**Duration** is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interest rates. A bond’s duration is easily confused with its term or time to maturity because they are both measured in years. However, a bond’s term is a linear measure of the years until repayment of principal is due. It does not change with the interest rate environment.

The duration of a bond can mean two different things. The Macaulay duration is the weighted average time until all the bond’s cash flows are paid. By accounting for the present value of future bond payments, the Macaulay duration helps an investor evaluate and compare bonds independent of their term or time to maturity.

The second type of duration is called “modified duration” and, unlike Macaulay duration, is not measured in years. Modified duration measures the expected change in a bond’s price to a 1% change in interest rates. In order to understand modified duration, keep in mind that bond prices are said to have an inverse relationship with interest rates. Therefore, rising interest rates indicate that bond prices are likely to fall, while declining interest rates indicate that bond prices are likely to rise.

Macaulay duration finds the present value of a bond’s future coupon payments and maturity value. Fortunately for investors, this measure is a standard data point in most bond searching and analysis software tools. Because Macaulay duration is a partial function of the time to maturity, the greater the duration, the greater the interest-rate risk or reward for bond prices.

Macaulay duration can be calculated manually as follows:

Where:

- f = cash flow number
- CF = cash flow amount
- y = yield to maturity
- k = compounding periods per year
- t
_{f}= time in years until cash flow is received - PV = present value of all cash flows

The previous formula is divided into two sections. The first part is used to find the present value of all future bond cash flows. The second part finds the weighted average time until those cash flows are paid. When these sections are put together, they tell an investor the weighted average amount of time to receive the bond’s cash flows.

**Modified Duration**

The modified duration of a bond helps investors understand how much a bond’s price will rise or fall if the YTM rises or falls by 1%. This is an important number if an investor is worried that interest rates will be changing in the short term. The modified duration of a bond with semi-annual coupon payments can be found with the following formula:

## 2 thoughts on “Duration of Bonds”