# It is also worth noting

It is also worth noting that the current relevance of cardinality-constrained portfolios is controversial. The main reason is straightforward: with the advent of new financial instruments such as exchange traded funds (ETF), nowadays investors can easily buy or sell a benchmark portfolio by investing only in a single asset, thus overcoming potential restrictions that arise when many assets have to be traded. Therefore, the practical relevance of the cardinality-constrained portfolio is questionable as the financial market provides specific instruments that replicates near-to-perfection alternative benchmark portfolios such as stock market indices. One potential concern that arises when addressing this question is the underdiversification of the resulting portfolios. That is, investors holding a cardinality-constrained portfolio might be subjected to a high level of unsystematic risk. In fact, Statman (1987) shows that for the US market a diversified portfolio would contain at least 30 stocks. However, the study of the relationship between portfolio size and risk in Statman (1987) is based on equally weighted randomly selected assets. In this sense, Jacob (1974) and Johnson and Shannon (1974) argue that it purchase Tozadenant is possible to obtain the same level of variation with far greater average portfolio returns and – more importantly – with fewer securities in the portfolio by using an alternative allocation scheme, thus enforcing the argument in favor of the optimal cardinality-constrained portfolios. In this paper we obtain cardinality-constrained minimum variance portfolios by implementing the reformulation proposed by Coleman et al. (2006) of the original (cardinality-unconstrained) problem. The advantages of this approach are threefold. First, it approximates the discontinuous counting function with a sequence of continuously differentiable non-convex piecewise quadratic functions which approaches the original non-differentiable counting function in the limit. Second, the approach can be easily implemented in most commercial packages and standard optimization algorithms such as Matlab\'s fmincon. Third, simulation results reported in Coleman et al. (2006) show that this approach outperforms other approaches such as the one proposed in Jansen and Van Dijk (2002) in yielding cardinality-constrained portfolios with lower tracking error. We provide empirical evidence involving a large data set consisting of daily returns of 45 stocks traded at the Brazilian equities markets from March/2009 to November/2011. We implement the cardinality-constrained minimum variance using the reformulation proposed in Coleman et al. (2006) along with a robust estimate of the covariance matrix of stock returns as proposed in Ledoit et al. (2004). We consider optimal portfolios with 3, 5, and 10 assets and a daily, weekly, and monthly re-balancing frequencies. Moreover, we formally test the differences in portfolio risk (standard deviation) and in risk-adjusted performance measured in terms of Sharpe ratios with respect to the Ibovespa index by employing a bootstrap approach proposed in Politis and Romano (1994). In this sense, the paper adds to the literature by providing a realistic out-of-sample implementation and evaluation of the cardinality-constrained portfolios and provide comparative analysis with respect to the main benchmark index under alternative re-balancing frequencies. The results are favorable to the cardinality-constrained portfolios considered in the paper. We find that optimal portfolios containing only 3 assets outperform the market portfolio in terms of lower risk and higher Sharpe ratios. Moreover, this result is also robust to the choice of portfolio re-balancing frequency. For instance, in the case of monthly re-balancing the portfolio standard deviation of the cardinality-constrained minimum variance portfolio with only 3 assets is 1.14, whereas the same figure for the Ibovespa is 1.43. The Sharpe ratio of the minimum variance portfolio with 3 assets is 0.068 whereas the same figure for the Ibovespa is -0.038. The results with daily and weekly re-balancing frequencies are even more favorable to the cardinality-constrained portfolios, although this is also accompanied by an increase in portfolio turnover. Overall, our results corroborate the evidence in Jacob (1974) and Johnson and Shannon (1974) as we find that Lysosomes is possible to obtain better risk-adjusted performance with fewer securities in the portfolio by using an alternative allocation scheme.