End-to-end risk budgeting portfolio optimization with neural networks

Traditional stochastic optimization in financial operations research applications consist of a two-step process: (1) calibrate parameters of the assumed stochastic processes by minimizing a loss function, and (2) optimize a decision vector by reference to the investor’s reward/risk measures. Yet this approach can encounter the error maximization problem. We combine the steps in a single unified feedforward network, called end-to-end. Two variants are examined: a model-free neural network, and a model-based network in which a risk budgeting model is embedded as an implicit layer in a deep neural network. We performed computational experiments with major ETF indices and found that the model-based approach leads to robust performance out-of-sample (2017–2021) when maximizing the Sharpe ratio as the training objective, achieving Sharpe ratio of 1.16 versus 0.83 for a pure risk budgeting model. Simulation studies show statistically significant difference between model-based and model-free approaches as well. We extend the end-to-end algorithm by filtering assets with low volatility and low returns, boosting the out-of-sample Sharpe ratio to 1.24. The end-to-end approach can be readily applied to a wide variety of financial and other optimization problems.