Under the Hood: What You Need to Know About Bond Duration and Convexity

Bond investments are at a unique crossroads today. When interest rates rise this year, which is a near certainty, the price of most fixed income securities will fall. This fact alone makes it a point of genuine interest for investors.

As we’ve discussed in the past, duration is required to determine the change in a bond’s price due to a change in interest rates. However, when the change in interest rates is more significant, duration alone is inadequate. In this case, convexity is required. Therefore, in this article, we’ll discuss duration and convexity. Both of these terms are highly relevant, especially when interest rates begin to ascend. 

Duration

The concept of duration was first developed by Frederick Macaulay in 1938. Known as Macaulay Duration, it is defined as the weighted average term-to-maturity of the cash flows from a bond. Simply put, it involves calculating the present value of future cash flows. A variant of this, known as Modified Duration, is another frequently used method. For our purpose, we’ll use Macaulay Duration. 

Let’s consider an illustration to clarify duration. Assume you have a zero-coupon bond (i.e. no coupon payments) with a maturity of 10 years (Example #1). Because there are no cash flows prior to maturity, the duration of a zero-coupon bond is the same as its maturity. If the same bond had a coupon of 5.0%, paid semi-annually, its duration would be 7.99 years (Example #2). This is because the bondholder would receive cash flows prior to maturity. Duration is the point where the cash flows are balanced on each side of the fulcrum.

Bonds with a longer duration are more price sensitive to changes in interest rates than bonds with a shorter duration. 

There are three factors that influence duration: coupon, maturity, and yield.

Given two bonds with the same maturity, the bond with the higher coupon will have a shorter duration than the bond with a smaller coupon. This is because a larger coupon accelerates the present value of the cash flows which moves the fulcrum to the left (farther from maturity).

When considering two bonds with the same coupon and payment frequency, the shorter duration will be found in the bond with the shorter maturity. Duration is useful in calculating the change in the price of a bond, providing the change in interest rates is relatively small. However, when interest rate changes are larger or when the bond contains an embedded option such as a call feature, convexity should be incorporated into the formula. Let’s look at convexity. 

Convexity

Convexity measures the sensitivity of duration when interest rates change. Moreover, it is derived from the price-yield curve of a normal bond which is convex. Because a change in interest rates will also change the weight that each cash flow receives when calculating their present value, duration will also change. This is why convexity should be used when interest rate changes are more significant.