Internal Rate of Return — Definition, Formula & Examples

Internal Rate of Return (IRR) is the interest rate at which an investment's total incoming cash flows exactly equal its total outgoing cash flows, making the net present value zero. It tells you the effective annual return an investment earns over its lifetime.

The internal rate of return is the discount rate $r$ that satisfies $\sum_{t=0}^{n} \frac{C_t}{(1+r)^t} = 0$ , where $C_t$ represents the net cash flow at time period $t$ . It is found by solving this polynomial equation, typically through numerical methods or financial calculators.

Key Formula

\sum_{t=0}^{n} \frac{C_t}{(1+r)^t} = 0

Where:

$C_t$ = Net cash flow at time period t (negative for outflows, positive for inflows)
$r$ = Internal rate of return (the unknown to solve for)
$t$ = Time period index (0, 1, 2, ..., n)
$n$ = Total number of periods

How It Works

You set up the net present value (NPV) equation with all cash inflows and outflows, then find the rate

r

that makes NPV equal zero. Since the equation is generally a higher-degree polynomial, you usually solve it by trial and error, interpolation, or a financial calculator. If the IRR exceeds your required rate of return (the minimum return you'd accept), the investment is considered worthwhile. If it falls below, the investment doesn't meet your threshold.

Worked Example

Problem: You invest $1,000 today and receive $600 at the end of year 1 and $600 at the end of year 2. Find the IRR.

Set up the NPV equation: Write the cash flows with the initial investment as a negative value and set NPV to zero.

-1000 + \frac{600}{(1+r)^1} + \frac{600}{(1+r)^2} = 0

Try r = 0.13 (13%): Substitute r = 0.13 and check whether the equation balances.

-1000 + \frac{600}{1.13} + \frac{600}{1.2769} \approx -1000 + 530.97 + 469.72 = 0.69

Try r = 0.1306: Since 0.69 is very close to zero, the IRR is approximately 13.07%. A financial calculator or spreadsheet confirms this.

r \approx 0.1306

Answer: The internal rate of return is approximately 13.07% per year.

Why It Matters

IRR is a standard tool in corporate finance and capital budgeting for comparing competing projects. Business and finance majors encounter it in courses on financial management, and professionals use it to evaluate bonds, real estate deals, and equipment purchases.

Common Mistakes

Mistake: Forgetting to make the initial investment negative in the cash flow equation.

Correction: Cash outflows (money you spend) must be entered as negative values, and cash inflows (money you receive) as positive. Without the correct signs, the NPV equation has no meaningful solution.