You need $200,000 to buy a new home. The bank offers a choice of a 30-year loan at an APR of 8% or a 15-year loan at 7.5%. Assume the closing costs are the same for both loans.
a) Compare the monthly payments for these two loan options.
b) Compare the total loan costs for these two loan options.
c) What are the pros and cons of each loan and which option would you choose?
To compare the monthly payments for the two loan options, we can use the formulas for calculating the monthly payment for a fixed-rate mortgage.
The formula for calculating the monthly payment for a fixed-rate mortgage is:
M = P * r * (1 + r)^n / ((1 + r)^n - 1)
Where:
M = Monthly payment
P = Loan amount
r = Monthly interest rate (APR divided by 12)
n = Total number of monthly payments (30 years for the 30-year loan and 15 years for the 15-year loan)
Let's calculate the monthly payment for each loan option:
For the 30-year loan:
Loan amount (P) = $200,000
Monthly interest rate (r) = 8% / 12 = 0.08 / 12 = 0.006667
Total number of monthly payments (n) = 30 years * 12 months = 360 months
Using the formula:
M = 200,000 * 0.006667 * (1 + 0.006667)^360 / ((1 + 0.006667)^360 - 1)
M ≈ $1,467.53
The monthly payment for the 30-year loan option is approximately $1,467.53.
For the 15-year loan:
Loan amount (P) = $200,000
Monthly interest rate (r) = 7.5% / 12 = 0.075 / 12 = 0.00625
Total number of monthly payments (n) = 15 years * 12 months = 180 months
Using the formula:
M = 200,000 * 0.00625 * (1 + 0.00625)^180 / ((1 + 0.00625)^180 - 1)
M ≈ $1,753.25
The monthly payment for the 15-year loan option is approximately $1,753.25.
Now let's compare the total loan costs for the two loan options.
The total loan cost can be calculated by multiplying the monthly payment by the total number of monthly payments.
For the 30-year loan:
Total loan cost = $1,467.53 * 360 = $528,911.80
For the 15-year loan:
Total loan cost = $1,753.25 * 180 = $315,585.00
Finally, let's discuss the pros and cons of each loan option and which option to choose.
Pros and cons of the 30-year loan:
- Lower monthly payment: The 30-year loan has a lower monthly payment, which can be more affordable for some borrowers.
- Higher total loan cost: The 30-year loan option has a higher total loan cost due to the longer loan term.
- More interest paid: Since the loan is spread out over a longer period, more interest is paid over time.
Pros and cons of the 15-year loan:
- Shorter loan term: The 15-year loan has a shorter loan term, allowing borrowers to pay off the loan faster.
- Higher monthly payment: The 15-year loan option has a higher monthly payment, which may be less affordable for some borrowers.
- Lower total loan cost: The 15-year loan option has a lower total loan cost due to the shorter loan term.
Which option to choose depends on individual circumstances and financial goals. If your priority is to have a lower monthly payment and greater affordability, the 30-year loan may be a better choice. However, if you can afford the higher monthly payment and want to save on interest in the long run, the 15-year loan may be the better option. It's essential to consider your current financial situation, long-term goals, and future income prospects when making a decision.
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