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Sagot :
To determine the difference in total cost between the two mortgage plans, we need to calculate the total cost of each plan and then find the difference. Here’s a detailed, step-by-step solution:
### Step 1: Calculate the Total Cost for the Given Adjustable-Rate Mortgage Plan
The given mortgage plan has different interest rates and monthly payments for different periods, as outlined in the table.
#### For Years 1-5:
- Interest Rate: 4%
- Monthly Payment: [tex]$2,506.43 - Total Months: 5 years * 12 months/year = 60 months Total cost for this period: \[ 2506.43 \, \text{dollars/month} \times 60 \, \text{months} = 150,385.80 \, \text{dollars} \] #### For Years 6-15: - Interest Rate: 6% - Monthly Payment: $[/tex]3,059.46
- Total Months: 10 years * 12 months/year = 120 months
Total cost for this period:
[tex]\[ 3059.46 \, \text{dollars/month} \times 120 \, \text{months} = 367,135.20 \, \text{dollars} \][/tex]
#### For Years 16-25:
- Interest Rate: 8%
- Monthly Payment: [tex]$3,464.78 - Total Months: 10 years * 12 months/year = 120 months Total cost for this period: \[ 3464.78 \, \text{dollars/month} \times 120 \, \text{months} = 415,773.60 \, \text{dollars} \] #### For Years 26-30: - Interest Rate: 10% - Monthly Payment: $[/tex]3,630.65
- Total Months: 5 years * 12 months/year = 60 months
Total cost for this period:
[tex]\[ 3630.65 \, \text{dollars/month} \times 60 \, \text{months} = 217,839.00 \, \text{dollars} \][/tex]
#### Summing up the Total Costs:
The total cost for the given adjustable-rate mortgage plan:
[tex]\[ 150,385.80 \, \text{dollars} + 367,135.20 \, \text{dollars} + 415,773.60 \, \text{dollars} + 217,839.00 \, \text{dollars} = 1,151,133.60 \, \text{dollars} \][/tex]
### Step 2: Calculate the Total Cost for the Fixed-Rate Mortgage
The fixed-rate mortgage has an interest rate of 4.45% for the full 30-year period.
To find the monthly payment for the fixed-rate mortgage, we use the standard mortgage payment formula:
[tex]\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
Where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the principal loan amount (ignored for comparison)
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( n \)[/tex] is the number of monthly payments
For a 4.45% annual interest rate:
[tex]\[ r = \frac{4.45\%}{12} = 0.0037083 \][/tex]
Since this is for a 30-year mortgage:
[tex]\[ n = 30 \, \text{years} \times 12 \, \text{months/year} = 360 \, \text{months} \][/tex]
Plugging these values into the formula gives us the monthly payment. Let's use [tex]\( P = 1 \)[/tex] (hypothetical principal, since we only care about the ratio):
[tex]\[ M = 1 \cdot \frac{0.0037083(1+0.0037083)^{360}}{(1+0.0037083)^{360} - 1} \approx 0.00495 \times P \][/tex]
Thus the total cost for the fixed-rate mortgage over 360 months is:
[tex]\[ 0.00495 \times P \times 360 \approx 1.813 \times P \][/tex]
Since we assumed [tex]\( P = 1 \)[/tex], this means the total cost (in our hypothetical units) is 1.813.
### Step 3: Compute the Difference in Costs
To get the difference, we subtract the total cost of the fixed-rate mortgage from the total cost of the given mortgage plan (using the actual costs for the given plan):
[tex]\[ 1,151,133.60 \, \text{dollars} - 1.813 \][/tex]
Since we took [tex]\( P = 1 \)[/tex], and in our computed difference (1.813), all units cancel out, what's left is:
[tex]\[ 1,151,133.60 - 0 \approx 1,151,133.60 \][/tex]
Let's round to the nearest dollar:
### Final Answer:
The difference in the total cost between the two mortgages, rounded to the nearest dollar, is \$115,113,179.
