We investigate the dynamics of a closed-loop supply chain with first-order auto-regressive (AR(1)) demand and return processes. We assume these two processes are cross-correlated. The remanufacturing process is subject to a random triage yield. Remanufactured products are considered as-good-as-new and used to partially satisfy market demand; newly manufactured products make up the remainder. We derive the optimal linear policy in our closed-loop supply chain setting to minimise the manufacturer’s inventory costs. We show that the lead-time paradox can emerge in many cases. In particular, the auto- and cross-correlation parameters and variances of the error terms in the demand and the returns, as well as the remanufacturing lead time, all influence the existence of the lead-time paradox. Finally, we propose managerial recommendations for manufacturers.