A number of variants of the classical Markowitz mean-variance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions taken in the portfolio, in an attempt to improve its performance and reduce transactions costs. However, one can regard cardinality as an objective function itself, thus adding another goal to those two traditionally considered (the variance and the mean of the return). In this paper, we suggest a new approach to directly compute sparse portfolios by reformulating the cardinality constrained Markowitz mean-variance optimization model as a biobjective optimization problem, allowing the investor to analyze the efficient tradeoff between mean-variance and cardinality, in a general scenario where short-selling is allowed. Since cardinality is a nonsmooth objective function, one has chosen a derivative-free algorithm (based on direct multisearch) for the solution of the biobjective optimization problem. For the several data sets obtained from the FTSE 100 index and the Fama/French benchmark collection, direct multisearch was capable of quickly determining (in-sample) the efficient frontier for the biobjective cardinality/mean-variance problem. Our results showed that a number of efficient cardinality/mean-variance portfolios (with values of cardinality not high) overcome the naive strategy in terms of out-of-sample performance measured by the Sharpe ratio, which is known to be extremely difficult.

## Citation

preprint 12-05, Dept. Mathematics, Univ. Coimbra