The Annual Interest Rate calculator computes the for a fixed rate loan.
INSTRUCTIONS: Choose the preferred currency units and enter the following:
Annual Interest Rate: The calculator return the annual % interest rate.
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,
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)`.