I took out a mortgage for $300,000 to buy a home. The bank charges me an annual interest rate of 5.2% computed monthly on the loan balance. If it takes me 30 years to pay off my loan, how much is my monthly payment?

To calculate your monthly mortgage payment, we can use the **loan amortization formula** for fixed-rate mortgages. The formula is:

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

Where:
- $M$ = monthly payment
- $P$ = loan principal (amount borrowed) = \$300,000
- $r$ = monthly interest rate (annual interest rate divided by 12)
- $n$ = total number of monthly payments (loan term in years × 12)

### Step 1: Calculate the monthly interest rate
The annual interest rate is 5.2%, so the monthly interest rate is:

\[
r = \frac{5.2\%}{12} = \frac{0.052}{12} = 0.004333
\]

### Step 2: Calculate the total number of payments
Since the loan term is 30 years, the number of payments is:

\[
n = 30 \times 12 = 360 \text{ months}
\]

### Step 3: Apply the values to the formula
Substitute $P = 300,000$, $r = 0.004333$, and $n = 360$ into the formula:

\[
M = \frac{300,000 \cdot 0.004333(1 + 0.004333)^{360}}{(1 + 0.004333)^{360} - 1}
\]

Now we compute the value step-by-step:

\[
M = \frac{300,000 \cdot 0.004333 \cdot (1.004333)^{360}}{(1.004333)^{360} - 1}
\]

Using a calculator, we find:
\[
(1.004333)^{360} \approx 4.4677
\]
Thus:
\[
M = \frac{300,000 \cdot 0.004333 \cdot 4.4677}{4.4677 - 1}
\]

Now simplify further:
\[
M = \frac{300,000 \cdot 0.01934}{3.4677} \approx \frac{5,802.46}{3.4677} \approx 1,673.75
\]

### Final Answer:
Your monthly mortgage payment is approximately **$1,673.75**.

Would you like further details or clarification on any step?

Here are 5 related questions for practice:
1. What happens to the monthly payment if the interest rate increases to 6%?
2. How much total interest will you pay over the life of the loan?
3. If you made an extra payment of $200 every month, how much would you save in interest?
4. How much is the outstanding loan balance after 10 years of payments?
5. What is the impact of reducing the loan term to 20 years with the same interest rate?

**Tip**: Making extra payments towards the principal can significantly reduce the total interest paid over the life of the loan.

Learn how to calculate the monthly mortgage payment for a $300,000 home loan with a 5.2% annual interest rate, compounded monthly over a 30-year term. This example explains key concepts like loan amortization, interest rates, and the formula used for monthly payments, providing a step-by-step solution.

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