# Modified Duration

## What is 'Modified Duration'

Modified duration is a formula that expresses the measurable change in the value of a security in response to a change in interest rates. Modified duration follows the concept that interest rates and bond prices move in opposite directions. This formula is used to determine the effect that a 100-basis-point (1%) change in interest rates will have on the price of a bond. Calculated as:

## BREAKING DOWN 'Modified Duration'

Duration measures the average cash-weighted term to maturity of a bond. It is a very important number for portfolio managers, financial advisors and clients to consider when selecting investments because, all other risk factors equal, bonds with higher durations have greater price volatility than bonds with lower durations. There are many types of duration, and all components of a bond, such as its price, coupon, maturity date and interest rates, are used to calculate duration.

## Modified Duration Calculation

Modified duration is an extension of something called Macauley duration, which allows investors to measure the sensitivity of a bond to changes in interest rates. In order to calculate modified duration, the Macauley duration must first be calculated. The formula for the Macauley duration is:

Macauley duration = Sum of (PV)(CF) * T / market price of the bond.

Here, (PV)(CF) is the present value of a coupon at period t and T is equal to the time to each cash flow in years. This calculation is performed and summed for the number of periods to maturity. For example, assume a bond has a three-year maturity, pays a 10% coupon, and that interest rates are 5%. This bond, following the basic bond pricing formula would have a market price of:

\$100 / (1.05) + \$100 / (1.05)^2 + \$1,100 / (1.05)^3 = \$95.24 + \$90.70 + \$950.22 = \$1,136.16

Next, using the Macauley duration formula the duration is calculated as:

Macauley duration = (\$95.24 * 1 / \$1,136.16) + (\$90.70 * 2 / \$1,136.16) + (\$950.22 * 3 / \$1,136.16) = 2.753

This result shows that it takes 2.753 years to recoup the true cost of the bond. With this number, it is now possible to calculate the modified duration.

To find the modified duration, all an investor needs to do is take the Macauley duration and divide it by 1 + (yield-to-maturity / number of coupon periods per year). In this example that calculation would be:

Modified duration = 2.753 / (1.05 / 1) = 2.621

This shows that for every 1% movement in interest rates, the bond in this example would inversely move in price by 2.621%.

## Duration Principles

Here are some principles of duration to keep in mind. First, as maturity increases, duration increases and the bond becomes more volatile. Second, as a bond's coupon increases, its duration decreases and the bond becomes less volatile. Third, as interest rates increase, duration decreases and the bond's sensitivity to further interest rate increases goes down.