Modified vs. Macaulay Duration
by David Harper, CFA, FRM, CIPM
Learning Outcome
- LO 23.9: Define, interpret, and calculate the yield-based DV01, the modified duration, and the Macaulay duration of a security.
This article contains:
- Formulas for each duration measure.
- English explanations
- EditGrid spreadsheet that solves for each. To make it easier, each term's calculations in the spreadsheet is color-coded: DV01, modified duration, and Macaulay duration
- A note about internal consistencies (reconciliations) in the spreadsheet
1. Formulas for each duration
In the course of FRM study, we price bonds without embedded options. For such plain-vanilla bonds, the distinction between modified and effective duration does not matter. (see Fabozzi for the difference: effective duration recognizes that cash flows dynamically change with the yield. An unnecessary nuance in our case).
Given that we assume a bond without embedded options, modified (or effective) duration is given by:
The dollar value of an zero (DV01) is given by:
And the Macaulay duration can be expressed as a function of the (modified/effective) duration above:
That is because Modified duration can be expressed as this expanded format:
And the term inside the braces above is the Macaulay Duration. I calculate this above formula in the spreadsheet below (see the blue section, which solves for the Macaulay)
2. English explanations
The modified (or effective) duration is arguably the most relevant risk measure: the approximate percentage change in bond price given a 1% (100 basis point) yield change.
The dollar value of an zero (DV01) is the approximate dollar price change given a one basis point (0.01%) change in yield. The difference between modified duration and DV01 is that the former measure a percentage change and the latter measures an absolute dollar change. The learning outcome calls for the yield-based DV01. That is what we have been doing! The yield-based DV01 is a special case of the DV01. A general DV01 is unspecific about the rate shift: it is agnostic about the type of interest rate change. A yield-based DV01 gives the dollar change for a change in the yield to maturity (YTM, or what we often just call 'yield').
And the Macaulay duration is the duration that can be expressed in time units. In the example below, the Macaulay duration is 8.17, so we could say "the weighted average time to receipt of the coupons and principal is 8.17 years." But some bond folks will grimace (although, it does have an intuitive meaning, it means our bond has roughly the sensitivity to rates of a zero-coupon bond with 8.17 years to maturity).
3. EditGrid spreadsheet
The EditGrid spreadsheet (below) can be easily uploaded into several formats, including MS Excel (Select File > Save As...).
The initial bond assumptions include the following (four bond inputs and one assumption about how much we will 'shock' the yield to estimate duration):
- Par value: $1,000
- Maturity: 10 years
- Coupon: 4.0%
- Yield: 6%
- Duration "shock:" 20 basis points
The layout contains four blocks, one input area and four solutions, as follows:
| Layout of the EditGrid Spreadsheet | |
| Input (Yellow) | Macaulay Duration (Blue) |
| Modified Duration (Green) | |
| DV01 (Orange) | |
4. Note about reconciliation
A great way to really understand these durations is to observe how they relate. Specifically, in the spreadsheet, notice the following:
We solve the DV01 by shocking the yield up one basis point. This gives the same result ($0.68) as (Price x Duration)/10,000:
The Macaulay Duration involves time weighting all of the cash flows. It produces a Macaulay duration of 8.17. The Modified Duration therefore equals 7.93 = 8.17/(1+6%/2). This is the same duration we get with the typical (the green block) modified duration where we shock the yield +/- 20 basis points.
Comments
In the fomula of Modified duration, the text indicate that ‘the term inside the braces is the Macaulay Duration’. I think you forgot to include the term 1/P for Macaulay duration.
JP - Yes, thanks for your careful eye. It is corrected - David
I know Macauley Duration has a unit of years but I never see a discussion of the units for modified duration. Is it unitless? Is the assumption that the yield is per year and cancels the macauley year units? Thanks
Hi Thomas,
Totally great question. Duration is a 1st derivative, like velocity is 1st derivative of distance (with respect to time). Distance is measured in feet, so velocity is measured in “feet per second” (feet/second, feet/time or dDistance/dTime). The bond curve is plotted Price versus Yield, so duration, as the Slope (1st derivative) is change in Price per change in Yield (dPrice/dYield). Because it is a straight tangent line on the P/Yield curve, it’s units are Price/Yield.
That’s just the math, maybe the way to think about it is (and i am just parroting Fabozzi) is “% change in bond price given a 1% increase in the yield.”. And regarding the period, no I don’t think an annual period is (should be) assumed for the yield. Yield can be annual, semiannual, monthly, etc. This is perhaps why most authors recommend thinking of duration not as “time to weighted average cash flow” (per Macaulay) and instead as a *sensitivity* concept. Hope that helps!
