Internal Rate of Return (IRR) — 1 period | Equation, Derivation and Rearrangements

Core idea

Overview

The single-period Internal Rate of Return (IRR) is the discount rate that equates the present value of a future cash inflow to the initial investment cost. In this simplified model, it represents the annualized percentage return on a project that lasts exactly one time period.

When to use: This formula is applied to short-term investments or capital budgeting scenarios where an initial cash outlay leads to a single terminal payoff. It is most effective when comparing simple, mutually exclusive projects with identical durations.

Why it matters: It provides a clear percentage-based metric to evaluate whether a project's return exceeds the cost of capital. In the real world, it serves as a 'break-even' benchmark, allowing managers to determine the maximum interest rate a project can sustain while remaining profitable.

Symbols

Variables

IRR = IRR, $C_{1}$ = Cash Flow Year 1, $C_{0}$ = Initial Cost

IRR

Variable

C_{1}

Cash Flow Year 1

$

C_{0}

Initial Cost

$

Walkthrough

Derivation

Understanding Internal Rate of Return (IRR)

IRR is the discount rate that makes NPV equal to zero. For one period, it is the rate that equates the initial cost with the discounted inflow.

Cash flows occur at the stated times (C0 at time 0, C1 at time 1).
The IRR decision rule uses a hurdle rate (e.g., cost of capital) for comparison.

1

Set NPV Equal to Zero:

By definition, IRR is the rate where the present value of inflows equals the initial outflow.

0 = - C_{0} + \frac{C _{1}}{1 + I R R}

2

Solve for IRR:

Rearranging gives the one-period IRR as the percentage return relative to the initial outlay.

C_{0} = \frac{C _{1}}{1 + I R R} ⟹ 1 + I R R = \frac{C _{1}}{C _{0}} ⟹ I R R = \frac{C _{1}}{C _{0}} - 1

Note: Sign convention matters: here $C_{0}$ is treated as a positive cost (an outflow written with a minus sign in NPV).

Result

C_{0} = \frac{C _{1}}{1 + I R R} ⟹ 1 + I R R = \frac{C _{1}}{C _{0}} ⟹ I R R = \frac{C _{1}}{C _{0}} - 1

Source: Standard curriculum — A-Level Business / Finance

Free formulas

Rearrangements

Solve for IRR

Make IRR the subject

I R R = \frac{C _{1}}{C _{0}} - 1

Rearrange the one-period Internal Rate of Return (IRR) formula to solve for IRR by isolating it step-by-step.

Difficulty: 2/5

Solve for $C_{1}$

Make C1 the subject

C_{1} = C_{0} (1 + I R R)

To make C1 the subject, first move C0 to the other side of the equation, then multiply both sides by (1+IRR) to isolate C1.

Difficulty: 2/5

Solve for $C_{0}$

Make C0 the subject

C_{0} = \frac{C _{1}}{1 + I R R}

To make $C_{0}$ the subject, add $C_{0}$ to both sides of the equation.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a hyperbolic curve because the initial cost appears in the denominator of the rearranged formula, causing the internal rate of return to decrease as the initial cost increases. For a finance student, this means that a smaller initial cost allows for a higher potential return, while a very large initial cost forces the internal rate of return toward zero. The most important feature of this curve is that the internal rate of return never reaches zero, illustrating that even with a very high initial cost, the investment maintains a positive, albeit diminishing, return relative to the future cash flow.

Graph type: hyperbolic

Why it behaves this way

Intuition

The equation visually balances the initial investment with the present value of future returns, showing the discount rate at which they exactly offset.

C_{0}

Initial cash outflow or investment made at the beginning of the period.

The upfront cost of the project.

C_{1}

Cash inflow received at the end of the single period.

The total money received back from the project after one period.

IRR

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

The effective annual percentage return generated by the investment, representing the maximum cost of capital the project can bear to break even.

\frac{C _{1}}{( 1 + I R R )}

The present value of the future cash inflow C_1.

How much the future cash inflow

C_{1}

is worth in today's money, given the discount rate IRR.

