Abstract: The portfolio selection problem with cardinality constraint is a hot issue in recent years. Much attention is paid to solve this problem because managing a portfolio with many assets often leads to high transaction costs and is a rather time-and energy-consuming experience in practice. However, the parameter uncertainty directly affects the effect of the model and makes it difficult to achive best performance of the portfolio. The parameters of the portfolio selection problem with cardinality constraint include not only the expected rate of return which was considered to be very important in previous studies, but also the sparsity controlling the size of the portfolio. Especially, there hasn't been much special research on estimation of portfolio's sparsity. In order to select optimal parameters better for investment decision, a sparse bi-level parameter estimation model is constructed for portfolio with cardinality constraint. The outer layer of the model is designed to maximize the utility of the portfolio which is measured by Sharpe ratio, while the inner layer of the model is designed to minimize the risk of a portfolio under a given expected return.The outer layer function is non-convex and non-smooth, so it is difficult to solve by the traditional gradient method. What's more, A framework of derivative-free optimization algorithm is built based on the model and we use ADMM to solve the sub problems. In particular, ADMM can get the closed-form solution of the sub-problem, which shows that the algorithm is effective.In this paper, numerical experiments are conducted by real-life data from OR-library and some Chinese stock markets(SSE 50、CSI 100)to estimate the expected rate of return and sparsity. The estimated sparsity is much smaller than the number of risky assets contained in each index,which will greatly ease the difficulty of portfolio management and decrease the transaction cost. The effectiveness of the model and algorithm is illustrated through comparison with the classical MV model with bounded constraints and the equal weight strategy(naive strategy).Specifically,our method improves the Sharpe ratio by at least 18.99% compared with 1/N strategy and by at least 39.03% compared with MV model which has upper and lower bounds out of sample. Finally, the proposed sparse bi-level parameter estimation model is extended to a general form, which can help estimate more accurate upper and lower bounds as well as other parameters. This is the direction of our future research.

Key words: cardinality constraints, parameter estimation, bi-level model, derivative-free optimization, optimal sparsity