You just borrowed $78,575. You plan to repay this loan by making X regular annual payments of $12,860 and a special payment of special payment of $48,600 in 9 years. The interest rate on the loan is 16.35 percent per year and your first regular payment will be made today. How many regular annual payments will be made?

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

To solve this problem, we need to calculate the number of regular annual payments $ n $ for the loan, taking into account that:

- The loan amount is $78,575.
- The regular payment made each year is $12,860.
- A special payment of $48,600 is made at the end of 9 years.
- The interest rate is 16.35% per year, and the first payment is made today.

This type of loan repayment problem involves the application of the **Present Value of Annuities** concept and the **Present Value of a Lump Sum Payment**. 

Let’s break down the approach:

### Step 1: Present Value of Regular Payments
Since the first payment is made today, this means we have an **annuity due**. The present value $ PV_{annuity} $ of these regular payments can be expressed as:

\[
PV_{annuity} = X \times \left( 1 + \frac{1}{(1 + r)} + \frac{1}{(1 + r)^2} + \cdots + \frac{1}{(1 + r)^{n-1}} \right)
\]

Which can be simplified using the formula for the present value of an annuity due:

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

Where:
- $ X = 12,860 $ (regular payment),
- $ r = 0.1635 $ (interest rate),
- $ n $ is the number of regular payments we need to solve for.

### Step 2: Present Value of the Special Payment
The present value of the special payment $ PV_{lumpsum} $ can be calculated using the present value formula for a lump sum:

\[
PV_{lumpsum} = \frac{48,600}{(1 + r)^9}
\]

### Step 3: Total Present Value
The total present value of the loan is the sum of the present values of the annuity payments and the lump sum payment, which equals the loan amount ($78,575):

\[
PV_{annuity} + PV_{lumpsum} = 78,575
\]

Now, let’s compute $ PV_{lumpsum} $ first and then solve for $ n $. I'll calculate this.The present value of the special payment is approximately \$12,437.66.

### Step 4: Set up the equation for regular payments
Now that we have the present value of the special payment, the total present value of the regular payments must account for the rest of the loan:

\[
PV_{annuity} = 78,575 - 12,437.66 = 66,137.34
\]

We now use the formula for the present value of an annuity due:

\[
66,137.34 = 12,860 \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]

We will solve for $ n $, the number of regular payments. Let me calculate this next.The number of regular annual payments required to repay the loan is approximately **8.47**.

Would you like more details on any of the steps? Here are five related questions you could explore:

1. How does the present value of an annuity differ from the future value?
2. What is the difference between an annuity due and an ordinary annuity?
3. How does compounding frequency affect the present value of a loan?
4. What are the practical applications of loan amortization in finance?
5. How does a balloon payment impact the structure of a loan?

**Tip**: When making loan payments, starting with an annuity due (paying at the beginning of each period) results in a lower total interest cost compared to paying at the end of each period (ordinary annuity).

You just borrowed $78,575. You plan to repay this loan by making X regular annual payments of $12,860 and a special payment of $48,600 in 9 years. The interest rate on the loan is 16.35 percent per year and your first regular payment will be made today. How many regular annual payments will be made?

Calculate the number of regular annual payments needed to repay a $78,575 loan with $12,860 annual payments and a $48,600 special payment in 9 years. The solution involves key finance concepts like the present value of annuities and lump sums at a 16.35% interest rate.

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