Time Value Calculator
Understand the power of compounding and discounting for your financial decisions.
Input Your Variables
Future Value (FV)
Present Value (PV)
Choose whether to calculate the future worth of an investment or the present worth of a future sum.
Please enter a valid positive number for the initial investment.
Please enter a valid positive number for annual contributions.
Please enter a valid positive number for the interest rate.
Please enter a valid positive whole number for the number of years.
Please enter a valid positive number for the desired future value.
Please enter a valid positive number for the interest rate.
Please enter a valid positive whole number for the number of years.
Future Value Result
—
—
Compounding Periods
—
Total Contributions
—
Total Interest/Growth
Formula: FV = PV * (1 + r)^n + PMT * [((1 + r)^n – 1) / r]
Investment Growth Over Time (FV Calculation)
| Year | Beginning Balance | Contributions | Interest Earned | Ending Balance |
|---|
Understanding the Time Value of Money
What is Time Value of Money?
The concept of the time value of money (TVM) is a fundamental principle in finance that states that a sum of money is worth more now than the same sum will be at a future date due to its potential earning capacity. This is primarily because money available today can be invested and earn returns, leading to a greater amount of money in the future. Our time value calculator is designed to help you quantify this growth and understand the impact of key financial variables. Essentially, money has a time value because of inflation, opportunity cost, and risk. A time value calculator helps users visualize these effects.
This principle is crucial for making informed financial decisions, including investments, savings, and loan evaluations. Anyone involved in personal finance, business management, or investment analysis can benefit from understanding and utilizing a time value calculator. Common misconceptions include believing that all money has the same value regardless of when it's received or thinking that interest rates are the only factor. While important, factors like compounding frequency and the duration of the investment also significantly impact the overall time value of money. A thorough grasp of the time value of money is essential.
Time Value Calculator Formula and Mathematical Explanation
The core of the time value calculator lies in its ability to compute either the Future Value (FV) or the Present Value (PV) of a series of cash flows. These calculations are based on fundamental financial formulas that account for compounding interest and discounting.
Future Value (FV) Formula
The future value represents the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. This is calculated using the compounding effect of money.
The formula for the future value of a single sum with periodic additional contributions is:
FV = PV * (1 + r)^n + PMT * [((1 + r)^n - 1) / r]
Present Value (PV) Formula
The present value represents the current worth of a future sum of money or stream of cash flows, given a specified rate of return. This calculation involves discounting future amounts back to the present.
The formula for the present value of a single future sum is:
PV = FV / (1 + r)^n
Note: For simplicity in this calculator, the PV calculation assumes a single future lump sum, not a series of cash flows. If there were periodic contributions in the past that led to the FV, a more complex PV calculation would be needed.
Variable Definitions Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| PV | Present Value (Initial Investment) | Currency (e.g., USD) | ≥ 0 |
| FV | Future Value (Desired Amount) | Currency (e.g., USD) | ≥ 0 |
| PMT | Periodic Payment/Contribution (Annual) | Currency (e.g., USD) | ≥ 0 |
| r | Annual Interest Rate (or Discount Rate) | Percentage (%) | 0.1% – 50%+ |
| n | Number of Years (Compounding Periods) | Years | ≥ 1 |
The time value calculator uses these formulas to project financial outcomes.
Practical Examples (Real-World Use Cases)
The time value calculator is versatile and applicable to numerous financial scenarios. Here are two practical examples:
Example 1: Saving for a Down Payment (Future Value)
Sarah wants to buy a house in 5 years. She has an initial savings of $10,000 and plans to add $500 annually to her investment account, which she expects to yield an average of 6% per year. She uses our time value calculator to see how much she might have.
- Initial Investment (PV): $10,000
- Annual Contribution (PMT): $500
- Annual Interest Rate (r): 6%
- Number of Years (n): 5
Calculation Result:
Using the calculator, Sarah finds the Future Value (FV) to be approximately $13,468.34. The total contributions amount to $12,500 ($10,000 initial + $2,500 annual), and the total interest earned is about $968.34. This projection helps Sarah gauge if her savings plan is on track for her down payment goal.
Example 2: Valuing a Future Inheritance (Present Value)
John is expecting to receive an inheritance of $50,000 in 10 years. He needs to understand its current worth today, assuming a reasonable discount rate of 8% per year, reflecting the opportunity cost and risk associated with waiting. He inputs these figures into the time value calculator.
