We propose a new affine term structure model to decompose inflation-linked swap (ILS) rates into genuine inflation expectations and inflation risk premia. The model accounts for both short-term macroeconomic factors and long-term economic trends, namely trend inflation and the equilibrium real interest rate. We estimate the model for the US and euro area and find that the inflation risk premium significantly drives ILS rates, with macroeconomic shocks playing different roles across maturities and regions. International factors, particularly oil prices, impact market-based inflation expectations during key global events like the COVID-19 pandemic and the Russian invasion of Ukraine.