Understanding Bonds

In finance, the duration of a financial asset that consists of fixed cash flows, such as 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 concerning 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 duration, a measure of interest rate risk. For large yield changes, convexity can be added to improve the duration's performance. A more important use of convexity is that it measures the duration's sensitivity 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, is often confusing. The Macaulay duration is the name given to the weighted average time until cash flows are received and is measured in years.

\(\begin{align*}MacD=\frac{\sum_{i=1}^n t_iPV_i}{V}=\sum_{i=1}^nt_i\frac{PV_i}{V}\end{align*}\)

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.

\(\text{ModD } = \dfrac{\text{MacD}}{(1+y/k)}\)

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 yield to maturity for an asset.

When yields are continuously computed, the Macaulay duration and the modified duration will be numerically equal. When yields are computed periodically, 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. Still, it is important to consider the conceptual distinctions between them. The Macaulay duration is a time measure with units in years and makes sense only for an instrument with fixed cash flows. For a standard bond, the Macaulay duration will be between 0 and the bond's maturity. It is equal to the maturity if and only if the bond is a zero-coupon bond.

Conversely, the modified duration is a derivative (rate of change) or price sensitivity and measures the percentage rate of price change concerning yield. The concept of modified duration can be applied to interest-rate-sensitive instruments with non-fixed cash flows, thus covering a wider range of instruments than 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.