To calculate the monthly payment for a mortgage, we use the loan amortization formula:

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

Where:

• $M$ = monthly payment
• $P$ = loan amount (principal)
• $r$ = monthly interest rate ($\text{annual interest rate} \div 12$)
• $n$ = total number of payments ($\text{loan term in years} \times 12$)

Step-by-Step Calculation:

1. Given values:

   • $P = 219,000$
   • Annual interest rate = $7\% = 0.07$
   • Loan term = $15 \, \text{years}$
   • Monthly interest rate: $r = \frac{0.07}{12} = 0.0058333$
   • Total number of payments: $n = 15 \times 12 = 180$

2. Substitute into the formula: $M = 219,000 \cdot \frac{0.0058333(1 + 0.0058333)^{180}}{(1 + 0.0058333)^{180} - 1}$

3. Simplify:

   • $(1 + r)^{180} = (1.0058333)^{180} \approx 3.058921$
   • Numerator: $0.0058333 \cdot 3.058921 \approx 0.017838$
   • Denominator: $3.058921 - 1 = 2.058921$

   So: $M = 219,000 \cdot \frac{0.017838}{2.058921}$

4. Final Calculation:

   • $\frac{0.017838}{2.058921} \approx 0.008667$
   • $M = 219,000 \cdot 0.008667 \approx 1,897.07$

Final Answer:

The monthly payment is approximately $1,897.07.

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Follow-up Questions:

1. How does the monthly payment change if the interest rate is reduced to 6%?
2. What is the total interest paid over the 15 years?
3. How would the monthly payment differ if the loan term were 30 years instead of 15 years?
4. What is the balance of the mortgage after 5 years of payments?
5. How much of the first payment goes toward interest versus principal?

Tip:

Always ensure the interest rate is converted to a monthly rate and the loan term to months when using the amortization formula!