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Why NPV Need Not Decrease with the Discount Rate for Mixed Cash Flows

Article Quant Q&A · Author: Brethlosze
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Summary

The post asks whether net present value (NPV) must fall as the discount rate rises, especially when cash flows include both positive and negative amounts. For a stream of positive future cash flows, the author says they can show NPV decreases with the rate, but asks whether the same relationship holds for arbitrary cash flows and multiple internal rates of return.

An answer supplies a cash-flow example with alternating signs and states that it has three IRRs at −5%, 0%, and 5%. It presents this as a counterexample to assuming a simple, globally decreasing relationship between NPV and the discount rate for mixed cash flows. The example illustrates why unconventional cash-flow patterns can produce multiple IRRs, so a comparison based only on which rate is lower may not determine which NPV is higher. The post gives a specific example rather than a general proof or a full account of conditions under which NPV is monotonic.

Key ideas

  • With only positive future cash flows, discounting makes their present values decline as the rate rises.
  • Mixed positive and negative cash flows can make NPV nonmonotonic in the discount rate.
  • The answer gives a cash-flow sequence with three stated internal rates of return.
  • Multiple IRRs make simple comparisons between a market rate and an IRR potentially misleading.

Tags

StatisticsFixed income
Full text
# Is the $NPV$ always a decreasing function in $r$


# 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 $r_m<r_p=IRR$, 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?

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/63223

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.