- Desired Future Value (FV): $50,000
- Annual Interest Rate (Discount Rate) (r): 8%
- Number of Years (n): 10
Calculation Result:
The time value calculator shows the Present Value (PV) is approximately $23,159.69. This means that $50,000 received in 10 years is equivalent to having about $23,160 today, given an 8% annual discount rate. This valuation is crucial for estate planning or comparing it against current investment opportunities.
How to Use This Time Value Calculator
Our time value calculator is designed for ease of use. Follow these simple steps:
- Select Calculation Type: Choose whether you want to calculate the Future Value (FV) of an investment or the Present Value (PV) of a future amount using the dropdown menu.
- Input Variables: Based on your selection, enter the relevant numbers into the input fields:
- For FV: Enter your Initial Investment (PV), Annual Contributions (PMT), Annual Interest Rate (%), and Number of Years (n).
- For PV: Enter your Desired Future Value (FV), Annual Interest Rate (%), and Number of Years (n).
- Review Input Requirements: Ensure you are entering positive numerical values where appropriate. Rates should be entered as percentages (e.g., 5 for 5%), and years should be whole numbers. Helper text is provided for each field.
- Click Calculate: Press the 'Calculate' button to see your results.
- Interpret Results: The main result (FV or PV) will be prominently displayed. You'll also see key intermediate values like total contributions made, total interest earned/discounted, and the number of compounding periods. The formula used is also shown for clarity.
- View Breakdown: For FV calculations, a year-by-year breakdown table and a growth chart are provided, illustrating how your investment grows over time.
- Reset or Copy: Use the 'Reset' button to clear all fields and start over, or 'Copy Results' to get a summary of your calculation.
Understanding the time value of money through these results empowers you to make better financial choices.
Key Factors That Affect Time Value Results
Several factors significantly influence the outcomes of any time value calculator, impacting both future and present values:
- Interest Rate (r): This is arguably the most significant factor. A higher interest rate dramatically increases future value due to compounding and decreases present value as future sums are discounted more heavily. It represents the return on investment or the cost of capital.
- Time Period (n): The longer the money is invested or the longer until a future payment is received, the greater the impact of compounding or discounting. The power of time is exponential due to the compounding effect.
- Initial Investment (PV) / Future Value (FV): The larger the initial principal or the target future amount, the larger the absolute gains or present value will be, assuming other factors remain constant.
- Periodic Contributions (PMT): Regular additional investments significantly boost future value. The more frequent and larger these contributions are, the faster the wealth accumulates. This is a key driver for long-term savings goals.
- Inflation: While not a direct input in this simplified calculator, inflation erodes the purchasing power of money over time. A nominal interest rate needs to outpace inflation to achieve real growth. Understanding the time value of money must consider this.
- Compounding Frequency: This calculator assumes annual compounding for simplicity. However, if interest compounds more frequently (e.g., monthly, quarterly), the future value will be slightly higher due to earning interest on interest more often.
- Risk and Uncertainty: Expected rates of return often correlate with risk. Higher risk investments might promise higher returns but carry a greater chance of loss. The discount rate used for PV calculations reflects this risk.
- Taxes and Fees: Investment gains are often subject to taxes, and investment products may have management fees. These reduce the net return and thus impact the actual time value of money realized.
Frequently Asked Questions (FAQ)
Q1: What is the difference between Future Value and Present Value?
A1: Future Value (FV) tells you what an investment made today will be worth in the future, considering growth. Present Value (PV) tells you what a future sum of money is worth today, considering the time it takes to receive it and the potential return you could earn elsewhere.
Q2: Can I use this calculator for more frequent contributions (e.g., monthly)?
A2: This calculator is simplified for annual contributions (PMT) and annual compounding. For more frequent calculations, you would need to adjust the interest rate (divide by the number of periods per year) and the number of periods (multiply by the number of periods per year), and use a more advanced formula or calculator.
Q3: What does a negative interest rate mean in the context of time value?
A3: While rare for investments, negative interest rates (or discount rates) can occur in certain economic conditions or for specific assets. They imply that money will be worth less in the future (for FV) or a future amount is worth more today (for PV) than the nominal amount suggests, reflecting a loss of purchasing power or a penalty.