David, great article! Thank you for actually including the formulas, for clearly explaining the distinction between Macaulay and modified duration, and providing great examples. This was the best resource I could find on the web regarding this subject.
I understand everything except L30 in this Excel table. What is “k” and why do you multiply Price by 2? It probably comes from the formula you are using, but I’ve tried to make this same table match with different (way simpler exercise) from Investopedia and they do not use “k” in their equation. In the end Investopedia durations are always twice as “big” as durations here. So my question is - which of these two equations is better and why and is using “k” really neccessary?
Link to my Investopedia exercise http://www.investopedia.com/university/advancedbond/advancedbond5.asp
Hi Marie,
I looked at investopedia’s formula. Actually, theirs is un-necessarily complicated; note the denominator in their Example #1 sums simply to the price of the bond ($1,000). So, their version is actually the same as mine but with an un-necessary denominator: the denominator only needs to be Price * k, where k = the number of periods per year (how many coupons paid per year?). So, in the investopedia example, an annual coupon bond implies: Price * (1). On the other hand, my example above illustrates a semi-annual coupon bond, so we use P*2 in the denominator.
But please note that this does *not* double the duration, the duration will be nearby regardless of the compound frequency. Because when you go from k=1 to k=2, the numerator is going to *roughly* (not exactly!) double due to the inclusion of the periods in the numerator (Investopedia: variable t; my example: column F).
Hope that helps, thanks for the question…David
Thank you for your answer, I kind of figured it out myself as well by now, but a reply still added some confidence. I think Investopedia should really add an example about semi-annual duration as well. Right now it is a bit confusing. The example here is way better.
Regarding the post by David Harper from Feb 5, 08.
Duration is NOT a first derivative of P as a function of yield.
It is the Negative of the first derivative, then multiplied by the ratio of (1+yield)/P. I.e., it is an absolute value of an elasticity. (In the case of modified duration, the ratio to multiply by is just 1/P instead.) The first derivative alone has units of $/% i.e., dollars per percent. This is since the change in prices is change in $, itself measured in $, like $1 change from $5 to $6, and the change in (gross) yield is a change in %, like a change of 1% say from 105% to 106% (though usually we would write those as decimals 1.05 and 1.06). Since the duration is the product of the 1st derivative times (1+yield)/P measured in %/$, the units all cancel out, % in the top and bottom, and $ in the top and bottom all cancel. This is a general result for all elasticities: they have no units (no dimension). We do say duration is X years because intuitively it makes great sense, especially for a STRIP, but in fact there is no unit. The intuition is correct: duration is the percent change in price as a result of a 1 PERCENT increase in yield. Modified duration (semi-elasticity) is the percent change in price as a result of 1 PERCENTAGE POINT increase in yield. (note here that we really mean gross yield (1+y) but since 1 percentage point in net and grow yield is the same, it doesn’t matter: 5% to 6% and 105% to 106% is for both just 1 percentage point change).
As for semi-annual duration, just calculate duration normally using half the yield and two times the periods, then the duration you obtain is measured in half-years, so multiply that number by 2 to get duration in years. Alternatively, one could calculate duration using the annual yield, but in the weighted average add not years 1,2,3 etc., but instead have a weighted average of .5, 1, 1.5, 2, 2.5 etc. Then the weighted average is already measured in years. The weights are as usual the PV(CFi)/price for cash flows in period i.
Alex,
You are so right, I gave the wrong answer on that. (I mistakenly gave units for so-called dollar duration. i.e., the 1st derivative).
But thank you for a really great, well-illustrated correction. You treat the ratios perfectly!
Thanks too for linking (Macaulay) duration to “elasticity” (i.e., unitless) and Modified duration to “semi-eleasticity” (i.e., gross yield). I had frankly forgotten those terms but they are very helpful…David
So getting back to my original question from Feb. 2008 are the traditional units for Macauley Duration years as the textbooks use and no units for modified duration? I am not clear after this explanation from Alex. Can someone provide a good book on this topic? Thanks David and Alex. Regards Tom Tallerico
Tnt, both duration and modified duration can be said to be in years because it makes sense (they are just the weighted average of years to maturity of the separate cash flows of a bond). That said, mathematically, duration has no units at all, and modified duration has messy units (the numerator has no units unless multiplied by 100 so it becomes %, and the bottom is percentage Points, not %).
Moorad Choudhry and Frank Fabozzi(CFA) both write a lot about bonds at different levels of difficulty.
Fabozzi is the best but his Fixed Income Analysis (CFA) is introductory and I don’t *think* he quite get to this. The book that really schooled me is Interest Rate Risk Modeling. The authors have a site/forum at http://www.fixedincomerisk.com/.
They go way in depth on duration (including over all the misconceptions) but you might preview it first because it has a lot of calculus. David
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