# The ambiguity of the term "duration"

This is a soft question about terminology.

Let B be a bond with coupon payments. There seem to be two uses of the word duration in finance:

• Sensitivity of B's log price to B's continuously compounded yield
• Sensitivity of B's log price to B's discretely (with the same frequency as the coupons) compounded yield

We also have three terms for duration: "duration", modified duration and Macaulay duration.

If I read Wikipedia correctly, then

• Modified duration: Sensitivity of B's log price to B's yield
• Macaulay duration: Weighted average time

The article specifically says that Macaulay duration equals the modified duration if you're working with continuously compounded yield.

Based on this: Macaulay duration is a special case of modified duration, and thus, adding the word "modified" to "duration" adds no extra meaning.

So, rather disappointingly, the 3 terms aren't able to disambiguate the 2 meanings.

How does one work around this issue? Is there better terminology?

The simplest workaround is actually to forget that Macauley duration exists. I actually feel very strongly about this: Macauley duration shouldn't be taught in school, should be mentioned only in passing in textbooks (if at all), and belongs only in the history section on Wikipedia.

This is because it's more or less useless in practice. When practitioners talk about duration, they're referring to modified duration, PV01, or effective duration, all of which are interest rate risk concepts. No one thinks about Macauley duration.

More specifically, modified duration tells you how much bond prices changes (in percentage terms) relative to a small change in bond yield. It can therefore be used for hedging and for duration targeting. Macaulay duration has no practical applications to speak of.

Macauley duration is also not a special case of modified duration. Under discrete compounding, it's easy to show that $$\text{Mod duration} = \frac{\text{Macaulay duration}}{(1 + y / f)},$$ where $f$ is the compounding frequency. As $f\to \infty$ (continuous compounding), the two become identical. It's a happy coincidence under a very specific scenario, but it doesn't make the two conceptually equivalent.

P.S. I'm all about having some intuitions behind mathematical formulas. Macauley duration might help with that. But I still firmly believe that teaching it, particularly to beginners, does more harm than good.

Macaulay duration is NOT a special case of modified duration. This can be read from Wikipedia

Both measures are termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. ... Modified duration, on the other hand, is a mathematical derivative (rate of change) of price and measures the percentage rate of change of price with respect to yield ... Modified duration is used more often than Macaulay duration.

You might be able to say that Macaulay duration and modified duration are both "duration"s. But these two definitely come from two distinct worlds, in spite of some common grounds

i have seen recently interest in the market for macaulay duration (% price change for a 1% change in instantaneous interest rate) , i think maybe because of the increasing focus on OIS, ie focus on overnight risk