What is compound interest?

The canonical formula

``` A = P × (1 + r/n)^(nt)

Where: A = final amount P = principal (starting amount) r = annual interest rate (as decimal: 7% = 0.07) n = compounding frequency per year (1 = annual, 12 = monthly, 365 = daily) t = time in years ```

Simple worked example: $10,000 at 7% annual return, compounded annually, for 10 years: - A = 10000 × (1 + 0.07/1)^(1×10) = 10000 × 1.967 = $19,672 - You earned $9,672 interest (interest growing on interest)

For the same 10 years at SIMPLE interest (no compounding): $10,000 × 7% × 10 = $7,000 (linear). Compound interest produces $2,672 more — that's the "interest on interest" effect.

The Rule of 72 (mental math shortcut):

`` Years to double money ≈ 72 / annual return % ``

Examples: - 6% return: doubles in 12 years - 7% return: doubles in ~10.3 years - 9% return: doubles in 8 years - 12% return: doubles in 6 years - 4% return: doubles in 18 years - 2% (HYSA in low-rate era): doubles in 36 years

Useful for rapid mental math on retirement planning.

Doubling examples (the power of time):

$10,000 invested at 7% annual return:

The non-linearity is striking: years 1-10 add $10k; years 40-50 add $145k. Compound interest's power is in the late years. This is why starting early matters disproportionately.

The "Einstein quote" myth:

The famous "Compound interest is the eighth wonder of the world... He who understands it, earns it; he who doesn't, pays it" — attributed to Einstein but no evidence Einstein said it. Origin unknown, likely 1900s financial press. Quote-investigator.com traces it to anonymous 1920s sources. The PRINCIPLE is real; the attribution is fake.

Long-term return benchmarks (used in retirement planning, NOT advice):

The "7% real return" baseline for S&P 500 over long timeframes is the canonical assumption in retirement math (Bengen 4% rule, Trinity Study).

Compounding frequency math:

Compounding more frequently barely matters at moderate rates:

Difference: <2.5% between annual and continuous. Don't pay extra fees for "daily compounding" — it's marketing, not meaningful.

The 5 biggest compound-interest applications:

The "starting early" advantage (real data):

Two scenarios, both ending at age 65 with same $300,000 total contributed:

Scenario A: Start at age 25, contribute $7,500/year for 40 years - Total contributed: $300,000 - At 7% return: ~$1,500,000 by 65

Scenario B: Start at age 45, contribute $15,000/year for 20 years - Total contributed: $300,000 - At 7% return: ~$650,000 by 65

Same money in. 2.3× the result for early starter. The 20 extra years of compounding more than doubles the outcome. This is why "start now" beats "save more later" almost always.

Common compound-interest mistakes:

• Confusing simple with compound — simple interest math underestimates long-term wealth dramatically
• Ignoring inflation — 7% nominal vs 4% real (after 3% inflation) makes 30-year projections 60% lower
• Linear thinking — assuming "twice the time = twice the money" — actually exponential
• Ignoring fees — 1% expense ratio over 40 years = 28% of final wealth lost. Use low-cost index funds (Bogle)
• Withdrawing during downturns — selling at -30% lock in losses; missing the recovery destroys decades of compounding
• Trying to time the market — "Time in the market beats timing the market" (Bogle); compound rewards consistency

This is NOT investment advice:

Returns vary. Past performance does not predict future results. Long-term S&P 500 returns include catastrophic periods (1929-1932 -89%, 2000-2002 -49%, 2008 -38%). The math assumes you stay invested through downturns. If you sell during crashes, the formula doesn't apply.

For personalized investment guidance, consult a fee-only fiduciary financial advisor (NAPFA.org, GarrettPlanning.com).