Intrinsic efficiency and multiple robustness are desirable properties in missing data analysis. We establish both for estimating the mean of a response at the end of a longitudinal study with drop-out. The idea is to calibrate the estimated missingness probability at each visit using data from past visits. We consider one working model for the missingness probability and multiple working models for the data distribution. Intrinsic efficiency guarantees that, when the missingness probability is correctly modelled, the multiple data distribution models, combined with data prior to the end of the study, are optimally accommodated to maximize efficiency. The efficiency generally increases with the number of data distribution models, except where one such model is correctly specified as well, in which case all the proposed estimators attain the semiparametric efficiency bound. Multiple robustness ensures estimation consistency if the missingness probability model is misspecified but one data distribution model is correct. Our proposed estimators are all convex combinations of the observed responses, and thus always fall within the parameter space.