Effective Duration: Definition, Formula, Example

Effective duration is a measure of how much the price of a bond will change in response to a change in interest rates. It is calculated as the percentage change in the bond's price for a 1% change in interest rates.

Effective duration is a more accurate measure of a bond's interest rate sensitivity than Macaulay duration, which is calculated as the weighted average of the time to maturity of the bond's cash flows. This is because effective duration takes into account the fact that the cash flows of a bond are not paid at a constant rate over time.

The formula for effective duration is:

```
Effective duration = -PV(?CF/?r)/PV(CF)
```

where:

* ?CF is the change in the bond's cash flows
* ?r is the change in interest rates
* PV(?CF/?r) is the present value of the change in the bond's cash flows due to a change in interest rates
* PV(CF) is the present value of the bond's cash flows

To calculate effective duration, you first need to calculate the change in the bond's cash flows due to a change in interest rates. This can be done by multiplying the bond's cash flows by the change in interest rates.

Once you have calculated the change in the bond's cash flows, you need to discount it back to the present value. This can be done using the bond's yield to maturity.

Finally, you need to divide the present value of the change in the bond's cash flows by the present value of the bond's cash flows. This will give you the effective duration of the bond.

Effective duration is a useful tool for investors who want to understand how a bond's price will be affected by changes in interest rates. It can also be used to compare the interest rate sensitivity of different bonds.

Here is an example of how to calculate effective duration. Consider a bond with a face value of $1,000, a coupon rate of 5%, and a maturity of 10 years. The bond is currently trading at a yield to maturity of 6%.

The first step is to calculate the change in the bond's cash flows due to a change in interest rates. This can be done by multiplying the bond's cash flows by the change in interest rates. In this case, the change in interest rates is 1%, so the change in the bond's cash flows is $50 ($1,000 x 0.01).

The next step is to discount the change in the bond's cash flows back to the present value. This can be done using the bond's yield to maturity. In this case, the present value of the change in the bond's cash flows is $47.62 ($50 x 0.9434).

Finally, we need to divide the present value of the change in the bond's cash flows by the present value of the bond's cash flows. This will give us the effective duration of the bond. In this case, the effective duration is 4.762 years.

This means that for a 1% change in interest rates, the price of the bond will change by 4.762%.

Effective duration is a useful tool for investors who want to understand how a bond's price will be affected by changes in interest rates. It can also be used to compare the interest rate sensitivity of different bonds.