Modified Duration
The modified duration of a bond is the price sensitivity of a bond. It measures the percentage change in price with respect to yield. As such, it gives us a (first order) approximation for the change in price of a bond, as the yield changes.
When continuously compounded, the modified duration is equal to the Macaulay duration.
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Modified Duration
The theoretical calculation of the Modified Duration is
\[ ModD = - \frac{1}{P } \cdot \frac{ \partial P } { \partial y } = - \frac{ \partial \ln P } { \partial y } ,\]
where \( P \) is the price of the bond.
The formula for the modified duration is
\[ Mod \, D(y) = - \frac{1}{P} \frac{ \partial P } { \partial y } \]
What is the reason for the negative sign?
\[ P = \sum_{t=1 } ^ T \frac{ C_t } { ( 1 + y ) ^ t } .\]
This gives us \[ \frac{ \partial P } { \partial y } = \sum_{t=1 } ^ T (- t) \times \frac{ C_t } { ( 1 + y ) ^ {t + 1 } } = \frac{ MacD \times P } { ( 1 + y ) } . \]
Thus, we can conclude that
\[ Mod D = \frac{ MacD } { (1 + y ) }. \]
What is the Modified Duration for a 10 year bond with fixed coupon payments of 5% and a face value of $1000, if the current price is $1100?
Continuing from the example in Macaulay Duration, we know that the YTM is \( 2.82 \% \) and the MacD is \(4.571 \). Hence,
\[ ModD = \frac{ Mac D } { 1 + y } = \frac{ 4.571 } { 1 + 2.82 \% } = 4.445. \]
More generally, if the yield is compounded \( k \) times a year, then
\[ Mod D = \frac{ Mac D } { ( 1 + \frac{ y}{k} ) } .\]
Thus, when the yield is compounded continuously, we have \( k \rightarrow \infty \) or that
\[ Mod D = Mac D \]
Calculating Modified Duration from Prices
From Calculus, we know that \( \frac{ \partial P } { \partial y } \) can be approximated by using \( \frac{ P ( y + \delta y ) - P ( y - \delta y ) } { 2 \delta y } \). As such, this gives us:
If we are given the bond prices across different yield rates, then we can estimate the modified duration by
\[ Mod D(y) \approx - \frac{ P ( y + \Delta y ) - P ( y - \Delta y ) } { 2 P \Delta y }. \]
This offers us a way to approximate the modified duration when we have a list of the price of the bond at different yields.
Effect of yield change on bond prices
From the definition of Modified duration, we can use it to estimate the change in price of a bond as interest rate changes.
Consider a bond currently priced at $1100 with a modified duration of 4.445. What would be the bond price as yields increase by 1%?
By substituting in the formula for Modified Duration, we get that
\[ 4.445 = - \frac{1}{1100} \times \frac{ \Delta P } { 1 \% }. \]
This gives us \( \Delta P = - 4.445 \times 1100 \times 1 \% = - $48.895 \). Thus, the new price would be
\[ P + \Delta P = $1100 - $48.895 = $1051.105. \]
This example shows how knowing the modified duration allows us to make a simple calculation to determine the (approximate) price of the bond. Of course, we could recalculate the price of the bond by accounting for the yield changes, but that is more complicated then the above approach.