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The Math Behind Compound Interest — And Why It Matters More Than You Think

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The Math Behind Compound Interest — And Why It Matters More Than You Think

Compound interest is one of those concepts that's easy to describe but hard to intuitively grasp. Let's build a calculator and explore what the numbers actually show.

The Formula 

A = P(1 + r/n)^(nt)
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Where:

  • A = final amount
  • P = principal
  • r = annual interest rate (decimal)
  • n = compounding periods per year
  • t = years 
function compound(principal, annualRate, years, periodsPerYear = 12) {
  const r = annualRate / 100;
  const n = periodsPerYear;
  return principal * Math.pow(1 + r / n, n * years);
}
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Adding Monthly Contributions

Most real-world scenarios involve regular contributions. The formula becomes:

A = P(1 + r/n)^(nt) + PMT × ((1 + r/n)^(nt) - 1) / (r/n)
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function compoundWithContributions(principal, annualRate, years, monthlyContribution, periodsPerYear = 12) {
  const r = annualRate / 100;
  const n = periodsPerYear;
  const nt = n * years;
  const rn = r / n;

  const principalGrowth = principal * Math.pow(1 + rn, nt);
  const contributionGrowth = monthlyContribution * (Math.pow(1 + rn, nt) - 1) / rn;

  return principalGrowth + contributionGrowth;
}
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Generating a Year-by-Year Table 

function generateYearlyBreakdown(principal, annualRate, years, monthlyContribution) {
  const rows = [];
  let balance = principal;
  const monthlyRate = annualRate / 100 / 12;
  let totalContributions = principal;

  for (let year = 1; year <= years; year++) {
    for (let month = 0; month < 12; month++) {
      balance = balance * (1 + monthlyRate) + monthlyContribution;
      totalContributions += monthlyContribution;
    }
    rows.push({
      year,
      balance: Math.round(balance),
      totalContributions: Math.round(totalContributions),
      totalInterest: Math.round(balance - totalContributions),
    });
  }

  return rows;
}
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The Rule of 72

A quick mental math trick: divide 72 by the annual interest rate to estimate how many years until your money doubles.

function yearsToDouble(annualRate) {
  return 72 / annualRate;
}
// 8% rate → ~9 years to double
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Compounding Frequency Matters Less Than You Think

Daily vs. monthly compounding has a surprisingly small effect:

Frequency $10,000 at 10% for 10 years
Annual $25,937
Monthly $27,070
Daily $27,179

The difference between monthly and daily is only $109 over 10 years.

Explore compound interest at toolzip.app/tools/compound-calculator.

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