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`r_(n+1) = [r_n - f(r_n)/(f'(r_(n+1)))] *12`

Enter a value for all fields

The **Annual Interest Rate** calculator computes the annual interest rate for a fixed rate loan.

**INSTRUCTIONS:** Choose the preferred currency units and enter the following:

- (
**P**) This is the original principal of the loan - (
**N**) This is the duration of the loan - (
**Pmt**) This is the monthly payment amount

**Annual Interest Rate:** The calculator return the annual % interest rate.

**Related Calculators:**

**Monthly Interest Rate:**: This computes the monthly interest rate on a fixed rate loan based on the monthly payment, duraton of the loan and principal amount.**Monthly Payment:**: This computes the monthly payment on a fixed rate loan based on the annual interest rate, duraton of the loan and principal amount.

A loan which has a fixed interest rate, **r**, and **N** equal monthly payments of the amount, **Paymnt**, can be characterized by the equation:

`P*r = "Paymnt" * ( 1 - 1/(1+r)^N )`

Rearranging we can define: `f(r) = P/"Paymnt" *r +1/(1+r)^N - 1`

Setting `f(r) = 0` to find the roots of this function, we can now use the iterative Newton Raphson Method to find the annual interest rate, **r**. *Newtonian Iteration Animation, Wikipedia / Ralph Pfeifer _{CC BY-3.0}*

The Newton Raphson Method is an iterative numerical analysis method for finding the roots of a real-value function. We want to find in this case the interest rate, **r**, that is where `f(r) = 0`.

The Newton Raphson Method makes successively better approximations of the value of **r** by using the derivative of `f(r)`, which we denote `f'(r)`, to define the tangent at the point `[r_n, f(r_n)]`. Each successive approximation uses the tangent thus derived for** **`r_n`, which will intersect the x-axis at the next value `r_(n+1)`.

If we pick a first guess interest rate, `r_0`, the Newton Raphson Method tells us that a next better approximation is given as follows:

`r_1 = r_0 - f(r_0) / (f'(r_0))`

If we look at the example graph at the right of the function, `f(x)`, the x-value approximation where the function crosses the x-axis at `x_1` is the intersection with the *x*-axis of the tangent to the graph which touches the curve `f(x)` at the point `(x_0, f (x_0))`.

The Newton Raphson Method generalizes the successive better approximations of the function's root (where the function intersects the x-axis) with the following equation:

`x_(n+1) = x_n - f(x_n) / (f'(x_n))` This equation iterates over some number of successive approximations of `x_(n+1)` and arrives quickly at a very close approximation of the interest rate, `r_(n+1)`.