# Bond Duration

## What does a bond duration measure?

**Duration** is the weighted average term to maturity of a bonds cash flows and therefore, is a valuable tool in assessing bond price sensitivity to interest rate shocks. It is the most common technique for quantifying this sensitivity and is generally used to approximate changes in the price of the bond for every 100 basis point change in yields(**modified duration**). As a general rule, the greater the value of duration, the more price volatility results from interest rate movements.

Let's take a look at the formula that Frederick Macaulay devised to calculate **bond duration**.

Remember, the present value of the cash flows can be derived by using the yield to maturity as the discount factor. Therefore, you can see that the lower the yield on the bond, the greater the duration on the bond.

Let's take an example on a 5 year bond yielding 5% and calculate the **Macaulay duration**.

Based on this example, the duration would be calculated as such: 8970.87 / 2 X 1000 = 4.49.

Now that we have calculated the duration, we can go on to define the price volatility that would result from a shift in market yields. The following formula will define this relationship between duration and bond price volatility and also account for compounding yield.

Assuming a 100 basis point shift in interest rates, the % change in bond price would be calculated as follows:

-(4.39 * .01 * 100) / (1 + .025) = 4.28% change in bond price.

Notice the negative sign in front of this equation and remember that bond prices move in the opposite direction as interest rate. Therefore, if interest rates are lowered by 100 basis points, we would insert a -.01 into the formula which would yield a positive price shift.

As you can see, duration is a useful measure in approximating interest rate risk; however, it does not work as well when there are large shifts in yields. The difference between the estimated change in bond price that we just calculated and the actual change in bond price is known as convexity and this must be included in the price change calculations when the yield change is large.

## How you can use the concept of Duration

A general rule is that a bond with a longer duration is far more volatile than a bond with a shorter duration. Additionally, zero coupon bonds have the same duration and maturity and therefore have the highest risk to interest rate changes. Zero coupon bonds aside, the duration of a bond will always be shorter than its term to maturity. One final generalization we can make is that lower coupon bonds will have higher durations than larger coupon bonds and therefore, larger coupon bonds will be less volatile when interest rates are changed. For example, if were looking at purchasing a bond and had three options (a discount bond, premium bond, or zero coupon bond) with the same yield to maturity, the premium bond would be the least volatile followed by the discount bond and zero coupon bond being the most volatile.

In conclusion, duration is a very effective means for determining interest rate risk for the individual investor. However, institutions may be more interested in looking at the bond convexity to be more precise with the estimates.