Let's break down your mortgage problem: ## 1. Mortgage Details - **Principal (\(P\))**: \$382,000 - **Annual Interest Rate (APR, \(r_{annual}\))**: 7.2% - **Monthly Interest Rate (\(r\))**: \( \frac{7.2\%}{12} = 0.6\% = 0.006 \) - **Loan Term (\(n\))**: 30 years × 12 months/year = 360 months --- ## 2. Monthly Payment Formula The monthly payment \(M\) for a fully amortizing loan: \[ M = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Plug in the values: - \(P = 382,\!000\) - \(r = 0.006\) - \(n = 360\) --- ## 3. Monthly Payment Calculation \[ M = 382,\!000 \cdot \frac{0.006(1 + 0.006)^{360}}{(1 + 0.006)^{360} - 1} \] Calculate \((1 + 0.006)^{360}\): \[ (1.006)^{360} \approx 8.138 \] Plug in: \[ M = 382,\!000 \cdot \frac{0.006 \times 8.138}{8.138 - 1} \] \[ = 382,\!000 \cdot \frac{0.048828}{7.138} \] \[ = 382,\!000 \cdot 0.006842 \] \[ \approx \$2,\!613.64 \] **Monthly Payment:** \[ \boxed{\$2,613.64} \] --- ## 4. Amortization Schedule (Structure) Each month, you pay: - **Interest:** Previous balance × monthly rate - **Principal:** Monthly payment - Interest - **Ending balance:** Previous balance - Principal paid **First 3 months as example:** | Month | Payment | Interest | Principal | Balance | |-------|----------|----------|-----------|-------------| | 1 | 2,613.64 | 2,292.00 | 321.64 | 381,678.36 | | 2 | 2,613.64 | 2,290.07 | 323.57 | 381,354.79 | | 3 | 2,613.64 | 2,288.13 | 325.51 | 381,029.28 | | ... | ... | ... | ... | ... | | 360 | 2,613.64 | 15.60 | 2,598.04 | 0.00 | --- ## 5. Full Amortization Schedule **To generate all 360 months:** You must use a spreadsheet (Excel, Google Sheets, etc.) as it is too large to write here. ### How to Create in Excel/Sheets **Columns:** - Month (1 to 360) - Payment (constant: \$2,613.64) - Interest = Previous Balance × 0.006 - Principal = Payment - Interest - Balance = Previous Balance - Principal **First row:** - Previous Balance = \$382,000 - Use formulas for each column as above - Drag down for 360 rows **Template:** | A | B | C | D | E | |-------|----------|----------|-----------|-----------| | Month | Payment | Interest | Principal | Balance | | 1 | 2613.64 | =E1*0.006| =B2-C2 | =E1-D2 | | ... | ... | ... | ... | ... | **Note:** - E1 is your starting principal (\$382,000). - Adjust cell references as needed for your software. --- ## 6. File Upload I'm unable to upload files directly. However, you can **copy the above structure into Excel or Google Sheets** and drag down the formulas to generate the full schedule. If you need a downloadable template, you must create it using the template above. If you encounter issues, let me know your preferred file format and I can provide step-by-step instructions or a CSV you can copy-paste.

# Introduction A mortgage loan is a type of amortized loan that allows individuals to finance the purchase of real estate through periodic payments over an extended period. The key features of such loans include the principal amount borrowed, the interest rate applied, and the repayment schedule. The annual percentage rate (APR) reflects the yearly cost of borrowing, including interest and other fees, expressed as a percentage. When the interest is compounded monthly, the periodic interest rate is derived by dividing the APR by 12. Understanding how to calculate the monthly mortgage payment involves applying the amortization formula, which accounts for the principal, interest rate, and loan duration. The amortization schedule provides a detailed breakdown of each payment, illustrating how the interest and principal components evolve over time until the loan is fully repaid. --- ## Explanation This introduction is crucial because it lays down the fundamental concepts necessary for understanding the subsequent calculations. Recognizing that a mortgage is an amortized loan helps in framing the problem—specifically, how payments are structured to gradually reduce the principal while covering interest. The mention of the APR and monthly compounding establishes the basis for calculating the periodic interest rate, which directly influences the payment amount. Additionally, grasping the purpose of the amortization schedule is essential for understanding how each payment contributes to reducing the outstanding loan balance over the loan term, thus providing a comprehensive view of the mortgage repayment process. --- # Presentation of Relevant Formulas Required To Solve The Question ### 1. Monthly Interest Rate \[ r = \frac{\text{APR}}{12} \] - **Description:** Converts the annual interest rate (APR) into a monthly interest rate, necessary because payments are made monthly. - **Relevance:** This rate is used in the amortization formula to determine interest accrued each month. --- ### 2. Amortization Payment Formula \[ M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \] - **Description:** Calculates the fixed monthly payment \(M\) needed