You just borrowed $329,620. You plan to repay this loan by making regular quarterly payments of X for 30 quarters and a special payment of $66,400 in 5 quarters. The interest rate on the loan is 4.07 percent per quarter and your first regular payment will be made in 3 months. What is X?

This is a loan amortization problem where we want to find the quarterly payment $ X $ given that there is a special lump sum payment after 5 quarters and the interest rate is 4.07% per quarter.

### Step-by-step solution:

1. **Break down the problem:**
   - Loan amount: $ P = 329,620 $
   - Number of quarters: $ n = 30 $
   - Quarterly interest rate: $ r = 4.07\% = 0.0407 $
   - Special lump-sum payment: $ L = 66,400 $ after 5 quarters.

2. **Present value of the payments:**
   The present value of all payments must equal the loan amount. This means:
   \[
   P = \text{Present value of regular payments} + \text{Present value of lump sum}
   \]

3. **Present value of the lump sum payment:**
   The lump sum payment is made after 5 quarters, so we need to discount it back to the present value using the formula:
   \[
   \text{Present value of lump sum} = \frac{66,400}{(1 + 0.0407)^5}
   \]

4. **Present value of regular payments:**
   We have 30 quarters of regular payments. Using the present value formula for an ordinary annuity (since payments are made at the end of each period), the present value of these payments is:
   \[
   \text{Present value of regular payments} = X \times \frac{1 - (1 + r)^{-30}}{r}
   \]
   where $ X $ is the regular payment and $ r = 0.0407 $.

5. **Set up the equation:**
   Now, we combine the present values and set them equal to the loan amount:
   \[
   329,620 = X \times \frac{1 - (1 + 0.0407)^{-30}}{0.0407} + \frac{66,400}{(1 + 0.0407)^5}
   \]

6. **Solve for $ X $:**

   First, calculate the present value of the lump sum:
   \[
   \frac{66,400}{(1 + 0.0407)^5} = \frac{66,400}{1.2219} \approx 54,336.73
   \]

   Now, subtract this from the loan amount:
   \[
   329,620 - 54,336.73 = 275,283.27
   \]

   Next, calculate the present value factor for the regular payments:
   \[
   \frac{1 - (1 + 0.0407)^{-30}}{0.0407} = \frac{1 - (1.0407)^{-30}}{0.0407} \approx 16.1247
   \]

   Now, divide the remaining loan amount by this factor:
   \[
   X = \frac{275,283.27}{16.1247} \approx 17,072.69
   \]

Thus, the regular quarterly payment $ X $ is approximately **$17,072.69**.

---

Would you like to see more detailed steps or have any further questions?

Here are 5 related questions you might find helpful:

1. How do changes in interest rate affect the value of $ X $?
2. How would the solution change if the special lump-sum payment was at a different time?
3. What is the total interest paid over the life of the loan?
4. Can we derive a formula for solving any such loan with a lump-sum payment?
5. How does the number of periods affect the present value of payments?

**Tip:** The earlier a lump-sum payment is made, the less interest it accrues, reducing its discounted value.

This loan amortization problem involves calculating the regular quarterly payments needed to repay a loan of $329,620 over 30 quarters. The solution includes a special lump sum payment of $66,400 after 5 quarters and an interest rate of 4.07% per quarter. Detailed step-by-step calculation for present value of payments and lump sum is provided.

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