# Bond Convexity

## What is Convexity?

Remember, as bond yields go higher, price goes lower.  This relationship between price and yield has a convex structure in nature.  The term used to describe this relationship is also known as convexity.  Notice in the diagram below, we have drawn a tangent line a yield Y*.  This tangent line is very similar to the concept of duration and represents the rate of change in price as interest rates change.  When the steepness of this tangent line increases, so does the duration.

bond convexity

For smaller changes in yield, duration does a good job in estimating the actual price; however, when yields move further away from the initial yield, Y*, duration becomes less reliable.  We can then go on to say that the more convex the price to yield relationship, the less accurate the price approximation that is derived from the tangent line, or duration.

You can calculate bond convexity using the following formula:

convexity formula

As we said earlier, duration does not give an entirely accurate picture of price changes due to interest rate changes.  This is due in part to the fact that duration does not take into account the convexity of the price to yield relationship.  You can calculate the percentage price change not explained by duration with the following formula:

Percentage Price Change not explained by Duration =  .5 X Convexity X (Yield Change)^2 X 100

## Conclusion

Bond convexity is a bit of a perplexing topic for many.  Some refer to convexity as the degree of curvature that exists in the price to yield relationship while others refer to convexity as the second derivative, or a more precise version of duration, which would be added to duration to get that much more precise.  Those who would use convexity to make a better price approximation would use the following formula:

Convexity basically measures the rate of change of duration as yields change.

Tim Ord
Ord Oracle

Tim Ord is a technical analyst and expert in the theories of chart analysis using price, volume, and a host of proprietary indicators as a guide...