Mortgage Payment Formula

The basic mortgage payment formula is:

$$ M = P \\frac{r(1 + r)^n}{(1 + r)^n - 1} $$

• \(M\) = Monthly payment
• \(P\) = Principal loan amount (home price - down payment)
• \(r\) = Monthly interest rate (annual rate divided by 12)
• \(n\) = Total number of payments (loan term in years multiplied by 12)

Additional Mortgage-Related Formulas

• Total Interest Paid: \( \text{Total Interest} = M \times n - P \)
• Total Principal and Interest: \( \text{Principal + Interest} = M \times n \)
• Monthly Cost with Extra Payments: \( M_{\text{total}} = M + \text{Monthly Extra Costs} \)

Advanced Mortgage Math Problems

1. Problem 1

   Question: Calculate the monthly mortgage payment for a $300,000 home with a $60,000 down payment, a 4% annual interest rate, and a 30-year term.

   Solution:

   Given:

   • Home Price (\(P\)) = $300,000 - $60,000 = $240,000
   • Annual Interest Rate = 4%
   • Monthly Interest Rate (\(r\)) = 4% / 12 = 0.003333
   • Term (\(n\)) = 30 years × 12 months = 360 months

   Using the formula \( M = P \\frac{r(1 + r)^n}{(1 + r)^n - 1} \):

   $$ M = 240,000 \\frac{0.003333(1 + 0.003333)^{360}}{(1 + 0.003333)^{360} - 1} $$

   Monthly Payment, \(M\) ≈ $1,145.80

2. Problem 2

   Question: For a $500,000 home with a $100,000 down payment, 3.5% interest rate, and a 20-year term, calculate the total interest paid over the loan term.

   Solution:

   Principal = $500,000 - $100,000 = $400,000

   Monthly Payment = Using the formula (calculated value: $2,312.87)

   Total Interest = Monthly Payment * Term (240) - Principal

   Total Interest ≈ $154,088

3. Problem 3

   Question: How much would the monthly payment be for a $750,000 loan at 5% interest over 25 years with an additional $200/month HOA fee?

   Solution:

   Monthly mortgage payment = $4,386.16, plus HOA = $4,586.16

4. Problem 4

   Question: For a $400,000 home loan with 3% interest, calculate the payoff date if $100/month extra payments are added.

   Using amortization calculations:

   New Term ≈ 25.6 years

5. Problem 5

   Question: A $350,000 home loan has a 4.5% annual interest rate with a 30-year term. Calculate the total interest paid if $150 extra is paid each month.

   Solution:

   Given:

   • Principal = $350,000
   • Monthly Interest Rate (\(r\)) = 4.5% / 12 = 0.00375
   • Extra Monthly Payment = $150

   Using an amortization formula with the extra payments applied, the new term is approximately 24.9 years.

   Total Interest ≈ $286,300

6. Problem 6

   Question: For a $600,000 mortgage at 5% interest, calculate the monthly payment and total amount paid if the loan term is 25 years.

   Solution:

   Given:

   • Principal = $600,000
   • Monthly Interest Rate (\(r\)) = 5% / 12 = 0.004167
   • Term (\(n\)) = 25 years × 12 months = 300 months

   Using the formula \( M = P \\frac{r(1 + r)^n}{(1 + r)^n - 1} \):

   Monthly Payment \(M\) ≈ $3,510.60

   Total Payment = \( M \times n \) ≈ $1,053,180

7. Problem 7

   Question: Calculate the impact of a one-time payment of $20,000 made after 5 years on a $500,000 mortgage at 4% interest with a 30-year term.

   Solution:

   With an amortization schedule, the extra payment reduces the term by approximately 1.8 years.

8. Problem 8

   Question: For a $200,000 loan at 3.75% interest over 15 years, calculate the monthly payment and total amount paid.

   Solution:

   Given:

   • Principal = $200,000
   • Monthly Interest Rate (\(r\)) = 3.75% / 12 = 0.003125
   • Term (\(n\)) = 15 years × 12 months = 180 months

   Using the formula \( M = P \\frac{r(1 + r)^n}{(1 + r)^n - 1} \):

   Monthly Payment \(M\) ≈ $1,454.44

   Total Payment = \( M \times n \) ≈ $261,799.20

9. Problem 9

   Question: What is the total interest paid on a $400,000 mortgage with a 6% interest rate if the loan is for 15 years?

   Solution:

   Given:

   • Principal = $400,000
   • Monthly Interest Rate (\(r\)) = 6% / 12 = 0.005
   • Term (\(n\)) = 15 years × 12 months = 180 months

   Monthly Payment \(M\) ≈ $3,376.60

   Total Interest = \( M \times n - P \) ≈ $208,788

10. Problem 10

    Question: A $450,000 mortgage at 5% interest over 25 years has an optional annual extra payment of $5,000. Calculate the payoff time reduction.

    Solution:

    Using amortization with the annual extra payment, the term is reduced by approximately 5 years, resulting in a 20-year term.