We investigate the effect of Eurosystem asset purchase programmes (APP) on the monthly yields of 10-year sovereign bonds for 11 euro area sovereigns during January-December 2020. The analysis is based on time-varying coefficient methods applied to monthly panel data covering the period 2004m09-2020m12. During 2020, APP contributed to an average decline in yields estimated in the range of 58-76 bps. In December 2020, the effect per EUR trillion ranged between 34 bps in Germany and 159 bps in Greece. Stronger effects generally display diminishing returns. Our findings suggest that a sharp decline in the size of the APP in the aftermath of the COVID-19 crisis could lead to very sharp increases in bond yields, particularly in peripheral countries. The analysis additionally reveals a differential response to global risks between core and peripheral countries, with the former enjoying safe-haven benefits. Markets' perceptions of risk are found to be significantly affected by credit ratings, which is in line with recent evidence based on constant parameter methods.