One popular procedure in risk management is to conduct stress tests, which should shed light on what happens in an extreme scenario. A particular stress scenario for credit portfolios is a credit spread widening. For CDS contracts an intuitive definition of such a scenario is to postulate that all par CDS spreads are blown out by a certain factor, leaving all other parameters unchanged. Even though two CDS contracts may have par CDS spreads of similar size, the present note explains why such a definition may affect the market value of one CDS more severely than the other when the two names have different recovery assumptions. More precisely, the upfront (i.e. market value) of the CDS with smaller recovery assumption is blown out further than the one of the CDS with bigger recovery assumption.