To calculate the periodic payment $PMT$ for a loan, we use the formula for the payment on an amortizing loan:

$PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}}$

Where:

• $P$ is the loan principal (the amount borrowed),
• $r$ is the periodic interest rate,
• $n$ is the total number of payments.

Given:

• Loan amount $P = 500,000$,
• Annual interest rate = 8%,
• Time period = 9 years,
• Payments are made quarterly (4 times per year).

Step 1: Convert annual interest rate to quarterly rate

$r = \frac{8\%}{4} = 0.08 / 4 = 0.02$

Step 2: Calculate the total number of payments

$n = 9 \times 4 = 36 \text{ payments}$

Step 3: Apply the formula

$PMT = \frac{500,000 \times 0.02}{1 - (1 + 0.02)^{-36}}$

Now, let’s calculate it:

$PMT = \frac{10,000}{1 - (1.02)^{-36}} = \frac{10,000}{1 - 0.487262}$

$PMT = \frac{10,000}{0.512738} \approx 19,506.35$

Thus, the periodic payment $PMT$ is approximately $19,506.35 per quarter.

Would you like a detailed breakdown of each step or have any questions?

Here are 5 related questions:

1. How would the payment change if the interest rate was 6% instead of 8%?
2. What would happen to the payment if the loan term was extended to 12 years?
3. How do bi-weekly payments affect the total cost of the loan?
4. What’s the total amount paid over the course of the loan?
5. How does the formula adjust for different compounding periods?

Tip: Increasing the frequency of payments (e.g., from quarterly to monthly) typically reduces the total interest paid over time.