# Units of Modified Duration and Macauley Duration

I know that the unit of the mod. Duration is % (actually no unit, because every number can be written as %) and the Macauley Duration has the unit time.

If you want to convert the Macauley Duration to the mod. Duration, you have to divide by $(1 + y/n)$, which is % aswell.

How is it possible that when you divide unit time by unit % the result is %? Shouldn't this only occur, when you divide time by time?

Your conversion from Macaulay Duration $$MacD$$ to the mod. Duration $$ModD$$ is correct, but your statement

[...] which is % as well.

is incorrect.

## Continuous time

$$MacD$$ is defined as $$MacD = \sum_{i=1}^n{t_i \frac{CF_i \cdot e^{-y t_i}}{V}}$$ where $$V= \sum_{i=1}^N{PV_i}$$ and therefor $$V$$ is the present value of all cash payments until maturity, which equals the current price.

$$PV_i$$ is the present value of cashflow (CF) $$i$$ and $$y$$ the yield to maturity. As you already mentioned, $$MacD$$ is expressed with unit (time).

The modified Duratio $$ModD$$ is defined as

$$ModD(y) = - \frac{1}{V} \cdot \frac{\partial V}{\partial y} = - \frac{\partial ln(V)}{\partial y}$$

As you can see, $$ModD(y)$$ is the percentage derivative of price with respect to yield (the first order derivative of bond price with respect to yield).

As stated in the Wikipedia article,

Macaulay duration is a weighted average time until repayment (measured in units of time such as years) while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield.

and further:

Modified duration can be expressed as the percent change in price per one percentage point change in yield per year (for example yield going from 8% per year (y = 0.08) to 9% per year (y = 0.09)). This will give modified duration a value close to the Macaulay duration (and equal when rates are continuously compounded).

## Discrete time

In discrete time, $$MacD$$ is defined as

$$MacD =\sum_{i=1}^n{\frac{t_i}{V(y_k)} \cdot \frac{CF_i}{(1+y_k)^{k \cdot t_i}}}$$

where $$k$$ is the compounding frequency per year and $$y_k$$ is the yield to maturity for an asset (periodically compounded). Taking the derivation of value $$V$$ with respect to $$y_k$$ of the above equation results in the mod. Duration $$ModD$$ for discrete time:

$$ModD = \frac{MacD}{1+\frac{y_k}{k}}$$

As mentioned in the comments, $$MacD$$ and $$ModD$$ are two different concepts:

$$MacD$$ is the weighted average time until cash flows are received, and is measured in unit (time). $$ModD$$ is the price sensitivity and therefore the percentage change in price for a unit change in yield.

### How about the units in detail?

$$MacD$$ is in unit (time), but technically, $$ModD$$ is also expressed in unit (time). Modified duration gives you the value of the percentage change in price per one percentage point change in yield per year. This is technically in unit (time)! As stated in the wikipedia article:

Formally, modified duration is a semi-elasticity, the percent change in price for a unit change in yield, rather than an elasticity, which is a percentage change in output for a percentage change in input. Modified duration is a rate of change, the percent change in price per change in yield.

Consider a simple example:

You have a 2-year bond with face value of \$100, a 20% semi-annual coupon, and a yield of 4% semi-annually compounded. $$MacD$$ is 1.777 years. $$ModD$$ is $$ModD = \frac{1.777}{1+ 0.4/2} = 1.742$$ The value of 1.742 is stated as %-change in price per 1 percentage point change in yield, i.e. $$\frac{\text{%-change in price}}{\text{1 percentage point change in yield}}=\frac{\%}{\frac{\%}{time}} = \text{time}$$ As $$ModD$$ expresses a (semi-)sensitivity, it is common to split up its unit (time) into "%-change per 1 percentage point change in yield" (with yield in % per time). • Your answer is correct in continuous time. But I believe the original question concerned the formulas in discrete time, where the pesky difficult to explain$\frac{1}{1+y/n}$term does occur... – Alex C Sep 14 '18 at 2:10 • Hello, at first thanks for your answers! Indeed, my question was stated in discrete time. Alex C, to summarize your comment: the reason for this "weirdness" is the difficult interpretation of the term$1/(1+y/n)\$ which is used? – Simon Sep 14 '18 at 7:23
• Well, that is my opinion. Macaulay came up with duration in 1938 as a weighted average of maturities (hence expressed in years). A year later in 1939 Hicks defined what he called "elasticity to interest rates" as a logarithmic derivative of price with respect to y. The formula is so similar to duration that it was renamed "modified duration". Perhaps we should not think of one as the modification of the other, but two different concepts. (More controversially: perhaps only the MD concept, under the name "Hicks elasticity" should be taught in schools from now on). – Alex C Sep 14 '18 at 8:56
• Thanks @AlexC for your helpful objection! I will edit my answer and provide details for discrete time as soon as possible. – skoestlmeier Sep 14 '18 at 9:06