In the aftermath of the financial crisis, attention has turned to reducing systemic risk in the derivatives markets. Much of this attention has focused on counterparty risk in the over-the-counter market, where trades are bilaterally executed between dealers and derivative purchasers. One proposal for addressing such counterparty risk is to mandate the trading of derivatives over a centralized clearinghouse. This paper lays out the advantages and risks to a mandated clearing requirement, showing how, in some instances, such a mandate can actually increase systemic risk and result in more financial bailouts. This paper also describes the dynamics of counterparty risk in the derivatives market. Discussing the relative importance of both the risk that arises from the price risk of the instrument at issue and the financial condition of the counterparty. The analysis then turns to an evaluation of how bilateral markets and clearinghouses manage these two risks. After demonstrating that resolving and replacing defaulted trades is the primary resolution problem facing both market structures, the paper lays out an auction alternative designed to address this issue.