Understanding the Units of Macaulay and Modified Duration

# Units of Modified Duration and Macauley Duration


# 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?

## Answer by skoestlmeier (score 1, accepted)

https://quant.stackexchange.com/a/41707

Short answer:

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).

## Answer by user66645 (score 1)

https://quant.stackexchange.com/a/74918

Wow - the dangers of using OTT maths to wrongly explain a simple question.

Original poster correctly stated that Mod Duration is unitless and wanted it explained yet this maths answer contradicts it.

Universally Mac Duration is measured in years.

Universally Mod Duration is Mac Duration/ (1 + yield x time)

As so many people learn formulae instead of understanding them it seems people dividing y by k ( or m or whatever letter you choose) think that they're are dividing by time. YOU ARE NOT - you are multiplying by time!!

(1+y/2) for a semi-annual bond's Mod Dur, means you are multiplying yield by 1/2 because 6 months is half a year. You are multiplying by 1/2 not dividing.

I repeat - you are multiplying by time not dividing!

Back to units: Mod Dur = Mac Dur / (1 + yield x time)

Mac Dur units = time (1 + yield x time units) are ( unitless + unitless x time) = time

QED time/time is unitless!

Apologies for making this so simple and please stop using wikipedia for your answers - its wrong!