Can anyone provide a mathematical proof that NPV is always negative when the rate of return is less than the discount rate? That is suppose that we have
- a single initial input, C0
- a constant known rate of return on the investment, r > 0
- and some constant discount rate, rd > r
- and letting Ct = cashflow in period t
The way I see it, the argument would be something like:
NPV = -C0 + summ( Ct / (1+rd)^t ) = -C0 + summ( (r^t*C0) / (1+rd)^t ) ... then some magic happens, then ... < 0
But how to get from beginning to end, I don't know.
Anecdotally, I see it as, since r < rd and both number will shrink exponentially with time, there would never be a periodwhere the returned cashflow would be greater than the return you could have gotten by investing at the discount rate rd, thus NPV < 0 (and if even this is a wrong way to think about it, please let me know).
Basically, asking for a proof that NPV always < 0 whenever the return on the investment is less than the discount rate for all periods.
Would appreciate a proof and explanation (or even a explanation about why this may be trying to prove something that is not necessarily true). Thanks.