In general, the higher the duration, the more sensitive the bond price is to the change in interest rates.
The more curved the price function of the bond is, the more inaccurate duration is as a measure of the interest rate sensitivity.
Using these assumptions, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question.
[2] Convexity does not assume the relationship between Bond value and interest rates to be linear.
However, these simplifying assumptions allow one to quickly and easily calculate factors which describe the sensitivity of the bond prices to interest rate changes.
[7] That is, their prices will be affected equally by small, first-order, (and parallel) yield curve shifts.
[8] For two bonds with the same par value, coupon, and maturity, convexity may differ depending on what point on the price yield curve they are located.
[12] However, bond price also declines when interest rate increases, but changes in the present value of sum of each coupons times timing (the numerator in the summation) are larger than changes in the bond price (the denominator in the summation).
[15] The positivity of convexity can also be proven analytically for basic interest rate securities.
For example, under the assumption of a flat yield curve one can write the value of a coupon-bearing bond as
Then it is easy to see that Note that this conversely implies the negativity of the derivative of duration by differentiating
[18] Effective convexity is a discrete approximation of the second derivative of the bond's value as a function of the interest rate:[18] where
These values are typically found using a tree-based model, built for the entire yield curve, and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates;[19][20] see Lattice model (finance) § Interest rate derivatives.