Certainly! Let's analyze each option step-by-step to find out which one will lower the monthly payment for a loan with an APR of 4% over 60 months.
### Original Loan Details:
- Principal: Let's assume the principal amount is [tex]$20,000.
- APR: 4% per year.
- Term: 60 months.
We can calculate the monthly payment using the amortization formula:
\[ M = \frac{P r (1 + r)^n }{(1 + r)^n - 1} \]
where:
- \( M \) is the monthly payment,
- \( P \) is the loan principal,
- \( r \) is the monthly interest rate,
- \( n \) is the number of payments (loan term in months).
First, let’s calculate the monthly interest rate:
\[ r = \frac{0.04}{12} = 0.003333 \]
### Monthly Payment for Original Loan:
For the original loan:
- \( P = \$[/tex]20,000 \)
- [tex]\( r = 0.003333 \)[/tex]
- [tex]\( n = 60 \)[/tex]
Plugging into the formula:
[tex]\[ M_{\text{original}} = \frac{20000 \times 0.003333 \times (1 + 0.003333)^{60}}{(1 + 0.003333)^{60} - 1} \approx \$368.33 \][/tex]
### 1. Loan Term of 36 Months:
For a 36-month term:
- [tex]\( P = \$20,000 \)[/tex]
- [tex]\( r = 0.003333 \)[/tex]
- [tex]\( n = 36 \)[/tex]
[tex]\[ M_{36} = \frac{20000 \times 0.003333 \times (1 + 0.003333)^{36}}{(1 + 0.003333)^{36} - 1} \approx \$590.36 \][/tex]
### 2. Higher APR of 4.5%:
For a higher APR of 4.5%:
- [tex]\( P = \$20,000 \)[/tex]
- [tex]\( r = \frac{0.045}{12} = 0.00375 \)[/tex]
- [tex]\( n = 60 \)[/tex]
[tex]\[ M_{\text{higher\_APR}} = \frac{20000 \times 0.00375 \times (1 + 0.00375)^{60}}{(1 + 0.00375)^{60} - 1} \approx \$373.86 \][/tex]
### 3. Downpayment of [tex]$2000:
For a downpayment of $[/tex]2000:
- New Principal [tex]\( P = 20000 - 2000 = 18000 \)[/tex]
- [tex]\( r = 0.003333 \)[/tex]
- [tex]\( n = 60 \)[/tex]
[tex]\[ M_{\text{downpayment}} = \frac{18000 \times 0.003333 \times (1 + 0.003333)^{60}}{(1 + 0.003333)^{60} - 1} \approx \$331.49 \][/tex]
### Conclusion:
Let’s compare the monthly payments:
1. Original loan: [tex]\( \$368.33 \)[/tex]
2. 36-month loan term: [tex]\( \$590.36 \)[/tex]
3. Higher APR of 4.5%: [tex]\( \$373.86 \)[/tex]
4. Downpayment of [tex]$2000: \( \$[/tex]331.49 \)
Among these options, a downpayment of [tex]$2000 reduces the monthly payment the most. Hence, making a downpayment of $[/tex]2000 will lower the monthly payment more than the other choices.
