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There is a growing list of examples for the existence of the signal-to-noise paradox, where in the ensemble-based climate prediction, the model ensemble mean forecast generally shows higher correlations with observations than with individual ensemble members. This seems to lead to a paradox that the model makes better predictions for the real world than predicting itself. Here we introduce a Markov model to represent the ensemble forecasts and reproduce the signal-to-noise paradox, which we argue is primarily dependent on the magnitude of the persistence and noise variance between the models and the corresponding observations. The monthly North Atlantic Oscillation indices based on uninitialized historical simulations of 40 CMIP5 models have been analyzed, suggesting that the signal-to-noise paradox is common in currently available coupled models, and the paradox is not due to problems with initialization processes used in the seasonal-to-decadal predictions in previous studies and is instead a general model problem.