modified duration vs effective duration vs duration

[dennisyap]

Hi David,

I am confused with the different kinds of duration and hope you can clarify my misunderstanding. I read on your website http://www.bionicturtle.com/learn/article/modified_vs_macaulay_duration/ that modified duration = effective duration for a non-callable bond and I try to compare Hull chapter 4 with Tuckman chapter 5.

1. On Hull, duration is defined as "average time until the cashflows on the bond are received" and on Tuckman, duration is defined as "% change in bond price when yield changes 1%". Are they both referring to the same thing?

2. On Hull, duration = Sum of t((ce^yt)/B) and Tuckman, duration = (Price"y-"- Price"y+")/(2*Price*change in yield). Are they also referring to same thing if I adjust duration in hull to semi annual compounding using duration/(1+y/2)?

3. When only the word duration is used, do it commonly refer to effective duration, modified duration or neither?



Dennis

 

[David Harper CFA FRM]

Hi Dennis,

I hope you get a chance to look at this learning XLS, which I created specifically to highlight the comparison between the durations and DV01:
http://www.bionicturtle.com/premium/spreadsheet/4.c.6_durations_modified_macaulay/
i..e, on the right, in light blue (col F+), is the Macaulay duration. Modified duration in green

1. These are only the same if compounding is continuous, otherwise they are different:

"average time until the cashflows" refers to Macaulay duration: if you note cols F+ in the XLS, like Hull's calculations, this Mac duration is *literally* a weighted average of time to cash (n * pv of cash flows)

"% change in bond price" is the *sensitivity* and refers to Modified duration. As a practical risk matter, this is the more useful duration.

You probably will recognize the key relationship: Mod duration = Mac duration / (1+ y/k) where k = compound periods per year (that's why, when continuous, k tends to infinite and, in this special case, the durations are equivalent)

2. Yes, exactly...again, the XLS will give you a concrete illustration of this. Under my default inputs, the modified duration generated by:
Price"y-”- Price"y+”)/(2*Price*change in yield)
= 7.931 (cell D26); that's a modified "sensitivity" duration so i would *not* say "years"

Then compare to my implementation that's equivalent to Hull's Sum of t((ce^yt)/B), final result in cell L35:
= 8.17 Mac duration so it's okay to say 8.17 years

and they reconcile with 8.17/(1+yield/2) = 7.931

3. First, a good question should be explicit. Second, since we don't really run calculations in the FRM (yet) on bonds with embedded options (i.e., we conceptually study MBS but we are not calculating MBS durations), in all cases, for us, effective = modified duration.

So, effective/modified duration (as the % duration) is the most important and it should be assumed if not specified.
The reason is simply that modified duration is used for dollar/value duration to aggregate bonds into a portfolio, so it really is the most useful
please take a look at this question: http://forum.bionicturtle.com/viewthread/2110/
...he didn't specify which duration, but to aggregate with dollar duration, we must assume the multiplier is modified duration!

Mac duration has one basic use, IMO: for a zero coupon bond with T years maturity, the Mac duration is 10 (then you can compute Mod duration from that, if you are given yield)
i.e., Mac duration gives you the one instance--in the case of a zero coupon bond--where you can eyeball duration w/o running calc

David

 

[dennisyap]

Hi David.

I understand the differences now. Thank you.


Regards,
Dennis