Answer to Question #212816 in Financial Math for phs

Question

Show that the total interest paid on the amortization of a loan with leveled monthly payments K for n years at an interest rate of i% per annum is equal

Answer 1

Monthly payments=k per month

Number of years=n years

Interest rate=$\frac{i}{100}$ Per annum

Let C be the total amount of loan after n years

$=K.12.n$

Let P be initial payment (loan amount)

$P=\frac{K[1-(1+\frac{i}{12×100})^{-12×n}]}{\frac{i}{12×100}}$

Therefore,

Total interest paid=$C-P$

$=K.12.n-\frac{K[1-(1+\frac{i}{1200})^{-12n}]}{\frac{i}{1200}}$

$=\frac{K.12n.\frac{i}{100}-K+K(1+\frac{i}{1200})^{-12n}]}{\frac{i}{1200}}$

$=\frac{K(\frac{i}{100}-1)+K(1+\frac{i}{1200})^{-12n}}{\frac{i}{1200}}$

$=\frac{1200K[(\frac{i}{100}-1)+(1+\frac{i}{1200})^{-12n}]}{i}$