Modified Duration

\frac{\frac{dPV}{dr}}{PV}=-\frac{1}{1+r}\times MacDur,

\frac{\frac{dPV}{dr}}{PV}=-\frac{MacDur}{1+r}

Without the negative sign, this is known as a bond’s modified duration, or ModDur:

ModDur=\frac{MacDur}{1+r}

The change in annualised yield-to-maturity:

\%\Delta PV^{Full}\approx-AnnMOdDur\times \Delta AnnYield

Approximate Modified Duration

AnnModDur\approx\frac{(PV_-)-(PV_+)}{2\times(\Delta Yield)\times (PV_0)}

AnnMacDur\approx AnnModDur\times (1+r)