# Annuity (finance Theory) - Amortization Calculations

Amortization Calculations

If an annuity is for repaying a debt P with interest, the amount owed after n payments is:

$frac{R}{i}- left( 1+i right) ^n left( frac{R}{i} - P right)$

because the scheme is equivalent with borrowing the amount to create a perpetuity with coupon, and putting of that borrowed amount in the bank to grow with interest . Conceptually, the payments to the perpetuity are being applied against an amount of of the debt, while the additional debt of compounds until retired in a single payment from the saved amount. What remains owed after the obligation to the excess borrowing is retired is the amortized amount of debt due to borrowing .

Also, this can be thought of as the present value of the remaining payments:

$Rleft = R times a_{N-n|i}$