Problem 15 - 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) Use the monthly payment formula. (b) Multiply monthly payment by 360. (c) Use the future value of annuity formula.

Step 1: Calculate Monthly Interest Rate and Total Number of Payments

The annual interest rate is given as 9.75%. To find the monthly interest rate, divide the annual rate by 12. Therefore, the monthly interest rate is \( \frac{9.75\%}{12} \) or \( \frac{0.0975}{12} = 0.008125 \). The total number of payments over 30 years is \( 30 \times 12 = 360 \) payments.

Step 2: Use the Amortization Formula to Calculate Monthly Payment (Part a)

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

Step 3: Calculate Total Amount Paid Over 30 Years (Part b)

Multiply the monthly payment by the number of total payments: \( M \times 360 \). This gives the total amount paid over the life of the loan.

Step 4: Compound Savings Calculation if Deposited (Part c)

For part (c), use the future value of an annuity formula which is:\[ FV = P \frac{(1 + r)^n - 1}{r} \]where \( P = M \) (the monthly payment from part (a)), \( r = 0.008125 \) (the monthly interest rate), and \( n = 360 \) (total number of payments). Calculate the future value \( FV \) to see how much will be in the account.

Key Concepts

Mortgage Calculation

When you buy a house, you might take out a mortgage. A mortgage is a type of loan specifically for purchasing real estate. To find out your monthly payment and how much you'll pay in total, you need to do some calculations.

The key tool for this is the amortization formula. This formula helps you figure out how much you need to pay each month to pay off the loan including interest. Here’s how it looks:

• \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]

To use it, you need:

• \( P \): the principal amount (the size of the loan);
• \( r \): the monthly interest rate, which is your annual rate divided by 12;
• \( n \): the total number of monthly payments.

If you have a $100,000 loan at an annual interest rate of 9.75%, you divide that rate by 12 to get your monthly rate. Over 30 years, you'll make 360 payments. Plugging these numbers into the formula tells you how much to pay each month. Then, multiply that monthly payment by the number of months to see how much you'll pay in total.

Compound Interest

Compound interest refers to interest calculated on the initial principal and also on the accumulated interest of previous periods. Unlike simple interest where only the principal earns interest, compound interest grows over time as it adds interest on interest.

With compounding, each month you have more money accruing interest. This can significantly increase your savings or the cost of borrowing over long periods.

For example, in a mortgage, the interest compounds monthly. So, each month's interest calculation adds to your total loan balance, making the next month's interest slightly higher than the first month.

• For loans, this means paying more over time, which makes knowing your monthly payments crucial.
• For savings, it's beneficial because your money grows faster the longer you leave it to compound.

Understanding compound interest can help you make better financial decisions, whether you're saving or borrowing.

Future Value of an Annuity

The future value of an annuity is how much a series of regular payments will be worth at a future date, taking interest into account. This is important if you’re saving for the future. Or, like in your exercise, finding out how much you’d save if you invested instead of took out a loan.

The formula to find this looks like this:

• \[ FV = P \frac{(1 + r)^n - 1}{r} \]

The letters stand for:

• \( FV \): the future value;
• \( P \): the regular payment amount each period;
• \( r \): the interest rate per period (monthly);
• \( n \): the total number of payments.

If you were to take your monthly mortgage payment and instead of paying off a loan, placed it into an account earning the same interest, you could see how much you'd have after 30 years. By using this formula, you can determine whether saving would be more beneficial than borrowing in the long run.