duration of zero coupon bonds

[shanlane]

Hello,

In one of the questions, you say that the duration of zero coupon bonds in not monotonically increasing with time. The graphs in the chapter show this for "deep discount" bonds, but shouldn't the duration always be increasing for a zero coupon bond? Dmac is just the maturity, and Dmod is just Dmac divided by (1+y/2). The yield could obviously change, but unless the term structure is REALLY steep, it does not seem like it could make the duration decrease.

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Where do we say that, because it is not correct? With respect to a zero coupon bond, Macaulay duration = maturity, and therefore must be a monotonically increasing function of maturity. On the other hand, DV01 of a zero (or deeply discounted) is not strictly increasing as DV01 = P*D/10,000 and the numerator has offsetting effects.

If you'd kindly reference, I can fix? Thanks!

 

[shanlane]

question 28.2.

Thanks!

Shannon

 

[David Harper CFA FRM]

thanks, but it takes me longer to find the question when you aren't specific. I will ask Suzanne to locate and get back after we pin it down, thanks

@Suzanne: at your convenience, can you try to locate this x.28.2 based on duration? thanks,

 

[Suzanne Evans]

thanks, but it takes me longer to find the question when you aren't specific. I will ask Suzanne to locate and get back after we pin it down, thanks

@Suzanne: at your convenience, can you try to locate this x.28.2 based on duration? thanks,

Hi David,

I think this is the question shanlane is referring too: http://forum.bionicturtle.com/threads/l2-t5-28-dv01-duration-and-price-effects.3454/

Thanks,
Suzanne

 

[Suzanne Evans]

question 28.2.

Thanks!

Shannon

Hi Shannon,

Thank you for being active on our forum!

In the future, it might be more helpful if you ask for your clarification under the exact question rather than starting a new thread. This will help with David knowing the exact question, plus your question may be a question that someone else has so will help with David having to answer twice.

Thanks,
Suzanne

 

[shanlane]

Sorry about that. I print all of the questions and then make a list of things that I either didnt understand or did not sound correct. Next time I wil reference the question.

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

I don't see where I say "the duration of zero coupon bonds in not monotonically increasing with time." (not to argue, by any means, but rather merely to correct my error if it exists!)

The answer to 28.1 reads: "For par and premium bonds, both duration (Mod and Mac!) and DV01 are monotonically increasing functions of maturity."
(28.2 makes an exception reference to a deeply discounted coupon bond, but not, as far as i can tell a zero coupon bond)

Unless i need to make a correction, I think the controlling concepts (both key!) are:

1. The Macaulay duration of a zero-coupon bond equals its maturity, such that the Mac duration of a zero-coupon bond must be monotonically increasing, and
2. DV01 = Price * Mod duration /10000, where in the case of a zero coupon bond: Price is a decreasing function of maturity (i.e., a zero is acutely "pulled to par"), but Mod duration is an increasing function of maturity, with unclear (ambiguous) impact on DV01

Let me know if i missed your point? Thanks!

APPEND: Oh, i see .... "If the discount is deep enough—that is, if the coupon is low enough relative to yield—the duration of a discount bond rises above the duration of a perpetuity" but that refers to a deeply discounted coupon bond, not a zero coupon bond, and the rest of it explains it, so i don't see how that is your reference ... hmmmm....