Problem 23 - Precalculus Mathematics for Calculus

Question

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) $860.51, (b) $309,783.60, (c) $3,101,095.17.

Step 1: Calculate the Monthly Interest Rate

The annual interest rate is given as \(9 + \frac{3}{4} = 9.75\%\). Since the interest is compounded monthly, we need to convert this into a monthly interest rate: \(r = \frac{9.75}{100 \times 12} = 0.008125\).

Step 2: Determine the Total Number of Payments

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

Step 3: Calculate Monthly Payment Using the Loan Formula

The formula for monthly payment \(M\) on an amortizing loan is given by:\[M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\]Where \(P = 100,000\), \(r = 0.008125\), and \(n = 360\).Substituting these values into the formula:\[M = 100,000 \frac{0.008125(1 + 0.008125)^{360}}{(1 + 0.008125)^{360} - 1}\]

Step 4: Use a Calculator to Find the Monthly Payment

Plug the numbers into a calculator to find the monthly payment \(M\). You should get approximately \(M = 860.51\).

Step 5: Calculate the Total Payment Over 30 Years

Multiply the monthly payment by the total number of payments to find the total amount paid:\[\text{Total Payment} = 860.51 \times 360 = 309,783.60\]

Step 6: Consider the Investment Scenario

If the monthly payments are invested instead of paying the loan, calculate the future value \(FV\) using the formula for the future value of a series:\[FV = PMT \times \frac{(1 + r)^n - 1}{r}\]Where \(PMT = 860.51\), \(r = 0.008125\), and \(n = 360\).

Step 7: Find the Future Value of the Investment

Using the same future value formula:\[FV = 860.51 \times \frac{(1 + 0.008125)^{360} - 1}{0.008125}\]Calculate this with a calculator to find that \(FV \approx 3,101,095.17\).

Key Concepts

Monthly Payment Calculation

Calculating the monthly payment of a loan is a crucial step in understanding loan amortization. For a \(100,000 loan with an annual interest rate of 9.75% compounded monthly over 30 years, we first need to find the monthly interest rate. Divide the annual rate by 12 months and convert the percentage to a decimal form:

• Annual interest rate: 9.75%
• Monthly interest rate: \( \frac{9.75}{100 \times 12} = 0.008125 \)

This rate helps determine how much interest will add up each month. Next, calculate the total number of payments, which, for a 30-year loan, is:

• 360 payments (30 years \( \times \) 12 months)

The formula to find the monthly payment \( M \) is: \[M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}\]Where:\(P \) is the loan principal, \( r \) is the monthly interest rate, and \( n \) is the number of total payments. Substituting the values, we calculate the monthly payment to be approximately \)860.51. This incorporates both the principal and the interest over the loan period.

Compound Interest

Compound interest is the interest calculated on the initial principal as well as the accumulated interest from previous periods. With our 9.75% interest rate compounded monthly, this is what transforms a regular payment scenario into a growing burden over time. Each month, the unpaid balance at month’s end becomes the new principal for calculating the next month’s interest.

• Interest computation: Each month's interest is based not just on the original principal but also on the past months' accumulated interest.
• Impact: For borrowers, it means paying interest on interest if not fully repaid within a specified time.

By compounding monthly, the actual annual interest paid ends up being slightly higher than 9.75%. Understanding this concept helps explain why loans can cost more over time than the borrower initially expects. It is crucial to keep this in mind when planning repayments.

Future Value of Series

The future value of a series is a powerful concept for understanding how savings can grow over time with regular investments and compound interest. In the scenario where the couple invests their monthly payment instead of financing a loan, we calculate this using the future value formula for a series:\[FV = PMT \times \frac{(1 + r)^n - 1}{r}\]Where:

• \( PMT \) is the monthly investment (\(860.51).
• \( r \) is the monthly interest rate (0.008125).
• \( n \) is the total number of contributions (360).

Substituting these values, you discover the future value of these invested payments would be approximately \)3,101,095.17 after 30 years. This highlights the power of compounded savings over time, where regular contributions can grow significantly more than the principal amount.