Convexity Formula

What is Convexity?

Convexity relates to the interaction between a bond’s price and its yield as it experiences changes in interest rates. By measuring the change in duration as interest rates fluctuate, convexity, a measure of the curvature of changes in the price of a bond in proportion to changes in interest rates, corrects this inaccuracy. Consider the case of convexity, which occurs when all spot rates change by the same amount.

Example of Convexity:

The formula is as follows:

$ C=\frac{d^{2(B)(r))}}{B*d*r^2} $

where

C=convexity

B=the bond price

r=the interest rate

d=duration​

For example, consider a 200-basis-point increase in all rates for the bond in our example. The bond’s price declines to USD 986,448.71 (a decrease in value of USD 51,473.32). Using duration alone indicates a price change of:

-2.56 * 1,037,922.03 * 0.02 = -53,141.61

But using the duration + convexity result it gives,

-53,141.61 + 1/2 * 8.246 * 1,037,922.03 *0.022 = -51,429.04 

which is reasonably accurate.