The paper is organized as follows In Section

The paper is organized as follows. In Section 2 we detail the formulation of the cardinality-constrained minimum variance portfolios and the methodology to evaluate portfolio performance. Section 3 presents the data used in the paper. Section 4 discusses the results of the empirical application. Finally, Section 5 brings concluding remarks.
Cardinality-constrained minimum variance portfolios In this section, we review the cardinality-constrained minimum variance portfolio and discuss the implementation strategy and methodology to assess out-of-sample performance. We start by defining a vector of portfolio returns R=R, …, R where N is the number of assets in the portfolio and t=1, …, T is the number of observations. Returns are computed as the first differences in log prices. The cardinality-constrained optimal portfolio problem is given bywhere is the vector of portfolio weights, is the portfolio objective function and if and otherwise. is a shortselling restriction. The cardinality constraint, , is a counting function and can be interpreted as an upper bound on the number of assets allowed to enter the portfolio. Coleman et al. (2006) point out that this problem is NP-hard and the existing methods for solving (1) are heuristic-based. The choice of the portfolio objective function in Eq. (1) will determine the nature of the problem. A common choice for is the mean-variance function according to Markowitz (1952), i.e. where Σ is a positive-definite covariance matrix, is the portfolio return, and δ is the risk-aversion coefficient. In this framework, individuals choose their allocations in risky assets based on the trade-off between expected return and risk. In order to implement the mean-variance optimization in practice, it orexins is common to obtain estimates of the vector of expected returns and its covariance matrix and plug these estimators in an analytical or a numerical solution to the mean-variance problem. This problem is known to be very sensitive to estimation of the mean returns (e.g. Michaud, 1989; Jagannathan and Ma, 2003). Very often, the estimation error in the mean returns degrade the overall portfolio performance and introduces an undesirable level of portfolio turnover. In fact, existing evidence suggest that the performance of optimal portfolios that do not rely on estimated mean returns is better. Following Fan et al. (2012), to avoid the difficulty associated with estimation of the expected return vector, from now on, we consider the minimum variance portfolio objective function, i.e. .
Data The data used in Vector paper consists of daily returns of N=45 stocks traded in the São Paulo\'s BMFBovespa stock exchange in Brazil from 02/03/2009 to 24/11/2011, yielding a total of T=677 observations. All selected stocks belonged to the composition of the Ibovespa stock market index during the sample period. Table 1 reports the ticker of each stock along with descriptive statistics. We observe that the data displays stylized facts in financial time series, such as mean returns close to zero and excess kurtosis with respect to that of the Gaussian distribution.
Results We report in Table 2 the performance statistics of the cardinality-constrained minimum variance portfolio and of the benchmark portfolios. We obtain cardinality-constrained portfolios with a target number of K= 3, 5, 10 assets in the portfolio. Also, we consider as benchmark portfolios the Ibovespa index and also an equally weighted portfolio that assigns equal weights in all the 45 stocks in the portfolio. Table reports the mean return, standard deviation of returns, Sharpe ratio, and portfolio turnover. All figures are based on daily returns. Portfolio compositions are re-balanced on a daily basis. However, the transaction costs involved in this re-balancing frequency might degrade the performance of the portfolios and hinder its implementation in practice. Thus, the performance of optimized portfolios is also evaluated in the case of weekly and monthly re-balancing frequencies. A potentially negative effect of adopting a lower re-balancing frequency is that the optimal compositions may become outdated.