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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 $r_m<r_p=IRR$, 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 $f_i$, $r_m<r_p$ 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 $|f_i|<f_0$. Perhaps I am missing some financial condition for a balanced Statement which allows this to be true always?

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    $\begingroup$ 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. $\endgroup$
    – nbbo2
    Commented Apr 4, 2021 at 0:23
  • $\begingroup$ 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 $f_n>0$ $\endgroup$
    – Brethlosze
    Commented Apr 4, 2021 at 2:12

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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%.

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  • $\begingroup$ 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. $\endgroup$
    – Brethlosze
    Commented Apr 6, 2021 at 3:23

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