Question

Assume that you take a L₀ loan to buy a house. Further assume that the loan comes with a monthly interest rate of r annual interest rate and is to be repaid over the course of n monthly payments. The loan does not charge any fees. Derive a repayment plan so that the loan is paid in full with n equal payments

Answer 1

Amount of loan = L0

Monthly rate of interest = r

no. of months of payment = n

Let, equal monthly payment = P

Then,

L0 = P/(1+r) + P/(1+r)^2 + P/(1+r)^3 --------------------- + P/(1+r)^n)

Or,

L0 = P*(1/(1+r) + 1/(1+r)^2 + 1/(1+r)^3 --------------------- + 1/(1+r)^n)

Let, 1/(1+r) = a

Then,

L0 = P(a + a^2 + a^3 ------------- + a^n) ------------------ (1)

Here, (a + a^2 + a^3 ------------- + a^n) is a geometric progression.

Sum of the geometric progression = a*(1-a^n)/(1-a)

Sum of the geometric progression = a*(1-1/(1+r)^n)/(1-1/(1+r))

Sum of the geometric progression = (1/(1+r)*(1-1/(1+r)^n)/r/(1+r)

Sum of the geometric progression = ((1+r)/(1+r))*(1-1/(1+r)^n)/r

Sum of the geometric progression = (1-1/(1+r)^n)/r

Putting the value of above sum in equation 1,

L0 = P*(1-1/(1+r)^n)/r

Above equation is the equation to be used for the payment of loan,

Amount of loan (P) = L0/((1-1/(1+r)^n)/r)