 # Duration

## What is Duration?

Duration is a measure of a bond's sensitivity to interest rate changes. The higher the bond's duration, the greater its sensitivity to changes in interest rates (also known as volatility) and vice versa.

## Modified Duration Formula and Example

The modified duration formula can produce more accurate results than the traditional Macaulay duration formula. To calculate the modified duration of a bond, use the formula for modified duration:

Modified Macaulay Duration = Macaulay Duration / (1 + y)

By plugging in the figures from our earlier example, we'd use the modified duration formula to come up with this:

Modified Macaulay Duration = 5.33 / (1 + 0.05) = 5.076

As we did with the last example, we apply this result to come up with our change in price (or discount rate in this case):

Approximate Change in Price = 5.076 x 0.002 = 0.010152

That means if the bond had originally sold for \$1,000 with a 5% yield, it would now be selling for \$1,000  x (1 - 0.010152) or \$989.85 -- a discount from its original \$1,000 price.

(Note: In terms of accuracy, this formula comes out only \$0.06 different from the actual price of \$989.91 calculated on a present value of a bond calculator.)

Effective Duration Formula and Example

The Effective Duration formula is another way to calculate bond duration. The formula uses the bond's currenty yield to maturity (YTM) along with two more present values (a slightly higher YTM and a slightly lower yield YTM).

The formula for Effective Duration is:

Effective Duration = (PL - PH) / (2 x P0 x Change in Yield)

Where:

PL = Price of the bond for the lower yield
PH = Price of the bond for the higher yield
P0 = Price of the bond at its current yield

Using the same example from our earlier table with interest rates rising 20 basis points, we can use discounted present values for each period to utlimately calculate effective duration:

PeriodCash FlowPV at 5%PV at 4.9%PV at 5.1%
1\$50\$47.62\$47.66\$47.57
2\$50\$45.35\$45.44\$45.27
3\$50\$43.19\$43.32\$43.07
4\$50\$41.14\$41.29\$40.98
5\$50\$39.18\$39.36\$38.99
6\$1,050\$783.53\$788.02\$779.06
Total \$1,000\$1005.09\$994.94

Using the sum of cash flows under each PV we calculated, we can plug in the numbers into our effective duration formula:

Effective duration = (\$1005.09 - \$994.94) / (2 x 1,000 x 0.001) = 5.075

As we did with the last examples, we apply this result to come up with our change in price (or discount rate in this case):

5.075 x 0.002 = 0.0105

Again, think of this number as a discount off the original bond price (If yields fell, this number would be a premium on the bond).

So,  \$1,000 x (1 - 0.0105) = \$989.85

Thus, the estimated price of the original bond using effective duration and assuming a 0.20% raise in interest rates (or yield) would be \$989.85 -- the same as our modified duration estimate.

Why Duration is So Important

Understanding the duration formula is not nearly as important as understanding that duration is a measure of risk because it has a direct relationship with price volatility. The greater duration of the bond, the greater its percentage price volatility.

By providing a way to estimate the effect of certain market changes on a bond's price, duration can help you choose investments that might better meet your future cash needs. Duration also helps income investors who want to take on minimal interest rate risk (that is, they believe interest rates might rise) understand why they should consider bonds with high coupon payments and shorter maturities.

[Use our Yield to Maturity (YTM) Calculator to measure your annual return if you plan to hold a particular bond until maturity.]