# Do Longer-Duration Bonds Have More Convexity?

Bond duration and convexity are crucial concepts that help investors assess the risks of investing in a bond. Both duration and convexity are only applicable to bonds and are not used for such assets as stocks, options or futures. While the calculation of these figures is somewhat complicated, you should be able to grasp their basic meaning and apply this data to your investments.

## Price vs. Yield

Duration and convexity result from the inverse relationship between bond prices and yields. The lower the bond price, the higher the yield and vice versa. Yield is the rate at which your money will grow. If, for example, you buy a bond with an original issue price of \$100 and annual interest of \$10, your yield is 10 percent. The \$100 investment will bring \$10 annually, and you'll receive \$100 back when the bond expires. However bonds change hands at various prices. If you were to buy the same bond for less than \$100, you will still receive the same \$10 annually, and the yield would be higher.

## Duration

While the price and yield of all bonds are inversely correlated, the precise relationship varies greatly among bonds. Duration measures the price sensitivity of a bond to changes in yield. The higher the duration, the more sensitive the bond's price is to interest rate variations. If, for example, a bond has a duration of two, the price will move twice as much as yield. Assume that the bond you purchased for \$90 has a yield of 12 percent and a duration of two. If the yield of this bond goes up to 13 percent, the price will decline by twice as much, or two percent, to \$88.20

## Convexity

The duration of a bond varies with the passage of time and yield changes. As the expiration date approaches, duration declines. Also, the duration of the bond when it's yielding 12 percent vs. 11 percent on the same date won't be the same. Convexity measures the sensitivity of duration to changes in yield. If the duration of a bond and yield increase together, it has positive convexity; if they move in opposite directions, the bond has negative convexity. Positive refers to a positive correlation between yield and convexity, which means they move in the same direction, while negative alludes to negative correlation, which in statistics means that two variables move in opposite directions.

## Factors Influencing Convexity

The convexity of a bond depends on various factors, but not on its duration. Most conventional, non-callable bonds have positive convexity. A bond is callable when the issuer can terminate the bond early by paying the bondholders the original issue price of the bond. Callable bonds, on the other hand, usually have negative convexity. The terms of the call also matter and vary from bond to bond. In some cases, calling the bond is only possible if the issuer pays a premium over and above the original issue price of the bond. This premium might make the callable bond's convexity positive.