# Is duration additive? $C_{newDur}=A_{fundDur}w_{a} + B_{fundDur}w_{b}$?

Suppose quantified duration (like Macaulay duration with changing intervals) $Dur = \frac{\sum t_{i} PV_{i}}{\sum PV_{i}}$ and two funds having durations $D_{a}$ and $D_{b}$. You own them in the proportion $w_{a}=0.4$ and $w_{b}=0.6$.

1. What is the duration of your portfolio?

2. Is it the following? $C_{newDur}=A_{fundDur}w_{a} + B_{fundDur}w_{b}$

3. Is duration combinations always sumproduct (like above, presupposing right not sure) or does it vary between different definitions of duration?

Resources

1. page 61 about parallel shift, page 73 about traditional immunization, page 79 about multivariate immunization (1990), here.

## 2 Answers

Yes, you are correct. Duration is additive, so your aggregate portfolio duration is the weighted average of your individual durations as you present in point 2.

That holds assuming a close to flat yield curve and parallel (additive) shifts.

If that's not the case, the situation gets a bit more complex. Unfortunately, right now I couldn't find any interesting and freely accessible paper that would deal with non-additive shifts or non-flat yield curve.

• Did you ever end up finding something ? – Alexandre Cassagne Mar 20 '16 at 21:49
• I don't remember much due to the time span, but I think most textbooks just blatantly assume this is the case, while non-additive shifts seem to pose a not so trivial research problem. I'm sorry I don't feel competent enough to delve into the subject right now, as I've mostly switched fields completely. Maybe someone more versed will be able to chime in. – Karol J. Piczak Mar 21 '16 at 16:10

Duration is also additive if you are dealing with key rate durations. In this case, Effective Duration is the weighted average of your key rate durations.