Q4: How does inflation affect the time value of money?
A4: Inflation reduces the purchasing power of money over time. A 5% nominal return might seem good, but if inflation is 3%, your real return is only 2%. Therefore, when considering the true time value of money, it's often useful to look at real rates of return (nominal rate minus inflation rate).
Q5: Is the time value of money concept only for investments?
A5: No, the time value of money is fundamental to many financial decisions, including valuing bonds, leasing vs. buying assets, calculating loan payments, retirement planning, and corporate capital budgeting. Understanding the time value of money is universally applicable.
Q6: Why is the Present Value of a future amount less than the future amount itself?
A6: Because money today can be invested to earn returns. To have a certain amount in the future, you need less than that amount today, as the difference will grow through interest or other investment returns over time. This process is called discounting.
Q7: What is the opportunity cost in relation to the time value of money?
A7: Opportunity cost is the potential benefit missed or forgone when choosing one alternative over another. When calculating PV, the discount rate often represents the opportunity cost – the return you could potentially earn on an investment of similar risk if you had the money today instead of receiving it in the future.
Q8: Does the calculator account for taxes on earnings?
A8: No, this is a simplified time value calculator. It does not account for taxes on investment gains or income. Actual realized returns will be lower after taxes are applied, which is an important consideration for long-term financial planning.
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var calculationType = document.getElementById('calculationType');
var fvInputsDiv = document.getElementById('fvInputs');
var pvInputsDiv = document.getElementById('pvInputs');
var presentValueInput = document.getElementById('presentValue');
var annualContributionInput = document.getElementById('annualContribution');
var interestRateFVInput = document.getElementById('interestRateFV');
var numberOfYearsFVInput = document.getElementById('numberOfYearsFV');
var futureValuePVInput = document.getElementById('futureValuePV');
var interestRatePVInput = document.getElementById('interestRatePV');
var numberOfYearsPVInput = document.getElementById('numberOfYearsPV');
var resultTitle = document.getElementById('resultTitle');
var mainResult = document.getElementById('mainResult');
var intermediateVal1 = document.getElementById('intermediateVal1');
var intermediateLabel1 = document.getElementById('intermediateLabel1');
var intermediateVal2 = document.getElementById('intermediateVal2');
var intermediateLabel2 = document.getElementById('intermediateLabel2');
var intermediateVal3 = document.getElementById('intermediateVal3');
var intermediateLabel3 = document.getElementById('intermediateLabel3');
var formulaDesc = document.getElementById('formulaDesc');
var growthTableBody = document.getElementById('growthTableBody');
var growthChartCanvas = document.getElementById('growthChart');
var growthChartCtx = null; // Chart context
// Initialize chart context if canvas exists
if (growthChartCanvas) {
growthChartCtx = growthChartCanvas.getContext('2d');
}
function showError(inputElement, message) {
var errorElement = inputElement.parentNode.querySelector('.error-message');
if (errorElement) {
errorElement.textContent = message;
errorElement.classList.add('visible');
}
}
function clearError(inputElement) {
var errorElement = inputElement.parentNode.querySelector('.error-message');
if (errorElement) {
errorElement.textContent = ";
errorElement.classList.remove('visible');
}
}
function isValidNumber(value) {
return !isNaN(parseFloat(value)) && isFinite(value);
}
function updateInputs() {
if (calculationType.value === 'fv') {
fvInputsDiv.style.display = 'block';
pvInputsDiv.style.display = 'none';
resultTitle.textContent = 'Future Value Result';
