To find the principal balance on Jeffrey's loan after 1 year, we need to follow these steps:

Step 1: Identify the relevant parameters

• Principal (P): $28,500
• Annual interest rate (r): 3.62% or 0.0362
• Compounding frequency: Quarterly (4 times a year)
• Loan term (n): 3 years
• Payment frequency: Monthly (12 times a year)

Step 2: Convert the annual interest rate to the quarterly interest rate

The nominal quarterly interest rate $i$ can be calculated as: $i = \frac{r}{4} = \frac{0.0362}{4} = 0.00905$

Step 3: Calculate the number of quarterly periods in 3 years

The number of quarters in 3 years is: $n_{\text{quarters}} = 3 \times 4 = 12 \text{ quarters}$

Step 4: Calculate the monthly payment

To find the monthly payment, we use the amortization formula: $PMT = \frac{P \times i}{1 - (1 + i)^{-n_{\text{months}}}}$ where:

• $P$ is the principal amount,
• $i$ is the monthly interest rate, and
• $n_{\text{months}}$ is the total number of payments (36 months).

However, since the interest is compounded quarterly, the effective monthly interest rate $j$ is: $j = \left(1 + \frac{r}{4}\right)^{\frac{1}{3}} - 1 = \left(1 + 0.00905\right)^{\frac{1}{3}} - 1$

Calculating $j$: $j \approx 0.003015$

Now calculate the monthly payment $PMT$: $PMT = \frac{28500 \times 0.003015}{1 - (1 + 0.003015)^{-36}}$ $PMT \approx 828.25$

Step 5: Calculate the principal balance after 1 year

After 1 year (12 payments), the remaining balance can be found using the amortization formula: $B = P \times (1 + j)^{12} - PMT \times \frac{(1 + j)^{12} - 1}{j}$ where $B$ is the remaining balance.

Substitute the known values: $B = 28500 \times (1 + 0.003015)^{12} - 828.25 \times \frac{(1 + 0.003015)^{12} - 1}{0.003015}$ $B \approx 28500 \times 1.0368 - 828.25 \times \frac{0.0368}{0.003015}$ $B \approx 29549.6 - 9,900$ $B \approx 19,649.60$

So, the principal balance on the loan after 1 year is approximately $19,649.60.

Would you like more details or have any questions?

Here are some related questions:

1. How would the remaining balance change if the interest rate were 4%?
2. What would be the remaining balance after 2 years?
3. How much total interest would Jeffrey pay over the life of the loan?
4. How would a higher payment frequency affect the remaining balance?
5. What would the monthly payment be if the loan term were extended to 5 years?

Tip: To quickly approximate the balance after a certain period, you can use online amortization calculators, which automate the complex formulas.