To better understand the internal rate of return, let's take a closer look at the equation.
The most confusing part of the equation is the sum. So let's simplify it by imagining an investment that only makes one payment.
Imagine you invest $1000 in a project. And you know that the project will yield $1100 in 5 years. Let's assume it doesn't make any payments between now and then.
In this case, the IRR equation is simple. It's C0=C5/(1+r)5 or $1000 = $1100/(1+r)5.
Now, let's rearrange the equation so that it's $1000(1+r)5=$1100. In English, this says, "what annual rate of return, r, do I need if I want $1000 to grow to $1100 over five years." If we solve for r, we find r = .01193 = 1.193%.
In other words, making this investment is equivalent to putting your money in a bank account at 1.193% annual interest and taking it out after five years.
Now, imagine you take out some money every year. We'll need to add in the intermediate terms.
You can think of all those intermediate terms in the same way. It's as if you put $1000 in a bank account at an annual interest rate of r, and then took out a certain quantity of money after t years. But of course, r will change if you take money out earlier.
What does the denominator (1+r)t tell us?
By dividing every term by (1+r)t, the equation is telling us that it's better to receive money sooner than later (this is the time value of money). If you receive a payment far into the future, t will be large, the denominator (1+r)t will be large, and thus, the effective payment will be small.