Derivation of convexity formula

Question

Let's say that I have a bond that pays coupon on a semi-annual basis. Therefore, the price of this bond can be calculated using the following formula:

$$ P = \sum_{i=1}^N \frac{CF_i}{(1 + YTM/2)^{2t_i}} $$

First derivative of the above is:

$$ \frac{\partial P}{\partial YTM} = \frac{1}{(1 + YTM/2)} \sum_{i=1}^N \frac{-2t_iCF_i}{(1 + YTM/2)^{2t_i}} $$

Second derivative (aka convexity) of the Price function is:

$$ \frac{\partial^2 P}{\partial YTM} = \frac{1}{(1 + YTM/2)^2} \sum_{i=1}^N \frac{({4t_i}^2+2t_i)CF_i}{(1 + YTM/2)^{2t_i}} $$

And the generalized form of the convexity formula for bonds that pay multiple coupons per year is:

$$ \frac{\partial^2 P}{\partial YTM} = \frac{1}{(1 + YTM/f)^2} \sum_{i=1}^N \frac{({(ft_i)}^2+ft_i)CF_i/f}{(1 + YTM/f)^{ft_i}} $$

I am getting slightly different results when I compare my results with Bionic Turtle. Is there any mistake in my derivation?

Thank you!

Answer 1

You have left out the chain rule term in the first derivative and second derivative.

First derivative should be: $$\frac{\partial P}{\partial YTM} = \frac{1}{2(1+YTM/2)} \sum_{i=1}^N \frac{-2 t_i CF_i}{(1+YTM/2)^{2 t_i}} $$

Second derivative should be: $$\frac{\partial^2 P}{\partial YTM^2} = \frac{1}{4(1+YTM/2)^2} \sum_{i=1}^N \frac{(4 t_i^2 + 2t_i) CF_i}{(1+YTM/2)^{2 t_i}} $$

With the "f" instead of "2": $$\frac{\partial^2 P}{\partial YTM^2} = \frac{1}{f^2(1+YTM/f)^2} \sum_{i=1}^N \frac{(( f t_i)^2 + f t_i) \cdot CF_i}{(1+YTM/f)^{f t_i}} $$

Hope this helps!