A debate has been raging for centuries regarding the effects of inter-annual storage on commodity prices. Most analysts consider storage to function as a price stabilizer, while others place it at the core of an explanation of intriguing features of commodity price series, such as skewed distributions. Most studies have been developed in the context of the theory of competitive storage where random shocks affect supply or demand. Recently, the endogenous chaotic behavior of markets has become another possible hypothesis regarding the origin of commodity price fluctuations. We develop a nonlinear cobweb model with intra- and inter-annual storage, risk averse agents and adaptive expectations. Like the theory of competitive storage, this nonlinear cobweb model with storage can reproduce some of the stylized facts of agricultural commodity prices (autocorrelation of first rank, low kurtosis and skewness). In addition, the effects of storage on price variation are mixed. In the presence of inter-annual storage, chaotic price series show less variation compared to a situation without inter-annual storage but we find that storage contributes to the endogenous volatility of prices by making chaotic dynamics more likely.