Which Equation Represents The Partial Sum Of The Geometric Series

Like stimulus packages, home mortgages and foreclosures also impact the economy. A problem for many borrowers is the adjustable rate mortgage, in which the interest rate can change (and usually increases) over the duration of the loan, causing the monthly payments to increase beyond the ability of the borrower to pay. Most financial analysts recommend fixed rate loans, ones for which the monthly payments remain constant throughout the term of the loan. In this exercise we will analyze fixed rate loans.

When most people buy a large ticket item like car or a house, they have to take out a loan to make the purchase. The loan is paid back in monthly installments until the entire amount of the loan, plus interest, is paid. With a loan, we borrow money, say (P) dollars (called the principal), and pay off the loan at an interest rate of (r)%. To pay back the loan we make regular monthly payments, some of which goes to pay off the principal and some of which is charged as interest. In most cases, the interest is computed based on the amount of principal that remains at the beginning of the month. We assume a fixed rate loan, that is one in which we make a constant monthly payment (M) on our loan, beginning in the original month of the loan.

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Suppose you want to buy a house. You have a certain amount of money saved to make a down payment, and you will borrow the rest to pay for the house. Of course, for the privilege of loaning you the money, the bank will charge you interest on this loan, so the amount you pay back to the bank is more than the amount you borrow. In fact, the amount you ultimately pay depends on three things: the amount you borrow (called the principal), the interest rate, and the length of time you have to pay off the loan plus interest (called the duration of the loan). For this example, we assume that the interest rate is fixed at (r)%.

To pay off the loan, each month you make a payment of the same amount (called installments). Suppose we borrow (P) dollars (our principal) and pay off the loan at an interest rate of (r)% with regular monthly installment payments of (M) dollars. So in month 1 of the loan, before we make any payments, our principal is (P) dollars. Our goal in this exercise is to find a formula that relates these three parameters to the time duration of the loan.

We are charged interest every month at an annual rate of (r)%, so each month we pay (frac{r}{12})% interest on the principal that remains. Given that the original principal is (P) dollars, we will pay (left(frac{0.01r}{12}right)P) dollars in interest on our first payment. Since we paid (M) dollars in total for our first payment, the remainder of the payment ((M-left(frac{r}{12}right)P)) goes to pay down the principal. So the principal remaining after the first payment (let’s call it (P_1)) is the original principal minus what we paid on the principal, or

As long as (P_1) is positive, we still have to keep making payments to pay off the loan.

1. Recall that the amount of interest we pay each time depends on the principal that remains. How much interest, in terms of (P_1) and (rtext{,}) do we pay in the second installment?

2. How much of our second monthly installment goes to pay off the principal? What is the principal (P_2text{,}) or the balance of the loan, that we still have to pay off after making the second installment of the loan? Write your response in the form (P_2 = ( )P_1 – ( )Mtext{,}) where you fill in the parentheses.

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   Show that (P_2 = left(1 + frac{r}{12}right)^2P – left[1 + left(1+frac{r}{12}right)right] Mtext{.})

4. Let (P_3) be the amount of principal that remains after the third installment. Show that

5. If we continue in the manner described in the problems above, then the remaining principal of our loan after (n) installments is

   This is a rather complicated formula and one that is difficult to use. However, we can simplify the sum if we recognize part of it as a partial sum of a geometric series. Find a formula for the sum

   and then a general formula for (P_n) that does not involve a sum.

6. It is usually more convenient to write our formula for (P_n) in terms of years rather than months. Show that (P(t)text{,}) the principal remaining after (t) years, can be written as

7. Now that we have analyzed the general loan situation, we apply formula (7.9) to an actual loan. Suppose we charge $1,000 on a credit card for holiday expenses. If our credit card charges 20% interest and we pay only the minimum payment of $25 each month, how long will it take us to pay off the $1,000 charge? How much in total will we have paid on this $1,000 charge? How much total interest will we pay on this loan?

8. Now we consider larger loans, e.g., automobile loans or mortgages, in which we borrow a specified amount of money over a specified period of time. In this situation, we need to determine the amount of the monthly payment we need to make to pay off the loan in the specified amount of time. In this situation, we need to find the monthly payment (M) that will take our outstanding principal to (0) in the specified amount of time. To do so, we want to know the value of (M) that makes (P(t) = 0) in formula(7.9). If we set (P(t) = 0) and solve for (Mtext{,}) it follows that

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      Suppose we want to borrow $15,000 to buy a car. We take out a 5 year loan at 6.25%. What will our monthly payments be? How much in total will we have paid for this $15,000 car? How much total interest will we pay on this loan?

   2. Suppose you charge your books for winter semester on your credit card. The total charge comes to $525. If your credit card has an interest rate of 18% and you pay $20 per month on the card, how long will it take before you pay off this debt? How much total interest will you pay?

   3. Say you need to borrow $100,000 to buy a house. You have several options on the loan:

      • 30 years at 6.5%

      • 25 years at 7.5%

      • 15 years at 8.25%.

      1. What are the monthly payments for each loan?

      2. Which mortgage is ultimately the best deal (assuming you can afford the monthly payments)? In other words, for which loan do you pay the least amount of total interest?

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