Overview
The convexity is defined similarly to the duration, but the time value is squared.
{% convexity = \sum t^2 w_t %}
The convexity in conjunction with the duration provides a better approximation to the curve risk.
{% \Delta P / P \approx -D \times \Delta y + 1/2 \times CON (\Delta y)^2 %}
The formula here can be seen as a variation of the
Taylor Formula
for approximating a function with a polynomial.
Implementation
Convexity is implemented in the duration library.
let du = await import('/code/duration/v1.0.0/duration.mjs');
let cashFlows = [
{t:0.5, value:10},
{t:1.0, value:10},
{t:1.5, value:10},
{t:2.0, value:10},
{t:2.0, value:100},
];
let rate = function(t){
return 0.05
}
let convexity = du.convexity(cashFlows, rate);