Zero-coupon yield to maturity ($R(t,T)=-\log P(t,T)/(T-t)$)

The continuously compounded yield of a unit-face zero-coupon bond is the constant rate producing its current discount factor over the remaining maturity. It equals the maturity average of the instantaneous forward rate. Differentiation gives $f(t,T)=R(t,T)+(T-t)R_T(t,T)$, and both rates tend to the short rate as $T\downarrow t$. This definition uses continuous compounding; quoted coupon-bond yields require a different cash-flow equation.