This study extends the maximal affine models of single assets to a multi-commodity setup. We show that the correlated version of maximal affine models for a single commodity is no longer maximal for multiple commodities. In the maximal model, the convenience yield of a certain commodity could depend on the prices of other commodities, which is consistent with the structural model in our companion study Casassus, Liu, and Tang [Review of Financial Studies, 26, 1324–1362, 2013]. This cross-commodity relationship is a feedback effect that may generate substantial co-movement among long-run commodity prices, a fact that is consistent with many empirical studies. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 35:75–86, 2015