To find the difference in the total cost between the two mortgage options, we'll break the problem into manageable steps and compute the total costs for both the adjustable rate mortgage (ARM) and the fixed-rate mortgage (FRM).
### Total Cost of the Adjustable Rate Mortgage (ARM)
1. Identify the payment periods and corresponding monthly payments:
- Years 1-5: [tex]$2506.43$[/tex]
- Years 6-15: [tex]$3059.46$[/tex]
- Years 16-25: [tex]$3464.78$[/tex]
- Years 26-30: [tex]$3630.65$[/tex]
2. Calculate the total payments for each period:
- For years 1-5:
[tex]\[
2506.43 \, \text{USD/month} \times 12 \, \text{months/year} \times 5 \, \text{years} = 150385.8 \, \text{USD}
\][/tex]
- For years 6-15:
[tex]\[
3059.46 \, \text{USD/month} \times 12 \, \text{months/year} \times 10 \, \text{years} = 367135.2 \, \text{USD}
\][/tex]
- For years 16-25:
[tex]\[
3464.78 \, \text{USD/month} \times 12 \, \text{months/year} \times 10 \, \text{years} = 415773.6 \, \text{USD}
\][/tex]
- For years 26-30:
[tex]\[
3630.65 \, \text{USD/month} \times 12 \, \text{months/year} \times 5 \, \text{years} = 213839 \, \text{USD}
\][/tex]
3. Sum up the total payments for all periods:
[tex]\[
150385.8 + 367135.2 + 415773.6 + 213839 = 1151133.6 \, \text{USD}
\][/tex]
### Total Cost of the Fixed Rate Mortgage (FRM)
1. Identify the mortgage parameters:
- Mortgage amount: [tex]$525,000$[/tex]
- Fixed interest rate: [tex]$4.65\%$[/tex] annually
- Term: 30 years
2. Calculate the monthly payment (M) using the formula:
[tex]\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\][/tex]
where [tex]\( P = 525,000 \, \text{USD} \)[/tex], [tex]\( r = \frac{0.0465}{12} \)[/tex], and [tex]\( n = 30 \times 12 = 360 \, \text{months} \)[/tex].
3. Calculate the fixed monthly payment:
[tex]\[
M = 525000 \times \frac{\frac{0.0465}{12} \times (1 + \frac{0.0465}{12})^{360}}{(1 + \frac{0.0465}{12})^{360} - 1} = 2718.20 \, \text{USD}
\][/tex]
4. Calculate the total payment over 30 years:
[tex]\[
2718.20 \, \text{USD/month} \times 12 \, \text{months/year} \times 30 \, \text{years} = 974553.57 \, \text{USD}
\][/tex]
### Difference in Total Cost
1. Calculate the difference between the total cost of the ARM and the FRM:
[tex]\[
1151133.6 \, \text{USD} - 974553.57 \, \text{USD} = 176580 \, \text{USD}
\][/tex]
The difference in the total cost between the adjustable rate mortgage and the fixed-rate mortgage, rounded to the nearest dollar, is:
[tex]\(\boxed{176580}\)[/tex].
