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Chapter 5: Problem 42

FINANCING A HomE Eight years ago, Kim secured a bank loan of \(\$ 180,000\) to help finance the purchase of a house. The mortgage was for a term of \(30 \mathrm{yr}\), with an interest rate of \(9.5 \% /\) year compounded monthly on the unpaid balance to be amortized through monthly payments. What is the outstanding principal on Kim's house now?

Short Answer

Expert verified
The outstanding principal on Kim's house after 8 years is approximately \$163,200.

Step by step solution

01

STEP 1: Check Initial Variables

The initial loan amount P is $180,000. The term of the loan t is 30 years, which is 360 months. The interest rate r per annum is 9.5%, which as a monthly rate is \(r_{m} = (9.5/100)/12\).
02

STEP 2: Calculate the Monthly Payment (M)

M can be calculated using the formula: \[M = P \times \frac{r_{m}(1 + r_{m})^t} {(1 + r_{m})^t - 1}\] where \(r_{m}\) is the monthly interest rate, \(t\) is the total number of payments or term, and \(P\) is the loan amount which is \$180,000.
03

STEP 3: Find the Outstanding Principal after 8 years

Now, to calculate the outstanding principal after 8 years, use the loan balance formula: \[B = M \times \left( \frac {(1 + r_{m})^t - (1 + r_{m})^{n}} {(1 + r_{m}) ^ t - 1}\right)\] where \(B\) is the outstanding balance, \(M\) is the monthly payment, \(r_{m}\) is the monthly interest rate, \(t\) is the total number of payments, and \(n\) is the number of payments made. Here \(n = 8 \times 12 = 96\) months. Compute the values to calculate the outstanding principal on Kim's house.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Loan Balance Calculation
Understanding how to calculate the remaining balance on a mortgage is crucial for homeowners like Kim, who need to manage their finances effectively. The loan balance equation provides a picture of how much is still owed after a certain number of payments have been made.

To determine the outstanding balance of a loan, it is important to know how much has been paid off already. In Kim's situation, after 8 years of monthly payments, we need to crunch some numbers. Using the formula for the outstanding balance, we consider the total number of monthly payments she's committed to and the number of payments she's actually made.

  • Total term of the loan: The mortgage term is 30 years, which translates to 360 monthly payments.
  • Number of payments made: With 8 years of payments under her belt, Kim has made 96 monthly payments.

Once we have these figures, we can plug them into the loan balance equation to find out how much Kim still owes. The formula shows the interplay between interest rates and the passage of time, reflecting how payments made towards the principal slowly reduce the balance owed over the years.
Monthly Mortgage Payments
Monthly mortgage payments are the regular amounts a borrower must pay to the lender, and they are a key component in the journey of home ownership.

These payments are determined by the loan amount, the interest rate, and the term of the mortgage. The formula to calculate the monthly payment, as seen in Kim's case, takes into account the compound interest effect because of the regular monthly payments. By dividing the annual interest rate by 12, we get the monthly interest rate, which affects the total amount repaid over the term when compounded.

Using the formula, we find that the monthly payment remains consistent over the life of the loan. This allows borrowers like Kim to plan their finances, as they will pay the same amount each month unless they decide to refinance or make additional payments. The formula's design ensures that by the end of the mortgage term, both the interest and the principal are fully paid off, assuming all payments are made on time.
Compound Interest
Compound interest is fundamental to understanding how financial products like mortgages work. Unlike simple interest, compound interest is calculated on both the initial principal and the accumulated interest from previous periods.

In Kim's mortgage scenario, the interest is compounded monthly, which means that each month's interest charge is based on the current outstanding balance of the loan, including the interest that has been added to the principal over time. This concept ensures that with each monthly payment made, a portion goes towards paying off the accumulated interest, and the remaining part reduces the principal balance.

The concept of compound interest can be daunting, but it fundamentally shapes the amortization of loans like mortgages. As homeowners pay off their mortgages over time, they not only reduce the principal balance but also decrease the amount of interest that can accrue on the remaining balance, illustrating the powerful effect of compound interest over the life of a loan.

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Most popular questions from this chapter

Find how much money should be deposited in a bank paying interest at the rate of \(8.5 \% /\) year compounded quarterly so that, at the end of \(5 \mathrm{yr}\), the accumulated amount will be \(\$ 40,000\).

A young man is the beneficiary of a trust fund established for him 21 yr ago at his birth. If the original amount placed in trust was \(\$ 10,000\), how much will he receive if the money has earned interest at the rate of \(8 \% /\) year compounded annually? Compounded quarterly? Compounded monthly?

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Determine which of the sequences are geometric progressions. For each geometric progression, find the seventh term and the sum of the first seven terms. $$ \frac{1}{2},-\frac{3}{8}, \frac{1}{4},-\frac{9}{64}, \ldots $$

A HomE The Johnsons have accumulated a nest egg of \(\$ 40,000\) that they intend to use as a down payment toward the purchase of a new house. Because their present gross income has placed them in a relatively high tax bracket, they have decided to invest a minimum of \(\$ 2400 /\) month in monthly payments (to take advantage of the tax deduction) toward the purchase of their house. However, because of other financial obligations, their monthly payments should not exceed \(\$ 3000\). If local mortgage rates are \(7.5 \%\) lyear compounded monthly for a conventional 30 -yr mortgage, what is the price range of houses that they should consider?
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