### Step 1: Calculate the Total Cost for the Given Adjustable-Rate Mortgage Plan
The given mortgage plan has different interest rates and monthly payments for different periods, as outlined in the table.
#### For Years 1-5:
- Interest Rate: 4%
- Monthly Payment: [tex]$2,506.43 - Total Months: 5 years * 12 months/year = 60 months Total cost for this period: \[ 2506.43 \, \text{dollars/month} \times 60 \, \text{months} = 150,385.80 \, \text{dollars} \] #### For Years 6-15: - Interest Rate: 6% - Monthly Payment: $[/tex]3,059.46
- Total Months: 10 years * 12 months/year = 120 months
Total cost for this period:
[tex]\[ 3059.46 \, \text{dollars/month} \times 120 \, \text{months} = 367,135.20 \, \text{dollars} \][/tex]
#### For Years 16-25:
- Interest Rate: 8%
- Monthly Payment: [tex]$3,464.78 - Total Months: 10 years * 12 months/year = 120 months Total cost for this period: \[ 3464.78 \, \text{dollars/month} \times 120 \, \text{months} = 415,773.60 \, \text{dollars} \] #### For Years 26-30: - Interest Rate: 10% - Monthly Payment: $[/tex]3,630.65
- Total Months: 5 years * 12 months/year = 60 months
Total cost for this period:
[tex]\[ 3630.65 \, \text{dollars/month} \times 60 \, \text{months} = 217,839.00 \, \text{dollars} \][/tex]
#### Summing up the Total Costs:
The total cost for the given adjustable-rate mortgage plan:
[tex]\[ 150,385.80 \, \text{dollars} + 367,135.20 \, \text{dollars} + 415,773.60 \, \text{dollars} + 217,839.00 \, \text{dollars} = 1,151,133.60 \, \text{dollars} \][/tex]
### Step 2: Calculate the Total Cost for the Fixed-Rate Mortgage
The fixed-rate mortgage has an interest rate of 4.45% for the full 30-year period.
To find the monthly payment for the fixed-rate mortgage, we use the standard mortgage payment formula:
[tex]\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
Where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the principal loan amount (ignored for comparison)
- [tex]\( r \)[/tex] is the monthly interest rate
- [tex]\( n \)[/tex] is the number of monthly payments
For a 4.45% annual interest rate:
[tex]\[ r = \frac{4.45\%}{12} = 0.0037083 \][/tex]
Since this is for a 30-year mortgage:
[tex]\[ n = 30 \, \text{years} \times 12 \, \text{months/year} = 360 \, \text{months} \][/tex]
Plugging these values into the formula gives us the monthly payment. Let's use [tex]\( P = 1 \)[/tex] (hypothetical principal, since we only care about the ratio):
[tex]\[ M = 1 \cdot \frac{0.0037083(1+0.0037083)^{360}}{(1+0.0037083)^{360} - 1} \approx 0.00495 \times P \][/tex]
Thus the total cost for the fixed-rate mortgage over 360 months is:
[tex]\[ 0.00495 \times P \times 360 \approx 1.813 \times P \][/tex]
Since we assumed [tex]\( P = 1 \)[/tex], this means the total cost (in our hypothetical units) is 1.813.
### Step 3: Compute the Difference in Costs
To get the difference, we subtract the total cost of the fixed-rate mortgage from the total cost of the given mortgage plan (using the actual costs for the given plan):
[tex]\[ 1,151,133.60 \, \text{dollars} - 1.813 \][/tex]
Since we took [tex]\( P = 1 \)[/tex], and in our computed difference (1.813), all units cancel out, what's left is:
[tex]\[ 1,151,133.60 - 0 \approx 1,151,133.60 \][/tex]
Let's round to the nearest dollar:
### Final Answer:
The difference in the total cost between the two mortgages, rounded to the nearest dollar, is \$115,113,179.
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