Question

1. Duration provides a good approximation of changes in the price of an option-free bond, when the change in yield is relatively small. However, for large changes in yield a convexity adjustment needs to be incorporated.

With the aid of your own fully annotated diagram, discuss the statement above and address why a convexity adjustment is necessary.

Answer 1

In the diagram above, we can see two bonds. Bond A has more convexity than bond B. From the graph we see that when the yield (interest rate) increases, the bond price comes down. The duration represents the slope of both the bonds. Hence, if we have to calculate the bond prices at the increased rates, we see that for a small change in rates, the tangential line (sloped line) can tell us the amount by which the bond prices have decreased but as we keep on increasing the rate, we see that there is a considerable difference between what is being shown by the duration line and what is actually the bond price. This difference becomes more pronounced when the convexity is higher. For e.g., we can see that bond A's price decreases less than bond B's price. Hence, convexity becomes important for a higher change in interest rate changes. For small changes, we can make use of duration.