intermediateLabel1.textContent = 'Compounding Periods';
intermediateLabel2.textContent = 'Total Contributions';
intermediateLabel3.textContent = 'Total Interest/Growth';
formulaDesc.innerHTML = 'Formula: FV = PV * (1 + r)^n + PMT * [((1 + r)^n – 1) / r]';
} else {
fvInputsDiv.style.display = 'none';
pvInputsDiv.style.display = 'block';
resultTitle.textContent = 'Present Value Result';
intermediateLabel1.textContent = 'Discount Periods';
intermediateLabel2.textContent = 'Discounted Amount'; // Placeholder for PV, might not be directly used
intermediateLabel3.textContent = 'Total Discounting Effect'; // Placeholder for PV, might not be directly used
formulaDesc.innerHTML = 'Formula: PV = FV / (1 + r)^n';
}
// Clear results and table when inputs change
mainResult.textContent = '–';
intermediateVal1.textContent = '–';
intermediateVal2.textContent = '–';
intermediateVal3.textContent = '–';
growthTableBody.innerHTML = ";
if (growthChartCtx) {
growthChartCtx.clearRect(0, 0, growthChartCanvas.width, growthChartCanvas.height);
}
}
function calculateTimeValue() {
var type = calculationType.value;
var inputsValid = true;
var pv, pmt, rate, years, fv;
var formattedRate, formattedYears;
// Clear previous errors
var allInputs = fvInputsDiv.querySelectorAll('input, select');
allInputs = Array.prototype.slice.call(allInputs).concat(Array.prototype.slice.call(pvInputsDiv.querySelectorAll('input, select')));
for (var i = 0; i < allInputs.length; i++) {
clearError(allInputs[i]);
}
if (type === 'fv') {
pv = parseFloat(presentValueInput.value);
pmt = parseFloat(annualContributionInput.value);
rate = parseFloat(interestRateFVInput.value) / 100;
years = parseInt(numberOfYearsFVInput.value);
if (!isValidNumber(pv) || pv < 0) { showError(presentValueInput, 'Please enter a valid positive number for the initial investment.'); inputsValid = false; }
if (!isValidNumber(pmt) || pmt < 0) { showError(annualContributionInput, 'Please enter a valid positive number for annual contributions.'); inputsValid = false; }
if (!isValidNumber(rate * 100) || rate <= 0) { showError(interestRateFVInput, 'Please enter a valid positive number greater than 0 for the interest rate.'); inputsValid = false; } // Rate must be positive for FV formula
if (!isValidNumber(years) || years <= 0 || !Number.isInteger(years)) { showError(numberOfYearsFVInput, 'Please enter a valid positive whole number for the number of years.'); inputsValid = false; }
if (inputsValid) {
formattedRate = rate;
formattedYears = years;
var fvResult = pv * Math.pow(1 + formattedRate, formattedYears) + pmt * (Math.pow(1 + formattedRate, formattedYears) – 1) / formattedRate;
var totalContributions = pv + (pmt * formattedYears);
var totalInterest = fvResult – totalContributions;
var compoundingPeriods = formattedYears;
mainResult.textContent = '$' + fvResult.toFixed(2);
intermediateVal1.textContent = compoundingPeriods;
intermediateVal2.textContent = '$' + totalContributions.toFixed(2);
intermediateVal3.textContent = '$' + totalInterest.toFixed(2);
populateTableAndChart(pv, pmt, formattedRate, formattedYears);
}
} else { // PV calculation
fv = parseFloat(futureValuePVInput.value);
rate = parseFloat(interestRatePVInput.value) / 100;
years = parseInt(numberOfYearsPVInput.value);
if (!isValidNumber(fv) || fv < 0) { showError(futureValuePVInput, 'Please enter a valid positive number for the desired future value.'); inputsValid = false; }
if (!isValidNumber(rate * 100) || rate <= 0) { showError(interestRatePVInput, 'Please enter a valid positive number greater than 0 for the interest rate.'); inputsValid = false; } // Rate must be positive for PV formula
if (!isValidNumber(years) || years <= 0 || !Number.isInteger(years)) { showError(numberOfYearsPVInput, 'Please enter a valid positive whole number for the number of years.'); inputsValid = false; }
if (inputsValid) {
formattedRate = rate;
formattedYears = years;
var pvResult = fv / Math.pow(1 + formattedRate, formattedYears);
mainResult.textContent = '$' + pvResult.toFixed(2);
