Chapter 4

Loans

Short- and long-term loans — auto loans, credit cards, student loans, and mortgages — plus an APA-style written analysis. The heaviest week of the course.

This week is loans. The flip side of savings: instead of money growing for you, money is growing against you on a balance you owe. By Sunday you'll be able to read a monthly payment, total cost, and total interest for any fixed-rate loan — auto, credit card, student, or mortgage.

The central formula is the amortization formula: M = P(r/12) / (1 − (1+r/12)^−12t). Once you can read it, every loan in your life becomes a known quantity. Topic 3 was the math of growth; Topic 4 is the math of paying it back.

By the end of this chapter, you'll be able to…

4.1Compute monthly payments and interest costs of short-term loans such as auto loans or credit cards.
4.2Find the interest, the balance due, and the minimum monthly payment for loans.
4.3Analyze the various aspects of long-term loans such as student loans and home mortgages.

Sections

Start with §4.1 →

Chapter summary

Formulas, key concepts, and the kind of one-page reference you'd want during a problem set.

This chapter is the math of money growing against you: the flip side of savings. You will learn the amortization formula, then walk through the four loan types every adult faces — auto loans, credit cards, student loans, and mortgages. By the end you can read a monthly payment, total cost, and total interest for any fixed-rate loan, and you will understand why a 30-year mortgage at 7% APR pays more interest than the loan itself.

Chapter glossary

All key terms introduced across this chapter, in the order they appear in the reading.

Principal (P): The amount you borrowed — the loan amount. Sticker price minus any down payment.
Annual rate (r): The annual interest rate as a decimal. 6.5% becomes 0.065. The amortization formula uses r/12 internally to convert to a monthly periodic rate.
Term (t): The length of the loan in years. Auto loans are usually 3-7; mortgages are 15 or 30; credit cards have no fixed term.
Monthly payment (M): The fixed amount you pay each month. The amortization formula in Lesson 2 computes this from the other three pieces.
Amortize: To pay off a loan in fixed payments over time. Each payment is interest first, principal second.
Loan amortization formula: The expression above. The only formula in the ALEKS dictionary not introduced in T3.
Periodic rate (r/12): The monthly interest rate. Annual rate divided by 12 because there are 12 monthly compounding periods per year.
Total payments (12t): The total number of monthly payments over the life of the loan. A 5-year auto loan has 60 payments; a 30-year mortgage has 360.
Negative exponent: (1 + r/12)^−12t = 1 / (1 + r/12)^12t. Most calculators handle the minus sign correctly; in Excel the syntax is ^(-12*t).
Total cost / total interest: Total cost of a loan = M × 12 × t (every payment summed). Total interest = total cost − principal. Both grow fast as t increases.
Auto loan: A fixed-term loan to buy a vehicle. Term typically 3-7 years; rate depends on credit score and current rates.
Down payment: Money paid up front. Reduces the principal you finance and the monthly payment that results. Usually 10-20% of sticker price.
Sticker price (total cost): The full price of the vehicle. Equals down payment plus the loan amount.
Amount financed (P): The principal of the loan. Sticker price minus down payment. This is the P that goes into the amortization formula.
Term length: The loan duration in years. Longer term means lower monthly payment but more total interest paid.
Revolving credit: A loan with no fixed term — you can borrow more, pay down, repeat. Credit cards and home-equity lines of credit are the common examples.
APR (credit card): The annual rate the card charges on unpaid balances. Typical range: 18-29%. Divide by 12 for the monthly periodic rate.
Minimum payment: The smallest payment the card issuer requires each month. Usually 1-3% of balance, plus the interest charged that month. Pays the loan off agonizingly slowly.
Interest portion: The part of your payment that covers the month's interest charge. Computed first; what's left goes to principal.
Principal portion: The part of your payment that actually reduces the balance. Equals payment − interest portion.
Subsidized loan: Federal loan where the government pays interest during in-school years and deferment. Only available to students with demonstrated financial need.
Unsubsidized loan: Federal loan where interest accrues from disbursement. Available regardless of need.
Capitalization: Accrued in-school interest gets added to the principal at the start of repayment, increasing the amount you'll pay back. ALEKS uses simple interest (I = Prt) for the in-school accrual; real federal loans accrue daily.
Deferment: A pause in repayment (e.g. while in school). Subsidized loans don't accrue interest during deferment; unsubsidized loans do.
Standard vs Income-Driven Repayment: Standard: 10-year fixed term, computed by the L2 formula. IBR (income-driven): monthly payment is a percent of discretionary income, term up to 25 years, with possible forgiveness of remaining balance.
Mortgage: A long-term loan secured by real estate. Standard terms are 15 or 30 years; the home itself is collateral.
Principal/interest split: How each monthly payment breaks down. Front-loaded toward interest in early years, back-loaded toward principal in later years.
Amortization schedule: A row-by-row table of every payment, the interest portion, the principal portion, and the remaining balance. The full visual record of how the loan gets paid off.
Escrow: An account the lender holds to pay your property taxes and homeowner's insurance. You pay 1/12 of the annual amount each month as part of your mortgage payment.
PITI: Principal + Interest + Taxes + Insurance — the four components of a typical mortgage payment. When people say "my mortgage" they usually mean PITI.
Refinancing: Replacing an existing mortgage with a new one, usually at a lower rate. Resets the amortization clock.