What is Compound Interest Derivation: A=P(1+r/n)^nt from First Principles, Continuous Compounding via Euler's Number, APR→APY Conversion & Rule of 72?
Compound interest is the process by which interest earned in one period is added to the principal before the next period's interest is calculated — producing exponential rather than linear growth. The formula A = P(1 + r/n)^(nt) can be derived from first principles: if $P is deposited at annual rate r compounded n times per year, after each sub-period the balance is multiplied by (1 + r/n). After n compounding periods (1 year), the balance = P × (1 + r/n)^n. After t years, A = P × (1 + r/n)^(nt). As n → ∞ (continuous compounding), the limit of (1 + r/n)^n = e^r, giving A = Pe^(rt) — the maximum achievable growth at rate r. The practical difference between monthly and continuous compounding is marginal (0.02–0.07% per year), making the choice of n far less important than the rate r and time t. Compound interest is also the engine of debt destruction: a 24% APR credit card balance doubles every 3 years at minimum payments.
Mathematical Foundation
Discrete Compounding: A = P(1 + r/n)^(nt)
= Principal (initial deposit or loan balance). The entire exponential growth multiplies on this base. A 10% difference in principal has identical impact on final balance as a 10% difference in the growth multiplier — but principal is in your control at day 0, while the growth multiplier accumulates passively over decades.
= Annual interest rate as a decimal (5% = 0.05). This is the nominal rate, NOT the effective rate. The effective annual rate (EAR/APY) = (1 + r/n)^n − 1, which is always ≥ r for n > 1. At r=10%, monthly compounding (n=12): APY = (1 + 0.10/12)^12 − 1 = 10.47%. The APY is what you actually earn.
= Compounding frequency per year. n=1: annual, n=4: quarterly, n=12: monthly, n=52: weekly, n=365: daily. The marginal benefit of increasing n diminishes rapidly. Going from n=1 (annual) to n=12 (monthly) adds significant real return. Going from n=12 to n=365 adds only a trivial fraction. The limit as n→∞ is continuous compounding: A = Pe^(rt).
= Time in years. The MOST powerful variable in the formula due to its position as an exponent. Doubling t more than doubles the interest earned because the exponent affects the entire accumulated balance. At 7% monthly: 10 years = 2.01×, 20 years = 4.04×, 30 years = 8.12×. Each additional decade MORE than doubles the previous decade's growth.
Continuous Compounding: A = Pe^(rt)
= Euler's number e (≈2.71828) raised to (r × t). This is the mathematical limit of (1 + r/n)^(nt) as n→∞. At r=10%, t=20: continuous = e^(0.10×20) = e^2 = 7.389×P. Monthly compounding at same rate: (1+0.10/12)^240 = 7.328×P. Difference: 0.84% — trivial vs. the 639% total gain. Continuous compounding is used in options pricing (Black-Scholes) and bond mathematics because it makes calculus tractable.
Rule of 72 — Derived from ln(2)
= Natural log of 2 = 0.6931. For continuous compounding: t = ln(2)/r exactly. For annual compounding at common rates (4–15%), the approximation 72/r% gives doubling time within ±1 year. At r=8%: exact = ln(2)/ln(1.08) = 9.006 years, approximation = 72/8 = 9 years. At r=12%: exact = 6.116 years, approximation = 6 years. The rule is most accurate between 6–12%.
Laws & Principles
- APR vs. APY: The Regulatory Distinction. The Truth in Lending Act (TILA, Reg Z) requires lenders to disclose APR; the Truth in Savings Act (TISA, Reg DD) requires banks to disclose APY on deposit accounts. APR = nominal rate r (does NOT include compounding effect). APY = (1 + r/n)^n − 1 (DOES include compounding). A credit card advertised at 24% APR compounded daily has an APY of (1 + 0.24/365)^365 − 1 = 27.11%. Always compare APY — not APR — when evaluating accounts. For debt, always ask for the effective APR including compounding frequency.
- The 1% Fee Rule: A 1% annual management fee applied to a compounding investment account appears small but creates massive opportunity cost over time. On $100,000 at 8% annual return over 30 years: without fee = $1,006,266; with 1% fee (earning 7%) = $761,226. The 1% fee costs $245,040 — 24.4% of your total terminal wealth. This is because the fee is applied to the ENTIRE growing balance, not just the original principal. Every basis point of fee is a basis point of compound growth permanently removed.
- Debt Compounding Asymmetry: Compound interest works identically on debt as on assets, but with asymmetric psychological impact. A 24% credit card balance compounds monthly: (1 + 0.24/12)^12 = APY 26.82%. $5,000 at 24% APR making minimum payments of 2% of balance ($100 initial) takes approximately 22 years to pay off and costs $9,000+ in interest — nearly 3× the original balance. The mathematically optimal strategy: always pay off the highest-APR debt first (avalanche method), because each dollar of high-rate debt eliminated provides a guaranteed, risk-free, tax-free 'return' equal to the APR.
Step-by-Step Example Walkthrough
" Compare two 25-year-old investors: (A) Invests $5,000/year for 10 years ($50,000 total, stops at 35), then lets it compound to age 65. (B) Waits until age 35, then invests $5,000/year for 30 years ($150,000 total) to age 65. Both earn 8% annually. Who wins? "
- A. Investor A's $50,000 invested age 25–35, compounding to 65 (30 more years after stopping): each annual $5k contribution at 8% grows differently based on when invested. Using FV of annuity for 10 years: FV_35 = $5,000 × [(1.08^10 - 1)/0.08] = $5,000 × 14.487 = $72,433 at age 35.
- Then compound $72,433 for 30 more years at 8%: $72,433 × (1.08)^30 = $72,433 × 10.063 = $728,765 at age 65.
- B. Investor B invests $5,000/year for 30 years starting at 35: FV = $5,000 × [(1.08^30 - 1)/0.08] = $5,000 × 113.283 = $566,416 at age 65.
- Investor A invested $50,000 and ends with $728,765.
- Investor B invested $150,000 (3× more capital) and ends with $566,416 — $162,349 LESS.
Final Result: Investor A, who invested $100,000 less and stopped contributing 30 years earlier, ends up with $162,349 MORE than Investor B at age 65. This is the power of time in compounding: the 10 early years of compounding from age 25–35 created so much base growth that 30 additional years of Investor B's contributions cannot overcome the head start. Every decade of delay in starting compound growth costs more in terminal wealth than the total contributions made during that decade.