A bond’s convexity refers to the sensitivity of the bond’s modified duration to changes in yield. Based on the price-yield curve:

§ at low yields, the modified duration changes very quickly as the yield changes (since the price-yield curve is very steep); therefore, the convexity is high

§ at higher yields, the modified duration changes very slowly as the yield changes (since the price-yield curve is relatively flat); therefore, the convexity is low

Convexity is positive for option-free bonds. Convexity can be negative if a bond contains an embedded call option. An embedded call option enables the issuer to repurchase the bond at a fixed price (known as the call price) at a specified time in the future. With very low yields, the likelihood of a bond being called is extremely high; as a result, investors will not pay more than the call price for a bond, no matter how low yields are. This results in a phenomenon known as “price compression”, in which the price of a bond is prevented from rising above the call price while the price can still fall due to rising yields. When yields are extremely low, the bond’s convexity can become negative as the price curve flattens out.