Let $t$ mean current time, let $T_0, T_n$ mean two times such that $T_0\le T_n$, and let $y_t[T_0, T_n]$ mean the forward swap rate of a swap starting at $T_0$ and ending at $T_n$. (I am ignoring $T_0+2$ issues, and assume that the swap starts at $T_0$.)
Then under the annuity numeraire $N_t = P_t[T_0, T_n]$, the forward swap rate $y_t[T_0, T_n]$ is a martingale under the risk-neutral measure associated with $N_t$. This follows from the fact that $y_t$ is a ratio of a portfolio of assets by $P_t[T_0, T_n]$. Indeed, $$ y_t[T_0, T_n] = (Z(t,T_0)-Z(t,T_n))/P_t[T_0,T_n], $$ where $Z(t, T_i)$ means the zero-coupon bond from $t$ to $T_i$.
Is there a corresponding numeraire for the yield of a bond?
I am guessing the answer is no under some mild assumptions because bonds are tradeable and their price has a non-zero second derivative with respect to yields, but cannot hack through the thicket of results at the moment.
Thanks in advance, any help appreciated!