Question

Sam needs to take out a personal loan for $8,900 to pay for a trip to Europe with his classmates. His bank has offered him the four loans listed in the chart below. If all of the loans are compounded monthly, which of the four loans will give Sam the lowest monthly payment?
Loan
Duration (Months)
Interest Rate
A
12
9.50%
B
24
8.75%
C
36
7.75%
D
48
6.60%

a.
loan A
b.
loan B
c.
loan C
d.
loan D

Answer 1

To determine which loan will give Sam the lowest monthly payment, we can use the formula for calculating the monthly payment on an amortizing loan:

\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

Where:
- \( M \) = monthly payment
- \( P \) = loan principal (amount borrowed)
- \( r \) = monthly interest rate (annual rate divided by 12)
- \( n \) = number of months (duration of the loan)

For each loan, we will calculate the monthly payment.

**Calculating for each loan:**

1. **Loan A:**
- Duration: 12 months
- Interest Rate: 9.50% annual = 0.095/12 monthly
- Monthly interest rate (\( r \)) = 0.00791667
- Monthly payment (\( M \)):
\[
M = 8900 \times \frac{0.00791667(1 + 0.00791667)^{12}}{(1 + 0.00791667)^{12} - 1}
\]
\[
M = 8900 \times \frac{0.00791667(1.100719)}{0.100719}
\]
\[
M \approx 8900 \times 0.071109 \approx 631.54
\]

2. **Loan B:**
- Duration: 24 months
- Interest Rate: 8.75% annual = 0.0875/12 monthly
- Monthly interest rate (\( r \)) = 0.00729167
- Monthly payment (\( M \)):
\[
M = 8900 \times \frac{0.00729167(1 + 0.00729167)^{24}}{(1 + 0.00729167)^{24} - 1}
\]
\[
M = 8900 \times \frac{0.00729167(1.192742)}{0.192742}
\]
\[
M \approx 8900 \times 0.062558 \approx 555.77
\]

3. **Loan C:**
- Duration: 36 months
- Interest Rate: 7.75% annual = 0.0775/12 monthly
- Monthly interest rate (\( r \)) = 0.00645833
- Monthly payment (\( M \)):
\[
M = 8900 \times \frac{0.00645833(1 + 0.00645833)^{36}}{(1 + 0.00645833)^{36} - 1}
\]
\[
M = 8900 \times \frac{0.00645833(1.250819)}{0.250819}
\]
\[
M \approx 8900 \times 0.06126 \approx 545.83
\]

4. **Loan D:**
- Duration: 48 months
- Interest Rate: 6.60% annual = 0.066/12 monthly
- Monthly interest rate (\( r \)) = 0.0055
- Monthly payment (\( M \)):
\[
M = 8900 \times \frac{0.0055(1 + 0.0055)^{48}}{(1 + 0.0055)^{48} - 1}
\]
\[
M = 8900 \times \frac{0.0055(1.304571)}{0.304571}
\]
\[
M \approx 8900 \times 0.04904 \approx 436.98
\]

**Monthly payments for each loan:**
- Loan A: $631.54
- Loan B: $555.77
- Loan C: $545.83
- Loan D: $436.98

Therefore, the loan that gives Sam the **lowest monthly payment** is **Loan D**.

**Answer:** d. loan D