Compound Interest Calculator

The compounding frequency dictates how often accrued interest is folded back into the principal. The discrete-compound formula is A = P(1 + r/n)^(nt), where P is the principal, r is the nominal annual rate, n is the number of compounding periods per year, and t is the time in years. As n grows toward infinity, the formula converges to its continuous form: A = Pe^(rt). The constant e ≈ 2.71828 is the natural exponential base; the limit comes from (1 + r/n)^n → e^r. Run the numbers on a 10-year, $10,000 deposit at a 7% nominal rate. With annual compounding (n=1) you finish at $19,671.51. With monthly compounding (n=12) you finish at $20,096.61. With daily compounding (n=365) you finish at $20,134.83. With continuous compounding you finish at $20,137.53. The gap between annual and continuous is $466 — meaningful. The gap between daily and continuous is $2.70 — essentially noise. This is why "compounded daily" is a marketing flag that delivers almost nothing extra over "compounded monthly," yet banks advertise it heavily because consumers perceive higher frequency as a better product. The Annual Percentage Yield (APY) folds frequency into a single comparable number: APY = (1 + r/n)^n - 1. A 5.00% nominal rate compounded daily yields APY ≈ 5.127%. The same 5.00% compounded annually yields APY = 5.000%. When comparing two savings accounts, do not compare nominal rates — compare APYs. The U.S. Truth in Savings Act (Regulation DD, 12 CFR Part 1030) requires depository institutions to disclose APY, precisely because nominal-rate comparisons mislead consumers. For loans, the analog is APR (annual percentage rate), and lenders are governed by Regulation Z. APR includes certain fees but generally does not compound — a 6% APR mortgage paid monthly has an effective annual cost slightly above 6%. APY for borrowers, sometimes called APR Yield or EAR (effective annual rate), reveals the real number. The arithmetic gap between APR and EAR widens with frequency: a 12% APR on a credit card compounded daily produces an EAR of about 12.747%. Educational only — this is not financial advice. Consult a licensed financial advisor for personalized guidance.

// $10,000 @ 7% for 10 years across compounding frequencies
const P = 10000, r = 0.07, t = 10;
const A_annual    = P * Math.pow(1 + r/1,   1*t);   // 19671.51
const A_monthly   = P * Math.pow(1 + r/12,  12*t);  // 20096.61
const A_daily     = P * Math.pow(1 + r/365, 365*t); // 20134.83
const A_continuous = P * Math.exp(r * t);           // 20137.53

// APY comparison for a 5% nominal rate
const APY_daily   = Math.pow(1 + 0.05/365, 365) - 1; // 0.05127 (5.127%)
const APY_annual  = Math.pow(1 + 0.05/1,   1)   - 1; // 0.05000 (5.000%)

The Rule of 72 says that money doubles in approximately 72 / r years, where r is the rate expressed as a whole number (so 8% gives 9 years). It is one of the most cited heuristics in personal finance, but few users know where the 72 comes from or where it fails. Start with the exact doubling-time formula. Setting 2P = P(1+r)^t and solving for t gives t = ln(2) / ln(1+r). For small r, ln(1+r) ≈ r — Taylor expansion truncated at first order. So t ≈ ln(2) / r ≈ 0.6931 / r. Multiplying numerator and denominator by 100 gives t ≈ 69.31 / R where R is the percentage. The "true" rule is the Rule of 69.3, but 69.3 has only one factor (3 × 23.1) and is awkward to divide mentally. 72 has six divisors that matter (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), so 72/4, 72/6, 72/8, 72/9 are all clean integer answers. The trade-off is small: 72 systematically over-predicts doubling time at low rates and under-predicts at high rates, but the error is acceptable in the 6%–10% range where it is most often used. Quantitative accuracy: At r = 8%, exact = 9.006 years vs Rule of 72 = 9.000 years (-0.07% error). At r = 10%, exact = 7.273 vs Rule = 7.200 (-1.0% error). At r = 1%, exact = 69.66 vs Rule = 72.0 (+3.4% error). At r = 25%, exact = 3.106 vs Rule = 2.880 (-7.3% error). The rule is quite accurate from 5%–11% and degrades at the extremes. For continuous compounding, the Rule of 69.3 is exact (since ln(2) = 0.6931). For typical savings products that compound monthly, the Rule of 72 is closer to reality. Variant rules exist. The Rule of 70 (closer to ln(2) × 100) is preferred by epidemiologists and demographers analyzing population doubling. The Rule of 114 estimates tripling time (ln(3) × 100 ≈ 109.86, rounded to a divisor-friendly 114). The Rule of 144 estimates quadrupling. All share the same Taylor-expansion logic. Important caveat: doubling time tells you nothing about real purchasing power. If your nominal return is 8% and inflation is 3%, the rule says nominal doubling at year 9 — but real doubling takes 14.4 years (using net 5%). Always pair Rule of 72 with the Fisher real-rate adjustment. Educational content only — not personalized financial advice.

