### Convexity

And the second derivative of price with respect to r, called Dollar Convexity, can be calculated as follow:

$$ \frac{\partial^2 Price}{\partial r^2} = \frac{\partial}{\partial r^2} \left[ \sum_{i=1}^{n}{\frac{c}{{\left( 1+r/f \right)}^i}}+\frac{F}{{\left( 1+r/f \right)}^n} \right] $$ $$ = \sum_{i=1}^{n}{\frac{\partial}{\partial r^2}\frac{c}{{\left( 1+r/f \right)}^i}}+ \frac{\partial}{\partial r^2} \frac{F}{{\left( 1+r/f \right)}^n} $$ $$ = c \sum_{i=1}^{n}{\frac{i (i+1)}{{\left( 1+r/f \right)}^{i+2}} \frac{1}{f^2}}+\frac{F n(n+1)}{{\left( 1+r/f \right)}^{n+2}}\frac{1}{f^2} $$ $$ = \frac{c}{f^2} \sum_{i=1}^{n}{i(i+1) z^{i+2}}+ \frac{F n(n+1)}{f^2} z^{n+2} $$ $$ = \frac{c z^2}{f^2} \sum_{i=1}^{n}{i (i+1) z^i} + \frac{F n(n+1)}{f^2} z^{n+2} $$ $$ = \frac{c z^2}{f^2} \left[ \sum_{i=1}^{n}{i^2 z^i}+ \sum_{i=1}^{n}{i z^i} \right] + \frac{F n(n+1)}{f^2} z^{n+2} $$ $$ = \frac{c z^2}{f^2} \left[z \frac{1+z-{(n+1)}^2 z^n +(2 n^2 + 2 n -1) z^{n+1} -n^2 z^{n+2}}{{\left(1-z\right)}^3} + z \frac{1-(n+1)z^n+n z^{n+1}}{{\left(1-z\right)}^2} \right]+ \frac{F n (n+1)}{f^2} z^{n+2} $$

We used the sum of series formulae:

$$\sum_{i=1}^{n}{i z^i}=z \frac{1-(n+1)z^n+n z^{n+1}}{{\left(1-z\right)}^2}$$ $$\sum_{i=1}^{n}{i^2 z^i}=z \frac{1+z-{(n+1)}^2 z^n +(2 n^2 + 2 n -1) z^{n+1} -n^2 z^{n+2}}{{\left(1-z\right)}^3}$$