Portfolios employing a (factor) risk model are usually constructed using a two step process: first, the risk model parameters are estimated, then the portfolio is constructed. Recent works have shown that this decoupled approach may be improved using an integrated framework that takes the downstream portfolio optimization into account during parameter estimation. In this work we implement an integrated, end-to-end, predict-&-optimize framework to the cardinality-constrained portfolio optimization problem. To the best of our knowledge, we are the first to implement the framework to a nonlinear mixed integer programming problem. Since the feasible region of the problem is discontinuous, we are unable to directly differentiate through it. Thus, we compare three different continuous relaxations of increasing tightness to the problem which are placed as an implicit layers in a neural network. The parameters of the factor model governing the problem’s covariance matrix structure are learned using a loss function that directly corresponds to the decision quality made based on the factor model’s predictions. Using real world financial data, our proposed end-to-end, decision based model is compared to two decoupled alternatives. Results show significant improvements over the traditional decoupled approaches across all cardinality sizes and model variations while highlighting the need of additional research into the interplay between experimental design, problem size and structure, and relaxation tightness in a combinatorial setting.

中文翻译：

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端到端、基于决策、基数约束的投资组合优化


采用（因子）风险模型的投资组合通常使用两个步骤构建：首先，估计风险模型参数，然后构建投资组合。最近的研究表明，这种解耦的方法可以使用一个集成框架来改进，该框架在参数估计过程中考虑了下游投资组合的优化。在这项工作中，我们实现了一个集成的、端到端的、预测-优化的框架，以解决基数约束的投资组合优化问题。据我们所知，我们是第一个将框架实现到非线性混合整数规划问题的公司。由于问题的可行区域是不连续的，我们无法通过它直接进行区分。因此，我们比较了三种不同的连续松弛，它们被放置在神经网络中作为隐含层。控制问题协方差矩阵结构的因子模型的参数是使用损失函数学习的，该损失函数直接对应于根据因子模型的预测做出的决策质量。使用真实的金融数据，我们提出的端到端、基于决策的模型与两种解耦的替代方案进行了比较。结果表明，在所有基数大小和模型变化方面，与传统的解耦方法相比，都有了显著的改进，同时强调了对实验设计、问题大小和结构以及组合环境中松弛紧密性之间的相互作用进行额外研究的必要性。