NPV and IRR
Net Present Value and Internal Rate of Return
Introduction
In this interactive presentation, we’ll cover the most widely used investment decision rules: Net Present Value (NPV) and Internal Rate of Return (IRR).

These rules are used to decide whether to invest in a project or asset.

It is important to note that, while NPV and IRR calculations give a number as an output, they are fed into a decision rule which is binary. The final output from NPV and IRR is either to reject or accept the investment.
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. If the NPV is greater than $0, the project is accepted. Otherwise the project is rejected.
The NPV is defined by its:
 `r`, which is the discount rate per period
 `n`, the number of periods
 `C`, the cash flow per period
 `C_0`, the initial investment
`NPV = C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} + ... + \frac{C_n}{(1+r)^n}`
NPV: Interpretation
NPV can be interpreted as the amount the market value of the firm will increase/decrease if the project is accepted.
In the following app, you can manipulate the variables we’ve introduced and see how they affect NPV.
Internal Rate of Return (IRR)
Internal Rate of Return is the discount rate (`r_{IRR}`) that makes the Net Present Value equal zero. It is normally used to compare projects. Projects with a higher IRR above a set threshold are accepted.
`C_0 + \frac{C_1}{(1+r_{\text{IRR}})^1} + \frac{C_2}{(1+r_{\text{IRR}})^2} + ... + \frac{C_n}{(1+r_{\text{IRR}})^n} = 0`
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`

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.
Credits and Collaboration
Click the following links to see the code, linebyline contributions to this presentation, and all the collaborators who have contributed to 5Minute Finance via GitHub.
Learn more about how to contribute here.
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