% annuity.m
%
% A = annuity (amount curently in account)
% P = payment per deposit period
% r = interest rate per period
% n = number of deposit periods
%
% Find the minimum interest rate assuming monthly compounding
% given the desired annuity amount A at the end of n monthly
% periods with given payment per period.
%
% The equation used is called the annuity due equation 
format long

A = 75000;
P = 150.0;
n = 20 * 12;

% Annuity due scalar function
f = @(r) (P/r)*( (1+r)^n - 1) - A;

% Find interval on which function changes sign
% by plotting f(r)

fplot(f, [0.001, 0.009])
grid on

% this gives the interval [a.b]

a = 0.005
b = 0.006

root = fzero(f,[a,b])
annualRateInPercent = (12 * root) * 100

% Try it using bisect

root = bisect(f, a, b, 1E-10, 30, true)