Problem 15 - Precalculus: Mathematical for Calculus

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) Monthly payment: \$859.35. (b) Total paid: \$309,366. (c) Account value: \$2,078,671.

Step 1: Understanding the Mortgage Loan Formula

To find the monthly payment on a mortgage, we use the formula:\[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \]where \( M \) is the monthly payment, \( P \) is the principal loan amount (\$100,000), \( r \) is the monthly interest rate, and \( n \) is the total number of payments. The given annual interest rate is \(9 \frac{3}{4}\% = 9.75\%\). We convert it to a monthly rate by dividing by 12, so \( r = \frac{9.75}{100 \times 12} \approx 0.008125 \). The total number of payments for a 30-year loan is \( n = 30 \times 12 = 360 \).

Step 2: Calculate Monthly Payment

Substitute the values into the formula to calculate \( M \):\[ M = 100,000 \times \frac{0.008125(1+0.008125)^{360}}{(1+0.008125)^{360} - 1} \]Calculate the exponent, \( (1+0.008125)^{360} \approx 20.706 \).Now, substitute back to find \( M \):\[ M = 100,000 \times \frac{0.008125 \times 20.706}{20.706 - 1} \approx 859.35 \]Thus, the monthly payment is approximately \$859.35.

Step 3: Calculating Total Payment Over 30 Years

To find the total amount paid, multiply the monthly payment by the number of months:\[ \text{Total payment} = 859.35 \times 360 \approx 309,366 \]The total amount paid over 30 years is approximately \$309,366.

Step 4: Future Value of Monthly Payments as Investments

Now, calculate how much would accumulate if the monthly payments were invested in an account with the same interest rate. Use the future value of annuity formula:\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]Substitute the known values:\[ FV = 859.35 \times \frac{(1 + 0.008125)^{360} - 1}{0.008125} \]Calculate:\[ (1 + 0.008125)^{360} - 1 = 20.706 - 1 = 19.706 \]Then:\[ FV = 859.35 \times \frac{19.706}{0.008125} \approx 2,078,671 \]Therefore, if deposited, the account would have approximately \$2,078,671 at the end of 30 years.

Key Concepts

Compounded Interest

Compounded interest is the fundamental concept behind most financial growth calculations, including our mortgage example. It's the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. Essentially, this means that interest is earned on interest. To understand compounded interest more clearly, consider the following:

• When interest is compounded monthly, the annual interest rate is divided by 12 to give the monthly interest rate.
• With compounding, the principal amount grows each month by the interest earned, which means each subsequent interest calculation is based on a slightly larger principal.

In our mortgage example, the annual interest rate is 9.75%. To find the monthly rate, we divide 9.75 by 12, which gives us approximately 0.8125%. This demonstrates how interest accumulates more quickly due to monthly compounding.

Monthly Payment Formula

When securing a loan or calculating installments for a mortgage, the monthly payment formula is crucial. This formula helps in determining what you need to pay each month to fully pay off a loan.The formula used is:\[M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\]Where:

• \(M\) - the monthly payment
• \(P\) - the principal loan amount, in our case, \(100,000.
• \(r\) - the monthly interest rate (annual rate divided by 12).
• \(n\) - total number of payments (months), so 360 for a 30-year mortgage.

This formula accounts for both the principal repayment and the interest, ensuring that the loan is fully paid by the end of the term. In our context, the couple's monthly payment was calculated to be approximately \)859.35.

Future Value of Annuity

The future value of an annuity formula is used when determining how much accumulated value regular payments will have at a future date. It is especially useful in setting savings goals or evaluating what an investment will grow into over time.The standard formula for the future value of an annuity is:\[FV = PMT \times \frac{(1 + r)^n - 1}{r}\]Where:

• \(FV\) - the future accumulated value of the investments.
• \(PMT\) - the regular payment amount, which was \(859.35 in our case.
• \(r\) - the monthly interest rate.
• \(n\) - the number of payments.

Through this formula, we discovered that if the couple opted to invest their monthly payments instead of paying the mortgage, those payments would grow to approximately \)2,078,671 after 30 years. This showcases the power of compounding and regular saving.