Use $$ \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $$ \$ 195,000 $$ mortgage. Which mortgage loan has the larger total cost (closing costs + the amount paid for points + total cost of interest)? By how much? Mortgage A: 15-year fixed at $$ 7.25 \% $$ with closing costs of $$ \$ 2300 $$ and 1 point. Mortgage B: 15-year fixed at $$ 5.25 \% $$ with closing costs of $$ \$ 2300 $$ and 4 points Choose the correct answer below, and fill in the answer box to complete your choice (Do not round until the final answer Then round to the nearest dollar as needed) A. Mortgage A has a larger total cost than mortgage B by $$ \$ $$. B. Mortgage B has a larger total cost than mortgage A by $$ \$ $$.

Question

UpStudy AI · Accepted Answer

To determine which mortgage loan has the larger total cost, we need to calculate the regular payment amount for both Mortgage A and Mortgage B using the formula provided:

$$
\text{Payment} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}
$$

Where:
- $$ P $$ is the principal amount (the mortgage amount).
- $$ r $$ is the annual interest rate (as a decimal).
- $$ n $$ is the number of payments per year.
- $$ t $$ is the number of years.

### Mortgage A:
- Principal $$ P = 195,000 $$
- Annual interest rate $$ r = 7.25\% = 0.0725 $$
- Number of payments per year $$ n = 12 $$ (monthly payments)
- Number of years $$ t = 15 $$

### Mortgage B:
- Principal $$ P = 195,000 $$
- Annual interest rate $$ r = 5.25\% = 0.0525 $$
- Number of payments per year $$ n = 12 $$ (monthly payments)
- Number of years $$ t = 15 $$

Next, we will calculate the monthly payment for both mortgages and then determine the total cost for each mortgage, including closing costs and points.

1. **Calculate the monthly payment for Mortgage A:**
   - Substitute the values into the formula.

2. **Calculate the monthly payment for Mortgage B:**
   - Substitute the values into the formula.

3. **Calculate the total cost for each mortgage:**
   - Total payments = Monthly payment × Total number of payments
   - Total cost = Total payments + Closing costs + Cost of points

Let's perform these calculations step by step.

### Step 1: Calculate Monthly Payment for Mortgage A
Substituting the values for Mortgage A into the formula:

$$
\text{Payment}_A = \frac{195000\left(\frac{0.0725}{12}\right)}{\left[1-\left(1+\frac{0.0725}{12}\right)^{-12 \times 15}\right]}
$$

### Step 2: Calculate Monthly Payment for Mortgage B
Substituting the values for Mortgage B into the formula:

$$
\text{Payment}_B = \frac{195000\left(\frac{0.0525}{12}\right)}{\left[1-\left(1+\frac{0.0525}{12}\right)^{-12 \times 15}\right]}
$$

### Step 3: Calculate Total Cost for Each Mortgage
- Total payments for Mortgage A = $$ \text{Payment}_A \times 12 \times 15 $$
- Total payments for Mortgage B = $$ \text{Payment}_B \times 12 \times 15 $$

Now, let's perform the calculations for both mortgages.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{195000\left(\frac{0.0725}{12}\right)}{\left(1-\left(1+\frac{0.0725}{12}\right)^{-12\times 15}\right)}$$
- step1: Remove the parentheses:
  $$\frac{195000\left(\frac{0.0725}{12}\right)}{1-\left(1+\frac{0.0725}{12}\right)^{-12\times 15}}$$
- step2: Divide the terms:
  $$\frac{195000\left(\frac{0.0725}{12}\right)}{1-\left(1+\frac{29}{4800}\right)^{-12\times 15}}$$
- step3: Add the numbers:
  $$\frac{195000\left(\frac{0.0725}{12}\right)}{1-\left(\frac{4829}{4800}\right)^{-12\times 15}}$$
- step4: Divide the terms:
  $$\frac{195000\times \frac{29}{4800}}{1-\left(\frac{4829}{4800}\right)^{-12\times 15}}$$
- step5: Multiply the numbers:
  $$\frac{195000\times \frac{29}{4800}}{1-\left(\frac{4829}{4800}\right)^{-180}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{9425}{8}}{1-\left(\frac{4829}{4800}\right)^{-180}}$$
- step7: Subtract the numbers:
  $$\frac{\frac{9425}{8}}{\frac{4829^{180}-4800^{180}}{4829^{180}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{9425}{8}\times \frac{4829^{180}}{4829^{180}-4800^{180}}$$
- step9: Multiply the fractions:
  $$\frac{9425\times 4829^{180}}{8\left(4829^{180}-4800^{180}\right)}$$
- step10: Multiply:
  $$\frac{9425\times 4829^{180}}{8\times 4829^{180}-8\times 4800^{180}}$$
Calculate or simplify the expression $$ \frac{195000\left(\frac{0.0525}{12}\right)}{\left[1-\left(1+\frac{0.0525}{12}\right)^{-12 \times 15}\right]}} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{195000\left(\frac{0.0525}{12}\right)}{\left(1-\left(1+\frac{0.0525}{12}\right)^{-12\times 15}\right)}$$
- step1: Remove the parentheses:
  $$\frac{195000\left(\frac{0.0525}{12}\right)}{1-\left(1+\frac{0.0525}{12}\right)^{-12\times 15}}$$
- step2: Divide the terms:
  $$\frac{195000\left(\frac{0.0525}{12}\right)}{1-\left(1+\frac{7}{1600}\right)^{-12\times 15}}$$
- step3: Add the numbers:
  $$\frac{195000\left(\frac{0.0525}{12}\right)}{1-\left(\frac{1607}{1600}\right)^{-12\times 15}}$$
- step4: Divide the terms:
  $$\frac{195000\times \frac{7}{1600}}{1-\left(\frac{1607}{1600}\right)^{-12\times 15}}$$
- step5: Multiply the numbers:
  $$\frac{195000\times \frac{7}{1600}}{1-\left(\frac{1607}{1600}\right)^{-180}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{6825}{8}}{1-\left(\frac{1607}{1600}\right)^{-180}}$$
- step7: Subtract the numbers:
  $$\frac{\frac{6825}{8}}{\frac{1607^{180}-1600^{180}}{1607^{180}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{6825}{8}\times \frac{1607^{180}}{1607^{180}-1600^{180}}$$
- step9: Multiply the fractions:
  $$\frac{6825\times 1607^{180}}{8\left(1607^{180}-1600^{180}\right)}$$
- step10: Multiply:
  $$\frac{6825\times 1607^{180}}{8\times 1607^{180}-8\times 1600^{180}}$$
The monthly payments for both mortgages are as follows:

1. **Mortgage A:**
   $$
\text{Payment}_A \approx 1780.08
$$

2. **Mortgage B:**
   $$
\text{Payment}_B \approx 1567.56
$$

### Step 3: Calculate Total Cost for Each Mortgage

**Total Payments Calculation:**
- Total payments for Mortgage A:
$$
\text{Total Payments}_A = 1780.08 \times 12 \times 15
$$

- Total payments for Mortgage B:
$$
\text{Total Payments}_B = 1567.56 \times 12 \times 15
$$

**Closing Costs and Points:**
- Mortgage A has 1 point (1% of $195,000 = $1,950) and closing costs of $2,300.
- Mortgage B has 4 points (4% of $195,000 = $7,800) and closing costs of $2,300.

**Total Cost Calculation:**
- Total cost for Mortgage A:
$$
\text{Total Cost}_A = \text{Total Payments}_A + 2300 + 1950
$$

- Total cost for Mortgage B:
$$
\text{Total Cost}_B = \text{Total Payments}_B + 2300 + 7800
$$

Now, let's calculate the total payments and total costs for both mortgages.
Calculate the value by following steps:
- step0: Calculate:
  $$1567.56\times 12\times 15$$
- step1: Multiply the terms:
  $$18810.72\times 15$$
- step2: Multiply the numbers:
  $$282160.8$$
Calculate or simplify the expression $$ 1780.08 * 12 * 15 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1780.08\times 12\times 15$$
- step1: Multiply the terms:
  $$21360.96\times 15$$
- step2: Multiply the numbers:
  $$320414.4$$
Calculate or simplify the expression $$ 282160.8 + 2300 + 7800 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$282160.8+2300+7800$$
- step1: Add the numbers:
  $$292260.8$$
Calculate or simplify the expression $$ 320414.4 + 2300 + 1950 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$320414.4+2300+1950$$
- step1: Add the numbers:
  $$324664.4$$
Now we have calculated the total costs for both mortgages:

1. **Total Cost for Mortgage A:**
   $$
\text{Total Cost}_A = 324664.4
$$

2. **Total Cost for Mortgage B:**
   $$
\text{Total Cost}_B = 292260.8
$$

### Step 4: Determine Which Mortgage Has the Larger Total Cost

Now, we can compare the total costs:

- Mortgage A: $$ 324664.4 $$
- Mortgage B: $$ 292260.8 $$

To find out by how much Mortgage A is more expensive than Mortgage B, we calculate:

$$
\text{Difference} = \text{Total Cost}_A - \text{Total Cost}_B
$$

Let's calculate the difference.
Calculate the value by following steps:
- step0: Calculate:
  $$324664.4-292260.8$$
- step1: Subtract the numbers:
  $$32403.6$$
The difference in total costs between Mortgage A and Mortgage B is:

$$
\text{Difference} \approx 32403.6
$$

### Conclusion
Mortgage A has a larger total cost than Mortgage B by approximately $$ \$ 32,404 $$ when rounded to the nearest dollar.

Thus, the correct answer is:
**A. Mortgage A has a larger total cost than mortgage B by $$ \$ 32,404 $$.**
Mortgage A has a larger total cost than Mortgage B by \$32,404.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Economics Questions