intermediateVal1.textContent = formattedYears; // Number of periods
intermediateVal2.textContent = '$' + fv.toFixed(2); // Just display FV for reference
intermediateVal3.textContent = '$' + (fv – pvResult).toFixed(2); // Total discount
// Clear table and chart for PV
growthTableBody.innerHTML = '';
if (growthChartCtx) {
growthChartCtx.clearRect(0, 0, growthChartCanvas.width, growthChartCanvas.height);
}
}
}
if (!inputsValid) {
mainResult.textContent = 'Error';
intermediateVal1.textContent = '–';
intermediateVal2.textContent = '–';
intermediateVal3.textContent = '–';
}
}
function populateTableAndChart(pv, pmt, rate, years) {
growthTableBody.innerHTML = ''; // Clear previous table data
if (!growthChartCtx) return; // Exit if chart context is not available
var labels = [];
var beginningBalances = [];
var contributions = [];
var interestEarned = [];
var endingBalances = [];
var currentBalance = pv;
for (var i = 1; i <= years; i++) {
var yearLabel = 'Year ' + i;
var interestForYear = (currentBalance + pmt) * rate;
var endingBalance = currentBalance + pmt + interestForYear;
labels.push(yearLabel);
beginningBalances.push(currentBalance);
contributions.push(pmt);
interestEarned.push(interestForYear);
endingBalances.push(endingBalance);
currentBalance = endingBalance;
}
// Populate table
for (var i = 0; i < labels.length; i++) {
var row = growthTableBody.insertRow();
row.insertCell(0).textContent = labels[i];
row.insertCell(1).textContent = '$' + beginningBalances[i].toFixed(2);
row.insertCell(2).textContent = '$' + contributions[i].toFixed(2);
row.insertCell(3).textContent = '$' + interestEarned[i].toFixed(2);
row.insertCell(4).textContent = '$' + endingBalances[i].toFixed(2);
}
// Update Chart
if (growthChartCtx) {
growthChartCanvas.width = growthChartCanvas.parentElement.clientWidth * 0.9; // Responsive width
growthChartCanvas.height = 300; // Fixed height
new Chart(growthChartCtx, {
type: 'bar', // Or 'line'
data: {
labels: labels,
datasets: [{
label: 'Ending Balance',
data: endingBalances,
backgroundColor: 'rgba(0, 74, 153, 0.6)',
borderColor: 'rgba(0, 74, 153, 1)',
borderWidth: 1,
fill: false // For line chart
}, {
label: 'Interest Earned',
data: interestEarned,
backgroundColor: 'rgba(40, 167, 69, 0.6)',
borderColor: 'rgba(40, 167, 69, 1)',
borderWidth: 1,
fill: false // For line chart
}]
},
options: {
responsive: true,
maintainAspectRatio: false,
scales: {
y: {
beginAtZero: true,
ticks: {
callback: function(value) {
return '$' + value.toFixed(0);
}
}
}
},
plugins: {
title: {
display: true,
text: 'Yearly Investment Growth Breakdown'
},
tooltip: {
callbacks: {
label: function(context) {
var label = context.dataset.label || '';
if (label) {
label += ': ';
}
if (context.parsed.y !== null) {
label += new Intl.NumberFormat('en-US', { style: 'currency', currency: 'USD' }).format(context.parsed.y);
}
return label;
}
}
}
}
}
});
}
}
function resetCalculator() {
document.getElementById('calculationType').value = 'fv';
document.getElementById('presentValue').value = '';
document.getElementById('annualContribution').value = '';
document.getElementById('interestRateFV').value = '';
document.getElementById('numberOfYearsFV').value = '';
document.getElementById('futureValuePV').value = '';
document.getElementById('interestRatePV').value = '';
document.getElementById('numberOfYearsPV').value = '';
mainResult.textContent = '–';
intermediateVal1.textContent = '–';
intermediateVal2.textContent = '–';
intermediateVal3.textContent = '–';
formulaDesc.innerHTML = 'Formula: FV = PV * (1 + r)^n + PMT * [((1 + r)^n – 1) / r]'; // Reset to default
var allInputs = fvInputsDiv.querySelectorAll('input, select');
allInputs = Array.prototype.slice.call(allInputs).concat(Array.prototype.slice.call(pvInputsDiv.querySelectorAll('input, select')));
for (var i = 0; i < allInputs.length; i++) {
clearError(allInputs[i]);
}
updateInputs(); // Ensure correct input fields are displayed
if (growthChartCtx) {
growthChartCtx.clearRect(0, 0, growthChartCanvas.width, growthChartCanvas.height);
