Question

Calculate the total payback for a $3,500 loan at 8% annual interest with monthly payments for two years. Show all of your steps.

Answer 1

To calculate the total payback for a loan, we will first determine the monthly payment amount using the loan amount, interest rate, and loan term. The formula for calculating the monthly payment for an amortizing loan is:

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

Where:

- \(M\) is the monthly payment
- \(P\) is the principal amount (the loan amount)
- \(r\) is the monthly interest rate (annual interest rate divided by 12)
- \(n\) is the total number of payments (number of months)

### Given data:
- \(P = 3500\)
- Annual Interest Rate = 8% = 0.08
- Monthly Interest Rate \(r = \frac{0.08}{12} = \frac{0.08}{12} = 0.00666667\)
- Loan term = 2 years = 24 months
- \(n = 24\)

### Step 1: Calculate the monthly payment \(M\)

Plugging the numbers into the formula:

\[
M = 3500 \times \frac{0.00666667(1 + 0.00666667)^{24}}{(1 + 0.00666667)^{24} - 1}
\]

Calculating \((1 + r)^{24}\):

\[
1 + r = 1 + 0.00666667 = 1.00666667
\]
\[
(1.00666667)^{24} \approx 1.17342
\]

### Now plug this back into our \(M\) formula:

\[
M = 3500 \times \frac{0.00666667 \times 1.17342}{1.17342 - 1}
\]

Calculating the numerator:

\[
0.00666667 \times 1.17342 \approx 0.00784281
\]

Calculating the denominator:

\[
1.17342 - 1 = 0.17342
\]

Now substituting values:

\[
M = 3500 \times \frac{0.00784281}{0.17342} \approx 3500 \times 0.0451873 \approx 158.15
\]

So, the monthly payment \(M\) is approximately **$158.15**.

### Step 2: Calculate the total payback

Now, we need to calculate the total payback over the 24 months:

\[
\text{Total Payback} = M \times n = 158.15 \times 24
\]

Calculating the total payback:

\[
\text{Total Payback} \approx 158.15 \times 24 = 3795.60
\]

### Final Result

Thus, the total payback for the $3500 loan at an 8% annual interest rate with monthly payments over two years is approximately **$3,795.60**.