to fully amortize a loan of principal \(P\) over \(n\) months with a monthly interest rate \(r\). - **Relevance:** Enables precise calculation of the monthly payment amount required to pay off the loan within the specified term. ### 3. Interest Portion of Each Payment \[ \text{Interest}_k = \text{Previous Balance} \times r \] - **Description:** Computes the interest component for the \(k^{th}\) payment based on the outstanding balance before the payment. - **Relevance:** Used to determine how much of each payment goes toward interest versus principal reduction. ### 4. Principal Portion of Each Payment \[ \text{Principal}_k = M - \text{Interest}_k \] - **Description:** The amount of the payment that reduces the principal balance after interest is accounted for. - **Relevance:** Shows the portion of each payment contributing to decreasing the loan balance. ### 5. Remaining Balance After Each Payment \[ \text{New Balance} = \text{Previous Balance} - \text{Principal}_k \] - **Description:** Updates the outstanding loan balance after each payment. - **Relevance:** Critical for constructing the amortization schedule and understanding the loan payoff process. --- ## Explanation Each formula serves a specific purpose in the process of calculating mortgage payments and constructing the amortization schedule. The conversion of APR to a monthly rate ensures interest calculations align with payment frequency. The amortization formula provides the fixed payment amount that guarantees the loan is paid off at the end of the term, considering compound interest. The interest, principal, and balance formulas work together iteratively to break down each payment, illustrating the evolving nature of loan repayment over time. These mathematical tools collectively facilitate a detailed understanding and precise calculation of mortgage amortization. --- # Step-by-Step Solution ### Step 1: Convert APR to Monthly Interest Rate \[ r = \frac{7.2\%}{12} = \frac{0.072}{12} = 0.006 \] - **Explanation:** Dividing the annual interest rate by 12 months yields the monthly interest rate, which is fundamental for monthly payment calculations. ### Step 2: Determine Total Number of Payments \[ n = 30 \text{ years} \times 12 \text{ months/year} = 360 \text{ months} \] - **Explanation:** The loan duration in months specifies the total number of payments, essential for amortization. ### Step 3: Apply the Amortization Formula to Calculate Monthly Payment \[ M = 382,\!000 \times \frac{0.006 \times (1 + 0.006)^{360}}{(1 + 0.006)^{360} - 1} \] Calculate \((1 + 0.006)^{360}\): \[ (1.006)^{360} \approx 8.138 \] Substitute into the formula: \[ M = 382,\!000 \times \frac{0.006 \times 8.138}{8.138 - 1} = 382,\!000 \times \frac{0.048828}{7.138} \] Calculate numerator: \[ 0.006 \times 8.138 = 0.048828 \] Compute the fraction: \[ \frac{0.048828}{7.138} \approx 0.006842 \] Final calculation: \[ M = 382,\!000 \times 0.006842 \approx \$2,\!613.64 \] - **Explanation:** The calculation yields the fixed monthly payment needed to amortize the loan over 30 years, considering monthly compounding interest. --- ### Step 4: Construct the Amortization Schedule For each month: - **Interest:** Multiply the previous balance by \(r = 0.006\). - **Principal:** Subtract interest from the fixed monthly payment. - **Remaining balance:** Subtract principal from previous balance. **First month:** - Previous balance: \$382,000 - Interest: \(382,\!000 \times 0.006 = \$2,\!292\) - Principal: \(2,\!613.64 - 2,\!292 = \$321.64\) - New balance: \(382,\!000 - 321.64 = \$381,\!678.36\) **Second month:** - Previous balance: \$381,678.36 - Interest: \(381,678.36 \times 0.006 = \$2,290.07\) - Principal: \(2,613.64 - 2,290.07 = \$323.57\) - New balance: \(381,678.36 - 323.57 = \$381,354.79\) **Third month:** - Previous balance: \$381,354.79 - Interest: \(381,354.79 \times 0.006 = \$2,288.13\) - Principal: \(2,613.64 - 2,288.13 = \$325.51\) - New balance: \(381,354.79 - 325.51 = \$381,029.28\) This process continues iteratively for all 360 months, with the interest decreasing and the principal portion increasing each month. --- ### Step 5: Generate the Full Schedule To fully detail all 360 payments, employ spreadsheet software such as Excel or Google Sheets: - Use the initial balance (\$382,000) in the first row. - Apply formulas for interest, principal, and remaining balance. - Drag formulas down for all months to complete the schedule. --- ## Conclusion The fixed monthly mortgage payment for the loan is approximately **\$2,613.64**. Constructing the amortization schedule involves calculating monthly interest based on the outstanding balance, deducting interest from the total payment to find the principal portion, and updating the remaining balance accordingly. This detailed schedule outlines how each payment reduces the principal over the 30-year term until the loan is fully paid.