# IRR Calculator - Internal Rate of Return

Created by Bogna Szyk and Tibor Pal, PhD candidate
Reviewed by Jack Bowater
Based on research by
Cipra T. Financial and Insurance Formulas (2006)See 1 more source
Wolfram Research
Last updated: Jul 20, 2022

This internal rate of return calculator (or the IRR calculator for short) is a helpful tool for determining whether a future investment will be profitable for you. As this metric is not always easy to understand and apply correctly, we prepared this handy guide to explain in detail how to calculate the IRR. We will also provide you with an IRR formula that illustrates the underlying principles.

## What is the internal rate of return?

By definition, an internal rate of return (IRR) is the interest rate at which all cash flows associated with a particular investment have a net present value equal to zero. In other words, a project's IRR is the discount rate that makes the present value of the expected future cash flows equal to the initial investment.

Practically, such an interest rate guarantees that the money you would invest in such a project today will earn you precisely $0. You will neither win nor lose if you chose this investment - the only consequence will be your money has remained constant. Why can estimating this particular value be helpful for a person dealing with finances? The reason is that the IRR corresponds to the project's rate of return. If this return exceeds the cost of the funds (for example, cost of a loan or APR) employed to finance the project, then the difference might be a helpful approximation for the profitability. On the other hand, if the IRR is lower than the cost of capital, the project is possibly unproductive. The same concept applies to the yield to maturity (YMT), where the discount rate forces the present value of the cash inflow to equal the price of the bond if you hold the bond to maturity. In both cases, the analysis of what is the time value of money constitutes the bottom line for the calculation. ## IRR formula The IRR formula is based on the equation used to calculate the net present value. As you may remember: $\footnotesize NPV = -C_0 + \sum_{i\ =\ 1}^n\left[\frac{C_i}{(1 + r)^i}\right]$ where: • $NPV$ – Net present value of the investment; • $C_0$ – Your Initial investment; • $C_i$ – Will be either positive (income) or negative (expenses). Each year, you have to increase the $i$ value by 1; • $n$ – Number of periods (typically years) between now and the moment when you will receive your money; and • $r$ – Discount rate (interest rate used in cash flow analysis). As mentioned earlier, $IRR$ is the discount rate, $r$, for when NPV equals zero. Hence, we need to use the following formula: $\footnotesize\! 0 = -C_0 + \frac{C_1}{(1 + IRR)} + \frac{C_2}{(1 + IRR)^2}\\[1em]\quad + \frac{C_3}{(1 + IRR)^3} + \dots + \frac{C_n}{(1 + IRR)^n}$ As you probably noticed, there is no direct IRR formula: you can find this value either by trial-and-error (guessing the value of IRR and adjusting it after each result) or by using an automated tool, such as this IRR calculator. ## How to calculate IRR: an example As the idea of IRR is hard to grasp at first glance, we have prepared an example to illustrate how you may use it. Let's say that a hairdresser wants to include perm maintenance in her hair salon. She estimates that the equipment will cost$6,000. She estimates that each year, she will earn $2,000 more than if she didn't purchase this equipment. After five years, the machine will be too old to use, so she will sell it for$1,000.

The hairdresser wonders whether it will be more profitable to buy all of that equipment or use her money to invest in her friend's coffee shop with an expected return of 12%.

To calculate the IRR, she needs to solve the following problem:

$\footnotesize\! 0 = -6000\! +\! \frac{2000}{(1 + IRR)}\! +\! \frac{2000}{(1 + IRR)^2}\\[1em]\quad + \frac{2000}{(1 + IRR)^3} + \frac{2000}{(1 + IRR)^4} \\[1em]\quad + \frac{2000}{(1 + IRR)^5} + \frac{1000}{(1 + IRR)^6}$

You can use this IRR calculator to confirm that the IRR metric, in this case, is 22.22%. It is substantially higher than her other investment option, which makes buying additional equipment the preferable choice.

## How to use the IRR calculator

You can quickly resolve the previously discussed mathematical problem by placing the values from the question into our calculator.

The initial investment is your expenses at time zero (with a positive sign) that precedes the yearly cash flows.

The default number of years in our IRR calculator is three, but if you want to analyze a more extended period, the additional lines will pop up automatically. Also, you will simultaneously see the estimated internal rate of return (IRR) as you input each year.

## IRR vs. MIRR: what is the difference?

Apart from the IRR metric, you can also determine the profitability of an investment with MIRR – the modified internal rate of return. The main difference between these two metrics lies in the approach to the cash inflows: in MIRR, we assume that each cash inflow is reinvested at a steady rate, called the reinvestment rate. This way, the profit you receive on your investment is used to generate additional income throughout your project.

Bogna Szyk and Tibor Pal, PhD candidate
Internal rate of return (IRR)
%
Initial investment
$Annual cash flows Year 1$
Year 2
$Year 3$
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