# 1.20 The Equivalent of the Multiple Cash Flows as A Singular Cash (Out-) Flow: “$1 to $2” The Rate of Return

You can readily see that indeed there exists a “zero” NPV solution for this – and any – NPV problem. That is to say that there exists a theoretical discount rate at which the project’s NPV equals zero. This abstract rate is called the “Internal Rate of Return.” The concept of the IRR shall occupy us more so soon.

Perhaps another, simpler manner in which example #4 may be understood better is the following. Let’s not look at a *series* of cash flows, instead let’s just look at *one* present- and *one* future-cash flow. Imagine you have one dollar, and that in five years it will grow to two dollars. That would imply a rate of (compound-) growth of (approximately) 15% per year. How did we get that? Solve for “R” below.

$1 (1 + R) ^{5 }= $2

R = .1487 (R ≈ .15)

$1 (1.1487) ^{5} = $2

*[See “Note 1” below, if you have trouble solving for “R.”]*

This is the same as saying that the present value of $2 at a discount rate of 15% is $1. To be precise:

$2 ÷ (1.1487) ^{5} = $1

At a discount rate of (approximately) 15%, the NPV would be zero because the cost is $1 and so is the present value of the one, $2 future cash inflow!

$2 ÷ (1.1487) ^{5}– $1 = $0

Clearly, this does not mean that you have made no money at all! Indeed, you have – 15%! You earned 15% on the dollar you invested at the start of the five-year period!

Thus, the IRR expresses a kind of rate of return (ROR) for the project; it does not mean that the NPV for the project is zero! It does not mean that you have made no money!

In contrast, the NPV provides a net dollar figure, given a pre-specified discount rate representing the firm’s cost of capital; the dollar figure, which the NPV presents, indicates the expected increase in wealth to the corporation from having invested in the project – in present value terms, and after accounting for the firm’s discounted, capital costs.

**Question**: Can you relate the NPV method to the various technical concerns discussed earlier? (We shall review this later as well.). For example, does it account for the time value of money? (Yes!)

**Note 1:**

If you have some difficulty with the solution for “R,” here are the algebraic transpositions, step by step:

$1 (1 + R)^{ 5} = $2

(1 + R) 5 = 2 / 1

1 + R = (2) ^{1/5 }

R = 1.1487 – 1

R = 0.1487

**Note 2:**

This corresponds with the general TVM formulae:

FV = PV (1 + R)^{n}(*Basic TVM Formula*)

AND

FV ÷ (1 + R)^{n}– PV = 0 = NPV(NPV Formula)

…and by doing simple algebra (with the first formula above), we get:

FV / PV = (1 + R) ^{n}

**R = (FV / PV) ^{1/n} – 1 **

Thus, we convert a series of cash flows into what looks like its equivalent Zero-Coupon Bond. This particular value for “R” is referred to as the “Internal Rate of Return,” or simply “IRR.”