"""
Exercise A.4 from "A primer on..."
Solve a system of difference equations describing the
remaining value of a loan (x) and the monthly payments (y).
As specified in the exercise, the solution is implemented as
a function, which takes the relevant model parameters
as arguments and returns the solution. 
"""

import numpy as np
import matplotlib.pyplot as plt

def loan(L, p, N):
    x = np.zeros(N + 1)
    y = np.zeros(N + 1)

    months = range(N + 1)

    x[0] = L
    y[0] = 0

    for n in months[1:]:
        y[n] = p / (100 * 12) * x[n-1] + L/N
        x[n] = x[n-1] + p / (100 * 12) * x[n-1] - y[n]

    return months, x, y

months, x, y = loan(5e6, 5, 25 * 12)  
#plt.plot(months, x)
plt.plot(months, y)
plt.show()