// Rule of 72 vs exact doubling time
function exactDoubling(r) { return Math.log(2) / Math.log(1 + r); }
function rule72(r)        { return 72 / (r * 100); }

console.log(exactDoubling(0.08), rule72(0.08)); // 9.006 vs 9.000
console.log(exactDoubling(0.10), rule72(0.10)); // 7.273 vs 7.200
console.log(exactDoubling(0.01), rule72(0.01)); // 69.66 vs 72.00
console.log(exactDoubling(0.25), rule72(0.25)); // 3.106 vs 2.880

A 7% nominal return in a 4% inflation environment is not a 3% real return — it is closer to 2.88%. The naive subtraction (nominal − inflation) is a first-order approximation; the exact relationship is the Fisher equation: (1 + r_nominal) = (1 + r_real)(1 + inflation). Solving for the real rate: r_real = (1 + r_nominal) / (1 + inflation) − 1. At 7% nominal and 4% inflation, the exact real rate is 1.07/1.04 − 1 = 0.02885, or 2.885%. The 0.115 percentage-point gap from the naive estimate is small in any single year but compounds dramatically over decades. Why this matters in practice: the Federal Reserve targets 2% inflation as measured by Personal Consumption Expenditures (PCE). The Bureau of Labor Statistics publishes the Consumer Price Index (CPI), which has historically run slightly above PCE due to methodology (CPI uses fixed weights, PCE adjusts for substitution). If you plan retirement on nominal returns alone, you systematically overstate your future purchasing power. A $1 million nominal portfolio in 2026 has roughly the purchasing power of a $552,000 portfolio in 1996, given the cumulative CPI rise of about 81%. Long-term equity data tells the real story. The S&P 500's total return (with dividends reinvested) from 1928–2024 averaged ~10% nominal annually, per data compiled by NYU professor Aswath Damodaran from CRSP and S&P Global. Over the same span, U.S. CPI averaged ~3% annually. Compounding these correctly gives a real annualized return near 7% — a number widely cited in retirement planning. But year-to-year volatility is huge (standard deviation ~19%), so the 7% real average is only meaningful over multi-decade horizons. Robert Shiller's online dataset (econ.yale.edu/~shiller/data.htm) provides month-by-month CPI-adjusted S&P returns from 1871, the gold standard for academic work. Treasury Inflation-Protected Securities (TIPS) provide a market-implied real rate. Subtracting the TIPS yield from the comparable nominal Treasury yield gives the "breakeven inflation rate" — what markets expect inflation to average over that horizon. As of 2026, 10-year breakeven hovers around 2.3%, suggesting markets price moderate inflation. The St. Louis Fed (FRED) publishes daily breakeven series at fred.stlouisfed.org/series/T10YIE. Rule of thumb when planning: use 4–5% real for diversified equity exposure, 0–1% real for high-grade bonds, and assume cash loses ~1–2% real per year. These are educational ranges, not guarantees of future results. Past performance does not predict future returns. Consult a licensed financial advisor before making investment decisions.

// Fisher equation: nominal-to-real conversion
function realRate(nominal, inflation) {
  return (1 + nominal) / (1 + inflation) - 1;
}

realRate(0.07, 0.04); // 0.02885 (2.885%, NOT 3%)
realRate(0.10, 0.03); // 0.06796 (6.796%, NOT 7%)

// Cumulative purchasing-power loss over 30 years at 3% inflation
const lossFactor = 1 / Math.pow(1.03, 30); // 0.4120
// $1,000,000 nominal in 30 years = $412,000 in today's dollars