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Mortgage — Definition, Formula & Examples

A mortgage is a loan used to buy a home or property, which you repay in equal monthly payments over a fixed number of years. Each payment covers that month's interest and pays off part of the amount you borrowed.

A mortgage is an amortized loan secured by real estate: if the borrower stops paying, the lender can take the property. With a fixed rate, the monthly payment MM is chosen so that the present value of all nn payments, discounted at the periodic rate rr, equals the principal PP. Mathematically, the payment stream is an ordinary annuity whose present value is the loan amount.

Key Formula

M=P⋅r(1+r)n(1+r)n−1M = P\cdot\frac{r(1+r)^n}{(1+r)^n - 1}
Where:
  • MM = Fixed monthly payment
  • PP = Principal: the amount borrowed (home price minus down payment)
  • rr = Monthly interest rate as a decimal (annual rate ÷ 12)
  • nn = Total number of monthly payments (years × 12)

How It Works

Start with the amount borrowed (the principal), the annual interest rate, and the length of the loan in years. Divide the annual rate by 12 to get the monthly rate rr. Multiply the years by 12 to get the number of payments nn. Substitute these into the payment formula to find the fixed monthly payment MM. Each month, the interest due is rr times the current balance. Whatever is left of MM after that interest reduces the balance, so early payments are mostly interest and later payments are mostly principal.

Worked Example

Problem: You take out a $200,000 mortgage at a fixed annual rate of 6% for 30 years. Find the monthly payment, the total interest paid, and how the first payment is split.
Step 1: Convert the annual rate to a monthly rate. Convert the years to a number of monthly payments.
r=0.0612=0.005,n=30×12=360r = \frac{0.06}{12} = 0.005, \qquad n = 30 \times 12 = 360
Step 2: Compute the growth factor (1+r)n(1+r)^n with a calculator.
(1.005)360≈6.022575(1.005)^{360} \approx 6.022575
Step 3: Substitute into the payment formula. Note that P⋅r=200,000×0.005=1,000P \cdot r = 200{,}000 \times 0.005 = 1{,}000.
M=1000×6.0225756.022575−1=6022.5755.022575≈1199.10M = \frac{1000 \times 6.022575}{6.022575 - 1} = \frac{6022.575}{5.022575} \approx 1199.10
Step 4: Find the total paid over 360 months. Then subtract the principal to get the total interest.
1199.10×360≈431,676,431,676−200,000=231,6761199.10 \times 360 \approx 431{,}676, \qquad 431{,}676 - 200{,}000 = 231{,}676
Step 5: Split the first payment. The interest is rr times the starting balance, and the rest goes to principal.
Interest=0.005×200,000=1000,Principal=1199.10−1000=199.10\text{Interest} = 0.005 \times 200{,}000 = 1000, \qquad \text{Principal} = 1199.10 - 1000 = 199.10
Answer: The monthly payment is about $1,199.10. The total interest over 30 years is about $231,676. In the first payment, $1,000 is interest and only $199.10 reduces the balance.

Another Example

This example adds a down payment, so you must find the principal first. It also shortens the term. The payment rises by less than half, but the total interest falls by more than half.

Problem: A house costs $250,000. You make a 20% down payment and borrow the rest at 6% for 15 years. Find the monthly payment and compare the total interest with the 30-year loan from the first example.
Step 1: Find the down payment and subtract it from the price to get the principal.
0.20×250,000=50,000,P=250,000−50,000=200,0000.20 \times 250{,}000 = 50{,}000, \qquad P = 250{,}000 - 50{,}000 = 200{,}000
Step 2: The monthly rate is still 0.005, but now there are only 180 payments.
n=15×12=180,(1.005)180≈2.454094n = 15 \times 12 = 180, \qquad (1.005)^{180} \approx 2.454094
Step 3: Substitute into the formula.
M=1000×2.4540942.454094−1=2454.0941.454094≈1687.71M = \frac{1000 \times 2.454094}{2.454094 - 1} = \frac{2454.094}{1.454094} \approx 1687.71
Step 4: Find the total paid and the total interest.
1687.71×180≈303,788,303,788−200,000=103,7881687.71 \times 180 \approx 303{,}788, \qquad 303{,}788 - 200{,}000 = 103{,}788
Step 5: Compare with the 30-year interest from the first example.
231,676−103,788=127,888231{,}676 - 103{,}788 = 127{,}888
Answer: The 15-year payment is about $1,687.71 per month. Total interest is about $103,788, which saves roughly $127,888 compared with the 30-year loan.

Visualization

Why It Matters

Mortgage calculations appear in Algebra 2 and precalculus units on exponential functions and geometric series. The payment formula comes from summing a geometric series. Loan officers, real estate agents, financial planners, and actuaries use these calculations daily. Buying a home is usually the largest debt a person ever takes on.

Common Mistakes

Mistake: Plugging the annual rate (0.06) and the number of years (30) directly into the formula.
Correction: Payments are monthly, so the rate and the count must be monthly too. Use r=0.06/12=0.005r = 0.06/12 = 0.005 and n=360n = 360.
Mistake: Estimating interest with simple interest: 200,000×0.06×30=360,000200{,}000 \times 0.06 \times 30 = 360{,}000.
Correction: That assumes you owe the full $200,000 for all 30 years. Your balance shrinks every month, so the actual interest is about $231,676. Use the amortization formula instead.
Mistake: Assuming that halving the term doubles the payment.
Correction: Twice $1,199.10 is $2,398.20, but the actual 15-year payment is $1,687.71. A shorter loan builds up less interest, so the payment grows by much less than double.

Check Your Understanding

A 20-year mortgage has an annual rate of 4.8%. What are rr and nn?
Hint: Divide the annual rate by 12 and multiply the years by 12.
Answer: r=0.048÷12=0.004r = 0.048 \div 12 = 0.004 and n=20×12=240n = 20 \times 12 = 240.
You borrow $150,000 at 6% per year. How much interest is charged in the first month?
Hint: Use the monthly rate times the starting balance.
Answer: $750, because 0.005×150,000=7500.005 \times 150{,}000 = 750.
You pay $1,000 a month for 30 years on a $150,000 loan. What is the total interest?
Hint: Total paid minus amount borrowed.
Answer: $210,000, because 1000×360=360,0001000 \times 360 = 360{,}000 and 360,000−150,000=210,000360{,}000 - 150{,}000 = 210{,}000.

Related Terms

  • Amortization — Process of paying a mortgage down over time
  • Annuity — Equal periodic payments; a mortgage is one
  • Principal — The amount borrowed on the mortgage
  • Interest — The cost of borrowing the principal
  • Compound Interest — Growth model behind the factor (1 + r)^n
  • Present Value — Loan amount equals present value of payments
  • Future Value — Opposite viewpoint used for savings plans
  • Per Annum — Mortgage rates are quoted per year
  • Semiannually — Compounding used for Canadian fixed-rate mortgages