Modified Duration

Modified duration $D_{Mod}$ measures the percentage change in price per unit increase in the Yield Rate $j$ :

$D_{Mod} = -\frac{1}{P}\frac{dP}{dj}$

$= \frac{D_{Mac}}{1+j}$

$\Delta P \approx -D_{Mod} \cdot P \cdot \Delta j$

It provides the 1st-Order Linear Approximation for price changes
Modified duration is less than Macaulay Duration whenever $j > 0$ (since then $1+j > 1$ )
For a more accurate approximation accounting for price-yield curvature, add the Convexity correction:

$\Delta P \approx -D_{Mod} \cdot P \cdot \Delta j + \frac{1}{2} \cdot \text{Convexity} \cdot P \cdot (\Delta j)^2$

Converting Between Duration Types A bond priced at $100{,}000$ has a Macaulay duration of 7.5 years at a yield of 6% effective annual. Find the modified duration and the approximate price drop if yields rise to 6.5%.
Answer $D_{Mod} = \frac{7.5}{1.06} = 7.075472 \text{ years}$ The approximation is applied to the bond's price, not its face amount:
 $\begin{align*} \Delta P &\approx -D_{Mod} \cdot P \cdot \Delta j \\ &= -(7.075472)(100{,}000)(0.005) \\ &= -3{,}537.74 \end{align*}$ 
The price falls by about $3.54\%$ , to roughly $96{,}462$ .