Although duration measures are simple and very useful for gauging a bond's interest rate risk, there's an important caveat.

Their prediction is not 100% accurate.

If we calculate the actual price of the bond for a yield to maturity of 3.43%, we get a bond price of 95.3403%, which is slightly different from the modified duration predicted value of 95.2091%.

So where does this prediction error come from? It arises because modified duration assumes a linear relationship between bond prices and yields.

In reality, the bond price yield relationship is convex.

This convexity effect explains the difference between the predicted and actual bond prices.

Let's now have a closer look at convexity and start with a question of what exactly convexity means.

In simple terms, convexity refers to the curvature in the bond price yield relationship.

Unlike modified duration, which assumes the bond price changes in a straight line as yields change, convexity accounts for the fact that bond prices react differently to small and large changes in interest rates.

This convexity works both ways.

When yields rise, bond prices fall, but due to convexity, the price does not drop as steeply as a linear calculation, using modified duration would predict.

Conversely, when yields fall, bond prices rise more than expected under a linear model.

So from a bond holder's perspective, convexity is beneficial.

It helps reduce losses when yields rise and amplifies gains when yields fall.

But what drives convexity? Different bonds have different levels of convexity and a few key factors influence this.

Bonds with longer maturities tend to have more convexity because their price is more sensitive to interest rate changes over a longer period.

Low coupon bonds or zero coupon bonds exhibit greater convexity.

This is because in such bonds, more of the bonds value is concentrated in the future, as there are fewer or no interim coupon payments, making the price more sensitive to changes in yield.

In contrast, high coupon bonds have less convexity since the bond holder receives more frequent payments, which makes the bonds price less sensitive to interest rate changes.

Convexity becomes especially important for large interest rate changes.

For small changes in yield, the modified duration approximation will be fairly accurate and the prediction error will be small.

However, for larger changes in yields, the prediction error increases because modified duration does not account for the curvature in the bonds price changes.