Therefore, technically defined, a bond’s convexity is the rate of change of its duration. If we were to unpack that russian doll of a statement, we would get: Convexity is the rate of change of the rate of change of its price with respect to a change in interest rates.

Which is more than a mouthful, and if that statement isn’t helpful to you, then perhaps this one will be: Convexity is how quickly a bond’s sensitivity to interest rates changes.

Understanding this will help you to see what is actually going on in a bond’s price when interest rates change. If we are in the convex part of the curve, we know that duration will change very quickly when interest rates change (this is what having high convexity means). This means that as interest rates change, the bond’s price will become even more sensitive to movements in interest rates, the result of which is larger changes in the bond’s price!

Hence, convexity is what is responsible for large asymmetric price moves when markets and risk appetites shift, which makes it extremely important for traders/investors to understand.

At this point, it is important to draw the distinction between the asymmetric nature of convexity and nonlinear dynamics in complex systems. While convexity, when described as a small change in one variable leading to a large change in another, can sound very similar to the butterfly effect, they are not exactly the same.

The difference between both really is one of scale. Change, in the context of convexity and complex systems, can lead to nonlinear and asymmetric outcomes in both, but, convexity is often contained to individual entities, not entire systems. Bond prices are a good example of this, a small change in interest rates that leads to a large change in a bond’s price only affects the bond’s price. On the other hand, small changes in one variable of a complex system can cause feedback loops across many different, but interrelated parts in the same system.

However, the asymmetric change in the bond’s price might go on to cause a cascading feedback loop through the rest of the financial system, simply because of how interconnected components of the financial system are. The Great Reflation of 2021 is a good example of this as the selloff in mainly the 10 and 30 year part of the UST curve went on to  spark rallies/sell-offs in other asset classes. 

Ultimately, convexity really is an expression of nonlinearity, which in turn can be observed on the micro, individual level, as well as the macro, system-wide one.

To be continued…