Risk budgeting is a portfolio strategy where each asset contributes a prespecified amount to the aggregate risk of the portfolio. In this work, we propose an efficient numerical framework that uses only simulations of returns for estimating risk budgeting portfolios. Besides a general cutting planes algorithm for determining the weights of risk budgeting portfolios for arbitrary coherent distortion risk measures, we provide a specialised version for the Expected Shortfall, and a tailored Stochastic Gradient Descent (SGD) algorithm, also for the Expected Shortfall. We compare our algorithm to standard convex optimisation solvers and illustrate different risk budgeting portfolios, constructed using an especially designed Julia package, on real financial data and compare it to classical portfolio strategies.

风险预算是一种投资组合策略，其中每项资产对投资组合的总风险贡献预先指定的金额。在这项工作中，我们提出了一个有效的数值框架，该框架仅使用回报模拟来估计风险预算投资组合。除了用于确定任意相干失真风险度量的风险预算投资组合权重的通用割平面算法之外，我们还提供了预期缺口的专用版本，以及也适用于预期缺口的定制随机梯度下降（SGD）算法。我们将我们的算法与标准凸优化求解器进行比较，并说明使用专门设计的 Julia 包构建的不同风险预算投资组合，基于真实的财务数据，并将其与经典投资组合策略进行比较。