fixed income - Derive Perpetual Bond Price

It is known that a perpetual bond with coupon $c$ has price

$$P=\frac{c}{r}$$ How do you get to this price? Is $r$ stated in discrete or continuous compounding?

Answer

A Consol Bond is a bond that pays an annual coupon of c every year. Therefore its price is $P=\frac{c}{1+r}+\frac{c}{(1+r)^2}+\cdots$. Factoring out the c and using the known formula for a geometric series, namely $u+u^2+u^3+\cdots = \frac{u}{1-u}$ we get $P=c[\frac{1}{1+r}/(1-\frac{1}{1+r})]=\frac{c}{r}$

Clearly this is a discrete compounding, not continous compounding formula.