Bond duration

In economics and finance, bond duration is the weighted average maturity of a bond or series of cash flows received.

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Price

It is useful as a measure of the price sensitivity of a bond to interest rate movements. It is approximately inversely proportional to the percentage change in price for a given change in yield. For example, for small interest rate changes, the duration is the approximate percentage that the value of the bond will lose for a 1% increase in interest rates. So a 15 year bond with a duration of 7 years would fall approximately 7% in value if the interest rate increased by 1%.

Macaulay duration

Macaulay duration is the weighted average maturity of a bond where the weights are the relative discounted cash flows in each period.

$\mbox{Macaulay duration} = \frac {\sum\ (\mbox{present value of cash flow}\times\mbox{time to cash flow})}{\mbox{price of the bond}}.$

Modified duration

Modified duration is calculated as follows:

$\mbox{Modified duration} = \frac {\mbox{Macaulay duration}}{1 + r}$

where r is the yield to maturity of the bond.

Convexity

Duration is a linear measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price is not likely to change linearly, but instead it would change over some curved function of interest rates. Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question. Then the convexity would be the second derivative of the price function with respect to the interest rate.

Average

The sensitivity of a portfolio of bonds such as a bond mutual fund to changes in interest rates can also be important. The average duration of the bonds in the portfolio is often reported. The average duration can be used similarly to the duration of a single bond to infer how the price of the portfolio would change in response to changes in interest rates.