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Liner shipping repositioning is the costly process of moving container ships between services in a liner shipping network to adjust the network to the changing demands of customers. Existing deterministic models for the liner shipping fleet repositioning problem (LSFRP) ignore the inherent uncertainty present in the input parameters. Assuming these parameters are deterministic could lead to extra costs when plans computed by a deterministic model are realized. We introduce an optimization model for the stochastic LSFRP that handles uncertainty regarding container demands and ship travel times. We extend existing LSFRP instances with uncertain parameters and use this new dataset to evaluate our model. We demonstrate the influence of uncertain demand and travel times on the resulting repositioning plans. Furthermore, we show that stochastic optimization generates solutions yielding up to ten times higher expected values and more robust solutions, measured against the CVaR90 objective, for decision-makers in the liner shipping industry compared to the application of deterministic optimization in the literature.