Modified vs. Macaulay Duration27 Sep 2007   by David Harper, CFA, FRM, CIPM

 

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:

David:

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.

Alex:

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.