The formulas behind the money calculators, each with a worked example. In every formula P is the principal, r the rate per period (annual rate ÷ 12 for monthly work), n the number of periods and t the years. Rates are written as decimals: 5% is 0.05. Related terms: Compound interest, APY, Principal and Amortization.
Interest and growth
| Name | Formula | Example |
|---|---|---|
| Simple interest | I = P × r × t |
$5,000 at 4% for 3 years: $600 |
| Compound interest | A = P(1 + r/n)^(nt) |
$10,000 at 5% for 10 years, yearly: $16,288.95; monthly: $16,470.09 |
| Effective annual rate (APY) | APY = (1 + r/n)^n − 1 |
5% compounded monthly: 5.12% |
| Rule of 72 | years ≈ 72 / rate% |
At 6%: about 12 years to double (exact: 11.90) |
| Compound annual growth rate (CAGR) | CAGR = (end / start)^(1/years) − 1 |
$10,000 to $16,000 in 5 years: 9.86% |
Try them: Simple Interest Calculator, Interest Rate Calculator, APY Calculator, Rule of 72 Calculator, Average Return Calculator and Compound Interest Calculator.
Loans and regular payments
| Name | Formula | Example |
|---|---|---|
| Monthly payment | M = P·r / (1 − (1 + r)^−n) |
$200,000, 6%, 30 years (r = 0.005, n = 360): $1,199.10 |
| Interest for one month | interest = balance × r |
First month of that loan: $1,000.00 |
| Future value of regular deposits | FV = D · ((1 + r)^n − 1) / r |
$500 a month, 7% a year, 30 years: $609,985 |
| Present value of regular payments | PV = C · (1 − (1 + r)^−n) / r |
$1,000 a month for 10 years at 6% a year: $90,073.45 |
The payment formula is the engine of the Auto Loan Calculator, Personal Loan Calculator and Student Loan Calculator; see How Loan Amortization Works for what each payment is made of. The deposit and annuity formulas drive the Annuity Calculator, Retirement Calculator and CD Calculator.
Investing
| Name | Formula | Example |
|---|---|---|
| Return on investment (ROI) | ROI = (value − cost) / cost |
Bought at $10,000, sold at $11,500: 15% |
| Dividend yield | yield = annual dividend / price |
$2.40 a year on a $60 share: 4% |
| Dollar-cost averaging | average cost = total spent / total shares |
$100 at prices 10, 8 and 12 buys 30.833 shares at an average of $9.73 (the average price is $10) |
| Real return (inflation-adjusted) | real = (1 + nominal) / (1 + inflation) − 1 |
7% with 3% inflation: 3.88% |
Try them: Dividend Calculator, Dollar-Cost Averaging (DCA) Calculator, Stock Profit Calculator and Bond Calculator.
Business and everyday money
| Name | Formula | Example |
|---|---|---|
| Markup | markup = (price − cost) / cost |
Cost $60, price $100: 66.7% |
| Profit margin | margin = (price − cost) / price |
Same sale: 40.0% |
| Break-even units | fixed costs / (price − variable cost) |
$3,000 fixed, $50 price, $30 variable: 150 units |
| Discount | price × (1 − d) |
20% off $80: $64.00; 20% then 10% more: $57.60 (28% off, not 30%) |
| Remove VAT from a gross price | net = gross / (1 + rate) |
$120 including 20% VAT: net $100, VAT $20 |
| Hourly to yearly pay | hourly × hours per week × 52 |
$25 × 40 × 52 = $52,000 (2080 hours) |
| Overtime at time and a half | regular + extra × rate × 1.5 |
45 hours at $20, 40 regular: $950.00 |
Try them: Markup Calculator, Break-Even Calculator, Discount Calculator, VAT Calculator, Sales Tax Calculator, Salary to Hourly Calculator and Overtime Calculator.
Common mistakes
- Mixing annual and monthly figures. Divide the annual rate by 12 and count months, or keep both annual. Using 6% with 360 months gives nonsense.
- Confusing markup and margin. The same sale is a 66.7% markup and a 40.0% margin. Always say which one.
- Adding percentages. Two successive discounts of 20% and 10% multiply (
0.8 × 0.9 = 0.72); they do not add to 30%. - Ignoring inflation and fees. Compare real returns, and remember that a yearly fee of 1% compounds against you too.