}
growthTableBody.innerHTML = '';
}
function copyResults() {
var type = calculationType.value;
var summary = '';
if (type === 'fv') {
summary = 'Future Value Calculation:\n' +
'Initial Investment (PV): ' + (presentValueInput.value ? '$' + parseFloat(presentValueInput.value).toFixed(2) : 'N/A') + '\n' +
'Annual Contribution (PMT): ' + (annualContributionInput.value ? '$' + parseFloat(annualContributionInput.value).toFixed(2) : 'N/A') + '\n' +
'Annual Interest Rate: ' + (interestRateFVInput.value ? interestRateFVInput.value + '%' : 'N/A') + '\n' +
'Number of Years: ' + (numberOfYearsFVInput.value ? numberOfYearsFVInput.value : 'N/A') + '\n' +
'——————————\n' +
'Future Value (FV): ' + mainResult.textContent + '\n' +
'Total Contributions: ' + intermediateVal2.textContent + '\n' +
'Total Interest Earned: ' + intermediateVal3.textContent;
} else {
summary = 'Present Value Calculation:\n' +
'Desired Future Value (FV): ' + (futureValuePVInput.value ? '$' + parseFloat(futureValuePVInput.value).toFixed(2) : 'N/A') + '\n' +
'Annual Interest Rate: ' + (interestRatePVInput.value ? interestRateFVInput.value + '%' : 'N/A') + '\n' +
'Number of Years: ' + (numberOfYearsPVInput.value ? numberOfYearsPVInput.value : 'N/A') + '\n' +
'——————————\n' +
'Present Value (PV): ' + mainResult.textContent + '\n' +
'Total Discounting Effect: ' + intermediateVal3.textContent;
}
// Use a temporary textarea to copy to clipboard
var textArea = document.createElement("textarea");
textArea.value = summary;
// Make the textarea out of screen
textArea.style.position = "fixed";
textArea.style.left = "-9999px";
textArea.style.top = "-9999px";
document.body.appendChild(textArea);
textArea.focus();
textArea.select();
try {
var successful = document.execCommand('copy');
var msg = successful ? 'Copied!' : 'Copy failed!';
showToast(msg);
} catch (err) {
showToast('Copy failed!');
}
document.body.removeChild(textArea);
}
function showToast(message) {
var toast = document.getElementById("toast");
toast.textContent = message;
toast.className = "toast show";
setTimeout(function(){ toast.className = toast.className.replace("show", ""); }, 3000);
}
// Initialize the calculator view on page load
updateInputs();
// Add event listeners for real-time updates if needed, or rely on calculate button.
// For real-time, uncomment the following lines and remove the 'Calculate' button.
var calculatorInputs = document.querySelectorAll('.calculator-section input, .calculator-section select');
for (var i = 0; i maxValue) maxValue = value;
});
});
if (maxValue === 0) maxValue = 1; // Avoid division by zero
ctx.clearRect(0, 0, canvas.width, canvas.height);
// Draw Y-axis scale if defined
if (options.scales && options.scales.y && options.scales.y.beginAtZero) {
// Basic scale drawing (simplified)
ctx.strokeStyle = '#ccc';
ctx.lineWidth = 1;
ctx.beginPath();
ctx.moveTo(40, canvas.height – 40); // Y-axis line
ctx.lineTo(canvas.width – 20, canvas.height – 40);
ctx.stroke();
}
// Draw bars
data.labels.forEach(function(label, index) {
var xPos = 40 + (index * (barWidth + gapWidth));
data.datasets.forEach(function(dataset, dsIndex) {
var value = dataset.data[index];
var barHeight = (value / maxValue) * scaleY;
ctx.fillStyle = dataset.backgroundColor;
ctx.fillRect(xPos, canvas.height – 40 – barHeight, barWidth, barHeight);
});
});
// Draw labels (simplified)
ctx.fillStyle = '#333';
ctx.textAlign = 'center';
data.labels.forEach(function(label, index) {
var xPos = 40 + (index * (barWidth + gapWidth)) + barWidth / 2;
ctx.fillText(label, xPos, canvas.height – 25);
});
};
window.Chart.prototype.drawLineChart = function() {
// Placeholder for line chart drawing logic
console.log("Line chart drawing not implemented in polyfill.");
};
// Add dummy Chart.js options structure if needed for the polyfill
if (!window.Chart.defaults) window.Chart.defaults = {};
if (!window.Chart.defaults.global) window.Chart.defaults.global = {};
window.Chart.defaults.global.responsive = true; // Simulate responsiveness
}
// — End Chart.js Polyfill —