We develop a systematic approach to bounding entropy by incorporating conditioning information. Our bounds feature a fixed-point solution to a dynamic asset-allocation problem, interpretable as generalized “Sharpe ratios” in the entropy space—our bounds balance exploiting physical return predictability and hedging risk-neutral higher-order moments. Applying our approach to various return predictors, we document enhanced entropy restrictions that more than double the benchmark equity risk premium. When incorporating higher-order return moments, our bounds are sharper than the corresponding optimally scaled Hansen-Jagannathan bounds over short horizons. We highlight our results’ implications in diagnosing leading macro-finance models and their consistency across different data.