A successful businessman is selling one of his fast food franchises to a close friend. He is selling the business today for $2,210,800.00. However, his friend is short on capital and would like to delay payment on the business. After negotiation, they agree to delay 4.00 years before the first payment. At that point, the friend will make quarterly payments for 19.00 years. The deal calls for a 6.32% APR “loan” rate with quarterly compounding. What quarterly payment will the friend make on the loan?

The problem involves determining the quarterly payment on a delayed payment schedule for a loan of $2,210,800 over 19 years with quarterly compounding at an annual percentage rate (APR) of 6.32%. Here's a step-by-step breakdown:

### Step 1: Understanding the terms
- **Loan amount (present value):** $2,210,800
- **Delay period:** 4 years
- **Payment period:** 19 years
- **Quarterly interest rate:** The annual interest rate is 6.32%, so the quarterly interest rate is:
  \[
  i = \frac{6.32\%}{4} = \frac{0.0632}{4} = 0.0158
  \]
- **Number of payments:** Since the payments are made quarterly for 19 years:
  \[
  n = 19 \times 4 = 76 \text{ payments}
  \]

### Step 2: Adjusting the loan for the delay period
Because the payments are delayed by 4 years, the present value must be compounded for this delay period. The future value $ FV $ of the loan after 4 years (with no payments made) is calculated using the compound interest formula:
\[
FV = PV \times (1 + i)^{4 \times 4} = 2,210,800 \times (1.0158)^{16}
\]

### Step 3: Calculate the quarterly payment
Once the future value after 4 years is known, we calculate the quarterly payment $ PMT $ using the formula for the present value of an annuity:
\[
PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}
\]
Rearranging to solve for $ PMT $:
\[
PMT = \frac{FV}{\frac{1 - (1 + i)^{-n}}{i}}
\]

Let me calculate the values for you.The quarterly payment that the friend will need to make on the loan is approximately **$64,476.10**.

Would you like further details on the calculation or have any other questions?

Here are some related questions you might explore:
1. How would the payment change if the delay period was longer or shorter?
2. What would happen if the interest rate was compounded monthly instead of quarterly?
3. How is the total interest paid over the life of the loan calculated?
4. What is the future value of the business after the delay period?
5. How would a balloon payment at the end of the loan affect the quarterly payments?

**Tip:** Always check how compounding frequency affects the total loan cost, as more frequent compounding results in higher interest.

Calculate the quarterly payment for a $2,210,800 business sale with a 6.32% APR and 4-year delayed payment. This finance problem involves compound interest, future value calculation, and annuity payments over 19 years with quarterly compounding.

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