Interpreting IRR and Recognizing Multiple-IRR Problems

# What is the correct interpretation of IRR?


# What is the correct interpretation of IRR?












The internal rate of return (IRR) is usually defined as the rate that sets the net present value of a sequence of cash flows to zero, and it is often described as a “discount rate.”

I would like to ask about IRR from a different angle: interpreting it as the yield earned by money while it is actually tied up in an investment. Under this interpretation, money earns a constant yield only while it remains committed; once distributions are paid out, that money leaves the investment and no longer earns within it. Decisions about what happens to distributed cash are separate.

My question is:

Is this interpretation of IRR — as the yield on money while it is tied up — commonly used or accepted? In much of the literature, the emphasis appears to be on discounted cash flows.

I elaborate this interpretation and provide a simple illustrative example in more detail here (if linking is acceptable):

One Yield to Rule Them All?

## Answer by João (score 0)

https://quant.stackexchange.com/a/85318

Internal rate of return (IRR): is the discount rate at which the present value of the free cash flows generated by the project equals the value of the initial investment required to implement it.

$$ II = \frac{FCF_1}{(1+\text{IIR})^1} + \frac{FCF_2}{(1+\text{IIR})^2} + \cdots + \frac{FCF_t}{(1+\text{IIR})^t} $$

Where:

$\text{IIR}$ – Internal rate of return

$t$ – Project (corporate finance example) lifetime (time horizon)

$FCF_t$ – Value of the $t$-th free cash flow generated by the project

$II$ – Initial investment made in the project

The concept of the internal rate of return is analogous to YTM (yield to maturity).

In other words, IRR indicates the rate of return generated by the project. Decision rule for IRR:

- If IRR > required rate of return for the project, accept the project (positive NPV).

- If IRR < required rate of return for the project, reject the project (negative NPV).

I would avoid saying the IRR is the discount rate. For example, when making a DCF project, someone questions you about which rate did you discount the CF at?

Now you can be think on IRR which is wrong, let's assume the base case scenario of the WACC as the correct one.

As the article mentioned about multiple IRR, I think deserves some attention.

A project always has the same NPV.

However, a project can have multiple IRRs

Example: Consider a company that is considering undertaking an investment with the following characteristics:

$II$: €1,600

$FCF_1$: €10,000

$FCF_2$: –€10,000

And following the IRR equation...:

$$ 1600 = \frac{10{,}000}{(1+\text{IIR})} + \frac{-10{,}000}{(1+\text{IIR})^{2}} $$

$$ \text{Let } z = 1+\text{IIR} $$

$$ 1600 = \frac{10{,}000}{z} - \frac{10{,}000}{z^{2}} $$

$$ 1600 z^{2} = 10{,}000 z - 10{,}000 $$

$$ 1600 z^{2} - 10{,}000 z + 10{,}000 = 0 $$

$$ z = \frac{10{,}000 \pm \sqrt{(-10{,}000)^2 - 4(1600)(10{,}000)}}{2(1600)} $$

$$ z = \frac{10{,}000 \pm 6{,}000}{3{,}200} $$

$$ z = 5 \quad \text{or} \quad z = 1.25 $$

$$ 1+\text{IIR} = 5 \quad \text{or} \quad 1+\text{IIR} = 1.25 $$

$$ \text{IIR} = 400\% \quad \text{or} \quad \text{IIR} = 25\% $$

Which one should be used?

None of them! Although there are two rates that satisfy the definition of IRR, the truth is that neither of them is truly informative about the return generated by the project. Instead you can use the modified internal rate of return (MIRR) -> the discount rate that equates the present value of the project’s future free cash flows with the terminal value of the cash inflows