When Discount Bond Modified Duration Stops Rising with Maturity
Summary
The document explains how modified duration can fail to increase monotonically with maturity for a discount bond. In the table discussed, duration for the lowest coupon rises through the shorter maturities but is lower at the longest maturity than at the preceding one. The accepted answer interprets this as an unusual case in which a bond’s yield to maturity is sufficiently above its coupon rate. Its illustrative calculations show duration rising and then falling at higher yields, with the turning point occurring sooner as the gap between yield and coupon grows.
The post distinguishes the table’s pattern from the more familiar increase in duration as maturity extends. It links the effect to discount pricing and points to bifurcation analysis for a formal treatment. The author says the original table values could not be replicated exactly, and the replacement figures are rough calculations under annual coupon payments. The examples therefore explain the possible shape of the relationship rather than verify the source table’s assumptions or numbers.
Key ideas
- Modified duration can decrease at longer maturities for some discount bonds.
- The table shows this pattern for the lowest coupon at its longest listed maturity.
- A yield substantially above the coupon rate can produce a rise and then fall in duration.
- A larger yield-to-coupon gap can move the duration maximum to a shorter maturity.
- The illustrative calculations are approximate and do not exactly reproduce the source table.
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# Modified Durations of Different Noncallable Bonds and function of Maturity
# Modified Durations of Different Noncallable Bonds and function of Maturity
I'm hoping someone could help me understand this subject better.
Basically I am reading a book and it shows a table
```
Coupon Rate | 10 yrs | 20 yrs | 30 yrs | 50 yrs
3% | 7.894 | 11.744 | 12.614 | 11.857
6% | 7.030 | 9.988 | 10.952 | 11.200
9% | 6.504 | 9.201 | 10.319 | 10.975
12% | 6.150 | 8.755 | 9.985 | 10.862
```
It then asks, How can you tell from the table that the modified duration is not an increasing function of maturity?
I don't really understand that. I know that as the coupon rate increases the loan is repaid faster because of the lower modified duration.
But it would seem as maturity increases then so does modified duration. So it would seem like its an increasing function just from looking at it. At least to me.
Can anyone tell me where this information comes from? Like how is it not an increasing function of maturity?
Thanks!
## Answer by Karol J. Piczak (score 3, accepted)
https://quant.stackexchange.com/a/1194
An interesting case you present here.
What they mean is that for discount bonds modified duration can decrease in value even if bond maturity increases.
That's indeed counter-intuitive and not that common.
In your example, when you look at modified duration values for `coupon rate: 3%`, you can see that it's value is rising with longer maturity (going from `10 yrs -> 20 yrs -> 30 yrs`), but for `50 yrs` it has decreased (`11.857` vs `12.614` for `30 yrs`).
The modified duration behaves "normally" with other coupon rates in your example.
I couldn't exactly replicate the values you have in the example, but instead I've made some rough modified duration calculations for annual payments on a `3%` coupon bond depending on the YTM you choose:
```
YTM | 10 yrs | 20 yrs | 30 yrs | 50 yrs
3% | 8.530 | 14.877 | 19.600 | 25.730
5% | 8.245 | 13.785 | 17.136 | 19.869
10% | 7.540 | 11.003 | 11.532 | 10.607
15% | 6.847 | 8.446 | 7.678 | 6.796
```
As you can see, if the YTM (current interest rate) is much higher than the actual coupon rate for our bond (which is `3%` in this example), the modified duration is no longer a monotonically increasing function of maturity (on the interval we're assessing). The bigger the difference, the sooner we get to the extremum.
You can have a look at "Bond duration, yield to maturity and bifurcation analysis" for a formal explanation of this subject.
The included chart is a really good explanation of what happens with discount bonds:Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.