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Sagot :
Certainly! Let's approach the given problem step-by-step.
### Part A: Total Cost of the Adjustable-Rate Mortgage (ARM)
To find the total cost of the adjustable-rate mortgage, we need to compute the total amount paid over the life of the loan given the different payment amounts over specific ranges of years.
We'll break this down by each period:
1. Years 1-5:
- Monthly Payment: \[tex]$1,074.18 - Number of Months: 5 years × 12 months/year = 60 months - Total Payment: \$[/tex]1,074.18 × 60 = \[tex]$64,450.80 2. Years 6-15: - Monthly Payment: \$[/tex]1,311.20
- Number of Months: 10 years × 12 months/year = 120 months
- Total Payment: \[tex]$1,311.20 × 120 = \$[/tex]157,344.00
3. Years 16-25:
- Monthly Payment: \[tex]$1,484.91 - Number of Months: 10 years × 12 months/year = 120 months - Total Payment: \$[/tex]1,484.91 × 120 = \[tex]$178,189.20 4. Years 26-30: - Monthly Payment: \$[/tex]1,555.99
- Number of Months: 5 years × 12 months/year = 60 months
- Total Payment: \[tex]$1,555.99 × 60 = \$[/tex]93,359.40
Summing these up, we get the total cost of the ARM:
[tex]\[ \text{Total ARM Cost} = \$64,450.80 + \$157,344.00 + \$178,189.20 + \$93,359.40 = \$493,343.40 \][/tex]
### Part B: Total Cost of the Fixed-Rate Mortgage (FRM)
To calculate the total cost of the fixed-rate mortgage, we'll use the given monthly payment and the fixed period of 30 years.
1. Fixed-Rate Mortgage Expectations:
- Monthly Payment: \[tex]$1,496.93 - Number of Months: 30 years × 12 months/year = 360 months - Total Payment: \$[/tex]1,496.93 × 360 = \[tex]$538,894.80 So, the total cost of the fixed-rate mortgage is: \[ \text{Total FRM Cost} = \$[/tex]538,894.80 \]
### Part C: Comparison of ARM and FRM
Using the values calculated in parts A and B, we can compare the advantages and disadvantages of the two mortgage types.
Advantages of ARM:
- Lower Total Cost: The total cost of the ARM is \[tex]$493,343.40, which is lower than the total cost of the FRM (\$[/tex]538,894.80). Thus, over the entirety of the mortgage, one pays less with the ARM.
- Lower Initial Payments: In the early years (1-5), the payments are significantly lower at \[tex]$1,074.18 compared to the fixed-rate monthly payment of \$[/tex]1,496.93. This can be advantageous for someone who might anticipate a rise in their income or expects to refinance or sell the property within the initial lower-payment period.
Disadvantages of ARM:
- Potential Increase in Payments: As the interest rate adjusts and increases over time, the monthly payments also increase. This can be a disadvantage for someone who may not expect an increase in their income or financial stability.
- Uncertainty: The nature of an ARM brings uncertainty with future payments potentially increasing, which can be riskier compared to the fixed payments of an FRM.
Advantages of FRM:
- Consistency and Predictability: Fixed monthly payments of \[tex]$1,496.93 over the 30-year period provide financial predictability and ease in budgeting. - Long-Term Stability: Without fluctuations in payment amounts, it's easier for homeowners to plan their long-term finances. Disadvantages of FRM: - Higher Total Cost: Over the life of the loan, the total cost is higher at \$[/tex]538,894.80 compared to the ARM.
- Higher Initial Payments: The fixed nature leads to higher initial monthly payments (\[tex]$1,496.93) compared to the ARM's initial payments (\$[/tex]1,074.18).
Each mortgage type has its own set of advantages and disadvantages. Borrowers should consider their financial situation, future expectations, and risk tolerance when deciding between the two.
### Part A: Total Cost of the Adjustable-Rate Mortgage (ARM)
To find the total cost of the adjustable-rate mortgage, we need to compute the total amount paid over the life of the loan given the different payment amounts over specific ranges of years.
We'll break this down by each period:
1. Years 1-5:
- Monthly Payment: \[tex]$1,074.18 - Number of Months: 5 years × 12 months/year = 60 months - Total Payment: \$[/tex]1,074.18 × 60 = \[tex]$64,450.80 2. Years 6-15: - Monthly Payment: \$[/tex]1,311.20
- Number of Months: 10 years × 12 months/year = 120 months
- Total Payment: \[tex]$1,311.20 × 120 = \$[/tex]157,344.00
3. Years 16-25:
- Monthly Payment: \[tex]$1,484.91 - Number of Months: 10 years × 12 months/year = 120 months - Total Payment: \$[/tex]1,484.91 × 120 = \[tex]$178,189.20 4. Years 26-30: - Monthly Payment: \$[/tex]1,555.99
- Number of Months: 5 years × 12 months/year = 60 months
- Total Payment: \[tex]$1,555.99 × 60 = \$[/tex]93,359.40
Summing these up, we get the total cost of the ARM:
[tex]\[ \text{Total ARM Cost} = \$64,450.80 + \$157,344.00 + \$178,189.20 + \$93,359.40 = \$493,343.40 \][/tex]
### Part B: Total Cost of the Fixed-Rate Mortgage (FRM)
To calculate the total cost of the fixed-rate mortgage, we'll use the given monthly payment and the fixed period of 30 years.
1. Fixed-Rate Mortgage Expectations:
- Monthly Payment: \[tex]$1,496.93 - Number of Months: 30 years × 12 months/year = 360 months - Total Payment: \$[/tex]1,496.93 × 360 = \[tex]$538,894.80 So, the total cost of the fixed-rate mortgage is: \[ \text{Total FRM Cost} = \$[/tex]538,894.80 \]
### Part C: Comparison of ARM and FRM
Using the values calculated in parts A and B, we can compare the advantages and disadvantages of the two mortgage types.
Advantages of ARM:
- Lower Total Cost: The total cost of the ARM is \[tex]$493,343.40, which is lower than the total cost of the FRM (\$[/tex]538,894.80). Thus, over the entirety of the mortgage, one pays less with the ARM.
- Lower Initial Payments: In the early years (1-5), the payments are significantly lower at \[tex]$1,074.18 compared to the fixed-rate monthly payment of \$[/tex]1,496.93. This can be advantageous for someone who might anticipate a rise in their income or expects to refinance or sell the property within the initial lower-payment period.
Disadvantages of ARM:
- Potential Increase in Payments: As the interest rate adjusts and increases over time, the monthly payments also increase. This can be a disadvantage for someone who may not expect an increase in their income or financial stability.
- Uncertainty: The nature of an ARM brings uncertainty with future payments potentially increasing, which can be riskier compared to the fixed payments of an FRM.
Advantages of FRM:
- Consistency and Predictability: Fixed monthly payments of \[tex]$1,496.93 over the 30-year period provide financial predictability and ease in budgeting. - Long-Term Stability: Without fluctuations in payment amounts, it's easier for homeowners to plan their long-term finances. Disadvantages of FRM: - Higher Total Cost: Over the life of the loan, the total cost is higher at \$[/tex]538,894.80 compared to the ARM.
- Higher Initial Payments: The fixed nature leads to higher initial monthly payments (\[tex]$1,496.93) compared to the ARM's initial payments (\$[/tex]1,074.18).
Each mortgage type has its own set of advantages and disadvantages. Borrowers should consider their financial situation, future expectations, and risk tolerance when deciding between the two.
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