# Alternate explanation of Duration

In many reputed sites such as, Investopedia, bond duration is explained as a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows. My understanding was duration is just the time weighted average of present value of future cash flows.

I am unable to see how duration explains when the bond is repaid. Please explain

Edit: Even reputed authors like John Hull mention this understanding

• That sounds Investopedia quote sounds like sloppy language to me? The two definitions with which I'm familiar are Macaulay duration which as you said is the "weighted average maturity of cash flows" and modified duration which is the negative of the "percent derivative of price with respect to yield," that is $-\frac{\partial \ln V}{\partial y}$. – Matthew Gunn Jul 23 '17 at 20:16
• @MatthewGunn I am referring to the Macaulay Duration, which is also explained in the investopedia link. – kasa Jul 23 '17 at 20:18
• For future readers, the definition given by Hull for Macaulay duration is correct and not the same as the Investopedia article. This Investopedia article is incorrect and misleading. – Matthew Gunn Jul 24 '17 at 21:45

The definition you quoted, "[duration is] a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows" is not a sensible definition of Macaulay duration.

The only place I could find an explanation behind that language is this previous version of the bond duration Wikipedia article that called that definition an inaccurate and confused notion.

Duration is sometimes explained inaccurately as being a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows. This quantity is the duration of a perpetual bond (assuming a flat yield curve at the coupon), and is simply $\frac {1}{r}$. For instance, if a bond pays 5% per annum and was issued at par, it will take 20 years of these payments to repay its price. Note the absurdity of interpreting duration this way: given a bond paying 5% per annum with a term of 5 years, the duration is approximately 4.37, whereas the price of the bond will not be repaid in full until maturity (at 5 years).

### Recreating the math on this?

Let's say you have a consol with a perpetual coupon $C$.

If interest rates are a constant $r > 0$ then the present value is given by:

\begin{align*} V &= \lim_{T \rightarrow \infty} \sum_{t=1}^T \frac{C}{(1 + r)^t} \\ &= \frac{C\left(\frac{1}{1+r} \right)}{1 - \frac{1}{1 + r}}\\ &= \frac{C}{r} \end{align*}

The Macaulay duration is given by:

\begin{align*} D_M &= \lim_{T \rightarrow \infty} \frac{1}{C/r} \sum_{t=1}^T t \frac{C}{(1+r)^t} \\ &= r \lim_{T \rightarrow \infty} \sum_{t=1}^T t \left( \frac{1}{1+r}\right)^t \end{align*}

The interior part is an arithmetico-geometric series. The infinite sum can be written as $\frac{ \left( \frac{1}{1+r} \right)}{\left(1 - \frac{1}{1+r}\right)^2}$ which simplifies to $\frac{r+1}{r^2}$ hence:

$$D_m = 1 + \frac{1}{r}$$

You'll get $\frac{1}{r}$ as the Macaulay duration and $\frac{C}{r} + C$ as the price if you included the undiscounted cashflow $C$ instead of starting with $\frac{C}{1+r}$.

• Essentially, you mean to say that even for perpetual bonds, the above definition wouldn't hold true? – kasa Jul 24 '17 at 11:47
• Also, I came to know that even reputed text books like John Hull mention this alternate understanding. books.google.co.in/… – kasa Jul 24 '17 at 11:55
• Thank you for the wonderful explanation, however 👌🏻 – kasa Jul 24 '17 at 11:56
• @kasa Hmm? I don't see where Hull says anything like that Investopedia quote. Hull writes, "... duration is therefore a weighted average of the times when payments are made... This explains where the term 'duration' comes from. Duration is a measure of how long the bondholder has to wait for cash flows. A zero-coupon bond that lasts $n$ years has a duration of $n$ years. However, a coupon-bearing bond lasting $n$ years has a duration of less than $n$ years, because the holder receives some of the cash payments prior to year $n$." That's perfectly correct. – Matthew Gunn Jul 24 '17 at 21:21
• yeah, makes sense. – kasa Jul 24 '17 at 21:22

Let's consider the simple case of a $T$-year zero coupon bond, whose continuously compounded yield is $y_t$. Then its price and yield are related by $$P = e^{-y_T \cdot T} .$$

By definition, the modified duration is $$-\frac{1}{P}\frac{dP}{dy} =\frac{e^{-y_T\cdot T} \cdot (-T)}{e^{-y_T\cdot T}} = T.$$

This demonstrates that for a zero coupon bond, assuming its yield to maturity is continuously compounded, then its modified duration and time to maturity are identical.

However, if a bond is not a zero coupon bond and the compounding convention is not continuous, then this intuitive result no longer holds.

With some relative simple math, we can show that the Macauley duration of a bond that pays semi-annual coupon and maturing in $T$ years is $$D_\text{mac} = \frac{1}{P}\left[ \sum_{t=1}^{2T} \frac{t}{2}\frac{c/2}{(1+y/2)^t} + T \frac{100}{(1 + y/2)^{2T}}\right] .$$

This shows that Macauley duration is the time-weighted present value of cash flows divided by price, or (roughly) the present-value-of-cashflow weighted time to maturity.

How useful is Macauley duration in real life? Frankly speaking, not very... From an analytical perspective, the only concepts that are used would be modified duration or effective duration. Unfortunately neither of them has a very nice interpretation (except in the simplest case shown above).

It is true that the unit of Macaulay Duration and Modified Duration is in years, if you look at the mathematical formula for them. However, it is a bad idea to interpret duration in general as a measurement for time.

The best interpretation for duration is that it is actually a measure of a bond's sensitivity to yield changes. For example, the effective duration of an Interest-Only (IO) MBS Strip is negative. In fact, when interest rate rises, mortgage prepayment is slower and investors will receive more interest payment, so the price of an IO Strip will go up. In the case, it makes zero sense to interpret a negative number as a measurement for time.