# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [FM] Bonds – III: Duration

Consider the following bond pricing equation using yield to maturity:

$\displaystyle P = \frac{C}{1 + y} + \frac{C}{(1 + y)^2} + \frac{C}{(1 + y)^3} + \frac{C}{(1 + y)^4} + ... + \frac{C + 100}{(1 + y)^n}$

where $C$ is the annual coupon rate, $y$ is the yield to maturity, and $n$ is the maturity of the bond.

This formula follows straight from the discounted cash flow model we encountered earlier. The only difference is that instead of the dividends, we have known coupons from the Government of India. That is, we are in a world of (default) risk-free interest rates.

The above can be written in a short-hand as:

$\displaystyle P =\sum_{t = 1}^{n} \frac{C_t}{(1 + y)^t}$

where $C_t = C$ for $t = 1, 2, ..., n - 1$, and$C_t = C + 100$ for $t = n$.

I know the summation sign causes that queasy feeling in the stomach for at least a few of us, but, again, it’s a very useful short-hand. I can only say that more you use it, the better/more comfortable you’ll get at it. But, still, in deference to those of us who are not so comfortable at using the summation sign, let’s take the example of a simple two period bond.

Example: Consider two separate zero coupon bonds maturing at time $t = 1$ and $t = 2$, with cash flows $C_1$ and $C_2$ and prices $P_1$ and $P_2$ respectively. Assuming that the yield to maturity $y$ is the same for both, we may write :

\displaystyle \begin{aligned} P_1 &= \frac{C_1}{1 + y} \\ P_2 &= \frac{C_2}{(1 + y)^2} \end{aligned}

Let’s say our RBI governor Dr. Raghuram Rajan, raises the interest rates, as for example, a couple of days back by $25$ basis points, i.e. by $0.25\%$. The question we ask then is, by how much the PV of each cash flow changes. The natural way to address this question is to use the technology of simple calculus. We have:

$\displaystyle P_1 = \frac{C_1}{1 + y}$

Then, the change in $P_1$ w.r.t change in $y$ is given by:

\begin{aligned} \displaystyle \frac{dP_1}{dy} &= -\frac{C_1}{(1 + y)^2} \\& = -\frac{C_1}{(1 + y)}\frac{1}{(1 + y)} \\ & = -P_1\frac{1}{(1 + y)} \\ \Rightarrow -\frac{dP_1/P_1}{dy} &= \frac{1}{1 + y} \end{aligned}

i.e. the percentage change in bond-price to small changes in $y$ of of a single cash-flow at time $1$ is $1/(1 + y)$. Since, this is also its time to maturity, the word duration is apt. That is,  the percentage change in a $1$ year zero coupon bond price to small changes in $y$ given by $1/ (1 + y) = \mbox{Duration} / (1 + y)$.

Similarly, one can write the change in the PV of the second zero coupon bond for small changes in $y$ as:

\begin{aligned} \displaystyle \frac{dP_2}{dy} &= -\frac{2C_2}{(1 + y)^3} \\& = -\frac{C_2}{(1 + y)^2}\frac{2}{(1 + y)} \\ & = -P_2\frac{2}{(1 + y)} \\ \Rightarrow -\frac{dP_2/P_2}{dy} &= \frac{2}{1 + y} \end{aligned}

i.e. the percentage change in bond-price to small changes in $y$ of of a single cash-flow at time $2$ is $2/(1 + y)$. Again, we notice that it is related to its time to maturity, or as can now start calling it, duration. The duration of a single cash-flow at the end of time $2$ is $2$, and then the percentage change in the bond price to small changes in $r$ given by $2/ (1 + y) = \mbox{Duration} / (1 + y)$.

That is, irrespective of the cash-flow at time $1$ or time $2$, the percentage change in the bond price is given by $\mbox{Duration} / (1 + y)$. We can now generalize this result to say that the percentage change in the bond price to small changes in $y$ (compounding $k$ times a year) is given by:

$\displaystyle -\frac{dP/P}{dy} = \mbox{Modified Duration} =\displaystyle \frac{\mbox{Duration}}{(1 + y/k)}$

We define $\displaystyle \frac{\mbox{Duration}}{(1 + y/k)}$ as the Modified Duration. The modified duration represents the percentage change in the bond price to small changes in $y$.

Duration for a coupon-bearing bond can be thought of being as that average time all the money gets received, i.e. as the weighted average time to maturity as:

\begin{aligned} \displaystyle \mbox{Duration} &= \frac{1 \times PV_1 + 2\times PV_2 + 3 \times PV_3 + ... n \times PV_n}{PV_1 + PV_2 + PV_3 + ... + PV_n} \\&= \frac{1}{P} \Big(\frac{1 \times C_1}{1 + y} + \frac{2 \times C_2}{(1 + y)^2} + \frac{3 \times C_3}{(1 + y)^3} + ... \frac{n \times C_n}{(1 + y)^n} \Big) \\&= \frac{1}{P} \sum_{t = 1}^{n} \frac{t \times C_t}{(1 + y)^t} \end{aligned}

The formula for Modified Duration holds not only for single cash-flow but also for a set of cash-flowsand we can write the Modified Duration for a coupon bearing bond also as:

$\mbox{Modified Duration} =\displaystyle \frac{\mbox{Duration}}{1 +y/k}$

And now we can measure the percentage change in the bond price to small change in yields using:

\begin{aligned} \displaystyle -\frac{dP}{P} &= \mbox {Modified Duration } * dy \\&= \frac{\mbox{Duration}}{(1 + y/k)} * dy \end{aligned}

That is, if we know the Modified Duration/Duration of our bonds, to see the percentage change in the value of the bonds we only need multiply the Modified Duration of our bonds by the change in yields and we are done. No need to re-evaluate the price of all our bonds for new changed yields. And since yield changes by the central banks are typically of the order of 25 – 50 basis points only, using Modified Duration/Duration is not a bad first approximation (why do I say approximation?).

As should be clear, if yields are small, using either Modified Duration and Duration is ok – of course, using Modified Duration is more precise.

Written by Vineet

September 25, 2014 at 2:03 pm

### 2 Responses

1. […] posts only contains a proof related to duration of a coupon bond which we skipped in the previous post. Those not interested in details may safely […]

2. […] posts only contains a proof related to duration of a coupon bond which we skipped in the previous post. Those not interested in details may safely […]