Every savings account, certificate of deposit, loan, and credit card balance runs on one of two engines: simple interest or compound interest. The difference looks trivial in year one — a few hundred dollars on a large deposit — but over five, ten, or twenty years it decides whether your money multiplies or merely grows. This guide states both formulas precisely, then works a single example — $10,000 at 8% per year for 5 years — through both methods, year by year, with every number shown. Run your own figures through the compound interest calculator to compare.
The two formulas, in plain words
Simple interest is charged (or paid) only on the original principal. The interest earned each period is identical, because the base never changes:
I = P × r × t
where I is total interest, P the principal, r the annual rate as a decimal, and t the time in years. The final amount is A = P + I.
Compound interest is charged on the principal plus all interest accumulated so far. Each period’s interest is added to the balance, so the next period’s interest is computed on a larger base:
A = P × (1 + r/n)^(n×t)
where n is the number of compounding periods per year (n = 1 for annual, 12 for monthly, 365 for daily). The interest earned is A − P.
The entire difference between the two methods is one phrase: interest on interest. Simple interest never pays it; compound interest always does.
Worked example: $10,000 at 8% per year for 5 years
Take a certificate of deposit of $10,000 earning 8% per year, left untouched for 5 years.
Simple interest: the straight-line case
I = 10,000 × 0.08 × 5 = 4,000
Total after 5 years: 10,000 + 4,000 = $14,000. Each year’s interest is exactly $800 — no more, no less, forever.
Compound interest (annual compounding): the accelerating case
A = 10,000 × (1.08)^5
Working through it: 1.08^2 = 1.1664; 1.08^3 = 1.259712; 1.08^4 = 1.36048896; 1.08^5 = 1.4693280768.
A = 10,000 × 1.4693280768 = $14,693.28
Interest earned: 14,693.28 − 10,000 = $4,693.28 — a full $693.28 more than the simple-interest result, from nothing but reinvesting the interest.
The year-by-year comparison
This table shows the balance and that year’s interest under each method. Watch the simple-interest column stay flat while the compound column accelerates:
| Year | Simple: interest | Simple: balance | Compound: interest | Compound: balance |
|---|---|---|---|---|
| 1 | 800 | 10,800 | 800.00 | 10,800.00 |
| 2 | 800 | 11,600 | 864.00 | 11,664.00 |
| 3 | 800 | 12,400 | 933.12 | 12,597.12 |
| 4 | 800 | 13,200 | 1,007.77 | 13,604.89 |
| 5 | 800 | 14,000 | 1,088.39 | 14,693.28 |
Two facts jump out. First, in year 1 the two methods are identical: $800 either way. The gap only opens once there is interest to compound — which is why short horizons barely show it and long horizons explode. Second, by year 5 the compound interest for that single year ($1,088.39) is more than a third larger than the simple interest ($800). That acceleration is the entire power of compounding.
What if it compounds monthly?
Many bank products compound more often than once a year. With monthly compounding, n = 12:
A = 10,000 × (1 + 0.08/12)^(12×5) = 10,000 × (1.0066667)^60 ≈ $14,898.46
Interest ≈ $4,898.46 — about $205 more than annual compounding on the same 8% “rate”. This is why two products quoting “8%” can pay different amounts: the one that compounds more often pays more. The true yearly yield here is roughly 8.30% — the difference between the quoted nominal rate (APR) and the effective rate (APY).
Where each method is actually used
Simple interest appears where the term is short or the product is deliberately flat:
- Short-term personal loans and many auto-finance products quote interest on the original amount.
- Flat-rate installment plans (“0% APR” offers aside) often use simple calculations.
- Late-payment penalties on utility bills and some fees are simple percentages of the outstanding bill.
Compound interest is the default everywhere money sits or owes for a long time:
- Savings accounts and CDs compound — typically monthly or quarterly.
- Mutual funds and 401(k) contributions compound through reinvested returns.
- Credit cards compound against the borrower, usually monthly (sometimes daily). Missed payments add interest to the balance, and next month’s interest is charged on that larger balance — the same math as the deposit example, running in reverse.