Signs and relationships

- C_0: Represents an initial cash outflow or investment, which is conventionally treated as a negative value in Net Present Value (NPV) calculations because it reduces the overall cash position.
(1+IRR) in the denominator: This term discounts the future cash inflow $C_{1}$ back to its present value. A higher IRR in the denominator results in a lower present value for $C_{1}$ , reflecting the time value of money.

Free study cues

Insight

Canonical usage

Monetary values for cash flows must be consistent in currency, while the Internal Rate of Return (IRR) is a dimensionless rate, typically used as a decimal in calculations and reported as a percentage.

Common confusion

A frequent mistake is using the percentage form of IRR (e.g., 10) directly in calculations instead of its decimal equivalent (0.10). Additionally, ensuring all cash flows are in the same currency and relate to the

Dimension note

The Internal Rate of Return (IRR) is a ratio representing a rate of return, making it a dimensionless quantity. While often expressed as a percentage for clarity in reporting, it should be treated as a decimal in calculations.

Unit systems

$C_{0}$ currency unit (e.g., USD, EUR) - Initial investment or cash outlay. Must be in the same currency as C_1.

$C_{1}$ currency unit (e.g., USD, EUR) - Cash inflow at the end of the period. Must be in the same currency as C_0.

IRRdimensionless - Internal Rate of Return. Used as a decimal in calculations (e.g., 0.10 for 10%) but often reported as a percentage.

One free problem

Practice Problem

An entrepreneur invests 5,000 in a product launch. Exactly one year later, they receive a payout of 6,250. Calculate the Internal Rate of Return (IRR) for this investment.

Initial Cost5000 $

Cash Flow Year 16250 $

Solve for: IRR

Hint: Rearrange the formula to solve for IRR: IRR = (C1 / C0) - 1.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When comparing project return against a bank loan rate, Internal Rate of Return (IRR) — 1 period is used to calculate IRR from Cash Flow Year 1 and Initial Cost. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Input the initial cost C0 as a positive value because the formula subtracts it.
Always convert percentages to decimal form before calculation.
If the resulting IRR is greater than the hurdle rate, the investment is generally considered viable.

Avoid these traps

Common Mistakes

Confusing with ROI/accounting return.
Multiple IRRs possible (not in this simple model).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

IRR is the discount rate that makes NPV equal to zero. For one period, it is the rate that equates the initial cost with the discounted inflow.

This formula is applied to short-term investments or capital budgeting scenarios where an initial cash outlay leads to a single terminal payoff. It is most effective when comparing simple, mutually exclusive projects with identical durations.

It provides a clear percentage-based metric to evaluate whether a project's return exceeds the cost of capital. In the real world, it serves as a 'break-even' benchmark, allowing managers to determine the maximum interest rate a project can sustain while remaining profitable.

Confusing with ROI/accounting return. Multiple IRRs possible (not in this simple model).

When comparing project return against a bank loan rate, Internal Rate of Return (IRR) — 1 period is used to calculate IRR from Cash Flow Year 1 and Initial Cost. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Input the initial cost C0 as a positive value because the formula subtracts it. Always convert percentages to decimal form before calculation. If the resulting IRR is greater than the hurdle rate, the investment is generally considered viable.

Yes. Open the Internal Rate of Return (IRR) — 1 period equation in the Equation Encyclopedia app, then tap "Copy Excel Template" or "Copy Sheets Template".

References

Sources

Brealey, Richard A., Myers, Stewart C., and Allen, Franklin. Principles of Corporate Finance.
Ross, Stephen A., Westerfield, Randolph W., and Jordan, Bradford D. Fundamentals of Corporate Finance.
Wikipedia: Internal rate of return
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. (2019). Fundamentals of Corporate Finance (12th ed.). McGraw-Hill Education.
Internal rate of return - Wikipedia
Brealey, Myers, and Allen Principles of Corporate Finance
Ross, Westerfield, and Jaffe Corporate Finance

Internal Rate of Return (IRR) — 1 period

Overview

Variables

Derivation

Set NPV Equal to Zero:

Solve for IRR:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Net Present Value (NPV) — 1 period

Frequently Asked Questions

Sources