Mimicking portfolios have many applications in the practice of finance. Here, we present a new method for constructing them. We illustrate its application by creating portfolios that mimic individual NYSE stocks. On the construction date, a mimicking portfolio exactly matches its target stock's exposures (betas) to a set of ETFs, which serve as proxies for global factors, and the portfolio has much lower idiosyncratic volatility than its target. Mimicking portfolios require only modest subsequent rebalancing in response to instabilities in target assets and assets used for portfolio construction. Although composed here exclusively of equities, mimicking portfolios show potential for mimicking non-equity assets as well.