Compound Interest Explained: Formulas, Examples, and Time Horizon

Simple interest pays a fixed rate on the original principal throughout the life of the investment. Compound interest pays the rate on the principal plus all previously earned interest, producing exponential rather than linear growth over time.

Over short periods the difference is small. Over long periods it dominates the outcome: $10,000 at 7% simple interest yields $80,000 after 100 years; at 7% compounded annually, it yields approximately $8.7 million.

The canonical formula is A = P × (1 + r/n)^(n × t), where P is principal, r is the annual rate (as a decimal), n is the number of compounding periods per year, and t is time in years. The exponent is where the exponential magic lives.

For daily compounding, n = 365. For monthly, n = 12. Continuous compounding — the theoretical limit — uses A = P × e^(r × t), which is marginally higher than daily compounding and rarely material at typical rates.

The rule of 72 is a mental shortcut: years to double ≈ 72 ÷ annual rate (in percent). At 6%, your money doubles in about 12 years. At 9%, about 8 years. The approximation is accurate within a percent or two for rates between 4% and 15%.

It is useful for quickly comparing opportunities. A 9% option doubles in 8 years; a 6% option needs 12. Over 24 years, the first triples to 8x while the second doubles twice to 4x. Small rate differences compound into large wealth differences.

Starting early is the most effective lever in compounding. $1,000 invested at 7% at age 25 becomes roughly $14,974 at 65. The same $1,000 invested at 35 becomes $7,612. At 45, $3,870. The first decade of compounding is worth more than the last two combined.

This is why personal finance writing emphasizes starting early over maximizing return. The math of compounding rewards consistency and time horizon more than chasing a slightly higher rate.

Compounding more often produces higher effective yields but with rapidly diminishing returns. On a 6% annual rate:

• Annual compounding: 6.00% effective
• Semi-annual: 6.09%
• Quarterly: 6.14%
• Monthly: 6.17%
• Daily: 6.18%
• Continuous: 6.184%

Most real-world savers don’t just deposit once — they contribute monthly. The future value of a series of regular contributions is FV = PMT × [((1 + r/n)^(n × t) − 1) / (r/n)], where PMT is the payment per period.

Example: $500 per month at 8% annual for 30 years. FV = 500 × [((1.006667)^360 − 1) / 0.006667] ≈ $745,180. Of that, contributions total $180,000 — the remaining $565,180 is pure compound growth.

Compounding cuts both ways. Credit card debt at 24% APR grows just as aggressively in the issuer’s favor. A $5,000 balance at 24% making only minimum payments takes decades to clear and ultimately costs many multiples of the original balance.

Paying down high-interest debt is mathematically equivalent to earning that rate risk-free. Before investing at expected 7%, clear any debt costing more than 7% — the guaranteed "return" from debt elimination beats most portfolios after tax.