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This paper explores the optimal allocation of government bond purchases within a monetary union, using a two-region DSGE model, where regions are asymmetric with respect to economic size and portfolio characteristics: the extent of substitutability between assets of different maturity and origin, asset home bias, and steady-state levels of government debt. An optimal quantitative easing (QE) policy under commitment does not only reflect different region sizes but is also a function of these dimensions of portfolio heterogeneity. By calibrating the model to the euro area, we show that optimal QE favors purchases from the smaller region (Periphery instead of Core), given that the former faces stronger portfolio frictions. A fully optimal policy consisting of both the short-term interest rate and QE lifts the monetary union away from the zero lower bound faster than an optimal interest rate policy alone, which entails forward guidance.