Mortgage Payment Formula

The basic mortgage payment formula is:

$$\begin{gathered} M=P \\ fracr(1+r{)}^n(1+r{)}^n-1 \end{gathered}$$

• $M$ = 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 $300,000homewitha$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 $$\begin{gathered} M=P \\ fracr(1+r{)}^n(1+r{)}^n-1 \end{gathered}$$:

   $$\begin{gathered} M=240,000 \\ frac0.003333(1+0.003333{)}^{360}(1+0.003333{)}^{360}-1 \end{gathered}$$

   Monthly Payment, $M$ ≈ $1,145.80

2. Problem 2

   Question: For a $500,000homewitha$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 $750,000loanat5$200/month HOA fee?

   Solution:

   Monthly mortgage payment = $4,386.16,plusHOA=$4,586.16

4. Problem 4

   Question: For a $400,000homeloanwith3$100/month extra payments are added.

   Using amortization calculations:

   New Term ≈ 25.6 years

5. Problem 5

   Question: A $350,000homeloanhasa4.5$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 $$\begin{gathered} M=P \\ fracr(1+r{)}^n(1+r{)}^n-1 \end{gathered}$$:

   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 $20,000madeafter5yearsona$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 $$\begin{gathered} M=P \\ fracr(1+r{)}^n(1+r{)}^n-1 \end{gathered}$$:

   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 $450,000mortgageat5$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.