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We consider the problem of obtaining predictors of a random variable by maximizing the correlation between the predictor and the predictand. For the case of Pearson’s correlation, the class of such predictors is uncountably infinite and the least-squares predictor is a special element of that class. By constraining the means and the variances of the predictor and the predictand to be equal, a natural requirement for some situations, the unique predictor that is obtained has the maximum value of Lin’s concordance correlation coefficient (CCC) with the predictand among all predictors. Since the CCC measures the degree of agreement, the new predictor is called the maximum agreement predictor. The two predictors are illustrated for three special distributions: the multivariate normal distribution; the exponential distribution, conditional on covariates; and the Dirichlet distribution.