NPV and IRR

1. Businesses Make Binary Decisions

Invest or Do Not Invest

So we need binary decision rules.

This presentation will provide an introduction to the two most important, Net Present Value (NPV) and Internal Rate of Return (IRR).

2. Maximize Shareholder Wealth

Early in your study of corporate finance you should have learned the Goal of Financial Management above. Clearly any decision rules we define must be consistent with this goal.

NPV will be the change in the value of the firm's equity if the project is accepted.
IRR will also generate accept decisions only if doing so increases the firm's equity.

Thus accepting all projects which have NPV > 0 will maximize shareholder wealth.

3. Net Present Value (NPV)

Net Present Value is the sum of the investment's expected cash inflows and outflows discounted back to their present value at a risk adjusted rate.

The NPV is defined by:

$NPV = -C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + ... + \frac{C_n}{(1+r)^n}$

where:

$r$, is the discount rate per period
$n$, the number of periods in the project's life
$C$, the expected incremental cash flows per period
$C_0$, the initial investment

4. NPV Desicion Rule

<center>$NPV ≥ \$0 \Rightarrow \text{Accept}$ $NPV < \$0 \Rightarrow \text{Reject}$</center>

5. The Discount Rate

The discount rate is the rate required on other projects of similar risk. You can think of it as an opportunity cost, because if the firm didn't use the funds for the project, then it could be paid as a dividend to shareholders. Shareholders then can invest the money in a financial asset earning that assets required return.

Often the project is as risky as a company's existing operations, in which case you use the company's weighted average cost of capital (WACC).
It is incorrect to use the firm's cost of borrowing, for a couple reasons. First the investment decision should be independent of the capital structure decision. Also, by increasing borrowing, you increase your cost of equity capital [Modigliani and Miller, Proposition II](https://micfm.shinyapps.io/MM_propositions_mobile/). So debt only appears cheap.

6. Internal Rate of Return (IRR)

Internal Rate of Return is the discount rate ($r_{IRR}$) that makes the Net Present Value equal zero. That is, we solve the following equation for the $IRR$, using an algorithm such as the [Newton-Raphson method](https://en.wikipedia.org/wiki/Newton%27s_method). $-C_0 + \frac{C_1}{(1+r_{IRR})^1} + \frac{C_2}{(1+r_{IRR})^2} + ... + \frac{C_n}{(1+r_{IRR})^n} = 0$

Note while NPV will have a unique solution, IRR may have multiple solutions depending on the signs of the cash flows. The interested reader can understand why by consulting the [Fundamental Theorem of Algebra](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra), [Descartes' Rule of Signs](https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs), and [Sturm's Theorem](https://en.wikipedia.org/wiki/Sturm%27s_theorem).

7. IRR Desicion Rule

<center>$IRR > \text{Discount Rate} \Rightarrow \text{Accept}$ $IRR < \text{Discount Rate} \Rightarrow \text{Reject}$</center>

8. Choosing Between Projects

When the accept/reject decision on one project does not affect another project, these projects are known as independent. If accepting one project means you must reject another, then these projects are mutually exclusive.

If a project has negative cash flows followed by all positive cash flows, then the project's cash flows are known as conventional.

If two projects are independent and have conventional cash flows, then NPV and IRR will lead to the same accept/reject decision, i.e. $NPV \iff IRR$. This is shown on the app in the following slide.
However, if you must rank projects – and projects are either mutually exclusive or cash flows are not conventional – then use NPV. NPV will generate the correct decision, though IRR may not.

<!– insert interactive app which shows differing ranked projects by npv and irr –>

9. Calculator

Stock Price Calculator (Constant Gowth)

Dividend Next Year:
Dividend Growth Rate:
Discount Rate:

Stock Price:

10. Self-Quiz Questions

1. A stock pays a constant annual dividend of $10. Its discount rate is 10%. What is the value of the stock today?

2. Assume McCauley Ski Inc. has a 12% ROA and 20% payout ratio. What is the maximum rate the firm can grow using only internal financing?

3. If a firm increases its profit margin, its IGR and SGR _______.

11. Reading Ease Score


"Flesch-Kincaid reading ease score: 81.35 Easy (6th grade)"


"Flesch-Kincaid grade level score: 4.35"