I decided to split the first part of my original question into this separate one as they are somewhat unrelated. The definition of carry of a (spot) starting swap I know is $$\frac{S(0, T) - F(0,\tau)}{DV01(Swap(\tau, T-\tau))}\tag{1}$$ where $S(0,T)$ is the swap rate for maturity $T$, $F(0,\tau)$ is the fixing with period $\tau$ and $DV01(Swap(\tau, T-\tau))$ is the dollar value of basis point of a swap starting in $\tau$ with time to maturity $T-\tau$, e.g. see this short research paper from Nordea. Why can $(1)$ be approximated by $S(\tau, T-\tau)-S(0,T)$ as it is done in the Nordea paper?



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