Problem 23 - Algebra and Trigonometry

Question

Mortgage A couple secures a 30 -year loan of \(\$ 100,000\) at 9\(\frac{3}{4} \%\) per year, compounded monthly, to buy a house. (a) What is the amount of their monthly payment? (b) What total amount will they pay over the 30 -year period? (c) If, instead of taking the loan, the couple deposits the monthly payments in an account that pays 9\(\frac{3}{4} \%\) interest per year, compounded monthly, how much will be in the account at the end of the 30 -year period?

Step-by-Step Solution

Answer

(a) $859.35 monthly payment. (b) $309,366 total paid. (c) $2,414,863.60 in savings account after 30 years.

Step 1: Calculate Monthly Interest Rate

First, convert the annual interest rate from a fraction to a decimal. The annual interest rate is given as \(9\frac{3}{4}\%\), which is equivalent to \(9.75\%\). Convert this to a decimal to get \(0.0975\). The monthly interest rate is then \(\frac{0.0975}{12}\) because there are 12 months in a year. Thus, the monthly interest rate \(r\) is approximately \(0.008125\).

Step 2: Calculate Number of Payments

Since the loan term is 30 years and payments are made monthly, the total number of payments \(N\) is \(30 \times 12 = 360\).

Step 3: Use the Amortization Formula

The formula for the monthly mortgage payment \(M\) is given by \(M = P \cdot \frac{r(1+r)^N}{(1+r)^N - 1}\), where \(P\) is the principal loan amount, \(r\) is the monthly interest rate, and \(N\) is the total number of payments. Plugging in the values: \(P = 100,000\), \(r = 0.008125\), and \(N = 360\), we calculate \(M = 100,000 \cdot \frac{0.008125(1+0.008125)^{360}}{(1+0.008125)^{360} - 1}\)."

Step 4: Calculate Monthly Payment

Calculate the value of \((1+0.008125)^{360}\), which is approximately \(34.719996405\). Then use this to find \(M\) using the formula from Step 3. This results in \(M \approx 859.35\). Thus, the monthly payment is approximately \$859.35.

Step 5: Calculate Total Payments Made

The total amount paid over the life of the loan is the monthly payment multiplied by the total number of payments. So, \(Total = 859.35 \times 360\), which equals \$309,366.00.

Step 6: Compute Future Value of Deposits

If the couple deposits the monthly payments into an account with the same interest rate, use the future value of an annuity formula: \(FV = M \times \frac{(1 + r)^N - 1}{r}\). Substitute \(M = 859.35\), \(r = 0.008125\), and \(N = 360\) into the formula to find \(FV = 859.35 \times \frac{34.719996405 - 1}{0.008125}\). Calculate \(FV\), which is approximately \\(2,414,863.60. Thus, the amount in the account at the end of 30 years will be approximately \\)2,414,863.60.

Key Concepts

Amortization Formula

The amortization formula is a crucial part of understanding mortgage calculations. It helps determine the monthly payment necessary to repay the loan over a specified term and interest rate.

The formula is: \[M = P \cdot \frac{r(1+r)^N}{(1+r)^N - 1}\]Where:

• \(M\) is the monthly mortgage payment.
• \(P\) is the principal amount of the loan.
• \(r\) is the monthly interest rate.
• \(N\) is the total number of payments.

This formula calculates your monthly payment by considering both the principal loan amount and how the interest accumulates over each payment period. Using this, you can ensure all loan costs are accounted for over the lifespan of the mortgage.

For example, using the principal of \(\\(100,000\), a monthly rate of approximately \(0.008125\), and a 30-year term, you'll find a monthly payment of about \(\\)859.35\). This ensures the loan is fully paid after 360 payments.

Monthly Interest Rate

Understanding the concept of the monthly interest rate can demystify how the interest on a mortgage is calculated. The annual interest rate on a loan is typically given as a percentage but must be converted to a monthly rate for mortgage calculations.

Given an annual interest rate, such as 9.75%, the conversion to a monthly rate means dividing the annual rate by 12. Therefore, the monthly rate is calculated as:\[\text{Monthly Interest Rate} = \frac{0.0975}{12} \approx 0.008125\]This small rate might seem negligible on its own, but over the course of many months, it accumulates into a significant sum. This underscores the importance of understanding how even a small interest rate can impact the total cost of a mortgage.

Calculating your monthly interest rate correctly is the first step to understanding overall loan costs and planning your financial commitments effectively.

Future Value of Annuity

The future value of an annuity is an interesting concept when considering what your regular deposits might grow to at the end of a specific period under a compound interest regime.

The future value of an annuity can be calculated using:\[FV = M \times \frac{(1 + r)^N - 1}{r}\]Where:

• \(FV\) is the future value of the annuity.
• \(M\) is the monthly deposit.
• \(r\) is the monthly interest rate.
• \(N\) is the total number of deposits.

If, instead of paying off a loan, you invest a sum equivalent to the loan payment (e.g., \\(859.35 monthly) into an account paying the same rate of 9.75% annually compounded monthly, you see the power of compound interest at work.

By the end of 30 years, this investment snowballs into a considerable sum of approximately \\)2,414,863.60. This illustrates the growth potential of steady investments over time.

Loan Repayment Schedule

A loan repayment schedule is the timeline for repaying a loan, indicated by regular (usually monthly) payments until the loan is fully settled. This schedule is an essential tool for both borrowers and lenders as it outlines the principal and interest amounts through each installment.

The schedule usually provides details such as:

• Number of payments, e.g., 360 for a 30-year loan.
• Each monthly payment amount, e.g., \(\$859.35\).
• Outstanding principal after each payment.
• Interest paid with each installment.

This transparency helps the borrower understand the reduction in debt and manage finances better. The structure of the mortgage ensures a clear visibility on the diminishing balance over time.

Understanding your loan repayment schedule can prepare you for financial planning and optimize debt management strategies, helping you stay on top of your obligations.