Second worked example: the borrower pays the difference
A credit card balance of $5,000 left unpaid for one year at 2.5% per month (a realistic US card rate, roughly 30% APR):
- Simple interest at 30% per year: 5,000 × 0.30 × 1 = $1,500
- Compounded monthly: 5,000 × (1.025)^12 = 5,000 × 1.34488882 = $6,724.44, so interest = $1,724.44
The compounding costs the borrower an extra $224.44 in a single year — about 15% more than the simple calculation. This is why minimum payments barely move a card balance: each month’s unpaid interest joins the principal and starts earning its own interest.
The Rule of 72: doubling time at a glance
A handy mental shortcut for compound growth: divide 72 by the annual rate to estimate how many years it takes money to double.
- At 8% compounded annually: 72 ÷ 8 = 9 years to double.
- Check against the example: 10,000 × (1.08)^9 = 10,000 × 1.999005 ≈ 19,990 — indeed just under double. The shortcut is remarkably close.
Under simple interest there is no such elegant rule: doubling requires earning 100% of the principal, which at 8% takes 100 ÷ 8 = 12.5 years. Compounding shaved three and a half years off the doubling time.
Traps that cost real money
- Comparing nominal rates without the frequency. “8% compounded monthly” beats “8% compounded annually” by about 0.3 percentage points of effective yield. Always compare effective annual rates (APY), not quoted rates (APR), when choosing between accounts.
- Withdrawing the gains. If you spend each year’s interest instead of leaving it in, you have converted a compound product into a simple one. The $693.28 gap in the worked example only exists because every dollar of interest stayed invested.
- Assuming loans are simple. Amortized loans (mortgages, car loans) charge interest on the reducing balance each month, which behaves like compounding on the unpaid portion. Paying extra toward principal early cuts total interest far more than the same extra paid late.
- Ignoring the sign. Compounding multiplies whatever it touches. At 8% it grows your deposit; at 30% it grows your debt. The math has no loyalty.
How to check your own numbers
Plug your principal, rate, compounding frequency, and horizon into the compound interest calculator and compare the result against the simple-interest formula I = P × r × t. The gap between the two answers is exactly the value of compounding over your chosen horizon.
Frequently asked questions
Is compound interest always better than simple interest?
For a saver or investor, yes — compound interest pays more than simple interest at the same rate and term, because it adds interest on interest. The only exception is a single period (one year with annual compounding), where both methods give identical results, as year 1 of the worked example shows. For a borrower, the reverse holds: you want simple interest on your debt, because compounding makes the debt grow faster.
What does “compounded monthly” actually mean?
It means the annual rate is divided into 12 slices, and each month the interest for that slice is calculated and added to the balance. At 8% nominal compounded monthly, each month adds 8% ÷ 12 = 0.667% of the current balance. Because each month’s addition raises the balance, the effective yearly return is about 8.30%, not exactly 8%.
Which method do banks use for savings accounts and CDs?
Compound interest, in almost all cases — savings accounts typically compound monthly or quarterly, and certificates of deposit usually compound quarterly or at maturity. Marketing materials quote the nominal annual rate, but the payout reflects the compounding frequency. Short-term Treasury bills instead use discount pricing, so read the terms.
Why do credit card balances grow so fast when unpaid?
Credit card interest compounds — typically monthly, at rates around 2–2.5% per month (24–30% APR) in the US. Each month’s interest is added to what you owe, and the next month’s interest is charged on the new, larger balance. The worked example above shows $5,000 becoming $6,724.44 in one year untouched — and that is before late fees, which are added to the balance and compound too.
How quickly will my money double at a given rate?
Use the Rule of 72: divide 72 by the annual rate (as a whole number) to estimate the doubling time in years. At 6% it takes about 12 years; at 8%, about 9 years; at 12%, about 6 years. The rule assumes annual compounding and works best for rates between 4% and 15%. For an exact figure, use the compound interest calculator and find the year the balance passes double.
What is the difference between APR and the effective (APY) rate?
APR is the quoted annual rate before compounding is counted; APY (annual percentage yield), also called the effective annual rate, includes compounding. They relate by: APY = (1 + r/n)^n − 1. At 8% compounded monthly, APY = (1 + 0.08/12)^12 − 1 ≈ 8.30%. When comparing accounts, compare APY to APY — comparing one product’s APR to another’s APY is comparing two different measurements.
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