Enter the initial cost as a positive number (we will treat it as an outflow).
Enter cash flows for each year separated by commas.
0.00%
Understanding the Formula to Calculate Internal Rate of Return (IRR)
The Internal Rate of Return (IRR) is a critical metric used in financial analysis, capital budgeting, and investment planning. Unlike simple ROI, IRR accounts for the time value of money, providing a more accurate picture of an investment's potential profitability over time.
The Logic Behind the Calculation
The IRR is technically defined as the discount rate that makes the Net Present Value (NPV) of all cash flows from a particular project equal to zero. In simpler terms, it is the compounded annual growth rate that an investment is expected to generate.
The mathematical formula used to solve for IRR is:
C₁, C₂, etc. = Net cash inflows during each period
n = Each specific time period
IRR = The rate of return we are solving for
Why is there no simple algebraic formula?
Unlike calculating simple interest, there is no closed-form algebraic equation to isolate the IRR variable on one side. Because the IRR variable is part of the denominator raised to increasing powers (t=1, t=2, etc.), calculating IRR requires numerical methods or trial-and-error iteration.
This calculator utilizes the Newton-Raphson method, an algorithm that iteratively estimates the root of the function until the margin of error is negligible.
Example Interpretation
Imagine you invest $10,000 today (Initial Investment) into a project. Over the next three years, the project returns:
Year 1: $3,000
Year 2: $4,000
Year 3: $5,000
If you sum these up ($12,000), you have a raw profit of $2,000. However, because money received in the future is worth less than money today, the IRR will calculate the specific percentage rate that equates these future flows back to your initial $10,000 cost. In this scenario, the IRR is approximately 8.9%.
When to Use IRR
Investors typically use IRR to compare the profitability of establishing a new operation or expanding an existing one. If the IRR of a new project exceeds a company's required rate of return (often called the hurdle rate), the project is generally considered desirable.
function calculateIRR() {
// 1. Get Inputs
var investmentInput = document.getElementById('initialInvestment').value;
var flowsInput = document.getElementById('cashFlows').value;
var errorDiv = document.getElementById('irrError');
var resultDiv = document.getElementById('irrResult');
var valueDisplay = document.getElementById('irrValueDisplay');
var explanationDisplay = document.getElementById('irrExplanation');
// Reset UI
errorDiv.style.display = 'none';
resultDiv.style.display = 'none';
errorDiv.innerHTML = ";
// 2. Validate Inputs
if (!investmentInput || investmentInput <= 0) {
errorDiv.innerHTML = 'Please enter a valid positive number for Initial Investment.';
errorDiv.style.display = 'block';
return;
}
if (!flowsInput) {
errorDiv.innerHTML = 'Please enter at least one cash flow.';
errorDiv.style.display = 'block';
return;
}
// Parse Cash Flows
var cashFlows = [];
var rawFlows = flowsInput.split(',');
for (var i = 0; i < rawFlows.length; i++) {
var val = parseFloat(rawFlows[i].trim());
if (isNaN(val)) {
errorDiv.innerHTML = 'Invalid format in cash flows. Use numbers separated by commas.';
errorDiv.style.display = 'block';
return;
}
cashFlows.push(val);
}
// Construct full flow array: [-Investment, Flow1, Flow2, …]
var allFlows = [-parseFloat(investmentInput)].concat(cashFlows);
// 3. Calculation Logic (Newton-Raphson Method)
var irr = 0.1; // Initial guess: 10%
var maxIterations = 1000;
var tolerance = 0.000001;
var found = false;
for (var i = 0; i < maxIterations; i++) {
var npv = 0;
var derivative = 0;
for (var t = 0; t 0) {
derivative -= (t * flow) / Math.pow(1 + irr, t + 1);
}
}
// Newton-Raphson step: New = Old – (Function / Derivative)
if (Math.abs(derivative) < tolerance) {
// Derivative too close to zero, break to avoid division by zero
break;
}
var newIrr = irr – (npv / derivative);
// Check for convergence
if (Math.abs(newIrr – irr) < tolerance) {
irr = newIrr;
found = true;
break;
}
irr = newIrr;
}
// 4. Display Results
if (found) {
var percentage = (irr * 100).toFixed(2);
// Check for realistic bounds (IRR usually between -100% and a high number)
if (irr 1000) {
errorDiv.innerHTML = 'The calculation could not converge to a realistic result. Ensure cash flows eventually turn positive.';
errorDiv.style.display = 'block';
return;
}
valueDisplay.innerHTML = percentage + '%';
var totalInflow = 0;
for(var j=0; j<cashFlows.length; j++) { totalInflow += cashFlows[j]; }
var profit = totalInflow – parseFloat(investmentInput);
explanationDisplay.innerHTML = 'With an initial investment of $' + parseFloat(investmentInput).toLocaleString() + ' and total future cash flows of $' + totalInflow.toLocaleString() + ' over ' + cashFlows.length + ' periods, the internal rate of return is ' + percentage + '%.Total nominal profit: $' + profit.toLocaleString();
resultDiv.style.display = 'block';
} else {
errorDiv.innerHTML = 'Calculation failed to converge. This often happens if cash flows do not offset the investment (no profit) or fluctuate wildly from negative to positive multiple times.';
errorDiv.style.display = 'block';
}
}