Internal rate of return (IRR)
The discount rate at which an investment's net present value equals zero; its effective annual return.
Find r such that 0 = −C0 + Σ CFt ÷ (1 + r)^t
Explanation
The internal rate of return is the rate that makes the present value of what you get back exactly equal to what you put in. Put another way, it is the annual compound return the investment delivers given its cash flows and their timing.
There is usually no neat formula for IRR. It is found by trial and error (spreadsheets and the app do this automatically): try a rate, compute NPV, and adjust up or down until NPV is zero.
IRR is intuitive, but use it with care:
- It assumes interim cash flows can be reinvested at the IRR itself, which may not be realistic.
- Cash flows that switch sign more than once (for example, invest, receive, invest again) can produce more than one IRR or none.
- A high IRR on a small amount can be worth less in dollars than a lower IRR on a large amount. NPV captures size; IRR does not.
In scenarios, an IRR-style figure is a natural way to summarize your personal rate of return across contributions and withdrawals, sometimes called a money-weighted return.
Example
You pay $10,000 today and receive $4,000 at the end of each of the next three years. Find r where the present value of the $4,000 payments equals $10,000.
- Try 8%: present value = $10,308.39 (too high, so r is above 8%)
- Try 12%: present value = $9,607.33 (too low, so r is below 12%)
- Try 9.70%: $4,000 ÷ 1.0970 = $3,646.31; ÷ 1.0970^2 = $3,323.89; ÷ 1.0970^3 = $3,029.98. Total = $10,000.18, essentially $10,000.
The IRR is about 9.70% a year. If your required return is 8%, the investment clears the bar; if it is 10%, it falls just short.