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We analyse the extent to which prices in the sovereign credit default swap (CDS) and bond markets reflect the same information on credit risk in the context of the current crisis of the European Monetary Union (EMU). We first document that deviations between CDS and bond spreads are related to counterparty risk, common volatility in EMU equity markets, market illiquidity, funding costs, flight-to-quality, and the volume of debt purchases by the European Central Bank (ECB) in the secondary market. Based on this we conduct a state-dependent price-discovery analysis that reveals that the levels of the counterparty risk and the common volatility in EMU equity markets, and the banks' agreements to accept losses on their holdings of Greek bonds impair the ability of the CDS market to lead the price discovery process. On the other hand, the funding costs, the flight-to-quality indicator and the volume of debt purchases by the ECB worsen the efficiency of the bond market.