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.
The theoretical calculation of the Modified Duration is
\[ ModD = - \frac \cdot \frac < \partial P > < \partial y >= - \frac < \partial \ln P > < \partial y >,\]
where \( P \) is the price of the bond.
To account for the fact that bond prices are negative This was defined historically, and we do not want to change the convention Negative duration implies that an increase in yield causes a decrease in price When you differentiate \( \frac<1> < x>\), you get \( - \frac<1> \)
The formula for the modified duration is
What is the reason for the negative sign?
Let's take this theoretical definition, and apply it to determine a calculation for the modified duration. If the yield is compounded annually, then the price of the bond is
Thus, we can conclude that
Continuing from the example in Macaulay Duration, we know that the YTM is \( 2.82 \% \) and the MacD is \(4.571 \). Hence,
\[ ModD = \frac < Mac D > < 1 + y >= \frac < 4.571 > < 1 + 2.82 \% >= 4.445. \]
Submit your answerWhat is the modified duration (in %) of a ten-year 5% par bond?
Note: Your answer should be positive.
More generally, if the yield is compounded \( k \) times a year, then
Thus, when the yield is compounded continuously, we have \( k \rightarrow \infty \) or that
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.
4.49 4.95 5.25 9.90A bond has the following prices at different yields
| Yield (in %) | Price (in $) |
| 7 | 1150.25 |
| 8 | 1100 |
| 9 | 1051.3 |
What is the modified of the bond at an 8 % yield?
From the definition of Modified duration, we can use it to estimate the change in price of a bond as interest rate changes.
By substituting in the formula for Modified Duration, we get that
\[ 4.445 = - \frac \times \frac < \Delta P >< 1 \% >. \]
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.
Submit your answerA fixed coupon that expires in 10 years with a face value of $1000 is currently priced at $1200. It has a modified duration of 2.5. What would be the bond price if yield increased by 1%?
Note: Ignore convexity considerations