You just borrowed $90,603. You plan to repay this loan by making X regular annual payments of $13,560 and a special payment of special payment of $32,300 in 7 years. The interest rate on the loan is 19.28 percent per year and your first regular payment will be made today. What is X?

 

Submission instructions: round your answer to 2 decimal places (such as 1.23 or 98.76).

We can solve this problem using the Present Value (PV) concept for cash flows, as it's a typical time value of money problem involving loan repayments. Here's the step-by-step breakdown:

### Step 1: Breakdown of cash flows
- You borrow $90,603 today (this is the present value of the loan).
- You will make regular payments of **$13,560** starting **today** (an annuity due situation).
- After 7 years, there will be a **special payment of $32,300**.
- The interest rate is **19.28% per year**, compounded annually.

### Step 2: Present Value (PV) of the loan

The total loan amount is the present value of the cash flows, which includes both the annuity payments and the special lump sum payment. The sum of these present values must equal the loan amount.

### Step 3: PV of the annuity due

For an **annuity due**, the formula to calculate the present value is:

\[
PV_{\text{annuity due}} = P \times \left(1 + r\right) \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $P = 13,560$ is the annual payment,
- $r = 0.1928$ is the annual interest rate,
- $n = 7$ is the number of years.

### Step 4: PV of the lump sum payment

For the **lump sum payment** in year 7, we discount it back to the present using the formula:

\[
PV_{\text{lump sum}} = \frac{F}{(1 + r)^7}
\]

Where:
- $F = 32,300$,
- $r = 0.1928$,
- $7$ is the number of years.

### Step 5: Sum of present values

The sum of the present value of the annuity due and the lump sum should equal the initial loan of $90,603. Thus, we set up the equation:

\[
90,603 = PV_{\text{annuity due}} + PV_{\text{lump sum}}
\]

Now let's solve this step-by-step.The regular annual payment $ X $ needed to repay the loan, along with the special payment of $32,300 in 7 years, is approximately **$18,514.35**.

Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:
1. How does the interest rate affect the present value of future payments?
2. What is the difference between an ordinary annuity and an annuity due?
3. How would the payments change if the special lump sum payment was made earlier?
4. How do loan amortization schedules work for varying payment plans?
5. How can you calculate future value instead of present value in loan scenarios?

**Tip:** An annuity due (payments starting today) has a higher present value compared to an ordinary annuity (payments starting in the future), given the same interest rate and terms.

You just borrowed $90,603. You plan to repay this loan by making X regular annual payments of $13,560 and a special payment of $32,300 in 7 years. The interest rate on the loan is 19.28 percent per year and your first regular payment will be made today. What is X?

Calculate the regular annual payment for a loan of $90,603 with a 19.28% interest rate, including a special lump sum payment of $32,300 in 7 years. This example covers key concepts such as the time value of money, present value of an annuity due, and discounting cash flows.

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