As discussed, duration is a first approximation of the expected price change for a small change in yield. As the yield changes grow large, the estimation error grows larger; this is particularly true for MBS, where the prepayment option introduces significant curvature into the bond’s price/ yield function. Convexity
is a second approximation of the expected price change. It represents how much the bond’s duration is expected to change, given changes in yields. An illustration of convexity is shown in Exhibit 11.2
. The exhibit highlights the change in the slope of the tangent lines at different points in the bond’s price/yield relationship.
Using the earlier example from physical science, since duration can be viewed as the equivalent of speed, i.e., the rate of change (in feet per second, for example), convexity is the equivalent of acceleration; it measures the rate of change of the rate of change (in feet per second per second).
Convexity is measured in a fashion similar to the effective duration calculation. (There is no meaningful convexity calculation that can be generated from a bond’s modified duration.) Using the term V
0 to indicate the security’s current value, convexity is calculated as follows:
is, as before, the change in yield in basis points. To illustrate the calculation, we return to the earlier example of a bond with a base-case price of ...