## Duration

In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield, or the percentage change in price for a parallel shift in yields. Since cash flows for bonds are usually fixed, a price change can come from two sources: The passage of time (convergence towards par) which is predictable and a change in the yield.

The yield-price relationship is inverse and investors would ideally wish to have a measure of how sensitive the bond price is to yield changes. A good approximation for bond price changes due to yield is the duration, a measure for interest rate risk. For large yield changes convexity can be added to improve the performance of the duration. A more important use of convexity is that it measures the sensitivity of duration to yield changes.

## Types of Durations

The dual use of the word "duration" in the Macaulay duration and the modified duration, as both the weighted average time until repayment and as the percentage change in price, often causes confusion. The Macaulay duration is the name given to the weighted average time until cash flows are received and is measured in years.

## Macaulay duration

The Macaulay duration is the name given to the weighted average time until cash flows are received and is measured in years.

Where: i indexes the cash flows, PV_{i }is the present value of the cash payment from an asset, t_{i} is the time in years until the payment will be received, and V is the present value of all cash payments from the asset.

The Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield.

## Modified duration

The modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield.

Where: k is the compounding frequency per year (1 for annual, 2 for semi-annual, 12 for monthly, 52 for weekly, and so on), y is the is the yield to maturity for an asset.

When yields are continuously-compounded the Macaulay duration and the modified duration will be numerically equal. When yields are periodically-compounded the Macaulay duration and the modified duration will differ slightly and in this case there is a simple relation between the two. The modified duration is used more than the Macaulay duration.

The Macaulay duration and the modified duration are both termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them. The Macaulay duration is a time measure with units in years and really makes sense only for an instrument with fixed cash flows. For a standard bond, the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond.

The modified duration, on the other hand, is a derivative (rate of change) or price sensitivity and measures the percentage rate of change of price with respect to yield. The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows and can thus be applied to a wider range of instruments than can the Macaulay duration. For everyday use, the equality (or near-equality) of the values for the Macaulay duration and the modified duration can be a useful aid to intuition.