A successful businessman is selling one of his fast food franchises to a close friend. He is selling the business today for $2,441,400.00. 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 16.00 years. The deal calls for a 7.36% APR “loan” rate with quarterly compounding. What quarterly payment will the friend make on the loan?

We can break this problem down into two parts:

1. **Finding the Present Value of the loan**: The businessman is selling the business for $2,441,400. Since payments are deferred for 4 years, we first calculate the present value of the payments after this 4-year period.

2. **Quarterly Payments**: After the 4-year delay, the friend will make payments for 16 years. We use the annuity formula to calculate the quarterly payment based on a 7.36% APR with quarterly compounding.

---

### Step 1: Adjusting for the Deferred Payment

The $2,441,400 is essentially the present value of the loan **4 years before** the first payment. The interest accumulates during this deferral period.

- **Interest rate per quarter**: Since the APR is 7.36%, compounded quarterly:
  \[
  \text{Quarterly interest rate} = \frac{7.36\%}{4} = 1.84\% = 0.0184
  \]

- **Number of quarters for deferral**: 4 years = 16 quarters.

Using the compound interest formula to find the future value after 4 years:
\[
FV = PV \times (1 + r)^n
\]
Where:
- $PV = 2,441,400$
- $r = 0.0184$
- $n = 16$ quarters

This future value will be the present value at the time the payments start.

### Step 2: Calculating the Quarterly Payment

Once we have the future value from step 1, we can calculate the quarterly payment for the annuity. The formula for the payment of an ordinary annuity is:

\[
PMT = \frac{PV \times r}{1 - (1 + r)^{-n}}
\]
Where:
- $PV$ is the present value found in step 1.
- $r = 0.0184$ is the quarterly interest rate.
- $n = 64$ is the number of quarters (16 years = 64 quarters).

Let me now calculate the required future value from step 1 and the quarterly payment in step 2.After the 4-year deferral period, the future value of the loan becomes approximately **$3,268,384.86**. 

Using this future value, the friend will need to make quarterly payments of approximately **$87,325.44** for 16 years to pay off the loan at the agreed 7.36% APR with quarterly compounding.

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How is the future value calculated when compounding interest is involved?
2. What if the deferral period was longer, such as 6 years instead of 4 years?
3. How would the payments change if the interest rate was lower, say 5% APR?
4. What is the effect of making additional payments during the deferral period?
5. How does the formula for an annuity differ from a perpetuity?

**Tip**: When dealing with deferred payments, the compounding during the deferral period can significantly increase the future balance owed.

A successful businessman is selling one of his fast food franchises to a close friend. He is selling the business today for $2,441,400.00. 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 16.00 years. The deal calls for a 7.36% APR 'loan' rate with quarterly compounding. What quarterly payment will the friend make on the loan?

Learn how to calculate the quarterly payment for a deferred business sale of $2,441,400, with a 7.36% APR compounded quarterly. After a 4-year deferral, the buyer will make quarterly payments over 16 years. This problem covers compound interest and annuity payment formulas.

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