The present value of a perpetuity (cash flows paid at the end of each year) is $PV = CF / r$ where $r$ is the interest rate. This formula is proved in the book that I'm studying, Principles of Corporate Finance.
However, then it is stated that if instead the cash flows are spread evenly throughout each year like a continuous stream of payments, we can use the same formula but replace $r$ with the continuously compounded rate $r_c$. This is not proved in the book and I cannot see how this would automatically hold. Can someone please show how this holds?
Best Answer
If we have continuous compounding the future value after one year is $C\cdot e^r$
For $n$ yearly payments we have the series
$FV=C+Ce^r+Ce^{2r}+Ce^{3r}+\ldots+Ce^{(n-1)r}$
Using the closed form of a geometric series we get
$FV=C\cdot \frac{e^{rn}-1}{e^r-1}$
To get the present value the FV has to be discounted $n$ times
$PV=C\cdot\frac{1}{e^{rn}}\cdot \frac{e^{rn}-1}{e^r-1}=C\cdot \left( 1-\frac{1}{e^{rn}}\right)\cdot \frac{1}{e^r-1}$
Now let $n$ go to infinity
$$PV=\lim_{n \to \infty} C\cdot\left( 1-\frac{1}{e^{rn}}\right)\cdot \frac{1}{e^r-1}$$
$=C\cdot(1-0)\cdot \frac{1}{e^r-1}=\boxed{C\cdot\frac{1}{e^r-1}}$