Modified convexity
(MC).
Broadly speaking, modified convexity measures the curvature of an instrument’s or a portfolio's price function, as yields change - from a given starting point - by a small amount.
More strictly, it is the rate of change of modified duration with respect to yield - at the given starting yield.
Modified convexity can be calculated from Convexity as follows:
Modified Convexity = CMOD = Convexity / (1+r)2
Where:
r = periodic yield = Nominal annual rate / compounding frequency per year
The estimation of price change for a given small change in yield can then be calculated as follows:
Price change estimation using Modified Duration (MD) only:
= - Price x MD x Change in yield
Price change estimation using Modified Convexity (CMOD):
= - (Price x MD x (Change in yield) ) + ½ x (Price x CMOD x (Change in yield)2 )
Because the value v yield relationship is a curve and not a straight line (values do not change linearly as yields change) the estimate of change in value using only modified duration will generally underestimate the new value (because the curve lies above its tangent).
Therefore the modified convexity adjustment is always positive - it always adds to the estimate of the new price whether yields increase or decrease.
It is also possible to estimate the MD and the CMOD from given observations of Price and Yield, by rearranging them to solve for MD and CMOD - effectively running the price change estimation formulae in the other direction.
