Simple vs Compound Interest Calculator
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How simple and compound interest are compared
Simple interest is calculated from the original principal, so equal periods add equal interest. Compound interest adds retained interest to the balance, allowing later interest to be calculated on earlier interest.
Formulas and cash-flow treatment
- Simple balance: \( A_s = P(1 + rt) \)
- Discrete compound balance: \( A_c = P(1 + r/n)^{nt} \)
- Continuous compound balance: \( A_c = Pe^{rt} \)
- Effective annual rate: \( EAR = (1 + r/n)^n - 1 \), or \(e^r - 1\) for continuous compounding.
For recurring cash flow \(C_j\) at time \(\tau_j\), the tool adds \(C_j[1+r(t-\tau_j)]\) to the simple side and \(C_j(1+r/n)^{n(t-\tau_j)}\) to the compound side. Withdrawals are negative cash flows. This definition keeps the comparison consistent, but it does not reproduce a lender’s repayment allocation or amortization rules.
APR, APY, AER, and EAR in this calculator
The input is a nominal annual interest rate: an annualized rate before the effect of within-year compounding. EAR is the calculator’s effective annual output; APY is a US deposit-account term and AER is a commonly used UK savings term for a comparable effective-year concept. The US CFPB definition of APY and interest rate explicitly distinguishes an annual rate that does not reflect compounding from APY, which does.
Do not enter an advertised APY or AER as though it were the nominal rate. For example, 5% nominal compounded monthly gives \((1+0.05/12)^{12}-1 = 5.116\%\) EAR. Entering 5.116% as the nominal input would count the compounding effect twice. Also note that a regulated loan APR can include fees, so it is not always identical to this tool’s fee-free nominal input; see the CFPB explanation of loan interest rate versus APR.
What frequency does—and does not—mean
Compounding frequency is how often retained interest is added to the interest-bearing balance. Daily accrual only means interest is calculated daily; it does not by itself prove that the account compounds daily. Product terms determine when accrued interest is credited or capitalized.
| Frequency | Effective annual rate | Ending balance |
|---|---|---|
| Annual | 5.000% | £10,500.00 |
| Quarterly | 5.095% | £10,509.45 |
| Monthly | 5.116% | £10,511.62 |
| Daily | 5.127% | £10,512.67 |
| Continuous | 5.127% | £10,512.71 |
Borrower versus saver
For a saver, a higher ending balance is beneficial; for a borrower with an untouched lump-sum debt, it means more is owed. A typical mortgage, personal loan, or credit card has payments and product-specific rules, so its balance cannot be modeled accurately as an untouched lump sum. Use a dedicated amortization calculator for repayment schedules.
How the difference grows over time
| Term | Simple balance | Compound balance | Compound − simple |
|---|---|---|---|
| 1 year | £10,500.00 | £10,511.62 | £11.62 |
| 5 years | £12,500.00 | £12,833.59 | £333.59 |
| 10 years | £15,000.00 | £16,470.09 | £1,470.09 |
| 20 years | £20,000.00 | £27,126.40 | £7,126.40 |
A lower compound rate can eventually overtake a higher simple rate, but the crossover must be calculated from the exact rates and frequency. For example, 6% compounded annually first exceeds 8% simple shortly after 10 years—not after 14 years.
Frequently asked questions
What is the difference between simple and compound interest?
Simple interest is calculated from the original principal, so equal periods add equal interest. Compound interest is calculated from principal plus retained interest, so later periods can add more.
Which is better for borrowers or savers?
With the same positive nominal rate and no cash flows, compound interest usually benefits a saver and costs a borrower more once interest compounds. Real products must be compared using their fees, effective rates, taxes, and payment rules.
What does compounding frequency mean?
It is how often retained interest is added to the balance. At a fixed positive nominal rate, more frequent compounding generally produces a slightly higher effective annual rate.
Is compound interest always higher than simple interest?
No. With the same positive rate and untouched principal, the short-term result can be lower, equal, or higher depending on frequency and the product's accrual convention. Over multiple years, annual or more frequent compounding generally becomes higher. Cash-flow timing can also change the comparison.
How are months and days converted?
The calculator converts months to years by dividing by 12 and days to years by dividing by 365. A provider may use actual calendar days or another day-count convention.
What is the difference between APR and APY, AER, or EAR?
This tool accepts a nominal annual rate that excludes the effect of within-year compounding. EAR, APY, and AER are effective annual measures that reflect compounding. Do not enter an advertised APY or AER as a nominal rate.
Are deposits or repayments included?
Only recurring cash flows entered in the advanced section are included. They use fixed timing and do not model changing repayments, fees, or a lender's amortization rules.
Why might the result differ from a bank statement?
Banks may use actual transaction dates, daily accrual without daily compounding, different day-count rules, variable rates, rounding, fees, taxes, minimum balances, or product-specific payment rules that this illustration excludes.
Methodology and review
Formula implementation and explanatory copy were created and internally reviewed by the Starlight Robotics editorial team on . No independent financial-professional review is claimed.
The calculator applies the standard simple, periodic compound, continuous compound, and effective annual rate equations shown above. Definitions were checked against the CFPB Regulation DD definitions and its official APY calculation appendix. The European Central Bank’s interest-rate explainer provides additional context on nominal rates and inflation.
Educational illustration only. Check the provider’s contract, disclosure standard, cash-flow dates, fees, and day-count convention before making a financial decision.