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

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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. 


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 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.

The table below contains the expected price change for a 30-Year U.S. Treasury Bond using the duration effect as well as the effect from convexity. Notice how the convexity effect is greater with a one percent change in interest rates than it is with a smaller change of only fifty basis points (1.43% to 0.36%).

It should also be noted that the effect from convexity is the same with a rise of one percent as it is with a decline of the same percentage. This reinforces the fact that, as interest rates rise, there is a deceleration in the rate of price decline for a bond with positive convexity. 

Expected Price Change Using Duration and Convexity

Convexity is a highly desirable feature. To further illustrate this concept, refer to the graph below. A typical bond – one without embedded options – has positive convexity and its price will rise as interest rates fall (point A). Conversely, when interest rates rise, the price of this bond will decline. However, as mentioned, its rate of descent will slow (point B).

Bond Convexity

The red line illustrates the price of a bond with negative convexity. The price of this bond will become less sensitive as interest rates fall and more sensitive as rates rise. As interest rates fall, the price of this bond will rise. But due to the uncertainty surrounding the coupon payments, especially if the bond is callable and is near or beyond its call date, its price is unlikely to rise much beyond par (point C).

When rates rise, the issuer is less likely to call the bond since it has a lower coupon than a new issue. These factors combine to work against the bondholder in favor of the issuer. Hence, negative convexity is highly undesirable. 


Bonds with longer maturities (and durations), lower coupons and embedded options are much less desirable in a rising interest rate environment. If you decide to invest in a bond today, be sure to check for embedded options such as call features, put options, etc. Anything that makes the coupon payments less predictable is a negative. In addition, you may not want to go very long on the maturity spectrum since you would be locking in a low rate for a longer period of time. 

The next decade may be one of the most difficult periods for bond investors within the past 50 years.

Although I suspect interest rates will remain low for a while longer, when they do rise, long-term, low-coupon bonds could experience a greater-than-usual price erosion.

Given the artificial suppression of interest rates over the past seven years, the potential economic slowdown in developed and emerging markets alike and the likelihood of higher interest rates, it seems wise to reassess any bonds you are holding.

If not, perhaps you should prepare yourself for a price decline when interest rates rise. I tend to dislike losing money if it can be avoided.