IRR Function (Internal Rate of Return)
Calculate the internal rate of return for a series of periodic cash flows
IRR Calculator
IRR Calculation
Calculates the internal rate of return (IRR) for a series of cash flows (payments and receipts).
Explanation
IRR Definition
Definition:
IRR = Interest rate where Net Present Value (NPV) = 0
Meaning:
The average annual return rate of an investment
Interpretation:
An IRR of 15.24% means the investment grows at an average rate of 15.24% annually
Purpose: Compare different investments
What is IRR?
- IRR = Internal Rate of Return = Internal rate of return
- Measure of investment profitability
- Considers timing of cash flows
- Higher IRR = better investment
- Calculated by iteration (Newton's method)
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Mathematical Foundations of IRR Calculation
The IRR (Internal Rate of Return) is the interest rate where the sum of all discounted cash flows equals zero:
IRR Formula (NPV = 0)
NPV = Net Present Value, CF = Cash Flow, t = Time period
Iterative Method
Newton's method, until NPV ≈ 0 (Tolerance: 0.00001%)
Description of Parameters
Cash Flow List
The cash flow list contains a series of payment streams occurring at regular intervals. The list must contain at least one negative value (cash outflow, e.g., investment) and at least one positive value (cash inflow, e.g., returns). Each value is entered on a separate line.
Important: The order of cash flows is critical! The first value is typically the initial investment (negative), followed by returns (positive).
Example:
-100000 (Investment)
30000 (Return Year 1)
30000 (Return Year 2)
20000 (Return Year 3)
Result (IRR)
The result is the internal rate of return (IRR) expressed as a percentage. This value indicates the average annual return of the investment.
Interpretation: An IRR of 15.24% means that the investment grows at an average rate of 15.24% per year.
Comparison: Investments with higher IRR values are more profitable. Comparing with the cost of capital or other investments helps with decision-making.
Calculation & Accuracy
The IRR is calculated by iteration. The calculator uses Newton's method to find the solution. If the method fails to find a solution after 20 iterations, an error message is displayed.
Accuracy: The result is calculated to 0.00001% precision.
Quick Reference
Formula Overview
\[\sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t} = 0\]
Finds IRR where NPV = 0
Decision Criteria
• IRR > Cost of Capital: Accept investment
• IRR > Other Projects: Preferred
• Higher IRR = Better return
• Compare with market rates
Use Cases
• Investment evaluation
• Project comparison
• Capital budgeting
• Real estate investment
• Private equity valuation
IRR Internal Rate of Return - Detailed Explanation
Fundamentals
The IRR (Internal Rate of Return) is a metric for evaluating investments and financial projects. It indicates the rate at which an investment grows on average each year.
IRR is the discount rate at which the Net Present Value (NPV) of all cash flows equals zero.
IRR vs. NPV
IRR and NPV are related concepts:
Differences
NPV: Calculates present value at a given interest rate
IRR: Finds the interest rate where NPV = 0
Benefit: IRR is independent of rate, easier to compare
Calculation & Methodology
IRR is not calculated by a simple formula, but by iteration:
1. Start with estimated value (e.g., 10%)
2. Calculate NPV for this value
3. Adjust NPV until NPV ≈ 0
4. Maximum 20 iterations
Practical Importance
IRR is an important decision criterion:
Decision Rules
- IRR > Cost of Capital: Accept investment
- IRR < Cost of Capital: Reject investment
- Higher IRR = Better investment (when comparing)
- Risk consideration is important
Practical Calculation Examples
Example 1: Small Investment
Scenario: Business plan
Investment: $-100,000
Year 1: $40,000
Year 2: $40,000
Year 3: $40,000
IRR: ≈ 9.7%
Example 2: Real Estate
Scenario: Real estate investment
Purchase: $-300,000
Rental Income/Year: $20,000
After 10 Years Sale: $400,000
IRR: ≈ 4.2% per year
Example 3: High-Growth
Scenario: Startup investment
Investment: $-50,000
Year 2: $20,000
Year 4: $150,000 (Exit)
IRR: ≈ 73% per year
Calculation Tips
- Time Points: Periods should be equal length
- Order: First line typically negative (investment)
- Minimum Requirement: Min. 1 negative + 1 positive value
- Error Handling: Convergence may not always occur
- Comparison: Compare with cost of capital
- Sensitivity: Test different scenarios
Key Insights
Time is Money
IRR accounts for the time value of money. Early cash flows are more valuable than later ones. An investment with quick returns has a higher IRR.
NPV vs. IRR
Both methods give similar results, but IRR is more intuitive (in percent), while NPV is an absolute value.
Multiple Solutions Possible
With unconventional cash flow patterns (multiple sign changes), multiple IRR values may exist. The calculator returns one value.
Not Everything is Percentages
A high IRR does not automatically mean a good investment. Risk, liquidity, and duration are equally important.
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