Mortgage Payment Problem: Suppose that a family borrows ...

Mortgage Payment Problem: Suppose that a family borrows $\$ 150,000$ to purchase a house. They agree to pay back this mortgage at $\$ 1074.65$ per month. But part of that payment goes to pay the interest for the month on the balance remaining. The interest rate is $6 \%$ per year, so they pay $0.5 \%$ per month. The balance $b_{n}$ remaining after month $n$ is given by the recursion formula $$b_{n}=b_{n-1}+0.005 b_{n-1}-1074.65$$ a. Explain the meaning of each of the three terms in the recursion formula. b. Find $b_{12}$, the balance remaining at the end of the first year of the mortgage. How much money did the family pay altogether for the year? How much of this amount went to pay interest, and how much went to reducing the balance of the mortgage? c. After how many months will the balance have dropped to zero and the mortgage be paid off?

Mortgage Payment Problem: Suppose that a family borrows $\$ 150,000$ to purchase a house. They agree to pay back this mortgage at $\$ 1074.65$ per month. But part of that payment goes to pay the interest for the month on the balance remaining. The interest rate is $6 \%$ per year, so they pay $0.5 \%$ per month. The balance $b_{n}$ remaining after month $n$ is given by the recursion formula
$$b_{n}=b_{n-1}+0.005 b_{n-1}-1074.65$$
a. Explain the meaning of each of the three terms in the recursion formula.
b. Find $b_{12}$, the balance remaining at the end of the first year of the mortgage. How much money did the family pay altogether for the year? How much of this amount went to pay interest, and how much went to reducing the balance of the mortgage?
c. After how many months will the balance have dropped to zero and the mortgage be paid off?

Key Concepts

Recursive Financial Models

A recursive financial model uses a formula that defines each term in a sequence based on its preceding term. In the context of loan repayment, such a formula will update the remaining balance after each period by accounting for new interest accrued and any payments made, thereby allowing an iterative calculation of the outstanding balance over time.

Mortgage Amortization

Mortgage amortization refers to the process of gradually repaying a loan through a series of scheduled payments. Each payment is divided between interest, which is the cost of borrowing, and principal, which reduces the outstanding balance of the loan. Over time, the proportion of each payment going towards principal increases as the interest on the decreasing balance declines.

Interest Rate Conversion

Interest rate conversion involves translating an annual interest rate into a rate that applies to shorter time intervals, such as monthly periods. This is essential in most financial calculations where payments or interest accrual happen at intervals shorter than one year, ensuring that the interest calculation accurately reflects the compounding period.

Principal and Interest Payment Breakdown

This concept involves understanding how each payment on a loan is split between two components: principal repayment and interest expense. Initially, a larger portion of the payment may cover interest due to the higher outstanding balance, while later payments contribute more significantly to reducing the principal as the interest portion declines.

Loan Term Calculation

Loan term calculation is the process of determining the number of periods required to fully repay a loan. This is achieved by evaluating how the recurring payments, which cover both interest charges and principal reductions, cumulatively decrease the outstanding balance to zero over time. It often involves solving equations based on geometric series or iterative methods.

Transcript

00:01 So in this problem, we were told that somebody bought a house that was worth $150 ,000, and that they're going to pay their mortgage of $1 ,074 .65 per month.

00:10 Now, they also are going to have to pay interest.

00:14 Well, if it's 6 % per year, as the problem described, that would mean that they're paying 0 .5 % per month.

00:20 So they give us this recursive formula to allow us to find the balance at the end of each month.

00:25 So in part a, what we're being asked to do is to represent what each term means.

00:30 Well, remember, b sub n represents the account balance.

00:34 So b sub n minus 1 is the previous month's balance.

00:38 So previous months balance.

00:42 So that's the first term.

00:44 The second term, 0 .005 times b sub n minus 1.

00:49 Well, remember, b sub n minus 1 is the previous month.

00:52 We're multiplying it by 0 .005 because that would be 0 .05 as a decimal.

00:57 So this is going to be the interest paid on last month's balance.

01:07 So that's how much interest is going to get paid.

01:11 Okay.

01:12 And then our last term, negative 1074 .65, first off, it's negative because our new balance is we're going to pay this mortgage off.

01:21 So this is our mortgage payment.

01:27 Okay.

01:28 So now let's go to part b.

01:30 In part b, what we're being asked to do is to figure out what would be b sub 12.

01:36 So remember, b sub 12 would represent how much they would have an e.

01:40 Now, we could keep doing this formula until we get to there, but we can also use our calculator.

01:44 So first off, make sure your mode is in sequence.

01:48 So then when you go to y equals, what you want to do is in u sub n.

01:54 So you're going to press y equals.

01:56 So then our use of n value or u of n, you're going to type in the letter u.

02:02 And then in parentheses, you're going to use your regular variable key, but it will show up as a at n minus 1 plus then we're going to do 0 .005.

02:11 So essentially we're just putting in our formula, you times your variable n minus 1, minus 1074 .65...

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