This cheat sheet covers the basic terms and calculations used when buying a home with a mortgage. Students need these skills to understand loan offers, compare monthly payments, and estimate the true cost of borrowing. Mortgage math connects percentages, exponents, budgeting, and long-term financial planning.
It also helps students recognize how small changes in rates or loan length can create large differences in total cost.
The core ideas are principal, interest rate, loan term, monthly payment, escrow, and amortization. The standard fixed-rate mortgage payment formula is M = P[r(1 + r)^n] / [(1 + r)^n - 1], where P is the loan amount, r is the monthly interest rate, and n is the number of monthly payments. Total paid equals monthly payment times number of payments, and total interest equals total paid minus principal.
A complete housing budget should also include property taxes, homeowners insurance, possible PMI, HOA fees, maintenance, and closing costs.
Key Facts
- Loan amount equals home price minus down payment, so P = price - down payment.
- Monthly interest rate equals annual interest rate divided by 12, so r = APR / 12 when APR is written as a decimal.
- Number of payments equals years times 12, so n = loan term in years x 12.
- For a fixed-rate mortgage, the monthly principal and interest payment is M = P[r(1 + r)^n] / [(1 + r)^n - 1].
- Total paid over the loan equals M x n, and total interest equals total paid - P.
- A larger down payment lowers the loan amount and may help the buyer avoid private mortgage insurance.
- Escrow payments usually include property taxes and homeowners insurance, so monthly housing cost is more than principal and interest.
- Closing costs are often about 2% to 6% of the home price and are paid at closing unless rolled into the loan.
Vocabulary
- Mortgage
- A loan used to buy real estate, with the property serving as collateral for the lender.
- Principal
- The amount of money borrowed or the remaining loan balance before interest is added.
- Interest
- The cost of borrowing money, usually shown as a yearly percentage rate.
- Amortization
- The process of paying off a loan through scheduled payments that gradually reduce the principal balance.
- Escrow
- An account used by the lender to collect and pay costs such as property taxes and homeowners insurance.
- Private Mortgage Insurance
- Insurance often required when the down payment is less than 20%, protecting the lender if the borrower defaults.
Common Mistakes to Avoid
- Using the annual interest rate directly in the monthly payment formula is wrong because the formula requires the monthly rate, r = APR / 12.
- Forgetting to convert the loan term into months is wrong because a 30-year mortgage has n = 30 x 12 = 360 payments, not 30 payments.
- Comparing only monthly payments is misleading because a longer loan may have a lower payment but much higher total interest.
- Ignoring taxes, insurance, PMI, and HOA fees gives an unrealistically low housing budget because principal and interest are only part of the monthly cost.
- Treating the down payment as the only upfront cost is incorrect because buyers often also need closing costs, moving costs, and emergency savings.
Practice Questions
- 1 A home costs $320,000 and the buyer makes a 10% down payment. What is the loan amount?
- 2 A borrower takes a $250,000 fixed-rate mortgage at 6% annual interest for 30 years. Using r = 0.06 / 12 and n = 360, estimate the monthly principal and interest payment with M = P[r(1 + r)^n] / [(1 + r)^n - 1].
- 3 A mortgage payment for principal and interest is 275,000 loan. What is the total paid, and how much is total interest?
- 4 Explain why a buyer should compare total interest, escrow costs, and closing costs instead of choosing a mortgage only by the lowest monthly payment.
Understanding Mortgage Basics & Calculations
A mortgage payment changes internally over time even when a fixed rate keeps the required principal and interest payment steady. This pattern is called amortization. Each month, the lender first calculates interest from the unpaid loan balance.
The rest of the payment reduces the balance. Early in the loan, the balance is large, so interest takes up most of each payment. Later, less interest is due and more of the same payment goes toward principal.
An amortization schedule lists this month by month. It shows why building ownership in a home can feel slow at first.
The interest rate is not the only number to inspect in a loan offer. The annual percentage rate, often called APR, can include certain lender fees as well as interest. It can help compare loans with similar terms, but students should read the details behind it.
Some loans have a fixed rate for the full term. Others have an adjustable rate that can change after an introductory period.
A low starting payment on an adjustable loan may rise later if market rates rise. Loan documents state how often the rate may change and the largest allowed increase.
A down payment affects more than the amount borrowed. It changes the buyer's starting equity, which is the share of the home value owned without debt. For example, a buyer who puts down twenty percent begins with more equity than a buyer who puts down five percent.
A smaller down payment can make buying possible sooner, yet it usually means a larger balance, more interest over time, and possibly private mortgage insurance. Home values can fall as well as rise. If a home loses value soon after purchase, a buyer with little equity may owe more than the home could sell for.
The amount sent to the lender each month may change even on a fixed-rate mortgage. Taxes and insurance are often collected through escrow. The lender estimates these bills, divides them across the year, then holds the money until payment is due.
If property taxes or insurance premiums increase, the escrow part of the monthly bill can increase. Buyers should separate the fixed loan payment from the total housing payment when making a budget.
They should leave room for repairs, utilities, moving costs, and savings. A mortgage lender may approve a payment that is technically affordable on paper but leaves too little flexibility in real life.
Closing is the point when the buyer signs final documents and receives ownership. Before this date, the lender provides a closing disclosure that lists the loan terms, cash needed, and fees. Students should compare it carefully with the earlier loan estimate.
Fees may include appraisal work, title services, government recording charges, prepaid insurance, and initial escrow deposits. Some costs are paid once, while others continue every year. Paying discount points is another choice.
A point is an upfront fee that can reduce the interest rate. It may save money only if the buyer keeps the mortgage long enough for the monthly savings to exceed the upfront cost.