APR Formula (Annual Percentage Rate)
The APR formula calculates the true annual cost of borrowing, including interest and fees.
Used to compare loans fairly.
The Formula
Simple APR = ((Interest + Fees) / Principal) / n × 100
Where n is the number of years of the loan.
Precise APR Formula
Loan Amount - Fees = Σ [Payment / (1 + APR/12)^k] for k = 1 to n
The precise APR is the rate that makes the present value of all monthly payments equal to the net loan amount (principal minus fees). This requires iterative calculation.
Variables
| Symbol | Meaning |
|---|---|
| APR | Annual Percentage Rate (the true cost of borrowing) |
| Interest | Total interest paid over the life of the loan |
| Fees | Upfront fees (origination, points, closing costs) |
| Principal | The original loan amount |
| n | Number of years (or total payment periods) |
APR vs Interest Rate
- Interest Rate: The cost of borrowing the principal only
- APR: The total cost including fees, spread over the loan term
- APR is always ≥ the interest rate (they are equal only when there are no fees)
- Federal law (TILA) requires lenders to disclose the APR
Example
$200,000 mortgage at 6% for 30 years with $4,000 in fees
Monthly payment = $1,199.10
Total interest over 30 years = $231,677
Total cost = $231,677 + $4,000 = $235,677
Simple APR ≈ ($235,677 / $200,000) / 30 × 100 ≈ 6.16%
The APR of 6.16% is higher than the 6% interest rate because it includes the $4,000 in fees.