# Is the $NPV$ always a decreasing function in $r$

I was able to prove that, for positive Cash Flows $$f_i$$ and any value of $$f_0$$, the $$NPV$$ function is decreasing in $$r$$, hence, for $$r_m, then $$NPV(r_m)>NPV(r_p)=0$$.

$$NPV(r,f_i)=\sum_{i=0}^n{1\over(1+r)^i}f_i$$

But I cannot prove or disprove with a proof or a simple counterexample that for any positive or negative Cash Flows $$f_i$$, $$r_m implies or not implies $$NPV(r_m)>NPV(r_p)$$.

In most cases, I have $$f_0<0$$ (for funding the project) and $$f_n>0$$ (for the final return), and the rest of cash flows reasonably less than the funding amount $$|f_i|. Perhaps I am missing some financial condition for a balanced Statement which allows this to be true always?

• No it is not always true. It depends on the number of changes of signs in the cash flows $f_i$ , doesn't it? Try cash flows with 2 changes of sign, positive, negative, positive again. There can be a hump in the NPV curve, and 2 rates where NPV = 0. Apr 4 at 0:23
• Sounds correct, though for some reason I have been unable to pass from the $NPV(IIR)=0$ case into a $NPV(r_m)>0$ case for $r_m<r_p$, $f_0<0$ and $f_n>0$ Apr 4 at 2:12

Try $$f_0=-0.9975, f_1=2.9975, f_2=-3, f_3=1$$. This should have 3i IRRs, namely -5%, 0 and 5% with the desired behavior between about -3% and +3%.
• But in this we do not have $|f_i|<|f_0|$, which is what we are looking for, for a initial funding greater than the cashflows. Apr 6 at 3:23