Compound Interest Explained: Formula, Examples & the Rule of 72

Compound interest is the mechanism behind every savings account, investment portfolio, and long-term loan. This guide explains the formula, shows how compounding frequency changes your returns, and teaches you the Rule of 72 for quick mental estimates.

Simple Interest vs Compound Interest

With simple interest, you earn interest only on your original principal. If you deposit $1,000 at 5% simple interest per year, you earn $50 every year, always on the original $1,000.

With compound interest, you earn interest on your principal plus on all previously earned interest. The interest becomes part of the principal, and then itself earns interest. This is the mechanism Albert Einstein allegedly called "the eighth wonder of the world", though whether he said it or not, the math is undeniably powerful.

Simple interest after 10 years: $1,000 × 5% × 10 = $500 total interest → $1,500
Compound interest after 10 years (annually): $1,000 × (1.05)^10 ≈ $1,629 → $629 total interest

That extra $129 on a $1,000 deposit may seem modest, but the gap widens dramatically over longer periods and larger sums.

The Compound Interest Formula

A = P × (1 + r/n)^(n×t)

Where:
A = final amount (principal + interest)
P = principal (initial deposit or loan)
r = annual interest rate as a decimal (e.g. 5% = 0.05)
n = number of times interest compounds per year
t = number of years

Variable Breakdown

Variable	Meaning	Common values
n = 1	Compounded annually	Bonds, annual savings
n = 4	Compounded quarterly	Many investment accounts
n = 12	Compounded monthly	Most savings accounts, mortgages
n = 365	Compounded daily	High-yield online savings
n → ∞	Compounded continuously	Theoretical max: A = Pe^(rt)

Worked Examples

Example 1, $5,000 for 10 years at 6% compounded monthly

P = 5,000  |  r = 0.06  |  n = 12  |  t = 10

A = 5,000 × (1 + 0.06/12)^(12×10)
A = 5,000 × (1.005)^120
A = 5,000 × 1.8194
A = $9,096.98

Interest earned = $9,096.98 − $5,000 = $4,096.98

Example 2, How much does compounding frequency matter?

Same setup: $10,000 at 8% for 20 years. Only the compounding frequency changes:

Annually  (n=1):   $10,000 × (1.08)^20    = $46,610
Quarterly (n=4):   $10,000 × (1.02)^80    = $48,010
Monthly   (n=12):  $10,000 × (1.00667)^240 = $49,268
Daily     (n=365): $10,000 × (1.000219)^7300 ≈ $49,530
Continuously:      $10,000 × e^(0.08×20)   ≈ $49,530

The jump from annual to monthly compounding adds about $2,658 over 20 years. Daily vs. monthly adds only another $262, the benefit of more frequent compounding diminishes rapidly as you increase frequency.

Example 3, Working backwards (present value)

You need $50,000 in 15 years. Your account earns 7% compounded annually. How much do you need to deposit today?

Rearranging:  P = A ÷ (1 + r/n)^(n×t)
P = 50,000 ÷ (1.07)^15
P = 50,000 ÷ 2.7590
P = $18,130.82

Deposit $18,131 today to have $50,000 in 15 years at 7%.

The Rule of 72

The Rule of 72 is a shortcut for estimating how long it takes to double your money at a given interest rate, no calculator needed:

Years to double = 72 ÷ annual interest rate (as a percentage)

At 6%:   72 ÷ 6  = 12 years to double
At 8%:   72 ÷ 8  = 9 years to double
At 12%:  72 ÷ 12 = 6 years to double

The rule works because ln(2) ≈ 0.693, and dividing by 100 × ln(2) ≈ 69.3 gives the exact doubling time for continuous compounding. The factor 72, slightly higher, gives a good approximation for annual compounding and is divisible by more numbers, making the mental arithmetic easier.

Reverse the rule: you can also find the interest rate needed to double in a target number of years. Want to double in 9 years? You need 72 ÷ 9 = 8% annually.

The Power of Starting Early

The single biggest factor in compound interest is time. Consider two investors who both earn 7% annually:

Investor A: invests $5,000/year from age 25 to 35 (10 years), then nothing.
Total invested: $50,000.  Balance at age 65: ~$602,000.

Investor B: invests $5,000/year from age 35 to 65 (30 years).
Total invested: $150,000.  Balance at age 65: ~$472,000.

Investor A invested 1/3 of the money but ended up with $130,000 more,
purely because of 10 extra years of compounding.

This is why financial advisors universally recommend starting to save and invest as early as possible, even if the amounts are small.

Compound Interest Against You: Debt

Compound interest works the same way when you are the borrower, except it works against you. Credit card debt is the most common trap:

Balance: $3,000  |  APR: 22.99%  |  Minimum payment: $60/month

Time to pay off: 87 months (over 7 years)
Total interest paid: ~$2,174
Total repaid: ~$5,174, 72% more than the original balance

Paying more than the minimum drastically reduces the total interest. Paying $150/month instead of $60 would clear the same debt in about 25 months and save over $1,500 in interest.

Use our free Compound Interest Calculator to model any scenario, savings, investments, or debt, with a year-by-year breakdown showing exactly how your balance grows or shrinks over time.