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What would the mathematical proof look like of the interpretation of negative internal rate of return shown here look like which says that "Negative IRR indicates that the sum of post-investment cash flows is less than the initial investment; i.e. the non-discounted cash flows add up to a value which is less than the investment"?

The steps I see are:

NPV = C0 + C1/(1+IRR) + C2/(1+IRR)^2 + ... + Ct/(1+IRR)^t = 0
==> C0 = -C1/(1+IRR) - C2/(1+IRR)^2 - ... - Ct/(1+IRR)^t
==> (some other stuff happens (using the relation: IRR < 0))
==> C0 > summation(C1, ..., Ct)

but what is the other stuff that happens in between? Thanks.

1 Answer 1

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The calculation in the example is 600 deposited with four withdrawals bringing the account balance to zero. In Excel the IRR is calculated like so.

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The hand-calculation for this is

enter image description here

∴ x = -0.0260109 or -1.51403

The solution closest to zero is taken.

The IRR is found by discounting all the cash flows (including the initial deposit) to net present value (NPV), equating their sum to zero and solving for x, as described here:

https://en.wikipedia.org/wiki/Rate_of_return#Internal_rate_of_return

enter image description here

In the example case the rate required to balance the NPV cash flows to zero is negative; the investment loses value.

The statement addressed is

Negative IRR indicates that the sum of post-investment cash flows is less than the initial investment; i.e. the non-discounted cash flows add up to a value which is less than the investment.

Taking the simple example

enter image description here

If - a + b + c + d + e = 0 then x = 0 is a solution.

If b + c + d + e > a then x has a positive solution.

If b + c + d + e < a then x has a negative solution.

This stands to reason: if an investment is drawn down to zero by amounts totalling less than the initial investment it must have lost value, hence a negative rate of return.

Mathematically there can be positive and negative solutions, e.g.

a = -246
b = 169
c = 63
d = -120
e = 181

has solutions x = -1.95615 or 0.0848535

Since b + c + d + e > a the solution should be positive.

If there were multiple positive solutions the smaller should be taken, as the first satisfactory value.

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