Introducing Loans and Amortizations

In this topic we detail how amortization is related to the present value of an ordinary annuity and we determine ways to (a) find the unpaid balance of a loan, (b) find the borrowing amount with a budget in mind, and (c) find the number of payments left needed to repay the loan.

Definition (Amortization) The process of repaying a loan that is to be repaid by a series of partial payments with interest charged on the unpaid balance at the end of each period is called amortization. If the debt of dollars, with interest rate per period, is amortized by equal periodic payments, the size of each payment is given by the formula

Example (Amortization) (a) A loan of $10,000 is to be amortized with 10 equal  quarterly payments. If the interest rate is 6%, compounded quarterly, what is the periodic payment?
    The payment is given by the formula where and We have
    (b) A woman buys a car for $15,000. If the interest rate on the loan is 12%, compounded monthly, and if she wants to make monthly payments of $500 for 3 years, how much must she have for a down payment?
    She wants the payment to be $500 which is given by the formula where and We have

Therefore, no down payment is needed.

Definition (Amortization Schedule) A table of values that reconstructs all the information of a loan that has been amortized is called an amortization schedule.

Example (Amortization Schedule) Develop an amortization schedule for a loan of $50,000 with interest at 10%, compounded semiannually, if it is to be repaid in years by making equal semiannual payments.
The amortization is and the payments are Therefore the schedule is:


1
2
3
4
5				$0

Definition (Unpaid Balance) For a loan of payments of dollars per period at interest rate per period, the unpaid balance after payments have been made is given by the formula,

Example (Unpaid Balance) Find the unpaid balance after 15 payments for a $150,000 loan that has been amortized at 12%, compounded quarterly, and has quarterly payments of $6489.36 for 10 years.
The unpaid balance is given by the formula where and We have

Example (Using Amortization) A couple purchasing a home wants their payment to be $500 per month. If they have $15,000 available for a down payment, and if the mortage rate on a 25-year loan is 12%, compounded monthly, how much can they spend on a house?
Each payment is $500 and is given by the formula where and so we have

So they can spend