Use PMT $$ =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest cent. The cost of a home is financed with a $$ \$ 190,000 $$ 20-year fixed-rate mortgage at $$ 4.5 \% $$. a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage.

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**a. Calculating the Monthly Payment and Total Interest**

Given the formula  
$$
\text{PMT} = \frac{P \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-n t}},
$$  
we have:

- Principal: $$P = 190{,}000$$  
- Annual interest rate: $$r = 4.5\% = 0.045$$  
- Number of compounding periods per year: $$n = 12$$  
- Term (years): $$t = 20$$

1. **Determine the monthly interest rate and total number of payments:**

$$
   \text{Monthly interest rate } = \frac{r}{n} = \frac{0.045}{12} = 0.00375.
   $$
   $$
   \text{Total payments } = n \times t = 12 \times 20 = 240.
   $$

2. **Substitute into the PMT formula:**

$$
   \text{PMT} = \frac{190{,}000 \times 0.00375}{1 - \left(1 + 0.00375\right)^{-240}}.
   $$

3. **Compute the denominator:**

First, compute $$1 + 0.00375$$:  
   $$
   1 + 0.00375 = 1.00375.
   $$
   Raise to the power $$-240$$:  
   $$
   \left(1.00375\right)^{-240} = \frac{1}{\left(1.00375\right)^{240}}.
   $$
   Using a calculator or logarithms, one finds that  
   $$\left(1.00375\right)^{240} \approx 2.456$$, so  
   $$
   \left(1.00375\right)^{-240} \approx \frac{1}{2.456} \approx 0.4071.
   $$
   Therefore,  
   $$
   1 - \left(1.00375\right)^{-240} \approx 1 - 0.4071 = 0.5929.
   $$

4. **Calculate the numerator:**

$$
   190{,}000 \times 0.00375 = 712.50.
   $$

5. **Determine the monthly payment:**

$$
   \text{PMT} \approx \frac{712.50}{0.5929} \approx 1201.30.
   $$

So, the monthly payment is approximately **\$1,201.30**.

6. **Calculate the total interest:**

The total amount paid over 240 months is:  
   $$
   \text{Total paid} = 1201.30 \times 240 \approx 288{,}312.
   $$
   Therefore, the total interest is:  
   $$
   \text{Total interest} = 288{,}312 - 190{,}000 \approx 98{,}312.
   $$

The total interest paid over the life of the loan is approximately **\$98,312**.

---

**b. Loan Amortization Schedule for the First Three Months**

For each month, the interest portion is  
$$
\text{Interest} = \text{Current Balance} \times 0.00375,
$$
and the principal repayment is  
$$
\text{Principal} = \text{PMT} - \text{Interest}.
$$

*Let the starting balance be $$B_0 = 190{,}000$$.*

**Month 1:**

- **Interest:**  
  $$
  I_1 = 190{,}000 \times 0.00375 = 712.50.
  $$
- **Principal:**  
  $$
  \text{Principal}_1 = 1201.30 - 712.50 = 488.80.
  $$
- **New Balance:**  
  $$
  B_1 = 190{,}000 - 488.80 = 189{,}511.20.
  $$

**Month 2:**

- **Interest:**  
  $$
  I_2 = 189{,}511.20 \times 0.00375 \approx 710.67.
  $$
- **Principal:**  
  $$
  \text{Principal}_2 = 1201.30 - 710.67 \approx 490.63.
  $$
- **New Balance:**  
  $$
  B_2 = 189{,}511.20 - 490.63 \approx 189{,}020.57.
  $$

**Month 3:**

- **Interest:**  
  $$
  I_3 = 189{,}020.57 \times 0.00375 \approx 708.83.
  $$
- **Principal:**  
  $$
  \text{Principal}_3 = 1201.30 - 708.83 \approx 492.47.
  $$
- **New Balance:**  
  $$
  B_3 = 189{,}020.57 - 492.47 \approx 188{,}528.10.
  $$

The following table summarizes the first three months:

| Month | Payment (\$) | Interest (\$) | Principal (\$) | Balance (\$)    |
|-------|--------------|---------------|----------------|-----------------|
| 1     | 1201.30      | 712.50        | 488.80         | 189,511.20      |
| 2     | 1201.30      | 710.67        | 490.63         | 189,020.57      |
| 3     | 1201.30      | 708.83        | 492.47         | 188,528.10      |

---

**Summary:**

- **Monthly Payment:** Approximately $$\$1,201.30$$.  
- **Total Interest Over 20 Years:** Approximately $$\$98,312$$.  
- **Amortization Schedule (First 3 Months):** As detailed in the table above.
**a. Monthly Payment and Total Interest** - **Monthly Payment:** \$1,201.30 - **Total Interest Over 20 Years:** \$98,312 **b. Loan Amortization Schedule for the First Three Months** | Month | Payment (\$) | Interest (\$) | Principal (\$) | Balance (\$) | |-------|--------------|---------------|----------------|-----------------| | 1 | 1,201.30 | 712.50 | 488.80 | 189,511.20 | | 2 | 1,201.30 | 710.67 | 490.63 | 189,020.57 | | 3 | 1,201.30 | 708.83 | 492.47 | 188,528.10 |

Use PMT to determine the regular payment amount, rounded to the nearest cent. The cost of a
home is financed with a 20-year fixed-rate mortgage at .
a. Find the monthly payments and the total interest for the loan.
b. Prepare a loan amortization schedule for the first three months of the mortgage.

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Month	Payment ($)	Interest ($)	Principal ($)	Balance ($)
1	1,201.30	712.50	488.80	189,511.20
2	1,201.30	710.67	490.63	189,020.57
3	1,201.30	708.83	492.47	188,528.10

Month	Payment ($)	Interest ($)	Principal ($)	Balance ($)
1	1201.30	712.50	488.80	189,511.20
2	1201.30	710.67	490.63	189,020.57
3	1201.30	708.83	492.47	188,528.10

Use PMT to determine the regular payment amount, rounded to the nearest cent. The cost of a home is financed with a 20-year fixed-rate mortgage at . a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage.