Rate from Annuity
Find the nominal annual interest rate implied by an annuity's present value when payments occur m times per year (PMT is per-period). Solved numerically via Newton-Raphson on the periodic rate, then annualized.
When to use: Use to back out the APR on a loan or annuity with sub-annual payments. Classic case: solve for the implied APR on a mortgage given the principal, monthly payment, and term — set m = 12, PMT = monthly payment, n = years.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| k | Interest Rate | Implied nominal annual rate (found iteratively) | % |
| PV | Present Value | Loan amount or annuity price | $ |
| PMT | Payment | Per-period payment (e.g. monthly payment when m = 12) | $ |
| n | Number of Years | Number of years | years |
| m | Periods/Year | Compounding and payment frequency per year | integer |
Real-Life Examples
Example 1: Mortgage APR
A $300,000 mortgage is repaid with $1,798.65/month for 30 years. What APR is implied?
Given
Step-by-Step
The mortgage carries a 6% APR (nominal annual, compounded monthly). Effective annual rate = (1 + 0.06/12)^12 − 1 ≈ 6.17%.
Example 2: Auto Loan
A $25,000 auto loan with monthly payments of $471.78 over 60 months. What APR?
Given
Step-by-Step
The implied APR is 5.00%, matching the standard auto-loan quote.
Frequently Asked Questions
Because m-times annuity formulas treat each period as its own compounding step. PMT is the per-period payment (e.g. the monthly mortgage payment when m = 12), not the annualized total. To go from an annual figure, divide by m before entering.
The output is the nominal annual rate (APR), compounded m times per year. To get the effective annual rate, plug the result and m into the EAR (m Times/Year Compounding) formula: EAR = (1 + k/m)^m − 1.
When inputs are inconsistent — for example, PMT × n × m ≤ PV implies a non-positive rate solution. The iteration may diverge or hit the safety floor at k = −1 and return NaN. In practice, sensible inputs converge in 5−15 iterations.