Given that the present value of a continuous money stream can be calculated by:
$\int_0^ \infty{\frac{C}{e^{rt}}}dx$ (1)
the solution I found when solving this perpetuity problem (1) was:
$\frac{C}{r}$ (2)
however, this is the same perpetuity solution formula found for discrete compounding perpetuities, which in turn returns the same value as in the discrete case, so I know something is not quite right.
Some textbooks and websites say that rather than using of the formula above (2), I should be using ln(1+r) as the denominator instead, as in:
$\frac{C}{ln(1+r)}$ (3)
My question is that I cannot find the reasoning for that adjustment (3) and more importantly, I cannot derive this formula from the initial setup (1)
Can you help?
You are confusing the continuous and the discrete interest rate. There is nothing wrong with your calculations, it's simply that the rates are not the same.
The continous interest rate, is by definition the rate that would provide, with continous compounding, the same re-payment after a given discrete period than the discrete rate. Meaning:
$$(1+r_{d}) = e^{r_{c}}$$
So that $r_{c} = ln(1 + r_{d})$ and the net present value of an infinite constant cash flow stream is
$$\int_{0}^{\infty} C.e^{-r_{c}t}dt = \frac{C}{r_{c}} = \frac{C}{ln(1 + r_{d})}$$