### Price

To compute derivatives of price with respect to coupon rate and face value, we take the analytical price formula:

$$ Price = c \frac{z-z^{n+1}}{1-z}+F z^n$$

and re-write it by expressing periodic coupon in terms of annual coupon rate as \( c = \frac{C}{f}F \), meaning periodic coupon is equal to annual coupon rate C times face value F divided by the number of coupons per year. Thus

$$ Price = \frac{C}{f}F \frac{z-z^{n+1}}{1-z}+F z^n$$

Which is easy to differentiate:

$$ \mbox{Coupon Delta}= \frac{\partial Price}{\partial C} = \frac{1}{f}F \frac{z-z^{n+1}}{1-z} $$ $$ \mbox{FV Delta}= \frac{\partial Price}{\partial F} =\frac{C}{f} \frac{z-z^{n+1}}{1-z}+ z^n$$ $$\quad \quad =\frac{c}{F} \frac{z-z^{n+1}}{1-z}+ z^n$$