Effective Duration

Effective duration is a duration calculation for bonds that have embedded options. This measure of duration takes into account the fact that expected cash flows will fluctuate as interest rates change. Effective duration can be estimated using modified duration if a bond with embedded options behaves like an option-free bond.

A bond with embedded options behaves like an option-free bond when exercise of the embedded option would offer the investor no benefit. As such, the security's cash flows cannot be expected to change given a change in yield. For example, if existing interest rates were 10% and a callable bond was paying a coupon of 6%, the callable bond would behave like an option-free bond because it would not be optimal for the company to call the bonds and re-issue them at a higher interest rate.

Effective duration calculates the expected price decline for a bond when interest rates rise by 1%. All else equal, the longer the maturity of a bond, the larger its effective duration. However, the value of the effective duration will always be lower than the maturity of the bond.

The formula for effective duration contains four variables. They are:

P(0) = the bond's original price per $100 worth of par value

P(1) = the price of the bond if the yield were to decrease by Y percent

P(2) = the price of the bond if the yield were to increase by Y percent

Y = the estimated change in yield used to calculate P(1) and P(2)

The complete formula for effective duration is: Effective duration = (P(1) - P(2)) / (2 x P(0) x Y)

As an example, assume that an investor purchases a bond for 100% par and that the bond is currently yielding 6%. Using a 10 basis-point change in yield (0.1%), it is calculated that with a yield decrease of that amount, the bond is priced at $101. It is also found that by increasing the yield by 10 basis points, the bond's price is expected to be $99.25. Given this information, the effective duration would be calculated as:

Effective duration = ($101 - $99.25) / (2 x $100 x 0.001) = $1.75 / $0.2 = 8.75

This effective duration of 8.75 means that if there were to be a change in yield of 100 basis points, or 1%, then the bond's price would be expected to change by 8.75%. This is an approximation. The estimate can be made more accurate by factoring in the bond's effective convexity.