You just borrowed $137,115. You plan to repay this loan by making X regular annual payments of $19,710 and a special payment of special payment of $44,200 in 10 years. The interest rate on the loan is 13.95 percent per year and your first regular payment will be made today. What is X?



This is a problem of loan repayment where you're making regular payments (annuity payments) with an additional special payment at the end. Since the first payment is made today, it involves an **annuity due** rather than an ordinary annuity.

### Step-by-Step Solution:

1. **Given Information:**
   - Loan amount $ P = 137,115 $
   - Regular annual payment $ A = 19,710 $
   - Special payment after 10 years $ S = 44,200 $
   - Interest rate $ i = 13.95\% = 0.1395 $
   - First regular payment is made today (annuity due).

2. **Breaking it down:**
   The total amount borrowed can be thought of as two parts:
   - The **present value of the regular payments** (annuity due for $ X $ years).
   - The **present value of the special payment** made at the 10th year.

   The present value of these two components must add up to the loan amount $ P $.

3. **Present Value of the Regular Payments (Annuity Due):**

   The formula for the present value of an annuity due is:

   \[
   PV_{\text{annuity due}} = A \times \left( 1 + \frac{1 - (1 + i)^{-X}}{i} \right)
   \]

   where:
   - $ A $ is the regular payment.
   - $ i $ is the interest rate.
   - $ X $ is the number of years.

4. **Present Value of the Special Payment:**

   The special payment occurs 10 years from today, so the present value of the special payment is:

   \[
   PV_{\text{special}} = \frac{S}{(1 + i)^{10}}
   \]

   where:
   - $ S $ is the special payment.
   - $ i $ is the interest rate.

5. **Total Present Value:**

   The total present value of the loan is the sum of the present values of both components:

   \[
   P = PV_{\text{annuity due}} + PV_{\text{special}}
   \]

   Substituting the formulas and known values into this equation, we can solve for $ X $, the number of regular annual payments.

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you have any questions or would like to review the formulas or logic used in the process, feel free to ask!

Here are five related questions to explore further:
1. How does the present value of an annuity change with different interest rates?
2. What are the key differences between annuities due and ordinary annuities?
3. How do special lump-sum payments affect loan repayment schedules?
4. How would changing the regular payment amount impact the loan term?
5. How does inflation impact loan repayments in real terms?

**Tip:** For loans, always clarify if payments are made at the beginning or end of periods—it can significantly affect calculations!

You just borrowed $137,115. You plan to repay this loan by making X regular annual payments of $19,710 and a special payment of $44,200 in 10 years. The interest rate on the loan is 13.95 percent per year and your first regular payment will be made today. What is X?

Learn how to calculate loan repayment with regular annuity payments and a special lump-sum payment at the end. This problem involves an annuity due, interest rates, and the present value of future payments. Perfect for undergraduate finance students or advanced high school learners.

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