To calculate per annum interest, you need two formulas. Simple interest is I = P × r × t: multiply the principal by the annual rate (as a decimal) by the number of years. Compound interest is A = P(1 + r/n)^(nt), where n is how many times per year interest is added to the balance. On a $10,000 balance at 5% over three years, simple interest produces $1,500; compounding monthly produces $1,616.17. Same rate, same time, different method, more than $100 apart. That spread widens as the balance, the rate, or the timeline grows.
The Four Inputs You Need
Every per annum calculation starts with the same handful of numbers. Get them wrong and the answer is wrong.
- Principal (P). The original amount: the size of the loan or the opening balance of the account.
- Annual interest rate (r). The percentage the lender or bank states. Convert it to a decimal before using it. 5% becomes 0.05. 4.25% becomes 0.0425.
- Time (t). The duration in years. Partial years need a quick conversion, covered below.
- Compounding frequency (n). Only needed for compound interest. Monthly is 12, quarterly is 4, daily is 365.
Lenders must disclose these terms clearly under the Truth in Lending Act so borrowers can compare credit products before signing.1Office of the Law Revision Counsel. 15 USC 1601 – Congressional Findings and Declaration of Purpose If the paperwork is vague about the rate, compounding frequency, or total cost, treat that as a reason to ask questions.
Calculating Simple Interest
Simple interest charges only on the original principal. Interest earned or owed sits separately; the balance the rate applies to does not grow.
Formula: I = P × r × t
Borrow $10,000 at 5% for three years:
$10,000 × 0.05 × 3 = $1,500
Total repayment is $11,500. If you pay down the principal early on a standard simple-interest loan, the total interest drops, because each payment shrinks the base the rate applies to. This method shows up in many short-term personal loans and some auto loans.
One caveat worth checking before you assume early payments help: some loans use precomputed interest, where the total is calculated upfront and locked into every payment. On those, extra payments do not reduce the interest already baked in.2Consumer Financial Protection Bureau. Whats the Difference Between a Simple Interest Rate and Precomputed Interest on an Auto Loan Ask which method your loan uses.
Calculating Compound Interest
Compound interest is where most real-world savings accounts, mortgages, and credit cards live. Interest earned in one period gets added to the balance, and the next period’s interest is calculated on that larger number.
Formula: A = P × (1 + r/n)^(n × t)
A is the final amount (principal plus all accumulated interest). P is the principal. r is the annual rate as a decimal. n is how often interest compounds per year. t is the time in years.
Same numbers as above: $10,000 at 5% for three years, compounded monthly (n = 12).
- Divide the rate by the compounding frequency: 0.05 ÷ 12 = 0.004167
- Add 1: 1 + 0.004167 = 1.004167
- Multiply the compounding frequency by the years to get total periods: 12 × 3 = 36
- Raise the growth factor to that power: 1.004167^36 = 1.16162
- Multiply by the principal: $10,000 × 1.16162 = $11,616.17
Total interest: $1,616.17. That is $116.17 more than simple interest on the same numbers, and the gap comes entirely from interest earning interest each month. Stretch the term to 20 years and the difference is no longer measured in dollars but in thousands. It is the same mechanism behind long-term investment growth and behind credit card balances that seem to grow no matter how much you pay.
A spreadsheet or financial calculator is the cleanest way to handle the exponent without rounding errors. Federal rules require lenders to disclose an APR that is mathematically accurate within one-eighth of a percentage point on standard loans.3eCFR. 12 CFR 1026.22 – Determination of Annual Percentage Rate
Adjusting for Partial Years
Loans and deposits rarely fall on tidy year boundaries. Convert the term to a fraction of a year and plug the decimal into t.
- Months: divide by 12. A six-month CD uses t = 0.5. Nine months is t = 0.75.
- Days: divide by 365. A 90-day bridge loan uses t = 90/365 = 0.2466.
On a 90-day, $10,000 loan at 5% simple interest:
$10,000 × 0.05 × 0.2466 = $123.29
Treating those 90 days as a full year would overstate the interest by nearly five times.
360-Day vs. 365-Day Years
Some short-term commercial loans and money market products use a 360-day year (ACT/360) instead of 365. Dividing days by 360 makes each day’s interest slightly larger. On the same 90-day, $10,000 loan at 5%, the 360-day method produces $125.00 versus $123.29 under the 365-day method. Loan documents specify which convention applies. Check when the numbers are large enough for the difference to matter.
APR vs. APY: Same Rate, Different Number
A 5% rate can mean two different things depending on how it is quoted, and mixing them up is one of the most common mistakes in comparing financial products.
APR (annual percentage rate) is the figure on loans and credit cards. It is the yearly cost of borrowing, including certain fees, but it does not account for compounding within the year. APY (annual percentage yield) is the figure on savings accounts and CDs, and it does factor in compounding, so it reflects what you actually earn over a full year.4eCFR. 12 CFR Part 1030 – Truth in Savings (Regulation DD)
To convert an APR that compounds during the year into an effective annual rate:
EAR = (1 + r/n)^n − 1
A credit card with a 24% APR compounded monthly has an effective rate of (1 + 0.24/12)^12 − 1 = 26.82%. That gap is real money over a year.
The rule of thumb: when borrowing, lower APR is better; when saving, higher APY is better. When comparing a loan rate to a savings rate side by side, convert both to effective annual rates so you are looking at the same thing.
The Rule of 72
For a quick mental estimate of how long money takes to double at a given compound rate, divide 72 by the annual rate. At 6%, roughly 12 years. At 8%, about 9 years. At 3%, around 24 years.
It works in reverse. To double your money in 10 years, you need roughly 72 ÷ 10 = 7.2%. The estimate is most accurate for rates between 5% and 12% and gets loose at the extremes, but it is fast and it is close. It also makes high-interest debt tangible: a credit card at 24%, unpaid, doubles the balance in about three years.
What Taxes Do to the Result
The formulas tell you what you earn or owe before taxes. Taxes then move the real number.
Interest you receive from savings accounts, CDs, bonds, and most other investments is taxable as ordinary income on your federal return. You owe tax on all taxable interest even if the amount is too small to trigger a Form 1099-INT from your bank.5Internal Revenue Service. Topic No. 403, Interest Received Financial institutions file that form when they pay $10 or more in interest during the year, but reporting the income is your obligation regardless of the amount.6Internal Revenue Service. About Form 1099-INT, Interest Income
Some interest you pay is deductible. Mortgage interest on up to $750,000 of home acquisition debt (for loans taken out after December 15, 2017) is deductible if you itemize.7Internal Revenue Service. Publication 936, Home Mortgage Interest Deduction Student loan interest is deductible up to $2,500 per year without itemizing, though income phase-outs reduce or eliminate the benefit at higher earnings.8Internal Revenue Service. Topic No. 456, Student Loan Interest Deduction These deductions lower the real cost of the interest you just calculated, and they belong in any honest side-by-side comparison of what a loan actually costs or what a savings account actually pays.