# Subject: Bonds - Duration Measure

Last-Revised: 19 Feb 1998
Contributed-By: Rich Carreiro (rlcarr at animato.arlington.ma.us)

This article provides a brief introduction to the duration measure for bonds. The duration measure for bonds is a invention that allows bonds of different maturities and coupon rates to be compared directly.

Everyone knows that the maturity of a bond is the amount of time left until it matures. Most people also know that the price of a bond swings more violently with interest rates the longer the maturity of the bond is. What many people don't know is that maturity is actually not that great a measure of the lifetime of a bond. Enter duration.

The reason why maturity isn't that great a measure is that it does not account for the differences in bond coupons. A 10-year bond with a 5% coupon will be more sensitive to interest rate changes than a 10-year bond with an 8% coupon. A 5-year zero-coupon bond may well be more sensitive than a 7-year 6% bond, and so forth.

Faced with the inadequacy of maturity, the investment gurus came up with a measure that takes both maturity and coupon rate into account in order to make apples-to-apples comparisons. The measure is called duration.

There are different ways to compute duration. I will use one of the common definitions, namely:

Duration is a weighted average of the times that interest payments and the final return of principal are received. The weights are the amounts of the payments discounted by the yield-to-maturity of the bond.

The final sentence may be alternatively stated:

The weights are the present values of the payments, using the bond's yield-to-maturity as the discount rate.

Duration gives one an immediate rule of thumb -- the percentage change in the price of a bond is the duration multiplied by the change in interest rates. So if a bond has a duration of 10 years and intermediate-term interest rates fall from 8% to 6% (a drop of 2 percentage points), the bond's price will rise by approximately 20%.

In the examples and formulas that follow, I make the simplifying assumptions that:

1. Interest payments occur annually (they actually occur every 6 months for most bonds).
2. The final interest payment occurs on the date of maturity.
3. It is always one year from now to the first interest payment.

It turns out that (especially for intermediate- and long-term bonds) these simplifications don't affect the final numbers that much -- duration is well less than a year different from its "true" value, even for something as short as a duration of 5 years.

Example 1:
Bond has a \$10,000 face value and a 7% coupon. The yield-to-maturity (YTM) is 5% and it matures in 5 years. The bond thus pays \$700 a year from now, \$700 in 2 years, \$700 in 3 years, \$700 in 4 years, \$700 in 5 years and the \$10,000 return of principal also in 5 years.

As you may recall, to compute the weighted average of a set of numbers, you multiply the numbers by the weights and add those products up. You then add all the weights up and divide the former by the latter. In this case the weights are \$700/1.05, \$700/1.05^2, \$700/1.05^3, \$700/1.05^4, \$700/1.05^5, and \$10,000/1.05^5, or \$666.67, \$634.92, \$604.69, \$575.89, \$548.47, and \$7,835.26. The numbers being average are the times the payments are received, or 1 year, 2 years, 3 years, 4 years, 5 years, and 5 years. So the duration is:

```    1*\$667.67 + 2*\$634.92 + 3*\$604.69 + 4*\$575.89 + 5*\$548.47 + 5*\$7,835.26
D = -----------------------------------------------------------------------
\$667.67 + \$634.92 + \$604.69 + \$575.89 + \$548.47 + \$7,835.26
```

D = 4.37 years

Example 2:
Bond has a face value of \$P, coupon of c, YTM of y, maturity of M years.

```    1Pc/(1+y) + 2Pc/(1+y)^2 + 3Pc/(1+y)^3 + ... + MPc/(1+y)^M + MP/(1+y)^M
D = ---------------------------------------------------------------------------
Pc/(1+y) + Pc/(1+y)^2 + Pc/(1+y)^3 + ... + Pc/(1+y)^M + P/(1+y)^M
```

We can use summations to condense this equation:

```        M
Pc*Sum i/(1+y)^i + MP/(1+y)^M
i=1
D = ------------------------------
M
Pc*Sum 1/(1+y)^i + P/(1+y)^M
i=1
```

We can cancel out the face value of P, leaving a function only of coupon, YTM and time to maturity:

```       M
c*Sum i/(1+y)^i + M/(1+y)^M
i=1
D = -----------------------------------
M
c*Sum 1/(1+y)^i + 1/(1+y)^M
i=1
```

It is straightforward to write a computer program to carry out the calculation. And those of you who remember how to find a closed-form expression for Sum{i=1 to M}(x^i) and Sum{i=1 to M}(ix^i) can grind through the resulting algebra and get a closed-form expression for duration that doesn't involve summation loops :-)

Note that any bond with a non-zero coupon will have a duration shorter than its maturity. For example, a 30 year bond with a 7% coupon and a 6% YTM has a duration of only 14.2 years. However, a zero will have a duration exactly equal to its maturity. A 30 year zero has a duration of 30 years. Keeping in mind the rule of thumb that the percentage price change of a bond roughly equals its duration times the change in interest rates, one can begin to see how much more volatile a zero can be than a coupon bond.