Rate from Annuity
Find the interest rate implied by an annuity's present value when payments occur once per year. Solved numerically via Newton-Raphson iteration.
When to use: Use when you know the present value, annual payment, and number of years but need the implied rate. Common in valuing fixed-payment loans and annuity products that pay annually.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| k | Interest Rate | Implied annual rate (found iteratively) | % |
| PV | Present Value | Loan amount or annuity price | $ |
| PMT | Payment | Annual payment | $ |
| n | Number of Years | Number of years | years |
Real-Life Examples
Example 1: Loan Rate
A $20,000 loan is repaid with $4,000/year for 6 years. What rate are you paying?
Given
Step-by-Step
The implied interest rate on the loan is about 5.47%.
Example 2: Investment Yield
Pay $50,000 for an annuity paying $8,000/year for 10 years. What is the yield?
Given
Step-by-Step
The annuity yields about 9.61% per year.
Frequently Asked Questions
The annuity present value equation is a polynomial in k with no closed-form solution when n > 4. Numerical methods like Newton-Raphson iteration are needed to find the rate that makes the equation balance.
Newton-Raphson is a numerical method that starts with an initial guess and repeatedly refines it using calculus. Each iteration gets closer to the answer until the error is negligibly small, typically converging in 5-15 iterations for financial problems.
The Newton-Raphson method in this calculator iterates until the change is less than one ten-billionth, giving precision far beyond what is needed for any practical financial decision. The result is essentially exact.
Common scenarios include finding the interest rate embedded in a loan (when you know the payment, amount borrowed, and term) or determining the yield on an annuity product (when you know the purchase price, periodic payout, and duration).