# 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!

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See the answers to this question about how to calculate the maximum duration for some more references and graphs: quant.stackexchange.com/questions/1624/… – Ari B. Friedman Aug 10 '11 at 14:17

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:

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yea i noticed it had a horizontal asymptote, as it goes to say 100 yrs it comes back down and its simply a horizontal asymptote. Not sure how i missed it, but that makes sense from what you said. I didn't notice that @ 3% in the table it actually decreases. – Matt May 21 '11 at 22:18
piczak i forgot to mention as well, it was a 9% yield rate. Could be why you didnt get the same answers. – Matt May 21 '11 at 22:21
The horizontal asymptote will always be at duration=(1 + 1/i) where i is the current market interest rate expressed in proportion terms (e.g. i=.05 for a 5% rate). – Ari B. Friedman Aug 10 '11 at 14:18