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Internal Rate of Return (IRR)

Calculates the discount rate at which the net present value (NPV) of all cash flows from a particular project equals zero.

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0 = t = 0 \sum n \frac{C F _{t}}{( 1 + I R R ) ^{t}}

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Core idea

Overview

The Internal Rate of Return (IRR) is a capital budgeting metric used to estimate the profitability of potential investments. It represents the discount rate that makes the net present value (NPV) of all cash flows from a particular project equal to zero. A project is generally considered acceptable if its IRR is greater than the company's required rate of return or cost of capital, as it indicates the project is expected to generate value.

When to use: Use IRR to evaluate the attractiveness of investment projects, especially when comparing multiple projects with different cash flow patterns. It's particularly useful for capital budgeting decisions where a clear hurdle rate (cost of capital) is established. Ensure all relevant cash inflows and outflows are accurately estimated over the project's life.

Why it matters: IRR is a crucial tool for investment appraisal, helping businesses and investors make informed decisions about allocating capital. It provides a single, easily understandable percentage return that can be compared against a company's cost of capital or other investment opportunities, guiding strategic growth and resource deployment.

Symbols

Variables

CF_t = Cash Flow at time t, IRR = Internal Rate of Return, t = Time Period, n = Total Number of Periods

C F_{t}

Cash Flow at time t

IRR

Internal Rate of Return

t

Time Period

years

n

Total Number of Periods

years

Walkthrough

Derivation

Formula: Internal Rate of Return (IRR)

The Internal Rate of Return (IRR) is the discount rate that makes the Net Present Value (NPV) of all cash flows from a project equal to zero.

Cash flows are known and occur at discrete intervals.
All intermediate cash flows are reinvested at the IRR itself.
The project has a conventional cash flow pattern (initial outflow followed by inflows).

Start with the Net Present Value (NPV) formula:

The Net Present Value (NPV) calculates the present value of all future cash flows, both positive and negative, discounted at a specific rate 'r'. $C F_{t}$ represents the cash flow at time $t$ , and $n$ is the total number of periods.

N P V = t = 0 \sum n \frac{C F _{t}}{( 1 + r ) ^{t}}

Define IRR as the rate where NPV is zero:

The Internal Rate of Return (IRR) is, by definition, the discount rate (denoted as IRR) that makes the NPV of a project's cash flows exactly equal to zero. This means the present value of cash inflows equals the present value of cash outflows.

0 = t = 0 \sum n \frac{C F _{t}}{( 1 + I R R ) ^{t}}

Note: For most projects, the initial investment ( $C F_{0}$ ) is a negative cash flow, representing an outflow, while subsequent cash flows ( $C F_{t}$ for $t > 0$ ) are typically inflows.

Result

0 = t = 0 \sum n \frac{C F _{t}}{( 1 + I R R ) ^{t}}

Source: Brealey, Myers, & Allen, Principles of Corporate Finance, 13th Edition, McGraw-Hill Education.

Visual intuition

Graph

Graph unavailable for this formula.

The graph displays a linear relationship where the cash flow at time t and the internal rate of return move in opposite directions, resulting in a line with a negative slope. For a finance student, this means that a higher cash flow at a specific time allows for a higher internal rate of return while still keeping the net present value at zero. The most important feature of this curve is that the linear relationship ensures a constant rate of change, meaning that any incremental increase in cash flow results in a predictable, proportional adjustment to the internal rate of return.

Graph type: linear

Why it behaves this way

Intuition

Visualize a financial equilibrium where the present value of all cash inflows precisely offsets the present value of all cash outflows.

C F_{t}

The net cash flow (inflow or outflow) occurring at time t.

Represents the actual money movement at a specific point in the project's life.

IRR

The specific discount rate that makes the Net Present Value (NPV) of all project cash flows equal to zero.

It's the effective average annual return a project is expected to yield over its lifespan.

t

The time period index, indicating when a cash flow occurs relative to the project start (t=0).

Determines how much a future cash flow is discounted, reflecting its distance in time.

n

The total number of periods over the project's lifespan.

Defines the overall duration for cash flow analysis, from initial investment to final return.

\sum_{t = 0}^{n}

The summation operator, aggregating the present values of all cash flows over the project's life.

Combines all present values of inflows and outflows to find the total Net Present Value.

Signs and relationships

(1+IRR)^t in the denominator: The denominator (1+IRR)^t discounts future cash flows to their present value. The exponent t ensures that cash flows further in the future are discounted more heavily, reflecting the time value of money.
0 on the left side: The 0 on the left side signifies that the sum of the present values of all cash flows (inflows and outflows) is zero, meaning the project's Net Present Value (NPV) is zero at the IRR.

Free study cues

Insight

Canonical usage

The Internal Rate of Return (IRR) is a dimensionless rate, typically expressed as an annual percentage, used to evaluate and compare the profitability of investment projects.

