Mathematical proof that NPV always negative when rate of return less than discount rate

Question

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.

Answer 1

The internal rate of return is the value of r which satisifies the equation

0 = C0 + C1 / (1 + r) + C2 / (1 + r)^2 + ... + Cn / (1 + r)^n

The net present value (NPV) is the value of the sum of discounted cash flows:

NPV = C0 + C1 / (1 + rd) + C2 / (1 + rd)^2 + ... + Cn / (1 + rd)^n

Since rd > r, for t = 1, ..., n, we have the simple inequalities:

(1 + rd) > (1 + r)

(1 + rd)^t > (1 + r)^t

Taking the inverse switches the direction of inequality:

1 / (1 + rd)^t < 1 / (1 + r)^t

Since infusions of cash are negative cashflow, C0 < 0. But presumably Ct > 0 for all subsequent time periods (i.e., for t = 1, ..., n). So, multiplying the inequality above by a positive quantity, Ct, preserves the direction of the inequality:

Ct / (1 + rd)^t  < Ct / (1 + r)^t

Now, looking at the sum term-by-term,

0 = C0 + C1 / (1 + r) + C2 / (1 + r)^2 + ... + Cn / (1 + r)^n
  > C0 + C1 / (1 + rd) + C2 / (1 + rd)^2 + ... + Cn / (1 + rd)^n = NPV

This shows that if we valued a cash flow (with an initial investment followed by positive cash flows) using a discount rate which is greater than the rate of return, then NPV would be negative.

Answer 2

Assuming the second line should be corrected to:

NPV = -C0 + summ( Ct / (1+rd)^t ) 
    = -C0 + summ( (r^t*C0) / (1+rd)^t )

We can continue:

      = -C0 + summ( (r/(1+rd)^t *C0 )

So clearly what we are interested in is, is:

summ( (r/(1+rd))^t ) 

necessarily < 1? All of the terms in the sum are positive, so the maximum value of the sum happens if we sum to infinity. Given that r < 1+rd, we know that (r/(1+rd)) < 1, so the sum to infinity has a finite value. Specifically, the sum to infinity is:

      1
------------  - 1
1 - r/(1+rd)

Let us multiply top and bottom of the fraction by 1+rd and expand 1

  1 + rd         1 + rd - r
----------   -   ----------
1 + rd - r       1 + rd - r

Simplify:

    r
----------
1 + rd - r

Which looks to me like it can be greater than 1. If rd == 5, r = 4, then the expression comes to 4 / 2 == 2 - which implies that the NPV is greater than zero.

This looks to me like it is wrong (but with luck, someone can point out my error)