanation is that high levels of basic interest rate cause the high levels of banking spread in the country. However, an important current of thought – pioneered by Bresnahan (1982) and Lau (1982) – considers that the microeconomic factors play a fundamental role in understanding the banking spread. This consideration gains more relevance when observing the measures recently adopted by the government of Dilma Rousseff in Brazil, which sought to promote greater competition between banks by using public banks as boosters of private banks.
To achieve the objectives and test the hypothesis of the paper, in addition to this introduction, the paper is structured in four sections. Section 2 presents the theoretical and empirical research about the determinants of banking spread as well as the mathematical model to substantiate the analysis. In sequence, the discussion turns to the methodology to be employed. In Section 4, the empirical analysis of the determinants of banking spread in the Brazilian economy will be conducted. Finally, we present the concluding remarks.