一类稀疏投资组合双层参数估计模型及其应用

带基约束的投资组合问题是近年来投资组合领域的热点问题，但是参数不确定性直接影响了模型的效果。带基约束的投资组合问题所涉及的参数不仅包括以往研究认为非常重要的预期收益率，还包括控制投资组合规模的稀疏度，尤其是最优稀疏度估计方面的专门研究还十分匮乏。为了使带基约束的投资组合模型更好地为投资决策服务，本文从投资者效用出发，用双层规划的思想构建了带基约束的投资组合双层参数估计模型。然后根据模型的特点，设计了无导数优化算法框架，并基于ADMM对算法子问题进行求解。本文实验针对真实的市场数据给出了预期收益率和最优稀疏度的估计，接着通过与等权重策略和含上下界约束的均值-方差模型进行比较，说明了模型及算法的有效性和实用性。最后，将本文提出的双层参数估计模型推广到了更一般的形式。

A Class of Bi-level Parameter Estimation Models for Sparse Portfolios and Its Application

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.