Convexity (finance mathematics)

Convexity is a key figure from financial mathematics that describes the behavior of a bond when interest rates change. It is an extension or improvement of the duration and - like this - only an estimate of the change in the present value.

The course of the present value of bonds in the event of changes in interest rates is convex . Since the duration only takes into account the first derivative - i.e. the slope - it can only be used for small changes in interest rates or becomes less precise the greater the change in interest rates.

The convexity also takes into account the second derivative - the curvature - and is therefore a more precise approximation of the actual change in value. The formula for calculating the convexity is:

${\displaystyle C={\frac {P_{0}(i_{0})''}{P_{0}(i_{0})}},}$

where P _{0 represents} the value of the bond at time 0 and i ₀ the corresponding interest rate.

For example, if P ₀ ( i ₀ )

${\displaystyle c\cdot (1+i)^{-1}+c\cdot (1+i)^{-2}+(c+N)\cdot (1+i)^{-3}}$

with N = nominal , c = coupon and i = interest rate , the first derivative in this case is

${\displaystyle -c\cdot (1+i)^{-2}-2c\cdot (1+i)^{-3}-3\cdot (c+N)\cdot (1+i)^{-4}}$

and the second derivative

${\displaystyle 2c\cdot (1+i)^{-3}+6c\cdot (1+i)^{-4}+12\cdot (c+N)\cdot (1+i)^{-5}.}$

The change in the present value of a bond according to the principle of convexity takes place as follows:

${\displaystyle \mathrm {d} P/P=-D\cdot \mathrm {d} Y+1/2\cdot C\cdot \mathrm {d} Y^{2},}$

in which

D = modified duration

P = price (including accrued interest, so-called "dirty price") of the bond

d P = change in this price

d Y = change in interest rate, e.g. B. 0.005 for a change of 50 basis points (100 basis points = 1%)

C = convexity