Duration is a measure of interest-rate risk. Or, stated differently, duration is a measure of how sensitive the price of a fixed-income instrument is to interest-rate changes.

### What it is:

Duration can be thought of as the weighted average of when the bondholder receives payment. Check that the way the weights react is consistent with the relationships on the previous slide. With the following app, you can set the maximum yield-to-maturity, and time to maturity, and see the resulting 3D duration surface. You can also change the coupon rate and see the effect on the duration surface. However, it is customary to first calculate what is called Maucaulay Duration, and then use this to calculate Modified Duration.

The Macaulay Duration is:. These are weights and sum to 1. Also note, we can calculate the duration of a bond portfolio as the weighted average of the duration of all of the individual the bonds in the portfolio. Calculate the Macaulay and Modified Durations for the following bonds.

## Macaulay Duration

The metric is named after its creator, Frederick Macaulay. The Macaulay duration can be viewed as the economic balance point of a group of cash flows.

Another way to interpret the statistic is that it is the weighted average number of years an investor must maintain a position in the bond until the present value of the bond's cash flows equals the amount paid for the bond. A bond's price, maturity, coupon and yield to maturity all factor into the calculation of duration. All else equal, as maturity increases, duration increases. As a bond's coupon increases, its duration decreases. As interest rates increase, duration decreases and the bond's sensitivity to further interest rate increases goes down.

Also, sinking fund in place, a scheduled prepayment before maturity and call provisions lower a bond's duration. The calculation of Macaulay duration is straightforward. The bond pays the coupon twice a year, and pays the principal on the final payment.

Given this, the following cash flows are expected over the next three years:. With the periods and the cash flows known, a discount factor must be calculated for each period.

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- How it works (Example):?
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Thus the discount factors would be:. Next, multiply the period's cash flow by the period number and by its corresponding discount factor to find the present value of the cash flow:. A coupon paying bond will always have its duration less than its time to maturity.

- Modified Duration;
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Interactive App With the following app, you can set the maximum yield-to-maturity, and time to maturity, and see the resulting 3D duration surface. You can move the 3D surface around, and zoom in and out, with your mouse.

### What is Convexity of a Bond?

Can you see the case where duration is decreasing with an increased time to maturity? Macaulay Duration From our definition above, we can make the following observations: Duration cannot exceed the number of periods to maturity of the bond. The Duration of a zero-coupon bond is the number of years until maturity. You can also use the following app to see duration decrease when maturity increases.

## Convexity of a Bond | Formula | Duration | Calculation

Modified Duration Calculation. Also in Fixed Income and Other Debt 1. Exchange Traded Notes notifications. Yield to Maturity notifications. Convexity notifications notifications notifications.

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