Common confusion

A common mistake is failing to convert percentage rates to decimals before using them in the formula (e.g., using '10' instead of '0.10' for 10%).

Dimension note

The Internal Rate of Return (IRR) itself is a dimensionless rate. In the formula, it is added to 1, making the discount factor (1+IRR)^t dimensionless.

Unit systems

IRR% - The IRR is a rate, mathematically treated as a decimal (e.g., 0.10 for 10%) in calculations. Its periodicity (e.g., annual, monthly) must be consistent with the cash flow periods.

$C F_{t}$ currency (e.g., USD, EUR) - All cash flows (inflows and outflows) must be expressed in the same currency unit.

$t$ periods (e.g., years, months) - Represents the number of discrete time periods. The length of each period must be consistent with the periodicity of the IRR.

One free problem

Practice Problem

A project requires an initial investment of $100, 000 (t = 0) . I t i se x p ec t e d t o g e n er a t ec a s h f l o w so f$ 30,000 in year 1, $40,000 in year 2, and a final cash flow in year 3. If the project's Internal Rate of Return (IRR) is 10%, what is the cash flow in year 3?

CF_0-100000

CF_130000

CF_240000

Internal Rate of Return0.1 %

Total Number of Periods3 years

Solve for: IRR

Hint: Remember to discount each cash flow back to time zero using the given IRR.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

A company evaluating whether to invest in a new manufacturing plant by comparing its expected IRR to the company's cost of capital.

Study smarter

Tips

IRR assumes cash flows are reinvested at the IRR itself, which may not always be realistic.
Be cautious with projects having non-conventional cash flows (e.g., alternating signs), as they might yield multiple IRRs or no real IRR.
Always compare IRR to the project's cost of capital (hurdle rate); accept if IRR > hurdle rate.
IRR can sometimes conflict with NPV for mutually exclusive projects; NPV is generally preferred in such cases.

Avoid these traps

Common Mistakes

Incorrectly handling initial investment as a negative cash flow at t=0.
Ignoring the possibility of multiple IRRs for non-conventional cash flow streams.
Comparing IRR to an inappropriate benchmark instead of the cost of capital.

Common questions

Frequently Asked Questions

The Internal Rate of Return (IRR) is the discount rate that makes the Net Present Value (NPV) of all cash flows from a project equal to zero.

Use IRR to evaluate the attractiveness of investment projects, especially when comparing multiple projects with different cash flow patterns. It's particularly useful for capital budgeting decisions where a clear hurdle rate (cost of capital) is established. Ensure all relevant cash inflows and outflows are accurately estimated over the project's life.

IRR is a crucial tool for investment appraisal, helping businesses and investors make informed decisions about allocating capital. It provides a single, easily understandable percentage return that can be compared against a company's cost of capital or other investment opportunities, guiding strategic growth and resource deployment.

Incorrectly handling initial investment as a negative cash flow at t=0. Ignoring the possibility of multiple IRRs for non-conventional cash flow streams. Comparing IRR to an inappropriate benchmark instead of the cost of capital.

A company evaluating whether to invest in a new manufacturing plant by comparing its expected IRR to the company's cost of capital.

IRR assumes cash flows are reinvested at the IRR itself, which may not always be realistic. Be cautious with projects having non-conventional cash flows (e.g., alternating signs), as they might yield multiple IRRs or no real IRR. Always compare IRR to the project's cost of capital (hurdle rate); accept if IRR > hurdle rate. IRR can sometimes conflict with NPV for mutually exclusive projects; NPV is generally preferred in such cases.

Yes. Open the Internal Rate of Return (IRR) equation in the Equation Encyclopedia app, then tap "Copy Excel Template" or "Copy Sheets Template". The corresponding spreadsheet function is: =IRR(values, [guess]). Note: Select a column or row containing all cash flows starting with the initial outflow as a negative number, e.g. {-10000, 3000, 4000, 6000}. The optional guess defaults to 0.1.

References

Sources

Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2022). Fundamentals of Corporate Finance (13th ed.). McGraw-Hill Education.
Wikipedia: Internal Rate of Return
Ross, S. A., Westerfield, R. W., & Jaffe, J. F. (2019). Corporate Finance (12th ed.). McGraw-Hill Education.
Brealey, Richard A., Myers, Stewart C., and Allen, Franklin. Principles of Corporate Finance. McGraw-Hill Education.
Ross, Stephen A., Westerfield, Randolph W., and Jaffe, Jeffrey F. Corporate Finance. McGraw-Hill Education.
Brigham, Eugene F., and Houston, Joel F. Fundamentals of Financial Management. Cengage Learning.
Brealey, Myers, & Allen, Principles of Corporate Finance, 13th Edition, McGraw-Hill Education.