Welcome to Section 4-2! Today, we are tackling a subject that impacts almost everyone: Borrowing. Whether you are planning to buy a car, saving for a house, or taking out a student loan, understanding the math behind the debt is the most powerful tool in your financial toolkit. In this lesson, we move beyond simple interest and dive into installment loans.

The Three Pillars of a Loan

When you take out an installment loan (like a car loan or a mortgage), three main factors determine how much you will pay each month:

  • Principal ($P$): The amount of money you are borrowing.
  • APR (Annual Percentage Rate): The interest rate you are charged per year.
  • Term ($t$): The length of time you have to pay back the loan (usually calculated in months).

The Monthly Payment Formula

Have you ever wondered how the bank figures out exactly what you owe? They use the Monthly Payment Formula. It looks complex, but we can break it down:

$$\text{Monthly Payment} = \frac{P \cdot r(1+r)^t}{(1+r)^t - 1}$$

Here is the trick to using this formula correctly:

  1. $t$ (time): Must be in months. If you have a 3-year car loan, $t = 36$.
  2. $r$ (rate): This is the monthly interest rate as a decimal. You calculate this by taking the APR and dividing by 12. For example, if your APR is 6%, $r = \frac{0.06}{12} = 0.005$.

How Much Can You Afford?

Sometimes you need to work backward. If you know you can afford exactly $250 a month for a car payment, and you know the interest rate, how expensive of a car can you buy? We use the Companion Formula (Borrowing Power):

$$\text{Amount Borrowed} = \frac{\text{Payment} \cdot ((1+r)^t - 1)}{r(1+r)^t}$$

Using the example from our class notes: If you can afford $250/month for 3 years at 5% APR, the math tells us you should shop for a car priced at $8,341.43 or less. This prevents you from overextending your budget!

Amortization and Equity

When you make a loan payment, not all of it goes toward paying back what you borrowed. A portion goes to interest, and the rest goes to the principal.

  • Amortization Schedule: A table that shows exactly how much of each payment goes to interest vs. principal. In the beginning of a loan, you pay mostly interest. By the end, you pay mostly principal.
  • Equity: This is the value of the item that you actually "own." It is calculated as:
    Equity = (Current Value of Item) - (Amount Remaining on Loan).

Mortgages: Fixed vs. ARM

Finally, we touch on home loans. A Fixed-Rate Mortgage locks in your interest rate for the life of the loan (e.g., 30 years). An Adjustable-Rate Mortgage (ARM) has an interest rate that can change. While ARMs might start cheaper, they can become much more expensive if rates rise!

Class Assignment

Please download the attached PDF assignment to practice these calculations. You will need to calculate monthly payments, total interest paid, and analyze amortization tables. Good luck!