Two interest formulas, one line of algebra apart, and they behave so differently over time that confusing them is expensive. The definitions are simple. What is not simple is knowing which one your own accounts use, because the answer determines whether a balance grows or shrinks — and by how much.
The two formulas
Simple interest is charged on the original principal only:
I = P × r × t
Compound interest is charged on the principal plus all interest accumulated so far:
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 the number of years.
Worked example: watching the gap open
$1,000 at 10% for three years.
Simple interest: interest = 1,000 × 0.10 × 3 = $300, so the balance is $1,300. Each year charges exactly $100, no matter what the balance is.
Compound interest, annually:
- Year 1: 1,000 × 1.10 = $1,100
- Year 2: 1,100 × 1.10 = $1,210
- Year 3: 1,210 × 1.10 = $1,331
The difference is $31 — modest, and on small balances irrelevant.
Now extend to twenty years, same $1,000 at 10%:
Simple: 1,000 + (1,000 × 0.10 × 20) = $3,000.
Compound: 1,000 × 1.10²⁰ = $6,727.
The gap is now $3,727 — more than the entire simple-interest result. That is the whole point: the gap is not a constant, it grows, because compound interest keeps charging on the accumulated total. The compound interest calculator and the simple interest calculator will show both curves against the same inputs.
Where simple interest is genuinely used
More than people expect, and mostly deliberately so:
Most consumer loans. Mortgages, auto loans and personal loans are typically simple interest with an amortising payment schedule. This is not a favour to you — it is standard. The interest charge in month 12 of a 30-year mortgage is barely half the charge in month 1, because the balance has fallen. Effective interest still compounds on a declining balance, but you never pay interest on previously-paid interest.
Short-term and fixed-term deposits. Some simple-interest accounts exist, and they are transparent by design — you know exactly what you will be paid.
Many payment-in-kind and promotional rates, where the headline figure assumes simple interest over the promotional period, which is why the APR is usually higher.
Where compound interest is used, and why
Credit cards are the big one. They charge interest daily on the full balance — genuinely compound, and at typical APRs of 20–29% the growth is spectacular. A $5,000 balance with no payments roughly doubles in about three years, which is the single most useful fact on this page for anyone carrying a card balance. The credit card guide works through the numbers.
Overdrafts and payday-style products use daily compounding at high rates, and the effective rate can exceed the advertised APR once fees are included.
Savings and investment accounts compound, usually monthly or daily. Here it works for you, and the rule of 72 guide covers how to reason about the growth rate.
APR: the number to actually compare
The interest rate is the price of borrowing the money. APR is the price of borrowing the money plus the cost of getting it — origination fees, closing costs, insurance, setup fees — expressed as an annualised percentage.
On a mortgage that is the difference between a genuinely great deal and a slightly worse one dressed up to look identical. A lender quoting "2.9% rate" against another's "3.1% APR" is not quoting the same thing, and in many markets only the APR is legally required to be displayed precisely because the rate alone is easy to misread.
One important exception: on credit cards, APR and interest rate are the same thing, and the APR is what appears on your statement.
Why the frequency question matters less than you think
The compounding frequency n in the formula attracts attention because it is the only variable you do not control. Its practical effect is small at normal rates:
At 7% annual, the effective rate is 7.00% compounded annually, 7.11% semi-annually, 7.18% monthly, and 7.29% daily. The whole spread from annual to daily is 0.29 percentage points — over a decade on $10,000, a difference of a few hundred dollars.
Two things follow. First, a bank advertising "compounded daily" is not offering you meaningfully more than one compounded monthly at the same quoted rate. Second, and far more important, the rate dominates everything else in the formula — moving from 6% to 7% is worth roughly 40% of the final balance over thirty years, while the frequency difference is worth under 1%. The rule of 72 exists precisely because people need a way to reason about the term that matters.
Three situations where the distinction is worth real money
Comparing savings products. Two accounts quoting the same rate with different compounding are nearly identical. But one quoting a higher rate with lower compounding can still be worse, and you have to compare effective annual rates, not the headline number.
Reading a loan offer. "7% simple interest" on a one-year loan is exactly 7%. "7% compounded monthly" on the same loan is 7.18%. Small, but on a large balance over a long term it is not nothing, and on a short loan it is usually worth a phone call to ask which applies.
Paying down debt. The order you choose between minimum payments changes the total cost far more than any rate difference. The debt-to-income guide covers the strategy question; the arithmetic point here is that on compound-interest balances, every dollar not paid compounds against you in the same way.
One shortcut for the comparison
To convert a quoted rate into an effective annual figure without a calculator, divide by the number of compounding periods and multiply by 365: an APR of 12% quoted as simple gives 12% APR-equivalent, while 12% compounded daily gives an effective rate of about 12.68%. The difference is small at conventional rates and large at card rates, which is exactly where it becomes worth doing properly.
Frequently asked questions
What is the difference between simple and compound interest?
Simple interest is calculated on the original principal only: I = P × r × t. Compound interest is calculated on principal plus accumulated interest: A = P(1 + r/n)^(nt). With one year's interest they are identical; the gap opens from the second year.
Which do savings accounts and loans actually use?
Most savings accounts compound, monthly or daily. Most consumer loans — mortgages, auto, personal — are simple interest with amortising payments, because the balance falls over time. Credit cards are the major exception: they compound daily on the full balance.
Does compounding frequency change my savings much?
Very little at typical rates. At 7% annual, annual versus monthly compounding changes the effective rate from 7.00% to 7.18% — a few weeks of extra growth over a decade. The rate you earn matters far more than how often it is credited.
Why do lenders advertise APR as well as interest rate?
Because the interest rate alone does not describe what you pay. APR includes interest plus most fees, annualised. On a mortgage this is the difference between two deals that look identical. On credit cards, APR and interest rate are the same thing.