Compound Growth Rate Calculator

**Compound Growth Rate (CAGR):** r = (End Value / Start Value)^(1/n) − 1 Where: - r = compound growth rate per period - End Value = final value after n periods - Start Value = initial value - n = number of periods **Express as percentage:** CAGR% = r × 100 **Worked example: investment doubling in 5 years** Start: $1,000. End: $2,000. Years: 5. r = (2000/1000)^(1/5) − 1 = 2^0.2 − 1 = 1.1487 − 1 = 0.1487 CAGR = 14.87% per year. **Verification:** 1000 × (1.1487)^5 = 1000 × 2.000 ≈ 2,000 ✓ **Rule of 72 (quick estimate):** Doubling time ≈ 72 / annual growth rate (%) - 10% growth: doubles in ~7.2 years. - 7%: ~10 years. - 5%: ~14.4 years. - 2%: ~36 years. - 0.5%: ~144 years. Quick mental math, accurate within 1% for typical rates. **Common CAGRs:** | Application | Typical CAGR | |---|---| | Bank savings (USA) | 0.1-2% | | 10-year Treasury | 2-5% | | S&P 500 (historical) | 7-10% | | Stock pickers (best) | 15-25% | | US GDP | 2-3% | | Inflation (long-term) | 2-3% | | Tech company revenue (mature) | 5-15% | | Tech company revenue (growth) | 25-100%+ | | Internet users (1995-2005) | 35% | | Moore's Law (transistors/chip) | ~41% (doubles every 2 yr) | | World population | ~1% | **Forward projection:** Future value = Start × (1 + r)^n For 10% CAGR over 30 years: $10,000 → $10,000 × 1.1^30 ≈ $174,494 **Reverse: time to reach target:** n = ln(End/Start) / ln(1 + r) For tripling at 8%: n = ln(3) / ln(1.08) = 1.099/0.0770 ≈ 14.3 years **Continuous compounding (limit):** A = Start × e^(rt) Where e ≈ 2.71828. For continuous 7% rate over 10 years: End = 100 × e^(0.7) ≈ 100 × 2.014 = $201.40 Compared to discrete annual: 100 × 1.07^10 ≈ $196.72. Continuous is upper bound for compounding effects. **Comparison: simple vs compound:** For 10% over 30 years starting at $10,000: - Simple interest (no compounding): 10,000 + 30 × 1,000 = $40,000. - Compound interest: 10,000 × 1.1^30 ≈ $174,494. Compounding adds $134,000 over simple. The longer the time, the more dramatic the difference. **Inflation-adjusted (real CAGR):** Real CAGR = (1 + nominal CAGR) / (1 + inflation) − 1 Often approximated: Real CAGR ≈ Nominal − Inflation. S&P 500 historical: ~10% nominal, ~7% real after 3% inflation. **Geometric vs arithmetic mean:** Compound (geometric) mean accounts for compounding: GM = (Π values)^(1/n) For 10%, 20%, -5% three-year returns: Arithmetic mean: 8.33%/yr. Geometric mean: (1.1 × 1.2 × 0.95)^(1/3) − 1 = 8.04%/yr. Geometric mean is always ≤ arithmetic mean. The difference grows with volatility. **Worked example: variable returns** Year 1: +30% → from $100 to $130. Year 2: -20% → from $130 to $104. Arithmetic mean: (30 - 20)/2 = 5%. Geometric mean (CAGR): (104/100)^(1/2) - 1 = 1.98%. Variation hurts compounding. Two equal-mean returns differ in compounded value if volatility differs. **Common CAGR pitfalls:** - **End point selection**: cherry-picking dates can mislead. - **Doesn't show volatility**: a smooth and a volatile path can have same CAGR. - **No path information**: 10% CAGR could mean 10% every year or 60% then -20%. - **Assumes reinvestment**: actual returns may not allow this. - **Survivorship bias** in historical data. **Other growth metrics:** - **Total return**: end/start - 1 (cumulative). - **Average annual return**: simple mean of yearly returns (different from CAGR). - **Internal Rate of Return (IRR)**: accounts for cash flows. - **Money-weighted return**: accounts for timing of investments. - **Time-weighted return**: removes effect of cash flow timing. **Growth contexts:** | Type | Typical | Example | |---|---|---| | Linear | Constant rate | $100/year salary raise | | Exponential | % rate | Population, investments | | Logarithmic | Diminishing | Customer acquisition | | Logistic | S-curve | Technology adoption | | Power law | Exponent | City sizes, wealth distribution | **Software:** - **Excel**: =(End/Start)^(1/Years) − 1 - **Python**: (end/start)**(1/n) - 1 - **R**: (end/start)^(1/n) - 1 - **Financial calculators**: dedicated CAGR functions.