Given the wealth of investment options available, and the many potential outcomes associated with each purchase, stock market investing can be daunting for everyone, especially for beginners and for communities historically underrepresented in finance who may have limited access to formal financial education. Stochastic Package Queries (SPQs) offer a user-friendly and efficient solution, allowing everyone to simply express their decision-making problems, and receive optimal recommendations under uncertain conditions in seconds. Under the hood, SPQs are processed by a highly scalable system comprising stock databases, linear program optimizers, market simulators, and customized algorithms that work in tandem to derive personalized and optimal investment strategies based on millions of simulated possible worlds. Unfortunately, SPQs can only optimize expected returns, and thus, when applied to portfolio construction, often recommend risky investments that look strong “on average”, but perform poorly in bear markets due to limited diversification.

We enrich the class of problems that can be solved scalably using SPQs, by allowing them to return risk-optimized portfolios that minimize expected losses in their worst case outcomes, or the Conditional Value-at-Risk (CVaR). We derive a mathematical formulation of the CVaR optimization problem based on numerous simulated possible worlds of future returns. Using auxiliary variables, our formulation represents tail-loss exceedances in a linear, optimization-friendly form. This enables CVaR to be optimized within the same Integer Linear Programming structure used by SPQs, preserving the original feasibility constraints and search process while explicitly prioritizing protection against extreme downside outcomes.